a thesis submitted for the degree of doctor of philosophy
TRANSCRIPT
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THE M E C H A N I C S OF THIN W A L L E D S T R U C T U R E S ? W I T H SPECIAL R E F E R E N C E TO F I N I T E R O T A T I O N S
A thesis submitted for the degree of Doctor of Philosophy in the Faculty of Engineering
of the University of London
by
David Nicholas Bates, MSc, DIC
Department of Civil Engineering Imperial College of Science and Technology
London, March 1987
To my mother and in memory of my father
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A B S T R A C T
This work is concerned with the mechanical behaviour of thin walled beam and shell structures. The formulation includes the effects of both geometric and material nonlinearities. Although no restriction is placed on the magnitudes of displacements and rotations, the strains are
0 assumed to remain reasonably small (< 0.04). Onlyconservative systems are considered.
The importance of finite rotations in geometrically nonlinear analysis is discussed, and it is argued that commonly used techniques of dealing with such rotations can be inadequate, especially with respect to the representation of certain forms of instability (e.g., lateral-torsional buckling). Starting with two fundamental theorems - Euler's theorem of rigid body rotation and the polar decomposition theorem of continuum mechanics - a rigorous finite rotation
C theory is developed. The implications of such a theory onthe incremental strain-displacement and constitutiverelations of thin walled beams and shells are examined. Consideration is also given to the problem of maintaining path-independent virtual work in the presence of externally applied moment couples and finite rotations.
A comprehensive technical theory of curved thin walled beams with arbitrary cross sectional shape (including mixed open/closed sections), is presented. The theory derives from the classical semi-inverse approach which allows thethree-dimensional equations of continuum mechanics to be expressed in terms of a single one-dimensional reference line. The reference line may be both curved and twisted in the initial state, and the model can be used either in stand-alone form or as a plate/shell stiffener. The foregoing finite rotation theory is fully integrated into the formulation. In addition, the model comprises a detailed treatment of plasticity and warping, and allows for the proper interaction of axial and shear stresses through the section walls.
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C O N T E N T S
ABSTRACT 3
LIST OF FIGURES 8
LIST OF TABLES 10
ACKNOWLEDGEMENTS 11
NOTATION 13
CHAPTER 1 Introduction
1.1 Background 221.2 Technical theories of thin walled structures 23
• 1.3 Objectives and scope 311.4 Finite element models 31
References 32
CHAPTER 2 Theory of rigid body rotation
2.1 Introduction 362.2 Generalized rotation coordinates 37
2.2.1 Euler angles 382.2.2 Modified Euler angles 39
2.3 Euler* s rigid body rotation theory 402.3*1 Spectral analysis of rotation matrix 402.3.2 The rotation matrix as a function of Euler
rotations 422.3.3 An improved second-order approximation for R 452.3.4 Compound Euler rotations 442.3.5 Extending the range of rigid body rotations 48
2.4 Displacements due to rigid body rotation 47References 51
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CHAPTER 3 Kinematics of thin walled structures
3*1 Introduction 533»2 Polar decomposition and the corotational method 533* 3 Trefftz buckling criterion 583*4 Finite rotations in beams 59
3.4.1 Convected coordinates 603.4.2 Convected displacements 61
• 3.4.3 Kinematic relations 643.4.4 Updating the nodal displacements 683.4.5 Strain-displacement equations 69
3.5 Finite rotations in shells 723*5.1 Convected coordinates 723.5.2 Convected displacements 743.5.3 Kinematic relations 743.5.4 Updating the nodal displacements 763.5.5 Strain-displacement equations 773.5.6 Five versus six degrees of freedom 79
• 3*6 Stress measures and stress accumulation 833.7 On the definition of conservative moments 89
3.7.1 Ziegler’s models 923.7.2 Argyris’s models 963.7.3 Exact models 973.7.4 Load correction matrix for GE-type moment
vectors 1023-7.5 On the application of concentrated moments
to shells 1063.8 Concluding remarks 107
References 110
CHAPTER 4 Nonlinear theory of curved thin walled beams
4.1 Introduction 1164.2 Geometry and kinematics 119
4.2.1 Cross sectional geometry and assumptions 1194.2.2 Cross sectional warping 1244.2.3 Geometrical description of motion 126
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4.2.4 Stress and strain transformations 1304.2.5 Differential fibre length ratios 1334.2.6 Curvatures 135
4.3 Strains 1384.3.1 Flexural warping and shear attenuation 1434.3.2 Generalized strain-displacement relations 1484.3.3 Incremental Green strain field 1494.3.4 Convected incremental strain field 151
• 4.4 Constitutive relations 1534.4.1 J2 flow theory 1564.4.2 J2 deformation theory 1574.4.3 Stress accumulation 158
4.5 Stress resultants and rigidities 1604.5.1 Total stress resultants 1604.5.2 Generalized stress-strain relations 163
4.6 Virtual work equilibrium equations 1684.6.1 Virtual work terms 8AW ex and SWlc 171
4.7 Finite element equilibrium equations 175# 4.7.1 Shape functions and transformation matrices 177
4.7.2 Numerical integration with respect to r 1824.7.3 Numerical integration with respect to s
and t 183References 191
CHAPTER 5 Summary and concluding remarks
5*1 Summary 1975*1.1 Theory of rigid body rotation 1975.1.2 Kinematics of thin walled structures 1985.1*3 Nonlinear theory of curved thin walled
beams 2035.2 Concluding remarks and suggestions for further
work 2042 0 8References
APPENDIX A
APPENDIX B
APPENDIX C
Shell finite element
Beam finite element
Geometric constants for a rectangular
2 1 1
223
segment 239
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LIST OF F I G U R E S
Chapter 2
2.1 Geometrical representation of rigid body rotation. 43
Chapter 3
3*1 Local convected axes based on continuum mechanics transformation of an infinitesimal region around P (after Biot [3]). 57
3« 2 Local convected axes based on rigid body motion of reference line or midsurface at P(after Wempner [4]). 57
3.3 Decomposition of the motion of the reference lineof a beam or midsurface of a shell. 62
3*4 Motion of the midsurface of a shell. 733*5 The nonconservative nature of a moment acting
about a fixed axis. 913.6 Alternative models for the application of a
conservative torque - after Ziegler [49]. 933.7 Two equivalent finite rotation paths. 953.8 Alternative models for the application of a
conservative torque - after Argyris [9]• 963.9 Alternative conceptual models for the application
of an exactly conservative moment vector. 98
Chapter 4
4.1 Geometry and coordinates of the beam crosssection. 120
4.2 Geometry of a curved beam in configurations °Cand C respectively. 127
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4.3 Shell forces associated with equilibrium alongthe normal n of a unit length of the contour. 147
4.4 Effective uniaxial stress-strain relationship fora von Mises material with isotropic hardening. 155
4.5 Stress resultants acting at the reference point. 1614.6 Elemental length of a curved rectangular
Timoshenko bar. 1674.7 Generalized conservative load systems for beam
• analysis. 171
Appendix A
A.1 Midsurface geometry, local coordinates, andnodal freedoms. 221
A.2 Stress resultants and Gauss point locations. 222A. 3 Sresses acting on an element of the shell. 222
Appendix B
B. l Geometry of reference line and definition oflocal axes r, s, t. 237
B.2 Definition of cross-sectional geometry. 237B.3 Stress resultants and nodal degrees of freedom. 238B.4 Typical integration station layout for a segment. 238B. 5 Local stress components crr and orm. 238
Appendix C
C. l Geometrical parameters for a rectangular segment. 239
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LIST OF TABLES
Chapter 3
3.1 Displacements corresponding to each stage of themotion depicted in Figure 3.3. 62
3.2 Moment vectors and work values for the two finite• rotation paths shown in Figure 3.7 (°M = m{l,0,0}). 95
Chapter 4
4.1 The role of shear attenuation in respect ofvarious categories of thin walled section. 146
4.2 Tangent rigidity matrix for a curvilinear thinwalled beam. 165
4.3 Lagrangian shape functions for 3-noded beamelement (see Appendix B). 178
4.4 Definition of section properties to I32for a curvilinear thin walled beam. 186
4.5 Elastic rigidity matrix for a curvilinear thinwalled beam. 187
4.6 Comparison of alternative quadrature rules forfunctions of order two and three respectively. 190
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A C K N O W L E D G E H E N T S
It is always difficult to identify the origin and to trace the evolution of ideas and interests. Clearly, a myriad of influences and personal contacts serve to alter and mould such ideas in time - a few of the more notable of these are acknowledged in what follows.
The author’s interest in structural mechanics in general, and in finite elements in particular, wasstimulated by attending the 1971/72 M.Sc. course in Steel Structures in the Department of Civil Engineering atImperial College. The lectures on stability by Professor E.H. Brown, and on finite elements and numerical methods by Dr A.C. Cassell and Dr R. Hobbs, undoubtedly fired the author’s wish to gain further insight in these fields. The help and encouragement afforded by fellow finite element ’enthusiasts* Keith Ward and Paul Lyons, in connection with the author’s M.Sc. project, is also warmly remembered.
The author is much indebted to his supervisor DrA.C. Cassell and to the Head of the Civil EngineeringDepartment, Professor P.J. Dowling, for arranging financial support, and for their interest and encouragement throughout the project. Thanks are also due to Dr N.C. Knowles of Atkins Research and Development for his collaboration during the developmental stage of the finite element work. The grants extended by the Science and Engineering Research Council in support of the project, are also gratefully acknowledged.
The author’s association with his colleague Urs Trueb during much of the project is warmly remembered. The efficient and innovative nature of the nonlinear finite element program FINAS, largely developed by Urs, created an ideal ’testbed* for the author’s beam and shell models, and his constructive suggestions concerning the coding and implemention of these elements, were most helpful.
Finally, thanks are extended to Dennis Hitchings for his support and interest during the early stages of the work, to
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Dr M.A. Crisfield for a number of stimulating discussions on various aspects of finite element modelling, to Chris Burgoyne and Professor E.H. Brown for some useful suggestions concerning finite rotations, and to Paul Davidson whose interest and assistance following the author’s decision to process this document using Wordwise and a BBC micro, were much appreciated.
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N O T A T I O N
All symbols are defined as the first occur in the text. However, for the convenience of the reader, the most commonly used symbols are listed below. The list has been divided into four main sections. The first section provides imformation on the general notation used for the
• representation of operators, vectors, matrices, tensors,etc., whereas the three remaining sections list the symbols occurring in Chapters 2, 3 and il respectively.
On the rare occasions when the same symbol has more than one meaning in different locations of the thesis, then each meaning is listed individually. Otherwise, symbols are generally listed only in the section within which they first occur. The order in which the notation is arranged within each section corresponds approximately to the order of occurrence within the text. This helps to provide continuity and facilitates cross referencing.
A number of the symbols can be referred to any one of three basic configurations of the body or structure, °C, C, and AC respectively. In the lists that follow, such symbols are written without a left superscript. Therefore, to refer the symbol to °C or AC, the left superscript o or a must be added.
The use of tensor notation has been kept to a bare minimum. Although, in many cases, this may not make for the most compact representation, in the author’s opinion, the alternative vector/matrix notation provides a much more immediate feel for the underlying mechanics. The latter notation also offers cosiderable advantages when it comes to developing numerical models, because error prone indicial expansions are not necessary. Where tensors have been used, the standard summation convention for repeated subscripts is implied.
in
General
»«
I I
{ >
x
r ji
trdet8
S u 0( )
° c cAcp
Q
o
approximationreplacementmuch greater thanmuch less thantends to (e.g. , x -*• 0)magnitude or modulusfirst-order derivative (e.g., y' = dy/dx)second-order derivative (e.g., y" = d2y/dx2)tilde under: vectorcomponents of a vectorscalar product of two vectorsvector productbar under: matrixdiagonal matrixunit matrix (3^3)right superscript: transposeright superscript: inversetracedeterminant variation indot over: finite increment inKronecKer deltaorder of (used for defining truncation errors) right subscript: value referred to local body-attached frame (r,s,t) of a beam or shell initial configurationcurrent (known) equilibrium configuration adjacent incremented configurationreference point lying on the reference axis of a beam or in the midsurface of a shell generic point lying in the cross section of a beam or along the midsurface normal of a shell at Pleft superscript: value measured in configuration °C
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A left superscript: value measured in configuration ACbar over: value measured at reference line (i . e. , at P)over: value attributable to rigid rotation only
A over: corotational value attributable to pure deformation (straining)
n total potential energy• U strain energy
V potential energy of applied loadsw worksw virtual workE elastic modulusG shear modulusV Poisson's ratiodA elemental areadV elemental volume
Chapter 2
i = 1,2,3: fixed rectangular framei = 1,2,3: body-attached rectangular frame
X position vector in X tX position vector in x t
■6 X ■e Euler anglesR orthogonal rotation matrixS first-order approximation to (R-I)
i = 1,2,3: eigenvalues of Re unit vector along axis of rotationa resultant rotation about the axis ea = ae Euler rotationa = ae alternative Euler rotation (see Section 2.3*3)0) = tan(a/2)e: parameter vector parallel to e0>* = sin(a/2)e: parameter vector parallel to eU i. s = au1/a°X1: displacement gradientG displacement gradient matrix
HI
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Chapter 3
x,y, z fixed global frame (rectangular)Xi i = 1,2,3: alternative global frame for
tensor variables (X2 = x, X 2 = y, X 3 = z)F deformation gradient matrix
H<1 left stretch matrixright stretch matrix
• p ® Eii Green's strain tensorp EiJ Almansi's strain tensorr* s, t local body-attached reference frame
(rectangular)mr,ms,mt local materially-attached frame*r auxiliary axis perpendicular to the deformed
shear plane ms-mt*r unit vector tangential to *r at Pr . s, t T
unit vectors tangential to r, s and t at P = {r,s,t>: orthogonal transformation matrix relating {x,y,z} to {r,s,t>
P position vector of PSi position vector of node i0 displacement (translation) of PHi displacement (translation) of node i0 Euler rotation at P (and Q)Hi Euler rotation at node i»i shape function associated with node iE r axial strain in a beamE r s • E r t average transverse engineering shear strains in
a beamX p i X j •X bending curvatures at the reference line of a
beamX = {Xr.Xs.Xt>: bending curvature vector in a beam? » natural curvilinear coordinates describing the
midsurface of a shell at P
E r * E s » E r scovariant base vectors to 5 and at P in-plane (membrane) strains in a shell
X p iX jiXp s bending curvatures in the midsurface of a shell
IO
to?
