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Doctoral Thesis
The finite twisting and bending of heated elastic lifting surfaces
Author(s): Bisplinghoff, Raymond L.
Publication Date: 1957
Permanent Link: https://doi.org/10.3929/ethz-a-000107351
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The Finite Twisting and Bending of
Heated Elastic Lifting Surfaces
THESIS
PRESENTED TO
THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY, ZÜRICH
FOR THE DEGREE OF
DOCTOR OF TECHNICAL SCIENCES
BY
Raymond Lewis Bisplinghoff
Citizen of the United States of America
Accepted on the recommendation of
Prof. Dr. M. Rauscher and Prof. Dr. G. Eichelberg
1\ '
Zürich 1957
Dissertationsdruckerei Leemann AG
Erscheint als Mitteilung Nr. 4
aus dem Institut für Flugzeugstatik und Leichtbau an der
Eidgenössischen Technischen Hochschule in Zürich
Verlag Leemann Zürich
Acknowledgement
Acknowledgement is due more persons than can be mentioned who aided
the author in bringing the work to completion. Professor M. Rauscher providedthe opportunities for undertaking the work and his aid and counsel duringits course is especially acknowledged. Dr. J. Schaffhauser gave valuable aid in
the experimental studies. Suggestions were received from Professors E. Reiss-
ner and G. Eichelberg and from Mr. J. Houbolt. The author is most gratefulto all of these people for their assistance.
Leer - Vide - Empty
Table of Contents
Summary 7
1. Introduction 8
2. Symbols 12
3. Basic Principles of Shallow Shell Theory 14
3.1. Stress Resultants and Couples 15
3.2. Equilibrium Conditions 16
3.3. Strain-Displacement Relations 17
4. Equilibrium and Compatibility Equations of Shallow Shells 19
5. Equations of State of an Elastic Shell Element 20
6. Equilibrium and Compatibility Equations of Heated Elastic Shallow Shells..
23
7. Variational Conditions of Equilibrium of Heated Elastic Shallow Shells... 27
7.1. Principle of Minimum Potential Energy 27
7.2. Principle of Minimum Complementary Energy 29
8. Energy Criteria for Stability of Heated Elastic Shallow Shells 31
8.1. Energy Criterion for Prescribed Displacements 32
8.2. Energy Criterion for Prescribed External Forces 32
8.3. The Role of External Disturbances 32
9. Finite Twisting and Bending of Rectangular Elastic Plates with Chordwise
Temperature Gradients 33
9.1. General Theory 33
9.2. Longitudinally Stiffened Fiat Plate of Constant Thickness 36
9.3. Longitudinally Stiffened Cambered Plate of Constant Thickness...
54
9.4. Longitudinally Stiffened Cambered Plate of Variable Stiffness....
58
10. Finite Twisting and Bending of Elastic Cylindrical Shell Beams with Chord¬
wise Temperature Gradients 72
10.1. General Theory 73
10.2. Pure Monocoque Beam with Parabolic Camber Distribution 75
11. Vibrations of Rectangular Elastic Plates in the Presence of Finite Deformations
and Temperature Gradients 90
11.1. Linearized Shallow Shell Theory in the Presence ofTemperatureGradients 90
11.2. Torsional Vibrations of an Initially Twisted and Heated Lifting Surface 93
Appendix A. Experimental Apparatus and Methods 96
A.l. Loading Device 96
A.2. Heating Devices 97
A.3. Stmctural Models 100
A.4. Measuring Devices 108
Appendix B. Functions of State of an Elastic Shell Element 109
References 114
Endliche Verdrehung und Biegung von geheizten Flügelflächen
Zusammenfassung
Die vorliegende Dissertation befaßt sich mit Kombinationen endlicher Ver¬
drehungen und Biegungen von geheizten Flügelflächen. Theorie und Experi¬mente beziehen sich auf einen langen Streifen konstanter Breite, so daß das
Problem im wesentlichen zweidimensional wird. Dafür erfährt die Veränder¬
lichkeit der Bedingungen über die Tiefe ein sehr gründliches Studium.
Die theoretische Betrachtung basiert auf Marguerres Theorie der schwach¬
gewölbten Schale, wobei der Einfluß des Temperaturgradienten über Ober¬
fläche und Dicke der Schale zusätzlich berücksichtigt wurde. Es wird ange¬
nommen, daß der Temperatureinfluß sich auf Wärmespannungen und eine
Veränderung des Elastizitätsmoduls beschränkt. Nach einer Rekapitulationder Grundgedanken der Theorie der schwachgewölbten Schale werden Zu-
standsgleichung und -funktion eines elastischen Elementes der Schale ent¬
wickelt. Dann folgt die Betrachtung des Gleichgewichtes, der Verträglichkeiten,der Randbedingungen und hierauf eine Zusammenfassung der Variations¬
bedingungen für das Gleichgewicht und der Energiekriterien für die Stabilität
einer geheizten, elastischen, schwachgewölbten Schale.
Zwei Grenzfälle von Bauausführungen werden betrachtet. Im ersten ist die
Flügelfläche eine hohle, längsversteifte Platte mit längslaufenden Stegen. Der
andere ist eine zylindrische Schale mit starken, längsversteiften Beplankungs¬
platten, aber ohne Rippen und Stege. Für beide Typen werden geschlosseneund approximative Lösungen für gleichzeitige endliche Verdrehung und Biegungunter Berücksichtigung des Temperaturgradienten über die Plattentiefe
gegeben.Zum Vergleich mit den theoretischen Ergebnissen werden experimentelle
Resultate vorgelegt. Die Daten beziehen sich auf sechs Modelle aus Aluminium¬
legierung bei verschiedenen Kombinationen von Biegemoment, Torsions¬
moment und Temperaturgradient. Alle Versuche wurden unter stationären
Temperaturbedingungen durchgeführt. Die Zu- und Abfuhr der Wärme
geschah durch längs der Modellfläche gelegte Linien von Quellen und Senken.
Abschließend wird die linearisierte Theorie der schwachgewölbten Schale
unter Berücksichtigung von Temperaturgradienten einer kurzen nochmaligenDiskussion unterworfen. Den Ausklang bildet eine Anwendung dieser Theorie
auf die kleinen Schwingungen einer ebenen Platte bei endlicher Verdrehungund Biegung in Gegenwart eines Temperaturgradienten über die Breite.
6
Summary
An investigation is described, and results are presented, on the topic of the
finite twisting and bending of heated elastic lifting surfaces. The investigation
encompasses both theory and experiment, and its scope is confmed to very
long elastic surfaces of rectangular planform. The problems investigated are
therefore essentially of a two-dimensional nature. Special attention is paid to
the character of the chordwise deformations and the role played by these
deformations in coupling the bending and twisting actions of the surface.
The theoretical approach is based upon the shallow shell theory of Mar-
guerre, modified to include stiffened shells with temperature gradients over
the surface and throughout the thickness. The effect of elevated temperatureis twofold: thermal stresses are produced by the temperature gradients, and
the modulus of elasticity is affected by the temperature level.
Following an introductory Statement of the assumptions of shallow shell
theory, a discussion is presented of the equations of state of an elastic shell
dement. There follows a statement of the equations of equilibrium and compa-
tibility and of the boundary conditions; then a resume of the variational
conditions of equilibrium and of the energy criteria for stability of heated
elastic shallow shells.
The theory is applied to two limiting cases of structural arrangement. In
the first, the lifting surface is assumed to be a longitudinally stiffened cambered
plate with varying stiffness along chordwise lines. The second assumes a
cylindrical shell with cambered longitudinally stiffened cover plates and
without internal structure. Solutions are presented in each case for the finite
twisting and bending of the surface in the presence of chordwise temperature
gradients.
Finally there is presented a brief discussion of linearized shallow shell
theory in the presence of temperature gradients with an application to the
problem of small vibrations of heated elastic lifting surfaces.
Experimental data are presented to confirm the theoretical results in each
of the important applications considered. A total of six aluminum alloy modeis
were tested under various combinations of bending moment, twisting moment
and chordwise temperature differential. The elevated temperature tests were
conducted under conditions of steady state thermal equilibrium, and tempera¬ture gradients were obtained by applying line sources and sinks of heat to
the surface of the specimen.
7
1. Introduction
Since the early 1930's, when the emergence of the new aluminum alloysmade it possible to construct strong and rigid full-cantilever aircraft liftingsurfaces, the latter have undergone continuous refinement in their detail
design and remarkable changes in their proportions. We can observe the nature
of the latter changes by studying the trends illustrated by figs. 1.1, 1.2 and 1.3.
Fig. 1.1 illustrates the trend of wing thickness ratio in fighter and transport
airplanes since 1920. The emphasis on drag reduction of fighters is evidenced
by a continuous reduction of thickness ratio from about 0.2 in 1930 to some-
thing of the order of 0.05 at the present time.
Fig. 1.2 depicts trends of wing slenderness. The latter is defined as the
ratio of wing structural semi-span to maximum thickness at the wing root
[l]1). The term structural semi-span denotes a distance measured along the
beam axis of the wing, as illustrated by the distance 1 in the sketch of fig. 1.2.
Lifting surfaces have become very slender in the principal bending direction
compared to other engineering structures, a trend which is responsible for the
growing importance of aeroelasticity.
Finally, fig. 1.3 depicts trends of wing structural aspect ratio [1], The latter
is defined as the ratio of the Square of the structural span to the wing area.
We find that lifting surfaces tended to a higher degree of slenderness in the
chordwise bending direction until the advent of supersonic fighters, at which
time a reversal of the trend appears. This reversal is evidently a result of both
aeroelastic and aerodynamic reasons.
It seems evident from these trends that lifting surfaces of future airplaneswill resemble more nearly flat plates than cylindrical shell beams. Whereas
aeronautical structural engineers have heretofore had comparatively little
interest in the theories of bending and stretching of plates, the latter may now
be used as a basis for predicting the behavior of thin lifting surfaces. We find
these theories well established with a history of development nearly as longas the recorded history of mechanics of materials. Simple experiments on the
vibrations of elastic plates were performed as early as 1787 by Chladni [2]. In
1811, while criticizing a paper submitted for a prize by Sophie Germain on the
subject of elastic surfaces, Lagrange stated for the first time the partial dif-
ferential equations of fourth order which govern the bending, including the
1) The numbers in [] refer to the numbered references at the end, page 114.
8
0.2 -
OL
(/)intu
z
I 0.1
I-
z
0. _1_
FIGHTERS
TRANSPORTS
PISTON TO
JET FIGHTER
PISTON TO
JET TRANSPORT
1900 1910 1920 1930 1940 1950 i960
YEAR
Fig. 1.1. Wing thickness ratio.
in
UJz(TUJQZUJ_l
o
30
26
22
18
14
10
FIGHTERS
TRANSPORTS
PISTON TO
JET FIGHTER
PISTON TO
JET TRANSPORT
I I
1900 1910 1920 1930 1940 1950 i960
YEAR
Fig. 1.2. Wing slenderness.
9
N
V)
o"
üUJ
<n<
_i
<
Oh-O
tEh-<n
10
B^fflS FIGHTERS
TRANSPORTS
PISTON T0 JET FIGHTEI
I
PISTON T0
JET TRANSPORT
1900 1910 1920 1930 1940 1950 i960
YEAR
Fig. 1.3. Wing structural aspect ratio.
vibrations, of elastic plates. The early contributors were, however, unable to
completely resolve all questions of boundary conditions, and it was not until
1850 that Kirchoff stated for the first time the elementary equations of elastic
plate bending together with the correct boundary conditions [3].It is assumed in the elementary theory of plate bending that the lateral
deflections are small compared to the thickness. For larger defiections, stretch¬
ing of the mid-plane must be considered, and equations taking the latter into
account were derived originally by Kirchoff and Clebsch. Since these equationsare not linear, Kirchoff applied them in only the simplest case in which
stretching of the mid-plane is uniform. The general equations for the largedeflections of "very thin" plates were simplified in 1907 by Föppl by the use
of an Airy stress function for the stresses acting in the mid-plane. The require-ment that the plates be "very thin" was removed by von Kdrmän in 1910,
and these equations have formed a useful basis for engineering studies since
that time.
During the latter portion of the nineteenth Century, we find parallel deve-
lopments in the theory of shells. In 1888, Love presented general equations of
the bending and stretching of elastic shells. The difficulty of obtaining Solutions
of these equations is great and we find complete Solutions in only a few cases
of practical interest involving certain symmetrical surfaces and loadings.
10
In 1938, Marguerre [4] presented a System of partial differential equationswhich permits the analysis of shallow shells; that is, shells differing not too
greatly from a flat plate. The equations obtained by Marguerre have the rela¬
tive simplicity of the von Kärmän flat plate equations yet take account of the
important arch effect of the shallow shell. This simplicity permits Solutions
to a wide variety of practical shell problems beyond those involving symmetri-cal surfaces and loadings.
The present paper describes some theoretical and experimental investiga-tions on the behavior of elastic lifting surfaces under conditions of finite
twisting, finite bending and heating. The theoretical studies are based upon
the shallow shell theory of Marguerre, modined to include the influence of
temperature gradients. The motivation for adding temperature gradients is
the well established fact that kinetic aerodynamic heating will be an importantfactor in the design of very thin wings for supersonic flight.
There are two principal effects of elevated temperature on structural
behavior. The first derives from the deterioration of the mechanical propertiesof materials at elevated temperature. This important effect, however, involves
a high degree of empiricism and we have omitted it with the exception that
the modulus of elasticity is assumed to be corrected for the influence of tem¬
perature. The second principal effect, which derives from the influence of
thermal stresses, has been incorporated in the present work. Thermal stresses
arise as a result of the fact that most engineering materials expand with
increasing temperature. Temperature gradients within a solid body producedifferent degrees of expansion in different parts of the body. Since the ele-
ments into which a solid body may be divided are interconnected and cannot
expand freely, stresses will ensue. The first important contribution to the
theory of thermal stresses was made by Duhamel in 1838 [3]. This was his
principal achievement in the theory of elasticity. Since Duhamel''s originalwork there have been numerous applications, but no essential changes in the
basic concepts of the theory.The object of the present work has been to demonstrate some of the features
of the non-linear behavior and the coupling action which characterize the
finite twisting and bending of thin heated lifting surfaces. We have restricted
the scope of the study to very long elastic lifting surfaces of rectangular plan¬
form; a desirable restriction at this stage in view of the mathematical diffi-
culties of applying the non-linear equations of shallow shell theory. The
problems investigated are therefore of two dimensional character, and we pay
particular attention to lateral deformations along chordwise lines.
In research studies on lifting surface behavior, questions of arrangementand stiffness of the internal strueture must be faced and suitable assumptionsmust be made. We have resolved these questions by concentrating attention
11
on two extreme cases. In the first, the entire lifting surface is assumed to be
a longitudinally stiffened plate of varying stiffness along chordwise lines. In
effect, we assume an infinite number of ribs of specined bending stiffness. In
the second case, we assume that the lifting surface is a hollow shell with
longitudinally stiffened cover plates and without internal structure. In this
case we have no ribs and the cross section is free to distort, constrained only
by the boundary conditions requiring continuity at the leading and trailing
edges. These two extreme cases are selected because of the unusual opportu-nities they present to study, by means of closed Solutions, the physical nature
of our problem. The intermediate cases of finite ribs, where approximatenumerical Solutions are required, are omitted from the present study.
The paper divides into three main parts: sections 3 through 8, sections 9
through 11 and the appendices. Sections 3 through 8 establish the theoretical
tools needed for the analysis of shallow elastic seolotropic shells with tempera-ture gradients. Sections 9 through 11 present applications, and the appendices
give details concerning the experimental apparatus and methods and some
brief remarks concerning thermodynamic functions of state for elastic shell
elements.
2. Symbols2)
A Isothermal mechanical energy per unit of shell surface in terms of
the strains (7.3). Cross sectional area (10.30).a Semi-length of lifting surface (9.5).B Isothermal mechanical energy per unit of shell surface in terms of
the stress resultants and couples (7.8).
Bijkl Influence coefficients (5.1).b Semi-chord of lifting surface (9.1).
C(x2,r]2} Influence function (9.76).
c Heat capacity per unit volume of material (7.7).
D =
12(1-y")Mexural "gidity (5.6).
Dijkl Influence coefficients (5.1).DR Distortion ratio of shell cross section (10.46).E Modulus of elasticity (5.3).F Airy stress function (4.3). Free energy per unit of shell surface (7.7).
F1(/ji),F2(fi.),F3(li.) Special functions of p (9.22), (10.34), (10.40).
/ Solidity function (5.6). Free energy per unit volume of material (B. 10).
2) The parentheses following each symbol indicate the equation in which the Symbolappears first.
12
G Modulus of rigidity (5.3). Gibbs' function per unit of shell surface
(7.15). Kernel function of integral equation (9.81).
GJS St. Venant torsional stiffness (11.16).G Jw Warping torsional stiffness (11.16).
g Non-dimensional temperature function (9.14). Gibbs' function per
unit volume of material (B.20).h Shell thickness (5.6). Enthalpy per unit volume of material (B.14).
I0 Mass moment of inertia per unit length about xx axis (11.16).
K =-—5- Elastic coefficient (5.6).
K1= |^J(1 -v2/2//i) Elastic coefficient (9.15).
K2 ='ia
(t — v /2//i) Thermoelastic coefficient (9.15).
K%)ld Influence coefficients (5.1).
k Curvature along x1 axis (9.7).
L Operator in differential equation (9.56).M Applied bending moment (9.6).
Mt Applied twisting moment (9.6).
MX) Stress couples (3.3).
NtJ Stress resultants (3.3).
Qi, Q2 Shear stress resultants (3.3).
q Intensity of transverse loading (3.4). Generalized coordinate (9.95).
Shear flow (10.7). Heat supplied to unit volume of material (B.l).
B1,B2 Effective edge stress resultants parallel to x3 axis (6.6).
r Radius of camber of lifting surface (9.36).
Slt S2 Shell boundaries (7.1).
s Entropy per unit volume of material (B.2).T Temperature (5.1). Kinetic energy (11.7).
tm Maximum thickness of shell wing (10.18).
U Isothermal mechanical energy of the shell (8.1).
u Internal energy per unit volume of material (B.l).
u1,u2,u3 Shell displacements (3.8).
V1,V2 Stress resultants parallel to x3 axis (3.6).
W Lateral bending deformation (9.7).
W Weighting numbers (9.43).
x1,xz,x3 Rectangular coordinates (3.1).
a Coefficient of thermal expansion (5.3).& [kTW
ß = yl
=Absolute value of root of characteristic equation (9.16).
Y Leading and trailing edge bluntness parameter (9.49). Cross section
distortion parameter (10.32).
13
y{ Assumed mode shapes in Rayleigh-Ritz analysis (9.95).A T Temperature differential (9.14).5 Indicates the variational process (7.1). Dial gage deflection (A.l).
8i;- Kronecker delta (3.3).
eit Strain (3.7).
£ Coordinate axis normal to shell surface (3.3).
7] Dummy Integration variable (9.76).6 Rate oftwist (9.7).
Ktj Curvature (3.7).
X =by—*-=- Non-dimensional twist rate parameter (9.21).
H = b 1/—^=- Non-dimensional curvature parameter (9.18).
v Poissons's ratio (5.3).
^,£2 Coordinate axes tangent to shell surface (3.3).
77 Functional employed in variational process (7.2).
p Parameter in differential equation (9.55). Mass per unit of shell
surface (11.6).
a{j Stress (3.3).
t Weighting parameter for stiffened sections (9.3).
>t Assumed mode shapes in Vibration analysis (11.9).
o) Frequency of Vibration (11.13).
!_J Row matrix (9.82).
{ } Column matrix (5.2).
[ ] Square matrix (5.2).
[I] Unit matrix (9.85).
[ ]_1 Inverted matrix (6.1).
3. Basic Principles of Shallow Shell Theory
We shall take as a basis for theoretical studies of the behavior of liftingsurfaces, the shallow shell theory of Marguerre [4]. This theory provides a
relatively simple avenue of approach to the non-linear phenomena associated
with the stretching and bending of shells which may be regarded as shallow.
It forms an ideal starting point from which to proceed to a study of finite
deformations of thin lifting surfaces. Applications of shallow shell theory have
been previously made by Meissner [cf., e. g., 5 and 6] and Flügge and Conrad
[7], who have also examined the validity and ränge of applicability of Mar-
guerre's assumptions.
14
We refer to fig. 3.1, which illustrates a segment of a shallow shell oriented
with respect to a set of reetangular co-ordinates xx, xz, x3. The equation for
the mid-plane reference surface of the shell may be taken as
A shallow shell is qualitatively one for which the slope of the mid-planereference surface is small at all points, or quantitatively one for which
1 +(8x3\ /dx3\
dxj [dxj1 (»,7=1,2), (3.2)
where i and j may take on the values 1 and 2. A rule for the latitude of Inter¬
pretation of (3.2) cannot be stated in general; however, Beissner [5] has sug-
gested that shallow shell theory will be more than aecurate enough as long as
dx3l8xi^ll8, and often aecurate enough for practical purposes as long as
dx3\dx^\\2.
Fig. 3.1. Segment of a shallow shell.
