A second order formulation for the analysis of slender, elasticbeamsCitation for published version (APA):Frenken, L. P. J. (1985). A second order formulation for the analysis of slender, elastic beams. (DCT rapporten;Vol. 1985.014). Eindhoven: Technische Hogeschool Eindhoven.

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A SECOND ORDER FORMULATION FOR THE ANALYSIS OF SLENDER, ELASTIC BEAMS

L.Ph.J. Frenken

WFW 85.014

Eindhoven University of Technology Department of Mechanical Engineering The Netherlands - January 1985

-2-

ACKNOWLEDGEMENT

The author wishes to take this opportunity to sincerely thank Dr. C.M. Menken for the patience and guidance he generously extended during the course of this investigation. The author also thanks Mrs. Marleen van Boxtel for skillful typing the manuscript.

Lambert Frenken

- 3 -

ABSTRACT

A second order formulation €or the analysis of elastic beams is presented. Shear deformation and distorsion of the cross section is not considered in the present theory. Based on energy considerations the analysis is able to predict the bifurcation-type buckling condition for slender beams taking into account the prebuckling deformation. As an example a closed form solution for the lateral buckling of a simply supported beam subjected to uniform bending is presented.

-4-

TABLE OF CONTENTS

Part

1 . INTRODUCTION

2 . ELEMENTS OF BEAM THEORY

3 . TOTAL POTENTIAL ENERGY

4 . THE MINIMUM POTENTIAL ENERGY CRITERION

5. THE TREFFTZ CRITERION

6 . APPLICATION OF THE THEORY

6 . 1 . Simply supported beam in uniform bending

7 . CONCLUSIONS

REFERENCES

FIGURES

APPENDIX A : THE INDERMEDIATE CLASS OF DEFORMATIONS

APPENDIX B : THE ROTATION MATRIX

APPENDIX C : THE TORSION PROBLEM

9

10

15

19

26

27

27

30

31

3 3

36

3 8

41

-5-

LIST OF SYMBOLS

A

by'bZ

1 r C 2

E

G

IO

I 1 Y'

J

- J

L

M

* 'a

* m a

= cross-sectional area

= monosymmetry parameters

= warping parameter

= amplitudes o f displacement variations

= Young's modulus o f elasticity

= shear modulus of elasticity

= polar moment o f inertia about the shear center

= moments of inertia about the Y and Z axis, respectively

= St. Venant's torsion section constant

= Jacobian

= beam length

= moment

= bimoment

= critical moment

= moments about the X , Y and Z axis, respectively

= bimoment per unit length

-6-

LIST OF SYMBOLS (cont'd)

mx,m rm Y Z

n

= moments per unit length about the X, Y and Z axis, respectively

= vector normal to the surface of the cross section

P

R

'ij

S

O

U

U

= total potential energy

= critical loads for buckling about the shear center, Y and Z

axis, respectively

= forces in the x, y and z-direction, respectively

= loads per unit length in the x, y and z direction, respectively

= rotation matrix

= Piola-Kirchhoff stress tensor o f the second kind

= surface coordinate of the cross section

= polar fourth moment of cross-sectional area about the shear center

= strain energy

= the x component of the displacement of the origin when no warping occurs

= the y and z components, respectively, of the displacement of the shear center

-7-

EIST OF SYMBOLS (cont'dj

u i

r

= displacements in the x, y and z direction respectively of any point in the cross section

= generalized displacement

= coordinate axes

= coordinates o f any point of the beam, prior to deformation

= coordinates o f the shear center prior t o deformation

= angles of rotation about the X , Y and Z axis respectively

= warping constant

yxy :yys_ i

~ X I E y l E Z = strain components

= first and second variations

E

'ij

= axial strain

= Green-Lagrange strain tensor

= linearized deformation tensor

= geometric constant of beam

V = Poisson's ratio

- 8 -

LIST OF SYM3ûLS (cont'd)

is ij

* - UI ij

Superscripts

*

= stress components

= Cauchy stress tensor

= warping function

= liiiearized rotation tensor

* & * i T x

= reference to the local coordinate system x r y z .

Subscripts

The summation index is adopted t o repeated lower case subscripts. Subscripts preceded by a comma denote differentiation with respect to these subscripts.

