Embed Size (px)
DESCRIPTION
Math
Citation preview
title: Selected Works in Applied Mechanics and Mathematicsauthor: Reissner, Eric.
publisher: Jones & Bartlett Publishers, Inc.isbn10 | asin: 0867209682print isbn13: 9780867209686
ebook isbn13: 9780585363530language: English
subject Mechanics, Applied--Mathematics.publication date: 1996
lcc: TA350.R45 1996ebddc: 624.1/7
subject: Mechanics, Applied--Mathematics.
Page iii
Selected Works in Applied Mechanics and Mathematics
Eric Reissner
Page iv
Editorial, Sales, and Customer Service OfficesJones and Bartlett Publishers40 Tall Pine DriveSudbury, MA 01776
Jones and Bartlett Publishers International7 Melrose TerraceLondon W6 7RLEngland
Copyright © 1996 by Jones and Bartlett Publishers.
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, includingphotocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner.
Manufacturing Buyer Dana L. Cerrito
Typesetting Doyle Graphics Limited
Cover Design Hannus Designs
Printing and Binding Hamilton Printing Company
Library of Congress Cataloging-in-Publication Data
(Not available at press time.)
Printed in the United States of America99 98 97 96 95 10 9 8 7 6 5 4 3 2 1
Page v
CREDITSThe following publishers graciously gave permission to reprint the articles listed.
AIAA: J. Aeron. Sc. 4, © 1937, p. 539; 8 © 1941, p. 8; 16 © 1949, p. 516; 18, © 1951, p. 579
ASME: Proc. 5th Intern. Congr. Appl. Mech. © 1938, p. 134, p. 542; Proc. 1st Nat'l Congr. Appl. Mech. © 1952, p. 584; Journal of Applied Mech. 12, ©1945, p. 155; 40, © 1973, p. 75; 41, © 1974, p. 343; 47, © 1980, p. 189, p. 375, p. 189; 59, © 1992, p. 211
Birkhauser: J. Applied Mathematics and Physics 17, © 1966, p. 298; 23, © 1972, p. 58; 30, © 1979, p. 96; 32, © 1981, p. 105; 33, © 1982, p. 389; 34, ©1983, p. 114; 35, p. 194
Elsevier Science, Ltd.: Computer Methods in Appl. Mechs. & Eng. 85, © 1991, p. 200; Pergamon Press, Inc.: Journal of Mechs. and Phys. of Solids 6, ©1957, p. 246; Int. J. of Solids Struct., 1, © 1965, p. 450; 11, p. 82; 13, p. 366; Int. J. Non-Linear Mechs. 17, © 1982; Int. J. of Solid Struct. 21, © 1983; p.121, p. 392; © 1995, p. 216.
Interscience: Commun. Pure & Applied Math., 7, © 1959, p. 264.
Macmillan Publishing Co.: Prog. Appl. Mech.; Prager Anniv. Vol. , © 1963, p. 275.
MIT Press: Journal of Math and Physics 23, © 1944, p. 147; 25, © 1946, p. 32; 27, © 1948, p. 435; 29, © 1950, p. 437; 37, © 1958, p. 264; Studies Appl.Math. 49, © 1970, p. 176; 52, © 1973, p. 66.
Oxford University Press: Qu. J. Mech. & Appl. Math., 21, © 1968, p. 300
Prentice Hall: Thin-Shell Structures: Theory, Experiment, and Design , Fung/Sechler eds., © 1974, p. 353.
The Quarterly of Applied Mathematics: Qu. Appl. Math. 4, © 1946, p. 21; 10, © 1953, p. 173; 20, © 1962, p. 43.
Society for Industrial and Applied Mathematics: J. Soc. Indust. Appl. Math. 4, © 1956, p. 237; 13, © 1965, p. 281.
Springer Verlag: Math Anal. 111, © 1935, p. 131; Ingenieur Archiv. 7, © 1936, p. 491; 40, © 1971, p. 321; Mechanics of Generalized Continua, IUTAMSymp. © 1967, p. 453; Acta Mechanica 56, © 1985, p. 463; Proc. Intern. Conf. Comp. Mech. © 1986, p. 397; © 1987, p. 407; Computational Mathematics1, © 1986; 5 © 1989, p. 478.
John Wiley & Sons, Inc.: Communications on Pure and Applied Mathematics, Vol. 7, © 1959, p. 264.
Page vii
CONTENTS
Preface xi
A Biographical Sketch xiii
Beams 1
Über Die Berechnung Von Plattenbalken 3
Least Work Solutions of Shear Lag Problems 8
Analysis of Shear Lag in Box Beams by the Principle of Minimum Potential Energy 21
Note on the Shear Stresses in a Bent Cantilever Beam of Rectangular Cross Section 32
On Finite Pure Bending of Cylindrical Tubes 35
Finite Pure Bending of Circular Cylindrical Tubes 43
Considerations on the Centres of Shear and of Twist in the Theory of Beams 54
On One-Dimensional Finite-Strain Beam Theory: The Plane Problem 58
On One-Dimensional Large-Displacement Finite-Strain Beam Theory 66
Upper and Lower Bounds for Deflections of Laminated Cantilever Beams Including theEffect of Transverse Shear Deformation 75
Improved Upper and Lower Bounds for Deflections of Orthotropic Cantilever Beams 82
Note on a Problem of Beam Buckling 93
On Lateral Buckling of End-Loaded Cantilever Beams 96
On Finite Deformations of Space-Curved Beams 105
On Axial and Lateral Buckling of End-Loaded Anisotropic Cantilever Beams 114
A Variational Analysis of Small Finite Deformations of Pretwisted Elastic Beams 121
Plates 129
Über Die Biegung Der Kreisplatte Mit Exzentrischer Einzellast 131
On Tension Field Theory 134
On the Calculation of Three-Dimensional Corrections for the Two-Dimensional Theory ofPlane Stress 143
On the Theory of Bending of Elastic Plates 147
Page viii
The Effect of Transverse-Shear Deformation on the Bending of Elastic Plates 155
Pure Bending and Twisting of Thin Skewed Plates 173
On Postbuckling Behavior and Imperfection Sensitivity of Thin Elastic Plates on a Non-Linear Elastic Foundation 176
On the Analysis of First- and Second-Order Shear Deformation Effects for Isotropic ElasticPlates 189
A Tenth-Order Theory of Stretching of Transversely Isotropic Sheets 194
On Asymptotic Expansions for the Sixth-Order Linear Theory Problem of TransverseBending of Orthotropic Elastic Plates 200
On Finite Twisting and Bending of Nonhomogeneous Anisotropic Elastic Plates 211
A Note on the Shear Center Problem for Shear-Deformable Plates 216
Shells 221
On the Theory of Thin Elastic Shells 225
A Note on Membrane and Bending Stresses in Spherical Shells 237
On Stresses and Deformations of Ellipsoidal Shells Subject to Internal Pressure 246
On the Foundations of the Theory of Thin Elastic Shells 253
The Edge Effect in Symmetric Bending of Shallow Shells of Revolution 264
On the Equations for Finite Symmetrical Deflections of Thin Shells of Revolution 275
Rotating Shallow Elastic Shells of Revolution 281
A Note on Stress Strain Relations of the Linear Theory of Shells 298
Small Strain Large Deformation Shell Theory 304
Finite Inextensional Pure Bending and Twisting of Thin Shells of Revolution 308
On Consistent First Approximations in the General Linear Theory of Thin Elastic Shells 321
On Pure Bending and Stretching of Orthotropic Laminated Cylindrical Shells 343
Linear and Nonlinear Theory of Shells 353
On Small Bending and Stretching of Sandwich-Type Shells 366
On the Transverse Twisting of Shallow Spherical Ring Caps 375
On the Effect of a Small Circular Hole on States of Uniform Membrane Shear in SphericalShells 385
A Note on the Linear Theory of Shallow Shear-Deformable Shells 389
A Note on Two-Dimensional Finite-Deformation Theories of Shells 392
Some Problems of Shearing and Twisting of Shallow Spherical Shells 397
On a Certain Mixed Variational Theorem and on Laminated Elastic Shell Theory 407
On Finite Axi-Symmetrical Deformations of Thin Elastic Shells of Revolution 416
Page ix
Variational Principles 433
Note on the Method of Complementary Energy 435
On a Variational Theorem in Elasticity 437
On a Variational Theorem for Finite Elastic Deformations 443
A Note on Variational Principles in Elasticity 450
A Note on Günther's Analysis of Couple Stress 453
On a Certain Mixed Variational Theorem and a Proposed Application 457
On a Variational Principle for Elastic Displacements and Pressure 460
On Mixed Variational Formulations in Finite Elasticity 463
Some Aspects of the Variational Principles Problem in Elasticity 470
On the Formulation of Variational Theorems Involving Volume Constraints 478
Vibrations 489
Stationäre, Axialsymmetrische, Durch Eine Schüttelnde Masse Erregte Schwingungen EinesHomogenen Elastischen Halbraumes 491
Forced Torsional Oscillations of an Elastic Half-Space I 511
Complementary Energy Procedure for Flutter Calculations 516
Reihenentwicklung Eines Integrals Aus Der Theorie Der Elastischen Schwingungen 518
On Axi-Symmetrical Vibrations of Shallow Spherical Shells 523
Aerodynamics 537
A Contribution to the Theory of Turbulence 539
Note on the Statistical Theory of Turbulence 542
On Compressibility Corrections for Subsonic Flow over Bodies of Revolution 547
Note on the Theory of Lifting Surfaces 551
Boundary Value Problems in Aerodynamics of Lifting Surfaces in Non-Uniform Motion 558
Note on the Relation of Lifting-Line Theory to Lifting-Surface Theory 579
A Problem of the Theory of Oscillating Airfoils 584
Bibliography 589
Page xi
PREFACEIt is a pleasure and an honor to write this brief preface to introduce our teacher, Professor Eric Reissner, some of whose works compose this volume. We hope toshed some light on not only Eric Reissner, as a contributor to the fields of applied mathematics and mechanics, but also on him as a generous and caring individual.
As a biographical sketch written by him follows this preface we will limit ourselves to some thoughts of ours having to do with his influence on us, with the recognitionwhich his work has received and with a brief personal assessment of what we believe to be the principal contributions of a man who is both Professor Emeritus ofApplied Mathematics of the Massachusetts Institute of Technology and Professor Emeritus of Applied Mechanics of the University of California.
All four of us were privileged to be taught the Theory of Elasticity and Theories of Plates and Shells by Professor Reissner at M.I.T. His lectures were always clear,incisive, and thorough, exposing both the subtlety of solid mechanics and the subtlety of his thinking. He demanded much of his students because he demanded somuch from himself. Yet, for us, on the other side of this keen professional was a generous and caring friend, colleague and mentor.
From the recognition which Eric Reissner has received in appreciation of his work we would like to mention the following:
He was elected a Fellow of the American Academy of Arts and Sciences in 1950, and received the Clemens Herschel Award of the Boston Society of CivilEngineers in 1955. He was a Guggenheim Fellow during 1962. He received the von Karman Medal of the American Society of Civil Engineers in 1964, "fornoteworthy contributions to the theory of elasticity and theory of plates and shells, and for outstanding papers on those subjects." Also in 1964, he received anHonorary Dr. Ing. degree from the University of Hannover, Germany, "in appreciation of his pathbreaking works in the field of elastomechanics."
Later, in 1973, he received the Timoshenko Medal from the American Society of Mechanical Engineers, "for distinguished research and exceptional teaching in solidmechanics, especially in the theory of elastic plates." On the occasion of Reissner's receiving this award, Professor J. P. Den Hartog, a former student of Timoshenko,and a friend and colleague of Reissner at M.I.T. for more than thirty years, congratulated him for having "surpassed the master in the value of his life's contributions."Professor Reissner was elected a Member of the U.S. National Academy of Engineering in 1976. He became a Corresponding Member of the InternationalAcademy of Astronautics in 1979, and a full Member in 1984. A symposium in his honor was held on the occasion of his 65th birthday at the University of Californiaat San Diego, and a volume of "Mechanics Today" (Pergamon Press) appeared in 1980, containing the papers presented at this symposium by his former students,colleagues and friends, from all over the world.
Page xii
After his retirement in 1978 he received the Structures, Structural Dynamics, and Materials Award from AIAA, in 1984, "in recognition of fundamental contributionsto the aerospace community as a teacher and researcher in applied mathematics and mechanics of aircraft structures, and for the establishment of the Reissnervariational principle," and a certificate of achievement from the Pressure Vessels and Piping Division of ASME, in 1987, "in testimony of his contributions to theDivision membership by his pioneering work in the theory of plates and shells." This was followed by the receipt of the ASME Medal in 1988, "for eminentlydistinguished contributions to the practice of engineering through his research on plates and shells, structures, theory of elasticity, turbulence, aerodynamics, wingtheory and mathematics and for his stewardship of numerous doctoral candidates," and by an Honorary Membership in ASME in 1991 "for his profound and lastingmark on international applied mechanics through over half a century of teaching and research and for wise counsel at the highest levels of ASME." In 1992, EricReissner became the seventh Honorary Member in the 70 year history of the German Gesellschaft für Angewandte Mathematik und Mechanik ''in recognition of hisexceptional accomplishments in Applied Mathematics and Mechanics."
It would be presumptuous of us to embark on a thorough assessment of Professor Reissner's work in this preface. His awards, and the citations thereof, indicate thespecific seminal contributions that the community of mechanicians appreciates him for. We can only cite, from our perspectives, what we think are the contributionsover a span of sixty years that will secure his place in the history of 20th century applied mathematics and mechanics: (i) the two-field variational theorem involvingindependent stress and displacement variations for linear as well as for finite elasticity; (ii) shear deformation plate theory, with resolution of the classical boundarycondition paradox of Kirchhoff; (iii) his contributions to the subject of the center of shear; (iv) his 1949 seminal contribution to the nonlinear theory of shells; (v) hisinsights concerning the asymptotics of edge-zone and interior solution contributions for plate and shell boundary value problems; (vi) his 1965 contribution tovariational theorems in elasticity, with rotations as additional independent variables. Apart from these contributions in the mechanics of solids his creative spirittouched on other topics as well. They included (i) statistical theory of turbulence; (ii) steady and unsteady aerodynamic lifting-surface theory; and (iii) analysis of finitespan effects for wing divergence and flutter speed computations.
This volume presents selected original research papers of Professor Eric Reissner in the various areas mentioned above. May it serve as a milestone and a beacon forfuture generations.
SATYA N. ATLURI, GEORGIA INSTITUTE OF TECHNOLOGY, ATLANTATHOMAS J. LARDNER, UNIVERSITY OF MASSACHUSETTS, AMHERSTJAMES G. SIMMONDS, UNIVERSITY OF VIRGINIA, CHARLOTTESVILLEFREDERIC Y-M. WAN, UNIVERSITY OF CALIFORNIA, IRVINE
Page xiii
A BIOGRAPHICAL SKETCHI was born January 5, 1913 in Aachen, Germany, the son of Hans Reissner, then Professor of Applied Mechanics and Founder of the Aerodynamics Institute at theAachen Technische Hochschule. That same year my father followed a call from his alma mater which meant that I grew up in Berlin during the period 19131936.
My secondary school years were scholastically without distinction as I preferred to work on improving athletic skills. I was a member of field hockey teams, ran inthe Potsdam-Berlin relay races, and for a while was the best shot putter among the 15 year olds in the Berlin area. Mathematics had always been my easiestacademic subject. I became truly interested in it upon exposure to the elements of the calculus. The new concepts fascinated me and I remember supplementarystudies from one of my father's old textbooks, authored by Serret and translated by Scheffers.
After graduation, in 1931, with average grades, except in mathematics, physics and physical education, I matriculated at the Technische Hochschule Berlin. I firstmajored in Applied Physics, as this seemed the safest subject from the point of view of future employment prospects. However, I soon found out that I was notparticularly disposed towards doing some of the things which went with becoming an applied physicist. On the other hand, I had no trouble at all with mathematicsand mechanics courses, and so I moved from Applied Physics to Applied Mathematics at the end of the second year.
In trying to combine ideas coming from different sources and courses I solved two problems which became published papers in 1934 and 1935 [1, 2]. My mostinfluential professors were Georg Hamel and my father who taught me Theoretical and Applied Mechanics. Aside from this I have good memories of learning aboutcomplex variables from Ernst Jacobsthal, about differential equations from Richard Fuchs and about theoretical physics from Richard Becker. A one-semester leavein 1934, to attend the Zürich Institute of Technology, provided a valuable opportunity to take courses from Enst Meissner, Wolfgang Pauli and Georg Polya.
Graduating with honors in the Fall of 1935 I spent the following six months expanding my Dipl. Ing. Thesis into a Dr. Ing. Dissertation, on the subject of forcedvibrations of a mass supported over a finite contact area by an elastic halfspace, expanding on some classical work by Lamb on the corresponding mass-less pointload problem [5, 78].
At that time the political developments in Germany became more and more unpromising. Several inquiries about opportunities abroad resulted in a one-yearMathematics scholarship at the Massachusetts Institute of Technology, and a one-year student visa allowing travel to the U.S.A. Before the year was over M.I.T.decided that it could use me as a research assistant in Aeronautics. This meant a permanent residence permit through the offices of the U.S. consulate in NiagaraFalls. The aeronautics appointment lasted from 1937 to 1939. It included an opportunity for a Ph.D. in mathematics with an analysis of the aeronautical
Page xiv
structures problem of tension field theory [15]. A subsequent instructorship in Mathematics was followed by promotions to Assistant Professor in 1942, AssociateProfessor in 1946, and Professor in 1949.
As I think back to my more than thirty years at M.I.T. I must begin by recalling the importance of friendships with my colleagues H. B. Phillips (who brought me intothe Department which he headed), Ted Martin, George Thomas and C. C. Lin. All of them contributed significantly to my development. In a temporal way, I havespecial memories of the period 1945 to 1960. It was during this period that those of my papers which are still often referred to were written. This period alsoincluded summer appointments with the Structures and the Dynamics Divisions of the Langley Field Laboratory of the National Advisory Committee for Aeronautics(1948, 1951), with Ramo-Wooldridge in Los Angeles (1954, 1955) to help solve problems in the design of the Atlas missile, and with Lockheed in Palo Alto (1956,1957) who were then concerned with the development of the Polaris. A highlight was a two months Symposium on Structures and Elasticity at the University ofMichigan in 1949. My assignment was to present the Theory of Elasticity, together with S. Timoshenko giving lectures on the Theory of Thin Plates, and R. V.Southwell on Advanced Airplane Structures. Also in 1949 I was asked to be Consulting Mathematics Editor for the Addison-Wesley Publishing Company, then avery small new organization. The fact that the ensuing series of books included a Calculus text by George Thomas and a text on Advanced Calculus by WilfredKaplan meant that this assignment, which lasted until 1960, resulted in significant economic benefits.
As the years went by I became more and more conscious of the fact that my research and teaching interests belonged to the Engineering Sciences rather than toMathematics. This being the case, I accepted in 1970 an appointment as Professor of Applied Mechanics to participate in the growth of this field at the new SanDiego campus of the University of California. There I joined in the efforts of Y. C. Fung, J. W. Miles, W. Nachbar, S. S. Penner and other younger, capable andfriendly colleagues. It turned out to be a truly uplifting and refreshing experience.
In conclusion, I would like to express thanks to my friends and one-time students Satya Atluri, Fred Wan, Tom Lardner and Jim Simmonds. Their help in bringing theproject of this Volume to fruition has been important. Besides, our personal contacts over the years and our joint studies from time to time have helped me to stayinvolved in the adventure of seeking new insights in the field of applied mechanics to this day. And, as far as keeping me involved personally and professionally isconcerned, I feel an obligation to acknowledge my long-ago students Bob Clark at Case Western, Millard Johnson at the University of Wisconsin, Jim Knowles atCal Tech, Sud Nair at Illinois Tech, W. T. Tsai at Long Beach State and the late Hubertus Weinitschke of the University of Erlangen-Nürnberg.
A word about the contents of this Volume. In selecting papers for inclusion I was guided by the wish to consider the significance of the work at the time it was done,relative inaccessibility except in this place, in some instances a preference for co-authored efforts, and finally a preference for brief articles, in order to be able toinclude as many of them as possible.
Finally, and foremost, I must express a sense of deep gratitude and affection to my wife Johanna. Her support and influence during our many years of harmonioustogethernesswhich included the raising of our son John, and our daughter Evahave been of inestimable value.
E.R.
Page xv
Page 1
BEAMSWhile still in high school and having just learned the geometrical significance of first and second derivatives of a function, I asked my father what significance theremight be to derivatives of higher order than two. Naturally, he mentioned that the fourth derivative of the deflection curve of a beam would be proportional to theintensity of the load distribution responsible for this deflection. I learned a good deal more about this subject during my first year at the university as a student in myfather's mechanics course.
As luck would have it, a student summer job involved the inadequacy of elementary beam theory for T-beams with very wide flanges. I read an analysis of thisproblem by von Karman who considered the behavior of the flanges as a problem of the theory of plane stress. However, instead of determining the constants ofintegration in a Fourier series solution by means of transition conditions between flange and web, von Karman pursued a more elaborate route by way of minimizingthe strain energy of the flange-web combination. Finding out that it was simpler to solve the problem without use of the minimum energy condition resulted in my firstpublished paper [1].
As a graduate student assistant three years later, I was asked to work on the related problem of shear lag in box beams. I recognized that it was a good enoughapproximation for the shear lag problem to replace the elementary uniform crosswise distribution of axial normal stress by a parabolic distribution and to look for anapproximate Rayleigh-Ritz type solution with the help of a minimum energy condition. I first used the principle of least work [24] and later the principle of minimumpotential energy [43].
After this I never lost my interest in beam problems of a non-standard nature. An unexpected result for the Saint Venant distribution of shear stress in plate-likerectangular cross-section beams [44,48] was followed by a shell-theoretical analysis of von Karman's problem of bending of curved tubes [60, 74, 223] and by ananalogous analysis of Brazier's non-linear effect of cross section flattening for the bending of straight tubes [83, 132, 138].
This in turn was followed by a consideration of the center of twist problem for the torsion of cantilevers [93], by a consideration of torsional vibrations of pre-twistedbeams [108], and by a solution of the torsion problem of circumferentially non-homogeneous tubes, which included as special cases the classical results for both openand closed cross section tubes [122, 173].
An interest by my doctoral student W. T. Tsai led me to the problem of how non-symmetrical beams should be loaded in order to have bending without twisting[187, 188]. While I had known of the text book solution for thin-walled open cross section beams, and also of attempts to deal with the problem in the context ofSaint Venant's flexure analysis by such people as G. I. Taylor and E. Trefftz, I had felt
Page 2
uncomfortable with the premise of these approaches, which involved either displacement specifications for an (in my mind) arbitrarily chosen point in the supportedcross section, or an energy specification without appeal to the form of the support condition. I came to the conclusion that for a rational definition of a center ofshear it was necessary to begin with a suitable system of mixed loading boundary conditions for a three-dimensional formulation of the problem. Subsequent to this itwas then possible to utilize the Saint Venant flexure assumptions in a Rayleigh-Ritz sense for an approximate determination of the coordinates of this center.
I returned to the problem once more fifteen years later by way of deducing beam-theoretical results as consequences of approximately analyzing elastic plates ofvariable thickness [263, 272, 280]. This included, in particular, the derivation of beam equations accounting for anti-clastic curvature effects in addition to warpingstiffness effects in the sense of Vlasov, with this permitting in particular a quantitative appraisal of the influence of Poisson's ratio on the location of the center of shear.
At about the same time I became interested, as a consequence of studying finite rotation shell theory, in a consideration of the one-dimensional space-curved beamproblem [190, 191, 225]. My approach here, the same as for the shell problem, was the reverse of what was usually done. Instead of beginning with the geometry offinite deformations I began with what was for me the easier problem of stating equilibrium equations for the deformed structure. After that I used the principle ofvirtual work in an inverse fashion to establish a system of virtual strain displacement relations. While the step from virtual to actual relations is elementary in the lineartheory, in finite-deformation theory this step required a judicious non-linear differential equations integration scheme. I arrived at a system of beam equations whichrepresented a generalization of Kirchhoff's rod theory, the generalization consisting in allowance for force-deformational effects, in addition to the classical moment-deformational effects.
An occasion to apply these equations was the appearance of a paper on lateral buckling in which the first Kirchhoff-rod-theory based solution of this problem, in a1904 paper by H. Reissner, was used once more, with a critical comment on the neglect of one of two pre-buckling deformation terms in the 1904 paper. I re-examined the problem [214], hoping to find fault with this comment but came to the conclusion that the criticism was in fact justified, although the effect of theneglected term was quite small compared to the effect which had not been neglected. In the course of this study, I then also used the force-deformational terms in myequations for a determination of the transverse shear effect on the value of the classical Michell-Prandtl buckling load.
A continuation of my concern with the problem of lateral buckling, led on to a note, jointly with my son John Reissner, on the consequences of constitutive coupling oftorsion and bending [237], and later on to results through use of the equations of three-dimensional finite-deformation elasticity concerning refined one-dimensionallateral buckling equations incorporating warping stiffness in addition to bending and twisting stiffness [240, 259, 265].
Page 3
Über Die Berechnung Von Plattenbalken[Der Stahlbau 7, 286288, 1934]
Einleitung
Die übliche Biegungstheorie der Träger mit gerader Mittellinie geht von der Voraussetzung aus, daß ein in einer Hauptträgheitsebene der Querschnitte wirkendesBiegungsmoment quer zu dieser Ebene konstante Spannungsverteilung erzeugt. Im allgemeinen führt diese Annahme auch zu keinen unzulässigen Widersprüchen mitden Ergebnissen der Elastizitätstheorie. Es ist jedoch seit langem bekannt, daß die erwähnte Annahme bei Plattenbalken und Kastenträgern einigermaßen breitemGurt auch näherungsweise nicht mehr zutrifft. Man hat in diesen Fällen den Begriff der "mittragenden Breite" eingeführt, worunter man diejenige Gurtbreite versteht,mit der bei der Annahme konstanter Spannung nach der Breite hin sich dieselbe maximale Biegungsspannung ergeben würde, wie diejenige des Plattenbalkens mitnach der Seite abklingenden Spannungen.
Eine rationelle Methode zur Berechnung der mittragenden Breite bei durchlaufenden T- Trägern hat zuerst Prof. v. Kármán angegeben 1). Vorausgesetzt wirddabeiwas auch hier geschehen solldaß die Plattenstärke klein ist im Vergleich zur Trägerhöhe, und daß die Biegungssteifigkeit der Gurtplatte senkrecht zu ihrer Ebenezu vernachlässigen ist gegen die des Steges2). Es wird also angenommen, daß in der Platte ein ebener Spannungszustand herrscht. Dieser Spannungszustand istoffenbar abhängig von der Belastung und von den Abmessungen des Systems. Den Zusammenhang zwischen Steg und Platte berücksichtigt v. Kármán mit Hilfe desPrinzips vom Minimum der Formänderungsarbeit. Zahlenbeispiele nach dieser Methode für verschiedene Lastverteilungen rechnete Dr. Metzner3). Es ergab sich ausdiesen Rechnungen, daß die tragende Breite längs der Trägerachse durchaus nicht immer konstant, sondern von der Momentenverteilung abhängig ist.
Erweiterungen der Theorie auf Kastenträger, auch auf Fälle nicht durchlaufender Träger finden sich in zwei Arbeiten von Prof. G. Schnadel4).
Im folgenden soll zunächst eine Methode angegeben werden, mit der ebenfalls der elastische Zusammenhang zwischen Steg und Gurt berücksichtigt, die Aufgabeaber auf ein reines Randwertproblem der Spannungsfunktion der Gurtplatte zurückgeführt wird. Auf diesem Wege können die formelmäßigen Ergebnisse derbisherigen
1Th. v. Kármán, Die mittragende Breite. A. FöpplFestschrift 1924. S. a. S. Timoshenko, Theory of Elasticity, S. 156. McGraw-Hill Book Comp. Inc. New York und London 1934.
2In einer späteren Mitteilung wird gezeigt werden, daß es möglich ist, die Aufgabe in gewissen Fällen auch ohne diese einschränkende Voraussetzung streng zu lösen.
3W. Metzner, Die mittragende Breite. Lufo IV, 1929.
4G. Schnadel, Die Spannungsverteilung in Flanschen dünnwandiger Kastenträger. Jahrb. d. Schiffbautechn. Ges. 1926.Die mittragende Breite in Kastenträgern. Werft, Reederei & Hafen 1928.
Page 4
Arbeiten mit sehr wenig Rechenaufwand erhalten werden. Weiter ergibt sich die prinzipielle Möglichkeit, diejenigen Näherungsverfahren zur Lösung vonRandwertaufgaben anzuwenden, welche die Angabe sämtlicher Randbedingungen durch die Randwerte der gesuchten Funktion und ihrer Ableitungen erfordern(Ritzsches Verfahren, Methode der Differenzenrechnung usw.).
In einem zweiten Abschnitt wird eine genauere Theorie aufgestellt, die insbesondere für Träger mit einer gegenüber der Spannweite nicht mehr kleinen Steghöhe vonBedeutung sein kann. Ferner wird gezeigt, wie man auch aus ihr durch Grenzübergang zu kleinen Steghöhen die alten Ergebnisse erhalten kann.
IEinfache Theorie. Steg Als Balken
Hier ist die folgende Aufgabe zu lösen: Gegeben nach Bild 1 ein Steg, in dem der Charakter der Spannungsverteilung nach der üblichen Näherungstheorie, und eineGurtplatte, in der ein ebener Spannungszustand vorausgesetzt werden soll. Die Berücksichtigung des elastischen Zusammenhangs erfolgt in der Weise, daß man ander Anschlußstelle StegGurt die Dehnung in der Platte derjenigen Dehnung gleichsetzt, die dort herrschen würde, wenn man einen Plattenbalken vor sich hätte vonder Gurtbreite 2 und der Breite nach konstanter Spannung.
Bild 1.
Nun lassen sich bekanntlich die Spannungen x, y, eines ebenen Spannungszustandes folgendermaßen als Ableitungen einer Spannungsfunktion F schreiben:
wobei F der folgenden Differentialgleichung genügen muß:
Den Zusammenhang zwischen den Dehnungen und den Spannungen gibt das verallgemeinerte Hookesche Gesetz, wenn man die Verschiebungen in der x- bzw. y-Richtung mit u bzw. v, die Winkeländerung mit bezeichnet, folgendermaßen:
wobei E den Dehnungsmodul, G den Schubmodul und m das Querkontraktionsver-
Page 5
hältnis bedeutet. Zwischen E und G besteht überdies die Gleichung
Das Koordinationssystem möge nach der in Bild 1 angegebenen Weise gewählt werden.
Die Randbedingungen für die Anschlußstelle StegGurt können ein für allemal angegeben werden. Aus Symmetriegründen folgt, daß die Verschiebung quer zurStegachse verschwinden muß.
Zur zweiten Bedingung werde die Aussage über die Dehnung längs der Trägerachse gemacht. Es ist unter den gemachten Voraussetzungen:
wobei M(x) das Biegungsmoment und W(x) das Widerstandsmoment des Trägers mit der vollmittragenden Breite 2 ist. Andererseits ist durch die folgendeGleichung definiert.
welche ausdrückt, daß der Inhalt der nach der Seite abklingenden Gurt-spannungsfläche einer ideellen rechteckigen Spannungsfläche gleichgesetzt wird. DasWiderstandsmoment wird, wie man leicht ausrechnet,
oder, wenn man nach auflöst,
Aus (8a) und aus der Definitionsgleichung (7) der tragenden Breite bekommt man das zugeordnete Widerstandsmoment
Wenn man (9) in die Randbedingung (6) einsetzt, erhält man schließlich als Randbedingung aus (6)
und unter Berücksichtigung der Beziehungen (3) und (1)
Page 6
Für die tragende Breite ergibt sich aus (7) unter Berücksichtigung von (8) die folgende Gleichung:
Durchführung Für Einen Besonderen Fall
Nimmt man als Spannungsfunktion F die M. Lévysche Lösung der biharmonischen Differentialgleichung
mit = n /l, so läßt sich durch sie einmal, wie v. Kármán gezeigt hat, der Spannungszustand in der Gurtplatte eines durchlaufenden Trägers von der Stützweite 2 l,der ein ebenfalls periodisches Moment von der Form
aufzunehmen hat, darstellen Man muß dann für den durchlaufenden Träger mit überall positiver Belastung an den Stützpunkten, d.h. für x = 0 und x = 2l ausSymmetriegründen fordern
Aus der Form der Spannungsfunktion ergibt sich damit, daß ebenda
Man kann aber auch, wie G. Schnadel zuerst bemerkt hat, den Spannungszustand in der Platte eines gelenkig gestützten Trägers von der Spannweite l darstellen,denn (10) erfüllt die Bedingung
für x = 1/2l und x = 3/2l in jedem Gliede der Spannungsfunktion für sich. Man erhältals zweite Randbedingung an denselben Stellen
d.h. die Lösung ist streng, wenn durch Versteifungen an den freien Rändern für die Erfüllung der Gl. (18) gesorgt wird, was in der Praxis oft der Fall ist (Man kannsich diesen gelenkig gestützten Träger auch als Teil eines durchlaufenden Trägers vorstellen mit periodischer, abwechselnd positiver und negativer Belastung.
Beschränken wir uns hier für die weitere Durchführung auf den Fall des unendlich breiten Gurtes, so werden wegen des Verschwindens der Spannungen für y =
Drückt man die Bedingung v(x, 0) = 0 mit Hilfe von (3) durch die Ableitungen der Spannungsfunktion aus, so erhält man folgenden Zusammenhang zwischen An undBn
Page 7
Damit nimmt die Spannungsfunktion die folgende Gestalt an
Die vierte Randbedingung, Gl. (11), des stetigen Überganges vom Gurt auf den Steg ist die folgende:
also:
damit erhalten wir aus Gl. (12) die folgende Bestimmungsgleichung für
welche also erlaubt, den Plattenbalken nach der elementaren Theorie mit x unabhängig von y zu berechnen, wenn die sich daraus ergebende ideelle Gurtbreite eingeführt wird.
Für M(x) = M cos x, eine Momentenverteilung, wie sie sich sehr angenähert für den gelenkig gestützten Träger unter gleichmäßiger Volllast ergibt, wird z.B., mit m= 10/3
Formel (24) findet sich bereits in der Arbeit von Herrn v. Kármán, der mit ihrer Hilfe feststellt, daß für eine einfach harmonische Momentenverteilung = const. wird(was man übrigens bei der gewählten Spannungsfunktion unmittelbar aus (6) und (7) ersehen kann, so daß dieses Resultat unabhängig von der Randbedingung (11)ist), und daß die tragende Breite durch die späteren harmonischen Glieder nicht unerheblich vermindert werden kann. Es ist möglich, aus Gl. (22) die folgendeschärfere und wie es scheint bis jetzt unbekannt gewesene Folgerung zu ziehen, daß es Momentenverteilungen gibt, für die im gefährlichen Querschnitt die tragendeBreite beliebig klein wird. Hinreichend dafür ist die genügende Kleinheit von
d.h. bei Spitzen in der Momentenfläche ist die Materialausnutzung besonders schlecht.
Page 8
Least Work Solutions of Shear Lag Problems[J. Aeron. Sciences 8, 284291, 1941]
Introduction
It is a well known fact that the distribution of bending stresses in thin-walled box-beams cannot be obtained from the customary theory of bending of beams when thelateral extension of such structures is of the order of magnitude of their spanwise extension. The elementary beam theory assumes a uniform distribution of spanwisenormal stress at any transverse section and consequently fully efficient chord members, but the shear deformation of these members leads to a distribution of normalstress which, at a given beam section, has its maximum in general at the side webs, decreasing toward the middle of the chord. Neglecting this effect amounts to anoverestimation of the strength of the beam. It has been found that the dimensions of box-beam-like airplane-wing structures are often such that this shear-lag effect isappreciable. It may happen that the stress in the middle of the sheet amounts to only 60 per cent of the edge stress. To determine the magnitude of the effect,theoretical and experimental investigations of the problem have been carried out, giving the desired information for some important cases and showing furthermorewhich mathematical problem is to be solved in any given case (see the references at the end of this paper). The main difficulty consists in the fact that the stressproblem is two-dimensional and attempts to solve its fundamental equations, with a variety of assumptions concerning the elastic properties of the cover sheets, havebeen successful for certain arrangements only.
It seems, therefore, desirable to find a way to reduce the shear-lag problem to a one-dimensional problem, in the sense that an equation be established for a quantityindicating the amount of shear lag at every cross-section of the beam. Such an equation will have to contain parameters depending on the dimensions of the structureand the distribution of the load. It need not be an exact result of the theory of elasticity so long as it is certain that the analysis retains the essential characteristics of theproblem and gives numerical results in close agreement with the exact results.
The purpose of this paper is to derive such an equation for the class of box-beams symmetrical about span-wise vertical and horizontal planes through the neutral axisof the beam. There is, however, no inherent difficulty in generalizing the results to include unsymmetrical beams as well, although the corresponding derivations will beless simple than the ones presented here.
Also given are applications of the fundamental equation to the actual solution of a series of shear-lag problems.
The starting point for the method developed here was the fact that in all symmetrical cases investigated so far the shape of the curve representing the
Page 9
distribution of normal stress across the beam seemed to be very nearly parabolic. If one makes the assumption that these curves should be true parabolas,distinguished from each other only through the values of their vertex curvature, all that remains to be done is to establish an equation for the spanwise variation of thisvertex curvature. The most convenient way to do this appears to the author to be the application of a minimum energy principle. A distribution of stress in thehorizontal (or nearly horizontal) cover sheets is assumed, at every cross-section parabolic in the spanwise normal stress and, moreover, satisfying the equilibriumconditions for every element of the sheet. The linear side-web normal stresses are determined in such a way that cover sheet and side-web normal stresses coincidealong the flanges. Furthermore, the condition is imposed that the resultant moment of the spanwise normal stresses at every section about a transverse horizontal axisequals the external bending moment at that section. When these conditions are satisfied there remains only one unknown quantity, the vertex curvature of the normalstress parabola, and this quantity may be determined by minimizing the internal work of the structure. This minimum condition is shown to reduce to an ordinarysecond order differential equation, with constant coefficients for beams of constant cross-section, and with variable coefficients for tapered beams.
Formulation of the Problem
A cantilever box-beam is considered, with rectangular doubly symmetrical cross-section, acted upon by a given distribution of bending moments (Figure 1).
Fig. 1.
The assumption of parabolic spanwise normal stress in the coversheets is expressed by writing,
For the normal stress in the side webs one has
Continuity of the stresses demands
Page 10
The condition of moment equilibrium is expressed by the equation
Introducing s, from Eqs. (2) and (3) into Eq. (4), there follows a relation involving sheet stresses only,
Eq. (5) serves to express s0 in terms of the vertex curvature 2s/w2 of the normal stress parabola, giving
It shall be assumed that the parameter m has the same value all along the span.
Thus one may write Eq. (1)
The least work condition will serve to determine the quantity s. To apply this condition, it is first necessary to find the remaining sheet stress components, and y,which must be in equilibrium with x as given by Eq. (7). Then it is necessary to establish the expression for the internal work W of the entire beam and finally thestresses have to be introduced into W and W be made a minimum.
The Distribution of Stress in the Sheets and the Internal Energy of the Bent Beam
The sheet stresses have to fulfill the following equilibrium conditions of generalized plane stress,
It is well known that Eqs. (8) and (9) may be satisfied in terms of stress functions in the following manner,
or
Eq. (11) is preferable when the transverse normal stress, y, need not be considered, and it can be shown that this is generally permissible in the shear-lag problem.
From Eqs. (7) and (8) one obtains by integration for the shear stress
Page 11
In this formula an arbitrary function of integration has been eliminated by the condition that must be antisymmetric about the axis y = 0.
With Eq. (11) for there follows from Eq. (9) for y,
*
In Eq. (13) an arbitrary function of x has been determined by the condition that y(x, ± w) = 0.
Eqs. (7), (11) and (12) are satisfied by taking as stress function
where
are the sheet stress and the sheet stress resultant of the elementary beam theory.
To obtain an expression for the internal work in terms of the stresses, it is necessary to agree on the elastic properties of the sheet material. It shall be assumed thatthe cover sheets are of non-isotropic material. In this way it is possible to account in a convenient way for the influence of closely spaced transverse and longitudinalstiffeners and it also shows that neglecting the work of the transverse normal stresses y corresponds exactly to a limiting case of the orthotropic stress-strainrelations.
Writing the stress-strain relations in the form
where vx and vy are Poisson ratios, the virtual work per unit of sheet area, t( x x + y y + ), gives for the elastic energy of the two sheets
For the existence of Ws it is necessary that the following relation is satisfied between the elastic constants,
Adding to the energy stored in the sheets, as given by Eq. (18), the energy of the two side webs,
which with Eqs. (2) and (3) becomes
*This equation for y may be used to estimate quantitatively the magnitude of the transverse normal stresses, associated with the parabolic distribution of the spanwise stresses determinedsubsequently.
Page 12
there is given in
the total energy of the box-beam, provided the conditions of support at the root section and the stiffening of the tip section are such that the edge stresses at thesesections are prevented from doing work during a virtual displacement. Such conditions at the root section require vanishing spanwise and transverse displacements,or, instead of the second condition, vanishing shear (which is the condition the available exact solutions fulfill).* At the tip section these conditions require vanishingspanwise normal stress and vanishing transverse displacement (as in the case of the available exact solution) or instead of the displacement condition, the condition ofvanishing shear. It is, however plausible that with the exception of the condition mentioned above* the possible contribution to the total work due to flexibility of tipand root ribs is relatively small and may therefore be disregarded.
The expression for the internal work will be further simplified by neglecting the work of the transverse normal stresses y compared with the work of x and . Itseems plausible that this omission is in general of no great influence in the final result** (see also references 6, 7, 21).
From Eqs. (18) and (19) it follows that neglecting y amounts to putting
in the stress-strain relations. This is exact for the case of a sheet rigid in transverse direction. In this connection it is noted that the presence of rather narrowly spacedtransverse stiffeners would in any event tend to give the sheet an effective Ey which is greater than Ex.
The Least Work Condition
Neglecting y in Eq. (18) and writing Ex = E the total work W is
and with x and from Eq. (11)
According to the rules of the calculus of variations the condition for a minimum of W is that the variation W, vanishes. Now
*For those cases where the spanwise displacement of the sheet at the root section is not completely restrained, due for instance to the presence of a retractable landing gear, it will be necessary toadd a term to the total work expression, to account for the bending energy of the transverse stiffener.
**It would be possible to find the solution without this simplifying assumption, by means of a fourth order equation instead of the second order equation derived in what follows, and thus to determinequantitatively its effect.
Page 13
and since (dH) = d( H) there follows, by integration by parts,
The second and the last integral in Eq. (25) may be combined to
Observing that from the equilibrium condition Eq. (5), which on account of Eq. (11) can be written in the form,
there follows
it is seen that the integral Eq. (26) vanishes.
The condition of least work is thus reduced to
From Eq. (14) follows
and
Since at the tip x(0, y) = 0 it follows further that
and with that
Page 14
Introducing Eqs. (30) and (31) into Eq. (29) one has
The integration with respect to y may be carried out, and since s(x) is arbitrary the integrand of the first term and the second term has to vanish separately, resulting,together with Eq. (32), in the following differential equation and boundary conditions for s(x):
With the solution of Eq. (36), subject to the boundary conditions Eq. (37), there is given the solution of the shear-lag problem for symmetrical box-beamsunder symmetrical (bending) loads. It is evident that the solution for antisymmetrical (torsional) loads for the same beam may be obtained in a completely analogousway. Some added considerations seem necessary to generalize the solution to the case of unsymmetrical beams. The nature of the analysis is, however, such that thepossibility of this generalization is apparent.
Of interest is the value of an effective sheet width weff. which, with the help of Eq. (7), is expressed in the form
In what follows x and weff. will be calculated for some typical beams of constant cross-section, for different loading conditions. These calculations permit one todraw some useful general conclusions, as will also be shown.
It should be noted that the integration of Eqs. (36) and (37) in closed form is also possible for tapered beams of the sort that w = w0(x + x0), t = t0(x + x0)n, that isfor beams with a law of sheet-thickness and beam-width variation such that if the beam were continued to the left of x = 0, w and t would simultaneously becomezero with exception of the case n = 0 where the sheet thickness is constant. The integration is possible by assuming the solutions of the homogeneous equation (36) inthe form s(x) = (x + x0)r.
A further noteworthy result consists in the fact, that when the sheet stress resultant S(x) = t b is constant, then s(x) = 0 satisfies Eqs. (36) and (37) and there is noshear lag. The explanation of this follows from the equilibrium condition Eq. (8) which shows that for ( t x/ x) = 0 the sheet shear stress vanishes and consequentlyno shear deformation occurs. Therefore, one can say that, in principle, it is possible to design box-beams of the type considered, with fully efficient chord members.
Page 15
Solution of the Shear-Lag Equation for Beams of Constant Width and Sheet Thickness
Assuming w and t constant,* Eqs. (36) and (37) become
where 2 and are defined by
The solution of the system of Eqs. (39) and (40) is found, by the method of variation of parameters, in the form
where = x/l.
This formula is valid for distributed as well as for concentrated loads if it is understood that a concentrated load has to be considered as limiting case of a distributedload, the limiting process to be carried out after the integrals have been evaluated.
The function s(x) and with that according to Eqs. (7), (12) and (13) the stress pattern in the coversheets will here be determined explicitly for the following typicalloading conditions:
1. A concentrated load P at the tip section
2. A uniformly distributed load p0
3. A concentrated load P at a distance l1 from the tip.
One obtains
*The solutions thus derived will be applicable also to the case of piecewise constant thickness t, if along the sections where t is discontinuous there is continuity of t x and t , which according toEqs. (7) and (12) means continuity of ts(x) and tds/dx.
Page 16
To represent the results of these formulas graphically it is convenient to write Eq. (7) for the spanwise stress in the form
Numerical Examples
Eqs. (43) to (49) shall be evaluated, assuming the following dimensions
From Eqs. (6), (38) and (41) results
With these values of the parameters the stresses in the middle of the sheet and at the edges of the sheet have been calculated and are represented in Figures 2 to 4,together with the corresponding curves for the stress without shear lag.
The graphs show that shear lag is most pronounced near the built-in end (the most highly stressed section) of the beam. It is further noteworthy that in the case ofconcentrated load application at midspan there occurs an appreciable sheet stress at the point of load application, which along the flanges is of opposite direction to
Fig. 2.
Page 17
Fig. 3.
the corresponding stress at the root section (see also reference 14). The elementary theory does not account for this stress.
The most important characteristics of these graphs are, however, that the shear lag at the built-in end is almost the same for the uniformly distributed load and for theconcentrated load at midspan, while for the beam with tip load there is considerably less shear lag. This suggests that the effective sheet width depends, for beams ofconstant cross-section, on the distance of the center of gravity of the load curve from the built-in end rather than on the span length of the beam.
A formula expressing this fact will now be derived. According to Eq. (38) weff. is, for beams with constant coversheet thickness t, given by
with, if it is furthermore assumed that the width w is constant, s(x) from Eq. (42). At the built-in end of the beam, weff. depends on s(l), which may be transformedinto
For values of in the practical range one may with good approximation neglect the term sinh under the intergral and put tanh = 1.
Thus
Page 18
Fig. 4.
Writing
it is seen that in the dimensionless ratio
the quantity L represents the distance of the center of gravity of the d2 b/dx2-curve from the built-in end of the beam. For beams of constant cross-section this isidentical with the distance of the center of gravity of the load curve from the built-in end.
Introducing Eqs. (54) and (41) into Eq. (50) there follows
*Instead of as indicated in Figure 5, equation should read: m = (Is + 3Iw)/I.
Page 19
Fig. 5.*
and in an analogous way
Eqs. (55) and (56) are general formulas for the amount of shear lag in beams of constant width and cover sheet thickness, with or without taper in height. Theyindicate in which way shear lag depends on the ratio of tension and shear modulus, on the relative magnitude of coversheet and side web stiffness and on the ratio ofsheet width 2w and the distance L of the center of gravity of the b -curve from the root section of the beam.
Figure 5 gives (weff./w) root as a function of w/L when E/G = 8/3 and for various values of the stiffness parameter m.
It may be added that solutions of the numerically discussed problems could also have been obtained by an exact method (references 7, 22), although only in the formof not very rapidly converging infinite trigonometric series. Corresponding exact solutions can also be obtained for beams with isotropic coversheets (see references1, 11, 13, 19, 21). These same exact methods are, however, not suitable for the treatment of tapered beams, while the present method, as shown, remains usable.Also, the general expressions, Eqs. (55) and (56), are mainly due to the fact that the present approximate solution has a considerably simpler form than obtainableexact solutions.
References
1Chwalla, E., Die Formeln zur Berechnung der ''vollmitragenden Breite" dünner Gurt und Rippenplatten , Der Stahlbau, 9, 7378, 1936.
2Cox, H. L., Smith, H. E., and Conway, C. G., Diffusion of Concentrated Loads into Monocoque Structures, R. and M. No. 1780, 1937.
Page 20
3Cox, H. L., Diffusion of Concentrated Loads into Monocoque Structures, III; General Considerations with Particular Reference to Bending LoadDistributions, R. and M. No. 1860, 1938.
4Cox, H. L., Stress Analysis of Thin Metal Construction, Journal of the Royal Aeronautical Society, 44, 231282, 1940.
5Duncan, W. J., Diffusion of Load in Certain Sheet-Stringer Combinations, R. and M. No. 1825, 1938.
6Ebner, H., and Koeller, H., Über den Kraftverlauf in längs und querversteiften Scheiben, Luftfahrtforschung, 15, 527542, 1938.
7Younger, John E., Metal Wing Construction, Part II, A.C.T.R. Series No. 3288, Material Division, U.S. Army Air Corps, 1930.
8Kuhn, P., Stress Analysis of Beams with Shear Deformation of the Flanges, N.A.C.A. Technical Report No. 608, 1937.
9Kuhn, P., Approximate Stress Analysis of Multi-Stringer Beams with Shear Deformation of the Flanges , N.A.C.A. Technical Report No. 636, 1938.
10Lovett, B. B., and Rodee, W. F., Transfer of Stress from Main Beams to Intermediate Stiffeners in Metal Sheet Covered Box Beams , Journal of theAeronautical Sciences, 3, 426, 1936.
11Metzner, W., Die mittragende Breite, Luftfahrtforschung, 4, 120, 1929.
12Reissner, H., Über die Berechnung der mittragenden Breite , Z. Ang. Math. Mech., 14, 312313, 1934.
13Reissner, E., Über die Berechnung von Plattenbalken, Der Stahlbau, 7, 282284, 1934.
14Reissner, E., Beitrag zum Problem der Spannungsverteilung in Gurtplatten, Z. Ang. Math. Mech., 15, 359364, 1935.
15Reissner, E., On the Problem of Stress Distribution in Wide-Flanged Box-Beams , Journal of the Aeronautical Sciences, 5, 295299, 1938.
16Reissner, E., The Influence of Taper on the Efficiency of Wide-Flanged Box-Beams , Journal of the Aeronautical Sciences, 7, 353357, 1940.
17Schade, H., Application of Orthotropic Plate Theory to Ship Bottom Structure , Proceedings, 5th International Congress, Applied Mechanics, pp. 144149,1938.
18Schapitz, E., Feller, H., and Koeller, H., Experimentelle und rechnerische Untersuchung eines auf Biegung belasteten Schalenflügelmodells,Luftfahrtforschung, 15, 563576, 1938.
19Schnadel, G., Die Spannungsverteilung in den Flanschen dünnwandiger Kastenträger, Jahrb. d. Schiffbaut, Ges. 27, 207291, 1926.
20Sibert, H., Effect of Shear Lag upon Wing Strength , Journal of the Aeronautical Sciences, 6, 418, 1939.
21von Kármán, Th., Die mittragende Breite, August-Foeppl Festschrift, Berlin, pp. 114127, 1924.
22Winny, H. F., The Distribution of Stress in Monocoque Wings , R. and M. No. 1756, 1937.
Page 21
Analysis of Shear Lag in Box Beams by the Principle of Minimum Potential Energy[Qu. Appl. Math. 4, 268278, 1946]
1Introduction
Let us consider a thin-walled box beam of web height 2h and cover sheet width 2w which is bent in such a way that one of the cover sheets is in tension while theopposite cover sheet is in compression (Figure 1). In elementary beam theory the assumption is made that the normal stress in the cover sheets does not vary in thedirection across the sheet. Because of the shear deformability of the cover sheets this assumption of elementary beam theory is often seriously in error for widebeams. In aeronautical engineering this effect is known under the name of shear lag.
Fig. 1.Sketch of spanwise element of box beam with doubly symmetric cross section.
Page 22
In recent papers, 1,2 shear lag in box beams has been analyzed by an application of the theorem of least work which is the basic minimum principle for the stresses.The present paper contains an application to the problem of shear lag of the theorem of minimum potential energy, which is the basic minimum principle for thestrains.3 It is shown that application of the theorem of minimum potential energy to the present problem leads to simpler and more general results than the applicationof the theorem of least work. While the least-work method furnishes the stresses in box beams with no cut-outs, application of the minimum-potential-energy methodfurnishes, in a simpler manner, the stresses in beams without or with cut-outs. It also furnishes beam deflections, and is equally convenient for beams supported instatically determinate or in statically indeterminate manner.
Application, in the manner described below, of the minimum-potential-energy principle to the problem of bending of thin-walled box beams leads to a differentialequation for the beam deflection which is a generalization of the relation z" = M/EI; this differential equation contains an additional term proportional to the fourthderivative of z which takes into account the shear deformability of the cover sheets. As the order of the differential equation in this theory is higher than the order ofthe differential equation of elementary beam theory, boundary conditions appear in addition to those of elementary beam theory. These additional boundaryconditions are different for beams with cut-outs and for beams without cut-outs.
The manner of application of the results obtained in the present paper is shown by solving explicitly the following four examples.
1. Simply supported beam. Load distributed according to a cosine law.
2. Cantilever beam with uniform load distribution. Cover sheets fixed at the support.
3. Cantilever beam with uniform load distribution. Cover sheets not fixed at the support.
4. Beam with both ends built in. Uniform load distribution.
For the sake of simplicity, it is assumed in what follows that the cross sections of the beams are rectangular and doubly symmetrical. It also is assumed that there is nospanwise variation of cross-sectional properties.
The author believes that the way in which the principle of minimum potential energy is here applied to the problem of shear lag will prove useful in other problems ofstructural mechanics. As an example of such future application, the theory for combined torsion and bending of beams with open or closed cross sections ismentioned.
2Formulation and Solution of Problem
In the following, we analyze a box beam of doubly symmetrical rectangular cross section, composed of cover sheets, sidewebs and flanges. A given distribution ofloads is applied to the sidewebs, acting normal to the plane of the cover sheets (Figure 1). To this load distribution there corresponds a distribution of bendingmoments M(x). The spanwise coordinate being x,
1E. Reissner, Least work solutions of shear lag problems, Journal of the Aeronautical Sciences, 8, 284291 (1941).
2F. B. Hildebrand and E. Reissner, Least work analysis of the problem of shear lag in box beams, N.A.C.A. Technical Note No. 893 (1943).
3For a formulation of these theorems see for instance I. S. Sokolnikoff and R. D. Specht, Mathematical theory of elasticity, McGraw-Hill Book Co., Inc., New York, 1946, pp. 275287.
Page 23
let y be the coordinate in the plane of the cover sheets perpendicular to the x direction, and z(x) the deflection of the neutral axis of the beam.
The potential energy of the bent beam may be considered as composed of three parts. The first part is the potential energy of the load system. This may be written inthe form
the integral being extended over the entire length of the beam.4 The second part is the strain energy of sidewebs and flanges. This may be written in the form
the quantity Iw denoting the principal moment of inertia of the two sidewebs and flanges.
The third part is the strain energy of the two cover sheets. If it is assumed that the normal strains in the chordwise direction in the sheets may be neglected, asdiscussed in the reference given in Footnote 1, then the strain energy of the two sheets is given by the integral
where the quantity t denotes the cover sheet thickness, and where E and G are the effective moduli of elasticity and rigidity. Spanwise normal strain x and shearstrain are then expressed in terms of the spanwise sheet displacement u as follows
The theorem of minimum potential energy states that the total potential energy
becomes a minimum for the correct displacement functions u and z, if only such displacement functions are compared which satisfy all conditions of support andcontinuity imposed on the displacements.
Direct application of this condition by means of the calculus of variations leads to a partial differential equation for u and to a complete system of boundary conditions.In what follows, an ordinary differential equation for the beam deflection z and boundary conditions for it are obtained instead. This is done by making a suitableapproximation for the sheet displacements u and by applying the rules of the calculus of variations to the resultant approximate expression for the potential energyfunction.
A reasonable assumption for the spanwise sheet displacements is
4Equation (1) implies that the beam is supported in such a manner that the end forces and moments can do no work. This restriction shortens the developments slightly.
Page 24
The second term on the right of Eq. (6) represents the correction due to shear lag. Instead of the vanishing chordwise variation of the sheet displacements ofelementary beam theory, we now assume a parabolic variation. The relative magnitude of the function U is a measure for the magnitude of the shear lag effect. Theform of the correction is such that continuity of the displacements along the flanges, that is along y = ±w, is preserved.
Denoting differentiation with respect to x by primes, we obtain the following expressions for the strains in the sheets from Eqs. (6) and (4):
On the basis of Eqs. (7) and (8) the following expression for the strain energy of the sheets is obtained:
In Eq. (9) the integration with respect to y is carried out. Setting
we have
Substituting Eqs. (11), (2) and (1) into Eq. (5), we obtain the following expression for the potential energy of the system
Differential equations and boundary conditions for z and U are obtained by making
Thus, with x1 and x2 denoting the ends of the interval of integration,
As z" and U are arbitrary in the interior of the interval (x1, x2) the terms multiplying them must vanish. This gives the following two differential equations
Page 25
The integrated portion of Eq. (14) defines the boundary and transition conditions for the function U. At a section where the sheet is fixed, U = 0 and
At a section where the sheet is not fixed and consequently U is arbitrary,
Transitions conditions for adjacent bays with different stiffness are:
The above boundary and transition conditions are in addition to those imposed on z and M in elementary beam theory, as may be verified by repeated integration byparts of the term containing z" in the integral of Eq. (14).
3The Modified Beam Equation and Its Boundary Conditions
By eliminating the quantity U from Eqs. (15) to (19), we obtain a system of relations containing the beam deflection z only.
The differential equation for z is derived by differentiating Eq. (16) and substituting U from Eq. (15). There follows
When the shear deformability of the sheets is neglected, that is when it is assumed that G = , Eq. (20) reduces to the well known result of elementary beam theory.
Equation (20) may be written in the alternate form
With the help of Eqs. (15) and (16), the boundary condition (17), which holds when the sheet is attached to the support, is transformed into
Similarly, the boundary condition (18), which holds when the sheet is not attached to the support, becomes
The continuity conditions (19) may be transformed in an analogous manner.
The values of the sheet stresses may be obtained from Eqs. (9) and (10). From Eq. (9) it follows that the flange stress is given by
For the application of the results it may be noted that the differential equation (21) can first be solved for the value of z" which, according to (24), gives directly the
Page 26
approximate value of the flange stress . The magnitude of the deflection z can then be found from the value of z" as in elementary beam theory.
For the evaluation of the solution we define the following two parameters
With (25) and (26) the differential equation (21) becomes
The boundary condition at an end section where the sheet is attached to the support becomes
and the boundary condition at an end section where the sheet is not attached to the support becomes
4Examples of Applications
(Figure 2)
1Simply Supported Beam. Load Distributed According to a Cosine Law
Designating the span length of the beam by l and assuming the origin of the coordinate system at the center of the beam, we consider the moment distribution
A particular solution of Eq. (27) is
As Eq. (31) satisfies the boundary condition (29) and the conditon of vanishing deflection at the ends of the beam, it is the complete expression for the deflectionfunction. When 1/k = 0, Eq. (31) reduces to the expression for z in the case where shear lag is not taken into account. The factor
expresses the effect of shear lag on deflection and flange stresses.
2Cantilever Beam with Uniform Load Distribution. Cover Sheets Fixed at Support
Assuming that, contrary to what is indicated in Figure 2, the free end of the beam has the coordinate x = 0 and the fixed end of the beam the coordinate x = l, wemay
Page 27
Fig. 2.Diagrammatic sketches of beams analyzed as
examples of application of the theory.
write the moment distribution in the form
The differential equation (27) then becomes
Page 28
Solving for z", we find
Satisfying the boundary condition (29) when x = 0 and (28) when x = l, we obtain
According to Eq. (24), the flange stress at the fixed end of the beam becomes
We take for a numerical example
so that according to Eqs. (25) and (26)
and we find
By application of the least work method1,2 a factor 1.186 is obtained instead of the factor 1.190 in Eq. (40).
The deflection of the beam is obtained from Eq. (36) by integrating twice and making z(l) = z (l) = 0. In the present case, the correction due to shear lag for themaximum deflection is about ten percent.
3Cantilever Beam with Uniform Load Distribution. Cover Sheets Not Fixed at Support
Moment distribution and differential equation are given by Eqs. (33) and (34). The constants of integration in (36) are determined by satisfying Eq. (29) for x = 0 andfor x = l. There follows
Taking again Is/I = .5, we should have, for the flange stress at the supported end, a value twice as large as the stress according to elementary beam theory for a beamwith sheet attached to the support. In the present solution the factor 2 is replaced by n = 1.714. This indicates that with the assumed parabolic chordwise variation ofsheet displacement the condition that at the support of the beam the sheet is free of stress is only approximately satisfied. The same difficulty arises in methods whichincorporate the ability of the sheet to carry normal stresses as effective width
Page 29
contributions to the strength of stiffners.5 This difficulty is not serious when the main purpose of such ''cut-out" calculations is the determination of the distance overwhich the cut-out is effective and its effect on the over all beam stiffness.6
The localization of the effect of the cut-out may be seen by writing (41) in the form
This equation indicates that the influence of the cut-out is small as soon as the distance l x satisfies the inequality
Thus, the wider the sheet and the smaller the value of the shear modulus G, the farther away does the effect of the cut-out extend in the spanwise direction.
The magnitude of the beam deflection is obtained from (41) in the form
which determines the constants of integration such that z(l) = z (l) = 0. For the deflection at the free end of the beam, we have
For a beam with dimensions as in (38) and (39), Eq. (45) becomes
This indicates that for a beam with dimensions as given shear lag due to lack of sheet restraint at the supported end of the beam is responsible for a thirty percentincrease of the maximum beam deflection as compared with the result of elementary beam theory for a beam fully restrained at the supported end. This increase ofdeflection of thirty percent compares with one of hundred percent which is obtained if the contribution of the cover sheets is neglected.
4Beam with Both Ends Built-In. Uniform Load Distribution
The distribution of bending moments may be written as
5P. Kuhn and P. Chiarito, Shear lag in box beamsmethods of analysis and experimental investigations, N.A.C.A. Technical Report No. 739 (1943).
6Exact solutions of problems of this kind have been obtained by F. B. Hildebrand, The exact solution of shear-lag problems in flat panels and box beams assumed rigid in the transverse direction,N.A.C.A., Technical Note No. 894 (1943).
Page 30
The value of M0 is determined by the load intensity, the value of M1 in this statically indeterminate problem has to be determined from the displacement boundaryconditions. The boundary conditions are
For these boundary conditions the moment distribution is not affected by shear lag, provided the moment distribution is symmetrical about the mid-span section of thebeam. Indeed, the differential equation (27) may be integrated to give
the limits of integration being so chosen that Eq. (51) satisfies the conditions of zero slope and zero vertical shear at the mid-span section. In view of (49) and (50),Eq. (51) implies
regardless of whether or not shear lag is taken into account. A considerably less simple proof of the same fact by means of the least work method has been given inthe reference quoted in Footnote 2. For the moment distribution of Eq. (47) there follows, from (52),
and hence
With this value of M and the requirement that z" be an even function of x, Eq. (27) is solved in the form
The constant C2 is determined from Eq. (50). There follows,
Taking a beam five times as long as wide, that is l/2w = 5, and assuming the remaining parameters as in (38) and (39), we obtain the following expressions for theflange stresses at the built-in section and at the center section of the beam
Page 31
These results agree to within a fraction of a percent with the corresponding results obtained by the least work method.2 It is worthy of note that, for this beam withboth ends built-in, shear lag is considerably larger than for a cantilever beam with the same load, same width and half the span of the beam with both ends built-in. Ifboth beams had the same span, the discrepancy would be even larger.
The deflection z of the beam is obtained from (56) and (48) in the form
Corresponding to the stresses of Eqs. (57) and (58) we find for the deflection at mid-span
Shear lag in this beam is thus responsible for an almost fifteen percent increase in deflection. This percentage increase of deflection, while appreciable, is considerablysmaller than the percentage increase of maximum flange stress.
Page 32
Note on the Shear Stresses in a Bent Cantilever Beam of Rectangular Cross Section*
[J. Math. & Phys. 25, 241243, 1946]
In this note we wish to report on some calculations which we have made concerning the distribution of shear stresses, according to the St. Venant theory, in bentcantilevers of rectangular cross section.1 Referring to Figure 1, our main result is the fact that for sufficiently wide beams (b/a >> 1) the component of stress yz(a, )not only is of the same order of magnitude as the stress xz(0, b) for which values have previously been calculated,1 but exceeds xz(0, b) in magnitude.2
We list first the exact expressions for xz(x, y) and yz(x, y) in a form convenient for our purposes and deduce from them simplified expressions for xz(0, b) andyz(a, y) which ensure accuracy for all decimals computed, when b/a 4. On the basis of these formulas we calculate for a number of values of b/a in the range (2,) (i) values of xz(0, b), (ii) values of the coordinate y = for which yz(a, y) is greatest, (iii) values of yz(a, ). The numerical results obtained are collected in
Table I.
Exact expressions for the shear stresses may be written in the following form, with 4ab = A,
*With G. B. Thomas.
1For an exposition of this theory see S Timoshenko's book on Theory of Elasticity (pp. 285288, 292298, McGraw Hill, 1934).
2We are indebted to Prof. S. Timoshenko and to Prof. H. Reissner for the information that this fact appears not to have been noted previously.
Page 33
Fig. 1.Diagram showing dimensions of cantilever beam and nature of shear stress
distribution.
From Eqs. (1) and (2) are derived the following approximate expressions which are exact within the accuracy of our calculations when b/a 4:
Substituting in Eq. (3) the series sums
this equation becomes
To obtain the maximum of yz(a, y) we set ( yz(a, y)/ y)y= = 0. According to Eq. (4) the maximum condition becomes
Page 34
Carrying out the summation in (6) we have
or
Equation (8a) may also be written in a form which gives directly the distance of the maximum location from the edge in units of the thickness 2a,
Introducing Eq. (a) in Eq. (4) we obtain
Equation (9) is less simple in appearance than Eq. (5) but is readily evaluated as the remaining series converges rapidly.
From (9) and (5) we obtain the following limit relation
which shows that for wide beams (plates) the component of shear parallel to the face of the plate reaches a value which is almost 35 percent higher than the valuereached by the component of transverse shear. The way in which this limit state is approached is apparent from the values given in Table I which has been computedon the basis of Eqs. (5), (8b), and (9). In the calculations the value of has been taken as .25.
Table I
0 1.000 0.000 0.000 0.0002 1.39(4) 0.31(6) 0.22(7) 0.31(4)4 1.988 0.968 0.487 0.5226 2.582 1.695 0.656 0.6498 3.176 2.452 0.772 0.73910 3.770 3.226 0.856 0.81015 5.255 5.202 0.990 0.93920 6.740 7.209 1.070 1.03025 8.225 9.233 1.123 1.10250 15.650 19.466 1.244 1.322
1.347
Page 35
On Finite Pure Bending of Cylindrical Tubes[Österreichisches Ing. Arch. 15, 165172, 1961]
Introduction
We are concerned in what follows with a rather general and simple formulation of the problem of finite pure bending of thin-walled cylindrical tubes of arbitrary crosssection. Our analysis contains as special cases Brazier's analysis of the flattening instability of circular cross section tubes1 and our own refinement of Brazier's theory,which in turn is a special case of results for toroidal tubes with circular cross section.2
The physical basis of the present work is similar to that of our earlier work. Certain simplifications arise through the consideration of initially straight tubes instead oftoroidal tubes. Further simplifications are due to an appropriate use of the variational theorem for displacements in elasticity.
Formulation of the Problem
We consider a cylindrical tube with cross section before deformation specified by middle surface equations x = x(s), y = y(s), where s represents circumferentialarclength, and by a wall thickness function h = h(s).
The originally cylindrical tube is deformed through the application of end moments Mx and My. These moments Mx and My result in a curving of the axial fibers of thetube, with curvature radii Rx and Ry. As long as elementary beam theory applies, the radii Rx, Ry and the moments Mx and My are related by equations of the form
In these formulas E = E(s) is the modulus of elasticity for axial stress, and the origin of the x, y-system of coordinates is chosen such that
Associated with the uniform curving of axial fibers is a deformation of the cross section of the tube which changes x into x + u and y into y + v and which isneglected in elementary beam theory. It is of the essence in what follows that this cross sectional deformation may be assumed to take place without meridionalextension of the middle surface of the tube (Figure 1).
1L. G. Brazier, Proc. Roy. Soc. A, 116, 104114, 1927.
2E. Reissner, Proc. 3rd U.S. Nat. Congr. Appl. Mech. 5169, 1958, and J. Appl. Mech. 26, 386392, 1959.
Page 36
Fig. 1.Element of tube cross section before and after
deformation.
The linear displacements u and v define an angular displacement which in turn defines a circumferential bending strain , given by
The components u, v and are connected through two relations which may be read from Figure 1, and which are
In addition to the circumferential bending strain we have an axial direct strain , which through a stress strain relation of the form = E , enters into the formulasfor moment components Mx and My. These moment components are defined with reference to the deformed cross section, as follows:
In order to express in terms of Rx, Ry, u, v we make use of the fact that pure bending of the tube takes place in such a way that plane sections perpendicular to theaxis of the undeformed tube are deformed into plane sections perpendicular to the curved axis of the deformed tube. This means that is given by the formula
Introduction of (7) into Eqs. (6) leads to expressions for Mx and My which are the extensions of (1) and (2) and which are
Page 37
In order that the cross sections of the tube are free of resultant forces, we must further have
which generalizes Eqs. (3) consistent with the step from (1) and (2) to (8) and (9).
In order to evaluate the basic formulas (8) and (9) it appears necessary to determine the functions u and v in their dependence on the geometry of the cross sectionand on the values of Rx and Ry. It will be seen in what follows that a somewhat simpler procedure is possible in which and are determined rather than u and v.
Derivation of Differential Equations
We base our derivation on the requirement that, for given values of Rx and Ry, the strain energy of the bent tube be a minimum. We take as expression for strainenergy, per unit of axial tube length
where and are given by (7) and (4), where C = Eh and where D is a circumferential bending stiffness function which for isotropic homogeneous tube materials isgiven by D = Eh3/12(1 v2).
The quantity s is to be made a minimum subject to the constraint Eqs. (5). Considering the form of and in (7) and (4) we may evaluate the minimum conditionwithout explicit use of the displacement components u and v by writing as constraint condition
Introducing from (4) and introducing the constraint condition (12) by means of a Lagrange multiplier F the condition of minimum strain energy assumes the followingform
The variational Eq. (13) is equivalent to two differential equations of the form
which must hold, together with the constraint Eq. (12). Boundary conditions for the system (12), (14) and (15) are the conditions of periodicity in s for , F, andd /ds. In view of one of these periodicity conditions we have so that the condition of no resultant force over the tube cross section is automaticallysatisfied.
Equations (8) and (9) for Mx and My may now be written, through the use of (6) and appropriate integration by parts, in the alternate form
Page 38
We may finally reduce the two first-order Eqs. (12) and (14) to the second order differential equation
and thereby reduce the problem to the two simultaneous second order Eqs. (15) and (17) for and F.
We note that these two equations may be written in the form
and
Equations (15 ) and (17 ) may be used as the starting point of an expansion procedure which will now be discussed.
Expansion in Powers of 1/Rx and 1/Ry
A formal expansion procedure for (15 ) and (17 ) which is suggested by the appearance of these equations upon writing cos = 1 1/2 2 ± . . ., sin = 1/6 3
± . . . consists in setting
where Fn and n are homogeneous of degree n in the quantities 1/Rx and 1/Ry. In particular
Introduction of (18) to (20) into (15 ) and (17 ), written as
Page 39
leads to the following system of successive differential equations
Solution of (21) to (23), and of the corresponding subsequent systems, is carried out by direct integration. We find from (21) and from the periodicity condition inview of the fact that cos = dx/ds and sin = dy/ds,
and
From (22) follows
with corresponding expressions for d 11/ds and d 02/ds. It is apparent that, in general, further integrations must be carried out numerically.
Corresponding expansions for applied moments, as defined by (16) are of the form
and
With (19), (20) and (25) it is found that Mx may be written as
Page 40
where the Xjk and Yjk are suitable constants. An analogous expression may be deduced for My.
The following observations may be made:
1. Equation (29), when specialized to the case of the homogeneous constant-wall thickness, circular-cross-section tube, and without any of the terms represented bydots or the terms with 1/Ry, reduces to the formula of Brazier for the moment Mx in terms of the curvature 1/Rx.
2. As long as the calculations of Mx and My do not go beyond third-degree terms in R-1 they may equally well be considered to be based on a linear system ofdifferential equations, of the form
Associated with this linear system are non-linear expressions for Mx and My of the form
3. Higher degree terms than those displayed in (29) may have an effect of the order of ten percent or more in the range of practical interest of the theory. The fact thatthis may be so becomes apparent even without explicit calculations upon introduction of appropriate non-dimensional variables and parameters.
Equations for the Circular-Cross Section Tube
We designate the radius of the circular cross section by b and write the coordinates of the middle surface of the tube in terms of a polar angle , as follows:
Therewith,
and Eqs. (15) to (17) become
Page 41
If we limit ourselves in what follows to the case that D and C are independent of then, because of symmetry, we may further limit ourselves to the consideration ofthe case 1/Ry = 0 and My = 0. Writing Rx = R we have then
and
If we further set
and indicate differentiation with respect to by primes, then the system (39), (40) assumes the form
and Eq. (41) becomes
where EI = Cb3 is the bending stiffness factor of the tube according to elementary theory. Equation (45) may be written in the alternate form
in which a dimensionless applied moment m appears as a function of the dimensionless curvature parameter .
Equation (45 ) may be simplified by introducing (44) and by integrating by parts. In this way there follows the relation
Equations (43) and (44) may be solved by expansion in powers of 2. We set
and expand both cos and sin in powers of 2. In this way we obtain a system of
Page 42
successive differential equations of which we list the first seven equations, as follows:
The solutions of (48) to (52) must be periodic of period 2 in . We list below the form of these solutions for Eqs. (48) to (50).
In terms of these expansions we have for the dimensionless moment m,
Introduction of (53) to (55) results in the following explicit formula for m,
Equation (57) is in agreement with our previous result for this problem and reduces, upon omission of all terms except the first two, to Brazier's result.
In order to delineate the range of values of which is of practical interest, we determine the value of for which flattening instability occurs. This value of isobtained by solving the equation dm/d = 0. From (57) follows for this value according to Brazier while if 5 is retained in (57) we obtain
. Corresponding values of m are mB = 1.086 and mC = 0.998. These numerical data indicate that the exact value of m for whichdm/d = 0 may differ from Brazier's value by an appreciable amount. They also indicate that it will be of interest to obtain numerically more accurate solutions of(43) and (44) than given here, for values of 2 in the range from two to three.
Page 43
Finite Pure Bending of Circular Cylindrical Tubes*
[Qu. Appl. Math. 20, 305319, 1962]
Introduction
The present paper is concerned with the determination of stresses, deformations and stiffness of originally straight circular tubes in pure bending. The non-linearproblem of determining the stiffness of such a tube as a function of the applied moment and the determination of a critical moment for which flattening instability occurshas originally been discussed by Brazier [1].
An alternate more precise formulation of the problem of flattening instability of circular cross-section tubes is contained in a recent paper by one of the presentauthors [2], as a special case of results for pure bending of general cylindrical tubes. In this same paper approximate solutions of the non-linear differential equationsof the problem were obtained as expansions in powers of a dimensionless parameter . It was found that the first terms of these expansions give the results of lineartheory and that consideration of two terms gave the results of Brazier [1]. It was further found that consideration of three terms lead to results which differed fromBrazier's to the order of ten per cent. Since the calculation of additional terms in the -series becomes progressively more complicated, an alternate determination ofthe results is of interest. The present paper presents such an alternate determination, involving the iterative solution of a system of two simultaneous non-linear integralequations. In addition to this, the previous three-term -series are extended by the calculation of fourth terms. Our calculations lead to the note-worthy conclusionthat Brazier's results for flattening instability are quite close to the results of precise calculations based on the equations given in [2], in the sense that consideration ofthree and even four terms in the -series lead to results which are further from the correct results (in the critical -range) than the results based on only two terms inthe -series.
In addition to these conclusions for the problem of the flattening instability, we obtain in what follows quantitative results for the non-linear behavior of stresses anddeformations in the tube. We find, in particular, that when the applied bending moment is of the order of the critical moment, the order of magnitude of the secondarycircumferential wall bending stressesassociated with the flattening of the cross-sectionis the same as the order of magnitude of the primary longitudinal direct fiberstresses in the tube.
Basic Equations
It has been shown previously [2] that the problem of pure bending of a tube with cross-section before deformation given by x = b sin , y = b cos for
*With H. J. Weinitschke.
Page 44
0 , 2 is associated with two simultaneous non-linear differential equations for a stress function variable F and an angular displacement variable of thefollowing form
In these equations R is the radius of curvature of the originally straight axis of the tube, D = Ebh3/12 is the circumferential wall bending stiffness factor and 1/A = Eshis the axial stretching stiffness factor of the tube.1
Fig. 1.Notation.
Equations (1) are to be solved in the interval 0 1/2 subject to the following boundary conditions,
To be determined are in particular the applied moment M given by
axial fiber stress and the circumferential bending stress b given by
1In our earlier paper [2] it had been assumed that D = Ebh3/12(1 2) which is the appropriate circumferential stiffness factor for small cylindrical bending. Considering that in the range of practicalinterest we will have cylindrical bending with relatively large deflections a stiffness factor D without the term (1 2) seems more appropriate.
Page 45
and cross sectional flattening and bulging displacements given by
Equations (1) to (3) are made non-dimensional by setting
Indicating differentiation with respect to by primes, we now have the differential equations
with boundary conditions
The dimensionless moment m becomes, after an integration by parts
and dimensionless stresses may be obtained in the form
Two alternative dimensionless stress quantities may be defined as follows. One definition makes use of the maximum fiber stress which would exist in the
tube bent to a radius R if there were no flattening effect. Introduction of into Eqs. (4) and (6) leads to the formulas
As flattening makes the tube more flexible than it would otherwise be, we expect that the ratio of will decrease as increases.
A second non-dimensionalization makes use of the maximum fiber stress , where I = b3h, which would exist in the tube subject to an applied moment M
if there were no flattening effect. Introduction of into Eqs. (4) and (6) leads to the formulas
In these equations we consider a function of m which is defined by means of Eq. (9). Whether or not will increase or decrease with increasing m will
depend on the shape of the curve for as function of and cannot be predicted without numerical calculations. Numerical calculations are also needed for acomparison of the magnitude of the secondary bending stresses b with the magnitude of the primary direct fiber stress , in their dependence on or m.
Page 46
Expansion in Powers of
The boundary value problem (7) and (8) may be solved, as in [2], by expansions
Expanding sin and cos in terms of 2 we obtain a system of successive linear differential equations, of which we list the first seven as follows:
We require that the boundary conditions (8) be satisfied identically in .
The functions 0, 2, 4, 2, 4 have been calculated in [2] and are listed here for completeness sake. The functions 6 and 6 as well as formulas for stresses anddisplacements have not been obtained before. We find
Insertion of Eq. (12a) into Eq. (9) gives an expression for the moment function m of the form,
Substitution of (17a), (17c), (18b) and (20) and subsequent integration gives the relation
The partial sum obtained by omitting the last listed term in Eq. (22) agrees with the result given in [2]. Retaining only the first two terms on the right side of Eq. (22)gives the result of Brazier. A quantitative discussion of the dimensionless moment curvature relation (22) is given further on in conjunction with a discussion of thecorresponding relation obtained from the numerical solution of the integral equation.
Page 47
For the calculation of stresses in accordance with Eqs. (9a, b), (10) and (11) we need the following expressions for the derivatives and :
which follow from Eqs. (12) and (17) to (20).
Expressions for the flattening and bulging displacements follow from Eqs. (5), (12) and (17) to (20) in the form
We note that while for sufficiently small we have , it is found that for increasing the flattening displacement u increases more rapidly than the bulgingdisplacement v.
4Integral Equation Formulation and Numerical Solution
The differential equations (7) with boundary conditions (8) may be reduced to the following system of integral equations
where x = /1/2 , , and G , G are Green's functions given by
(in what follows the bars on , will not be written, for simplicity's sake.) Numerical solutions of adequate accuracy of these integral equations may be obtained by acombination of iteration and numerical integration.
Values of the dimensionless moment m and of the displacements u and v are calculated by introducing the solutions (x), (x) into the integrals in Eqs. (3) and (5).Dimensionless stresses in accordance with Eqs. (9a, b), (10) and (11) are obtained by calculating d /dx and df/dx in terms of the integrals which follow, rather thanby numerical differentiation of the discrete values of and obtained from the solution
Page 48
of Eqs. (27) and (28). We find, in dimensionless form
where
In order to describe the iteration scheme to be used, we write Eqs. (27) and (28) in the form
where the integral operator K1 depends on only and where K2 is linear in . The most straightforward iteration of (39) is expressed by n+1 = K1[ n], n+1 =2K2[ n, n]. Using the iteration n+1 for the calculation of n+1, a more rapidly converging iteration scheme for (39) is
The two equations (40) can be written as one equation as follows
Numerical calculation shows convergence of (41) for values of 2 up to about 5. For larger 2, examples of both oscillations and steady increase in magnitude ofsuccessive iterates were obtained, that is, the iteration scheme diverges.
In order to obtain solutions for larger values of 2, the iteration scheme is modified by introduction of two ''relaxation parameters" and as follows2
Clearly, if the sequences n, n converge, they converge to a solution of (39). The relaxation parameters and are allowed to depend on 2. With this schemeand with appropriate choice of , , the speed of convergence was considerably increased as compared with the iteration (40), and convergence was induced forvalues of 2 for which (40) diverges. In this way solutions up to were obtained; the range
2A similar iteration scheme using one relaxation parameter was employed by Keller and Reiss in solving a system of non-linear difference equations [3].
Page 49
could probably be extended to still larger 2 by proper choice of , , although solutions beyond a critical value (see Section 5) are physically less interesting.Table 1 shows some numerical results. The special case = = 1 is identical with the iteration (40). A proof for the convergence of the modified scheme (42) forvalues of 2 > 1.8 has yet to be obtained.
Table 1. Number of iterations for 0.1% accuracy of solutions of (39)2 = = 1 = 1, = 3/4 = 3/4, = 1 = 3/4, = 3/4 = .65, = .60
2.6 6 8 96.0 34 10 189.0 oscill. 32 20 1613.0 div. div. 32 2720.0 div. div. oscill. 33
Some numerical solutions of Eqs. (39) have previously been calculated by G. L. Brown [4], who used Simpson's rule for numerical integration combined with aniteration equivalent to (40). He found, using interval lengths x = 1/5, 1/9 and 1/16 that the latter was not small enough to draw conclusions on the accuracy of theresults obtained.
5Discussion of Results
The integral equations (39) have been solved for values of 2 up to 25. These solutions are in the following referred to as "numerical solutions." A comparison of the-expansion with the numerical solutions shows that the -expansion solutions are accurate almost up to the critical value c, defined by dm/d = 0 (see Figures
24). For larger values of , they become quite inaccurate. As a check on the numerical solutions, Eqs. (7) were approximated by a finite difference system
which was solved for i, i i = 2, . . ., N, the values 0, 1, 0, 1 being given from the solution of the integral equations. If the solutions of Eqs. (43) differed bymore than one unit in the third figure from the solutions of Eqs. (39), the latter solutions were recalculated with a smaller spacing h, and the check via Eqs. (43) wasrepeated for that spacing.
The moment curvature relation is shown in Figure 2. What is of particular interest is the value c, for which flattening instability occurs. For this value of thedimensionless curvature, the moment m attains its maximum value mc. The numerical values for c, and mc when retaining 2, 3, or 4 terms in Eqs. (22), and thecorresponding values c, mc from the numerical solution are given below. The numbers of the last column of Table 2 were obtained by interpolation from a largescale plot of the dimensionless moment-curvature relation near the critical point ( c, mc).
Figure 3 shows the maximum values of the dimensionless direct stress b/Esh and bending stress bb/Ebh as defined by Eqs. (9a, b). The maximum bending stress
Page 50
Fig. 2.Dimensionless moment curvature relation.
Fig. 3.Dimensionless maximum direct and bending stresses.
occurs at the neutral plane, that is b,m = b(0). The bending stress b(1/2 ) at the farthest distance from the neutral plane is slightly less in absolute value than b(0)and is also displayed in Figure 3.
For small bending stresses are negligible with respect to direct fiber stresses. For values of approaching c, the bending stresses become of the same order of
Page 51
Table 2. Critical curvatures and moments2 terms 3 terms 4 terms numerical solution
c 1.513 1.422 1.468 1.526mc 1.086 0.998 1.034 1.063
Fig. 4.Dimensionless maximum flattening and bulging displacements.
magnitude as the direct stresses. It is interesting to note that the maximum direct stress is not attained at but for the somewhat larger value .
Values of the dimensionless maximum flattening and bulging displacements of the cross section are displayed in Figure 4. For small values of , the two displacementsare nearly identical. As in the -expansions (Eqs. (25) and (26)), flattening increases faster with increasing than bulging, the rate of increase being strongest near
c, for both flattening and bulging.
Next we compare our results with those of elementary linear beam theory. In Figure 5 the ratios and with = Es/Eb, according to Eqs. (11)3 are
plotted against the dimensionless moment m/mc. In Figure 6, the stress ratios and are plotted against . We conclude from these graphs that fora given moment, the direct stress produced according to the nonlinear theory is
3The calculations were carried out on the IBM-709s at Western Data Processing Center, Univ. of Calif., Los Angeles, and at Computation Center, M.I.T., Cambridge.
Page 52
Fig. 5.Stress ratios as functions of dimensionless
moment m/mc.
Fig. 6.Stress ratios as functions of dimensionless
curvature .
slightly larger than the value given by the linear theory which neglects the flattening of the cross section. On the other hand, for a given curvature of the central axis ofthe tube, the nonlinear theory stress is less than that given by the elementary beam theory.
Finally, we consider briefly the stress distribution over the cross-section of the tube and its deviation from the elementary (linear) stress distribution. As is seen from
Page 53
Fig. 7.Dimensionless direct stress vs. dimensionless distance d/b from neutral axis for
several values of .
Fig. 7, this deviation is quite small for values of c. For larger values the distribution becomes more markedly nonlinear, in fact, for > 2.5 the maximum fiberstress is no longer attained in the outermost fiber.
References
1. L. G. Brazier, "On the flexure of thin cylindrical shells and other thin sections," Proc. Roy. Soc. A, 116, 104114 (1927).
2. E. Reissner, "On finite pure bending of cylindrical tubes," Österr. Ing. Arch., 15, 165172 (1961).
3. H. B. Keller and E. L. Reiss, "Iterative solutions for the nonlinear bending of circular plates," Comm. Pure Appl. Math., 11, 273292 (1958).
4. G. L. Brown, "On the numerical solution of two simultaneous non-linear differential equations arising in elasticity," M. S. Thesis, Mass. Inst. of Technology, June1960.
5. L. Collatz, "Numerische Behandlung von Differentialgleichungen" 2. Auflage, Springer 1955.
Page 54
Considerations on the Centres of Shear and of Twist in the Theory of Beams*
[Muskelisvili 80th Anniversary Volume, pp. 403408, Moscow 1972]
1Introduction
The considerations which follow resulted from endeavours to understand, appreciate and reconcile different statements in the literature on the subject of the centre ofshear and of the centre of twist, and on the conditions under which these centres would or would not coincide. The outcome of our considerations was an approachwhich to us seemed quite different from what had been done previously, and rather more appropriate to the question, as we hope to make evident in what follows.
Specifically, our first object is to show that coincidence of the centres of twist and of shear may be taken to be no more than a natural consequence of a reasonableformulation of the problems of torsion and of flexure in the theory of beams.
Our second object is to show that an explicit, approximate determination of the location of these centres may be based on the Saint-Venant solutions of the problemsof torsion and flexure (which by themselves are known to leave these centres arbitrary), in conjunction with a direct-methods-of-the-calculus-of-variations-type useof the principle of minimum complementary energy.
2A Formulation of the Problems of Torsion and Flexure
We consider a linear elastic prismatical body, with boundaries defined, in an (x, y, z) co-ordinate system, by means of a cylindrical surface f(x, y) = 0 and twoparallel planes z = 0 and z = L.
We designate displacements by ux, uy, uz and stresses by x, y, z, xy, xz, yz and we assume that the usual three-dimensional homogeneous equations of linearelasticity hold. We further assume that the cylindrical boundary portion of the body is free of tractions and that the plane boundary portion z = 0 is fixed.
In regard to the plane boundary portion z = L we assume the absence of normal tractions while at the same time its tangential displacement components correspondto a plane rigid body translation and rotation, i.e., we stipulate
where U, V, and are given constants.
It is evident that we may, in conjunction with the above set of prescribed boundary conditions, consider a set of overall tractions consisting of transverse
*With W. T. Tsai.
Page 55
forces P and Q and of a torque T defined by
In view of the linearity of the problem, P, Q and T will come out to be linear combinations of U, V and and it may be concluded that, inversely, U, V and arerelated to P, Q and T in the form
It may further be concluded that Eqs. (3) retain their form for such generalizations of the stated problem as are obtained upon replacing the given conditions of endfixity at z = 0 by homogeneous linear relations of the form gi(ux, uy, uz, z, xz, yz) = 0 for i = 1, 2, 3, and by replacing the condition z = 0 for z = L by anyhomogeneous linear relation g(uz, z) = 0 for z = L.
3The Centre of Twist and the Centre of Shear
The possibility of defining points in the cross-sections of a prismatical beam which may be designated as centre of twist and as centre of shear, respectively,depends on the rationality of the concept of the cross-sections of the beam rigidly translating and rotating in their own planes, at least approximately. We proposehere to sharpen these definitions by confining them to cross-sections which are prescribed to translate and rotate rigidly in their own planes, i.e., to the end cross-section of the prismatical beam for which the boundary conditions (1) and its mentioned generalizations are stipulated*.
Definition of Centre of Twist
With end cross-section displacements prescribed in the form ux = U y , uy = V + x we define the co-ordinates xT, yT of the centre of twist at those values of xand y for which ux = uy = 0, while at the same time P = Q = 0, i.e.,
Introduction of Eq. (3) into Eq. (4) expresses yT and xT in terms of influence coefficients, as follows:
*Given that the end cross-section translates and rotates rigidly, interior cross-sections may or may not also translate and rotate rigidly, exactly or approximately. To the extent that they do, byvirtue of geometrical and material properties of the beam, one may ask for the dependence of the location of the centres of twist and of shear on distance from the end section of the beam, as hasbeen done sometime earlier, in 1955, by the first-named author, in regard to the centre of twist.
Page 56
Definition of Centre of Shear
We define co-ordinates xs, ys of the centre of shear as the co-ordinates of the point of intersection of the lines of action of the end forces P and Q for the case that(1) there is no rotation of the end section, and (2) the total torque about the point x = y = 0 (and any other point in the cross-section) is due to the forces P and Q.
Setting in accordance with the above definition, in Eqs. (3)
we obtain from the third equation in (3) the relation
Since (7) must hold for arbitrary ratios P/Q we have as expressions for ys and xs
Conditions for Coincidence of Centre of Twist and Centre of Shear
Inspection of Eqs. (8), (5) and (3) reveals that a sufficient condition for the coincidence of the centre of twist and the centre of shear is that the influence coefficientmatrix C in Eqs. (3) be symmetric.
Since this matrix will be symmetric provided a strain energy function exists for the beam problem, coincidence of the two centres is established in the foregoing formost cases of practical interest.
Conversely, we may expect that the centre of twist and the centre of shear, as defined in the present account, will in general not coincide with each other for beamswith material and/or support condition properties of such nature that a strain energy function does not exist for them.
4The Principle of Minimum Complementary Energy for the Problems of Torsion and Flexure
We know that among all states of stress and displacement which satisfy the differential equations of equilibrium and the given surface traction conditions for f(x, y) =0 and z = L, the state which also satisfies the stress-displacement differential equations and the given displacement boundary conditions for z = 0 and z = L isdetermined by a variational equation = 0 where
Here W is the complementary energy density of the material of the beam, and Eq. (9) may be appropriately generalized for more general boundary conditions for z =0 and z = L, of the kind noted in the paragraph following Eqs. (3).
In accordance with our earlier discussion it is our object to use the equation = 0, on the basis of suitable approximative assumptions for the state of stress, in
order to obtain approximate values of the integrals , in terms of the parameters U, V and in (9).
Page 57
To utilize (9) in a practical sense it suggests itself that we limit ourselves to cases for which part of the approximate assumptions for the state of stress consists instipulating, as in Saint-Venant's theory of torsion and flexure, that
We further assume that the material of the beam is homogeneous and transversely isotropic so that, with (10),
where E and G are independent of x, y, z.
Page 58
On One-Dimensional Finite-Strain Beam Theory:The Plane Problem[J. Appl. Math & Phys. (ZAMP) 23, 795804, 1972]
Introduction
The following is concerned with a consistent one-dimensional treatment of the class of beam problems dealing with the plane deformation of originally plane beams.Our principal result is a system of non-linear strain displacement relations which is consistent with exact one-dimensional equilibrium equations for forces andmoments via what is considered to be an appropriate version of the principle of virtual work.
Having a consistent system of equilibrium and strain displacement equations it is further necessary to stipulate, or rather to establish by means of an appropriate set ofphysical experiments, an associated system of constitutive equations. We discuss the nature of this aspect of the problem, including a solution of its linearized version,but without arriving at the solution of the general problem.
The principal novelty of the present results is thought to be a rational incorporation of transverse shear deformation into one-dimensional finite-strain beam theory. Acase may be made that the theory, with this effect incorporated, is of a more harmonious form than the corresponding classical theory, where account is taken of finitebending and stretching, while at the same time it is postulatedfollowing Euler and Bernoullithat the transverse shearing strain is absent, with the corresponding forcebeing a reactive force.
As an application of the general work a solution is given of the problem of circular ring buckling, including consideration of the effects of axial normal strain and oftransverse shearing strain on the value of the classical Bresse-Maurice Lévy buckling load.
Kinematics of Beam Element
We consider an element ds of a one-dimensional beam with equations x = x(s) and y = y(s) before deformation. We designate the tangent angle to the beam curveby 0 and write cos 0 = x (s) and sin 0 = y (s), where primes indicate differentiation with respect to s. We note that 0 is also the angle between the normal to thebeam curve and the y-axis.
Due to deformation the points x = x(s) and y = y(s) of the undeformed beam curve are changed to x(s) + u(s) and y(s) + v(s). We now assume that transverseelements which were originally normal to the beam curve do not necessarily remain so but end up enclosing an angle 1/2 with this curve. At the same time wedesignate the angle enclosed by such an element and the y-axis by . We then have a geometrical situation as shown in Figure 1. We note in particular, in addition tothe angle , the relative change of length e of the beam curve element ds, and the
Page 59
Fig. 1.
change of the angle 0 into an angle , and we read from the deformed beam element, as relations between , , e, u and v,
Dynamics of Beam Element
We now consider the deformed beam element, with normal and shear forces N and Q and with a bending moment M, in accordance with Figure 2. Together with thiswe assume force load intensities px and py and a moment load intensity m, per unit of undeformed beam curve length, also in accordance with Figure 2.
Fig. 2.
Page 60
We then read from Figure 2 as component equations of force equilibrium in the directions of x and y,
At the same time we obtain as equation of moment equilibrium
We note, for future use, the possibility of deducing from (2a, b) the relations
where n = pxcos + py sin and q = pycos pxsin are components of load intensity in the directions of N and Q, respectively.
Constitutive Equations
We postulate that the material of the beam is elastic and that we have the existence of axial and transverse force strains and and of a bending strain , in such away that constitutive equations for beam elements may be written in the form
We are ignorant, at this point, not only in regard to the form of the functions in (4), but also in regard to definitions for the components of strain , and whichenter into the constitutive equations (4).*
Defining Equations for Strain
In order to obtain equations for strain we consider a virtual work equation of the form
and we stipulate, as Principle of Virtual Work, that equation (5) be equivalent to the dynamic equations (2) and (3) in the interior of the interval (s1, s2), given that , and are appropriate expressions for virtual strains.
Since we know the form of the dynamic equations but do not at this point know expressions for virtual strains we use equation (5), in conjunction with (2) and (3), todeduce expressions for virtual strains.
Introduction of (2) and (3) into equation (5) gives a relation of the form
*However, we expect that e, and 0, for sufficiently small strain.
Page 61
and in this we may now consider N, Q and M as arbitrary differentiable functions of s.
In order to utilize (6) we integrate by parts, thereby eliminating all derivatives of N, Q and M as well as the boundary terms on the right. In this way we obtain
The arbitrariness of N, Q and M means that (7) implies the virtual strain displacement relations
It remains to take the step from virtual strain displacement relations to actual strain displacement relations.
One of these actual strain displacement relations follows directly from equation (9) in the form
A correspondingly simple derivation of expressions for and is clearly not possible through direct use of (8a, b). Remarkably, we may obtain and by using (8a,b) in conjunction with the geometrical relations (1). To do this we observe that equations (1) imply the following relations between virtual quantities
We now use (11a, b) in order to eliminate u and v in (8a, b). In this way we obtain
The form of (12a, b) is such that we can now go from virtual strains to actual strains. The results are
Having (13a, b) we can further express and in terms of u, v and . Introduction of (13a, b) into (1a, b) gives first
and then, by inversion
Page 62
We finally note the possibility of rewriting the moment equilibrium equation (3) somewhat more simply with the help of the strain components and as in (13), in theform
Observations on the Problem of Experimentally Derived Constitutive Equations
In order to see the nature of the problem of experimentally establishing the nature of the functions in equations (4) we consider the problem of an originally straightbeam, with x = s, y = 0 and 0 = 0, fixed at the end x = 0 and subject to given displacements u(a) = ua, v(a) = va and (a) = a at the other end. We assumeabsent distributed loads and have then from equations (2a, b)
where Xa and Ya are two constants of integration the mechanical significance of which is evident.
To proceed further we consider the moment equation (3*) as a differential equation for , by writing
and by considering the constitutive equations involving N and Q partially inverted in the form
so that M = fM ( , , ) = fk(N,Q, ') = g( , ').
The resultant second-order equation for must be solved subject to the boundary conditions (0) = 0 and (a) = a, with which = (x; Xa, Ya, a).
Having we find u and v from (14a, b). The boundary conditions for u and v are satisfied upon setting
We now measure Xa, Ya and Ma as functions of ua, va, a, and of a, giving a set of three relations , etc. The remaining task then is to deducefrom the form of these three experimentally determined functions , and the form of the desired three functions N, Q, M in equations (4).
The Linear Case
We consider a range of stresses and strains within which
with a view towards determining the elements , . . ., of the three by three matrix [C].
From equations (17) follow the linearized relations
Page 63
and the moment equation (3*), again with boundary conditions (0) = 0 and (a) = a, is reduced to
Equations (19) for the translational edge displacements become
In order to solve the problem as stated in (20) to (23) we partially invert (20) in the form
and write (22) in the form , with solution
We then have further, from (24),
and, upon making use of (23a, b),
We now stipulate knowledge of a matrix [B], as a result of experiment, such that
Having (26) to (28) we may then successively determine the elements of the matrix [C*] in terms of the elements of [B]. To see this we write
and have then from the relation a = B NNa + B QQa + B MMa that
from which , and follow in succession in terms of elements of [B].