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e r t• Est average transverse engineering shear strains in a shell
tp. ~ L {a.3«0>: local Euler rotation about an axis in the initial midsurface
<P magnitude of cgL++ ++ r, s. t convected reference axes for five degree of
freedom shell model (see Section 3.5.6)e i j small (linear) strain tensor
♦ a i5 Cauchy stress tensorT u 2nd Piola-Kirchhoff (2nd P-K) stress tensorC i J k 1 symmetric elastoplastic constitutive tensorC i JR1
C
modified elastoplastic constitutive tensor (see Section 3.6)modified elastoplastic constitutive matrix
c— <r constitutive correction matrix (see eqns. (3.61) and (3.62))
SWex external virtual work® i i linear part of incremental Green strain tensor
(referred to C)nonlinear part of incremental Green strain tensor (referred to C)
Pm mass densityM externally applied moment vector (couple)m magnitude of M2a lever arm associated with M*i i = 1,2,3: rectangular body-attached follower
frame used for specifying the orientation of Mi. d . k unit vectors along x lf x 2 and x 3 respectivelya = {aj^.aj.ag}: Euler rotation of lever(s)M~ s moment vector applied according to model STE
moment vector applied according to model GE-lc load correction stiffness submatrix*1 linear stiffness matrixKni geometric stiffnessK,— 1 c load correction stiffness matrixA incremental nodal displacements
i = 0,1,2,3: Euler rotation parameters (see Section 3.8 )
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Chapter 4
h wall thicknessm, n transverse contour-attached axes, with m
tangential and n normal, to the contour at Pa angle between s and nP distance between P and Qp distance between P and Q
^ Pm * Pn components of p along m, nA#Pn component of p along nR—a orthogonal matrix relating ra, n to s, t0) warping function<P torsional function$ average shear flow over the wall thickness hP position vector of Pq position vector of Qv ,v .V,.~ r ~ s ~ t unit orthogonal vectors tangential to r, s,
and t at PXi.x 2,x3 material (fibre) frame at QG , G . G ~ 1 ~ 2 ~3 covariant base vectors tangential to x 1# x 2,
and x 3V ftv f,v+r~rf ~sf ~tf deformed fibre frame at P
initial curvatures of reference line at PKr. K s.K t current curvatures of reference line at Pe~ V (Ep» E r t}e~ G {e i»e j j•£ij} - convected (Eulerian) strainsE~ c {Et» Eim»<7~ V { a r . ar s . crr t }a~ G {ct1.ct12.ct13> - Cauchy stressescr~ c E G~ VEC~ GE° — c
{ctx . cr1Tn» a in }_
{e g .e® .e g >“r r s rt{e ®»e g 2.e g3 > - Green strains {e 2» e 1tt1» Ein}^{'Cr*Trs»Trt}-j
5 g {Tj.Ti2.Tig} r- 2nd P-K stresses}
Sc {Ti»Tim»Tin>Ji m
Iffi
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3 curvature parameter (see eqn. (4.27))J— G Jacobian transformation matrix: V-system to
G-systemJ— C Jacobian transformation matrix: V-system to
C-systemQ curvature factor (see eqn. (4.38))K resultant curvature of reference line at P$ radius of gyration corresponding to K^tn^n torsional shear strain factors
^ n flexural shear strain factorsflexural warping factor
rshear attenuation factor generalized corotational strains at P
H
9n•mm •m im
transformation matrix relating e to r and+* Ce to E ~ c ~shear force, moment and twisting moment acting on a unit length of the contour
A area of beam cross sectionmoment of inertia of cross section about an axis through P parallel to m
tf-m radius of gyration corresponding to Ime/v, v {er,ers,ert>: linear part of e®
(referred to C)k~ V {k r »k rs.k r t>: nonlinear part of
(referred to C)e c {e1(e im,ein}: linear part of e *
(referred to C)E/V
c i 1 k 1P0-L iJkl
generalized incremental strains at P elastic constitutive tensor constitutive correction tensor
5 plastic state indicatorC— e elastic modulus matrix
plastic modulus matrixSi J deviatoric stresses^y o * CTy initial and current yield stress respectivelya = CTy
Heffective stress isotropic hardening modulusH
- 2 0 -
EP plastic modulus accounting for isotropic hardening
F— L {Fr,Fs « F t}: local internal force vector at PM~ L {Mr,Ms,Mt>: local internal moment vector at PMw warping moment (bimoment) at PS {F ,M .M >: stress resultants (generalized
stresses) at PQ s • Qt linearized values of Fs and Ft
• T1 r linearized value of M r or first-order St Venant torque
V Wagner coefficient£A / incremental stress resultants corresponding
to increments of 2nd P-K stressD symmetric tangent rigidity matrix© analytic curvature parameter for a plane
curved bar (see eqn. (4.139))SW1C virtual work associated with the application
of GE-type moment vectorsC curvilinear axis, approximately parallel to r,
used for applying an eccentric line load2 { *)x . y)y » n2 > : lever arm (eccentricities) of
C-linemagnitude of tj
k r w initial twist of c-lineQw curvature factor for C-linew {wx,wy#w z >: distributed line loadS, concentrated eccentric loads applied via
rigid levers to discrete points J on the reference line
S a { *)x i » j • s } J lever arm (eccentricities) of loads P3
I ? S R STE-type moment vectors aplied to the reference line at discrete points k
xw .xj.xk scalar load factors corresponding to loads w, Pj and M jk respectively
T— s transformation matrix associated with STE-type moment vectors (see eqn. (3*77))
Ki OCl
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load correction stiffness submatrix corresponding to distributed line load w/Vload correction stiffness submatrix corresponding to concentrated loads
A nodal displacementsU generalized reference line displacments0 translational components of U0 rotational components of UA/
• N shape function matrix connecting A to UN— u shape function matrix connecting A to uN— o shape function matrix connecting A to 0fii linear strain-displacement matrix—„ i nonlinear strain-displacement matrixG vector of incremental displacement derivatives
appearing in eqn. (4.147)R~ex vector of equivalent external nodal loadsR„~in vector of internal nodal forces equivalent to
the internal stress field5 natural curvilinear coordinate along r:
-1 ^ g < 1J = dr/dg: one-dimensional JacobianNi standard Lagrangian shape functions of order
n-1, where n is the number of nodesNi J i Lagrangian shape functions of order 2n-l for
torsional-warping freedomsQ iniitial stress matrix (see eqn. (4.183))a~ce elastic part of Cauchy stressesa~ c p S~ e
plastic part of Cauchy stresses stress resultants corresponding to ctcc
S~PD— e
stress resultants corresponding to g cp elastic rigidity matrix corresponding to Cfi
D-PD— a
plastic rigidity matrix corresponding to C- prigidity correction matrix corresponding to C— <T
Ii 1 = 1 - 32: section properties for a curvilinearthin walled beam
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C H A P T E R 1
I N T R O D U C T I O N
!■! Introduction
This thesis evolved from a research project in which the primary objective was to formulate a finite element model suitable for the nonlinear analysis of thin walled structures, including stiffened plates and shells. To facilitate comparison with experimental results obtained from models of ring and stringer stiffened stiffened cylinders, the ability to operate with several thousand degrees of freedom, was required.
Whilst such an ambitious project clearly lies within the capabilities of the finite element method and of modern computing power, it does present some special difficulties. Inevitably, in such a testing environment, any errors, inconsistencies, inaccuracies, etc., will sooner or later show up. For example, 'bugs' may be eliminated directly from the computer code, and conceptual errors eliminated from the numerical model. But, in the author's opinion, a much more challenging, and ultimately more rewarding, difficulty arises when unsatisfactory behaviour persists even after the removal of errors of the aforementioned kind. Evidently, progress on this level requires considerable insight into the underlying theoretical basis of the model. Only in this way is it possible to create a framework within which to implement constructive changes.
Although the original plan was to present a thesis that was mainly concerned with the finite element formulation and its applications, in the event, the focus has clearly shifted onto the underlying structural mechanics. To a large extent, this change came about through a process that is similar to that described in the last paragraph. For example, the poor convergence characteristics of a curved beam undergoing finite rotation in three dimensions,
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eventually led to a detailed study of the kinematical implications of finite rotations. This, in turn, led to a much clearer understanding of polar decomposition and the corotational method.
Thus, the decision to concentrate on the mechanics can be seen to have come about in a natural way. At the same time it was also a positive decision made in the hope that a presentation of this form will provide some fresh insight into a conceptually difficult subject.
1»2 Technical theories of thin walled structures
The process by which the technical theories of flexible bodies are derived from the three-dimensional relations of continuum mechanics, is known as the inverse or semi-inverse procedure. The inverse procedure involves the substitution of an assumed form for the displacement field and/or stress distribution into the governing differential equations and boundary conditions. The semi-inverse procedure is similar, but consists of a partial specification of the displacement field and/or stress distribution, leaving unspecified functions to be determined by the governing equations and boundary conditions. In either case, the problem to be solved may be considerably simplified. Similar techniques can be used to formulate finite element models for the analysis of beams, plates, and shells, and, in this context, the process has generally been referred to as 1 degeneration *.
During the last fifty years, the theory of shells has received the attention of many distinguished mechanicians and mathematicians. In a short space, it would be practically impossible to provide a meaningful review of this work, and therefore, no attempt will be made to do so. However, some fundamental observations, which set the scene for the more detailed considerations of later chapters, may prove useful.
Because a shell is curved, the fibre lengths vary
through the wall thickness, and this leads to interaction between bending and stretching. This effect is of lower order than the corresponding interaction occurring in plates. Indeed it is usually sufficient in the latter case to assume that the bending and stretching are uncoupled. The average deformation due to tranverse shear can easily be accounted for using the methods introduced by Reissner [1] and Mindlin [2]. Still more advanced theories can be developed to include stretching and torsion of 'normal* fibres. At this level of refinement however, in the case of a thin to medium-thick shell, the resulting model is essentially equivalent to a full three-dimensional continuum theory.
Although they all derive from a common origin (i.e., the three-dimensional relations of continuum mechanics), numerous different technical shell theories have been proposed and are used, each reflecting a slightly different set of simplifying assumptions. Such theories are often labelled as 'elementary', 'advanced', 'refined', etc., suggesting a sort of league table of absolute performance. However, in the present case, where we are only concerned with thin plates and shells, it would be quite wrong to attach much significance to this kind of implied superiority of one theory over another. Actually an examination of why this is so, is rewarding because it reveals some profound ideas concerning the relationship between different theories and between theory and reality.
First consider the views of Novozhilov [3] concerning the validity of Kirchhoff's hypothesis in the context of thin walled plates and shells, and slender beams. Novozhilov writes:
Kirchhoff's hypothesis in the theory of plates and shells rests on simplifications which result when elongations and shears are neglected in comparison with rotations in determining the direction of fibres of the strained body.Since thin plates and shells are flexible bodies whose angles of rotation under a deformation ordinarily are large in comparison with the elongations and shears, the adoption of this hypothesis usually introduces negligible errors into
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the calculations. Hence it is clear that the simplifications proposed by Kirchhoff, and subsequently extended by Love to shells, can hardly be called a hypothesis as is ordinarily done. For, in essence, we are dealing with purely geometrical approximations whose error can always be estimated.Kirchhoff’s 'hypothesis' does not include any assumptions about the properties of the materials of which the plates and shells are made. Thus, a theory based on this 'hypothesis' can be used with equal effectiveness both for bodies which obey Hooke's law and for bodies which do not. It is only important that the basic condition be satisfied, namely, that the strains be small in comparison with the angles of rotation.When the 'hypothesis' of plane sections is applied to beams which are subjected to deformations in the elastic-plastic domain, it is generally considered necessary to refer to an experiment which confirms the 'hypothesis' in this case. It is clear from the above that the designing of specialexperiments to verify the 'hypothesis' of plane sections for bending in the elastic-plastic domain must be ascribed to an insufficiently clearunderstanding the the nature of this hypothesis.
Now consider a related discussion by Malvern [4], who addresses the wider issue of whether or not it is right toassume, per se, that a general three-dimensional theory willnecessarily beam, than writes:
lead to a more realistic model of say a slender the alternative 'elementary' theory. Malvern
It is sometimes said that solutions to engineering problems obtained through the use of simplifying assumptions of the the engineering theory of beams are only approximations of the 'exact' solutions of the theory of elasticity. The foregoing paragraphs should have made it clear that the adjective 'exact* is not Justified. There may be exact mathematical solutions to the equations formulated in elasticity or in other branches of continuum mechanics, but the equations themselves are not exact descriptions of nature. In this respect the difference between the elementary theory and the advanced theory is one of degree rather than kind. When the elementary theory is formulated consistently and logically, it is Just as respectable as the advanced theory from a mathematical or logical point of view. And from a practical point of view it is Just as good in those areas where its predictions agree closely enough with experience. The bounds of applicability of these elementary theories are determined by experience, either from experimental verification, or from comparisons with predictions of the more advanced continuum theories.