-9-

1 . INTRODUCTION

Mechanical nonlinearities, such as plasticity effects, more than geometric effects are usually considered to be important in the load-carrying capacity of beams. Nevertheless, in the case of slender beams, buckling occurs in the elastic range and the nonlinear stability problem has not been widely investigated.

In the classical buckling analysis of beams, it is assumed that the prebuckling displacements are small enough to be neglected in the derivation of the governing differential equations [ l - 4 1 . However, for example in aluminium extrusions with open cross sections where the ratios of the major axis flexural stiffness to the minor axis flexural stiffness and the torsional stiffness may be less than three, the actual flexural-torsional buckling load may exceed the classical predictions by up to 25%, due to the prebuckling displacements [5-111. Also for thin walled beams, interaction phenomena between local and global modes of buckling may affect the load- carrying capacity of beams considerable [12-171.

The purpose of this paper is to present a set of nonlinear equilibrium equations for slender beams with undeformed cross section under various loading and support conditions. A straightforward way for deriving the equations governing the bifureation-type buckling condition taking into account the prebuckling deformation is also presented. The analysis is used to obtain a closed form solution for the lateral buckling of a simply supported beam subjected to uniform bending.

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2. ELEMENTS OF BEAM THEORY

We consider an initially straight, prismatic, homogeneous beam of length L and cross-sectional area A, subjected to end, surface and volume loading. The beam is referred to rectangular Cartesian coordinates xIyIzI where the X-axis is the longitudinal axis, as shown in Fig. 1 .

The object of beam theory is to reduce a three-dimensional problem to an approximate one dimensional one. Slender beam theory may be derived in terms of the following simplifying approximations.

1 . During bending and/or stretching, cross sections normal to the undeformed longitudinal axis are assumed to remain plane, normal and undeformed, so

that transverse normal and shearing strains may be neglected in deriving the beam kinematic relations.

2. During twisting the cross sections of the beam rotate about the shear- center axis, while the normal displacement of any point in the cross section is equal to the product of the angle of twist per unit length, and the so called warping function $(y,z), which is a function of the cross section geometry only.

3 . Transverse normal stresses are assumed to be small compared with the other normal stress component, so that they may be neglected in the stress-strain relations.

These assumptions are known as the Bernoulli-Euler-Vlasov assumptions. Let us now consider the beam in a slightly deformed configuration, as

shown in Fig. 2. Let u denote the x-component of the displacement of the origin when no warping occurs, and let v and w denote the y and z

components, respectively, of the displacement of the shear center. In the following we will restrict ourselves to the intermediate class of deformations, which is defined by the limitation that the strains be small compared with unity, and rotations moderately small (see appendix A ) . As a consequence rotations may be described by a vector, letting a , $, y denote the x, y and z components of the rotation o€ the shear center respectively. Then, as a consequence of the first two approximations, the displacement

-1 1-

- - - components at any point in the beam u, v, wr may be expressed by the relations

- * 1 2 w = w t (y-y )U - -(z-z )a O 2 0

Here y Subscripts preceded by a comma denote differentiation with respect to these subscripts. The superscript

and z o denote the location of the shear center prior to deformation. O

* is used to denote quantities with respect to * * *

the local coordinate directions x , y , z of the beam in deformed configuration. These quantities are related to the global coordinate directions x, y, z through the rotation matrix [RI, i.e.

in which cos ( , ) indicates the directional cosine of the two axes. For small angles of rotation the matrix [RI may be presented in the linear form (see also appendix BI

’i Y -B U

f3 -a 1

so

( 2 . 3 )

and

-

* vr X

* X W,

Furthermore, neglecting the shear deformation,

p = -w, X Y = VIx ( 2 . 6 )

Substituting Eqs ( 2 . 6 ) into Eqs (2.4) and ( 2 . 5 ) and neglecting terms in- volving products of u r X , since u, =

appendix A ) , yields is small compared with unity (see * xx X

* vIX = v, t w, a

X X (2 .7 )

* - X

wIX - -vrxa t w,

The second order equations ( 2 . 1 ) may be considered as an attempt to construct a general theory of deformation of slender beams, based on Ref. [18] Eqs (VI.49) and (VI.83). The linearized form of the deformation field ( 2 . 1 ) is used in a majority of the earlier studies (Bleich, 1952;

Vlasov, 1961; Timoshenko and Gere, 1961; Galambos 1968) . However, in contrast with the linearized form, the present nonlinear deformation field ( 2 . 1 ) makes it possible to investigate the stability of slender beams, taking into account the prebuckling deformation.