We next introduce (26 ) into (27a, b) and compare the resultant relations with corresponding relations in (28). In this way we obtain the remaining six elements ,etc. of the matrix [C*] in terms of the elements of [B].
Finally, having [C*] we find the elements of [C] by returning from (24) to (20).
Buckling of Circular Rings
As an application of the foregoing we consider the classical problem of in-plane buckling of a circular ring of radius R, subject to a uniform normal pressure p. Wewish to obtain a buckling-load formula which
Page 64
incorporates the effects of (1) the symmetrical deformation of the ring prior to the onset of buckling, (2) axial strain associated with the buckling mode, (3) transverseshearing strain associated with the buckling mode. We will be concerned, in particular, with the question of appropriate constitutive equations.
Inspection of Figure 2 indicates that for uniform normal pressure p, per unit of deformed beam curve, we have as expressions for the load intensity components q andn in the force equilibrium equations (2*a, b)
together with an absent moment load intensity m in equation (3*).
We further have, with as in equation (10) and with Rd 0 = ds, that = R1 + . Therewith the equilibrium equations (2*a, b) and (3*) may be written in the form
In complementing (31) by constitutive equations we have no difficulty in deciding that suitable relations involving and are of the form
In stipulating a relation involving we find it necessary to concern ourselves with the question whether would be determined by the force Q tangential to thedeformed cross section or by a force Q* normal to the deformed centerline. Evidently, we have Q* given in terms of Q and N by the relation Q* = Q cos N sin or, approximately, by Q* = Q N . If we stipulate that = BQ* we arrive at a relation for in terms of Q and N, of the form = BQ/(1 + BN).* If we use Q instead ofQ* at the outset we have instead that = BQ. We may subsume both relations to one of the form
and consider in the end the two limiting cases = 0 and = 1.
Having equations (31) and (32) we now consider the stability of the state
for which, evidently, in view of (31b) and (32b)
We now write
and linearize (31) and (32) in terms of Q, M, , , N1 and 1 so as to have.
*This, together with (32b), is effectively equivalent to constitutive equations of the form Q = ( /B) + ( /C) and N = ( /C) + ( 2/2C).
Page 65
Equation (32a) remains as is and equations (32b) and (32c) become*
We now use (32a), (34) and (37) to write (36a, b, c) as a system of equations for N1, Q and , as follows
It is evident that (38b), differentiated once, my be written with the help of (38a) and (38c) as one second-order differential equation for Q.
Appropriate solutions, for a complete ring, will be of the form Q = cos ns/R where n = 2, 3, . . . From this follows as the equation for possible values of P,
Equation (39) may be written as a cubic equation for PR2/D, involving axial-strain and transverse shear-strain parameters k = CD/R2 and k = BD/R2. We will herelimit ourselves to a discussion of the case k = 0, with k k, for which the cubic equation reduces to a quadratic of the form
The smallest positive value of P follows from this for n = 2. We consider in particular the cases = 1 and = 0.
When = 1 we have from (40), in agreement with a recent result by Smith and Simitses**
When = 0 the solution is
*We note the possibility that C and B, as well as D in equation (32a), may be considered to depend on P.
**J. Eng. Mech. Div., ASCE 95, EM 3, 559569 (1969).
Page 66
On One-Dimensional Large-Displacement Finite-Strain Beam Theory[Studies Appl. Math. 52, 8795, 1973]
Introduction
In what follows we consider once more the classical problem of large displacements of thin curved beams. As regards the literature on this problem prior to about1930 we may refer to the Historical Introduction and to Chapters 18 and 19 of the 4th Edition of Love's Mathematical Theory of Elasticity. Subsequently, furtherattempts at improving the theory have been made by various investigators, but as far as could be ascertained none have dealt with the problem in the mannerdescribed in what follows.
Briefly stated, we consider a large-deformation theory of space-curved lines, with the cross sections of the lines acted upon by forces and moments. We take asbasic the differential equations of force and moment equilibrium for elements of the deformed curve. We then stipulate a form of the principle of virtual work, and usethis principle so as to obtain a system of strain displacement relations, involving force strains and moment strains in association with the assumed cross-sectionalforces and moments. Having a one-dimensional description of stress states and strain states we complete the formulation of the problem by postulating a system ofone-dimensional constitutive equations which, in the end, must incorporate the consequences of suitably designed experiments or of the one-dimensionalconsequences of a theory of beams treated as a three-dimensional problem.
The advantages of the present development, in comparison with work by others, seem to this writer to consist in the explicit consideration of axial and transverseforce strains in place of the usual early introduction of inextensibility and of the Euler-Bernoulli hypothesis. We have previously considered linear beam theory [2], aswell as the nonlinear theory of plane deformations of plane beams [5] in this manner. The present step to the general nonlinear theory suggested itself in the course ofrelated work on thin shells [4], which in turn was suggested by work of Simmonds and Danielson [6] and by earlier work on the nonlinear symmetrical problem of theshell of revolution [3]. It is to be emphasized that the present theory, as far as it goes, is strictly along classical lines without consideration, for example, of Vlasov'sconcept of bi-moments. It remains to be seen in what way the present approach may be generalized so as to apply to more general non-classical one-dimensionalnonlinear beam theories.
Geometry and Statics of Beam Elements
We consider a space curve with equation r = r(s) before deformation. We take the parameter s to be the length of arc, measured from a given point on the curve, andhave then a tangent unit vector
Page 67
t = r (s), where the prime here and in what follows indicates differentiation with respect to s. Having t we introduce two mutually perpendicular unit normal vectors n1
and n2, such that t × n1 = n2, n1 × n2 = t and n2 × t = n1, with generalized Frenet formulas
where 1, 2 and t are given functions of s.
Due to deformation the curve r = r(s) changes into a curve R = r(s) + u(s), with tangent vector R (s) where |R (s)| = 1 + , with 0, in general.
Having the curve R = R(s) we now stipulate the existence of external forces and moments p ds and m ds acting on the element |dR|. We further stipulate the existenceof internal forces and moments P(s) and M(s) acting over ''cross sections" of the curve. Consideration of the changes of P and M in going from s to s + ds, as well asof the principle of action and reaction then gives as equations of force and moment equilibrium of the elements of the deformed beam the two vector relations
Equations (2) are of course equivalent to six scalar component equations. In order to obtain such component equations, we introducefollowing Simmonds andDanielson's idea in treating thin-shell theory [6]a triad of as yet unspecified mutually perpendicular unit vectors T, N1 and N2 in terms of which
and
with corresponding decompositions for p and m.
In reducing (2) to scalar form through introduction of (3a, b) we make use of differentiation formulas for T, N1 and N2 of the form
with r1, r2 and rt depending on how the triad (T, N i) is defined.
The Equation of Virtual Work and Virtual Strain Displacement Relations
We introduce in association with the external forces and moments p ds and m ds, virtual translational and rotational displacements R = u and . At the sametime we introduce in association with the internal forces and moments P and M virtual force and moment strains . With this we write as equation ofvirtual work for any segment (s1, s2) of the beam
and we designate as principle of virtual work the statement that with and suitably given in terms of arbitrary u, and their derivatives, Eq. (5) is equivalentto the two vector equilibrium Eqs. in (2).
Page 68
For present purposes Eq. (5) is used, not to deduce (2) on the basis of given and , but rather to deduce and , on the basis of knowing the form of (2). Todo this we introduce p and m from (2) in terms of P and M and write (5) in the form
where now P and M may be considered as arbitrary quantities. Integration by parts to eliminate P and M , and use of the relation R × P · = R × · P, thenresults in two relations, which may be designated as virtual strain displacement relations. These are
where R = t + u .
In order to translate (7) into a system of scalar relations we now define scalar components of virtual strain by means of the following two expressions
These, in conjunction with (3a, b) imply as expressions for virtual strains,
and it remains to use Eqs. (7) for the determination of t, i, t and i, in such a way that from these the determination of t, i, t and t becomes possible.
Derivation of Implicit Strain Displacement Relations
The crucial step in what follows is the tentative assumption of an implicit representation of the force strains t, and i, in terms of the triad (T, N i), as follows
From this follows as relation for ( u) = ( R) = (R ),
Introduction of (10), (11) and (9a) into the first Eq. in (7) gives after some cancellations,
Equation (12) is satisfied, identically in t and i, by further setting
Equations (13) in turn are solved for by considering the cross products T × T and Ni × Ni in conjunction with the canonical expansion formulas for vectorial
Page 69
triple products. In this way we obtain
Having Eq. (14), we now determine ( ) in order to determine from the second Eq. in (7). The nature of the calculations which are required is as follows
The introduction of (17) and (18) into (15), with corresponding expressions for the terms (N1 × N1) and (N2 × N2) , gives after a remarkable series ofcancellations the simple relation
A comparison of (19) and (9b) gives further
Inasmuch as t and i should vanish for the case of no deformation, that is for the case that rt = t and ri = i, we deduce from (20) as implicit expressions for thecomponents of moment strain,
Intrinsic Equations of Beam Theory
Having Eqs. (21) together with (2), (3a, b), (4) and (10) it is now easy to state an intrinsic form of the beam problem, that is a complete system of differentialequations without reference to displacement variables, as soon as it is agreed to stipulate a system of six constitutive equations of the form
Besides these six scalar equations, we have another set of six scalar equations upon introducing the component representations (3a, b) and (10), in conjunction withthe differentiation formulas (4), into the vectorial equilibrium Eqs. (2).
These six additional scalar equations are
Page 70
with ri, and rt in terms of i and t as in (21), and
The form of Eqs. (23) and (24) is as expected, except for the appearance of the nonlinear P-terms in (24).
Derivation of Explicit Strain Displacement Relations
Considering that all components of strain depend on the choice of the orthogonal triad (T, N i) we must, in order to obtain an explicit set of strain displacementrelations, connect this triad with the triad (t, ni), through a suitable set of angular displacement parameters.
Geometrically speaking, the step from (t, ni) to (T, N i) involves an angle of rotation , about an axis defined by a unit vector . The associated transformationformula, for any vector A in terms of the corresponding vector a is, in accordance with Hamel [1], the Rodriguez formula*
Having (25) for T and Ni in terms of t and ni, respectively, we find components of force strain t and i on the basis of Eq. (10) in the form
in their dependence on and on the components of u and .
In order to obtain corresponding expressions for t and i, we need the coefficients rt and ri in the differentiation formulas (4). In order to obtain these we nowconsider the problem of differentiating T and Ni as given by Eq. (25). It will be sufficient to describe the details for the case of the vector T.
Straightforward differentiation of the formula for T which follows from (25) gives
In the first line on the right we may introduce t from Eqs. (1) and make use of the fact that (25) also gives the Ni in terms of the ni. Furthermore we take account ofthe fact that T must be perpendicular to T. In this way we see that (27) is equivalent to a formula
*We note that this formula plays an important role in Simmonds' and Danielson's treatment of shell theory [6], and also in Tameroglu's treatment of small-strain EulerBernoulli type curved beamtheory [7].
Page 71
where
It is now necessary to determine . We do this, essentially in the manner in which Hamel [1] determines the "angular velocity" of A, by considering two specialchoices of t, namely t = and t perpendicular to .
When t = we also have T = and furthermore t· = 1, t· = 0, and × t = 0. Equation (29) becomes then
To proceed from (30) we use the unit-vector identity × ( × ) = . In this way (30) may be written as
Equation (31) shows that must be of the form
with a scalar function of and which remains to be determined.
To find we now consider the case that t and T are perpendicular to . When this is the case Eq. (25) implies as relation between t and T
We introduce (33) as well as (32) into Eq. (29) and obtain as equation for the determination of
We transform this equation with the help of appropriate expansion formulas for triple and quadruple vector products on the left, and obtain after a number ofcancellations the remarkably simple relation
Introduction of (35) into (32) gives
It would now be possible to introduce Eq. (36) as it stands into Eq. (28) and then proceed with the determination of rt and ri. A considerably more convenient resultfollows upon first transforming Eq. (36) through the introduction of a modified (non-unit) axis-of-rotation vector , given by
Insertion of (37), together with = + , into Eq. (36) gives
We now determine ( ) so as to make the term with in (38) disappear. This gives
Page 72
and therewith, after some further transformations
where = 1n2 2n1 + tt, with 1, 2, t being three independent angular displacement parameters, and c an arbitrary constant. It is convenient, in order to establishdirect contact with known results of linear theory to set
We now introduce Eqs. (40) and (41) into Eq. (25) for T and Eq. (28) for T and obtain after some further simple transformations
and
with analogous formulas for Ni and .
Having Eq. (42) for T and the corresponding equations for N1 and N2 we could now, upon writing u = uini + utt obtain expressions for the components of forcestrain t and i in terms of ui,ut, i and t in accordance with (26). Inasmuch as all that we need, for the satisfaction displacement boundary conditions, areexpressions for T and the Ni as well as an expression for u we limit ourselves here to writing as expression for u, on the basis of (10)
with T and the Ni taken from Eq. (42).*
In order to evaluate (42) and (44) we need of course to determine the angular displacement vector and for this purpose we now deduce expressions for themoment strains t, 1, 2 in terms of the components of the angular displacement vector .
Combining the first of Eqs. (4) with Eq. (43) and the second relation in (21) we have
Combination of the second equation in (4) with the expression for which corresponds to (43) and with the first relation in (10) gives further
A simple direct calculation shows that (45a, b) may be written, alternately, in the
*For cases for which the effect of force strains on the deformation of the beam is negligible Eq. (44) reduces to the simple form .
Page 73
form
Equation (46) suggests expressing i and t, in terms of the corresponding expressions of linear theory. Setting
we will have,
which, except for notation and sign conventions, agrees with the expressions for moment strains of linear theory in [2].
Introduction of (47) into (46) gives as expressions for the components of moment strain of the present nonlinear theory
Moment Strain for In-Plane Deformation of Plane Beams
For this case we may set 1/ 2 = 1/ t = 0 and 2 = t = 0. Therewith and and therewith, according to (49a)
In order to express 1 in terms of the angle of rotation , as in the independent treatment of the plane problem [5], we consider that for this case the unit rotationvector equals the unit vector e2. Furthermore, = 1n2 = 1e2 and therefore, in accordance with Eqs. (40) and (41),
and then
which is in agreement with the result in [5].
References
1. G. Hamel, Theoretische Mechanik , 103107, Springer-Verlag, 1949.
2. E. Reissner, Variational considerations for elastic beams and shells, J. Eng. Mechanics Division, Proc. Amer. Soc. Civil Eng., EMI, 2357, 1962.
3. E. Reissner, On finite symmetrical deformations of thin shells of revolution, J. Appl. Mech. 36, 267270, 1969.
Page 74
4. E. Reissner, Linear and non-linear theory of shells, Thin Shell Structures, pp. 2944, edited by Y. C. Fung and E. E. Sechler, Prentice-Hall, 1974.
5. E. Reissner, On one-dimensional finite strain beam theory: The plane problem, J. Appl. Math. and Phys. (ZAMP) 23, 795804, 1972.
6. J. G. Simmonds and D. A. Danielson, Non-linear shell theory with finite rotation and stress function vectors, J. Appl. Mech. 39, 10851090, 1972.
7. S. Tameroglu, Finite theory of thin elastic rods, Techn. and Scientific Res. Council of Turkey, Appl. Math. Division, Rpt. No. 4, July 1969.
Page 75
Upper and Lower Bounds for Deflections of Laminated Cantilever Beams Including the Effect of Transverse ShearDeformation[J. Appl. Mech. 40, 988991, 1973]
1Introduction
The considerations which follow were originally motivated by the semi-elementary treatment of plane-stress solutions of the problem of the isotropic, homogeneouselastic cantilever beam of narrow rectangular cross section, fixed at one end and acted upon by a transverse force at the other end, as in [1]. Of specific interest hereis the tip deflection formula
where the term having the numerical factor k accounts for the influence of transverse shearing strains on the deflection of the beam.
In deriving a formula of the foregoing type by semi-elementary means, that is, by use of simple polynomial solutions of Airy's differential equation as in [1], it isnecessary to make certain assumptions concerning the approximate formulation of the conditions of end fixity of the cantilever, with the value of k depending on theassumptions which are made.
In what follows we propose to remove this ambiguity by the use of upper and lower-bound results for the ratio V/P, obtained through suitable application of theprinciples of minimum potential and complementary energy of the theory of elasticity.
Having once formulated a procedure for deriving such bounds for homogeneous beams it is readily evident that the same procedure is applicable for non-homogeneous, laminated beams. The results which follow are therefore stated for this more general class of problems.
2Formulation of Problem
We consider states of plane stress in a rectangular region with boundaries x = 0, x = a, and y = ± c, this region representing a beam of unit width. We assume thatthe boundary portion x = 0 is loaded by a force P in the negative y-direction, that the boundary portion x = a is fixed, and that the boundary portions y = ± c aretraction-free.
Writing as differential equations for stress and strain the usual relations
and
Page 76
the associated system of boundary conditions is here taken in the following form:
At the same time the force P is given by the integral
In the semi-elementary formulation of this problem, referred to previously, the displacement boundary condition for x = 0 is replaced by a stress boundary condition(0,y) = (3P/4c)(1 y2/c2), in order to conform with the type of solution which is used. With this type of solution it is also not possible to satisfy the displacement
conditions (4) as they stand. Instead, all remaining arbitrariness in the semi-elementary solution is removed by postulating and satisfying a system of conditions of theform
*
The formulation of the boundary-value problem is completed by the statement of stress-strain relations which for a linear isotropic homogeneous medium are of theform
where E = 2(1 + )G, with E, , and G being constants [1].
In what follows we consider the problem subject to a system of relations of the form:
where B is a given function of x, y, (and of x and y), and we obtain explicit results for the case that (9) reduces to
where E, , Ey, and G are given even functions of the coordinate y.
For the formulation of the solution procedure which is to be used, we require a statement of the relations inverse to (9) as well, in the form
*Alternately, consideration is given to the possibility of using, instead of ( u/ y)a,0 = 0, a condition ( v/ x)a,0 = 0, although, strictly speaking, a condition of this kind would not appear to beprescribable "from the outside." We note, as a possible alternate condition which would be prescribable, the relation
Page 77
with a corresponding inverse statement of (10).
3Upper and Lower Bounds for the Work Done by the Force P
Certain transformations of a functional I which enters into the statement of both the principle of minimum potential energy and the principle of minimumcomplementary energy lead to the conclusion that, with conditions as stated in Eqs. (1)(6), (9), and (9 ), we have as inequalities for the work quantity PV,
In this where and are differentiable functions which satisfy the displacement boundary conditions (4) and (5b), and where , , and are differentiable functions which satisfy the equilibrium differential Eqs. (1) and the stress boundary conditions (3) and (5a), with .The equal signs in these inequalities apply whenever , , and , , , respectively, are the actual solution functions of the boundary-value problem stated in (1)(5), together with (9) and (9 ).*
4Application of Formula for Bounds
In order to apply the system of inequalities we first express the force P as a function of the given displacement (which it must be considering Eq. (6), and the fact thatthe term V in (5b) is the only nonhomogeneity in the equations of the boundary-value problem). We will not consider the general case, and assume instead that theconstitutive equations are the linear Eqs. (10). This will make P a linear function of V, and we write
with a view toward obtaining upper and lower bounds for the flexibility coefficient C, with the help of the system (11), written now in the form
where, consistent with (10),
*Equation (1) may be obtained in the same way as a related, more general Eq. in [2]. A direct specialization of some interest of the result in [2] is possible for the case that the stress boundarycondition (5a) is replaced by a displacement condition u(0, y) = y where is a given constant. [We note that the signs in front of the last integrals in Eqs. (3.5) and (7.3), and in front of thesecond integral in (6.3) in [2] should be reversed]. Inequalities of a similar nature have been stated earlier by others, in particular by C. Weber, but as far as we know not for the problem which ishere under consideration.
Page 78
and
with , , and given in terms of and as in (2).
Lower-Bound Calculation
We now choose, guided by the results of the semi-elementary theory, as expressions for and
where is an arbitrary constant. We have then
so that all three displacement boundary conditions are satisfied by the differentiable functions and .
Introduction of (16) and (17) into (2) gives as approximations for strains to be used in Ã,
and therewith
where
We now determine from the condition , in the form
Introduction of (22) into (20) gives as the smallest value of which is compatible with the assumed state of displacement
Page 79
Introduction of (23) into (13) gives
as the desired lower-bound formula for C.
Upper-Bound Calculation
We now choose, again guided by the results of the semi-elementary theory, and in accordance with Eqs. (1), (3) and (5a),
with a constant parameter which remains to be determined.
Introduction of (25) into Eq. (14) gives
We now determine from the condition in the form
and, with this, obtain as the largest value of which is compatible with the assumed state of stress,
Introduction of (28) into Eq. (13) gives as the desired upper bound for C
In comparing the inequalities (24) and (29) with the approximate formula stated in the Introduction it is found that both inequalities provide appropriate generaliz-
Page 80
ations of the factor Pa3/3EI in the semi-elementary result, provided we agree to omit the effect of Poisson's ratio in the definition of E*.
We are interested, particularly, in the terms with G in Eqs. (24) and (29), which are a measure of the relative importance of the effect of transverse sheardeformation, and list explicit results for two special cases:
Homogeneous Beams
We assume that E and G are independent of y and have then
These values may be compared with the result of the semielementary calculation based on the substitute boundary conditions (7), which gives a factor 3/2 in place ofthe numerical factors on the right of (30a) and (30b).
We note in particular, in view of certain important consequences of this observation, which however will not be discussed in this paper, that the shear correctionterms (30) become of relative order of magnitude unity for the case that G is so small relative to E as to make G/E of the same order of magnitude as the ratio c2/a2.
Sandwich-Type Beam
The simplest way to appraise the effect of transverse shear deformation is to consider a beam consisting of a uniform shear-resisting core layer with negligibleresistance to longitudinal normal stress, and two face layers sufficiently thin to assume negligibility of their bending stiffness [3]. Mathematically, this means that weconsider a beam with the moduli E and G given as functions of the thickness coordinate y by the relations
where t << c.
Introduction of (31) into the bound formulas (29) and (24) leads to identical expressions for C, that is to the exact solution for this case, with the right-hand side of(30a, b) now being of the form 3ctEf/a2Gc.
*Elementary considerations overcome this difficulty by concluding from the smallness of y relative to x that y y x, and by using this relation to replace à in (15) by the simpler expression
, before carrying out calculations. The difficulty with this approach is that now the associated displacement state will not satisfy the boundary condition , andconsequently, while a rational approximate result is obtained, it can no longer be taken to supply a lower bound for C. We may improve the bound (24) for 0, within the present context, by
adding, for example, a term of the form Vy2 (x) in Eq. (16) for , with (0) = (a) = 0, and by then determining and (x) from the variational equation .
Page 81
5Concluding Remarks
We note the possibility of analogous bound determinations without the assumption that the elastic moduli are even functions of the thickness coordinate y orindependent of the spanwise coordinate x. It is also possible to consider other conditions of loading and support, as for example, given by the case of a beam whichis clamped at both ends, symmetric about midspan, and carrying a concentrated load at midspan.
References
1. Timoshenko, S., and Goodier, J. N., Theory of Elasticity, 3rd ed., McGraw-Hill, New York, 1970, pp. 4146.
2. Reissner, E., ''On Some Variational Theorems in Elasticity," Problems in Continuum Mechanics (Muskelisvili Anniversary Volume ), Soc. Industrial andApplied Math., Philadelphia, 1961, pp. 270281.
3. Reissner, E., "Small Bending and Stretching of Sandwich-Type Shells," NACA TN No. 1832, Mar. 1949; also NACA Report 975, 1950.
Page 82
Improved Upper and Lower Bounds for Deflections of Orthotropic Cantilever Beams*
[Int. J. Solids Structures, 11, 9611971, 1975]
Introduction
In what follows we extend our earlier work on upper and lower bounds for the deflection of end-loaded laminated cantilever beams of narrow rectangular crosssection through use of the principles of minimum potential and maximum complementary energy [1,2].
Specifically, we propose to improve the bounds
for the flexibility coefficient C of an orthotropic homogeneous beam, where C0 = a3/2Ec3 is the value of C in accordance with elementary theory and where
and
with
effectively, when , in such a way that the value of the factor 6/5 in CU1 as well as the occurrence of the negative, linear, c/a-term in CL3 areestablished as quantitatively significant aspects of the behavior of the actual values of C/C0.
Going beyond this we will show that while the term (6E/5G)(c2/a2) originates from what may be designated as the interior solution contribution of the boundary valueproblem of the end-loaded cantilever, we have that the terms linear in c/a which occur in CL3/C0, as well as the corresponding terms in the quantities CL4/C0 andCU2/C0 which are obtained in what follows, are in fact boundary layer solution contributions of our boundary value problem. Insofar as these boundary layer solutioncontributions are concerned we mention in particular our detailed analysis of the structure of the layer, with the number and the magnitudes of distinct
*With S. Nair.
Page 83
characteristic lengths which are found being well-defined functions of certain moduli ratios of the material of the beam.
In regard to the upper and lower bound formulas used in this work, we have previously noted that "inequalities of a similar nature have been stated earlier by others,in particular by C. Weber, but as far as we know not for the problem which is here under consideration." A review of the literature undertaken by us since hasestablished that the above statement should be amplified by referring to a specific publication by Weber [3], which considers in particular an isotropic homogeneousbeam on two simple supports with a concentrated load at midspan, in such a way that the formulation for one-half of this beam is effectively equivalent to theformulation of our cantilever beam problem. When interpreted in this manner the work of Weber includes a formula CL2 C CU1, where CU1/C0=1 + 12(1 +
)c2/5a2 and CL2/C0 = 1 2 + 2(1 + )c2/a2, consistent with the formulas for laminated orthotropic beams given in [1].
Formulation
We restate our plane stress problems once more in the form of differential equations
for the domain c y c, 0 x a, together with boundary conditions of the form
In this we assume that Em = (EEy)1/2 and that E, Ey, and G are given constants, with 2 < 1 as condition for strain energy positive definiteness.
Evidently the uniform end deflection V will be associated with an end force P given by
in the form
and our objective is the determination of upper and lower bounds for C, as a function of the given parameters E, Ey, , G, c and a.
In order to obtain these bounds we make use of the fact that the work quantity PV is bounded in terms of potential and complementary energy approximation Id andIs,
Page 84
where
and
In this x, y and must satisfy the equilibrium differential equations together with the prescribed stress boundary conditions, and u and v must be differentiablefunctions which satisfy the displacement boundary conditions of the given problem. Our procedure from here on is then to minimize Id and maximize Is with referenceto certain systems of functions x, y, , u and v, and to discuss the bound relations
where
which follow from (8) upon eliminating P through use of equation (7).
Upper Bound Calculation
We consider an equilibrium stress system of the form
where = x/a and = y/c are dimensionless coordinates, = c/a, primes indicate differentiation with respect to , and the constant as well as the function F( )are arbitrary except for the condition F(0) = 0.
We introduce equations (1315) into (9) and perform the indicated integrations with respect to . In this way we obtain an expression for Is of the form
where
Page 85
Maximization of Is with respect to , for fixed F, gives as expression for ,
and therewith
Introduction of (19) into equation (12) gives
where it remains to choose the function F( ), with the simplest choice, F( ) = 0, evidently leading to the previously obtained bound CU1/C0 = 1 + (6E/5G) 2.
In what follows we will obtain a value Imin by determining the function F( ) from the variational equation I = 0, together with the constraint condition F(0) = 0.
Application of the standard rules of the calculus of variations now leads to the Euler differential equation
and to the Euler boundary conditions
Appropriate integrations by parts in the expression for I, in conjunction with equations (21) and (22), now permit a simplification of the expression for Imin, asfollows:
It remains to solve the differential equation for F, subject to the associated four boundary conditions, and to introduce the value of Imin into equation (20) for CU/C0.In this way we obtain an expression for CU2/C0, which can be written in the form
where = (E/G), , and
Page 86
with K1 and K2 being the two roots with positive real parts of the characteristic equation
that is,
when [2 + (90/11)]1/2 < , and
when < [2 + (90/11)]1/2 .
Lower Bound Calculation
We assume as expressions for displacements
where
We introduce (29) into equation (10) and carry out all -integrations. This gives
We now set Id = 0 and obtain the Euler differential equations
as well as the Euler boundary conditions
Page 87
Integration by parts in (31), with the use of equations (30) and (32) to (36), gives as expression for Id,min,
With this there follows from equation (12) as expression for the new lower bound,
In order to evaluate (38), we must solve the system (32) to (35) subject to the boundary condition (30) and (36). The result can be written in the form
where
with k1, k2 being the two roots with positive real parts of the characteristic equation
that is
when [28(1 2)/3]1/4 < , and
when < [28(1 2)/3]1/4 .
Asymptotic Bound Formulas
Consideration of the expressions (27), (28), (42), and (43) for the roots Ki and ki of the characteristic equations (26) and (41) indicates the existence of the followingorder of magnitude relations,
Furthermore, the hyperbolic tangent functions in (25) and (40) can be effectively replaced by unity for sufficiently large real parts of their arguments, say when
Page 88
and the bound formulas (24) and (39) then can be written in the form
where,
We note that with and both proportional to we have that Ci as well as ci are independent of so that CU2/C0 as well as CL4/C0 come out to be third-degreepolynomials in .
Replacement of the hyperbolic tangent functions by unity is readily seen to be equivalent to the assumption that the functions F, f i and gi describe a boundary layerphenomenon in the region adjacent to = 1. The form of equations (44) and (45) gives as conditions for the existence of a boundary layer the relations
In view of the defining relations for and , these conditions may be written in the alternate form
We next use equation (44) to obtain information concerning the width of the boundary layers which are associated with the solution of the given problem. It isapparent that there exist either one or two boundary layers, in accordance with the following pattern
Page 89
As (E/G) (Ey/E) approaches unity from above the two layers of width b1 and b2 coalesce into the one layer of width b.
Asymptotic Formulas for Isotropic Beams
Setting Ey = E and E = 2(1 + )G, we will have = , and = 2(1 + ) and, in accordance with equation (54a), the asymptotic formulas (46) to (53) now applyas long as .
We will refrain from rewriting equation (46) to (53) for this special case and instead refer to Figure 1 which shows the values of CU2/C0 and CL4/C0 for = 1/3 inthe range 0 < 1/2 together with the previously obtained less accurate bounds. The curves in the inset show the behavior of the bounds in the neighborhood of =0. We see that the effect of the "dominant" linear terms in , which make C/C0 < 1 in a small neighborhood of = 0 is in fact of no practical significance whatsoeverfor the case of the isotropic beam.
Fig. 1.Dimensionless flexibilities
CU1/C0, CU2/C0, CL3/C0 andCL4/C0 as functions of = c/a
for isotropic beams with = 1/3.
We supplement the results shown in Figure 1 by a short table which gives numerical values in the range 0 1.0, and which shows the smallness of the errorassociated with the approximation C 1/2(CU2 + CL4), up to values of the depth-span ratio for which it is no longer appropriate to use the word "beam."
Results for Orthotropic Beams
Figure 2 shows values of the bounds as a function of for several values of the auxiliary parameter = 2/ 2, on the basis of the exact bound expressions (24) and(39), as well as on the basis of the asymptotic expressions (46) and (47). We see that the exact and the asymptotic values coincide effectively in the entire range of
-values shown, for small values of . As the values of increase,
Page 90
Table 1. Upper and lower bounds for influencecoefficients for isotropic beams with = 1/3.
CL4/C0 CU2/C00.0 1.0000 1.00000.1 1.0261 1.02900.2 1.1142 1.12080.3 1.2633 1.27490.4 1.4729 1.49080.5 1.7423 1.76800.6 2.0709 2.10610.7 2.4579 2.50460.8 2.9027 2.96300.9 3.4046 3.48091.0 3.9629 4.0577
the range of validity of the asymptotic formula extends over a smaller and smaller -range. Specifically, if we write / = and note that according to (54a) we
must now have we find that we must also have , or , as condition for the validity of the asymptotic formula. For = 100 this means that theasymptotic results should deviate from the exact bound results as soon as , a conclusion which is in fact substantiated by the dots in Figure 2.
Fig. 2.Dimensionless flexibilities CU2/C0 andCL4/C0 as functions of = (E/G)c/a
for orthotropic beams with = 1/2 and = G/ (EEy) = 0, 1.5, 100, together
with values of CL3/C0 and CL2/C0for = 100.
Page 91
Our calculations also show that the effect of the prevented end-section transverse normal strain, which is insignificant for isotropic beams, can be quite significant forbeams with strong orthotropy.
We supplement our discussion with an explicit statement of the exact bound results (24) and (39), in comparison with results of exact solutions of the givenboundary value problem, for the limiting cases Ey = 0 and Ey = .
We deduced in [2] that when Ey = 0, then
We now find from equation (24) and (39),
Numerical calculations for = 1/2 show that now CL4 is closer to C than is CU2. As shown in Table 2, we have that when 0 0.5 CL4 agrees with C to within1/100 of one percent. We also note that CU2 as well as CL4 are discontinuous at = 0, just as C comes out to be, but that the magnitude of the discontinuity of CU2
does not agree with that of C, whereas there is agreement for CL4 and, incidentally, also for CL3, as may be readily deduced from the formula on page 82 whichleads to the result CL3/C = 1 2 + 2 + O( 4) for the limiting case Ey = 0.
Table 2. Comparison of bounds with the exact flexibility coefficient whenEy = 0 and = 1/2.
C/C0 CL4/C0 CU2/C00.0 0.7500 0.7500 0.86570.1 0.7619 0.7619 0.87820.2 0.7979 0.7979 0.91520.3 0.8578 0.8578 0.97700.4 0.9414 0.9414 1.06330.5 1.0486 1.0486 1.17380.6 1.1792 1.1791 1.30850.7 1.3329 1.3328 1.46690.8 1.5096 1.5092 1.64880.9 1.7089 1.7082 1.85401.0 1.9306 1.9294 2.0821
Page 92
When Ey = the bound formulas (24) and (39) reduce to the following form
The exact solution for this case can be obtained by extending results in [4]. We find
with kn as the sequence of positive roots of the transcendental equation tan kn = kn.
A comparison of the bound results (58a) with the exact result (58b) shows that the bounds obtained here are in agreement with the exact formula insofar as terms upto the order 2 are concerned. For small , the remaining terms in equations (58a, b, c) are of the order 3. Numerical results obtained using (58) when = 1/2 aregiven in Table 3. In view of the closeness of these bounds a calculation of the exact expression (58b) which involves summation of an infinite series, is not undertaken.
Table 3. Upper and lower bounds for flexibilitycoefficients for orthotropic beams when Ey = and =1/2.
CL4/C0 CU2/C00.0 1.0000 1.00000.1 1.01196 1.011970.2 1.0476 1.04780.3 1.1069 1.10730.4 1.1895 1.19030.5 1.2951 1.2967
References
1. E. Reissner, Upper and lower bounds for deflections of laminated cantilever beams including the effect of transverse shear deformation. J. Appl. Mech. 40, 988(1973).
2. S. Nair and E. Reissner, An improved lower bound for deflections of laminated cantilever beams including the effects of transverse shear deformation. J. Appl.Math. Phys. (ZAMP) 25, 89 (1974).
3. C. Weber, Veranschaulichung und Anwendung der Minimalsätze der Elastizitätstheorie. Z. ang. Math. Mech. 18, 375 (1938).
4. F. B. Hildebrand, On the stress distribution in cantilever beams. J. Math. & Phys. 22, 188 (1943).
Page 93
Note on a Problem of Beam Buckling*
[J. Appl. Math. & Phys. (ZAMP), 26, 839841, 1975]
Introduction
In what follows we wish to record briefly some results for the static buckling loads of a non-uniform beam loaded at midspan by a concentrated force P (i) parallel tothe axis of the undeflected beam, (ii) tangential to the axis of the deflected beam. Our interest in this problem emanated from a wish to construct a beam problemwhich contained as special limiting cases the case where the two types of forces gave identical buckling loads and the case where no tangential static buckling loadexisted while at the same time a lowered buckling load for the 'parallel' load case would occur.
Our problem concerns a built-in beam of span 2a, with stiffness D1 in the interval (a,0) and stiffness D2 in the interval (0,a). It is assumed that the force P = PB
causing buckling is applied at midspan, in the direction of negative x, and that the support at x = +a has axial freedom, so that the entire force P is taken by the beamsegment a x < 0, with the segment 0 < x a supplying a variable degree of lateral constraint. Evidently, the limiting case D2 = corresponds to the ordinarybuckling case of a beam of stiffness D1, with ends x = a and x = 0 clamped, and with the distinction between parallel and tangential force ceasing to exist. On theother hand, the limiting case D2 = 0 becomes the problem of the end-loaded cantilever of length a, with the buckling load for the parallel force being one-sixteenththat for the case D2 = , while no such (static) buckling load exists for the tangential force. In view of these results for the two limiting cases we expected PB to comeout as a decreasing function of the parameter D1/D2 for the case of the parallel load, and an increasing function of D1/D2 for the case of the tangential load. Ourcalculations confirm this expectation for the former but not for the latter case.
Statement of the Problem
We have the differential equations
with boundary conditions
and with transition conditions
*With G. E. Lee.
Page 94
together with
for the case of the parallel force, and
for the case of the tangential force.
Derivation of Stability Equations
We take
where 2 = P/D1, to satisfy (1) and (2). Introduction of (5a) and (5b) into the transition conditions (3) and (4a) or (4b) gives four simultaneous linear homogeneousequations for the four constants ci. The conditions of a non-vanishing determinant of these systems for a non-trivial solution can be reduced to the following forms.
For the parallel force case, with = a, and = D1/D2,
For the tangential force case,
When = 0 both equations reduce to the form cos + 1/2 sin 1 = 0, with the smallest non-zero root = B = 2 . When = equation (6a) becomes cos =0, with B = /2, and equation (6b) reduces to the impossibility 0 = 1.
Discussion of Numerical Results
A numerical determination of , where P0 = 4 2D1/a2, as a function of = D1/D2 results in the static stability curves as shown (Figure 1). As expected,the values of PB/P0 for the parallel-force case decrease monotonously with from the value unity when = 0 to the value 1/16 when = . On the other hand, thevalues of PB/P0 for the tangential-force case are in the range smaller than the corresponding values for the parallel-force case, contrary to what we hadexpected. Beginning at about = 1, the values of PB/P0 for the tangential-force change their decreasing trend to an increasing trend, in such a way that when 1.5the values of PB for both cases coincide. As is increased further the stability curve for the tangential-force case approaches a point with vertical tangent, for 2.16and PB/P0 = 0.62. A further increase in results in a doubling back of the PB/P0-curve. We conclude from this that the beam is unable to buckle statically due to theeffect of a tangential force when , long before the known result of the non-existence of a static buckling load for the case = is reached.
Page 95
Fig. 1
Page 96
On Lateral Buckling of End-Loaded Cantilever Beams[J. Appl. Math. & Phys. 30, 3140, 1979]
Introduction
The history of the problem of lateral buckling of transversely loaded beams begins with two fundamental papers, written independently and nearly simultaneously, byA. G. M. Michell [1] and L. Prandtl [2]. Michell as well as Prandtl used appropriate geometrical ad hoc considerations to arrive at a correct physical understandingof the problem and at a buckling load formula which is correct, except for the analysis of certain secondary effects which are of small numerical significance in almostall practical circumstances.
Five years after the publication of Michell's and Prandtl's work it was observed by H. Reissner [3] that the equations of the problem of lateral buckling could bededuced in a straightforward manner, without ad hoc considerations, by an appropriate specialization of Kirchhoff's general theory of space-curved beams, with theanalysis of the (two) secondary effects, being automatically included in the analysis of the problem. Beyond making the above important advance in the analysis of thelateral buckling problem, H. Reissner went on to reduce the problem of the end-loaded cantilever beam to a boundary value problem for a third-order lineardifferential equation, and to a buckling load equation of the form c1P + c2P2 + . . . = 1. In arriving at the above result H. Reissner neglects one of the two secondaryeffects and indicates that he intends to give the numerical consequences of his formula in a different place.
Ten years after the publication of H. Reissner's note, in 1914, his assistant M. K. Grober reconsiders the problem [4] ''based on some calculations which Mr.Reissner turned over to me for further development." The principal result of Grober's work is the derivation of a third-order differential equation with full inclusion ofsecondary effects, and with a buckling formula given by the vanishing of a second-order determinant, with each of the four terms in the determinant a power series inthe buckling load. Grober concludes his analysis with the statement that he will "as soon as possible evaluate some numerical cases and compare the results withexperiment." Grober was killed in action in World War I, very soon after completion of this paper, and thus was prevented from carrying out his intentions.
Thereafter the number of publications on the problem of lateral buckling increases steadily. From among this literature two contributions should be mentionedspecifically. One of these is a paper by K. Federhofer [5], in 1931, which includes the numerical evaluation of H. Reissner's buckling equation for the case of anarrow rectangular cross-section beam (with the secondary effect amounting to 7.8 percent for a width-depth ratio of 1:5). The other is a recent paper by D. H.Hodges and D. A. Peters [7] which undertakes to re-examine the problem once again ab
Page 97
initio on the basis of H. Reissner's approach. In the process of doing this the authors rederive Grober's general third-order differential equation, unaware of this earliercontribution to the subject. However, in addition, the authors reconsider the problem of systematically determining first-order approximations for secondary effects.They find, in what must be considered a significant advance in the field of this problem, that a systematic first-order analysis of both second-order effects comes outto be actually simpler than the analysis in which one of the two effects is neglected, with the basic third-order differential equation of the problem now having anexplicit first integral, leaving the problem in the form of a boundary value problem for a second-order differential equation, just as for the case of the problem withoutconsideration of any secondary effects.
The main purposes of the present paper are the following. 1. We wish to reconsider the problem on the basis of Kirchhoff's equations of equilibrium for finitelydeforming rods in such a way that full advantage is taken of the fact that the boundary conditions of the problem allow a complete solution of the equations of anintrinsic form of the theory, that is of a formulation of the theory without any regard to the form of strain displacement relations. 2. We wish to show that a suitablenon-dimensionalization of the equations of the theory indicates the evident possibility of a straightforward perturbation expansion, in such a way that the results ofthe theory without secondary effects appear as the leading terms in the expansion, with both secondary effects appearing systematically in second and still higher-order terms. 3. We use the equations of a recently developed extension of Kirchhoff's equations, which takes account of axial extension and transverse sheardeformation effects [6], for the purpose of determining the effect of transverse shear deformation on the lateral buckling load of the cantilever beam.
Formulation of Problem
We write Kirchhoff's Equations for Finite Deformations of Originally Straight Beams in the Form
In these equations primes indicate differentiation with respect to an axial coordinate x, Pt, P1 and P2 are forces acting over the cross-section of the beam, tangent andnormal to the center line, with P1 and P2 being in the directions of the principal axes y1 and y2 of the cross-section, and with pt, p1 and p2 being the correspondingsurface force intensities. Furthermore, Mt, M1 and M2 are cross-sectional twisting and bending moments, with mt, m1 and m2 being surface moment intensitiescorresponding thereto, and with t, 1 and 2 being twisting and bending strains which are here taken to be related to the twisting and bending moments byconstitutive equations
Page 98
of the form
In what follows we restrict attention to the problem of a cantilever beam which is free of distributed surface loads, so that pt = p1 = p2 = 0 and mt = m1 = m2 = 0,and which is acted upon by a force and by a moment at the unsupported end of the beam. Of the various possible loading conditions which may be subsumed underthe above description we will be concerned specifically with the problem of a transverse end force P oriented in a direction which coincides with the principaldirection y1 of the cross-section of the undeformed beam. It is evident that the problem as described is of such nature that one possible state, the unbuckled state,involves no more than the two forces P1, Pt, the moment M1 and the bending strain 1, with P2 = 0, M2 = Mt = 0 and t = 2 = 0, and with the associated bucklingproblem being the problem of determining the smallest value of P which allows the existence of alternate, buckled, states with some or all of the quantities P2, M2,Mt, 2, t non-vanishing.
Boundary conditions for the system of differential equations (1) to (3) which correspond to the prescribed loading condition may be formulated as follows.
We evidently have at the loaded end, x = 0, the conditions
Since we do not know the orientation of the loaded end of the beam, we cannot say anything about the values of P1, P2 and Pt for x = 0. However, the conditionthat the direction of P remains the same, no matter what the orientation of the loaded end cross-section might be, in conjunction with the condition that the supportedend of the beam, x = L, is assumed to be built-in, means that we know the values of P1, P2 and Pt for x = L, as follows
We note that the number of boundary conditions corresponds to the number of first-order differential equations for P1, P2, Pt, M1, M2 and Mt, which is obtainedupon eliminating 1, 2 and t from Eqs. (1) and (2) by means of Eq. (3). We further note specifically that the above formulation of the problems holds, without anyreference to relations which exist between the strain components and whatever description we might choose for translational and rotational displacementcomponents of the elements of the beam. In other words, our problem has been stated entirely within the framework of the intrinsic equations of one-dimensionalbeam theory.
Derivation of Buckling Differential Equations
We begin by considering the unbuckled state, with overbars designating forces, moments and strains of this state.
We have then, from Eqs. (1)
and from Eqs. (2) and (3),
Page 99
with boundary conditions
We obtain equations governing the onset of buckling, upon setting
and upon linearizing Eqs. (1) and (3) in terms of Pt, P1, M1 1 and P2, M2, Mt, 2 and t. Of the six equations obtained in this way only three are needed,those following from Eqs. (1a), (2a) and (2c) in the form
The associated boundary conditions are
Note that the singly underlined terms in (10) describe the effect of initial deformations on the process of buckling while the doubly underlined term describes the effectof finite deformation in the analysis of the initial state. For practical applications both effects in the analysis of the given problem are generally negligible. In thework of H. Reissner [3] the effect of the is taken into account and the effect of the is explicitly neglected. In the work of Hodges and Peters [7], it isobserved that both effects are of the same order of magnitude in terms of appropriate dimensionless parameters but that, numerically, the effect of the is onlyabout one-fifth the effect of the .
Non-Dimensionalization and Perturbation Expansion for Equations of Unbuckled State
We set in Eqs. (6) to (8)
and we indicate differentiation with respect to by dots. With this the differential equations (6) and (7) may be written in the form
and the boundary conditions (8) become
We now consider
as a small parameter and expand the solution of (13) and (14) in powers of 1. The result of the simple calculation, to the degree needed in what follows, comes outto be
Page 100
where
and
For what follows it is important to note that the small parameter 1 may also be written in the form
or, with
as
where, as is known from previous work, the value of for which buckling occurs is of the order of magnitude unity.
Non-Dimensionalization of Equations for On-Set of Buckling
We begin by rewriting Eqs. (10), in conjunction with Eqs. (3), in the form
We next introduce into these the contents of Eq. (12), and furthermore write
with the choice of the factor PL2/D2 in the expression for Mt being of particular importance. With this Eqs. (22) become
with the boundary conditions for this system following from Eqs. (4) and (5) as
and with the coefficient functions p, q and m following from (16) to (20) as
Page 101
We specifically note from the appearance of Eqs. (24) that, as apparently first observed by Hodges and Peters [7], there are altogether three terms which determinethe effect of the small parameters 2 and t, on the smallest possible non-vanishing value of . We further note that it appears likely that the third-order eigenvalueproblem (24) and (25), with coefficient functions given in accordance with (13) and (14) without any assumptions concerning the smallness of 2 and t, would offerno particular difficulties in regard to a direct numerical solution. However, we will limit ourselves here to seeing in which way a direct solution, without use of suchcomputational facilities, becomes possible upon explicit utilization of the assumptions t << 1 and 2 << 1.
Perturbation Expansion for the Solution of the Characteristic Value Problem
It is clear from (24) to (26) that the functions g, t and 2 may be expanded in powers of the parameters t and 2. We limit ourselves here to a determination of thezeroth and first degree terms in these expansions.
As long as we restrict attention to first degree terms, Eqs. (24) may be written in the simplified form
again with the boundary conditions (25).
The third-order problem (27) and (25) may be reduced to a second-order problem, through recognition of the existence of a first integral of the system, as follows.We multiply (27a) by a factor and add the resulting relation to Eq. (27c). In this way there follows first
Having the factor t in front of the bracket in (28), we may now utilize Eqs. (27) without 2 and t, in order to transform the contents of the bracket in (28)
advantageously. Using (27b) we obtain . Therewith, and with the first two conditions in (25) we deduce from (28)the first-integral relation
Having (29) we obtain a differential equation for g alone by first combining (27a, b) and (29) in the form
and by setting in this 2 = g and in the terms multiplied by t. The resulting differential equation for g comes out to be
with the two associated boundary conditions being, in accordance with (25) and (27), the conditions .
Page 102
In order to solve the problem as stated, including first-order effects in 2 and t, we use ordinary perturbation expansions of the form
It is evident, without any calculations, that c2 = 1. In order to obtain the values of and ct, we deduce from (31) the differential equations
where h( ) = 1/2(1 3 2), with boundary conditions .
The well-known appropriate solution of the zeroth-order equation is , where 0 4.0126. Solution of the first-order equation in (33) by themethod of variation of parameters and satisfaction of the boundary conditions for gt, then gives as the value of ct,
and therewith 0(1 + 0.5 2 + 0.64 t) where, it should be noted, the correct numerical value of the coefficient of t has first been obtained by Hodges and Peters[7], with the corresponding value of ct, which follows upon omission of the doubly underlined terms in (27) and (28) being 1.64.*
Effects of Transverse Shear Deformation on Lateral Buckling Load
A determination of the effect of transverse shear may be based on an extension of Kirchhoff's equations for beams, in which the effect of transverse sheardeformations 1 and 2 and of a longitudinal extensional strain t is taken in account of by replacing the three Kirchhoff moment equilibrium equations (2) by equationsof the form [6],
in conjunction with additional constitutive equations involving the quantities and P. For what follows we take these additional constitutive equations in the form
With the above the equations corresponding to Eqs. (6) to (8) for the unbuckled state remain as before except that Eq. (7a) is replaced by
*In order to see the numerical significance of the improvement in [7], we note that for a homogeneous narrow rectangular cross section, with Poisson's ratio = 1/3 we have, when t = 0.1, that 1.097 0, whereas the corresponding result without consideration of the doubly underlined terms comes out to be 1.115 0.
Page 103
At the same time Eq. (10a) for the buckled state remains as before while Eqs. (10b, c) are replaced by
We now non-dimensionalize, as in Eqs. (12) and (23), and introduce two transverse shear deformation parameters 1 and 2 of the form
We then have that Eqs. (13a, b) remain unchanged while Eq. (13c) is replaced by
Of the three non-dimensionalized buckling equations (24) we have that one of them, (24a), remains unchanged while the remaining two are replaced by
In considering the problem of solving the system (13) and (24), subject to the boundary conditions (14) and (25) we note the possibility that the transverse shearparameters t may, for sandwich-type beams, be quantities of order of magnitude unity. In what follows we will limit ourselves to a solution of the problem for thecase that both 1 and 2, as well as t, and 2, are small compared to unity. We may then neglect products of these parameters and evaluate the effect of non-vanishing i as one which is additive to the effects of t and 2. With this we have that Eqs. (24a) and (24b*, c*) may be simplified to
We now obtain, as before, a first integral relation, g + 2 = 0, and we use this relation to transform the second equation in (38) to a second-order differentialequation for g, of the form
again with the boundary conditions .
We again expand the solution of this, in the form g = g0 + 2g + . . . and , and then obtain in the same way as in going from (31) to (34)
and therewith a reduction of the value of the buckling load parameter due to the effect of transverse shear deformability, in accordance with the relation 0(12.785 2). Given a beam with homogeneous narrow rectangular cross section of thickness 2c the value of 2 comes out to be (2E/5G)(c2/L2). For a narrowrectangular sandwich cross section, with shear resistant core of thickness 2c enclosed
Page 104
between two face sheets of thickness t, the parameter 2 is given by the expression (E /Gc)(ct/L2), with the evident possibility of a significant 2-effect for sufficientlylarge values of E /Gc.
References
[1] A. G. M. Michell, Elastic Stability of Long Beams under Transverse Forces , Phil. Mag. (5th Series) 48, 298309 (1899).
[2] L. Prandtl, Kipperscheinungen, Dissertation der Universität München, 75 pages (1900).
[3] H. Reissner, Über die Stabilität der Biegung, Sitz.-Ber. der Berliner Math. Gesellschaft 3, 5356 (1904).
[4] M. K. Grober, Ein Beispiel fur * die Kirchhoff'schen Stabgleichungen, Phys. Z. 15, 889892 (1914).
[5] K. Federhofer, Berechnung der Kipplasten gerader Stäbe, Sitz.-Ber. Akad. Wiss. Wien 140, 237270 (1931).
[6] E. Reissner, On One-Dimensional Large-Displacement Finite-Strain Beam Theory, Studies in Appl. Math. 52, 8795 (1973).
[7] D. H. Hodges and D. A. Peters, On the Lateral Buckling of Uniform Slender Cantilever Beams, Intern. J. Solids and Structures 11, 12691280 (1975).
Page 105
On Finite Deformations of Space-Curved Beams[J. Appl. Math. & Phys. (ZAMP) 32, 734744, 1981]
Introduction
We are concerned in what follows with the manner of derivation and with an application of large-displacement finite-strain theory of space-curved beams, aspreviously considered in [5].
In regard to the manner of derivation we have two objects. One of these is of an expository nature, with a clarification of the way in which our descriptions of thestate of strain and of the state of stress are shown to be consistent without the necessity of a "tentative assumption of an implicit representation of force strains." Theother is an approach to the problem of relations for components of moment strain in terms of components of rotational displacement, without use of Rodriguez'formula, in a way which involves a symmetric treatment of the two components of bending strain without a participation in this of the one component of twisting strain.
As an example of application of the general theory we consider the problem of helical deformations of a helical rod for the case of a simply symmetric cross sectionwith unequal principal bending stiffnesses and with non-coincident centroid and shear center locations, in generalization of an analysis in Love's Treatise [3].
Vectorial One-Dimensional Equilibrium Equations and Virtual Strain Displacement Relations
We have as equations of equilibrium for a cross sectional force P and a cross sectional moment M the two vectorial relations
with primes indicating differentiation with respect to arc length s along the undeformed "center" line of the rod, and with R = R(s) being the radius vector to points ofthe deformed center line.
We obtain vectorial virtual strain displacement relations for a force strain vector and a moment strain vector in terms of virtual translational and rotationaldisplacement components R and , as in [5], through use of the virtual work equation
in conjunction with the equilibrium Eqs. (1), in the form
Page 106
In this we have that ( R) = (R ), but we cannot write ( ) in place of ( ) , inasmuch as we do not have the existence of a function in association with thestipulated .
Derivation of Scalar Strain Displacement Relations and Equilibrium Equations
Given the radius vectors r(s) and R(s) to the undeformed and the deformed center lines, respectively, we introduce in association with these two radius vectors twotriads of mutually perpendicular unit vectors (t, n1, n2) and (T, N1, N2). In this t is tangent and n1, n2 are perpendicular to the curve r(s) but no such stipulation ismade relative to the triad (T, N1, N2) and the curve R(s), with the determination of R and T, N1, N2 being part of the problem of the rod, in a manner which willbecome apparent.
In order to derive from the vectorial virtual strain displacement relations (3) actual scalar strain displacement relations we now take and in the form
and R in the form
with the choice of at, ai left open. In addition to this we define virtual triad vectors T, Ni in terms of the virtual rotational displacement in the form
with (6) implying the supplementary relation
Introduction of (4a), (5) and (6) into (3a) leaves, after appropriate cancellations,
and therewith,
In order to reduce Eq. (3b) in a corresponding manner we first deduce from (7) the relation
To proceed further we make use of Frenet-type differentiation formulas
Introduction of (4b), (10) and (11), with
into Eq. (3b) leaves, after some cancellations, three scalar relations of the form
Page 107
Having Eqs. (9a, b), in conjunction with the relations t = r (s), and
we conclude, on the basis of the fact that at = 1 and ai = 0 when R = r, and t = i = 0 when T = t and Ni = ni, that we will have, as expressions for the coefficientsat and ai in (5) and for the coefficients 1/rt and 1/ri in (11),
and
At the same time we have from Eqs. (1a, b) in conjunction with the representations
and the corresponding representations for p and m, as scalar equations of equilibrium
with these differing from the corresponding equations of Kirchhoff [2] by way of the presence of the force-deformational terms P.
Strain Components in Terms of Rotational Displacement Measures
In order to take account of the duality properties of the two unit normal vectors ni which sets these apart from the one unit tangent vector t we proceed as follows tointroduce three scalar rotational displacement parameters 1, 2 and t.
We first introduce two mutually perpendicular unit vectors , symmetrically in terms of 1 and 2, by writing
in conjunction with the defining relations
We next take the triad vector T in the form , by writing
Page 108
We thereafter introduce the third rotational displacement parameter t by writing as expressions for N1 and N2
Having Eqs. (19) and (20) we may obtain expressions for r1, r2 and rt, for arbitrarily large i and t, by comparing the expressions for T and N i which follow from(19) and (20) with the corresponding expressions in (11), in conjunction with Eqs. (13). We are limiting ourselves in this account for simplicity's sake to the case ofsmall finite i and t by stating the appropriate results including all first and second degree terms but neglecting third and higher degree terms in 1, 2 and t andthe derivatives of these quantities.
With this we have then as expressions for T, N1 and N2
and
and from this follows, upon again omitting all third and higher degree terms
As regards the components of force strain t and i we obtain, on the basis of Eqs. (5), (14a), (19 ) and (20 ) and with R = t + u , the vectorial relation
for the determination of components of translational displacement in terms of force strains and rotational displacement measures. Alternately, we may obtain forcestrains in terms of translational and rotational displacement measures in the form
with the form of the final formulas depending on the nature of the component representation for u.
Helical Deformations of a Helical Rod
We now consider a slightly generalized version of the classical problem of a helical rod, acted upon by forces P the line of action of which coincides with the axis ofhelix, and by moments M turning about this axis, with this condition of loading sometimes being designated as a ''wrench", with the axis of the wrench coinciding withthe axis of the helix. We observe that for this
Page 109
system of loading we have as explicit solution of the equilibrium Eqs. (1a, b) the expressions
where it remains to determine the shape of the deformed rod, in terms of the geometrical parameters describing the undeformed rod, and in terms of the loads P andM.
With r and being polar coordinates in the plane perpendicular to the axis of the helix and with a and b indicating radius and rise of the center line curve, we have asvector equation of the center line of the undeformed rod
and from this we deduce as expression for the tangent unit vector t, with the help of the relation ds = (a2 + b2)1/2 d = c d , and with a = c cos , b = c sin ,
As regards the normal unit vectors n1 and n2 we first introduce a special set and of unit vectors by writing
with t, , evidently being mutually perpendicular, and by then writing in terms of an angle ,
with the directions of n1 and n2 coinciding with the principal axes in the plane of the cross section of the rod .
Given Eqs. (26) and (28) as defining relations for the triad t, n1, n2 it is then readily established that the coefficients in the differentiation formulas (13) are
Given Eq. (25) for the undeformed center line we now write the corresponding relation for the deformed center line as
with being given in terms of in the form = k . We then have from this
and it now remains to define unit vectors T, N1, N2 associated with the radius vector R as given by (30).
We will in what follows restrict attention to cases for which T is tangent to the R-curve by writing
Page 110
where cos = kA/C and sin = kB/C, with C = k(A2 + B2)1/2, and therewith
so that, in accordance with (5) and (14a),
Having T as in (32) we now define vectors and Ni, consistent with (27) and (28), in the form
and
With T as in (32) and Ni as in (36) we obtain as expressions for the coefficients in the differentiation formulas (11),
and therewith, in accordance with Eqs. (14b), as expression for bending strains i and twisting strain t
Formulation of Stress Strain Relations
We assume that the cross section of the rod in its deformed state will be symmetric with respect to an axis parallel to N1 and we will designate cross sectionalcoordinates in the directions of N1 and N2 by x1 and x2. We further assume that the origin of the x1, x2-system defines the center line of the rod which is taken to bethe line of shear centers of the cross sections, with the centroids of the cross sections being on the x1-axis, at a distance xc from the shear center. Limiting attention tothe case of linear stress strain relations we may then immediately write two of the stress strain relations in the form
Two additional one-dimensional stress strain relations follow from the integral relations
in conjunction with the defining relations
Page 111
where , in the form
The four Eqs. in (39) in conjunction with the four defining relations in (34) and (38) will become a system of four simultaneous equations for the determination of thefour quantities , , k, C in terms of the given geometrical quantities , , c and the given loads P and M, upon expressing Mt, M2, M1 and Pt in terms of P and Mthrough use of the defining relations which follow from (15), in conjunction with (24), (32) and (36), in the form
With A = k1 C cos , in accordance with the defining relations in the text which follow Eq. (32), we then have altogether as equations for the determination of , ,k and C,
Among the various special cases of the system (43) we mention the following.
Rod with Doubly Symmetric Cross Section
Setting xc = 0 in Eqs. (43a, b) we may use Eq. (43a) in order to reduce (43b, c, d) to a system of three equations for , and k, upon setting in these equations C= c + S1 P sin . In general, this latter relation will be effectively equivalent to C = c, as implied by the original Kirchhoff form of the theory. Strictly speaking, we willhave C = c upon stipulating S = , with P then being reactive. Aside from the fact that it is possible to imagine cases for which this would not be justified (as forexample for a rod with helical spring cross sections) it should be noted that, on the basis of Eqs. (41), the stipulation S = should by rights be associated with astipulation D1 = , that is with a stipulation of complete circumferential fiber inextensibility. This difficulty, however, may be by-passed by considering the problemwith C = c and D1 < as the first step of a perturbation expansion in powers of P/S.
Page 112
Rod with Kinetically Symmetric Cross Section
By kinetic symmetry we mean, in accordance with Love [3] that, in addition to xc = 0, we have D2 = D1 Db. We obtain the results stated in [3], and there creditedto Kelvin and Tait [1], within the present context by recognizing that when D2 = D1 then part of the solution of the system (43) is given by the relation
With this Eqs. (43b) and (43c) are both equivalent to the one relation
with (45) and (43d) now being two equations for the determination of and k, in terms of M and P, [with C = (P/S) c sin ]. We will limit ourselves here to using(45) and (43d) for the derivation of the set of relations
which may readily be recognized to be equivalent to Eqs. (40) on page 415 in [3], upon setting C = c.
Finite Pure Bending of a Circular Ring
We obtain equations for this problem upon setting = = 0, c = a and P = 0 in the system (43). We then have (43d) satisfied automatically and Eqs. (43a, b, c)assume the form
Equations (47b, c) imply as implicit relation for k in terms of M, for given values of a, , D2 and D1,
with given in terms of k and M in the form
Bending and Twisting of a Partially Rigid Rod
Given a rod with narrow cross section, such that D2 << D1, we may consider the problem of bending and twisting approximately, by considering Eqs. (43) subjectto the assumptions D1 = and S = . We
Page 113
now have, from (43a, b), as constraint conditions
with k, and to be determined, in terms of M and P, by means of Eqs. (49a) and (43c, d). We note that the special case = 0 of this problem is the one-dimensional analogue of a problem of inextensional bending in two-dimensional shell theory which has been considered in [4].
References
[1] Kelvin and Tait, Treatise on natural philosophy , Part II, 1895-Edition, pp. 136145.
[2] G. Kirchhoff, Über das Gleichgewicht und die Bewegung eines unendlich dünnen elastischen Stabes, J. reine u. angew. Math. 56, 285313 (1859).
[3] A. E. H. Love, A treatise on the mathematical theory of elasticity, 4th Ed., pp. 413417, Cambridge, 1934.
[4] E. Reissner, Finite inextensional pure bending and twisting of thin shells of revolution , Quart. J. Mech. & Appl. Math. 21, 293306 (1968).
[5] E. Reissner, On one-dimensional large-displacement finite-strain beam theory, Studies Appl. Math. 52, 8795 (1973).
Page 114
On Axial and Lateral Buckling of End-Loaded Anisotropic Cantilever Beams*
[J. Appl. Math. & Phys. (ZAMP) 34, 450457, 1983]
Introduction
In what follows we consider, in extension of the classical Euler and Michell-Prandtl calculations, the problems of axial and lateral buckling of end loaded cantileverbeams for the case of material anisotropy. By material anisotropy we here mean that bending strains as well as twisting strains depend on both bending moments andtwisting moments acting over the cross section of the beam. We assume that this dependence is linear but it will be evident that more general cases could also beconsidered. We further assume that the cross sections of the beam have an axis of symmetry in such a way that an elementary unbuckled state of the beam prevailsfor loads smaller than the critical loads which are to be determined.
Buckling Differential Equations and Loading Conditions
With the notation for forces and moments as indicated in Figure 1 and with quantities pertaining to the unbuckled state indicated by overbars we have as equilibriumdifferential equations for the process of buckling, in accordance with [2]
In this primes indicate differentiation with respect to the axial coordinate x, is the internal axial force prior to the onset of buckling, with being the center line
curvature associated with the bending moment , and with the force strains and which are associated with the forces and being considered negligible.
Fig. 1.
*With J. E. Reissner.
Page 115
For present purposes it is appropriate to consider the effect of also negligible and to stipulate that , and for the case of an axial compressive force F,a transverse force Q and a bending moment M acting at the end x = 0 of the beam are given by
Equations (1a, b, c) are complemented by two constitutive equations which are here taken in the form
With (2a, b, c) and (3a, b) and with equations (1a, b, c) become a third order system of differential equations for the quantities P2, M2, Mt
with the remaining task being the stipulation of boundary conditions for this homogeneous system, and the derivation of relations for the determination of the criticalvalues of F, Q and M in their dependence on the parameters Db, Dt, Dbt and L.
Buckling Due to Applied End Bending Moment
The assumption F = Q = 0 in conjunction with the stipulation that P2(0) = 0 makes P2(x) = 0 and leaves as equations for Mt and M2,
As regards the boundary conditions for this system we assume that in the process of buckling the load moment vector of magnitude M remains perpendicular to theaxis of symmetry of the undeflected cross section x = 0, as well as perpendicular to the deflected cross section x = 0 itself. In view of an additional, geometrical,stipulation that the end x = L of the beam is built-in we then have as two boundary conditions of an intrinsic formulation of this buckling problem the two conditions
For the solution of this problem we use equation (5a) to eliminate M2 in (5b) and (6b) so as to have as a second-order boundary value problem for Mt,
Page 116
with (7) and (8a, b) reducing to the well known classical form of this problem when 1/Dbt = 0.
The form of the system (7) and (8a, b) suggests the introduction of the two parameters
In terms of these we obtain as an equation for the critical values of M( )
with M(0) = /2 and with numerical values for M( )/ M(0) as in Table I.
Table I0 0.1 0.2 0.3 0.5 0.7 0.9 0.95 1.0
M( )/ M(0) 1 0.941 0.891 0.845 0.770 0.709 0.659 0.647 0.637M( )/ M(0) 1 1.069 1.151 1.252 1.540 2.092 3.930 5.758
Buckling Due to Applied End Force
We now set M = 0 in (4a, b, c) and stipulate as three boundary conditions for this third order system
with equation (12c) being a consequence of the stipulation that the plane defined by the force components F and Q does not change in the process of buckling andthat the end x = L of the beam is again built-in.
An inspection of the boundary value problem (4a, b, c) and (12a, b, c) now reveals the existence of a first-integral
The special case Q = 0 of this is of course trivially evident on the basis of (4b) and (12a). The special case F = 0 has previously been derived in this manner in [2].
We now use (13) in conjunction with (4a) in order to express Mt and M2 in terms of P2, in the form
The introduction of (14a, b) into (5b) gives as a second order equation for P2
Page 117
with the boundary conditions for (15) following from (12a), (12c) and (14a) in the form
Non-Dimensionalization
We introduce two dimensionless buckling load parameters
and with as in (9b), a dimensionless independent variable
Indicating differentiation with respect to by dots and writing P2(x) = ( ) the boundary value problem in (15) and (16a, b) then becomes
where
and where, in view of (18) and (16b),
Given a determination of the function ( ) in accordance with (19a, b, c) we then have from (20a, b) as a parametric representation of a system of ( F, Q) criticalvalue curves
Equations (19a) to (21) may be seen to imply, in particular, the special case relation
upon observing that as approaches + the product approaches /2. The corresponding relation for the case of no axial force F follows directly from (21)and (20a) in the form
Numerical Results
The smallest positive values of , in the range 10 10, for which ( ) = 0, were calculated through use of twenty five term partial sums of a power series solution
= 1 + C2 2 + C4 4 + . . . . The results of these calculations are listed in Table II, together with the asymptotic expression which
Page 118
Table II
( ) ( ) ( ) ( )
0.0 2.0065 2.0065 1.4 2.721 1.240 1.3280.1 2.072 1.933 1.6 2.787 1.176 1.2420.2 2.139 1.864 1.8 2.849 1.120 1.1710.3 2.204 1.795 2 2.907 1.071 1.1110.4 2.266 1.729 3 3.162 0.892 0.9070.5 2.324 1.665 4 3.381 0.777 0.7850.6 2.379 1.605 5 3.577 0.698 0.7020.7 2.431 1.548 1.877 6 3.758 0.638 0.6410.8 2.479 1.494 1.756 7 3.927 0.592 0.5940.9 2.525 1.444 1.656 8 4.09 0.554 0.5551.0 2.568 1.397 1.571 9 4.24 0.523 0.5241.2 2.648 1.313 1.434 10 4.38 0.496 0.497
Table III0 0.1 0.2 0.3 0.5 0.7 0.9 0.95 1.0
Q( )/ Q(0) 1 0.936 0.881 0.833 0.753 0.690 0.637 0.626 0.615Q( )/ Q(0) 1 1.076 1.167 1.279 1.612 2.287 4.893 8.029
holds for sufficiently large positive values of , with the asymptotic value differing from the exact value by less than one percent when 4 < .
Given the numerical values of ( ) in Table II we obtain the effect of the parameter for the Michell-Prandtl problem of lateral buckling due to a transverse force Q,in accordance with equation (23), as listed in Table III, where Q(0) = 4.013.
It is apparent that these results are qualitatively similar to the corresponding results in Table I for the problem of lateral buckling due to an applied end moment.
Finally, the results in Table II, in conjunction with equation (21), lead to a system of ( Q, F) curves for various values of in the range (0.9, 0.9) as shown inFigures 2a, b. We note that the result for = 0 has previously been obtained by Lensing [1]. As regards the qualitative aspects of these results we limit ourselves toobserving the physically unanticipated effect of an increase of the axial buckling load in the range < 0 under the influence of a moderate simultaneous transverseload, with this bringing to mind the possibility of an unanticipated case of imperfection sensitivity.
A remarkable property of a mathematical nature may be seen in the appearance of an envelope to the family of ( Q, F)-curves when 0 < and F < 0. The nearly
straight line behavior of the curves for sufficiently large positive values of is consistent with the fact that in the asymptotic range equation (21) impliesthe relation F = ( 2/4) Q.
Addendum
Analogous results are possible for systems of ( Q, M) and ( F, M) curves, and also for a system of ( Q, F, M) surfaces.
Page 119
Fig. 2a.
Fig. 2b.
Page 120
References
[1] J. Lensing, Die verallgemeinerte linearisierte Theorie dritter Ordnung des geraden [elastischen Balkens, Dissertation Braunschweig, 100 pp., 1976.
[2] E. Reissner, On lateral buckling of end-loaded cantilever beams, J. Appl. Math. Phys. (ZAMP) 30, 3140, 1979.
Page 121
A Variational Analysis of Small Finite Deformations of Pretwisted Elastic Beams[Int. J. Solids Struct. 21, 773779, 1983]
Introduction
The analysis which follows attempts a synthesis of results for stretching, bending, twisting, and warping of prismatical beams as obtained in [3, 5], and of results byKrenk [2] and Hodges [1] for the linear and nonlinear theory of pretwisted beams.
While there are similarities in our approach and in the approaches in [1, 2], insofar as use of the principle of minimum potential energy is concerned, and insofar asuse of the St. Venant torsional warping function for the introduction of the warping stiffness effect into the ensuing one-dimensional theory is concerned, there are alsodifferences, as will be apparent from a comparison of the respective publications.
The present study limits itself to the discussion of two specific examples of application. The first of these is the problem of finite stretching and twisting of beams withdoubly symmetric cross-section, including consideration of the effect of end section warping restraint. The second is the problem of cantilever torsion and flexurewithin the range of applicability of linear theory, with a view towards establishing the influence of pretwist on twist and shear center locations.
Energy Functionals and Displacement Modes
We begin as in [3] with the stipulation that an adequate three-dimensional strain energy expression for a beam with originally straight z-axis and cartesian crosssectional coordinates x, y is of the form
In this, we assume that the restriction to problems of small finite deformations for sufficiently slender beams implies the appropriateness of the use of the abbreviatedGreen strain formulas
In order to be able to use the variational equation for displacements which is associated with eqns (1) and (2) for the derivation of an approximate one-dimensionalbeam theory we assume as before as approximations for the cartesian displacement components , ,
Page 122
with u, v, , w, , , being seven functions of z only, and with being a suitably assumed function of x, y, and z.
In our earlier analysis of prismatical beams, was taken to be the St. Venant warping function for torsion, with differential equation
and boundary condition ( ,x y) dx ( ,y + x) dy = 0. Here we modify our definition of by first introducing rotated cross-sectional coordinates , , involving anangle of pretwist (z), through the relations
and by then stipulating that be the warping function for the rotated cross-section, with differential equation
and boundary condition
with G = G( , ) a given non-negative function and with (5a, b) implying the integral relations
and
We note, for subsequent use, that the determination of , as in (5a, b), allows us to set, without loss of generality,
with a Young modulus function E = E( , ).
Having the above definition of , it follows that the effect of pretwist will manifest itself in the one-dimensional theory which is to be established by way of modifyingthe approximate normal strain expression of prismatical beam analysis, through the appearance of one additional term, involving a factor ,z, as follows:
At the same time, we have as before [5], except for negligible terms x and y ,
In writing (6a, b) we take account of eqn (4) by observing that
Page 123
and
with an associated area element change of dx dy into d d in the defining relation (1).
In order to proceed further, we now stipulate as expression for the potential energy of external distributed or concentrated loads, consistent with the displacementapproximations in eqn (1),
and we assume, for the sake of definiteness, a specific strain energy density function U of the form
With eqns (1) to (10), the general results which are to be obtained now follow as a consequence of the variational equation
with arbitrary w, u, , , , , and for all values of z, excepting the effect of prescribed displacement boundary conditions.
Derivation of One-Dimensional Differential Equations and Boundary Conditions
We write on the basis of eqns (1) and (2)
and, on the basis of eqns (6a, b, c),
where
An introduction of eqns (13a, b), in conjunction with appropriate defining relations for one-dimensional cross-sectional stress measures, into eqn (12) reduces thisrelation to the one-dimensional form
Page 124
Equation (15) in conjunction with eqns (9), (11) and (14a, b, c) leads to a system of seven one-dimensional differential equations of equilibrium
and to a system of one-dimensional stress boundary conditions
or alternately, displacement boundary conditions
Given the form (12) and (13), in conjunction with eqn (15), we have as defining relations for the one-dimensional stress measures in eqns (16) to (22)
Equations (24) and (25) together with eqns (5e), (6a, b, c), and (14a, b, c) imply, as a system of one-dimensional constitutive relations, the two matrix equations
and
Page 125
with the elements of the constitutive matrices in (26) and (27) being given by
and
with x and y as in eqn (4), and with the ensuing relations
and
We note that the system of differential equations involving (14a, b, c), (16) to (19), (26), and (27) reduces to our earlier results for prismatical beams [4, 5] uponstipulating that = 0, identically, whereupon H = 0 in (14b) and H = 0 in eqn (19), with this entailing a reduction of the 6 × 6 system in eqn (26) to a 5 × 5system, by way of a deletion of the last row and the last column in the coefficient matrix in eqn (26).
We further note that we may, as for the case = 0[4], obtain a somewhat simpler theory by stipulating that transverse shear deformability is negligible, insofar as theeffect of the stress measures Qx and Qy is concerned, by stipulating in eqn (27) that AG = and
with Qx and Qy then being reactive, and with the remainder of the system (27) reducing to
As also done in [1, 2], when 0, we do not make the further assumption here of neglecting transverse shear deformability in relation to the magnitude of S.Instead, we retain the distinction between T and S as in eqn (35).
Stretching, Twisting and Warping of Doubly Symmetric Cross-Section Beams
We consider a uniform pretwisted beam of length 2L with a doubly symmetric cross-
Page 126
section, acted upon by forces Fz and moments Mz at the ends z = ±L, with Mx, My, Fx, and Fy stipulated to vanish. Because of the assumed double symmetry wehave then that Mx, My, Qx, Qy, , , u, and v vanish throughout, with the equilibrium equations (16)(19) reducing to the three relations
In these F, T, N, R, S , and H follow from eqns (26) and (27), with
in the form
An inspection of eqn (36) in conjunction with eqns (38)(41) indicates that the first two relations in eqn (36) are in effect two nonlinear equations for the determinationof w and in terms of Fz, Mz, and . The introduction of this result into the third relation in eqn (36), with R, S, and H, as in eqns (41) and (40), then leaves asecond-order differential equation for the determination of . As regards the solution of this differential equation, the following two cases will be of particular interest,
It is evident that for case (i) it will be necessary to determine as solution of a nonlinear second-order boundary value problem, explicitly or by numericalprocedures.
As regards case (ii), one finds the simpler result that both the differential equation and the boundary conditions are satisfied upon setting = 0 throughout, with thethree relations in eqn (36) then becoming three simultaneous ordinary equations for w , , and , of the form
Upon setting = 0, this system reduces to the corresponding result in [4]. Upon linearization, the consequences of eqns (43)(45) are consistent with thedevelopments in [2]. For the nonlinear case with 0, we can use eqn (45) so as to express in terms of w and , with eqns (43) and (44) then becoming asystem of two simultaneous nonlinear equations for the determinations of w and in terms of Fz and Mz.
Torsion and Flexure of a Pretwisted Cantilever Beam
In an extension of earlier work [3], we now use the contents of eqns (14), (16) to (23), (26), and (27) for a consideration of the problem of a cantilever, fixed at z =L and acted upon by forces
Page 127
Fx0, Fy0, and a torque Mz0 at the end z = 0. While it is feasible to obtain results for this problem on the basis of the complete nonlinear system of equations as stated,we will here limit ourselves to the consideration of its linearized version. Further-more, we assume for simplicity's sake that transverse shear deformability is negligibleand that the origin of the x-, y-axis system coincides with the centroid of the cross section.
Given the above stipulations, we immediately obtain from eqns (16) to (22) the same as for the problem of the beam without pretwist
with the equilibrium differential equation (19), and one of the stress boundary conditions in eqn (22) remaining in the form
With eqn (46) and with the assumed choice of axes and eqns (14a, b), we then have, as the linearized version of the constitutive system (26),
In view of the assumption of absent transverse shear deformability, the associated system (27), with eqns (35) and (14b, c), reduces to
For the problem as stated, the sixth-order problem (47) to (51) for the five variables w, , , , is associated with the five displacement boundary conditions
in addition to the one stress boundary condition in (47).
With the solution of the above, we can subsequently determine u and v from u + = 0, v + = 0, which follow from eqns (34) and (14c), in conjunction with theconditions u(L) = v(L) = 0.
The problem as it stands now requires that we solve eqns (49a, b) for and , as linear combinations of Fx0z, Fy0z, , and , with the substitution of theseexpressions into eqns (48) and (50), giving H and R as linear combinations of Fx0z, Fy0z, , and . The introduction of these into eqn (47), with IpGS = JG(DtG
Mz0), where DtG = IpG JG in accordance with eqn (51), altogether leaves a second-order differential equation for with the two boundary conditions R(0) = (L) =0 and with the solution coming out as a linear combination of Mz0, Fx0, Fy0. Having determined , we will then have further, on the basis of eqn (51), an expressionfor of the form
with the functions depending on cross-sectional properties, as well as on the rate of pretwist function .
Page 128
The Influence of Pre-Twist on Shear and Twist Center Locations
Given eqn (53) we can, as for the problem without pretwist [3], determine coordinates xT, yT of cross-sectional centers of twist, or equivalently, centers of shear xS,yS upon setting Mz0 = Fy0xT Fy0yT for (0) = 0, in the form
References
1. D. H. Hodges, Torsion of pretwisted beams due to axial loading, J. Appl. Mech. 47, 393397 (1980).
2. S. Krenk, A linear theory for pretwisted beams, J. Appl. Mech. 50, 137142 (1983).
3. E. Reissner, Further considerations on the problem of torsion and flexure of prismatical beams, Int. J. Solids Structures 19, 385392 (1983).
4. E. Reissner, On a simple variational analysis of small finite deformations of prismatical beams, J. Appl. Math. Phys. (ZAMP) 34, 642648 (1983).
5. E. Reissner, On a variational analysis of finite deformations of prismatical beams and on the effect of warping stiffness on buckling loads, J. Appl. Math. Phys.(ZAMP) 35, 247251 (1984).
Page 129
PLATESI learned about bending of plates and about solutions of plate problems in my father's statics of structures course. By way of having taken, at the same time, acomplex variables course I knew that the singular function r2 ln r, for a point load at r = 0 could be written in the form |z|2ln|z| where z = x + iy. I also knew about theconformal map of the unit circle onto itself, with the point z = 0 moving to z = a, by way of the mapping function (z a)/(1 az). I used this knowledge to construct thesingularity |z a|2 ln |(z a)/(1 az)| for a point load at z = a, and to synthesize a deflection function Re{|z a|2 ln |(z a)/(1 az)| + (1 |z|2) (z)}, such that this solution satisfiedthe condition of vanishing edge deflection. I then wrote the second boundary condition, involving edge slope and edge bending movement, as a differential equationfor (z) and in this way obtained a closed form solution which generalized Michell's solution for the clamped-edge plate so as to include, in particular, the solution fora simply supported plate [2]. To my pleasant surprise this solution became, soon thereafter, a section in Grammel and Biezeno's new Technical Dynamics treatise.
My next involvement with plates came by way of having to correct student solutions for simple cases of Wagner's theory of parallel tension lines, as an idealizeddescription of post buckling behavior of rectangular plates. I used what I had just then learned about differential geometry to generalize the problem so as to allow fornon-parallel tension lines and applied this generalization to obtain information on the behavior of circular ring plates, with edge shears applied through the intermediaryof elastic edge rings [15].
Upon having to teach a course in elasticity and having to explain the difference between plane stress and plane strain I was led next to a procedure for thedetermination of three-dimensional Poisson's ratio corrections for the two-dimensional theory of plane stress [30]. A concluding remark that ''it appears feasible toextend this analysis to problems of plate bending theory . . ." was followed up two years later, the delay being due in part to requests for work concerning cementedjoints [33], non-uniform motion aerodynamics [34] and shallow shell theory [41,46].
Taking account of transverse stress effects was actually simpler for the problem of plate bending than it had been for the problem of stretching. In addition, the resultsof doing so turned out to be more significant. In particular, taking account of transverse shear deformation meant a step from the classical fourth-order theory with itscontracted boundary condition paradox to a sixth-order theory, allowing for a description of boundary layer effects and for a resolution of the paradox [36,38].Fortuitously, restricting attention to homogeneous isotropic plates meant a form of the sixth-order system of differential equations which made it possible to derive anexplicit solution for the benchmark problem of stress concentrations due to a small
Page 130
circular hole. It was only much later that additional, computational, advantages were found to be associated with the use of this sixth-order theory, as a consequenceof the fact that its strain-displacement relations did not involve the second derivative terms of the fourth-order theory.
The subject of plates stayed with me from then on. An analysis of finite deflections of sandwich-type plates [51] was followed by the concept of displacement-basedhigher-order theories [57], by considerations of finite deflections of circular plates [64,77], by a simple explicit Kirchhoff-type solution for the problem of twisting dueto corner forces of a rhombic plate [81], by results for combined finite twisting and bending, with complex instability characteristics [105], by an introduction of theconcept of constitutive coupling of bending and stretching for the analysis of laminated anisotropic plates [133], and by someregrettably incorrectconclusions on thevariational derivation of boundary conditions in connection with Goodier's parametric interior expansion procedure, without regard to boundary layer effects [143].
Wishing to understand the essence of Koiter's asymptotic analysis of post-buckling behavior and imperfection sensitivity next led me to consider the problem of finitedeflections of plates on a non-linear elastic foundation [173]. Aside from some physically interesting observations I concluded that I could obtain Koiter-type resultsby a differential-equation based parametric expansion procedure, with sequential elimination of secular terms.
After this I concerned myself again with various aspects of sixth and higher order theories. A self-consistent entirely two-dimensional formulation of the sixth-ordertheory with conjugate stress and strain couples and resultants, became the starting point of two specific observations [204]. The first of these was that the fortunateproperty of the isotropic case to be associated with uncoupled differential equations 4w = 0 and 2 2 = 0 did not hold for most orthotropic plates. Theirbehavior was governed instead by two simultaneous third-order equations for w and which could not be decoupled. The second observation was that it waspossible to deduce from the equations at the sixth-order theory, a Kirchhoff-type interior solution portion, with modified contracted boundary conditions accountingfor first-order transverse shear effects (which was pursued further in [221]), consistent with Goldenveiser's later result as a consequence of three-dimensionalasymptotics.
Subsequent to this I derived a two-dimensional twelfth-order theory for transversely isotropic layers [236]. Remarkably, this twelfth-order theory involved again afourth-order interior solution problem, but now with a boundary layer governed by two second-order shear deformation equations and one fourth-order equationwhich accounted for transverse normal stress effects [260].
Finally, I returned once again to the sixth-order theory for orthotropic plates [264,270] in order to deal with the asymptotic expansion problem, this time on the basisof three simultaneous equations for w and for the two shear stress resultants Qx and Qy, with all three of these variables now having interior and boundary layerportions, in contrast to the isotropic case for which there was no boundary layer portion for w. One of my aims in this analysis was to contribute to the understandingof the importance of distinguishing "soft" and "hard" support, in the spirit of the questions then raised and dealt with in nearly simultaneous papers of Babuska andBathe.
Page 131
Über Die Biegung Der Kreisplatte Mit Exzentrischer Einzellast[Math. Annalen 111, 777780, 1935]
Das Problem, die Biegungsflächen einer dünnen elastischen Kreisplatte zu bestimmen, die längs des Randes gestützt und in einem Punkte innerhalb des Randes durcheine Einzelkraft P beansprucht ist (Greensche Funktionen), lautet in mathematischer Formulierung bekanntlich1) folgendermaßen:
Es ist eine Lösung der Gleichung
zu bestimmen, die folgenden Bedingungen genügt:
Dabei sind , 0, 0 a < 1 und N Konstanten, z = x + iy = rei und eine Lösung von (1), die für r 1 keine Singularitäten aufweist.
Diese Aufgabe ist in geschlossener Form bisher nur für den Fall = , und zwar von J. H. Michell2) durch Inversion des zentralsymmetrischen Ansatzes gelöstworden, während für , wo das Verfahren nicht zum Ziele führt, A. Föppl3) vermittels Reihenentwicklungen nach einer klassischen Methode von Clebsch4) eineLösung gegeben hat.
Die vorliegende Note enthält eine Lösung der von Föppl behandelten Aufgabe in geschlossener Form.
1A. E. H. Love, Mathematical Theory of Elasticity, 4. ed. 1927; S. 489491.
2J. H. Michell, London Math. Soc. Proc. 1902; S. 223.Die Annahme = bedeutet, daß die Platte starr eingespannt ist. Beim Problem der gelenkig gestützten Platte ist die Querkontraktionszahl (0 0,5). Wenn man der Zahl > 0 eine andere Bedeutung zulegt, kann man (3) auch als Randbedingung für eine elastisch eingespannte Platte ansehen.Es sei noch die Bedeutung der Konstanten in (4)
angegeben: 0 ist der Plattenradius, N die Biegungssteifigkeit der Platte, a 0 der Belastungsradius [das Koordinatensystem ist so gewählt, daß (P) = 0] und w 0 die Biegungsfläche in wahrer Größe.
3A. Föppl, Ber. d. Kgl. Bayr. Ak. d. Wiss. 1912; S. 155.
4A. Clebsch, Theorie der Elastizität fester Körper. Leipzig 1862.
Page 132
Um diese zu finden, wird ausgegangen von der bekannten Tatsache, daß sich jede Lösung von (1) in der Form
mit
darstellen läst. Also ist auch
eine Lösung von (1). Man erkennt, daß sie die Bedingungen (2) und (4) bereits im Ansatz erfüllt, wenn (z) eine für r 1 regulär analytische Funktion ist. DieBedingung (3) dient dazu, diese Funktion zu bestimmen.
Es ist zweckmäßig, die Bedingung (3) etwas umzuformn. Wegen (2) ist
Damit wird
Beachtet man, daß
so erhält man mit (5) aus (3) die folgende Gleichung zur Bestimmung von (z)
(3a) ist in bekannter Weise als eine Differentialgleichung für (z) aufzufassen5). Ihr allgemeines Integral ergibt sich mit einer willkürlichen Konstanten c und unterWeglassung einer bedeutungslosen rein imaginären additiven Konstanten zu
Wegen der vorausgesetzten Regularität von (z) ist c = 0.
Wenn man (8) in (5) einsetzt, wird die gesuchte Biegungsfläche
5Auf diesen direkten Schluß hat mich freundlicherweise Herr Prof. G. Hamel hingewiesen. Ursprünglich 'war die Lösung gefunden worden durch Ansetzen einer Potenzreihe für (z), derenKoeffizienten sich aus (3a) bestimmen, und Identifizierung der Reihe mit der Funktion (8).
Page 133
Nun läßt sich das Integral in (9) in allen den Fällen geschlossen ausintegrieren, in denen eine rationale Zahl ist, also praktisch immer, und damit hat man in (9) diegesuchte Lösung der Aufgabe in geschlossener Form.
Für den Grenzfall = 0 z. B. erhält man in reeller Schreibweise
Für = 1/3 ist
Für die Durchbiegung unter dem Lastangriffspunkt (Biegungspfeil) ergibt sich daraus
Man kann als einfache Verallgemeinerung des vorstehenden Lösungsgedankens den folgenden Satz aussprechen:
Die Randwertaufgabe der Bipotentialtheorie für den Kreis lä t sich im Falle gegebener Funktionsrandwerte auf eine Randwertaufgabe der Potentialtheorie fürden Kreis zurückführen, deren Art von der zweiten Randbedingung für die Bipotentialfunktion abhängt.
Dieser Satz kann mit Vorteil noch für andere technisch wichtige Aufgaben aus der Theorie der durch eine oder mehrere Einzelkräfte und stetige Belastungbeanspruchten Kreisplatte verwendet werden. Es ist beabsichtigt, dies an anderer Stelle auszuführen.
Page 134
On Tension Field Theory[Proc. 5th Intern. Congr. Appl. Mech. pp. 8892, 1938]
1Introduction
Tension field theory has been developed by H. Wagner1 to describe the state of stress in a certain type of thin-walled structures after buckling has set in. It substitutesfor the non-linear problem of a large-deflection theory of elasticity a simplified linear theory which in a number of practically important cases yields results in goodagreement with experimental data.
The reasoning leading to tension field theory is best explained by means of a definite example. Consider a strip of thin sheet supported perpendicular to the plane ofthe sheet and acted upon by uniform shear along the edges in the plane of the sheet.
Up to a certain intensity of the shear load, a uniform plane state of stress is produced in the sheet. If the load is increased beyond that intensity buckling occurs.However if the distance of opposite edges of the sheet is kept constant the shear load can have an intensity many times that of the buckling load without failure of thestructure as such. What happens is that wrinkles are formed in the sheet, the wave length of which decreases with increasing load, and that the sheet is mainly stressedin tension in the direction along the wrinkles while the compressive stress perpendicular to the wrinkles which causes the wrinkling becomes small compared with thetensile stress. Neglecting this compressive stress and also the bending stresses induced by the deformation out of the plane of the sheet against the tensile stress, onehas to study types of plane states of stress for which at each point the only principal stress component which is different from zero is a tension, while the straincomponent perpendicular to the direction of the non-vanishing principal stress component is not uniquely determined by the stress at that point.
In the present paper Wagner's formulation of the problem is modified in such a way, that it appears as a special case of a more general problem in plane stress, initself of interest and hitherto not considered. It deals with the theory of elasticity of anisotropic media, the curvilinear anisotropy of which depends on the boundaryconditions of the problem.
In this new form the tension field is analyzed by straightforward calculus avoiding the lengthy geometrical considerations of Wagner.
As an application of the theory a solution of the following problem is here given: A flat sheet of circular ring form, stiffened along inner and outer edges, is stressed bytwisting one edge with respect to the other, the axis of the applied torque being perpendicular to the plane of the sheet. This is the first solved problem in tension fieldtheory where non-parallel tension lines occur.
1H. Wagner, Zeitschr. Flugtechnik u. Motorluftschiffahrt 1929.
Page 135
2Formulation of the General Problem
In this paragraph is given the mathematical formulation of the following problem: To determine the state of stress in a material of such curvilinear anisotropy that theaxes of elastic symmetry are always tangent to the lines of principal stress. The anisotropy of such a material thus depends on the conditions which are prescribedalong its boundaries.
Tension field theory is obtained as a particular case of this problem in which one of the two different moduli of elasticity has the value zero.
Introducing as system of coordinates , the orthogonal system of the lines of principal stress in which the line element has the form
the following system of equations has to be solved.
1. The equations of equilibrium for the principal stress components and
2. The stress-strain relations. Calling u and v the displacements in and -direction, E and E the moduli of elasticity and G the modulus of rigidity and assuming nolateral contractions, these relations are
3. The relation between h1 and h2 which expresses the fact that the system ( , ) is orthogonal
It seems difficult to solve this system if not either E = E (isotropy) or one of the E's say
The latter case shall here be discussed.
3Solution of the Equations of Tension Field Theory
Introducing the assumption (5) into Eqs. (3) there follows
and with that from Eqs. (2) h1/ = 0, h1 = h1( ). It amounts only to a change of scale along the curves = const. if one puts
Geometrically the relation (7) means that the lines = const. are straight (Figure 1).
Page 136
Fig. 1.
Furthermore it follows from (2) that
where g is an arbitrary function.
From (4) follows
where 1 and 2 are arbitrary.
For the calculation of the displacements u and v two cases are conveniently distinguished.
(a) The case that the straight lines = const. are parallel, and hence also the lines = const. are straight. Then 1( ) = 0 and by scaling appropriately one may put2( ) = 1.
The displacements result from Eqs. (3) in the form
Here h and k are two more arbitrary functions.
(b) The case that the lines = const. are not parallel. In this case a change of scale makes possible to put 1( ) = 1 which means that the variable is identical withthe varying angle between the lines = const. and a fixed straight line.
From (3) results then
Page 137
In (8), (9) and (10) the four unknown quantities , h2, u and are expressed in terms of four arbitrary functions g, 2, h and k.
Since these arbitrary functions depend on the variable whereas the boundary conditions are given in terms of a fixed coordinate system, say a Cartesian system (x,y), relations must be established between the system ( , ) and the system (x, y). For this purpose the following equations must be used
The integration of this system of equations is effected by a geometrical consideration. Since the lines = const. are straight, one may write
This introduced into (11) gives for m and n
and from (13)
Hence
where d /d = 1( )
In (8), (9), (10) and (15) there is given the general solution of the tension field problem in terms of four arbitrary functions which have to be determined byboundary conditions.
4The Stresses in a Sheet of Circular Ring Form Wrinkled under the Influence of Shear Stresses Acting along the Edges
(Figure 2) As an application of the preceding general results this problem is solved rigorously. The tension lines start from the inner edge, and because of symmetry,each makes along the inner edge the same angle with the radius from the origin. This angle cannot be assumed but must be determined by means of the boundaryconditions.
Introducing as independent variables and the distance along the tension lines from an origin whose position depends on the angle of this line with the x-axis,one finds the following relations between the coordinates x, y, the polar
Page 138
Fig. 2.
coordinates r, , and the coordinates ,
For the stresses one obtains from (8) the principal stress
and the radial and tangential stresses
Page 139
For the displacements in radial and tangential direction ur and u results
Equations (18) and (19) are the most general expressions for stresses and displacements compatible with the assumed tension lines. If one is only interested in astress distribution independent of the angle variable = + /2, that is, independent of , these expressions are simplified to
and
For the determination of the four constants R0 = r0 sin , g0, h0 and k0 serve two boundary conditions at the inner edge r = r0 and two conditions at the outer edge r= r1. It shall be assumed that the shear load is applied by means of two stiffening rings of cross sectional area A0 and A1 which have moduli of elasticity E0 and E1.The two rings are necessary to prevent the sheet from collapsing after wrinkling has started.
Assuming that the tangential displacement of the outer edge is zero and of the inner edge equal to and expressing the fact that the radial deformation of thestiffening rings under the influence of the uniformly distributed radial load has to equal the radial deformation of the sheet along its edges, one has the conditions
Further discussion of the problem is here restricted to the case of rigid stiffening rings, so that in (23) and (25) one puts
Page 140
That is, ur(r1) = 0 and ur(r0) = 0. Conditions (22) and (23) lead with (21) to
From (24) and (25) follows
Equations (27), (28) and (29) are three linear homogeneous equations for g0, h0, k0 which cannot all be zero. Therefore the determinant of this system has to vanish,that is
Fig. 3.
Page 141
Developing and introducing the expressions for h2(r) and R0 from (16) this becomes the following equation to determine the angle
which angle thus is seen to depend on the ratio r0/r1.
Since (32) cannot be solved explicitly, has been determined numerically as function of r0/r1 and the result plotted in Figure 3. It is seen that varies between 45°and 90°; if the diameter of the hole approaches the diameter of the outer edge, the sheet behaves approximately like a straight strip and equals 45°; if the diameterof the hole is small compared with the diameter of the sheet the tension lines depart under right angles from the radius vector. For very small r0/r1 where the numerical
Fig. 4.
Page 142
evaluation of (32) is inconvenient the following approximate relation holds
Having obtained , one also knows g0, and with that the stresses in terms of the tangential edge displacement .
Of especial interest is the ratio between radial stress and tangential stress along the edges since the radial stress for prescribed shear determines the dimensions of thestiffening rings. This ratio is, according to (20),
The values of the ratios (34) are plotted as function of r0/r1 in Figure 4.
Page 143
On the Calculation of Three-Dimensional Corrections for the Two Dimensional Theory of Plane Stress (Excerpt)[Proc. 15th Semi-Annual Eastern Photoelasticity Conf. pp. 2331, 1942]
Formulation of the Problem
In this note a method is developed which permits the approximate solution of the following class of boundary value problems of the theory of elasticity. A body,bounded by one or more cylindrical surfaces with parallel generators and by two parallel planes perpendicular to the axes of the cylindrical surfaces, is acted upon bytractions applied to the cylindrical portion of the boundary, the boundary tractions being parallel to the bounding planes and moreover non-varying in the direction ofthe axes of the cylinders.
Mathematically the problem consists in finding solutions of the following system of nine equations:
for the nine unknowns x, y, z, xy, xz, yz, u, v, w, subject to the following boundary conditions
In these conditions 2h is the distance between the bounding planes, i(x, y) = 0 the equation of the ith cylindrical bounding surface, s the arc length along thecircumference of the cylinders and px and py the x and y-components of the boundary tractions.
The two-dimensional theory of plane stress can be described as a method of approximately solving the system of Eqs. (1)(4), based on the following
Page 144
assumptions:
It is known that these assumptions are in general consistent only with the system of differential equations (1)(3) when the value of Poisson's ratio, , is zero.Otherwise it is necessary to disregard two of the shear stress strain relations, Eqs. (2f) and (2g), since it is impossible to satisfy them for a state of stress obtainedfrom Eqs. (1), (2a)(2d) and (4)(6).
The usefulness of the approximate theory lies in the fact that experimental evidence and mechanical intuition indicate that in many cases the results of this approximatetheory are very close to the corresponding exact results. However, the same evidence and intuition indicate also that the results of the approximate theory may beseriously in error in such regions of the body where appreciable changes in stress occur over distances which are of the same order of magnitude as the thickness 2hof the body. Such changes occur for instance in the edge zone of holes (cut-outs) when the diameter of the holes is of the order of magnitude of the thickness 2h.