3.1. Stress Resultante and Couples
Following the usual procedure of shell theory, we formulate the equationsin terms of stress resultants and couples. The stress resultants and couples
pertaining to shallow shell theory may be defined with reference to fig. 3.2,
which illustrates an infinitesimal element of a stiffened shell. The edges of the
element are acted upon by stress vectors aii and a^ which are respectivelytangential and perpendicular to the deformed reference surface. The tangentialand normal axes are the ^,^,1 axes, as shown by fig. 3.2. We assume that
a complete macroscopic description of the statics of a stiffened shallow shell
is provided by three stress resultants in the reference plane, Ntj, two lateral
shear stress resultants, Qt, and three stress couples, M^. The stress resultants
15
M„
%*/^M„
Fig. 3.2. Element of a shallow shell.
and couples, which are respectively forces and couples per unit projectedwidth on the xx—xz plane, are defmed by
ST< SK
Mit = hti Ja,y^C+ Saytdt (,? = 1,2), (3.3)ST( SK
Qi= $°it,d>LSK
where 8i}- is the Kronecker delta and where 8 Ti represents the stiffener area
normal to the i-th axis, and SK represents the skin.
Since 0^=
0^, it is evident that Nij = Nji and Mii = Mii, and that (3.3)therefore represent eight independent quantities. The positive directions of
the stress resultants and couples are determined by the positive directions of
the stresses, <jy, and the definitions (3.3). The positive directions of <ri;- are
taken according to the usual Convention of the theory of elasticity [8], and the
resulting positive directions of Ntj and M^ are as illustrated by fig. 3.2.
3.2. Equilibrium Conditions
We assume that the shell is acted on by a transverse load of intensity, q,
parallel to the xA axis, as illustrated by fig. 3.2. Such a load may be represen-
tative of a lateral pressure or inertial load. Then the three scalar equations of
16
force equilibrium which are consistent with the approximations of the generalshallow shell theory are [4]
Wy_
dx~
8Q. d r n
'
(i,j,k = 1,2)*), (3.4)
dxi dXj N]kj^-(x3 + u3)\+q = 0,
where u3 is the lateral displacement of the mid-plane reference surface. The
two equations of moment equilibrium are
^-^ = 0 (i,j= 1,2). (3.5)
A third moment equation is assumed to be satisfied identically. It may be
seen from the form of the equilibrium equations that the shallow shell theoryis essentially a theory in which the difference between tangential stress resul¬
tants and stress resultants in the directions xx and x2 are negligible, whereas
differences between transverse stress resultants and stress resultants in the
direction of x3 are recognized. In fact, the stress resultants in the direction
of x3 are taken as
Vl = Ql + Nl]J^(x3 + u3) (m=1,2). (3.6)
It is apparent that the essential function of the curvature is to provide an
arch effect which assists in carrying the vertical load.
3.3. Strain-Displacement Relation»
The assumptions are completed when a relationship is stated which con-
nects the strains throughout the shell thickness with the deformations of the
mid-plane reference surface. This relationship is a key element in shell theory,since it permits a basically three-dimensional problem to be reduced to one
of two dimensions. The fundamental assumption to be made is that the dis-
placements are linear across the shell thickness. At the outset we assume
that the normal and transverse shear stresses cause a line normal to undeformed
reference surface to be displaced angularly but remain straight. The strains,
t], at any point within a layer parallel to the reference surface, are approxi-mated by the first two terms of their power series in £, and can be expressedin the form
«w=
«», + £"« (i,j =1,2), (3.7)
where itJ are the strains in the plane of the reference surface and ktj describe
3) A subscnpt which is repeated mdicates summation as the index that is repeatedtakes the values 1,2.
17
the additional curvature of the reference surface due to deformation. Within
the order of the approximations of shallow shell theory, the reference surface
strains and curvatures are related to the deformations of the reference surface
%, u2 and u3, and the transverse shear strains, by
du^ 8x3 du.3 8u3
1 /8u3\2
18x1 2\8xJ '
e12
8xx 8x
8u2|dx3 8u3
[1 /8usy
8x2 8x2 8x2 2\8 x2J '
8u1 8u2 8x3 8u3 8x3 8u3 8u3 8u3
8x2 8xx 8xx8x2 8x28xx 8xx8x2
Kn~ dxj»+dx^82u3 8e2t;
K" =
-c^+Jx^>
(3'9)
„ _o
d2%,
861{|8^
ox1ox2 8x2 8x1
where c4j are the transverse shear strains. An essential feature of the reduction
of the strain-displacement relations of shallow shells to the relatively simpleforms stated above is the assumption that bending displacements are signi-
ficantly larger than stretching displacements [4]. In the applications to be
considered, it will be assumed that the Euler-Bernoulli-Navier hypothesis
applies. This implies that the normal to the undeformed reference surface is
rotated without extension into the normal to the deformed reference surface,
and that the transverse shear and normal strains are zero.
«{{=
*<£= 0 (» = 1,2). (3.10)
Equations (3.1) through (3.10) summarize the basic hypotheses of the
shallow shell theory. They are concerned with the negligibility of certain stress
components and with the character of the deformation of the shell. Our objectin subsequent parts of the present work is to obtain Solutions which are con-
sistent with these assumptions to some problems of interest to engineers. It is
important to observe that the assumptions involve, at least explicitly, no
consideration of the properties of the material from which the shell is con-
structed, and they may therefore be considered applicable to a ränge of ma¬
terial behavior and stress levels.
18
19
8xx8x28xx8x28xx28x2~8x228xx2
u382x382u382x382u382x38%
8x28x2\8xx8x2)Sxj2'
8x18x28x%2\8x18x2J
vs2u3
1 82u382u3\2
82u3
/e228282e12en82
equation.compatibilitysingle
aintocombinedbecansurface,referencetheofplanetheindisplacements
andstrainsbetweenrelationsthedescribing(3.8),equations,threeThe
shell.shallowtheofcurvatureinitialtheofinfluencetherepresenttermsthree
finalTheobtained.isplatesflatofstretchingandbendingtheforequation
equilibriumtheincluded,are(4.4)ofsideright-handtheontermsthreefirstthe
When[8].platesflatofdeflectionsbendingsmalloftheorytheinconsidered
normallythoseare(4.4)ofsideleft-handtheontermsthethatobserveWe
8xl8x28xl8x28x-f8x28x228x-f8xx8x2"8xx8x2
(4.4)4-5+T,0.0
—
5-7,oööT,7,£-R+82x*82F82F82x382F82x382u382F
8x^8x228x228x28x28xl8x28xx2
d2Fd2u382F32u382M2282M12&MU
follows:asequations,equilibriumsingleaintocombinedbemay(4.2),
and(4.1)equations,equilibriumthreethe(4.3),dennitionstheofmeansBy
(4-3)"w
=ffu^7'
=^aW'
=N»82F82F82F
[8]bydefined
is(4.1)satisfieswhichfunctionstressTheequation.singleatofunction
stressAiryanofmeansbyfurtherreducedbemaytheyhowever,state,plastic
theinvolvingapplicationscertaininusefulbemay(4.2)and(4.1)Equations
(4.2)1,2),(i,j,k=0=q+U3)+(X3Njki^r8x4dXidXi-+
""e8x,
82Mij
forcesverticalofequilibriumofequationsingleaand
(4.1)1,2),=(»,?0=^resultantsstressmid-plane
theofequilibriumofequationstwotheareTheyfive.ofinsteadequations
threebysummarizedbecanstateequilibriumthethatsocombinedbecan
(3.5)and(3.4)thatobserveweequations,equilibriumthefirstConsidering
equations.simultaneousof
numberlesserato(3.9)and(3.8)bygivenrelationsdisplacementstrainsix
theand(3.5)and(3.4)bygivenequationsequilibriumfivetheofreduction
theistheoryshellshallowtheofapplicationspracticalinstepinitialAn
ShellsShallowofEquationsCompatibilityandEquilibrium4.
When the right-hand side of (4.5) is put equal to zero, we obtain the compa-
tibility equation for plane stress problems. Including the first two terms on
the right-hand side gives the compatibility equation for the bending and
stretching of flat plates [8]. The final three terms on the right-hand side bringin the influence of the initial curvature of the shallow shell.
Equations (4.4) and (4.5) thus represent in two equations, all of the equi-librium equations and strain-displacement relations of section (3) except (3.9).When equations (3.9), (4.4), and (4.5), which are applicable over a ränge of
material behavior, are supplemented by the equations of state of an element
of the shell, we obtain a mathematically complete set of equations.
5. Equations of State of an Elastic Shell Element
Three sets of natural variables describe the state of an element of an elastic
body; stress, strain and temperature. Equations expressing a relation among
these three are called equations of state. The equations of state of a shell
element are those equations which relate the stress resultants and coupleswith the strains in the plane of the reference surface, the curvatures and
temperature. A general form of the equations of state may be written as
a) Nt} = KXJkl ekl + Bt]kl Kkl— 8W Tx,
b) Mll = BvM-ekl + Dl]klKkl-hllW, (i,j,k,l= 1,2), (5.1)
c) Ql = GleH.In these equations, Kl}kl, Bljkl, Dl]kl and Gx represent symmetrical arrays of
influence coefficients for isothermal straining. Each coefficient may vary with
its location on the surface of the shell. Tx and Ti are functions of the tempera¬ture distribution T{x1,x2,0- The Tt reflect the temperature distribution over
the surface and the T% the temperature distribution throughout the thick-
ness. T (x1, x2, £) is an increment of temperature above a reference absolute
temperature T0 for a state of zero stress and strain. The total absolute tem¬
perature, denoted by Tx, is the sum of T0 and T. In those cases in which the
shell element is sufficiently thin so that transverse shear strains can be neg-
lected without appreciable error, the third equation in (5.1) is not required. The
present work is restricted to applications in which (5.1) can be simplified to
the extent that they represent the state equations for an orthogonally seolo-
tropic elastic shell element, and we assume that they may be represented bythe matrix forms
{Nn N22} = [K] {en l22) + [B] {Kll k22} - {T, TJ,{Mn M22} = [B] {e-u e-22} + [D] {*u k22} - {Tx T2},
{N12M12} = [KJ{i12K12}, (5.2)
20
21
(5.5)
SK
-"2211~-"1122
SK
"21-^1J
SK
STa
ST<iE—t>2222
SK
ST,
=-ßllll
SK
l-"l^2J-"-2211—All22
SKST,
SKST,
K*
=a'hii
(5.2):inquantitiestheforformulasfollowingthefindwe(5.4)and(5.3)(3.7),
(3.3),Combiningstiffeners.theofexpansionthermalofcoefficienttheand
modulusYoung'srespectivelyareaST.andEST.andskin,theofexpansion
thermalofcoefficienttheandratio,Poisson'srigidity,ofmodulusthelus,
modu¬Young'srespectivelyarea.tandG,vi,Ei,whereandv2E1=v1E2where
T),xST2-(e22ESTi=o22T),asTi
—(nEsti=°n
are,stiffenerstheofstateofequationscorrespondingtheand
(5.4)
(5.3)<?-,=
v1x1)T],+e22-(oi2+[v1e11
v2ac2)T],+v2e22-(cc1+[en
v\vi
E,
l-vxv2Jii
Ei
[10],arestateofequationspertinentthe
material,orthotropicelasticanfromconstructedisskintheIfwork.present
theininterestofstructuresandmaterialsofcombinationsthefor(5.2)in
coefficientsthermoelasticandelastictheforformulasexplicitstateshallWe
inversion.
matrixofprocessthebyrelatedarecoefficientselasticofformstwoThe
coefficients.influenceflexibilityasis,thatform,inversetheirincoefficients
thesemeasuretoconvenientmoreoftenisitconducted,areexperimentsWhen
coefficients.influencestiffnessofformthein(5.2)inappearpropertieselastic
Thetypes.complexmoreforelementshellauponexperimentsperformto
necessarybemayitwhereas[9],methodsanalyticalbycomputedbemaythey
constructionoftypessimpleInshell.theofformstructuraltheandproperties
materialtheupondepend(5.2)incoefficientsthermoelasticandelasticThe
-^1212.-"1212
^»1212^1212
-°2222J-°2211L
-^1122-^1111=[5]
-^'2222jL^2211
-^1122^1111
2222.A2211
A
^1122*1U1=[K]where
J J L~vlv2 J * ~ vl v2ST, SK SK
#2222 = [ESTi1?dl+ f _A_W; (5.5)J J 1 — vl v2
ST, SK
K1212 = jGdi;, b1212= jöeds, D1212= KW;SK SK SK
ST, SK
T.^JE^^TdC +j^^Td^ST, SK
ST, SK
ST, SK
Formulas (5.5) are useful when the temperature varies throughout the shell
thickness and when the elastic properties of the material are a function of the
temperature. Other special cases may be derived from (5.5). For example, in
the important practical case when the material is isotropic and its propertiesare unaffected by the temperature Variation throughout the thickness, and
when the shell structure is symmetrical about its mid-plane and stiffened onlyin the longitudinal or x1 direction, (5.5) reduce to
-"-1111 = -Kfl> -"-1122 = -"-2211 = -"-v/2' -"-2222 ~ -"- /2 >
-"llll = -"1122 = -"2211 = -"2222 = 0;
-Dllll = Dfl> -°1122 = Ai211 = Dvfi> -°2222 = D U! (5-6)
v -ohi n _n n Gh3J-
Dh(l-V).-"1212
~
tr,('/2> -"1212—
u> -^1212~
\2 2'
T1 = *(l+v)K]1T, f2 = oc(l+v)K]2T;
Tf1 = *(l+v)Df1T, f2 = oc(l+v)Df2T;
where
ST, SK SK
h-1^* /{«+£ Je«, h-g /P«;ST, SK SK
22
ST, SK
IlT ~
K =
12(A3
Eh
STi SK
f2T=\ JTdt;SK
1D =
SK
Eh3
12(l-v2)'
h is the local thickness and the quantities ft, J{ and fiT, fiT are termed solidityfunctions and solidity-temperature functions respectively. All these quantities
may vary with their location on the surface of the shell. When the shell is
completely solid we obtain the appropriate coefficients from (5.6) by putting
ST^O, /« = /,= !, (t=l,2), (5.7)
and finally when the temperature is constant throughout the thickness we
have in addition
1<t=T, /=iT = 0, (* = 1,2). (5.8)
6. Equilibrium and Compatibility Equations of Heated Elastic
Shallow Shells
In Section 4 we stated an equilibrium equation (4.4) and a compatibilityequation (4.5) which are applicable over a ränge of material behavior. When
these equations, together with (3.9), are supplemented by equations of state,
we obtain a mathematically complete set. For example, if we combine (5.2)with (4.4) and (4.5), we obtain a mathematical statement of the problems of
orthogonally geolotropic elastic shallow shells with lateral loading and with
arbitrary temperature Variation. We proceed by introducing (3.8), (3.9), and
(4.3) into eqs. (5.2), and reducing them to the forms
dx-f 8x2' \8x22 dxj2
+ [B][K]-i{T1T2}-{f1T2},
{i12M12} =
2 -D1212
-"-1212
2/D 51212\L 1 ^1212 v 1
.\ A1212/
1"
-"-1212
-"1212
-"1212.
f 82u3 82F
\8x18x2 8x18x2
(6.2)
(6.3)
23
When (6.1), (6.2), and (6.3) are substituted into (4.4) and (4.5), we obtain the
equilibrium and compatibility differential equations in terms of the dependentvariables u3 and F. Since later applications of the present work are restricted
to cases in which the elements of the matrix [B] are zero, the equilibrium and
compatibility equations derived explicitly below are confined to this case.
Putting [B] = B1212 = 0 in (6.1), (6.2), and (6.3), and substituting the result-
ing equations into (4.4), we obtain for the equilibrium equation
8x2 \ 11U8x2+ 11228x2)+ 8x,8x2\ m28xx8xj
+ —¥ U>2211—|+D __| = q-—l--—l + ———— {x3 + u3) 6.4
8x2*\ 8xy* 8x2*/ 8xxl 8x2* ox-f 8x2l
,82F 82
, ,82F 82
,
8x18x28x18x2 8x2l 8xx2
The corresponding compatibility equation derives from (6.1), (6.2), (6.3),and (4.5).
82 1= 82F = 82F\ 82 ( 1 82F \
8x1*\Kma8zl*+ mi8xa*) +8xx8x2\K1212 8xx8x2)
82 / - 82F - 82F\ 82 - - - ~
+ '8x^[Kn228~x^ + KllllIx^j=
~Jx~* {K*211 Ti + K*222 T*>
82<K T a-K
Tu( 8*U* V d^382u3(6'5)
-^(Kllllll+ Klli) +
\8x^8x~J ~8x^8x/
82x3 82u3 82x3 82u3 82x3 82u3
8xx2 8x22 8x22 8xx2 8x18x28x18x2'
where Kijkl are the elements of the [K]-1 matrix.
Equations (6.4) and (6.5) constitute two partial differential equations in the
unknown quantities u3 and F, which apply to elastic orthogonally seolotropicShells of varying stiffness with lateral loading and with temperature gradientsover the surface and through the thickness. Membrane and flat plate theories,
with and without temperature gradients, can be derived as special cases of
(6.4) and (6.5). Because of their similarity to the von Kärmän equations for
the bending and stretching of flat plates, there are numerous known techniquesof Solution, and a comparatively simple approach is thus provided to the ana-
lysis of shells which can be regarded as shallow.
In order to complete our statement of the partial differential equations of
the general shallow shell theory, we state the boundary conditions pertainingto rectangularly shaped shells. Boundary conditions along a shell edge fall
into two categories; conditions on displacements and on applied forees. We
24
Fig. 6.1. Boundary conditions on shell edge.
refer to fig. 6.1 to establish the additional notation needed to specify boundaryconditions. The three components of applied force per unit length along a
boundary are designated by Nu, Ny, and Bt, and the applied moment per
unit length by Mu. Similarly, the three components of prescribed linear
displacement are designated by üu, ü^, and üi3, and the prescribed slope by8 ü3\8 xt. The applied forces Nu and N^ and the prescribed displacements uu
and My are in planes parallel to the xx — cc2 plane, and the force and displace¬ment components R{ and üi3 are parallel to the x3 axis.
In accordance with the classical theory of plate boundary conditions in the
absence of transverse shear strains, as formulated by Kirchoff [8], we construct
the boundary conditions on the edge stress resultants parallel to the x3 axis
by means of
Ra = Va +^, (6.6)R^V1 +8M^dx„ Bx1
where Rt are effective edge stress resultants in the x3 direction, and Vi are
given by (3.6). The complete boundary conditions for edge stress resultants
parallel to the x3 axis are
N-8
11dx1
8
' 8x9
N12—(x3 + u3) + N22— (x3 + u3) +
8xx 8x9
22+ 2^M^=Rtf
(6.7)
8x2 8xx
where we have made use of (3.5) as well as (3.6). If the plate has free corners,
25
it is necessary to consider, in addition to (6.7), corner forces of magnitude2Mi2.A summary of the various possible edge boundary conditions may be
represented by the following table:
Force
boundary conditions
Displacement
boundary conditions
Face 1
Nu = Nn
N12 = N^
y,+d^-Bi8x2
Mu = Mu
or
or
or
or
ul — ^ll
^2 = ^12
M3 = M13
8u3 8ü3
8xx 8xx
Face 2
N22= ^22
»«. = »„
8xx
M22= M22
or
or
or
or
M2 = M22
Ul = U21
U3 = M23
8v3 8ü3
8x2 8x2
There may be mixtures of the force and displacement conditions listed in
the table. However, for a given boundary, either a force or its correspondingdisplacement must be prescribed, but never both. There are, of course, other
possibilities, such as elastically supported edges, and prescribed forces or dis-
placements on curvilinear boundaries. The mathematical forms appropriateto these may be deduced from the above.
For completeness, we state formulas for the stresses in skin laminates
parallel to the mid-plane. The normal and shear stresses, expressed in terms
of the stress resultants and couples, are respectively
Ki *ffl} = [E] [Z]-1 ({#11 ^22} + (Ä T2}) + £ [E] [D]-1 ({Jfu M22} + {Ti T2})T
,„ , . „ ..
.,(6-8)
1-VjV;ÄK+"2«2) E2 («2 + "l al)}>
and
where
"I2 = ö( "12 1 r -^"12K
1212 -^1212/(6.9)
EEi Ei v2
E2 "1 E2
26
The equations of this section, which are restricted to deformations in the
elastic ränge, are applicable only when the combined stresses are contained
within a suitably defined yield surface. When the Mises-Hencky theory of
yielding [11] is applied, the elastic theory is valid so long as
^i + <32-»ii"ffl + 3tä« + «'i{ + «l£)^o(Z,i). (6-10)
where ^(Tj) is the yield point of the material in a uni-axial tensile test con-
ducted at an appropriate temperature Tx.
7. Variational Conditions of Equilibrium of Heated Elastic Shallow Shells
It will be useful for reference in the future to adapt the variational con¬
ditions of elasticity to the problems of heated elastic shallow shells. We assume
that the shell planform is rectangular, that it is loaded by surface tractions
q(x1,x2), and that it is subjected to a temperature distribution T(x1,x2, £)variable over the surface and throughout the thickness. The edge boundaryconditions are divided in two categories. Over certain regions designated byS-l, the edge forces and moments per unit length are prescribed, and over
certain other regions designated by S2, the edge geometrical constraints are
prescribed.