Neglecting the shear deformation (Wagner-hypothese), then for the intermediate class of deformations the EX, i,,, and y strain-displacement relations for a three-dimensional medium are (see appendix A):

components of the xz

- - 1 - 2 E X = u,x + 2(VIX + i , ; )

-13-

- - Y = u, + VI + VlxVI + WIxW,

YxY Y X Y (2.8)

- - are extensional and shearing strain components at any

XI Yxy' where E point through the cross section. Introduction of Eqs (2.1) and (2.5) gives

- E = u, X - y(v,,,+ wlxxa) - z(wlxx- vlx,,a)

X

+ $ ( a , - VI xwI xxx+ WI xvI xxx) xx

1 2 2 2 x X o x x t -tv, + w, + 22 v, a, - 2y0w,xalx

2 2 2 +[(y - Yo) f ( 2 - zo) larx}

- = [$, - (z - z o ) l ( a I x - vlx~lxx+ wlxvIxx) si XY Y

$,,= c*, z+ (Y-Y,) 1 (a, x- VI xwl xx+ WI -

xx)

(2.9)

where higher order terms have been neglected. Eqs (2.9) are the kinematic relations for the beam. - -

in a XZ The generalized Hooke's law for the strain components E X , yxyt

three dimensional isotropic medium hac the form

- 1 - - - T 'xz- G xz

where

E 2(1tv) G =

(2. I O )

(2.11)

E is the modulus of elasticity, G is the shear modulus and v i s Poisson's ratio. The symbols 6 T

- etc. denote stress components at any point x' xy

-14-

through the cross section. As a consequence of

and iz are neglegibly small. Omission from Eqs

the relations

the third (2.8) and

approximation rearrangement

- 0 Y

gives

(2.12)

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3. TOTAL POTENTIAL ENERGY

The total potential energy P o f a beam subjected to end, surface and volume loading is the sum o f the strain energy V and the potential energy of the applied conservative load R [20] :

P = U + R (3.1)

The strain energy €or a three dimensional isotropic medium referred to arbitrary orthogonal coordinates may be written

- - - - I u = - J J J ( o e t o e + o E x x y y z z 2 v

- - - - - - y )dxdydz

+ T xyyxyt TXzyxzf Tyz yz (3.2)

- Omission of o and in accordance with the basic approximations o f

slender beam theory, introduction of Eqs (2.12) and rearrangement gives Y' z YZ

(3.3)

Taking the principal axes for the cross section coordinates (y,z), and the shear center O as the pole of the normalized warping, then, by definition :

JJydydz = JJzdydz = JJJidydz = A A A

jfyzdydz = JJyqiiydz = JJzJidydz = o A A A

Substituting Eqs (2.10) into Eq. (3.31, integrating with respect to y and z,

and making use of Eqs (3.4) yields

1 2 2 2 f ;ivIx + w r X + 22 o x v, atx - 2yOwIxaIX)

-16-

T 1 0 4 A x + - - a 1 4 ~

- 2 + w'xv'xxx) v ' x w ~ x x x vfxxx t b a * 'X) ( ' IXX

XW' xxx i- ET(a .xx -

where

2 2 - - - 'i J f ( y + z )zdydz - 2Z0 by 'y A

n

I = S $ zLdydz ' A

I z = S $ y'dydz A

2 T = S f JI dydz A

-17-

The warping function JI i s determined by the well-known Neumann-problem (see appendix C)

2 *IYY+ *IZZ = v * = o (3.7a)

within the cross section, and

on the surface o f the cross section. The normal vector n is positive outward, and the surface coordinate s is positive as indicated in Fig. 3.