It is for a quantitative analysis of such effects that it is desirable to calculate three-dimensional corrections for the two-dimensional theory of plane stress.
Since it is believed that an exact solution of the three-dimensional problem presents very great analytical difficulties, an approximate method for its solution is heredeveloped which reduces the three-dimensional problem mathematically to one in two-dimensions while retaining the three-dimensional mechanical characteristics ofthe solution.
The method employed here is an appropriate application of the Principle of Least Work.
Method of Solution
The approximate analysis of the three-dimensional effect may be based on replacing the assumptions, Eqs. (6), of two-dimensional stress by the followingassumptions:
while the assumptions, Eqs. (5), of plane stress are replaced by
In Eqs. (7) and (8) the functions S, T, s and t depend on x and y only while the functions g, which are to be determined suitably, depend on z only.
The first set of conditions here imposed on the stresses of Eqs. (7) and (8) is that they have to satisfy the equilibrium conditions (1) and the boundary conditions (3)and (4).
From the equilibrium conditions (1) follows that g1 and g2 depend on g and that txz, tyz and sz depend on sx, sy and sz in the following way,
Page 145
(where dashes denote differentiation with respect to z) and
The boundary conditions (3) are satisfied by prescribing
and the boundary conditions (4) imply the following set of five conditions
In view of the special form of the expressions (7) and (8) it is still not in general possible to satisfy all the stress strain relations exactly. Since, however, Eqs. (7) and(8) are more general expressions than Eqs. (5) and (6) imply, it will be possible to satisfy the stress strain relations more nearly than is done in the two-dimensionaltheory of plane stress.
The way in which this closer approximation is obtained is the following. Use is made of the basic minimum principle for the stresses to determine the functions of Eqs.(7)(10) by an application of the direct methods of the calculus of variations. The minimum principle can be stated in the following form: ''Among all possible states ofstress which satisfy the equilibrium conditions in the interior of the body and the stress boundary conditions on the surface of the body, the correct state of stressmakes the difference of the strain energy and of the work of the unprescribed boundary stresses a minimum." In the case that all boundary conditions are stressboundary conditions the strain energy itself is a minimum for the correct state of stress. The more general minimum principle permits, however, extension of thepresent results to problems in which displacement as well as stress boundary conditions are prescribed.
On the basis of the equilibrium state of stress (7) and (8) a "best" approximation will here be obtained by determining the arbitrary functions sx, sy and txy such that thestrain energy of the body becomes as small as is possible with expressions for the stresses of the form Eqs. (7) and (8).
For the function g(z) which determines the shape of the stress curves over the thickness 2h of the body the following assumption is made
This expression for g satisfies the surface conditions (11).
It will be shown that application of the minimum energy principle leads to a system of simultaneous partial differential equations for a stress function from which theaverage stresses Sx, Sy and T are derived and for the stress corrections sx, sy and txy. The advantage which this system of equations possesses compared with thebasic system of Eqs. (1) and (2) is that the number of independent variables is
Page 146
reduced from three to two and that it is of a form which permits an explicit solution of the boundary value problem for the cases that the solid is bounded by twoparallel planes (problem of the infinite strip) or by two concentric circular cylinders (problem of the annulus). The latter case includes the problem of a circular hole inan infinite sheet which seems to be of the greatest practical interest among those which may be analyzed by the method of this paper.
Page 147
On the Theory of Bending of Elastic Plates[J. Math. & Phys. 23, 184191, 1944]
Introduction
In this paper there is established a system of differential equations of the sixth order for the linear problem of bending of thin plates. The form of these equations issuch that results obtained by their application coincide with the results of the classical theory of thin plates except for narrow edge zones. On the basis of the presentequations it is possible and necessary to satisfy three boundary conditions along the edges of a plate while, as is well known, the classical theory leads to twoboundary conditions only. The history and significance of the problem has recently been discussed by J. J. Stoker. 1
Formulation of the Problem
We consider an elastic body bounded by two parallel planes and by a cylindrical surface perpendicular to the two planes. The distance h between the two parallelplanes is assumed to be small compared with the remaining linear dimensions of the body, which because of this order of magnitude relation may be called a "plate."A coordinate system (x, y, z) is chosen such that the faces of the plate are the planes z = ± h/2. The cylindrical boundary of the plate may be given by equations ofthe form x = x(s), y = y(s) where s stands for the circumferential arc length. It is assumed that the two faces of the plate are free of shear stress while the normaltraction z is a given function of x and y. The resultant of the surface stresses z is balanced by stresses distributed over the cylindrical boundary of the plate.
We begin by assuming that the bending stresses are distributed linearly over the thickness of the plate,
By means of the differential equations of equilibrium we obtain as expressions for the transverse shear stresses
The shear stress resultants V are given in terms of the stress couples M and H,
1Mathematical problems connected with the bending and buckling of elastic plates. Bulletin of the American Mathematical Society, 48, pp. 247261 (1942).
Page 148
For the remaining normal stress component there results
if it is assumed that the vertical loads p are distributed over both faces of the plate as follows
Comparison of equations (5) and (4) gives
Substituting equations (3) in equation (6) we have
To obtain further relations for the three stress couples it is necessary to make use of the stress strain relations. This can be done in various ways. For the presentpurpose it is convenient to do this by means of Castigliano's Theorem of Least Work. For simplicity's sake it may be assumed that there are prescribed along thecylindrical portion of the boundary either the values of n, and or ns, and or zs (in such a way that equations (1) and (2) remain satisfied) or vanishing of the work ofthe stresses n and or ns and or zs. The Theorem of Least Work then states that among all statically correct states of stress the state of stress which also satisfiesthe stress strain relations and the displacement boundary conditions is characterized by the condition that the variation of the strain energy vanishes.
Taking isotropic materials the strain energy is given by
Substituting the values of the stresses from equations (1), (2) and (4) the integration with respect to z may be carried out. With
2The difference between this condition and the condition that the surface loads are applied to one face of the plate only is not important for the present purposes.
Page 149
there results
The variation of according to equation (10) is to be made equal to zero in such a way that the equilibrium equation (7) remains satisfied. Introducing according tothe rules of the calculus of variations a Lagrangian multiplier w(x, y) we than have
Equation (11) is the basic relationship from which every other result will be deduced. It may be remarked that the results of the classical theory of plate bending areobtained if in equation (11) one neglects the strain energy of the transverse shear and normal stresses. The main point of the present work is recognition of the factthat application of the variational principle without this neglect leads to a system of equations for which three conditions can and have to be satisfied along the edgesof the plate. This is due to the fact that retention of the transverse shear stress terms increases the order of the resultant system of differential equations.
The Differential Equations and Boundary Conditions of the Theory
Carrying out the variations in equation (11),
Page 150
Integrating by parts and rearranging,
In equation (13) the line integral is taken along the cylindrical boundary of the plate, s and n referring to the tangential and normal direction, respectively. It is seen thatthe variational equation (13) implies three differential equations and three boundary conditions. The first two differential equations may be solved for Mx and My. Ifthis is done there follow, with the notation,
as differential equations,
and as boundary conditions
In addition to the differential equations (15) a fourth differential equation is the equilibrium condition (7).
Before showing that the system of equations (15) and (7) is indeed of the sixth order, the significance of the three displacement boundary conditions may beexplained. The second parts of equations (16) express the fact that the plate is supported in such a manner that the edge moments and forces can do no work. Forthis to be so it is necessary that the linear elements perpendicular to the middle
Page 151
surface of the plate do not change direction when the plate is stressed. Because of the effect of shear deformation this is not equivalent to the condition of no changeof slope of the middle surface of the plate. Equation (16a) states what the change of slope of middle surface has to be in order that Mn may do no work. Equation(16b) states that when w = 0 along the boundary the distribution of edge twisting moments has to be such that .
Transformation of the Differential Equations (7) and (15)
Differentiating equation (15a) twice with respect to x, equation (15b) twice with respect to y and equation (15c) twice with respect to x and y there follows byaddition, because of equation (7),
where
Observing equation (6) we obtain as differential equation for the deflection w,
Clearly, the second term on the right of equation (17) is insignificant unless p changes appreciably within distances of the order of the plate thickness.
As the deflection w satisfies a fourth order equation as in the classical theory the additional two orders must be contained in equations (15). Adding equations (15a)and (15b) there follows, in view of equation (6),
With the help of equations (18) and (7) one may transform equations (15) into three equations each one of which contains only one of the three quantities Mx, My andH. Taking first equation (15a) in the form
and substituting 2H/ x y from equation (7)
Taking now My from equation (18) and rearranging there results
Page 152
A corresponding equation (19b) is obtained for My.
The equation involving H only follows from equation (15c) which may be written
Substituting Mx + My from equation (18) there follows
The three equations (19) together with equations (18), (17) and (7) may for the present be considered as the final form of the system of differential equationsgoverning the problem.
The "Edge Effect" Equations of Plate Theory
For the analysis of the edge effect under consideration various terms in the differential equations are unessential. Omitting these terms there remains as the relevantsystem of equations
The edge effect terms in these equations are those having h2 as a factor. If they are omitted the customary equations of plate theory remain.
The integration of the above system of equations is effected by first finding w from (20), by substituting the result in (22) to (24) whence Mx, My and H are obtainedas combinations of particular solutions and complementary functions, Mx = Mx,hom + Mx,part, and so on. Equations (20) and (7) are two equations connecting thethree complementary functions,
As an illustration consider a plate of the shape of a semi-infinite rectangular strip (0 y a, 0 x). Assume that the edges y = 0, a are simply supported and that theedge x = 0 is acted upon by a distribution of bending and twisting moments and of
Page 153
vertical shear force in the following manner.
From equation (20) there is obtained as a suitable expression for w,
As we assume
we deduce from equations (22) to (24), except for negligibly small terms,
From equation (25) follows
From equation (26) follows
whence in view of equation (29)
Thus, there are three constants of integration by means of which the three boundary conditions (27) may be satisfied.
Connection with the Theory of Moderately Thick Plates3
Considering in equations (19) the terms with h2 as small correction terms one might approximate these small terms by substituting in them the relations between Mx,My, H and w which hold if
3See A. E. H. Love, The Mathematical Theory of Elasticity, Cambridge, 1934, pp. 465487.
Page 154
the correction terms are omitted. If this is done there follows
Corresponding formulas may be found in Love's Treatise on page 473 when there is no surface load p. The only difference between Love's formulas and the presentformulas is a factor (1 + /8) in the small fourth derivative terms.4
The important difference between equations (31) and (19) is that by the simplification of equations (19) to equations (31) the sixth order problem is reduced to afourth order problem with resultant loss of the possibility to satisfy the three boundary conditions of the problem.
It may be remarked that the approach to the theory of moderately thick plates and to more general problems which has been initiated by G. D. Birkhoff5 and whichconsists in series developments of the solutions in terms of a thickness parameter does not appear to be suitable for the analysis of the edge effect6 which is the mainconcern of the present paper. It may be of considerable interest to determine the inner reason for this difference between the variational approach as employed hereand the approach by way of the thickness-parameter series solutions.
In conclusion it may be stated that it is entirely possible by means of the variational method to obtain more accurate solutions than the one here obtained. One may forinstance use instead of equations (1) more general equations of the form
where now Mx and x and the corresponding quantities occurring in y and xy are to be determined by application of Castigliano's theorem. Instead of the two-termexpressions of equation (32) one can, in principle, also use suitable n-term expressions. It is probable, however, that the calculations necessary for such extensions ofthe present results soon become quite involved.
4Note that in Love the plate thickness is 2h while here it is h.
5Circular Plates of Variable Thickness , Phil. Mag. (Ser. 6), 43, pp. 953962 (1922).
6J. N. Goodier, On the problems of the beam and the plate in the theory of elasticity. Trans. Roy. Soc., Canada, 33, Sect. 3 (1938).
Page 155
The Effect of Transverse-Shear Deformation on the Bending of Elastic Plates[J. Appl. Mech. 12, A69A77, 1945]
Introduction
It is well known that the classical theory of bending of thin elastic plates normal to their original plane permits the satisfaction of fewer boundary conditions along theedges of a plate than can in reality be prescribed. For instance, along a free edge, one has the three conditions of vanishing vertical force and of vanishing bending andtwisting couple. Kirchhoff (1) has shown that the assumptions underlying the classical theory are responsible for a contraction of the three conditions mentioned intotwo conditions, which are vanishing bending couple, and vanishing of the sum of vertical force and edgewise rate of change of twisting couple. The meaning of thisreduction in the number of boundary conditions has been explained by Thomson and Tait (2). The history of the problem has been discussed by Love (3) andrecently again by Stoker (4).
Because of the simplifying assumptions made in the development of the classical theory, it may lead to consequences such as the following.
1. There occur concentrated reactions at the corners of simply supported plates of polygonal shape.
2. Treatment of the St. Venant torsion problem of a rod with narrow rectangular cross section by means of plate theory, while leading to a fairly accurate torque-twistrelation when the width of the plate is larger than, say, 10 times the thickness of the plate, does furnish insufficient information regarding the distribution of stress overthe cross section.
3. Results for the magnitude of the stress concentration at the edge of holes in transversely bent plates become uncertain when the diameter of the hole is so small asto be of the order of magnitude of the plate thickness (5,6).
In the present paper, a theory of bending of plates is developed which, to a considerable extent, is free of the limitations just described. In this theory, three boundaryconditions can and must be satisfied along the edges of a plate. The theory is applied (a) to the torsion problem of the rod with rectangular cross-section where verygood agreement is reached with the results of the exact theory; (b) to the stress-concentration problem of the plate with circular hole. Here considerable deviationsfrom the results of classical plate theory are obtained as soon as the diameter of the hole is less than 3 times the thickness of the plate.
The manner in which the equations of the theory are obtained consists in an application of Castigliano's theorem of least work, combined with the Lagrangianmultiplier method of the calculus of variations. The physical basis of the present
Page 156
results forms the device of not discarding the energy of the transverse shear stresses, in contrast to what is done in the different derivations of classical plate theory.
The results here obtained for flat plates may be extended so as to apply to curved shells.
Derivation of Fundamental Equations
As in the standard theory of thin plates, it is assumed that the bending stresses are distributed linearly over the thickness of the plate
In Equations [1] Mx and My are the bending couples, Hxy the twisting couple, h the thickness of the plate (which in what follows is assumed to be uniform), x, y areco-ordinates in the middle plane of the plate, and z the thickness co-ordinate (Figure 1).
Fig. 1.Orientation of stress resultants and stress couples, and stress
variation over plate thickness.
From Equations [1], there are obtained, by means of the differential equations of equilibrium, expressions for the transverse shear stresses which satisfy the conditionsthat the faces of the plate are free from shear stress
The shear stress resultants Vz and Vy depend upon the stress couples:
Substituting Equations [2] in the remaining differential equation of equilibrium, there results for the transverse normal stress the expression
which satisfies the loading conditions
Page 157
The shear-stress resultants and the intensity of the vertical loading p are related by the equation
Combination of Equation [3] and Equation [6] results in one equation for the three quantities Mx, My, Hxy. To obtain further equations, use has to be made of thestress-strain relations. In view of the simplifying assumptions made in the writing of expressions for the stresses, this cannot be done in an exact manner. It is done in arational manner in what follows by means of Castigliano's theorem of least work. This theorem states that, among all statically correct states of stress, the state ofstress which also satisfies the stress-strain relations and the displacement boundary conditions is characterized by the condition that the variation of the followingexpression vanishes:
In Equation [7] the double integral extends over the cylindrical portion of the surface of the elastic body under consideration, un and us are displacement componentsparallel to the plane of the plate in normal and tangential direction, and w is the displacement component normal to the plane of the plate.
Substituting for the stresses from Equations [1], [2], and [4], and carrying out the integration with respect to z where possible there follows, with
It is consistent with the assumption of linear bending stress distribution to assume that the displacements un and us vary linearly over the thickness of the plate and thatw does not vary over the thickness of the plate
Page 158
The line integral in Equation [8] then becomes
where for a plate with built-in edges
For a plate with free edges , and are unprescribed.
The variation of according to Equation [8] is to be made equal to zero in such a way that the equilibrium Equation [6] remains satisfied. According to the rules ofthe calculus of variations, this is accomplished by multiplying Equation [6] by a Lagrange multiplier and by combining Equations [8] and [6] in the following manner
Carrying out the variations
Integrating by parts in the last integral we have further
Substituting this in Equation [11], it is seen that, when Vn is arbitrary, then
As the same result can be obtained for any interior curve, it may be concluded that the Lagrange multiplier is to be identified with the plate-deflection function w.
The variations of Vx and Vy, according to Equation [3], depend upon the variations of Mx, My, and Hxy. Integrating in Equation [11] further by parts there
Page 159
follows
Equation [13] is the fundamental relationship of the present theory. From it follow the differential equations of the theory which hold in addition to the equilibriumequations and also the ''natural" boundary conditions of the problem, which under the assumptions made are either stress- or displacement-boundary conditions.
The stress-boundary conditions, which make the variations in the line integral vanish, are
The displacement-boundary conditions are
There are to be prescribed either Equation [14a] or Equation [15a], either Equation [14b] or Equation [15b], and either Equation [14c] or Equation [15c]. Thesignificance of Equations [14] is evident. The same is true for Equation [15c]. Equations [15a] and [15b] indicate that, due to the effect of shear deformation, normaland tangential line elements in the middle surface do not remain perpendicular to the linear element which was before deformation perpendicular to the middlesurface.
The double integral in Equation [12] is equivalent to three differential equations. They are, if the first two are solved for Mx and My
Page 160
In addition to the three equations just given for the six unknowns Mx, My, Hxy, Vx, Vy, w, there hold the three equilibrium Equations [3] and [6]. In its present form,this system of equations is not readily solved. It will next be shown that it can be transformed in such a manner that the way to its solution is clear.
The first of the equations in its final form is the equilibrium Equation [6]
By means of Equation [I], Equations [16] are changed into
Equations [II] to [IV] will be used to determine Mx, My, Hxy when Vx, Vy, and w are known. From Equations [II] to [IV], there is next derived a system of twoequations for Vx, Vy, and w. According to Equations [3], Vx and Vy are combinations of derivatives of Mx, My, and Hxy. In view of this, there follows from Equations[I] to [IV], by differentiation and combination:
In the foregoing equations
Equations [I], [V], and [VI] may be solved simultaneously for Vx, Vy, and w. Once this is done the stress couples Mx, My, and Hxy are obtained from Equations [II]to [IV] without further integrations. The equations of the standard theory of thin plates are obtained by disregarding on the left of Equations [II] to [VI] all but the firstterms. The possibility of satisfying three instead of two boundary conditions in the present theory derives from the presence of the V terms in Equations [V] and[VI].
Page 161
Introduction of Stress Function
Considering now Equations [I] to [VI] when p = 0, it is seen that Equation [I] can be satisfied by means of a stress function
Substituting Equations [17] in Equations [V] and [VI] these may be written in the form
Equations [18] are Cauchy-Riemann equations for the functions D w and (h2/10) . Consequently, with two conjugate harmonic functions and , there isobtained
From
there follows
where 1 is the general solution of
Thus, the stress function is a combination of a harmonic function and of a function defined by Equation [22]. And if the harmonic contribution to is taken as theimaginary part of a complex function (x+ iy) then D w is the corresponding real part of the same function .
From
it follows further that w itself is a biharmonic function, the same as in the classical theory of plates without surface loading. Once the solution of the homogeneoussystem of equations is found, it is only necessary to obtain a particular integral to take care of the load function p.
The fact that in this formulation of the problem the only differential operators which occur are the invariant operators and 10/h2 indicates that explicit solutions ofthe theory may also be found in terms of plane polar and elliptical co-ordinates.
Before doing this, there will first be discussed, as an example of the scope of the theory, a relatively simple example of a plate problem with rectangular boundary.
Page 162
Torsion of a Rectangular Plate
A rectangular plate of length 2l and width a is considered. The two sides y = ± a/2 are free of stress while the two sections x = ±l are assumed to rotate withoutdistortion and to be free of normal stress. The condition of distortionless rotation means that line elements perpendicular to each other before deformation remain soafter deformation so that along the rotated end sections . With this stipulation, the boundary conditions, Equations [14] and [15], become
To satisfy Equation [24a] let
As it is expected that the stresses will come to be independent of x and odd in y, the solution of Equation [V] is taken in the form
and as Vy vanishes all along the edges, the solution of Equation [VI] is taken as
From Equations [II] to [IV] follows then
As yet unsatisfied is the boundary condition, Equation [25b]. Substituting Equation [30] in Equation [25b]
The only nonvanishing stress couples and resultants are then
From Equations [32] and [33], there are obtained the values of the shear stresses by means of Equations [1] and [2], substituting the value of D and the
Page 163
relation E = 2(1 + v)G
According to customary thin plate theory, corresponding expressions for the stresses would be
except at the edges y = ± a/2 where the stresses xz may be assumed to become infinite in such a way as to be equivalent to concentrated forces.
In the present theory, according to Equations [34] and [35], the stress xy is substantially constant over the width of the plate, except near the edges y = ±a/2 whereit decreases to zero within a distance of the order of magnitude of the plate thickness h. The stress xz has its largest values when y = ±a/2 and drops down nearly tozero values a distance away from the edges y = ±a/2 which is again of the order of magnitude of the thickness h.
The results of Equations [34] and [35] may be compared with the results of the St. Venant torsion theory, and the agreement is remarkably close even for plates sothick that the designation of "plate" is no longer appropriate.
Taking first the case of a square cross section, there follows
In the exact theory, the numerical factors would be equal to each other and have the value 1.35 (ref. 7). It is of some interest to note that the average of the twovalues, 1.33, is remarkably close to the exact value. If one did not know the exact value, it would have suggested itself to consider this average as the trueapproximation rather than either one of the values in Equation [36].
For a plate twice as wide as thick, there results for the maximum shear stress
which differs by less than 2 per cent from the exact value 1.86.
The limiting values of the stresses for very large values of a/h are xy(0, h/2) = 2G (h/2) and xz(a/2, 0) = 1.58G (h/2). An expression for the resultant torque isobtained from
Page 164
As
it follows that, as in the exact theory, the stresses xy and xz contribute in equal measure to the value of T. With Hxy from Equation [32]
The values of k1, according to Equation [40] compare with the exact values of k1 and the values of k1 according to customary plate theory as follows:
a/h 1 2k1 0.139 0.228 0.333k1,ex 0.1406 0.229 0.333k1,pl 0.333 0.333 0.333
For values of a/h which are larger than three, k1 of Equation [40] becomes (1/3)(1 .63h/a) which is a well known approximation formula.
Polar-Co-Ordinate Solutions of Equations of the Theory
Introduction of a stress function, according to Equations [17], and the subsequent integration of the system of equations in terms of the functions , , and 1, asdefined by Equations [19] to [23], inclusive, indicates the way to obtain explicit solutions in polar co-ordinates r, .
The shear-stress resultants are now expressed in terms of the stress function as follows
As before
and
where + i = (x + iy) and now
For D w may be written
Page 165
The conjugate of this is
Suitable solutions of Equation [22] are
In Equation [44], In and Kn are the modified Bessel functions (8). The functions In become rapidly large for large values of their arguments, while the functions Kn
become rapidly small for large values of their argument. For small values of the argument, In stays finite and Kn becomes infinite.
Equation [42] is integrated to
The starred constants depend upon the unstarred constants as follows
For each term in the trigonometric series, Equations [43], [44], and [45], there are six constants of integration, so that three boundary conditions can be satisfiedalong both edges of a circular-ring plate.
In order to evaluate the foregoing solutions, it is necessary to obtain expressions for the stress couples Mr, M , and Hr , which correspond to Equations [II] to [IV]for Mx, My, and Hxy. This may be done as follows: Observe that, in Equations [II] to [IV], the shear-stress resultants Vx and Vy occur in the same manner in whichthe displacement components u and v occur in the components of strain x, y, xy. This suggests that, in the equations for Mr, M , and Hr , the quantities Vr and Vmay occur in the same manner in which radial and circumferential displacement components occur in the expressions for r, , and r . The correctness of thisstatement may be proved by deriving the equations of the theory directly, introducing curvilinear co-ordinates before applying Castigliano's theorem. This calculationis omitted in the present paper.
Page 166
Equations [II] to [IV] then have the following analogues in polar co-ordinates
Substituting the stress function from Equations [41], there may be written, for the case of absent surface loads
The foregoing results permit a number of applications to problems involving circular boundaries. Two of these are made in what follows.
Bending and Twisting of an Infinite Plate with Circular Hole
The solution of these problems by means of classical thin plate theory has been given by Goodier (5). They have been investigated by Bickley (9) as problems of thetheory of moderately thick plates. Experimental results (10) have confirmed the results of thin plate theory for a hole diameter plate thickness ratio of about seven.Taking first the case of plain bending, the boundary conditions in the present theory are
Instead of Equations [47] and [48], there may be written
The conditions at infinity suggest that the following expressions for w and be taken
Page 167
From Equation [52] there follows for the shear-stress resultants with, according to (8)
For the stress couples, there results after some calculations
Substituting Equations [54], [56], and [58] in the boundary conditions, Equations [49], and [50], and with the notation
the following expressions for the constants in Equations [51] to [58] are obtained
Page 168
The stress-concentration factor of the problem is obtained from the value of the tangential edge stress couple
The value of this function is greatest for = /2
For large values of the following asymptotic expressions for K2 and K0 may be used (8)
Hence
which is in agreement with the result obtained by means of standard thin plate theory (5,6).
For small values of the function K2 becomes infinite of a higher order than the function K0 and consequently
It is noteworthy that in the limit of vanishing hole diameter the value of the stress-concentration factor in bending becomes almost twice as large as in the limit ofvanishing plate thickness, and moreover equal to the value of the stress-concentration factor in plane stress.
For intermediate values of a/h the value of kB has been calculated by means of tables for the functions K2 and K0 (8). Figure 2 contains a graph of kB versus 2a/h. Itis seen that even for holes 3 times as wide as the plate is thick the value of kB, according to the present calculations, is still more than 10 per cent greater than thevalue obtained by the application of standard plate theory.
That taking into account the shear deformability of the plate leads to higher values of the stress-concentration factor than not taking into account this effect becomesphysically evident when it is recognized that the assumed loading condition of the plate would lead to independent states of plane stress in every layer of the plate foran ideal orthotropic material offering no resistance to the transverse shear stresses rz, z, that is, for a material for which Grz = G z = 0. In contrast to this the resultsof the customary theory may be thought of as exact results for a material for which Grz = G z = .
Page 169
Relatively simple expressions are obtained for the shear-stress resultants Vr and V , by substituting the constants B2 and D2 in Equations [54] and [55]
For large values of a/h, for which the first terms of the asymptotic expressions of K2 and K0 may be used, this reduces to
Letting r = a + nh and consequently , it follows from the asymptotic formulas for Vr and V that, for very thin plates and for adistance of the order of magnitude of the plate thickness away from the edge of the hole, the shear-stress resultants have the values
These expressions coincide with the expressions which according to the standard theory of thin plates hold throughout the plate (5, 6).
Comparison of Equations [71] and [73a] shows that the resultant Vr increases from its true edge value zero very rapidly to the edge value of thin platetheory.
Also noteworthy is the behavior of the function representing V . From Equation [72] it follows
This shows that, in the present theory, the edge value of V is of opposite sign from the value according to Equation [73b]. Moreover, for thin plates, V (a, ),according to the present theory, is of an entirely different order of magnitude than according to the usual plate theory. For given plate thickness, its value no longerdecreases
Page 170
with increasing hole diameter. Substituting
there results for the maximum transverse edge shear stress
Thus, with transverse-shear deformation taken into account, there are portions of the plate where the transverse shear stress is of the same order of magnitude as theprimary bending stress 0, no matter how thin the plate may be.
From Equation [70] there follows for the variation of edge shear with diameter-thickness ratio
which compares with the constant value 4M0/(3 + )a, according to the standard theory. Figure 2 contains a graph of this function.
Plate Subject to Pure Twist
The results for this case may be obtained from the preceding results by superposition of a state of plain positive bending about the y-axis, as given, and a state of plainnegative bending about the x-axis. Hence, for
Fig. 2.Stress-concentration factors and edge shear-stress resultant
versus ratio of hole diameter to plate thickness.
Page 171
pure twist, every stress and displacement quantity T is expressed as follows in terms of the corresponding quantity B for plain bending about the y-axis:
From Equation [66] there follows
Values of kT as function of the ratio 2a/h are plotted in Fig. 2. Limiting values of kT are
which agrees with the result of standard thin plate theory, and
which is the same value which occurs when adjacent layers of the plate can slide freely with respect to each other. It is evident that, for the twisted plate, the effect oftransverse shear deformability is still greater than the same effect for the plate subject to plain bending.
Inspection of Equations [69] and [70] reveals that the transverse-shear-stress resultants for the plate subject to pure twist have exactly twice the magnitude of thecorresponding stress resultants for the plate subject to plain bending. Consequently, the value of the maximum transverse shear stress is now
where 0 now represents the undisturbed maximum shear stress parallel to the plane of the plate.
Remarks on Further Stress-Concentration Problems
It is apparent that, by an application of the general results of the present paper, still further stress-concentration problems may be solved, for which there will besignificant deviations from the results obtained by means of the classical plate theory. Of these may be mentioned (a) the plate subject to uniform transverse shear atinfinity and the couples necessary for equilibrium (5, 6); (b) the same problems for a plate with rigid or elastic circular inclusion (6); (c) the same problems for a platewith circular hole reinforced by an elastic ring; (d) the plate with elliptical hole (5, 6). For this the additional problem arises to calculate the solutions of the equation
in elliptical co-ordinates which take the place of the modified Bessel functions in the solution for the polar co-ordinate system; (e) a circular plate with a small hole atthe center and a linear (hydrostatic) load distribution.
Page 172
Remarks on Accuracy of Solutions of Stress-Concentration Problems
It is not possible to make with certainty statements regarding the accuracy of the numerical results obtained. The results are as accurate as it is possible under theassumed variation of stress over the thickness. For the problem of torsion of a rectangular plate, comparison with the known exact solution indicates a surprisinglyhigh degree of accuracy of the result here obtained. For the problem of the plate with the circular hole, such a comparison is not possible as the exact solution of thethree-dimensional problem is not known. A way to determine the accuracy of the present solutions would be the following: Instead of the linear bending-stressdistribution take a more general expression for the bending stresses containing third powers of the thickness co-ordinate z. Determine the corresponding transverseshear and normal stresses and again apply Castigliano's theorem. If the more accurate results thus obtained are in good agreement with the results based upon thelinear bending-stress distribution, those may be assumed to be final from a practical point of view. However, the author would like to state as his belief that thepresent results regarding stress-concentration factors and transverse-shear forces are such that a more accurate analysis would indicate changes in values which areno more than 20 per cent of the difference of the values obtained here and the values obtained from standard thin plate theory. Thus, if the standard plate theory givesa value of 1.5 and the present theory a value 2.0 then it is believed that the actual value lies in between 1.90 and 2.10.
A version of the contents of the first third of the present paper, which differs formally from the present improved version, together with the discussion of some pointsnot considered here, has been published elsewhere (11).
Bibliography
1. ''Über das Gleichgewicht und die Bewegung einer elastischen Scheibe," by G. Kirchhoff, Journal für reine und angewandte Mathematik , vol. 40, 1850, pp.5188.
2. "Treatise on Natural Philosophy," by W. Thomson and P. G. Tait, Oxford University Press, 1867.
3. "A Treatise on the Mathematical Theory of Elasticity," by A. E. H. Love, fourth edition, The Macmillan Company, New York, N. Y., 1927, pp. 2729.
4. "Mathematical Problems Connected With the Bending and Buckling of Plates," by J. J. Stoker, Bulletin, American Mathematical Society, vol. 48, 1942, pp.247261.
5. "The Influence of Circular and Elliptical Holes on the Transverse Flexure of Elastic Plates," by J. N. Goodier, Philosophical Magazine, series 7, vol. 22, 1936,pp. 6980.
6. "The Influence of the Shape and Rigidity of an Elastic Inclusion on the Transverse Flexure of Thin Plates," by M. Goland, Trans. A.S.M.E., vol. 65, 1943, pp. A-69 to A-75.
7. "Theory of Elasticity," by S. Timoshenko, McGraw-Hill Book Company, Inc., New York, N. Y., 1934, p. 248.
8. "A Treatise on Bessel Functions and Their Application to Physics," by A. Gray and G. B. Mathews, second edition, prepared by A. Gray and T. M. MacRobert,London, Eng., 1931.
9. "The Effect of a Hole in a Bent Plate," by W. G. Bickley, Philosophical Magazine, series 6, vol. 48, 1924, pp. 10141024.
10. "Stress Concentration Around an Open Circular Hole in a Plate Subjected to Bending Normal to the Plane of the Plate," by C. Dumont, National AdvisoryCommittee for Aeronautics, Technical Note No. 740, December, 1939.
11. "On the Theory of Bending of Elastic Plates," by E. Reissner, Journal of Mathematics and Physics, vol. 23, 1944, pp. 184191.
Page 173
Pure Bending and Twisting of Thin Skewed Plates[Qu. Appl. Math. 10, 395397, 1953]
1Introduction
The following is concerned with the problems of pure bending and twisting in the theory of transverse bending of thin plates. Known results for rectangular plates ofuniform thickness will be extended to skewed plates.
It was shown by Kelvin and Tait that the problem of St. Venant torsion of a thin rectangular plate with edges parallel to axes x and y has the solution w = xy wherew is the deflection function of the plate and the (constant) angle of twist per unit length. Within this theory, which neglects transverse shear deformation, the torque isapplied to the plate by means of concentrated forces of suitable direction which act at the corners of the plate.
We will show here that Kelvin's and Tait's solution is readily extended to skewed plates of uniform thickness. In so doing we obtain, in particular, the influence of theangle of skew on the torque-twist relation for the plate. We also obtain a solution for the problem of pure bending of a skewed uniform plate. We find that purebending of the skewed plate is associated with a twisting deformation the relative magnitude of which depends on the angle of skew.
2Formulation of the Problem
Let x, y be mutually perpendicular directions in the undeflected middle surface of the plate. The differential equation for a uniform plate bent by edge forces andmoments only is of the form
where w is the deflection of the plate. Stress couples Mx, My and Mxy, defined in the usual way, are
For what follows we have no need for the corresponding expressions for the transverse stress resultants.
We consider plates bounded by two straight lines x = ±l and by two straight lines y = ±1/2c x tan . The angle is the angle of skew of the plate, 2l is the span ofthe plate and c is the chord of the plate.
Expressions for bending moment Mn and twisting moment Mnt acting along the edges y = ±1/2c x tan follow by means of the usual transformation formulas for
Page 174
plate bending couples,
In addition to this we need Kelvin's and Tait's result that there occur concentrated forces P at the corners of the plate given by
and that the effective transverse edge stress resultant Rn is of the form Vn + Mnt/ s.
3Choice of Deflection Function
We shall find that for the problems of twisting and bending which are here considered it is sufficiently general to assume a deflection function
where A, B and C are constants. We then have a uniform distribution of stress couples
vanishing transverse stress resultants Vx and Vy, and vanishing effective edge stress resultants Rn.
4Twisting of Skewed Plates
The following boundary conditions must be satisfied
In addition to this we have that the applied torque T is given by
Equations (7), (8) and (9) are three simultaneous equations for the determination of the three constants A, B and C. We obtain
and therewith
The meaning of this result becomes somewhat more transparent if we introduce a new chordwise coordinate counted from the centerline y = x tan of the plate,by setting
Page 175
We then have
We may define an effective angle of twist per unit length by means of the expression
Combination of (13) and (14) gives the following result for this effective angle of twist
Equation (15) indicates that the skewed plate has a smaller torsional rigidity than the unskewed plate and in which way the torsional rigidity varies with the angle ofskew .
5Pure Bending of Skewed Plate
We denote the applied moment by M. The following boundary conditions must be satisfied
In addition to this we have the condition of vanishing torque, or of vanishing corner forces P, which becomes
From equations (16) to (18) we obtain the following expressions for the coefficients A, B and C in w
The deflection w is now
In terms of the chordwise variable defined by (12) this becomes
We see from (21) that the skewed plate has a smaller bending stiffness than the unskewed plate and that moreover the applied bending moment produces also atorsional deformation.
Page 176
On Postbuckling Behavior and Imperfection Sensitivity of Thin Elastic Plates on a Non-Linear Elastic Foundation[Stud. Appl. Math. 49, 4557, 1970]
1Introduction
The considerations which follow are presented as a contribution to the understanding of the asymptotic theory of postbuckling behavior of imperfection sensitivenonlinear elastic structures, in particular for cases for which a multiple mode linear buckling analysis applies. Specifically, a study of Hutchinson's paper onimperfection sensitivity of externally pressurized spherical shells [1], treated within the scope of Koiter's general theory [2,3], suggested to the author that he shouldlook for a qualitatively similar problem of a simpler nature than the spherical shell problem, and that he should endeavour to derive the relevant results by asymptoticexpansion considerations applied to a differential equation formulation without making use of potential energy concepts.
Our simplified version of the spherical shell problem concerns a uniform infinite elastic plate, on a non-linear elastic foundation, and in a state of uniform two-dimensional hydrostatic compression prior to the onset of buckling in the event that the plate is without imperfection. It is found that in order that this plate problem bequalitatively similar to the spherical shell problem the non-linear foundation should be a quadratic foundation. It is further found that with such a quadratic foundation itis allowable to base the work on the simple Kirchhoff form of plate theory, with no account being taken of the non-linear interaction between stretching and bendingof the middle plane of the plate.
In addition to the plate with quadratic foundation support, we also consider the case of a plate on a cubic foundation. We find that the shift from quadratic to cubicfoundation has two important consequences. The first of these is that now postbuckling behavior is effectively of the single-mode type. The second is that now it isgenerally necessary to base the analysis on the non-linear von Karman type plate equations in which interaction between midplane stretching and bending is taken intoconsideration.
2The Linear Problem
We are concerned with the differential equation
where D, N and k1 are positive constants where is a given function of x and y and w is to be determined, subject to suitable boundary or periodicity conditions.
Page 177
We consider for the problem of buckling, that is for , solutions of the form
where A is an arbitrary constant and have then that N must have a value NB given by
We define the buckling load of linear theory NBL as the smallest value of NB as a function of a and b and we designate the corresponding values of a and b by aL andbL. We then have, from NB/(a2 + b2) = 0
In regard to the problem of imperfection sensitivity, we consider the case that is given by
and that w is again given by (2.2). We then have as relation between A and
which is meaningful provided N is sufficiently small compared to NB.
Of importance for what follows in relation to the above are the usual superposition possibilities associated with linearity and also that for a plate of given finitedimensions the effect of boundary conditions becomes insignificant provided the foundation constant k1 is large enough to cause buckling in ripples the wave lengthsof which are small compared to the linear dimensions of the plate.
3Non-Linear Buckling. Quadratic Foundation
We consider the equation
where D, N and k1 are as before and where k2 and h are additional constants, k2 having the same dimension as k1 and h having the same dimension as w. Weassume that k2 is of the order of magnitude of k1 and positive. The constant h may be taken to be the thickness of the plate.
Having the results (2.2) to (2.4) of linear buckling theory, we wish to obtain information on the effect of the non-linear term in (3.1). The simplest way that this effectcould manifest itself would be by making the arbitrary amplitude constant A of linear theory a function of NB and k2. The actual situation comes out to be considerablymore complicated.
Page 178
Equation (3.1) may be modified by introducing non-dimensional independent and dependent variables in the form
where L remains to be determined. Inspection of the resulting equation for suggests setting
whereupon equation (3.1), with (2.4), becomes
Since we are concerned with values of N near to NBL, we rewrite (3.4) with a parameter defined by
to read as
Inspection of equation (3.6) indicates that results of interest for small values of should be obtainable from an expansion of the form
where 1, 2, etc. and their derivatives are O(1) (at most).
Introduction of (3.7) into (3.6) leads to the sequence of equations
etc.
We propose to show that the first two equations of this sequence directly lead to the desired results concerning initial postbuckling behavior.
Evidently, equation (3.8) has solutions of the form
where the Ai are arbitrary and the i and i are required to satisfy the relation
A comparison of (3.10) and (3.11) with (2.2) to (2.4) indicates that breaking off the expansion of the solution of the non-linear problem at this stage has led to resultsequivalent to those obtained by direct consideration of the linear buckling problem.
Introduction of equation (3.10) into (3.9) and observation of (3.11) gives next
Page 179
where
The decisive step in our procedure is now to stipulate that the right hand side of (3.12a) must in the end not contain any of the terms cos i cos i with
, or in other words that the solution 2 must be free of secular terms.
To see the meaning of this statement we consider successively the effect of assuming a one-mode, a two-mode or a three-mode combination for 1.
One-Mode Solution
We have on the right of (3.12a)
The only possible way in which all additive terms cos 1 cos 1 can be made to disappear is by setting A1 = 0. This means there is no one-mode solution of thelinear problem which can be extended into the non-linear domain by our procedure.
Two-Mode Solution
We now have on the right of (3.12a)
where both A1 and A2 are different from zero and where .
In trying to match terms coming from the non-linear portion in the above to the terms of the linear portion we find one possibility, upon setting
whereupon
so that
Furthermore, equation (3.14) may be written in the form
The conditions of no secular terms in 2 are then two in number, namely
giving
Page 180
Introduction of (3.20), (3.17), (3.7) and (3.5) into (3.2) leads to the conclusion that possible two-mode solutions of the nonlinear problem are such that
representing a severely limited subset of the class of solutions A1 cos 1 cos 1 + A2 cos 2 cos 2 of the corresponding linear problem.
We note finally that associated with (3.21) is a formula for the dependence of N/NBL on the value wc for which w is numerically largest. We find from (3.21) that
Three-Mode Solution
We now have
with A1A2A3 0, to insure the three-mode property.
In order to make possible the disappearance of secular terms in 2 we must now find terms among the non-linear portion of g which annul all three linear portions.
The evident symmetry properties of the expression on the right of (3.23) suggest that we should try to balance the linear A3-term by all or part of the non-linearA1A2-term. In order that this be possible, we must equate 3 to either 1 + 2 or 1 2 and at the same time equate 3 to either 1 + 2 or 1 2. Considering thatat the same time
we find that we must use once the plus sign and once the minus sign, i.e.
Introduction of 1 and 1 from (3.25) into (3.24) leads to the result that the possible solutions of (3.24) and (3.25) are a one-parameter set which may be written inthe form
Introduction of (3.26) into the terms with A1A2 in (3.23) shows that one of the conditions for the elimination of secular terms in 2 comes out to be
Page 181
Having taken care of the linear A3-term in (3.23) it is next necessary to take care of the linear A2 and A1 terms. Inspection of the A1A3 and A2A3-terms in (3.23)together with the relations (3.26) shows that elimination of all cos 2 cos 2 -terms and cos 1 cos 1 -terms in g is accomplished upon setting
The system (3.27) and (3.28) has the solutions
which are to be substituted in
Equation (3.30) allows the derivation of a formula for N/NBL, analogous to (3.22), differing from (3.22) only insofar as the numerical factor is concerned.
Four-or-More-Mode Solutions
Evidently the most general solution 1 of the form (3.10), with (3.11), may be written as
It would seem to be of interest to establish the possible forms of the function A( ) such that combination of (3.31) with (3.9) does not result in the production ofsecular terms in 2.
4Imperfection Sensitivity. Quadratic Foundation
The differential equation for w is now
In addition to non-dimensional variables as in (3.2) we introduce a non-dimensional initial deflection in the form
Equation (4.1) can now be written as
In attempting an asymptotic expansion of the form (3.7) for the solution of (4.3) an important element of the procedure comes out to be the necessity of associating
the effect of with the determination of 2 rather than 1. This means that in undertaking the start of an asymptotic expansion we must stipulate that
(rather than ) where and are O(1) (at most).
Page 182
Introduction of (4.4) and (3.7) into (4.3) leaves equation (3.8) for 1 as before while equation (3.9) for 2 is changed to
In solving (4.5) we distinguish between two types of contributions to the imperfection function . One contribution does not contain any of the terms which can occurin 1 and therefore the question of secular terms in 2 does not arise. The other contribution does contain terms also occurring in 1 and consequently secular termsmust be eliminated. Presumably, it is this second contribution which is critical insofar as imperfection sensitivity is concerned and in what follows consideration islimited to this second contribution, in conformity with the previous work in this field.
Writing then
and taking 1 as in (3.10) we now have as equation for 2
Having (4.7) the analysis from here on proceeds as for the non-linear buckling problem. Specifically we will have a two-mode solution where
and we will have a three-mode solution where
with the i and i as in (3.15,16) and (3.26), respectively.
The nature of the further discussion of the problem of the imperfect plate is illustrated for the two-mode case with . We now have from the second relation in(4.8)
and from the first relation in (4.8)
To see the physical meaning of this result we revert back from the dimensionless functions and to and w by writing in accordance with (4.4) and (4.2)
and in accordance with (3.7) and (3.2)
We further set
Page 183
and we may then consider the variation of 1 and 2 as a function of N/NBL, for given .
We take first the case A2 = 0 and . For this we find
We then take the case A1 = 2 and . For this we find
From equations (4.15) we obtain the existence of a critical bifurcation-stress N* and of an associated dimensionless deflection , given by the intersection of thetwo N versus 1-curves, that is by
It is apparent that the above results for the quadratic-foundation plate problem are analogous to Hutchinson's results for the externally pressurized spherical shell [1].The analogy becomes even more evident if we write equations (4.8) and (4.9) in terms of the quantities i and as defined in (4.14). Equations (4.8) then become
and equations (4.9) become
For a complete analogy with Hutchinson's equations we should on the right of (3.8 ) and (4.9 ) have rather than . This can be accomplished formally by
changing the definition of in equation (4.4) by writing instead of (4.4) the relation . Since, however, the entire analysis is based on using the first stepin an expansion in powers of the quantity , the results obtained in this manner would not seem to be more accurate than the results based on (4.8 ) and (4.9 ), in thatregion of N/NBL-values for which the analysis based on the first step of the expansion procedure is relevant.
5Postbuckling Behavior and Imperfection Sensitivity of Plate on Cubic Foundation. Elementary Theory
We now consider the equation
and in it set in extension of equations (3.2) and (4.2)
Page 184
This transforms (5.1) into the relation
We take L again as in (3.3) and in addition dispose of w0 in a manner consistent with what was done for the quadratic-case by setting
With as in (3.5) we can now write equation (5.3) in the form
which differs from (4.3) in the exponent of the non-linear term only.
Postbuckling Behavior.
Setting we attempt an expansion in powers of analogous to equation (3.7). In order to obtain a system of successive equations as in (3.8) and (3.9) we mustnow set
This gives
We may now proceed in the same manner as in equations (3.10) to (3.12) for the quadratic case by considering successively one-mode, two-mode and three-modecombinations for 1. We limit ourselves here to the one-mode case because in contrast to what happens for the quadratic foundation, a one-mode solution is possiblefor the cubic foundation.
One-Mode Solution
The right-hand side of the equation , as g is of the form
In this all additive terms cos 1 cos 1 can now be made to disappear by setting
The relation between deflection w and load N is obtained upon introducing 1/2 1 and w0 from (5.4) into equation (5.2). We find
and from this
Imperfection Sensitivity
In analogy to the step from (4.3) to (4.4) we now set
Page 185
giving as equation for 2
Restricting consideration again to the one-mode case, we set in (5.13)
and have then as condition for the absence of secular terms in (5.13) the relation
In view of (5.12), (5.14) and (5.2) this is associated with the representation
If we designate the coefficients on the right of (5.16) by 1 and , respectively, we obtain from (5.15) as load deflection relation, as affected by the initialimperfection
From this follows for the critical load N* and the associated dimensionless deflection ,
We add the remark that a one-mode solution also applies for the case of a ''quadratic" foundation with k2w2 replaced by k2|w|w.
6A Remark on Mixed Quadratic-Cubic Foundation Effects
We set in the non-linear buckling problem, as governed by an equation of the form
w = (k1/k2)h ( , ), as in (3.2), and = 1 N/NBL, as in (3.5). This transforms equation (6.1) into
The conclusion which follows concerns itself with the effect of foundations for which
Page 186
Assuming, as in (3.7), that = 1 + 2 2 + . . ., we obtain from (6.2) a sequence of equations of which we list the first three,
It is apparent from (6.4) that for the mixed quadratic-cubic foundation, with the restriction (6.3), initial postbuckling behavior is effectively governed by the quadraticcontribution to the foundation effect, the cubic contribution having an influence which is small of higher order.
7Postbuckling Behavior of Plate on Cubic Foundation. Rational Theory
In place of equation (5.1) we consider the von Karman finite deflection equations
We introduce dimensionless variables as in (5.2) and write further
Choosing
and setting
equations (7.1) and (7.2) may be written in the form
Expanding again in powers of a sequence of systems of equations is obtained, of which the first two are
Further consideration will be limited to the case of a one-mode solution
Page 187
of the first equation in (7.8). From the second equation in (7.8) it follows then that
Introduction of (7.10) and (7.11) into (7.9a) gives as equation for 2
Avoidance of secular terms in 2 now requires that A1 be subject to the condition
or
which reduces to (5.9) when = 0.
Introduction of (7.14) into (7.10) and (7.3) gives further, in modification of (5.10),
and then
It is evident from (7.16) that consideration of the non-linear finite plate deflection terms results in an effect which counters all or part of the effect of the cubic
foundation term in (7.1) and that this countering effect depends on the aspect ratio of the wave pattern through the factor , which varies between one halfand one, and on the value of the parameter . For a homogeneous plate with differing stretching and bending moduli Es and Eb we have from (7.5) with D = 1/12(1
2)Ebh3 and C = 1/Esh,
Therewith equation (7.16) becomes
Page 188
showing imperfection sensitivity whenever k3/k1 > 4/3(1 2)Es/Eb, and absence of such sensitivity when k3/k1 < 4/3(1 2)Es/Eb.
References
1. J. W. Hutchinson, Imperfection Sensitivity of Externally Pressurized Spherical Shells. Journal of Applied Mechanics 34, 4955, 1967.
2. W. T. Koiter, On the Stability of Elastic Equilibrium, Thesis Delft, 1233, 1945.
3. W. T. Koiter, General Equations of Elastic Stability for Thin Shells. Donnell Anniversary Volume, 187228, Houston, Texas, 1967.
Page 189
On the Analysis of First- and Second-Order Shear Deformation Effects for Isotropic Elastic Plates[J. Appl. Mech. 47, 959961, 1980]
Introduction
In what follows we consider once more briefly the problem of transverse shear deformations for isotropic plates, within the framework of the two-dimensional sixth-order theory as derived from three-dimensional theory by a variational method [2] or, alternately, by means of self-contained two-dimensional considerations [3].
Specifically, we are here concerned with the fact that it is possible to distinguish between first and second-order shear deformation effects, with the determination ofthe first-order effects depending on a rational analysis of edge-zone behavior and with the second-order effects requiring no such analysis [5]. As regards the natureof the two kinds of effects we note, in particular, that the second-order effect is a natural generalization of Timoshenko's analysis of shear deformation in beams whilethe first-order effect disappears in a specialization of the plate problem to the corresponding problem of the beam.
As regards the objects of this Note, these are as follows.
Recent considerations by Simmonds [4], including a description of results by Goldenveiser [1] concerning the asymptotic derivation of a fourth-order plate theory inwhich first-order shear correction terms are accounted for by a modification of the classical Kirchhoff boundary conditions, make it seem worthwhile to indicate thatresults of the same nature are in fact implied by the writer's sixth-order two-dimensional plate theory.*
Our results on modified Kirchhoff boundary conditions in [3] were stated for the case of straight edges only. It seems desirable to present a derivation of thecorresponding results for the case of curved edges inasmuch as edge curvature brings with it a significant supplementary term in the relevant formulas.
Two-Dimensional Plate Equations in Polar Coordinate Form
We depart from our earlier Cartesian-coordinate statement of sixth-order two-dimensional theory for plates which are two-dimensionally isotropic and homogeneous[3] and rewrite these (for the case of absent distributed surface loads) with reference to polar coordinates r, in the form
*We note that we did not think of this possibility in the presentation of our earlier work [2], and that our later analysis by a direct two-dimensional approach [3] led us to this possibility withoutconsciousness of its relation to Goldenveiser's results.
Page 190
In this we have
with D and B being transverse bending and shear deformation factors, and w and being solutions of the differential equations
The factors D and B are, for the case of a plate which is also homogeneous in thickness direction
Therewith, for this case
and then for a three-dimensionally isotropic material as considered in [2].
Asymptotic Analysis
Given a circular ring plate with inner edge r = a we consider the system of stress boundary conditions,
and, alternately, the system of displacement boundary conditions,
The possibility of an asymptotic analysis is given upon making the fundamental assumption
*The restatement of this expression involves a useful transformation with the help of the differential equation (7b).
Page 191
and upon making use of the order-of-magnitude relations
with these depending on restrictive assumptions of the form , etc.
It follows from (12) and (6) that now
and, if we designate -contributions to Mrr, etc., by a superscript i, the boundary conditions (9) may be written in the form
for r = a, while the boundary conditions (10) may be written in the form
Having the systems (14) and (15) we now proceed to deduce from them a system of abbreviated relations, in such a way that terms of relative order 1/ a areretained, while terms of relative order 1/( a)2 are being disregarded . To accomplish this reduction, it is necessary to stipulate at the outset a suitable order-of-magnitude relation between the dependent variables and .
Inspection of the system (14) indicates that a reduction of this system is accomplished upon stipulating that
Therewith equations (14a, b) become, except for terms of relative order 1/(a )2,
with the relevant expressions for , , now involving w rather than directly, as a consequence of equation (13), and with equation (14c) remainingunchanged.
The corresponding order-of-magnitude stipulation regarding and for the system (15) is
Page 192
Therewith the -term in (15c) is of the same order of magnitude as the -term, with the -contribution in (15b) now being of relative order 1/ a. At the same time,because of (13), equations (15b, c) may be written in terms of w rather than in terms of , as follows:
Having the systems (17a, b) and (14c), and (19a, b) and (15a), we now use these for the derivation of equivalent systems which are of such nature as to allow asequential determination of w and , with the -problem being the desired generalization of Kirchhoff's problem in which first-order transverse shear deformationterms are taken into account entirely by modification of Kirchhoff's boundary conditions.
In order to derive from the given systems of three boundary conditions for w and separate systems of two conditions for and one condition for we make use ofthe differential equation 2 2 = 0 in the asymptotic form ,rr 2 = 0, from which it follows that = ( ) e- (ra) and therewith, except for terms of relative order1/ a,
The introduction of (20) into (17a, b) changes these relations into
for r = a. Equation (21b) may be rewritten in the form
A substitution of this in (21a) and (14c) then gives as modified Kirchhoff boundary conditions, involving w alone
for r = a.* It is evident from (23a, b) and (22), in conjunction with the differential equations (7a, b), that the asymptotic determination of w and up to terms ofrelative order 1/ a is now in fact of a sequential nature. The previous result [3] for the case of a straight boundary follow from (23a, b) by first setting (), /a = (),2 andby then going to the limit a , with the term 2/ a in (23b) disappearing in this process.
*We note that for the case of a three-dimensionally isotropic plate, which is the case considered by Goldenveiser [1, 4], we have in the foregoing . The numerical factor
corresponds to a factor 0.630 . . . in Goldenveiser's three-dimensional asymptotic analysis.
Page 193
The analogous reduction of the displacement boundary conditions (19a, b) comes out as follows. We first combine (20) and (19b) in the form
and then use this relation in equation (19a) so as to obtain as second displacement boundary condition for w alone, in complementation of (15a),
for r = a. Equations (25), (24), and (15a) reduce directly to the corresponding conditions in [3] for the case of a straight boundary.
References
1 Goldenveiser, A. L., "The Principles of Reducing Three-Dimensional Problems of Elasticity to Two-Dimensional Problems of the Theory of Plates and Shells,"Proceedings of the 11th International Congress on Applied Mechanics , 1966, pp. 306311.
2 Reissner, E., "The Effect of Transverse Shear Deformation on the Bending of Elastic Plates," J OURNAL OF APPLIED MECHANICS, Vol. 12, 1945, pp. A69A77.
3 Reissner, E., "On the Theory of Transverse Bending of Elastic Plates," International Journal of Solids and Structures, Vol. 12, 1976, pp. 545554.
4 Simmonds, J. G., "Recent Advances in Shell Theory," Proceedings of the 13th Annual Meeting Soc. Eng. Science, 1976, pp. 617626.
5 Timoshenko, S., and Woinowsky-Krieger, S., Theory of Plates and Shells, McGraw-Hill, 1959, pp. 98104.
Page 194
A Tenth-Order Theory of Stretching of Transversely Isotropic Sheets *
[J. Appl. Math. & Phys. (ZAMP) 35, 883889, 1984]
Introduction
In what follows we return once more to a problem which concerned us a long time ago [1,2], having to do with the approximate determination of three-dimensionalPoisson's ratio corrections to the two-dimensional generalized plane stress theory of stretching of sheets.
Our reasons for returning to this problem are in part due to a belated recognition that a 1949 joint manuscript on this subject, which had been accepted forpublication, subject to stylistic revisions, had in fact never been resubmitted with these revisions. Going beyond this, subsequent variational developments [4] andmore effective insights concerning the nature of interior and edge-zone solution contributions in the two-dimensional analysis of transverse bending of plates [5] madeit likely that a reconsideration of the problem at this time would result in an analysis both simpler and more insightful than our earlier work. We add to these reasonsthe observation that the first paper on the subject, which may be considered effectively as a preparatory effort to the second named author's early analysis oftransverse bending of shear-deformable plates [3], appears in a now nearly unavailable publication [2].
The present paper limits itself to the formulation of the general problem, and to its reduction to a system of Laplace operator differential equations. A subsequentpaper (by R.A.C.) will be a reconsideration of a specific half-plane boundary-value problem previously considered in [1], with results going somewhat beyond whathad been obtained earlier.
Derivation of System of Two-Dimensional Differential Equations
We consider a uniform homogeneous transversely isotropic layer bounded by two parallel planes, z = ±c, and by cylindrical surfaces, i(x, y) = 0, with there beingtwo such surfaces for the problem of an infinite layer with a single hole. We assume that the boundary portions z = ±c are traction free and that the cylindricalboundary portions are acted upon by an equilibrium system of prescribed tractions, , , , with and being even in z and being odd in z.
*With R. A. Clark.
Page 195
Differential equations and boundary conditions of the problem as stated are known to be the Euler equations of a variational equation I = 0, where
with E, G, Ez, positive, with 2 < 1 and < 1 to ensure strain energy positive-definiteness, and with independent interior and boundary stress anddisplacement variations [4].
We use this variational problem to derive a tenth-order two-dimensional system of sheet equations by stipulating approximate stress distributions
where
We note that the function Z has been chosen in such a way as to ensure satisfaction of the stipulated traction conditions for z = ±c and that, moreover, equations (2)and (3) are consistent with the homogeneous equilibrium equations for stress, provided the nine coefficient functions in (2) and (3) are subject to the two-dimensionaldifferential equations
An introduction of (2) to (4) into (1), with the supplementary stipulation that the z-dependence of the traction functions and is consistent with the stipulations in(2) to (4), and with defining relations
Page 196
for weighted displacement averages Ux, Uy, Vx, Vy and W, gives, upon integration with respect to z
In this the constitutive coefficients C, B, A are given by
The variational equation I = 0 now implies as Euler differential equations the five equilibrium equations (5) and (6), together with nine constitutive equations of theform,
and, as Euler boundary conditions,
We note that the above results, except for the introduction of specific displacement averages, are in essence equivalent to the results in [2], upon setting Ez = E, z =
and 2(1 + )G = E, and equivalent to the results in [1], upon replacing 1/E = z/Ez in [1] by the constitutive parameter in equation (1).
Reduction to a System of Laplace Operator Equations
Then tenth-order system (5), (6), (10) to (13) may be uncoupled in terms of three functions , and so as to have one second-order equation for and twofourth-order equations for and , respectively, with all three equations having derivatives in Laplace operator form
Page 197
only. It is found that of these three functions represents the interior solution contribution and and edge-zone solution contributions to the complete solution ofthe given system of differential equations, in the range of parameter values for which it is appropriate to make this kind of distinction.
The first step of the reduction is satisfaction of the three equilibrium conditions in (5) in terms of stress functions K, and , as follows:
The second step involves writing the constitutive equations (11) in the inverted form,
with a corresponding expression (17c) for Ryy, and then to use the equilibrium equations in (6) to deduce as expressions for Sx and Sy,
A comparison of (18a, b) with (16) then leads to the conclusion that and depend on Vx, Vy and T in the form
The third step depends on the use of the constitutive equations in (12), in conjunction with equations (19b) and (16), to deduce a differential equation for , of theform,
where, on the basis of (9), CRBs = (2c2/21)(E/G).
The fourth step consists in deducing from the constitutive equations in (10), in conjunction with the stress-function relations in (15) and (16) through use of theconventional compatibility equation involving Ux,x, Uy,y and Ux,y + Uy,x, the further differential equation,
Equation (21) is evidently equivalent to a representation for K in the form
where and where is a solution of the biharmonic equation
Page 198
The fifth and final step is based on the relation,
which is a ready consequence of (12), (19a) and (16). To make use of (24), we note that equation (17a) and its permutation (17c), together with (19a), result in therelation
Introducing (25) as well as T from (16), Nxx + Nyy from (15) and K from (22) into equation (13), we have, after some rearrangement,
Eliminating W between equations (26) and (24) and taking (23) into account, we obtain as a second fourth-order differential equation, which, remarkably, involves only,
The coefficients Ai, in this are given by
Given the way in which the half thickness c enters into the coefficient functions in (20) and (27), it is evident that, as long as E, G and Ez are of the same order ofmagnitude, the functions and represent edge-zone solution contributions provided that all characteristic in-plane linear dimensions which enter into theformulation of boundary-value problems are large compared to the length c. At the same time it is evident from (23) that under the same circumstances the function represents the interior solution contribution.
We complement our reduction of the differential equations of the problem by the observation that, while equation (16) directly expresses the transverse stressmeasures Sx, Sy and T in equation (3) in terms of and , equation (15) in conjunction with (22) directly expresses the in-plane stress measures Nxx, Nyy, Nxy interms of and .
Expressions for Rxx, Ryy and Rxy in terms of the stress functions are more complicated. They may be obtained by first using (19a) to write (17a,c) as
with Vx and Vy following from (12), (16), (26) and (22). The results are
Page 199
where
We note that the terms with 2 in (31) include as special cases known results for the exact theory of plane stress for isotropic materials [6]. This means that theapproximate results as obtained here include exact results of three-dimensional elasticity subject to the limitation that the form of the prescribed boundary conditions issuch as to be compatible with the stipulation that = 0 and = 0 throughout.
References
[1] R. A. Clark, On the theory of generalized plane stress. M. S. Thesis. Massachusetts Institute of Technology, 1946.
[2] E. Reissner, On the calculation of three-dimensional corrections for the two-dimensional theory of plane stress . Proc. 15th Semi-Annual EasternPhotoelasticity Conference, 2331 (1942).
[3] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, A69A77 (1945); 13, A252 (1946).
[4] E. Reissner, On a variational theorem in elasticity. J. Math & Phys. 29, pp. 9095 (1950).
[5] E. Reissner, On the analysis of first and second-order shear-deformation effects for isotropic elastic plates . J. Appl. Mech. 47, 959961 (1980).
[6] S. Timoshenko and J. N. Goodier, Theory of elasticity, 2nd Ed., 241244 (1951).
Page 200
On Asymptotic Expansions for the Sixth-Order Linear Theory Problem of Transverse Bending of Orthotropic ElasticPlates[Computer Methods in Applied Mechanics and Engineering 85, 7588, 1991]
1Introduction
Given the existing interest, from a computational point of view, in the boundary layer aspects of the sixth-order theory of transverse bending of shear deformableelastic plates [13] we are concerned in what follows with asymptotic expansions, in extension of earlier results for the leading terms of such expansions, which werededuced in a less systematic manner for isotropic [4, 5] and orthotropic [6] plates.
The starting point of our analysis is a system of three simultaneous differential equations for the deflection W of the plate in conjunction with two transverse shearstress resultants Qx and Qy, with this system having been established, for isotropic plates, in [7, 8], and for orthotropic plates in [9].
With our earlier work having been concerned with appropriate versions of the canonical stress boundary condition problem and the canonical displacement boundarycondition problem we here consider a mixed problem which has the two canonical problems as limiting cases. As regards our method of analysis we note inparticular our approach to the task of appropriate scaling of the equidimensionalized version of the boundary layer portions of our three dependent variables, for thepurpose of obtaining a better understanding of the importance of distinguishing 'soft' and 'hard' support relative to the matter of a smooth transition from the sixth-order theory to the fourth-order Kirchhoff theory, in the limit of vanishing plate thickness. In this connection we also note our introduction of the concept of 'almostsoft' support as a consequence of the present approach to this transition problem.
For convenience sake our considerations are limited to the problem of a semi-infinite plate with a straight edge, with this limitation enabling us to dispense with roadblocks in the path of clarity without affecting the significance of our results.
2The Sixth-Order Boundary Value Problem
With x and y coordinates in the plane of the undeflected plate, the differential equations for transverse shear stress resultants Q, stress couples M, rotationaldisplacements and deflection W are, for a plate with linear orthotropic behavior,
Page 201
For present purposes it is assumed that the values of the coefficients D, A and B are independent of x and y and that the load intensity P is a given function of x, y.
For a solution of (1)(4), in the domain x 0, y , we here stipulate a system of boundary conditions of the form
As regards the statement of conditions for y = ± , it is sufficient in the present context to note the possibilities of decaying or periodic behavior. The barredquantities in (5) are given functions of y, subject to the requirement that the semi-infinite plate be in overall equilibrium. The quantity b is a characteristic length such
that P ,y = O(P/b), , etc., and the quantities C are weighting factors with constraints
When Cw = Cx = Cy = 0 the mixed conditions (5a, b, c) reduce to a system of stress conditions. When CQ = CM = CT = 0, the mixed conditions reduce to a systemof displacement conditions. Of particular interest in what follows are the limiting cases CT = 0 and Cy = 0, with the former representing 'hard' support and the latter'soft' support in the sense of the considerations in [13].
To facilitate our approach to the solutions of the boundary value problem (1)(6) we rewrite (3) and (4) in the form
and we introduce (9a, b, c) into (2) so as to have as two equations for the three dependent variables W, Qx, Qy:
In place of using (10) and (11) as they stand we use (1) to eliminate Qy,xy in (10) and Qx,xy in (11) so as to finally have
Page 202
Equations (12) and (13), in conjunction with (1), are our basic system of three simultaneous differential equations for W, Qx and Qy.1 The associated boundaryconditions in terms of W, Qx and Qy follow upon introducing (8) and (9a, c) into (5a, b, c), and (9a, c) into (6).
Remark
For the case of isotropy with Dx = Dy = D, Dv = vD, Dt = 1/2(1 v)D, Bx = By = B and Ax = Ay = A, eqs. (12) and (13) reduce to the compact form
and these, with V written in place of Q, become eqs. (V) and (VI) in [7], upon setting
3Equi-Dimensionalized Form of the Boundary Value Problem
We retain the variables Qx and Qy and we introduce equi-dimensionalized variables w, x, y, mx, my, mt and p as follows:
We further write
with , and being dimensionless quantities of order unity and with h as the thickness of the plate.
Equations (12), (13) and (1) therewith take on the form
The introduction of (17)(19) into (8) and (9a, b, c) gives
1Equations (12) and (13) have previously been derived in [9]. However, in place of using them in conjunction with (1), they are there used in conjunction with a third equation of higher order whichfollows from (1) upon the introduction of the differentiated Q-terms from (12) and (13) into (1).
Page 203
while the use of (17) and (18) in (5a, b, c) changes these conditions into
with a corresponding change in (6).
4Interior and Edge Zone Solution Contributions
In order to effect a solution decomposition into interior and edge zone contributions we introduce three dimensionless independent variables , , , and adimensionless small parameter by writing
With this we assume that the solution of the system (20)(22) will be of the form
At the same time we consider the possibility that the load intensity function p may be of the form
in such a way that there are no changes of orders of magnitude in connection with differentiations with respect to , and .
The introduction of (27) and (28) into the system (20)(22) leaves as differential equations for the determination of interior and edge zone solution contributions twoseparate systems of the following form:
and
The introduction of (27) and (28) into (23a, b) and (24a, b, c) gives, in analogy to (27),
Page 204
where
and
Given (29)(39) the remaining task is to establish a rational procedure for the formulation of boundary conditions for interior and edge zone solution contributions, onthe basis of the conditions in (25a, b, c) for = = 0 and the corresponding conditions for = = .
5Parametric Expansions for Interior and Edge Zone Solution Contributions
An inspection of the system (29)(31) shows the evident possibility of parametric expansions of the form
with the leading terms in these expansions being equivalent to the consequences of the classical fourth-order theory. However, it turns out that it is necessary, inconnection with the nature of the boundary conditions (25a, b, c), to use in place of (40) the more general expansions
with distinct recursion relations following from (29)(31) for terms involving even and odd powers of .
Corresponding expansions for the solutions of (32)(34) are of a somewhat less elementary nature and the rational resolution of the difficulty in connection with thenegative powers of represents an essential element of our analysis.
Assuming, for simplicity's sake, that pe = 0 in what follows it is necessary to take account of two salient facts, as follows. 1. We do not at this stage know the ordersof magnitude of the edge zone contributions relative to the interior contributions. 2. The orders of magnitudes of we, and relative to each other must be such as
to result in a sequence of second-order problems, which complement rationally the sequence of fourth-order problems for wi, and .
The form of (32)(34) makes it appear that expansions in accordance with the stated two requirements should be
with the exponent n remaining at our disposal.
Page 205
The introduction of (42a, b, c) into (32)(34), with pe = 0, indicates that in order to accomplish our objective it is essential that we first consider (33), written in theform
With following from this sequentially we next obtain sequentially, on the basis of writing the homogeneous equation (34) in the form
After this it remains to obtain on the basis of (32), written in the form
We complement (43)(45) by a restatement of (38) and (39a, b, c) in the form
In conjunction with the above we have, upon introducing (41) into (36) and (37a, b, c),
Page 206
With this and with (27) we now rewrite the boundary conditions (25a, b, c) for x = 0 as conditions for = = 0, in the form
In order to utilize the above it is necessary to appropriately dispose of the edge zone solution contribution exponent n. An appropriate disposition will be one which isassociated with the possibility of having self-consistent systems of conditions for the successive terms of the fourth-order interior expansions as well as for thesuccessive terms of the second-order edge zone expansions. We expect that such a self-consistent system will be unique, but we will not, here, furnish a mathematicalproof of the validity of this expectation.
Restricting, for the present, attention to the question of the leading terms in the two types of expansion we see that, in general, the appropriate choice for n will be
With n = 1 the boundary conditions for the leading expansion terms become
To see that the system (52a, b, c) is not always self-consistent it suffices to consider the special case CT = CQ = 0. For this case we have that (52a, b, c) becomes asystem of three conditions for the solution of the fourth-order interior problem, with no condition for the second-order edge zone problem.
An inspection of (50c) suggests that when CT = 0 we should replace (51) by the stipulation
With this value of n and with CT = CQ = 0 the leading term version of (50a, b, c) reduces to the form
In connection with the special case n = 2 and CT = CQ = 0 we note the following generalization of this case. If we assume
with cT = O(1) and cQ = O(1) then it follows from (7) that
Page 207
With this we obtain, in place of the two conditions in (54a),
with (54c) in conjunction (54b) again being of a self-consistent nature.
Remark
When n = 1 we will have the order of magnitude relations
with these being consistent with a previously observed stress concentration result [6]. At the same time it is seen, on the basis of (42c), (46a, b) and (47a, b) that theedge zone contributions to displacements and bending stress couples come out to be of a smaller order of magnitude than the associated interior contributions.
For the exceptional cases with n = 2, and dominate and , respectively, in addition to dominating , while and come out to be of the same
order of magnitude as and .
6Explicit Determination of the Leading Terms of the Edge Zone Solution Expansions
We deduce from (43) and the prescribed solution behavior for = that
with K0 and K1 being arbitrary functions.
The introduction of (56) into (44) gives
The introduction of (57) into (45) gives
with this being consistent with the known result that we 0 for the case of isotropy, for which vy = v xx and 2 ty = (1 v) xx.
With (56)(58) we then have further that
and
and the introduction of (59) and (60) into (46a, b) and (47a, b, c) gives
with (59)(62) being of the essence in what follows.
Page 208
7Kirchhoff and Non-Kirchhoff Interior Solution Results
We depart from a restatement of the here relevant portions of (50a, b, c) with , etc., as follows:
The results which are the principal purpose of our analysis are obtained by solving (63c) for , in the form
and by using this relation to obtain from (63a, b) as a system of two boundary conditions for the interior solution contribution
Given that the nature of the stipulated parametric expansions implies that is of the same order of magnitude as and , we have to consider separatelythe following possibilities.
It follows from (64) that for this case
with (65a, b) reducing to the well-known conditions
for the fourth-order theory of Kirchhoff.
Given the designations in [13] of the boundary conditions with Cy = 0 as conditions of 'soft' support, it suggests itself to designate the boundary conditions with Cy =O( ) as conditions of 'almost soft' support.
We now have from (64) that, necessarily,
with (65a, b) reducing to
Page 209
It follows from (71a) that in this range of values of the weighting factors Cy and CT the interior solution contribution of the sixth order theory is not in agreement withthe Kirchhoff solution of the problem, unless CQ = O( ).
We now have from (64)
and from (65a, b)
For this 'almost hard' case of support to be self-consistent we must have CQ = O( ). If we stipulate that CQ = cQ then (74a) becomes, with Cw = 1,
with (74b) remaining unchanged.
For (74a , b) to determine an interior solution of the Kirchhoff type it is evidently necessary to have cQ = 0 and CM = O( ), whereupon, in place of (74a , b)
The contents of (75) in conjunction with (73) represent a generalization of the corresponding result in [4, 5], by way of the presence of the parameter cT in (73).
8Explicit Inclusion of First-Order Transverse Shear Deformation Effects
Given the fact that the differential equations for the determination of w0, Qx0, Qy0 and w1, Qx1, Qy1 are of the same form, it is possible to deduce, on the basis of(50a, b, c), a reduced system of boundary conditions for the direct determination of
with
We here limit ourselves to showing this for the case n = 1 for which we can write (50a, b, c) in the form
where, in accordance with (61) and (62), , etc. We limit ourselves further by requiring that our results be consistent with the conditions of Kirchhoff, byagain
Page 210
stipulating almost soft support conditions, now in the form
We then have, on the basis (77c), that
for = = 0, and therewith from (77a, b) as a system of generalized Kirchhoff conditions for the determination of the interior solution
Equations (80a, b) are a generalization of our results in [4, 5] which were obtained for the special case Cy = Cw = Cx = 0.
References
[1] D. N. Arnold and R. S. Falk, The boundary layer for the Reissner-Mindlin plate model, SIAM J. Math. Anal. 21 (1990) 281312.
[2] I. Babuska * and S. T. Scapolla, Benchmark computation and performance evaluation for a rhombic plate bending problem, Internat. J. Numer. Methods Engrg.28 (1989) 155179.
[3] B. Häggblad and K.-J. Bathe, Specifications of boundary conditions for Reissner/Mindlin plate bending finite elements, Internat. J. Numer. Methods Engrg. 30(1990) 9811011.
[4] E. Reissner, On the theory of transverse bending of elastic plates, Internat. J. Solids and Structures 12 (1976) 545554.
[5] E. Reissner, On the analysis of first and second-order shear deformation effects for isotropic elastic plates, J. Appl. Mech. 47 (1980) 19591961.
[6] E. Reissner, Asymptotic considerations for transverse bending of orthotropic shear deformable plates, J. Appl. Math. & Phys. (ZAMP) 40 (1989) 543557.
[7] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech. 12 (1945) A69A78, 13 (1946) A252.
[8] E. Reissner, On the theory of bending of elastic plates, J. Math. & Phys. 23 (1944) 184191.
[9] K. Girkmann, Flächentragwerke, 5th Edition (Springer, Wien, 1959) 583610.
Page 211
On Finite Twisting and Bending of Nonhomogeneous Anisotropic Elastic Plates[J. Appl. Mech. 59, 10361038, 1992]
Introduction
In what follows, we utilize an intrinsic form of the equations for small finite deflections of plates, based on the original Kirchhoff displacement version rather than onthe von Karman deflection-stress function version, in order to obtain generalizations of known exact solutions for the problem of twisting and bending of rectangularplates. This problem was first stated and solved, on the basis of von Karman's equations, for isotropic transversely homogeneous plates of constant thickness byReissner (1957). Subsequently, solutions were obtained for isotropic plates of variable thickness of the symmetrical double-wedge type by Bisplinghoff (1957) andSimmonds (1958), and of the symmetrical lenticular type, independently by Mansfield (1959) and Simmonds (1958). A solution for orthotropic plates of constantthickness has been given by Chen (1974). We now consider anisotropic transversely nonhomogeneous plates with constitutive coupling of stretching and transversebending and show that for this class of problems, it is again possible to effect a reduction of the problem to a system of two simultaneous linear second-order ordinarydifferential equations, with the significant nonlinear effects coming from the coefficients as well as from the right-hand sides of these equations.
In the present analysis, the boundary conditions at the loaded ends of the plate are, as before, prescribed in a global rather than in a local sense, so as to makepossible a one-dimensional solution procedure. We complement our one-dimensional analysis by the statement of a two-dimensional problem with a suitable systemof local boundary conditions, where it remains to establish that the one-dimensional result does in fact represent the interior portion of an asymptotic solution of thetwo-dimensional problem.
Differential Equations and Boundary Conditions
We begin with a statement of the equilibrium and strain displacement equations for small finite deflections in the form
In this, x and y are coordinates in the undeflected midplane of the plate; N and M are midsurface parallel stress resultants and couples; u, v, and w are base plane
Page 212
parallel and perpendicular displacement components; and and are bending and stretching strains, respectively.
We here associate these equations with constitutive relations of the form
For what follows it is of importance to transform the above system to an intrinsic form by deducing from the strain displacement relations (3) and (4) the threecompatibility equations
The differential equation system (1), (2), and (5) to (8) is to be solved for a rectangular region |x| a, |y| b, with the edges y = ±b being traction-free and with theedges x = ±a acted upon by twisting moments T and transverse bending moments M.
In the statement of these boundary conditions, account must be taken of the nature of the effective transverse edge stress resultants
and of the transverse corner forces 2Mt,(±a, ±b), in accordance with Kirchhoff. The transverse corner forces are equivalent to base-plane perpendicular forces. Thetransverse resultants Qe are distinct from base-plane perpendicular resultants P which are
The conditions for the traction-free edges are
and the conditions for the loaded edges are here stipulated to consist of three integrated moment conditions
and of three integrated force conditions
Page 213
Remark
The analysis which follows turns out to be not applicable to the more general problem for which
in place of the conditions with F = 0 and Ms = 0 in (12) and (13).
The One-Dimensional Semi-Inverse Problem
We restrict attention to plates for which the elements of the constitutive matrices are independent of x and we attempt a solution of the given boundary value problemwith stresses and strains also independent of x.
We then have, on the basis of (1) in conjunction with (11), and on the basis of (7),
with k and as arbitrary constants.
With (15), and with the omission of all x-derivative terms, Eqs. (2) and (8) take on the form
where the primes indicate differentiation with respect to y.
With Py as in (10), and with (15) and (9b), the solution of (16) has to satisfy the boundary conditions
Of the six global traction conditions in (12) and (13), the four homogeneous conditions are readily shown to be automatically satisfied on the basis of the contents of(16) and (17). The two remaining conditions may be transformed in a similar way, so as to come out in the intrinsic form
In order to solve (16) and (17), for the purpose of determining M and T as functions of k and , use has to be made of a semi-inverted form of the constitutive Eqs.(5) and (6), in conjunction with (15), as follows:
The last of these relations is needed only for the purpose of an eventual determination of the displacement components u and v in terms of the solution of the intrinsicproblem.
Page 214
Explicit Form of the Semi-Inverse Problem without Constitutive Coupling
With the stipulation of a vanishing C-matrix in (5) and (6), and with (15), we now have as constitutive relations
where Byx = Bxy, etc., and
where Dyx = Dxy, etc.
The differential equations in (16) become two simultaneous equations for My and x upon deducing from (23) that
and upon writing (24b) in the form
The elimination of x then leaves as a fourth-order differential equation for My:
The introduction of (26) into (24a, c) gives an expression for Mx and Mt to be used in the determination of M(k, ) and T(k, ) on the basis of (18) and (19)
Remark
The same as for the orthotropic case, with Bxt = Byt = 0 and Dxt = Dyt = 0, we have that (27) is an equation with constant coefficients for plates with constant valuesof the constitutive coefficients.
The same for the isotropic case in Mansfield (1959) and Simmonds (1958), it is found that for lenticular cross-section plates with
Page 215
Eq. (27) has the particular solution
where now
For this case the solution of the homogeneous Eq. (27) is not needed, inasmuch as Myp by itself satisfies (17).
A Local Boundary Value Problem
The physical significance of the one-dimensional solution of the problem with the global boundary conditions (12) and (13) depends on its being an asymptotic interiorsolution portion, for sufficiently small values of b/a, of a suitably stated problem with all boundary conditions of the local kind. Inasmuch as this aspect of the problemhas not been discussed in the previous literature, we here formulate a system of local boundary conditions for an eventual use in conjunction with an asymptoticanalysis of the two-dimensional problem.
We replace the global conditions on Nx and Nt in (12) and (13) by the two local conditions:
Given that the expressions for x and t in (15) imply that w = W(y) 1/2kx2 xy, we replace the global conditions on Mx, Px, and Mt in (12) and (13) by two localconditions:
It remains to be established that the boundary value problem with (31) and (32) in place of (12) and (13) does in fact have a solution which asymptotically coincideswith the solution of the one-dimensional problem, except in zones of order b adjacent to the edges x = ±a.
References
Bisplinghoff, R. L., 1957, ''The Finite Twisting and Bending of Heated Elastic Lifting Surfaces," D.Sc. Dissertation, Zurich Institute of Technology.
Chen, C. H. S., 1974, "Finite Twisting and Bending of Thin Rectangular Orthotropic Plates," JOURNAL OF APPLIED MECHANICS, Vol. 41, pp. 315316.
Mansfield, E. H., 1959, "The Large-Deflection Behavior of a Thin Strip of Lenticular Section," Quarterly Journal of Mechanics and Applied Mathematics, Vol.12, pp. 421430.
Reissner, E., 1957, "Finite Twisting and Bending of Thin Rectangular Plates," J OURNAL OF APPLIED MECHANICS, Vol. 24, pp. 391396.
Simmonds, J. G., 1958, "Finite Bending and Twisting of Thin Wings," B.S. and M.S. Thesis, Massachusetts Institute of Technology, Cambridge, MA.
Page 216
A Note on the Shear Center Problem for Shear-Deformable Plates[Int. J. Solids Structures, 32, 679682, 1995]
Introduction
Recent results for the shear center problem in the framework of Kirchhoff plate theory have left open the extent to which the effect of transverse shear-deformabilitybecomes significant with increasing thickness-width ratio and with decreasing transverse shearing stiffness. In the following this problem is considered for anorthotropic plate, using the principle of minimum complementary energy in conjunction with Saint-Venant type stress assumptions. In an earlier application of thisapproach to non-shear-deformable plates (Reissner, 1991), the numerical consequences were found to be quite close to corresponding results obtained by a moreaccurate and more complex analysis in which account was taken of anti-clastic curvature constraints by Reissner (1989) and Gu and Wan (1993). There is no reasonto suppose that the same would not be true when transverse shear-deformability is taken into account.
The present approximate analysis reduces the problem to an ordinary second order differential equation. It is found that this equation can be solved explicitly forplates with linear widthwise-thickness variation, with a resultant closed-form expression for the shear center coordinate in terms of an appropriate dimensionlessparameter.
Formulation
Consider a rectangular cantilever plate of span L and width a, with edges at y = 0, a and x = 0, L. The edge x = 0 is clamped and the edges y = 0, a are tractionfree. The edge x = L is stipulated to deflect uniformly by an amount W, in conjunction with two conditions of absent bending moments and edgewise rotationaldisplacements.
The minimum complementary energy formulation for a shear-deformable plate is, for this problem, given in terms of stress couples Mx, My, Mt and stress resultantsQx, Qy by the variational equation
The complementary energy density V is for a linearly elastic orthotropic plate, to which attention is restricted in what follows, of the form
Page 217
where
for a transversely isotropic homogeneous plate of thickness h = h(y).
Equation (1) is associated with constraint differential equations
and with constraint boundary conditions, which in this case are
Equations (1) and (5) will be used in conjunction with the Saint-Venant type assumptions
for an approximate determination of a force Q and a torque T,
and a shear center coordinate
Reduction
The three relations in eqn (4), in conjunction with eqns (5) and (6), give the following as expressions for Qx, Mt, and Mx:
with the prime indicating differentiation with respect to y.
The introduction of eqns (6) and (9) into eqns (2) and (1) leads to the one-dimensional variational equation
with constraint boundary conditions
and with .
The Euler differential equations of (10) are
While it would be possible to reduce eqn (12) to one second order equation for Mt, it is preferable to proceed as follows. Introduction of from the firstrelation in (12) into the second gives the following as an expression for Mt in terms of Qx:
Page 218
With eqn (13) the first relation in eqn (12), together with eqn (11), leaves the boundary value problem
It is allowable, for simplicity's sake, to set 3W/L3 = 1. Furthermore, except for terms of relative order h2/L2, eqn (14) may be replaced by
With this and upon observation of eqns (15) and (13), the expressions for Q and T become
Upon setting Bx = 0, the non-shear-deformational plate theory result
becomes an immediate consequence.
A Closed-Form Solution
It is possible to obtain an explicit solution in closed form for plates for which
In view of eqn (3), this includes the case of a homogeneous orthotropic plate of linearily varying thickness h = h0 .
Upon setting
eqns (14) and (15) assume the equi-dimensional form
with dots indicating differentiation with respect to .
An inspection reveals that the differential equation in (19) is explicitly solvable in terms of suitable powers of . Upon satisfaction of the two boundary conditions thesolution comes out to be
and
Page 219
The introduction of eqns (17) and (21) into eqn (16) gives, after some transformations,
and therewith, in accordance with eqn (8),
An impression of the significance of the effect of transverse shear deformation may be gained by considering a homogeneous orthotropic plate for which
with G = Gt = E/2(1 + ) for the case of isotropy. As increases, the value of ys/a decreases, at first approaching the cross-sectional centroid value yc/a = 2/3 fromabove. For sufficiently large values, ys/a becomes smaller than yc/a. For example, when = 1 then ys/a = 0.59. In this connection it is worth noting that for a"plate," for which the cross-section is an equilateral triangle with and for which a plate theoretical analysis is clearly not rational, eqn (23) gives ys/a =0.647 when G/Gt = 1 and 2 = 4/15, in place of the correct value ys/a = 2/3.
It seems reasonable to limit the applicability of eqn (23) by the stipulation that h0/a 1/2. This does not preclude the possibility of significant numerical effects, forsufficiently large values of G/Gt.
References
Gu, C. H. and Wan, F. Y. M. (1993). Approximate solutions for the shear center of orthotropic plates. Arch. Appl. Mech. 63, 513521.
Reissner, E. (1989). The center of shear as a problem of the theory of plates of variable thickness. Ing.-Arch. 59, 325332.
Reissner, E. (1991). Approximate determinations of the center of shear by use of the Saint Venant solution for the flexure problem of plates of variable thickness.Arch. Appl. Mech. 61, 555566.
Page 221
SHELLSMy first contribution to the literature dealing with shells was motivated by the need to explain the subject to students of Applied Mathematics. Knowing three-dimensional elasticity and having been a student in Dirk Struik's course on Elementary Differential Geometry, I found a relatively straightforward derivation of a two-dimensional linear shell theory, involving the Kirchhoff-Love hypothesis [21,28]. While the equilibrium equations of this theory were unambiguous, the same could notbe said about the associated strain displacement relations. It was only later that I learned to use the principle of virtual work to obtain consistent strain displacementrelations, with the Kirchhoff-Love condition as a consequence of constitutive stipulations [136].
Participation in an industry-sponsored research project dealing with pre-stressed spherical domes next led to an analysis of shallow spherical shells. I derived asystem of differential equations, without realizing that these could have been obtained by transforming Marguerre's cartesian system to polar coordinate form [41]. Ithen applied these equations to obtain quantitative data for a variety of axi-symmetric loading and support conditions [46]. This was followed much later by a study ofcertain unsymmetrical load problems, this time in connection with some guided missile problems [120].
Returning to the period 19451950, two efforts stand out. The first was a two-dimensional theory for the behavior of sandwich-type shells [58]. It was clear that therelative core softness required that allowance be made for the effect of transverse shear deformability, the same as for the sixth-order generalization of Kirchhoff platetheory. I found that for shells with soft cores it was also necessary to consider the effect of transverse normal stress deformability, with this being analogous in aphysical sense to the cross-sectional flattening effect in the bending of curved tubes. A much later return to the transverse normal stress deformation effect [208] wasto show its significance relative to Naghdi's considerations of "Cosserat-type" shells.
The second effort was an attempt to deal with nonlinear finite deflection problems of shells. It came to me that it should be possible to generalize the well known H.Reissner-Meissner linear-theory equations for the symmetrical bending of shells of revolution so as to have equations for finite deformations by considering meridionalslices of the shell as plane elastica resting on an elastic circular ring foundation [56, 68]. I later extended this work by a consideration of transverse shear deformation[141] and, as part of a final comprehensive account, of transverse normal stress in conjunction with anti-symmetric transverse shear, with this resulting in a symmetricsystem of three simultaneous second order equations in place of the usual two-equation system [262].
Page 222
After the first two non-linear shell-of-revolution papers, I soon considered a variety of linear and non-linear problems concerned with shallow helicoidal shells [85,90], buckling of hyperbolic-paraboloidal shells [91], vibrations of shallow shells [94, 95, 107] and corrugation effects for circular cylindrical shells [97].
A membrane-theoretical treatment, in Wilhelm Flügge's pioneering text, of the problem of non-axisymmetric edge loads acting on spherical shells led me next [102] toa boundary layer analysis of this problem on the basis of the shallow shell equations [41]. I established the extent to which the elementary membrane solution cameout as an 'interior' solution, and what the form of the 'contracted' boundary conditions for this interior solution would be. I furthermore discussed how the interiorsolution could be either a membrane solution or an inextensional bending solution, depending on the nature of the prescribed edge conditions. The seeds of theanalysis in [102] would prove to be useful, later on, in a number of other plate and shell studies.
Further work from this time period concerned symmetrical shell of revolution problems with intriguing solution properties. Bob Clark and I considered the problem ofa pressurized ellipsoidal shell [109] to establish the range of validity of a membrane solution due to H. Lorenz, and to obtain information on bending stiffness effects.An analysis of the effect of polar orthotrophy showed unexpectedly strong consequences [110]. A study of the finite-deflection equations of shallow isotropic shells[121] brought out interesting connections between co-existing linear bending and non-linear membrane boundary layers. A General Lecture for the 3rd NationalCongress of Applied Mechanics [115] presented an opportunity to summarize these results, as well as the results for some other problems, in particular for toroidalshells.
A project of a different nature which had been on my mind for a while was to derive asymptotically a system of two-dimensional shell equations from threedimensional elasticity, for the case of a symmetrically deforming transversely isotropic circular cylindrical shell. The basic thought was to non-dimensionalize the
equations of the three-dimensional theory for a shell of radius a and wall thickness h by the introduction of a third length , based on a knowledge that thetwo-dimensional theory to be obtained could be expected to involve this length b. An outline of this procedure led to Millard Johnson's 1957 Ph.D. dissertation, andsubsequently to a joint publication [117]. It fell to others, in particular to E. L. Reiss, to complement our expansion involving the small parameter h/b by a secondexpansion in terms of the smaller parameter h/a, for a more rational consideration of the boundary conditions problem than we were able to entertain without thissecond expansion.
In a later attempt to derive shell theory from three-dimensional elasticity I pursued the thought that since the two-dimensional theory was concerned with forces andmoments, there might be an advantage to start out from a 3D theory involving moment stresses in addition to force stresses [181]. From an analytical point of viewone of the attractions of this approach was that now a system of first order equilibrium differential equations became associated with a system of first ordercompatibility equations, rather than one of the second order as in the theory without moment stresses. I found that this approach allowed a particularly simplederivation of 2D equilibrium and compatibility equations, and that the absence of second derivatives in the 3D theory made it appealing to attempt an asymptotic
Page 223
derivation of 2D constitutive relations from a given 3D system by transforming the 3D relations into a set of integro-differential equations through elimination ofderivatives with respect to the shell thickness coordinate. I believe that my attempt was, in essence, successful but it left a feeling that I might not, in fact, have "dottedall the i's" in deducing the consequences of my integro-differential equation system. I am hoping that some day someone else will go back to this work for furtherprogress.
I would like to mention some other concerns with problems of shells, alone or jointly with Jim Simmonds, Fred Wan, and W. T. Tsai.
1. Asymptotic expansions for the solutions of the two-dimensional equations for circular cylindrical shells involving 'long and short' characteristic lengths [148, 154].
2. Interior discontinuity aspects for rotating shells of revolution [150].
3. An 'elementary' version of finite-deformation shell theory [157].
4. An inextensional dislocation solution for finite bending and twisting of conical ring sector shells [162].
5. Inversion problems in connection with the form of 2D constitutive relations [155, 158].
6. A 'complete' formulation of the linear version of the symmetric shell of revolution problem, including the possibility of constitutive coupling of bending and twisting[166, 182].
7. Studies of the behavior of laminated anisotropic cylindrical shells [183, 185, 195, 205].
There were finally two particularly meaningful efforts. The first of these was the formulation of a completely self consistent two dimensional finite deformation shelltheory, including transverse shear deformation and drilling moments [196, 232]. In this I was influenced by related earlier work of Simmonds and Danielson. Theessence of my analysis was to start with a readily established system of vectorial equilibrium equations, with the subsequent use of virtual work for the establishmentof a vectorial system of virtual strain displacement relations. It was the step from virtual to actual strain displacement relations, by way of the introduction of a specialtriad of unit vectors, where I needed to know what Simmonds and Danielson had done (although I had earlier independently resolved an analogous task for thesymmetric shell of revolution problem).
The second final effort was a serendipitous consequence of wanting to show to the students in my course on shells a simple example of the use of linear shallow shelltheory for an explicit solution of a problem where the step from plate to shell would be associated with qualitative consequences the nature of which could not beforeseen intuitively. I found two such examples by considering the effect of a small circular hole on states of otherwise uniform transverse twisting or membraneshearing of a spherical cap. While for the plate the corresponding classical problems of bending and stretching are no more difficult than the problems of twisting andshearing, the same turned out not to be the case for these shell problems, due to the terms in the shell equations responsible for the coupling of tangential andtransverse action. The two circular-hole problems were solved first [217, 218] and these solutions were then supplemented by corresponding solutions for a rigidinsert [219, 228]. Remarkably, two of these four problems showed a very substantial effect
Page 224
of shell curvature while for the other two the effect was relatively minor. An asymptotic analysis involving a boundary layer adjacent to hole or insert revealed thefollowing circumstances. While the solutions for twisting and shearing were, in the absence of the hole or insert, of the inextensional bending or membrane type,respectively, the effect of an insert or of a hole, in contrast, induced a membrane state or an inextensional bending state, respectively, just outside the boundary layer.This meant a conflict for the combinations hole and membrane shear, and of insert and transverse twisting, with no such conflicts for the other two problems. Itbecame apparent that the substantial effects of shell curvature on stress concentrations were associated with the conflict situations and not with the other two. Ofparticular interest was the observation that the indicated conflicts resulted in interior solution portions with inextensional bending or membrane far-field behavior, andopposite near-field behavior, and with a transition zone in which the two types of behavior were of equal importance. Given the long standing notions concerningmutually exclusive inextensional bending and membrane states it was intriguing to have come upon a physically meaningful situation with no such exclusivity.
Page 225
On the Theory of Thin Elastic Shells[Contr. Appl. Mech., Reissner Anniv. Vol., 231247, J. W. Edwards, 1949]
Introduction
The present paper is concerned with the subject of rotationally symmetric deformations of thin elastic shells of revolution.
First a self-contained formulation of the problem of finite symmetrical deflections of shells of revolution is given. From this the equations of the small-deflection(linearized) theory are obtained by specialization. An essential step in the treatment of the small-deflection problems is their reduction to two simultaneous second-order differential equations. This reduction was first given by H. Reissner [11] for spherical shells of constant thickness. Subsequently E. Meissner [6] published thecorresponding reduction for general shells of revolution. Here this reduction is carried out in a slightly modified manner, which is believed to possess certainadvantages, which will be indicated.
From the general equations of the small-deflection theory a simplified system of equations is obtained in a systematic manner which applies for shallow shells. It isshown that the solution of this system of equations can be expressed in terms of Bessel functions for the entire class of paraboloidal shells of constant thickness. Thisgeneralizes known results for the case of a shallow spherical shell [9] for which the meridian curve is equivalent to a second-degree parabola. It is also shown that thesolution can be given in terms of elementary functions for a class of shallow shells with varying thickness, such that the problem of conical shells with linearly varyingthickness is included as a special case.
Finally, some observations are added on the subject of the asymptotic integration of the differential equations of shell theory, which was first shown in H. Reissner'swork [11] to be an appropriate method for this class of problems [5] These observations concern the effect of significant changes of thickness and curvature of theshell over distances of the same order of magnitude as those associated with the edge effect in shell theory.
Formulation of the Problem of Finite Symmetrical Deflections of Shells of Revolution
Geometrical Relations
(Figure 1) The equation of the middle surface of the shell is written in the parametric form
so that together with the polar angle in the x, y-plane are the coordinates on the middle surface. The sloping angle of the tangents to a meridian curve is givenby
Page 226
Fig. 1.Middle surface of shell, showing coordinates , on middle
surface and unit vectors associated with middle surface.
From Eq. (2) follows that
where primes denote differentiation with respect to and where is given in the form
Let i, j, and k be unit vectors in the x, y, and z-directions, respectively. The radial and circumferential unit vectors jr and j , are then defined by
and tangential and normal unit vectors j and n by
The radius vector R to any point of the shell may now be written in the following form:
where represents the distance of the point from the middle surface.
Page 227
The quantities , , define a system of orthogonal curvilinear coordinates in space. The linear element for this coordinate system is of the following form:
Finally it is noted that
where R and R are the principal radii of curvature of the middle surface of the shell.
Analysis of Strain
(Figure 2) The quantities referring to the undeformed middle surface are indicated by a subscript o, and the equation of the deformed middle surface is written in theform
The quantities u and w are then the components of displacement in the radial and axial directions, respectively. Moreover
Fig. 2.Side view of element of shell in undeformed and in deformedstate. Also shown are visible stress resultants and couples
and load intensity components.
Page 228
where is the angle enclosed by the tangents to the deformed and the undeformed meridian, at one and the same material point.
The customary assumption is now made that the normal to the undeformed middle surface is deformed without extension into the normal to the deformed middlesurface.* Then
and dS2 as given by Eq. (8) refer to the same material element in the body.
Comparison of Eqs. (12) and (8) gives for the components of strain and in the tangential and circumferential directions the following expressions:
In what follows attention is restricted to thin shells in the sense that the thickness h is small compared with the magnitudes of the radii of curvature R and R asdefined by Eq. (9). The terms with may then be neglected in the denominators of Eq. (13) and instead of Eq. (13) may be written,
where
Note that and as given do not involve the axial displacement component w. This component is obtained from the relation z = sin in the form
Finally a relevant compatibility relation is set down which follows without calculation from a comparison of Eqs. (15) and (16) in the form
The present modification of the customary procedure consists up to this point in using radial and axial displacement components rather than normal and tangentialdisplacement components. This gives the possibility of obtaining the formulas of the finite-deflection theory with no more difficulty than the formulas of the linearizedsmall-deflection theory. It also permits a simpler derivation of the compatibility equation than is otherwise the case.