7.1. Principle of Minimum Potential Energy
The principle of minimum potential energy has its origin in the first law
of thermodynamics expressed in the form of the principle of Virtual work. The
latter states that if a body is in equilibrium under the action of prescribedexternal forces, the work (virtual work) done by these forces in a small additio-
nal displacement compatible with the geometrical constraints (virtual dis¬
placement) is equal to the change of internal strain energy. We assume that
the shell is initially in a state of equilibrium, and that infinitesimal virtual
strains 8ei;- and S/cy are imposed which are compatible with the prescribeddisplacement boundary conditions on S2. Then we obtain by the principle of
virtual work the result that
a b
$ $(N118in + N228£22 + N128e12 + Mn8K11 + M228K22 + M128i<l2)dx1dx2 =—a —b
ab a
J $q8u3dx1dx2+ J-a—b ~a
f) Ä qii
N218u1 + N228u2 + R28u3-M22[
8x2
b
dx1 (7.1)-b
b
+ J-6
Nn 8ux + N12 8 u2 + i?x S u3 -Mn——^
a
—a
27
The principle of minimum potential energy is formed from (7.1) by rewritingit as
8ij, = 0, (7.2)
where ttp is termed the potential energy of the System. It is defined by
__ ßu b
N21u1 + N22u2 + R2u3-M22^ dxx
du*
ab a
"p= J" \{A-quJ)äxxdx2- \~a ~b —a
b
f-6
Nu ux + N12 u2 + R1u3-Mn
(7.3)
dx2,
where we assume the existence of a function A of the strains ei3- and k^ with
the special properties of a perfect differential that
8A-Kr
dA
and 8A=N1heu + Nn&im + N12hiu + M11SKU + MnSK„ + MubKia. (7.5)
The function A may be regarded physically as the isothermal mechanical
energy per unit of shell surface area expressed in terms of the strains. Byintroducing the equations of State (5.2) it can be verified that A is a positivedefinite quadratic function of the strains which must have the form of
A =
^ (Äull eu + 2 K1122 en i22 + a2222 e22 + K1212 12 + D1VYl ku + 2 x^1122 Kn k22
+ D222242 + D121242-2T1in-2T2i22-2T1K11-2T2K22), (7-6)
if (7.4) are to be satisfied. It is evident that the above statement of the princi-pal of potential energy depends upon the existence of a function A having the
special properties required by (7.4). Such functions can be constructed for
certain equations of State of conservative Systems; however, their existence
cannot always be assured for other more general equations of State [12].
Equation (7.2) states symbolically the principle that of all the admissible
displacement functions ux, u2 and u3 which satisfy the prescribed displacementboundary conditions on S2, the displacements which also satisfy equilibriumand the force boundary conditions on 81, are selected by the extremum con-
dition of a functional irp. If we introduce (3.8), (3.9) and (4.3) it can be verified
that the Euler differential equations of (7.2) are the previously derived equa¬
tions of equilibrium (4.1) and (4.4). In addition, there is produced as a by-product, the force boundary conditions summarized in the boundary condition
table of section 6. It is apparent that in the principle of minimum potential
energy, the assumed quantities are the strain-displacement relations (3.8) and
(3.9), the equations of state (5.2), and the displacement boundary conditions
on S2. The derived quantities are the differential equations of equilibrium(4.1) and (4.4), and the force boundary conditions on 81. Besides serving as
28
another means of deriving the differential equations of equilibrium and the
force boundary conditions, equation (7.2) can play a very important role in
obtaining approximate Solutions by the Ritz method.
Although the above development of the principle of minimum potential
energy is based upon the principle of Virtual work, it can be derived also from
a more fundamental theorem within the framework of the thermodynamics of
reversible Systems. It has been pointed out by Hemp [13] that in the case of
three dimensional linear elastic Systems with temperature gradients, the strain
energy density in terms of the strains usually employed in the principle of
minimum potential energy can be replaced by the free energy. This is an
application of the thermodynamic theorem which states that in the case of
infinitesimal isothermal reversible changes of state with respect to a positionof equilibrium, the work done by the external forces is equal to the change in
free energy [14]. The latter theorem differs from the principle of Virtual work
in that it embodies the second law of thermodynamics as well as the first, and
it Substitutes the free energy for the internal strain energy. A necessary con-
dition on its application is the requirement that the rate of loading be suffi-
ciently slow so that there is no sensible change in the original temperaturedistribution which existed before loading. The latter requirement is consistent
with the employment of elastic coefficients of isothermal straining as defined
by (5.1)*).It is shown in Appendix B (equation B.12) that the free energy F per unit
of surface area of an elastic shell is given by
F = A + j [J,Cedri-riJ'ee^]d£, (7.7)
-Ä/2 n t0
It is apparent that when A is replaced by F in (7.2), the result is the same as
previously indicated since we assume that T is not varied in the variational
process.
7.2. Principle of Minimum Complementary Energy
A second variational principle, the principle of minimum complementaryenergy, involves variations in the stresses instead of the strains. In derivingthis principle we postulate the existence of a positive deflnite quadratic func-
tion B of the stress resultants and couples Ntj and Mi3, with the properties
SB dB,
.
dNit' M8Mit
4) As contrasted, for example, with the elastic coefficients of adiabatic straining which
are only slightly different.
29
The function B may be regarded physically as the isothermal mechanical
energy per unit of shell surface area expressed in terms of the stress resultants
and couples. By applying the equations of state (5.2) it is evident that B must
have the form of
B =
1
Kmi N*, + 2 Zllffl N„ N22 + Jf2222 NL +1
L2222iY 22 T rr ^'12^1212
N*+Dim m 11
+ 2Dn22MnM22 + D2222Ml2 + Ti Ml2 + 2(KullT1 + K1122T2)Nn (7.9)-^1212
+ 2 (iT2211 7\ + K2222 T2) N22 + 2 (Dlin f1 + D1122 T2) Mn
+ 2 (-^2211 -L l+-^'2222 * 21 ^22
if (7.8) are to be satisfied. KiiU and Dijkl are the elements of the [K]_1 and
[D]_1 matrices respectively.
If we make arbitrary small changes in the stress resultants and couples,we have the following change in the function B :
BB = e11hN11 + i228N22 + i12hN12 + K11hM11 + K228M22 + K128M12. (7.10)
Integrating (7.10) over the surface, we have for the entire shell
3 J $Bdx1dx2 = f J(6USi^u + ea28^M + euS^12 + #eu8Jfu—a —b —a —b
+ k22 8 M22 + k12 8 M12) dx1dx2,(7.11)
We introduce (3.8) and (3.9), integrate by parts, and make use of equilibrium
equations (4.1) and (4.2) and the force boundary conditions on 8X. By this
process, we can reduce (7.11) to the form of
8 (-*c) = 0 (7.12)
where irc is a functional termed the complementary energy of the systemdefmed by
a b
ttc =— J" J"Bdx1dxz + J"—a —b —a
N„ U„ + Ni2 Wo, + RnUo Mmdx9 dxv
+ fcu.
Nuun + Nu u12 + Rx un - Mu-
(7.13)
dx2.
Equation (7.12) states the principle that of all the admissible stress func-
tions Ni} and My which satisfy equilibrium and the prescribed force boundaryconditions on St, the stress functions which also satisfy compatibility and
the displacement boundary conditions on S2 are selected by the extremum
condition of a functional ttc . Thus, in the principle of minimum complementary
30
energy, the assumed quantities are the differential equations of equilibrium(4.1) and (4.2) and the force boundary conditions on £1; and the derived
quantities are the stress-displacement relations (3.8), (3.9) and (5.2), and the
displacement boundary conditions on S2. The principle of minimum comple-
mentary energy may serve a useful purpose in verifying the correct stress —
displacement relations and displacement boundary conditions. However, like
the principle of minimum potential energy, it has practical utility when used
in conjunction with the Ritz method, where one now assumes stress modes
instead of displacement modes.
The transformation of the functional ttp into the functional ttc is known as
Friedrichs' transformation [15], and when a function A is assumed to exist,
the function B is related to it by the formula
B = N11e11 + N22i22 + N12i12 + MuKn + M22K22 + M12K12-A. (7.14)
It has been stated also by Hemp [13] that in the case of three dimensional
linear elastic Systems with temperature gradients, the strain energy densityin terms of the stresses usually employed in the principle of minimum comple-
mentary energy can be replaced by the Gibbs' function. It is shown in Appen¬dix B (eq. B.22) that the Gibbs' function G per unit of surface area of an elastic
shell is given byA/2 r, T.
= -£ + J [jcd^-T^c,^G
h/2
dt,. (7.15)
If we use (7.15) to replace B by G in (7.12) the results are seen to be unchangedsince we assume that T is invariant in the variational process.
8. Energy Criteria for Stability of Heated Elastic Shallow Shells
In examining the behavior of elastic shallow shells with temperature gra¬
dients when subjected to large deformations there is often found more than
one equilibrium position for a given loading and temperature distribution.
The differential equations of equilibrium and compatibility can determine the
several equilibrium positions, but cannot distinguish which position the shell
will actually assume. Such selection can sometimes be made by physical
reasoning or it can be made in a more formal way by applying the energy
criterion of Karman and Tsien [16]. This criterion asserts that the most prob¬able equilibrium state is the state with the lowest possible energy level. We
adapt this criterion to heated elastic shallow shells by considering the two
limiting cases of prescribed displacements and prescribed external forces.
31
8.1. Energy Criterion for Prescribed Displacements
We apply the criterion of Kärmän and Tsien to heated elastic shallow
shells by identifying the energy U of a System having prescribed displace¬ments with
U = [ \Adxxdx2, (8.1)— a — b
where A is the isotermal mechanical energy per unit of shell surface area
expressed in terms of the strains (cf. 7.6).
8.2. Energy Criterion for Prescribed External Forces
When the shell is loaded by external forces, the potential energy of the
loading must be included in calculating the total energy level. Thus, the
energy level U' for prescribed external forces is given by
_ _
gu b
N21u1 + N22u2 + E2u3-M22-ab a
U' = U — J" §qusdx1dx2— J*—a —b —a
*3
''8x9dxx
-b
b
~ i-b
_ du a
N11u1 + N12u2 + R1u3-Mn^ dx2.v%l -a
(8.2)
U' may also be expressed in the alternative form
U' = - f jBdx1dx2, (8.3)-a — b
where B is the isothermal mechanical energy per unit of shell surface area
expressed in terms of the stress resultants and couples (cf. 7.9)5).
8.3. The Role of External Disturbances
Questions of selecting the most probable equilibrium state from among
several possible equilibrium states are of importanee in evaluating the design
buckling load for elastic shells subjeeted to temperature gradients and largedeformations. The classical buckling load may be of only -academic interest
since it is possible for the unbuckled strueture to jump, during the loading
process, to an equal or lower energy level buckled equilibrium position before
the classical load is reached. The probability of such a jump depends on the
magnitude of the external disturbances. The classical or upper buckling load
assumes the existence of infinitesimal disturbances, a condition not usually
6) The analogous forms of U, the free energy, and total internal energy on the one
hand and U', the Gibbs' funetion, and enthalpy on the other, are evident by comparing(8.1) and (8.2) with the development in Appendix B.
32
realized in practice. On the other hand, there may be a lower buckling load
defined by the principle of minimum total energy. Such a buckling load requiresthe existence of finite disturbances which cause the structure to jump from an
unbuckled to a buckled State when a condition is reached where the energy
levels in the two states are the same. The order of magnitude of the requireddisturbance depends upon the "energy hump" separating the two "energy
troughs" or equilibrium positions, however, it is not possible to State preciselywhat this magnitude is.
9. Finite Twisting and Bending of Rectangular Elastic Plates with
Chordwise Temperature Gradients
We consider a class of two-dimensional problems involving the finite
twisting and bending of long cambered elastic plates of rectangular planformwith properties which are invariant along the spanwise axis, but permitted to
vary chordwise. Although such problems do not take account of the finite
length features of low-aspect ratio lifting surfaces, they are nevertheless im-
portant initial steps in understanding the behavior of lifting surfaces under
conditions of heating and finite deformation.
9.1. General Theory
The general theory for the case of a longitudinally stiffened cambered plateof length 2 a and width 2 6 is considered first. The plate is constructed of
elastic isotropic material, and it is subjected to a chordwise Variation in tem¬
perature. The temperature distribution throughout the thickness is assumed
constant. The plate is loaded by pure twisting and bending moments, desig¬nated by Mt and M respectively. Fig. 9.1 illustrates the axis System and other
dimensional notation. The plate thickness, camber and temperature, repre-
sented respectively by h(x2), x3(x2), an(i T (x2), are assumed invariant with
respect to the lengthwise or xx direction, but vary in the chordwise or x2
direction. In fact, these quantities are taken as even function of x2 so that
h(x2) = h(-x2), x3(x2) = xz(-~x2), T(x2) = T{-x2), {-b^x2^b). (9.1)
We seek Solutions for the internal stresses as well as formulas which express
relations among the twist rate, curvature, twisting moment, bending moment,
and the temperature distribution. This application is an extension of the case
of an unheated uniform solid flat plate treated by Meissner [17] and, in fact,
some of the special functions derived there are applicable here.
33
Fig. 9.1. Orthogonally stiffened cambered plate of variable thickness.
The pertinent differential equations derive from (6.4) and (6.5), and when
it is assumed, in addition to (9.1), that the stress distribution is of a two
dimensional character, that is F = F(x2), they are
11dx^
K 'dx2 8xx28x2
+ Dfa(2-v)d^u»
8 Xj2 8 x22
+82
8x22
8x„2
( = 82u3 r,j82u3\ B2F B2us
[vDf28x7+Df^)^q
+8^8xi;
<x(l+v
(9.2)
82F
Mhh(\-^Uh)8x<
82
8x22 i-"2/»//,(r-»hlh)T
+ ( S2u3 \
\8 xx 8 x2J
2 82u3 82u3 82x3 82u3(9.3)
Sxj2 8x22 8x22 8xx2'
l-v
where? \d^l 1^
ST, SK
-ST, SK
The boundary conditions on (9.2) and (9.3) along the free edges can be
satisfied exactly, whereas along the loaded edges they must be satisfied in
only an average way. The boundary conditions are the following:
34
For x2 = ±b,
a) N22 = 0, b) N21 = 0, c) R2 = 0, d) Jf8g = 0. (9.4)
For xx = ±a,
b
a) \Nlxdx2 = 0,~b
c) $N12dx2 = 0,-b
b) § Nux2dx2 = 0,
d) $R1dx2-4:M12(b) = 0;
(9.5)
6 &
a) $R1x2dx2-4:bM12(b) = Mt, b) $[Mn + (u3 + x3)Nn]dx2 = M. (9.6)-b -b
Foliowing Reissner, [17], the form of the deflection shape is taken as
u3(xt, x2) = d x1 x2 - \ k xx2 + W (x2), (9.7)
where 6 is the rate of twist, Je is the curvature in the xx direction, and W (x2)is a function to be determined which describes the chordwise deformation. An
explicit aecounting for the chordwise deformation is one of the main features
of the present analysis which extends it beyond the linear bending and twistingof beams with rigid cross sections. When (9.7) is introduced into (9.2) and
(9.3), we obtain
d:
d2 (n,d2W\ hc
w[Dhdx7)-vk- Ca Üb JCn(9.8)
d2
dx22
1 d2F 1 d2
lEhf1(l-v2f2lf1)dx22j \-v dx2
1 /„ ,ePW 1d2x3\1 — vz \ d x2l d x2J
Integrating (9.9) twice gives
l (i-^Mi) J(9.9)
—t=Nn + kW + lcx„
(9.10)
where A0 and Ax are constants of integration. Substituting (9.10) into (9.8)
yields a total differential equation in W analogous to that of a beam on an
elastic foundation, of the form
35
-Ehf1(l-v2^\(A0 + A1x2)k,
(9.11a)
or alternatively a differential equation in Nn
d2 f^= d2
dx22
+
VEhhV-^UjfJ"
d2 (r)?d2x3\ d2 \ =
+ jfcaj^u = (vjfc2 + 0!dx22
d2 *(l+v)(T-vfal]1)T
l-"*/.//l
(9.11b)
We can work with either (9.11a) or (9.11b), and their explicit Solutions depend
upon the form of the functions h(x2), x3(x2), and T(x2), as well as the space
variations of the solidity functions and the nature of the dependence of the
material properties on the temperature. The Solution to (9.11a), or alterna¬
tively the Solution to (9.11b), contains four additional constants, and the
total of six constants, including A0 and Alt can be evaluated by boundaryconditions (9.4) and (9.5). When explicit Solutions of either (9.11a) or (9.11b)are available, relations among rate of twist, curvature, twisting moment,
bending moment and temperature distribution can be computed from bound¬
ary conditions (9.6). The latter can be rewritten in the following forms more
convenient for Integration:
Mt = 0 $Nnx22dx2 + 4:b(l-v)Df2(b)6,-b
M
0
-b
v/2XT2+ Nn (w + x3)\dx2.ldx2
(9.12)
(9.13)
Formulas (9.12) and (9.13) may be evaluated by means of the Solution to
either (9.11a) or (9.11b) and with the aid of (9.10). The subsections which
follow illustrate the evaluation of these formulas for some specific structural
arrangements.
9.2. Longitudinally Stiffened Fiat Plate of Constant Thickness
As an initial application, we select a flat longitudinally stiffened plate of
constant thickness. In addition to the assumption of constant thickness, we
assume that the solidity functions are constants and that the physical pro-
36
perties of the material are independent of temperature. A parabolic chordwise
temperature distribution is taken, as follows:
T(x2) = Tmc + ATg(x2), (9.14)where
-(*)"Tmc is the mid-chord temperature, and A T is the temperature differential
between the mid-chord and the outer edges. This particular temperature distri¬
bution is selected for convenience in analysis and because it is representativeof that which can be expected in a supersonic lifting surface in accelerated
night. The methods described here may, however, be applied to any reasoablychosen temperature distribution.
a) Solution of the differential equation
The general differential equation (9.11a) now becomes
= d*W 1
D'Uj^+K^BW = --K1k0*x2* + K2kT-K1(l-v*)(Ao + A1x2)k, (9.15)«)
where
Kl - EKk^Jf), k2 = EkjlX{^myWhen T has the form of (9.14), the Solution of (9.15) is
W — Cx cosh ß x2 cos ßx2 + C2 sinh ß x2 sin ßx2 + C3 cosh ß x2 sin jS x2
+ Cisinhßx2cosßx2--—^ + j^T ~^-(Aü +A^x2),(9.16)
where
ßm4"
The constants in (9.16) are evaluated by means of boundary conditions (9.4)and (9.5). Boundary conditions (9.4c) and (9.4d) can be rewritten in the
equivalent forms
a) aW{±b) = 0' b) dx^{±b)==vk'' (9J7)
providing N12 = N22 = 0, as required by boundary conditions (9.4a) and (9.4b).
6) The differential equation in terms of the stress resultant, Nn, has the simpler form
^11+ I^1N -o
37
In view of the nature of the boundary conditions and the fact that T is an
even function of x2, we conclude that W must be an even function of x2 and
that C3 = Ci = A1 = 0. The constant A0 may be evaluated from boundary con-
dition (9.5 a), however, its explicit evaluation will not be required. The remain-
ing constants Cx and C2 are computed from (9.17). Substituting (9.16) (with
Cä = Ci = A1 = 0) into (9.17) yields the following simultaneous equations in
C, and C9:
Cx (cosh fj, sin /x + sinh xt cos fi) + C2 (cosh xt sin /x— sinh /x cos ix) = 0,
(71sinh/xsin/x-C'2cosh/xcosxt =
-^\T+vk~K~khA ]
'
(9.18)
where p, = ßb. Solving (9.18), we find
Cx = -+vk-2K2
c U^wk 2K* at\
sinh fj, cos /x— cosh /x sin /x
sin 2 /x + sinh 2 /x
sinh ti cos /x + cosh xi sin xt
(9.19)
sin 2 ti + sinh 2 xx
All boundary conditions are satisfied by the combination of (9.16) and (9.19),with C3 = Gi = A1 = 0, except those corresponding to the loaded edges.
An insight into the nature of the lateral bending deformation may be
obtained by Computing W for the case of pure bending of a solid plate without
twist or temperature gradient. Fig. 9.2 illustrates curves of W for various
0.08
0.06
0.04
0.02
-0.02
-0.04
Fig. 9.2. Lateral bending deformation of solid plate (0 = T = 0, v— 1/3).
38
values of /jl, computed from (9.16) with 9=T = 0 and v=l/3. For very low
values of jj,, the plate assumes an anticlastic curvature of approximately vk,
as predicted by the elementary theory of beam bending. However, for largevalues of fi, the anticlastic curvature is virtually cancelled over the central
portion of the plate and the lateral deflection is concentrated near the edge.In fact W (b) approaches a constant value of approximately 0.1 h for largevalues of fi. The cancellation of the anticlastic curvature is due to the flat-
tening effect of the radial component k Nn of the stress resultant N1X. This
feature of the lateral bending behavior of plates has recently been discussed
by Ashwell [18] and by Fung and Wittrick [19].
Substituting (9.16) into (9.10), gives the following result for the spanwisestress resultant:
Nn = K1k(C1 cosh ß x2 cos ßx2 + G2 sinh ß x2 sin ß x2), (9.20)
where C1 and C2 are given by (9.19) and C3 = Ci = A1 = 0. Inserting (9.20) into
boundary condition (9.12), we obtain the following relation among twist rate,
curvature, temperature differential and twisting moment:
M,
where
- =W1 + T^iM +i^-^)' (9-21)
(\ l-v l-v /x4 /
^jcoshl^x-cos^
±Df2
is a function of the curvature and where
L 4 A A t\
A T is the temperature differential required to produce thermal buckling when
the chordwise temperature gradient is parabolic.
AT =
«AVi^OL.. (9.23)24«6*(1+v)/1(t-v/2//1)
Mt is a reference twisting moment defined by
Mt= tDß''h2^=. (9.24)K3 6/^/137/^
The quantity Mt has the physical significance of twisting moment per unit
of twist rate according to the St. Venant torsion theory.