Finally, in his definitive work on the foundations of linear and nonlinear thin shell theory, Koiter [5,6] was able to demonstrate in a rigorous manner that the simpliflying assumptions employed by Kirchhoff-Love provide
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an adequate basis for the so-called ’intrinsic* thin £ theories. (An intrinsic theory is one that excludes -fche rigid body motion of the particles.) At the same Koiter takes great care to always accompany ^ n yapproximations made in respect of the overall geometr of the motion, with a rigorous error analysis leadini -toprecise limitations as to the magnitude of such motioi' inthis way, various formulations valid for large, mode^.-te,
• small, and infinitesimal rotations are distinguished.It is clear from the foregoing discussions tha' -the
various intrinsic kinematical assumptions, which sire commonly used when deriving the technical theories oftrhin walled structures, actually provide a rather poor l ^ s i s from which to compare the absolute precision of s u c h theories. The essential reason is that, although "the assumptions have a direct bearing on the deformation, t h e y have no effect at all on the preceding rigid body motion. (Note that in this thesis the word deformation specifiiaiiy excludes the rigid body component of the overall motion. ) But in a slender thin walled structure, it is the rigid t o d y motion that dominates the response. This suggests t^&t a meaningful comparison between alternative theories cat in fact be made by comparing the accuracy with which the^ are capable of representing such motion.
Consider the polar decomposition theorem of continuum mechanics. The theorem states, quite simply, that any general motion of a deformable body can be decomposed in -to a rigid body motion followed by a pure deformation (the order can be reversed but this is irrelevant to the present argument). An alternative, but exactly equivalent viewpoint, is that a pure deformation is frame indifferent - in other words, the deformation is the same regardless of the frame of reference of the observer. These concepts are so closely related to our sensory perception of reality that they might Justifiably be described as self-evident.
Now observe the nature of the two parts of the overall motion. Rigid body motion is an axiomatic process - th£-t is to say, it can be represented exactly without making any
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assumptions other than the presumed validity of Euclidean and differential geometry. In contrast, the deformation itself is non-axiomatic because it depends directly on the use of a constitutive (material) model. No matter how sophisticated such a model is, it clearly can never exactly replicate the real behaviour of the material. Based on these observations, the following principle is easily deduced:
If two theories describing the overall motion of a flexible body differ only in respect of the rigid body component of that motion, then the theory that best approximates this component will always provide a more accurate description of reality, both qualitatively and quantitatively.
An alternative, but equivalent, principle is:
Any approximation made in respect of the rigid body component of the overall motion of a flexible body must reflect equally as a spurious component of the actual deformation.
Evidently, rigid body motion provides a crucial and definitive yardstick against which different formulations can be measured. Furthermore, such formulations can be classified as ’geometrically consistent* or ’geometrically inconsistent’ according to whether or not they admit exact (strain free) rigid body motion. As pointed out by Novozhilov, thin walled structures are characterized by their ability to undergo finite rotation whilst at the same time sustaining only small strains. Therefore geometrical consistency has relatively greater importance to slender structures than it does to stocky structures. Moreover, as observed by Thompson and Hunt [7], the quest for structural efficiency leads inevitably to components that are relatively more slender and more imperfection sensitive. Thus it seems that the price that must be paid for structural optimisation is an increase in behavioural complexity. Correspondingly, if slender optimized components are to function safely in both the pre- and post-buckling range, then it is essential that they are modelled in a way that allows for the proper interaction of all significant
28
structural parameters but excludes any spurious effects.In the context of linear finite element modelling, it is
well-known that strain free rigid body motion is a necessary condition for convergence to correct solutions under mesh refinement. However, according to the principles established above, the same condition must apply to the general nonlinear case involving arbitrary finite rotations. To put it another way, the strain field that is predicted within a given element should be invariant with respect to any preceding rigid body rotation, however large. An element that is unable to meet this condition will inevitably be contaminated in the sense that other constant strain states (e.g., inextensional bending, or pure membrane action) cannot be represented exactly. In this case, the structural response as a whole will not converge to the correct solution, irrespective of mesh refinement and/or step size.
A well established property of shells is that, for certain geometries, exact inextensional bending deformation is possible. Equivalently, but less importantly, pure membrane stretching should also be admitted. One of the most common difficulties with finite element models is that they fail to reproduce these properties exactly. Even in cases where independent bending and membrane modes are available under linear conditions, spurious interactions frequently occur in the presence of general nonlinear finite motions. Such spurious behaviour can be attributed, yet again, to inaccurate representation of the rigid body component of the overall motion.
Against this backround it is somewhat surprising to find that relatively few degenerate finite element models are geometrically consistent. Certainly, most models do provide strain free rigid body motion for certain element geometries in the linear, small rotation range. However, this condition is often not satisfied for arbitrary element geometries. Perhaps more seriously, in a nonlinear environment, many existing models fail to provide genuine strain free rigid body motion in the presence of finite rotations,irrespective of element geometry.
29
Perhaps the most notable geometrically consistent finite element formulation is the 'natural approach* evolved by Argyris et al. [8]. Significantly, their one-dimensional beam element appears to be the only publicized model with the proven ability to give the correct results for beam and frame problems that buckle in a general (three-dimensional) lateral-torsional mode. This last contention is Justified because, as far as the author is aware, no results have been published, that would either confirm or refute, the instability levels and post-buckling behaviour of a simple right-angle frame analysed by Argyris et al. in references [9.10]. This is in spite of the fact that the correctness of their results can be independently verified using shell, or even, solid elements, and in spite of their stated misgivings concerning the inadequacy of the conventional, virtual work based, form of the geometric stiffness. One of the major objectives of the present research is to provide a geometrically consistent, virtual work based, alternative to the natural approach.
Degeneration normally leads to finite element models whose generalized freedoms are referred to a single reference line (beam) or reference surface (plate or shell). Any geometrical description of the motion of such a line or surface necessitates the introduction, either explicitly or implicitly, of generalized rotational freedoms. In fact, in the common case where the average effects of transverse shear deformation are to be included, the explicit introduction of rotational freedoms cannot be avoided. Therefore, in practice, a detailed investigation of the theory of finite rigid body rotation must be considered a prerequisite to the development of geometrically consistent finite element models.
Finally, a few comments concerning the rather specialized nature of thin walled beam formulations may be useful. Thin walled beams have great intrinsic complexity. This complexity arises because a beam of say I section is actually a composite structure involving the intersection of three separate shells. Still further complexity arises when
30
the beam is curved and/or* twisted in the initial state. Taking a somewhat wider view, it is also apparent that, of all structural components, the one-dimensional formulation of a curved thin walled bar, involves the greatest level of degeneration and, equivalently, the greatest choice with respect to suitable simplifying assumptions. Not surprisingly, the derivation of a numerical model capable of accurately reproducing all the significant modes of behaviour, is a most demanding task.
The formulation of refined technical theories for the nonlinear analysis of arbitrary curved beams, has received relatively little attention in the litrature. The main developments can, however, be traced in references [11-15]. The very recent paper by Wunderlich et al. [15], undoubtedly represents the 'state of the art' in this field. This paper was published when the author's own work was near to completion. Therefore, although, in a broad sense, there are some significant similarities, the material presented here was developed quite independently. The only exception concerns the description of nonlinear constitutive models (see Section ll.ll). In this case, motivated by [15] » the author has included the so-called Jz deformation theory, as well as the conventional J 2 flow theory.
In contrast to shells, the distinction between elementary and advanced beam theories, has much greater significance. Thus, for example, the basic Bernoulli-Euler theory is quite inadequate when applied to the elastoplastic buckling of a curved bar of arbitrary cross section. To handle problems of this generality requires a theory that includes at least the first four of the following enhancements:
1. Warping strains2. Flexural and torsional shear strains3. Effects of initial curvature and twistH. Wagner effect5. Weber effect6. Cross sectional distortion
31
7. Differential warping (shear lag)
An advanced theory, following these guidelines, is presented in Chapter 4.
1-3 Objectives and scope
The main objectives of this thesis are:
1) to derive a rigid body rotation theory both in an exact and second-order incremental form, with due consideration of the twin conditions of uniqueness and symmetry that pertain to the total potential function of a flexible body subject to finite rotation.
2) to discuss the implications of the aforementioned theory on the strain-displacement and constitutive relations of thin walled structures.
3) to investigate the problem of maintaining path independent virtual work in the presence of externally applied moment vectors, and to derive the corresponding load stiffness correction matrix.
H) to formulate a comprehensive nonlinear theory of curved thin walled beams of arbitrary cross section (including mixed open/closed sections), incorporating the ideas gleaned from 1), 2), and 3).
To simplify the presentation, the strains are assumed to remain small (< 0.04), and only conservative systems are considered.
1-4 Finite element models
Two finite element models, for shells and beams respectively, have been developed by the author to study the behaviour of thin walled structures. These elements are
32
incorporated in the Department of Civil Engineering (Structures Section) program FINAS developed by Trueb [16]. Recently, FINAS has been extended to include the nonlinear transient analysis of structures, and fluid elements have been added to model dynamic problems involving fluid-structure interaction [17].
Apart from extensive testing and verification by the author, the beam and shell elements have proved very versatile and successfull in the context of small and large scale nonlinear modelling of thin walled structures. Moreover, they have made an important contribution to many of the research projects undertaken by the Structures Section over the past five years [16-31]. As a brief glance at these references shows, many of the projects were concerned with the theoretical and experimental collapse behaviour of ring and stringer stiffened cylindrical shells. The pernicious nature of the pre- and post-buckling response of such structures provides a particulary testing environment in which to demonstrate the combined capabilities of the shell and beam elements.
For the convenience of the reader, technical summaries of the shell and beam finite elements are included in Appendices A and B respectively. It should be pointed out however, that the elements include most, but by no means all, of the features and refinements discussed in this thesis. The author hopes to publish more detailed information on the innovative aspects of the formulations in due course.
References
[1] REISSNER, E. , ’’The effect of transverse sheardeformation on the bending of elastic plates”, J. Appl. Mech., ASME, Vol. 12, 19^5, A69-A72.
[2] MINDLIN, R.D., ’’Influence of rotatory inertia and shearon flexural motions of isotropic, elastic plates”, J. Appl. Mech., ASME, Vol. 18, 1951, PP. 31-38.
33
[3] NOVOZHILOV, V.V., "Foundations of the nonlinear* theory of elasticity", Graylock Press, New York, 1953, PP. 197-198.
[4] MALVERN, L.E., "Introduction to the mechanics of continuous medium", Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969, PP. 2-3.
[5] KOITER, W.T., "A consistent first approximation in the general theory of thin elastic shells", Proc. IUTAM Symp. on Theory of Thin Elastic Shells, North-Holland Publishing Co., Amsterdam, i960, pp. 12-33.
[6] KOITER, W.T., "On the nonlinear theory of thin elastic shells”, Proc. Kon. Ned. Ak. Wet., Series B, Vol. 69 , 1966, pp. 1-54.
[7] THOMPSON, J.M.T. and HUNT, G.W., "Dangers of structuraloptimisation", Eng. Optim., Vol. 1, 1974, pp. 99-110.
[8] ARGYRIS, J.H., BALMER, H., DOLTSINIS, J .St., DUNNE,P.C., HAASE, M., KLEIBER, M., MALEJANNAKIS, G.A.,MLEJNEK, H.-P., MULLER, M . , and SCHARPF, D.W., "Finite element method - the natural approach". Comp. Meth. Appl. Mech. Engrg., Vols. 17/18, 1979, PP. 1-106.
[9] ARGYRIS, J.H., DUNNE, P.C., and SCHARPF, D.W., "Onlarge displacement - small strain analysis of structures with rotational degrees of freedom", Comp. Meth. Appl. Mech. Engrg., Vol. 15, 1978, pp. 129-134.
[10] ARGRYRIS, J.H., HILPERT, O., MALEJANNAKIS, G.A., and SCHARPF, D.W., "On the geometrical stiffness of a beam in space - a consistent V.W. approach", Comp. Meth. Appl. Mech. Engrg., Vol. 20, 1979, PP. 123-131.
[11] BARSOUM, R.S. and GALLAGHER, R.H., "Finite elementanalysis of torsional and torsional-flexural stability problems", Int. J. Num. Meth. Engrg., Vol. 2, 1970, pp.335-352.
[12] RAJASEKARAN, S., "Finite element method for plasticbeam-columns", in: Theory of Beam Columns, Vol. 2 -Space Behavior and Design (W.F Chen and T. Atsuta, eds. ), McGraw-Hill Book Co. , 1976, pp. 539-608
34
[13] WUNDERLICH, W. and OBRECHT, H., "Large spatialdeformations of rods using generalized variational principles", in: Nonlinear Finite Element Analysis inStructural Mechanics, W. Wunderlich, E. Stein, and K.-J. Bathe, eds.), Springer-Verlag, Berlin, 1981, PP. 185- 2 1 6 .
[14] RAMM, E. and OSTERRIEDER, P., "Ultimate load analysis of three-dimensional beam structures thin-walled cross sections using finite elements", Int. Conf. on Stability of Metal Structures, Paris, Nov. 16-17, 1983*
[15] WUNDERLICH, W., OBRECHT, H., and SCHRODTER, V.,"Nonlinear analysis and elastic-plastic load-carrying behaviour of thin walled spatial beam structures with warping constraints", Int. J. Num. Meth. Engrg., Vol. 22, 1986, pp. 671-695.
[16] TRUEB, U., "Stability problems of elasto-plastic plates and shells by finite elements", Ph.D. Thesis, University of London, 1983.
[17] CHELGHOUM, A., "Dynamics of structures including fluid interaction”, Ph.D. Thesis, University of London, 1986.
[18] DOWLING, P.J., HARDING, J.E., AGELIDIS, N., and FAHY,W., "Buckling of orthogonally stiffened shells used in offshore engineering", Buckling of Shells, AState-of-the-Art Colloquium, Institut fur Baustatik Universitat Stuttgart, May 6-7, 1982.