Definition of Stress Resultants and Couples
(Figure 3) Since rotational symmetry has been assumed, the non-vanishing components of stress are the four components ,
*It is recalled that this means that deformations due to transverse shear stress and transverse normal stress are neglected compared with the deformations due to the remaining stresses, and that thisway may be justified for thin shells by a study of the equations of the three-dimensional theory of elasticity.
Page 229
Fig. 3.Element of shell showing stress resultants and stress couples.
, , . The stress resultants and couples are defined as follows:
In writing Eqs. (21) and (22) terms of the order h/R, compared with unity, have again been neglected, that is, attention has again been restricted to thin shells.
Resultants and couples may be combined to resultant and couple vectors as follows:
In addition to this a load intensity vector p is introduced in the form
For what follows it is convenient to write N and p in the alternate form
The radial (horizontal) stress resultant H and axial (vertical) stress resultant V are related to N and Q as follows:
Page 230
Differential Equations of Equilibrium
Force and moment equilibrium conditions for elements of the shell are, in vector form,
With N from Eq. (23), M and M from Eq. (24), and N and p from Eq. (26), Eqs. (28) and (29) imply the three scalar equations
Stress Strain Relations
As the effect of transverse shear and transverse normal stress on the deformations is neglected, the relevant stress strain relations for an isotropic medium are
In Eq. (32) and are taken from Eq. (14) and the result is introduced into Eq. (21) and (22). This leads to the system
where
Eqs. (30), (31), (32), (34), and (35) represent seven equations for the seven quantities u, , H, V, N , M , M and thus form a complete system of equations for theproblem. The subject of the finite deflection theory will not be pursued further here, but rather from now on attention will be restricted to the linearized theory.
The Equations of the Small-Deflection Theory
The small-deflection (linearized) theory follows from the foregoing by referring the differential equations of equilibrium (30) to (32) together with Eq. (27) to theundeformed shell and by omitting in the expressions for the strains as given by Eqs. (16), (17), all non-linear terms. The resultant system of equations was shown inRefs. 6 and 11 to be reducible to two simultaneous second order differential equations for and R 0Q of a remarkably symmetrical appearance. In what follows thisderivation is modified slightly by choosing as one of the two basic variables the quantity r0H rather than R 0Q. In so doing it is possible to pass directly from theequations of shell theory to the equations of stretching and bending of circular plates, while with and Q as variables the equation for stretching of plates is certainlyno immediate consequence of the shell equations. While this advantage of ready transition to a special case might be thought to be of no practical consequence, it willbe shown that it does permit a ready discussion of the problem of the shallow shell, with results which go beyond those known heretofore.
Page 231
It may further be stated that with the selection of and r0H as basic variables the problem of finite deflections may be reduced in a corresponding manner to twosimultaneous second order equations. Discussion of this latter aspect of the problem is, however, left for a future occasion.
In terms of and r0H and in terms of the components pV and pH of external load intensity the following expressions obtain for all other quantities:
The two simultaneous equations for and r0H are obtained by introducing M and M from Eqs. (41) and (42) into the moment equilibrium equation (32) and byintroducing M and M from Eq. (34), with N and N from Eqs. (38) and (40), into the compatibility equation (20 ). The results can be written as follows:
If this is desirable Eqs. (45) and (46) may be reduced to one differential equation of fourth order for either or r0H. It was found by Meissner [6] that in some casesthis fourth order equation may be factorized into two independent second order equations. This factorization is of advantage if the solutions of the uncoupled secondorder equations can be expressed in terms of tabulated functions. It seems to the writer that for the purpose of obtaining solutions in series or asymptotic form thecoupled equations (45) and (46) are as convenient as the uncoupled equations. A case in point is the problem of the spherical shell of constant thickness where theuncoupled second order equations are hypergeometric equations.
The relative order of magnitude of the various terms in (45) and (46) becomes more readily apparent after a transformation which eliminates the first derivative termsin Eqs. (45) and (46). To this end two new functions X( ) and Y( ) are introduced,
Page 232
The differential equations for X and Y may be written in the form
The quantities , , , and are given by
In Eqs. (50) and (51) the subscript r indicates the value of at a suitably chosen reference station. In general the quantity 2 is a large number, as, for
instance, for the case of the spherical shell of radius a for which .
Stresses and Deflections of Shallow Shells
The equation of the middle surface of the shell is written in the form
where a is a reference length, ( ) is of order unity, and a number small compared with unity. From Eqs. (50) to (55) are obtained
Page 233
A shallow shell is defined by the stipulation that for it terms of order 2 may be neglected compared with terms of order unity. If this is done Eqs. (48) and (49)assume the following form:
In what follows the question of finding particular integrals for Eqs. (62) and (63) will not be considered but rather it will be shown that Eqs. (62) and (63) with rightsides equal to zero can be solved in terms of elementary or tabulated functions for certain classes of shells.
Shallow Shells of Uniform Thickness
When h = 0 and F = G = 0 Eqs. (62) and (63) reduce to the following form:
Eqs. (64) and (65) can be written as one complex equation
Eq. (66) is solvable in terms of Bessel functions, whenever
that is, for the case of a parabolic shell of nth degree. When n = 1 this case is contained in more general results for conical shells [7]. When n = 2 the case is that ofthe shallow spherical shell [9]. No previous solutions are known to the writer for other values of n.
The solution of Eqs. (66) and (67) can be written in the form
where Z is the symbol for the general cylinder function. The application of this solution to the treatment of specific problems will not be considered here.
Among such specific problems may be mentioned the problem of the shell with clamped edge as treated earlier for the case n = 2, and for n < 0 the problem of theinfinite shell with circular hole with uniform radial tension at infinity.
A still simpler case arises when
as then the solution of Eq. (66) is composed of powers of .
A Class of Shallow Shells of Variable Thickness
For shallow shells with thickness variation variation
Page 234
Eqs. (62) and (63), with F = G = 0, become
It is apparent that Eqs. (71) and (72) become a system of equidimensional equations and therewith have solutions which are composed of powers of when
The case m = 1, which gives a conical shell with linearly varying thickness, is included in E. Meissner's result [7] for the conical shell of any opening angle. Note thatm = 2 gives a shallow spherical shell with quadratically increasing thickness away from the apex. Evidently, applicability of this solution for m = 2, and, in fact for allvalues of m, is restricted to ring shells only, so that the apex is not part of the actual shell. A problem of practical interest here may be the problem of a ring shellrotating about its axis with thickness being largest at the inner edge, which is the case when the exponent m is negative.
Note on Asymptotic Solutions
Reverting now to the general differential equations (48) and (49) it can be seen that in general the quantity 2 is a number large compared with unity, while thefunctions , , and are of order unity. Under such circumstances the solutions of (48) and (49) may be expressed, for F = G = 0, in the following form:
The expansions (74) are not convergent, but rather the error committed in using the first n terms of the series instead of an exact solution tends to zero as tends toinfinity. The possibility of a development of this nature was first indicated by H. Reissner [11]. The results for the case of a spherical shell of constant thickness,including the determination of bounds for the error, were obtained by O. Blumenthal [1], and first applied in a dissertation of E. Schwerin [12].
Corresponding results for the cylindrical shell with linearly varying wall thickness, which had previously been treated by power series methods [10], were given by E.Meissner [8]. Subsequently Steuermann [14] gave the first term in each series for the case of arbitrary , , and subject to the aforementioned order of magnitudeconditions. A concise and complete version of the first-term approximation for the spherical shell of constant thickness together with examples of application waspublished by M. Hetényi [2], while M. F. Spotts [13] worked out the first-term approximation for a spherical shell with a specific law of wall thickness variation.
In all these cases the two second order equations (48) and (49) were first reduced to one fourth order equation before applying the asymptotic integration scheme.
It may not be superfluous to indicate that the first-term approximation may be obtained in the following manner. Omit the terms with and and write
Page 235
Then, except for terms small of order 1/ , the solution of (75) is
Solutions of this kind, irrespective of the field of application, are also discussed in a recent book by H. and B. Jeffreys [3].* In Eq. (76) the constants C1 and C2 arecomplex and thus equivalent to four real constants of integration. Separation of real and imaginary parts on the right of Eq. (76) leads to separate expressions for Xand Y and then also for the quantities defined by Eqs. (37) to (44).
For what is to be added to this subject the following two known facts are of significance:
(1) The accuracy of the first-term solution increases as increases and is in most practical cases, except for shallow shells, sufficient.
(2) The quantity is of order of magnitude (a/h)1/2 where a is a representative linear dimension of the shell which, for instance, for sphere and cylinder is equal to theradius of the middle surface of the shell. From the form of the approximation (76) it follows that the state of bending in the shell is effectively contained within an edgezone the width of which is of order (ah)1/2 and thus small compared with a.
The question may then be considered of reducing the stresses due to bending by using shells with appropriately varying thickness and radii of curvature. As the regionover which these stresses are important is of order (ah)1/2 it appears that it might be of advantage to admit significant changes of thickness and of the radii ofcurvature over distances of this same order of magnitude. The observation to be made here is that in such cases the approximation (76) is no longer valid. The reasonfor this is that in obtaining Eq. (76) the assumption has been made that the functions and which occur in Eqs. (48) and (49) themselves are of order unity, whichis the case when significant changes of h and R occur over distances of the order a and not over smaller distances. In the situation which is now contemplated it isevident that
where now the functions and are of order unity. On this occasion no more can be done than to call attention to this problem which requires furtherinvestigation.
References
[1]. Blumenthal, O., Zeitschrift für Mathematik und Physik, Vol. 62, 1914, p. 343.
[2]. Hetényi, M., Intern. Assoc. Bridge and Structural Engineering. Vol. 5, 1938, p. 173.
[3]. Jeffreys, H., and Jeffreys, B. S., Methods of Mathematical Physics, Cambridge University Press, 1946, p. 491.
[4]. Lohmann, W., Ingenieur Archiv, Vol. 6, 1935, p. 338.
[5]. Love, A. E. H., Treatise on the Mathematical Theory of Elasticity, Cambridge University Press, 1934, p. 589.
[6]. Meissner, E., Physikalische Zeitschrift, Vol. 14, 1913, p. 343.
*In Eq. (76) it has been assumed that . The case requires the changing of +i into i in the exponents. The approximation (76) evidently ceases to hold in the neighborhood
of regions in which . A discussion of this case, which is also of importance in certain problems of quantum mechanics, may be found in Ref. 3.
Page 236
[7]. Meissner, E., Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich, Vol. 60, 1915, p. 23.
[8]. Meissner, E., Vierteljahrschrift der Naturforschenden Gesellschaft in Zürich, Vol. 62, 1917, p. 153.
[9]. Reissner, E., Journal of Mathematics and Physics, Vol. 25, 1946, pp. 80, 279.
[10]. Reissner, H., Beton und Eisen, Vol. 7, 1908, p. 150.
[11]. Reissner, H., Festschrift Mueller-Breslau, 1912, p. 181.
[12]. Schwerin, E., Dissertation Technische Hochschule Berlin, 1917, and Armierter Beton, Vol. 12, 1919, pp. 25, 54, 81.
[13]. Spotts, M. F., Journal of Applied Mechanics, Vol. 6, 1939, p. A97.
[14]. Steuermann, E., Proc. 3rd Intern. Congr. Appl. Mech., Stockholm, 1930, Vol. 2, p. 60.
[15]. Tölke, F., Ingenieur Archiv, Vol. 9, 1938, p. 282.
Page 237
A Note on Membrane and Bending Stresses in Spherical Shells[J. Soc. Indust. Appl. Math. 4, 230240, 1956]
1Introduction
The problem which forms the starting point of this note is the following. A segment of a thin spherical elastic shell is acted upon by edge forces which are tangent tothe shell surface. Since these edge forces have no component normal to the shell surface and since no bending moments are applied at the edge of the shell, it may bethought that for this problem the state of stress in the interior of the shell is such that bending stresses are negligible in comparison with membrane stresses. If oneattempts to determine the stress distribution on the basis of the assumption that bending stresses are absent, one finds that the general expressions for stresses do notcontain a sufficient number of constants of integration for the satisfaction of the two conditions of prescribed normal stress and of prescribed shear stress along theedge of the shell segment. While this difficulty disappears of course in a theory which takes bending into account, it is nevertheless desirable to be able to solve theproblem within the framework of the much simpler membrane theory. In earlier work1,2,3 this is done by satisfying part of the boundary conditions only, namely thecondition which refers to the prescribed normal stresses. In regard to the remaining shear stresses, the remark is made that these will be carried by a reinforcingring.1,2
The original object of the present note was to show that a more satisfactory answer than this is possible, and that it is possible to determine the membrane state ofstress in the interior of the shell by one single boundary condition which involves both the normal stress distribution and the shear stress distribution along the edge ofthe shell, without explicit reference to bending. The membrane stress in the interior of the shell obtained in this manner will in general differ from the state of stressdetermined by satisfying the normal stress edge condition exactly, even though the remaining edge shears are carried by a reinforcing ring.
It is found as is expected that, subject to certain restrictions, deviations from the membrane state determined on the basis of the single boundary condition areconfined to a narrow edge region within which bending action is of importance. However, it is also found that the situation is somewhat less simple than indicatedabove, in the following sense. The state of stress outside this edge region is not necessarily a membrane state. It may happen that the state of stress in the interior alsois primarily bending rather than membrane. Specifically, this is found to be the case for the shell segment subjected to tangential edge forces, in the absence of
1W. Flügge, Statik und Dynamik der Schalen, pp. 4344, Berlin 1934.
2S. Timoshenko, Theory of plates and shells, pp. 380383, New York 1940.
3R. L'Hermite, Resistance des materiaux, pp. 645646, Paris 1954.
Page 238
transverse edge forces and edge moments. On the other hand, if the condition of no transverse edge forces is replaced by the condition of no transverse edgedisplacement then the state of stress in the interior of the shell is primarily a membrane state.
In order to simplify the analysis, the present derivations are carried out within the framework of the theory of shallow spherical shells. It will be evident that similarresults may be obtained for spherical shells without the assumption of shallowness, and also for other shells.
2Differential Equations and Boundary Conditions
The system of differential equations of the linear theory of shallow spherical shells of constant thickness which we shall use here can be reduced to two simultaneousequations for axial deflection W and for a stress function F, as follows.4
In these equations R stands for the radius of the middle surface of the sphere, D is the bending stiffness factor Eh3/12(1 2), C is the membrane stiffness factor Eh, Eis Young's modulus, is Poisson's ratio, h the wall thickness of the shell, p the axial surface load intensity and 2 the Laplace operator 2/ r2 + r1 / r + r2 2/ 2.
The stress function F represents the stress resultants Nr, N , Nr which are tangential to the middle surface,
The deflection W represents the stress resultants Qr, Q which are perpendicular to the middle surface and the stress couples Mr, M , Mr ,
We consider a shell segment extending over the region 0 r r0 and acted upon by a distribution of edge loads, tangential to the middle surface of the shell.Designating the normal component of this edge load by N0( ) and the shear component by S0( ), we have the following two boundary conditions
4E. Reissner, Stresses and small displacements of shallow spherical shells I, J. Math. Physics, 25 (1946), pp. 8085.
Page 239
Additional boundary conditions involve transverse forces and moments and the deformations associated with them. We consider in particular two cases. For the firstof these we assume that transverse forces R0 and moments M0 are also applied to the edge of the shell. This means that we have
For the second case we assume that the edge is moment free and that the transverse forces are of such magnitude as to make the transverse deflection vanish,
In addition to the four conditions for r = r0 the solutions of the differential equations (1) and (2) must also satisfy suitable regularity conditions for r = 0.
3Differential Equation and Boundary Conditions of Membrane Theory
The equations of membrane theory follow from (1) to (12) by setting in them
This means that we are left with the differential equation
and the boundary conditions
for the determination of the stress function F. It is clear that there is in general no solution of this problem which also satisfies the regularity conditions for r = 0. Whatthis means is that the given load distribution cannot, in general, be supported by the shell without appealing to its bending stiffness. However, it may be that thisbending stiffness is needed in a narrow edge zone only and that in the interior beyond this edge zone we have a state of membrane stress. So far as this interiormembrane state is concerned the problem then is to determine the form of the one boundary condition which replaces the two incompatible conditions (17) and (18).
4General Solution of Differential Equation
We will show that the separation into interior states and narrow-edge-zone states occurs for sufficiently small values of bending stiffness D and that from this it ispossible to deduce appropriate boundary conditions for the interior state.
The differential equations (1) and (2) allow the following representation of F and W,
The functions and are harmonic, Fp and Wp are particular solutions which take account of p, and is the general solution of
Page 240
where
The solutions (19) and (20) allow satisfaction of the complete system of boundary conditions (11) to (14).
In order to have a separation of the solution into a part which describes what we call the interior state and into a part which describes what we call the narrow-edge-zone, or boundary layer, state, we are limiting ourselves to problems for which
If this condition is satisfied, we have that the -contributions to the solution which are necessary to satisfy the boundary conditions for r = r0 decay rapidly withdistance away from the edge and are such that differentiation with respect to r changes their order of magnitude in the vicinity of r = r0. Typical relations inmathematical form are
Before introducing the solutions (19) and (20) into the boundary conditions we list stress resultants and couples once more in explicit form as follows. We set as anabbreviation,
we indicate the fact that the functions , , Fp and Wp are associated with the interior state by a superscript i and we indicate partial differentiation by a commapreceding the appropriate subscripts. In this way we have
Page 241
The quantities with superscript i follow from (3) to (10) if in these expressions we replace F by + Fp and W by + Wp.
5The Problem of Prescribed Edge Forces and Moments
If we introduce (26) to (34) into the boundary conditions (11) to (14a), these assume the following form
for r = r0. Now, since 1 << kr0, we have that in each of the four equations (35) the highest -derivative terms dominate the others. This means that in the last twoequations the explicitly appearing -terms are small compared to the -terms and we may neglect these -terms. With only -contributions remaining the boundaryconditions (35) may be reduced to boundary conditions for the interior state, as follows. The first two of equations (35) are used to express and ,r in the form
Introducing (36) into the abbreviated form of the last two of equations (35) we are left with the following system of two boundary conditions for the interior state
for r = r0.
Explicit Form of Boundary Conditions for the Case of Absent Surface Loads
Setting Fp = Wp = 0 we may write (37a, b) in terms of and , as follows:
Page 242
For the special case in which the tangential forces N0 and S0 are absent the boundary conditions (38) may be replaced by an integrated system of the form
Equation (39) may be recognised as a specialisation of earlier results for inextensional deformations of general shallow shells.5 In this sense we recognise that thephysical nature of the interior state for the boundary value problem (35), rather than being that of a membrane state, may be that of a state of bending without middlesurface extension. Our present results are more special than the results of the earlier work,5 inasmuch as they are restricted to spherical shells; and they are moregeneral than the earlier results since they include not only the action of transverse edge forces and edge bending movements but also the effect of tangential edgeforces which were not considered in the earlier work.
6Prescribed Tangential Edge Forces and Vanishing Transverse Edge Deflection and Moment
For this problem the first two of equations (35) remain unchanged. The last two equations are replaced by the following:
We may again neglect the explicitly occurring -terms and then (40) is replaced by
In the first of equations (41) we eliminate by means of the first of equations (36). In this manner we arrive at the following system of boundary conditions for theinterior state.
In the event that Wp = 0, which is in particular the case if surface loads are absent, we conclude from the second of equations (42) that Wi = 0 throughout and the firstof equations (42) reduces to the following simple form
It is evident that for the present problem the interior state is a membrane state and equation (43) is the one boundary condition for the determination of this state interms of given tangential edge loads, which was the original aim of the present discussion. We note that in terms of the potential function which represents theinterior membrane state the boundary condition (43) assumes the form,
5M. W. Johnson and E. Reissner, On inextensional deformations of shallow elastic shells, J. Math. Physics, 34 (1955), pp. 335346.
Page 243
and this condition, with replaced by F, takes the place of the two boundary conditions (17) and (18).
Similar considerations involving the explicit determination of interior states without explicit determination of the associated boundary layer state are possible for othersystems of boundary conditions, for instance when instead of prescribed Nr and Nr0 we have prescribed displacements u and v in radial and circumferential direction.
7Some Explicit Solutions
In this section we consider the formal solutions of some specific load and support condition problems on the basis of the following expressions for the harmonicfunctions and which occur in (19) and (20),
Simply supported shell with tangential edge loads
The second of equations (42) shows that B = 0 and equation (43 ) determines the coefficient A in (44) as follows
With this we have
and the membrane stress resultants in the interior assume the following form:
Equations (48), for the special case Q = 0, may be compared with the corresponding results which follow from use of the boundary condition Nr(r0, ) = P cos n .Instead of the factor n/(n + 1) in (48) there occurs then a factor unity.
Shell Carrying Tangential Edge Loads and Free Otherwise
We take N0 and S0 as in (45) and in addition to this set in (38),
Introduction of (44) into (38) leads to the following expressions for stress function F and deflection W,
Introduction of (50) and (51) into (3) to (10) leads to the corresponding equations for stress resultants and couples. It is readily seen that the extreme fiber
Page 244
stresses B associated with the couples M through the relation B = ± 6M/h2 are large compared with the average stresses D associated with the resultants Nthrough the relation D = N/h, as long as the basic order of magnitude relation (23) is satisfied. In this sense we have that for the present problem membrane action inthe interior is negligible compared with bending action in the interior and the latter is as if the middle surface of the shell were inextensible.
Shell Carrying Radial Edge Loads and Otherwise Free
In order that the edge loads shall be radial rather than tangential we set in the boundary conditions (38) S0 = 0 and introduce a transverse force R0 of such magnitudethat the resultant of it and of the tangential normal force N0 is transformed into a radial normal force N0. The appropriate value of R0 is
Therewith and with M0 = 0 the boundary conditions (38) assume the following form,
With N0 as in (45) the functions and are now
Again, we will have that bending stresses associated with will be large compared to direct stresses associated with , as long as 1 << kr0.
It is evident that by suitable combination of solutions of the form (54) and (55) we may solve such problems as the problem of the shell subjected to twoconcentrated radial forces acting at the opposite ends of a diameter of the boundary of the shell. It is also evident that the series occurring in such a solution areexpressible in closed form by means of elementary functions.
Among other problems deserving of further consideration we may mention the following three:
(1) A shell acted upon by an equilibrium distribution of axial edge forces, there being no other loads.
(2) A shell with free edge, acted upon by an equilibrium distribution of surface loads. Here it will be of interest to see to what extent the interior state will be amembrane state or a state of inextensional bending or a combination of the two.
(3) Returning to the problem of the shell with given edge forces and moments, as discussed in Section 5, we may ask the question to determine a distribution oftransverse forces R0, for given N0 and S0 and for M0 = 0 such that the solution has
Page 245
the property of vanishing edge deflection W0. The solution of this problem should be the same as the solution of the problem described in Section 6. One can showthat this is in fact the case. Accordingly, the problem of Section 5, for suitable specialisations of the edge loads, may lead to interior states which are either states ofinextensional bending or membrane states. It would seem to be of interest to pursue this subject further by analysing in greater generality possible classifications ofboundary conditions with respect to the physical nature of the interior stress states to which they give rise.
Page 246
On Stresses and Deformations of Ellipsoidal Shells Subject to Internal Pressure*
[J. Mech. Phys. Solids 6, 6370, 1957]
1Introduction
In the following we consider a thin elastic ellipsoidal shell of revolution with the major axis of the elliptic cross section perpendicular to the axis of revolution. Weassume that the only load is a uniform internal pressure (Figure 1).
Existing solutions (Lorentz 1913; Timoshenko 1940) for the stresses are obtained using the membrane theory of shells which neglects effects due to bending. For thepresent problem membrane theory is general enough to allow one to satisfy all prescribed boundary conditions and, as there are no discontinuities either in the loadfunction or in the radii of curvature of the shell, membrane theory yields a continuous stress distribution with no singularities.
Fig. 1.
A qualitative consideration of the problem indicates that some bending does occur and that effects due to bending become progressively more important as theelliptical cross section of the shell becomes flatter and flatter. Our object in the present note is to investigate this aspect of the problem quantitatively. We find thatsuitable results may be obtained if appropriate solutions of the differential equations of the bending theory of shells are represented by expansions in powers of a smallparameter. In contrast to some problems of shell theory concerned with bending effects, we do not encounter any solutions of the boundary-layer type in the range ofparameter values which we consider.
Our principal result of a qualitative nature may be described as follows. We know the range of validity of thin-shell theory is restricted by the basic assumption thath/Rmin << 1, where h is the wall thickness of the shell and Rmin is the least value
*With R. A. Clark
Page 247
of the radii of curvature of the middle surface. For an ellipsoidal shell of revolution this order of magnitude relation is equivalent to the relation h/b << b/a, where aand b are the major and minor semi-axis of the elliptic cross section of the middle surface. Our result for the present problem is that bending effects are negligible, ormembrane theory is valid, only if the more restrictive relation h/b << (b/a)3 is satisfied.
2Basic Equations
The middle surface of a closed ellipsoidal shell of revolution may be represented in cylindrical co-ordinates r, , z by parametric equations of the form
where 0 .
Assuming that the thickness h of the shell is uniform and that a uniform internal pressure p is the only load, the basic differential equations of the linear bending theorymay be written as follows (Reissner 1949)
where the operators L are defined by
and where
The quantity E is Young's modulus, is Poisson's ratio, is the meridional angle of rotation due to deformation, is a stress function, and primes indicatedifferentiation with respect to .
Stress resultants and couples are given in terms of and by the following relations:
Displacements u and w in radial and axial directions are given by the formulas
In order to determine stress and displacement quantities, as given by (6) and (7), it is necessary to solve the two simultaneous differential equations (2) and (3) in theinterval 0 subject to suitable boundary conditions at the ends of this
Page 248
interval. Such boundary conditions are the symmetry conditions of vanishing angular deflection and vanishing transverse shear force Q. In view of the form of Q asgiven by (6) the system of boundary conditions may be written as follows:
3Membrane Solution
Membrane-theory formulation of the problem is obtained from the preceding equations by setting, in (2) and (6),
Indicating the corresponding expressions for and by a subscript M, we have
and these expressions do in fact satisfy the boundary conditions (8).
Introduction of (9) and (10) into expressions (6) for stress resultants shows that the couples and the transverse stress resultant Q are identically zero while the directstress resultants N and N are given by the known expressions
Of particular interest is the fact that when the circumferential stress resultant N M changes sign as varies from zero at the apex to the value 1/2 at pointsfarthest removed from the axis of revolution of the shell.
4Corrections to Membrane Solution
In order to obtain corrections to the membrane solution (10) and (11) which account for non-zero values of the bending stiffness factor D, we introduce non-dimensional variables and parameters, as follows:
The quantity Rmin is the meridional radius of curvature for = 1/2 and is the least value of the two principal radii of curvature when b < a. The basic equations beingused here are meaningful only if the parameter is large compared to unity.
With (13) to (15) differential equations (2) and (3) assume the following more symmetrical form
Page 249
Inspection of the system (16) and (17) suggests expansions of the form
in terms of inverse powers of the large parameter . Equating coefficients of corresponding powers of on the right and on the left of (16) and (17), afterintroduction of (18) and (19) into these equations, leads to the following expressions for the coefficient functions gn and n:
It is readily seen that the leading terms of expansions (18) and (19) are identical with the corresponding membrane solutions (10) and (11). Accordingly, theremaining contributions to the series, beginning with g1 and 1, represent corrections to the membrane solution due to the finite bending stiffness of the shell wall.
The following observation is of importance. While the expansions (18) and (19), with gn and n given by (20) to (23), represent particular solutions only of thesystem of differential equations (16) and (17), these expansions satisfy the boundary conditions (8) term by term. Accordingly, it is unnecessary to consider thegeneral solution of the homogeneous system of differential equations.
For the purposes of this note we may limit ourselves to the explicit evaluation of 0 and g1 in addition to g0. We find, making use of the relation
the following expressions
5Formulae for Stresses
From (6) and (13) we obtain for direct stresses D and D,
From (6) and (14) we obtain for bending stresses B and B,
Page 250
We consider in particular the values of stresses at the apex = 0 and at the points = 1/2 farthest removed from the apex of the shell.
From equation (12) the direct stresses have the following values according to membrane theory
From equations (26) to (29) we have according to bending theory
We are mostly interested in shells for which a/b is reasonably large compared to unity. According to membrane theory the circumferential hoop stress D(1/2 ) isthe critical stress provided a/b 2 and we shall limit ourselves to this range in the following discussion. In order to compare the results of bending theory with those ofmembrane theory we consider the ratio of D(1/2 ) and B(1/2 ) to the membrane solution M(1/2 ). Dividing (35) and (37) by (32) and introducing from(15) we may write
where
Page 251
For a/b greater than 3 or 4, coefficients A and B, given by (40) and (41), vary relatively slowly as a/b increases. In particular, for = 0.3 we have the followingTable:
a/b 2 3 4 5 6A 1.66 1.14 1.03 0.98 0.96 0.92B 1.44 0.84 0.70 0.64 0.61 0.55
Using this Table we may easily draw curves, shown in Fig. 2, representing (38) and (39). Since we are limiting ourselves to the calculation of first-order corrections,we expect that the curves drawn are quantitatively reliable only as long as the corrections amount to no more than, say, 10 per cent of the critical membrane stress
M(1/2 ), as is indicated by the solid portion of the curves in Figure 2. The broken portions of the curves are probably qualitatively correct, but eventually theresults become meaningless as the corrections, or the combination (a/b)2 (h/Rmin), increase.
The following observations may be made on the basis of equations (38) and (39) or Fig. 2. Membrane theory is valid or bending effects may be neglected only if
which for b/a << 1 is a much more restrictive condition than the basic assumption of thin shell theory that h/Rmin << 1. Thus, for a shell with fixed dimensions a and b,there will always be a range of shell thickness h, satisfying (42), for which membrane theory is valid, but this may be only a small part of the range for which bendingtheory is valid. Also, for a fixed value of h/Rmin, no matter how small, there will always be shells or ratios a/b for which membrane theory is not valid.
Fig. 2.
Page 252
6Formulae for Displacements
Combining relations (7), (13), (26) and (27) we obtain the following general expressions for the displacements:
The axial displacement w is of greatest interest. The integrations involved may be carried out explicitly in terms of elementary functions. Retaining only those termswhich correspond to the membrane solutions (10) and (11), we find that the total axial expansion = w( ) w(0) is given by
where e is the eccentricity of the elliptic cross section given by
The first term in (45), involving the factor (a/b)2, represents the contribution due to the angular rotation and is seen to dominate for large values of a/b. Using (46)we may also write (45) in the somewhat more compact form
For a/b = 1 and e = 0, expression (47) reduces to the value of the deflection for a spherical shell. But as a/b increases and e 1, the expression for eventuallybecomes meaningless, as does the membrane solution for stresses when the thickness h is held fixed.
References
1. Lorentz, H. 1913. Technische Elastizitätslehre p. 34. (Oldenbourg, München und Berlin).
2. Reissner, E. 1949. Reissner Anniversary Volume , pp. 231247. (Edwards , Ann Arbor, Mich.)
3. Timoshenko, S. 1940. Theory of Plates and Shells p. 265. (McGraw Hill, New York).
Page 253
On the Foundations of the Theory of Thin Elastic Shells*
[J. Math. & Phys. 37, 371392, 1958]
1Introduction
In the following we are concerned with the fundamental problem of establishing two dimensional systems of differential equations for stresses and displacements ofthin elastic shells. Our object is to derive such a system of equations as a logical consequence of the known system of differential equations of three-dimensionalelasticity, with the possibility of establishing shell theories of varying degrees of accuracy. While ultimately results of this kind should be obtained for shells of arbitraryshape and for arbitrarily large deformations the present paper is limited to a consideration of the problem for the circular cylindrical shell of constant thickness, underthe assumption of small rotationally symmetrical deformations.
The circular cylindrical shell is characterized by two lengths, the radius a and thickness h. The main point of the present paper is that by introducing a suitable thirdaxial length in the differential equations of the three-dimensional theory it becomes possible to expand the solutions of the equations of the three-dimensional theory interms of powers of a suitable dimensionless parameter. The first terms in these expansions are found to represent the results of conventional shell theory as based onthe Euler Bernoulli hypothesis and on the assumption that h/a is negligibly small compared to unity. The second and higher order terms represent the three separateeffects of (i) finite values of h/a, (ii) transverse shear stress deformation, (iii) transverse normal stress deformation. A preliminary order-of-magnitude analysis of theequations of the three-dimensional theory establishes that the characteristic axial length is the geometric mean (ah)1/2 of radius and wall thickness of the shell.Associated with this length are boundary layer phenomena typical for thin shell theory. A different class of boundary layer phenomena for the equations of the three-dimensional theory is associated with the length h, taken in the axial direction. The essential point in our procedure is that it incorporates the consequences of the oneboundary layer while being insensitive to the consequences of the other boundary layer.
2The Boundary Value Problem
We consider axially symmetric deformations of a semi-infinite cylindrical shell of constant thickness. We start with the three-dimensional equations of the linear theory.With the usual notation, illustrated in Fig. 1, the equilibrium equations have the following form
*With M. W. Johnson.
Page 254
Fig. 1.Semi-infinite circular cylindrical shell showing stress components
z, r, , rz, displacement components ur, uz and coordinatesr, z for axially symmetric deformations.
We assume that the material is orthotropic with the direction normal to the middle surface an axis of elastic symmetry and write the stress strain relations as follows:
We assume that stresses and deformations are due to a system of tractions applied at the end z = 0 of the semi-infinite shell.
We introduce dimensionless coordinates and and dimensionless parameters and by
Dimensions a and h are the radius and thickness respectively (Figure 1) while b is a typical length in the z-direction to be determined presently. In fact, the properdetermination of b is the crucial step of the non-dimensionalization process. With (2.3) and (2.4) the equilibrium equations (2.1) become
We now non-dimensionalize stresses so that all terms in (2.5) except the additive terms are of the same order in . This is accomplished by setting
and by relating the two parameters and as follows:
Page 255
The quantity , which has the dimension of stress, will be determined later from the boundary conditions such that the dimensionless variables are O(1).
We note that the stresses (2.6) agree in so far as the order in is concerned with the results of conventional shell theory. Introduction of (2.6) and (2.7) into (2.5)gives dimensionless equilibrium equations of the following form:
Relation (2.7) fixes the length scale in the axial direction as b = (1/2ah)1/2 so that this formulation has the possibility of giving results corresponding to the conventionalbending theory. However we cannot expect it to be capable of representing effects associated with a length scale h in the axial direction.
Next we non-dimensionalize the stress-strain relations (2.2). In order that the displacements appear with the proper order of magnitude we set
This results in the following dimensionless form of the stress-strain relations:
where
The factor (1 2) is added in (2.9) in order to simplify some formulae which follow.
We next arrange equations (2.8) and (2.10) in the order in which integration with respect to will be possible.
Page 256
The system (2.12) will be considered in conjunction will conditions of vanishing surface stress over the cylindrical boundary portions, i.e.
3Expansion with Respect to Small Parameter
We assume that for sufficiently small all quantities in (2.12) can be expanded, asymptotically, for each value of and in the form
We are interested in the solution for small and will study in detail only the first two systems obtained by substituting (3.1) into (2.12) and equating coefficients of 0
and 1 on both sides of the equation.
First system
Second system
Systems (3.2) and (3.3) can now be integrated with respect to in a step by step fashion since the right sides of each equation are known functions of at each step.
Introducing macroscopic displacements and stress functions , etc., system (3.2) yields
where primes indicate differentiation with respect to .
Page 257
It is clear from the form of (2.12) that this integration can also be carried out in all higher order systems. For the second system integration of (3.3) gives
Expressions (3.4), when the proper differential equations and boundary conditions are given for and , are those obtained in the conventional theory of
Page 258
thin shells. Expressions (3.5) and (3.6), which are of higher order than (3.4), are not obtained in the usual theory.
4Equations for Macroscopic Quantities
Satisfaction of boundary conditions at = ± 1 gives equations for the macroscopic quantities appearing in (3.4) through (3.6). Introducing the expansion in powers of (3.1) into the boundary conditions (2.13), we find that these assume the following form
Satisfaction of conditions (4.1) for k = 0 by stresses (3.5) leads to the following expressions:
A convenient form of the macroscopic equations is obtained by reducing (4.2) to the following differential equations for middle surface displacement functions and ,
plus expressions giving and in terms of these middle surface functions,
In a similar manner, satisfaction of (4.1) when k = 1 leads to the following differential equations for middle surface displacement functions and ,
and the following expressions for and ,
Equations (4.3) are two ordinary differential equations with constant coefficients for and . They are, in fact, of the same form as the equations of conventionalshell theory for the displacements of the middle surface. Their solution can be found by elementary means. After solving these equations subject to suitable boundary
Page 259
conditions we can find and by (4.4). Then, microscopic displacements and stresses are given by (3.4) and (3.5).
Knowing the first approximation, we can calculate the right sides of (4.5). The left sides are of the same form as in (4.3) and these equations can again be solved byelementary means. The microscopic quantities of the second approximation are then determined from (3.6) with and given by (4.6).
We note that at the k + 1st step one must solve a system of the form
where G(k) and H(k) are known from the previous steps. One then obtains and from expressions corresponding to (4.6) and the microscopic quantities fromexpressions corresponding to (3.6).
5Definitions of Stress Resultants and Couples
For the macroscpic formulation of our problem it is necessary to introduce stress resultants and couples defined as follows
Expressions (5.1) are non-dimensionalized and expanded in a -series using (2.3), (2.4), (2.6) and (3.1). The results can be written in the form
where
6End Conditions
It remains to consider the form of suitable boundary conditions at the ends = 0 and = of the semi-infinite shell. It will be assumed that these
Page 260
conditions are conditions of prescribed normal and shear stress, as follows
In accordance with the stress expansions in the interior it is necessary that the given functions and are expanded in powers of , as follows:
Inspection of equations (3.4), (3.5) and (3.6) indicates that the functions and must depend on the variable in certain specified ways. must be a linearfunction of , must be a quadratic function of , and so on. Moreover, the coefficients of the various powers of in the different functions may not necessarily allbe assumed to be independent of each other.
The resolution of the problem of the form of the boundary conditions (6.1) and (6.2) is effected here by the following consideration. The boundary stress functions and are considered to be composed of two portions,
such that the portions and have form (6.3) which allows their identifications with the boundary values of sz and srz and such that the portions and donot give rise either to forces or moments applied to the elements of the edge of the shell.
The boundary value problem pertaining to the boundary stresses and is then solved by the expansions proposed in this paper. For an exact solution of thecomplete problem it would be necessary to solve in addition the problem pertaining to the boundary stresses and . At this point the assumption is made thatthe solution of the second problem does not in fact have any bearing on the solution of the originally given problem of obtaining two-dimensional shell theoryequations inasmuch as the stresses associated with the solution of the second problem are significant only in an edge region with axial extension of the order of thethickness h. Whether this assumption is based on an appeal to St. Venant's principle or on an actual solution of the second problem is not material for the purposes ofthe present work. It suffices for the present purposes to define the problem of obtaining two-dimensional shell theory equations in such a way that the stressesassociated with the second problem are ignored.
We now divide boundary conditions (6.1) and (6.2) pertaining to the problem as defined above into two different fundamental sets with the following properties
Page 261
It will be seen that (6.5) and (6.6) lead to boundary conditions appropriate to differential equations (4.3) and (4.5). That is, the middle surface displacement functionsare uniquely determined so that equations (4.3), (4.5) and boundary conditions (6.5), (6.6) are satisfied to order 1. This fact shows that it is indeed possible to writethe boundary stress functions and in the form (6.4) where and are identified with boundary values of sz and srz and and do not give rise toforces or moments as far as terms up to order 1 are concerned. The form of equations (4.7) and boundary conditions (6.5) and (6.6) indicates that this can be doneup to any order in .
In terms of the quantities defined by (5.2) and (5.3) boundary conditions (6.5) and (6.6) can be written as
where, in order that dimensionless quantities be O(1) we have chosen in the following manner,
7Stress-Strain Relations
In order to write end conditions (6.5) and (6.6) in terms of middle surface displacement functions for use with equations (4.3) and (4.5) we must derive macroscopicstress-strain relations.
We relate stress resultants and couples as defined in section 5 to the middle surface displacement functions by substituting stresses from (3.4) and (3.6) into (5.3)performing the integrations and using expressions (4.3) through (4.6). The results are
Page 262
We note that relations (7.1) are of the same form as those obtained in the conventional theory.
8The Boundary Value Problem for the First Two Approximations
By using the results of section 7, we write end conditions (6.7) and (6.8) for k = 0 and k = 1 as
where
Integrating the first equation of (4.3) and using conditions (8.1) and (8.4) we obtain
Similarly, the first equation of (4.5) together with end conditions (8.2), (8.5), (8.7) and (8.10) yields
We see from (7.1) and (7.2) that (8.14) and (8.15) indicate that and are identically zero. We could have obtained these conditions directly from thefact that Nz( ) must vanish identically due to the form of the boundary conditions.
If we eliminate from the second equation of (4.3) with the aid of (8.14) and eliminate from the second equation of (4.5) using (8.15), we obtain thefollowing system for and ,
Page 263
where
The problem is now reduced to solving equations (8.16) and (8.17) subject to boundary conditions (8.2), (8.3), (8.5), (8.6), (8.8), (8.9), (8.11) and (8.12). and
are then to be obtained from (8.14) and (8.15).
With equation (8.14) the relevant stress-strain relations (7.1) and (7.2) simplify to
Supplementary Note
The second half of this paper, with analytical and numerical details, is here omitted, as modifications are required in the determination of , upon consideration ofthe effect of the boundary layer of width h, which has not been part of this work.
Page 264
The Edge Effect in Symmetric Bending of Shallow Shells of Revolution[Commun. Pure & Appl. Math. 7, 385398, 1959]
1Introduction
The present paper generalizes earlier work on the edge effect in shallow spherical shells subjected to internal pressure to general shallow shells of revolution andgeneral symmetric load distributions, for such shells and load distributions as behave qualitatively similar to the uniformly loaded spherical shell. Our previous work [1,2] consisted in a boundary layer analysis of the spherical shell with simply supported edge, and included as limiting cases the results of linear bending theory [3] and ofnon-linear membrane theory [4]. In what follows we consider shells with simply supported edge as well as shells with built-in edge.
The present analysis makes explicit a property of the problem which previously has not been stated in such fashion, although it was implicit in our earlier work. It maybe described as follows: For certain well-defined ranges of parameter values the boundary layer at the edge of the shell is such that within one boundary layer ofspecified width there is to be found a second boundary layer, the width of which is of a different order of magnitude.
2Basic Equations
The problem of symmetric deformations of shallow shells of revolution subjected to loads in the direction of the axis of the shell may be reduced to two simultaneousdifferential equations of second order for a stress function variable and an angular deformation variable , cf. [1]. In the following it will be assumed that the shell isisotropic, homogeneous and of uniform thickness. The differential equations are then of the form
where
In equation (2.3) h is the wall thickness of the shell, E is Young's modulus and is Poisson's ratio. The variable r is the radial distance from the axis of the shell ofpoints of the middle surface before deformation, and the quantity is the sloping angle of the meridians of the middle surface before deformation. For a shallow shell
Page 265
this sloping angle is related to the middle surface equation z = z(r) in the form
where the prime indicates differentiation with respect to r. The quantity L is a differential operator defined by
Stress resultants and stress couples are defined as follows:
The quantity pV represents the axial surface load intensity.
To the degree of approximation implied by the use of shallow shell theory axial and normal surface loads are equivalent, and it is assumed that no radial surface loadsare present. Radial and axial displacement components u and w are given by
In what follows equations (2.1) and (2.2) will be considered for systems of boundary conditions of the form
where S stands for simply-supported and C for clamped.
3Linear Membrane Theory
An approximate solution of the problem which is known to be incompatible with the boundary condition u(a) = 0 may be obtained by assuming D = 0 and byignoring non-linear terms in the differential equations (2.1) and (2.2). This solution is
where and V are restricted in such a way that LM and LM satisfy the symmetry conditions
In what follows we are interested in determining the circumstances in which the solution (3.1) is effectively correct except in a narrow edge zone and in determiningthe nature of the actual solution which is valid within the narrow edge zone.
Page 266
4Differential Equations and Boundary Conditions for Supplementary Solution
We write
Introducing (3.1) and (4.1) into equations (2.1) and (2.2) we obtain for the supplementary solutions s and s the following differential equations:
and the following system of boundary conditions:
and
5Non-Linear Membrane Supplementary Solution
If in equations (4.2) and (4.3) we assume D = 0, then they are equivalent to one equation for s which may be written in the form
The appropriate boundary conditions which remain from (4.4) to (4.7) are those pertaining to s, namely
For the purpose of determining the existence of a solution of the boundary layer type we write in (5.1) to (5.3)
where (1) = (1) = 1 and where it is assumed that and and its derivatives with respect to r/a are of order of magnitude unity.
We further set
and
Page 267
where x is a new dimensionless variable and where and 0 are parameters which are to be chosen suitably.
Introduction of (5.5) and (5.6) into (5.1) gives as a new form of the differential equation of the problem
where
The boundary conditions (5.2) and (5.3) assume the form
and
We now dispose of the parameters 0 and by setting
Therewith equation (5.7) becomes
The form of (5.12) and (5.9) indicates that in order to have a boundary layer phenomenon we must have
When (5.13) is satisfied we may take (5.12) effectively in the simpler form
with the boundary conditions
Equations (5.14) to (5.16) are the boundary layer equations of non-linear membrane theory of the problems as formulated in equations (5.1) to (5.3). The form of(5.12) shows that the condition of validity of this boundary layer theory is the order of magnitude relation (5.13). The width of the boundary layer is given in theform
Page 268
For a spherical shell of radius R with uniform load pV = p, which is the case treated by Bromberg and Stoker [4], we have
and therewith
while
6Boundary Layer Analysis of Supplementary Solution
We make use of the substitutions (5.4) to (5.6) of non-linear membrane theory, together with the additional relations
where B(r/a) and its derivatives with respect to r/a are O(1).
Introduction of (5.4) to (5.6) and (6.1) into (4.2) and (4.3) gives
The boundary conditions (4.5) to (4.7) for r = a become
and
or
The boundary conditions for r = 0 become conditions for x = and need not be listed explicitly.
Page 269
We now assume the following two relations involving the three parameters 0, 0 and :
which may be written in the form
Introducing (6.8) into equations (6.2) to (6.6) we obtain the following form of these equations:
and
or
In addition to the system (6.9) to (6.13) contains the two parameters and .
In view of the form of the first term of equation (6.9) we now take as
and set
In order to have the existence of a narrow edge zone, we assume that >> 1. The system (6.9) to (6.13) then reduces to the following form:
Page 270
together with the regularity conditions
which take the place of the boundary conditions for x = . We note that in order that these regularity conditions can be satisfied certain regions of k-values are notadmissible. The meaning of this is that the present boundary layer theory is restricted to load conditions for which the question of buckling of the shell in a rotationallysymmetric manner does not arise.
A special case of (6.16) to (6.20) is given when
For this range of surface load intensities the system (6.16) to (6.20) reduces to that for the boundary layer in the linear theory of bending [3].
In contrast to this we have that, when
both linear bending and non-linear membrane action are of importance.
The system (6.16) to (6.20) ceases to be suitable when 1 << k. In view of the form of the third term in (6.9) we take for the analysis of this latter case in the form
Assuming again that 1 << we see that the system (6.9) and (6.10) reduces to the form
while the boundary conditions are again given by (6.18) to (6.20).
We note that, when
the system (6.24) and (6.25) contains the same type of terms as the system (6.16) and (6.17). The case k << 1, which was covered in (6.16) and (6.17), does not
concern us within the framework of the normalization .
There remains the special case
When (6.27) holds the system (6.24) and (6.25) reduces to the form
This, however, is equivalent to equation (5.14) of non-linear membrane theory. Moreover, for this case the normalization (6.23) of is exactly the same as that givenby (5.13).
Page 271
We have then that the boundary layer case of non-linear membrane theory is a special case of the general boundary layer case which is applicable when both
Since the order of the system (6.28) is halved in comparison with the order of the system (6.24) and (6.25) the same must be assumed for the boundary condition(6.18) to (6.20) and these reduce to the form (5.15) and (5.16).
If we wish to retain all boundary conditions (6.18) to (6.20) we must retain in (6.24) the term k2 '' even though k2 << 1. Since the system (6.24) and (6.25) is alinear system with constant coefficients, this can be done in an explicit fashion and the results so obtained for the support condition (S) have been given earlier [1,2]. Itis, however, more in the spirit of the remainder of the analysis, to consider the case k2 << 1 as one where there occurs a secondary boundary layer, which lies inthe interior of the previous boundary layer of non-linear membrane theory.
7Analysis of the Secondary Boundary Layer
Assuming that k is in the range (6.27), the system (6.28) together with the boundary conditions (5.15) and (5.16) is valid except in an edge region . Settingas an abbreviation
we have then the interior solution
which does not satisfy the boundary condition (6.19). This interior solution is to be corrected in the vicinity of x = 0, in order that (6.19) be satisfied. To find the formof the correction to i = gi we introduce a new independent variable y and new dependent variables F and G in the form
where r, s and t are suitable constant exponents.
Introduction of (7.3) into (6.24) and (6.25) gives
The boundary conditions (6.18) and (6.19) assume the form
Instead of the boundary conditions (6.20) we have the transition conditions
where y = krx.
In order to preserve the order of (7.4) and to retain condition (7.6) we now set
Page 272
so that
With this choice of exponents the system (7.4) and (7.5) becomes
with the boundary conditions
and
The solution of (7.11) and (7.12) is readily carried out in the form
where G0 and F0 are constants of integration. Equation (7.13) furnishes the additional relation
It remains to satisfy the transition conditions (7.8). Since in the range 1 << y the term with F0 in (7.15) is negligible, the transition conditions for both F and G reduceto the form
which, since y/k = x, is satisfied by setting
The secondary boundary layer solution is then
and
Equations (7.19) to (7.21) may be written in the equivalent form
and
which holds in the region . For larger values of x the solutions (7.22) to (7.24) are to be replaced by (7.2).
We note that while the width of the primary boundary layer was of the order , the width of the secondary boundary layer is of the order
Page 273
which, in as much as k2 << 1, is in fact a smaller order of magnitude than the width of the primary boundary layer.
8The Order of Magnitude of Bending Stresses in the Secondary Boundary Layer
We consider the meridional bending stress rB in comparison with the direct stress D of linear membrane theory.
The order of magnitude of D is
The order of magnitude of rB is
With 0 from (6.8), this is
and, with from (6.23) and from (7.23) and (7.24),
In view of the form of (7.23) and (7.24) this means that for the shell with built-in edge
and for the shell with simply supported edge
Equations (8.5) and (8.6) are in accordance with statements made earlier for the shell with simply supported edge [1] and, quite briefly, for the shell with clampededge [2].
9Summary of Boundary Layer Results
We may summarize our results in the following manner:
Deviations from the results of linear membrane theory are confined to a narrow edge zone provided the parameters k and as defined by (6.14) and (6.15) are suchthat
When k << 1, the edge zone state is described by linear bending theory and the width of the edge zone is of the order a/ .
When k = O(1), determination of the edge zone state involves both linear bending and non-linear membrane action, and the width of the edge zone is again of ordera/ .
Page 274
When 1 << k, the edge zone state is effectively separated into two distinct states, with different orders of magnitude for the widths of the two edge zones. The wider
of the two edge zones is associated with non-linear membrane action and its width w is of order . The narrower of the two edge zones is associated with
non-linear bending action and its width N is of order . Stresses associated with this second narrower edge zone become relatively insignificant as k increasesfor the shell with simply supported edge, but retain their significance for the shell with clamped edge.
For the spherical shell of radius R with uniform load distribution p, the parameters and k have the following form
The linear dimensions , W and N associated with the respective edge zones of the spherical shell are given as follows
References
[1] Reissner, E., On axi-symmetrical deformations of thin shells of revolution, Proc. Symp. Appl. Math., Vol. 3, 1950, pp. 2752.
[2] Reissner, E., Symmetric bending of shallow shells of revolution, J. Math. Mech., Vol. 7, 1958, pp. 121140.
[3] Reissner, H., Spannungen in Kugelschalen (Kuppeln), Müller-Breslau Festschrift, Leipzig, 1912, pp. 181193.
[4] Bromberg, E., and Stoker, J. J., Non-linear theory of curved elastic sheets, Quart. Appl. Math., Vol. 3, 1945, pp. 246265.
Page 275
On the Equations for Finite Symmetrical Deflections of Thin Shells of Revolution[Progr. Appl. Mech.; Prager Anniv. Vol., pp. 171178, Macmillan Co. 1963]
Introduction
In this note we complement earlier work on the problem of finite symmetrical deformations of shells of revolution [1] in several ways. We first incorporate into it theeffect of transverse shear deformation by an appropriate generalization of our earlier formulas for components of strain. We then use the principle of virtual work toshow that these formulas are consistent with our earlier equilibrium equations, under the assumption of small strain which is made in our formulation. As a by-product,we obtain a system of equilibrium equations which remains applicable without the assumption of small strain. We next state a variational equation for our shellproblem which has stress displacement relations as well as equilibrium equations as Euler equations. From this general variational equation we derive a mixedvariational equation in which there remain two independent variations, one of a stress function variable and the other of a displacement variable. We finally use thismixed variational problem to effect a simplification of the two basic simultaneous differential equations which were established in [1].
Deformation and Strain
We designate the points of the shell before deformation by coordinates and , in accordance with Figure 1. We assume that the deformation of the shell ischaracterized by middle surface displacement components u and w and by a shear strain component , again in accordance with Figure 1. We then read from Figure1 the following formulas for circumferential strain and meridional strain ,
where
and
Page 276
Fig. 1.
Equation (1) reduces to
while (2) may be written as
Assuming normal and transverse shearing strain to be small compared to unity, we may linearize in terms of , as follows
Page 277
A second relation involving follows from the identity
in the form
We use (9) to show that, except for terms of relative order 2, equation (7) may be replaced by
Using (10) we obtain that (9) may be replaced to the same degree of approximation by
We may use (10) and (11) to derive our earlier formula for m in terms of and u for the case m = 0 by eliminating w between the two relations, and so obtain
We may also use (10) and (11) in conjunction with (5) to establish a compatibility equation of the form
When = 0 this formula reduces to our earlier result given in [1].
A further consequence of (10) and (11) is the formula
which again generalizes an earlier result for the case = 0.
It is easy to see that we may replace the factors cos and sin of m in (13) and (14) by cos and sin , respectively, as long as m is assumed to be smallcompared to unity. In this way we have from (13) and (14)
and
We feel that it should be possible to modify the factors of m in (13) and (14) in the same manner as the factors of m but do not have a similarly simple reasoning forso doing.
Page 278
Variational Derivation of Equilibrium Equations
We require that equilibrium equations and stress strain relations are Euler equations of the variational problem
where m = W/ N , etc., are the stress strain relations of the problem. We find, with
the following three Euler equilibrium equations
Equations (19) and (20) agree with the usual form of these equations, upon setting U,w = pV and U,u = pH. The moment equilibrium equation (21) reduces to theusual form of this equation upon introducing the assumption that the effect of strain (but not necessarily of deformation) on the form of the equilibrium equations is tobe ignored.
Variational Derivation of Two Simultaneous Second-Order Differential Equations
We assume linear stress strain relations and an isotropic material such that
where , B, C, and D may be functions of .
We express M and M in terms of through the relations
and N , N , and Q in terms of a stress function H and a load function V which ensures satisfaction of the two force equilibrium equations, through the relations
We introduce (22) to (25) into (15), together with U = upH wpV, eliminate derivatives of u and w by integration by parts and arrive at the following variational
Page 279
equation
In this and H are to be varied independently. The Euler differential equations of this variational problem are the two simultaneous differential equations for andH which have been derived earlier [1], supplemented by the effect of the terms with B in (26) and of the term mN mQ in the moment equation (21).
As these latter terms should not arise to the degree of approximation inherent in the present theory, we modify (26) to prevent their occurrence. The appropriatemodification consists in replacing in the terms with C and B in (26) the quantity by . The qualitative a posteriori justification for this step consists in the observationthat upon writing
the terms involving (cos cos )/C, etc., are small of the order of strains compared with the terms cos cos and sin sin inside the last bracket in (26).*
The variational equation I = 0, where
*Strictly speaking, this argument applies only to the terms with 1/C and not to the terms with 1/B. We nevertheless treat the terms with 1/B in the same way as the terms with 1/C, since ourincorporation of the transverse shear effect is an approximate one. We note that our analysis is readily extended to the case where (22) contains mixed terms NM as well.
Page 280
is associated with the following two Euler differential equations
Equation (28) is identical with a corresponding equation in [1]. Equation (29) differs from a corresponding equation in [1] by the inclusion of transverse shear strainterms and by the fact that all terms with in the coefficients of H, V, and pH previously were terms with . Formally, this change from to in (29) represents aconsiderable simplification of the earlier result. In previous applications of the equation corresponding to (29) the possibility of this simplification had shown itself inthe course of the analysis of the differential equation.
Differential Equations for Plates
Setting r = , z = 0 and = 0 we have from (28) and (29)
and
where again equation (31) is considerably simpler than a corresponding equation in [1].
Reference
[1] Reissner, E., "On Axisymmetrical Deformation of Thin Shells of Revolution," Proc. of Symposia in Appl. Math. 3, 2752 (1950).
Page 281
Rotating Shallow Elastic Shells of Revolution*
[J. Soc. Indust. Appl. Math. 13, 333352, 1965]
Introduction
In what follows, we wish to show that the problem of the rotating thin elastic shell of revolution is intriguing in the following sense. It is possible to obtain a solution bymeans of a linear theory which looks so reasonable that one expects the nonlinear effects to be of secondary nature. However, when one considers the problem onthe basis of a nonlinear theory, one finds that nonlinear effects are not at all of a secondary nature. Moreover, while a straightforward elementary analysis of theprincipal aspects of the nonlinear problem is possible, a consideration of the finer structure of the problem leads to an interior-layer problem in which not only thenature but also the location of the layer has to be determined in the course of the analysis.
For the sake of simplicity, our analysis is limited to the class of shells which are generally designated as shallow shells. Stresses and deformations in such shallowshells are governed by differential equations which are similar to the differential equations for finite deflections of flat plates as first obtained by von Kármán.
Statement of the Problem
We consider a thin elastic shell of revolution with middle surface equation z = z(r) which rotates with angular velocity about its axis. We designate radial, axial, andangular displacements by u, w, and , stress resultants and couples by N, N , Q, M, M , and load intensity components by pr and pz in accordance with Figure 1.We assume that the shell is shallow in the sense that sin( + ) + , where = z and = w , primes indicating differentiation with respect to r.
The differential equations of the problem consist of three equilibrium equations which may be written in the form
and of four stress-displacement relations which are taken as
and
*With F. Y. M. Wan.
Page 282
Fig. 1.A rotating shell of revolution.
where
and
We will limit ourselves here to problems for which axial surface and edge loads are absent and for which the only radial loads are the inertia forces of the rotating shellso that
where is the volume mass density and is the constant angular velocity of the shell.
To reduce the problem further, we introduce a stress function , in terms of which
and we assume for the purpose of the present discussion that D and A are independent of r. We then obtain from the moment equation (1c) a differential equation for and of the form
A second equation for and follows through the use of the compatibility equation = (r ) + + 1/2 2 which is implied by (3a, b) and the stress strainrelations (2a, b)
Page 283
in the form
The nonlinear system (7) and (8) is to be solved in an interval ri r r0, where 0 ri, subject to the boundary conditions of absent edge forces and moments
for r = ri, and r = r0. For the special case ri = 0, we may alternately be interested in boundary conditions u = = 0 for r = 0. The difference between the two sets ofconditions is the difference between the cap with or without a zero diameter hole at the apex.
When = 0, the system (7) and (8) reduces to a special case of the differential equations for finite bending of flat plates as first formulated by von Kármán. In whatfollows, we are particularly interested in the problem of the conical shell for which = constant, and in the problem of the toroidal cap for which = (r a)/R.
Membrane Solutions
Experience with related problems of shell theory indicates that for sufficiently thin shells, the state of stress and deformation in the rotating cap or ring should beeffectively as if the shell had no bending stiffness; that is, the solutions of the complete problem (7) to (9) should be effectively the same as those for the special caseD = 0. Setting D = 0, one of the two second order equations, (7), reduces to a zeroth order equation
while the other, (8), remains as is. Concurrently, only one of the two sets of boundary conditions (9) remains, namely
as long as ri 0. When ri = 0, the first of these conditions is to be replaced by limr 0 r1 = 0, or alternately by limr 0 (r ) = 0.
Linear Membrane Theory
Omitting the nonlinear term in the dependent variables and in (10), we have
except where = 0, that is, except in regions where the shell is a flat plate, and where the linearized equation (10) supplies no information concerning . Evidently,the solution (12) satisfies the boundary conditions (11).
Having (12), we find from the linearized form of (8) that
Page 284
as long as 0. In view of (6), we have further that meridional and circumferential stress resultants are independent of and are given by
In the region where = 0, we have from (8) that
so that
and
which is the well-known solution for the rotating uniform disk.
The above analysis suggests in particular the consideration of a problem where a flat disk, say for 0 r rt, is joined to a shell for which 0, say for rt r r0,with the constants of integration c1 and c2 to be determined from one condition for r = 0 and from appropriate transition conditions for r = rt. Evidently, there will betwo such transition conditions, one expressing the fact that N must be continuous at r = rt and the other that must be continuous. As it is not in general possible tosatisfy all three conditions by means of the two constants c1 and c2 it follows that the problem of the shell with discontinuous meridional slope does not in general havea solution within the framework of linear membrane theory.
Before discussing the nonlinear membrane problem, we note the radically different stress distributions as given by the linear membrane solution (14), and by (17) forthe corresponding problem of the flat disk with the boundary condition (r0) = 0 and with
in particular, when ri 0. Case (i) of the disk without hole at the apex gives
while case (ii) for the disk with a hole of radius ri gives
The distributions of stress in accordance with (14), (19) and (20) are illustrated in Figure 2.
The following question now suggests itself in connection with these results. Given a rotating shallow cap, for which the rotating disk is a limiting case, what
Page 285
Fig. 2.Stresses in rotating shallow shells according to linear membrane
theory compared with the corresponding stresses in rotating disks(a) continuous at center, (b) with small circular hole at center.
significance, if any, is associated with the membrane stress distribution as given by (14)?
Nonlinear Membrane Theory
We now consider that (10) may be solved in two different ways:
When = 0, we find from (8) and with LM defined by (13), that 2 + 2 2 LM = 0, so that
The plus sign in front of the square root is chosen as we expect that for sufficiently small , NLM should be nearly the same as LM.
From (22), we find that real values of NLM are obtained only as long as
or, in view of (13), as long as
Equation (24) shows in particular that, for the toroidal cap with = (r a)/R, there is no real solution of the equations of nonlinear membrane theory with = 0 in a
Page 286
finite neighborhood of r = a. This is a qualitative distinction from the corresponding result of linear membrane theory where there is such a solution.
Equation (24) shows further that for any rotating shallow shell of revolution there is a critical angular velocity er above which the solution (21a) and (22) ceases tobe possible throughout. This critical value of is given by
The condition (25) may alternately be expressed as a condition of a maximum possible hoop strain in conjunction with the solution = 0. Since when = 0, =AN = 2r2/E, we have from (24) that
and, in particular, for a conical shell that ( )max = 2/(6 + 2 ).
An additional observation of interest is that when LM = 1/2 , which is the condition of criticality, we have NLM = , so that use of the linear theory for this caseresults in a one hundred per cent underestimation of the value of .
We now consider the alternate solution = of (10). Introduction of this result into (8) leaves as differential equation for ,
The difference in signs of the two terms of the right side of (27) indicates that the deflection of the given surface in space into the base plane is associated withcompressive circumferential strains while the effect of the rotational motion is to set up tensile circumferential strains.
The solution of (27) may be written in the form
In summary, the differential equations of the nonlinear membrane theory of the rotating shell have the two classes of solutions
and
with meridional and circumferential stress resultants given by
Page 287
for the first class of solutions, and
for the second class of solutions.
It remains to determine for any given problem in which range of values of r either the one or the other solution applies.
Conical Shells
For the class of shells for which is constant, the solution (29) is surely inapplicable for r rc, where
and therefore if the problem has a solution for rc < r it must be given by (30). Evidently, it is possible that the correct solution is given by (30) for values of r whichare less than rc, say for rt < r, where rt rc. If ri < rt then a transition will occur at r = rt, the solution (29) being valid for r < rt and the solution (30) being valid for rt
< r. Setting constant, we have as expressions for N and N in the region rt < r:
while for r < rt, N and N remain given by (31).
In the region where (34) is valid the slope function is equal to , expressing the fact that the portion rt < r of the shell is deformed into a flat disk. In thecomplementary region r < rt, we have from (29) and with the definition (33) for rc,
This means that at the junction r = rt, we have a discontinuity in of the amount
For the formulation of a system of boundary and transition conditions we begin by assuming that rc < r0; we have then from (34a), as boundary condition for r = r0,
Page 288
As N = 0 for r < rt, in accordance with (31a), we have from equilibrium considerations that N as given by (34a) must also vanish for r = rt, so that
It remains to establish a third condition for the determination of the three quantities c1, c2, and rt. This third condition follows from the observation that thecircumferential strain must be continuous at r = rt. In view of the fact that N = 0 for r = rt, this means that N must be continuous for r = rt. From (31) and (34b)there follows then the further relation
We may use (37a) and (37b) to express c1 and c2 in terms of rt and in terms of the parameter E 2/ 2, which according to (33) may also be written as .In this way we obtain from (37a) and (37b),
Introduction of (38) into the remaining condition (37c) leads to the equation
from which rt/r0 is to be determined as a function of rc/r0, or equivalently rc/r0 as a function of rt/r0, as shown in Figure 3. We see that, indeed, rt/r0 is smaller thanrc/r0 for all 0 < rc/r0 < 1.
With rt/r0 as a function of rc/r0 in accordance with (39) and with c1 and c2 from (38), we obtain the following explicit expressions for N and N in the region rt r.
Equations (40) are supplemented in the region r < rt by N and N distributions in accordance with (31).
Considering the fact that when rc = r0 we also have rt = r0, we conclude, by means of a continuity argument, that when rc > r0 the solution everywhere in themembrane is given by (29) and (31); that is, the stress distribution as given by nonlinear membrane theory is the same as that given by linear membrane theory.
We have then that, for a given conical membrane, stresses and deformations according to nonlinear membrane theory are similar to those of linear membrane theoryas long as the rotational speed is less than er. When is larger than er then the nonlinear membrane solution differs in an essential way from the solution bymeans of linear membrane theory. A flattened-out portion of the membrane
Page 289
Fig. 3.The critical and transition radii for a conical ring membrane.
appears, for rt r, with a discontinuity of the angular displacement at r = rt. As increases, rtdecreases. When rt = ri then the membrane is completely flattenedout. It persists in this flattened-out state for still further increases of . As increases towards infinity, the state of stress in the originally conical membraneapproaches the state of stress in a flat disc rotating with the same angular velocity.
In order to see the nature of these results, we have in Figure 4 shown the distributions of = N /h and = N/h for various values of rc/r0 for the case of a shell withri/r0 = 0.2. We note that the result of linear membrane theory is the same for all values of rc/r0, and that it coincides with the results of nonlinear membrane theory forrc/r0 1. We note in particular that an analysis of the problem by means of the linear membrane theory in general leads to much higher stresses than the correspondinganalysis of the problem by means of nonlinear membrane theory. Figure 5 presents values of max and max according to nonlinear membrane theory.
Toroidal Cap
We consider next the class of shells for which
For this problem, it is clear that the solution (29) is inapplicable in a neighborhood of r = a, no matter how small is. In order to delineate the region of inapplicability
Page 290
Fig. 4.Stress distributions in a shallow conical ring shell for different rotating
speeds according to nonlinear membrane theory.
of (29), we first determine for fixed the range of values of r for which (29) is surely not usable. The endpoints rci and rc0 of the interval in question are given by
Combining (41) and (42), we find
where
Having established that (29) cannot be applicable for rci r rco, we anticipate the possibility that the alternate solution (30) must be used in an interval rti r rt0,where rti rci and rc0 rt0, with transition to the solution (29) occurring at r = rti and r = rt0. Without determining the values of rti and rt0, we can state that as the
Page 291
Fig. 5.Maximum direct stresses for a shallow conical membrane.
angular velocity increases, we will encounter two different situations as follows. For sufficiently small values of , rci as well as rc0 will be inside the interval (ri, r0).With increasing values of either one or both of these will be outside (ri, r0). When both values are outside, the rotating toroidal cap will be completely flattened out.As the angular velocity increases further and further, the stress distribution will approach more and more the distribution for a rotating disk. Unlike the conicalmembrane, the toroidal cap membrane begins to flatten out in the neighborhood of the crown rather than of its outer edge, and does so for arbitrarily small values of
.
Introducing from (41) into (32a,b) we have the following expressions for N and N in the region rti < r < rt0.
while for ri < r < rti and rt0 < r < r0, N and N remain given by (31).
As N = 0 for r < rti and r > rt0 N as given by (44a) must also vanish at r = rti and r = rt0, so that
Page 292
The remaining two conditions needed to determine the four quantities c1, c2, rti, and rt0 again come from the continuity of the hoop stress resultant N at rti and rt0.From (31) and (44b) we have
We use (45a) and (45b) to express c1 and c2 in terms of rti and rt0 and in terms of the parameter k2 defined by (43c) as follows:
Introduction of (46a, b) into (45c, d) leads to two transcendental equations of the form
and
for rt0/a and for
The solution of (46c, d) leads to values of rt0/a and rti/a as functions of the load parameter k which is defined by (43c). This solution is obtained by first using (46c)to eliminate rt0/a from (46d). The resulting equation is then solved to give as a function of k2. Having this, we next use (46c) to obtain rt0/a as function of k2. Afterthat, (46e) is used to obtain rti/a. Values of rt0/a, rti/a together with values rc0/a and rci/a as given by (43a,b) are shown in Figure 6, which also includes an alternatedefinition of k. We note that rc0 < rt0 and rti < rci for all k2 > 0; that is, the flattened-out portion emanating from the crown of the toroidal cap is indeed wider than isrequired by the criticality condition (42).
We may introduce (46a, b) into (44a, b) to get explicit expressions for N and N with rt0 and rti given by Figure 6. The corresponding distributions of and forvarious values of k2 and for a shell with ri/a = 0.5 and r0/a = 1.5 are shown in Figure 7. Figure 8 shows how the maximum stresses vary as functions of k2. It isinteresting to note that while nonlinear membrane theory must be used for all 2 > 0 in order to obtain real , linear membrane theory (with nonexistent real ) givesthe correct maximum hoop stress for sufficiently small values of k2, say k2 less than 0.05.
Page 293
Fig. 6.The critical and transition radii for a toroidal cap membrane.
Interior Layer Analysis
Nonlinear membrane analysis, which is based on setting D = 0 in the differential equation (7) and in the boundary condition (9b), has been shown to lead to a solutionof the problem for which may be discontinuous for one or more values rt of r. No such discontinuities can occur when D does not vanish. This suggests that forsufficiently small values of D, the smoothing out of the discontinuities for D = 0 may be confined to relatively narrow layers surrounding r = rt. In the following wedetermine the circumstances under which this narrow layer will exist and the order of magnitude of the bending stresses in it when it does exist.
We begin by nondimensionalizing the differential equations of the problem by setting
We have then from (7) and (8):
Page 294
Fig. 7.Stress distributions in a shallow toroidal cap for different rotating speeds
according to nonlinear membrane theory.
where
and primes now indicate differentiation with respect to x.
The following qualitative conclusions appear from (48) and (49):
(1) When 4 << 1, membrane theory should in general be adequate. If, in addition, << 1, then linear membrane theory should in general be adequate.
(2) When 4 = O(1), the combined effect of nonlinear membrane and linear bending action in the shell must be considered. When << 1, linear bending theoryshould be adequate.
(3) When 4 >> 1, we have from (48) that effectively = 0 and (49) becomes the equation of the rotating disk for all values of the load parameter .
Knowing that nonlinear membrane theory may lead to a discontinuity in at t, we now consider the case 4 << 1 in relation to this result. To determine the effect ofbending stiffness near r = rt, we set
Page 295
Fig. 8.Maximum direct stresses for a shallow toroidal membrane.
where and t are yet to be chosen. We require that differentiations with respect to y do not change the order of magnitude of and g, and we anticipate that willturn out to be small compared to unity. We may then write (7) and (8) approximately in the form
where now primes indicate differentiation with respect to y.
We require that and g approach the appropriate nonlinear solution as y goes from zero to values of order of magnitude unity, that is, as y tends to ± . In order tohave a truly fourth order problem with the assumed order of magnitude relation concerning differentiation with respect to y, we set
From this,
Page 296
Since rt and t are of the same order of magnitude as r0 and 0 respectively, we have = O( ) << 1 as anticipated. The differential equations for and g nowbecome
with t defined by
We know from the earlier nonlinear membrane consideration (see (24)) that t is at most unity. In fact, for a conical shell,
Equations (57) and (58) are supplemented by the limiting conditions
Together they form a boundary value problem for a layer of shell in the neighborhood of r = rt.
Even without the explicit solution to this boundary value problem, we can readily obtain the order of magnitude of the stresses near rc/rt. Introducing (51a, b), (52a,b) and (55) into (6a, b) and (2c) we find
Equations (62) show that in the region where bending action is of importance, the bending stress is of a smaller order of magnitude than the maximum direct stresses if << t.
Concluding Remarks
In addition to considering the problems of the conical ring membrane and the toroidal cap membrane as reported here, we have also considered a number of othercases. These are
(i) a spherical cap, for which = r/R,
(ii) a shell the meridian of which is a quartic parabola, so that ,
(iii) a shell for which / 0 = r0/r.
The spherical cap is of particular interest insofar as the criticality condition (25) applies to the entire membrane at once, /r being constant. Accordingly, when <er, we have the possibility of two distinct solutions over the entire membrane, while for er < only the completely flattened-out state is possible. The question
remains as to which of the two possible solutions for < er is the appropriate one for all or part of the spherical cap. We think that the unflattened state is the
Page 297
physically correct state as it goes continuously over into the flattened-out state as goes beyond er.
For the case we find, in contrast to what happens for the conical ring and the toroidal cap, that the discontinuity moves outward from the apex r = 0 asthe rotational speed increases. Moreover, the flat region of the deformed shell is now r rt rather than r rt as in the case of the conical ring, or rti r rt0 as in thecase of the toroidal cap.
The problem of the shell with / 0 = r0/r attracted our attention originally because the equations of the linear bending theory could be solved in terms of elementaryfunctions. We have obtained numerical data for both the nonlinear membrane problem and the linear bending problem for this shell. It turns out that the nonlinearmembrane behavior is similar to that of the conical shell while the results for the linear bending problem, though explicit, do not seem to be of sufficient interest towarrant including them in this account.
Our analysis of the nonlinear bending problem has been limited to one interesting aspect of it, the formulation of an interior-layer analysis together with resultsconcerning various orders of magnitude involved in this analysis. We have also undertaken a narrow-layer analysis of the flattening out phenomenon at the apex of thetoroidal cap shell, which is not included in this account.
While asymptotic considerations such as these are of intrinsic interest, it should however be mentioned that insofar as the symmetric problem of the rotating shell ofrevolution is concerned, it is possible today to obtain quantitative data by purely numerical methods.
Added in Proof
Subsequent to the presentation of this paper, the authors became aware of a report by W. Flügge and P. M. Riplog entitled A large-deformation theory of shellmembranes, designated as Technical Report No. 102 of the Engineering Mechanics Division of Stanford University and dated September, 1956. The contents ofthis report anticipate the contents of our work insofar as the formulation of transition conditions for the nonlinear membrane solutions is concerned, and insofar as arotating conical membrane is considered. The work of Flügge and Riplog assumes general shells of revolution rather than shallow shells of revolution. The limitation toshallow shells means that we can explicitly evaluate the transition conditions of nonlinear membrane theory whereas without this limitation the evaluation is tied in witha proposed numerical solution of the differential equations of the problem.
Page 298
A Note on Stress Strain Relations of the Linear Theory of Shells*
[J. Appl. Math. & Phys. (ZAMP) 17, 676681, 1966]
In a recent paper [1]we have shown, among other things, that the lines-of-curvature stress strain relations of Flügge, Lurje, and Byrne,
etc., where
could, in inverted form, be written as a system of relations
where the jk and jk, in addition to the usual translational displacement components, involved an angular displacement component , such that , ,while
The inversion of part (1) of the FLB system is straightforward and does not lead to unexpected results. The inversion of part (2) of the system is not straightforwardto the same degree and leads to results of a form somewhat simpler than expected, namely
where 2 = 1 + ( h)2/24.
*With F. Y. M. Wan.
Page 299
In the present note, we show that a corresponding inversion, with somewhat less simple final results, is possible for a system of stress strain relations which has beenproposed as one combining simplicity and adequacy [2]. This system follows upon omitting terms with in (1) and upon writing in (2)
with corresponding modifications of and . By this, Equations (2) are changed to
N21 being given analogously. Equation (9) and the corresponding equation for N21 may alternately be written as
We note particularly, that now M12 = M21, while N12 and N21 as given in (9 ) are consistent with the moment equilibrium equation
While it is possible to simply state the final results of our inversion procedure and then verify their correctness, it is of interest here to give the steps leading to theseresults. We first solve (8) and (9) in the form
where and are arbitrary functions. We then use the compatability relation
to express in terms of ,
Page 300
With this, we write
We now see that it is consistent to identify with the angular displacement component in (4) and write, in place of (14) and the first part of (11)
The system (15) and (16) is not yet in such a form that we can write it in terms of a strain energy function W, as in (3). The appropriate modification consists inmaking use of the symmetry relation M12 = M21 in order to write 12 and 21 as
In this and in (16) the relation M12 = M21 is now to be considered not as an identity but as a consequence of the finite compatibility Equation (12).
With (16), (17) and the inverted form of the abbreviated version of Equations (1), we now have for the function W in (3)
The step from lines of curvature coordinates to general orthogonal coordinates may be carried out in analogy to what has been done in [1] for the relations of Flügge,Lurje, and Byrne. Using the invariance relations for the states Njk and Mjk, together with the further invariant
we obtain
Page 301
From this follows as stress strain relations which generalize (16) and (17) to general orthogonal coordinates
where the quantities jk and jk are, as in [3],
with 1 + w,1/ 1 u1/R11 u2/R12 = 0, and 2 defined analogously. Since M12 M21 is an invariant, the symmetry condition M12 = M21 continues to hold for generalorthogonal coordinates and the terms involving (M12 M21) may again be omitted in the expressions for jk.
The following further consideration may also be of some interest. Returning again to lines of curvature coordinates and to the stress strain relations (8) and (9) writtenin the form
we note that we may write these relations together with the relations N11 = C( 11 + 22), M11 = D( 11 + 22), etc., in the form
Page 302
provided we first observe that we may replace (22) by
in view of the fact that (18) implies 12 21 = 0. With this, we have for lines of curvature coordinates
In the resulting stress strain relations , 12 = 21 is to be considered as a consequence of the moment equilibrium Equation (10).
The corresponding function U for general orthogonal coordinates follows from the invariants for the jk and jk together with the supplementary invariant
as
Introduction of this form of U into (24) gives as stress strain relations which generalize (23) and (22 ) to the case of arbitrary orthogonal coordinates
Page 303
while the remaining relations come out to be
with analogous expressions for N22 and M22.
Equations (27) to (30) can be shown to coincide with the relations stated by Koiter [2] upon setting and upon observing that our 12 + 21, 12 +21, , M12 = M21, and 1/2(N12 + N21) correspond to Koiter's , , , W, and S, respectively. A point of interest in this derivation of a system of relations
analogous to Koiter's is that in the present formulation N12 and N21 come out as two separate quantities directly, instead of getting the sum N12 + N21 from a stressstrain relation, to be combined subsequently with the moment equilibrium equation giving N12 N21.
References
[1] E. Reissner and F. Y. M. Wan, On Stress Strain Relations and Strain Displacement Relations of the Linear Theory of Shells, The Folke Odquist Volume,pp. 487500, John Wiley & Sons, New York, 1967.
[2] W. T. Koiter, A Consistent First Approximation in the General Theory of Thin Elastic Shells, Proc. of IUTAM Symposium on the Theory of Thin ElasticShells, pp. 1233, Delft 1960.
[3] E. Reissner, Variational Considerations for Elastic Beams and Shells , Proc. of ASCE, J. Eng. Mech. Div. 88, 2357 (1962).
Page 304
Small Strain Large Deformation Shell Theory[Proc. 9th Midwestern Mech. Conference, pp. 5558, 1965]
In speaking of small strain large deformation shell theory, we mean a theory in which the equilibrium equations are affected by the state of deformation but not by thestate of strain, while the stress strain relations involve the states of stress and strain without reference to the state of deformation.
We consider here a relatively elementary self-contained vectorial approach to the problem of deriving small strain large deformation shell theory, omitting referencesto earlier more general formulations of this problem. We conclude our discussion with some results for shells of revolution which, as far as we know, have not beengiven previously.
Let r = r( , ) be the equation of the middle surface of the shell before deformation, and , lines of curvature coordinates. Due to deformation r changes into . Interms of and r and their derivatives with respect to and we have as middle surface normal and shearing strains (Fig. 1).
which, for small strains is equivalent to
where = |r |, , with a corresponding definition for .
We next define the corresponding components of strain E , E and for surfaces which before deformation are at a distance from the middle surface. We write R= r + n where and we introduce the hypothesis of negligible transverse shear and normal strains by postulating that the change of R due todeformation results in a vector P = + where is a normal vector to the deformed middle surface given by
It is readily seen that differs from a unit vector by additive terms of order .
We now have, in analogy to Equ. (1 )
Page 305
Fig. 1.
With |R | = (1 /R ), P = + and
where
there follows from Eq. (3), except for terms which are small of higher order
where
Equations (6) for the relevant components of strain as a function of the thickness coordinate imply shell stress strain relations involving stress resultants and coupleswhich are the same as for linear shell theory. In writing these stress strain relations, we make use of the fact that the , -coordinates on the deformed middle surfaceareexcept for terms of the relative order orthogonal coordinates. This means, for example, that a suitable system of stress strain relations for an isotropic
Page 306
homogeneous shell is of the form
where C = Eh and D = Eh3/12(1 2).
In writing equilibrium equations for stress resultants and couples, we may use, as we here disregard the effect of strain, the well known form of these equations forgeneral orthogonal coordinates on the middle surface of the shell, with curvature radii P , P and P . It is the appearance of P , P , P , instead of principalcurvature radii R , R for the undeformed middle surface, which represents the effect of deformation on the form of the equilibrium differential equations.
In vector form, with resultant vectors N and couple vectors M, these equilibrium equations are
where
with corresponding definitions for N and M .
In reducing Eqs. (10) and (11) to scalar form for , and -components, we need in addition to the differentiation formula (4) for the associated formulas
with an analogous relation for .
Of the six scalar equilibrium equations, the one which represents moment equilibrium about the normal to the deformed middle surface, together with the seven stressstrain relations (8) and (9), expresses the eight stress resultants and couples N , M , N , M , N , M , N , M in terms of derivatives of the radius vector . Theremaining five equilibrium equations then are a system of five equations for the two transverse shear stress resultants Q , Q and for three scalar components of ,which through elimination of Q and Q are readily reduced to a system of three equations for the three components of .
In deriving a small strain large deformation theory, we have made use of the small strain property of the theory in two distinct instances. One of these has to do withthe equilibrium equations and the other with what we may call the strain displacement equationsupon writing = r + u in Eq. (6). If we had been completelyconsistent in our procedure, we should now be able to ascertain the consistency of the equilibrium equations and of the strain displacement equations
Page 307
via an appropriate variational principle. In attempting to do this, one finds that the approximate theory as formulated is approximately consistent rather than exactly so.For an exact consistency, we should have to modify the form of the equilibrium equations, in a manner which is not difficult to establish, or we should have to modifythe form of the strain displacement equations, in a manner which seems less easy to establish. We will not concern ourselves here with this question.
We conclude our account with some remarks on the subject of displacement representations. In a general theory it seems easiest to represent the displacement vectorin terms of components referred to in the geometry of the undeformed shell, i.e.,
For more special theories, such as the theory of shells of revolution, including the still more special case of the circular plate, an alternate representation seems to beas good or better. We may write for a shell of revolution
and
where u, w and are functions of and .
We have some years earlier considered the problem of a large deformation small strain theory of shells of revolution for the special case of axisymmetricaldeformations given when u( , ) = U( ), w( , ) = W( ) and ( , ) = . It may be shown that, as they should be, the present formulas are consistent with thosein our earlier work.
A somewhat more general class of states of displacementnon-rotationally symmetricleading to a rotationally symmetric state of strain in shells of revolution may beshown to be of the form
where c1 and c2 are arbitrary constants. Equations (19) may be taken as the starting point of a large deformation theory of the two dislocation theory problems ofpure bending and pure twisting of shells of revolution, which problems we have, some time ago, considered within the framework of the linear theory of shells, andwithin the framework of the nonlinear theory of circular plates.
References
1. J. L. Sanders, Nonlinear theories for thin shells, Q. Appl. Math., 21, 2136, 1963.
2. P. M. Naghdi and R. P. Nordgren, On the nonlinear theory of elastic shells under the Kirchhoff hypothesis, Q. Appl. Math., 21, 4959, 1963.
3. E. Reissner, On axi-symmetrical deformations of thin shells of revolution, Proc. Sym. Appl. Math., 3, 2752, 1950.
4. , On bending of curved thin-walled tubes, Proc. Nat. Ac. Sci., 35, 204208, 1949.
5. , Note on the problem of twisting of a circular ring sector, Q. Appl. Math., 7, 342347, 1949.
6. , On finite twisting and bending of circular ring sector plates and shallow helicoidal shells, Q. Appl. Math., 11, 473483, 1954.
Page 308
Finite Inextensional Pure Bending and Twisting of Thin Shells of Revolution[Qu. J. Mech. Appl. Math. 21, 293306, 1968]
1Introduction
We are concerned in what follows with the solution of a class of problems involving large deformations of thin elastic shells. We consider sectors of shells ofrevolution, the sector of opening angle 2 corresponding to a complete shell of revolution with radial slit. The radial edges of the sector shell are taken to be actedupon by tractions of such nature as to be equivalent to axial moments and axial forces (Figure 1). Special cases of this class of problems have previously been treatedwithin the framework of the linear theory of thin shells of revolution. Specifically, the linear problem involving axial moments has been shown to include a shell-theoretical formulation of von Kármán's problem of pure bending of curved tubes [5], and the linear problem involving axial forces (the problem of pure twisting) hasbeen shown, for closed cross section tubes, to be a membrane-theoretical generalization of the Batho-Bredt problem of torsion of thin-walled closed-cross-sectioncylindrical tubes [6].
Fig. 1.Pure bending and twisting of ring sectorshell of revolution (shown for the case ofa conical shell of constant wall thickness).
Page 309
The present non-linear treatment of the problem of the combined effects of axial moments and forces is based on a recent result concerning states of rotationallysymmetric strain in shells of revolution, which are associated with states of displacement without rotational symmetry [8, 9]. We do not attempt here to make use ofthis result in the most comprehensive way possible. Rather we limit ourselves to an application of the result subject to the restrictive assumption of inextensional(bending) deformations in the (incomplete) shell of revolution. As to the significance of the results, which are based on the assumption of inextensibility of the middlesurface of the shell, it may be considered intuitively evident that the results will be significant for open-cross-section shells, such as conical shells. However, theexample of the flat circular-ring plate shows that care has to be taken not to apply the results based on the assumption of inextensibility beyond their range of physicalsignificance. While for the ring plate the solution of the axial force problem is of an inextensional character [7], the solution of the axial moment problem assuredly isnot, in an elementary sense, although there, too, conclusions of great interest follow through use of the concept of inextensibility, as shown by Ashwell [2].
As for the problems of bending of rectangular and rhombic strips [1, 3, 4], the exact solution of the finite deformation problem involves besides an inextensionalbending state, a superposed more general edge zone state. This problem, which has recently been investigated for the case of pure bending of the ring shell [9], willnot be considered here beyond the observations that determination of the superposed edge zone state leaves, in essence, a linear problem, and that for sufficientlylarge deformations the non-linear inextensional-bending solution by itself provides the major portion of the results which are of physical interest.
Additional physical interest in the solution which is obtained in what follows may be ascribed to the fact that this solution can be considered, within the framework of atheory of elastic dislocations, to represent the state of stress and strain for non-linear combinations of ''edge" and "screw" dislocations, for the class of bodies whichmay be treated as thin shells of revolution.
2Description of Large-Deformation States of Small Strain
In accordance with a recent elementary presentation of the subject [8] let r = r( , ) be the equation of the middle surface of the shell before deformation and ,lines-of-curvature coordinates. Let = ( , ) be the corresponding equation of the deformed middle surface (Figure 2). Components of normal and shearing strainin the middle surface of the deformed shell are taken in the form
with a corresponding expression for . Components of middle surface curvature changes are taken in the form
Page 310
Fig. 2.Shell surface coordinates for deformed and undeformed shell
element.
with a corresponding expression for . In this n and are normal vectors to the undeformed and deformed middle surface, respectively, defined by
3Rotationally Symmetric Strain in Shells of Revolution
The radius vector r to points on the undeformed middle surface is taken in the form
where r = r( ), z = z( ) are parametric equations of the meridians of the surface and is the longitudinal angle measured in the base plane of the shell (Figure 3). Wefurther write
where
and, in accordance with (3),
In analogy to equation (4) the radius vector to points on the deformed middle surface of the shell is taken in the form
It has previously been shown [8, 9] that the state of strain associated with the radius vectors r and is rotationally symmetric for the following class of functions , ,:
Page 311
Fig. 3.Position variables for deformed and undeformed shell of
revolution element.
In this cz and c are arbitrary constants. It is their presence which generalizes the known theory of a symmetric bending and torsion of shells of revolution so as toinclude the problems of pure bending and twisting of sectorial shells.
Introduction of equations (9) into equations (1) gives as expressions for the components of rotationally symmetric mid-surface strain
In (10) and (12) it will be convenient to introduce, in analogy to the tangent angle to the meridian curves on the undeformed middle surface, the tangent angle tothe meridian curves on the deformed middle surface, by means of the relation
Corresponding somewhat more lengthy expressions follow from (2) and (3) for components of curvature change. These are not listed here as they are not needed intheir generality for the purpose of the present analysis.
4Inextensional Deformations
States of inextensional deformation are defined by the relations
Page 312
Introduction of equations (14) into equations (10) to (13) leads to the following representation of the possible states of inextensional deformation
The special case cz = 0 of equations (15) and (17) has been stated previously [9]. When both cz and c are zero then equations (15) to (17) reduce to the statement = r, = 0, = which implies that the only inextensional deformations for this case consist of axial rigid body translation and rotation.
5Inextensional Curvature Changes
We will calculate and directly in accordance with equation (2) and then use Gauss's "theorem egregium," which for inextensional deformations of surfaces ofrevolution is of the form
with
in order to obtain .
Equations (8) and (9) give
and, in accordance with (3),
From this follows
Introduction of the inextensibility conditions (16) and (15) into (23) reduces (23) to
Page 313
Introduction of (15) to (17) into (24) reduces this equation after various transformations to the remarkably simple form
Having P and P we find from (18),
and then, in accordance with (2),
In this the factor tan follows explicitly from equations (15) and (17).
6Moments and Strain Energy
It will be assumed that the shell is orthotropic in such a way that bending and twisting couples M , M and T are given in terms of the curvature changes , and as follows:
We may account for meridional thickness and material property changes by considering D , D , D and D = D to be functions of .
The stress-strain relations (30) are associated with a strain energy function U of the form
there being no mid-surface strain contribution to U in view of the assumed mid-surface inextensibility.
7Load-Deflection Relations
Considering the force F and the moment M of the tractions applied to sections = constant of the shell as prescribed, we may establish the relations between themand the associated displacement measures cz and c through use of the condition of minimum potential energy.
The potential energy of the shell, per unit of circumferential angle , is given by
Page 314
The condition of minimum potential energy evidently reduces to the two derivative relations / cz = / c = 0. Accordingly, we have
as load-deflection relations, with U given in terms of cz and c through equations (31), (28) and (29).
8Determination of Stress Resultants
Because of the assumption of mid-surface inextensibility the normal stress resultants N and N and the shear stress resultants N and N take on the character ofreactive quantities which must be taken from the equations of equilibrium rather than from stress-strain relations. The expressions for N , N , N , N which areobtained in this way will not, in general, be such as to allow satisfaction of the conditions of no stress along edges = constant of the shell. It may be expected that inorder to satisfy these conditions the state of inextensional bending as determined above must be perturbed, effectively within narrow edge zones only, by a moregeneral state of bending and stretching. It may further be expected that the analysis of these narrow edge zone states is associated with a linear system of perturbationequations, as has previously been shown to be the case for the problem of pure bending of the shell [9], and for analogous problems of flat plates [1, 3, 4], in thelatter cases subject to the provision of sufficiently large deformations.
9Pure Bending and Twisting of Conical Shells
A relatively simple application of the general results for inextensional pure bending and twisting of shells of revolution is to the problem of the conical shell. Setting
we have from (15) and (17)
and therewith
Introduction of (37) into (29) and (28) gives as expressions for curvature changes
Page 315
and
while remains as in (29).
With this the strain energy function U may be written as
Three relatively simple special cases will be discussed further.
10Pure Bending (without Twisting)
It is indicated by the symmetry of the problem that states of deformation are possible for which simultaneously cz = 0 and F = 0. Setting cz = 0 reduces U in equation(40) to a quantity which may be designated by U0 and which is given by
From this and (33) follows as load-deflection relation
and, upon developing in powers of ,
When D = constant the load-deflection relation (42) reduces to a previously given result [9].
For the special case of a circular cylindrical shell equation (42), upon setting sin = 1 and r = a, reduces to the relation
which, in contrast to what happens when sin 1, is linear for all values of c .
In comparing (44) and (43) it is observed that the bending stiffness of the conical shell for sufficiently small c is of the same order of magnitude as the corresponding
Page 316
stiffness of the circular cylindrical shell, the difference between the two corresponding quantities being caused by a factor 1/sin2 . This is in marked contrast with theelementary result for the limiting case r of a straight strip which deforms extensionally.
We further discuss the case of a uniform conical shell. Setting D = constant, we write first
Setting
and
we have from (45)
In introducing an extra factor sin2 on both sides of (48), we have that the right-hand side of the equation becomes independent of , for sufficiently small values of|c |. Figure 4 shows the dependence of Ma sin3 /D b on c for several values of . It is seen that, except when = 1/2 , a closing of the gap is associated with areduction in stiffness and an opening of the gap is associated with an increase in stiffness as measured by the function dM/dc .
As an additional result of interest we determine the distribution of the circumferential bending moment M in the form
An alternative form of M is
where, in accordance with (17),
The special case = 0 of the result expressed by (50) and (51) has previously been established by Ashwell [2] through application of the von Kármán equations forfinite deflections of flat plates.* However, in extending the application of the concept of inextensional deformation to the case = 0, consideration must be given tothe fact that for very small c the plate will surely not behave in accordance with inextensional bending theory but rather in accordance with the theory of plane stress,with a bifurcation phenomenon occurring for some critical value of c .
*It is remarkable that in this way the correct result was found for arbitrarily large values of , although the validity of the von Kármán equations is limited to the range tan2 << 1.
Page 317
Fig. 4.Load-deflection relations for pure bending of conical ring sector shells of
constant thickness.
11Pure Twisting, Linear Case
Setting c = 0 and disregarding higher than second degree terms in cz in equation (40), we obtain from (40) and (33) the force-deflection relation
The associated distribution of twisting couples over the cross-section of the shell in accordance with equations (29) and (30) is
while, with = = 0, there are no associated bending couples.
It is apparent from equations (52) and (53) that the solution of the linear pure twisting problem for the conical shell is the same as the corresponding previouslyestablished result [7] for the case of the flat plate.
Page 318
12Pure Bending and Twisting of Cylindrical Shells
The limiting case of a circular cylindrical shell is obtained by setting in (34) and in the subsequent relations
With this the strain energy function U reduces to
Introduction of equation (55) into equation (33) leads to load-deflection relations of the form
and
where = cz/a.
Some specific consequences of (56) and (57) are as follows.
Setting M = 0, we have from (56), for relatively small cz/a,
From (57) follows then further
Specifically, for an isotropic shell for which
there follows from (58)
and from (59)
Page 319
Setting c = 0 we have from (57), for relatively small cz/a,
and for an isotropic shell
The associated values of M are
Setting c = 1, which corresponds to the deformation of the cylinder with radius a into a flat strip, we have from (57)
independent of the values of cz/a (excepting cz/a = 1). From (56) follows further
Equation (66) indicates that as long as , as is the case for isotropic shells with < 1/2, the magnitude of M decreases with so that the configurationwith cz = 0 would not be stable. Note that for this case the assumption of inextensional deformations cannot be expected to give a physically meaningful result.
Setting c = 2, which means that the cylinder of radius a is bent into a cylinder of the same radius with reverse curvature, we have, from (56),
and, for the isotropic shell,
Equation (57) for F becomes
and for the isotropic shell
It is evident that a general discussion of load-deflection relations for conical shells, on the basis of the strain energy function (40), will be considerably more involvedthan the foregoing discussion of special cases. It is apparent, however, that
Page 320
such a discussion is feasible and that it will be somewhat easier for the two sets of cases for which the stiffness factors are independent of r and proportional to r3,respectively, the latter case corresponding to a uniform-material shell with wall thickness varying linearly with distance from the apex of the shell.
References
1. D. G. Ashwell, Jl. R. aeronaut Soc. 54 (1950) 708.
2. , Q. Jl. Mech. Appl. Math. 16 (1963) 163.
3. Y. C. Fung and W. H. Wittrick, J. appl. Mech. 21 (1954) 351.
4. , Q. Jl. Mech. appl. Math. 8 (1955) 191.
5. E. Reissner, Proc. natn. Acad. Sci. U.S.A. 35 (1949) 204.
6. , Q. appl. Math. 7 (1949) 342.
7. , Q. appl. Math. 11 (1954) 59.
8. , Proc. ninth Midwestern Conference on Mechanics (Madison, Wisc., 1965), 55.
9. W. A. Smith, Pure bending of shells of revolution: A nonlinear dislocation problem . Ph.D. Dissertation, Massachusetts Institute of Technology, July 1966.
Page 321
On Consistent First Approximations in the General Linear Theory of Thin Elastic Shells[Ingenieur-Archiv 40, 402419, 1971]
1Introduction
The present paper considers once more the subject of deriving consistent rational systems of linear two-dimensional differential equations for the approximate analysisof three-dimensional problems of stresses and strains in ''thin" elastic layers surrounding a given surface in space. The basic assumption in the present considerationsagain is the assumption of thinness, in the sense that significant changes of stress and strain along the layer are assumed to require distances large compared to thethickness of the layer. It is characteristic of this approach that with it the well-known Love-Kirchhoff hypothesis, when applicable, ceases to have the character of ahypothesis or assumption, and becomes a consequence of an analysis based on the assumption of thinness.
The present writer finds it difficult to attribute the basic idea of the thinness approach for the derivation of two-dimensional shell theory from three-dimensionalcontinuum mechanics formulations to a specific source. Evidently, Goodier's parametric expansion approach to deriving equations of plate theory [5], and Chien'scomprehensive considerations of shell theory [1] made use of the concept of thinness. The object of deriving two-dimensional shell theory as the first term of anasymptotic expansion of suitable solutions of three-dimensional elasticity theory may have been stated for the first time in a paper by Johnson and the present writerwhich deals with this problem for the case of symmetric deformations of circular cylindrical shells [11]. Subsequently, relevant work in this area was done in particularby Reiss [14, 15], Green [6, 7, 8], Goldenveiser [3, 4], John [9], Rutten [21], and by the present writer [17, 18, 19], in part jointly with Lardner [13].
In one of the earlier papers [18] the point was made that in deriving two-dimensional shell theory, involving the concepts of stress resultants and stress couples, or offorces and moments, it might be of advantage to start from a three-dimensional theory in which forces and moments play an equally important role, in the sense thatforce stresses as well as moment stresses are assumed to be supported by the medium. This earlier consideration did in fact indicate the reality of the anticipatedadvantages, in various ways, but in particular in regard to the derivation of two-dimensional equilibrium equations and compatibility equations.