The position of the shear center O according to the energetic definition (see appendix C ) , is given by the equations:

RIL - 111e i i û t a t i u n aiid ciytì cûnventiûnc for positive fûïces âi?U mûments âct iny ûi?

a beam element are illustrated in Fig. 4, where Qx, Qy and Q, are the longitudinal and transverse components of the force acting on the cross section and M x t M

cross section. The bimoment M is not a real moment, its dimension is [AL] . q,, cj , qz, mx, m r m,Z and m acting on the beam. The potential energy of the applied loads for a

conservative system is the negative of the work done by the loads as the structure is deformed. Consequently, the potential energy for the surface loads as shown in Fig. 4 may be written

and M, are the components of the moment acting on the Y * 2

a Y Y a are the forces and moments per unit length,

L Q = - j [qxu t q v t qiw t mxa - m w t rnZvfx Y Y 'x O

-18-

* L + Mxa - M w, + Mzv, + Ma(aI x- V I xW, xx+ W, x V ~ xx) 1 I (3.9)

o Y Z

assuming that longitudinal and transverse loads are applied at. the centroid and shear center o f the cross section, respectively. For other load application points, the relations (2.1) have to be introduced.

-19-

4 . THE MINIMUM POTENTIAL ENERGY CRITERION.

For equilibrium the total potential energy P must be stationary; i.e. its first variation 6P must equal zero. Substituting Eqs (3.5) and ( 3 . 7 ) into Eq. ( 3 . 1 ) and applying the minimum potential energy criterion yields

1 2 2 L J EA[uIX+ ?(vlx t w l X + 22 v, alx- 2 ~ ~ ~ ~ ~ a ~ ~ 6P = o x

O

+ -b 1 a , 2 ~~w'xxx6v' - w,xov'xxx~ vfxw'xxx+ w'xv~xxx 2 3, x X - Er ( c a I x x -

2 VI xxx6wlx 1 t -b 1 a , )(vrx6wlxXx- - Er (arxx- v,xw,xxx+ wfxvIxxx 2 3, x

-20-

t w,: t 22 v, a, - 2y0wIXaIx + J IEA[U,~ + y(Vtx o x x 1 2 L

O

f GJ(aIx - ~~~w~~~ + ~ I ~ ~ t ~ ~ ) 6 a ~ ~ } d x

L J isöu t 4y6v t qz6w t mt6a - m 6wI t mZ6vIv -

O Y X

* -+ ma(6aIX - v,x~wIxx- wIxXfivIx t w, x frv, xx t ~ ~ ~ ~ 6 ~ , ~ ) } d x

- [Qx6u t Q 6v t QZ6w t Mx6a - M 6wIX+ MZ6vfx Y Y

* - 6VIX -I- w, 6v, i- v,xx6w,x)ll = 0 t Ma(6ax - VIx8Wrxx w'xx x xx

O

( 4 . 1 )

By integration by parts and by application o f the lemma of Dubois-Reymond one obtains the equilibrium equations for a beam element

f

-21-

-22-

1 2 2 - { E A [ u I x + j í v , + w , t 22 o x x v, a, - 2y0wf x ~ f x

- I E A [ U , ~ + T ( v , 2 1 t w f 2 t 22 v I x a . - 2yowlxafx X X O X

i- p ~ a r x ) l r x x 1 2 t W ' v +[Er ( a fxx - vr xw xxx x 'xxx

-[ErlalXx - v'xw,xx + w,xv'xx + % a 2 ) b a 2 * 'x * 'x 1 'x

* - m t m = o

X a'x

With t h e dynamic boundary cond i t ions :

1 2 + 22 v f a , - 2yowfxafx Q, = E A [ U t x + j ( V r X W I X o x x

1 Q = EA[^, t ? ( V I 2 X t w , X -i- 22 o x x v, a , - 2yow, x o L f x Y

I O t - A a f : ) l ( v f x + 2 o x a f I

(4 .28)

(4 .3a )

+Er (aIxx - xw r xxx

-24-

t E A [ u I x t ? (Vr : 1 t w I 2 + 22 v r a I x - 2YowIxaIx X o x

tGJ(a, - V f xx + W I x"I xx)

* - m a

( 4 . 3 d )

( 4 . 3 e )

* * - maw'x + M a W I x x (4.3f)

-25-

1 t -b a 2, * M a = Er(a, xx - xw ' xxx w'xvfxxx 2 * 'x

-26-

5 . THE TREFFTZ CRITERION.