[19] TSANG, S.K., HARDING, J.E., WALKER, A.C., andANDRONICOU, A., "Buckling of ring stiffened cylinders subject to combined pressure and axial compressive loading”, ASME 4th. Nat. Congress of Pressure Vessals and Piping Technology, Portland, Oregon, 1983.
[20] ESTEFEN, S.F. and HARDING, J.E., "Ring stiffenerbehaviour and its interaction with cylindrical panel buckling", Proc. Instn. Civ. Engrs., Part 2, Vol. 75, June 1983, PP. 243-264.
[21] DOWLING, P.J., HARDING, J.E., and AGELIDIS, N.,"Collapse of box girder stiffened webs”, Proc. Horne Conf. on Instability and Plastic Collapse of Steel Structures, Manchester, Sept. 20-22, 1983.
35
[22] AGELIDIS, N., "Buckling of stringer stiffened shells under axial and pressure loading", Ph.D. Thesis, University of London, 1984.
[23] FAHY, W.G., "Collapse of longitudinally stiffened cylinders subject to axial and pressure loading", Ph.D. Thesis, University of London, 1984.
[24] ESTEFEN, S.F., "Collapse of ring stiffened cylinders", Ph.D. Thesis, University of London, 1984.
[25] DOWLING, P.J., OWENS, G.W., and CHUNG, K.F., "Stabilityof tapered frames", Proc. Conf. on Stability Aspects of Industrial Buildings, Structural Research Council, Cleveland, Ohio, April 16-17, 1985.
[26] TSANG, S.K. and HARDING, J.E., "Buckling behaviourunder pressure of cylindrical shells reinforced by light ring stiffeners", Proc. Instn. Civ. Engrs., Part 2, Vol. 79, June 1985, PP. 365-381.
[27] TSANG, S.K., "Collapse of ring stiffened cylindrical shells under combined external pressure and axial compression", Ph.D. Thesis, University of London, 1985.
[28] BURGAN, B.A. and DOWLING, P.J., "Interaction between shear lag and buckling in compression flanges", Proc. Annual Technical Session, SSRC, Washington, D.C., April 15-16, 1986.
[29] BURGAN, B.A., "Special problems in wide and narrowstiffened compression flanges", Ph.D. Thesis,University of London, 1987*
[30] ONOUFRIOU, A., "Collapse of damaged ring stiffened cylinders", Ph.D. Thesis, University of London, 1987.
[31] BURGAN, B.A. and DOWLING, P.J., "Effect of shear lag onthe collapse of stiffened compression flanges", Proc.ECCS International Colloquium on Stability of Plate and Shell Structures, Ghent, Belgium, April, 1987, PP. 163-171.
36
C H A P T E R 2
THEORY OF R I G I D B O D Y R O T A T I O N
2-1 Introduction
It has been pointed out in Chapter 1 that, in contrast • to continuum mechanics theory, ’degenerated' beam and shell
theories which include the effects of transverse shear, require the introduction of generalized rotation coordinates. These coordinates enable the motion of a general point in the beam or shell cross section to be described in terms of the corresponding motion of a parallel reference line or surface (e.g., centroidal axis or midsurface).
Although infinitesimal rotations in space may be treated as vectors the same is not true for finite rotations. In particular, finite rotations are in general non-commutative,i.e ., they do not obey the parallelogram rule for thesynthesis of two vectors. Equivalently, it is easilyverified that the final orientation of a rigid bodysubj ected to two or more successive rotations aboutnon-parallel axes, depends on the sequence in which they are applied.
Against this background, there is evidently a need for a refined theory which admits the exact calculation of spatial point-wise orientation of a body subjected to arbitrary finite rotations. Failure to utilise such a theory, or at least a good approximation, can lead to spurious self straining and inaccurate structural response in the presence of finite rotations. Furthermore, in nonlinear analysis, geometrical inconsistencies can develop which render the iterative determination of the equilibrium path difficult or even impossible. Finally, and perhaps most importantly, certain buckling modes involving general spatial deformation (e.g., lateral-torsional buckling of beams, frames and stiffeners) cannot be accurately modelled unless rotations
37
are approximated to at least second-order, even though the pre-buckling displacements may be very small.
2-2 Generalised rotation coordinates
In order to specify the angular orientation of a rigid body in a fixed frame X 4 (i = 1,2,3) it is sufficient to specify the corresponding orientation of a body-attached follower frame x A (i = 1,2,3). Assuming that initially x A are coincident with X t, then the familiar direction cosine transformations apply
X = Rx , x = R tX (2.1)
Alternatively, the matrix R can be thought of as transforming any point with position vector °X to a new position X where °X and X are both measured in the fixed space Xj
X = R° X , °X = R tX (2.2)/ v — / v r / v —
The nine components of the matrix R are often referred to as generalized rotation coordinates. Only three of these are linearly independent, the remaining six being related by the ortho-normality constraints
k = irikI>Jk = = 1,2»3 (2.3)
where 8 t j denotes the Kronecker delta.Burgoyne [1] derived equilibrium equations for a
beam-column using direction cosines as generalized coordinates. The rotation matrix was represented both in an exact and a second-order form, although, in the latter case,only three generalized coordinates were used. The maindrawback of this approach is that the rotation coordinates cannot easily be identified with the usual engineering notion of rotations. Furthermore, the use of constraint
38
conditions is an inconvenience that can be avoided.There are several useful systems of three coordinates
without constraint equations. Two of these, Known as Euler angles and modified Euler angles, will be introduced briefly in order to demonstrate their weaknesses. The method actually adopted for this study is based on Euler’s theorem of rigid body rotation, and leads to three, symmetric, linearly independent, rotation coordinates. It is
• demonstrated below, that the theorem is directly linked tospectral analysis of the rotation matrix R. Consequently, the associated rotation coordinates can be thought of as the natural or intrinsic measures of spatial rotation.
2-2-1 Euler angles
In this method the angular orientation of the body is assumed to result from three successive rotations about the follower axes x 4. The angles associated with each rotation are termed Euler angles and the sequence in which they are applied is characterised by the six permutations i, d* i (i,d =1,2,3* i * J). Thus if the three angles are labeled cp, X, 4> then the orientation of the body results from a rotation cp about the axis x t = , followed by a rotation X about the convected position of x jp followed by a rotation <l> about the convected position of x 1(
Equations which utilise Euler angles as generalized coordinates will become ill-conditioned if any two of the rotation axes are nearly colinear. Evidently, this occurs in the above scheme when the angle X is small as well as for various combinations of approximately quarter and half turns. A further disadvantage of Euler angles, particularly for structural applications, is their complete lack of symmetry.
Despite these deficiencies, Euler angles were adopted by Love [2], Novozhilov [3] and Frisch-Fay [U] for describing the spatial orientation of initially straight beams. The use of Euler angles in shell analysis is more recent, and
39
employs two generalized coordinates to describe theorientation of the midsurface normal [3-9] .
A good example of the difficulties that can arise if Euler angles are adopted as generalized coordinates for general purpose finite element analysis, is provided by the work of Surana [8]. He shows that the response of a finite element model which utilises a specific sequence of Euler angles as generalized coordinates, may not always be
• independent of this sequence. For example, suppose two localEuler angles, say cp and X, are used to define theorientation of the normal to a shell, and consider a deformation process in which X = 0 throughout. In such a case, as Surana shows, no special difficulties arise provided the sequence tp, X is used, but severe convergence difficulties can result from the alternative sequence X, cp.
Clearly, Euler angles lack the necessary uniqueness that is a prerequisite to general purpose finite element analysis. It is of interest to note, that in order to remove this lack of uniqueness, Surana, somewhat arbitrarily, introduces an alternative scheme obtained by taking the average of the rotation matrices which correspond to the two sequences.
2-2-2 Modified Euler angles
These coordinates are quite similar to Euler angles except that the sequence of application is altered from i, d, i to i, d, k (i,d,k = 1,2,3, i * d * k). Thus a rotation cp is applied about the axis x* = X if followed by a rotation X about the convected axes x J# followed by a rotation 4> about the convected axis x k.
Symmetry is somewhat improved compared to Euler angles and no numerical difficulties arise if all three angles are close to zero. However, there still exist critical cases in which two of the axes are colinear.
Modified Euler angles are also referred to as Bryant or Cardan angles [10] (the latter name derives from the fact that the three angles correspond to the three degrees of
freedom of a two-point Cardanic suspension). Examples of their application to the large displacement analysis of beams in space is given in references [11-13].
2-3 Euler’s rigid body rotation theory
An alternative representation of finite rotations in space will now be considered. This is based on Euler’s theorem (c.1776) which asserts that the change in orientation of a rigid body fixed at any point can be achieved by a rotation about some axis through this point.
Euler’s theorem is equivalent to the statement that if a rigid body is rotated into a new configuration about a fixed point, then there is only one line passing through this point which remains invariant during the motion. Stated mathematically, this theorem implies
e = Re (2.H )
where e is a unit vector directed along the fixed axis of rotation. (Obviously, this relation remains valid if e is replaced by a colinear vector of arbitrary length).
- l x O -
2-3-1 Spectral analysis of rotation matrix
Equation (2. IX) can be rewritten in the form
(R-I)e = 0 (2.5)
so that comparing with the spectral problem
(R-\I)e = 0 (2.6)
it can be seen that Euler’s theorem is true provided \ = 1 is an eigenvalue of R. Noting that R is a proper orthogonal matrix of order three and that detR = 1, equation (2.6)
leads to the characteristic polynomial
(X-l)[X2+X(l-trR)+l] = 0 (2.7)
where trR denotes the trace of R. Therefore X = 1 is indeed an eigenvalue of R and Euler’s theorem is confirmed.
Assuming that the elements r tj of R are known, it is now possible to deduce both the rotation and the axis of rotation to which R corresponds. Equation (2.7) indicates that for trR lying in the range -1 < trR < 3, the two remaining roots defined by the term in square brackets are complex conjugate. Thus,
\ x = 1 , X2 = exp101, X 3 = exp"i(X, 0 < a < it (2.8)
where a is the resultant rotation about the eigenvector e. But
trR = Xj+Xj+X3 (2.9)
so that introducing the identities
exp1® = cosa+isina (2.10)exp-1“ = cosa-isina
one obtains immediately
cosa = H(tr*R-l) = H(i'n+r 2Z+r33-l) (2.11)
Substitution of X = 1 into (2.6) leads after some manipulation to
e = 2slnar 32 3
r 2 1 r i 2
( 2.12 )
The above analysis leads naturally to the conclusion that the matrix R can be expressed in terms of three
42
generalized rotation coordinates a, where
a = ae (2.13)
It follows that a is the resultant or magnitude of the Euler rotation a, that is,
a = (a*+a|+a§)^ (2.14)
Since finite rotations are in general non-commutative it is important to emphasize that a is not a true vector. Argyris [14] uses the terminology ’rotational pseudovector * whilst others prefer the simpler, but nevertheless incorrect, term 'rotation vector*. In this work, the generalized rotation coordinates a lf a 2, and a 3 will be simply called Euler rotations.
2-3-2 The rotation matrix as a function of Euler rotations
An explicit definition of R in terms of a can be obtained by a straightforward geometrical argument. The derivation that follows is similar to that given by Argyris in his definitive work on finite rotations [14].
In Figure 2.1, the vector OP = °X is transported to OQ = X by a rotation a about the fixed axis OC. Let N be the foot of the perpendicular from Q on to CP, then
X = °X + NQ + PN/v
By definition, e is a unit vector directed along OC. Thus, denoting the radius of the circular arc PQ by a, and unit vectors along PN and NQ by PN and NQ respectively, we have
a(NQ) = ex°X , a(PN) = ex(ex°X)
so that
4 3
NQ = asina(NQ) = sina(ex°x)PN = a(1- c o s a )(P N ) = (1- c o s a )(ex(ex°X))
and therefore
X = °X+sina(ex °X) + (1-cosa) (ex (ex °X) ) (2.15)/v a/ /s/ a/ ^
CP = CQ = a
F I G U R E 2 . 1 G e o m e t r i c a l r e p r e s e n t a t i o n o f r i g i d b o d y r o t a t i o n .
Evaluating the vector product terms in equation (2.15) and using the second of equations (2.2) now leads to the required definition of R
c + e * (1 - c ) I1 e x e 2 (1-c)- e 3 s 1
1 eie 3 (1 - c )+ e 2se ^ j (1-c) + e 3s 1
1 c + e 2 (1-c) 1| e 2e 3 (1-c J-ejS (2.16)e 1e 3 (l-c)-e2s 1
1 e 2 e 3 (1-c)+ex s 1i c + e | (1 - c )
where c cosa, s sina, and e 1, are components of
zu
the axis of rotation e = a/a. It is of interest to note that equation (2.16) provides immediate confirmation of the results (2.11) and (2.12) obtained previously by spectral analysis.
Another useful form of R is obtained by noting that
ex °X = — S°X (2.17)~ ~ a - ~
where S is the familiar skew-symmetric matrix
S =0 -a 3 a“3 0 -a
-a2 0(2.18)
Hence
ex(ex °X) = 2 s s ° xa — ~ (2.19)
Substitution of (2.17) and (2.19) into (2.15) results in
sina _R = I + a1-cosa
a 2 SS
or in matrix form
( 2 . 2 0 )
R =l-s(oc|+a|) I -foCg+goCjOCz fa3+ga1a 2 j l-g(aj+a§)
-fa2+gaxa3 l fa1+ga1a z
fa2+ga1a3 -fa1+sa1a2 l-e(a?+a2)
( 2 . 2 1 )
where f = sina/a and g = (1-cosa)/a2.The first and second-order approximations of equation
(2.20) will be found useful in later applications. Thus, putting sina = a, cosa = 1 gives
R =1 "«3 aa 3 1 -a
— a2 1( 2 . 2 2 )
= a,whereas, sina cosa 1-^a2 leads to
45
R1 - % ( olI+(x Z ) I
, I« 3+^ 1a z |- a z + ^ a ^ g |
- a 3+J$a1a 2 I1-H(aj + a | ) I
!a 1+^a2a 3 j
a 2 + J$0Cia 3 -a* +)£(x2a 3 1-MocJ +a2 )
(2.23)
Although, these approximations violate the constraint condition (2.14), it is interesting to note that they have the same eigenvector e as the exact form (2.20).