Having subsequently considered this approach further for the case of symmetrical deformations of shells of revolution [19] it was felt that the additional insight gainedmade it appropriate to consider once more the case of the general shell.
What follows is then another treatment of the problem of a derivation by asymptotic methods of a rational approximate system of two-dimensional constitut-
Page 322
ive equations of linear shell theory, to be associated with an exact system of two-dimensional equilibrium and compatibility equations. The method of approach is as inour Kopenhagen IUTAM presentation [18]; starting with the equations of linear force and moment stress elasticity theory in three dimensions we derive exact two-dimensional equilibrium and compatibility equations from the given three-dimensional formulation of the shell problem. Furthermore, we incorporate all otherconsequences of the three-dimensional equilibrium and compatibility equations into the three-dimensional constitutive equations, thereby transforming these into asystem of integro-differential equations. We then use these integro-differential equations as the starting point of an asymptotic expansion procedure, and we show thatthe first term of the asymptotic solution, supplemented by one consequence of a start towards the determination of the second term, does in fact supply the desiredrational system of two-dimensional constitutive equations.
The present work differs from our earlier presentation [18] of the basic ideas in two respects. One of these is expository, in as much as the present somewhatlengthier scalar derivations are believed to be more easily interpreted than the previous vectorial derivations. Another modification of our earlier work is ofconsiderable technical importance. It consists in a significant rearrangement of the system of integro-differential constitutive equations, in such a way that the feasibilityof the asymptotic expansion procedure becomes explicit beyond what it had been before.
Having these modifications of the earlier work we first apply our method to the derivation of a system of two-dimensional constitutive equations for a class oforthotropic shells, which include the constitutive equations of Koiter [12] and Sanders [22] for isotropic shells without moment stresses.
Secondly, to give some further indication of the scope of the present approach, we derive two-dimensional constitutive equations for a class of shells for which thenormals to the middle surface are not directions of elastic symmetry for the material of the shell. We find that for this class of shells the Love-Kirchhoff hypothesisceases to be valid, while at the same time the transverse shear stress resultants remain reactive quantities.
Finally, we consider shells which are sufficiently soft in transverse shear to make transverse shear deformation a first order effect. We find that the asymptoticprocedure as described remains applicable, with the constitutive relations now including expressions for transverse-shearing strains in terms of derivatives of twodimensional membrane and bending strain components. For transversely isotropic and homogeneous shells it is possible to write these expressions in terms of theassociated transverse shear stress resultants.
2Formulation of Three-Dimensional Linear Shell Theory
We assume a three-dimensional orthogonal coordinate system 1, 2, with position vector R where the coordinate curves 1, 2 are the lines of curvature on themiddle surface of the shell with position vector r, and where the -curves are the straight lines perpendicular to this middle surface.
We write
Page 323
and
With the stipulation that 1 and 2 are lines of curvature coordinates we have the Gauss-Weingarten formulas
and therewith
and where
We utilize force and moment stress vectors , and , , and force and moment load intensity vectors p* and q*, with equilibrium equations
Having these, we define force and moment strain vectors , and , in terms of translational and rotational displacement vectors and , in such a way thatthe relations between strains and displacements are consistent with the equations of equilibrium via an appropriate version of the principle of virtual work, that is, inthe form
The defining equations (8) and (9) imply a system of six vectorial compatibility equations
Of the six compatibility equations in (10) and (11) only four involve -differentiations. These four together with the two equilibrium equations (6) and (7) are a systemof six equations for the twelve vector variables *, *, e*, and k*. This system of six vector equations for twelve vector variables is complemented by a system ofconstitutive equations which we write here in the symbolic vector form
where U* is a given function of the six strain vectors e* and k*, [20].
Page 324
A further complementation of equations (6), (7) and (10) to (12) consists in stating boundary conditions of given tractions for the faces = ±c( 1, 2) of the shell. Inwhat follows we assume for simplicity shells of uniform thickness, that is, we assume c to be a constant. We then have as boundary conditions the system
where the four vector quantities and are given functions of 1 and 2.
In deriving equations of a two-dimensional shell theory from the three-dimensional system (6) to (13), we postulate as a property of the results to be obtained thatsignificant changes of the solution of these equations, either of the three-dimensional system or of the two-dimensional system, along the curves i = const. will requiredistances which are large compared with the thickness 2c of the shell. Furthermore, we accept that with this limitation on the type of solution of the given problem thequestion of boundary conditions on the lateral surfaces ( 1, 2) = 0 may be considered subsequent to the treatment of the part of the problem stated by means ofequations (6) to (13).
Definition of Pseudo-Stresses, Pseudo-Strains, and of Stress Resultants and Stress Couples
The form of the equilibrium equations (5) and (7) together with equation (8) for the linear-element coefficients Ai suggests the introduction of pseudo stress vectors and and of pseudo load intensity vectors p and q defined by1
Therewith equations (6) and (7) assume the form
In connection with the introduction of pseudo stresses and load intensities as in (14), and as proposed earlier in [17], the following may be noted. Referring to well-known definitions for stress resultant vectors Ni and stress couple vectors Mi in terms of the pseudo stress vectors and we have as expressions for Ni and Mi
in terms of the pseudo stress vectors i and i
and the well-known vector equilibrium equations of two-dimensional shell theory follow, as shown in [18], very simply from an integration with respect to of thetwo pseudo stress equilibrium equations (15) and (16), in conjunction with the face boundary conditions (13).
1Contrary to what has been done in [18] we here use the *-sign for actual quantities, instead of for the pseudo quantities, in order to simplify the writing of a larger number of mathematicalrelations involving the pseudo quantities.
Page 325
Just as equations (6) and (7) suggest the introduction of pseudo stress vectors and so the form of the compatibility equations (10) and (11) suggests theintroduction of pseudo strain vectors k and e, defined by
Therewith equations (10) and (11) may be written as
Having equilibrium and compatibility equations in terms of pseudo stresses and pseudo strains, it is appropriate to rewrite the constitutive equations (12) in terms ofthese quantities. It is readily verified that we may write
provided we define the function U appropriately, that is,
3Scalar Formulation of Three-Dimensional Linear Shell Theory
While both vectorial equilibrium equations and vectorial compatibility equations of two-dimensional linear shell theory may be derived directly from the vectorial formof the corresponding set of equations of three-dimensional theory [18], the derivation of two-dimensional constitutive equations and the determination of thedependence of stresses and strains on the thickness coordinate require the consideration of a scalar version of the equations of three-dimensional theory.
Differential equations of equilibrium and compatibility in scalar form are obtained with component representations
The premultiplication by n in (24) and (26) turns out to be convenient for the transition from three-dimensional shell space to two-dimensional shell surfaceformulation.
Introduction of (23) and (24) into equations (15) and (16) and observation of the differentiation formulas (3) leaves the following system of six scalar equilibrium
Page 326
equations
and
Equations (27) to (30) are to be considered in conjunction with a system of scalar boundary conditions which follows from (13) in the form
Introduction of (25) and (26) into equations (19) and (20) and observation of the differentiation formulas (3) leaves the following system of eighteen scalarcompatibility equations
Page 327
and
It would now be possible to use the nine equations (36) to (39) to express the nine pseudo moment strain components k in terms of the nine pseudo force straincomponents e and their first derivatives with respect to i and , and by introduction of these expressions into equations (32) to (33), obtain a system of nine second-order compatibility differential equations for the quantities eij, ei , e i and e . This system in turn could then be shown to be equivalent to the conventional system of
nine compatibility equations for the six conventional strain components , , and e . It is a significant point of the present considerations that thisstep is not taken, even for the case of a conventional medium which is unable to support moment stresses, and that, instead, a system of first-order compatibilitydifferential equations is used in conjunction with the first-order system of equilibrium differential equations (27) to (30).
4Some Transformations of Three-Dimensional Equilibrium and Compatibility Equations
Assuming for simplicity's sake the absence of volume force and moment intensities p and q and of surface traction and we deduce from the force equilibriumequations (27a, b), (28) and (31) the relations
Page 328
and we write the moment equilibrium equations (29a, b) and (30) in the alternate form
In addition to this we rewrite the twelve compatibility equations (34), (35), (38), and (39), which are the ones containing -derivatives, with the help of functions ofintegration ij, i, ij, i, and with the help of operators D(k) and D(e) in the integrated form
5Two-Dimensional Equilibrium and Compatibility Equations
We may, essentially as in [18], deduce a system of six two-dimensional equilibrium equations as a rigorous consequence of (40) to (43) and (31), as follows.
Defining scalar stress resultants Nij and Qi, in accordance with
we have from (40a, b) and (41), with ( i) =c = 0 and ( ) =c = 0, as two-dimensional force equilibrium equations
Page 329
In order to obtain the corresponding equations of moment equilibrium we deduce from equation (42a) and the boundary conditions (31) the relation
with a similar formula following from (42b).
At the same time, equation (43), with (31), gives
Defining force stress couples and moment stress resultants and Pi, in accordance with
and setting
it is evident that (53) is the appropriate equation of equilibrium of moments turning about the normals to the middle surface, in the form
In order to obtain the corresponding version of the moment equation (52a) we transform the term with 1 in it as follows
Introduction of (56a) and of the definitions (54a, b, c, d) into (52a) gives
The analogous relation following from (52b) is
While it is of course to be expected that exact two-dimensional equilibrium equations are a rational consequence of exact three-dimensional equilibrium equations, it isnot equally evident that the same should be the case for the system of two-dimensional compatibility equations which is associated with the equilibrium equations(50a, b), (51), (55) and (57a, b), when consistent strain displacement relations for two-dimensional shell theory are obtained through use of the principle of virtualwork in conjunction with (50), (51), (55) and (57). This, however, is in fact true, as previously found in [18]. For completeness sake, and for subsequent use we hererestate the argument and the results in scalar form which in [18] were developed vectorially.
Page 330
We obtain one set of three scalar two-dimensional compatibility equations upon substituting equations (44) and (45) for kij and ki into the three -derivative freethree-dimensional compatibility equations (32a, b) and (33). If this is done, the terms involving the operators D(k 1, k 2, k ) cancel out exactly and the resultingequations are
and
Next, introduction of (47a, b) and (48) into (36a, b) and (37), after appropriate cancellations, leads to a further set of three exact two-dimensional compatibilityequations,
and
Having now a complete set of two-dimensional equilibrium and compatibility shell equations it remains to establish a set of two-dimensional constitutive equations, insuch a way that this set follows as a rational consequence of a given set of three-dimensional constitutive equations.
6Constitutive Equations for a Class of Orthotropic Shells
In order to show the essence of our procedure for the derivation of two-dimensional constitutive equations we consider a system of three-dimensional constitutiveequations which somewhat generalizes the case of a conventional transversely isotropic medium.
We assume relations between force stresses and force strains of the form
Page 331
At the same time we assume that moment stresses and moment strains are related in the form
Equations (62) to (67) reduce to a conventional system of constitutive equations for a medium unable to support moment stresses upon setting = = = = 0,
whereupon the equilibrium equations (42a, b) and (43) imply the symmetry relations and and these in turn, upon introduction into (63) and (64),imply as part of the constitutive relations the symmetry conditions and .
Equations (62) to (67), when written in terms of pseudo stresses and strains become, consistent with (21) and (22),
Page 332
7Integro-Differential Constitutive Equations
Consideration of the system (68) to (75), in conjunction with equations (40) to (43) for stresses, and equations (44) to (48) for strains, indicates that it is possible towrite (68) to (75) as a system of integro-differential equations for the stresses ij, i , ij, i , together with the strains e i, e , k i, k , there being altogether eighteensuch equations for the above eighteen dependent variables. The solutions of these eighteen equations will involve the twelve functions ij, i, ij, i and these in turnare then subject to the six two-dimensional equilibrium equations (50a, b), (51), (55), (57a, b), together with the six two-dimensional compatibility equations (58a,b), (59), (60a, b) and (61).
Introducing e11 and e22 from (46) into equation (68a), the first of the eighteen integro-differential equations becomes, with the abbreviation /Ri = i,
A corresponding equation, (76b), for 22 follows from (68b).
Equations (76a) and (76b) are taken in conjunction with a relation which follows from equations (72) for , with taken from (41),
Introduction of equations (47a, b) into equations (69a, b) gives
Equation (70a), combined with (48), becomes
and equation (70b) is changed analogously into (79b).
Equation (71a) becomes, with the help of (40),
and equation (71b) is changed analogously into (80b).
Page 333
Equation (73a), together with (44), becomes
with a corresponding equation (81b) for 2j following from (73b).
Equation (74a), together with (45), becomes
with a corresponding equation (82b) for 2 .
Next, combining (75a) with (42a), we have
and a corresponding equation (83b) holds for 2 and k 2.
Finally, equation (75b) together with (43) may be written in the form
Equations (83a, b) and (84) are complemented by the boundary conditions
and of this set of six conditions only three are independent, in view of the side conditions (52a, b) and (53).
So far, all that has been accomplished is the exact reduction of the three-dimensional shell problem from a differential equation problem for thirty-six dependent stressand strain variables to a problem consisting of eighteen three-dimensional integro-differential equations for eighteen stress and strain variables, together with a systemof twelve two-dimensional differential equations for the determination of twelve two-dimensional parameter functions in the solution of the three-dimensional integro-differential equation problem. While this reduction may be considered of interest by itself, its significance within the present context is that of a starting point of arational iterative or asymptotic expansion procedure for the derivation of a two-dimensional system of constitutive equations.
8Derivation of Two-Dimensional Constitutive Equations
Our object is to use the three-dimensional system of constitutive equations in the form (76) to (85) for an asymptotically rational derivation of a system of two-dimensional relations involving the two-dimensional strain measures ij, i, ij, i in conjunction with the two-dimensional stress measures Nij, Qi, Mij, Pi, for whichquantities we already have a two-dimensional system of twelve equilibrium and compatibility differential equations.
Page 334
Insofar as this derivation is concerned, we depend at all times on the following basic order of magnitude relations
We designate L as the smallest characteristic length and we stipulate that the two-dimensional shell equations which are to be obtained must be such that its solutionsare consistent with the a priori stipulated behavior (86).
We further stipulate for the purposes of the present derivations the order of magnitude relations
and these again must be consistent with the consequences of the systems of two-dimensional equations which are to be obtained.
Additionally, we here take account of the compatibility equations (60a, b) and (61) by noting that with (86) to (88),
We now introduce the contents of equations (86) to (89) into the constitutive equations (76a, b) and (77), taking in (77) at the same time account of equations (79a,b). It is then evident that equation (76a) may be written in the form
A corresponding relation for 22 follows from (76b), and equation (77) together with (79a, b) gives
Turning to equations (78a, b) for 12 and 21 we first deduce, in analogy to equations (90a, b), that
Prior to deducing a corresponding relation for 12 21 we consider equations (81) for ij and equations (82) for i .
Equations (81a, b) become
In reducing equations (82a, b) we take account of (89). Therewith,
We now return to (78a, b) and write first
Page 335
In this we use (89) for 12 21 and, with 12 21 as in (84), equations (92), (93) and (94) for 12 + 21, ij and i . Therewith equation (95) becomes
We next write equations (79a, b), with (88a, b), in the form
and equations (80a, b) in the form
Having equations (90) to (98) we are now in a position to obtain rational asymptotic consequences of the three-dimensional constitutive equations in regard to acompletely two-dimensional formulation of shell theory. It is apparent from equations (90) to (98) that rational asymptotic consequences consist in the results whichare a consequence of the limiting process
carried out in this system of equations.
In performing this limiting process we will here assume that the coefficient ratios E/E , E/G, /E, etc. are of order of magnitude unity, at most, leaving aside for thepresent the consideration of such cases as those for which it is assumed that (E/G) = O(1) while << 1.
Letting tend to zero in (90a, b) and (91) in the above sense, we obtain
Letting tend to zero in (92), we obtain
Letting tend to zero in (93) and (94), we obtain
Letting tend to zero in (96), we obtain
Page 336
together with the boundary conditions (85) for k . From this we conclude readily the further relations
In the same manner, equations (97) and (98), together with the boundary conditions ( k i) = ±c = 0, upon letting tend to zero, lead to the relations
The contents of equations (105a, b, c) complete the information which is derived by the first-step asymptotic considerations of the system of three-dimensionalintegro-differential constitutive equations. It is evident that what has been obtained are explicit expressions for the components of stress 11, 22, 12 + 21, ij, i ,and for the components of strain e , e 1, e 2, k 1, k 2, k . What has not been obtained are expressions for the components of stress 12 21 and i which also formpart of the solution of the system of integro-differential equations. The reason for 12 21 and i not being determined at this state of the asymptotic analysis is thatthese quantities come out to be small of order relative to the stresses 11, 12, and 12 + 21. That this is so may be seen most simply from equation (84),according to which we have now, except for terms small of higher order,
and from equation (83a), according to which,
with a corresponding equation (107b) for 2 .
Determination of 12 21 and i from (106) and (107), with 11, 12 + 21, 22 and ij given by equations (100), (101), and (102), must then be considered in thenature of the beginning of a second step of the asymptotic procedure.
Insofar as the establishment of two-dimensional shell theory is concerned it turns out that use of equations (106) and (107) is almost unnecessary, as will appearbelow.
9Two-Dimensional Constitutive Equations for a Class of Orthotropic Shells
Having in equations (50a, b), (51), (55), (57a, b), and (58a, b), (59), (60a, b), (61) altogether twelve two-dimensional equilibrium and compatibility equations fortwenty-four two-dimensional stress and strain measure components it remains to obtain twelve additional relations which would be expected to be constitutiveequations.
Referring to equations (100a, b), (101), and (102a, b), and to the definitions (49a), (54a, b, c, d) for stress resultants and couples, and assuming for simplicity's sakethat all coefficients in the constitutive equations (100) to (102) are independent of , we obtain eight two-dimensional constitutive equations of the following form
Page 337
In addition to this we have from (104a) and (105a), also in the nature of two-dimensional constitutive equations, the three constraint relations
Since equations (108a) to (111c) are a set of eleven, rather than twelve as required, it is necessary to look for an additional relation between the dependent variablesof the two-dimensional theory. As indicated in the preceding section, such an additional relation must be in accordance with the three-dimensional equilibriumconditions (106) and (107a, b). We find that only one of the possible four deductions from (106) and (107a, b) implies information not already included in the two-dimensional equations obtained so far and it is in this sense that use of (106) and (107a, b) is ''almost" unnecessary.
Use of equations (107a, b) for the calculation of expressions for Q1 and Q2, does no more than reproduce the two-dimensional moment equilibrium equations (57a,b), with P1 = P2 = 0 as in (110a, b). Similarly, use of (106) for the calulation of N12 N21 reproduces the moment equilibrium equation (55), except for terms small ofhigher order.
However, use of (106) in conjunction with (100c) for the calculation of M12 M21 does produce an additional relation, namely,
With (108c), equation (112) may be written in the alternate form
and we note that this relation had been stated earlier [16] to be in the nature of a (seventh) two-dimensional shell equilibrium equation, rather than in the nature of aconstitutive equation.
Both equations (112) and (112 ) may, to the degree of approximation implied by the present two-dimensional shell theory formulation, be replaced by the simpler
2Concurrent with (111a, b, c) we have that neither the transverse shear stress resultants Qi nor the stress resultant difference N12 N21 enter into the constitutive equations, and these stressmeasures become reactive quantities in the system of equilibrium equations. The fact that the Qi are reactive quantities (subject to the assumed order of magnitude restrictions on elastic moduli) isof course well-known. It is perhaps less well-known, but has been stated clearly before in [2], that N12 N21 is a reactive quantity in the same sense as the Qi.
Page 338
relation
To do this we use (112) in conjunction with (109c) for obtaining M12 and M21 separately, and we neglect the terms of the order ( 12 + 21)/Ri in comparison with theterm 12 + 21.
Adding (112*), or (112), to the eleven equations (108a) to (111c) we have now completed the derivation of a consistent system of two-dimensional shell equations.
10Constitutive Equations for Transversely Isotropic Shells
We define a transversely isotropic shell as one for which the three-dimensional constitutive equations (62a, b) and (65) are derived from the inverse relations
that is, for which the coefficients E*, *, , are given by
Introduction of (115) into the two-dimensional constitutive equations (108a, b) and (109a, b) reduces these, with the relations
(and upon assuming = 0) to the usual constitutive equations for an isotropic medium (which is as it should be in view of the assumption E/E = O(1), which wasmade in the derivation).
We may note that with (116a, b), and with = 0, our system of relations (108a) to (111c), together with (112*), is precisely that system of relations which haspreviously been proposed by Koiter [12] and Sanders [22].
11On Constitutive Equations for Anisotropic Shells
Among the many possible generalizations of the system of constitutive equations (62a) to (67b) we consider briefly a system in which (62a, b) together with (64a, b)are replaced, with = o(1), by relations of the form
Page 339
At the same time we assume, for simplicity's sake, the absence of moment stresses , by means of stipulating that = 0 in (67a, b).
Proceeding now in analogy to what has been done in going from equations (62) and (64) to equations (90) and (97), we find that in the limit 0 the system (117a)to (118b) reduces to
Having (119) we obtain, as for the case = 0, the system (108) and (109) for the Nii and Mii, with coefficients as in (116a, b). At the same time we no longer havei, k i, and e i as in (105) and in (111), but rather,
together with .
We note in particular that with the assumed anisotropy we now do not have a condition of absent transverse shear deformation. However, the magnitudes of thetransverse shear deformation terms are not given in terms of the magnitudes of the transverse shear stress resultants. These resultants remain reactive quantities, as inthe theory with = 0.
12Two-Dimensional Constitutive Equations Including the Effect of Transverse Shear Deformation
We now consider the derivation of two-dimensional constitutive equations from the three-dimensional system (62a) to (67b), subject to the assumption
As noted before, the assumption (123) precludes the previous conclusions from equations (97) and (98) and, consequently, the two-dimensional constitutiveequations (105a) no longer hold.
In what follows we will, in obtaining two-dimensional results under the assumption (123), assume additionally and for simplicity's sake the absence of momentstresses, and the case of transverse isotropy, as per equations (113) to (116).
Observing (123) we first conclude from (79a) and (80a) that, except for terms small of higher order,
Page 340
In order to be able to use (124) for the derivation of a two-dimensional relation involving 1 we next consider equation (76a) together with (77) in the form
A corresponding equation (125b) holds for 22, and, except for terms small of higher-order, eii is given by ii + ii.
We now note, on the basis of (115), the relation
and we assume that /(1 ) = O(1).
Having (127), and as long as G/E = O(1), which we will assume to be the case, we may in (124), through use of the basic order of magnitude relations (86) and(87), approximate 11 by
Corresponding approximations hold for 12, 21 and 22.
We will not concern ourselves here with the form of k 1 and e 1 but will consider only the expression
where we have (128) for 11, and corresponding relations for 12, 21 and 22. For simplicity's sake we will limit ourselves here again to -independent values of Eand and write, consistent with (128),
Introduction of this into equation (129) gives
In view of the two-dimensional equilibrium equation (50a) the first term on the right of (131) inside the braces equals 1/2Q1/R1. The two-dimensional equilibriumequation (57a) gives for the second term the value 3Q1/4c. Accordingly, the first term is negligible and equation (131) is effectively equivalent to
with an analogous relation between 2 and Q2.
Having (132) we see that, indeed, the asymptotic approach can be used to obtain constitutive equations for shells including the effect of transverse shear deformation,and these are meaningful within the framework of the theory as long as transverse
Page 341
shear deformation is in fact a first-order effect as a consequence of the relative smallness of the shear coefficient G.
The result expressed by equation (132) is, however, less far reaching than might be thought. While 1 is indeed given by (132) for the case consideredand for moregeneral systems of constitutive equations follows from (129) as a combination of derivatives of the ij and ijthe resultants Qi may be shown not to loose their reactivecharacter as long as E/G = O(L/c). We find that with this restriction on the order of magnitude of E/G, elimination of the reactive property of Qi requires theconsideration of a transverse shear boundary layer, just as for the case E/G = O(1). The situation changes if it is assumed that E/G = O(L2/c2). However, when this isdone then the iterative procedure ceases to work, and it will then in general not be possible to reduce the three-dimensional problem to a twelfth or lower-order two-dimensional problem.
References
1. W. Z. Chien: The intrinsic theory of thin shells and plates. Quart. Appl. Math. 1 (1943) p. 29; 2 (1944) p. 43 and 120.
2. J. W. Cohen: The inadequacy of the classical stress-strain relations for the right helicoidal shell. Proc. 1st IUTAM Symposium on Shell Theory (Delft, 1959), p.415 Amsterdam 1960.
3. A. L. Goldenveiser, Construction of an approximate thin shell theory by means of asymptotic integration of the equations of the theory of elasticity. Prikl. Mat.Mekh. 27 (1963) p. 593.
4. A. L. Goldenveiser: On two-dimensional equations of the general linear theory of thin elastic shells. Problems of Hydrodynamics and Continuum Mechanics(Sedov Anniversary Volume) (1969) p. 334.
5. J. N. Goodier: On the problems of the beam and the plate in the theory of elasticity. Trans. Roy. Soc. Canada (3), 32 (1938) p. 65.
6. A. E. Green and W. Emmerson: Stresses in a pipe with a discontinuous bend. J. Mech. Phys. Solids 9 (1961) p. 91.
7. A. E. Green: On the linear theory of thin elastic shells. Proc. Roy. Soc. Ser. A, 266 (1962) p. 143.
8. A. E. Green: Boundary layer equations in the linear theory of thin elastic shells. Proc. Roy. Soc. Ser. A, 269 (1962) p. 481.
9. F. John: Estimates for the derivatives of the stresses in a thin shell and interior shell equations. Comm. Pure Appl. Math. 18 (1967) p. 235.
10. M. W. Johnson: A boundary layer theory for unsymmetric deformations of circular cylindrical elastic shells. J. Math. Phys. 42 (1963) p. 167.
11. M. W. Johnson and E. Reissner: On the foundations of the theory of thin elastic shells. J Math. Phys. 37 (1958) p. 371.
12. W. T. Koiter: A consistent first approximation in the general theory of thin elastic shells. Proc. 1st IUTAM Symposium on Shell Theory, (Delft, 1959),Amsterdam p. 12. (1960).
13. T. J. Lardner and E. Reissner: Symmetrical deformations of circular cylindrical shells of rapidly varying thickness. Symposium on the Theory of Shells (DonnellAnniversary Volume) p. 47. University of Houston, Texas, (1967).
14. E. L. Reiss: A theory for small rotationally symmetric deformations of cylindrical shells. Comm. Pure Appl. Math. 13 (1960) p. 531.
15. E. L. Reiss: On the theory of cylindrical shells. Quart. J. Mech. Appl. Math. 15 (1962) p. 325.
16. E. Reissner: On some problems in shell theory, Proc. 1st Symposium on Naval Structural Mechanics (Stanford University, 1958), p. 74, Oxford 1960.
Page 342
17. E. Reissner: On the derivation of the theory of thin elastic shells. J. Math. Phys. 42 (1963) p. 263.
18. E. Reissner: On the foundations of generalized linear shell theory, Proc. 2nd IUTAM Symposium on Shell Theory (Kopenhagen, 1967), p. 15, 1969.
19. E. Reissner: On the derivation of two-dimensional shell equations from three-dimensional elasticity theory. Studies in Applied Mathematics 49 (1970) p. 205.
20. E. Reissner and F. Y. M. Wan: A note on Günther's analysis of couple stress. Proc. IUTAM Symposium on Mechanics of Generalized Continua (Freudenstadt-Stuttgart, 1967), p. 83, 1968.
21. H. S. Rutten: Asymptotic approximation in the three-dimensional theory of thin and thick elastic shells. Proc. 2nd IUTAM Symposium on Shell Theory(Kopenhagen, 1967), p. 115, 1969.
22. J. L. Sanders: An improved first-approximation theory for thin shells. NASA Report 24, 1959.
Page 343
On Pure Bending and Stretching of Orthotropic Laminated Cylindrical Shells*
[J. Appl. Mech. 41, 168172, 1974]
Introduction
In what follows we consider the problem of pure bending and stretching of orthotropic elastic cylindrical shells as a special case of a recent general study [1], in whichno attention was payed to this specific, relatively simple case. Aside from wishing to indicate once more, in somewhat simpler form, the nature of our analysis in [1],we are motivated in the work which follows by the discovery of a significant effect which does not exist as long as the analysis is limited to shells which arehomogeneous in the wall thickness direction. Briefly, we have found that for shells which are nonhomogeneous in thickness direction, the magnitude of the overallbending and stretching stiffness factors of a closed-cross-section shell may be altered to a significant degree if the closed-cross-section shell is made into an open-cross-section shell by means of a longitudinal slit. The effect of a longitudinal slit on the torsional stiffness of a cylindrical shell is of course well known. One of ourobjects here is to show that an analogous effect exists for the problem of bending and stretching of laminated shells, which does not manifest itself as long as theproblem is considered for the class of "ordinary" shells.
Formulation of Problem
We consider a cylindrical shell with circumferential and axial coordinates s and z, with middle surface equations x = x(s), y = y(s), and with curvature function k =k(s) = where cos = x (s) and sin = y (s). We have as equilibrium and compatibility equations a system
and as constitutive equations a system which we write in the form
*With W. T. Tsai.
Page 344
In this Nz, Ns, Q are stress resultants and Mz, Ms, and P are stress couples, and z, s, , z, s, and are the corresponding strain resultants and couples. Wespeak of a "conventional case" of the shell problem when P = 0 and = 0, identically, with the 6 × 6 matrix relation (3) then reducing to a 4 × 4 relation.
We have shown earlier [1] that equations (1) and (2) can be solved explicitly, in terms of six constants of integration Nx, Ny, M, z, y, and , in the form
where
and that the six constants of integration are determined by means of six boundary conditions of the form
where
and where Mx, My, and N are overall moments and axial force acting over the cross section of the shell. Equation (6b), as it stands, expresses the conditions ofcircumferentially univalued expressions for displacements of a closed-cross-section shell. The limiting case of a longitudinally slit open-cross-section shell can beshown to be included in (6b), and to lead to the compatible conclusions of vanishing Ns, Q, and Ms.
Expressions for displacements, in terms of membrane and bending strains have been shown previously [1] to be of the form
with equations (9) once more indicating the appropriateness of the univaluedness conditions (6b).
Page 345
Form of Exact Solution
Considering the form of the boundary conditions (6a, b) and the choices of signs in (4a) and (6b), we semi-invert equations (3) in the form
where
We then introduce (4a, b) on the right of (10), thereby obtaining the relations
Introduction of (12a, b) into (6a, b) gives as a system of six equations for the six constants of integration
The open-cross-section case of the longitudinally slit shell can be considered as a special case of (13a, b), upon recognizing that across the slit the elements of B22
are infinite, implying as a consequence of (13b) the relations M = Nx = Ny = 0, and leaving directly, as a system of overall constitutive equations for the longitudinallyslit shell, the system
Page 346
We do not, in what follows, consider the numerical consequences of a direct solution in accordance with the exact procedure but instead consider a rationalapproximate solution for a class of special cases of some technical interest.
Constitutive Equations for Shell of Laminated Cross-Ply Material
We consider a shell of wall thickness h, with thickness coordinate , and with constitutive equations
where E1, E2, and E are constants.
We introduce the assumption of a linear strain distribution in the form e = + , and use this in conjunction with defining equations . In this waywe obtain as a special case of (3)
together with = 0 and P = 0. The coefficients in (16) are
We particularly note the order of magnitude relations C = O(Eh), B = O(Eh2), D = O(Eh3), = O(1), and the presence of the cross coupling terms with B, whichare due to the orthotropy of the layers, with B = 0 when E1 = E2.
We also note that for a 2n-ply plate, with (15a, b) holding in alternate layers of thickness h/2n, expression (17c) for B is changed into B = [( 1)n/8n](E2 E1)h2.
Approximate Analysis of Cross-Ply Laminated Shells
Considering the form of the constitutive equations (16) we need not consider Q and in equations (3)(6). As may be verified a posteriori, the terms with Mz and s in(6a, b) give contributions which are small of higher order* so that (6a, b) may be approximated by
It follows next from the form of (4a, b) that Ns and z contribute terms small of higher order in the constitutive equations (16), so that these may be approximated inthe form
* In making this statement we limit ourselves to the consideration of shells with cross-sectional diameters of uniform order of magnitude and we also omit consideration of the special case of a flatplate which is included in the general results of the preceding sections.
Page 347
To evaluate (18a, b) we first solve (19b) and (19d) in the form
and then introduce (20b) into (19a), to get as expression for Nz,
In this we have from (4a, b)
We now introduce (20c) with (21a, b) into (18a, b). In this way we obtain as a system of six equations for the six unknowns Nx, Ny, M, x, y, , the relations
The six equations in (22a, b) decouple into three pairs of two equations for two unknowns each, upon assuming , C, D, and B to be independent of s, and upon
choosing the coordinates x(s), y(s) in such a way that . In what follows we assume that these conditions are given. We then obtain from (22b)
and therewith from (22a)
We note that for the closed-cross-section shell, the influence of the coupling coefficient B is eliminated in the overall constitutive relations, giving a result in agreementwith the result of ''elementary" theory.
We now consider an open-cross-section shell as one having a longitudinal slit for some value s0 of s. We account for this slit by assuming that over a distance s wehave that the thickness h vanishes. Since C/(CD B2) = O(E1h3) and CB/(CD B2) = O(h1) we may conclude from the homogeneous equation (22b) that then,necessarily,
Page 348
(which is in agreement with the direct consequence of setting Ms = Ns = Q = 0 for s = s0) and therewith equation (22a), after a slight transformation, leads to theresult
A comparison of equations (24) and (26) indicates that, to the extent that there is a difference in the stretching and bending stiffnesses of the closed and of the open-cross-section shell, the open-cross-section shell is the one which is less stiff, since when B2 0 smaller values of N, Mx, My are needed to produce specified values of, x, and y than when B2 = 0.
Expressions for Stresses
In view of the assumed symmetry it will be sufficient to consider the case of an axial load N, setting Mx = My = 0 and therewith x = y = 0 and Ny = Nx = 0, andfrom (21), z = and Ms = M. At the same time we also have z = 0 and Ns = 0.
In order to evaluate (15a, b), we further use the relations
and
and therewith, from (20)
Introduction of the foregoing results into (15a, b) gives as expressions for z and s, for the closed shell
Page 349
where = 0h/C(1 2), and for the open shell
where = 0h(1 B2/CD)/C(1 2 B2/CD).
Numerical Example
We consider the case
and have then from (17),
and therewith
Equations (24) and (26) then show that for this case the stiffness of the open shell is approximately 92 percent of the stiffness of the closed shell.
Equation (31) for the stresses in the closed shell can now be written in the form
Equations (32) for the open shell become
The distribution of stress in accordance with the foregoing is indicated in Figure 1.
Ratio of Stiffness of Open and Closed-Cross-Section Shells
Equations (24) and (26) give an expression for the ratio r of the stiffness of the open-cross-section shell to that of the closed-cross-section shell which is
Page 350
Fig. 1.Stress distributions across thickness for
longitudinally slit and unslit cylindrical shell,for the case of the numerical example.
From this may be deduced values of B2/CD as a function of , with the value of the ratio r as a parameter. The results are as indicated in Figure 2 by means of thesolid lines.
Insofar as the composite structure is concerned it appears as if positive values of r, no matter how small, might be encountered. Consideration of the strain-energydensity of the shell, in accordance with the constitutive equations (16), shows that the condition for the positive-definiteness of the associated strain-energy function isof the form
and, in the event that (34) is the only restrictive condition, all curves in Fig. 2, down to r = 0 (which is given by the equation B2/CD = 1 2) are admissible.
However, a more stringent condition is encountered, upon imposing the requirement that the strain-energy functions of the individual layers, with constitutiveequations as in (15), be positive-definite. An analysis of (15a) or (15b), in conjunction with a consideration of an appropriate three-dimensional orthotropic medium,for which = 0, as indicated in the Appendix, gives as condition of positive-definiteness the relation
It remains to see what the restriction (35) means as a restriction on B2/CD, as a function of . Introduction of the limiting condition into the definingrelations
Page 351
Fig. 2.Ratio r of stiffnesses of longitudinally slit to
un-slit shell as a function of constitutive coefficientsin equation (16).
gives as limiting relation between B2/CD and the relation
As a consequence of this only those portions of the solid curves which lie beneath the curve defined by equation (37), which in Fig. 2 is drawn as a dotted line, arephysically admissible. This means in particular that the smallest possible value of the ratio r for the two-ply shell is given by 0.25 rather than by zero.
References
1. Reissner, E., and Tsai, W. T., "Pure Bending, Stretching, and Twisting of Anisotropic Cylindrical Shells," Journal of Applied Mechanics, Vol. 39, 1972, pp.148154.
2. Washizu, K., Variational Methods in Elasticity and Plasticity , Pergamon Press, 1968, p. 278.
Appendix
The following considerations are stated here insofar as the authors have not been able to locate an equivalent discussion in the literature known to them.
Given an orthotropic medium with constitutive equations
and with an associated strain-energy density,
Page 352
we have as necessary and sufficient conditions for W to be positive-definite the three relations [2]
In what follows we wish to determine the effect of (40) on the coefficients of a system of constitutive relations
which follows from (38) upon introducing the assumption of plane stress, in the form = 0.
Elimination of e from the first two equations in (38), by means of the third equation with = 0, gives as expressions for the constitutive coefficients in (41),
Introduction of Ess, Esz, and Ezz in terms of Es, E , and Es, and in terms of Es , Ez , and E transforms equations (40) into
and
Remarkably, equation (44) is equivalent, as may be shown by direct expansion of the determinant, to the simple relation
Introduction of (45) into (43) gives the further simple relations
which complete the statement of restrictions imposed on the coefficients in the plane-stress constitutive relations (41).
Page 353
Linear and Nonlinear Theory of Shells[Thin Shell Structures (Sechler Anniv. Vol.) 2944, Prentice-Hall 1974]
Introduction
Among a number of different possible approaches to the theory of elastic shells, we here select one which appears to us to be preferable to others in that it leads toresults of considerable generality, in relatively short order and with the use of relatively simple methods of analysis.
In developing shell theory we consider as basic the dynamics of stress resultants and couples, in a space-curved two-dimensional continuum, rather than thekinematics of stretching and bending of surfaces. We furthermore consider as basic the concept of work, and in particular the concept of virtual work of equilibriumsystems of forces and moments, in such a way that a "rational" formulation of a principle of virtual work is acceptable, intuitively, provided such a formulation cannotbe shown to be flawed in some specific sense by a knowledgeable student of the subject.
Having the concepts of dynamics and of virtual work we furthermore accept as reasonable the concept that from this must flow, in a convincing fashion, all that isneeded to describe the associated kinematics of the two-dimensional continuum via a system of strain resultants and couples and the associated translational androtational displacements. It is the principal task of this paper to describe, based to a considerable extent on quite recent studies, primarily by Schaefer, Günther,Reissner, and Simmonds and Danielson, how by proceeding in this fashion a consistent and rational theory of two-dimensional stress and strain, that is, of a two-dimensional theory of shells, does, in fact, emerge.
Having a two-dimensional theory of stress and strain in shells, via the concepts of stress resultants and couples and the associated measures of strain, the problem ofstress-strain relations or constitutive equations, relating stress measures and strain measures, remains to be considered. We limit ourselves in this account to statingcertain formal, but nontrivial, aspects of this problem. However, we remark that in its essence this problem may be considered in at least two distinct ways. One ofthese deals with the problem of devising suitable systems of physical experiments for elements of the two-dimensional continuum in order that a system of two-dimensional constitutive equations be established directly. The other deals with the problem of devising suitable mathematical methods to deduce constitutiveequations for the shell as a two-dimensional continuum, as exact or asymptotic, or otherwise rationally approximate consequences of a given system of constitutiveequations for the shell considered as a three-dimensional continuum.
Dynamics of Stress-Resultant and Stress-Couple Vectors
We consider a given surface in space with the equation r = r( 1, 2) where 1, 2 are curvilinear coordinates
Page 354
Fig. 1.Notations.
which, for simplicity's sake, will be considered orthogonal, with linear element . We also consider a second, associated surface with theequation = ( 1, 2). On this second surface the coordinate curves i = constant will in general not be orthogonal. We assume that the two surfaces represent thesame aggregate of particles, the first one being the undeformed state of the aggregate and the second one being its deformed state.
We now take an infinitesimal element of the surface = ( 1, 2), with sides given by the vectors ,1d 1 and ,2d 2. For this surface element to be an element of ashell we assume that its four sides are acted upon by vector forces and moments N1 2d 2, M1 2d 2, etc., in accordance with Figure 1. We also assume theexistence of vector surface forces and moments p 1 2d 1d 2 and q 1 2d 1d 2. We note that N i and Mi are forces and moments per unit of undeformed length,and that p and q are forces and moments per unit of undeformed area. Consideration of increments in going from sides i = const to sides i + d i = const, andobservation of the principle of action and reaction, then leads to two dynamic vector equations, one concerning the balance of forces and the other the balance ofmoments, of the form
Page 355
Each of these two vector equations is, of course, equivalent to three scalar component equations in some suitable manner, as will presently be considered.
In addition to the balanced equations (1) and (2) which hold in the interior of the shell we also have balance equations for elements ds of the edges of the shell. WithN and M being edge force and moment intensities we have as expressions for N and M in terms of the edge values of the stress resultants and stress couples Ni andMi
In this, n indicates the outward normal direction of the boundary curve, tangent to the shell surface, and the summation convention involving repeated subscripts isbeing invoked.
Virtual Work and Virtual Strain Displacement Relations
We define virtual displacements and virtual strains as sets of (infinitesimally small) kinematically possible arbitrary displacements and strains which, upon beingproperly associated with a set of forces and moments, give rise to a quantity of virtual work in such a way that the dynamic equations (1)(3) are equivalent to anequation of the form
In this, dS = 1 2d 1d 2, and are virtual translational and rotational displacement vectors, and i and i are virtual strain resultant and strain-couplevectors.
The way in which Eq. (4) is usually considered is to assume that i and i are known in terms of , and suitable derivatives thereof, whereuponby means ofintegration by parts in order to eliminate derivatives of and and by then considering and arbitrary in the interior as well as along theboundarysatisfaction of Eq. (1)(3) appears as a necessary and sufficient condition for the validity of Eq. (4).
In the present circumstances we make use of Eq. (4) in an inverse fashion as follows. We consider Eq. (1)(3) as given and utilize them to eliminate p, q, N, and M inEq. (4), with Ni and Mi now being arbitrary functions.
Introduction of Eqs. (1)(3) into Eq. (4) gives first
We now integrate by parts to eliminate derivatives of N i and Mi in Eq. (5) and at the same time write ( ,i × Ni)· = ( ,i × )·Ni. In this way Eq. (5) assumes themuch shortened form
Page 356
With N i and Mi being arbitrary vector functions in Eq. (6) we deduce from this equation the virtual strain displacement relations
We note that since by definition there exists a function = ( 1, 2) we may write ( ),i = ( ,i) in Eq. (7). However, as long as we have not actually establishedthe existence of a function corresponding to the quantity in Eq. (7) we are not in a position to replace ( ),i by ( ,i).
We show in what follows that this difficulty is easily overcome as soon as the foregoing relations of nonlinear theory are reduced to the corresponding relations oflinear theory. We shall also show how to overcome this difficulty, in a much less obvious way, for the given nonlinear relations (7).
Strain Displacement Relations and Compatibility Equations of Linear Shell Theory
We obtain equilibrium and strain displacement relations of linear theory by the simple device of replacing in Eq. (2) the unknown radius vector to points on thedeformed shell surface by the given vector r to points on the undeformed surface.
Changing ,i into r,i in Eq. (2) changes ,i into r,i in Eq. (7), and therewith i and i are given as linear combinations of ( ),i = (r + u),i = u,i, andnow( ),i
= ,i. With this we can go directly from virtual strain displacement relations to actual strain displacement relations, of the form
We note, before considering component representations of i, i, u, and , that the vectorial strain displacement relations (8) evidently imply two vectorialcompatibility equations which read
and
We further note that Eq. (9) and (10), which are due to Günther (1961), are in a well-defined manner, with which we shall not concern ourselves here, the static-geometric duals of the (homogeneous) equilibrium equations (1) and (2). From this it then follows further that, in analogy to the strain displacement relations (8), wehave a system of relations expressing stress resultants and stress couples in terms of stress functions, by means of which the (homogeneous) equilibrium equations(1) and (2) are identically satisified. In crediting Günther with Eq. (8)(10) and with the associated stress-stress function relations we remark that Güntheracknowledges that equivalent scalar-form results for the special case of the circular cylindrical shell had earlier been established by Schaefer (1960).
Component Representations
We list in what follows the system of scalar strain displacement relations which follows from the virtual relations (8) upon writing
Page 357
and
where r,i = iti and n = ti1 × t2, with the Gauss-Weingarten differentiation formulas
In this S1 and S2 are in-plane radii of curvature given by 1/S1 = 1,2/ 1 2 and 1/S2 = 2,1/ 1 2, and the Rij = Rij are the usual out-of-plane curvature radii of thesurface.
Introduction of Eq. (11)(13) into Eq. (8) gives as expressions for in-plane strain resultants ij, transverse (shearing) strain resultants i, transverse (bending andtwisting) strain couples ij, and in-plane strain couples i,
We note that all six strain-couple components are given in terms of components of rotational displacement only, while four of the six strain-resultant components aregiven in terms of both translational and rotational displacement components.
Constitutive Equations of Linear Theory
For the purpose of establishing constitutive equations it is evidently necessary to have in conjunction with the component representations (11) for strain resultants andcouples corresponding representations for stress resultants and couples. We write
Equations (18) and (11) in conjunction with Eq. (4), indicate that it is appropriate to stipulate that constitutive equations should be a system of 12 relations involvingNij, Qi, Mij, Pi, ij, i, ij, and i in the form Nij = Nij( 11, 12, . . ., 2), etc.
Page 358
It is remarkable that Günther's work (1961) as well as Schaefer's work (1960) does not allude to this point. Instead, the general considerations of dynamics andkinematics of the two-dimensional problem are complemented by a consideration of constitutive relations for the three-dimensional problem, with attention restrictedto conventional three-dimensional mediums, unable to support moment stresses, and with negligible transverse shear deformability, thereby precluding the possibilityof obtaining constitutive equations which involve the quantities Pi, Qi, i, and i in addition to the other measures of stress and strain. The appropriate statement ofsufficient generality appears to have been made subsequently only, in a report by Reissner (1966).
Among possible general forms of constitutive equations we mention specifically the case where a strain energy function A exists, depending on all 12 measures ofstrain 11, 12, . . ., 2, such that
We also mention the partially inverted case where in terms of a function B = B(N11, . . ., Q2; 11, . . ., 2),
Evidently, ''conventional theories" in which i = 0 as well as Pi = 0, throughout, will be given whenever the function B in Eq. (20) does not involve the four argumentsQi, and i. Additional restrictions of practical significance are given when N12 and N21 enter into B in the form N12 + N21 only and/or when 12 and 21 enter in theform 12 + 21 only, implying constitutive relations 12 = 21 and/or M12 = M21, respectively.
We have considered elsewhere the way in which two of the best-known and most rational systems of conventional constitutive equations, known as the Flügge-LurieByrne relations and the Koiter-Sanders relations, respectively, may be written in terms of the strain measures ij and ij as defined by Eqs. (14) and (16)(Reissner and Wan, 1966, 1967). As an example of less conventional possibilities which may arise within the present context, we mention the case of a sandwich-type plate, constructed in such a way that in addition to transverse shear deformation being significant, by way of constitutive-relation portions i = CQi, we have thatM12 M21, with constitutive-relation portions (M12, M21) = D1( 12 + 21) ± D2( 12 21) (Reissner, 1972b).
We conclude this discussion of constitutive relations for two-dimensional (linear) shell theory with two general observations.
The first is to recommend, as a potentially fruitful and as yet essentially unexplored area of study, the problem of designing programs for the experimentaldetermination of constitutive relations, without explicit reference to the three-dimensional nature of the structure.
The other is the further consideration of systematic analytical procedures for the derivation of the forms of appropriate two-dimensional constitutive equations, givena knowledge of the form of these relations for the structure considered as a
Page 359
three-dimensional continuum. Our own approach here is through the use of formal asymptotic expansion procedures (Reissner, 1971), which in various differingways have also been used by Goodier, Friedrichs, Johnson, Reiss, Goldenveiser, Green, Rutten, and Novotny, to mention the names of some of the investigators inthis area.
Solution of the Virtual Strain Displacement Relations of Nonlinear Theory
We may consider the central difficulty of extending the linear two-dimensional theory of shellsas expressed by means of Eqs. (8)(18), together with the appropriatescalar versions of the linearized equilibrium equations (1) and (2)to consist in going from the nonlinear vectorial virtual strain displacement relations (7) to consistentnonlinear, scalar actual strain displacement relations which generalize the linear relations (14)(17).
We have ourselves obtained an explicit solution of this problem for the special case of rotationally symmetric bending of shells of revolution (Reissner, 1969, 1972a).Going beyond this, Simmonds and Danielson have considered the general case of the problem (Simmonds and Danielson, 1970, 1972).* In doing this they have,through use of Rodrigues' vector representation of finite rigid-body rotation, obtained one possible generalization of Eqs. (16) and (17) for strain-couple components.Insofar as expressions for strain-resultant components are concerned, Simmonds and Danielson limit themselves to the statement that "these can easily be expressedin terms of the components of [the finite rotation vector] and the [translational] displacement vector" (Simmonds and Danielson, 1972).
In attempting to deal with the nonlinear virtual strain displacement relations (7) it is reasonable to expect that it will be necessary to use component representationsinstead of dealing directly with vectorial relations, as in going from the linearized version of Eq. (7) to the vectorial results in Eq. (8). However, inasmuch as now finitedeformations are anticipated in association with the change from the radius vector r to the radius vector it is no longer reasonable to use the triad ti, n in expressingstress vectors and strain vectors in terms of scalar components. We could use, instead of ti, and n, the non-orthogonal vectors ,i/| ,i| and the unit vector perpendicular to the plane of ,1 and ,2. Instead, we follow the suggestion of Simmonds and Danielson (1970, 1972) to use an independent, and for the time beingotherwise undefined, mutually perpendicular triad Tj, N in the representation of both stress and strain resultants and couples, as follows:
A crucial step in the procedure now is to assume that the triad Ti, N may be chosen in such a way that the difference between it and the deformed tangent and normalvectors ,i and ,1 × ,2 comes out to be a measure of the state of strain ij, i, in the form
*The difference between Simmonds and Danielson's approach and our approach is indicated by the titles of their publications as well as by the absence of a reference to Günther's work.
Page 360
with ij being the Kronecker delta. Equations (23) evidently imply the formulas
upon writing ,i = r,i + u,i = iti + u,i. Returning to Eq. (23) we next deduce the relation
Introduction of Eqs. (25) and (23) and of the first of equations (22) into the first of equations (7) gives
We now note that the terms with ij and i on the left and on the right of Eq. (26) cancel each other. This leaves a system of relations which is valid no matterwhat the values of ij and i, of the form
We next show that the system (27) may be used to express in terms of Tj, N, Tj and N. For this purpose we consider the vector products T1 × T1, T2 ×T2, and N × N, together with the canonical decomposition formula for expressions a × (b × c). In this way we obtain T1 × T1 = (T1· )T1. etc., and then,
by addition of the three vector equations,
Having Eq. (18) and the second of equations (22), we now have as an equation for the determination of the strain couples ij and i
To proceed further we evidently will need expressions for Tj,i and N,i, in terms of Tj and N, analogous to Eqs. (13) for the derivatives of tj and n. We write, inanalogy to Eq. (13),
*Equation (28) is analogous to a formula for the "angular velocity vector of the rigidly rotating triad (Ti, N)" in Simmonds and Danielson (1972). Its validity is indeed "easy to show," as aconsequence of Eq. (27). It is less easy to see in Simmonds and Danielson, where Eq. (28) is equivalent to an Eq. (2), while Eq. (27) is equivalent to an Eq. (25) which appears later on.
Page 361
From Eq. (30) follows
Introduction of Eqs. (31) and (30) into the righthand side of Eq. (29) is readily seen to result in a cancellation of all the terms with Tj and N. What remains can bewritten in the convenient form
Having Eq. (32) we conclude that, evidently, ij = (1/rij) and i = (1/si).
In going from virtual strains to actual strains we must observe that the actual strain couples ij and i should vanish in the event that the triad Ti, N remains coincidentwith the triad ti, n. Comparing Eq. (30) with Eq. (13) we find that this will be the case if we use the coefficients in the differentiation formulas (13) as functions ofintegration, as follows:
We now have in Eqs. (24) and (33) expressions for six strain-resultant and six strain-couple components, in terms of the displacement vector u and in terms ofparameters which describe the rotation of the triad ti, n into the triad Ti, N, that is, in terms of three translational and three rotational displacement parameters. Wemight choose as these parameters the components of u in the directions of ti and n, together with the components of a Rodrigues' finite rotation vector. However, weprefer to make no choice of such a nature at this point but instead proceed to the statement of an intrinsic form of the theory, consisting of a complete system of scalarequilibrium and compatibility equations for strain resultant and couple components and stress resultant and couple components.
Compatibility Equations for Finite Strain
Familiarity with the results of linear theory suggests that there should be altogether six scalar compatibility equations for strain resultants and couples.
It is possible to obtain three such compatilibity equations, in accordance with Simmonds and Danielson (1972), by using the two equations in Eq. (23) in the form( ,1),2 = ( ,2),1, that is, by writing
and by making use of the differentiation formulas (30) for Ti and N. In this way there follows, upon making use of the defining equations (33) and of the Gauss-Codazzi
Page 362
equations involving the i, Si and Rij,
The linear terms in Eqs. (35)(37) are identical with the contents of known scalar component decompositions of the compatibility equations (10). We also note thatsuitably specialized forms of Eqs. (35)(37) reduce, upon introduction of a restriction to small and , to previously given equations of a proposed nonlinear shallowshell theory (Reissner, 1969b).
Having obtained three compatibility equations expressing the fact that ,12 = ,21 we now obtain another set of three such conditions, on the basis of thedifferentiation formulas (30). Equations (30) imply altogether three vectorial compatibility relations, of the form Ti,12 = Ti,21 and N,12 = N,21. Of the associated ninescalar relations there are only three which are not identically satisfied. These three can be written in the following form:
Again, the linear terms in Eqs. (38)(40) are identical with the contents of the known scalar version of Eq. (9). Equations (38)(40) reduce to previously stated shallowshell theory equations for the case that the i are considered small compared with the ij (Reissner, 1969b).
We add the observation that the six first-order partial differential compatibility equations (35)(40) reduce to a "conventional" system of one second-order, two first-order, and one zeroth-order equation, upon setting 1 = 2 = 0 and upon using Eqs. (35) and (36) as equations of definition for 1 and 2. With this, Eq. (40)becomes a second-order equation, Eq. (38) and (39) remain first-order equations, and Eq. (37) becomes a zeroth-order equation, all in the eight variables ij and
ij.
Equilibrium Equations for Finite Strain
We complement the six scalar compatibility equations (35)(40) by six scalar equilibrium equations which follow upon introduc-
Page 363
tion of the representations (21), together with the differentiation formulas (30) and the defining equations (33), into the two vector equations (1) and (2).
From Eq. (1) follows
From Eq. (2) follows
A reduction to a more conventional system of equations again becomes possible upon assuming Pi = 0 and i = 0 and upon using Eqs. (44) and (45) as definingequations for Q1 and Q2 in conjunction with Eqs. (41)(43).
A general observation which may be made at this juncture is this. Both equilibrium equations and compatibility equations display the nonlinearizing effects of possiblelarge deformations and strains in the same relatively mild fashion, by an occurrence of second-degree dependent-variable terms only. It would seem that this fact willbe of significance in limiting the validity of direct applications of constitutive equations which have been established in connection with linear theory in the treatment ofproblems requiring the use of nonlinear equilibrium and compatibility equations.
Page 364
Equations for Rotationally Symmetric Deformations of Shells of Revolution
We now compare the form of Eq. (35)(46) for the special case of rotationally symmetric deformations of shells of revolution with previously given results of the lineartheory (Reissner and Wan, 1969) and also with previously given results of the nonlinear theory for the problem of symmetric bending (Reissner, 1969a, 1972a).Setting
and
we obtain from Eqs. (35)(40) as compatibility equations
At the same time the equilibrium equations (41)(46) assume the form
Page 365
Equations (49)(60) are in complete agreement with the contents of Eqs. (3)(6) in Reissner and Wan (1969) upon omission of all nonlinear terms. At the same time thesix nonlinear equations (49), (51), (54), (56), (57), and (59) become identical with a set of six such equations in Reissner (1969a)* upon setting N = N = Q =M = M = P = 0, = = = = = = 0, and
We note in particular that while the derivation of these equations in Reissner (1969a) was for the case of rotationally symmetric displacements, the present derivationincludes the case of nonrotationally symmetric displacements, as long as these are compatible with the assumption of rotationally symmetric stress and strain.
References
Günther, W., 1961. Analoge Systeme von Schalengleichungen. Ingenieur-Archiv 30: 160186.
Reissner, E., 1966. On the foundations of the theory of elastic shells. Proc. 11th Intern. Congre. Appl. Mech. , Munich, 1964, pp. 2030.
Reissner, E., 1969a. On finite symmetrical deformations of thin shells of revolution. J. Appl. Mech. 36: 267270.
Reissner, E., 1969b. On the equations of non-linear shallow shell theory. Studies Appl. Math. 48: 171175.
Reissner, E., 1971. On consistent first approximations in the general theory of thin elastic shells. Ingenieur-Archiv 40: 402419.
Reissner, E., 1972a. On finite symmetrical strain in thin shells of revolution. J. Appl. Mech. 39: 11371138.
Reissner, E., 1972b. On sandwich-type plates with cores capable of supporting moment stresses. Acta Mechanica 14: 4351.
Reissner, E., Wan, F. Y. M., 1966. A note on stress strain relations of the linear theory of shells. J. Appl. Math. Phys. (ZAMP), 17: 676681.
Reissner, E., Wan, F. Y. M., 1967. On stress strain relations and strain displacement relations of the linear theory of shells. The Folke-Odqvist Volume , pp.487500.
Reissner, E., Wan, F. Y. M., 1969. Rotationally symmetric stress and strain in shells of revolution. Studies Appl. Math. 48: 117.
Schaefer, H., 1960. Die Analogie zwischen den Verschiebungen und den Spannungsfunktionen in der Biegetheorie der Kreiszylinderschale. Ingenieur-Archiv 29:125133.
Simmonds, J. G., Danielson, D. A., 1970. Nonlinear shell theory with finite rotation vector. Proc. Koninkl. Ned. Akad. Wetenschap. B, 73: 460478.
Simmonds, J. G., Danielson, D. A., 1972. Nonlinear shell theory with finite rotation and stress-function vectors. J. Appl. Mech. 39: 10851090.
*Correcting a misprinted sign in Eq. (10a ), the correct sign being as in (10a).
Page 366
On Small Bending and Stretching of Sandwich-Type Shells[Int. J. Solids Structures, 13, 12931300, 1977]
Introduction
Recent work on the subject of direct formulations of two-dimensional theories for three-dimensional problems, with emphasis on the concept of a Cosserat surface[1], suggested a reconsideration of the author's earlier work on sandwich-type shells [2].
In what follows we show that our earlier derivation of a two-dimensional sandwich-type shell theory from a suitably idealized three-dimensional formulation containsin a natural way what appears to be the essence of the difference between Cosserat-type elastic surface theory and ordinary two-dimensional shell theory, to wit anincorporation of the effect of transverse normal stress deformation into the equations of the two-dimensional theory, over and above the incorporation of the effect oftransverse shear stress deformation.
The developments which follow are in the main equivalent to our earlier developments. However, in reviewing our earlier work the possibility of certain improvementsand simplifications became apparent. It is these improvements and simplifications, in addition to recognition of the fact that sandwich-type shell theory as formulatedby us does in fact contain the essence of Cosserat elastic surface theory, which led to the writing of the present paper.
To indicate the nature of the following considerations it may be worthwhile to quote (with some slight modifications in wording) from the Introduction to our earlierwork [2], as follows. "In this report an extension of the classical theory of small bending and stretching of thin elastic shells is considered. Instead of a homogeneousshell we consider a shell constructed in three layers: A core layer of thickness 2c with elastic constants Ec, Gc, c and two face layers of thickness t with elasticconstants Et, Gt, t. In the developments certain restrictive assumptions are made, which somewhat limit the applicability of the results. In so doing formulas areobtained which are as compact as possible while still describing the essential characteristics of the sandwich-type shell." (Our reconsideration shows that the earlierformulas, while being "compact", were not in fact "as compact as possible".)
"The thickness ratio t/c is assumed small compared to unity; at the same time the ratio tEt/cEc is assumed large compared to unity. This latter assumption means thatthe face material is so much stiffer than the core material that the contribution of the core layer to stress couples and tangential stress resultants is negligible. It isknown that for flat plates this assumption necessitates the taking account of the effect of transverse shear deformation. The same would be expected to be true forcurved shells."
Page 367
"A further effect. . . . is the effect of transverse normal stress deformation. We show that this effect arises in a manner which is typical for shells and has nocounterpart in plate theory. It may be likened, roughly, to what happens in the bending of curved tubes."
Statics of Sandwich-Type Shell
In order to derive a complete system of equations we first consider the statics of the face layers and of the core layer. Combination of the results obtained for thesetwo components will lead to those equations of equilibrium which hold for elements of a shell regardless of the constructional nature of the elements, and in addition torelations which are associated with the sandwich-type nature of the elements.
Coordinate System on Shell
In formulating differential equations we use a curvilinear coordinate system 1, 2, , such that 1 and 2 are lines-of-curvature coordinates on the middle surface ofthe composite shell, and the normal distance from this
Fig. 1.
Page 368
middle surface (Figure 1). The linear element in this system of coordinates is of the form
Statics of Face Layers
The face layers are treated as shells of thickness t, with negligible bending stiffness about their own middle surfaces. Because of this they are designated in whatfollows as face membranes.
The middle surfaces of the face membranes are given with reference to the three-dimensional system of coordinates by = ±(c + t/2) ±c, with the linear elementson these two middle surfaces given by
The components of external load intensity on the upper and lower face membranes are designated by piu, qu and pil, ql, respectively. The core layer stresses whichact upon the two membranes are designated by i u, u, i l, l, and the stress resultants acting over the cross sections of the membranes by Niju = Njiu and Nijl =Njil (Figure 2).
Fig. 2.
There are then three equations of force equilibrium for the elements of each of the two membranes. Writing nu = n(1 + c/Rn), anl = n(1 c/Rn), we have twotangential component equations for forces in the direction 1,
two analogous equations for forces in the direction 2, and two equations for forces in the normal direction,
Page 369
Statics of Core Layer
Assuming that the components of stress 1, 2, 12 are negligible so that only the transverse stresses , and i need to be retained we have three differentialequations of equilibrium of the form
Integration of these gives
with m and i m being the values of and i for = 0.
Statics of Composite Shell
In view of the fact that all face-parallel core-layer stresses are neglected we have as expressions for stress couples Mij and middle surface parallel stress resultantsNij,
etc., with evident differences between N12 and N21 as well as between M12 and M21. In the same way we have as expressions for components of external force andmoment load intensity
Furthermore, a load term of the following form is encountered,
with this term representing the average transverse normal stress at any station of the shell, in the event that the loads qu and ql alone were responsible for this stress.
The above is complemented by expressions for transverse shear stress resultants Qi. We find, with the help of eqn (5),
Page 370
with a corresponding expression for Q2, and we note that eqn (9) in conjunction with the first relation in (5) makes it possible to express the shear stress values i u
and i l in terms of Qi.
Differential equations of equilibrium for the composite shell are now obtained by suitable combination of the above results.
Addition of the two relations in (2) gives as one equation of force equilibrium
Subtraction of the same two relations gives as one equation of moment equilibrium
Two analogous equations follow upon interchange of subscripts.
Two additional equations are obtained by adding and subtracting, respectively, the two normal component equilibrium relations in (3).
Addition of the equations in (3), and observation of eqns (5)(7), gives the conventional equation of transverse force equilibrium,
A further equation, which is required for the sandwich-type shell, is obtained by subtracting the second relation in (3) from the first relation. We find,making use of eqns (5) and (9),
We note that this sixth equilibrium equation for the elements of the composite shell has no relation to the conventional sixth equilibrium equation for shellelements which expresses the condition of moment equilibrium about the normals to the middle surface.*
We obtain the middle surface normal moment equilibrium equation, now as a seventh equilibrium equation, together with what amounts to an eighth equation ofequilibrium, by using the exact expressions for N12, N21, M12, M21, which correspond to (6), in conjunction with the fact that N12u = N21u and N12l = N21l. Theresultant relations are
We note that while it is often assumed that the second of these relations is effectively equivalent to a ''constitutive" relation M12 = M21, there is no need and noobvious advantage, in making such an assumption in this place. Beyond this we can say that, while the first relation in (14) is entirely a statement of two-dimensionalstatics, the
*The corresponding relation in [2] is there written without the Qi-terms which appear above. The reason for this is the "provisional" definition of m in eqn (33) in [2] which is not consistent withthe significance of m in eqn (40). However, the considerations which follow show that the Qi-terms in (13) and (15) may in fact be considered negligible.
Page 371
second may be thought of as a consequence of a mixture of constitutive and equilibrium considerations, inasmuch as the form of this relation does depend oninformation on the three-dimensional nature of the state of stress, as previously discussed for the problem of the homogeneous shell [3].
Stress Strain Relations for Composite Shell
We derive stress strain relations through the use of the theorem of minimum complementary energy, as first employed by Trefftz for homogeneous shells withoutconsideration of the effect of transverse stresses [4].
Designating the complementary energy of face layers and core layers by t and c, respectively, we obtain stress strain relations through the device of extremizingt + c, with the constraint differential equations of equilibrium for the composite shell incorporated into the variational equation through the device of Lagrange
multipliers which then may be identified with the appropriate displacement components, as follows.
In this, the ui, w, i and are readily identified as effective components of translational and rotational displacements. As regards the multiplier k it was noted in [2]that "there is no immediate simple geometrical interpretation, [although] such an interpretation in terms of an average transverse normal strain might be deduced."Similarly, there is no immediate simple geometrical interpretation for the multiplier .
It remains to express the complementary energy contributions t and c in terms of stress resultants and couples and in terms of the transverse normal stressmeasure m.
Assuming the face membranes to be isotropic we have an expression for t,
Page 372
Equation (16) is transformed into an expression involving stress resultants and couples for the composite shell by writing, on the basis of eqn (6),
In what follows we restrict attention to cases for which c/R << 1.* Therewith, and with the definitions,
for stiffness coefficients, we have
Next, with the face-layer-parallel core stresses 1, 2, 12 assumed negligible, we have as expression for c,
In this we now take the stresses in accordance with eqn (5), and we again neglect terms of the order /R in comparison with unity. Therewith and with the help of thedefining relation (9), we now have
In our earlier work [2] eqn (21) had been transformed by elimination of the Qi-derivative terms through use of the transverse force equilibrium eqn (12), therebyintroducing a term N11/R1 + N22/R2 q into c. We now undertake a significant simplification of the results to be obtained by showing that it is, in effect, rational toneglect the Qi-derivative terms in the above in comparison with the Qi-terms themselves, as long as Ec is of the same order of magnitude as Gc, and as long as it isassumed that significant changes of stress resultants and couples of the composite shell require distances of an order L which are large compared to 2c, that is, aslong as it is required that the solutions to be obtained are such as to justify the use of a two-dimensional theory. For a proof of the correctness of our statement weneed only observe that with Qi,j/ j = O(Qi/L) the Qi-derivative terms in (21) are in fact small of relative order (c/L)2 in comparison with the Qi-terms, and so mayrationally be neglected in the expression for c.
Remarkably, it is possible to justify neglect of the Qi-derivative terms in the last term of the variational eqn (15) in a manner which is in every way consistent with
*Except for evaluation of the term in (16). Remarkably, this term reduces to the form 2N12N21 + 2M12M21/c2, without any restriction onthe magnitude of c/R. Our statement here represents a correction of the corresponding result in [2].
Page 373
the above. To see this we take account of the moment eqn (11) in order to establish that Qi = O(M/L). From this it follows that the Qi-derivative terms in the last termin (15) are in fact small of relative order (c/L)2 in comparison with the Mii-terms, consistent with the formulation in [2].
We refrain from rewriting eqn (15) with the appropriate expressions for t and c, and with the two aforementioned simplifications involving neglect of Qi-derivativeterms, and proceed to state the stress strain relations for the composite shell which follow as a consequence of this variational equation
In these we have
While 11, 22, 11, 22 are the usual expressions for midsurface normal strains and bending strains, the quantities 12, 21, 12, 21 have no such direct geometricalsignificance, whereas 12 + 21 and 12 + 21 are midsurface shearing strain and twisting strain, respectively.
We make two observations in regard to the form of the stress strain relations (22). The first of these is as follows.
We may use the two equilibrium relations (14) in order to eliminate and from the expressions for N12, N21, M12, M21. When this is done we obtain, except forterms of relative order c2/R2,
Page 374
and therewith
with corresponding expressions for N21 and M21.*
Our second observation concerns the appearance of the terms with k in the two relations involving M11 and M22, and the relation between k and m in eqns (22).
We may, in order to eliminate the explicit appearance of the Cosserat-concept from the above, proceed as follows. We combine the last equation in (22) with thesimplified version of the Cosserat-type equilibrium eqn (13), in order to obtain the relation k = (M11/R1 + M22/R2)/Ec. We introduce this result into the equationsinvolving M11 and M22 in (22) and have therewith as "conventional" stress strain relations
It is evident, as first noted in [2], that there will be significant effects of the transverse normal stress deformability of the core layer on the bending stiffness of thecomposite shell whenever Ec is small enough to result in the order of magnitude relation (tc/R2) (Et/Ec) = O(1).
References
1. P. M. Naghdi, Direct formulation of some two-dimensional problems of mechanics. Proc. 7th U.S. National Congr. Appl. Mech. , 321 (1974).
2. E. Reissner, Small bending and stretching of sandwich-type shells. NACA TN 1832 (1949). (Also NACA Report 975, 126, 1950).
3. E. Reissner. On some problems in shell theory. Proc. 3rd Symp. Naval Struct. Mech. 8794. Pergamon Press, Oxford (1960).
4. E. Trefftz, Ableitung der Schalenbiegungsgleichungen mit dem Castigliano's chem Prinzip. Z. f. ang. Math. & Mech. 15, 101108 (1935).
*We note that in the event that the second relation in (14) is replaced by the statement that M12 = M21 we shall then have
and the terms with in the expressions for N12 and N21 will be absent. From this it follows that (1 + ) (N12 + N21) = C( 12 + 21). and furthermore, with (1 + ) (N12 N21) = C( 12 21 + 2 ) =(1 + ) (M21/R2 M12/R1) we may eliminate from the expressions for M12 = M21 in such a way that, except for terms of relative order c2/R2,
Page 375
On the Transverse Twisting of Shallow Spherical Ring Caps[J. Appl. Mech. 97, 101105, 1980]
Introduction
The original aim of this paper was to formulate a nonrotationally symmetric stress-concentration problem for thin shells which could be solved in closed form, and toobtain the solution of this problem. It appeared in the course of the analysis that this stress-concentration problem was also a particularly fitting example for theapplication of an asymptotic solution method for unsymmetric shell problems, involving the concepts of interior and edge zone solution contributions and of theconcept of contracted boundary conditions for the separate determination of these contributions, which had been proposed sometime earlier [4].
The problem is as follows. We consider an isotropic shallow spherical shell with the edges defined by two pairs of mutually perpendicular planes perpendicular to abase plane, with the corners of the rectangle in the base plane which is determined by the two pairs of mutually perpendicular planes coinciding with the corners of theshell boundary curve. Given this configuration, we assume that the edges of the shell are free of stress, except for the action of equal and opposite concentratedcorner forces, as indicated in Figure 1. Our object is the state of stress in the shell, without or with a small concentric circular hole at the apex.
Fig. 1.
Page 376
It is evident that a limiting case of the foregoing problem is the corresponding problem of a flat plate, with the solution of the problem without the circular hole being aspecial case of the problem of Saint-Venant torsion of narrow rectangular cross section beams, and with the solution of the circular-hole problem being included insolutions by Goodier for a class of transverse plate flexure problems [2].
In the present analysis the plate flexure problem appears upon assuming the value of a certain parameter to be zero. At the same time the asymptotic analysiscorresponding to the procedure described in [4] is appropriate for values of which are large compared to unity. In the interim region of finite values of it isnecessary to obtain appropriate solutions of the equations of shell theory, which in this instance may be taken from shallow-shell theory.
Regarding the physical aspects of the problem we find, as expected, a dominance of bending stresses over membrane stresses in the interior of the shell region. Onthe other hand, we also find that for sufficiently large values of we have membrane stresses in an edge zone which are of the same order of magnitude as thebending stresses in this zone, in such a way that the value of the stress-concentration factor for this problem of transverse bending involves both bending andmembrane stresses in a significant manner.
Equations for Isotropic Homogeneous Shallow Spherical Shells
We consider a shallow spherical shell with middle surface equation z = H r2/2R, where R is the radius of the shell, H the distance of the apex from the base plane ofthe shell, and r and are polar coordinates in the base plane. We assume that the shell is free of distributed surface forces and have then that tangentional stressresultants N, stress couples M, and transverse stress resultants Q are expressed as follows in terms of a stress function K and a transverse displacement function w,[3].
Use of appropriate equations of equilibrium and compatibility in conjunction with the foregoing and in conjunction with stress-strain relations of the form rr =B(Nrr N ), etc., leads to differential equations for K and w of the form
where 2 = ( ),rr + r1( ),r + r2( ), .
It is readily verified that the solution of the system (4) may be expressed in terms of three functions , , and in the form [4],
provided that
where 4 = 1/R2BD.
Page 377
We note for what follows as expressions for resultants and couples in terms of , , and
and we also note the designations of and as inextensional bending and membrane (interior) solution contributions, respectively, and the designation of asedge zone solution contribution, with the physical significance of the latter designation depending on an appropriate relation between the length-parameter 1/ andan appropriate linear dimension of the shell.
The Boundary-Value Problem
We start out with the observation that the classical solution w = Pxy/2(1 )D for Saint Venant twisting of a flat rectangular plate as produced by an arrangement ofconcentrated corner forces P, in conjunction with an assumption of no in-plane stress, that is, in conjunction with the stipulation K = 0, also satisfies the differentialequations (4) for shallow spherical shells. Furthermore, this solution of (4) satisfies the same corner force conditions for a spherical cap with otherwise free edges, inthe event that the projection of these edges onto the base plane of the shell happens to be rectangular.
Having the aforementioned simple solution for transverse twisting of a spherical cap, we ask for the way in which this solution is modified by the presence of a circularhole of radius a, concentric with the apex of the shell, given that a is small compared to the overall dimension of the cap. Evidently, the boundary conditions for theedge of this hole are of the form
As regards the boundary conditions along the outer edges of the cap we make the stipulation that for large r we will have a homogeneous state of stress withCartesian couple and resultant components Mxy = P/2, Mxx = Myy = 0, Qx = Qy = Nxx = Nyy = Nxy = 0. This is transformed, in an elementary manner, into four
Page 378
conditions of the form
Closed-Form Solution
The form of the boundary conditions (14) and (15), in conjunction with the form of the differential equations (4) indicates that suitable expressions for w and K will beproduct solutions (r) sin 2 . Considering that w and K must be as in (5) and (6), and deleting at the outset terms not compatible with the prescribed boundarycondition at infinity, we have then that w and K will be of the form
with four arbitrary constants cn, and with the Kelvin functions ker and kei2 subject to the two ordinary second-order differential equations
In deriving expressions for stress resultants and couples from (16) and (17), it will be convenient to introduce the abbreviations
Therewith, and with (18a, b), we obtain from equations (1) and (3)
Introduction of (20) to (23) into the boundary conditions (14) then leads to the following set of four simultaneous equations for the determination of the four
Page 379
constants of integration cn,
where now ki ki( ), etc.
Upon suitable transformations, this system of equations can be written in a somewhat simpler form. To begin with, equations (24) and (25) are readily shown to beequivalent to the set1
Having (24 ) and (25 ), we may use (26) and (27) so as to obtain in place of these two equations the set
Before evaluating the system (24 ) to (27 ), it is useful to establish the analytical form of the quantities which are of principal physical interest. These quantities are theedge values of the couple M and of the resultant N . We obtain a particularly convenient form of these expressions by making use of equations (1) and (2), inconjunction with two of the boundary conditions in (14), so as to have
An introduction of (16) and (17) into (28) gives, with the help of (18a, b).
Having (29a, b) we see, with the help of (24 ) and (27 ), the possibility of the further relations
1 Corresponding to the fact that the conditions Nrr = Nr = 0 for r = a can be shown to be equivalent to conditions K = K,r = 0.
Page 380
and it remains only to determine the coefficients c1 and c2 from equations (24 ) to (27 ). We do this by first expressing c3 and c4 in terms of c2, from (24 ) and (25 ),in the form
and by then using (26 ) and (27 ) in order to obtain the relations
It is possible to simplify the form of (32a, b) somewhat by making use of certain identities involving Kelvin functions of various orders. In this way we obtain,2 uponintroducing (32a, b) into (30a, b), as expressions for the significant edge moment and the significant edge resultant, in terms of zeroth-order Kelvin functions,
where
and
where
Stress-Concentration Factors for Bending Stresses and Membrane Stresses
We define a bending stress-concentration factor kb as the ratio M (a, /4)/M0 where M0 = M ( , /4) = P/2. Therewith kb is directly given by the right-hand sidein (33a).
In order to obtain the corresponding membrane stress-concentration factor km, it is necessary to be more specific about the nature of the two-dimensionally isotropicshell medium. We shall assume in what follows that the shell is homogeneous in thickness direction and have then the relation
2 See equations (9.9.14) to (9.9.17) in [1].
Page 381
We write further
and therewith obtain from (34a)
Stress-Concentration Factors for Small and for Large Values of
Given that = a = , the limiting case of a flat plate corresponds to the assumption = 0. We find, from equations (33b) and (34b),that 1(0) = 1/4 and 2(0) = 0 and therewith from (33a) and (35c),
with this result coinciding, as it should, with Goodier's result for plates, without consideration of transverse shear deformation [2].
For the case of large , corresponding to a shell problem with distinct interior and edge zone solution contributions use may be made of appropriate asymptoticformulas. We find, by making use of certain known cross-product expansion formulas3 that
and therewith,
Inasmuch as bending and membrane stresses superimpose the relevant stress-concentration factor for the most highly stressed face of the shell comes out to be, forsufficiently large values of ,
It may be noted that the numerical values of k for = 0 and for = are not greatly different, but that while for = 0 the stress concentration is due entirely tobending, a significant fraction of it is, for 1 << , due to membrane rather than due to bending action. Numerical values for 1, 2, kb, and km, as a function of and
, may be found in Table 1.
Interior Solution Stresses for Large
The form of the expressions (16) and (17) for w and K indicates that for large values of the effect of the terms with c3 and c4 is significant in a narrow edge zoneonly and that outside this zone the remaining
3 Equations (9.10.32) to (9.10.34) in [1].
Page 382
Table 1kb km
1 2 = 0 = 1/3 = 1/2 = 0 = 1/3 = 1/20 0 250 0 1 333 1 600 1 714 0 0 00.1 0.0070.3 0.0360.5 0.0690.8 0.1161 0.1432 0.2413 0.2994 0.3365 0.362
0 1.000 1.000 1.333 1.500 0.577 0.544 0.500
expression for w is as if bending occurred without stretching and the remaining expression for K is as if the state of stress of the shell was a pure membrane state.
We obtain information on the state of stress outside the narrow edge zone, and in particular on the relative significance of bending and membrane stresses, bydetermining the values of M and N in accordance with (16) and (17) and the defining relations (1) and (2), by setting c3 = c4 = 0 in (16) and (17) and by thenderiving the relations
We evaluate (40a) by taking c1 from equation (26), with c3 and c4 as in (31) and (32a). Therewith we obtain, except for terms small of higher order
A corresponding evaluation of (40b) leads to the relations
A comparison of (41a, b) with (38a, b) shows that the order of magnitude of the bending stress in the interior is the same as the order of magnitude of this stress inthe edge zone, in such a way that the dimensionless edge zone value 1 + decreases to a value 1 in the interior. At the same time the interior membrane stresscomes out to be small of relative order 1/ 2 so that, effectively, the interior state of the shell is a state of inextensional bending.
Direct Asymptotic Solution for Interior and Edge Zone States
We proceed as in [4] to solve the given boundary-value problem, for values of which are sufficiently large compared to unity, through use of equations (5)(13).Introduction of (7) and (13) into the two sets of boundary conditions (14) and (15) then leaves as conditions for
Page 383
the determination of the two harmonic functions and and of the ''plate on an elastic foundation" function , for r = a,
with equations (42), (43), and (45) also holding for r = , and with the right-hand side of (44) being replaced by (P/2D) sin 2 for r = .
We now note that when 1 << we have the order-of-magnitude relations,
etc. We use these for an asymptotic solution of the problem, by retaining in (44) and (45) the highest and second highest order-of-magnitude terms in , ( 2 ),r and2 , only, that is, we replace equations (44) and (45) by the abbreviated equations
An introduction of this into (42) and (43) then leaves as two conditions for the determination of the two harmonic functions and ,4
Having determined and , we subsequently determine the associated approximation for the edge zone function with the help of equations (48),5 and we use theresults obtained in this way in order to obtain from equations (8) and (12) as approximate expressions for the relevant edge values of circumferential stress resultantand stress couple
for r = a.
4Note that upon writing equations (7)(13) in the form , etc., so as to distinguish between interior and edge zone solution contributions, equations (49) and (50) are equivalent to the
previously derived contracted boundary conditions for the determination of the interior state [4], of the form .
5We note that these equations may be written, equivalently, as and , for r = a.
Page 384
In order to carry out the remaining simple calculations we write, consistent with (16) and (17), in order to assure satisfaction of all conditions at infinity
and we further write
and
We now introduce (52) and (53) into the boundary conditions (49) and (50) and obtain as two equations for the determination of c1 and c2
Equations (56) imply, consistent with (32), that
Having c2 and c1 as in (57), we obtain C3 and C4 from (47) and (48) in the form
and therewith, from (51),
The above expressions for the edge values of N and M may be compared with the interior values of these same two quantities, and, which follow from (52), (53), and (57), consistent with the contents of equation (41).
References
1 Abramowitz, M., and Stegun, J. A., Handbook of Mathematical Tables, Dover Publications, Inc., New York, 1965.
2 Goodier, J. N., "The Influence of Circular and Elliptical Holes on the Transverse Flexure of Elastic Plates," Philosophical Magazine, Series 7, 22, 1936, pp.6980.
3 Reissner, E., "Stresses and Small Displacements of Shallow Spherical Shells-I," Journal of Mathematics and Physics, 25, 1946, pp. 8085.
4 Reissner, E., "A Note on Membrane and Bending Stresses in Spherical Shells," Journal of the Society of Industrial and Applied Mathematics, 4, 1956, pp.230240.
Page 385
On the Effect of a Small Circular Hole on States of Uniform Membrane Shear in Spherical Shells[J. Appl. Mech. 47, 430431, 1980]
Introduction
In extension of a recent analysis of the problem of stress concentration due to a small circular hole in a transversely twisted shallow spherical shell [1], we consider inwhat follows the problem of stress concentration in the same shell which, without the hole, would be in a state of uniform membrane shear. The method of analysis issimilar to the method described in [1]. However, the nature of the results which are obtained in what follows is significantly different from the nature of the results forthe problem of transverse twisting. While for the former problem the effect of the parameter a2/Rhwhere a is the radius of the hole, R the radius of the shell, and h thewall thickness of the shelldid not greatly affect the numerical value of the stress-concentration factor, it is found that for the present problem not just the numericalvalue but the order of magnitude of the stress-concentration factor depends on the value of the parameter a2/Rh.
Differential Equations and Boundary Conditions
With polar coordinates r, in the base plane of the shallow spherical shell with middle surface equation z = H r2/2R, and with the radius of the small circular holebeing at r = a, we have that stress resultants and stress couples may be expressed in terms of a transverse deflection function w and an Airy stress function K, wherew and K are given in the form
In this and are harmonic functions and satisfies the differential equation 4 + 4 = 0, where 4 = 1/R2BD. For a shell which is uniform in thickness direction,so that D = Eh3/12(1 2) and B = 1/Eh, the defining relation for takes on the form 4 = 12(1 2)/R2h2.
For the determination of the functions , , and use is made of the boundary conditions of the problem. The condition that at an "infinite" distance from the hole thestate of stress is a state of uniform membrane shear of magnitude S, means that as r approaches infinity, w must tend to zero, and K must approach Sxy = 1/2Sr2 sin2 . The stipulation of a traction-free edge of the hole is expressed by means of the four conditions
Page 386
In order to satisfy these conditions, expressions are needed for Nrr, Nr , Mrr, Qr, and Mr in terms of , , and . These are [1]
For the determination of stress-concentration factors, the edge values of the resultant N and the couple M are needed. In view of the form of the boundaryconditions, it is convenient to take these, for an exact solution, in the form
For an asymptotic solution, for large values of the dimensionless parameter a, it is preferable to write N and M in the form
Closed-Form Solution
The boundary conditions at infinity indicate that w and K should be products of the form (r) sin 2 . Considering the representation (1) and deleting at the outsetterms which are inconsistent with the existence of a uniform membrane stress state at infinity, K and w come out to be
In this the cn are four arbitrary constants and ker2 and kei2 are Kelvin functions.
Having equations (6), it is possible to obtain as edge values of N and M , on the basis of equations (4) and (2) and with a,
In view of the relative complexity of determining the constants cn, we limit ourselves here to the observation that the results obtained in this way are consistent
Page 387
Fig. 1.
with the flat-plate results
and that in the range 1 << , where the form of the solution (6) indicates the distinction between the "interior" solution contribution, involving the constants c1 and c2
and an "edge zone" solution contribution involving the constants c3 and c4, there is consistency with the more simply derived asymptotic results which follow.
Asymptotic Solution for Large Values of
The solution (1) is now written in the form
where we = , Ke = RD 2 and, consistent with (6),
In applying the boundary conditions (2) with the help of (3), an asymptotic solution of the problem is obtained by using the fact that when 1 << then /a = o( ,r)etc. This allows writing the conditions of vanishing edge values of Mrr and Qr + r1Mr , in the asymptotic form
for r = a. A substitution of (11) into the conditions of vanishing Nrr and Nr for r = a then leads to the relations
for r = a. The introduction of (10) into (12) gives
Page 388
Having determined and it remains to determine the asymptotic form of the edge zone solution contribution , with the help of (11). For this purpose we write asasymptotic expression for , consistent with (6),
and as asymptotic expression for 2
A use of (15) and of the corresponding formula for ( 2 ),r in (11) gives
Having determined c1, c2, C3, and C4, we now obtain the relevant values of N and M , on the basis of (5). Outside the edge zone these values are
Inside the edge zone we are limiting ourselves here to giving the corresponding values at the edge,
Equations (18a, b) imply as the asymptotic values of membrane and bending stress-concentration factors
The foregoing results are noteworthy for the following reasons:
1 While the state of stress in the shell is, in the absence of a hole at the apex, a pure membrane state, the existence of the hole changes the "interior" state into a statewhich is predominantly a state of inextensional bending, in spite of the fact that the loading conditions for the shell involve tangential stress resultants only.
2 The state of stress in the edge zone adjacent to the hole involves membrane and bending stresses of the same order of magnitude. More importantly, as the value of
increases, a more and more pronounced stress concentration occurs, with the stress-concentration factors km and kb becoming largecompared to the value "4" of the factor km for the limiting case = 0 of a flat plate.
Reference
1. Reissner, E., "On the Transverse Twisting of Shallow Spherical Caps," Journal of Applied Mechanics, 47, 1980, pp. 101105.
Page 389
A Note on the Linear Theory of Shallow Shear-Deformable Shells*
[J. Appl. Math. Phys. 33, 425427, 1982]
Introduction
The starting point of what follows in an earlier consideration of the equations of linear shallow shell theory including the effects of transverse shear deformability and ofmoment components normal to the shell surface [2]. Given the relatively greater technical importance of the first-named effect, and given certain recent advances inthe formulation of the problem of shear-deformable plates [3], we now undertake a reconsideration of the appropriate special case of the results in [2], in such a waythat the final form of the equations of this tenth-order theory shell theory contains both the eighth order classical results, and the sixth order theory of shear-deformable plates as special cases. While we limit consideration to the problem of shells which are two-dimensionally homogeneous and isotropic it will be apparentthat analogous results may be obtained without these assumptions of homogeneity and isotropy.
Basic Equations
We have as equilibrium equations of linear shallow shell theory
and we take stress strain relations in the form
with corresponding expressions for 22, 2 and M22. Equations (1) to (5) are complemented by strain displacement relations,
with corresponding expressions for 22, 2, 21, 22.
In the above z = z(x1, x2) represents the middle surface of the shell, the ui and w are tangential and transverse displacement components, the i are rotationaldisplacements, the Nij and Qi are tangential and transverse stress resultants and the Mij are stress couples.
*With F. Y. M. Wan.
Page 390
We note that while the transverse component w and the tangential components Nij are effectively equivalent to base plane perpendicular and parallel components,respectively, there is no such equivalence for the transverse components Qi and the tangential components ui.
Reduction of Differential Equations
We satisfy (1) in terms of an Airy stress function by setting
and we note that (6) and (7) imply a compatibility equation of the form 11,22 ( 12 + 21),12 + 22,11 + Lw = 0 where
Therewith, and with (4), we obtain as one of three differential equations governing the problem of the shallow shear-deformable shell, the same as in the theorywithout transverse shear deformability,
In order to obtain the remaining two differential equations we begin by combining (5) and (7) in the form
with a corresponding expression for M22, and we use (11), in conjunction with (3), so as to have
Equations (12) evidently imply the relation (1 AD 2)(Q1,1 + Q2,2) = D 2 2w and this, in conjunction with (2), (8), and (9), gives as the second of three differentialequations
We may note, as before [2], that equations (10) and (13) in conjunction with (12) have earlier been given by Naghdi [1], and that it remains to reduce the twelfthorder system (10), (12), and (13) to one of tenth order. We here accomplish this reduction, essentially as in [2] and in [4 (p. 52)], by the introduction of a function with defining relation
Having (14) we obtain from (12) as differential equation for , and as the third differential equation of the system of three,
where it remains to express the quantities Qi and Mij in terms of w, K and . We accomplish this by deducing from (2), in conjunction with (8) and (9),
Page 391
Q1,11 = Q2,21 (LK + q),1 and therewith 2Q1 = (Q1,2 Q2,1),2 (LK + q),1, with a corresponding expression for 2Q2. This, in conjunction with the defining relation(14) then gives, with the further defining relation
in place of equations (12),
and in place of equations (11)
where we can, with the help of (13), replace 2v by 2w + A(LK + q) and in this way avoid the appearance of fifth derivatives of w in the expressions for Qi.
We recover the previously known results for shear-deformable plates upon setting L 0, and for shells with absent transverse shear deformability upon setting A0 and therewith 0.
References
[1] P. M. Naghdi, Note on the equations of shallow elastic shells. Quart. Appl. Math. 14, 331333 (1956).
[2] E. Reissner and F. Y. M. Wan, On the equations of linear shallow shell theory. Studies Appl. Math. 48, 133145 (1969).
[3] E. Reissner, On the theory of transverse bending of elastic plates. Intern. J. Solids Structures 12, 545554 (1976).
[4] S. Lukasiewiecz, Local loads in plates and shells. PWN-Polish Scientific Publishers (1979).
Page 392
A Note on Two-Dimensional Finite-Deformation Theories of Shells[Int. J. Non-Linear Mech. 17, 217221, 1982]
Introduction
In what follows we are concerned with a simplification and extension of two-dimensional shell theory as previously considered in [1].
Our simplification has to do with the avoidance of an a priori assumed implicit relationship between components of force strain and tangent vectors to the deformedshell surface. Our extension has to do with a description of the state of moment strain in a way which involves a symmetric treatment of two rotational displacementmeasures for the description of rotations about the two mutually perpendicular surface tangent vectors, without a participation in this of the one rotationaldisplacement measure which describes rotation about the normal vector to the shell surface.
Vectorial Equilibrium Equations and Virtual Strain Displacement Relations
We consider an undeformed surface in space with equation r = r( 1, 2), where 1 and 2 are orthogonal coordinates, with linear element ,and we write = ( 1, 2) for the equation of the corresponding deformed surface. We write N i and Mi for stress resultant and stress couple vectors associatedwith the deformed shell surface and p and q for surface force and moment load intensity vectors, in accordance with Figure 1. We then have as equations of forceand moment equilibrium
As in [1] we obtain virtual strain displacement relations for force strain vectors i and moment strain vectors i in terms of virtual translational and rotationaldisplacement components and through use of the virtual work equation
in the form
where it is permissible to write ( ,i) in place of ( ),i in (4a), with no equivalent exchange in (4b) inasmuch as we do not necessarily have the existence of a function in association with the in (4).
Page 393
Fig. 1.
Derivation of Scalar Strain Displacement Relations
Given the radius vectors r and to the undeformed and the deformed surfaces, we introduce, in association, two triads of mutually perpendicular unit vectors (t1, t2,n) and (T1, T2, N). In this t1 and t2 are tangent vectors to the coordinate curves on the undeformed surface and n the corresponding normal vector, but nocorresponding stipulation is made in relation to the triad (T1, T2, N), so that the determination of and of T1, T2 and N is part of the problem of the shell.
In order to derive scalar strain displacement relations from the vectorial virtual relations (4) we now take ( i and i in the form
and we write ,i in the form
with the choice of aij and bi remaining to be made.
In addition to the defining relations (5) and (6) we further define virtual triad vectors Ti and N in terms of the virtual rotational displacement vector in the form
Page 394
The introduction of (5a), (6) and (7) into (4a) leaves, after some cancellations,
and therewith aij = ij and bi = i.
Considering the fact that when = r and therewith ,i = iti we will have ij = 0 and j = 0 we conclude that the derived relations involving the virtual straincomponents ij and i are equivalent to relations involving actual strain components ij and i of the form
consistent with the a priori expression for ,i in [1] and in [2].
In order to derive corresponding relations involving scalar moment strain components from equation (4b) we make use of the fact that equation (7) implies asexpression for
and we write as expressions for the derivatives of Ti and N
Introduction of (10), (11) and (5b) into (4b), with ( Ti),j = (Ti,j), etc. leaves, after some cancellations
and therewith ij = (1/rij) and i = (1/si).
In order to derive from these relations involving virtual moment strain components relations involving actual moment strain components, we use in addition to thedifferentiation formulas (11) the Gauss-Weingarten formulas
Inasmuch as ij = 0 and i = 0 for coincident triads (Ti, N) and (ti, n) we have then on the basis of (11) and (13) as relations involving the moment strain componentsij and i,
with the possibility of deriving from (9) and (14), in conjunction with (6) and (11), a system of six scalar compatibility equations, as in [1], without the need for anexplicit consideration of translational and rotational displacement measures.
Page 395
In order to derive expressions for moment strains involving three scalar rotational displacement measures we here proceed as follows.
We first define two mutually perpendicular vectors , depending on two parameters i, in the form
where = 1 2/2 and .
We then define N through the relation which gives
and we finally define Ti with the use of a third scalar rotational measure in the form
where B2 = 1 + 2.
Having equations (16) and (17) we may use equations (14) and (15) in order to derive expressions for 1/rij and 1/si, and therewith in accordance with (14) for ij
and ij, in terms of 1, 2 and . We will, in this note, limit ourselves to a derivation of such results for the case of small finite deformations, up to and including firstand second degree terms in i and and their derivatives.
With a restriction to second and no higher-degree terms equations (16) and (17) assume the following form:
Having (16 ) we have further,
and then, with (17a ), up to and including second degree terms,
so that
Page 396
Expressions for the remaining ij and i come out to be in an analogous manner,
Expressions for force strains ij, i may be deduced, in terms of translational and rotational displacement components, by writing ,i = r,i + u,i with, for example, u =uiti + wn, with this leading to expressions of the form
where and represent the corresponding expressions of linearized theory, in accordance with equations (14) and (15) in [1]. We note that as far as the solutionof specific boundary value problems is concerned it may be of greater utility to determine, rather, the displacement vector u, in terms of force strain and rotationaldisplacement components, by suitably integrating the two relations u,i = i[( ij + ij)Tj + iN ti].
References
1. E. Reissner, Linear and non-linear theory of shells. Thin Shell Structures (Sechler Anniversary Volume) pp. 2944. Prentice Hall, New York (1974).
2. J. G. Simmonds and D. A. Danielson, Non-linear shell theory with finite rotation and stress-function vectors. J. Appl. Mech. 39, 10851090 (1972).
Page 397
Some Problems of Shearing and Twisting of Shallow Spherical Shells[Proc. Intern. Conf. Comp. Mech., pp. I.3I.12 Springer Verlag, 1986]
Summary and Introduction
The present account has four separate purposes. The first of these is a concise rederivation of some results for two problems of stress concentration in shallowspherical shells due to the effect of a circular hole, or rigid insert. Aside from their numerical significance these results have a special analytical significance for shelltheory, for the following reason. They involve the asymptotic analysis of shell problems for which the interior solution contribution is not either of the membrane typeor of the inextensional bending type, but is such that a far-field membrane solution goes into a near-field inextensional bending solution, or vice versa, with a transitionregion in which membrane stresses and bending stresses are of the same order of magnitude. Of particular interest is that this kind of interior solution complexity isassociated with stress concentrations which are an order of magnitude higher than analogous concentrations which are encountered for related problems with interiorsolution contributions which are of the membrane type, or of the inextensional bending type, throughout.
The results as described above were obtained in 1980 [69, 11], in extension of earlier considerations in 1956 [3], 1959 [4] and 1967 [12, 13]. We here propose tohigh-light the indicated aspects, independent of other aspects of the more comprehensive earlier considerations.
The second purpose of the present account is a complementation of the linear-theory formulation in [69] by a statement, in dimensionless form, of the associatedfinite-deflection problems. On the basis of this formulation we describe a perturbation analysis for the effect of small finite deflections on the values of the factors ofstress concentration, including interaction effects which are due to nonlinearity.
Our third purpose is to use the nondimensional nonlinear shallow spherical shell equations for the solution of a problem of buckling due to membrane shear, to theextent of deriving formulas for the critical stress and for the wavelength of the associated wrinkles, in accordance with the classical elementary theory of buckling,(without here considering the associated problem of initial post buckling behavior and imperfection sensitivity, in analogy to the corresponding analysis for the problemof uniform surface pressure, as considered by John Hutchinson, and by the writer [5]).
We conclude by a formulation of the bifurcation problem associated with the effect of a system of equal and opposite transverse forces applied to the corners of asquare-based shell with edges which are otherwise traction free. We discuss, in particular, results for the associated special-case problem of a square plate, asconsidered by Lee and Hsu with the use of a finite-difference method within the
Page 398
framework of the present formulation [1], and by Ramsey within the framework of an elastic Cosserat surface theory [2], with widely differing numerical results.
Linear-Theory Equations for Isotropic Homogeneous Shallow Spherical Shells
Assuming the absence of distributed surface loads the differential equations for an Airy stress function K and for the transverse deflection w are
In this R is the radius of the shell, and 2 = ( ),rr + r1( ),r + r2( ), , D = Eh3/l2(1 2), C = 1/Eh, and the associated expressions for stress resultants and couples interms of K and w are the same as those for the limiting case 1/R = 0 of a circular plate.
The solution of (1) can be written, effectively as in [3], in the form
where
with and being inextensional bending and membrane solution contributions, and with being an edge zone solution contribution, for shell problems in the range ofparameter values which is associated with the possibility of such behavior.