According to the Trefftz criterion for loss of stability the critical load for a continuous structural system is defined as the smallest load for which the second variation of the total potential energy of the system is no longer positive definite. At this load the equilibrium changes from stable to unstable [20-221. The linear differential equations for determination of the bifurcation-point load are obtained by integrating the expression €or 6 P by parts and by application of the lemma of Dubois-Reymond. 2

The linear stability equations may also be obtained by application of the equivalent adjacent-equilibrium criterion. Let

I u = u t u O

i- v = v o w = w

O a = a + a O

+ w1 1

(5.1)

where u v w a denotes the configuration whose stability is under investigation, and where the variations uII vII wII a l are admissible virtual increments. introduction into Eqs. (4.1) is seen t o give terms that are linear and nonlinear in the uoI voI wo' a. and ui, vq1 wII a, displacement components. In the new equations, the terms in uo, wo' wo, aO alone add to zero because u v w a is an equilibrium configuration and terms that are nonlinear in u,' vII wl, a l may be omitted because of the smallness of these incremental displacements. Thus the resulting equations are homogeneous and linear in u

o' o' o' o

o' o' o' o

vl1 wl1 ai with variable coefficients in uoI voI wgI ao. 1'

-27-

6. APPLICATION OF THE THEORY.

The governing differential equations for buckling of slender beamp are complicated and closed form solutions can only be obtained for single members with simple axial loading arrangement and boundary conditions. For some other cases it is possible to make simplifying assumptions, so that approximate closed form solutions can be obtained. However, for more complicated structures under general in-plane loading, a numerical technique must be used.

6.1 Simply supported beam in uniform bendins.

An approximate closed Lorm solution can be obtained from the flexural- torsional buckling of a simply supported double-symmetric beam subjected to two equal and opposite end moments (see Fig. 5 ) [ ? I ] . For the specified loading conditions the prebuckling displacements are

1 2 2 o'x

= - - w U O' x

v = o

W M

o'xx EI O - - -

Y (x- = o

U

Critical conditions occur when 6'P = o . The boundary conditions in terms of variations in u, v, w and OL are :

34 f wolXwq f x = o

vi = v

w1 - 'xx OL1 = OL

IlX = o

= o

= o

1 'xx -

1 'xx

Since the in-plane displacements components u I an w definite form in the second variation and can therefore only increase tiie calculated value of the critical moment, these displacement will be zero in the buckling mode. A solution for the buckling mode can be obtained by

only occur in positive 1

-28-

assuming displacement functions for v boundary conditions. Therefore assuming

and a 1 1 which satisfy the prescribed

v = c sin :x 1 1

al = c sin x 2 ( 6 . 3 )

where C

variations, substituting Eqs ( 6 . 2 ) and ( 6 . 3 ) into Eqs ( 5 . 1 ) and following the procedure described in section 5 , the resulting equation can be

and C 1 2 define the absolute magnitudes of the displacement

expressed in matrix form as

= [:j ( 6 . 4 )

where

2 TI 2 - - I (2) [GJ t ET(-) ] 2 EI Y “3 = -

For a nontrivial solution to this homogeneous equation system the determinant o f the coefficients of C1, C must equal zero. Hence, neglecting 2

2 a3 since it is small compared with other terms, the critical moment is given by

-29-

where

2 Io lT Po (;) 1 pz p = (1 - p ) [ 1 - P Y Y

2 T i P = EI (-) Y Y L

When I > > I the solution reduces to the well known classical solution Y 2

I (Po Pz TI (6.8)

When I is not very much greater than Iz, the dominant term in the Y T

I 2 denominator is ( 1 - -1 and the percentage increase in the critical moment

is almost independant of the span. I Y

For a monosymmetric beam the critical moment is found to be [ 7 ] :

T

- 30-

7 . CONCLUSIONS.

The presented second order formulation for the analysis of slender beams makes it possible to predict the bifurcation-type buckling condition under various loading and support conditions, taking into account the prebuckling deformation. A closed form solution, obtained for the lateral buckling of a simply supported beam subjected t o uniform bending, agrees with earlier analyses: if the ratios of the major axis flexural stiffness to the minor axis flexural stiffness is less than three, the in-plane deformations may effect the buckling load considerable.

-31-

REFERENCES

1.

2 .

Bleich, F.: "Buckling strength of Metal Structures", McGraw-Hill, New York, 1952.