2-3-3 An improved second—order approximation for R
Whilst considering alternative ways of representing the rotation matrix R, the author realised that a second-order approximation of higher precision than (2.23) can be obtained by the simple expedient of using a change of variable.
The new approximation arises through the use of an alternative set of generalized rotation coordinates, that is,
a = ae = sinae = (sina/a)a
which in turn implies
(2.24)
(sina/a)S = S[ (1-cosa)/a2 ]SS = Js(SS)+0(a4)
Here, the skew-symmetric matrix S has the same form as S, but is expressed in terms of the new coordinates a. Substitution of (2.25) into (2.20) now leads to
R = l + s+Ji(SS)+0(a*) ( 2 . 2 6 )
Noting that the previous approximation, (2.23)» can be written
R ( SS ) +0 ( a 3 ) (2.27)
- U 6 -
it is immediately apparent that the replacement of a by a leads to an order of magnitude improvement in the degree to which the orthogonality conditions are satisfied.
In the context of finite element software, the steps involved in adopting a, rather than a, as generalized coordinates, are trivial. For the author*s beam and shell finite elements, the necessary changes involve only three lines of coding. Observe that, because the generalized coordinates that are actually used in the nonlinear equilibrium equations of the structure are incremental quantities, it is unecessary (and inconvenient) to alter the meaning of the total accumulated nodal rotations. In other words, although 5 replaces a, the total rotations associated with a given equilibrium state, retain their original meaning. One obvious advantage of this approach is that the changes are completely transparent to the user - thus, a total rotation of 90° about the x-axis will produce the familiar output values {Tt/2,0,0} rather than the less familiar result {1,0,0}.
The changes that occur in the total potential function when (2.26) is used instead of (2.27), will be very small if measured purely in terms of the relative energy levels at a given point. However, in the context of bifurcation instabilities, the local topology of the potential function is of prime importance. Consequently, it is quite possible that topological changes associated with terms in a 3 can be significant in certain buckling problems. Moreover, in cases where the incremental rotations within a given step are large or moderate, the new scheme can improve the rate of convergence. Note that although a cannot exceed ± Tt/2, this limit is sufficiently high to mean that it has little or no relevance to the vast majority of practical problems.
To simplify the presentation, and to avoid confusion, the foregoing change of variable, is not actually used in the theoretical developments of later chapters. In this way, Euler rotation components (both incremental and total) are retained as generalized coordinates throughout the present work. The alternative approach discussed above can
4 7
therefore be thought of as an optional, and easily implemented, numerical refinement.
2-3-4 Compound Euler rotations
In Section 2.2 the transformation rules for rotations about fixed and follower Cartesian axes, were derived. The
• same rules can be applied to two (or more) Euler rotations.Thus, for instance,
X = R(a)°X = R(a. )R(a )°X (2.28)~ ~ ~ ~ d — ~ a. ~
is the transformation that results from an Euler rotation ot~ Qfollowed by an Euler rotation aw.~ D
Equation (2.28) confirms the physically self-evident fact that two or more Euler rotations can always be reduced to an equivalent single Euler rotation. But, knowing a& and aw how can a be found? One way is by spectral analysis of^ D ~the resultant matrix R(a). Alternatively, the compound rotation relations [14] may be used, i.e.,
(2.29)
vectors parallel to the unit respectively, i.e.,
* a ato = tan—^e~ a 2 ~ a(2.30)
to. = tan^e.
Another useful formula relating a explicitly to aa, otb is [14,15]
a a a ab aa abcos" = cos— cos— -sin— sin— coscp 2 2 2 2 2
to = --------(to +to -to xuv )~ 1 — ti) • to ~ a ~ b ~ a ~ b~ a ~ b
20)a = tan~ to to
where toQ, tofe are parameter eigenvectors e , ew of R , R.~ a ~ b — a.' — b
(2.31)
- a s -
where <p is the angle enclosed by eQ and efe.Finally, using equations (2.29) and (2.30), it is
straightforward to show that a * a +aw except in the special~ ~ Q ~ Dcase when e and ew are parallel, and this confirms theassertion made earlier that the associated rotation matricesR(a ) and R(aw ) are, in general, non-commutative.— ~a — ^ □
2-3-5 Extending the range of rigid body rotations
In contrast to the rotation matrix definitions of Section 2.3.2, which are valid for an Euler rotation a of arbitrary magnitude, the associated relationships derived in Sections 2.3*1 and 2.3.H are effectively limited to the range -Tt < a < Tt. The reason is that equations such as (2.12), (2.29) and (2.30) evidently break down when theresultant rotation a is an integral multiple of Tt. Mathematically, this breakdown occurs because theantisymmetric part of the rotation matrix vanishes, resulting in three real eigenvalues, and an apparent loss of uniqueness.
The rotation range quoted above will clearly beadequate for many practical engineering problems. However, finite element techniques are increasingly being used to address complex problems involving the general motion of mechanical linkages, mechanisms and multibody systems which require a unique and continuous description of arbitrary rigid body rotations. Consequently, a method of dealing with the troublesome case when a = nrt, n = 0, 1,2,3 • • • , oo will be of considerable value.
In accordance with equations (2.7) and (2.11), which are true for all a, the eigenvalues for n even and n odd respectively are
a = mt, n = 0, 2, U. . . , oo, = X2 = \3 = 1a = nTT, n = 1, 3, 5. . . ,oo, = 1, \ 2 = Xg = —1
In the first case (n even), the associated rotation
49
matrix reduces to the identity matrix I, so that any eigenvector e will satisfy equation (2.6). In physical terms this corresponds to the fact that when a rigid body is subject to one or more complete turns about an arbitrary axis, it will return to its original position. Clearly the case is trivial and need not be considered further.
In the second case (n odd), the rotation matrix of equation (2 .1 6 ) reduces to the symmetric form
R = 2e, e.el-H
sym.
e, e1 3e, e.e|-H
Consequently,
e x = ±[)$(r11+l)J^ e2 = ± [H(rz 2+l) e3 = ± [)$(r3 3+l) ]
(2.33)
(2.34)
The signs of e lf e 2, e 3 can now be selected so that they are consistent with the signs of the off-diagonal coefficients r 12, P 131 r 2 3 in (2.33). If the resulting vector is denoted by e*, then clearly
e ±e (2.35)
This result reflects the fact that, in the special case when a is an odd multiple of n, the final orientation of the body is independent of the sign of e. In practice, however, the choice of sign can usually be based on the physical attributes of the problem in hand. Thus in the present context, the motion of a structure is traced incrementally, and provided the rigid body rotations within a given step are less than Tt, the appropriate choice of sign is immediately rendered unique.
The breakdown of the conventional compound rotation formulae (2 .2 9 ) is evidently caused by the use the tan function in the definition of the parameter vectors coq and
50
. The associated singularities can however be avoided by+ * Dintroducing an alternative definition based on say the sin function, i . e . ,
to* = s i n ^ ? e ~ a 2 ~ a( 2 . 36 )
* . a b= s i n 2 6 S b
Utilising equation (2.31), which is true for all values of aa and ocb, the relationship between t»)* and to can be written
00* = co ( 1 - 0)*- oo* )h = ±oo ( f f -oo*- oo* )~ ~ ~ ~ ~ q b ~ a. ~ b
where
fa = ( l-oo*« oo* )**
(2.37)
(2.38)
Consequently, equations (2.29) can be replaced by
a/
aA /
± ( o o * f w +oo* f - C 0 * X O ) * ) ~ a b ^ b a ~ a ~ b
2 0 0*|W* Isin Iw* |
(2.39)
Note that in the first of equations (2.39), the sign is chosen so that it agrees with the sign of (f f -to** oo*).a b ~ q ~ b
2-4 Displacements due to rigid body rotation
Returning once again to the basic transformation equations (2.2), and deducting °X from both sides leads immediately to
u = X-°X = (R-I)°X (2.40)
where u is a vector of the total displacements of the point X relative to its original position °X. Later on this
51
equation will be used to deduce the kinematical relations for beams and shells which are valid for rotations of arbitrary magnitude.
Over regions within which the displacements are continuous (i.e.# free from singularities), the matrix (R-I) can be written in the alternative form
ui, 1 ui. * ui. 3(R-I) = U2 , 1 U2, 2 U 2 . 3 = G (2.41)
u 3 ♦ 1 U3 , 2 U3 * 3
where u 1 , 1 = auj / a ° x 1 etc. . The matrix on the right-handside of this equation can be recognized as the displacement gradient matrix of continuum mechanics, where it is used to describe the general motion of deformable bodies, rather than the restricted case of rigid body rotation considered here.
References
[1] BURGOYNE, C.J., "The nonlinear behaviour of elastic beams and beam-columns", Ph.D. thesis, University of
.London, 1981.[2] LOVE, A.E.H., "A treatise on the mathematical theory of
elasticity", 4th ed., Dover Publications, New York, 19^4.
[3] NOVOZHILOV, V.V., "Foundations of the nonlinear theory of elasticity", Graylock Press, New York, 1953-
[4] FRISCH-FAY, R., "Flexible bars", Butterworths, London, 1962.
[5] RAMM, E., "A plate/shell element for large deflectionsand rotations", in: Formulations and ComputationalAlgorithms in Finite Element Analysis, (K.-J. Bathe, J.T. Oden and W. Wunderlich, eds.), M.I.T. Press, Cambridge, Mass., 1977
52
[6] PARISCH, H. , '•Geometrical nonlinear analysis ofshells", Comp. Meth. Appl. Mech. Engrg., Vol. 14, 1978,PP. 159-178.
[7] PARISCH, H., "Large displacements of shells includingmaterial nonlinearities", Comp. Meth. Appl. Mech. Engrg., Vol. 27, 1981, pp. 183-214.
[8] SURANA, K.S., "Geometrically nonlinear formulation for curved shell elements", Int. J. Num. Meth. Engrg., Vol. 19. 1983, PP. 581-615.
[9] OLIVER, J. and ONATE, E., "A total Lagrangianformulation for the geometrically nonlinear analysis of structures using finite elements. Part I.Two-dimensional problems: shell and plate structures",Int. J. Num. Meth. Engrg., Vol. 20, 1984, pp.2253-2 2 8 1 .
[10] WITTENBURG, J., "Dynamics of systems of rigid bodies",B.G. Teubner, Stuttgart, 1977.
[11] BESSELING, J.F., "Derivatives of deformation parametersfor bar elements and their use in buckling andpostbuckling analysis", Comp. Meth. Appl. Mech. Engrg., Vol. 12, 1977, PP. 97-124.
[12] BATHE, K.-J. and BOLOURCHI, S., "Large displacementanalysis of three-dimensional beam structures", Int. J. Num. Meth. Engrg., Vol. 14, 1979, pp. 961-986.
[13] WUNDERLICH, H.O., "Large spatial deformations of rodsusing generalized variational principles", in:Nonlinear Finite Element Analysis in Structural Mechanics, (W. Wunderlich, E. Stein, and K.-J. Bathe, eds.), Springer-Verlag, 1981.
[14] ARGYRIS, J.H., "An excursion into large rotations",Comp. Meth. Appl. Mech. Engrg. , Vol. 32, 1982, pp.85-155.
[15] PARS, L.A., "A treatise on analytical dynamics", Ch. 7: The Theory of Rotations, Heinemann, 1965, pp. 90-107.
53
C H A P T E R 3
K I N E M A T I C S OF THIN W A L L E D S T R U C T U R E S
3-1 Introduction
In this chapter the kinematical behaviour of thin walled structures is discussed. As emphasized earlier, an accurate prediction of the ’real* behaviour of such structures is crucially dependent on the high precision modelling of the overall motion. It is well-known that thin walled structures can undergo large rigid body motion without sustaining large strains. Therefore, since rigid body motion makes no contribution to the potential energy of the structure, it seems quite natural that a method of focussing directly on the deformation itself, should be sought.
It turns out that the foregoing objectives can be met quite elegantly by utilizing the polar decomposition theorem of continuum mechanics, and the closely related concept of corotational coordinates. However, in order to successfully apply the method, it is essential that the rigid body component of the motion is defined with high precision in terms of the adopted generalized freedoms. Observing that the exact representation of rigid body translation is trivial, it is clear that this goal cannot be achieved without first developing an unique and objective method of dealing with finite rigid rotation such as that detailed in Chapter 2.
3-2 Polar decomposition and the corotational method
In structural and continuum mechanics it is obvious that a more general form of motion than that discussed in Chapter 2, is needed. However, according to the polar decomposition theorem [1], the motion of any infinitesimal volume within a continuous deformable body can always be decomposed into
5 4
either
or
1, a r i g i d t r a n s l a t i o n 2 m a r i g i d b o d y r o t a t i o n 3. a p u r e d e f o r m a t i o n
1 m a p u r e d e f o r m a t i o n 2 m a r i g i d b o d y r o t a t i o n 3 m a r i g i d t r a n s l a t i o n
The exact interface between rigid body motion and deformation is usually defined by the condition that the deformation itself produces no rotation of the lines of principal strain. Mathematically, the decomposition is given by
F = A R = RA (3.1)— — i — --- r
where F is the deformation gradient matrix, R is an orthogonal rotation matrix, and A , A r are the left and right symmetric stretch matrices respectively.