Elementary Solutions for Transverse Twisting and Membrane Shearing
These elementary solutions are
The associated expressions for stress couples and resultants are
Boundary Conditions for the Problems of a Circular Hole or Rigid Insert
We have as conditions at the edge of a circular hole of radius r = a,
where Rr = Qr + Mr , /r, and at the edge of a rigid insert of radius r = a,
Conditions at the outer boundary r = are, for the problem of membrane shear,
and for the problem of transverse twisting
Page 399
The nonhomogeneous portions of (8) to (11) indicate that all solutions of (1) which are here needed will be of the form
From the four problems in accordance with (8) to (11) we here consider two only, these being the effect of a hole on the otherwise uniform state of membrane shear,and the effect of a rigid insert on the otherwise uniform state of transverse twisting. In connection with the latter problem we have as expressions for the tangentialdisplacement components ur and u , consistent with formulas derived in [4],
Boundary Conditions Written in Terms of , and
With the introduction of a dimensionless coordinate = r/a, and in terms of a parameter
we have from (8) and (2) as conditions at the boundary = 1 of a hole
The corresponding conditions at the boundary = 1 of the rigid insert follow from (9), (13) and (14) in the form
We omit an analogous rewriting of the conditions for = , and shall instead use (10) and (11) directly.
Asymptotic Boundary Conditions
It follows from the form of the differential equations in (3) that as long as we stipulate that differentiation with respect to leaves orders of magnitude unchanged, thesame is the case for differentiation with respect to , insofar as the solution contributions and are concerned. In contrast to this, the relation 4 + 4 = 0, with
, = O( ), implies that , = O( ) and 2 = O( 2 ).
Page 400
Accordingly we have that when 1 << equations (15c, d) are asymptotically equivalent to
The introduction of (18) into (16a, b) leaves as a system of two conditions for and
when = 1. It is evident from (18) and (19a, b) that when 1 << the simultaneous determination of , and in accordance with (16ad) becomes in effect asequential determination, with the interior solution contributions and being obtained first and the edge zone solution contribution following subsequently, interms of the results for , with the second relation in (18) now being, effectively, of the form ( 2 ), = 0.
The corresponding reduction of the rigid-insert conditions (17ad) proceeds even more simply, as follows. It is concluded from (17c, d) that, necessarily, = O( ),and therewith 2 = O( 2 ) and ( 2 ), = O( 3 ) in (16a, b). This means that, except for terms of relative order 1/ , (17a, b) is equivalent to the set of relations
for = 1, with again being determined subsequently, on the basis of the two conditions
for = 1.
Effect of a Circular Hole for the Problem of Membrane Shear
Given the boundary conditions for = in (10) and (11), with T = 0, we have as appropriate choices of , and in accordance with (3) and (12)
and, with ,
The introduction of (22) in (19a, b) gives
The introduction of 2, with c2 as in (24), into the reduced form of (18) gives
Page 401
With this and with (2) we have as expressions for K2 and w2
With N = a2K, we obtain from (26) for the ''interior" values of that stress resultant which is responsible for the membrane stress concentration at the edge of thehole
The corresponding quantity at the edge of the hole is, when 1 << ,
Equations (29) and (28) imply a membrane stress concentration factor km = 2/2, in agreement with the result in [7], and consistent with the numerical results for thefull range of -values in [11].
The corresponding circumferential stress couple values follow from (27) in the form
With b = 6M/h2 and with 0 = S/h we then have from (31) a bending stress concentration factor , again in agreement with the results in[7] and [11].
With equations (28) to (31) it is now possible to make the following qualitative statements.
(i) Stresses in an edge zone 1 << < e, where e = 1 + O(1/ ), are such that the membrane stress m decreases rapidly from its edge value 0 2/2 to 0. At the
same time the bending stress b decreases slightly, from to .
(ii) The interior domain e < is subdivided into three sub-regions, a "near field" region, a "far field" region and a transition region, as follows. When e < <<1/2 then m << b, and we have inextensional bending. When 1/2 << then b << m and we have membrane behavior. When = O( 1/2) then m and b
are of the same order of magnitude.
It is evident that the far field membrane behavior of the shell is a consequence of the boundary conditions for = . It is not evident, but may be concluded from theanalysis, that the condition of no edge tractions for = 1 "forces" an adjacent portion of the interior domain into a state of inextensional bending. Apparently, this"conflict" between far field and near field behavior is responsible for a much enhanced stress state in the edge zone, with membrane and bending stress concentrationfactors which are both O( 2), in contrast to the flat plate results km = 4 and kb = 0.
Effect of a Rigid Insert for the Problem of Transverse Twisting
The boundary conditions (10) and (11), with S = 0, and with , and as in (2) and (3) are
Page 402
satisfied upon stipulating
The introduction of (32a, b)( into (20a, b) gives
The introduction of (32a), (33) and (34) into (21) gives
With this, and with (3), we now have
With Mr = Da2[(1 ) , + , ], except for terms of relative order 1/ , we have from (36)
With Nr = a2( 1K, + 2K, ) we have from (37), except for terms of relative order 1/ ,
Equation (38) implies a bending stress concentration factor kb = 2/2(1 ).* Equation (40) gives, with 0 = 3T/h2, a membrane stress concentration factor
, consistent with the results in [8] and [9].
With (38) to (40) we now make the following qualitative statements, analogous to the statements made for the problem of the effect of a hole on the state ofmembrane shear.
(i) In the edge zone 1 < e, where e = 1 + O(1/ ), b decreases from 0 2/2(1 ) for = 1 to 0 for = e. The value of m does notchange.
(ii) The interior domain is again subdivided into a near field e < << 1/2, a far field 1/2 << , and a transition region = O( 1/2). In the near field we now haveb << m, and in the far field m << b, in contrast to the reverse relations for the problem of the effect of a hole on the state of membrane shear.
It is again evident that the far field behavior is a consequence of the boundary conditions for = . It is now concluded from the analysis, again with no clear physicaladvance insight, that the condition of no edge displacements when = 1 forces the adjacent portion of the interior domain into a state of membrane stress, with thisconflict between near field and far field behavior being responsible for an
*Consistent with the result in [9], which corrects an earlier formula with 1 + in place of 1 in [8].
Page 403
enhanced stress state in the edge zone, compared with the stress state for the case of a flat plate, for which Kb = 4/(1 ) and km = 0.
Finite Deflection Effects
The effect of small finite deflections is introduced into equation (1) by replacing the zeros on the right hand side by quantities L(w, K) and L(w, w)/2, respectively,where L(w, K) = [w,xxK,yy + w,yyK,xx 2w,xyK,xy]R = [(r1w,r + r2w, )K,rr + (r1K,r + r2K, )w,rr 2(r1w,r r2w, )(r1 K,r r2 K, )] × R in the sense of von Karman andMarguerre.
A dimensionless form of the ensuing system of nonlinear equations, which is appropriate in the present context, follows upon introducing independent variables =/a, = y/a, = r/a, and dependent values and g in the form
With as in (15) and with the additional parameters
we then have as a dimensionless nonlinear generalization of (1)
with the possibility of perturbation expansions
It is evident that the homogeneous differential equations for the zeroth order terms in the expansions (46), with the nonhomogeneous boundary conditions (10) and(11), represent the linear uncoupled version of the results associated with either one of the two loading conditions which are here considered. The subsequentdetermination of first and higher order terms depends on the solution of systems of nonhomogeneous differential equations, with homogeneous boundary conditions,and with a coupling of the effects of the loads S and T.
Given the formal expansions in (46) for the solutions of (44) and (45) there remains the question of the range of -values, in their dependence on the value of , forwhich such expansions are practical. A reasonable expectation is that the procedure will be feasible, when = O(1), as long as t = O(1) and s = O(1). A widerrange of admissable -values may be surmised when 1 << , on the basis of the following argument. We may expect, as for the linear problem, interior and edgezone solution contributions. For the interior solution contributions we will have, effectively, expansions in powers of / 2 in place of the expansions in powers of .For the edge zone solution contribution, with a radial edge zone of width a/ , we will have that the left hand sides in (44) and (45) are O( 4) while, in view of thepolar coordinate form of L( , g), the right hand sides will be O( 2 3). Accordingly,
Page 404
as long as = O(1), the right hand sides will be of order 1/ relative to the left hand sides, and asymptotically negligible in first approximation, with the possibility ofmeaningful expansions for second and higher order approximations, in powers of / .
Given the above qualitative discussion of expansion procedures in the context of the indicated three-dimensional parameter space, we here briefly mention the effectsof a fourth parameter which becomes part of the problem upon incorporation of the effect of transverse shear deformability into the equations for shallow shells. Ananalysis of the linear-theory problem of the shallow spherical shell with a circular hole, for the two loading conditions considered in the foregoing, may be found in[10].
Shear Wrinkling of Spherical Shells
We consider a problem of buckling due to membrane shear in spherical shells, as follows. The stipulation gs = + k, s = , s = and t = gt = 0, withsubsequent linearization of (44) and (45) in terms of k and , reduces these equations to the form
Inasmuch as we anticipate that we will be concerned, primarily, with wrinkles caused by the principal compressive stress we stipulate at the outset a buckling mode
, . The introduction of this into (47) leads to the relation ( 2 ) 2 + 4 4 = 0 for the determination of a critical valuec and a critical wave length parameter c. Setting / 2 = 0 we find and c = 2 2. With as in the second relation in (43) and as in (15), and with c =, we then obtain as expressions for a wrinkling half wave length ac and for the associated stress c
It remains to complement these classical linear-theory results by considerations of initial postbuckling behavior and imperfection sensitivity, in analogy to the earlieranalysis of the mathematically related problem of the effect of radial pressure [5].
Finite Transverse Twisting Due to Corner Forces
We consider a shell with edges defined by = ± 1, and = ± 1 and with the corners (1, 1), (1, 1), (1, 1), (1, 1) acted upon by an equilibrium system of forces ± T,with the solution of the linear version of the problem as in (4).
Given equations (44) and (45) we have that a dimensionless formulation of the nonlinear version of this problem follows, with vanishing s, s, gs, and upon omittingthe subscript t, from the system
with edge conditions
Page 405
for = ±1 and = ± 1, respectively, and with corner conditions
Writing, in a manner which is consistent with (46)
we then have, from (49) and (50)
etc., with (51) and (52) satisfied by the expansions in (54) term by term, and with the corner conditions (53) understood to require 1, = 2, = 0, etc.
While the expansions in (54) contain even and odd powers of for both and g, when 0, the case = 0 of a flat plate is special, inasmuch as for this case 2n+1
= g2n = 0, so that (52) and (53) reduce to the form 4g1 = 1, 4 2 = 2g1, , with the next two equations following from (49) and (50) as 4g3 = 2, and 4 4 =L( 2,g1) g3, .
Given the possibility of expansions as in (54) it remains to be seen up to what values of it is practical to obtain solutions in this way. Independent of this, there is thequestion concerning the existence of a smallest critical value c for which solutions which evolve continuously as increases from zero bifurcate for = c. If wedesignate the continuously evolving functions and g which are associated with c by c and gc and if we write = c + and g = gc + for the determination of
c by means of a characteristic value problem we will have from (49) and (50) as a linear system of buckling differential equations
with edge conditions for and in accordance with (51) and (52), and with the homogeneous corner conditions , (±1, ±1) = 0.
The special case = 0 of this problem has been considered by Lee and Hsu [1] by the method of finite differences. The numerical results in [1] include the relationsc 10 and c = wc(1, 1) wc(0, 0) 4h, for the case = 0. The number of subdivisions which were used in [1] was thought to be sufficient to make the value of c
reliable, but insufficient to make the value of c entirely reliable, with the given data suggesting that the actual value of c might be larger by about ten percent.
An alternate analysis of the problem by Ramsey [2] "using the restricted form of Naghdi's nonlinear shell theory, along with kinematic results of Green and Naghdi forsuperposed small deformations on a large deformation of an elastic Cosserat surface" includes numerical results which may be written in the form
and c 0.75h.
References
1. Lee, S. L.; Hsu, C. S.: Stability of saddle like deformed configurations of plates and shallow shells. Int. J. Non-Lin. Mech. 6 (1971) 221236.
2. Ramsey, H.: A Raleigh quotient for the instability of a rectangular plate with free edges twisted by corner forces. J. Mec. Theor. & Appl. 4 (1985) 243256.
Page 406
3. Reissner, E.: A note on membrane and bending stresses in spherical shells. J. Soc. Ind. Appl. Math. 4 (1956) 230240.
4. Reissner, E.: On the determination of stresses and displacements for unsymmetrical deformations of shallow spherical shells. J. Math. & Phys. 38 (1959) 1635.
5. Reissner, E.: A note on postbuckling behavior of pressurized shallow spherical shells. J. Appl. Mech. 37 (1970) 533534.
6. Reissner, E.: On the transverse twisting of shallow spherical ring caps. J. Appl. Mech. 47(1980) 101105.
7. Reissner, E.: On the effect of a small circular hole on states of uniform membrane shear in spherical shells. J. Appl. Mech. 47 (1980) 430431.
8. Reissner, E.: On the influence of a rigid circular inclusion on the twisting and shearing of a shallow spherical shell. J. Appl. Mech. 47 (1980) 586588.
9. Reissner, E.; Reissner, J. E.: Effects of a rigid circular inclusion on states of twisting and shearing in shallow spherical shells. J. Appl. Mech. 49 (1982) 442443.
10. Reissner, E., Wan, F. Y. M.: On the effect of a small circular hole in twisted or sheared shallow shear deformable spherical shells. J. Appl. Mech. 53 (1986)597601.
11. Reissner, J. E.: Effects of a circular hole on states of uniform twisting and shearing in shallow spherical shells. J. Appl. Mech. 48 (1981) 674676.
12. Wan, F. Y. M.: Membrane and bending stresses in shallow spherical shells. Intern. J. Solids Structures 3 (1967) 353366.
13. Wan, F. Y. M.: On the displacement boundary value problem of shallow spherical shells. Intern. J. Solids Structures 4 (1968) 661666.
Page 407
On a Certain Mixed Variational Theorem and on Laminated Elastic Shell Theory[Refined dynamical theories of beams, plates and shells, Proc. Euromech Coll. 219, pp. 1727, 1987]
1Summary and Introduction
We are concerned in this paper with the derivation of a two-dimensional system of shell equations by means of the direct methods of the calculus of variations, for anelastic layer the material of which is non-uniform in thickness direction. We have on earlier occasions considered this problem for transversely isotropic layers withuniform material properties in thickness direction, by means of the classical variational theorem for displacements [1], as well as by means of a variational theorem fordisplacements and stresses [3, 4].
We now consider the problem of the non-uniform shell, in particular the problem of the laminated shell, as the problem of a three-dimensional an-isotropic elasticlayer, through use of a variational equation which has recently been established for displacements and transverse stresses [6, 7]. In connection with this equation wenote in particular that by its use we avoid the necessity of making approximations for stresses which may be discontinuous functions of the thickness coordinate,because of material property discontinuities, while retaining the capability of using approximate expressions for stresses which must be continuous in thicknessdirection.
Among the results which are obtained in what follows we mention in particular those which concern the effects of transverse shear and normal stresses, for layers withmaterial properties which are not symmetric about the middle surface of the shell.
2Strain Displacement Relations and the Variational Equation for Displacements and Stresses in Three Dimensions
We use lines of curvature coordinates on the middle surface of a layer of thickness 2c, with a third coordinate along the normals to the middle surface. The elementof volume which is associated with this orthogonal system of coordinates is given by the expression
with 1 and 2 as the coefficients of the linear element and R1 and R2 as the principal radii of curvature of the middle surface.
Page 408
For this coordinate system we have as components of normal and shearing strain, in terms of displacement components U1, U2, U , the expressions
with corresponding expressions for 2 and 2 .
Given the above components of strain, in conjunction with the corresponding components of stress i, 12, i , , we have as the volume integral portion of thefunctional which enters into the variational theorem for displacements and stresses
with the function W being the complementary energy density and with the constitutive equations of the layer being
Given ISD as in (5), with i, 12, i , as in (2) to (4), the variational equation ISD = 0 has, with arbitrary variations i, 12, i , , Ui, U , as Eulerequations the six stress displacement relations in (6) together with the three (homogeneous) differential equations of force equilibrium.
We do not, in what follows, make use of the variational equation ISD = 0 in connection with the problem of deriving two-dimensional shell equations. We noteinstead that earlier work of this nature which was based on a generalization of the functional ISD, so as to have a moment equilibrium condition 12 = 21 asadditional Euler equation of the variational equation [4], should be considered superceded by what follows, even for the problem of the layer with material propertieswhich are uniform in thickness direction.
3A Variational Equation for Displacements and Transverse Stresses
A wish to avoid the complications of having to approximate the discontinuous distributions of membrane and bending stresses in laminated shells, coupled with a wishto avoid the necessity of having to obtain the associated transverse stresses from quantities which involve differentiations with respect to the thickness coordinate,suggests the formulation of a variational equation for displacements and transverse stresses, as follows. We change the character of the first three relations in (6)from that of Euler equations to constraint equations and we consider these three constraint equations solved in the form
Page 409
With this we define a semi-complementary energy density through the Legendre transformation
with (8) implying the system of inverted constraint constitutive relations
The introduction of W from (8) in ISD in (5) results in a new functional
which is such that the variational equation ITSD = 0 has, with arbitrary variations i , , Ui, U and with the constraint equations (9) and (2) to (4), a systemof Euler equations consisting of the three constitutive relations
in conjunction with the three (here homogeneous) equations of equilibrium for stress.
In connection with the intended applications of this variational equation for transverse stresses and displacements we have, as previously noted [7], that when W is ofthe form
where W2 is homogeneous of the second degree and W1 homogeneous of the first degree in terms of the arguments i, 12 then the three constraint equations (6)become a system of linear equations for i, 2 and 12, of the form
and the semi-complementary energy V in (8) becomes
In this W0 remains as in (12) and the arguments i and 12 in W2 are now functions of i, 12, i , , in accordance with (7).
For the special case of a transversely isotropic material with
Page 410
We have, on the basis of (13) and (14)
For the general linear case for which, in addition to the indicated homogeneity properties for W2 and W1, we further have that W0 is homogeneous of the seconddegree, the semi-complementary energy function V will be of the form
upon determination of the coefficients Eij and Fij in terms of the corresponding coefficients in W2 and W1 by means of (13). We note that in order to make theforegoing result applicable to cases of body forces in the equilibrium equations for stress it is necessary to add in the defining relation (10) for ITSD a term P(Ui, U ),
with for problems of simple harmonic motion.
4Derivation of a System of Two-Dimensional Shell Equations
We proceed on the basis of an assumption that we may obtain a rationally approximate two-dimensional theory of laminated shells without considering the effect ofmaterial changes in thickness direction on the nature of the dependence of the components of displacement on the thickness coordinate . Accordingly, we herestipulate as approximations
In regard to these approximations we note in particular the fact that their use in conjunction with the variational equation ITSD = 0 will, in a natural way, introducestress resultants and stress couples into the two-dimensional theory which is to be established. We furthermore note that for many purposes the approximation U =w( 1, 2) will be adequate but that the more flexible approximation as in (19) will make possible some observations of interest.
With (19) we next write, on the basis of (2) to (4)
and
Page 411
In this
with corresponding expressions for 22, 21, 22, 21, and
In association with this we stipulate as approximations for components of transverse stress
and
together with normalization conditions
for the even functions e, ge and the odd functions 0, g0. We need, at this stage, to be no more specific concerning the form of the functions and g. We note,however, the possible choices
or, as none of the traction conditions for = ±c are constraint conditions, even more simply
We know that for a transversely uniform shell a choice of e as in (28), in conjunction with a stipulation Si = 0 in (25) will be appropriate, whereas for a sandwich-type shell with uniform core and thin facings, and with material properties which make non-transverse core stresses negligible, a choice as in (29) is preferable [5].
We next write the variational equation ITSD = 0, with the indicated supplementary function P, in the form
Page 412
and in this introduce i, 12, i , , Ui, U , i , from (19) to (21), (25) and (26). We further write as a system of two-dimensional constraint constitutive relations,on the basis of the three-dimensional relations in (9),
and we observe that, as a consequence of (25) to (27)
and
Finally, we write analogously
With this equation (30) takes on the form
The arbitrariness of Qi, Si, T, K in (35) implies as Euler constitutive equations
and
Page 413
The arbitrariness of ui, i, w, , in conjunction with equations (22) to (24), implies as Euler equations of equilibrium
Equations (38) and (39) are the conventional force and moment equilibrium relations. Equation (40) is not a third moment equilibrium condition but, rather, a relationshowing the dependence of the transverse normal stress measure T on the stress couples M11 and M22 and on the rates of change of the antisymmetric transverseshear stress measures Si, similar to an earlier result for sandwich shells, with Si = 0 because of a material symmetry assumption [5].
Satisfaction of the third moment equilibrium equation
is assured as a consequence of the form of the constitutive relations (31).
Conventional Theory
We are left with a system of ''conventional" shell equations upon stipulating at the outset that = 0 in (19). With this the seventh equilibrium equation (40) does notarise and furthermore we have i = 0 in (36) and = 0 in (37). In this connection we note specifically that the associated constitutive relations remain a part of theconventional theory, so as to account rationally for the deformational effects of the transverse components of stress in shells with significant material unsymmetriesrelative to their middle surfaces.
5Two-Dimensional Constitutive Relations for Transversely Isotropic Shells
The introduction of V from (17) and of the expressions for strain from (20) and (21) and for stress from (26) into (31), and the neglect of terms of relative order c/Rleads to the
Page 414
following system of relations
with
Analogously we have from (32) and (33)
with
The following observations may be made in regard to this system of constitutive relations.
(i) The approximation = 0 in (48) implies the relation
The effect of the term with CQS in (48 ), for shells for which G is not an even function of , may not have been considered previously.
(ii) The special case = CTT T of equation (49) in conjunction with 11 and 22 in accordance with (23), and in conjunction with the equilibrium equation (40), withSi = 0, has previously been encountered in the analysis of sandwich-type shells [5].
(iii) When C2 0 then the assertion that, in first approximation, the strain energy is the sum of the stretching and of the bending strain energy does not apply.
(iv) Equations (43) and (45) are consistent with the recommendation in [2] to allow for the non-negligibility of terms ( 12 21)/Ri while considering terms
Page 415
( 12 + 21)/Ri as negligible, the same as terms 11/Ri and 22/Ri which could have been retained in (42) and (44).
6On Two-Dimensional Constitutive Relations for the General Linear Case
With V as in (18) the most general linear system of constitutive relations comes out to be, on the basis of equations (31), (36) and (37), of the symbolic form
in conjunction with a relation ( kl, kl, Qk, Sk, T, K) = 0 which comes from (37). We will here not concern ourselves with consequences or special cases of thisresult except for noting the possibility of a first-order importance of transverse shear stress effects, in connection with the analysis of shells of such construction thatsome or all of the coefficients F11, F12, F21, F22 in equation (18) do not vanish.
References
1. Hildebrand, F. B., Reissner, E. and Thomas, G. B., Notes on the Foundations of the Theory of Small Displacements of Orthotropic Shells, NACA TechnicalNote No. 1833, 1949.
2. Koiter, W. T., A Consistent First Approximation in the General Theory of Thin Elastic Shells, IUTAM Symp. on Thin Elastic Shells, pp. 1233, North-HollandPublishing Company, Amsterdam, 1960.
3. Reissner, E., Stress-Strain Relation in the Theory of Thin Elastic Shells, J. Math. & Phys. 31, 109119 (1952).
4. Reissner, E., On the Form of Variationally Derived Shell Equations, J. Appl. Mech. 31, 233238 (1964).
5. Reissner, E., On Small Bending and Stretching of Sandwich-Type Shells, Intern. J. Solids Structures 13, 12931300 (1977).
6. Reissner, E., Reflections on the Theory of Elastic Plates, Appl. Mech. Rev. 38, 14531464 (1985).
7. Reissner, E., On a Mixed Variational Theorem and on Shear-Deformation Plate Theory, Intern. J. Num. Math. Engg. 23, 193198 (1986).
Page 416
On Finite Axi-Symmetrical Deformations of Thin Elastic Shells of Revolution[Computational Mechanics 4, 387400, 1989]
1Introduction
In what follows we return once more to the problem of formulating a system of two simultaneous second order ordinary differential equations for the analysis of finitetorsionless axi-symmetrical deformations of thin elastic shells of revolution, including the effects of transverse shear deformation and of membrane drilling moments.
Given the existing literature on this subject, the reasons for the present paper may be described as follows.
We generalize a previously used procedure for the problem of plane deformations of plane beams (Reissner 1972b) so as to have a new and more transparentderivation of our analogous earlier results (Reissner 1969, 1972a) for the shell-of-revolution problem.
We show that the introduction of a certain semi-complementary energy density function leads to a particularly simple form of the two simultaneous second orderequations, as well as to a particularly simple variational formulation of the torsionless axi-symmetrical finite deformation problem of the shell of revolution.
The foregoing general considerations are complemented by a consideration of some aspects of the problem of formulating linear constitutive relations, byconsideration of some critical comments (Naghdi and Vongsarnpigoon 1985; Schmidt and DaDeppo 1975) on an early version (Reissner 1950) of the finite-deformation analysis of the shell-of-revolution problem, and also by a consideration of some recent concerns (Koiter 1980; Simmonds 1975) with an elimination oflateral-contraction effects from the basic system of two simultaneous second order differential equations.
Finally, we introduce the effect of transverse normal stress deformability and we show that this leads to a system of three simultaneous second order equations, inplace of the two equations without the effect of transverse normal stress deformability.
2Kinematics of Shell Element
We consider an arc length element ds of the meridians of the undeformed shell-of-revolution surface, with r = r(s) and z = z(s) as parametric equations. We designatethe associated tangent angle by , and write r = cos and z = sin , with the primes indicating differentiation with respect to s.
Due to deformation the radial distance r(s) is changed into r(s) + u(s) and the elevation z(s) is changed into z(s) + w(s). At the same time, the tangent angle ischanged into + , and the arc length ds of the undeformed meridian curve is
Page 417
Fig. 1.Side view of shell element before and after deformation.
changed into (1 + e) ds such that
For the purpose of a kinematic account of the effect of transverse shear deformability we consider the tangent angle change to be composed of two parts,
In this the angle represents the change, due to deformation, of the material normals to the meridian curves, and the angle is a measure of the effect of transverseshear deformability, as indicated in Figure 1.
3Dynamics of Shell Element
We complement the meridional arc length coordinate s by a circumferential angular coordinate and we consider an undeformed surface element r d ds which isdeformed into an element (1 + e)(r + u) d ds. For this deformed element we introduce stress resultants and stress couples as forces and moments per unit ofundeformed length, and body forces and body moments as forces and moments per unit of undeformed area. We further stipulate that the meridional membranestress resultants Ns and the meridional transverse shear stress resultants Q are perpendicular and tangential, respectively, to the deformed material surface normal(and not tangential and perpendicular, respectively, to the deformed meridian curves, except when = 0). The stress resultants Ns and Q are complemented by acircumferential membrane stress resultant N , parallel to the base plane of the shell. With these three stress resultants, and with radial and axial body force
Page 418
Fig. 2.Side view of deformed shell element, showing orientation of stress resultants,
stress couples, and load intensity components.
components pr and pz we have as axial and radial component equations of force equilibrium
In order to write the associated equation of moment equilibrium we introduce meridional bending stress couples Ms and circumferential bending stress couples M .While, evidently, the axis about which Ms turns coincides with the direction of N we stipulate that the axis about which M turns coincides with the direction of Ns, inaccordance with Figure 2. For the sake of mathematical symmetry we introduce a drilling stress couple P, the axis of which coincides with the direction of Q. Thestress couples Ms, M and P are complemented by body moments m and with these we read from Figure 2 as equation of moment equilibrium
4Strain Displacement Relations
Given the six stress resultants and stress couples Ns, N , Q, Ms, M , P we postulate the existence of six associated strain resultants and strain couples s, , , ,, with these measures of strain and of stress being
Page 419
conjugates via an equation of virtual work of the form
We use (6) for the purpose of obtaining virtual strain displacement relations, by eliminating pr, pz and m by means of (3)(5), and by then integrating by parts so as toeliminate derivatives of stress resultants and couples, as well as the boundary terms on the right. Inasmuch as the remaining relation can then be considered to be validfor arbitrary Ns, N , Q, Ms, M , P we arrive at the following system of virtual strain displacement relations
The step from these virtual relations to actual relations is an obvious one for Eqs. (9a, b) and (10a, b), as follows
The determination of s and is less simple and is carried out as follows. We write, on the basis of (1a, b)
and we use these two relations to eliminate u and w in (7) and (8). The subsequent observation of (2) then gives
with these two virtual relations implying as actual relations
In order to deduce expressions for s and in terms of u, w and we first deduce from (1) and (2), in conjunction with (17a, b), that
A consideration of (18) and (19) as a system of two simultaneous equations for s and then gives as expressions for the remaining two strain resultants s and
Page 420
5Compatibility Equations
The first, and from a practical point of view most significant compatibility equation follows from (18) in conjunction with (11a) in the form of a relation which is not acompatibility equation in the usual sense of the word. Evidently, an introduction of u from (11a) into (18) gives the relation
Equation (22) becomes a true compatibility equation upon making use of (11b) and (12b) in order to transform (22) into
Two additional compatibility equations follow from (11b) and (12a, b) in the form
Given the three first order compatibility equations (23)(25) we note, for historical reasons, that they are equivalent to two equations, one of the first order and one ofthe second order, without appearance of the strain couple . With taken from (23) the second order equation evidently follows from (25). We omit listing thissecond order equation which, upon stipulating that = 0, coincides with a special case of a corresponding relation in (Koiter 1966). We will, however, list theremaining first order equation which follows from (24) in the form
and we note that an earlier observation in (Reissner 1969) that the generally non-negligible term s on the right of (26) had not been taken account of in Koiter(1966), appears to be not well known.
6Constitutive and Variational Equations
Given the conjugacy of the stress resultants and couples in (3) to (5) relative to the strain resultants and couples in (11), (12), (20) and (21) we have as a system ofconstitutive equations for elastic shells
Given the virtual work equation (6) we expect, and readily verify, the possibility of a variational equation for displacements in the event that the six relations in (27)are specialized so as to read, in terms of a strain energy density function U( s, , , s, , ; s)
If we limit ourselves, as we shall do in this account, to the consideration of problems with displacement-independent load intensity functions pr, Pz, and m then theform of this variational equation is
with the boundary condition functional Ib depending on the nature of the prescribed boundary conditions.
Page 421
The fact that for most problems in shell theory it is of practical advantage to have strain resultants expressed in terms of stress resultants instead of the other wayaround suggests the formulation of a variational equation for displacements and stress resultants, as follows. We invert the first three of the six relations in (28) so asto have
and we define a semi-complementary energy density function V(Ns, N , Q, s, , ; s) through the partial Legendre transformation
in terms of which
The introduction of (31) into (29) results in a variational equation of the form
In this we now have the strain displacement relations, in conjunction with the partial set of constitutive relations
as constraint conditions and, in view of the fact that now u, w, and N, N , Q are arbitrary, the complementary set of constitutive relations (32) as well asthe differential equations of equilibrium as Euler equations.
For completenessake we also list, in connection with our analysis in (Reissner 1963), the variational equation for displacements, stress resultants and stress coupleswhich obtains upon inverting all six of the constitutive relations in (28) and upon introducing a conventional complementary energy density function W(Ns, N , Q, Ms,M , P; s) through the complete Legendre transformation
in terms of which
and with which we then have as a canonical variational equation for displacements, stress resultants and stress couples
We finally note that a change of the character of the strain displacement relations from constraint equations to Euler equations, in the sense of Hu and Washizu, isagain possible, as is a deduction from this of a variational equation for displacements and strain resultants and couples, upon changing the character of the constitutive
Page 422
relations from that of Euler equations to constraint equations.
7The General Two Simultaneous Second Order Equation Form of the Problem, and the Associated Variational Equation
We here proceed as in earlier less general considerations to reduce the (pseudo) compatibility Eq. (22), in conjunction with the moment equilibrium Eq. (5) to asystem of two simultaneous second order differential equations. Of particular interest, in connection with the present reduction, is the fact that the step to greatergenerality has as a consequence a result of unexpected simplicity.
In order to accomplish the proposed reduction we introduce alongside the angular displacement variable a stress function variable
and we integrate the axial force equilibrium equation (3) in the form
where . With (39) and (40) we have then that
and the introduction of (39) into the radial force equilibrium equation (4) gives
With (41)(43) and with (11b) and (12a, b) the semi-complementary energy density function V, as defined in (31) and (30), may be seen to depend entirely on , ,, and s.
In order to reduce (22) and (5) to two simultaneous equations for and we now proceed as follows. We first conclude from (32) in conjunction with (43), from(35) in conjunction with (12a), and with V(Ns, . . .) = V* ( , , , ; s), that
The appearance of (44a, b) suggests that we next endeavour to obtain expressions for V*/ and V*/ , with this being the crucial step in the derivation of thesimple general result of our analysis. Again with V as in (30) and (31), and now with Ns and Q as in (41) and (42), s and as in (11b) and (12b), and with ccos( + ) and s sin( + ) we then have
Page 423
The introduction of (45) and (44a) into (22) gives as the first of the two simultaneous equations for and
The introduction of (46) and (44b), in conjunction with a use of the relation (1 + e) (Ns sin Q cos ) = Ns (1+ s)Q which follows from (17a, b), into (5) gives asthe second of the two simultaneous equations
With Ns, Q and N given directly in terms of the solutions and of (47) and (48) we recall that determination of Ms, M and P requires the use of (31) inconjunction with (11b) and (12a, b), and that determination of the translational displacements u and w may be accomplished by means of the relations
which follow from (11a) and (19).
Given the differential equations (47) and (48) it is possible to derive a variational equation which has these as Euler equations, as follows. We depart from (34) with Vas in (31), with N , Ns, Q as in (41)(43) and with , s, as in (11a), (20) and (21). In this we eliminate derivatives of u and w by integrations of part. In this waywe arrive at a variational equation of the form
with arbitary and and with a suitable combination of Ib and of the boundary terms which come from the indicated integrations by part.
8The Two Simultaneous Second Order Equations for Sheardeformable Two-Dimensionally Isotropic Shells with Linear Constitutive Equations
We stipulate
where A, B, D and are given functions of s.
Page 424
We now have, with Ns, Q and N as in (41)(43) and with s and as in (12a) and (11b) as expressions for the various V-derivative terms in (47) and (48)
The introduction of (52) and (54) into (47) leaves as the first of the two simultaneous equations for and ,
The introduction of (53) and (55) into (47) gives as the second of the two equations for and
As regards these two somewhat formidable appearing simultaneous differential equations we note the following.
(i) The terms having A and B as factors in Eq. (57) come from considering the terms s Q Ns alongside the term Q in the condition of moment equilibrium. They arehere included, it is believed for the first time, because they are a natural consequence of the compact equations (48) and (51). It is questionable whether there areproblems of practical interest concerning thin shells where these terms would ever play a significant role. They must, however, be considered as long as one wishes tohave the possibility of dealing with problems where s and/or would come out to be order of magnitude unity.
Page 425
(ii) The terms with on the left sides of (56) and (57) may be transformed as follows
As regards the presence of these terms in equations (56) and (57), a case has been made by (Simmonds 1975), that it would be legitimate to omit these termsaltogether within the range of validity of the present theory with this conclusion subsequently having been supported in (Koiter 1980). In light of this concern in(Simmonds 1975) and (Koiter 1980) it seems worth stating that when there is no s-independence of , A and D these terms must certainly be retained, as may beseen most easily on the basis of an inspection of Eqs. (56) and (57) for the linear-theory case of a flat plate for which (56) and (57), with = 0 and r = s, reduce tothe uncoupled form1
(iii) Given the remarkably compact and symmetrical appearance of (47) and (48), for all possible choices of the semi-complementary energy density function V it istempting to conjecture that whatever simplication of these equations would be physically reasonable should consist entirely in a modification of the form of thisfunction V. For example, if there are reasons to expect that Ns and Q are small compared to N , and small compared to s, so that , Eqs. (56)and (57) would appear in a greatly simplified form. An a posteriori validation of the appropriateness of this simplication would then consist in deducing reactive valuesof Q and Ns in terms of the values of and as obtained on the basis of the simplified problem from Eqs. (41) and (42), and to show that these are in fact smallcompared to the values of N which follow from (43), with a corresponding a posteriori verification concerning the stipulated smallness of .
(iv) A recently proposed (Simmonds and Libai 1987) simplified version of the two simultaneous differential equations for and , subject to an assumption of smallstrain and subject to the further assumptions that = 0, A = 0 and D = 0, suggests a comparison with the corresponding version of (56) and (57), for simplicity'ssake with pr = 0 and Nz = 0.
We have from (56)
1For an example of the quantitative significance of the terms with we consider Eq. (60) with pr, = 0, Nz = 0, N = , rNs = , in the interval a r , for the case A = Ao (r/a)m, Ns(a) = Ns(a) =
No and Ns( ) = 0. It is readily established that the solution of this problem comes out to be = Noa (r/a)p with and therewith Ns/No = (r/a)p1 and N = pNs.
When m = 0 we have p = 1, for all values of . When m = 2 we have and therewith p(0) 2.41 and p(1/2) = 2.73, which indicates a significant quantitative effect of thePoisson's ratio term.
Page 426
and from (57)
The corresponding equations in Simmonds and Libai (1987) are
Aside from the non-occurrence of the terms with in (62) and (63) we see that the term r1[sin( + ) sin ] cos( + ) in (63) is replaced by the -independentterm r1 sin in (65), while the term r1 cos2( + ) in (62) is replaced by a term r1 in (64). Given the present direct two-dimensional analysis we here do not havethe possibility of order of magnitude considerations in connection with admissable strain energy properties as in (Simmons and Libai 1987) and we think thereforethat it may be preferable in the analysis of specific problems to use (62) and (63) in place of (64) and (65), in particular in view of the statement in (Simmonds andLibai 1987), that there is ''no proof that the terms which we neglect will have a negligible influence on the solution of the equations; rather we merely assert that oursimplifications are consistent with those that are unavoidable in any first-approximation theory that adopts an uncoupled quadratic strain energy density in whichtransverse shear strains are ignored."
9Historical Comments
With the history of the linear-theory problem of axi-symmetric deformations of shells of revolution being securely associated with the names of H. Reissner and E.Meissner we here limit ourselves to an outline of the developments concerning the corresponding finite-deflection problem.
The first attempts towards a non-linear formulation of the symmetric shell of revolution problem is thought to be due to V. I. Feodosiev (1945). The approach inFeodosiev (1945) is to depart from the two simultaneous H. Reissner-Meissner equations, by way of a reference to a Swedish monograph which appeared in 1933,with the introduction of the finite-deflection effect depending on a replacement of the angle in the coefficients of the two simultaneous second order differentialequations of the linear theory by the corresponding tangent angle + for the deformed shell.
Without awareness of (Feodosiev 1945) it subsequently occurred to the present author that the symmetrical finite-deflection problem of the shell of revolution shouldbe treated as a problem of finite (meridional) bending of a system of variable cross section beams resting on an elastic (circular ring) foundation, and that, proceedingin this fashion, it would be possible to establish directly a finite-deflection analogue of the linear H. Reissner-Meissner formulation, without needing to consider a thennon-existent general finite-deflection theory of elastic shells. A first report on this analysis, including the derivation of all relevant strain displacement and equilibriumequations, but without reduction of the problem to two simultaneous second order differential equations, was completed in time for it's publication in a particularlyappropriate volume (Reissner 1949). The derivation of the two simultaneous differential equations, as well as of a number of significant consequences therefrom, wasshown shortly therafter in (Reissner 1950).
Page 427
With the analysis in (Reissner 1949, 1950) having been subject to the classical constraint hypothesis of undeformed middle surface normals being deformed intonormals to the deformed middle surface we subsequently reconsidered the problem of deriving strain displacement relations without this constraint hypothesis in(Reissner 1963). Additionally, in place of the formally unrelated derivation of strain displacement equations and equilibrium equations in (Reissner 1949, 1950) wederived in (Reissner 1963) a formally consistent system of equilibrium equations, based on an appropriate version of the finite-elasticity variational theorem forstresses and displacements in (Reissner 1953). The principal consequence of this alternate approach is, as already suggested in (Reissner 1950), that the variationallyconsistent stress resultants and couples had to be understood as forces and moments per unit of undeformed length measured along lines on the middle surface of theshell, in place of the earlier definitions of forces and moments per unit of deformed length.2
We returned to the finite symmetrical deflection problem of the shell of revolution problem again in (Reissner 1969), for a simpler and more direct derivation of theresults in (Reissner 1963) and also for the purpose of incorporating into the theory the effect of a drilling moment P. In this derivation an assumption of small strainwas made and while direct geometrical considerations were again employed for a rederivation of the strain displacement relations in (Reissner 1963) the nowrequired supplementary definition of a drilling strain couple was accomplished through an appropriate virtual work consideration. A subsequent brief note (Reissner1972a) showed that "removal of the restrictive assumption of small strain is a relatively simple matter" but did not concern itself with the problem of the twosimultaneous second order differential equations. Instead we used the insights gained in (Reissner 1969, 1972a) for an ab initio analysis of the finite-strain problem ofplane deformations of originally plane beams (Reissner 1972b), with virtual work instead of geometric derivations of all strain displacement relations, and withemphasis on the concept of force-deformational effects in beam theory.
Given the analysis in (Reissner 1972b), the present account of the symmetric shell of revolution problem represents a long planned ab initio analysis of this latterproblem, in complementation of the work in (Reissner 1969, 1972a).
10A Comment on Linear Constitutive Relations
An earlier observation in Reissner (1972b) on the subject of transverse shear deformation effects for a problem of circular ring buckling suggests the followingobservations concerning the formulation of constitutive relations.
2The indicated difference manifests itself by way of the appearance of a term [(1 + )rMs] in place of the term (rMs) and of a term (1 + s)M in place of M in Eq. (5), with corresponding
modifications in (3) and (4). Given that the thought occurs that the second term on the right might not always be negligible compared to the first term sothat it might not always be safe to assume that [(1 + )rMs] is effectively equivalent to (rMs) as had been done in (Reissner 1950). While no examples of application are known where the abovehas been of importance, this formal difficulty with the analysis in (Reissner 1950) is here mentioned in light of the fact that it has twice been commented upon in the literature, in (Schmidt andDaDeppo 1975) and in (Naghdi and Vongsarnipigoon 1985). Inasmuch as the analysis in (Reissner 1963) disposed of this difficulty, one may regret that neither of the two criticisms referred to theanalysis in (Reissner 1963), or to the related later developments in (Reissner 1969, 1972a).
Page 428
While the semi-complementary energy function V in (51) implies the linear relations
an inspection of Figure 2 suggests that we might have, equally well or perhaps even more appropriately, stipulated a system of physically linear relations of the form
In transforming (67a, b, c) into expressions for s, and in terms of Ns, Q and N we here limit ourselves to an assumption of small strain with e and related tos and , as a consequence of (17a, b), in the form
The introduction of (68) into (67a, b, c) then gives, except for terms which are small of higher order
or, equivalently,
Equations (70a, b, c) are, as they should be, consistent with the existence of a stress resultant portion VR of the semi-complementary density function V. It is readilyverified that when
(70a, b, c) is consistent with (32).
Furthermore, (70c) is consistent with a beam theory relation = BQ/(1 + BN) BQ B2QN in (Reissner 1972b), which had there been stated, on the basis of animplied order of magnitude stipulation that A << B.
11An Analysis of the Deformational Effect of Transverse Normal Stress
Given the surface-theoretical concepts of stress and strain in accordance with Figures 1 and 2 we now introduce, in accordance with Figure 3, a transverse normalstress measure T in conjunction with a supplementary anti-symmetrical transverse shear stress measure S, and an antisymmetric transverse load intensity measure q.
In order to take account of these three additional dynamic variables in a statement of a generalization of the virtual work Eq. (6) we further introduce additionalconjugate strain measures t and and an additional conjugate displace-
Page 429
Fig. 3.Visualization of transverse normal stress measure T, antisymmetric transverse shear
stress measure S, and antisymmetric transverse load intensity measure q.
ment measure . With the help of these we add a term
on the right of Eq. (6).
In order to make use of the augmented virtual work equation it is necessary to stipulate appropriately modified equilibrium relations or appropriately modified straindisplacement relations We here decide, in contrast to the procedure utilized in the analysis without the effects of transverse normal stress deformation, to stipulateappropriately modified strain displacement relations involving the supplementary displacement variable .
An inspection of Figures 13 suggests that the strain resultant components s, and , and the strain couple component should be unaffected by . At the sametime, the strain couple components s and should now be of the form
As regards the form of the additional strain measures t and we stipulate, on the basis of Figure 3, the relations
Page 430
With (11a), (12b), (20), (21), (66a, b) and (67a, b) we then deduce from the augmented virtual work equation a system of four differential equations of equilibrium,as follows. The two force equilibrium equations (3) and (4) remain unchanged. The moment equilibrium equation (5), with e and as in (17a, b), is changed and nowinvolves the displacement variable ,
An additional fourth equilibrium equation comes out to be of the form
The above kinematic and dynamic modifications require associated constitutive modifications. The introduction of the supplementary strain measures t and leads toa supplementation of the six constitutive relations in (28) by two relations of the form
The eighth order problem as stated in equations (3), (4), (11a), (20), (21), (28), and (66)(70) can be reduced, as follows, to a surprisingly symmetric sixth orderform.
We again introduce a semi-complementary energy function V, in accordance with (31), and we again observe (32) and (35), with the latter three relations nowsupplemented by two relations of the form
We again make use of the stress function representations in (41)(43) and we again use the pseudo-compatibility relation (22), now in conjunction with the modifiedmoment equilibrium equation (74), for a re-derivation of the two fundamental second order Eqs. (47) and (48). We then use the supplementary equilibrium Eq. (75)in conjunction with (77) and (72a, b) to derive a third second order differential equation, as follows.
We conclude from (77) and (73b) that
and we conclude from (72a, b) and (73a), in conjunction with (35) and (77), that
The introduction of (78) and (79) into (75) gives as the third of three simultaneous second order equations the remarkably harmonious relation
Given the sixth order system (47), (48) and (80) we add the following observations.
(i) The variational Eq. (50) which has (47) and (48) as Euler equations will also have (80) as Euler equation upon introduction of the terms with and , inaccordance with (76) and (77), and upon adding a term rq to the integrand in (50).
Page 431
(ii) We have the physically rational possibility of a reduced fourth order problem for the analysis of transverse normal stress deformation effects, by assuming that thedeformational effect of S is negligible. Negligibility of the effect of S means the non-occurrence of the argument in the energy density functions U and V. Equation(80) then reduces to the zeroth-order form V*/ = q, which determines as a function of , , and . The introduction of this into (47) and (48) results inthe indicated fourth order system including the effect of transverse normal stress deformation.
(iii) The foregoing analysis of the effect of transverse normal stress deformation is consistent with our earlier analysis, within the framework of infinitesimal deformationtheory, of the problem of sandwich-type shells by way of an exact solution of a suitably idealized layer problem (Reissner 1977), and of the problem of arbitrarilylaminated shells by way of an approximate solution through use of a variational theorem for displacements and transverse stresses (Reissner 1987).
References
Feodosiev, V. I. (1945): Large displacements and stability of a circular membrane with fine corrugations. Prikl. Mat. Mekh. 9, 389412 (in Russian).
Koiter, W. T. (1966): On the nonlinear theory of thin elastic shells. Proc. R. Netherlands Acad. Sci. B69, 152.
Koiter, W. T. (1980): The intrinsic equations of shell theory with some applications. Mech. Today 5, 139154, Pergamon Press.
Naghdi, P. M.; Vongsarnpigoon, L. (1985): Some general results in the kinematics of axi-symmetrical deformation of shells of revolution. Q. Appl. Math. 43, 2336.
Reissner, E. (1949): On the theory of thin elastic shells. In: Contributions to applied mechanics (Reissner anniversary volume), pp. 231247. Ann Arbor/MI: J. W.Edwards.
Reissner, E. (1950): On axi-symmetrical deformations of thin shells of revolution. Proc. Symp. Appl. Math. 3, 2752.
Reissner, E. (1953): On a variational theorem for finite elastic deformations. J. Math. & Phys. 32, 129135.
Reissner, E. (1963): On the equations for finite symmetrical deflections of thin shells of revolution. In: Progress in Applied Mechanics (Prager anniversary volume),pp. 171178. New York: MacMillan
Reissner, E. (1969): On finite symmetrical deflections of thin shells of revolution, J. Appl. Mech 36, 267270.
Reissner, E. (1972a): On finite symmetrical strain in thin shells of revolution. J. Appl. Mech. 39, 11371138.
Reissner, E. (1972b): On one-dimensional finite strain beam theory: the plane problem. Z. Angew. Math. Phys. 23, 795804.
Reissner, E. (1977): On small bending and stretching of sandwich-type shells. Int. J. Solids Struct. 13, 12931300.
Reissner, E. (1987): On a certain mixed variational theorem and on laminated elastic shell theory. In: Elishakoff, I.; Irretier, H. (eds.) Refined dynamical theories ofbeams, plates and shells and their applications. Proc. Euromech. Colloq. 219. Berlin, Heidelberg, New York: Springer, pp. 1727.
Schmidt, R.; DaDeppo, D. A. (1975): On finite axi-symmetric deflections of circular plates. Z Angew. Math. Mech. 55, 768769.
Simmonds, J. G. (1975): Rigorous expunction of Poisson's ratio from the Reissner-Meissner equations. Int. J. Solids Struct. 11, 10511056.
Simmonds, J. G.; Libai, A. (1987): A simplified version of Reissner's nonlinear equations for a first-approximation theory of shells of revolution. Comput. Mech. 2,15.
Page 433
VARIATIONAL PRINCIPLESAs far as I remember, my first exposure to the use of variational methods came from an assignment to consider the effective width problem of beams with wideflanges, on the basis of von Karman's least work analysis of this problem. Von Karman's results and the method of analysis evidently had been of considerableinterest to Timoshenko, as he devoted several pages to it in his Theory of Elasticity which had appeared just then. Somewhat later, I used the theorem of least workfor a Rayleigh-Ritz type analysis of the related problem of shear lag in box beams [24], together with Francis Hildebrand of the problem of the flanged built-incantilever as a problem of plane stress [27], and of the problem of three- dimensional corrections for the two-dimensional theory of plane stress [30].
After having seen the usefulness of the theorem of least work for approximate reductions of the dimensionality of three different two- and three-dimensional problemsit occurred to me to inquire what a similar approach would accomplish for the derivation of a two-dimensional theory of transverse bending of plates. The outcome ofthis were the equations of the 6th order theory of shear-deformable plates in [36].
With four applications of a variational theorem for stresses, and with the knowledge of Kirchhoff's use of a variational theorem for displacements for a fourth ordertheory of plates, I became interested in seeing what I could do using the displacement theorem. It seemed natural that the first such effort should be a reconsiderationof the box beam shear lag problem [43]. A judicious choice of displacement approximations showed that there could in fact be certain advantages in the use of thedisplacement principle, in place of the stress principle. This insight was then utilized, with the help of my friends Hildebrand and Thomas, for the formulation of aprocedure to derive two-dimensional plate and shell theories of varying orders [57].
Having used the variational theorem for stresses and the variational theorem for displacements, I began to wonder whether this had to be an either-or proposition.The first consequence was a generalization of the variational theorem for stresses, to make this theorem applicable to linear problems of simple harmonic motion [52].The possibility of this generalization depended on the simultaneous introduction of stress and displacement variations, which had to be interdependent to retain thedynamic constraint stipulations.
With the concept of interdependent stress and displacement variations, a natural next step was to think about the possibility of a variational theorem with independentstress and displacement variations. Once that question was asked in this fashion, a quite simple answer was found, essentially by experimentation with variouspossible combinations of terms, first within the framework of geometrically
Page 434
linear elasticity [66], and subsequently for the general case [84] in terms of Green-Lagrange strains and Kirchhoff-Piola stresses.
Soon afterwards, my friend Washizu, who at that time was spending two years at M.I.T., came one day to my office to say that he had a variational theorem withindependent variations not only of stresses and displacements, but also of strains, in such a way that not only equilibrium and stress-strain, but also strain displacementrelations came out as Euler equations. I first objected that since only stresses and displacements would be encountered in the boundary conditions of problems, it wasnot natural to consider strain displacement relations in ways other than as defining relations. I was, however, soon persuaded that the "three-field" theorem whichWashizu, and independently Hu, had proposed was a valuable advance which I wished I had thought of myself.
Some years later, in 1963, I eventually had another good idea. I asked what would happen if one considered the symmetry property of the stress tensor not as adefinition but as an explicit consequence of moment equilibrium, in such a way as to have the moment equilibrium equations as Euler equations of a variationaltheorem, the same as the force equilibrium equations [149]. I submitted the answer to this question, within the framework of a two-field as well as a three-fieldvariational theorem, as a brief note to a journal which had a special section for such notes. The manuscript was returned to me with the opinion of two referees whothought this result to be "somewhat trivial" and "a complication rather than a simplification."
Having to submit the note to a second journal caused some delay in its publication. The usefulness of the thought became eventually apparent, in particular throughFrayes de Veubeke's publication in 1972 of a deeper result for geometrically non-linear elasticity in terms of Biot stresses, and through work by Tom Hughes in 1988on drilling degrees of freedom.
I then lost interest in variational principles, except in connection with their use for the exact or approximate solution of various specific problems, until 1982, whenSatya Atluri suggested that I write an article on the subject for a planned finite-element handbook. I declined at first, saying that I had been away from this field fortoo long. However, after reconsidering I accepted, with the idea that it would perhaps be a good experience to collect once more one's thoughts about an area ofwork that had been personally meaningful for quite a time, a long time ago. The article appeared in print in 1987 [258] and it did, I believe, throw additional light onsome aspects which before had not been understood as well as I had hoped.
Re-thinking old thoughts sometimes leads to new results. Mine included a generalization of Leonard Herrmann's theorem for incompressible or nearly incompressiblematerials [242], an alternate approach to Frayes de Veubeke's 1972 theorem [243, 248], and after that what might be called a one and-a-half field theorem fordisplacements and transverse stresses, intended for use with problems of laminated plates and shells [241, 256]. Finally, Satya Atluri and I jointly formulated a five-field theorem, with independent pressure and volume change variations in addition to the variations of displacements, distortional strains and deviatoric stresses[266].
Page 435
Note on the Method of Complementary Energy[J. Math. & Phys. 27, 159160, 1948]
In the following lines we propose to show that the well known minimum principle for the stresses in the theory of elasticity (the principle of minimum complementaryenergy) may be extended in such a way that it applies to dynamic as well as to static problems. While the proof of this extended principle is quite simple, we have notbeen able to find a statement of it in the literature. The subject is considered in quite a different way in a recent paper by H. M. Westergaard.1
We do not here intend to give the most general theorem possible2 but rather restrict attention to problems in which loads, stresses and displacements are simpleharmonic functions of time. We use the customary engineering notation for stresses ( , ) and displacements (u, v, w), and omit the common time factor cos t in allequations of the theory. (In so doing we exclude consideration of materials with non-linear stress strain relations).
We have as differential equation of equilbrium in the x-direction
and two corresponding equations for equilibrium in the y and z-directions. As in the static case, which corresponds to = 0, we have a stress energy function Wwhich is determined by the relations
where the components of strain x, xy etc. are appropriately expressed in terms of the stresses x, xy etc., but are not at this stage required to satisfy the equationsof compatibility.
We further introduce a kinetic energy function K given by
If we assume for simplicity's sake that on the entire surface of the body we have prescribed stresses then the extended theorem may be formulated as follows.
Theorem
Among all states of stress and displacement which satisfy the differential equations of equilibrium in the interior of the body and the condition of prescribed stress onthe surface, the state of stress which also satisfies the compatibility relations
1On the Method of Complementary Energy. Proc. American Soc. Civil Engs., February 1941, pp. 199227.
2Such generalizations should also include the three-dimensional theory of elastic stability.
Page 436
for strain is determined by the variational equation
Proof
We have
According to (2) the compatibility relations are equivalent to
The conditions that stresses and displacements satisfy the differential equations of equilibrium are of the form
with corresponding equations involving v and w.
We put (5) into (4), integrate by parts, and take into account that because of the conditions of prescribed surface stress we have three relations of the form
so that the remaining surface integrals vanish. Eq. (4) then assumes the form
The right hand side of (8) vanishes because of (6) and thus the theorem is proved.
Remark
As for the problem of statics a more general form of the theorm states that if over a part S1 of the surface the stresses are prescribed while over the remaining part S2
the displacement components have prescribed values , , then the appropriate variational equation is
where px, py and pz are the unknown x, y and z-components of the surface stress.
Page 437
On a Variational Theorem in Elasticity[J. Math. & Phys. 29, 9095, 1950]
1Introduction
There are in the theory of elasticity two well known variational theorems or principles, one of them being for displacements and the other one being for stresses [1].The former is also called the principle of minimum potential energy while the latter is often referred to as the principle of minimum complementary energy. Withreference to the differential equations of the theory of elasticity, which consist of equations of equilibrium for the components of stress and of the stress-displacementrelations, a possible characterization of the difference between the two variational theorems is as follows. In the theorem for displacements the stress-displacementrelations are taken as equations of definition for the components of stress in terms of appropriate displacement derivatives and the variational equation is equivalent tothe system of differential equations of equilibrium. In the theorem for stresses the differential equations of equilibrium serve to restrict the class of admissible stressvariations and the variational equation is equivalent to the system of stress-displacement relations.
Both variational principles have been found valuable for the purpose of obtaining approximate solutions of boundary value problems. When viewed in the above lightthese approximate solutions are such that part of the complete system of differential equations (either the stress-displacement relations or the equations of equilibrium)is satisfied exactly while the remaining equations are satisfied approximately only.
It is natural to ask whether it might not be possible to use the calculus of variations for the purpose of obtaining approximate solutions in such a manner that there is nopreferential treatment for either one of the two kinds of differential equations which occur in the theory. In what follows this question is answered in the affirmative. Avariational problem will be formulated which has both the equation of equilibrium and the stress-displacement relations as appropriate Euler equations.
It is also shown that application of the theorem to a problem which had previously been treated by means of the method of complementary energy leads to the resultsobtained by the complementary-energy method in a manner which represents some simplification over the earlier derivations.
2The Boundary Value Problem
Let x, xy, etc. be components of stress in the usual notation and let u, v, w be components of displacement. Define quantities x, xy, etc. in the usual way in termsof displacements by x = u/ x, xy = u/ y + v/ x, etc.
Page 438
The differential equations to be considered are the three equations of equilibrium
(where for simplicity's sake body forces are not considered), and the six stress-displacement relations
In Eqs. (2) W is a given function of the six arguments x, xy, etc.
Let
be the x-component of the stresses acting on an element of area with normal direction n, with corresponding definitions for py and pz.
The system (1) and (2) is to be solved for a region bounded by a surface S subject to the following boundary conditions on S,
A bar indicates a given function and equations (4) are to be understood in the sense that on that part of S where px is prescribed u is not prescribed while on theremaining part of the surface the reverse is true. Corresponding statements hold for the other two formulas in (4).
3The Variational Theorem
We introduce a function F which is defined in terms of twelve arguments x, xy, . . ., x, xy, . . . by the relation
The variational theorem may be stated in the following form.
Among all states of stress and displacement which satisfy the boundary conditions of prescribed surface displacement the actually occurring state of stressand displacement is determined by the variational equation
The symbol S1 in equation (6) indicates that each of the three surface integrals is to be taken over that part of the surface only where the appropriate surface stress isprescribed.
For the proof of the theorem it must be shown that the Euler equations of the variational problem (6) are the differential equations (1) and (2) and that the naturalboundary conditions corresponding to (6) are equations (4).
Page 439
Writing equation (6) in the form
there follows first
Elimination in (8) of partial derivatives of displacement variations by integration by parts leads to the final form of the variational equation,
where S2 is the part of S which does not belong to S1.
The integrals over S2 vanish because in them u = 0, etc. The vanishing of the volume integral requires satisfaction of the equilibrium equations (1) and of the stress-displacement relations (2) and the vanishing of the surface integrals over S1 requires satisfaction of the stress boundary conditions. It has thus been shown that thevariational equation (6), under the assumed restrictions concerning variations of boundary displacements, is equivalent to the complete system of the differentialequations (1) and (2) and of the boundary conditions (4). This is the result which was to be proved.
We may note the fact that extensions of the theorem are possible to more general systems of boundary conditions, to thermo-elastic and to dynamic problems, and toproblems of finite strain.
4An Application of the Variational Theorem
As an example of application consider the problem of deriving a system of two-dimensional equations for transverse bending of plates. Let z = ± h/2 be the equationsof the faces of the plate. Assume a distribution p(x, y) of normal stress and no tangential stress over the face z = h/2 and let the face z = h/2 be unloaded. Forsimplicity's sake assume further that all boundary conditions at the cylindrical boundary (x, y) = 0 of the plate are displacement boundary conditions.
Page 440
For an isotropic plate material which obeys Hook's Law the variational equation (6) reads then as follows
In order to obtain a two-dimensional system of equations the following expressions are used as approximations to the true stresses
Equations (11) are of the same form as those used in an earlier consideration of the same problem by means of the method of complementary energy [2]. Substitutionof (11) into the variational equation (10) permits explicit integration with respect to z in all terms which are quadratic in the stresses and suggests introduction of thefollowing weighted displacement averages
The resulting variational equation is of the form
After carrying out the variations, integrating by parts and taking account of the fact that for the assumed boundary conditions the variations of the displacementaverages vanish at the boundary, the following equation is obtained
Page 441
The contents of every bracket in (14) must vanish. The first three of the resulting equations are, as was to be expected, the differential equations of equilibrium ofplate theory. The remaining equations are the appropriate stress-displacement relations of plate theory when deformations due to transverse shearing and normalstresses are taken into account [2].
In comparing the present with the earlier derivation of the same results the fact that the equilibrium equations are now obtained from the same variational equation asthe stress-displacement relations is of little importance for the present problem since direct derivation of the equilibrium equations is simple. A considerablesimplification may, however, be thought to be the fact that the present derivation avoids the use of the Lagrangian multiplier method which was basic in the earlierwork and which made necessary special considerations in order to determine the physical significance of these multipliers.
The foregoing derivation has in common with the derivation by means of the method of complementary energy the consequence that approximations are obtained forweighted averages of the displacements rather than for the displacements themselves. In order to obtain approximations for the displacements themselves one mayuse equations (12) and in them introduce the approximations
If this is done it is found that * = , * = and .
In contrast to this it is found that if the problem is analyzed by means of the method of potential energy, approximations such as equations (15) may be made fordisplacements and the energy method furnishes equations for weighted averages of the stresses [3]. In view of the fact that the various terms in the expressions for thestrain energy are obtained in the potential energy method by differentiations of expressions such as (15) while no such differentiations are required with the use of thecomplementary energy method it is likely that the complementary energy method results in a more accurate system of equations than does the potential energymethod.
In addition to this a certain difficulty appears in the use of the potential energy method for the problem of plate bending in connection with the approximation. It is found that in order to be able to use this approximation the further unrelated and formally incompatible assumption must be made that in the general
expression for the strain energy the terms with z are first eliminated by setting in the stress strain relations z = 0, making for an isotropic medium z = [ /(1 )] ( x
+ y).
No such difficulty occurs in the application of the present variational theorem. It is in fact possible to use simultaneously in the variational equation (10) theapproximations (11) for the stresses and the approximations (15) for the displacements. The result is a variational equation of the same form as (14) except that in it
Page 442
, and w0 are replaced by the corresponding starred quantities. This apparent greater flexibility of the variational theorem obtained here may possibly be founduseful in connection with still other problems of the theory of elasticity.
References
[1] See for instance E. Trefftz in Handbuch der Physik, vol. 6, p. 73, Berlin 1927, or R. Courant and D. Hilbert, Methoden der mathematischen Physik, vol. 1,p. 228, Berlin 1931, or I. S. Sokolnikoff, Mathematical theory of elasticity, p. 284, New York 1946.
[2] E. Reissner, On bending of elastic plates, Quarterly of Applied Mathematics 5, 5568, (1947).
[3] F. B. Hildebrand, E. Reissner, and G. B. Thomas, Notes on the foundations of the theory of small displacements of orthotropic shells, NACA T.N. No.1833 (1949).
Page 443
On a Variational Theorem for Finite Elastic Deformations[J. Math. & Phys. 32, 129135, 1953]
1Introduction
In the following we formulate a variational theorem of the theory of finite elastic deformations, which is characterized by the fact that the Euler equations of thevariational problem consist of the differential equations of equilibrium and the stress-displacement relations and for which stress and displacement boundaryconditions are natural boundary conditions.
A corresponding result for infinitesimal deformations has been indicated earlier [1, 2, 3]. It is found here that generalization to finite deformations is quite direct if oneworks with the notion of stress introduced by Trefftz [4] and further developed by Kappus [ 5].*
On the basis of the theorem for three-dimensional elasticity it is possible to state analogous relations for beams, plates and shells. We do this here for the problem offinite bending of plates. The appropriate variational equation is such that from it one may obtain by specialization results previously given by Kirchhoff [6], Marguerre[7], and Wang [8].
2Differential Equations for Finite Elastic Deformations
Let xm be the cartesian coordinates of a point of a body before deformation and let r be the corresponding radius vector, written in the form,
Let r + u be the radius vector to the same material point after deformation and write the displacement vector u in the form
An infinitesimal rectangular parallelepiped with edge vectors ( r/ xj) dxj = ijdxj in the undeformed state is deformed into a parallelepiped which in general is notrectangular. Its edge vectors are [ (r + u)/ xj] dxj = [ij + ( u/ xj)] dxj = gjdxj where
*This was first reported in a colloquium at Brown University on March 14, 1952. The same result was found independently by B. Frays de Veubeke.
Page 444
Components of finite strain ejk may be defined in terms of the lattice vectors gj by means of the relations 2ejk = gj.gk jk, or
Let sj be the force, per unit of undeformed area, acting on an element of area which before deformation was perpendicular to the xj-direction. Components of(pseudo) stress are defined, following Trefftz [4], by writing
Introduction of (5) and (3) into the force equilibrium equation sj/ xj + f = 0 leads to the following three scalar equilibrium equations
The conditions of moment equilibrium for the deformed parallelepipedon are of the form
We shall assume in the following that the components of body force intensity m may be written in terms of a function (x1, . . . ; u1, . . .) in the form
The system of equations (6), (7) and (8) is completed by the stress strain relations which for the present purposes are taken in the form
where W is a given function of its arguments smn.
For the formulation of boundary conditions we require a component representation for surface forces. Let pm be the xm-component of surface force, per unit ofundeformed surface area, acting on an element of area which before deformation was perpendicular to a vector . The condition of force equilibrium for aninfinitesimal tetrahedron before deformation with sides perpendicular to the vectors and im gives
Let S1 be that part of the surface of the body where stresses are prescribed and S2 that part of the surface where displacements are prescribed. The boundary valueproblems of the theory of elasticity with finite deformation which we consider here consist of the differential equations (6) and (9) together with the boundaryconditions
where and are given functions.
Page 445
3The Variational Theorem
The following theorem will be proved. The state of stress and displacement which satisfies the differential equations of equilibrium and the stressdisplacement relations in the interior of the body, and the conditions of prescribed stress on the part S 1 and of prescribed displacement on the part S2 ofthe surface of the body, is determined by the variational equation
In order to see the validity of (12) it is noted that according to the rules of the calculus of variations and in view of the definitions of the functions W and thevariational equation (12) is equivalent to,
From (4) we have for the variations of the components of strain ejk,
With (14) we obtain by suitable integration by parts,
Since sjk = skj equation (15) may also be written in the form
The surface integral in (16) may be rewritten in terms of pm according to (10). Introduction of (16) and (13) gives then
Page 446
Since in the interior of the body we have sjk and um arbitrary, on the surface S1 um arbitrary and on S2 pm arbitrary there follows that the variational equation(17) is equivalent to the differential equations (6) and (9) and to the boundary conditions (11). This proves the theorem.
4Stress Strain Relations
For many problems for which non-linearity due to finite deformations must be considered although the strains themselves are very small, it is adequate to assumelinear relations between the components of pseudo stress sjk and the components of finite strain ejk. If in addition it is assumed that the material is isotropic then wemay write
where E is the modulus of elasticity and is Poisson's ratio. For a material with the stress strain relations (18) the function W introduced through equation (9) is of theform
As an example of stress strain relations for finite strain we may consider the stress strain relations for an incompressible isotropic material which has been termed neo-Hookean. According to Rivlin [9] the relations between components of principal strain and principal stress for this material are of the form
where p is of the nature of a hydrostatic pressure the magnitude of which follows by application of the condition of incompressibility,
The determination of p from (20) and (21) requires the solution of a cubic equation.
In view of the definition of the components of pseudo stress smn the relation between the component of principal stress tI and the component of principal pseudostress sI is of the form
with corresponding relations between tN and sN. For an incompressible material equation (22) may be written alternately as
Therewith the stress strain relations (20) assume the form
Combination of (24) with the incompressibility relation (21) next gives the following explicit expression for p
Page 447
and with this the stress relations (24) may be written as
Since eN = W/ sN we have then for the function W
In order to obtain relations between ejk and sjk from (27) we introduce in (27) the invariants
In terms of these,
If (29) is developed according to powers of s/E the result is, up to third-degree terms,
Retention of only second degree terms leads back to the linear relations (18) between sjk and ejk, with having the value 1/2.
5Finite Bending of Plates
The variational equation corresponding to (12) for finite bending of plates as discussed by Kirchhoff [6] and von Karman [10] is of the following form,
Page 448
In (31) u, v, w are components of displacement of points of the middle surface, Nx, Ny, Nxy, Q are stress resultants and Mx, My, Mxy stress couples, is the directionin the plane of the undeflected middle surface normal to its boundary, and
and Mx and My are defined correspondingly. The expression for Q is
and Q is a force in the direction of z, while Vx and Vy are transverse shear stress resultants in the direction of the normal to the deformed middle surface, given by Vx
= Mx/ x + Mxy/ y and Vy = Mxy/ x + My/ y. Note that although is obtained from Q by putting bars on V and N but not on w.
Elimination of all stress resultants and couples from (31) by means of the stress strain relations and comparison of only such displacements as satisfy all prescribeddisplacement boundary conditions, together with the assumption that = X u + Y v + p w leads to the minimum-potential-energy equation for finite bending of platesas given by Kirchhoff [6].
Introduction in (31) of an Airy stress function F for Nx, Ny, Nxy, together with the assumption that = pw, elimination of the displacement components u and v andcomparison of only such states as satisfy the boundary condition for Nx, Ny, and Nxy and w leads to a variational equation given by Marguerre [7]. The Eulerequations of this variational equation are the two simultaneous differential equations for F and w given by von Karman [10].
Finally, by restricting admissible states to those satisfying all equilibrium equations, equation (31) may be reduced to a recent result by Wang [8].
Equation (31) may be generalized in such a way that the effect of transverse shear stress deformation is taken into account. To this end we must replace in (31) theexpression Mx 2w/ x2 My 2w/ y2 2Mxy 2w/ x y by
and make corresponding changes in the boundary integrals. Furthermore we must add in the double integral the terms
The quantities and are projections on the x, z and y, z-planes, respectively, of the angle which the deformed normal encloses with the z-axis.
References
1. Reissner, E., On a variational theorem in elasticity. J. Math. and Physics 29, 9095 (1950).
2. Reissner, E., Stress-strain relations in the theory of thin elastic shells . J. Math. and Physics 31, 109119 (1952).
3. Reissner, E., On Non-uniform torsion of cylindrical rods. J. Math. and Physics 31, 214221 (1952).
4. Trefftz, E., Uber die Ableitung der Stabilitätskriterien des elastischen Gleichgewichts aus der Elastizitätstheorie endlicher Deformationen. Proc. IIIIntern. Congr. Appl. Mech., vol. III, 4450 (1931).
Page 449
5. Kappus, R., Zur Elastizitätstheorie endlicher Verschiebungen . Z. ang. Math. and Mech. 19, 271285, 344361 (1939).
6. Kirchhoff, G., Vorlesungen über Mechanik, Dreissigste Vorlesung . Leipzig, Teubner 1876, 1883 and 1897.
7. Marguerre, K., Die mittragende Breite der gedrückten Platte. Luftfahrtforschung 14, 121128 (1937).
8. Wang, C. T., Principle and application of complementary energy method for thin homogeneous and sandwich plates and shells with finite deformations.N.A.C.A., T.N. 2620 (1952).
9. Rivlin, R. S., Large elastic deformations of isotropic materials . Phil. Trans. Roy. Soc. 240, 459490 (1948).
10. von Karman, T., Festigkeitsprobleme im Maschinenbau, Encyclopädie der mathematischen Wissenschaften 4, 1910, p. 349.
Page 450
A Note on Variational Principles in Elasticity[Int. J. Solids Structures, 1, 9395, 1965]
We are conerned in what follows with a class of extensions of known variational principles for boundary value problems of the differential equations of elasticity
together with
or
In this Ad and As are given functions of the six variables x, xy, etc. or x, xy, etc., respectively, these variables being defined in such a way that
in terms of displacement components, and
Omitting for the sake of brevity a discussion of boundary conditions we have that equations (1) and (2b) are the Euler equations of a variational equation of the form where
with x, xy, . . ., xy, yx, . . . defined in accordance with (3) and (4) and with the six components of stress and the three components of displacement being variedindependently. Imposition of appropriate partial constraint on the nine variation variables reduces the general variational equation involving F1 to the minimumprinciples of complementary and potential energy, respectively [1].
A more general variational equation than the one for stresses and displacements has been formulated by Washizu [2], by changing the status of the defining equations
for strains (3) into additional Euler equations of the variational equation. Washizu's equation is of the form where
and where now the six components of strain, the six components of stress and the three components of displacement are varied independently.
Page 451
The principal purpose of the present note is the formulation of two variational equations in which the three moment equilibrium equations (4) for components of stressare Euler equations of a variational principle, rather than equations of definition. This purpose is accomplished by defining nine components of strain in terms of threecomponents of linear displacement and three additional components of angular displacement, as follows
and by replacing the six stress-strain relations (2) by nine stress-strain relations of the form
or
It is now readily seen that the variational equation involving F1 may be generalized to a variational equation where
with x, xy, etc., defined by (7) and with the nine components of stress x, xy, yx, etc., and the six components of displacement ux, x, etc., being variedindependently.
Analogously, the variational equation involving F2 may be generalized to a variational equation where
and where now there are altogether nine independent strain variations, nine independent stress variations and six independent displacement variations.
Further generalizations of the variational equations involving G1 and G2, so as to take account of body forces, finite deflections, time dependence, couple stresses,etc. may be obtained by following the same procedures as with F1 and F2.
It remains to establish the form of the functions Bd( x, xy, yx, . . .) and Bs( x, xy, yx, . . .) which corresponds to the form of the functions Ad( x, xy, . . .) andAs( x, xy, . . .) for the case that the moment equilibrium equations (4) are equations of definition rather than Euler equations.
In order to determine Bd we take account of the fact that the stress-strain relations (8a) should be compatible with the Euler moment equilibrium equations.Accordingly we have the relations
and these, as partial differential equations in the independent variables xy, yx, etc., imply that Bd depends on the variables xy, yx, etc., as a function of the sums xy
+ yx, etc. Since xy + yx = xy we may then identify the function Bd with the customary function Ad, except that in this xy needs to be considered explicitly as thecombination xy + yx, with xy and yx as independent variations.
In order to determine the form of Bs we take account of the fact that the angular displacement quantities z, etc., and the linear displacement quantities ux, etc.,should satisfy the relations
Page 452
In view of equation (7) this means that the stress-strain relations (8b) should be compatible with the relations yx xy = 0, etc., or with
The partial differential equations (11b) indicate that Bs must depend on the six independent variables xy, yx, etc. as a function of the sums xy + yx, etc. Accordinglywe have that Bs is obtained from As by replacing xy, xz, yz in it by 1/2( xy + yx), 1/2( xz + zx), 1/2( yz + zy).
In order to illustrate the meaning of this requirement we consider the case of a linear isotropic medium. For this case xy, etc. occurs in As in the combination
A corresponding invariant term in Bs is
with k an arbitrary constant.
The requirement that Bs be a function of the sums xy + yx, etc. means that, necessarily, we must have
Possible applications of the variational equations or depend on the idea that it may be of advantage, in the approximate solution ofproblems by the direct methods of the calculus of variations, to satisfy moment as well as force equilibrium equations approximately only, instead of satisfying one set
exactly and the other approximately. One such application of a special form of the equation is to the derivation of approximate stress-strain relations forthin elastic shells [3].
References
[1] E. Reissner, On some variational theorems in elasticity, Problems of Continuum Mechanics (Muskhelisvili Anniversary Volume), pp. 370381, Philadelphia(1961). This paper also contains references to earlier work on the subject.
[2] K. Washizu, On the Variational Principles of Elasticity and Plasticity , Report No. TR2518, Aerospace and Structures Research Laboratory, M.I.T. (1955).
[3] E. Reissner, On the form of variationally derived shell equations, J. Appl. Mech. 31, 233238 (1964).
Page 453
A Note on Günther's Analysis of Couple Stress*
[Mechanics of Generalized Continua, IUTAM Symp. Freudenstatt-Stuttgart, pp. 8386, Springer Verlag 1967]
The following considerations are concerned with Günther's form of couple stress theory [1]. While the observations which follow were made without knowledge ofthe earlier work, they are offered here as a supplement to it.
We assume an orthogonal coordinate system xi, with position vector x and with coordinate tangent unit vectors ti = x,i/ i where x,i·x,j = i j ij. We designate forcestress vectors by i, moment stress vectors by i and body force and moment intensity vectors by p and q. We take as basic relations the two equations of force andmoment equilibrium
where V = 1 2 3, S1 = 2 3, etc.
We next introduce force and moment strain vectors i and i, and translational and rotational displacement vectors u and . We obtain expressions for i and i interms of u and through use of the principle of virtual work, written in the form
In this p and q are taken from (1) and n and n are surface traction vectors. Integration by parts to eliminate derivatives of i and i in (2) and observation that i
and i are arbitrary functions in the remaining volume integral leads to the vectorial strain displacement relations
which are equivalent to relations stated by Günther [1]. The present approach and Günther's approach differ from each other inasmuch as Günther departs from (3)and obtains (1) by use of the principle of virtual work. It seems to us easier to depart from (1), which is in accordance with elementary principles of dynamics, andavoid the less elementary geometrical considerations which are involved in stipulating (3).
From (3) follows by inspection, as noted previously in [1], a system of six vectorial compatibility equations
*With F. Y. M. Wan.
Page 454
We complement Günther's formulation by phenomenological stress strain relations as follows. Writing
we stipulate the existence of functions A( , ) and B( , ) such that
An appropriate definition of vectorial derivatives allows us to write (7) in the form
Observing (1), (3) and (7) we readily have as variational principles, generalizing corresponding principles of elasticity without couple stresses, a variational principlefor stresses and displacements, ISD = 0, and a variational principle for stresses, displacements and strains, ISDS = 0, where
and
In ISD stresses and displacements are varied independently and the Euler differential equations consist of the equations of equilibrium and the stress strain relations.In ISDS stresses, displacements and strains are varied independently and the Euler differential equations are equilibrium equations, strain displacement relations andstress strain relations1.
A recent result [4] on a static-geometric analogue of the two-dimensional elastic-shell theory version of the variational principle for stresses and displacements [3]suggests the formulation of a variational principle for strains and stress functions in couple stress elasticity, such that the compatibility Eqs. (4) and (5), together withthe stress strain relations, are Euler differential equations.
1A restricted form of the variational principle ISDS = 0 has previously been stated by Naghdi [2]. The restriction consists in assuming symmetry conditions ij = ji as equations of definition. Asa consequence of this, an additional Euler differential equation = 1/2 curl u is obtained, the force stress strain relations become 1/2( ij + ji) = A/ ij = A/ ji and the number of associated Eulerboundary conditions is five instead of six.
Page 455
A small amount of mathematical experimentation indicates that a suitable stress function representation of the solutions of the homogeneous Eqs. (1) is
Taking Eqs. (10) and (11) as equations of definition, we find that the variational equation
where the Fi, Hi, i and i are varied independently does in fact have the compatibility Eqs. (4) and (5) and the stress strain relations (7) as Euler equations. Inaddition, it is found that the associated Euler boundary conditions are displacement boundary conditions expressed in terms of tangential strain vectors. Theseconditions are homogeneous conditions which may be made non-homogeneous conditions upon adding in (12) an appropriate surface integral.
We state two further variational principles which we think will eventually be found useful. The first of these is a mixed principle, in the sense that it has parts of theequilibrium equations and compatibility equations as Euler equations, while the complementary parts are equations of definition.
Restricting attention to the case of absent body forces and moments, we take as equations of definition the conditions of force equilibrium in (1), via the stressfunction Eqs. (10), and the equations for bending strains in (3), these being equivalent to the compatibility Eqs. (4). Additionally, we now take the relations betweenstresses and strains in the mixed form
as equations of definition.
We then have that the equation IM = 0 where
and where and the Fi are varied independently has as Euler differential equations the conditions of moment equilibrium in (1) together with the force straincompatibility Eqs. (5).
The second additional variational principle is for boundary values. It is a generalization of a principle previously stated for elasticity without couple stresses [5]. Definea functional
Page 456
where u, , n and n are the surface values of states of displacement and stress which in the interior satisfy equilibrium and stress displacement relations, subject tothe limitation of absent body forces and moments and subject to the limitation that the stress energy function B in (7) is homogeneous of the second degree in thesense that i· B/ i + i· B/ i = 2B. It can then be shown, in extension of what is done in [5] for elasticity without couple stresses, that the Euler equations of IB
= 0 are the boundary conditions.
References
[1] Günther, W.: Abh. d. Braunschweigischen Wiss. Ges. 10, 195213 (1958).
[2] Naghdi, P. M.: J. Appl. Mech. 31, 647652 (1964).
[3] Reissner, E.: Proc. American Soc. Civil Eng. 88 (EM), 2357 (1962).
[4] Wan, F. Y. M.: J. Math. and Phys. 47, 429431 (1968).
[5] Reissner, E.: Problems of Continuum Mechanics (Muskhelishvili Anniversary Volume), Philadelphia 1961, pp. 371381.
Page 457
On a Certain Mixed Variational Theorem and a Proposed Application[Int. J. Num. Meth. Eng. 20, 13661368, 1984]
Introduction
In this note we report briefly on the formulation of a variational equation for displacements and some stresses, with the thought that this equation will prove useful forthe approximate analysis of isotropic and anisotropic laminated elastic plates [1]. The basic thoughts in the formulation of this equation are (1) that it is inconvenientto use approximations for the primary bending and stretching stresses in the approximate analysis of laminated plates, and (2) that it is allowable to make relativelycrude approximative assumptions concerning transverse stresses, in view of the fact that these are of a smaller order of magnitude than the primary bending andstretching stresses. The result of our consideration is a variational theorem for the three components of displacements and for three of the six components of stressin three-dimensional geometrically linear elasticity.
Derivation
We begin with a statement of the classical variational equation for displacements, for simplicity's sake subject to the assumptions of absent body forces and traction-free boundary portions z = ± h/2, and of displacement boundary conditions over all cylindrical boundary portions (x1, x2) = 0. With a view towards our ultimatepurpose, we write this equation in the form
where ij = 1/2(ui,j + uj,i), = uz,z and i = ui,z + uz,i, for i, j = 1, 2.
To obtain the intended result we begin by rewriting (1), with the help of Lagrange multipliers and i, as
and we separate the function U in this into two parts
such that U0( ij) = U( ij, 0, 0) and U1 = U U0.
We next use three of the Euler equations associated with (2),
as three simultaneous equations for the determination of , 1, 2 in the form = ( , i, ij), j = j( , i, ij), and we define a complementary function W through
Page 458
a partial Legendre transformation
where then, in the usual way, = W/ and i = W/ i.
Introduction of (5) and (3) into (2) gives as the wanted mixed variational theorem the equation
with arbitrary ui, uz, i, , and with ij = 1/2( ui,j + uj,i).
An alternative version of (6), with the variations and i restricted so as to be consistent with constraint conditions ( , i)z =± h/2 = 0 and i,i + ,z = 0, evidentlymay be deduced from (6) so as to read
The Physically Linear Problem
We have for this that U0 is a second-degree polynomial in the ij and that U1 may be written in the form
with U11 being linear homogeneous in ij, , i, and U12 a homogeneous second-degree polynomial in and i. An introduction of (8) and (4) into (5) andobservation of the consequences of the indicated homogeneity properties then gives as expression forW,
For the example of a transversely isotropic material with
and
the function W in the variational equation (6) follows from (9), with
in accordance with (9), (8) and (11) in the form
The same result also follows, somewhat less conveniently, by a direct evaluation of (5). For the general case, with U1 in (8),
equations (4a, b) take on the form
Page 459
The function W in (6) is now
with , 1 and 2 expressed in terms of , 1, 2 and ij by way of the solution of the three by three system (15a, b, c).
Derivation of an Approximate Linear Two-Dimensional Sixth-Order Laminated Plate Theory
We assume material symmetry relative to the x1, x2-plane* and we stipulate as expressions for displacements
At the same time we stipulate as expressions for transverse stresses
where the even function may, in the absence of reasons for a better choice, be approximated by 1 (2z/h)2, and we carry out the indicated integration in (6), with Was in (9), with respect to the thickness variable z. With appropriate defining relations for stress couples and resultants, the remaining two-dimensional variationalequation will then have the wanted sixth-order theory as its system of Euler equations.
Concluding Comment
The possibility of obtaining more general higher order results, involving plate surface loads, absent material symmetry relative to the plane z = 0, as well as bodyforces, in particular inertia forces associated with simple harmonic motion for physically linear theory, is apparent. Similarly, a corresponding curvilinear co-ordinateformulation, to be used in connection with the derivation of approximate two-dimensional shell theories, is an evident possibility.
It is conceivable that the variational equation (6), and its generalization to the case of curvilinear co-ordinates, may also prove useful in connection with thedevelopment of finite element procedures.
References
[1] E. Reissner, 'Note on the effect of transverse shear deformation in laminated anisotropic plates', Computer Meth. Appl. Mech. Eng. 20, 203209 (1979).
*As regards equations (14), this means that Cij, Bij and B must be even functions of z, while Cijk and Bi must be odd functions of z.
Page 460
On a Variational Principle for Elastic Displacements and Pressure[J. Appl. Mech. 51, 444445, 1984]
Introduction
In what follows we obtain a generalization of a variational principle in geometrically linear elasticity, for displacements ui and for a suitably defined pressure variablep, which is known to be of technical significance for incompressible or nearly incompressible materials. Recognition of the possibility and importance of such aprinciple for the case of physically linear isotropic elasticity, as well as its formulation for this case, is due to L. R. Herrmann [1]. A generalization of Herrmann'sresult to the case of anisotropic physically linear materials has been given by S. W. Key [2]. Our purpose here is a simple and direct derivation of the correspondingresult for physically nonlinear materials.
Derivation
We depart from a statement of the variational principle for displacements, in the form
where ij 1/2(ui,j + uj,i) and where, for simplicity's sake, we assume that body forces are absent and that all boundary conditions are displacement conditions. As iswell known, equation (1), with the further defining (or constraint) relations ij = U/ ij, has as its Euler equations the differential equations of equilibrium for stress.
To retain the technical significance of (1) for cases for which the form of U is such as to make the sum kk exactly or nearly equal to zero we introduce, followingHerrmann and Key, the alternate strain variables
where
and we write with this
Having (3) and (4) we rewrite (1), with the introduction of a multiplier p, in the form
Page 461
We will transform (5) into what is wanted by stipulating p to be given by
and by using the inversion = ( ij, p) of this to define a complementary function
An introduction of U from (7) into (5) transforms (5) into
where now, in view of (2) and (3), ij = ij 1/3 ij kk, with independent ui and p, and with this being in essence the desired result.
Given that (8) is readily shown to be equivalent to the relation
we, incidentally, have as a ''parametric" version of the constitutive relations ij = U/ ij,
Determination of the Complementary Function
Determination of W for the case of a physically linear material, with U = U1( jj) + U2( ij) where U1 and U2 are homogeneous first and second-degree polynomials,respectively, is a simple matter. We can now write, in accordance with (4)
where U10 and U20 may depend on xi, but not on the ij, and, in accordance with (6),
The introduction of (12), (11) and (4) into (7) gives
and therewith
Page 462
A corresponding explicit determination of W, in accordance with (6) and (7), for physically nonlinear materials, is in general not possible. An obvious exception tothis difficulty is given by the class of materials for which U = U1( ij) + U2( ij) + ( kk)2F( 12, 13, 23).
References
1. Herrmann, L. R., "Elasticity Equations for Incompressible or Nearly Incompressible Materials by a Variational Theorem," AIAA J., Vol. 3, 1965, pp. 18961900.
2. Key, S. W., "A Variational Principle for Incompressible or Nearly Incompressible Anisotropic Elasticity," Int. J. Solids Structures, Vol. 5, 1969, pp. 951964.
*This result is consistent with the contents of equation (16) in [2], upon writing , and discarding the additive quantity in (14),which is of no relevance for the variational statement.
Page 463
On Mixed Variational Formulations in Finite Elasticity[Acta Mechanica 56, 117125, 1985]
1Introduction
In what follows we place on record two variational formulations for the equations of finite elasticity, with arbitrary variations of translational and rotationaldisplacements and arbitrary variations of some stresses. These formulations have the property that for them force and moment equilibrium equations as well as someof the constitutive equations come out to be Euler equations, with the remainder of the constitutive equations, together with the strain displacement relations, beingequations of constraint. The nature of our results is such that they may be considered natural generalizations of two recent results, one of them dealing with variationalformulations of finite elasticity, with arbitrary variations of displacements and all stresses [5], and the other with a variational formulation of infinitesimal elasticity,with arbitrary variations of displacements and some stresses [6].
2Differential Equations and Boundary Conditions
We write
for the radius vectors to material points in their initial and finite positions and we stipulate the existence of vectors of stress i, as forces per unit of undeformed area,acting on elements of area with unit normal vectors ei in their initial state. We further stipulate that the state of stress in the deformed medium is subject to theequations of force and moment equilibrium
For present purposes we assume that the vectors i have a component representation
with the tj being a triad of mutually perpendicular unit vectors which we take in the form
We designate the ij as generalized Piola components of stress, given the designation "Piola components" for the special case ij = ij. While we may consider thecoefficients jk as given functions of xi, in what follows they will be considered as dependent "rotational displacement" variables, to be determined in conjunctionwith the translational displacement variables ui and the stress variables ij as a consequence of a constitutive stipulation.
Page 464
We further write, on the basis of Eqs. (1) and (4),
and then, on the basis of (2) to (5), with = tn n and mk jk = emjn n,
We take (6) to imply conjugacy of the ij and ij and we designate the generalized displacement gradient components ij as components of strain, conjugate to thecomponents of stress ij.
Given the above statical and geometrical relations we here limit ourselves to a consideration of the class of materials with single-valued constitutive relations
in terms of a complementary strain energy density function
with this form of the constitutive equations selecting from the class of generalized Piola stress components a system of distinguished generalized Piola components.We take our assumption of single-valuedness of the derivatives in (7) as an assumption concerning material properties, with this assumption not necessarily holding for"more general cases" for which relations ij = W/ ij with ij ji, are deduced, by a Legendre transformation, from single-valued relations ij = U/ ij.
The nine scalar constitutive relations (7) are associated with the nine scalar strain displacement relations in (5), and with six scalar equilibrium relations which follow(2) to (5), with , in the form
Altogether (9), (7), and the second sets of relations in (5) and (4) represent a system of thirty first and zeroth order differential equations for the determination of thirtydependent variables ij, ij, ij and ui.
Boundary conditions which are associated with this system are here taken to be conditions of prescribed surface displacements
over boundary portions Su, and conditions of prescribed surface stresses
over boundary portions S .
The traction components k are given in terms of the surface values of the stress components ij and in terms of the components i of the unit normal vectors toundeformed surface elements. With kek = = i i we have, with (3) and (4)
Having earlier noted that for the ij to be dependent variables requires a supplementary constitutive statement we may now say that this requirement consists in thestipulation that W depends on the six quantities 11, 1/2( 12 + 21), . . . rather than on the nine quantities 11, 12, 21, . . ., with the physico-geometrical aspects of
Page 465
this assumption having been discussed in [5]. We also note the equivalence of our distinguished generalized Piola components ij, with components as introduced
by Biot [2], with Biot designating the combinations as "alternative" stress components. Related considerations, differing in various details may be found in[3], [4] and [7].
3A Variational Formulation for Stresses and Displacements
We have, upon appropriate synthesis, in complete formal analogy to our original formulation in terms of Kirchhoff-Trefftz stresses and Green strains [1], that thevariational equation
with W as in (8), with ij as in (5), with k as in (11), and with the ij constrained in accordance with (4), is such as to have as Euler differential equations the sixequilibrium equations in (9) and the nine constitutive equations in (7), and as Euler boundary conditions the conditions in (10) and (11).
To see this we first deduce from (13) that
with the variations ij, uk and k arbitrary and with the variations ij having to be such that mk jk + jk mk = 0.
It is readily evident, on the basis of the form of (14), that the constitutive relations (7) as well as the force equilibrium conditions in (9) and the boundary conditions in(10) and (11) are Euler equations of this variational equation. To see that the moment equilibrium conditions in (9) are also Euler equations associated with (14) wehave earlier made use of the constraint relations ik jk = ij through the use of Lagrange multipliers [5]. The same conclusion follows, without use of such multipliers,upon observing that the antisymmetry amj = ajm of the quantity amj mk jk implies the possibility of writing, in terms of three arbitrary variations n, mk jk
= emjn n. In as much as we can, in (14), write
This then implies that the moment equilibrium equations in (9) are in fact also Euler equations of (13).*
*I am indebted to H. Bufler for suggesting this alternate derivation of the equations of moment equilibrium as part of the Euler equations of (13).
Page 466
4A Variational Formulation for Displacements and Reactive Stresses [5]
For the derivation of one version of the intended mixed variational formulation we make use of a variational equation which involves a strain energy density functionU( 11, 12 + 21, . . .) in terms of which
with ij as in (5), and with Euler equations which consist of the differential equations of equilibrium (9), of the boundary conditions (11), and of the kinematicalconditions ij = ji. This variational equation is, with the reactive stress quantities ij ji,
In evaluating (17) we stipulate ui, ( ij ji), and n in mk jk = emjn n to be arbitrary in the interior, with ui arbitrary over S , and with and ui = 0over Su.
The correctness of our statement is readily verified upon showing that (17), in conjunction with (16), is equivalent to the relation
with ij following from (5), in the same way as in going from Eq. (13) to Eq. (14).
We complement the above by noting the possibility of a formal derivation of (17) (with the more general boundary conditions in Eq. (60) in [5]), in place of thesynthesizing procedure used in [5]. This formal procedure consists in the use of a Legendre transformation relation of the form
with
and with the reactive stress measures ij ji then appearing in the resulting variational equation automatically.
5A Mixed Formulation for Displacements and Some Stresses
In generalization of our consideration for infinitesimal elasticity [6] we now derive a variational formulation of our finite-elasticity boundary value problem whichinvolves independent variations of displacements and of the "transverse" stresses 13, 31, 23, 32 and 33, as follows as a consequence of (17).
We separate the function U in (17) into two parts
such that
Page 467
We then use the three constraint equations
as three equations for the determinations of 13 + 31, 23 + 32, 33 in terms of 13 + 31, 23 + 32 and 33, and thereafter define a partial complementary energyfunction W1 through the partial Legendre transformation
Upon taking account of the dependence of 13 + 31, 23 + 32, 33 on 1/2( 13 + 31), 1/2( 23 + 32), 33 in accordance with (23) we then find as the desired partialset of inverted constitutive relations
Our new mixed variational formulation follows upon substituting U from (21), with U1 as in (24), into Eq. (17). This leaves, after appropriate cancellations, as thedesired formulation
In evaluating (26) we again consider ij given in terms of uk and jk, with arbitrary uk and with jk such that mk jk = emjn n with arbitrary n. Wefurthermore take ( 12 21), 13, 31, 23, 33 and 33 as arbitrary in the interior, and uk arbitrary over S .
As regards examples of the determination of the function W1 for a given function U we here limit ourselves to making reference to the work in [6]. As in [6], weanticipate useful consequence of our formulation, now for problems of finite deflections of laminated plates.* Also as in [6], we note the evident possibility ofanalogous results in terms of curvilinear coordinates, for prospective applications to problems of finite deflections of shells.
6An Alternate Version of the Variational Formulation for Displacements and Some Stresses
We depart from (13) instead of from (17) and, with the notation introduced in (20), invert the three relations
*The first application of our formulation in [6] is contained in recent work by H. Murakami [8].
Page 468
in the form
With this we then define a new "semi-complementary" function V( 11, , 22; , , 33) through the relation
which implies, in the usual way
The introduction of W from (29) into (12) then gives as the alternate version of (26)
It is of interest to note that for the class of problems for which
with W0 homogeneous of the second degree in its arguments, and W1 homogeneous of the first degree, the function V becomes, upon making use of (24),
In this the arguments 11, , 22, are obtained, in accordance with (27), from a system of three simultaneous linear equations
For the special case of a transversely isotropic medium, with
where E = 2(1 + )G, and
Eq. (34) is the same as
Page 469
together with . The introduction of the latter relation, together with
into (35) then gives as the form of the function W0 in Eq. (32)
with W2 in (32) remaining in the form (36).
References
[1] Reissner, E.: On a variational theorem for finite elastic deformations. J. Math. & Phys. 32, 129135 (1953).
[2] Biot, M. A.: The mechanics of incremental deformations, pp. 482485. New York: John Wiley 1965.
[3] Frayes de Veubeke, B.: A new variational principle for finite elastic deformations. Intern. J. Eng. Sc. 10, 745763 (1972).
[4] Bufler, H.: On the work theorems for finite and incremental elastic deformations. Comp. Meth. Appl. Mech. Eng. 36, 95124 (1983).
[5] Reissner, E.: Formulation of variational theorems in geometrically nonlinear elasticity. J. Eng. Mech. (ASCE) 110, 13771390 (1984).
[6] Reissner, E.: On a certain mixed variational theorem and a proposed application. Intern. J. Num. Meth. Eng. 20, 13661368 (1984).
[7] Atluri, S. N.: Alternate stress and conjugate strain measures and mixed variational formulations involving rigid rotations, for computational analyses of finitelydeforming solids, with applications to plates and shells-1, theory. Comp. & Struct. 18, 93116 (1984).
[8] Murakami, H.: A mixture theory for wave propagation in angle-ply laminates. J. Appl. Mechanics 20, 331337 (1985).
Page 470
Some Aspects of the Variational Principles Problem in Elasticity[Comp. Mech. 1, 39, 1986]
1Introduction
In speaking of elasticity, we here mean the statement of physically meaningful boundary value problems for deformations due to applied loads in bodies which behaveelastically. By elastic behavior, we mean the local interdependence of stress and strain, with suitable definitions for what is meant by stress and what is meant bystrain. We assume in what follows an acquaintance with these definitions and with the elements of the associated analysis.
Given the computational usefulness of variational formulations of appropriately stated boundary value problems, it is our objective in what follows to survey the matterof such formulations in elasticity, in a way which reflects our long-time concern with it, emphasizing that which seems to be of the essence and, in the process,indicating certain pitfalls.
While our concern here is on matters historical, primarily, some new insights are thought to have been included. Specifically, the statement of a generalized Hellingerformulation, in terms of generalized Piola stresses, belongs in this category.
2Conventional Infinitesimal Elasticity
With the usual definitions for components of stress ij = ij and components of displacement ui, and with cartesian coordinates xi, the differential equations ofconventional infinitesimal elasticity are here taken in the form of a system of first-order differential equations, involving material-property functions P(u) and W( ), asfollows
This system is associated with boundary conditions which are here taken, with functions S(u) and D(t), in the mixed form
In this the quantities tk = mk m are cartesian surface traction components, and the boundary portions aS and aD together, without overlapping, represent the surfacea which bounds the volume v in which (1a) and (1b) hold.
Given Eqs. (1a, b) and (2a, b), one finds that the variational equation which has both these sets of relations as Euler equations is of the form
with ij and ui being altogether nine independent variations.