When the explicit expression for Ay is substituted into (9.21), the latter
contains four principal terms. The first term is a linear function of twist rate
according to the St. Venant torsion theory. The second term represents the
39
.3. Bending moment vs. curvature curves of uniform solid rectangular platesfor various temperature differentials.
40
#-" V = 1/3AT
U
6
4
2
n.
M
\2 sA 2.8S323.
5.7854 V
-2
-4-
-6
2 -8
0 \\
6 ^
(h)
M-2.0 v - X
0 12 3 4 5 6
.3. Bending moment vs. curvature curves of uniform solid rectangular platesfor various temperature differentials.
41
effect of the spanwise stress resultant due to bending, and the third, the effect
of the spanwise stress resultant due to twisting. Both of the latter effects are
non-linear in character. The final term represents the effect of the spanwisestress resultant induced by the temperature differential.
A second relation among curvature, twist rate, temperature differential
and bending moment is obtained in a similar manner from (9.13).
M^
l-^Ulhl hlh \ 4 d* I V du
where M is a reference bending moment defined by
jgMklÄHi-'tM. (9i2fl)
The latter has the physical significance of bending moment per unit of cur¬
vature according to the Bernoulli-Euler bending theory. Equation (9.25) is
divided into four principal terms. The first term is a linear function of cur¬
vature according to the Bernoulli-Euler bending theory. The second representsthe effect of the spanwise stress resultant due to bending. The third and
fourth terms are a combination of two effects. One is the effect of the span¬
wise stress resultant due to twisting, and the other is the effect of the spanwisestress resultant due to the temperature differential. It is apparent that the
latter three terms owe their existence to the chordwise bending degree of
freedom, W.
In order to depict the nature of the Solutions represented by (9.21) and
(9.25), we consider their behavior when applied to a solid plate Fig. 9.3 illustra-
tes families of curves which show the Variation of M/M with /la2 for various
values of A2 and A TjA T. Since bending curvature is proportional to /u,2, these
curves, which are odd functions of ja2, represent the Variation in applied bendingmoment with curvature. Fig. 9.3a gives the results for ATjAT = 0, which
correspond to the curves of Meissner [17]. In fig. 9.3a we see that a very
small addition of A2 (proportional to twist rate, d) stiffens the plate slightly,but that additions of large values of A2 reduce the stiffness for small values of
curvature and increase it for large values. Fig. 9.3b through 9.3h show the
influence of temperature differential, represented by the parameter A TjA T.
Here we find a tendency for the curves corresponding to small values of A2 to
exchange places with those corresponding to large values of A2, as the para¬
meter A T\A T is increased. It can be observed that regardless of the value of
A TjA T, there can be found a twist rate which will cause the plate to have
42
a bending stiffness equal to the value possessed under conditions of pure
bending without twisting or heating.
Beyond certain critical values of A2, there occurs a transition in the platebehavior from a monotonically increasing moment-curvature relation to a
jump phenomenon. These critical values, Ac2, are given by the following for¬
mula obtained by putting the first derivative of M/M with respect to n2,evaluated at p,2 = 0, equal to zero.
Formula (9.27) yields a Single real value of Ac2 for AT/AT<1 and two real
values for ATjAT^ 1. The plate has theoretically no bending stiffness at the
origin for values of Ac2 defined by (9.27). Above the critical values of Ac2, there
may occur three possible equilibrium positions corresponding to a given value
of MjM. The stability of the plate in these various positions is examined in
the next subsection.
Fig. 9.4 gives results for the solid plate which illustrate the Variation of
MtjMt with A2 for various values of fj.2 and A T\A T. Here we find a Single
equilibrium position for all values of \x2 when A T\A T^=l. At A T\A T = 1. the
curve of MJMt versus A2, corresponding to /j? = 0, has a horizontal tangent at
the origin; that is, the torsional stiffness vanishes at the origin. At highervalues of A T\A T, the torsional stiffness at the origin vanishes at higher values
of ju.2 and, in fact, the critical values, p2, required to produce vanishing torsional
stiffness are given by the transcendental equation
4 AT H-c !-"
Above the critical values, fxc2, we find a torsional jump phenomenon analogousto that of the bending phenomenon. At very smail values of A2 and A T\A T,
there is little influence of bending curvature on torsional stiffness. For largervalues of A2, there is a reduction of torsional stiffness with added curvature.
The effect of temperature differential is to reduce the torsional stiffness
appreciably at small values of A2 and jx2, but to exert only a small influence
at large values of A2 and p,2. The departure of the present results from the
linear theory may be observed by comparing the solid and dotted lines of
fig. 9.4.
b) Stability of the various equilibrium positions
We have seen in the preceding subsection that under certain conditions of
loading and temperature gradient, there are three possible equilibrium posi-
43
32
M,
TT,
28 to—1 /ir—W / / / / / ""
24 f' 12 jf-j13 —ff-H j// AT
20IS -\\l44-jt6urrrr1 >•%
16
12^
y
8
4
0 ^
£
1 1 1 1
"
Linear Jheory
±i\ i i i i
10
32
28
24
20
16
12
8
4
0
M,
10—// /
u—ff / //// ""
M* i3 ;/t7u —il-l j
III §"»iS 4L i:« |7//
^
- ^Linear 0)9
S *-'*'
±\210
2 4
Twisting moment vs. twist rate curves of uniform rectangular plates for
various temperature differentials.
Fig. 9.4. Twistmg moment vs. twist rate curves of uniform rectangular plates for
various temperature differentials.
45
46
tocorrespondpointsminimumAe2,A2>ofvaluesforthatseewe9.5a,fig.to
example,forReferring,Um.ofvaluesminimumbyrepresentedareequilibrium
stableofConditions9.5.fig.byillustratedasju2,vs.Umplottingbyportrayed
becanstabilityforconditionsTheUm.symbolthebyrepresentwewhich
length,unitperenergymodifiedarepresents(9.31)ofsideleft-handThe
45
(9.31)A8
+A4
^\\AT!\l+"/W
(l-vX/^ATyyFAn)+
T
45+FM
A4
jy1+v
2vF1(Vl)
(ATv45
(AT\245/l-v\
:a.+JL^a.145
A41+v
T2dx„J1-v2
fEhe212
+UWh2
633=U„
yieldtorearrangedbecan(9.30)Equation
(9.30)2A4|
+/u4+
/**
FiM
ATIAT-")^A4+(x(1-,')W,AT\2'/45„,AT^»-£<.
1-+
FiM+x9d2
J1-1/CT
Eh«2U
,AT~\.,45
..
,.f,1Dh2\,
f_9lEhoc2
result:followingtheplate,rectangularsolidtwisted
andbentoflengthunitaforobtainwe(9.29),inintegraltheEvaluating
(9.29)—6a—
Adx1dx2.J"J—U
isexpressionenergyapplicablethe
displacements,prescribedareconditionsboundarythewhichincasetheFor
(8.1).Sectionofcriterionenergytheapplyingbyplatetheofpositionslibrium
equi¬varioustheofstabilityrelativethewayformalaindeducecanWe
disturbance.aofapplicationby
otherthetoonefrom"snap"tocausedbefact,inmay,andpositionsthose
ofoneeitherassumemayplateTheslope.positiveahascurvethewhere
curvatureofvaluesminusandplustocorrespondingpositionsequilibrium
possibletwobemaytheremoment,bendingappliedofvaluegivenafor
However,region.thiswithinequilibriumstableofpositionaassumetoplate
theforpossiblephysicallynotisItcurves./x2vs.MjMtheofslopenegativea
bycharacterizedregionforbiddenaisthereA2>AC2,forthatseewe9.3a,fig.
toexample,forrefer,weIfstable.arepositionsequilibriumwhichcase,
simplerelativelythisindeduce,toreasoningphysicalbypossibleisIttions.
finite values of /x2. It can be verified that the values of /j,2 represented by these
minimum points are identical with those values of fi2 where the M\M vs. /j,2curves intersect the abscissa in fig. 9.3 a. Since the plate tends to seek a positionof minimum energy, it is thus apparent that for values of A2 > Ac2, it can be
caused to "snap" alternately from one "energy trough" to the other. The
order of magnitude of the disturbance necessary to cause "snapping" is
measured by the depth of the "energy trough". The influence of chordwise
temperature differential on the stability can be observed by studying figs. 9.5b,
9.5 c, and 9.5 d. In general, this influence is to reduce the possibility of "snap¬
ping" at high twist rates and to increase it at low twist rates.
c) Theory and experiment
Experiments were conducted on three modeis in order to verify the theo-
retical results described in subsections a) and b) above. Descriptions of the
experimental apparatus and of the three modeis, designated as modeis 1,2,
and 3, are given in Appendix A.
Fig. 9.6 illustrates the comparison between theory and experiment obtained
from model 1. Figs. 9.6a and 9.6b plot applied twisting moment versus twist
rate for temperature differentials of A TjA T = 0 and 1, and for applied bendingmoments of MjM = 0 and 5. Figs. 9.6 c and 9.6d plot applied bending moment
versus curvature for the same temperature gradients and for applied twistingmoments of MJMt = 0 and 7. The theoretical curves in fig. 9.6, which are repre¬
sented by the solid lines, are not directly readable from figs. 9.3 and 9.4, since
the test conditions are such that the loads are prescribed, whereas figs. 9.3
and 9.4 represent conditions in which the deformations are prescribed. How-
ever, the solid curves of fig. 9.6 can be computed from formulas obtained by
rearranging and combining (9.21) and (9.25) or by a graphical process appliedto figs. 9.3 and 9.4. The latter method was employed in the present investiga-tion. The agreement shown in the plots of applied twisting moment vs. twist
rate (figs. 9.6a and 9.6b) is good, whereas the agreement in the plots of applied
bending moment vs. curvature (fig. 9.6c and 9.6d) is fair.
Fig. 9.7 illustrates other comparisons between theory and experimentobtained from model 2. Fig. 9.7a shows plots of applied twisting moment vs.
twist rate for the three temperature differentials of AT/AT = 0, 0.4, and 1.0,
in the absence of applied bending moment. The agreement is comparable with
that shown in figs. 9.6a and 9.6b. Figs. 9.7c and 9.7d show the response of
the plate to two different load paths, both having the same end point. As with
model 1, the trends of the theory are supported by the experimental data
obtained from model 2.
Initial imperfections in the plate can play an important role in the experi¬mental results, and they are probably the principal reason for differences
47
Fig. 9.5. Modified energy per unit length vs. p? for four temperature differentials.
48
Fig. 9.6. Theory and experiment. Model 1. (Deflections measured with respect to heated
plate.)
49
fa} Torsional Moment vs Twist Rate for
Vanous Temperature Differentials
v (b) Experimentally Estabtished Curves
Showmg the Growth of Twist Rate with
Temperature Differential
(c) Response to a Given Load Path C
°oo o o o o o ° °
4 Theory —»—
Expenmen t o
(Defleetions Measuredwith Respect to
Unheated Plate.1
äM
B
/ 1/.6 4T
%/>:_ V
/D/&T
Load Path
0 5 10 20 30 40 50 60 70
Temperature Differential,Degrees Centigrade
(d) Response to a Given Load Path
Theory ——
Experiment o
(Defleetions Measured with Respect to
Unheated Ptate)
-o u u iu o—u~
„" O o o
00° w Ob—u
4 K
Fig. 9.7. Theory and experiment. Model 2.
50
between theory and experiment. As an indication of the influence of initial
imperfections on the purely torsional action of the plate, we can refer to the
following formula derived from shallow shell theory:
M,
41AT
1 +m4 A4
45 1-v1+3(« (9.32)
where Af represents the initial twist per unit length. It can be inferred from
this formula that as long as A/ is less than approximately +0.15, the results
are within the ränge of expected experimental error of those that would be
obtained if there were no initial twist. In conducting tests on modeis 1 and 2,
the initial twist rate was kept well within this ränge. However, the bending
Fig. 9.8. Idealized temperature distribution.
imperfections could not easily be controlled because of the influence of the
dead weight of the dial gage measuring device. The presence of initial bendingimperfections is probably the principal reason for the relatively poor agree-
ment between theory and experiment shown by figs. 9.6 c and 9.6d.
In preparing figs. 9.6 and 9.7, AT was established both by theory and
experiment, and reasonable agreement was found to exist between the two
results. The theoretical results were based upon the general formula
bh*f2(l-v)AT =
3«/i(1+")(t->'/2//1)[ jg(x2)x22dx2-^- j g(x2)dx2]-b -6
(9.33)
applied to the actual temperature distributions in the modeis. The latter are
reasonably simulated by the idealized temperature distribution diagram of
fig. 9.8. Formula (9.33), applied to the diagram of flg. 9.8, reduces for
Jt = /2 = f2 = 0 and t = 1 to
AT =
h2
GC1-2C2 62a(l+v)!(9.33a)
51
where
1 / 2 \2 / l\2a3a3 a
Ci =2K +fsl +a3[ai + a2 + ^a3\ + ^L + -±-, C2=-£ + a3,
and where ol5 a2, and a3 are the non-dimensional distances shown in fig. 9.8.
The experimental results for A T were based upon simple heating and twisting
experiments. For example, the experimental curves illustrated by fig. 9.7 b
were taken as the basis for the determination of A T for model 2. These curves
are plots of experimentally obtained twist rate versus temperature differential
for various small values of applied twisting moment and for zero-applied
bending moment. It would be expected that a knee would evidence itself in
these curves near AT =AT, and that the rate of growth of twist rate with
temperature differential would differ before and after AT—AT. The sharpnessof this knee depends upon the magnitude of the initial twist imperfection and
the applied twisting moment, with the knee becoming sharper as the imper¬fection and the twisting moment are reduced. The experimental technique
employed was to nullify the influence of the initial twist imperfection bymeans of a small twisting moment arrived at by a cut and try process. It is
shown in fig. 9.7 b that for Mt\Mt = Q.2, the application of temperature differen¬
tial has no influence on twist rate until such time as there is a relatively sharpknee in the curve, which is followed by a nearly straight line of constant finite
slope. The position of this knee may be used to locate approximately the criti-
cal temperature differential AT. A. second method of establishing A T from
the experimental data of fig. 9.7 b makes use of (9.32), rearranged in the
following way:
AT = ATj, (9.34)where
The temperature differential A T is thus a straight line function of the para-
meter, /, and the slope of the line is the critical temperature differential A T.
In fig. 9.9, we have used test data from fig. 9.7 b to construct the functional
relationship of (9.34), and a mean line has been drawn through the data
points. There is fair agreement between the slope of this line and the results
obtained by other methods.
It is appropriate to make a brief remark concerning the effect of tempera¬ture on modulus of elasticity, and its influence on the theoretical and experi¬mental comparisons of modeis 1 and 2. Fig. 9.10 illustrates the effect of tem¬
perature on the modulus of elasticity, E, of 7075-T6 material from which
Models 1 and 2 are constructed. The Variation in E is of the order of magni¬tude of 5% over the temperature ränge of the tests. A mean value of E was
52
400
3.
Model
experiment.
and
Theory
9.11.
Fig.
0.6
0.5
04
03
0.2
0.1
0
AT
oExperiment
Theory
allo
y.minum
alu-
7075-T6
city
,
elasti-
of
modulus
on
temperature
of
Effect
9.10.
Fig.
data.
mental
experi-
from
TA
of
determination
for
construction
Graphical
9.9.
Fig.
300
200
wo
6
6Model
on
Tests
of
Range
Temperatur*
Ol
1UjI«
«I
o
•4M
5Uj
scOl
38
'£
Apb
\y
=Ät
Slope
X\
AO/o
>^
oL/.84
Mt
_
Mt
y°
aro.56
4_
20
40
60
AT°C,
selected in Computing the theoretical curves, however, the Variation of E
over the semi-chord was regarded as negligible.
Fig. 9.11 compares theory and experiment for model 3 tested under con-
ditions of pure twisting at room temperature. Although several elevated tem-
perature tests were conducted with this model, no useful data were obtained
since the bonded joints creeped excessively at the relatively high stresses and
temperatures of interest in the present investigation. The agreement shown
by fig. 9.11 is of some value in substantiating the application of solidity func-
tions to partially soüd sections.
9.3. Longitudinally Stiffened Cambered Plate of Constant Thickness
We select as a second application, a longitudinally stiffened plate of cons¬
tant thickness with parabolic camber distribution, as follows:
(*)" (9.35)
and with the chordwise temperature distribution specified by (9.14). The maxi-
mum camber is represented by ±x3m where the plus sign denotes an upwardand the minus sign a downward camber. Since in shallow shell theory, the
Square of the slope is assumed small compared to one, parabolic and circular
camber are regarded as identical, and we may put also
,2
where r is the radius of camber. This application is an extension of the case
of pure bending of a solid unheated cambered plate treated previously byAshwell [18].
a) Solution of the differential equation
The differential equation (9.11a) assumes the following form when appliedto a cambered plate of constant thickness:
d*W„ ,„Tir
1
-K1k2x3 + K2kT-K1(l-v2){A0 + A1x2)k.
DUj^ +ZiW = -2W*t (9.37)
The Solution is
W = C1cosh.ßx2co8ßx2 + C2$\nhßx2sm.ßx2 + C3co$h.ßx2smßx2 (9.38)
1 -V2
—u (A0 + A1x2),
where we have assumed a positive or upward camber. Applying the boundary
+ Cismhßx2cosßx2—^- + ~^cT-x3m * ~ (y)
54
conditions in a manner similar to that of the previous subsection^ we find
that the constants are
C -—(—+ h2K*
AT2x<>m\ sinh/^cos^-coshMsiPM
1ß*\k Kikb* 62 } sinh2M + sin2/x
n -1 (°2 a. h 2Kz Arr 2x3m\ sinh/xcos^ + coshMsiny.Li-J*\k+VIC-YJWA1 W~) sinh2/Ll + Sin2/i
' (^^}
C3 = C, = A1 = 0.
Substituting (9.38) into (9.10) gives for the spanwise stress resultant
Nn = K1k (C± cosh ßx2 cos jßa;2 + C2 sinh ß x2 sin ßxz), (9.40)
where Cx and C2 are given by (9.39). Putting (9.40) into (9.12), we~obtain the
following relation among twist rate, curvature, temperature differential and
twisting moment:
= X*(l + -^-F1(H.) +^-^), (9.41)\ l—v 1 — v H- /
where
^-»[i-^m^^&JF1*)./SS
A second relation among curvature, twist rate, temperature differential
and bending moment is obtained in a similar manner from (9.13).
M=
2hlh ^iW
(9.42)
The critical temperature differential A T is the same as that of a flat plate,and all other parameters, excepting ATC, are identical to those defined in the
previous subsection.
The nature of the Solutions represented by (9.41) and (9.42) is illustrated
by fig. 9.12. The curves of the latter figure apply to a solid cambered plate of
constant thickness, having maximum camber to thickness ratios, x3mjh, of
0, 1, 2, and 3. Referring to fig. 9.12 a, we see that the influence of camber is to
produce a non-linearity in the curves of bending moment versus curvature,
even in the absence of twist and temperature differential. The degree of the
non-linearity is increased with increasing camber. The non-linearity is pro-
duced by a gradual flattening out of the camber which proceeds with increasing
bending moment and curvature until such time as an instability occurs. The
55
(a)-£L v.s. fi
Fig. 9.12. Curves showing the behavior of a solid cambered plate of constant thickness.
56
Fig. 9.13. Theory and experiment. Model 4.
instability is followed by a period of rapid reduction of bending moment with
curvature and then by a later period in which the behavior is essentially like
that of a flat plate. The effects on bending behavior of adding twist and tem-
perature differential are illustrated by figs. 9.12b, 9.12c, and 9.12d. One
significant effect is to move the bending moment-curvature curves away from
the origin. Other effects are similar to those already observed with the flat
plate. The twisting moment vs. twist rate curves of figs. 9.12e and 9.12f show
that camber has no influence on the twisting behavior as long as the beam
curvature remains zero. For significant values of curvature, the influence of
camber is to reduce the torque required to maintain a given value of twist.
b) Theory and experiment
A comparison of theory with experimental results obtained from model
4 is shown by flg. 9.13. The solid curve in fig. 9.13, representing the theory, is
identical to the curve of fig. 9.12 a, which corresponds to a maximum camber
to thickness ratio, x3mlh, of 2. The circles in fig. 9.13 represent experimental
points obtained from pure bending tests of model 4. The agreement is, on the
whole, satisfactory with one exception. Referring to the upper right-hand
57
quadrant of the figure, we see that theory and experiment are in reasonable
agreement until such time as the theoretical curve has a horizontal tangent.
Beyond this point there is relatively poor agreement, although the two curves
appear to be converging in the upper right-hand corner of the figure.
9.4. Longitudinally Stiffened Cambered Plate of Variable Stiffness
Previous applications of the theory in subsections (9.2) and (9.3) are limited
to plates of constant thickness and homogeneous material properties. However,
actual lifting surfaces have airfoil contours, and hence vary in stiffness alongtheir chord. Moreover, the dependence of material properties on temperature
may produce further variations in stiffness. In many practical problems, the
thickness, camber, material properties, solidity functions, and temperature
distribution are known only in a numerical sense. It is necessary that the
methods of analysis employed in such problems also be numerical or approxi-mate in nature. In a numerical or approximate analysis of long lifting surfaces
subjected to finite bending and twisting, the basic problem to be treated is
one of Computing approximate forms of the chordwise deflection shape W (x2)and the spanwise stress resultant Nn (x2). Once these approximate forms are
available, relations among bending moment, twisting moment, curvature,
twist rate and temperature differential can be obtained by numerical eva-
2luationofboundary conditions (9.12) and (9.13). We suppose that W, d2W/dx22
and N±1 are known numerically at » + 1 stations across the semi-chord of a
lifting surface, the properties of which are all even functions of x2. Then we
may express boundary conditions (9.12) and (9.13) respectively in the follow-
ing approximate forms:
Ml = 26Wiz*lNlli + 4b(l-v)D7a(b)e (t = l...»+l), (9.43)
_ _- d2W- = -
<9-44)M =2kWiDifli-2vWtj^Difat + 2Wi(Wi + xai)Nlu (i = l...»+l),
where Wt are weighting numbers [20] which depend upon the method of
numerical Integration employed. Equations (9.43) and (9.44) can be restated
in the matrix forms
Jf( = 20LVjr^J{^ii} + 46(l-v)£/2(6)0, (9.45)
M =2k[H[^W^]{D]1}~2vd2W _ _
(9-46)rirj{i)/2}+2Lif+^]rifj{^11}.