Timoshenko, S.P. and J.M. Gere: "Theory of Elastic Stability", 2nd ed., McGraw-Hill, New York, 1961.

3 .

4.

5 .

6 .

7 .

8 .

9.

10

11

Vlasov, V . A . : "Thin Walled Elastic Beams", 2nd ed., Israel Program for Scientific Translation, Jeruzalem, 1961.

Galambos, T.V.: "Structural Members and Frames", Prentice-Hall, Englewood Cliffs, New York, 1968.

Michell, A.G.M.: "Elastic Stability of Long Beams under Transverse Forces", The London Edinburgh and Dublin Philosoph. Mag. J. Sci. 48(5th ser.) p. 298, 1899.

Prandtl, L., "Kipperscheinungen", Dissertation, Nuremberg, 1899.

Baker, J.F., Home, W.R. and Heyman, J.: "The Steel Skeleton", Vol. II ,

Cambridge üniversity Press, Cambridge, 1956.

Woolcock, S.T. and Trahair, N.S.: "Effect of Major Axis Curvature of I-

beam Stability", J. Eng. Mech. Div., ASCE 99 (ENI), pp. 85-98, 1973.

Vacharajittphan, P., Woolcock, S.T. and Trahair N.S.: "Effect of In- plane Deformation on Lateral Buckling", J. struct. Mech., 3 ( 1 ) ,

pp. 29-60, 1974.

Roberts, T.M.: "Second Order Strains and Instability of Thin Walled Bars of Open Cross Section", Int. J. Mech. Sci., Vol. 23, pp. 297-306, 1981.

Roberts, T.M. and Azizlan, Z.G., "Influence of Pre-buckling Displacements on the Elastic Critical Loads of Thin Walled Bars of Open Cross Section", Int. J. Mech. Sci., Vol. 25, No. 2, pp. 93-104, 1983.

-32-

12. Cherry, S.: "The Stability of Beams with Buckled Compression Flanges", Struct. Eng., 38(9), pp. 277-285, 1960.

13. Van der Neut, A.: "The Interaction of Local Buckling and Column Failure of Thin Walled Compression Members", Proc. of the Twelfth Int. Congr. of Appl. Mech., Stanford University, 26-31 August, 1968, Springer-Verlag, 1969.

14. Wang, S.T., Yost, M.I. and Tien, Y.L.: "Lateral Buckling of Locally Buckled Beams Using Finite Element Techniques", Comput. Struct., J.(7), pp. 467-475, 1977.

15. Graves-Smith, T.R. and Shridharan S.: "A Finite Strip Method for the Post-Locally-Buckled Analysis of Plate Structures", Int. J. Mech. Sci., Vol. 20, pp. 833-842, 1978.

16. Roberts, T.W. and Jhita P.S.,: "Lateral, Local and Distorsional Buckling of I-beams". Thin walled structures, pp. 289-308, 1983.

17. Bradford, M.A. and Hancock, G.J.: "Elastic Interaction of Local and Lateral Buckling in Beams", Thin Walled Structures, pp. 7-25, 1984.

18. Novozhilov, V.V.: "Foundations of the Nonlinear Theory of Elasticity", Greylock Press, Rochester, N.Y., 1953.

19. Novozhilov, V.V.: "Theory of Elasticity", Pergamon Press, London, 1961.

20. Langhaar, H.C.: "Energy Methods in Applied Mechanics", Wiley, New York, 1961.

21. Koiter, W . T . : "Purpose and Achievements of Research in Elastic Stability", Recent Advanc. Eng. Sci., Vo1.3, pp. 197-181, 1966.

22. Koiter, W.T.: "Thermodynamics in Elastic Stability", Proc. Third Canadian Congr. Appl. Mech., Calgary, 1971.

- 3 3 -

Fig. 1: Prismatic beam

F ig . 2: Beam in deformed configuration

Fig. 3: Orientation o f the normal vector n and the surface coordinate s

Mz t

Fig. 4 : Loading arrangement

-35 -

* t

Fig. 5: Beam subjected to uniform bending

- 36 -

APPENDIX A : The intermediate class of deformations.