Naturally, rigid body translation does not appear in the relations (3.1), because it affects neither the rigid body rotation nor the deformation. The left and right stretch tensors are referred to the rotated and unrotated states respectively, so that although they are both measures of the same relative deformation, they are not in general equal. Noting that an Eulerian rather than Lagrangian description of motion will be used in this work, it seems natural to adopt the first form of decomposition as the basis for further discussion. Thus, for present purposes, it will be assumed that any general motion is equivalent to a rigid body motion followed by a pure deformation.
The stretch tensors are inconvenient to use in practice because their components are complicated irrational functions of the displacement gradients [1]. Instead, the deformation is usually specified directly using either Green’s (Lagrangian) strain tensor
e?j= ^(ulf j+uJ#1+uktluktJ) (3.2)
or Almansi’s (Eulerian) strain tensor
55
EiJ" ^(u i,J+uJ,i-uk,iuR,J) (3.3)
where the displacement gradients u ltl etc. are referred to the initial (Lagrangian) or convected (Eulerian) coordinates respectively. These tensors are valid for arbitrary values of the displacement gradients and vanish under rigid body motion.
The motion of thin walled structures such as beams and shells is frequently described by assigning rotational as well as translational degrees of freedom to a finite number of points (eg., nodes lying along the longitudinal reference line of the beam or in the midsurface of the shell). In such cases, the displacements of points lying away from the reference line or midsurface will be complicated irrational functions of the generalized (nodal) degrees of freedom, and after substitution in the appropriate strain tensor, the resulting expressions will become completely intractable. Significant simplification can then only be achieved by specifying that all the components of the displacement gradient tensor are small, which in turn restricts the method to the small strain, small rotation range.
This work is concerned with the large displacement analysis of thin walled structures, where small strains certainly do not imply small rotations. Furthermore, as will be demonstrated later, the use of kinematic relations which are simplified with respect to the role of finite rotations, can lead to erroneous predictions of critical instability levels, even in cases where the rotations that precede the instability are very small.
An alternative approach leading to strains that are practically exact within the small strain, large rotation range, is achieved by using the decomposition concept. Thus, in order to find the generalized strains at any point on the reference line of the beam or in the midsurface of the shell, the deformation may first be isolated by removing the rigid body rotation of this point from the total nodal displacements. This process is equivalent to transforming the nodal displacements to a set of convected axes which are
56
attached to and rotate with the point in question.Displacements which have been transformed in this way
are termed in the literature, convected, rigid-convected, or corotational. Here, the notation convected displacements will be used. (To avoid any possible confusion, total displacements that have been transformed back to the local convected axes by using the standard direction cosine relations, will be referred to simply as localdisplacements, whereas convected displacements that have been similarly transformed will be termed local convected displacements).
In accordance with the polar decomposition theorem, the convected displacement gradients determined after the removal of the local rigid body rotation of the material, must all be of the same order as the strains themselves. Consequently, the quadratic terms in the strain tensors can be neglected and one is left simply with the familiar linear strain-displacement relations, but now expressed in terms of the local convected coordinates. A formal proof of this physically intuitive result has been given by Belytschko and Hsieh [2].
Figures 3.1 and 3.2 depict the continuum [3], and corotational [tt], definitions respectively of the rotation of an infinitesimal region around any point P lying on the reference line of the beam or shell. It can be seen that the two definitions differ by an amount that is equal to one half of the average transverse engineering shear strain. In the context of small strain, large rotation analysis, this discrepancy is clearly negligible and does not invalidate the conclusion that a simple linear strain definition can be used provided the displacement gradients are referred to the proposed convected axes. Indeed, the adoption of asecond-order definition, would introduce terms proportional to the square of the engineering shear strains, and this is obviously inappropriate if one elects to work in the small strain range.
In order to distinguish the proposed approach from the standard updated Lagrangian method, it will be referred to
57
F I G U R E 3 ml L o c a l c o n n e c t e d a x e s b a s e d o n c o n t i n u u m m e c h a n i c s t r a n s f o r m a t i o n o f a n i n f i n i t e s i m a l r e g i o n a r o u n d P ( a f t e r B i o t [ 3 J > .
F I G U R E 3 . 2 L o c a l c o n n e c t e d a x e s b a s e d o n r i g i d b o d y m o t i o n o f r e f e r e n c e l ine o r m i d s u r f a c e at P ( a f t e r W e m p n e r ( C l ) .
58
here as a corotational formulation. Corotational finite element formulations have not been as widely exploited as their unconvected counterparts, although a number of examples can be found in the literature [2,4-18].
3-3 Trefftz buckling criterion
Consider a deformable body that is moved from a fundamental equilibrium state to a closely adjacent state that is not necessarily in equilibrium. The change in potential energy is given by
f i = u + v = n 1 + f i 2 + n 3 + ____ ( 3 . 4 )
where U denotes the internal strain energy, and V the potential energy of the applied loads. The components n 1 » n 2 , . . . , represent the aggregate of all terms that are linear, quadratic, etc., in the displacement changes that characterize the incremental motion.
Since the fundamental state is in equilibrium, then in accordance with the principle of virtual work, the linear term of equation (3*4) must vanish, that is,
rTi = 0 (3.5)
According to the stationary criterion of Trefftz [19], a critical instability develops when the first variation of n 2
vanishes, i.e.,
S ( n 2 ) = 0 ( 3 . 6 )
Thus, a necessary condition for the proper modelling of buckling problems is that n 2 should include all those terms that are quadratic in the generalized displacements. Provided Green’s strain tensor is used in unabridged form,then this condition is satisfied automatically for continuum
59
problems in which the generalized displacements consist only of the three translations ux, uy, uz of various points inside the body. However, in the case of the so-called degenerate theories of plates, shells and beams, the generalized displacements are the translations and rotations of the midsurface or reference line, and in order to obtain the motion of points lying away from this surface or line, a kinematic transformation is required. It follows that, in order to satisfy the Trefftz criterion, the kinematic relations that arise implicitly out of the degeneration process must also be quadratic. Specifically, when deriving the kinematic relations, the incremental rotation matrix R should be approximated to at least second-order.
If linear kinematic relations are used, or if significant quadratic terms are excluded, then, inevitably, the ability of the model to represent certain forms of buckling may be precluded or, at least, severely curtailed. This conclusion is important to the present work, since it provides a background against which to judge existing numerical models. In the finite element field, most degenerate models have been based on linear kinematics and consequently their ability to accurately predict instabilities across a broad spectrum is problematical.
3-4 Finite rotations in beams
The beam theory to be developed later will be based on the well-known Timoshenko model, i.e., an Euler beam coupled with a generalized transverse shear deformation theory [20]. The beam is also designed to model cross sectional warping due to twisting of the longitudinal reference line. However, it is reasonable to assume that transverse shear and warping are uncoupled, so for present purposes, one need only consider a standard Timoshenko beam. The basic assumptions are therefore that the cross section remains plane and undistorted but, in the presence of transverse shear, it may rotate out of the plane which is originally normal to the
60
longitudinal reference line.For curved or pretwisted beams, the assumption of plane
sections does not imply that the strains vary linearly over the cross section. In such cases, the actual distribution is influenced by the variation in both the length, and orientation, of longitudinal fibres over the cross section. However, these are second-order deformational effects, and, for present purposes, a planar strain distribution will be
• assumed. A refined theory that is not subject to thisrestriction is detailed in Chapter U.
3-4-1 Convected coordinates
Referring to Figure 3.2, it is clear that the transformation from the unconvected state (°r,°s,°t) to the convected state (r,s,t) must be considered in two stages, each involving a rigid Euler rotation. Now in accordance with the polar decomposition theorem, the total rotation 0 can be decomposed into a rigid body rotation § followed by a small transverse shear deformation j|, that is,
R( 0 ) = R(e)RCf') (3.7)
Hence, the required two-stage transformation can be written
T = R t (0 )R( 0 )°T = R(g)° T (3-8)
where
° T = {°r,°s,°t>~ ~ ~ (3.9)T = {r,s,t>
In the first stage, the orthogonal triad o o (0 o t ) iscarried to (*r,ms,mt ) by an Euler rotation 0 = ( 0 x * 0 y , 0Z > .Within the context of finite element analysis, thecomponents of 0 are appropriately adopted as generalized rotation coordinates. Therefore, the corresponding rotation
6 1
m a t r i x R ( 0 ) is d e t e r m i n a t e .In t h e s e c o n d stage, t h e t r i a d (* r ,m s ,m t ) is c a r r i e d to
( r ,s,t) by an E u l e r r o t a t i o n -j|. T h e l o c a l c o m p o n e n t s of J c a n be i d e n t i f i e d w i t h t h e a v e r a g e s h e a r d e f o r m a t i o n o f t h e c r o s s s e c t i o n , t h a t is,
eL = <o,es ,et > = {o E }rt r s ( 3 . 1 0 )
w h e r e E r s » e rt a r e e n g i n e e r i n g s h e a r s t r a i n m e a s u r e s . T h e v a n i s h i n g f i r s t c o m p o n e n t of § i m p l i e s t hat § is o r t h o g o n a l to b o t h r and *r. C o n s e q u e n t l y a n e x a c t d e f i n i t i o n of § is
AeA / 0e cos 1(r • *r) yx *r |rx *r| (3.11)
Here, *r is k n o w n f r o m t h e f i r s t s t a g e t r a n s f o r m a t i o n , a n d r can be f o u n d d i r e c t l y f r o m t h e c u r r e n t g e o m e t r y of t h e r e f e r e n c e l i n e u s i n g
r P » r ( 3 . 1 2 )
w h e r e p is t h e p o s i t i o n v e c t o r of t h e r e f e r e n c e p o i n t P w i t h/v
r e s p e c t to t h e f i x e d space.
3-4-2 Convected displacements
The decomposition of the overall motion of a thin walled beam or shell is illustrated in Figure 3-3. For clarity, the motion is depicted in the two-dimensional spaces x, z and r, t, but the physics of the actual three-dimensional motion is obtained by treating the various vectors shown in the figure as three-dimensional.
In t h e f i g u r e , t h e p o r t i o n o f the r e f e r e n c e l i n e o r m i d s u r f a c e l y i n g b e t w e e n t h e r e f e r e n c e p o i n t P a n d a t y p i c a l n o d e i, is s h o w n in f o u r d i f f e r e n t c o n f i g u r a t i o n s °C, aC, 2C a n d C r e s p e c t i v e l y . In t h i s way, t h e t o t a l m o t i o n is d e c o m p o s e d in a c c o r d a n c e w i t h the p o l a r d e c o m p o s i t i o n
62
z *rL
F I G U R E 3 . 3 D e c o m p o s it i o n o f t h e m o t i o n o f t h e r e f e r e n c e line o f a b e a m o r m i d s u r f a c e o f a s h e l l .
S t a g e Symb o lR e f e r e n c e p o i n t P No de i
D e s c r i p t i o nT r a n s l a t i o n R o t a t i o n T r a n s l a t i o n R o ta t i o n
”io___1
LJ LJJJ 0 u
H i 0 r i gi d t r a n s l a t i o n
’ C - 2 C /-N 0 f St Si r i gi d r o ta t i o n
1 /%y
M ^ u r e d e f o r m a t i o n ^
° c - C• UlU = U £ H i •0-. ** t total m o ti o n
T A B L E 3.1 D i s p l a c e m e n t s c o r r e s p o n d i n g to e a c h s t a g e o f t h e m o t i o n d e p i c t e d in F i g u r e 3 . 3 .
63
theorem into the stages detailed in Table 3.1.The convected nodal displacements are those associated
with the pure deformation shown shaded in Figure 3.3 and also in Table 3.1. Working from the triangle P-i-2i and using equation (3.8 ), one can write
u . = (p .-p ) - R ( 0 ) ( ° p .-° p )~ 1 1 /W - ~ ~ 1 <v (3.13)
Note that the rotation matrix R(0) is the same matrix thatis required to define the current convected coordinatesr, s, t (see equations (3.7) and (3.8)). Assuming that thecurrent coordinates p j( p and current displacements u,, u~ 1 ^ ~ 1 *+are known, the determination of the initial coordinates ° P i » °p is straightforward, i.e.,
°£i = P — u ~ i a i° p = P - U
(3.14)
The convected nodal rotations are obtained by removing the rigid body rotation 'o' from the total nodal rotations 9,.~ ~ 1This is best achieved by applying the compound rotation formulae (2.39) to each node in turn, using,
co = -sin(?/2)e~ Q ~ Q (3.15)0Jb = sin(0i/2)eb
where 'O' = | 0* | , 04 = |0 1, e = '§70', ew = 0^/0^. Anothermethod is to apply spectral analysis (see equations (2.11) to (2.13)) directly to the compound rotation matrix
R(0i) = R(0i)RT(§) (3.16)
Evidently, the use of the approximation
0, = 0 .-§ (3.17)
is appropriate only when the residual rotations § are small enough to be treated as vector quantities (say, < 0.02).
6 ll
In most existing finite element applications of the decomposition method, the approximation (3*17) is used with respect to a single reference point within the element (say the centroid). Clearly, this approach is valid only if the relative rotations within the element are small. An improved method, applicable to numerically integrated elements, is to introduce local convected coordinates at each integration station [21]. This expedient alleviates the error caused by
• the approximation (3.17) - thus, for example, the error willbe roughly halved in the case of a two-point Gauss rule. For maximum precision, however, it is quite feasible to apply the exact decomposition (3.16) at each integration station. In this case, the relative rotations within the element will be limited only by assumptions concerning the magnitude of material strains.