If we apply Simpson's rule, for example, the weighting matrix, [^JT^J, takes
on the simple form
58
r^o =
1 0 0.
0 0
0 4 0.
0 0
0 0 2
2 0 0
0 0 0 4 0
0 0 0 0 1
(9.47)
where it is assumed that the semi-chord is divided into an even number, n,
of equal intervals A.
The remainder of the present subsection is concerned with one closed form
Solution applicable to an uncambered double-wedge airfoil and two approxi-mate methods of treating cambered plates of variable stiffness along the chord.
a) Closed form Solution for symmetrical double-wedge airfoil
We consider a lifting surface of rectangular planform having a double-
wedge cross section, as illustrated by fig. 9.14. The airfoil is assumed doubly
symmetrical about the x2 and x3 axes, and the thickness is taken as
h = hmh(x2), (9.48)
where hm is the maximum thickness and h (x2) describes the thickness Variation.
In the case of a double wedge airfoil, we have
h(x2) = l-yxjb, (9.49)
hm «egnTTTT—---TTrnpD3—xb . b J
Fig. 9.14. Double wedge airfoil.
where y is a parameter which determines the bluntness of the leading and
trailing edges. The flexural rigidity is
D=EhUl -yx2/b)3
12(l-v2)(9.50)
and we assume that the modulus of elasticity, E, is unaffected by tempera-
ture. Introducing (9.50) into (9.11a), the pertinent differential equation in
W reads
59
60
(9.55)n4Wa^^V/
<J2
„d2JF/<J2
1
is(9.53)offormhomogeneousThe
kyY2\Kimk}*\y/
$]•A+A0Alm"'
x_y2*Dj2v(ff+AT
^lra*
DjsyY2\Klmk)
\KZm/1
"
+\Klmkf2\y}\Klmk2\bj
\[K^ß2(b\2]1[6AJ2(r\2_
+
k=A
1
where
(9.54)Cy2,+By+A=WP
is(9.53)ofSolutionparticularthe(9.14),
bygivenformtheofT,distribution,temperatureparabolicatakeweIf
12(1!
j_va'ß''lm/ll=Ä2mI'1_l/2
-ß/lm/ll=Alm(1_„2)'
j*y.(1-2/)Axb
+A0
where
2
(9.53)
K2mlcyT~{\-v2)Klm+y2)ylce2+\Klm{^\\-2y
toreducesequationdifferentialthe
(9.52)i-y-=y
variableofchangethemakeweWhenb.<:x2^0rängethein(9.51)ofSolutions
ininterestedarewethickness,thein0=x2atdiscontinuitytheofBecause
constants.as/2,and/2,/x,functions,soliditytheassumedhavewewhere
7f^(T-f)(1_''?)ir-Ä^(1-^)(1-''?)(i'+^i'+
(
^7+
(9.51)
-yx2/6)a
12(l-rv'2*da;s"Ldx22i2(l-v2)t*j~»\ldx2[
rd2
\Eh3m(l-Yx2lb)3d2Wd2
?
where
V4^ lliDJ,
We recognize (9.55) as a form similar to that treated by Kirchoff [3] in con-
nection with vibrations of tapered beams and Timoshenko [8] in the analysisof a cylindrical tank with tapered walls subjected to internal pressure. Equa-tion (9.55) can be written in the Operator form
L[L(W)]+PiW = 0, (9.56)
where
^-iu^\Employing a method of reduction due to Kirchoff [3] we rewrite (9.56) in the
following equivalent forms:
L[L(W) + iP*W]-iP*[L(W) + iP2W] = 0,
L[L(W)-ip2W] + iP2[L(W)-iP2W] = 0. (9.57)
Thus (9.56) is satisfied by the Solutions of the two equations
L(W) + ip*W = 0, (9.58)
L(W)-ip*W = 0, (9.59)
and we have reduced the fourth-order equation to two second-order equations.
Introducing the new variables t, = Wyla and r) = 2pyla, we can reduce (9.58)and (9.59) respectively to
fJ2 r AY
^4+r,^-{l-i7,2n=0' (9-60)
^4+7,d~-{1+wn=0- (9-61)
Fundamental Solutions of (9.60) are ViJ^Vir)) and Vi H^ (Vir)) [21] where Jxand .ff]1' are respectively Bessel and Hankel functions of the first kind of
order one. The two linearly independent Solutions of (9.60) are therefore
^ = Re ViJ1(\/iv) + nm Vu^Vi-n),
U = Re ViH^WivJ + iln ViH^(Vir)).
It may be shown [8] that the two linearly independent Solutions of (9.61)
are the complex conjugates of the complex quantities £t and £2, and no new
functions are obtained by solving (9.61). Thus the complementary Solution of
(9.55) can be written in the form
61
W = y-'l- [Ö1 Re fiJx (2 P]/iy) + Ca Im ViJ, (2PYiy)
+ CsRe ]/iH[1)(2p Vij) + ClIm \/iH[1)(2P ]/iy~)l(9.63)
which can be reduced to
(9.64)W = y-1'*[-C1ReJ0'(2p Viy) -C2Im J0' (2p Viy)
-CiBeH$Y(2PYiy')-ClImH$r(2pViy~)],
where J0 and H^ are respectively Bessel and Hankel functions of the first
kind of order zero. The primes denote differentiation with respect to the
argument (2p]/y). Finally, we express (9.64) in terms of Kelvin functions in
the following way:
W = y-HC1bev'(2pyy) + Czbei'(2P\/y)
+ C3ker'(2pyy)+Cikei'(2pyy)].
The functions ber and bei are related to the modified Bessel function of the
first kind of order zero, I0, by
70 (Vi x) = ber x + i bei x (9.66)
and the functions ker and kei to the modified Bessel function of the second
kind of order zero, K0, by
K0(Vix) =kerx + ikeix. (9.67)
The functions I0 and K0 are related to the Bessel function of the first kind of
order zero, J0, and the Hankel function of the first kind of order zero, .ff0(1>, by
70 (Vix) = J0 (i fix), K0 (fix)=~#0(1> (* Vi*)- <9-68)
Combining (9.54) and (9.65), we have for the complete Solution of (9.53)the following:
W = y~'l> [Cx ber' (2 p ]/y) + C2 bei' (2 p Vy) + C3 ker' (2 p Vy)
+ Cikei'(2p\/y)]+A + By + Cy2,
where A, B, and C are the constants defined by (9.54) and Cx through C4 are
constants yet to be determined from the boundary conditions.
We take the boundary conditions, expressed in terms of the original inde-
pendent variable x2, as
dW(0)_Q
3yd2W(0) d3W(0)_
3vyk
dx2'
b dx22 dx23 b'
d2W(b) .
,. 3yd2 W(b)„ ,d3W(b) 3vyk
(9'70)
c)-JxT
= vk' d) -F-nr~{1-r)~d^r=
~T-
62
In terms of the new independent variable y, the boundary conditions are the
following:
a) ü*m-o
c)
dy
d2W(l-y) vb2k
,^d2W{l) d3W(l) 3vb2kb) 3——5 h
dy2 dys
dy2 Td)3 J-r-^ + V-v)
(9.71)
rfy2 iy3
Substituting (9.69) into (9.71), we find that the constants Ct through C4 are
defined by the set of simultaneous equations portrayed in matrix form by
fig. 9.15. In a practical application it would be necessary to invert the Square
matrix shown on the left side of fig. 9.15 numerically for specified values of
the parameters p and y. When explicit values of Ct through C4 have been
computed by means of this inverted matrix, the result for W follows from
(9.69).
W(x2) = (l -y|P [^ber' (2/>j/l -y |) +C2bei' (2p|/l -y °g)+ C3ker'^-|/l-y^ + C4kei'(2/,-|/l-y|)+ A + B(l-y%)+c(l-y^\
The stress resultant iVn is obtained from (9.10) with the following result:
Nu = Klmk^-y^y^C1bev'{2p^l-yfj+C2bei'{2P^l-yfj
(9.72)
+ Co ker
+
'(2"]/1-rf) + CW'(2(>)/i-v?)'.JU^fi-r?)
(9.73)
6* 2K2mATV +
k2 Klmk2b2\
It should be noted that the constants A0 and At (cf. 9.53) are implicitly con-
tained in (9.72) and (9.73). These constants can be evaluated by inserting (9.73)into boundary conditions (9.5a) and (9.5b). Finally, relations among bendingmoment, twisting moment, curvature, rate of twist, and temperature differen-
tial are obtainable by substituting (9.72) and (9.73) into (9.12) and (9.13).Such a complete closed Solution, although possible in principle, would be very
tedious. In a practical case, it would be desirable to evaluate (9.72) and (9.73)
numerically at several chordwise stations and Substitute the results into (9.45)and (9.46).
Other Information relating to the internal stress distribution follows from
(9.72). For example, the result for the spanwise bending moment per unit
63
airfoil.
wedge
double
of
Solution
for
equations
Simultaneous
9.15.
Fig.
(1~y)/a
„2
(£*-
,*)<
..6C
3vb2k2
B-2C
^2
p3(l-y)3/»kei(2p/r^)]
+
6kei
'(2p
l/l-
y)+
Vl-y)
l/T^ker(2p
-6p
1/l-y)]
p3(l-y)3/»ker(2p
+
[4p2
(l-y
)ker
'(2p
/TIy
)
2kei
'(2p
)/T^
;)]
+
^1-y)
j/l-yker(2p
-2p
[p2(l-y)ker'(2pj/l-y)
6kei'(2p)+p3kei(2p)]
+
f4p2ker'(2p)-6pker(2p)
[pker(2p)-kei'(2p)]
+p3(l-y)Vsbei(2pl/l-y)]
j/l-
y)p
bei'
(24-6
/F^y
")|/l-yber(2p
-6p
her'(2pVf^y)
(1-y)
[4p2
Vl-y)]
bei'(2p
2+
Vl-y)
/l-yber(2p
-2p
[p2(
l-y)
ber'
(2/3
/r^)
6bei'(2p)+p3bei(2p)]
+
[4p2ber'(2p)-6pber(2p)
[pber(2p)-bei'(2p)]
Vl-y)
6ker'(2p
+
y)—
)/l
)/l-
ykei
(2p
6p
+
(2p|/l-y)
-4p2(l-y)kei'
[
Vl-y)]
2ker'(2p
+
\'\-y)
Kl-ykei(2p
2p
+
Vl-y)
p(2
-p2(l-y)kei'
[
6ker'(2p)+p3ker(2p)]
+
6pkei(2p)
+[-4p2kei'(2p)
[-pkei(2p)-ker'(2p)]Vl
^y)]
p3(l-y)3/°ber(2p
+_
/l-y)
6ber'(2p
+
^1-y)
|/l-ybei(2p
6p
+
KT^)
[-V(l-y)bei'(2p
Vl-y
)]2ber'(2p
+
l/l-
ybei
(2/o
l/l-
y)2p
+
[-P2
(l-y
)bei
'(2p
ir^y
-)
6ber'(2p)+p3ber(2p)]
+
6pbei(2p)
+[-4p2bei'(2p)
"[-pbei(2p)-ber'(2p)]
4»Ol
width is
M11 = Df1k-Dvf2(£}'{jt-/.{Cx [-p*y bei' (2p Vy) + 2p Vybei (2p |/y)
+ 2ber'(2p ^)] + Ca[p»yber' (2pYy)-2p l^ber(2p J^) + 2bei'(2P |/£)]+ (73[-p2?/kei'(2p ^) + 2p /^kei(2p ^) + 2ker'(2p Vy)] (9.74)
+ <74[p22/ker'(2p Vy)-2p |^ker(2p ^) + 2kei' (2p ]/y)}}
where y = 1 ~yx2jb. In the evaluation of the stress and deformation formulas
recorded above, tables such as those of Dwight [22], are available for the
Kelvin functions and their derivatives. For small and large values of the
argument, 2p ]/y, simplified approximate formulas may also be employed to
evaluate these functions [cf., e.g., 22]. It may also be mentioned that other
simplifying assumptions can be introduced. For example, among these is the
introduction of C1 = C2 = Ci = A1 = 0 in the Solution for W (cf. 9.54 and 9.72),
together with appropriate assumptions relative to the required boundaryconditions.
b) Approximate Solutions derived from the integral equation
We consider next a method of approximate Solution which applies to
cambered cross sections with arbitrary geometric properties that are even
functions of x2. The applied temperature distribution is assumed also to be
an even function of x2, and constant throughout the thickness. The physical
properties of the material may vary with temperature in an arbitrary manner.
In such shells, with inhomogeneous material properties, we can expect thermal
stresses to be induced even under conditions of uniform heating; whereas the
presence of such stresses in shells with homogeneous material properties
requires the existence of temperature gradients. In the previous applicationswe have used the differential equation of equilibrium as a starting point in
obtaining closed form Solutions for the chordwise defiection shape, W, and the
spanwise stress resultant, Nn. However, the integral equation of equilibriumis, in general, a better basis for proceeding to approximate Solutions. This is
due essentially to the fact that the basic Operation performed on the approxi¬mate Solution is an Integration rather than a differentiation. The appropriate
integral equation of equilibrium is derived from (9.8) by performing successive
integrations and introducing the boundary conditions
wm-w.. b> y^ä-o.d*W(b)
_vh _d_ DUfW^ vkdDf2(b)
(9.75)
65
where W0 denotes a rigid translation of the cross section measured to the
center of the cross section. The integral equation obtained in this way is
W-W0 = Dfa(b)vkoL(xs)~Ddf*{b)vkß(xt)+ !C(x2,r}2)(-kNn + vk^—mdr]2,
where <x (x2) and ß (x2) are functions defined by
<x(x2)=xj ^- [xl^l>2 2J DU J Dfa
ß (x2) = x2 j ^=p)dx2 - j ^^x2dx2
and where C (x2, rj2) is an influence function of the following form:
C(x2,V2) = $ l*-Vp-Vd\ („ ^ x2),
0^2
Vt
C(x2,V2) = f(^-A)(x2-A)rfA (^
_
^_
(9.76)
(9.77)
(9.78)
f (^2~A)(a
In the employment of the integral equation, applied to sections with proper-
ties that are even functions of x2, we can restrict our attention to the interval
0 7=x2^.b. By introducing (9.10) into (9.76), we can eliminate W and obtain
thereby an integral equation in the dependent variable Nn, or alternativelyeliminate jVu ,
and obtain an equation in the variable W. The choice is made
here to eliminate W, and the resulting integral equation in jVu has the form
1 6
Nu(x2) = -k2jC(x2,7]2)Nn{r)2)drl2 + vk2Df2(b)<x(x2)Klmh o
_vk2dDj^)ß{xj + vkibc^ ,V2)d2DjMdr]2 (979)ax2 o a7]2
+ f(x2) + [kW0 + (l-v*)A0],where
Ö2r 2 v
i(x2)=°-^ + kx,-^T,
and where we have dropped the constant Ax because of the even nature of the
Solution. The combined constants [kW0 + (l—v2)A0] can be lumped togetherin a single constant which is evaluated by applying boundary condition (9.5a).
66
This yields the result
-[kW0 + (l-v*)A0]=lri—\-k*Sh(z2)$C(z2,V2)Nn(rl2)drl2dx2ßdx2[ ° °
0
b- d DF (b) b-
+ vkiDf2(b)fh(x2)a(x2)dx2-vk2—-^^-jh(x2)ß(x2)dx2 (9.80)0 «#2 0
b_ b d2Df(v) b-
+ vk*\h(x2)$C(x2,yl2)— !2y2'dri2dx2+ß(x2)f(x2)dx20 0 a7l2 0
Substituting (9.80) into (9.79), the integral equation reduces to
1
Nn (x2) = - t»/ö (x2, V2) Nu (r)2) dV2 + F (x2), (9.81)
where
1 b-
0(x2,7]2) = C{x2,7]2)-T-_ jC{r,r,2)h(r)dr$hdx2
F(x2)=v^\Df2(b)ä(x2)-^^ß(x2)+jO(x2,rl2)dL^lAdrl2L ax2 0 arl2
,
+/(*«),
1 6-
a(x2) = cl{x2)-- jh(x2)x(x2)dx2,
jhdx,o
2
1 6-
ß(x2) =ß(x2)--T: Hh(x2)ß{x2)dx2,\hdx2ö
1 *-
/ (*2) = / ix2) - -J2 .f h (x2) f(x2)dx2.\hdx2ö
Equation (9.81) is a Fredholm integral equation in the dependent variable Nnwith a Symmetrie kernel funetion, G(x2, rj2). Closed Solutions can be obtained
in simple cases, however, its usefulness lies principally in its application to
approximate Solutions. An equation of this type can be regarded as the limit-
ing form of a System of linear algebraic equations [23]. We can obtain approxi¬mate Solutions by dealing with this System of equations, and the degree of
approximation will depend upon the number of such equations and the method
by which they are derived. Numerous devices may be employed to reduce
integral equations to simultaneous linear algebraic equations. Among these
may be mentioned collocation methods, Galerkin's method, and numerical
integration by weighting matrices. We select the latter method for the present
application because of its simplicity and precision. Dividing the semi-chord
67
into n intervals, integrating (9.81) numerically, and expressing the result in
matrix form yields
(r-h.
+ F[G][^J){^n} = {n (9.82)
where the elements of the [G] matrix are given by
GU Cij-Lijr^jw
(k=l...n).
A possible form of the weighting matrix, f^W^J, has been reoorded earlier
by (9.47). The following explicit result for the column matrix, {iVn}, is obtained
by a process of matrix inversion:
Fn} = ( dDf2(b)m
(9.83)
where
{*} = ([/]-
{ß} = (uY
{/} = ([/]
){ßh
lij rtrjw
)Otw-fei)'Equation (9.83) can be used as a basis for direct numerical computation of the
spanwise stress resultant. The corresponding result for the chordwise defor-
mation shape is obtained by substituting (9.83) into the following matrix
formula derived from (9.10):
[W} =
lf N,
k\K h jsM-m+IItFJ (l-^2)^ok
(9.84)
Approximate relations among bending moment, twisting moment, curvature,
twist rate, and temperature differential can be derived by substituting (9.83)and (9.84) into (9.45) and (9.46).
The precision of (9.83) can be tested by applying it to the simple case
treated exactly in subsection (9.2). If we divide the semi-chord into two inter¬
vals and employ a two-interval Simpson's rule weighting matrix, (9.83) reduces
to
68
{N11} = K1k(vwhere
[I] =
fl2 2K2AT+T Kxkb2 )4([/]
+ Klk*[G]^W^)-Ax, 62)31
(9.85)
ri 0 0"
0 1 0
0 0 1
[G]=,b3
"0
0
242)
0
1
12
1
"2
[X^J =
b
6
"1 0 0"
0 4 0
0 0 1
The results of evaluating (9.85) for /x2= 1 are illustrated by the three plotted
points in fig. 9.16. The solid line in fig. 9.16 represents the exact Solution for
Nn, as computed from (9.20). Thus, in a test with /x= 1, involving a parabolic
temperature distribution where only two intervals are employed, there is
satisfactory agreement between the approximate and exact results. For largervalues of ja, more intervals would be required, especially in the neighborhoodof the edge.
N„
K,b
03
02
01
0
01
-02
b/2
Exact Solution by Eq (9-20)
Approximate Solution by Eq
(9-831 Based on Two Interval
Simpsons Rule Weighting
Matrix
Fig. 9.16. Approximate calculation of the stress resultant Nu.
c) Approximate Solution by the Rayleigh-Ritz method
Another useful approximate method of Solution is based upon the principleof minimum potential energy applied in conjunction with the Rayleigh-Ritz
process. We consider, as in the previous subsection, cambered cross sections
with geometric properties that are even functions of x2, but otherwise arbi-
trary in character. The applied temperature distribution is also taken as an
even function of x9.
69
The principle of minimum potential energy is applied in the form stated
by (7.2). Considering a unit cross-sectional slice of a long lifting surface of
constant cross section, we write
8ttp= ?8Adx2-Mt8d-M8k = 0. (9.86)—6
The Variation of the function A is expressible in the form
8A = Nn8in + N228i22 + N128i12 + M118K11 + M228K22 + M128Kl2. (9.87)
In the present application we can put
8eu = x2*08d + k8 W+ W8k + x38k, N22 = N12 = 0;
d2WS*ii = 8 k,
8k22 = 8d*wdx22
8*c12 = -280,
Mn = Df1k-Dvf,idx22
= d2WMn = Dvftk-Dft-r-T;
(9.88)
'dx2
M12 = -Dj2(l-v)6.