An elastic body is deformed so that a generic point P in the reference state displaces to P'. Let x. be the Cartesian coordinates of the reference state and let xi be the Cartesian coordinates of the deformed state, then the displacement field is given by

1

- i u = X I - x i i I A . 1)

In what follows, derivatives are with respect to the original state, and are indicated by ( ),i = a ( )/axi, while a repeated index will imply a

summation. The Green-Eagrange strain-tensor is given by [' is]

while the corresponding Piola-Kirchhoff stress tensor of the second kind is

- - 1 - oij = =pi, -t Uj, i)

and the linearized rotation tensor as

- 1 - - w =-(u - u 1 ij 2 j'i i'j

the Green-Lagrange strain tensor may be written

- 1 - E ij = Bij + pTki - Wki)(Bkj - W k j ' (A.5)

-37-

For most stability problems 6 as well as while class of deformations the Green-Lagrange strain tensor may be written approximately [19]

are small compared with unity, is o f the same order as, or higher than i2. For this intermediate

1 - - - - Eij = Bij i - 2 w ki 'kj (A.6) -

Written out the EXX, E and EXZ components, f o r a three-dimensional medium XY are

- - E = t q w t w t w 2 ) xx OXX 2 xx xy xz

- - - - 1 - - - - E = e i-(w w t w + w w ) xy xy 2 xx xy xywyy xz yz (A. 7)

- - - - 1 - - - - t -(w + w + w w ) Ox2 2 xxwxz xy wyz xz 22

E = XZ

Substituting Eqs (A.3) and (A.4) into Eq. (A.7) and neglecting the shear deformation (i = = o ) yields

- xy exz

1 -2 -2 - - E X = UIx 4- prx + WiX)

where

= S E yXY XY

- 3 8 -

APPENDIX B : The rotation matrix.

* * * If the rotation of the Cartesian coordinate system x , y , z to the Cartesian coordinate system x, y, z is relatively large, it can not be described by a vector and it is treated by means of modified Euler angles. If it is described by a finite rotation y about the Z axis, followed by a

rotation $ about the Y

fig. B-I), then the rotation matrix [RI is given by E201

I * axis followed by a rotation a about the X axis (see

* * cos(x ,Y) cos(x , z )

cos@ cosy -cosa siny + sina sin@ cosy sina siny + cosa sin$ cosy

1 -sin$ sina cos$ cosa cos@

cos$ siny cosa cosy t sina sins cosy -sina cosy t cosa sin@ siny

The rotation matrix [RI can be approximately written in antisymmetric form as

cos$ cosy

s in$

siny cosa cosy -sina i -sin@

sina cosa cos$

* * Noting that the Y

deflection curve and taking the effect of axial elongation into consideration, we have

and Z axes are perpendicular to the shear center

-Wfx t g @ = - 1 +E

v,X t g y = - I + & fB . 3 )

(B.2)

where E is the axial strain.

-39-

[RI =

Making use of the relations

X VI 1 -

tqa sina =

( lttg2a)

1 cosct =

( íttg2a)

the rotation matrix is expressed in the form

2 ( I t & )

SinCC

4 . -

cos a I T E

[(1td2t WI2] X

( B . 4 )

cos a l t k :

-sinol

( B . 5 )

If we assume that the deformations are small, then, Eq. ( B . 5 ) reduces to the simple form

[RI =

1 vi x

1

w 1 x

a

1

-Y 1

-B

a

1

( B . 6 )

and the rotation may be treated as a vector.

-40-

\

ì a

Fig. B-1: Orientation of the X , Y , Z and X': Y,'Z" coordinate sy s terns.

-41-

APPENDIX C : The torsion problem.

The total potential energy P for a prismatic beam in uniform torsion is (see section 31

L

Application of the minimum potential energy criterion by variation o f $I

yields

By application of Green's theorem and the lemma of Dubois-Reymond one obtains :

2 * i y y + * I Z Z = v * = 0

within the cross section, and

(C. 3a)

(C. 3b)

on the surface o f the cross section. The normal vector n is positive outward, and the surface coordinate s is positive as indicated in Fig. 3. Multiplying Eq. (C.3b) by z, integration over the surface and application o f

Stoke's theorem yields

This equation can also be obtained by varying Eq. (C.1) with respect to y and applying the lemma o f Dubois-Reymond.

O

-42-

Analogous one finds