3-4-3 Kinematic relations
In practice it is necessary to cast the nonlinear equilibrium equations of the beam or shell in an incremental form. This leads to the consideration of three distinct configurations denoted by °C, C, and AC. As before, °C and C represent the initial (unloaded) and the currently known configurations respectively, whereas AC is the neighbouring, equilibrium configuration occupied after the application of the next load increment. Thus, the position vectors of the three configurations are related through
( 3 . 18 )
Consider now a single beam element having n nodes, andassume that the six convected nodal displacements u.» 0. (i^ 1 ** 1= l,2,...,n) are known for the current configuration C.Introducing a set of appropriate interpolation functions N t(r), the local displacements and displacement gradients at
6 5
any point P on the reference axis can be found using
<§L ’S L > = l Ti5 1N i ^ i » S 1 >
{SL,r’§L.r> = IT(3.19)
where T contains the direction cosines of the local axes r, s, t with respect to x, y, z respectively (see equation (3.8)). In accordance with the adopted definition of convected coordinates, the convected displacements and displacement gradients are
Au =~ L
ooo
%~ L, A A .{o.es,et}
Au = - l , r (Sr,r,0,o>0 ♦ r <6r.r>8s.r
With these preliminaries, the local convected displacements and displacement gradients at any point Q(0,s,t), can now be defined. Following the arguments of Section 3*2, and using the linear relation (2.22) for the local rotation R(©L) in conjunction with equation (2.40), gives the convected displacements
u l = {10s-s01,0,0>
and corivected displacement gradients (c.f
(3 .2 1 )
equation (2.4l))
ur,r + ^s,r-S§t, 1r 1 1 ® s
IQ
> r II -tBr.r 1 0 1 0 (3.22)S0r, r 1
1 0i ° -
where the four components A
u s . S » Ut,t» u s , t, and ut,s vanishdue to the assumption that the normal plane s-t remains undistorted.
In order to find the tangential stiffness of the element in the current configuration, it is necessary to calculate the internal increment of virtual work associated with the
66
incremental motion that carries C to AC. The corresponding incremental motion of the point Q(0,s,t) provides the basis for these calculations. Noting that the orthogonal axes *r, ms, mt should be used as the datum for the local incremental Euler rotation 0^, and applying equation (2.40), leads to the exact definition
u r 0= u s + [<r (|l )-i >r (0l )] s
• •Ut u t t
where ur, u s, ut are the incremental displacements at s = t = 0. The presence of the residual (shear) rotation matrix R(j§l) implies that, strictly speaking, there is a non-vanishing initial displacement contribution to thetangent stiffness. However, for small strain incremental analysis, this contribution will be negligible andconsequently the approximation R(,§L) = I can be used. This approximation is exploited in the works of Novozhilov [22], Wempner [4] , and Hughes and Liu [23] .
Putting R(|l) = I, and using the first-orderapproximation (2.22) to define R(0l), leads to
ur = ur+t0s-30tUS = O s-t0r (3.24)Ut = u t +10 r
On the other hand, the second-order approximation (2.23) gives
ur = ur + t ( 0s+^0r0t )-s ( 0 t-i$6r0s )us = us-t(0r-^0s0t)-%s(0r+0t) (3.25)U t = U t + S ( 0 r +j$0 s 0 t ) — i$t ( 0 * + 0 | )
Equations (3.25) can be simplified by taking into account the assumption that the cross section remains undeformed. For present purposes this assumption is taken to imply that <js = crt = CTst = 0 (actually, unless Poisson's ratio is zero,
67
the corresponding strains es, et» Est cannot also vanish, but the conflict is of little practical consequence). Hence, the incremental virtual work terms a s5es» <7tSet. crst&Es-t vanish, and it follows that the underlined quadratic terms in equations (3.25) can be dropped.
The nine displacement gradients at the point Q can be found by differentiating equations (3.25) with respect to r, s, and t respectively. Substitution of these gradients into Green’s strain tensor, leads, after the application of the principle of virtual work, to the linear and geometric stiffness matrices of the element.
The quadratic terms appearing in equations (3*25) are important, since they show that in finite element formulations that are degenerated from continuum models, geometrical nonlinearity arises not only from the quadratic terms of the Green strain tensor, but also from the nonlinearity of the kinematic relations themselves. Clearly, if the nonlinear terms in (3.25) are dropped, then the geometric stiffness will be incomplete and certain forms of behaviour will be precluded, or at least, poorly represented. Furthermore, as discussed in Section 3»3t "the use of linear kinematics violates the Trefftz buckling criterion, and the ability to accurately predict instabilities cannot then be guaranteed.
Experience with the curved beam element developed by the author has provided clear evidence that, for geometrically nonlinear problems in which flexure and torsion interact, use of the linear kinematic relations (3.24) can seriously impair the convergence characteristics of the numerical model. In addition, spectral analysis of various lateral and lateral torsional buckling models using (3.24) has been found to result in errors ranging from approximately 40% to 60X in the critical load levels. The results agree closely with those obtained in a similar study carried out by Argyris et al. [9].
In spite of these deficiencies, most existing finite element models for space beams having rotational degrees of freedom are based on a linear set of kinematic relations
68
such as (3.2/1), and therefore their ability to handle geometrically nonlinear problems is inevitably restricted.
The fact that the geometric stiffness of a space beam derived by the conventional virtual work method is incomplete, and can lead to serious errors in the stability analysis of simple beams and frames, is discussed in detail by Argyris et al. [9|24]. They point out that the internal moment vector predicted by the conventional method consists of a semitangential torque and two quasitangential moments, and discuss the consequent difficulties. In contrast, their own method, known as the natural approach, is based on three semitangential moments and leads to the complete geometric stiffness.
In reference [24], the corrections that should be applied to the conventional geometric stiffness are defined. These corrections are, however, linked to a-priori assumptions concerning the nature of both the rotations and internal moments, and do not appear to admit an explicit definition of the internal strain field that corresponds to the corrected stiffness. This contrasts with the present approach in which the only a-priori condition is that the relative motion of two points inside a rigid body subject to finite rotations, is represented uniquely to at least second-order. The internal strain field, tangential stiffness, and semitangential nature of the internal moments, then follow directly from a straightforward application of the principle of virtual work.
3-4-4 Updating the nodal displacements
Within each load or time step the equilibrium equations of the structure are solved iteratively, the process being discontinued only when the residual out-of-balance forces are sufficiently small. After each iteration, the total nodal displacements must be updated using the current incremental values. Evidently, updating the nodal translations is trivial, i.e.,
69
u ± =s y i+u1 (3. 26)
where the symbol =* Indicates replacement.It has already been shown that, in general, Euler
rotations are non-commutative. Therefore, in this case, the additive relation
6 l 0 +0 ~ i (3.27)
is inconsistent. The consistent approach, and one which generally improves numerical convergence, is to determine the accumulated value of Sj by spectral analysis of the compound rotation matrix R(01)R(01). In other words,
B(9i) - BCjjJRCej) (3-28)
should be used instead of the usual accumulation rule (3.27). However, it is worth noting that, in practice, the difference in convergence rate resulting from the use of the accumulation rules (3.27) and (3.28), is very often quite small.
3-4-5 Strain—displacement equations
The total strains at any point Q(0,s,t) are given by the linear corotational strain tensor
ei J = ^(ui,j+u3,i) (3*29)
Under the assumption of a planar strain distribution, equations (3*22) can be substituted directly in (3.29)i giving
70
6 r = ur , r + s , rE r s =
A A-0t-t0r> rE r t = A A0 S+S0r,r
Consequently, the sixare
A
£ r - Ur, r/SE r s = -0tAErt = 0s
A A
x r — ® r , rA A
* s = ® s , rA^ t = 9t,r
Inspite of theirallow the generalizedwith high precisionand rotations. It is :
equations (3.30) do
— s 0t, r(3.30)
(3.31)
of torsion because they do not comply with the well-known St. Venant condition that the shear stresses should vanish on the lateral surfaces of the beam. To overcome this difficulty, it is necessary to include cross sectional warping in the kinematical description. As mentioned earlier, a refined theory that accounts for sectional warping, along with other second-order deformational effects, is detailed in Chapter U .
The incremental strain field is calculated from Green's strain tensor (3*2), using the current configuration as reference. This gives
E r • = «r. r + H ( U r , r +Us,r + U t ,r )
(3.32)e r s = u r ,s + u s tr + u r #r^r . s "*"US ,r u s ,s + u t ,rU t , S• # • • • • #Ert = u r ,t+ u t #r+ u r #ru r ,t - u Sf ru s ,t + u t ,ru t , t
where, in accordance with current assumptions, the underlined terms may be omitted. Substitution of the nonlinear kinematic relations (3*25) into (3.32), leads to
71
the following explicit definition of the six generalized incremental strains:
= S r , r ^ ( U s , r + i S , r ) grs = U s , r - Q t ^ r , r Q t ^ t , r 8 r ^ r Q s g r t = 5 t , r + 0 s + G r , r 0 S - ^ , r 0 r + ^ r 0 t----- (3.33)X r - 9r t r 9 1Q s , r )X s = 6 s ,r+^ ( t ,r~®t®r,r )Xt — 0 t ,r+^ (®s®r,r-®r® s ,r )
To arrive at these results, terms in e2 and terms that are quadratic functions of the lateral coordinates, have been neglected, and use has been made of the approximations (4.88) (see Section 4.3.3). The symmetry of equations (3.33) is self-evident. Moreover, the fact that the beam curvatures can be written in the compact form:
X = 9 +J4(0, x0, ) (3.34)
turns out to have special significance to the virtual work equations of the beam (see Section 4.6).
If the linear kinematic relations (3.24) are used instead of the nonlinear relations (3.25), then the terms shown underlined in (3.33) vanish. The resultingstrain-displacement equations then coincide with those that are conventually used in nonlinear beam formulations. In this case, the corresponding potential will be incomplete in the sense that it does not fully comply with the Trefftz buckling criterion. As shown by Argyris [9,24], this deficiency leads to a formulation that is unable to represent common cases of general three-dimensional buckling with acceptable accuracy.
For finite element analysis, substitution of (3.19) into (3.33) leads to the relationship between the generalized strains and nodal freedoms, and this in turn allows the tangent stiffness of the element to be found.
72
3-5 Finite rotations in shells
For present purposes it is convenient to adopt a shell model that is kinematically consistent with the beam discussed in the foregoing section. Thus a Mindlin-type model is assumed, in which the cross sectional fibres that are originally normal to the undeformed midsurface, remain plane and unextended, but may deviate from the normal in the
• presence of transverse shear. In the finite element context,these assumptions lead naturally to a ’degenerate continuum' based formulation. Shell elements of this type have proved extremely successful in applications involving both geometric and material nonlinearity.
In order to achieve full compatibility, the shell element will be based on the same six nodal freedoms u 1> 01that were used for the beam. In this way, the two element types can be used together to represent stiffened plates and shells at a fraction of the cost of a full shell-shell idealization. Moreover, in addition to plates and smooth shells, the element can be used to model folded plates, multiple shell Junctions, box-sections etc..
3-5-1 Convected coordinates
The motion of the midsurface of a shell is illustrated in Figure 3.4. The natural curvilinear coordinates K* "n are introduced to provide a parametric description of the midsurface geometry. Normally the coordinates r and s can be chosen arbitrarily as long as they remain tangential to the midsurface at the reference point P. In this case, however, compliance with the polar decomposition theorem renders the choice unique. Thus the orientation of r and s is dictated by the continuum mechanics definition of the rigid body motion of an infinitesimal region of the midsurface at P. This condition requires that r, s are symmetrically placed with respect to tj (see Figure 3.4). Consequently, the convected coordinates are defined as
73
t = Cdexdn )/I (d^xd,)Is = (d>, + tx J e ) / I (dy,+txd e ) I (3-35)r = sxt
where d r , 1 are the unit covariant base vectors tangential'w Q ^7to ^ respectively.
F I G U R E 3 . 4 Hotiort o f t h e m i d s u r f a c e o f a s h e l l .
Note that equations (3.7), (3.8) and (3.9) that weregiven previously for the beam are also valid for the shell. However, equations (3*35) evidently render the more involved two-stage process that was needed in the former .case, redundant.
3-5-2 Convected displacements
In Section 3.4.2, explicit equations were derived for the calculation of the convected displacements in beams. These equations are also applicable to the shell case - indeed the entire section can be taken as valid for beams or shells.
It is worth noting that the local Euler rotation component 6t corresponds to the rigid body rotation of the shell element about the local axis t ( which is normal to the midsurface at the reference point P). Consequently, the classical conditions
are satisfied exactly at the Gauss points of the element. By implication, equation (3.10) becomes
3-5-3 Kinematic relations
Conceptually, the determination of the kinematics of a point Q(0,0,t) in the shell wall follows completely the procedure given in Section 3.4.3 for finding the kinematics of the point Q(0,s,t) in the cross section of the beam.
Replacing the beam interpolation functions (r) by N^rjS), the local displacements and displacement gradients become (c.f. equations (3.19))
? t = 0 t (3.36)01 = 0
L {er,es,o> = {-Est,Ert.o> (3.37)
(3.38)
and the corresponding convected displacements and
75
d i s p l a c e m e n t gr a d i e n t s are (c.f. e q u ations (3.20))
u L = {0.0.0}g L = {0r ,^s .O>Il.t = <«r.r.3Str.0>H l . s = ^u r . s «u s .s •0 } 5 L ,r = ^ ® r ,r *® s ,r •0} l L .s " <9r.s»9s.s.0>
(3.39)
The local convected displacements of the point P are (c.f. equation (3.21))
^ f A A .U L = { t 0 s , - t 0 r ,O} (3.40)
and the corresponding convected displacementgiven by (c.f. equation (3.22))
U r . r ^ s . r u r , s+t9s.s V)«D
n<0| “ s, r-^r. r u s , s-tOr,s A“Or0 0 0
gradients are
(3.41)
where u tft vanishes due to the assumption that normal fibres parallel to t are inextensional. The linear, incremental kinematic relations (3-24) become
ur = Or+t0su s = O s-t0r (3-42)u t =
and the second-order equations corresponding to (3.25) are
• I • • •ur = ur + t(0s+)$0r0t)u s = s- 1 ( 0 r - J$0s 0 t ) (3.43)u t = u t-i$t ( 0^ + 0| )
On account of the assumption a t = 0, the underlined term in the third of equations (3*43) can be dropped.