Combining (9.86), (9.87), and (9.88), we obtain, because of the arbitrary nature
of the quantities 0, k, and W, the following results:
6 bj[Nnx2* + 2Df2(l-v)]dx2 = M„ (9.89)-6
6
j^N^W+xJ +Dhk-DvJt^^dx^M, (9.90)
•>
j[Nnk8W-(Dvf2k-Dj2^y^\dx2 = 0. (9.91)
-b
We observe that (9.89) and (9.90) coincide with the end boundary conditions
(9.12) and (9.13). Equation (9.91) expresses a variational condition for Com¬
puting W. For example, if we expand (9.91), we obtain
-6
b
d2
dx2
{K3?-BH*^L-ft (9-93)
^(D^)-vk-ä^\iwlr0' (9-94)
which correspond to the differential equation (9.8) and the edge boundaryconditions given by (9.75). However, our principal interest is in applying (9.91)
70
to obtain approximate Solutions for W. To this end, we put
W = yt{x2)qt (i=l...n), (9.95)
where yi (x2) are even functions of x2 which satisfy boundary conditions (9.93)and (9.94), and qt are generalized co-ordinates. It is convenient to select the
functions yi (x2) such that
$Klmhyidx2 = 0 (i = l...n). (9.96)-6
Introducing (9.10) and (9.95) into (9.91) and taking (9.96) into consideration,
we obtain the following simultaneous equations:
Ao9j = bi (M = l...n), (9.97)
where we have put Ax = 0 because of the even character of the function iVnand where
b= d2v d2 v b
Aa- SDf2jvijvidx2 + k2 SKimhyiyidx2,
ax9 ^ -b -b
bt = kv \Dj2T-\dx2——- \KXmhyix2dx2-k2 $Klmhx3Yidx2
b
+ & ^K2mhTyidx2.-b
Simultaneous numerical Solution of (9.97) for specined values of k and 8 yieldsthe generalized co-ordinates, qt, and when the latter are substituted in (9.95),we obtain an approximate result for W. The stress resultant, jVu, is obtained
by putting W into (9.10). The result obtained in this way, however, contains
the constant A0. The latter can be eliminated by means of boundary condition
(9.5a). We have as a final result for Nlt,b
a2j .1 &lm"'X2 dx2
Nn = Klmhkyiqi + Klmh— [ x22-=^~b\KXmhdx2
-b
J " l m" XZ d x2
+ Klmhk\ x3-=±-b | (9.98)
\Klmhdx2
\K2mhTdx2
KtnhlT-^=^-b ) (i = l...n).2m
JKlmhdx2
71
Finally, relations among bending moment, twisting moment, curvature,
twist rate, and temperature differential are obtained by substituting the above
approximate expressions for W and jVn into (9.12) and (9.13), or alternativelyinto (9.45) and (9.46). In selecting the functions, yi(x2), there are two general
requirements. They should satisfy the boundary conditions, and they should
be linearly independent. Exact satisfaction of both the free edge boundaryconditions is not absolutely necessary, since the tendency in the minimization
process will be to superpose the functions in such a way as to satisfy these
boundary conditions in the final results. Perhaps the simplest useful functional
form is that of a polynomial expression in x2.
10. Finite Twisting and Bending of Elastic Cylindrical Shell Beams with
Chordwise Temperature Gradients
We consider next a class of two-dimensional problems involving the finite
twisting and bending of long elastic cylindrical beams with cross sections
composed of two identical shallow shell segments joined along a horizontal
plane of symmetry of the cross section. The upper and lower shell segments
are longitudinally stiffened, of uniform thickness, and constructed of elastic
isotropic material. Such beams are representative of thin symmetrical liftingsurfaces with heavy cover plates.
Fig. 10.1. Cylindrical shell beam.
72
10.1. General Theory
We assume that the beam is loaded by pure twisting and bending moments,
denoted respectively by Mt and M, and that the cover plates are subjectedto a temperature distribution constant throughout their thickness and variable
over their chord. The shell thickness, camber, and the chordwise temperaturedistribution are denoted by h, x3(x2), and T (x2) respectively; the latter two
functions being assumed even functions of x2. Fig. 10.1 illustrates the positionof the beam with respect to its axis System and other dimensional notation.
We seek final Solutions, similar to those of the plate problem, for internal
stresses as well as relations among twist rate, curvature, twisting moment,
bending moment, and temperature distribution.
The applicable differential equations derive from (6.4) and (6.5), and when
we introduce (5.6) they read as follows:
1. Top cover
= 8 u3T on7o u3T = 8 u3T
(8*FT 8* 8*FT 8* 8*FT 8* \~
\8x* 8x* 8x18x28x18x2 8x* aV/3 3T
^,g/i-y/,/Ä\ pFt ,U^ft_ -i yttWl^^A1<x*+
\ \-v )8x1*8x%*+f1 Bxf h\ f1)[\8x18xj
(10.1)
&*F,
8x
(10.2)
(10.3)
82u3T 82u3T 8*x3 8*u3T] <xEhf2f_
/a\ g2 T
8x? 1x^~8~x^ 8x^\ 1-v \~VJj8xf
2. Bottom cover
(8*FB 8* 8*FB 8* 8*FB 8* \
\8x{ 8x* 8x18x28x1dx2 8x2* 8x12)y 3 3B'
g%l2/l - v/,//i\ g" Fb,
/, **'b=k11\ . f,\ 17 8*u3B V
8xf+
\ l-v )8x1*8xi*+
f1 8xf h\ 'fJWx^xJ8*u3B 8*u3B 8*x3 8*u3B] «Eh]2(
_
J2\8*T8xS 8x22 8x* 8x* J l-v X Vf1J8x22'
The subscripts T and B refer to the top and bottom covers respectively.The boundary conditions require satisfaction of equilibrium and continuity
along the leading and trailing edges, where they are satisfied exactly. In addi-
tion, they require satisfaction of equilibrium between the internal stress resul-
tants and couples and the external twisting and bending moments at the
(10.4)
73
74
(10.12)2dx22'
d2T.
(10.11)
(10.10)2dx22'
d2T
(10.9)
dx22)dx22*\dx24R
hd2x3\L^hd2WBdiFB_K
JUa
Q
QCadl2q6,=k^f+Vfz^f
*CaQj
Ua.
d^WB]]cd2Fr
+ICdx2)dx2+k
-Kl\°dxfk
diFT-Kle2ikd2WT\kd2xA
-2qe,=
k^f+Dh:^fFTd2diWT-
equations:differentialtotalfollowingtheproducescoversbottom
andtoptheofequationsdifferentialpartialtheinto(10.7)Substituting
(10.8)N12B.=q=-N12T
byresultantsstressshearthetorelatediswhichflowshearaisqwhere
^B(x2),+-gx1x2=i^B
i^y^),+^XjXa=JT
-l-PXjXg-)-lrB(x2),—I^Xj=w3B
j13/nJrpW~\~U/n3C-tv\uC-ttCS"—==
rpIfrq
assectionprevioustheofthosetoanalogousSolutionstakeWe
-b-b
M.=u3B)]dx2+NUB(-x3+u3T)+j[N11T(x3MiiB)dx2++S(MuTe)bb
-b
Mt=M12B)dx2+-$(M12Tb
-b-6
(10.6)V1B)x2dx2+j(V1Tu3B)]dx2++N12B(-x3+u3T)+-5[N12T(x3d)bb
-ft
0;R1B)dx2=+$(R1Tc)b
-b-6
0;=N11B)x2dx2+(NiiT$b)0;=NnB)dx2+j(N11Ta)b6
(10.5)0.=M22B+M22Td)
u3B;—u3Tb)
±a,=xxFor
8x2
8u3B
8x2
8u3Tc)
^11B'—N11Ta)
±b,==x2For
following:theareconditionsboundaryTheway.average
aninonlyhowever,satisfied,areconditionsboundarylatterTheends.loaded
Integrating (10.10) and (10.12) twice yields
d2F
i^If = NnT = K1(^^+kWT + kx3)-K2T +AJ + AJx2, (10.13)
= ^(—l+kWs-kx^-K.T +A^+ A^x,, (10.14)d2 Fn „
dxt*-*11B
(10.15)
where A0 and Ax are constants of integration. Substituting the latter results
into (10.9) and (10.11) gives the following differential equations:
^f2^+K1k^WT = -2qe-K1k(^+ kx^+ K2kT-k(A0T + A1Tx2),
Dhd^+K1k^WB = 2qe-K1k^-kx^+K2kT-k(A0s + A1Bx2).
Solutions to (10.15) depend upon the nature of the functions x3(x2) and T (x2),and each Solution contains four additional constants. All of the constants can
be evaluated by boundary conditions (10.5), (10.6a), and (10.6b). When expli-cit Solutions of (10.15) are available, relations among rate of twist, curvature,
twisting moment, bending moment, and temperature distribution foUow from
boundary conditions (10.6d) and (10.6e). The former can be writtenin aform
more convenient for integration, as follows:
Mt = 2Aq + 2q j(WT-WB)dx2
b
~"
(10.16)
+ 6 j(N11T + N11B)x22dx2 + %bGh3]26,-b
where A is the area enclosed by the top and bottom Covers. .Boundary con-
dition (10.6e) can also be rewritten as
M = ID^bk-DvUdWT dWBY>
dx2 dx2 J_66
(10.17)
+ f[N11T(WT + xa) + N11B(WB-xa)]dxa.-b
10.2. Pure-Monocoque Beam with Parabolic Camber Distribution
In order to obtain some explicit results from our theory, we assume that
the beam is pure-monocoque, that is, without webs, ribs, or other internal
structure, and that the camber distribution is parabolic.
75
^3 -2 "(?)' (10.18)
where tm is the maximum thickness of the section. In addition, we assume a
parabolic temperature distribution identical to that given in the previoussection by (9.14).
a) Solutions of the differential equations
When the camber and temperature distributions are as stated above, the
Solutions to (10.15) are the following:
WT = C1Tcoshßx2cosßx2 + C2Tsinhßx2smßz2 + CsTcosh.ßx2smßx2 /jq ^q\
+ C/smhßx2co8ßx2-j^-^--x3 + j^cT-^-jc(A0T + A1Tx2),
WB = CXBcoshßx2cosßx2 + C2Bsmhßx2sinßx2 + C3Bcoshßx2aiaßxs ,^o 2q\
+ CiBsinhßx2cosßx2 +1^-^^ + xi + 1^lcT-irjc(A(B + A1Bx2).
The boundary conditions needed to evaluate the constants in (10.19) and
(10.20) are (10.5), (10.6a), and (10.6b). Boundary conditions (10.5) can be
rewritten in the following equivalent forms:
a) NUT(±b) = N11B(±b); b) WT(±b) = 0; c) WB(±b) = 0;
d)dWT(±b)
=
dWB(±b)_ d*WT(±b)]d*WB(±b)
= ^L(10.21)
In view of the nature of the boundary conditions, and the fact that x3 (x2)and T (x2) are even functions of x2, we conclude that WT and WB must also
be even functions of x2, and that C3 =Cf = Cf = CB = A[r — AB = 0. The six
remaining constants, Cf, C2, Cf, CB, A^, and AB can be evaluated by the six
boundary conditions (10.6a) and (10.21). Applying the latter in conjunctionwith (10.13), (10.14), (10.19), and (10.20), we obtain the following simultaneous
equations:
-^o ~~ ^o = 0,
(C^+Cj-8) (sinh fj, cos /x+cosh fi sin fi)+(C2T+C2B) (cosh^ sin ^ — sinh /x cos ;u) = 0,
C/cosh/,cos/, + C/sinh/xsin^-(^^ = |^ + ^|2--^(Tm(; + J7'),
CjS cosh /ix cos jLt + C2B sinh p sin /x
^-.lii + ü^j^Mr +AT) (l022)
76
Ui*)
77
relation
theapplyingbyaccomplishedisThis6.rate,twisttheoffavorinq,flow,
sheartheeliminatetoadvantageousisitresults,finaltheformingtoPrior
(10.29)+j^-r2)-ßx28mßx2sinhC2BK1klC1Bcoshßx2cosßx2+=N11B
(10.28)C22'sinhi8x2sin/3x2--=^2-),+Ä'1fc[C,12'coshJ8x2cos^x2=NUT
forms
theassumetheyfact,inandpoint,thisatcomputedbealsocanresultants
stressspanwiseThe(10.27).through(10.23)bygivenconstantstheofmeans
byexplicitlyevaluatedbecan(10.20)and(10.19)functionsdisplacementThe
.^l^+jsr^T^+jr).8012/*+jsinh2/4b2KxkV*+ifc2p.2
~
°0
sinh2p.-sin2p,AT\2K26^h_K1b2k(nA_TA
2/J2"Zjl2"7"
(10.26)2K2AT\\62_;coshfisinp.(
bPLp.p,)cosusmh—
p.sinp.(cosh—
—p-ß—
<mfcoshp-cosp.
(10.27)
+-
2K2AT\~\62
lp,sinp.coshp.+cosp.sinh2q6
coshp.+sinhp.cosp.sin2 ^=
C,B
(10.25)Jr\i2Z202,pcosp,/
t)Pp.)cosp.smh+p.sinp.(cosh—~•^
,...tm
[sinhpsinp.
Ä^ifc2
sinh—
p,sinp,cosh2gö
coshp,sinh+p.cosp,sin=Cx*
+~k~TLk~~b2~)\'\V2~ß2'K^¥+2Ä"2JJ1\]ö2/coshpsinp+sinhp,cosp2q6 (10.24)
p.)cosp,sinh—
p.sinp(cosh+-^J8
£p.cospcosh
p,coshp.+sinhp.cospsin2_r(7
(10.23)
-k-Kfklf)+
\vkAT\2K262
./
2jS2Ä\Ä;2
p)cosp,sinh+p,sinp(cosh+:
p.cosp.sinh—
p.sinp,cosh2q8
ß[p.coshpsinh+pcospsin
iwp,sinp.sinh=<V
areequationsofsetthistoSolutions
-^-.^-|+-^
-&v-=p.coscoshp.ß2C2B)+(0/-
psinpsinhß2CXB)+[CXT2Ä'9J7702
—7-=p)cosp.sinh+psinp(coshß02B)—(G2T+
2L
p.)sinp,cosh—
p,cosp,(sinhßGXB)—(Cj7
§q~ = 2ÄG6, (10.30)
where A is the net cross-sectional area, taking account of distortion. When
the cover deformations are introduced, eq. (10.30) reduces to
ab 6 b
Ohf2 2L 4K T (10.31)
Substituting into eq. (10.31) the explicit expressions for WT and WB givesthe result
AGhf226
!y0, (10.32)
where y is a quantity which reflects the magnitude of cross-section distortion'
as follows:
7 =-i (¥)'.«*>
1+2-1-v
h
tx
A4FiW
(10.33)
F1{fi.) is a function of /x previously defined by (9.22), and F2 (/x) is a second
function of ju. defined by„ .
, ,31 sinh 2 a — sin 2 u,
(10.34)2 /n2 sinh 2/X + sin 2 /x
We observe in (10.33) that y — 1 for no cross-section distortion.
We apply (10.16) to derive a relation among twisting moment, rate of
twist, curvature, and temperature differential. Substituting into (10.16) the
quantities WT,WB, NnT, and JV11B from (10.19), (10.20), (10.28), and (10.29)
respectively, together with (10.32) and appropriate constants, yields the
following result:
where
3/2M\2l
(10.35)
4,_>._«(, _„4£
ia 8 'jji1 +
JsM\2lf, W J
(10.36)
J T is the temperature differential required to produce thermal buckling when
the cross section is assumed distortion-free and when the chordwise tempera¬
ture distribution is parabolic.
AT =
45
321 + 2h(bhV A*(l-v)h
kw*{\+v){T-vuiU)(10.37)
78
Mt is a reference twisting moment defined by
7,i/ h
1 ' h(i-*Mi)'M GWA»jty lt
(1038)
There is strong similarity between the forms of (10.35) and (9.21); the princi-
pal new features of (10.35) being the distortion quantity y and the parameters
bh/A and tmb\A. When (10.36) is substituted into (10.35), the right-hand side
of the latter contains five terms. The first term reflects the influence of cross-
section distortion. When distortion is prevented, y = 1, and this term is simplyBredt's Solution of the St. Venant torsion problem for thin wall tubes. The
quantity y also introduces a non-linear bending-torsion coupling action throughthe mechanism of cross-section distortion. The second term represents the
influence of twisting of the upper and lower plates, and is a finite wall thick-
ness correction effect according to the linear theory. The third term representsthe effect of the spanwise stress resultant due to bending, and the fourth term
the stiffening effect of the spanwise stress resultant induced by finite twist.
Finally, the fifth term represents the effect of the spanwise thermal stresses.
When the curvature and the temperature differential are assumed zero, and
when the cover plates are assumed solid, (10.35) reduees to a result previouslyobtained by Meissner [6].A relation among bending moment, curvature, rate of twist, and tempera¬
ture differential derives from (10.17) by introducing the explicit expressionsfor WT, WB, N11T, and N11B, in addition to the camber distribution given
by (10.18). This relation has the form
^_„2f^)|2MW2 l-v A'[/iWa 1-, A*
jf M /*' l&U7/; (i-v*/2//i) /** l>U7/"i (i-va7.//i) ^
L /2//i \ 4 d^ 1 ^ d>-
1 / 1 dF^
where F3 (ju.) is a third funetion of p defined by
„ .
— 4xt(cosh 2fj.cos2fi+ 1) cosh2it + cos2/xnn 4.0^
3{(l' =(sinh2^ +sin 2^)2
+sinh 2 /x + sin 2 /x
( ''
and M is a reference bending moment as follows:
>)
in
(10.39)
79
M-^ftäV-fM (10.41)2/36 l-»'2
Equation (10.39), as written above, is divided into three principal terms. The
first term, proportional to Fs (ju.)//u.3, represents the effect of the spanwisestress resultant due to bending, and it takes account of the flattening of the
cross-section. The latter phenomenon is well known, and it is often referred
to as the "Brazier effect", since it was discussed first by Brazier in connection
with circular cylinders [24]. The first term is also identical to the result obtained
by Fralich, Mayers, and Reissner, who considered the behavior in pure bendingof a long, monocoque beam of circular-arc cross section [25]. When fi appro¬
aches zero, the first term approaches the Bernoulli-Euler bending Solution of
a thin wall beam with closely spaced rigid ribs. The second principal term,
involving the parameter Ajbtm, represents the effect of distortion of the cross
section due to twisting and it introduces a non-linear bending-torsion couplingaction. The third term, involving the parameter hb/A, represents in part the
influence of the finiteness of the cover skin thickness, that is, the contributions
of the plate actions of the Covers. This term also contributes the effects of the
thermal stresses.
We refer to figs. 10.2 and 10.3 to show the nature of the Solutions repre-
sented by (10.35) and (10.39). Fig. 10.2 shows curves of bending moment
versus curvature for a shell beam having the parameters A\btm = ±\%, bhjA =
0.16, and r=l/3. Figs. 10.2a, 10.2b, and 10.2c are for speeified values of the
twist parameter, A, and for the three temperature differentials, A T\A T, of 0,
0.05, and 0.10. Like the cambered plate of the previous section, we find a
non-linear relation between bending moment and curvature, even in the case
of pure bending. This non-linearity arises from a continual flattening of the
cross section with curvature, which continues until a condition of instabilityis reached. Following the instability, the beam acts essentially like two flat
plates attached together at their leading and trailing edges. The influence of
adding twist is also to Hatten the cross section, which reduces the critical
bending stress required for instability. The addition of a chordwise tempera¬ture differential produces a further flattening of the cross section and reduetion
in critical bending stress. Fig. 10.2d shows the results obtained when the
twisting moment is prescribed instead of the twist rate. The different character
of these curves from those of prescribed twist rate, especially for large values
of twisting moment and twist rate, is evident by comparison.
Fig. 10.3 shows curves of twisting moment versus twist rate for the same
beam parameters. The general character of these curves is similar to those of
flg. 10.2. All the curves are non-linear since there is gradual flattening of the
cross section as twisting moment is added. For small values of curvature, the
80
Parameters
Beam
beam.
shell
pure-monocoque
of
curvature
vs.
moment
Bending
10.2.
Fig.
co
M/M
U/M»
beam.shellpure-monocoqueofcurvesratetwistvs.momentTwisting10.3.Fig.
3-
'"
A'3
btmA3
i.ÜÜ-OKS.V.ParametersBeom
A
18
IS -
M, «^= 02
H
fiio
12
/ 10
8
s 6
i
1// 2
x2
0 1 2 3 i
Fig. 10.4. Influenae of Variation in the parameter bhjA on the behavior of a pure-
monocoque shell beam. AT/AT= 0 A/b t = l/3 v = 1/a.
83
\-0
Fig. 10.5. Curves spanwise stress resultant vs. curvature for pure-monocoque shell beam
A _4btm 3
bh0.16
Ibi DR vs X
'rTTTTTTTTn
Fig. 10.6. Curves of distortion ratio for pure-monocoque shell beam
A _46«?re
_
3
_6äA= 0.16
AT
ät'
84
m/m = o
Twisting Moment vs Twist Rate (AT/AT=0)
Theory
ExperimentoM/rvl=0
dm/R. 07
4 K'
Bendinq Moment vs Curvature (AT/AT = 0)
Theory
Experiment'o Mt/Mt = 0
a Mr/Mt=07
DR vs Twist Rate (,u?=0,AT/AT=0)
Theory
Experiment °
DRvs Curvature (X2 =0, AT/AT=0)
Theory
Experiment o
Fig. 10.7. Theory and experiment at room temperature. Model 6,
85
twisting moment versus twist rate curves exhibit an instability, whereas for
large values of curvature, their behavior is similar to that of a flat plate.The effect of varying the parameter bh/A is illustrated by fig. 10.4. Since
the parameter Ajbtm has a constant value of 4/3 for a beam with parabolic
camber, variations of b h/A are equivalent to variations of h/tm, that is, varia-
tions in the ratio of skin thickness to maximum beam thickness. These results
show that an increase in b h/A has the effect of causing the beam bending and
twisting behavior to approach that of a plate. For example, referring to fig.10.4c, we see that for bh/A = 0A, the bending instability due to flattening is
nearly eliminated when A2 = 0, and when A2 = 4, there has emerged the jump
phenomenon which is so evident in fig. 9.3.