Page 471
For a discussion of the origin of this two-field variational equation for stresses and displacements, supplementing earlier variational equations for stresses ordisplacements, we here refer to Reissner (1983).
In connection with the associated one-field formulations for stresses or displacements, the following consequence of (3) is thought to be of particular interest. If weconsider (1a) and (2a) to be equations of constraint, we find, through an appropriate integration by parts, the modified variational equation
Equation (4) is still a variational equation for stresses and displacements, although with interdependent rather than independent variations ij and ui. A reduction to
an equation for stresses alone requires the restrictive stipulations and , with and given functions of position.
Alternately, if we take (1b) and (2b) as constraint equations and if we introduce U( ), with ij (ui,j + uj,i)/2, and B(u) through the Legendre transformations
we will have from (3), with ij = U/ ij and tk = B/ uk, as a variational equation for displacements alone
The conventional form of (6), with B = 0, results from this upon setting and .
A Four-Field Equation for Stresses, Displacements, Strains and Body Forces
Given the well-known three-field formulation, associated with the names of Hu and Washizu, which obtains upon writing (1b) in the form
in conjunction with the Legendre transformation (5a), we here note the possibility of a four-field formulation, upon writing (1a), analogously, as
An inversion of (8b), in circumstances where this is meaningful, in conjunction with the introduction of a function Q(p) = pjuj(p) P[u(p)], with which uj = Q/ pj, thenresults in a variational equation
Equation (9), with arbitrary variations ij, ij, ui, and pi, is readily seen to have (7a), (8a) and the inverted form of (7b) and (8b) as Euler differential equations,with (2a, b) remaining as Euler boundary conditions.1
Page 472
A Two-Field Equation for Strains and Displacements
If in the two-field equation (3) for stresses and displacements we write ij = U/ ij, and if we eliminate W( ) through use of (5a), we obtain from (3) as avariational equation for strains and displacements
where tj = ( U/ jk) k. Evidently, we now have that the equilibrium and strain displacement relations are Euler equations, with the constitutive relations beingconstraints, instead of having equilibrium and constitutive relations as Euler equations, with the strain displacement relations as constraints.
From the vantage point of the three-field Hu-Washizu formulation, with equilibrium, constitutive and strain displacement relations as Euler equations, (10) is animmediate consequence, upon changing the Euler constitutive relations into constraints.
We finally note that when U = (1/2)Eijkl ij kl, , , and then equation (10) reduces to a result which has previously been stated by Oden andReddy (1976).
3Infinitesimal Elasticity and the Effect of Body Moments
Given that the effect of a body moment intensity with components qk results in a de-symmetrization of the components of stress, on the basis of the vectorial equationof moment equilibrium, we now write, in terms of nine components of stress ij, as component equations of force and moment equilibrium
The change from six distinct components of stress as in (1) to the set of nine components in (11) necessitates the introduction of nine components of strain, ij, inplace of the six components ij = (1/2)(ui,j + uj,i). It has earlier been shown by Reissner (1965) that a suitable set of nine such components, involving rotationaldisplacement components i in addition to the translational components ui, is of the form
As before, the suitability of this introduction is made to depend on the possibility of obtaining the moment Eqs. (11b) in addition to the force Eqs. (11a) as Eulerequations of a variational statement which may be considered as a natural extension of the statement in (3).
We now reason, somewhat more effectively than before, where there was no introduction of moment load components qk, as follows. Inasmuch as (11b) indicatesthat the tangential stress component differences ij ji have a reactive character,
1The above was written with the thought that the concept of the four-field equation had not previously been considered. Subsequently, the author was made aware of such a consideration in Bufler(1979). This, in turn, reawakened memories of the contents of a set of hectographed lecture notes, written sometime between 1960 and 1965, which also contained the variational Eq. (9).
Page 473
these differences should not enter into relations of constitutivity, that is, the material behavior function W( ) should again depend on six measures of stress, 11, 12 +21, etc. It turns out to be convenient to write
and then to stipulate, in place of (1b), as a set of constitutive equations
Given (11a, b) as a point of departure, it is possible to write, in analogy to (1a), as a more general set of equilibrium equations
where now P is a given function of the uj and k. However, with the step from (11b) to (15b), we will no longer be able to say that the ij ji are reactive; and we canthen no longer pre-suppose, on the basis of the same reasoning, that W must be restricted as in (13).
With boundary conditions as in (2a, b) except for writing mk in place of mk, it is then possible to stipulate a variational equation for stresses, translationaldisplacements and rotational displacements, of the form
with ij, uj, and k being independent variations. The verification that (15a, b) and (14) are Euler differential equations of (16), and (2a, b) are Euler boundaryconditions, can be carried out readily.
4Differential Equations for Finite Elasticity
In establishing equations for finite-deformation elasticity, we begin with expressions
for position vectors before and after deformation. We then note that the vectors z,1dx1, z,2dx2, z,3dx3 are the edge vectors of deformed elements of volume which intheir undeformed state have edge vectors e1dx1, e2dx2, e3dx3.
We next define stress vectors i acting on the faces of the deformed element of volume, as well as body force and body moment vectors p and q associated with thiselement. We take stress vectors as forces per unit of undeformed area, and we take body force and moment vectors as forces and moments per unit of undeformedvolume. With these stipulations we have vectorial differential equations of force and moment equilibrium which read
For an elastic body (18a, b) must be supplemented by relations which involve the stress vectors i, with the rates of change u,j of a displacement vector u = z x.This will be done here in two distinct ways, one of them being the classical procedure associated with the names of Kirchhoff and Trefftz, and the other being amodification of a classical procedure generally associated with the name of Piola.
Page 474
5Kirchhoff-Trefftz Components of Stress and the Associated Variational Equation for Stresses and Displacements
Following Kirchhoff and Trefftz, we restrict consideration to problems for which q = 0 and we stipulate a component representation
The introduction of (19), in conjunction with component representations,
into (18a, b) then gives as scalar equations of equilibrium the symmetry relations ij = ji, in conjunction with the differential equations
In order to have stress strain relations which take the place of (1b) for infinitesimal elasticity, one then defines components ij of finite strain in the form
and with this one stipulates, in place of (1b), as constitutive relations
Given that the above choice of ij = ji for the description of states of finite strain is, effectively, a natural one, in view of the geometrical significance of thesequantities, one has to think of the component representation (19) for the vectors of stress as an inspired one. The reason that this is so depends on the dual symmetryproperty ij = ji, in conjunction with the fact that the work expression i· z,i = ijz,j· z,i can, with ij = ij and with (22), be written in the form ij ij, therebyestablishing the conjugacy of ij and ij, and the possibility to think of a variational equation which has (21) and (23) as Euler differential equations.
As regards the form of this variational equation, it turns out that it is the same as equation (3), except for a replacement of the volume integral term (ui,j + uj,i) ij by aterm (ui,j + uj,i + um,ium,j) ij, with a corresponding change of the expressions for tk so as to now read tk = ( jk + uk,j) mjvm.
Given that the essence of Eq. (3) for infinitesimal elasticity was in effect synthesized by visualizing a calculus of variations problem which would have (1a, b) as Eulerequations [Reissner (1950)], the validity of the analogous result for the problem of finite elasticity, with (21) and (23) as Euler equations, was subsequentlyconjectured, and then proved by a verification of the correctness of this conjecture [Reissner (1953)].
6Generalized Piola Components and a Generalized Hellinger Theorem
We define generalized Piola components of stress ij, as in Reissner (1984), by writing in place of (19)
in terms of a triad of mutually perpendicular unit vectors tj, where
Page 475
An introduction of (24a, b) and (20b) into (18a) then gives, in place of (21), the force component equations
In writing the scalar consequences of the associated moment condition (18b), it is of advantage for what follows to consider components in the direction of thevectors tj. In this connection we here also limit ourselves by now setting again
in (18b) and we further write, with em = nmtn,
With this and with tn × tj = enjktk, we than have as component moment equilibrium equations
In order to have components of strain ij with the property that i· z,i = ij ij we here introduce these strain components by writing
Equation (28 ) in conjunction with (24a) allows a ready verification of the desired property of the ij, as long as the direction cosines ij are given quantities, thatis, as long as ij = 0. Consequently, the strain components
are, for given jk, the conjugates of the stress components ij in (24a).
The generalized Piola components ij reduce to the conventional Piola components upon setting ij = ij, with this reducing Eq. (30) for the conjugate straincomponents ij to the usual displacement gradient form uj,i. Furthermore, an assumption of very small rotations, by way of stipulating that jk = jk + ejkm m, withsubsequent linearization in terms of uj,i and m, ensures the consistency of Eqs. (30) and (12).
A complementation of the force equilibrium Eqs. (26) and the strain displacement Eqs. (30) by constitutive equations
may be considered to result in a generalized Hellinger formulation of the finite elasticity problem in the following sense. The variational equation
has, with independent variations ij and uk, and with (30) as constraint conditions, the constitutive Eqs. (31) and the force equilibrium Eqs. (26) as Euler differentialequations. The formulation of Hellinger (1914) results from the above upon stipulating that ij = ij.
Equations (32) and (30), with the consequences (31) and (26), are subject to the same complications as the special-case result in Hellinger (1914). Thesecomplications are of two kinds. In the first place it is, in general, not possible to write a physically reasonable system of constitutive equations in the form (31). In thesecond place, no account is taken of the moment equilibrium Eqs. (29) which must be satisfied.
Page 476
As in the case for Hellinger's result, Eq. (32) may be transformed into a pseudo-variational equation for stress alone, upon considering the force equilibrium Eq. (26)as conditions of constraint. This allows the deduction of a variational equation , with the ij having to be such that ( ik ij),i = 0.
It does not seem to have been remarked before in connection with Hellinger's result that a frequently referred-to analysis by Levinson (1966) is no more or no lessthan a reformulation of Hellinger's analysis in terms of curvilinear coordinates. As regards the two complications mentioned above, Levinson notes that it may notalways be possible to write constitutive equations for Piola stresses and their conjugate strains in the form (31). However, it is not noted that non-satisfaction of themoment Eq. (29) means the effective non-validity of the result, with the absence of a possibility to be concerned with the conditions of moment equilibrium beingmentioned as a possible advantage in connection with the solution of specific problems by direct methods procedures.
7On Fraeijs De Veubeke's Theorem
Given that the variational Eq. (32) is, for fixed values of the ij, in general associated with states of stress and displacement which imply non-satisfaction of themoment equilibrium conditions (29), we find that this difficulty disappears upon removing the indicated constraints on the ij. To see that this is so, we now write asexpression for ij in the developed form of (32), with m in place of k,
The terms with um,i in this lead, as before, to Euler equations of force equilibrium. To see that the terms with jm in (33) lead to Euler equations of momentequilibrium, we take account of the fact that equation (25) implies that
with this in turn implying the possibility of writing
A multiplication on both sides in (35) by nm, and observation of (25), gives the further relation
Equation (36) in conjunction with (33) now allows us to state that a variational equation of the form
with ij as in (30) has, with independent variations ij, uk and k, and with jm as in (36), a system of Euler equations consisting of the stress displacementrelations (30) in conjunction with the force equilibrium conditions (26) and the moment equilibrium conditions (29).
Equation (37), in conjunction with (33), is now free of one of the two complications associated with the generalized Hellinger Eq. (32). There remains the othercomplication which concerns the physical reasonableness of a system of constitutive
Page 477
relations of the form (31). Our relatively simple way of dealing with this complication is as follows. We stipulate, on intuitive grounds, that the contents of (32) will bephysically reasonable in the event that the function W of the nine arguments ij does in effect depend on six arguments ( ij + ji)/2 only. With the form of W restrictedin this fashion, we then have the strain symmetry condition, ij = ji, as a consequence of the variational equation. As previously noted by Reissner (1984), this strainsymmetry condition can be thought of as a condition that the directions of the stress components ij are parallel to the edges of that rectangular element of volumewhich is ''most nearly" congruent with the deformed, in general oblique element of volume, and with this in turn meaning that these distinguished generalized Piolastresses are the same stresses as those first introduced, in a different context, by M. A. Biot.
With the basic idea of a finite-elasticity variational equation with rotational as well as translational displacement variations, including the step from Eq. (34) to (36)being, as far as is known, due to Fraeijs de Veubeke (1972), we note here that extensions and clarifications of this idea and of the ensuing analysis have since beenconsidered by others, in particular by Atluri (1983), Murakawa and Atluri (1978), Bufler (1983, 1985) and the present writer (1984, 1985).
References
Atluri, S. N. (1983): Alternate stress and conjugate strain measures, and mixed variational formulations involving rigid rotations, for computational analyses of finitelydeformed solids, with application to plates and shells Part: Theory, Comp. & Struct. 18, 93116.
Bufler, H. (1979): Generalized variational principles with relaxed continuity requirements for certain nonlinear problems, with an application to nonlinear elasticity.Comp. Meth. Appl. Mech. Eng. 19, 235255.
Bufler, H. (1983): On the work theorems for finite and incremental elastic deformation with discontinuous fields: A unified treatment of different versions. Comp.Meth. Appl. Mech. Eng. 36, 95124.
Bufler, H. (1985): The Biot stresses in nonlinear elasticity and the associated generalized variational principles. Ing. Arch. 55, 450462.
Fraeijs de Veubeke, B. (1972): A new variational principle for finite elastic deformations. Int. J. Eng. Sci. 10, 745763.
Hellinger, E. (1914): Die allgemeinen Ansätze der Mechanik der Kontinua. Enzyklopädie der Mathematischen Wissenschaften, 4, Art. 30, pp. 654655.
Levinson, M. (1966): The complementary energy theorem in finite elasticity. J. Appl. Mech. 33, 826829; ibid. 34, 714.
Murakawa, H.; Atluri, S. N. (1978): Finite elasticity solution using hybrid finite elements based on a complementary energy principle. J. Appl. Mech. 45, 539547.
Oden, J. T.; Reddy, J. N. (1976): Variational methods in theoretical mechanics, p. 115. Berlin, Heidelberg, New York: Springer.
Reissner, E. (1950): On a variational theorem in elasticity. J. Math. & Phys. 29, 9095.
Reissner, E. (1953): On a variational theorem for finite elastic deformations. J. Math. & Phys. 32, 129153.
Reissner, E. (1965): A note on variational principles in elasticity. Int. J. Solids Struct. 1, 9395.
Reissner, E. (1983): Variational principles in elasticity, manuscript of a chapter in: Handbook of finite element methods. London: McGraw-Hill.
Reissner, E. (1984): Formulation of variational theorems in geometrically nonlinear elasticity. J. Eng. Mech. 110, 13771390.
Reissner, E. (1985): On mixed variational formulations in finite elasticity. Acta Mechanica 56, 177125.
Page 478
On the Formulation of Variational Theorems Involving Volume Constraints *
[Comp. Mech. 5, 337344, 1989]
1Introduction
A continued concern with variational theorems which are suitable for numerical implementation in connection with the analysis of incompressible or nearlyincompressible materials has led us to the formulation of five-field, and in one case seven-field, theorems for displacements, deviatoric stresses, pressure, distortionalstrains and volume change. In essence these theorems may be thought of as generalizations of the Hu-Washizu three-field theorem for displacements, stresses andstrains and of the earlier two-field theorem for displacements and stresses.
For ease of exposition, what follows is divided into three parts. The first part deals with geometrically linear elasticity. The second part deals with the effect ofgeometric nonlinearity in terms of Kirchhoff-Trefftz stresses and Green-Lagrange strains. The third part is concerned with results involving generalized Piola stressesand conjugate strains, as well as with results about distinguished (Biot) generalized stresses and their conjugate strains. Also for ease of exposition, attention is limitedto statements about volume integral portions, omitting body force and boundary condition terms.
In addition to formulating five-field theorems, as well as one seven-field theorem, we use these theorems, through the introduction of various constraints, for thededuction of alternate six, five, four, three, and two-field theorems for incompressible or nearly incompressible elasticity.
2A Five-Field Generalization of the Hu-Washizu Theorem for Geometrically Linear Elasticity
With ui as components of displacement, with the abbreviation,
and with a strain energy density function U(uij) we have for the volume integral portion, in the absence of body forces, of the classical one-field principle of minimumpotential energy
With components of stress ij, with constraint constitutive equations ij = U/ uij and with arbitrary variations ui, Eq. (2) implies the Euler differential equations ofequilibrium ij,i = 0. Furthermore, with components of strain ij, and with constraint
*With S. N. Atluri.
Page 479
strain displacement relations ij = uij, Eq. (2) may alternately be written in the form
Given (2 ) we obtain, following Hu and Washizu, a three-field variational theorem for displacements, stresses and strains upon changing the character of the constraintrelations ij = uij, and ij = U/ ij into Euler equations, with this three-field theorem having the form
Given the step from the one-field theorem (2) to the three-field theorem (3) we now propose to establish a five-field theorem for displacements ui, distortional strains
, a volume change variable = kk, deviatoric stresses and a mean stress p = (1/3) kk, as follows.
We write, with the introduction of a deviatoric displacement gradient tensor ,
We further write
and with this we assert the validity of the five-field variational theorem
It is evident that the Euler equations of (6) include the relations , = ukk, and therewith the strain displacement relations ij = uij.
The further Euler equations , p = U*/ also imply the constitutive relations ij = U/ ij as Euler equations, inasmuch as
Finally, in order to verify that (6) also implies the equilibrium equations ij,i = 0 as Euler equations it is sufficient to observe, on the basis of (4), that
Remarks
Equation (6) may also be obtained upon writing the principle of minimum potential energy in the form , with an introduction of the side
conditions and = ukk by means of Lagrange multipliers and p.
Still another way of deriving (6) is to introduce the decompositions (4) into the Hu-Washizu Eq. (3), with the desired result following upon observing the relations.
We further note that the linear isotropic materials case of (6) was included in an unpublished 1978 manuscript "Notes on an analysis of nearly or preciselyincompressible behavior of elastic-plastic solids" by the first named author in collaboration
Page 480
with H. Murakawa. The present deduction of (6) evolved on the basis of a manuscript "On a modification of the Hu-Washizu variational equation in elasticity"concerning the four-field theorem in Eq. (10) which the senior author submitted in July 1986 to the journal Computational Mechanics.
3Reductions of the Five-Field Theorem of Geometrically Linear Theory
Our first reduction is to a four-field theorem for displacements, distortional strains, deviatoric stresses and pressure. We change the character of the relation p =U*/ from Euler equation to constraint equation and, with the inversion of this relation, define a semi-complementary energy density function V*
through the partial Legendre transformation
Equation (9) implies in the usual way that = V*/ p and the introduction of U* from (9) into (6) gives as the desired four-field theorem
As regards the problem of determining V* we note the ease of doing this for materials for which at the outset , and also for materials forwhich U(uij) = U2(uij) + (ukk)2F(u12, u13, u23), with U2 as a second degree polynomial. For other more general cases, and in connection with numerical applicationsof (10), we expect that it will be possible to combine the discretization of (10) with a determination of the function V* in an incremental sense as in [1].
Our second reduction of (6), to a three-field theorem for displacements, volume change and pressure, is obtained upon changing the character of the relations
and from Euler equation to constraint equation. With this, Eq. (6) evidently reduces to the form
Equation (11) has been stated previously in [5]. An implicit version of (11) occurs also in [2], not for it's own sake but as a stepping stone towards a two-fieldtheorem for displacements and pressure which is obtained by again changing the Euler equation p = U*/ into a constraint equation where now andtherewith
The introduction of U* from (12) into (11) leaves the two-field theorem
with (13) being equivalent, upon writing and so that , to the result in[2].
Page 481
4The Five-Field Theorem of Geometrically Nonlinear Theory for Green-Lagrange Strains and Kirchhoff-Trefftz Stresses
We again depart from a statement of the volume integral portion of the classical variational principle for displacements (1), with the uij now being the components ofthe Green-Lagrange strain tensor
and with the principle of Hu and Washizu for displacements, strains and (Kirchhoff-Trefftz or second Piola-Kirchhoff) stresses being again Eq. (3).
The additive decomposition of geometrically linear theory in (4) into distortional and dilatational contributions is known to be replaceable in geometrically nonlineartheory by a multiplicative decomposition which is dependent on the determinantal relation
where is the relative change of volume due to deformation, so that Ju = 1 for an incompressible material.
Given Eq. (15) we have the possibility of defining quantities which correspond to the quantities in (4) by writing
inasmuch as, evidently,
We now again make use of the artifice of replacing the basic Eq. (2) by (2 ), with constraint strain displacement relations ij = uij, and we define distortional straincomponents through the relations
where
Having (18) and (19) we next write, in analogy to Eq. (5) for the geometrically linear case,
and we stipulate, in analogy to the result for the geometrically linear case, that the five-field theorem of geometrically nonlinear theory, involving Lagrange multipliers
and p, be of the form
with as a function the choice of which remains at our disposal.
Given Eq. (21) we note in particular the Euler constitutive equations
Page 482
In view of (20) and in view of the relation ij = U/ ij we have as expressions for and p in terms of Kirchhoff-Trefftz stresses and Green-Lagrange strains:
and, alternately, as expressions for the ij, in terms of the , and p
In order to verify that (21) has the appropriate equilibrium equations as Euler equations as well, it is only necessary to verify that
This is readily accomplished on the basis of (16) in conjunction with the Euler strain displacement equations of (21).
5Reductions of the Five-Field Theorem (21)
As for the geometrically linear problem we obtain a four-field theorem which no longer involves the volume change measure J by inverting (22b) and by then defininga function
with which the desired four-field equation follows from (21) in the form
Our second reduction, to a three-field theorem, again depends on considering the relations and as constraint equations rather than Eulerequations. With this we obtain from (21) as a three-field theorem involving ui, J , and p
which corresponds to an equivalent theorem in [5], upon stipulating (Ju) = Ju and (J ) = J .
Given Eq. (27) we may further deduce a two-field theorem for p and the ui, by changing the Euler relations into constraints so as to have this two-fieldtheorem, in generalization of the result in [2] for the geometrically linear problem, in the form
6A Five-Field Theorem of Geometrically Nonlinear Theory in Terms of Generalized Piola Stresses and Displacement Gradient Components
With stress vectors i as forces per unit of undeformed area acting over the surfaces of deformed material elements
Page 483
defined by position vectors z = x + u the equations of equilibrium for deformed material elements are, except for body force terms which are omitted in this account,
Given that, in terms of the unit vectors ei in the directions of the coordinate axes xi, we have as defining relations for Piola components of stress
we have earlier [3] defined generalized Piola components of stress ij by writing
with the unit vectors tj given by
with (33) implying that
With (33) and (34) we define generalized displacement gradient components
on the basis of deducing with the help of (34a) that,
Given the representations (32) and (36) we have from (30a, b) as component equilibrium equations
As long as we stipulate that ij = 0, we have on the basis of (32) and (36) that
which assures the conjugacy of ij and wij. If, with (38) and with
we stipulate constitutive relations of the form
we then have that the variational equation
with arbitrary ui, has the force equilibrium Eq. (37a) as Euler equations.
For (41) to be meaningful, it is necessary to restrict the form of U in such a way that the moment equilibrium Eq. (37b) is also satisfied. It is readily shown that Eq.(37b) will be satisfied, identically, upon stipulating that
Page 484
with uij as in (14), where we note that on the basis of the relations uij = 1/2(z,i·z,j ij), we have as expression for the uij in terms of the wij,
With (42) and (43) it is then established that (41) is a one-field variational theorem of geometrically nonlinear theory in terms of generalized displacement gradientcomponents as defined by (35) which directly corresponds to the theorem in Eq. (2) for geometrically linear theory.
From (41) to (43) we next have as a three-field Hu-Washizu theorem, corresponding to Eq. (3) of the linear theory
where now U ( ij) = U( ij), with ij = ij + ji + ik jk.
The generalization of (44) to a five-field theorem involving volume change and pressure, in addition to deviatoric stresses and strains, involves the relation
where
with corresponding formulas for , ij and J . In this way we then have in analogy to (21) with
as a five-field theorem involving displacements, deviatoric strains and stresses, and pressure and volume change variables,
From (48) we may again deduce a three-field theorem which corresponds to a result in [5] upon introducing the relations and as constraints.This three-field theorem is, in analogy to (28)
The possibility of deducing four- and two-field theorems corresponding to (10) and (13) is questionable to the extent that the invertibility of the constitutive relation
is questionable within the framework of the concept of generalized Piola components of stress.
While the practical usefulness of the theorems in (48) and (49) is in doubt for other than the special case of ij = ij, the considerations leading to them furnish aparticularly convenient approach to a broadened formulation where the status of the ij is changed from that of given quantities to that of additional dependentvariables.
Page 485
7A Seven-Field Theorem Involving Distinguished Generalized Piola or Biot Stresses and Displacement Gradient Components
We now consider the directions of the vectors tj in (33a,b) not as given but as dependent variables which are to be determined as part of the solution of the problem.We then have, as a consequence of (35)
An observation of the fact that, as a consequence of (33b), ik jk = jk ik and therewith, in terms of the quantities m, ik jk = eijm m leads, with (34b)and with a consistent change of subscripts, to the relation ij = kj ekim m. This, in conjunction with (35), transforms (50) into
With equation (39) replaced by (51), and with ij again as in (40) we will now have that the variational equation (41), with arbitrary uj and m, has the characterof a two-field principle with not only the force equilibrium Eq. (37a) but also the moment equilibrium Eq. (37b) as Euler equations.
It is necessary at this point to decide to what extent the form of the function U (wij) cannot be arbitrarily stipulated. We do know that if U is as in (42) that then themoment equilibrium equations are satisfied automatically and that therewith the variational Eq. (41), with or without ij = 0, is a valid one-field equation. Analternate disposition concerning U , leading to a three-field variational equation, is as follows. We stipulate, as a physically reasonable restriction, to be satisfied by anappropriate determination of the ij, the strain symmetry conditions
and we write
After this we change the constraint relations (52) into Euler equations with the help of Lagrange multipliers k, and therewith state as a three-field variational equationwhich has force and moment equilibrium conditions as well as the same kinematic relations (52) as Euler equations
The physical significance of the multipliers k is obtained upon deducing from (54), with the abbreviation
or
Page 486
For (56), with wik as in (50), to have the equilibrium Eqs. (37a, b) as Euler equations it is evidently necessary to have
Equation (57), with the further abbreviation implies that
and accordingly we have that only the sums of the stresses 12 and 21, etc., enter into the constitutive relations.
Having expressions for the k in accordance with (58), it is possible to write the three-field theorem (54) in the form
with independent variations ui; i; and ( ij ji), consistent with a result in [4].
Given equation (59) it is again possible to increase the number of fields and now obtain a five-field principle in the sense of Hu-Washizu, by introducing componentsof strain ij, with strain displacement relations ij = wij as additional Euler equations. In this way we then have, in place of (59)
with independent variations 11, , . . ., 11, , . . ., ( 12 21), . . ., ui, and i. Finally we may use (60) to again deduce a theorem with supplementaryvolume change and pressure variables, where we now write in analogy to (16)
and
with corresponding formulas for 11, , . . ., and J and with
With (63) and again with a pressure function p, we now deduce a seven-field theorem
Page 487
with independent variations , , . . ., J , , , . . ., , . . ., p, ui, and i.
As before, we may deduce from this, by introduction of suitable constraints, lower field relations including a five-field theorem
corresponding to the three-field theorem in [5], and a four-field theorem
corresponding to the two-field theorem in terms of Green-Lagrange strains and pressure in Eq. (29).
References
1. Atluri, S. N. (1978): On rate principles for finite strain analysis of elastic and inelastic nonlinear solids. In: Recent research on the mechanical behavior of solids,pp. 79109. University of Tokyo Press.
2. Reissner, E. (1983): On a variational principle for elastic displacements and pressure. J. Appl. Mech. 51, 444445.
3. Reissner, E. (1984): Formulation of variational theorems in geometrically nonlinear elasticity. J. Eng. Mech. 110, 13771390.
4. Reissner, E. (1985): On mixed variational formulations in finite elasticity. Acta Mechanica 56, 117125.
5. Simo, J. C., Taylor, R. L. and Pister, K. S. (1985): Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comp.Meth. Appl. Mech. Eng. 51, 177208.
Page 489
VIBRATIONSMy concern with problems of vibration came from an interest of my father and his colleague A. Hertwig, in the possibility of utilizing Lamb's solution for an oscillatingpoint force on the surface of an elastic halfspace as a tool for the interpretation of machine vibration tests. I volunteered for a first step in such an attempt by selectingas Dipl. Ing. Thesis topic the problem of a uniform oscillating pressure distribution acting over a circular surface portion of the halfspace. The thesis was accepted inDecember 1935 and led to a temporary appointment by the Institute for Soil Mechanics (Degebo), with the assignment to apply the results in it to the problem offorced vibrations of a system consisting of a body of given mass and given base radius, resting on the elastic halfspace. I was able to deduce the equations whichwere needed to obtain data of engineering interest and discovered, in particular, the importance of radiation damping for the solution of this problem. The report onthis work was well received by Professor Hertwig, to the point of his deciding, in April 1936, that it could serve as a Dissertation for the degree of Dr. Ing. Excerptsfrom this were published in [5].
An opportunity for further graduate studies at M.I.T. meant that I lost contact with the later work at the Degebo, and so failed to be aware that my numerical resultsdid not well agree with experimental data. A number of years later it was discovered by others that there was a sign mistake in one of the terms of my halfspacesolution. Upon correcting this mistake, theory and experiment were no longer inconsistent. It is gratifying to know that in spite of this known flaw the contents of [5]are still being appreciated, with F. E. Richart's text on Vibrations of Soils and Foundations referring to this day to [5] as ''the classic paper in this field." It should beadded that when thought was given to a re-publication of this paper, my colleague Enrique Luco generously volunteered to obtain new versions of the numerical data,including the preparation of new corrected figures. I feel greatly indebted to Professor Luco for his unselfish effort, which made inclusion of the paper in this volumepossible.
I no longer remember why very soon thereafter I considered the problem of torsional oscillations of the halfspace [9]. The analysis came out much simpler, with aclosed-form solution for the case of forced oscillations due to a radially linear distribution of circumferential surface shear. Also considered, briefly, was the mixedboundary value problem with displacements prescribed for r r0 and absent tractions for r0 < r, by means of a system of dual integral equations.
It later occurred to me that it would be useful to approach this mixed boundary value problem with the help of an oblate spheroidal coordinate system. My notes onthis approach were utilized by H. F. Sagoci who was then a Turkish army officer, on leave to obtain a Ph.D. in geophysics. We found a closed-form solution for the
Page 490
static case [35], and an exact series solution in terms of Mathieu functions for the case of simple harmonic motion. As the responsibility for the analysis for this latercase were entirely Dr. Sagoci's, this part of the work was published by him as a follow-up report without my being co-author.
In addition to interests in nonsteady aerodynamics [62], and in making the minimum complementary energy principle applicable to vibration problems [52, 59] themain other effort which is to be mentioned here concerns the vibration problem of shallow elastic shells. Beginning with a note on axisymmetric vibrations of sphericalshells [45], as a straightforward extension of the work on the static problem [46], I later realized, by means of systematic order of magnitude considerations, that inmany cases the problem could be much simplified by neglecting the effect of horizontal inertia in comparison with the effect of transverse inertia [94]. This observationwas first applied to a problem of beam vibrations [88], and then to obtain quantitative results for the problem of axi and non-axisymmetrical vibrations of sphericalshells [95, 107], partly in cooperation with Millard Johnson.
Aside from this there was a paper on torsional vibrations of shallow helicoidal shells, jointly with K. Washizu [108], and a paper on breathing vibrations ofpressurized cylindrical shells, jointly with J. G. Berry [113], where we reported on work done in conjunction with the Atlas missile project.
Page 491
Stationäre, Axialsymmetrische, Durch Eine Schüttelnde Masse Erregte Schwingungen Eines Homogenen ElastischenHalbraumes[Ingenieur Archiv 7, 381396, 1936]
1Aufgabestellung
Die axialsymmetrische Schwingung des elastischen Halbraumes bei gegebenen Oberflächenspannungen ist zuerst in einer grundlegenden Arbeit von H. Lambbehandelt worden*.
Schon der Titel der Lambschen Arbeit zeigt, daß die Theorie im Hinblick auf die seismologischen Anwendungen entwickelt werden sollte.
Als daran gegangen wurde, die Untersuchungen Lambs und seiner Nachfolger auf die dynamische Bodenuntersuchung anzuwenden, zeigte es sich, daß zwar dievorhandenen Ansätze für diese Problemstellung benutzt werden konnten, daß aber sowohl bei der formellen Durchführung als der numerischen Auswertung nocherhebliche Weiterarbeit notwendig war, da in allen diesen Arbeiten auf die Erscheinungen in unmittelbarer Nähe des Erregungszentrums bei Erregung durchzusätzliche Massen (Resonanzverhältnisse usw.) nicht eingegangen worden ist.
Bei der Untersuchungsmethode, die den Anlaß zu dieser Arbeit gegeben hat, sollen die Eigenschaften eines Bodens aus seinem Verhalten bei periodischer Belastungeines begrenzten Oberflächenstückes erschlossen werden**.
Der erregende Mechanismus, ein schüttelndes Massensystem, der sog., "Schwinger", versetzt den Boden in eine nach kurzer Zeit stationäre Schwingung, bei derAmplitude, Phasenwinkel zwischen erregender Kraft und Verschiebung sowie Leistungsbedarf in Abhängigkeit von der erregenden Kraft und der Frequenz beigegebener Schwingermasse und Belastungsfläche gemessen werden können.
Um eine Theorie dieses Verfahrens der Bodenuntersuchung aufzustellen, hat man bisher das System Schwinger-Erdboden als ein System eines oder zweierMassenpunkte betrachtet, die untereinander und gegen ein starres Fundament elastisch und reibend verbunden gedacht sind***.
In der vorliegenden Arbeit wird die Annahme zugrunde gelegt, daß das System aus einem elastischen, homogenen, mit Masse behafteten, isotropen Halbraumbesteht, an welchem periodisch wirkende Oberflächenspannungen angreifen. Insbesondere wird auf den Fall eingegangen, daß diese Spannungen erzeugt werdendurch rotierende Massen, die an einem schweren auf dem Halbraum ruhenden Körper angebracht sind.
*H. Lamb, On the Propagation of Tremors over the Surface of an Elastic Solid. Philos. Trans. Roy. Soc., Lond. (A) 203 (1904), 142.
**A. Hertwig, G. Früh, H. Lorenz, Die Ermittlung der für das Bauwesen wichtigsten Eigenschaften durch erzwungene Schwingungen. Veröff. Inst. Dtsch. Forsch. Ges. Bodenmech. (Degebo), Heft 1,Springer, Berlin 1933.
***H. Lorenz, Diss. Techn. Hochsch. Berlin 1934, sowie A. Hertwig, G. Früh, H. Lorenz, a.a.O.
Page 492
Da die vorstehende Annahme sich dem wirklichen System mehr annähert als die vorhergenannten Ansätze, werden die dabei auftretenden Abweichungen zwischenTheorie und Versuch noch sicherer auf die Erweiterung sowohl der experimentellen wie der theoretischen Untersuchung hinweisen.
Für die Natur des den Boden darstellenden Halbraumes sind Elastizitätsmodul, Schubmodul oder Querkontraktionszahl, Massendichte, innere Reibung und beisandförmigen Körpern deren jeweilige Sackungsgrenze der inneren Reibung maßgebend. Jedoch wird hier von der Berücksichtigung der inneren Reibung abgesehen,was um so mehr angebracht ist, als der zunächst immer versuchte Ansatz einer von der Geschwindigkeit abhängigen Reibung für feste und sandförmige Massen sicherwesentlich falsch ist und über andere Ansätze noch keine Klarheit herrscht.
Die allgemeine Behandlung der erzwungenen rotationssymmetrischen Schwingungen des elastischen Halbraumes erfolgt unter Zugrundelegung der erwähnten Arbeitvon Lamb, welche unter Benutzung eines Vortragsmanuskriptes von H. Reissner etwas vereinfacht werden konnte. Des Zusammenhangs halber wird die Ableitungdieser Ergebnisse nochmals angedeutet.
Es möge noch genauer gesagt werden, worin der Unterschied der vorliegenden Arbeit gegenüber den bisherigen durch seismologische Fragen angeregten Unter-suchungen besteht.
Auf Grund der Fragestellungen der Erdbebentheorie ist von Lamb und seinen Nachfolgern hingearbeitet worden auf die Darstellung der Erschütterungswellen ingroßer Entfernung vom Erregungszentrum. Man konnte sich dabei auf den Fall gegebener linien- bzw. punktförmiger Erregungsquelle beschränken und die in diesemFall auftretende unendliche Amplitude im Erregungszentrum in Kauf nehmen.
Dagegen war es für die oben angegebene Fragestellung erforderlich, die Methode auf den Fall einer flächenhaft verteilten, durch zusätzliche Massen verursachtenErregung zu erweitern und hauptsächlich die Oberflächenverschiebungen innerhalb der Belastungsfläche zu betrachten.
Große Schwierigkeiten bereitete dabei die Auswertung der durch uneigentliche Integrale gegebenen formelmäßigen Resultate, die auf einem Wege erfolgte, der,soweit festgestellt werden konnte, neu ist und an anderer Stelle dargestellt wird.
2Theorie Der Rotationssymmetrischen Schwingungen in Einem Elastischen Halbraum
Seien die Dichte, und die Laméschen Elastizitätskonstanten des homogen und isotrop vorausgesetzten Halbraumes und t die Zeit, so gelten für dieFortpflanzung der Dichteänderung und des Rotationsvektors die Gleichungen
Bei axialer Symmetrie ist in Zylinderkoordinaten r, und z, wenn u, v, w die entsprechenden Verschiebungen bedeuten,
Page 493
Zwischen und einerseits und den Verschiebungen u und w andererseits besteht folgender Zusammenhang:
Partikularlösungen der Gleichungen (1) und (2), von denen später der reelle Teil zu nehmen ist, sind
worin J0 und J1 Besselsche Funktionen bedeuten. Für die nachherige Erfüllung der Oberflächenbedingungen muß zwischen und unter Einführung einer Größe die Beziehung
gelten, worin im folgenden zwecks Abkürzung
gesetzt werden soll. Für die Verschiebungen u und w erhält man dann aus (4)
Als nicht verschwindende Spannungen treten r, , z und rz auf, von denen im folgenden nur z und rz benötigt werden. Diese letzteren ergeben sich wiefolgt:
(a)Freie Schwingung
Die freie Schwingung, die von Rayleigh* zuerst behandelt worden ist, ist definiert durch die Forderung
Unter Benutzung von (11) und (12) erhält man aus diesen Forderungen als Frequenzengleichung
*Rayleigh, Proc. Lond. Math. Soc. 17 (1885), 411.
Page 494
F( ) besitzt, wie Rayleigh gezeigt hat, nur zwei Nullstellen, die mit ± bezeichnet werden, und zwar ist für
Unter Einführung einer neuen Konstanten D ergibt sich, wenn 1 und 1 die Werte von und an der Stelle 2 = 2 bedeuten, bei der freien Schwingung
(b)Erzwungene Schwingung
Für die Behandlung der durch gegebene Oberflächenkräfte erzwungenen Schwingung wird der folgende durch Summation aus (9) bis (12) hervorgehende Ansatz
gemacht, in dem A und B willkürliche, durch die Oberflächenbedingungen zu bestimmende Funktionen sind.
Ist die Oberfläche schubspannungsfrei, ein Fall, der allein hier betrachtet wird, so folgt aus (20)
und für die Oberflächennormalspannung ergibt sich aus (19)
*Für den Fall, daß weitere -Werte interessieren, geht man zweckmäßigerweise so vor, daß man annimmt und aus (14) dann die Querkontraktion m = /2( + ) bestimmt, da die Auflösung derGleichung (14) nach bei gegebenem m bei der erforderlichen, sehr großen Genauigkeit wesentlich umständlicher ist.
Page 495
Andererseits ist das gegebene z(r, 0) folgendermaßen durch ein Fourier-Besselsches Integral darstellbar:
so daß man durch Vergleich von (22) und (23)
erhält. Für den Fall einer gleichmäßig über einen Kreis mit dem Halbmesser r0 verteilten Normalbelastung Peipt
ergibt sich aus (24)
so daß sich bei dieser Belastungsverteilung, die für die weiteren Betrachtungen zugrunde gelegt wird, aus (17) und (18) folgende Ausdrücke für die weiterhinbenötigten Oberflächenverschiebungen u und w ergeben, wobei noch nach Gleichung (14) wieder die Funktion F( ) eingeführt wird,
Die Integrale (27) und (28) sind in gewissem Sinne unbestimmt, da ja die Funktion F( ) auf dem Integrationswege (für = ) einmal von erster Ordnungverschwindet, in welchem Falle man dem Integral verschiedene Werte zuerteilen kann, je nachdem in welcher Weise, man von links und von rechts bei der Integrationan die Polstelle des Integranden herangeht.
Dieser Willkür in der Wahl der Inte grale entspricht die Tatsache, daß die vorstehend gegebene Lösung des Problems keine vollständige ist, da ihr noch ein Systemfreier Schwingungen überlagert werden kann, ohne die Oberflächenbedingungen des Problems zu verletzen und überlagert werden muß, um die gleich zu nennendeBedingung im Unendlichen zu erfüllen. Unter Berücksichtigung dieser Tatsache wird so vorgegangen, daß man den sog. Cauchyschen Hauptwert* der auftretendenIntegrale als partikulare Lösung nimmt und dieser Lösung eine freie Schwingung so überlagert, daß die resultierende Schwingungsform aus divergierenden Wellenbesteht, was die Bedingung dafür darstellt, daß aus dem Unendlichen keine Energieeinstrahlung stattfindet.
*Hauptwert:
Page 496
Diese Bedingung, daß die Wellen nach außen fortschreitende sind, ist also eine aus physikalischen Gründen hinzukommende Forderung an die richtige Lösung, diedadurch zu einer eindeutigen gemacht wird.
Um sie zu befriedigen, ist es nötig, den Anteil der stehenden Schwingungen in der hergestellten Lösung (27) und (28) in Evidenz zu setzen. Dies geschieht nach demVorgang von Lamb mit dem Hilfsmittel der Integration im Komplexen. Es werden in (27) und (28) an Stelle von J0 und J1 Hankelsche Zylinderfunktionen eingeführt*,und zwar ist
wobei
Damit ergeben sich aus (27) und (28) die jetzt von bis + erstreckten Integrale
Nach Einführung einer komplexen ( = + i )-Ebene können, für r > r0 die Integrale (31) und (32) durch Integration längs eines geeigneten Weges in die folgendeForm gebracht werden:
Die darin zur Abkürzung eingeführten Zeichen H, K und ( ) sind definiert durch
*Dabei ist für das folgende wesentlich, daß im Gegensatz zu J die Eigenschaft besitzt, für Werte des Arguments mit großem positiven Imaginärteil exponentiell abzunehmen.
Page 497
Nun stellen bei Zylindersymmetrie Andrücke von der Form wie bekannt* divergierende Wellen dar, so daß also bereits erreicht ist, daß diestehengebliebenen Integrale Aggregate von divergierenden Wellen darstellen, während die ausintegrierten Teile den Anteil der stehenden Schwingung an derhergestellten Partikularlösung angeben.
Durch Überlagerung einer freien Schwingung nach (15) und (16), in der für die willkürliche Konstante D der Wert
festgesetzt wird, erreicht man, daß die gesamte Schwingung, deren Verschiebungen durch einen Stern gekennzeichnet werden mögen, aus divergierenden Wellenbesteht. Man erhält insbesondere an der Oberfläche
dabei muß jedoch beachtet werden, daß diese Darstellung abgeleitet ist unter der Voraussetzung r > r0, und daß für die Berechnung der Verschiebungen innerhalbdes Belastungskreises die ursprünglichen Formeln (27) und (28), zu denen ebenfalls die freie Schwingung nach (15), (16) und (38) hinzuzufügen ist, benutzt werdenmüssen.
Läßt man in (39) und (40) r0 0 gehen, nimmt also eine punktförmige Last an, ein Schritt, den Lamb a.a.O. bereits zu Anfang seiner Rechnungen vornimmt, sogehen die Ergebnisse in die Lambschen über, bei denen die Verschiebung im Lastangriffspunkt unendlich ist.
Für die Aufgaben der Baugrundmechanik ist dieser Grenzübergang nicht zulässig, da es, wie sich zeigen wird, hierbei gerade auf die Verschiebungen innerhalb derLastangriffsfläche ankommt.
Führen wir, um dimensionslose Größen zu erhalten, eine neue Integrations-variable ein, definiert durch
und setzen wir
*Es ist dies ersichtlich mit Hilfe der Integraldarstellung
Page 498
so erhalten wir für die vertikale Oberflächenverschiebung w*(r, 0) dié für alle r gültige Darstellung
was für r > r0 nach (39) identisch ist mit
Formel (44) ist zwar für die hier durchgeführte Untersuchung der Erscheinungen in der Nähe des Erregungszentrums nicht brauchbar, dagegen wäre sie dann vonVorteil, wenn es sich darum handeln würde, eine asymptotische Entwicklung für große a (d.h. also für große r) zu gewinnen.
Da das Integral von bis läuft, ist es für > 0 (d.h. für alle Fälle, in denen das Mittel kompressibel ist) ohne weiteres möglich, diese Rechnung zu vereinfachen,indem bereits unter dem Integral für die Hankelsche Funktion ihre asymptotische Entwicklung eingesetzt wird. Dadurch wird man auf die asymptotische Entwicklungeiner Reihe von Integralen geführt, deren jedes dieselbe Form hat wie die Formel für die Oberflächenverschiebung beim ebenen Problem, für welches Lamb a.a.O.diese Entwicklung angegeben hat, während er sich für den rotationssymmetrischen Fall auf Angabe ds ersten Gliedes beschränkt. Eine solche Rechnung müßte nochdurchgeführt werden, wenn Beobachtungen über Abnahme der Amplituden mit der Entfernung, über Wellenlängen u. dgl. zur Beurteilung der Bodeneigenschaften mitherangezogen werden sollen.
Die allgemeine Theorie der durch gegebene Oberflächenkräfte erzwungenen Schwingungen des elastischen Halbraumes ist vorstehend soweit geführt worden, wie sieim folgenden benötigt wird.
3Das System Schwinger-Halbraum
Um die allgemeine Theorie auf den Fall einer schwingungserzeugenden Schüttelmaschine anwenden zu können, ist es nötig, gewisse die Grundplatte des Schwingersbetreffende Idealisierungen einzuführen, da andernfalls die Frage des Gleichgewichts zwischen Oberflächenspannungen z(r, 0) und den Trägheitskräften desSchwingers auf ein gemischtes Randwertproblem für den Halbraum führt, das bisher selbst in dem wesentlich einfacheren statischen Fall nur unter Annahme einervollkommen starren Platte gelöst ist.
Die vereinfachende Annahme besteht darin, daß man die elastische Nachgiebigkeit der Grundplatte so festsetzt, wie sie sich bei einer von vornherein angenommenenDruckverteilung unter der Platte ergibt und nur die Größe der resultierenden Kraft, die durch die Platte auf den Boden übertragen wird, aus derGleichgewichtsbedingung für die vertikalen Kräfte entnimmt.
Page 499
Abb. 1. Schema des Systems.
Als Druckverteilung unter der Platte wurde die in Abschnitt 1 betrachtete gleichmäßige angenommen.
Wenn Verschiebung und Beschleunigung entsprechend Abb. 1 positiv in Richtung wachsender z, die am Halbraum angreifende Kraft entsprechend positiv alsZugkraft angenommen wird, so lautet die Gleichgewichtsbedingung für die vertikalen Kräfte
in welcher m0 die Gesamtmasse der Schüttelmaschine und C die Amplitude der Erregungskraft bedeutet. Für den besonderen bei der Schüttelmaschine der Degebovorliegenden Fall der Erregung durch zwei umlaufende Massen m1 vom Umlaufsradius l ist
In (45) soll so bestimmt werden, daß man von der in Abschnitt 2 abgeleiteten Lösung den reellen Teil nimmt, also so, daß
ist. Dann wird (45)
worin nach (43)
mit
Page 500
und
Damit schreibt sich (48)
woraus man
und
erhält. Aus (54) und (49) folgt als Formel für die Schwingungsamplitude
Für die Maschinenleistung bzw. die ins Unendliche ausgestrahlte Energie L erhält man
Der Winkel ist Phasenwinkel zwischen Erregungskraft C und Bodendruck . Der Messung zugänglich und als eigentlicher Phasenwinkel zu bezeichnen ist jedochder Winkel zwischen Erregungskraft C und Verschiebung w*(0, 0), für den man die folgende Gleichung erhält:
Es erweist sich für die Diskussion als zweckmäßig, in den Gleichungen (53) bis (57) das Massen verhältnis
einzuführen. Es ist dann
Page 501
und man erhält aus (55) bis (57)
Durch die Formeln (55) bis (57) ist die Aufgabe nun zurückgeführt auf die Berechnung der durch uneigentliche Integrale gegebenen Funktionen 1(a0, ), und 2(a0,), weleche den größten Arbeitsaufwand in dieser Untersuchung erforderte. Unter Verzicht auf die Wiedergabe der Methode sowie der Rechnungen werde jedoch
hier nur das Resultat angegeben. Man erhält
worin, um daran zu erinnern, zu setzen ist. Die Funktionen 1 und 2 sind für den benötigten a0-Bereich in Abb. 2 dargestellt.
Addendum (1995)
Es war schon lange bekannt, durch die Arbeit von T. Y. Sung (ASTM Special Technical Publication No. 156, ''Symposium on Dynamic Testing of Soils" pp. 3564,1953), dass die Funktionen 2 in den Gleichungen (60a, b, c) keine Minuszeichen vor den Anteilen mit J1 haben sollten, im Gegensatz zu den Formeln in derOriginalform dieser Arbeit. Es folgte daraus dass die numerischen Daten in den Abbildungen 2 bis 13 nicht richtig waren.
Als die Frage einer Neu-Publikation der Arbeit in einem Sammelband aktuell wurde schlug mein Kollege Enrique Luco vor dass es gerechtfertig wäre, in Hinblick aufdie seminale Bedeutung der Arbeit auf dem Gebiet der Bodenmechanik, die numerischen Resultate in der Originalarbeit zu korrigieren mit Hilfe der verbessertenWerte der Funktion 2. Die folgenden Abbildungen enthalten die korrigierten Resultate.
Page 502
Abb. 2. Bild der Funktionen 1 und 2.
Ich schulde Professor Luco und seinem Mitarbeiter Dr. Francisco Barros herzlichen Dank für ihre selbstlosen Bemühungen die neue Daten zu erhalten. Weiterhinschulde ich ihnen Dank für die Herstellung der dazugehörigen Figuren.
4Ergebnisse
Nach den Formeln (55a) bis (57a) wurden eine Reihe von Phasen-, Amplituden- und Leistungskurven berechnet und in den Abb. 3 bis 7 dargestellt, aus denen der
Einfluß der dimensionslosen Größe , der Querkontraktionszahl m und der Größe auf Phase, Leistung und Amplitude hervorgeht. In Abb. 8ist der Zusammenhang zwischen b und der kritischen Frequenz p , bei der die Phasenverschiebung 90° beträgt, für einige Werte der Querkontraktionszahl mdargestellt.
Für die Rechnung erwies es sich als zweckmäßig, die Einhüllenden der Amplituden- und Leistungskurven zu berechnen, die sich aus A und A/ b = 0 bzw. L undL/ b = 0 ergeben, wenn daraus die Größe b eliminiert wird. Sowohl A/ b = 0
Page 503
Abb. 3. Einfluß der Schwingermasse auf die Phasenkurven.
als auch L/ b = 0 führt auf die Gleichung
welche gerade auch die Bedingung dafür darstellt, daß die Phasenverschiebung 90° beträgt. So erhält man den wichtigen Satz, daß Phasen- und Leistungskurven ihreEinhüllenden bei derjenigen Frequenz berühren, bei der die Phasenverschiebung 90° beträgt. Es ist
welche Kurven ebenfalls in Abb. 6 und 7 sowie gesondert in Abb. 9 und 10 aufgetragen sind, und welche also den Zusammenhang zwischen kritischer Frequenz,kritischer Amplitude bzw. Leistung sowie Apparat- und Bodenkonstanten darstellen.
Page 504
Abb. 4.Einfluss der Querkontraktionszahl m
auf die Phasenkurven.
Abb. 5.Einfluss der Querkontraktionszahl m
auf die Amplituden bei konstanter Erregung C.
Page 505
Abb. 6.Amplitudenkurven A bei
frequenzunabhängiger Erregung Cfür den Fall m = 0.
Abb. 7.Einfluß der Schwingermasse auf die
Leistungskurven bei konstanter Erregung.
Page 506
Abb. 8.Abhängigkeit der kritischen Frequenz
pk von Apparat- und Bodenkonstanten(Phasenwinkel = 90°.
An Hand der Kurvenblätter 8 und 9 ist es grundsätzlich möglich, Eigenfrequenzen und Schwingungsamplituden von Motorprüfständen oder ähnlichen Einrichtungenbei Kenntnis der elastischen Eigenschaften des Bodens im voraus zu berechnen.
Dies werde an einem Beispiel gezeigt. Seien für einen Boden (Sandstein):
Schubmodul = 29000 kg/cm2
Querkontraktionszahl m = 0.25
Bodendichte g = 2.38 · 103 kg/cm3
Auf ihm stehe eine Maschine mit folgenden Konstruktionsdaten:
Fundament-plus Maschinengewicht m0g = 3000 kg
Erregungskraftamplitude C = 5000 kg
Auflagerfläche
Massenverhältnis b = 7.0
Page 507
Abb. 9.Einhüllende der Amplitudenkurven.
Daraus folgt aus Abb. 8
und aus Abb. 9
Also ergibt sich für die kritische Frequenz
und für die kritische Verschiebungsamplitude
Man kann dann untersuchen, ob diese Frequenz und Amplitude in der Nachbarschaft Störungen hervorrufen können und ob man die in manchen Fällen der Praxis
Page 508
Abb. 10.Einhüllende der Leistungskurven.
bewährten Federzwischenschaltungen vornehmen muß, bei deren Vorhandensein die Rechnung allerdings etwas abzuändern wäre.
5Eine Beziehung Zwischen Der Massenpunkttheorie Und Der Theorie Des Schwingenden Halbraumes
Folgende Frage ist noch von Interesse und kann beantwortet werden: Gibt es einen gedämpft elastisch schwingenden Massenpunkt, bei dem Masse, Federkonstanteund Dämpfungskonstante so gewählt werden können, daß Phasen-, Amplituden- und Leistungskurven übereinstimmen mit denjenigen, die vorstehend für das SystemSchwinger-Halbraum abgeleitet worden sind?
Die Antwort ist die, daß es ein für einen Schwinger und einen Boden einmal gewähltes Massenpunktsystem nicht gibt, daß man aber zu jeder Erregungsfrequenz einSystem mit passender Masse, Dämpfung und Federkraft angeben kann. Die genannten Größen wären also frequenzabhängig zu wählen.
Für die Schwingung des Massenpunktes gilt die Differentialgleichung
Ihre Lösung ist
Page 509
Abb. 11.Dämpfungskonstante als Funktion von a0 und m.
worin
und
ist, während man für die Maschinenleistung
hat.
Fordert man Übereinstimmung von A*, * und L* mit den entsprechenden Größen A, und L, die durch die Gleichungen (55a) bis (57a) gegeben sind, so erhältman für den Dämpfungskoeffizienten
Page 510
Abb. 12.Feder und Massenkonstante als Funktion von
a0 wenn m = 0.
und zwischen Federkonstante c und Masse M die Beziehung
Damit ist eine elastizitätstheoretische Deutung der Größen und c Mp2 gefunden. Man könnte Versuchsreihen daraufhin studieren, inwieweit die Relationen (69) und
(70) der Wirklichkeit entsprechen. und c Mp2 sind für einige Werte von m als Funktion von in Abb. 11 und 12 dargestellt. Für genügend langsameSchwingungen (a0 hinreichend klein) erhält man aus (69) und (70) mit f1 = c0(m) + c2(ma02 + ... und f2 = c1(m)asub>0 + ... in erster Näherung
Ubrigens muß bemerkt werden, daß bei dem vorliegenden System c und M nur in der Verbindung c Mp2 vorkommen und daß es nicht möglich scheint,eine,,mitschwingende Bodenmasse" M m0 und eine Federkonstante c getrennt voneinander zu erhalten.
Page 511
Forced Torsional Oscillations of an Elastic Half-Space. I*
[J. Appl. Phys. 15, 652654, 1944]
1Introduction
The subject of this and a subsequent note is the investigation of the torsional oscillations in a homogeneous, isotropic elastic half-space under the influence of periodicshear stresses applied in rotationally symmetric manner to a circular portion of the surface of the half-space. This same problem was studied some time ago1 underthe supposition that the law of variation of shear stresses over the surface is given. Under these circumstances it is possible to obtain an explicit solution of theproblem by means of Fourier-Bessel integral methods and this solution was evaluated for the case of shear stresses increasing linearly from the center of the stressedsurface region to the edge of the stressed surface region. Of greater practical interest, as was previously pointed out,1 is the case that the law of variation ofdisplacements over the loaded portion of the surface is prescribed. Specifically, the question arises for the distribution of surface shear if the load is applied by meansof a rigid disk which means that the torsional displacement varies linearly from the center of the disk to the edge of the disk. Mathematically this is a mixed boundaryproblem and Fourier-Bessel methods only reduce the problem to an integral equation problem which in turn may be reduced to the problem of solving an infinitenumber of linear equations for an infinite number of unknowns.1
In this and the following paper it is shown that an explicit solution of the mixed boundary problem can be obtained by introducing in a suitable manner a system ofoblate spheroidal coordinates. In the present paper the formulation of the problem is given and a solution in closed form is obtained for the static case of torsionaldeformation. In the subsequent paper the characteristics of the vibration problem are determined by means of spheroidal wave functions.
Results of interest of the present calculations are expressions for the angle of rotation of the loaded surface in terms of the applied torque, for phase differencebetween angle of rotation and applied torque, and for the shear stress distribution under the rigid disk.
2Formulation of the Problem
To be obtained are the stresses and displacements in a homogeneous isotropic elastic half-space (z 0), a circular portion of whose
*With H. F. Sagoci.
1E. Reissner, Ing. Arch. 8, 229245 (1937).
Page 512
Fig. 1.
surface (r r0) is forced to rotate an angle , the axis of rotation being perpendicular to the surface of the half-space. The remaining portion of the surface (r0 < r) isassumed to be free of stress (Figure 1).
On the basis of the general equations of the theory of elasticity2 it had been shown previously1 that in the solution of this problem only the circumferential displacementcomponent v occurred and that all components of stress are zero except the following two components of shear stress
The differential equation for v is, if internal damping is neglected
The boundary conditions of the problem are
*
The applied torque T is given by
It is of importance to notice that the solution of Eq. (2) can be reduced to the solution of the wave equation. Introducing a function W defined as
there follows that the equation for W is
2See, for instance, A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity (Cambridge University Press, London, 1927), pp. 144145.
*In what follows it would be possible to obtain the solution of the more general problem in which r is replaced by (r).
Page 513
Fig. 2.
3Introduction of Oblate Spheroidal Coordinates
Looking upon the infinitely thin circular disk as the limiting case of an ellipsoid of revolution suggests introduction of the system of curvilinear coordinates in which thedisk becomes one of the coordinate surfaces. Such coordinates , are3 the following
The surface = const. represent ellipsoids of revolution, the surfaces = const. represent hyperboloids of one sheet (Figure 2). The half-space z 0 is defined in thenew coordinates by 0 while the portion r r0 of the surface of the half-space is characterized by = 0.
In terms of the coordinates , the boundary conditions (3) and (4) become
Equation (10) indicates that the solution for the half-space 0 may be thought of as contained in the solution for the space exterior to the limiting ellipsoid = 0, theadditional requirement being that this solution is an even function of .
4Solution for the Static Case ( / t = 0)
The equation of motion (7) is solved by separation of variables
The equations for and g are
3See, for instance, Stratton, Morse, Chu, and Huttner, Elliptic Cylinder and Spheroidal Wave Fuctions (John Wiley & Sons, Inc., New York, 1941).
Page 514
The values of the separation constants dl, insuring periodicity in are4
Solutions of Eqs. (12) and (13) are the associated Legendre polynomials
As becomes infinite as , approaches infinity and becomes singular for = 1, it must be required
The condition that v and W are even functions of requires that only even values of the subscript l occur. A series soloution for v possessing appropriate behavior isthen
We note that the first terms of this series are the functions5
It can be seen, according to Eq. (9), that the first term of the series (18) is sufficient to satisfy the boundary conditions of the problem. There follows
and the expression for the displacement v becomes in terms of the curvilinear coordinates and
From Eq. (22) there follows that in cylindrical coordinates the surface displacements are given by
4Whittaker and Watson, Modern Analysis (Cambridge University Press, London, 1940), pp.323326.
5Hobson, Spherical and Ellipsoidal Harmonics (Cambridge University Press, London, 1931).
Page 515
Fig. 3.
The shear stress distribution under the plate is given by
and in cylindrical coordinates
At the edge of the disk (r = r0) the shear stress becomes infinite. Surface displacements and shear according to Eqs. (23) and (25) are plotted in Figure 3.
The relation between torque and rotation is obtained by substituting Eq. (25) in Eq. (5). The result is
This formula may be compared with the expression for the angle of rotation at the center of the disk under the assumption of a linear shear stress distribution.1 In thatcase Eq. (26) holds when the factor 16/3 is replaced by a factor . This indicates that when the results of the theory are to be applied to the interpretation of soil-mechanical data, no quantitative agreement can be reached unless rather accurate information regarding the distribution of the stresses acting between the actuatingdisk and the foundation is known. This fact must also be taken into consideration for the solution of the problem of the vibrations of an elastic half-space due tostresses applied normal to the surface of the half-space. It suggests itself that earlier work6 on this problem should be extended with this observation in mind.
6E. Reissner, Ing. Arch. 7, 381396 (1936).
Page 516
Complementary Energy Procedure for Flutter Calculations[J. Acron. Sc. 16, 316, 1949]
In the following there is outlined a procedure for flutter calculations which bears the same relation to the well-known Rayleigh-Ritz procedure1 as does the theorem ofminimum complementary energy in elasticity (minimum principle for the stresses) to the theorem of minimum potential energy (minimum principle for thedisplacements).
The procedure is stated here for the case of torsion-bending flutter of a cantilever wing. The possibility of its extension to more general flutter problems will beapparent.
The differential equations for torsion-bending flutter of a cantilever without sweep may be written in the form
where the coefficients anm depend in a known way1 on reduced frequency and other parameters.
Boundary conditions may be taken in the form
The variational principle that forms the basis of the proposed procedure may be stated in the following way: Among all states of stress (M, T) and displacements(y, ) that satisfy Eqs. (1), (2), and (5), the state that also satisfies Eqs. (3) and (4) is determined by the variational equation,
The proof of this theorem consists in showing that Eq. (6) is satisfied if one substitutes in it Eqs. (3) and takes account of the fact that M, T, y, and mustsatisfy Eqs. (1), (2) and (5), the same as M, T, y, and .
To use the theorem for the approximate determination of flutter speeds, set
1See, for instance, U.S. Air Forces Technical Report No. 4798 (1942) by B. Smilg and L. S. Wasserman.
Page 517
where Y and are assumed modesfor instance, the fundamental uncoupled bending and torsion modes of the cantilever beam or, more simply, Y = (x/L)2 and =(x/L)and where A and B are constant amplitude factors. Substitute Eq. (7) in Eqs. (1) and (2) and calculate
such that M and T satisfy Eqs. (5) for all values of A and B. Corresponding to Eqs. (7) and (8) one has y = ( A) Y, etc. The results are substituted in Eq. (6), andthis equation reduces them to the following two simultaneous equations for A and B
The coefficients k are given by the following expressions
The determinantal equation determining the flutter speed follows from Eqs. (9) as
It is believed that the use of Eqs. (10) and (11) may lead, for the same modes Y and , to more accurate results than the Rayleigh-Ritz procedure, because in thepresent procedure the strain energy is expressed in terms of functions that are obtained from the assumed modes by integration rather than by differentiation as in theRayleigh-Ritz procedure. This may permit the working with less close approximations to the actual modes than is possible with the Rayleigh-Ritz procedure; or itmay, in some instances, permit to take account of fewer degrees of freedom than is necessary with the Rayleigh-Ritz procedure. These advantages will be somewhatoffset by the fact that evaluation of the coefficients k involves a larger number of integrations than does evaluation of the corresponding coefficients in the Rayleigh-Ritz procedure.
Page 518
Reihenentwicklung Eines Integrals Aus Der Theorie Der Elastischen Schwingungen[Math. Nachr. 8, 149153, 1952]
Im folgenden geben wir die Herleitung eines Resultates wieder, welches in der Theorie der erzwungenen Schwingungen eines elastischen Halbraumes gebrauchtwird1). Dieses Resultat ist senierzeit in weniger allgemeiner Form und ohne Herleitung angegeben worden2).
Es handelt sich um die Auswertung des Integrals
von dem der Cauchysche Hauptwert zu nehmen ist. Die feste Zahl 2 ist kleiner als 1; insbesondere sind die Werte 2 = 0, 1/3, 1/2 von Interesse. Die Auswertungwird benötigt für reele Werte von a0.
An Stelle von (1) kann geschrieben werden3):
oder auch
wo
Herrn Georg Hamel zum 75. Geburtstag gewidmet.
1Siehe E. Reissner, Stationäre, axialsymmetrische durch eine schüttelnde Masse erregte Schwingungen eines homogenen elastischen Halbraumes. Ingenieur-Arch. 7, 381396 (1936).
2Loc. cit., 388389. Die folgenden Betrachtungen sind einem unveröffentlichten Abschnitt der Dissertation des Verfassers (Techn. Hochsch. Berlin 1936) entnommen.
3Loc.cit., Gleichung (60).
Page 519
und
In beiden Integralen (2) und (3) ist der Cauchysche Hauptwert zu nehmen.
Statt (2) und (3) können wir auch schreiben:
und wir beachten, daß für genügend große die folgenden konvergenten Reihenentwicklungen bestehen:
die Werte der ersten drei Koeffizienten in diesen Reihen sind
Das Resultat, welches wir herleiten wollen, ist die folgende Reihenentwicklung für das Integral (1):
Insbesondere wird dann, wie man mit Hilfe von (5) sieht,
Um die Entwicklung (6) zu erhalten, schreiben wir mit einer endlichen reellen Zahl , welche größer ist als die größte Nullstelle von M( ):
Page 520
Die Singularität des Integranden steckt dann in den S . Wir betrachten vorerst S und S gesondert. Dabei beachten wir folgendes: S und S können nach Potenzenvon a0 und entwickelt werden. Zusammenfassung von S und S ergibt dann eine Potenzreihe in a0 mit Koeffizienten, welche Potenzreihen in sind. Da dasIntegral, das durch diese Reihe dargestellt wird, unabhängig von ist, wird der Koeffizient jeder Potenz von a0 unabhängig von sein. Wir behalten in derEntwicklung von S und S von vornherein nur diejenigen Glieder bei, welche von frei sind, da sich die anderen nach der Zusammenfassung wegheben.
Wir beginnen mit der Berechnung von S . Auf Grund der Annahme über die Größe von können wir unter Beachtung von (2a), (3a) und (5) schreiben:
Wir setzen weiter
Für die Funktion G wird durch partielle Integration die Rekursionsformel
erhalten, so daß sich alle G durch Besselsche Funktionen und durch G0 ausdrücken lassen. Für G0 erhält man
Wir brauchen von G nur diejenigen Bestandteile, welche von frei sind. Wenn wir diese Bestandteile durch einen Stern kennzeichnen, haben wir
und damit
Für die von freien Bestandteile von haben wir dann
Wir betrachten nun S . Wir werden zeigen, daß keinen von freien Bestandteil hat und daß somit . Um dies zu sehen, verfahren wir
Page 521
in folgender Weise. Wir schreiben
Wie man zeigen kann, liegt nur eine Polstelle des Integranden in dem Intervall ( , ). Es sei 3 diese Polstelle, die der Gleichung
genügt. Weiterhin machen wir Gebrauch von der Tatsache, daß 1 und 2 die folgende Gleichung befriedigen:
Wir betrachten nun Integrale von der Form
und
Um in dem letzteren den von freien Bestandteil zu erhalten, setzen wir
womit
wird. Beachtet man, daß
ist, so sieht man, daß in (23) außer dem Integral
nur Integrale von der Form vorkommen, welche zu Snv Beiträge von der Form ( 2 2)(2k + 1)/2 liefern. Da
Page 522
kein von freies Glied besitzt, bleibt also nur der von freie Bestandteil von
übrig.
Für n = 3 haben wir
und da die Entwicklung dieses Ausdruckes nach Potenzen von 1/ keinen von freien Bestandteil besitzt, folgt dasselbe für s3 und somit
Für n = 1, 2 haben wir
Der von freie Bestandteil dieses Austruckes hat den Wert , und damit ist
Zusammenfassung der vorstehenden Rechnungen ergibt
Nun verschwindet aber der Ausdruck in der eckigen Klammer, wie man folgendermaßen sieht. Vergleich von (16) und (17) mit (2), (3) und (4) zeigt, daß
ist und infolgedessen
Unter Berücksichtigung von (19) folgt daraus
und damit
was wir beweisen wollten.
Kombination von (15) und (35) ergibt die Reihenentwicklung (6) für das Integral (1).
Page 523
On Axi-Symmetrical Vibrations of Shallow Spherical Shells[Qu. Appl. Math. 13, 279290, 1955]
1Introduction
The present note may be considered as a sequel to an earlier paper on the same subject [4]. In this earlier paper the solution of the differential equations for axi-symmetrical vibrations of shallow spherical shells was given in terms of certain Bessel functions. The problem of the frequency determination for a shell segment withclamped edge was considered as an example of application of this solution. It led to the vanishing of a third-order determinant each element of which involved Besselfunctions and the solution of a certain cubic (Eqs. 32 and 35 of Ref. 4). Similar results, by a somewhat different method, had earlier been obtained by Federhofer [2].The Bessel-function frequency determinant being difficult to evaluate, no numerical results have yet been obtained by its use. Instead, an approximate solution for thelowest frequency was obtained by means of the procedure of Rayleigh and Ritz (Eq. 40 of Ref. 4 and a similar result in Ref. 2).
The Bessel-function solution of Ref. 4. was obtained on the basis of assumptions which had previously been made for the problem of static deformations of shallowspherical shells [3]. It was not observed at that time that an additional approximation would be appropriate for the problem of transverse vibrations. This additionalapproximation is based on the fact that for transverse vibrations of shallow shells the magnitude of the longitudinal inertia terms is negligibly small compared with themagnitude of the transverse inertia terms [5]. Upon omission of longitudinal inertia terms it becomes possible to reduce the differential equations of dynamics to thesame form as the equations of statics except that the transverse load function must include the d'Alembert term hwtt.
The present paper contains applications of this result to three specific problems of axi-symmetrical vibrations of spherical shells.
(1)Determination of the Lowest Frequency of Free Vibrations for a Shell Segment with Clamped Edge
The numerical results are compared with the corresponding results from the earlier Rayleigh-Ritz formula.
(2)Determination of the Frequencies of Free Vibrations of a Shell Segment with Free Edge
The relation between the present frequency equation and the corresponding frequency equation for a flat plate is of such nature that known numerical results for theflat plate can be translated with little difficulty so as to furnish the corresponding results for the shallow spherical shell.
Page 524
(3)Forced Vibrations Due to Point Load at Apex of the Shell
We determine first the point-force singularity which occurs regardless of the form of the boundary conditions at the edge of the shell. We then consider further thecase for which the boundary is sufficiently far removed to assume it at infinity and use the condition that sufficiently far from the point of load application the solutionmust represent outward travelling waves. Our results generalize corresponding results of H. Cremer and L. Cremer [1] for the unlimited flat plate.
2Differential Equations for Transverse Vibrations of Shallow Spherical Shells
Let the equation of the shell surface be given by
In the absence of longitudinal loads we have as differential equations for the displacement w in the direction of z and for an Airy stress function F,
where = density of shell material, h = wall thickness (assumed constant), E = modulus of elasticity, D = Eh3/12(1 2) and 2 = 2/ r2 + r1 / r + r2 2/ 2 (Figure1).
We shall further need the relations
which are the same as in problems of statics.
3Axi-Symmetrical Solutions
We set
Restricting attention to the case q = 0 it is found that solutions of the homogeneous equations (2) and (3) are of the form
Page 525
Fig. 1.Spherical shell segment, showing notations for geometrical
dimensions.
The constants Cn in (8) and (9) are arbitrary, the usual notation for Bessel functions and modified Bessel functions is employed, and furthermore
Taking account of the fact that
there results for the constants Bn
Of the eight constants Cn only six are physically significant. The constant C7 has no effect on stresses and displacements. The condition of vanishing circumferentialdisplacements as in problems of statics [3], is of the form (hE/R)W = 2 + const. From this (hE/R)C6 = 4B6. With B6 from (12) this implies 4C6 = 0. Since 0we have
and therewith altogether
Page 526
The solution (13) and (14) will be applied to the three problems indicated earlier.
4Frequency Equation for Shell Segment with Clamped Edge
Let r = a be the edge of the shell segment. Since W and must be regular for r = 0 we have
The conditions of vanishing edge displacement W and edge slope W are
where = a.
We assume that the third edge condition stipulates vanishing horizontal displacement or, equivalently, vanishing circumferential strain . With the help of (4) thisrelation becomes (F r1F )a [ 2F (1 + )r1F ]a = 0. Introduction of (14) gives as third relation for the constants C1, C3, C5,
The frequency equation of the problem is the condition of vanishing of the determinant of the system (16), (17) and (19). Setting , , this frequencyequation may be brought into the following form
The parameter 4 is
and is given in terms of and , as follows
The quantity H in (21) is the rise of the shell. It is related to the radii R and a through the formula,
The frequency equation (20) furnishes as a function of . Approximate values of for the lowest frequency may be found in Table 1.* Having as function of weobtain further from (22)
*For these and all other computations in this paper the author is indebted to Millard W. Johnson.
Page 527
Table 1. Numerically smallest solution ofEq. (20)
40 3.19610 3.23520 3.27350 3.380100 3.537300 4.000500 4.328800 4.6801000 4.8551400 5.1121800 5.2852500 5.4823000 5.5705000 5.72310000 5.828
5.910
Let 0 be the value of corresponding to = 0 and = 0, that is the value of the frequency for a flat plate and for vanishing Poisson's ratio. We have
where 0 = 3.195 and . It is convenient to express the actual values of in units of 0, as follows,
Numerical values of / 0 as a function of and H/h may be found in Table 2 and in Figure 2.
When H/h is large enough so that the second term under the square root in (26) dominates the first term, practically when H/h is larger than about 25, then Eq. (26)may be replaced by the approximation
It is interesting to note that the thickness h which occurs in the flat-plate frequency formula (25) has been replaced by the shell rise H and the frequency has becomeindependent of wall thickness. It should be emphasized, however, that the simple formula (26*) depends on two limitations. On the one hand we must have that
and on the other hand we must have in order that the theory of shallow shells is applicable. We may note that it is this latter restriction whichensures that the transverse vibrations of the shell take place at frequencies which are low compared with the frequencies of longitudinal vibrations of flat plates, theselatter being of order (E/ )1/2(1/a).
Page 528
Table 2. Frequency of clamped-edge shell in units of corresponding flat-platefrequency for = 0, as function of and H/h
/ 0
= 0 = .3 = .5
0 1.000 1.0483 1.1547.5 1.08 1.149 1.281.0 1.31 1.40 1.591.5 1.61 1.75 2.002.0 1.94 2.16 2.422.5 2.31 2.57 2.873.0 2.67 2.99 3.323.5 3.06 3.40 3.734.0 3.43 3.78 4.124.5 3.81 4.16 4.495 4.18 4.51 4.896 4.90 5.17 5.457 5.59 5.80 6.018 6.22 6.41 6.609 6.86 7.01 7.2010 7.50 7.62 7.8211 8.13 8.25 8.4412 8.76 8.88 9.0514 10.05 10.15 10.2816 11.35 11.42 11.5418 12.67 12.73 12.8220 14.00 14.1 14.2
Comparison with Rayleigh-Ritz Formula
Equation (40) of Ref. 4 may be written in the following form, which is equivalent to (26)
Figure 2 contains values of ( / 0)RR as dotted lines. It is seen that the Rayleigh-Ritz solution agrees remarkably well with the differential equation solution forsufficiently small values of H/h, practically up to values of H/h of about three. As H/h increases the error becomes larger, the percentage error approaching a finitelimiting value of from 60 per cent to 25 per cent as decreases from 0.5 to 0.
5Frequency Equation for Shell Segment with Free Edge
The boundary conditions for a free edge are
where Mr, Vr and Nr are given in (4) to (6). To this are again added regularity conditions for r = 0 which again mean that Eq. (15) must hold. We observe furtherthat (28) contains two relations involving W , W and W but neither W itself nor
Page 529
Fig. 2.Lowest frequency for shell segment with clamped edge, in units of corresponding
frequency for flat plate with zero Poisson's ratio.
and its derivatives. This means that the frequency equation for the shell with free edge does not involve the constant C5 in Eq. (13) for W. In other words theadmissible values of are the same as those for a flat plate with free edge.
The frequency equation for axially symmetric vibrations of a flat plate with free edge, according to Kirchhoff, is
Examples of the numerically smallest values of are = 0, = 2.87; = .3, = 3.00; = .5, = 3.07. Having we obtain by means of (10), (18) and (23) inthe form
Let 0 and 0 be appropriate values for the flat plate and for = 0. We may then write, in analogy to the result (26) for the shell with clamped edge,
Equation (31) is the same as Eq. (26) except that for the shell with free edge the quantity is independent of the values of H/h while for the shell with clamped edge
Page 530
Table 3. Frequency of free-edge shell in units of corresponding flat-platefrequency for = 0, as function of and H/h
/ 0
= 0 = .3 = .5
0 1 1.148 1.315.5 1.084 1.224 1.3801.0 1.305 1.423 1.5621.5 1.610 1.705 1.8212.0 1.96 2.035 2.1352.5 2.33 2.40 2.483.0 2.71 2.77 2.853.5 3.11 3.16 3.224.0 3.51 3.55 3.624.5 3.91 3.96 4.015 4.32 4.36 4.416 5.15 5.17 5.217 5.98 6.00 6.038 6.81 6.82 6.869 7.64 7.66 7.7010 8.47 8.49 8.5311 9.31 9.32 9.3512 10.15 10.15 10.1814 11.82 11.82 11.8216 13.50 13.50 13.5018 15.20 15.20 15.2020 16.86 16.86 16.86
was found to be a function of H/h. Values of / 0 for the numerically smallest solution of (29) as function of H/h and may be found in Table 3 and Figure 3.
We note that when H/h is greater than about 10, Eq. (30) may be simplified to = 2(E/ )1/2(H/a2) which is the same limiting expression which was previouslyobtained for the shell segment with clamped edge.
We further note that very probably for this problem of a shell with free edge the lowest frequency of axi-symmetrical free vibrations is higher than the frequencies ofcertain non-symmetrical vibrations. This fact is concluded from corresponding known results for the special case of the flat plate.
6Oscillating Point Load at Apex of Shell
The following conditions must be satisfied for an oscillating point load P exp (i t) at the apex of the shell
Page 531
Fig. 3.Lowest frequency of axi-symmetrical vibrations for shell segment with
free edge, in units of corresponding frequency for flat plate withzero Poisson's ratio.
In view of (4), (5), (6), (7) these conditions assume the following form for the solution functions W and as given by (13) and (14)
We begin by omitting terms from the solutions (13) and (14) which automatically satisfy the conditions (32 ) and (33 ) and retain the singular portion
Equations (34) and (35) are valid, with real values of the constants C2 and C4, as long as is real. According to (10) this means as long as
When
Page 532
the representations (34) and (35) are no longer convenient. It will, however, be shown that the results obtained in the range of frequencies (36a) are readilytransferred to the range (36b).
We note that the frequency which divides the ranges (36a) and (36b) is exactly that lowest frequency of free vibrations which occurs in the range of sufficiently largevalues of H/h [see Eq. (26*)].
In order to determine the constants C2, C4 and C8 we observe the following relations in which use has been made of (11),
Equations (38) being valid when r << 1.
Introduction of (34), (35), (37) and (38) into the finiteness conditions (33 ) shows that these conditions are satisfied provided
The load condition (32 ) leads to the further relation
Introduction of C2, C4 and C8 from (39) into (34) and (35) leads to the following expressions for the singular solutions Ws and s,
It will now be shown that (40) and (41) remain valid, but are conveniently written in different form, in the range of small given by (36b). For values of in therange (36b) we write
and we introduce Kelvin functions through the following known relations
Page 533
If (42) and (43) are introduced into (40) and (41) there follow as expressions for Ws and s which are valid when < R1(E/ )1/2,
Equations (44) and (45) contain as special case, when = 0, the previously given formulas [3, Eq. (48)] for the corresponding problem of static deflection. Writing = (hE/R2D)1/4 = [12(1 2)]1/4(Rh)1/2 Eqs. (44) and (45) become for = 0
We finally note the following important fact. As long as the solution (44) applies, that is as long as < (2H/a2)(E/ )1/2, we have that the deflection amplitude Wdecreases exponentially at large distances r from the point of load application. When is larger than (2H/a2)(E/ )1/2 so that the solution (40) applies then thedeflection amplitude W decreases as r1/2, that is, much more slowly than exponentially. In this latter case an important distinction can be made between standing waveand travelling wave solutions.
7Travelling Wave Solution for Effectively Unlimited Shell Segment
We inquire for a solution in which the deflection w = exp(i t)W(r) behaves for sufficiently large values of r as r1/2 exp[i( t r)]. For a solution with this type ofbehavior we have that the wave produced by the pulsating point load travels outward, with energy being dissipated through radiation.
In view of the asymptotic behavior of the Hankel function of the second kind
an appropriate result is obtained by adding in (40) and (41) a suitable multiple of the non-singular solution J0( r) as follows,
We note that the solution (47) and (48) contains as a special case the corresponding result for a flat plate, as given by H. Cremer and L. Cremer [1]. The case of theflat plate follows if we set R = in the expression for , making = ( h 2/D)1/4. We note further that while for the spherical shell the solution (47) and (48) is valid,subject to the restriction that > (E/ )1/2R, this relation ceases to be a restriction for the case of the flat plate.
In order that the travelling wave solution (47) have a meaning it is necessary that the boundary r = a be sufficiently far removed from the point of load
Page 534
application. Let us assume that sufficiently far means
Equation (49) may be written as a restriction on frequencies as follows
8Two Formulas of Acoustical Significance
We consider the ratio of velocity at the point of load application to the force causing this velocity, ( w/ t)r=0/P exp(i t). According to Eq. (47)
Since J0(0) = 1 and, according to (38), limr 0[1/2 Y0( r) + K0( r)] = 0 there follows,
For H = 0, Eq. (50a) reduces to a result in Ref. 1. We note that a ratio which is independent of frequency for the flat plate is a function of frequency for the shallowspherical shell. In the range of applicability of (50a), which is given by (49 ), this ratio decreases with increasing frequency towards a limiting value which is theconstant flat plate value.
When < (2H/a2)(E/ )1/2 so that Eq. (44) applies we have instead of (50a)
In view of the fact that kei(0) = 1/4 and with defined in (42) this may be written in the following form
We finally observe, on the basis of (50a) and (50b), that the following result holds for the work of the force P exp (i t) per cycle
It is recalled that Eq. (51) is derived without consideration of damping sources other than radiation damping. In addition to this, while formally the second part of Eq.
Page 535
(51) holds for all greater than (2H/a2)(E/ )1/2, for the solution to have physical meaning the stronger restriction (49 ) actually applies.
References
1. H. and L. Cremer, Theorie der Entstehung des Klopfschalls, Z. f. Schwingungs-und Schwachstromtechnik 2, 6172 (1948).
2. K. Federhofer, Zur Berechnung der Eigenschwingungen der Kugelschale, Sitzber. Akad. Wiss. Wien 146, 5769 (1937).
3. E. Reissner, Stresses and small displacements of shallow spherical shells, J. Math. and Phys. 25, 8085, 279300 (1945); 27, 240 (1948).
4. E. Reissner, On vibrations of shallow spherical shells, J. Appl. Phys. 17, 10381042 (1946).
5. E. Reissner, On transverse vibrations of thin shallow elastic shells, Q. Appl. Math. 13, 169176 (1955).
Page 537
AERODYNAMICSI owe my involvement with aerodynamics to the friendship shown to me during my graduate student days at M.I.T. by two young Associate Professors, Heinz Petersand Manfred Rauscher.
At the suggestion of Peters I studied G. I. Taylor's papers on the statistical theory of turbulence, in order to report on them in his aerodynamics course. As oneconsequence of this we jointly wrote a brief note on the spatial decay of turbulence generated at the boundary of a half space [11]. The other consequence combinedwhat I had learned from G. I. Taylor and from Norbert Wiener's Fourier Analysis course, for a rigorous solution of a linear ''model" problem concerning the decay ofhomogeneous isotropic turbulence [16].
At the suggestion of Rauscher I later studied the problem of an oscillating lifting surface, in order to help with his aero-elastic concern with finite-span corrections forthe two-dimensional Glauert-Theodorsen flutter coefficients. I found this problem to be very difficult but managed to derive an approximate one-dimensional integralequation for obtaining such corrections, as a generalization of Prandtl's lifting line equation for the corresponding steady state problem [34]. This result wassubsequently extended, simplified and applied [47,49] with the cooperation of Maxwell Hunter and John Stevens, who were two outstanding M.I.T. graduatestudents, both destined to become highly successful aerospace engineers.
In preparation for this work Francis Hildebrand and I had undertaken to analyze the aerodynamic span effect for the limiting-case problem of torsional divergence[32]. This in complementation of my father's pioneering solution, in 1926, in which two-dimensional aerodynamic coefficients had been used.
After some attempts to extend our work into the subsonic range [65,71,72] I ended my concern with problems of lifting surfaces in non-uniform motion with theformulation of the more general problem of an impenetrable deformable surface in space, set in motion in a perfect fluid of infinite extent [62]. With this surfacerepresenting a discontinuity of pressure and tangential velocity it was then assumed, as a generalized Kutta condition, that there would be a trailing surface oftangential velocity discontinuities, in such a way that the fluid velocity at the trailing edge of the impenetrable surface would not become infinite. I proceeded from thisto a linearized version of the problem which then was specialized to the case of a plane impenetrable surface. This in turn served as a starting point for a review of thecontents of [34,47,49].
It remains to mention three results for steady state problems, which I believe have remained significant. The first of these was to show that the Prandtl-Glauertcompressibility correction for two-dimensional flow and the Goethert correction for axi-symmetrical flow would both be contained as limiting cases in the solu-
Page 538
tion of the problem of axi-symmetrical flow along a circumferentially corrugated cylinder [55].
The second was to use the two-dimensional integral equation for the pressure distribution over a rectangular surface with given downwash distribution as a startingpoint for a quite simple derivation of Weissinger's refinement of Prandtl's lifting line equation, and also for the derivation of a system of two simultaneous one-dimensional integral equations for the spanwise variation of lift and section moment [61].
The third result concerned the derivation of a one-dimensional integral equation for the chordwise distribution of pressure for an infinite lifting strip, on the basis ofassuming a spanwise sinusoidal distribution of downwash. The form of this one-dimensional integral equation allowed the deduction of some not previously knowndetails concerning the nature of aspect ratio effects for wings of finite span [70].
Page 539
A Contribution to the Theory of Turbulence*
[J. Aeron. Sc. 4, 384385, 1937]
Studies of the problem of Wind-Tunnel Turbulence have recently been published by Taylor, 1 von Kármán,2 and Dryden.3 These studies are concerned with thedecay of homogeneous isotropic turbulence. The scale and intensity of the turbulence is assumed to be known at a given instant throughout the fluid and the decaywith respect to time is asked for.
In this note the authors investigate the turbulence produced at a constant rate in the boundary, and decaying in a direction normal to the surface of generation. Thisproblem, so far as the authors are aware, has not been treated before. The problem becomes especially simple if one considers a fluid in a semi-infinite spacebounded by a plane along which the turbulence is produced. The mean motion is assumed to be constant throughout and parallel to the plane.
This investigation is not only of theoretical interest but should be useful in the study of the behavior of turbulent boundary layers, and might be of help in the study ofthe influence of surface roughness. The stated problem can be treated by means of the equations of the statistical theory of turbulence recently proposed by vonKármán. It furthermore might serve as the basis for an experiment in which the consequences of the theory can be tested.
The considered turbulent motion has statistically an isotropic character due to the absence of mean shear stresses, i.e., the correlations of the velocity components ata certain point in the space with respect to a set of orthogonal axes are independent of the rotation of this system. The turbulence, however, is not homogeneous,since the correlations are functions of the distance from the boundary.
Consider a space with the orthogonal axes x1, x2, x3, in which the fluid is bounded by the (x1,x3) plane and has a constant mean velocity U in the x1 direction. Thecomponents of the turbulent velocity u and vorticity may be denoted by u1, u2, u3, and 1, 2, 3, respectively. The following equations for the dissipation ofenergy and the dissipation of vorticity have been derived by von Kármán:
*With H. Peters.
1G. I. Taylor, Proceedings of the Royal Society, London, Vol. 151, 1935.
2Th. von Kármán, The Fundamentals of the Statistical Theory of Turbulence, Journal of the Aeronautical Sciences, Vol. 4, 131138, 1937.
3Hugh L. Dryden, The Theory of Isotropic Turbulence, Journal of the Aeronautical Sciences, Vol. 4, 273280, 1937.
Page 540
Neglecting with von Kármán the first terms on the right-hand side of Eqs. (1) and (2) as small compared with the other terms and substituting
where is a length characteristic for the scale of the turbulence, Eqs. (1) and (2) with
take on the form
where c1 = 2 2k2/k4, and c2 = 2 2k3/k5.
The solution of Eqs. (4) and (5) is determined if the following boundary conditions are prescribed:
It seems to be difficult to obtain the general solution of the non-linear system Eqs. (4) and (5). It is, however, possible to give a certain manifold of solutions which are
likely of importance. Furthermore, the general relation between and which must exist under the stated assumptions can be deduced from Eqs. (3) and (4).
To obtain a solution, assume
where A, B, n, and m are constants. This introduced into Eqs. (4) and (5) gives
Hence
In order to satisfy the assumption , it is necessary that m > 2 and hence n < 2.
If the assumed power law is correct the constants m and n can easily be determined experimentally. With known values for n and m the constants c1 and c2 can beevaluated by means of Eqs. (5) and (6). If furthermore, the values of k2 and k3 are assumed to be known (for instance, equal to the values for isotropic homogeneousturbulence) the ratio of the two mixing lengths l1 and l2 can be calculated.
Page 541
Of interest is the relation between the "mixing length" and the distance from the wall, which can be obtained from Eq. (3).
It thus can be seen that in this case the "mixing length" is proportional to the distance from the wall.
The general relation between and is
which is obtained by subtracting Eq. (5) multiplied by 2 from Eq. (4), with this resulting in a differential equation of the first order for with coefficients dependingon and = d /dx2.
In the case of a flow along a flat plate with no pressure gradient, the above prescribed boundary conditions are approximately satisfied at the outer edge of theturbulent boundary layer, due to the slow change of the layer thickness and of the scale and the intensity of the turbulence in the boundary layer in this direction. Arough experimental check on = f(x2)indicates that in this case the assumed power law is at least a good approximation. These tests have been carried out in arecently completed wind tunnel for the study of the behavior of boundary layers.
Page 542
Note on the Statistical Theory of Turbulence[Proc. 5th Intern. Congr. Appl. Mech., pp. 359361, 1938]
Recent investigations by G. I. Taylor, Th. v. Kármán, H. L. Dryden and L. Prandtl have dealt in different ways with the problem of the decay of homogeneousisotropic turbulence. In this problem a viscous incompressible fluid is considered for which at time t = 0 a statistically homogeneous and isotropic distribution ofvelocity and pressure is given over all space. The initial distribution is characterized by two averages: the mean value of the kinetic energy per unit of volume, , anda certain length , indicating the scale of the turbulence. By introducing assumptions of different kinds into certain averaged transformations of Navier-Stokes'equations, results have been obtained concerning the change with time of the quantities and .
Now it would be possible to give a rigorous solution of the problem if its basic equations were linear. If one neglects the quadratic inertia terms in the equations ofmotion one has such a linear problem. The values of and would then result in terms of the initial distribution of u and hence in terms of the initial values of and without using assumed interrelations between these mean values.
In this note it will only be shown how the method works for the simplest "model" of the corresponding turbulence problem, which has already been considered by v.Kármán (1937a).
The simplified problem consists in finding a solution of the equation
which satisfies the initial condition
In order to carry through the analysis with the help of comparatively simple mathematics the assumption will be made that u0(x) = 0 for |x| > x0. By an appropriatelimiting process x0 this restriction will be removed in the results.1 For the sake of simplicity it will also be assumed that u0(x) is an even function that is u0(x) = u0(x). By doing so the necessary calculations are shortened without essential loss of generality.
Equation (1) may be satisfied by
1Methods for mathematically rigorous solutions of such problems of analysis may be found in papers of N. Wiener.
Page 543
A ( ) is determined, in view of (2), from
Inversion of (4) gives
The correlation between simultaneous values of u at different points is expressed in the following form
With Eq. (3) one has
Integrating first with respect to x leads to
For the integration with respect to one has to observe that for large values of x0 only those values of the integrand contribute substantially for which either or. Hence one may write
Since A( ) = A( ) and
I becomes
Introducing (7) into (6) results in
Page 544
where A( ) is given in terms of u0 and x0 by Eq. (5). The mean square of u is then
and the "correlation function" R becomes
This expression is solution of the differential equation which v. Kármán (1937a) derived for R
For results
In Eqs. (9), (10) and (12) there are given the statistically significant quantities of the problem in terms of the initial values u0 of u.
These expressions become simpler and clearer if one asks only for their behavior for sufficiently large values of t. In (9) only small values of contribute appreciablyto the value of the integral if t is large. Therefore, one may write
and with (5)
Equation (14) shows how depends asymptotically on time t and on the initial values u0. It shows how important the randomness of the initial distribution of the
"velocity" u0 is. For a periodic, and thus not random, initial distribution for instance follows from (14) that . The point is that for such an initial distribution thepreceeding analysis is not appropriate. Indeed when u0(x) = cos ax then cos ax and
To the same degree of accuracy as in (14), there follows from (10) for the correlation function R
This means simply that for increasing t the correlation curve flattens out more and more. In order to follow this process quantitatively the development of (10) has tobe carried at least one step further.
Page 545
For 2 one obtains, taking into consideration that has its peak at = (2 t)1/2 and that this peak becomes more pronounced with increasing t
It is noteworthy that in this linear problem is found to be asymptotically independent of the initial state of the system and also not related to the value of .
In the same way many further results can be obtained. To mention one, in the three dimensional case
one derives
This is interesting in view of the fact that in this case the equation for the mean energy dissipation is identical with the equation for the decay of the kinetic energy in thecorresponding turbulence problem
Here one has
whereas Taylor derived in an entirely different way
and v. Kárman
with an undetermined value of n.
2A check on this result (16) is obtained in the following way: Multiplying Eq. (1) by u and taking means leads to . With from (16) this is and consequently =
const./ in accordance with (14).
Page 546
In summary the problem of the decay of homogeneous isotropic turbulence becomes explicitly solvable if the equations of motion are linearized. The velocitycorrelations may be obtained as function of the initial conditions of the system. The analysis is here carried out for the simplest "model" of isotropic turbulence. Themean square of the velocity, , and the scale length are asymptotically developable with respect to time t. The result is that is asymptotically independent ofthe initial conditions and not related to the values of and that is asymptotically proportional to a certain integral over the initial values u0 and to a certainpower of t. It is seen to be of primary importance for the decay law whether the initial distribution is at random or not.
References
1. Dryden, H. L., 1937: J. Aero. Sc. 4, p. 273.
2. Kármán, Th. v., 1937a: Proc. Nat. Acad. Sc. Wash. 23, p. 98.
3. , 1937b: J. Aero. Sc. 4, p. 131.
4. Kármán, Th. v. and Howarth, L., 1938 Proc. Roy. Soc. A, 164, p. 190.
5. Prandtl, L., 1938: J. Appl. Mech. 5, p. 122.
6. Taylor, G. I., 1935: Proc. Roy. Soc. A, 151, p. 421.
7. , 1937: J. Aero. Sc. 4, p. 311.
8. , 1938a: Proc. Roy. Soc. A, 164, p. 164.
9. , 1938b: Proc. Roy. Soc. A, 164, p. 476.
Page 547
On Compressibility Corrections for Subsonic Flow over Bodies of Revolution[Nat. Adv. Comm. Aeron. TN No. 1815, 1949]
Introduction
The present paper is concerned with the form of the compressibility corrections for subsonic flow which follow from linear-perturbation theory. It is now well knownthat there are essential differences between the compressibility corrections for two-dimensional flow and the corresponding corrections for flow about slender bodiesof revolution [1,2].
The analysis presented herein shows that the relation between two-dimensional and axisymmetrical flow can be clearly demonstrated in the solution for the flow pastan infinitely long corrugated cylinder. In fact, a solution is obtained which contains as limiting cases both the Prandtl-Glauert correction for two-dimensional flow andthe Göthert correction for flow past slender bodies of revolution. Although the results for these two limiting cases are already known, the result obtained in the presentpaper shows the nature of the transition from one limiting case to the other. The nature of this transition has been treated from a different point of view in [3], wherethe bodies considered consisted of a family of ellipsoids ranging from the ellipsoid of revolution to the infinitely long elliptic cylinder. It is of interest that the presentexample is a natural extension of the two-dimensional wavy wall treated by Ackeret in a classical paper [4].
It should be mentioned that the results presented herein were obtained in June 1948, while the author was associated with the National Advisory Committee forAeronautics at Langley Air Force Base, Virginia.
Axisymmetrical Linear-Perturbation Flow past a Corrugated Cylinder
Let u + U and v be components of fluid velocity in the axial and radial directions, respectively. Let (x,r) be the perturbation velocity potential in terms of which
where x and r are the axial and radial directions, respectively. The linearized differential equation for is
where M is the undisturbed stream Mach number.
Let r = a + (x) be the equation of the meridian profile of the body of revolution such that | (x)| << a and | (x)| << 1. The boundary condition at the
Page 548
surface of the body of revolution is then of the following form:
Consider now the particular case of a ''ripple"
with and l being amplitude and wave length of the ripple. An appropriate solution of equation (2) is
where I0 and K0 are modified Bessel functions of order zero and A and B are arbitrary constants. The following properties of I0 and K0 are needed
For the body of revolution in an unlimited air stream, the asymptotic behavior of the functions In requires that the coefficient A vanish. From equations (5), (4), and(3) then follows that the form of the perturbation potential caused by the ripple is
Equation (7) leads to the following expression for the axial velocity u at the surface of the body of revolution
The ratio of u(x, a) for compressible and for incompressible flow is then
It follows from (9) and (6) that when ,
and when ,
Page 549
Equation (10a) is of the form of the Prandtl-Glauert correction while equation (10b) is of the form of the Göthert correction. The transition between the two issupplied by equation (9). Figure 1 shows the relation between the results of equations (9) and (10) when the undisturbed stream Mach number M has the value0.866.
The validity of the foregoing formulas is governed by the restrictions that << a, as well as << l.
Fig. 1.Comparison of compressibility corrections.
M = 0.866.
Velocity Correction Formula for Cylinder with Bump
It is possible to deduce from the above corresponding results for a body of revolution having a meridian profile given by
in the form
Page 550
The corresponding solution for incompressible flow is
A comparison of equations (13), (12), and (11) shows that
which is in accordance with the general results in [2] for this case. The velocity correction formula which follows from equations (12) and (13) is of the form
Velocity Correction Formula for Ripple in the Presence of Tunnel Walls
If, again, a ripple of the form of equation (4) is taken with the boundary condition (3) at the surface of the body of revolution, there is now an additional condition atthe boundary of a tunnel of radius b
The function as in (5) which satisfies the conditions (3) and (16) is
where = /l and .