We have indicated by (10.28) and (10.29) expressions for Computing the
spanwise stress resultants in the top and bottom Covers. It will be of interest
to evaluate these expressions more explicitly. At the points of maximum beam
thickness, that is, at the mid-chord, we have the following expressions for the
spanwise stress resultants in the cover plates:
NUTIN,,) 4u.sinhusinu f_
2 lhb\ n//» v . ,. ,.
—.]+-{-,— I J y _ _
2 u (cosh ix — COS a) (csch a — CSC a)
Ul-„)[l+imf]AT1 ,10.42)
A4+
//i/=2(l-v2/2//i) AT*
where Nn is a reference stress resultant defined by
ju.(coth|u —cot/x)']}
N =
EWtJhUl-^hlh) (10 43)11
2)/362 (l-^a)
The shear flow in the beam is given by the simple relation
q = qyX2, (10.44)
where q is a reference shear flow defined by
GAh*f2J /, (10.45)4/36 VUl-^Mi)
86
87
twistingofplotstheinexperimentandtheorybetweendisagreementsimilar
producewillfactorssameTheseränge.post-bucklingtheinexperimentand
theorybetweenagreementpoortheforresponsiblepartlyleastatevidentlyare
theory,theinconsiderednotfactorsjoints,rivetedtheofslippageandsheets,
coverbottomandtoptheofInterferencetogether.pressedaresheetscover
lowerandupperthethatextentthetoflattenedisbeamtheränge,buckling
post-theinpoint,thisBeyondinstability.ofpointthetoupsatisfactory
iscomparisonstheseinagreementTheexperiment.withratiodistortion
theofcurvestheoreticalcompare10.7dand10.7cFigs.data.experimental
thebelowfalltotheorytheoftendencygeneralawithfair,isagreement
Therespectively.curvatureversusmomentbendingandratetwistversus
momenttwistingforexperimentandtheorycompare10.7band10.7aFigs.
tests.temperature
elevatedandtemperatureroombothfromdata10.8fig.andteststemperature
roomfromobtaineddatagives10.7Fig.6.modeloftestsfromobtaineddata
experimentalwiththeoryofcomparisonsshow10.8and10.7Figs.subsection.
previoustheofresultstheverifyingofpurposesfor3e),A.See.(cf.6and
5modeisasdesignatedmodeis,twowithconductedwereExperiments
experimentandTheoryb)
curvature.
ofvaluesseveralforratetwistwithVariationthe10.6bfig.andratetwistof
valuesseveralforcurvaturewithDRofVariationtheillustrates10.6aFig.
pcoshp.sinh+p.cosp.sin
p,cosp,sinh+p.sinp.cosh47)(10\bU^(l-v2]/^)^sinh/*coshpO+p(sinp.cosp.
,4\btJh{i-v2Uh)
Mi-")
M\y+
r—r--~~
—.=DRA4
fi(l—v)
(A\
2sinhpsinp,
asexpressedbecanratiodistortionthethatfindwe
(10.20),and(10.19)deformations,covertheforexpressionstheofmeansBy
'
SectionUndistortedofDepthMaximum
SectionDistortedofDepthMaximum
follows:asdefinedDR,ratio,distortionthetermed
parameteraofmeansbysectioncrosstheofflatteningtheontwistingand
bendingofinfluencetheexaminewetheory,theofconsiderationfinalaAs
resultant.stressspanwisetheondifferentialtemperatureandrate
twistofinfluenceadditionalconsiderableratherthecurvature,withresultant
stressspanwiseofVariationthetoadditioninshow,curvesThese0.2.and
0=TTjAAofdifferentialstemperaturetheforandA2ofvaluesseveralfor
(10.42)fromcomputedpßversusN11Tofcurvesillustrate10.5band10.5aFigs.
AT/CT - 0
oM/M
°-
1.0-
a
0.9-
a
0.8-
0.7-
0.6-
0.5-
0.4-
(a) Bending Moment vs Curvature (Mt/Mt =0) o,3-
Theory
Experiment
AT/ÄT = 0
AT/CT - 0.05
AT/AT =0.10
0.2-
O.i
0
-at/ät = o
AT/5T = 0.05
(b) Bending Moment vs Curvature (Mt/Mt = 0.7)
Theory
Experiment
o AT/ÄT = 0
a AT/AT=0.05
8 |iz
AT/AT=0 °
AT/AT = 01
(M/M)cr (M,/M,),
Ic) Bending Moment vs Curvaturet Mr/M, =0 5)
Theory
Experiment
o AT/ÄT = 0
AT/AT = 0 I
-T—
Bp.'
(d) Influence of Temperature Gradient
on Bending and Twistmg Instability
Ol AT/AT 02
Fig. 10.8. Theory and experiment at elevated temperature. Model 6.
88
moment versus twist rate and bending moment versus curvature. Since the
theoretical results have apparently little value in the post-buckling ränge,
the theoretical curves in figs. 10.7a, 10.7b, 10.8a, 10.8b, and 10.8c are termina-
ted shortly after the point of instability. In figs. 10.8a, 10.8b, and 10.8c, the
quality of agreement between theory and experiment is about the same as in
fig. 10.7. Again we find a general tendency of the theory to underestimate the
experimental data, especially in those cases where twisting moments or tem¬
perature gradients are involved. One possible explanation for these differences
is the fact that the parameters Mt and A T are especially sensitive to the area
A and the semi-chord b. Because of the nature of the riveted construction of
model 6 it was not possible to estimate these quantities with great precisionand errors of the order of 5 percent or less can account for the observed
differences.
Finally, fig. 10.8 d compares the theoretical and experimental influence of
temperature gradient on the critical twisting and bending moments. Here we
find approximate agreement between theoretical and experimental trends.
Because of the limitations of the Mark I heating device, it was not possible to
obtain experimental data beyond A TjA T = 0.2.
In preparing fig. 10.8, A T was established by theory. An experimentalcheck was beyond the capabilities of the heating device. The following generalformula gives the temperature differential A T for an elastic pure monocoque
shell with parabolic camber:
s
Ui
oG 4
Co
UJ
10
.3"3o
Temperature Range of
Tests on Models 1 &2
0 WO 200 300 400
Fig. 10.9. Effect of temperature on modulus of elasticity, 2024-T 3 aluminum alloy.
89
AT = "•+§wn
46a/1(l+v)(r-v/2//1)-b
ö-b
(10.48)
When g(z2)=x22jb, (10.48) reduces to the result for a parabolic temperaturedistribution given previously by (10.37). When we assume the idealized tem¬
perature distribution of fig. 9.8, (10.48) reduces to the foUowing:
AT = 1 + m l*A*(l-v)
W«j1{\+v)(T-Vf2!f1){C1-Ciß)(10.49)
where Cx and C2 are defined by (9.33a). Formula (10.49) was used to computeA T in preparing the theoretical curves of fig. 10.8.
Finally, we refer to fig. 10.9 to show the effect of temperature on the
modulus of elasticity, E, of 2024-T3 aluminum alloy material from which
model 6 is constructed. The Variation in E over the temperature ränge of the
tests was of the order of magnitude of 7 percent. A mean value of E was
selected in Computing the theoretical curves, however the Variation over the
semi-chord was not taken into aocount.
11. Vibrations of Rectangular Elastic Plates in the Presence of Finite
Deformations and Temperature Gradients
We consider as a final topic, small vibrations of rectangular elastic plateswhich are initially deformed by a finite amount and, in addition, subjected to
temperature gradients over their surface.
11.1. Linearized Shallow Shell Theory in the Presence of Temperature Gradients
Prior to taking up a specific application, we consider first a linearization
process applied to (6.4) and (6.5). We set in these differential equations
w3 = %m +^%> F=Fm + AF, q = qm + Aq, (11.1)
where Au3< <u3m, AF< <Fm and Aq< <qm. Linearizing in terms of the
incremental functions Au3, AF and Aq, we obtain the foUowing differential
equations:
90
91
inequationdifferentialaproducetou3AoffavorinreplacedisFAwhere
(11.4)inplacedisresultThisAu3.oftermsinFAfor(11.5)solvingfirstby
SolutionthiswithproceedwouldWequantites.incrementaltheforequations
linearlattertheofSolutionthethenremainsThere(11.5).and(11.4)into
substitutedareresultstheandsections,previousindiscussedalreadylinesthe
alongseparatelysolvedbemay(11.3)and(11.2)Equationsquantities.mental
incre¬theofbehaviorthegovern(11.5),and(11.4)equations,linearizedThe
measured.areFAandu3Aquantitiesincrementalthewhichfromposition
equilibriummeanadefineequationsTheserespectively.qandFu3,replacingqmandFmu3m,with(6.5)and(6.4)tosimilarare(11.3)and(11.2)Equations
W3m)+8Tx^){X38x2
\8282Au3
8 '"(ZJx^dx~28~x~8x~2
+8~x^2~8x2\""^VUMä^V\
(115)8%2d%Aw*I
8"82AU*d2\AF-(+K8"(K82
FAd^tej\K^X2Jx~Jx~2
+FJxjr
Ä2211+8~x^2l^2222"^
\821t82\82—82l—82
U3m'+~8x~Jr~dx^2){X3+8x18x28x18x23+\Jx~^cTx^2~Sie,2}
\8282AF8282AF82182AF\82
8x8x22
82
S2Fm,
(11.4)
8282Fm„
82Fm
\u3
8x18x28x18x2*'
x28x28[+qA~WȊx2)8V2222+
x28r22nx28
ä8~x~Jx~2)rma4r8x^8x~2+ßUs8~x^2)Vl122+Jx~}V^11118~x~2
\82(82\8282(82
8x18x28xx8x2x28x28x28Xj28
u3m8x38u3m8x38u3m8xa8
(11.3)
8xJ~~8xJ\8x^8x~2)+Kll22l2)+
8x22(KllllllU3m8U3m8\U3md/i\7p~v,~mtlr
"
T*>K22+TlJx~}{K22n~=Fm8~x~})Knn+
Jx~}(Zl1228~x~}+
————-82\82—82I—82
Fm~8x~Jx~2)\K^2Jx^8x~2+FmJxl2)K2211+
Jx~}(^2222Jx~2
\821I82\82—82I—82
W3m,•+(^38x2)8x28x18x28x18x2+3x2\8xx2q
\8282Fm8282Fm82(82Fm
(1L2)I>2222~8x^2)U3m
+Jx^2[D2211Jx^+\8282/82
U*mIx^8x~J^12124:8~x~Jx~2+U3mJx~})D+Jx~}r11118~x~}
\82/82\8282I82
terms of the single dependent variable A u3. Solution of the latter equation
completes the problem.
Equations (11.4) and (11.5) may be used as a basis for studying the bucklingor the small Vibration behavior of shallow shells. In the case of the latter,
when it is assumed that longitudinal inertial forces are negligible comparedto lateral inertial forces, we put into (11.4)
Aq = ~lp~8^- (1L6)
where p is the mass per unit of surface area.
In the approximate treatment of small vibrations, it may be desirable to
apply the Eayleigh-Ritz method, in which case we replace (11.4) by a varia-
tional condition. The latter condition is Hamilton 's principle, which we express
in the case of free vibrations by
&J(T-U)dt=*0. (11.7)
where T is the kinetic energy and U the potential energy of the shell. The
latter are defined respectively by
ab ab
T = \[ \p^A2dx1dx2, U = jJAdx1dx2. (11.8)
~~a —b —a ~-b
where A1) is the isothermal mechanical energy per unit of shell area given
by (7.6).The Rayleigh-Ritz method is applied by putting
Au^Qiix^xJqiit) (i = l...»), (11.9)
where ^i {xx, x2) are linearly independent functions of xx and x2 which satisfythe geometric boundary conditions and qt (t) are generalized coordinates. When
we express the displacements in terms of generalized coordinates, Hamilton's
principle reduces to Lagrange's equation which, in the case of free vibrations, is
8_Udt\dqj
'
dqi
Substituting (11.9) into (11.8) and making use of (3.9), (4.3) and (5.2), we can
write the kinetic and potential energies in terms of the stress functions and
displacements in the following forms:
b
£(H)+^=°(i=i-n) (1L10)
T = ^j jp^MJdx.dx, (i = l...n) (11.11)
—a —b
') It is not rigorously correot to employ the quantity A when the changes in strain
are rapid; however the error involved is negligibly small in problems of the type con-
sidered here.
92
TJ-1! [\k (82F-4.82AFY4.9W (8*F J*AF\(PFm d2AF\
2j J L 1111\dxJ+ ~dx^f) +
ZAll22\8xJ+ ^xj)\dxJ + ~8xJ')
~"i (8Hm d2AIY 1 / &Fm 82AF\+
sm\dx1*+
dxj*)+K1212\8x18x2 +Sx^xJ (11.12)8)
+D [-exj- + 8x7qi) + 2D [TxJ-+ 8x^qi) VdxJ-
+ Jx^qi)
\8x22 8x2l ) i\8xl8x2 8xx8x2 jCt OC-t Cb Jüa
(i—l...n)
We proeeed with the Bayleigh-Ritz method by substituting (11.9), in addition
to the known values of uSm and x3, into (11.5). This gives a linear partialdifferential equation in A F, the Solution of which provides a coupling con-
dition between A F and the generalized coordinates qt (t). The latter condition,
together with Fm and uSm, is substituted into (11.12). We introduce (11.11)and (11.12) into (11.10) and linearize the result in terms of the qi{t). The
natural frequencies and mode shapes are obtained from this set of equations
by putting
qj = q.ei<»t (j=l...n) (11.13)
and solving the resulting determinantal equation. In (11.13), qt is the complex
amplitude and a> the frequency of Vibration.
11.2. Torsional Vibrations of an Initially Twisted and Heated Lifting Surface
The problem of the small torsional vibrations of heated lifting surfaces
with uniform initial twist provides a simple application of the approach dis-
cussed in the previous subseetion. We introduce (11.6) into (11.4) together with
x3 + u3m = ox1x2 ._ ..
Au3 = A 6(x1)x2^
where 8 is the initial uniform twist rate and A 8 (x-^ is a function describingthe small additional twisting displacement during Vibration. Assuming that
the plate is stiffened longitudinally with geometric properties that are even
funetions of x2, we obtain
82 / = 8*A8\a
82 fn/1 ,f^01 82 /n ? d2A6\
»tSAe
,tPA 6
„ „ „ „82A8 (11.15)
8) A term in the expression for U, which is a function of T only, has been omitted
since it makes no contribution in the final result.
93
(11.16)
-26$AN12x2dx2 + I0-^r = 0.
Multiplying (11.15) by x2dx2 and integrating over the chord yields9)
^r^^j~
^r s^7/~
^ teJ UmXa 2
üb'
where
EJW = §Df1x22dx2 = warping torsional stiffness
GJS = 2(1 — v)$Df2dx2 = St. Venant torsional stiffness
70 = $ px22dx2 = mass moment of inertia per unit length about the~b
xx axis.
Equation (11.16) is coupled with (11.5) through the term containing A N12,and an exact Solution requires this coupling action to be taken into account
by simultaneous Solution of the two equations. We can obtain an unusually
simple result by assuming that the plate is very long, of constant cross section,
and rewriting (11.16) in the following approximate form:
82A 9,d2A9
QJ» ' +/if | iVllma;2 dx2
dx?" et2V^r
= o. (n.17)tJa\Nllt-b
The quantity within the Square brackets represents a correction to the St.
Venant torsional stiffness which takes account of the influence of the mean
spanwise stress resultant, NUm.
Putting A 6 = A 6eiwt we can obtain from (11.17) the result that the torsional
frequency under conditions of initial twist and temperature gradient is simply
o» 1/1 +
(Tj~ jbNumx22dx2 (11.18)
where con is the frequency of the nth torsional mode in the presence, and w0n
the frequency in the absence of initial twist and temperature gradient. If we
assume, for example, a parabolic chordwise temperature distribution accordingto (9.14), and uniform thickness, we can make use of (9.12) and (9.21) to
deduce that (11.18) has the form of
u0m
AT WKl-t£ + b-^^r92. (11.19)AT 15(l-v)Df2
where A T is the temperature differential and 9 is the initial twist rate.
9) In order to make this reduction we require the additional assumption that
8DhV>
dx., \~bl^D%lh =
[^T =o.
94
Fig. 11.1. Theoretical and experimental Vibration frequencies. Model 2.
A comparison of (11.19) with experimental data is shown by fig. 11.1. The
experimental data were obtained from torsional Vibration tests of model 2.
Electronic shakers attached to the "L" shaped units of the static loadingdevice were used to oscillate the model in torsion about its xt axis. By this
means, a Vibration mode was excited which was for all practical purposes the
lowest fixed-end torsion mode of the plate. The plotted points in fig. 11.1
show the experimentally determined influence of twist rate and temperaturedifferential on this mode. The reference frequency w01 was obtained by experi-ment. The theoretical trends shown by the solid lines are roughly confirmed
by the experimental data although the quality of agreement is inferior to
that obtained from static tests of the same model. The lack of agreement is
probably due in considerable measure to the influence of initial imperfectionswhich proved exceptionally difficult to eliminate because of the effect of the
inherent friction in the shaker devices.
95
APPENDIX A
Experimental Apparatus and Methode
A.l. Loading Device
In order to conduct experimental studies of the behavior of infinite aspectratio heated lifting surfaces under conditions of finite bending and twisting,a special loading device was designed and erected in the laboratory of the
Institut für Flugzeugstatik und Flugzeugbau at E.T.H. The principle of
Operation of the device is illustrated by the schematic diagram of fig. A.l,
and a photograph is shown by fig. A.2. It consists essentially of two "L"
shaped structural steel units, each attached rigidly to the test specimen, and
each supported by a flexible steel cable. The steel supporting cables are sus-
pended from a heavy steel tubulär frame. Loading is accomplished by four
load pans, each attached at the ends of the "L" shaped units. By loading the
load pans symmetrically, any desired combination of pure bending and pure
torsional moments can be applied to the specimen. Since abnormally large
bending and torsional deflections are a feature of the experiments in the
present investigation, it was necessary to carefuUy prevent the induction of
spanwise axial loads in the specimen due to the loading device. This was
accomplished by suspending one of the supporting steel cables from a shaft
inserted through a roller bearing with freedom to roll parallel to the longi-tudinal axis of the test specimen along a grooved track. As a result of the
extra degree of freedom provided by this device, both steel cables remained
vertical during loading. It is visible in the upper central portion of the photo¬
graph of fig. A.2.
The bending and torsional moments applied to the specimen are a function
not only of the weights in the load pans, but also of the angular displacementsof the arms of the "L" shaped units. The angular displacements affect the
loading in two ways. The first, is a result of the shortening of the effective
bending and torsion arms as they displace angularly downward. The second,
is a result of the static unbalance of the "L" shaped units. In order to make
the latter statically stable in a horizontal attitude, they were suspended from
a point above their center of gravity, and balanced in a horizontal attitude
(with the specimen removed) by means of auxiliary balancing arms upon
96
which additional masses were suspended. The latter may be seen in the lower
central portion of the photograph of fig. A.2. As the arms displace angularlydownward, they are no longer statically bälanced, and it was necessary to
apply a correction proportional to the static unbalance. The latter correction,
although insignificant at high load levels, was important in tests on very
flexible specimens. Since the effective bending and torsion arms, and the
corrections due to static unbalance, are dependent on the angular dispositionof the arms, it was necessary to measure these angles carefully during the
course of each experiment. This was accomplished by means of an optical
System in which the image of the point of a needle was projected on a mirror
attached to the loading arm. The reflection from the mirror was directed into
a calibrated scale on the wall of the laboratory. Since it was necessary to
simultaneously record two angles, two of these Systems were employed.
'F.
Fig. A.l. Schematic diagram of loading device.
A.2. Heating Devices
The elevated temperature experiments were all conducted under conditions
of steady-state thermal equilibrium. Temperature differentials were obtained
under steady-state conditions by applying line sources and sinks of heat to
the surface of the specimen. This approach was taken in order to avoid the
necessity of employing the complex heating and data recording equipment
normally required to conduct experiments under conditions of transient
heating.Two devices were used to apply line sources of heat. Both employed the
principle of electrical resistance heating, and both used nickel-chromium alloy
97
o
°>CD
$'503O
t-J
<
98
p~a*«»* i.<*»»«»s*.*i&, t«M?-*v zww v**Z3>,
Fig. A.3. Mark I. Heating element.
strips as the heating elements. The latter is a commercially available alloytermed "nichrome 5", containing 80% Ni and 20% Cr. The first device,
Mark I, is illustrated by the photograph of fig. A.3. It consists of a spirallywound 0.17 mm X 3 mm nichrome strip enclosed in a heat resistant tetra-
fiuoroethylene resin insulating tube. The inside diameter of the spiral windingis 2.5 mm, and the nominal outside diameter of its surrounding tube is 4 mm.
The latter was fabricated from Teflon, a commercially available E. I. du Pont
Co. material normally used for purposes of electrical insulation. The Mark I
device was employed to heat the leading and trailing edges of the lifting sur-
faces. In these applications, the Teflon tubes were held in place by a number
of steel clips, as illustrated by figs. A.3 and A.5. These clips produced negli-
gible stiffness additions to the basic model and provided a simple method of
attachment and replacement of the heating unit. It was found during the
course of the experiments that it was necessary to keep the clips snugly
together in order to prevent local overheating of the Teflon tube.
The second heating device, Mark II, consists of a flat nichrome strip bonded
to a Fiberglas cloth insulating layer. The latter provides electrical insulation
between the nichrome strip and the specimen. The device is illustrated sche-
matically by the sketch of fig. A.4. The nichrome strip is 0.13 mm by 12 mm
in cross section, and the insulating Fiberglas cloth is 0.03 mm thick. Some
difficulty was encountered in developing a bonding technique which providedsufficient strength under the elevated temperature and high strain conditions
of the present experiments. A satisfactory Solution was obtained by applyinga thermo-setting synthetic casting resin to bond the Fiberglas cloth to the
specimen and a thermo-setting synthetic adhesive resin to bond the nichrome
strip to the Fiberglas cloth. Both the casting and adhesive resins are ethoxy-
99
line-class resins manufactured by the CIBA Company of Basle. The former is
Araldite Casting Resin F used in conjunction with Hardener 972. The latter
is Araldite Bonding Resin 1 applied in the form of a stick. In using the Aral¬
dite 1 to bond tlu? nichrome strlp to the Fiberglas cloth, moderate pressure
was applied. The curing times and temperatures which produced a satisfac-
tory bond are summarized below.
a) Fiberglas cloth bonded to specimen with Araldite F and Hardener 972;
cured for one hour at 150° C.
b) Nichrome strip bonded to Fiberglas cloth with Araldite 1; cured for four
hours at 140° C.
Nichrome Strip —-^
Araldite / —
Fiberglas Cloth
Araldite F
\ Specimen 1/
Fig. A.4. Mark II. Heating element.
A.3. Structural Models
Six structural modeis were employed to verify the theoretical studies. Since
these studies are for the most part of a two-dimensional nature, the lengthof each model is considerably in excess of its width. The aspect ratios of all
the modeis lie within the ränge of 4 to 5. The modeis are each described in
some detail by the following paragraphs:
a) Model 1
The first model is simply a flat aluminum alloy 7075-T6 plate used in
conjunction with the Mark I heating device. A photograph of Model 1 is shown
by fig. A.5. Since this model was intended to verify theoretical studies of
elastic behavior under conditions of large bending and twisting displacements,a low thickness ratio (1.68%) and a material having a high yield stress were
selected. A chordwise temperature differential was produced by attaching the
Mark I heating elements to the leading and trailing edges of the plate and by
applying a line heat sink along the surface of the plate at its center line. The
line heat sink was obtained by passing tap water through a slotted rubber
100
Fig. A.5. Model 1.
\ \ \ \ \ s/
Fig. A.6. Cross section of cool-
ing water Channel.
r-4"\AA/\/V\A/WWVl-|
Mark I
Heoting Elementst-l'VWVWWVWI-
Fig. A.7. Wiring diagram of model 1.
Fig. A.8. Chordwise tem-
perature distribution of
model 1.
tube bonded to the specimen with Araldite D. The tube may be seen in the
photograph of fig. A.5, and its cross section is illustrated by the sketch of
fig. A.6.
The Mark I heating elements were wired in parallel with the secondarywinding of a step-down transformer, as illustrated by the wiring diagram of
fig. A.7. The primary winding was attached in series with a manually adjus-table auto-transformer and an ammeter to a 220 volt 50 cycle line.
All of the elevated temperature tests of model 1 were conducted with an
edge to center-line temperature differential of approximately 50° C, the criti-
101
Fig. A.9. Model 1 in extreme bending condition.
Fig. A.IO. Model 1 in extreme condition of bending and twisting.
102
Fig. A.ll. Model 2.
Ma-k I
Heoiing Elements
Fig. A.12. Wirmg diagram of model 2.
0 12 3
Current in Primary Wmding of Transformer
Fig. A.13 Calibration curve for model 2 temperature differential.
103
cal buckling temperature differential. The temperature distribution over the
semi-chord is illustrated by fig. A.8. This distribution was, within the limits
of precision of the temperature measuring device, symmetrical about the
center-line of the model. The power input to the transformer required to
maintain the 50° C temperature differential was 580 watts, and the voltageacross the heating elements was 30 volts. Figs. A.9 and A.10 illustrate model 1
under extreme conditions of bending and bending and twisting respectively.The important basic data of model 1 are the following:
b = 75 mm D =- 1117 kg-cm M = 40.6 kg-cmh = 2.52 mm Mt= 61.5 kg-cm Ä~f = 50°C
/i = 7a = /i = /2=l " = 1/3
b) Model 2
The basic properties of model 2 are identical to those of model 1 exceptthat it employed the Mark II heating elements. A photograph is shown by
fig. A.ll. The heating elements were wired in series with the secondary windingof the transformer, as illustrated by the wiring diagram of model 2 in fig. A.12.
Since the model was used in tests where several heating and loading pathswere followed, it was operated over a ränge of temperature differentials from
0° to 70° C. Temperature control was maintained by manual adjustment of
the auto-transformer used in conjunction with the ammeter and with a cali-
bration curve of current in the primary winding versus temperature gradient
(cf. fig. A.13).The basic data of model 2 are identical to those of model 1 except that the
critical temperature differential of model 2 was 40° C. The critical tempera¬ture differentials differ because the heating devices of the two modeis producedifferent temperature distributions. The stiffness properties of modeis 1 and 2
are for all practical purposes the same. A careful examination of the test data
shows a negligible addition of torsional stiffness and less than 5 percent increase
in bending stiffness due to the Mark II heating elements.
c) Model 3
Model 3 consists of a flat built-up sandwich beam fabricated of aluminum
alloy 2024-T3 sheet and strips, as illustrated by the photograph of fig. A.14.
The aluminum alloy sheets are 1.6 mm thick, and the spanwise strips are
4.8 mm square. Eighteen spanwise strips were bonded between the sheets bymeans of Araldite Bonding Resin 1. Mark I heating elements were inserted in
the outermost cells of the beam, and casting resin was poured around them
to provide uniform heat transfer to the model. Cooling along the model center-
line was obtained by passing tap water through the middle cell of the model.
104
The inlet and outlet taps of the cooling Channel are visible in fig. A.14. Ele¬
vated temperature tests were conducted on model 3 at various temperaturedifferentials up to 100° C. A power input to the transformer of 2000 watts
was required to produce the 100°C temperature differential.
The important basic data of model 3 are the following:
b = 87.5 mm D = 36,900 kg-cm M = 5,550 kg-cmh = 8.1 mm Mt= 7,210 kg-cm A~T = 476°C
fx = 0.689 /i = 0.885 v = 1/3
f2 = 0.395 /2 = °-781
d) Model 4
Model 4 is a cambered solid aluminum alloy 2024-T 3 plate with a thickness
of 1.61 mm. This model, which was used to verify a portion of the theoretical
cambered plate studies of Section 9.3, is shown by the photograph of fig. A.15.The testing of model 4 was restricted to room temperature experiments. The
basic data of model 4 are the following:
b = 83 mm D_= 287 kg-cm M = 5.91 kg-cmh = 1.6 mm J/,= 9.00 kg-cm xSmjh = 2
/i=/>7i = /2=l " = 1/3
e) Models 5 and 6
Models 5 and 6 are identical within manufacturing tolerances, and theyconsist of a pure-monocoque cylindrical shell fabricated from 1.6 mm 2024-T 3
aluminum alloy sheet. The covers of the shell are formed in the shape of
circular arcs and they are joined at their leading and trailing edges by 2.4 mm
diameter A17S-T4 rivets spaced 7 mm apart. Two such modeis were cons-
tructed. The first, designated as model 5, was used in early experiments to
assess the model behavior and its strength. The second, designated as model 6,
was tested to provide the final data given in See. 10.1b). Models 5 and 6 are
shown by figs. A.16 and A.17 respectively. The Mark I heating device was
used in the elevated temperature tests of model 6, and two cooling Channels
were employed, as shown by the photograph of fig. A.17. Mid-chord to edge
temperature differentials slightly in excess of 100° C were achieved with this
arrangement.The important basic data of modeis 5 and 6 are the following:
b = 85 mm bh/A =0.160 D = 287 kg-cm M = 382 kg-cmh = 1.6 mm Ajbtm = ±ß_ 3f,= 508 kg-cm AT = 464°C
tm= 7.42mm ]1=]t = f1 = fs=l v =1/3
105
I t
&
•3O
106
i *$*
0>
TSO
> 7/T
•8o
3
107
A.4. Measuring Devices
Since the experiments conducted during the course of the present investiga-tion were of a steady state nature, simple measuring devices could be employedto measure deformation and temperature.
a) Devices for measuring curvature and twist
Two principal devices were employed to measure beam curvature and
twist rate. The first, which was applied to modeis 1, 2, and 3, consists of an
assembly of four dial gages mounted on a post which was attached rigidly to
the center of the model. The photograph of fig. A.18 shows this assemblymounted on the underside of model 3. Earlier figures, e. g. A.9 and A.10,
illustrate other views of the dial gage assembly. The dial gage readings are
interpreted as deflections measured with respect to a plane tangent to the
deformed surface of the model at the origin of coordinates xx and x2 Thus, if
the dial gages are numbered as shown by fig. A.18, the curvature and the
twist per cm of length are given respectively by
k~49
-
49* (All)
6=hzh =?izli. (A.2)98 98
K '
where 8^ is the deflection of the ith gage and where the coordinate gage loca-
tions (in cms) with respect to the xx and x2 axes are as follows:
Gage 1, (-7, 7), Gage 3, ( 0, -7),
Gage 2, ( 7, 7), Gage 4, (-7, -7).
The second device used for measuring curvature and twist rate of modeis 4,
5, and 6, employed finely graduated glass scales suspended from the leadingand trailing edges of the modeis. The curvature and the twist per cm of lengthwere computed from the relative deflections of the scales, the latter beingmeasured by means of an engineer's transit.
Various other methods were employed to check the curvature and twist
results measured by the devices described above. Eor example, the reflected
light beam optical System used to measure the angular disposition of the
loading arms provided an accurate check on the curvatures and twist rates of
the unheated specimens. In the pure bending tests of model 6, curvature was
measured by measuring the angle between two long rods rigidly attached to
the edge of the model 30 cm apart. These rods may be seen in the photographof model 6 shown by fig. A.17.
ioa
Fig. A.18. Dial gage assembly.
6j Device for measuring temperature
Steady state temperatures were measured by a portable probe type NiCr-Ni
direct reading thermocouple device manufactured by the Elmes Staub and
Company of Richterswil.
APPENDIX B
Functions of State of an Elastic Shell Element
Besides the equations of state, we can obtain some additional Information
by considering the behavior of an elastic shell element in terms of thermo-
dynamic variables. This Information relates to the application of energy prin-
ciples to the conditions for equilibrium and to the nature of the thermoelastic
coupling between the processes of heat transfer and elastic deformation.
Specifically, the free energy and the thermodynamic potential or Gibbs' func-
tion are of interest in this connection. Hemp [13] has previously stated expres-
sions for these functions applicable to three-dimensional elastic bodies.
When heat and external forces are applied simultaneously to an elastic
body, there is a change of internal energy. We consider first, a unit volume
of the shell material. The first law of thermodynamics requires that
du = dq + atJdei:/ (i,j =1,2, 3), (B-i)
109
where du is the increment of internal energy and dq is the mechanical equi-valent of the heat supplied, both in the time interval d t. The final term on the
right hand side of the equation represents the work done on the unit volume
by the surrounding medium1). For the change of entropy ds we have, also in
the time d t,
ds=Y[--T(Jiidii (*»?' = 1.2,3) (B.2)
where T1 is the absolute temperature of the volume element. Since u is a
function of the strains and the absolute temperature, d s can also be expressedin the form
de{j (»,,-=1,2,3) (B.3)
The second law of thermodynamics requires that cisbea total differential in
Tx and ei;-, and as a result
Introducing (B.4), together with
(B.5)(ft).-Vwhere cf is the heat capacity per unit volume under conditions of zero strain,
into (B.3) yields
6)
Integrating and applying the initial conditions s = 0 for Tt = T0 and ei}= 0,
we obtain for the entropy per unit volume
2\, 0
The increment of internal energy per unit volume may also be regardedas a perfect differential in the variables Tx and ey. It is computed from
du=T1ds + aijdeij (4,7 = 1,2,3), (B.8)
Introducing (B.6), integrating and assuming w = 0 for T1 = T0 and e^= 0, we
obtain for the internal energy per unit volume
x) The strain components are defined such that the expanded form of the final term is
"ij dcij = °iideu + o22d e22 + <r33 df33 + cr12d en + a13 d <r13 + ct23 d ei3
110
Ti Uj «(/
= jctdT1-T1j(^jdeii + jaijdeij (i,? = l,2,3) (B.9)
In the usual treatment of three dimensional elasticity, the strain energy is the
last term of the right hand side of (B.9). This term is clearly not equal to the
internal energy, even if the body is strained at constant temperature. In fact,
by inserting (5.3) into (B.9) we observe that the strain energy can be equatedto the internal energy only if the straining is isothermal and if a and d E^d Txare zero. The strain energy is, however, equal to the external work requiredto strain the body isothermally. The other terms in the expression for internal
energy represent the heat required to keep the temperature constant.
A third function of state, called the free energy /, is defined by
f = u-T1s. (B.IO)
It is regarded also as a function having the independent variables of absolute
temperature and strain. Inserting (B.7) and (B.9) into (B.IO), we obtain for
the free energy per unit volume
T\ Tx U)
f=jcfdT1-T1jcf^+jaiidetj (m = 1,2,3) (B.ll)
T. T, 0
It is evident from (B.ll) that the strain energy is equal to the free energy
provided the straining process is isothermal. The free energy, F, per unit of
shell surface area is obtained by integrating (B.ll) over the thickness and
introducing (3.7), (5.3) and (5.4). This yields
F=j [ j cdTi-T^ ct^]dt+A (B.12)
-A/2 T„ T.
where A is the isothermal mechanical energy per unit of shell surface area
previously defined by (7.6). We perform the Integration leading to (B.12) by
making use of the fact that d W =
atj d etj is a perfect differential where
"-1 ell T, e22 + i
.. ..
lle22+Cre12"1"2
1,[^i(a1 + v,«,)eil + ^,(ag + v1a1)6|J + ^[Äsri6j1 + ^srsj8] (B.13)
— T [ESTl a.STl en + EST'i asr2 e22]
The functions of state of entropy, internal energy and free energy derived
above, are functions of the variables of absolute temperature and strain. Other
111
functions of state can be derived in which the independent variables are abso¬
lute temperature and stress. One such function, is the enthalpy or heat content,
h, defined byh = u-aiieij (t\/=l,2,3). (B.14)
Combining (B.2) with (B.14) gives
dh=T1äs-iidoij (i,j = l,2,3). (B.15)
If we express the differential of the entropy in the form
d8 = ea^+^jdaii (»,? = 1,2,3) (B.16)
we find that
dh = c0dT1 + T1(~^doij-*ijd<jij (*,,- = 1,2,3) (B.17)
where c„ is the heat capacity per unit volume under conditions of zero stress.
Integrating (B.16) and (B.17) we obtain expressions for the entropy and
enthalpy per unit volume in terms of the absolute temperature and the stresses.
T\ an
S=SCa^7 + i {^)ad<Jii (M = 1.2,3) (B.18)
T„ 0
Ti oh otj
h=je0dT1+T1j^jdati-jeiidatj (m = 1,2,3) (B.19)
Finally, another function of state in the variables of absolute temperature and
stress, called the thermodynamic potential or Gibbs' function [14], is formed by
g = h-T1s, (B.20)
where g is the Gibbs' function per unit volume. Inserting (B.18) and (B.19)into (B.20) yields
g=jcadT1-T1j c0^-j eijdatj (»,7 = 1,2,3) (B.21)
T0 T0 0
The Gibbs' function G per unit of shell surface area, obtained by integrating
(B.21) over the thickness, has the form of
ft/2 Ti 3\
G=\[ \ c.dTt-Tij ca^d£-B (B.22)
-Ä/2 T„ T„
where B is the isothermal mechanical energy defined by (7.9).
112
From the previous discussion we can derive some Information concerningthe mutual infiuence of straining and heating. From (B.7), for a small changeof temperature T, the change in entropy per unit volume is
s = c<%-\ (ä?)/ey ^?=1'2'3) (R23)
0
and the heat absorbed by a unit volume is
|^)de« (»,? = 1,2,3) (B.24)
o
Under conditions of adiabatic straining, which is approximated by rapid
loading, the infiuence of strain on temperature can be derived from (B.24) as
Making use of (5.3), we observe that if the elastic moduli are unaffected by
temperature, rapid positive strains produce a slight cooling and rapid negativestrains a slight warming. If the terms involving a are neglected, and if the
moduli of elasticity Ei are affected by temperature, we find that the sign of
8 T/8 etj is the same as the sign of 8 EJ8 T1.The equations of state (5.1) through (5.4) apply by definition only in the
case of isothermal straining, that is, when the rate of straining is sufficientlylow so that the temperature of the element is continuously adjusted to that
of its surroundings. For other rates of straining, the equations of state assume
a different form. For example, the equations of state for adiabatic strainingof an elastic orthotropic material can be derived by substituting the value of
T obtained from (B.24) by putting q — 0 into (5.3). The results obtained in
this way are, however, only slightly different from the isothermal equationsof state.
References
1. Bisplinghoff, R. L., Some Structural and Aeroelastic Considerations of High-Speed
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April 1956.
2. Westergaard, H. M., Theory of Elasticity and Plasticity, Harvard University Press,
Cambridge, Mass., 1952.
3. Todhunter, I. and Pearson, K., A History of Elasticity and Strength of Materials,
Vol. II, Part II, Cambridge University Press, 1893.
4. Marguerre, K., Zur Theorie der gekrümmten Platte großer Formänderung, Pro-
ceedings of the 5th International Congress of Applied Mechanics, pp. 93—101, 1938.
113
q = ceT-T0 j (
5. Reissner, Eric, On Some Aspects of the Theory of Thin Elastic Shells. Journal of
the Boston Society of Civil Engineers, Vol. XLII, No. 2, pp. 100—133, April 1955.
6. Reissner, Eric, On Finite Torsion of Cylindrical Shells. Proceedings of the 1 st Mid-
western Conference on Solid Mechanics, pp. 49—51, 1953.
7. Flügge, W. and Conrad, D. A., Singular Solutions in the Theory of Shallow Shells.
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8. Timoshenko, S., Theory of Plates and Shells. McGraw-Hill Book Company Inc.,
New York, 1940.
9. Dow, N., Libove, C. and Hubka, R., Formulas for the Elastic Constants of Plates
with Integral Waffle-Like Stiffening. NACA Technical Report 1195, 1954.
10. Sokolnikoff, I. S., Mathematical Theory of Elasticity. McGraw-Hill Book Company,Inc., New York, 1956.
11. Hill, R., The Mathematical Theory of Plasticity. Oxford at the Clarendon Press, 1950.
12. Washizu, K., On the Variational Principles of Elasticity and Plasticity. Aerolastic
and Structures Research Laboratory, Massachusetts Institute of Technology, Tech¬
nical Report 25—18, March, 1955.
13. Hemp, W. S., Fundamental Principles and Theorems of Thermoelasticity. The
Aeronautical Quarterly, VII, pp. 184—192, August 1956.
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15. Courant, R. and Hubert, D., Methods of Mathematical Physics. Interscience Publishers,
Inc., New York, 1953.
16. von Kärmän, Th. and Tsien, Hsue-Shen, The Bückling of Spherical Shells by Exter-
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20. Benscoter, S. U. and Gossard, M. L., Matrix Methods for Calculating Cantilever-
beam Deflections. NACA TN 1827, 1949.
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114
Biographical Sketch
R. L. Bisplinghoff was born Feb. 7. 1917 and lived until age 18 in the mid-
western town of Hamilton, Ohio in the United States. After completing his
elementary and high school education in 1934, he attended the University of
Cincinnati for a total of six years where he was awarded the degrees of Aero¬
nautical Engineer and Master of Science in Physics. During school vacation
periods he worked as draftman and stress analyst for the Aeronca Aircraft
Corporation in Cincinnati, Ohio. Following graduation he worked for one
year in Vibration and flutter research for the U.S. Army Air Corps at WrightField in Dayton, Ohio and two years as Instructor of Aeronautical Engineeringat the University of Cincinnati. During the latter period he taught aircraft
structures and aerodynamics, and conducted research on aircraft structures.
During the war period from 1943 to 1946, Mr. Bisplinghoff was a Naval
Officer in the Engineering Division of the Naval Bureau of Aeronautics where
he engaged in research on aircraft flutter, aircraft loads and structural ana-
lysis. Following the war in 1946 he was appointed Assistant Professor of
Aeronautical Engineering at the Massachusetts Institute of Technology in
Cambridge, Massachusetts. In 1948 he was promoted to Associate Professor
and in 1952 to Professor of Aeronautical Engineering. Since 1952 he has been
the Professor in charge of the structures and aeroelasticity divisions of Instruc¬
tion in the Dept. of Aeronautical Engineering, with the additional duty of
Director of the Aeroelastic and Structures Research Laboratory. In 1957 he
was appointed Deputy Head of the Dept. of Aeronautical Engineering.
During the academic year 1956/57 he was granted sabbattical leave for a
year of study at the Swiss Federal Institut of Technology, Zürich. He also
received during this period a National Science Foundation Post-Doctoral
fellowship from the National Science Foundation of the U.S. government.