When b , equation (17) reduces to equation (7). The axial perturbation velocity u at the surface of the body of revolution follows from equation (17) in the form
where it remains to obtain numerical values of uc/ui in their dependence on a/l,(b a)/l and M .
References
1. Lees, Lester: A Discussion of the Application of the Prandtl-Glauert Method to Subsonic Compressible Flow over a Slender Body of Revolution. NACA TN No.1127, 1946.
2. Sears, W. R.: A Second Note on Compressible Flow about Bodies of Revolution. Quarterly Appl. Math., 5, 1947, pp. 8991.
3. Hess, Rovert V., and Gardner, Clifford S.: Study by the Prandtl-Glauert Method of Compressibility Effects and Critical Mach number for Ellipsoids of VariousAspect Ratios and Thickness Ratios. NACA TN No. 1792, 1949.
4. Ackeret, J.: Über Luftkräfte bei sehr grossen Geschwindigkeiten insbesondere bei ebenen Strömungen. Helvetica Physica Acta, 1, 1928, pp. 301322.
Page 551
Note on the Theory of Lifting Surfaces[Proc. Nat. Ac. Sc. 35, 208215, 1949]
1Introduction
In this note we consider the problem of the steady motion of a rectangular lifting surface of finite span in incompressible flow. We are concerned with two separateobjects. One is an attempt to derive somewhat more clearly than in the original paper Weissinger's improvement of Prandtl's lifting-line theory [1]. The other is togeneralize the one-dimensional integral equation of lifting-line theory for the spanwise variation of lift intensity to two simultaneous equations for lift intensity andmoment intensity. In so doing, certain difficulties of lifting-line theory and of its improvements by Weissinger with reference to behavior at and near the tips of the wingare substantially ameliorated.
While we restrict attention here to a rectangular surface and to incompressible flow, the extension to subsonic compressible flow is a simple matter, and also theextension to tapered wings with or without sweep is feasible.
2The Integral Equation of the Rectangular Lifting Surface
We consider a nearly plane wing with chord 2b and span 2sb. The integral equation of the linearized theory may be written in the following dimensionless form
In equation (1) w is proportional to the slope h/ x of the lifting surface, and p is the pressure jump across the lifting surface. Of all singular integrals the Cauchyprincipal value is to be taken. The function K is of the form
The problem of lifting surface theory consists in the solution of (1) for p(x, y) in the region (-1 x 1; -s y s) subject to the trailing edge condition that p(1, y)remains finite.
3Lifting-Line Theory
Lifting-line theory may be thought of as an approximate solution of (1), in the following sense. The term K in the integral equation is neglected, essentially on the basisof the fact that for sufficiently large values of the aspect ratio s K behaves like 1/(y - )|y - | and is therefore small compared with 1/(y - ) over most of the rangeof integration.
Page 552
To obtain the lifting-line equation one may multiply the abbreviated equation (1) by a factor [(1 + x)/(1 - x)]1/2 and integrate as follows:
Let
and use the formula
Equation (3) then assumes the following form:
One disadvantage of this theory consists, for wings with finite tip chord, in the fact that, while the lift intensity assumes the correct values zero at the tips, the pressuresp(x, ±s) do not, in general, and the results of the theory will be of doubtful validity in regions adjacent to the tips, presumably up to distances inward from the tips ofthe order of magnitude of the tip chord.
This behavior of the lifting-line theory may be seen by a consideration of the section moment m, defined as
It can be shown as follows that, in general, m(±s) does not assume the value zero which it would have in the solution of the complete equation (1). We multiply theabbreviated equation (1) by a factor (1 - x2)1/2 and integrate again across the chord,
With (7) and with the relation
Page 553
we may write (8) in the form
Combination of (10) and (6) gives the following alternate expression for m,
In general the second and third terms in (11) do not cancel each other and consequently we have in general m(±s) 0. A known exception forms the case ofconstant w(x, ±s) that is the case of a wing with uncambered tip chord.
4Weissinger's Extension of the Lifting-Line Theory
This extension takes into account in an approximate manner the term with K in equation (1). It may be derived in the following way. Corresponding to (6) and in thesame manner, the following exact relation is first obtained
The last term on the right of (12) is to be taken into account approximately only and this is done by stipulating that in it one may set
which is the exact solution when w is constant and the flow is two-dimensional. Taking as an abbreviation for the triple integral in (12) the symbol I we have then
Equations (12) and (14) are not yet Weissinger's results but rather, in principle, an improvement of his results for the rectangular plan-form wing.
Weissinger's first result, called the F-method, is obtained by introducing the following approximation
Page 554
Note that equation (15) would be exact if K were a linear function of x. With (15) we have further
The function KF can be expressed in terms of elliptic integrals and has been tabulated by Weissinger.
Evidently, once the approximation (15) has been made it is only consistent to make a corresponding approximation in (16). This leads to the so-called L-method which in Weissinger's paper is based on lifting-line considerations. The result is
Calculations have shown [1] that results based on either KF or KL are practically indistinguishable. One advantage of KL besides its simplicity is the fact that ananalogous result may be obtained without too much difficulty for wings with sweepback [1].
With the approximation (17) the integral equation (12) may be written in the following form
Equation (18) shows somewhat more clearly than the corresponding result in Weissinger's paper the relation between this theory and the original lifting-line theory. Italso shows, since [1 + (y - )2]1/2 > |y - |, that the improved theory leads to more pronounced aspect ratio effects than the original theory.
We may note that with
the integrals on the right are expressible in terms of elliptic integrals. It is questionable, however, whether to do so will be of computational advantage.
We remark finally that here also, as in the lifting-line equations (6) and (11), it can be shown that in general the tip moments m(±s) do not vanish.
5A System of Simultaneous Equations for Lift and Moment Distribution
We begin with the exact equation (12) and add to this the following exact equation which takes the place of the approximate equation (10)
We must now take into account approximately the triple integrals both in (12) and (20). As we wish to express these triple integrals both in terms of l and m we can
Page 555
now no longer use the simple approximation (13) for p. Instead we set
With (21) we obtain from (12) and (20) the following two fundamental equations
The kernel functions Kll, etc., are seen to be of the form
The two functions Kll and Klm may be tabulated, just as the function KF of equation (16) has been tabulated and just as the function in (14) may be tabulated.Once this has been done one may solve equations (22) and (23) by means of series of the form
where
Equations (22) and (23) then become systems of simultaneous equations for the coefficients lk and mk. For specific applications it will be necessary to tabulate thefunctions
and corresponding expressions for Klm, for various values of k and s, whereupon the usual collocation procedure can be applied; or to tabulate the correspondingquantities which occur in an application of the Multhop procedure to equations (22) and (23).
Page 556
Considerable simplification is introduced into the foregoing results if again as in going from (15) to (16) and (17) use is made of the following formulae forapproximate numerical integration
Equations (24) and (25) then become
More accurate approximations to Kll and Klm may be obtained if instead of equations (34) the following approximate numerical integration formulae are used
While equations (34) are exact when (x) is any linear function of x, equations (37) are exact when (x) is any third-degree polynomial in x.
We will omit listing the approximations to Kll and Klm which replace (35) and (36) when (37) is used.
We may note that when m = -l/2, which would follow from the assumption that the (1 - x2)1/2-term in (21) is negligible, equation (22) contains the kernel [1/2(y - )]+ Kll - 1/2Klm. As
equation (22) is then reduced to a combination of equations (12) and (14) of the Weissinger procedure, which is as it should be.
It is to be expected that the solution of equations (22) and (23) is nearer to the exact solution of lifting surface theory than is the solution of the combined equations(12) and (14).
An example where the system of two simultaneous equations will give a meaningful answer whereas the one-equation procedure will not is given by a wing formedsuch that w = const. h/ x = C(y)(1 - 2x), that is, a wing for which according to the two-dimensional theory every section has zero lift but non-zero moment.
Page 557
The one-equation procedure predicts that the effect of finite span does not modify the zero lift property. It says nothing about the effect of finite span on the momentdistribution. The present two-equation procedure permits determination of the spanwise variation of moment distribution and indicates that the effect of finite spanmodifies the zero lift properties which the two-dimensional theory predicts.
It is thought that the idea underlying the proposed two-equation procedure, which may be extended to an n-equation procedure, will also be useful in other problemsof lifting-surface theory.
Reference
1. Weissinger, J., "The Lift Distribution of Swept Back Wings," NACA T.M. No. 1120, March, 1947 (Translation of German ZWB F.B. 1553 (1942)).
Page 558
Boundary Value Problems in Aerodynamics of Lifting Surfaces in Non-Uniform Motion *
[Bull. Amer. Math. Soc. 55, 825850, 1949]
1Introduction
In the present paper we propose to discuss certain aspects of the theory of lifting surfaces in non-uniform motion. Briefly, lifting surface theory is concerned with themotion of an impenetrable, deformable surface through an incompressible or compressible non-viscous fluid. In general the impenetrable surface is intended torepresent approximately an airplane wing, a tail surface, or a propeller. The adjective lifting indicates the nature of the interaction desired between the impenetrablesurface and the surrounding fluid.
The mathematical nature of the problems arising in this theory is that of boundary value problems of partial differential equations. Our principal object here isformulation of these boundary value problems and presentation of some of the methods, exact or approximate, which have been used in the solution of some of theseproblems. As may be seen from the list at the end of this paper the amount of work done in this field is considerable and the following account is restricted to thoseaspects of the theory which have been of particular interest to the writer.
Lifting surface theory as developed may be designated as a perturbation theory in the following sense. Because of the assumption of no viscosity there are evidentlytypes of motion of an impenetrable surface which proceed without disturbing the surrounding fluid at all. One now asks for such motions which proceed nearlywithout producing any disturbances and one uses the assumption of small disturbances to simplify the differential equations and boundary conditions of the theory. Ingeneral this simplification leads to a linearized theory and it is this linearized theory which will here be discussed. The main reason for the considerable literature on thesubject is the fact that the range of applicability of the linearized theory has been found adequate for many problems arising in engineering, and in particular inaeronautical engineering.
Evidently one may, if one wishes, consider separately problems of uniform and non-uniform motion in lifting surface theory. Historically, uniform-motion theory, asinitiated by Prandtl, precedes non-uniform motion theory by about ten years. Solution of problems of non-uniform motion theory has turned out to be of considerablygreater mathematical complexity than solution of uniform-motion problems.
*An address delivered before the New York Meeting of the Society on February 28, 1948, by invitation of the Committee to Select Hour Speakers for Eastern Sectional Meetings.
Page 559
In what follows we shall formulate the problems of non-uniform motion in somewhat greater generality than has heretofore been done. After this we shall discuss insome detail various aspects of the theory of nearly plane lifting surfaces in incompressible flow, and in particular the step from two-dimensional to three-dimensionaltheory.
2The General Problem
The actual problem of linearized lifting surface theory will be considered as an approximation to the following nonlinear problem. An impenetrable, deformable surfaceof given extent moves in a prescribed manner through a compressible perfect fluid. From part of the edge of the impenetrable surface emanates a surface of velocitydiscontinuity in such a manner that the fluid velocity remains finite along this part of the edge,1 henceforth called the trailing edge. Along the remainder of the edge,called leading edge, the fluid velocity will on account of the sharpness of the edge in general assume infinite values for an incompressible fluid. For a compressiblefluid the assumption of a sharp leading edge will in general make impossible a continuous solution in the region exterior to the surfaces of discontinuity. We need notfor the present purposes consider this difficulty as it disappears in the linearized form of the problem.
Let X, Y, Z be the axes of a fixed frame of reference and let x, y, z be the axes of a frame of reference moving with the impenetrable surface (Figure 1). Let U(t) bethe velocity of the origin of the moving system with reference to the fixed system and let (t) be the angular velocity of the moving system with reference to the fixedsystem. Let ur be the velocity of a fluid particle relative to the moving system and let u be the velocity of the same particle relative to the fixed system. The velocityvector u may be written in the form u0 + ui, where u0 exists without being caused by the presence of the impenetrable surface and where ui is induced by the motionof the impenetrable surface. Correspondingly we have a pressure p = p0 + pi and a density = 0 + i.
We then have the following kinematical relations involving velocity u and acceleration a
The differential equations of the problem are of the following form
Equations (3) to (5) are to be solved in the space exterior to two surfaces FL = 0 and FT = 0, where FL represents the given surface of pressure and velocitydiscontinuity and where FT represents a surface of velocity discontinuity, determination of which is part of the problem. On FL we have the condition of no relativenormal flow. On FT we have the two conditions that the normal velocities of points of the surface are
1This condition of finite trailing edge velocity, first introduced in two-dimensional airfoil theory by Kutta and Joukovsky in order to obtain a definite lifting action, was subsequently found torepresent rather well the effect of viscosity of actual fluids in a perfect-fluid theory of airfoils. The fact that in a three-dimensional theory enforcement of this condition necessitated introduction ofa trailing surface of discontinuity was first observed by Prandtl.
Page 560
Fig. 1.
given by the corresponding velocities of the surrounding fluid and that the pressure is continuous across this surface. Thus
where the subscripts + and - distinguish the two sides of FT.
The surfaces FL and FT are connected along a line CT which is part of the edge of FL = 0 and which is sufficiently described for the present purposes by thedesignation ''trailing" edge. Along CT we have the additional condition that u remains finite.
In addition to the boundary condition (6) and (7) and the trailing edge conditions there are needed conditions at infinity. The form of these conditions will evidentlydepend on the form of the motion and on whether the fluid is compressible or incompressible.
For incompressible flow these conditions are roughly vanishing of all disturbances at an infinite distance from lifting surface and trailing surface. For compressible flowno such general statement can be made. In some cases all that is required is to superimpose on the conditions for incompressible flow a condition stating that radiationenergy is not created or reflected at infinity. In other cases the disturbances caused by the motion of the lifting surface cannot be required to vanish at infinity. Generaldetermination of these conditions is outside the scope of this report.
When the velocity distribution u0 which exists without the presence of the surface FL is such that × u0 = 0, equations (3) to (5) may be reduced to one scalarequation by means of the introduction of a velocity potential which, for incompressible flow, satisfies the Laplace equation but which for compressible flow is of amore general type.
Page 561
We postpone here introduction of the velocity potential and first linearize the problem.
3The Linearized Form of the General Problem
Basic assumption for a linearized theory is that the lifting surface moves through the fluid nearly without disturbing the fluid such that powers and products of thequantities ui, pi and i and their derivatives may be neglected.
The equation of motion (3) becomes then
The equation of continuity (4) becomes
and the equation of change of state becomes
We shall in some of what follows write as abbreviations
The quantity a0 is the velocity of sound at a point of the undisturbed medium.
Turning now to the boundary conditions (6) and (7) we begin by establishing the condition for the motion of a surface FLP = 0 without any disturbance of thesurrounding fluid. Equation (6) indicates that this condition is as follows
We have chosen the subscript P to indicate this surface because we wish to refer to it henceforth as the projection of the lifting surface. As the actual lifting surfacemust deviate only slightly from this projection in order to move nearly without causing disturbances we may write
where L is small in the same sense as ui, i, and pi are small. Then, considering (12), and except for quantities small of higher order, the boundary condition at thelifting surface becomes
Note that in satisfying the boundary condition at the projection of the lifting surface rather than at the lifting surface itself we again depend on the perturbationproperties of the solution to be obtained.
The next step is the determination of the form of the trailing surface of discontinuity FT. As the equation of this surface is one of the unknowns of the problem wemust, in order to have a linear problem, omit the term ui in (7) and the shape of FT is then such that the equation
Page 562
is satisfied. In addition to this we have the condition that at the trailing edge CT the surface FT is connected to the surface FLP. The meaning of (15) is that within thelinearized theory the shape of the trailing surface of discontinuity is independent of the velocity distribution ui induced by the motion of the lifting surface.
Having the equation of FT we then obtain from (17) the two conditions of continuous normal velocity and pressure across the surface in the form
To the formulation of the problem as contained in equations (8) to (16) we must add the condition of finite ui along CT and appropriate conditions at infinity.
We remark that previous formulations of this problem of non-steady motion in their most general form are based on the assumptions u0 = 0, = 0, U = Ui. Underthis assumption, and under the additional assumptions that dU/dt = 0 and FLP/ t = 0, Kuessner [44]2 has obtained an integral equation for the pressure distributionat the lifting surface.
4Velocity Potential Formulation of the Linearized Problem
In what follows we shall assume that the fluid is at rest except for the motion induced by the lifting surface, that is, we put
With these assumptions we have the existence of a velocity potential in terms of which
Combination of (18), (17), and (8) gives for the pressure pi the following expression
Combination of (19), (11), (10), and (9) gives the following differential equation for ,
The boundary condition (14) becomes
The transition conditions (16) become
and the trailing edge condition is that along CT we have finite.
The problem from here on is the solution of the mixed boundary value problem (20) to (22), with appropriate conditions at infinity. The object of such solutions isprimarily the determination of the pressures pi on both sides of the lifting surface.
2Numbers in brackets refer to the references cited at the end of the paper.
Page 563
Solutions obtained so far are all for nearly plane lifting surfaces such that FLP z = 0. Predominant among these are solutions for the two-dimensional problem, whichmay be characterized by the requirement that / y 0.
For two-dimensional incompressible flow essential contributions are due to H. Wagner [78], W. Birnbaum [3], H. Glauert [22], T. Theodorsen [73], H. G. Kuessner[42], P. Cicala [7], and G. Ellenberger [14].
For two-dimensional subsonic compressible flow which is less completely solved than the problem of incompressible flow, one must mention the work of C. Possio[53], R. Timman [75], D. Haskind [24], and that given in [60]. Various approximate methods for the solution of Possio's integral equation of the problem aredescribed in [39].
The corresponding problem in the supersonic range has been dealt with by Possio [52], S. von Borbely [6], H. A. John and G. Temple [72], I. E. Garrick and I.Rubinov [19], and I. A. Panichkin [51].
The perturbation theory of non-steady two-dimensional transonic flow has recently been considered by C. C. Lin, H. S. Tsien and the writer [47].
In the three-dimensional theory one has the solution of Schade and Krienes [41, 62] for the lifting surface of circular plan-form in incompressible flow and, also forincompressible flow, a number of developments for an approximate analysis of the three-dimensionality of the flow for surfaces whose span is appreciably greaterthan their chord. We shall in what follows describe a particular approach to this problem based on earlier publications on this subject [56, 57]. A discussion ofvarious other methods of analysis for this problem by Cicala [8, 9], W. R. Sears [69], R. T. Jones [32, 33], and Kuessner [44] can be found in [56].
Finally, we mention work by Garrick and Rubinov [20] and by E. A. Krasilschikova [40] on three-dimensional supersonic flow and a forthcoming publication onthree-dimensional subsonic flow [59]. In both problems it appears that further work is required before all difficulties inherent in the problem are overcome.
5Motion of Nearly Plane Lifting Surface in Incompressible Flow
Further discussion will be carried out for this subclass of the general problem. Our object is to indicate the particular nature of the boundary value problem in questionand to outline one of the possible methods of solution.
We assume that the projection of the lifting surface lies in the x, y-plane and that the motion of the lifting surface is in the direction of negative x. We further assumeincompressible flow. We have then
and, in accordance with (12) and (15),
We shall designate the region occupied by the projection of the lifting surface by RL and the region occupied by the trailing surface of discontinuity by RT (Figure 2).
From (20) follows that when a0 = the differential equation is, for steady as well as for non-steady motion,
Page 564
Fig. 2.
From (19) follows, for the pressure induced by the motion of the lifting surface,
If the instantaneous distance of a point of the lifting surface from the x, y-plane is designated by Z(x, y, t) we have FL = FL + L = z - Z(x, y, t) = 0. Consequentlythe boundary condition (21) becomes
The transition conditions (22) become
To equations (27) and (28) is to be added the condition of finite trailing edge velocities
and conditions at infinity which for incompressible flow may be taken in the form
The above problem is to be understood as a boundary value problem for the exterior of an infinitely thin semi-infinite tube surrounding the regions RL and RT, in thesense that (27) holds for z = +0 and for z = -0. It may be recalled that the main object is the determination of pi+ and pi- in RL, with pi- - pi+ being the lift intensityproduced by the motion of the impenetrable surface FL.
The form of the boundary conditions (27) to (30) is such that the problem for the exterior of the semi-infinite tube may be transformed by a symmetry considerationinto a mixed boundary-value problem for one of the half spaces z > 0 or z < 0. This is done by observing that equations (27) and (30) are compatible with theassumption that is an odd function of z. If we define a region RR as the x, y-plane minus the regions RL and RT and take into account that is continuous except
Page 565
across RL and RT we may replace the boundary conditions (27) and (28) by the following system of conditions at z = 0:
We then must determine in one of the half-spaces, say z > 0, with the conditions (29) and (31) at the boundary z = 0 and with equations (30) giving the conditionsat infinity.
Let us remark that explicit solutions of the problem thus formulated are possible in the two-dimensional case by the use of elliptic cylinder coordinates, while for thecircular plan form wing the use of spheroidal coordinates is appropriate [41, 62]. The problem would be of a standard nature if the conditions in RT were the same asin RR. The main difficulty of obtaining an explicit solution comes from the particular form of the boundary condition holding in RT inasmuch as all that can be said onthe basis of (31) about the values of in RT is
where is an arbitrary function of its two arguments. Compensating for this arbitrariness is, as will be seen, the finiteness condition (29).
6Integral Equation Formulation of the Problem
As one is interested primarily in the values of pi in RL it suggests itself to derive an integral equation for this quantity. This procedure, adopted by Birnbaum [3],Possio [53], Kuessner [42], and others at the suggestion of Prandtl, and known under the name acceleration potential method, has the advantage that it can bedeveloped without explicit introduction of the trailing surface of discontinuity. It does however have the disadvantage of leading to an integral equation with a distinctlymore complicated kernel than the corresponding integral equation for the values of / x in RL which we propose to discuss here. The main advantage of the latterformulation is that it permits immediate recognition of the explicit solvability of the problem of non-steady motion in terms of the solution of the corresponding problemof steady motion.
Setting as an abbreviation
we have the following representation for / x in terms of the boundary values ,
where
Page 566
Equation (34) may be converted into an expression for / z, by appropriate differentiation and integration, of the following form
Note that when / = 0 which corresponds to the assumption of two-dimensional flow the -integration in (36) can be carried out with the result that
which equation can of course be obtained directly in a simpler manner.
To separate two-dimensional from three-dimensional effects the following transformation is useful. We write (omitting for brevity the t in )
Appropriate integration by parts then gives the following relation
We must now in equation (38) let z tend to zero and substitute the boundary conditions (31). It is advantageous that (38) is in such a form that, as can be proved, thetwo limiting processes of integration and of letting z tend to zero can be interchanged, provided the integrals are defined where appropriate as Cauchy principalvalues. Thus from (31) and (38)
In (39) xL and xT indicate the coordinates of the leading and trailing edge, T is still to be determined as far as possible from the boundary conditions and K is of thefollowing form
We shall from now on assume for simplicity's sake that the region RL is the rectangle |x| b, |y| sb and that the velocity U which occurs in (31) is constant.
Page 567
We further introduce dimensionless variables,
and a dimensionless parameter k of the form
For the case of simple harmonic motion k is referred to as the "reduced frequency" of the motion. We may again for simplicity's sake omit in what follows the primesdesignating the dimensionless variables.
We then have from (32) that
Furthermore on account of the finiteness condition (29)
Then
and with
we have from (43)
We introduce (46) and (47) into (39) and obtain the following form of the integral equation of the problem
For uniform motion we have / = 0 and the second and fourth integrals in (48) are absent. The form of equation (48) indicates clearly the manner in which fornon-uniform motion the values of the solution (x, y, t) depend on the past history of the motion through the cumulative effect of successive changes of
The quantity is one-half of what is usually referred to as the circulation intensity at a station y = const. of the lifting surface.
Equation (48) is now to be solved for , in terms of wL and . Once this is done is found by integration in terms of wL and therewith is expressed in terms of wL
Page 568
only. The pressure pi at the lifting surface is then, according to (26), of the form
The advantage of (48) compared with the corresponding equation for the values of pi lies in the form of the kernel 1/(x - ) in the first term. This permits solution of(48) in a manner analogous to what is done for the problem of uniform motion.
7Solution of the Two-Dimensional Problem
In what follows we wish to describe briefly one of the possible methods of solution of this problem, namely that by L. Schwarz [67]. We shall subsequently indicatehow to utilize this method for an approximate solution of the three-dimensional problem.
Introducing the (unessential) restriction of simple harmonic motion we set
where the barred quantities are functions of the space coordinates at most. Equation (48) can then be written in the following form
Equation (52) is solved by means of a pair of inversion formulas of the form
which may be considered as a result of two-dimensional potential theory, as discussed most fully by H. Söhngen.3
Application of (53) to (52) leads to the following expression for
The main difficulty from here on is the calculation of the pressure which according to (50) is given by
Before listing the result of this lengthy and somewhat devious calculation we may indicate the nature of the equation for which occurs in (54). If we integrate bothsides of (54) as follows:
3Math. Zeit. vol. 45 (1939) pp. 245264.
Page 569
and take account of the formulas
then equation (56) can be written in one of the forms
It is at this stage that a combination of Bessel functions makes its appearance in the theory. The integral in the denominator of (59) is expressible in terms of Hankelfunctions, as follows:
In view of (60) can be written in the alternate form
Our purpose in outlining in some detail the steps leading from (52) to (61) has been to indicate the nature of some of the more simple transformations in thecalculation of the pressure distribution on oscillating airfoils. Considerable care is necessary to arrange the analysis in such a manner that advantage is taken of allpossible simplifications. In this way there is found the following expression for of equation (55)
In (62) the function is given by
and the function C, first introduced by Theodorsen [73], is of the form
Page 570
The results outlined in this section find their main application in the analysis of airplane flutter.4 For this purpose explicit expressions have been obtained by Cicala [7],Kuessner [42], Theodorsen [73], and others for lift and moment amplitudes and defined by
and for control-surface hinge moments defined by
for various appropriate forms of .
Plots of representative pressure distributions for various values of the reduced frequency k and for some of the more important forms of wL may be found in a recentpaper by Postel and Leppert [55].
We may conclude this section with some remarks concerning the solution of the problem for non-oscillatory motion.
It may readily be seen that the results for simple harmonic motion may be used for Laplace transform analysis by replacing wherever it occurs ik by -q, whereuponequations (61) and (62) become relations between Laplace transforms. For applications of the Laplace transform method in this field reference may be made to workby I. E. Garrick [17, 18] and W. R. Sears [70].
Another form of the results consists in integro-differential equations for , L and Ma, without any assumption concerning the form of the solution. The nature of theseresults may be seen from the simplest of them, the equation determining . Omitting all but the first two of the integrals on the right of (48) we may obtain thefollowing relation
After evaluating the inner integrals on the right of (66) we are left with an equation of the form
A formal solution of (67) by Fourier series or Laplace transform methods is again readily obtained. A special case of such a solution is given by (61). Practicalapplications, especially of the Laplace transform solution, are however not a simple matter, the reason for this being the occurrence of Hankel function combinationsin the denominator of the transforms to be evaluated, and the possibility of solving (67) directly by machine methods would be of considerable advantage.
4Briefly, the problem of flutter is the determination of those flight speeds at which self-sustained oscillations of a component of the airplane become possible due to the aerodynamic forcesproduced by an oscillatory motion of this component.
Page 571
8Remarks on the Problem of Tunnel Wall Interference in the Two-Dimensional Theory
A further problem of some interest concerning the two-dimensional theory of oscillating airfoils is the effect of tunnel walls on the pressure distribution . For a wingof chord 2b located at the center of a tunnel of height 2h one finds, using the method of images, that (52) is replaced by the following equation [58]
where the parameter is given by
Equation (69) may be transformed into an equation of the form (61) by means of the following substitutions5
The result is of the form
The solution of (71), which involves elliptic integrals, has not yet been given. An approximate solution, valid for sufficiently small values of , has been obtained in thefollowing way [58]. We set in the interval | | 1
The limits of integration in the second integral on the right of (68) preclude the direct use of (72). This difficulty is overcome by writing
and by splitting the second integral on the right of (73) into two integrals as follows, . The integral leads to the function
which has been tabulated. In the integral one may introduce the approximation (72). In this way the following approximate equation, which takes the place of (68),is obtained:
5This transformation has been used in a study of the corresponding problem of steady flow, where the second integral on the right of (68) is absent, by L. Lees and H. S. Tsien, Journal of theAeronautical Sciences vol. 12 (1945) pp. 173187, 202.
Page 572
It is of some interest to observe that the effect of the presence of tunnel walls may, for moderate values of k, be appreciably larger than the corresponding effect in thesteady-state solution for which k = 0. It was found in one representative example that when k = 0.25 the wall effect was twice what it was when k = 0. This increaseof the wall effect will occur when at the same time k is (1) large enough for the trailing surface of discontinuity to be of importance and (2) small enough for thecharacteristic wavelength in the trailing surface of discontinuity to be appreciably greater than the wing chord.
9Approximate Theory for Lifting Surfaces of Finite Span
We now propose to discuss a method of approximate solution of the integral equation (48) for three-dimensional flow from the following point of view. Our object isto reduce (48), which contains double integrals, to such a form that a solution of single integral equations only is required. We shall show that this is possible,provided the ''aspect ratio" s is sufficiently large, in such a way that what remains to be found is the solution of a problem of the kind encountered in the two-dimensional theory and the solution of a problem of the kind encountered in the determination of the spanwise lift distribution for a wing in uniform motion accordingto Prandtl's lifting-line theory. In this analysis we shall restrict attention to the case of simple harmonic motion in the sense of (51). Equation (48) then assumes thefollowing form
Our first step is to observe that the function K as given by equation (40) behaves, for |(x - )/(y - )| << 1, like 2-1(x - )/|y - |(y - ) and is therewith in this regionsmall compared with the remaining part of the kernel, 1/(y - ). If it is now assumed that 1 << s, then K is small compared with 1/(y - ) over most of the region ofintegration and may over this part of the region be disregarded. There remains the immediate neighborhood of the line = y where this order of magnitude relationdoes not hold. In order to disregard the contribution due to K in this region it is observed that K is an odd function of y - and that we expect to be a slowlyvarying function of . In view of this we expect the contribution due to both K and 1/(y - ) to be negligible. We thus assume that
and assume that the approximation is justified for "sufficiently" large values of s.
Obviously the above argument is not very satisfactory, from a mathematical point of view, but no more rigorous argument has yet been given to justify this from apractical standpoint rather satisfactory result. Possibly, equation (77) represents the first step in an asymptotic development in powers of 1/s but this has not yet beenproved.
Page 573
The argument leading to (77) cannot directly be applied to the last integral in (76) since in this integral we do not have the fact that |(x - )/(y - )| is small comparedto 1 over most of the region of integration. We proceed instead as follows. Write, with - x = ,
In the second integral on the right we can again neglect the contribution due to K, and thus we may write
In view of (79) we now define a function F by the relation
and combine (77), (79), and (80) in order to obtain from (76) the following approximate integral equation of the problem:
Equation (81) is the result which it was intended to obtain. The solution of (81) proceeds as follows. One first obtains , just as in the two-dimensional theoryexcept that now also depends on . One then obtains, by integration of , an equation for the determination of which is, as was previously stated, of thetype of the equation for the determination of the spanwise lift-distribution for a wing in uniform motion according to the lifting-line theory. Finally, just as in the two-dimensional theory, an expression is obtained for the pressure distribution pi.
It is readily shown, for instance by the procedure leading from (66) to (67), that the equation for is of the following form:
Page 574
where is the value of according to the two-dimensional theory as given by (61) and where the function is defined by
An equation corresponding to (82) had been obtained by Cicala [9] on the basis of entirely different considerations involving the effect of lifting lines and horseshoevortices. Equation (82) as it stands is different from Cicala's to the extent of a difference in the expression for . It turns out that Cicala's result can be obtained from(81) by omitting the factor e-ikx in front of the last integral in (81). In aerodynamical language this means that "the downwash induced by the spanwise variation ofcirculation" is assumed uniform across the chord. This is indeed correct, as has long been known, for the case of uniform motion for which k = 0. Evidently lifting lineconsiderations do not permit us to determine the chordwise variation of downwash referred to above and in this respect the integral equation method as outlined goesfurther.
Further work along the lines indicated leads to the result that the effect of three-dimensionality of the flow as determined by the foregoing approximate theory may beincorporated into equation (62) for the pressure distribution pi of the two-dimensional theory by merely changing the function C(k) into C(k) + . The term depends on the data of a given problem in the following manner:
The foregoing theory, up to equation (83) and including expressions for lift and moment as defined by (65a, b), is essentially that of [56]. In this referencethere are also included discussions of earlier work of a related nature by R. T. Jones [33], Kuessner [44], von Borbely [5], and Sears [69]. Extension of the theoryto surfaces of non-rectangular plan form and modification of pi by means of the function was first presented in [57]. Prior to this M. W. Hunter had established thatthe results of [56] permitted incorporation of the effect of finite span into the expressions for , , and by means of the function [26].
A considerable simplification of the present developments as compared with those in [56] and [57] is due to the fact that the basic integral equation of the problem ishere taken in the form (76), which is a direct consequence of (38), rather than in that form which corresponds to equation (36).
Methods of analysis and numerical examples of application of this theory may be found in [61]. It may further be mentioned that the analogue of equation (81), forsubsonic compressible flow, has recently been obtained [59].
A shortcoming of the approximate theory as discussed is the following. One would expect from an exact solution of the problem that the pressure pi tends to zero,and therewith also circulation , lift L, and moments M, as the tip sections are approached. This is indeed the case for lifting surfaces with zero tip chord, such as theelliptical surface. For lifting surfaces with finite tip chord, such as the rectangular surface, one finds however that only tip vanishes as it should whereas Ltip and Mtip,
Page 575
while smaller than according to the two-dimensional theory, can not in general be made to vanish. The reason for this difficulty is to be found in the form of theapproximate equation (81) in which the effect of spanwise variation of appears solely by way of the average of . A more refined approximate theory
undoubtedly requires inclusion of the effects of weighted averages of as well, such as and , in the integral equation of the problem. With such arefined theory one has reason to expect that tip conditions can be satisfied to a greater degree of approximation than by the present theory. In particular, it will bepossible to ensure that Ltip and Mtip vanish, as shown for the problem of uniform motion in Proc. Nat. Ac. Sc. vol. 35 (1949) pp. 208215.
References and Literature
1. M. A. Biot, Some simplified methods in airfoil theory, Journal of the Aeronautical Sciences vol. 9 (1942) pp. 185190.
2. M. A. Biot and C. T. Boehnlein, Aerodynamic theory of the oscillating wing of finite span, GALCIT Report No. 5 to the Army Air Forces (1942).
3. W. Birnbaum, Das ebene Problem des schlagenden Flügels, Zeitschrift für Angewandte Mathematik und Mechanik vol. 4 (1924) pp. 277292.
4. S. von Borbely, Mathematischer Beitrag zur Theorie der Flügelschwingungen, Zeitschrift für Angewandte Mathematik und Mechanik vol. 16 (1936) pp. 14.
5. , Über einen Grenzfall der instationären räumlichen Tragflügelströmung , Zeitschrift für Angewandte Mathematik und Mechanik vol. 18 (1938) pp. 319342.
6. , Über die Luftkräfte die auf einen harmonisch schwingenden zweidimensionalen Flügel bei Überschallgeschwindigkeit wirken, Zeitschrift fürAngewandte Mathematik und Mechanik vol. 22 (1942) pp. 190205.
7. P. Cicala, L'azione aerodinamiche sui profili di ala oscillante in presenza di corrente uniforme , Memorie della Reale Accademia delle Scienze di Torino (2)vol. 68 (19341935) pp. 7398.
8. , Su moto non stazionario di un'ala di allungamento finito, Rendiconti della R. Accademia Nazionale dei Lincei vol. 25 (1937) pp. 97102.
9. , La teoria e l'espirienza nel fenomeno della vibrazioni alari, L'Aerotechnica vol. 18 (1938) pp. 412433.
10. , Zuschriften an den Herausgeber, Zeitschrift für Angewandte Mathematik und Mechanik vol. 19 (1939) p. 384 and vol. 20 (1940) p. 184.
11. , Lo stato attuale delle ricerche sul moto instazionario di una superficie portante, L'Aerotechnica vol. 21 (1941) pp. 671685, 759773.
12. F. Dietze, Zur Berechnung der Auftriebskraft am schwingenden Ruder, Luftfahrtforschung vol. 14 (1937) pp. 361362.
13. , Die Luftkräfte der harmonisch schwingenden in sich verformbaren Platte (Ebenes Problem ), Luftfahrtforschung vol. 16 (1939) pp. 8496.
14. G. Ellenberger, Bestimmung der Luftkräfte auf einen ebenen Tragflügel mit Querruder, Zeitschrift für Angewandte Mathematik und Mechanik vol. 16(1936) pp. 199226.
15. , Luftkräfte bei beliebiger instationärer Bewegung eines Tragflügels mit Querruder und bei Vorhandensein von Böen , Zeitschrift für AngewandteMathematik und Mechanik vol. 18 (1938) pp. 173176.
16. I. E. Garrick, Propulsion of a flapping and oscillating airfoil, NACA T.R. No. 567 (1936).
17. , On some Fourier transforms in the theory of non-stationary flows, Proceedings of the Fifth International Congress on Applied Mechanics, 1938, pp.590593.
18. , On some reciprocal relations in the theory of non-stationary flows , NACA T.R. No. 629 (1938).
Page 576
19. I. E. Garrick and S. I. Rubinov, Flutter and oscillating airforce calculations for an airfoil in two-dimensional supersonic flow , NACA T.N. No. 1158(1946).
20. , Theoretical study of airforces on an oscillating or steady thin wing in a supersonic main stream , NACA T.N. No. 1383 (1947).
21. R. Glaser, Über die Berechnung der Koefficienten einer in der instationären Tragflügeltheorie auftretenden unendlichen Matrix , Zeitschrift fürAngewandte Mathematik und Mechanik vol. 23 (1943) pp. 279289.
22. H. Glauert, The force and moment on an oscillating airfoil, British A.R.C., R. and M. No. 1242 (1929).
23. J. M. Greenberg, Some considerations on an airfoil in an oscillating stream , NACA T.N. No. 1372 (1947).
24. M. D. Haskind, Oscillations of a wing in a subsonic gas flow (in Russian), Prikl. Math. i Mekh. vol. 11 (1947) pp. 120146.
25. H. Hoenl, Über das Schallfeld einer gleichformig-translatorisch * bewegten punktförmigen Schallquelle, Annalen der Physik vol. 43 (1943) pp. 437464.
26. M. W. Hunter, Jr., Calculation of the aerodynamic span effect in flutter analysis, M.S. Thesis, Massachusetts Institute of Technology (1944).
27. R. Isaacs, Airfoil theory for flows of variable velocity, Journal of the Aeronautical Sciences vol. 12 (1945) pp. 113117.
28. , Airfoil theory for rotary wing aircraft , Journal of the Aeronautical Sciences vol. 13 (1946) pp. 218220.
29. K. Jaeckel, Über die Kräfte auf beschleunigt bewegte veränderliche Tragflügelprofile , Ingenieur-Archiv vol. 9 (1938) pp. 371395.
30. , Über die Bestimmung der Zirkulationsverteilung fur* den zwei-dimensionalen Tragflügel bei beliebigen periodischen Bewegungen, Luftfahrtforschungvol. 16 (1939) pp. 135138.
31. , Zur Theorie der tragenden Linie im Instationären, Luftfahrtforschung vol. 19 (1942) pp. 5763.
32. R. T. Jones, The unsteady lift of a finite wing, NACA T.N. No. 682 (1939).
33. , The unsteady lift of a wing of finite aspect ratio, NACA T.R. No. 681 (1940).
34. W. P. Jones, Aerodynamic forces on wings in simple harmonic motion , British A.R.C., R. and M. No. 2026.
35. , Theoretical airload and derivative coefficients for rectangular wings , British A.R.C., R. and M. No. 2145.
36. , Wind tunnel interference effect on the values of experimentally determined derivative coefficients for oscillating airfoils , British A.R.C., R. and M. No.1912.
37. W. P. Jones and S. S. Skan, Calculations of derivatives for rectangular wings of finite span by Cicala's method, British A.R.C., R. and M. No. 1920.
38. T. von Kármán and W. R. Sears, Airfoil theory for non-uniform motion, Journal of the Aeronautical Sciences vol. 5 (1938) pp. 379390.
39. S. N. Karp, S. S. Shu and H. Weil, Aerodynamics of the oscillating airfoil in compressible flow , Air Material Command T.R. No. F-TR-1167-ND (1947).
40. E. A. Krasilschickowa, Disturbed motion of air caused by vibration of a wing moving at supersonic speed (in Russian), Prikl. Math. i Mekh. vol. 11 (1947)pp. 147164.
41. H. Krienes and T. Schade, Theorie der schwingenden kreisformigen* Tragfläche auf potentialtheoretischer Grundlage , Luftfahrtforschung vol. 19 (1942)pp. 282291.
42. H. G. Kuessner, Zusammenfassender Bericht über den instationären Auftrieb von Tragflügeln , Luftfahrtforschung vol. 13 (1936) pp. 410424.
43. , Das zwei-dimensionale Problem der beliebig bewegten Tragfläche unter Berücksichtigung von Partialbewegungen der Flüssigkeit , Luftfahrtforschnungvol. 17 (1940) pp. 355361.
44. , Allgemeine Tragflächentheorie, Luftfahrtforschung vol. 17 (1940) pp. 370378.
45. , Zuschriften an den Herausgeber, Zeitschrift für Angewandte Mathematik und Mechanik vol. 19 (1939) p. 384 and vol. 20 (1940) p. 184.
46. H. G. Kuessner and L. Schwarz, Der schwingende Flügel mit aerodynamisch ausgeglichenem Ruder, Luftfahrtforschung vol. 17 (1940) pp. 337354.
Page 577
47. C. C. Lin, E. Reissner, and H. S. Tsien, On two-dimensional non-steady motion of a slender body in a compressible fluid, Journal of Mathematics andPhysics vol. 27 (1948) pp. 220231.
48. J. W. Miles, The aerodynamic forces on an oscillating airfoil at supersonic speeds , Journal of the Aeronautical Sciences vol. 14 (1947) pp. 351357.
49. , Harmonic and transient motion of a swept wing in supersonic flow, Journal of the Aeronautical Sciences vol. 15 (1948) pp. 343346.
50. R. M. Morris, The two-dimensional theory of moving airfoils, IV, Proc. Roy. Soc. London Ser. A vol. 188 (1947) pp. 439463.
51. I. A. Panichkin, Forces acting on an oscillating profile in a supersonic gas flow (in Russian), Prikl. Math. i Mekh. vol. 11 (1947) pp. 165170.
52. C. Possio, L'azione aerodinamica sul profilo oscillante alle velocità ultrasonore , Acta Pontificia Accademia delle Scienze vol. 1 (1937) pp. 93106.
53. , L'azione aerodinamica sul profilo oscillante in un fluido compressible a velocità iposonora , L'Aerotechnica vol. 18 (1938) pp. 441458.
54. , L'azione aerodinamica su una superficie portante in moto oscillatoria, Atti dell'Accademia Nazionale Lincei, Rendiconti (VI) vol. 28 (1938) pp. 194200.
55. E. E. Postel and E. L. Leppert, Jr., Theoretical pressure distributions for a thin airfoil in incompressible flow , Journal of Aeronautical Sciences vol. 15(1948) pp. 486492.
56. E. Reissner, On the general theory of thin airfoils for non-uniform motion, NACA T.N. No. 946 (1944).
57. , Effect of finite span on the airload distributions for oscillating wings I, NACA T.N. No. 1194 (1947).
58. , Wind tunnel corrections for the two-dimensional theory of oscillating airfoils , Cornell Aeronautical Laboratory Report No. SB-218-5-3 (1947).
59. , Oscillating airfoils of finite span in subsonic compressible flow , NACA T.N. No. 1953 (1950) [also NACA T.R. No. 1002 (1951)].
60. E. Reissner and S. Sherman, Compressibility effects in flutter, Curtiss-Wright Research Lab. Report No. SB-240-S-1 (1944).
61. E. Reissner and J. E. Stevens, Effect of finite span on the airload distribution for oscillating wings II, NACA T.N. No. 1195 (1947).
62. T. Schade, Theorie der schwingenden kreisformigen * Tragfläche auf potentialtheoretischer Grundlage I, Luftfahrtforschung vol. 17 (1940) pp. 387400.
63. W. Schmeidler, Schwingenflug mit Wirbelablösung, Zeitschrift für Angewandte Mathematik und Mechanik vol. 17 (1937) p. 361.
64. , Vortrieb und Widerstand , Zeitschrift für Angewandte Mathematik und Mechanik vol. 19 (1939) pp. 6586.
65. , Zur Theorie des Schwingenfluges, Sitzungsberichte der Berliner Mathematischen Gesellschaft vols. 38, 39 pp. 6471.
66. L. Schwarz, Berechnung der Funktionen V1(s) und V2(s) fur* grössere Werte von s , Luftfahrtforschung vol. 17 (1940) pp. 362369.
67. , Berechnung der Druckverteilung einer harmonisch sich verformenden Tragfläche in ebener Strömung , Luftfahrtforschung vol. 17 (1940) pp. 379386.
68. , Untersuchung einiger mit den Zylinderfunktionen nullter Ordnung verwandter Funktionen, Luftfahrtforschung vol. 20 (1943) pp. 341372.
69. W. R. Sears, A contribution to airfoil theory for non-uniform motion, Proceedings of the Fifth International Congress on Applied Mechanics, 1938, pp.483487.
70. , Operational methods in the theory of airfoils in non-uniform motion, Journal of the Franklin Institute vol. 230 (1940) pp. 95111.
71. H. Söhngen, Bestimmung der Auftriebsverteilung fur beliebige instationäre Bewegungen (Ebenes Problem ), Luftfahrtforschung vol. 17 (1940) pp.401419.
72. G. Temple and H. A. Jahn, Flutter at supersonic speeds, British A.R.C., R. and M. No. 2140.
73. T. Theodorsen, General theory of aerodynamic instability and the mechanism of flutter, NACA T.R. No. 496 (1935).
Page 578
74. T. Theodorsen and I. E. Garrick, Non-stationary flow about a wing-aileron tab combination including aerodynamic balance , NACA T.R. No. 732(1942).
75. R. Timman, Beschouwingen over de luchtkrachten op trillende vliegtuigvleugels, Dissertation, Technische Hogeschool Delft, 1946, 154 pp.
76. O. (Taussky) Todd, On some boundary value problems in the theory of the non-uniform supersonic motion of an aerofoil , British A.R.C., R. and M. No.2141.
77. , A boundary value problem for a hyperbolic differential equation arising in the theory of the non-uniform supersonic motion of an aerofoil , CourantAnniversary Volume, 1948, pp. 421435.
78. H. Wagner, Über die Entstehung des dynamischen Auftriebs von Tragflügeln, Zeitschrift für Angewandte Mathematik und Mechanik vol. 5 (1925) pp.1735.
Page 579
Note on the Relation of Lifting-Line Theory to Lifting-Surface Theory[J. Aeron. Sc. 18, 212213, 1951]
The following is concerned with linearized lifting-surface theory for the pressure distribution on wings in subsonic motion and its relation to lifting-line theory for thespanwise variation of lift. It is shown that in certain cases lifting-line theory may be obtained by means of systematic developments with regard to aspect ratio fromlifting-surface theory. It is also indicated by means of a typical example that such developments are not possible in all cases where lifting-line theory is considered tofurnish an approximation to the results of lifting-surface theory.
Consider a lifting-surface of infinite span, of uniform chord 2b, moving with subsonic velocity U in a fluid of undisturbed density . The integral equation of thelinearized theory for this problem may be written in the form
where
The dimensionless coordinates x and y are related to the physical coordinates X and Y in the form
and the dimensionless downwash function w is expressed in terms of the local lifting-surface slope (X, Y) by
The function p gives the pressure jump across the lifting surface, M is the Mach Number corresponding to the velocity U, and principal values are to be taken of theintegrals in Eq. (1).
Assume a downwash function w of the form
The aspect ratio of a portion of the lifting surface between consecutive nodal lines of the downwash distribution follows from Eqs. (5) and (1) as
Page 580
The double integral equation [Eq. (1)] may be reduced to a single integral equation by setting
Combining Eqs. (7) and (1) requires evaluation of two infinite integrals with respect to . With a new variable of integration , defined by y - = s , these integralsmay be written as follows:
Setting as an abbreviation
the resultant one-dimensional integral equation becomes
The function F may be expressed in terms of tabulated functions. From
follows by appropriate differentiation and integration by parts, for z > 0,
In Eq. (13) K0 is a modified Bessel function of the second kind. From this there is obtained, by integration and by making use of some properties of the modifiedBessel functions, the representation
For sufficiently small values of z, this implies the approximation
which is based on the leading terms in the power series expansions for K0 and K1.
For large values of z one has the limiting relation
Table 1 contains numerical values of F and Fa.
Page 581
Table 1s 0 0.1 0.2 0.5 1.0 2.0 3.0 4.0 5.0F 0 0.196 0.32 0.58 0.84 1.11 1.23 1.31 1.35Fa 0 0.196 0.32 0.58 0.81 0.92
It may be noted that the approximation (15) is certainly adequate as long as z is less than 1, and this means that as long as |x - |/s is less than 1; or, since |x - | 2,
as long as A.R. , the integral equation [Eq. (11)] may be written in the form
Equation (17) is a special case of a class of integral equations which Eichler [1] has shown to be explicitly solvable.
The following observations may be made: Omission of the last of the three integrals on the right of Eq. (17) leaves an equation the solution of which corresponds toPrandtl's lifting-line theory.
A development of the solution of the complete equation [Eq. (17)] for sufficiently large values of s will be of the form
The term P0 is the exact solution of the two-dimensional theory. The term P1 is correctly taken account of by lifting-line theory. The term P2 is not taken account ofby lifting-line theory, since this theory does not introduce logarithmic terms in s.
In order to obtain a result corresponding to Weissinger's improvement of lifting-line theory [2, 3] we may proceed as follows: Write
and integrate Eq. (11) in the form
In Eq. (20)
Page 582
and in the term multiplied with F the function P is approximated by
There remains a double integral over F which is approximated as follows [3]:
Equation (20) now reads
In view of Eq. (15) we have, for sufficiently large values of s,
and thus, in this improvement of lifting-line theory, we do have a logarithmic term in s, as in the solution of the lifting-surface theory problem. However, it may beobserved that the numerical coefficient of the In s-term in Eq. (25) will not, in general, agree with the corresponding coefficient following from lifting-surface theory,since in lifting-surface theory as formulated in Eq. (17) the value of L depends on the function W(x) in a less simple manner than predicted by Eq. (24). In this regard,an improved approximation will be obtained by using as in reference 3, instead of one equation for L, two simultaneous equations for L and
The result (25) for sufficiently large values of s may be contrasted to the result for small values of s. In view of Eq. (16), we have, in this case,
and, consequently, the value of L in this case is one-half of what it is according to the original lifting-line theory. This result is in accordance with a recent observationby Lawrence [4].
If we denote the value of L according to lifting-line theory by Lp, we have, for the ratio of L of Eq. (24) to Lp,
Table 2 contains numerical values of the ratio L/Lp as function of s and, in view of Eq. (6), as function of A.R. and M.
While it is seen to be possible to obtain lifting-line theory for the infinite lifting surface with spanwise periodic downwash distribution by means of systematicdevelopments in terms of the aspect ratio parameter s, it may also be seen that a corresponding procedure is not possible for a finite lifting-surface with finite tipchord. We need only consider the semi-infinite lifting surface of constant chord, with
Page 583
Table 2s 0 0.2 0.25 0.33 0.5 1.0 2.0 5.0 10.0L/Lp 0.5 0.565 0.58 0.605 0.65 0.755 0.86 0.955 0.985 1.0
constant downwash function w(x, y) for y 0. In this case the aspect ratio is infinite. Lifting-line theory may be used to obtain an approximate result for the liftdistribution that is zero at the tip y = 0 and approaches a finite limiting value for increasing values of y. This distribution will, however, be different from the distribution
obtained by solving the double integral equation [Eq. (1)] with the integral replaced by the integral .
References
1. Eichler, M., Mathematische Zeitscrift, 48, 503526, 19421943.
2. Weissinger, J., Mathematische Nachrichten, 2, 45106, 1949.
3. Reissner, E., Proc. Nat. Acad. Sci. (U.S.A.), 35, 208215, 1949.
4. Lawrence, H. R., Cornell Aeronautical Laboratory Report No. AF-673-A-1, August, 1950.
Page 584
A Problem of the Theory of Oscillating Airfoils[Proc. 1st Nat. Congr. Appl. Mech. pp. 923925, 1952]
1Introduction
We consider an airfoil of infinite span and uniform chord in the path of an incompressible fluid flowing with velocity U. The airfoil deforms periodically, both in timeand in the direction of the span. To be determined are the spanwise variation of lift and moment as a function of the frequency of oscillation and of the wavelength ofthe deformations in the direction of the span of the airfoil.
W. R. Sears has presented an approximate solution of this problem during the Vth International Congress of Applied Mechanics [1]. We here present anotherapproximate solution of the problem, based on general results for the oscillations of airfoils of finite span which were obtained previously [2]. In these earlier generalresults the problem of the oscillating airfoil of finite span is reduced to an integral equation for the spanwise variation of circulation. The solution of this integralequation furnishes a three-dimensional correction to the basic function C(k) of the two-dimensional theory of oscillating airfoils.
In view of the difficulties of solution of most problems in the theory of airfoils of finite span it is worthy of note that the solution of the present problem can be givenexplicitly on the basis of the general theory of reference 2, in terms of relatively simple elementary functions.
2Formulation of the Problem
Let the chord of the airfoil be denoted by 2b and the frequency of oscillation by . Let y be a spanwise coordinate and let (y) be the amplitude of the circulationintensity at the station y of the airfoil.
According to reference 2 the integral equation for is of the form
where (2) is the value of according to the two-dimensional theory and where the functions and F are defined as follows
The function F, which was first introduced by Cicala into the theory of oscillating airfoils of finite span, and the function are both tabulated [2]. The
Page 585
parameter k is defined by the relation
Three-dimensional corrections of this theory for lift and moment distribution of the two-dimensional theory consist in a modification of the function C(k) of the two-dimensional theory to C(k) + , where is given by the expression
A table of values of the function in the first bracket may be found in ref. 3.
3Solution of the Integral Equation for Spanwise Periodic Circulation Distribution
If we assume that the amplitude of deformation of the airfoil is of the form (x)(cos y/2bs) then the circulation distribution of the two-dimensional theory is of theform
The distance between two successive nodal lines of (2) is equal to 2bs and since 2b equals the chord of the airfoil the value of s is the aspect ratio of a portion of theairfoil bounded by the successive nodal lines.
It is readily seen that when (2) is given by (6) a solution of the integral equation (1) is of the form
Introduction of (6) and (7) into (1) after some slight transformations gives the following equation for the amplitude constant 0,
In an appendix to this note it is shown that the infinite integral in (8) can be expressed in terms of elementary functions and we obtain the formula
With the help of equation (9) we have calculated / (2), and C + for various values of aspect ratio s and reduced frequency k and the results are listed in Table1. It may be noted that the behavior of the function C + is very similar to the behavior of this function for the elliptical plan form wing, as calculated previously [3,Figure 22]. As expected, the effect of finite aspect ratio becomes progressively less important as the value of the frequency parameter k increases. The data may,
Page 586
Table 1C(k) +
k s = 3 s = 6 s = 3 s = 6 s = 3 s = 6 C(k)0 .549 .709 -.451 -.291 .549 .709 1
.04 .584+.072i .754+.073i -.379+.106i -.221+.092i .548-.010i .706-.025i .926-.116i
.06 .601+.097i .776+.091i -.344+.131i -.189+.107i .548-.012i .703-.036i .892-.143i
.10 .637+.126i .816+.111i -.286+.149i -.139+.115i .546-.024i .693-.057i .832-.172i
.20 .716+.166i .890+.118i -.186+.150i -.066+.096i .542-.039i .661-.092i .728-.189i
.30 .781+.175i .931+.103i -.136+.119i -.041+.068i .530-.060i .624-.111i .665-.179i
.40 .829+.168i .954+.086i -.105+.093i -0.29+.049i .520-.072i .596-.116i .625-.165i
.50 .865+.154i .967+.0.71i -.086+.070i -.024+.035i .512-.081i .534-.116i .598-.151i
.75 .917+.116i .982+.046i -.061+.030i -.018+.015i .498-.091i .541-.106i .559-.121i1.00 .943+.084i .987+.031i -.045+.006i -.014+.005i .494-.095i .525-.095i .539-.101i1.50 .966+.047i .992+.016i -.019-.015i -.007-.004i .502-.089i .514-.077i .521-.074i2.00 .979+.027i .995+.009i -.002-.015i -.001-.005i .511-.073i .512-.062i .513-.058i
however, also be interpreted in still another way, namely as follows. For wings with finite aspect ratio the circulatory components of lift and moment are much lessdependent on the value of the reduced frequency k than is predicted by the two-dimensional theory of oscillating airfoils.
4Appendix. Evaluation of an Integral
In order to go from equation (8) to (9) we must evaluate the integral
The value of the imaginary part of this double integral follows directly if use is made of the basic property of the sine transform. The complete integral may beevaluated as follows. We introduce in the inner integral a new variable of integration , defined by = z/x . This changes (10) into
We now consider as a complex variable and integrate (assuming x positive) instead of along the real -axis along a straight line enclosing a small negative angle with the real -axis. This changes the value of the integral by an amount proportional to the value of the small angle Since now Im{ix } < 0 we may interchangethe order of integration in (11) and this gives us
Page 587
We now let tend to zero and go around the singularity at = 1/x along the semicircle = x-1 + ei , where - 0 and where tends to zero. This gives us
the symbol indicating that the Cauchy principal value of the integral is to be taken. The remaining integral in (13) can be evaluated by elementary means and wehave then for positive values of x,
Introduction of (14) into equation (8) leads to the result (9) of this note.
References
1. W. R. Sears, ''A Contribution to Airfoil Theory for Non-uniform Motion", Proceedings of the Fifth International Congress of Applied Mechanics , 1938, pp.483487.
2. E. Reissner, "Effect of Finite Span on the Airload Distributions for Oscillating Wings I," NACA T.N. No. 1194 (1947).
3. E. Reissner and J. E. Stevens, "Effect of Finite Span on the Airload Distributions for Oscillating Wings II," NACgvfgtgA T.N. No. 1195 (1947).
Page 589
BIBLIOGRAPHY
The Scientific Papers of Eric Reissner
*1. Über die Berechnung von PlattenbalkenDer Stahlbau 7, pp. 286288, 1934
*2. Über die Biegung der Kreisplatte mit exzentrischer EinzellastMathematische Annalen 111, pp. 777780, 1935
3. Beitrag zur Theorie der PlattenbalkenKong. Norske Vidensk. Selsk. Forh. 8, pp. 1316, 1935
4. Beitrag zum Problem der Spannungsverteilung in GurtplattenZ.f. ang. Math. und Mech. 15, pp. 359364, 1935
*5. Stationäre, axialsymmetrische durch eine schüttelnde Masse erregte Schwingungen eines homogenen elastischen HalbraumesIngenieur-Archiv 7, pp. 381396, 1936
6. Allgemeine Integration der Plattengleichung bei linear veränderlicher SteifigkeitIngenieur-Archiv 7, pp. 8082, 1936
7. Über die Berücksichtigung der Gurtsteifigkeit bei der Berechnung der "mittragenden Breite"Schweizerische Bauzeitung 108, pp. 191194, 1936
8. Bemerkung zur Theorie der Biegung kreisförmiger PlattenZ.f. ang. Math. und Mech. 17, pp. 5758, 1937
9. Freie und erzwungene Torsionschwingungen des elastischen HalbraumesIngenieur-Archiv 8, pp. 230245, 1937
10. On the Theory of Beams Resting on a Yielding FoundationProc. National Acad. Sci. 23, pp. 328333, 1937
11. A Contribution to the Theory of TurbulenceJ. Aeronautical Sciences 4, pp. 384385, 1937 (with H. Peters)
12. Remark on the Theory of Bending of Plates of Variable ThicknessJ. Math. and Phys. 16, pp. 4345, 1937
13. Remark on the Theory of Bending of Plates of Variable Thickness IIKong. Norske Vidensk. Selsk. Forh. 10, pp. 9799, 1937
14. On the Problem of Stress Distribution in Wide-Flanged Box BeamsJ. Aeronautical Sciences 5, pp. 295299, 1938
*15. On Tension Field TheoryProc. Fifth Int. Congr. Appl. Mech. , pp. 889892, 1938
*16. Note on the Statistical Theory of TurbulenceProc. Fifth Int. Congr. Appl. Mech. , pp. 359361, 1938
17. Remark on the Theory of Bending of Plates of Variable Thickness IIIJ. Indian Math. Soc., pp. 199201, 1939
18. A Problem of Buckling of Elastic Plates of Variable ThicknessJ. Math. and Phys. 19, pp. 131139, 1940 (with R. Gran Olsson)
19. Note on the Problem of the Distribution of Stress in a Thin, Stiffened Elastic SheetProc. Nat'l Acad. Sci. 26, pp. 300305, 1940
*An asterisks indicate inclusion in this volume
Page 590
20. The influence of Taper on the Efficiency of Wide-Flanged Box BeamsJ. Aeronautical Sciences 7, pp. 353357, 1940
21. A New Derivation of the Equations for the Deformation of Elastic ShellsAmer. J. Math. 63, pp. 177184, 1941
22. A Contribution to the Theory of Elasticity of Non-Isotropic Materials (with applications to problems of bending and torsion)Phil. Mag. Ser. 7, 30, pp. 418427, 1941
23. Shear Lag in Corrugated Sheets Used for the Chord Member of a Box BeamNACA Techn. Note No. 791, 1941 (with J. S. Newell)
*24. Least Work Solutions of Shear Lag ProblemsJ. Aeronautical Sciences 8, pp. 284291, 1941
25. On a Class of Singular Integral EquationsJ. Math. and Phys. 20, pp. 219223, 1941
26. A Method for Solving Shear Lag ProblemsAviation 40, pp. 4849, 140, 1941
27. Distribution of Stresses in Built-in Beams of Narrow Rectangular Cross SectionJ. Appl. Mech. 9, pp. A108A116, 1942 (with F. B. Hildebrand)
28. Note on the Expressions for the Strains in a Bent, Thin ShellAmer. J. Math. 64, pp. 768772, 1942
29. Note on Some Secondary Stresses in Thin-Walled Box BeamsJ. Aeronautical Sciences 9, pp. 538542, 1942
*30. On the Calculation of Three-Dimensional Corrections for the Two-Dimensional Theory of Plane StressProc. 15th Semi-Annual Eastern Photo-elasticity Conference pp. 2331, 1942
31. Least-Work Analysis of the Problem of Shear Lag in Box BeamsNACA Techn. Note No. 893, 1943 (with F. B. Hildebrand)
32. The Influence of the Aerodynamic Span Effect on the Magnitude of the Torsional-Divergence Velocity and on the Shape of the Corresponding Deflection ModeNACA Techn. Note No. 926, 1944 (with F. B. Hildebrand)
33. The Stresses in Cemented JointsJ. Appl. Mech. 11, pp. A17A27, 1944 (with M. Goland)
34. On the General Theory of Thin Airfoils for Non-Uniform MotionNACA Techn. Note No. 946, 1944
*35. Forced Torsional Oscillations of an Elastic Half-Space IJ. Appl. Phys. 15, pp. 659661, 1944 (with H. F. Sagoci)
*36. On the Theory of Bending of Elastic PlatesJ. Math. and Phys. 23, pp. 184191, 1944
37. Note on the Theorem of the Symmetry of the Stress TensorJ. Math. and Phys. 23, pp. 192194, 1944
*38. The Effect of Transverse Shear Deformation on the Bending of Elastic PlatesJ. Appl. Mech. 12, pp. A69A77, 1945; 13, p. A252, 1946
39. Buckling of Plates with Intermediate Rigid SupportsJ. Aeronautical Sciences 12, pp. 375377, 1945
40. Solution of a Class of Singular Integral EquationsBull. Amer. Math. Soc. 51, pp. 920922, 1945
41. Stresses and Small Displacements in Shallow Spherical Shells IJ. Math. and Phys. 25, pp. 8085, 1946
42. Glue Line Stresses in Laminated WoodTrans. Amer. Soc. Mech. Engrs. 68, pp. 329335, 1946 (with A. Dietz and H. Grinsfelder)
*43. Analysis of Shear Lag in Box Beams by the Principle of Minimum Potential EnergyQuart. of Appl. Math. 4, pp. 268278, 1946
Page 591
*44. Note on the Shear Stresses in a Bent Cantilever Beam of Rectangular Cross SectionJ. Math. and Phys. 25, pp. 241243, 1946 (with G. B. Thomas)
45. On Vibrations of Shallow Spherical ShellsJ. Appl. Phys. 17, pp. 10381042, 1946
46. Stresses and Small Displacements in Shallow Spherical Shells IIJ. Math. and Phys. 25, pp. 279300, 1946
47. Effect of Finite Span on the Airload Distributions for Oscillating Wings IAerodynamic Theory of Oscillating Wings of Finite SpanNACA Techn. Note No. 1194 , 1947
48. On Bending of Elastic PlatesQuart. of Appl. Math. 5, pp. 5568, 1947
49. Effect of Finite Span on the Airload Distributions for Oscillating Wings IIMethods of Calculations and Examples of ApplicationNACA Techn. Note No. 1195 , 1947 (with J. E. Stevens)
50. Note on the Membrane Theory of Shells of RevolutionJ. Math. and Phys. 26, pp. 290293, 1947
51. Finite Deflections of Sandwich PlatesJ. Aeronautical Sciences 15, pp. 435440, 1948
*52. Note on the Method of Complementary EnergyJ. Math. and Phys. 27, pp. 159160, 1948
53. Contributions to the Problem of Structural Analysis of Sandwich-type Plates and ShellsTheory and Practice of Sandwich Construction in Aircraft , pp. 2148, Institute of Aeronautical Sciences, 1948
54. On Two-Dimensional Non-Steady Motion of a Slender Body in a Compressible FluidJ. Math. and Phys. 27, pp. 220231, 1948 (with C. C. Lin and H. S. Tsien)
*55. On Compressibility Corrections for Subsonic Flow over Bodies of RevolutionNACA Techn. Note No. 1815, 1949
*56. On the Theory of Thin Elastic ShellsContributions to Appl. Mech. (Reissner Anniversary Volume) pp. 231247, J. W. Edwards, Ann Arbor, Mich., 1949
57. Notes on the Foundation of the Theory of Small Displacements of Orthotropic ShellsNACA Techn. Note No. 1833, 1949 (with F. B. Hildebrand and G. B. Thomas)
58. Small Bending and Stretching of Sandwich-type ShellsNACA Techn. Note No. 1832, 1949 (Also NACA Report 975, 1950)
*59. Complementary Energy Procedure for Flutter CalculationsJ. Aeronautical Sciences 16, p. 317, 1943
60. On Bending of Curved Thin-Walled TubesProc. Nat'l Acad. of Sciences 35, pp. 204208, 1949
*61. Note on the Theory of Lifting SurfacesProc. Nat'l Acad. of Sciences 35, pp. 208215, 1949
*62. Boundary Value Problems in Aerodynamics of Lifting Surfaces in Non-Uniform MotionBull. Amer. Math. Soc. 55, pp. 825850, 1949
63. Note on the Problem of Twisting of a Circular Ring SectorQuart. Appl. Math. 7, pp. 342347, 1949
64. On Finite Deflections of Circular PlatesProc. Symposia Appl. Math. 1, pp. 213219, 1949
65. Oscillating Airfoils of Finite Span in Subsonic Compressible FlowNACA Techn. Note No. 1953, 1950 (Also NACA Report 1002, 1951)
*66. On a Variational Theorem in ElasticityJ. Math. and Phys. 29, pp. 9095, 1950
67. On the Theory of Beams on an Elastic FoundationBeiträge zur angewandten Mechanik (Federhofer-Girkmann Festschrift), pp. 87102, F. Deuticke, Wien, 1950
Page 592
68. On Axi-symmetrical Deformations of Thin Shells of RevolutionProc. Symposia Appl. Math. 3, pp. 2752, 1950
69. Deformations and Stresses in Bourdon TubesJ. Appl. Phys. 21, pp. 13401341, 1950 (with R. A. Clark)
*70. Note on the Relation of Lifting-Line Theory to Lifting-Surface TheoryJ. Aeronautical Sciences 18, pp. 212213, 1951
71. Extension of the Theory of Oscillating Airfoils of Finite Span in Subsonic Compressible FlowNACA Techn. Note No. 2274, 1951
72. On the Application of Mathieu Functions in the Theory of Subsonic Compressible Flow Past Oscillating AirfoilsNACA Techn. Note No. 2363, 1951
73. Torsion and Transverse Bending of Cantilever PlatesNACA Techn. Note No. 2369, 1951 (with Manuel Stein)
74. Bending of Curved TubesAdvances in Applied Mechanics, Vol. II, pp. 93122 Academic Press, New York, 1951 (with R. A. Clark)
75. Stresses and Deformations of Toroidal Shells of Elliptical Cross SectionJ. Appl. Mech. 19, pp. 3748, 1952 (with R. A. Clark and T. I. Gilroy)
76. Stress Strain Relations in the Theory of Thin Elastic ShellsJ. Math. and Phys. 31, pp. 109119, 1952
77. A Problem of Finite Bending of Circular Ring PlatesQuart. Appl. Math. 10, pp. 167173, 1952
*78. Reihenentwicklung eines Integrals aus der Theorie der elastischen SchwingungenMath. Nachrichten 8, pp. 149153, 1952
79. On Non-Uniform Torsion of Cylindrical RodsJ. Math. and Phys. 31, pp. 214221, 1952
80. A Problem of Finite Bending of Toroidal ShellsQuart. Appl. Math. 10, pp. 321334, 1952 (with R. A. Clark)
*81. Pure Bending and Twisting of Thin Skewed PlatesQuart. Appl. Math. 10, pp. 395397, 1952
82. Behaviour in Pure Bending of a Long Monocoque Beam of Circular Arc Cross SectionNACA Techn. Note 2875, 1953 (with R. W. Fralich and J. Mayers)
*83. A Problem of the Theory of Oscillating AirfoilsProc. 1st Nat'l Congr. Appl. Mech. , pp. 923925, 1953
*84. On a Variational Theorem for Finite Elastic DeformationsJ. Math. and Phys. 32, pp. 129135, 1953
85. On Finite Twisting and Bending of Circular Ring-Sector Plates and Shallow Helicoidal ShellsQuart. Appl. Math. 11, pp. 473483, 1953
86. On Finite Torison of Cylindrical ShellsProc. 1st Midwestern Conf. on Solid Mechanics, pp. 4951, 1953
87. On Axi-Symmetrical Vibrations of Circular Plates of Uniform Thickness, Including the Effects of Transverse Shear Deformation and Rotary InertiaJ. Acoustical Soc. Amer. 26, pp. 252253, 1954
88. Note on the Problem of Vibrations of Slightly Curved BarsJ. Appl. Mech. 21, pp. 195196, 1954
89. Torsion of a Circular Cylindrical Body by Means of Tractions Exerted Upon the Cylindrical BoundaryStudies in Mathematics and Mechanics (R. von Mises Memorial Volume), pp. 262273, Academic Press, 1954 (with H. Reissner)
90. Small Rotationally Symmetric Deformations of Shallow Helicoidal ShellsJ. Appl. Mech. 22, pp. 3134, 1955
Page 593
91. On Some Aspects of the Theory of Thin Elastic ShellsJ. Boston Soc. Civil Engrs. 42, pp. 100133, 1955
92. Stresses in Elastic Plates over Flexible SubgradesProc. Amer. Soc. Civil Engrs. 81, pp. 129, 1955 (Also Trans. Amer. Soc. Civil Engrs. 122, pp. 627653, 1957)
93. On Torsion with Variable TwistÖsterreichisches Ingenieur-Archiv 9, pp. 218224, 1955
94. On Transverse Vibrations of Thin Shallow Elastic ShellsQuart. Appl. Math. 13, pp. 169176, 1955
*95. On Axi-Symmetrical Vibrations of Shallow Spherical ShellsQuart. Appl. Math. 13, pp. 279290, 1955
96. On Finite Sum Equations for Boundary Value Problems of Partial Difference EquationsJ. Math. and Phys. 34, pp. 286297, 1955 (with H. Glantz)
97. On Inextensional Vibrations of Shallow Elastic ShellsJ. Math. and Phys. 34, 335346, 1955 (with M. W. Johnson)
98. Elementary Differential EquationsAddison-Wesley Publ. Co., Inc., Reading, Mass., xi + 260 pp., 1956 (with W. T. Martin)
99. On Axially Symmetric Bending of Nearly Cylindrical Shells of RevolutionJ. Appl. Mech. 23, pp. 5967, 1956 (with R. A. Clark)
100. Note on Torsion with Variable TwistJ. Appl. Mech. 23, pp. 315316, 1956
101. Stress Distribution and Design Data for Adhesive Lap Joints Between Circular TubesTrans. Amer. Soc. Mech. Engrs. 78, pp. 12131221, 1956 (with J. Lubkin)
*102. A Note on Membrane and Bending Stresses in Spherical ShellsJ. Soc. Indus. and Appl. Math. 4, pp. 230240, 1956
103. A Derivation of the Equations of Shell Theory for General Orthogonal CoordinatesJ. Math. and Phys. 35, pp. 351358, 1956 (with J. K. Knowles)
104. Bounds on Influence Coefficients for Circular Cylindrical ShellsJ. Math. and Phys. 36, pp. 119, 1957 (with M. B. Sledd)
105. Finite Twisting and Bending of Thin Rectangular Elastic PlatesJ. Appl. Mech. 24, pp. 391396, 1957
106. Pure Bending of Pretwisted Rectangular PlatesJ. Mech. and Phys. of Solids 5, pp. 261266, 1957 (with L. Maunder)
107. On Transverse Vibrations of Shallow Spherical ShellsQuart. Appl. Math. 15, pp. 367380, 1957 (with M. W. Johnson)
108. On Torsional Vibrations of a Beam with a Small Amount of PretwistJ. Japan Soc. Aeronautic Eng. 5, pp. 330335, 1957 (with K. Washizu)
*109. On Stressses and Deformations of Ellipsoidal Shells Subject to Internal PressureJ. Mech. and Phys. of Solids 6, pp. 6370, 1958 (with R. A. Clark)
110. Symmetric Bending of Shallow Shells of RevolutionJ. Math. and Mech. 7, pp. 121140, 1958
111. A Note on Deflections of Plates on a Visco-Elastic FoundationJ. Appl. Mech. 25, pp. 144145, 1958
112. On Variational Principles in ElasticityProc. Symp. Appl. Math. 8, pp. 16, 1958
113. The Effect of an Internal Compressible Fluid Column on the Breathing Vibrations of a Thin Pressurized Cylindrical ShellJ. Aeronautical Sci. 25, pp. 288294, 1958 (with J. G. Berry)
114. Contributions to the Theory of Thin Elastic ShellsProc. 9th Int'l Congr. Appl. Mech. (1956), pp. 290296, 1958
Page 594
115. Rotationally Symmetric Problems in the Theory of Thin Elastic ShellsProc. 3rd Nat'l Congr. Appl. Mech. , pp. 5169, 1958
116. Note on Stress Strain Relations for Thin Elastic ShellsJ. Math. and Phys. 37, pp. 269282, 1958 (with J. K. Knowles)
*117. On the Foundations of the Theory of Thin Elastic ShellsJ. Math. and Phys. 37, pp. 375392, 1958 (with M. W. Johnson)
118. On Influence Coefficients and Nonlinearity for Thin Elastic ShellsJ. Appl. Mech. 26, pp. 6972, 1959
119. Upper and Lower Bounds for the Stiffness of Transversely Bent Circular PlatesJ. Appl. Mech. 26, pp. 142143, 1959
120. On the Determination of Stresses and Displacements for Unsymmetrical Deformations of Shallow Spherical ShellsJ. Math. and Phys. 38, pp. 1635, 1959
*121. The Edge Effect in Symmetric Bending of Shallow Shells of RevolutionComm. Pure Appl. Math. 12, pp. 385398, 1959
122. On Torsion of Thin Cylindrical ShellsJ. Mech. and Phys. of Solids 7, pp. 157162, 1959
123. On Finite Bending of Pressurized TubesJ. Appl. Mech. 26, pp. 386392, 1959
124. On the Solution of a Class of Problems in Membrane Theory of Thin ShellsJ. Mech. and Phys. of Solids 7, pp. 242246, 1959
125. Torsion and Extension of Helicoidal ShellsQuart. Appl. Math. 17, pp. 409422, 1959 (with J. K. Knowles)
126. On Stress Strain Relations and Strain Energy Expressions in the Theory of Thin Elastic ShellsJ. Appl. Mech. 27, pp. 104106, 1960 (with J. K. Knowles)
127. On Some Problems in Shell TheoryProc. First Symp. Naval Structural Mech. (1958), pp. 74114, 1960
128. Parametric Expansions for a Class of Boundary Value Problems of Partial Differential EquationsJ. Soc. Indus. Appl. Math. 8, pp. 389396, 1960, (with M. W. Johnson)
129. On Twisting and Stretching of Helicoidal ShellsProc. IUTAM Symp. on Thin Shell Theory (1959), pp. 434466, 1960
130. Three-Dimensional Theory of Elastic Plates with Transverse InextensibilityJ. Math. and Phys. 39, pp. 161181, 1960 (with J. L. Boal)
131. On Some Variational Theorems in ElasticityProblems of Continuum Mechanics (Muskhelishvili Anniversary Volume), pp. 370381, 1961
*132. On Finite Pure Bending of Cylindrical TubesÖsterreichisches Ingenieur Arch. 15, pp. 165172, 1961
133. Bending and Stretching of Certain Types of Heterogeneous Aeolotropic Elastic PlatesJ. Appl. Mech. 28, pp. 402408, 1961 (with Y. Stavsky)
134. Elementary Differential Equations, 2nd EditionAddison-Wesley Publ. Co., Inc. XIII + 331 pp., 1961 (with W. T. Martin)
135. Note on Finite Inextensional Deformation of Shallow Elastic ShellsJ. Math. and Phys. 40, pp. 253259, 1961
136. Variational Considerations for Elastic Beams and ShellsProc. Amer. Soc. Civil Engrs. 8 (EM), pp. 2357, 1962
137. Note on Axially Symmetric Stress Distributions in Helicoidal ShellsMiszellaneen der Angewandten Mechanik (Tollmien Anniversary Volume), pp. 257266, Akademie Verlag, 1962
*138. Finite Pure Bending of Circular Cylindrical TubesQuart. Appl. Math. 20, pp. 305319, 1962 (with H. J. Weinitschke)
Page 595
139. On Pure Bending of Pressurized Toroidal MembranesJ. Math. and Phys. 42, pp. 3846, 1963
140. A Problem of Shearing and Transverse Bending of Shallow Hyperbolic Paraboloidal ShellsJ. Appl. Mech. 30, pp. 295296, 1963 (with D. P. O'Mathuna)
*141. On the Equations for Finite Symmetrical Deflections of Thin Shells of RevolutionProgress in Applied Mechanics (Prager Anniversary Volume) pp. 171178, Macmillan Publishing Co., 1963
142. On Stresses and Deformations in Toroidal Shells of Circular Cross Section which are Acted upon by Uniform Normal PressureQuart. Appl. Math. 21, pp. 177188, 1963
143. On the Derivation of Boundary Conditions for Plate TheoryProc. Royal Society, A, 276, pp. 178186, 1963
144. On the Derivation of the Theory of Thin Elastic ShellsJ. Math. and Phys. 42, pp. 263277, 1963
145. Note on the Problem of St. Venant FlexureJ. Appl. Math. and Phys. (ZAMP) 15, pp. 198200, 1964
146. On the Form of Variationally Derived Shell EquationsJ. Appl. Mech. 31, pp. 233238, 1964
147. On Asymptotic Expansions for Circular Cylindrical ShellsJ. Appl. Mech. 31, pp. 245252, 1964
148. On Asymptotic Solutions for Nonsymmetric Deformations of Shallow Shells of RevolutionInt. J. Engrg. Sci. 2, pp. 2743, 1964
*149. A Note on Variational Principles in ElasticityInt. J. Solids Structures 1, pp. 9395, 1965
*150. Rotating Shallow Elastic Shells of RevolutionJ. Soc. Indus. Appl. Math. 13, pp. 333352, 1965 (with F. Y. M. Wan)
151. A Note on Stress Functions and Compatibility Equations in Shell TheoryTopics in Applied Mechanics (Schwerin Memorial Volume) pp. 2332, Elsevier Co., 1965
152. Application of a Variational Theorem for Boundary Values in Shell TheoryJ. Strain Analysis 1, pp. 8385, 1965 (with T. J. Lardner)
153. On the Foundations of the Theory of Elastic ShellsProc. 11th Int. Congr. Applied Mechanics (1964), pp. 2030, 1966
154. Asymptotic Solutions of Boundary Value Problems for Elastic Semi-Infinite Circular Cylindrical ShellsJ. Math. and Phys. 44, pp. 122, 1966 (with J. G. Simmonds)
*155. A Note on Stress Strain Relations of the Linear Theory of ShellsJ. Appl. Math. and Phys. (ZAMP) 17, pp. 676681, 1966 (with F. Y. M. Wan)
156. Symmetrical Deformations of Circular Cylindrical Shells of Rapidly Varying ThicknessDonnell 70th Anniversary Volume , pp. 4773, University of Houston, 1967 (with T. J. Lardner)
*157. Small Strain Large Deformation Shell TheoryDevelopment in Mechanics 3 (Proc. 9th Midwestern Mechanics Conference, Madison, Wisconsin, 1965), pp. 5558, 1967
158. On Stress Strain Relations and Strain Displacement Relations of the Linear Theory of ShellsThe Folke-Odqvist Volume , pp. 487500, 1967 (with F. Y. M. Wan)
159. On the Nonlinear Theory of Thin PlatesProc. 3rd Southeastern Conference on Theor. and Appl. Mech. (Columbia, South Carolina, 1966), pp. 165175, 1967
Page 596
160. A Note on the Formulation of the Problem of the Plate on an Elastic FoundationActa Mechanica 4, pp. 8891, 1967
161. On Axial Extension and Torsion of Helicoidal ShellsJ. Math. and Phys. 47, pp. 131, 1968 (with F. Y. M. Wan)
*162. Finite Inextensional Pure Bending and Twisting of Thin Shells of RevolutionQuart. J. Mech. and Appl. Math. 21, pp. 293306, 1968
*163. A Note on Günther's Analysis of Couple StressMechanics of Generalized Continua, IUTAM Symposium, Freudenstadt-Stuttgart, 1967, pp. 8386, 1968 (with F. Y. M. Wan)
164. On St. Venant Flexure Including Moment StressesPrikl. Mat. i Mekh. 32, pp. 923929, 1968
165. On the Foundations of Generalized Linear Shell TheoryTheory of Thin Shells, Proc. IUTAM Symposium , Copenhagen, 1967, pp. 1530, 1969
166. Rotationally Symmetric Stress and Strain in Shells of RevolutionStudies in Appl. Math. 48, pp. 117, 1969 (with F. Y. M. Wan)
167. On Generalized Two-Dimensional Plate Theory-IInt. J. Solids Structures 5, pp. 525532, 1969
168. On the Equations of Linear Shallow Shell TheoryStudies in Appl. Math. 48, pp. 133145, 1969 (with F. Y. M. Wan)
169. On the Equations of Non-Linear Shallow Shell TheoryStudies in Appl. Math. 48, pp. 171175, 1969
170. On Finite Symmetrical Deflections of Thin Shells of RevolutionJ. Appl. Mech. 36, pp. 267270, 1969
171. On Generalized Two-Dimensional Plate Theory-IIInt. J. Solids Structures 5, pp. 629637, 1969
172. On Axially Uniform Stress and Strain in Axially Homogeneous Cylindrical ShellsInt. J. Solids Structures 6, pp. 133138, 1970
*173. On Postbuckling Behavior and Imperfection Sensitivity of Thin Elastic Plates on a Non-Linear Elastic FoundationStudies in Appl. Math. 49, pp. 4557, 1970
174. A Note on Postbuckling Behavior of Pressurized Shallow Spherical ShellsJ. Appl. Mech. 37, pp. 533534, 1970
175. On Oblique Coordinates and Shallowness in Shell TheoryAdvanc. Math. 3, pp. 264276, 1970
176. Variational Methods and Boundary Conditions in Shell TheoryStudies in Optimization 1, pp. 7894, 1970
177. On the Derivation of Two-Dimensional Shell Equations from Three-Dimensional Elasticity TheoryStudies in Appl. Math. 49, pp. 205224, 1970
178. A Note on Pure Bending and Flexure in Plane Stress Including the Effect of Moment StressesIngenieur-Archiv 39, pp. 369374, 1970
179. On Stretching, Twisting, Pure Bending and Flexure of Pretwisted Elastic PlatesInt. J. Solids Structures 7, pp. 625637, 1971 (with F. Y. M. Wan)
180. A Note on Imperfection Sensitivity of Thin Plates on a Non-linear Elastic FoundationIUTAM Symposium Herrenalb , 1969, pp. 1518, Springer Verlag, Berlin, 1971
*181. On Consistent First Approximations in the General Linear Theory of Thin Elastic ShellsIngenieur-Archiv 40, pp. 402419, 1971
182. On Rotationally Symmetric Stress and Strain in Anisotropic Shells of RevolutionStudies in Appl. Math. 50, pp. 391394, 1971 (with F. Y. M. Wan)
183. Pure Bending, Stretching, and Twisting of Anisotropic Cylindrical ShellsJ. Appl. Mech. 39, pp. 148154, 1972 (with W. T. Tsai)
Page 597
184. A Consistent Treatment of Transverse Shear Deformations in Laminated Anisotropic PlatesAIAA J. 10, pp. 716718, 1972
185. On Reduction of the Differential Equations for Circular Cylindrical ShellsIngenieur-Archiv 41, pp. 291296, 1972
186. On Sandwich-Type Plates with Cores Capable of Supporting Moment StressesActa Mechanics 14, pp. 4351, 1972
*187. Considerations on the Centres of Shear and of Twist in the Theory of BeamsContinuum Mechanics and Related Problems of Analysis (Muskelisvili 80th Anniversary Volume), pp. 403408, 1972 (with W. T. Tsail)
188. On the Determination of the Centers of Twist and of Shear of Cylindrical Shell BeamsJ. Appl. Mech. 39, pp. 10981102, 1972 (with W. T. Tsai)
189. On Finite Symmetrical Strain in Thin Shells of RevolutionJ. Appl. Mech. 39, pp. 11371138, 1972
*190. On One-Dimensional Finite-Strain Beam Theory: the Plane ProblemJ. Appl. Math. and Phys. (ZAMP) 23, pp. 795804, 1972
*191. On One-Dimensional Large-Displacement Finite-Strain Beam TheoryStudies in Appl. Math. 52, pp. 8795, 1973
192. On Kinematics and Statics of Finite-Strain Force and Moment Stress ElasticityStudies in Appl. Math. 52, pp. 97101, 1973
193. A History of the Center-of-Shear ConceptMaillart's Work and RamificationsProceedings 2nd National Conference on Civil Engineering: History, Heritage and the Humanities , Princeton University, pp. 7796, September 1973
*194. Upper and Lower Bounds for Deflections of Laminated Cantilever Beams Including the Effect of Transverse Shear DeformationJ. Appl. Mech. 40, pp. 988991, 1973
*195. On Pure Bending and Stretching of Orthotropic Laminated Cylindrical ShellsJ. Appl. Mech. 41, pp. 168172, 1974 (with W. T. Tsai)
*196. Linear and Nonlinear Theory of ShellsThin Shell Structures, Prentice-Hall, Inc., pp. 2944, 1974
197. An Improved Lower Bound for Deflections of Laminated Cantilever Beams Including the Effect of Transverse Shear DeformationJ. Appl. Math. and Phys. (ZAMP) 25, pp. 8998, 1974 (with S. Nair)
198. On Transverse Bending of Plates, Including the Effect of Transverse Shear DeformationInt. J. Solids Structures 11, pp. 569573, 1975
199. Note on the Equations of Finite-Strain Force and Moment Stress ElasticityStudies in Appl. Math. 54, pp. 18, 1975
*200. Improved Upper and Lower Bounds for Deflections of Orthotropic Cantilever BeamsInt. J. Solids Structures 11, pp. 961971, 1975 (with S. Nair)
*201. Note on a Problem of Beam BucklingJ. Appl. Math. and Phys. (ZAMP) 26, pp. 839841, 1975 (with G. E. Lee)
202. On the Determination of Stresses and Deflections for Anisotropic Homogeneous Cantilever BeamsJ. Appl. Mech. 43, pp. 7580, 1976 (with S. Nair)
203. Transverse Bending of Laminated Anisotropic PlatesJ. Engg. Mech. Div. (ASCE) 102, EM3, pp. 559563, 1976
204. On the Theory of Transverse Bending of Elastic PlatesInt. J. Solids Structures 12, pp. 545554, 1976
205. On Stretching, Bending, Twisting and Flexure of Cylindrical ShellsInt. J. Solids Structures 12, pp. 853866, 1976
206. On Asymptotic Expansions and Error Bounds in the Derivation of Two-Dimensional Shell TheoryStudies in Appl. Math. 56, pp. 189217, 1977 (with S. Nair)
Page 598
207. A Note on Generating Generalized Two-Dimensional Plate and Shell TheoriesJ. Appl. & Phys. (ZAMP) 28, pp. 633642, 1977
*208. On Small Bending and Stretching of Sandwich-Type ShellsInt. J. Solids Structures 13, pp. 12931300, 1977
209. Hans Reissner, Engineer, Physicist and Engineering ScientistThe Engineering Science Perspective 2, pp. 97105, 1977
210. On Bounds for the Torsional Stiffness of Shafts of Varying Circular Cross SectionJ. Elasticity 8, pp. 221225, 1978
211. A Note on Finite Deflections of Circular Ring PlatesJ. Appl. Math. & Phys. (ZAMP) 29, pp. 698703, 1978
212. Two and Three-Dimensional Results for Rotationally Symmetric Deformations of Circular Cylindrical ShellsInt. J. Solids Structures 14, pp. 905924, 1978 (with S. Nair)
213. Some Considerations on the Problem of Torsion and Flexure of Prismatical BeamsInt. J. Solids Structures 15, pp. 4153, 1979
*214. On Lateral Buckling of End-loaded Cantilever BeamsJ. Appl. Math. & Phys. (ZAMP) 30, pp. 3140, 1979
215. Note on a Nontrivial Simple Example of Higher-Order One-Dimensional Beam TheoryJ. Appl. Mech. 46, pp. 337340, 1979
216. Note on the Effect of Transverse Shear Deformation in Laminated Anisotropic PlatesComputer Meth. Appl. Mech. & Eng. 20, pp. 203209, 1979
*217. On the Transverse Twisting of Shallow Spherical Ring CapsJ. Appl. Mech. 47, pp. 101105, 1980
*218. On the Effect of a Small Circular Hole on States of Uniform Membrane Shear in Spherical ShellsJ. Appl. Mech. 47, pp. 430431, 1980
219. On the Influence of a Rigid Circular Inclusion on the Twisting and Shearing of a Shallow Spherical ShellJ. Appl. Mech. 47, pp. 586588, 1980
220. On Torsion and Transverse Flexure of Orthotropic Elastic PlatesJ. Appl. Mech. 47, pp. 855860, 1980
*221. On the Analysis of First- and Second-Order Shear Deformation Effects for Isotropic Elastic PlatesJ. Appl. Mech. 47, pp. 959961, 1980
222. On the Effect of Shear Center Locations on the Values of Axial and Lateral Cantilever Buckling Loads for Singly Symmetric Cross-Section BeamsJ. Appl. Math. & Phys. (ZAMP) 32, pp. 182188, 1981
223. On Finite Pure Bending of Curved TubesInt. J. Solids Structures 17, pp. 839844, 1981
224. On a One-Dimensional Theory of Finite Torsion and Flexure of Anisotropic Elastic PlatesJ. Appl. Mech. 48, pp. 601605, 1981
*225. On Finite Deformations of Space-Curved BeamsJ. Appl. Math. & Phys. (ZAMP) 32, pp. 734744, 1981
226. A Note on Bending of Plates Including the Effects of Transverse Shearing and Normal StrainsJ. Appl. Math. & Phys. (ZAMP) 32, pp. 764767, 1981
227. On the Derivation of Two-Dimensional Strain Displacement Relations For Small Finite Deformations of Shear-Deformable PlatesJ. Appl. Mech. 49, pp. 232234, 1982
228. Effects of a Rigid Circular Inclusion on States of Twisting and Shearing in Shallow Spherical ShellsJ. Appl. Mech. 49, pp. 442443, 1982 (with J. E. Reissner)
Page 599
*229. A Note on the Linear Theory of Shallow Sheardeformable ShellsJ. Appl. Math. & Phys. (ZAMP) 33, pp. 425427, 1982 (with F. Y. M. Wan)
230. On Lateral Beam Buckling and Finite-Deflection Plate TheoryProc. IUTAM Symposium , Numbrecht 1981, pp. 2334, 1982
231. Some Remarks on the Problem of Column BucklingIngenieur-Archiv 52, pp. 115119, 1982
*232. A Note on Two-Dimensional Finite-Deformation Theories of ShellsInt. J. Non-Linear Mechanics 17, pp. 217221, 1982
233. Stress Couple Concentrations for Cylindrically Bent Plates with Holes or Rigid InclusionsJ. Appl. Mech. 49, pp. 8587, 1983
234. On a One-Dimensional Formulation of the Problem of Torsion and Flexure of Shear-Deformable PlatesJ. Appl. Mech. 49, pp. 225227, 1983
235. Further Considerations on the Problem of Torsion and Flexure of Prismatical BeamsInt. J. Solids Structures 19, pp. 385392, 1983
236. A Twelfth Order Theory of Transverse Bending of Transversely Isotropic PlatesZ.f. ang. Math. & Mech. 63, pp. 285289, 1983
*237. On Axial and Lateral Buckling of End-Loaded Anisotropic Cantilever BeamsJ. Appl. Math. & Phys. (ZAMP) 34, pp. 450457, 1983 (with J. E. Reissner)
238. On a Simple Variational Analysis of Small Finite Deformations of Prismatical BeamsJ. Appl. Math. & Phys. (ZAMP) 34, pp. 642648, 1983
239. On Some Problems of Buckling of Prismatical Beams under the Influence of Axial and Transverse LoadsJ. Appl. Math. & Phys. (ZAMP) 34, pp. 649667, 1983
240. On a Variational Analysis of Finite Deformations of Prismatical Beams and on the Effect of Warping Stiffness on Buckling LoadsJ. Appl. Math. & Phys. (ZAMP) 35, pp. 247251, 1984
*241. On a Certain Mixed Variational Principle and a Proposed ApplicationInt. J. Num. Meth. Eng. 20, pp. 13661368, 1984
*242. On a Variational Principle for Elastic Displacements and PressureJ. Appl. Mech. 51, pp. 444445, 1984
243. On the Formulation of Variational Theorems in Geometrically Non-linear ElasticityASCE J. Eng. Mech. 110, pp. 13771390, 1984
244. On the Derivation of the Differential Equations of Linear Shallow Shell TheoryFlexible Shells, Euromech-Colloquium No. 165, pp. 1221, Springer Verlag, 1984
*245. A Tenth-Order Theory of Stretching of Transversely Isotropic SheetsJ. Appl. Math. & Phys. (ZAMP) 35, pp. 883889, 1984 (with R. A. Clark)
246. A Problem of Unsymmetrical Bending of Shear-Deformable Circular Ring PlatessIngenieur-Archiv 55, pp. 5765, 1985 (with J. E. Reissner)
*247. A Variational Analysis of Small Finite Deformations of Pretwisted Elastic BeamsInt. J. Solids Structures 21, pp. 773779, 1985
*248. On Mixed Variational Formulations in Finite ElasticityActa Mechanica 56, pp. 117125, 1985
249. Reflections on the Theory of Elastic PlatesApplied Mechanics Reviews 38, pp. 14531464, 1985
250. On a Mixed Variational Theorem and on Sheardeformable Plate TheoryIntern. J. Num. Meth. Eng. 23, pp. 193198, 1986
251. Some Problems of Shearing and Twisting of Shallow Spherical ShellsProc. Intern. Conf. Computational Mech. (ICCM86-Tokyo) pp. I-3I-12, Springer Verlag 1986
*252. Some Aspects of the Variational Principles Problem in ElasticityComputational Mechanics 1, pp. 39, 1986
Page 600
253. On the Effect of a Small Circular Hole in Twisted or Sheared Shallow Sheardeformable Spherical ShellsJ. Appl. Mech. 53, pp. 597601, 1986 (with F. Y. M. Wan)
254. On Small Finite Deflections of Shear-Deformable Elastic PlatesComputer Methods in Appl. Mech. & Eng. 59, pp. 227233, 1986
255. On Finite Deflections of Anisotropic Laminated Elastic PlatesIntern. J. Solids Structures 22, pp. 11071115, 1986
*256. On a Certain Mixed Variational Theorem and on Laminated Shell TheoryRefined Dynamical Theories of Beams, Plates and Shells Proc. Euromech-Colloquium No. 219, pp. 1727, Springer Verlag, 1987
257. A Note on the Derivation of Higher Order Two-Dimensional Theories of Transverse Bending of Elastic PlatesProc. Euromech-Colloquium No. 219 , pp. 2831, 1987
258. Variational Principles in ElasticityFinite Element Handbook, pp. 2.32.19, McGraw Hill Book Co., 1987
259. On Lateral Buckling of End-Loaded Cantilever Beams Including the Effect of Warping StiffnessComputational Mechanics 2, pp. 137147, 1987 (with J. E. Reissner and F. Y. M. Wan)
260. On a Generalization and on the Meaning of Some Exact Formulas of the Theory of ''Moderately Thick" Elastic PlatesIntern. J. Solids Structures 23, pp. 711717, 1987
261. A Further Note on Finite-Strain Force and Moment Stress ElasticityJ. Appl. Math. & Phys. (ZAMP) 38, pp. 665673, 1987
*262. On Finite Axi-Symmetrical Deformations of Thin Elastic Shells of RevolutionComputational Mechanics 4, pp. 387400, 1989
263. The Center of Shear as a Problem of the Theory of Plates of Variable ThicknessIngenieur-Archiv 59, pp. 325332, 1989
264. Asymptotic Considerations for Transverse Bending of Orthotropic Sheardeformable PlatesJ. Appl. Math. & Phys. (ZAMP) 40, pp. 543557, 1989
265. Lateral Buckling of BeamsComputers and Structures 33, pp. 12891306, 1989
*266. On the Formulation of Variational Theorems Involving Volume ConstraintsComputational Mechanics 5, pp. 337344, 1989 (with S. N. Atluri)
267. On a One-Dimensional Theory of Finite Bending and Stretching of Elastic PlatesComputers and Structures 35, pp. 417424, 1990
268. On Buckling of Crosswise Rigid Elastic PlatesActa Mechanica 82, pp. 175184, 1990
269. A Note on the Boundary Layer in the Sixth-Order Theory of Transverse Bending of Anisotropic Sheardeformable Elastic PlatesCommunications in Appl. Num. Meth. 6, pp. 519524, 1990
*270. On Asymptotic Expansions for the Sixth-Order Linear Theory Problem of Transverse Bending of Orthotropic Elastic PlatesComputer Methods in Appl. Mechs. & Eng. 85, pp. 7588, 1991
271. A Mixed Variational Equation for a Twelfth-Order Theory of Bending of Nonhomogeneous Transversely Isotropic PlatesComputational Mechanics 7, pp. 255260, 1991
272. Approximate Determinations of the Center of Shear by Use of the St. Venant Solution for the Flexure Problem of Plates of Variable ThicknessArch. Appl. Mech. 61, pp. 555566, 1991
273. Some Ramifications of the Center of Shear ProblemZ.f. ang. Math. & Mech. 72, pp. 315319, 1992