The truncation errors in equations (3.42) and (3.43) are
7 6
O(t02) and O(t03) respectively, which in turn implies that an accurate prediction of incremental strains can be achieved provided the range of application of the two equation sets is limited to small (0 < 0.04) and moderate (0 < 0.2) incremental rotations respectively. (Note that these limits are specified only in order to provide an approximate quantitative guide to the commonly employed terminology 'small* and 'moderate* rotation.)
3-5-4 Updating the nodal displacements
The relationships given previously in Section 3.4.4 are also applicable to the proposed shell model. This is true because both the shell and beam formulations are based on the same six global degrees of freedom.
As discussed later, it is also possible to develop an alternative shell model with five degrees of freedom, in which only the two local rotations of the midsurface normal, say a, 0, contribute to the potential energy. It is shown that a geometrically consistent theory is then only possible if the moving frame in which the rotation coordinates are measured, is also the convected frame that corresponds to the application of an Euler rotation {a,0,0} to the original frame (°r,°s,°t). A unique feature of this scheme is that accumulation of the nodal rotation parameters 0 t can be carried out in the static frame (°r,°s,°t). Consequently, in this case, the standard accumulation rule (3.28) can be used with 0i and 04 replaced by {ait0 itO} and {a^ B lf0} respectively.
The latter observations, provide another example of how uniqueness in the total potential function of structural components is crucially dependent on the proper representation of the changes in geometry that occur in any given motion.
77
Following the classical nonlinear thin shell theories of Novozhilov [22], Sanders [25]* Koiter [26], and Budiansky [27]» the internal strain field can be assumed to take the form:
E r = Er + tXr + tzT)r
3-5-5 Strain-displacement equations
Here, the parameters T)r, ns, and nrs account for the effects of fibre length variation over the wall thickness. However, for small strain analysis of thin shells, it is generally agreed that these effects are negligible. Therefore the underlined terms can be ignored. The resulting strain field thus consists of axial and transverse shear components that are linear and uniform respectively in the thickness coordinate t. The eight generalized strain components describing the incremental deformation of the midsurface evidently comprise stretching (er.Es»Ers), bending (Xr .Xs.Xrs), and tranverse shear (Ert.Est).
Discarding the quadratic terms in (3.44) is equivalent to the assumption of a linear variation in the displacement gradients. Consequently, as was the case for the beam, the corotational strain tensor (3.29) can be substituted directly into the displacement gradient matrix (3*41). This gives
(3.44)
uu r , r+ * e S ,r
A
s , s-^®r,s(3.45)
78
Comparing (3*44) and (3.45), the eight generalized strains can evidently be identified as
E r = A
u r , rE s = /\
s | sA A
E r s u r , s +S s ,rX r = A
0 s, r*s = A
- 0 r, sX r s = A
0 s,/s
s“ 0 r ,rE r t = A
0sE s t = A
(3-46)
Bearing in mind that, provided the strains remain small, these results are valid for displacements of arbitrary magnitude, the remarkable simplification that arises through the corotational method, is again in evidence.
In accordance with (3*2), the incremental Green strains referred to the current configuration, take the form:
E r = r + M u ? , r +U 2 +1*12 s , r + u t , rE s = “ s , s + H ( u J t S +U 2 + u 2s , s + u t , sE r s = s + u S f r + u r t r U r , S +1 s• • a * • •£ r t = u r , t + u t # r + U r t r u r , t + U s• • # # • •e s t ss u s . f * - u t , s + u r , s u r , t + U s
r U St s+ u t , r U t ,s (3-47)r^s.t+Ut,rUt,t s^s, t+u t ,Su t#t
Here, in accordance with the Mindlin-Reissner model, the underlined terms can be neglected.
Substitution of the nonlinear kinematic relations (3.43) into (3.47), now leads to an explicit definition of the eight generalized incremental strains. After dropping terms that are quadratic in the thickness coordinate t, one finds
79
E r E se'rsXr* s X r s
E r tS*t
u r
u,
“ u s
r+ u i fr+^ t . r )S +i$( U J ( S + U s, S + U t , S )
s + u s , r + u r , r u r , s + u s , r u s , s + u t , r u t , s r + u r t r ® s , r " u s , r ® r , r + % ^ r ^ t , r + t ^ r t r ) s + u r , s ® s » s — u s , s ® r , s ( ® s ® t , s + ® t ® s , s ) s + ® s , r + ^ r , r ® s , s + u r , s® s , r “ u s , s ® r , r » r ® r , s +% ^ r ^ t i r ^ t ® r , s + 8 1 ® s , r )r + 0 s+Gr .r©s-Qs, r0r+^0rQjs - e t^ u r t S e s- u S t se r+fee5e t
(3.48)
Note that the ±n-plane quadratic membrane terms (second term of Er and third term of e s ) are 0 (e 2 ). Within the context of small strain analysis of thin shells, these terms are clearly negligible and can be safely omitted.
It is interesting to note that geometrically nonlinear finite element models are usually based on linear kinematic relations of the form (3.42). This is equivalent to the a-priori assertion that 8 t « 0 r »0s , and leads to a five freedom strain field in which the terms shown underlined in(3.48) vanish. The corresponding potential will be incomplete in the sense that it does not fully comply with the Trefftz buckling criterion.
For finite element analysis, substitution of (3.38) into(3.48) leads to the relationship between the generalized strains and nodal freedoms, and this in turn allows the tangent stiffness of the element to be found.
3*5-6 Five versus six degrees of freedom
Following the discussion at the end of the previous section, it is of interest to investigate the possibility of formulating a five freedom incremental strain field of the same precision as the six freedom form (3.48). To achieve this, the motion of the midsurface normal can be decomposed into a rigid translation followed by an Euler rotation, say <Pl = {a, 0,0}, about an axis lying in the undeformed midsurface. Thus, <p = cpe, where <p is the resultant rotationrv, L /v 9
80
tp = (a2+p2)^ = cos-1(0t •t ) (3.49)
and e is the axis of rotation
e 0 txt (3.50)I°txt|
where t denotes the orientation of the ’normal* in the current configuration.
These definitions imply that cp must be referred to the convected orthonormal frame (r,s,t) given by
where it has been assumed, without loss of generality, that (r,s,t) coincides initially with (°r,°s,°t). Coincidence of (r,s,t) and (r,s,t) in C, occurs only in the special case when 0t, and the shear angles frs, Ert, are zero. Moreover, the new frame is unique in the sense that it is the only frame with respect to which the rotation component about t vanishes identically.
To within terms in tp3, the kinematic relationscorresponding to our five degree of freedom model are
ur = ur+tpus = u s-ta (3-52)u t = u t - h t ( « 2 +| 3 2 )
where, as before, the underlined term can be neglected. Comparing this result with equations (3.43), it is not difficult to see that a consistent five degree of freedom model can be evolved in which the incremental strain field is simpler than, but has the same precision as, the six freedom form (3.48).
An important corollary to these observations is that five degree of freedom shell models whose rotational coordinates are not referred to the convected triad (r,s,t), are strictly valid only for small incremental rotations, and lead to a potential which will not fully satisfy the Trefftz
(3.51)
81
criterion.Many nonlinear finite element formulations for the
analysis of shells can be found in the litrature. Many of these have five degrees of freedom and are of the so-called ’degenerate* type, that is, they include independent rotational parameters similar to a, 0, and admit the modelling of transverse shear deformations by the Reissner-Mindlin theory [23,28-35]. However, as far as the author is aware, none of these models make use of the convected frame (r,s,t), and therefore, they lead to equilibrium equations that will not be fully in accord with the requirements of the Trefftz criterion.
The practical consequences of using linear kinematic relations in frames other than (r,s,t) can be judged by recalling the fact that the resulting error corresponds to second-order flexural (bending and transverse shear) terms in the overall potential. The influence of such terms on the pre-buckling response of most shells is likely to be relatively small. The reason is simply that, prior to buckling, the potential is usually dominated by membrane energy. However, in the post-buckling range, which is characterized by the transfer of membrane into flexural energy, the influence of higher-order flexural terms can be significant. Furthermore, not all shells are dominated by membrane action in the pre-buckling regime. For example, the dominant action in helicoidal shells and open tubes is bending and twisting [36,37].
The generalized strain-displacement equations (3*48) can evidently be used to develop shell finite elements that have six degrees of freedom at each node. An important advantage of such elements is that, in principle, they are not limited to the analysis of smooth shells but can also be used to model discontinuous and stiffened systems. Basically, the need for six freedoms arises because the process of stiffness accumulation at any node must be carried out in a single reference frame. If a significant discontinuity exists accross the node, then it is clearly impossible to select a reference frame that exactly matches the
82
midsurfaces of all the elements meeting at this node. The natural choice in such cases is to cast the equilibrium equations in terms of six global freedoms, as advocated in Section 3.5•
The disadvantage of six degree of freedom shell models is that a rational method must be devised for dealing with the singularity that arises when adjacent elements of zero Gaussian curvature are exactly co-planar. The most common method of dealing with this problem is to add a small ficticious rotational spring orientated along the midsurface normal at each node. Unfortunately, this technique has the drawback that it interferes with the ability of the element to undergo strain free rigid body motion. A better method, and one that admits strain free rigid body motion, is to use the penalty function technique (see [38]) to link the incremental rotation 0t to the average incremental rotation of the midsurface, that is
0t = ^(u s,r—ur,s) (3*53)
This equation is valid for moderate incremental rotations [/l] . Note that in practice, in order to avoid singularity, the penalty constraint should be applied in such a way that that the associated 'drilling* stiffness is never identically zero. The technique achieves good compatibility (i.e., approaching C 1) along shell-shell and shell-stiffener interfaces.
Equations (3.35-3*41) and (3.4-3-3.48) were used by the author to develop an 8-noded shell element having six degrees of freedom at each node. To avoid singularity in the case of plate and shell regions that are initially flat, the element comprises a non-vanishing drilling stiffness that operates via the penalty function method.
Membrane and shear locking are eliminated by the use of reduced integration in conjunction with the so-called residual energy balancing method [39-4-2]. The generalized strains are interpolated in such a way that geometric truncation errors vanish identically at the 2x2 Gauss
83
quadrature positions, irrespective of the element shape. This leads to an exact representation of rigid body motion and constant strain states even in the case of curved or distorted elements. Moreover, the corotational method advocated in this chapter, ensures that the potential is represented with uniform precision for displacements and rotations of arbitrary magnitude.
The element can be used to analyse thin to medium-thick shells of diverse type, including stiffened plates and shells, folded plates, shell Junctions, box-sections, etc.. Although a full description of the element is not included here, it is expected that details of the formulation will be made available in the form of published papers.
3-6 Stress measures and stress accumulation
Stress predictions are obviously of major importance to any finite element formulation. In order that the predicted nonlinear response should be of high precision, it is essential that the adopted stress measures are objective with respect to rigid body rotation and that the stress and strain measures are work conjugate. As is well-known, the material response law is usually derived experimentally from an uniaxial test. Consequently, in order to avoid ambiguity, the stress and strain measures to which the results of such a test are referred, should also be defined. These guidelines represent the minimum requirements for large displacement finite strain analysis.
Provided the displacement gradients are everywhere small, the stress-strain law for an isotropic elastoplastic material, takes the classical form
c i J k 1 e k 1 (3. 54)
where is the Cauchy (true) stress tensor, ekl is thesmall (linear) strain tensor, and C ljkl is the symmetricelastoplastic constitutive tensor whose coefficents are
based on the Cauchy stress-logarithmic strain relation in simple tension.
Explicit definitions of C i3kl under plastic conditions are well established in the litrature and will not be repeated here. For metal plasticity, the so-called J2 deformation and J2 flow theories have found wide acceptance, although the latter theory is generally held to be the most realistic and reliable.
It is important to emphasize that the restriction to small displacement gradients is a stronger condition than small strains, since it also implies small rotations. Note, however, that the restriction arises only through the adoption of non-objective stress and strain measures, and that the coefficients of C i3kl actually remain valid even under finite strain conditions.
In practice, the constitutive equations (3-54) are frequently applied in the small strain, finite rotation, regime. It is well known that this approach is Justified for elastic and hyperelastic materials [33.^3], However, the same is not true for materials undergoing plastic straining. In particular, it is shown in ] , that specializationof finite strain formulations to the small strain case, does not lead to the conventional law C 1Jkl. Moreover, the additional terms are shown to be important in the common case of materials with a plastic hardening modulus that is small relative to current stress levels. According to McMeeking and Rice [4-4] , the terms also play a significant role in the elastoplastic buckling of slender structures.
Following [/iZi,45]t an objective constitutive lawsuitable for small strain and finite rotation analysis of elastoplastic materials, can be written
Tij = [C t j k ! ~h(a t k 5 j 1 +<J j k 8 i x 1 5 j k+a3iSik)]eki = c i3klEkl (3.55)
Here, the increments of 2nd Piola-Kirchhoff stress t 13 and Green strain ekl are objective with respect to finite rigid rotation of the material, and are work conjugate. Under
8 5
conditions of plastic flow, the coefficients of C iJkl are of the same order* as the hardening modulus of the material. Therefore, the Cauchy stress terms appearing in (3.55) can significantly influence the behaviour of materials with small or vanishing hardening modulus. Needless to say, most existing small strain and finite rotation formulations are based on the material tensor C iJkl rather than the modified tensor Jkl.
In references [44.A-5] it is shown that, under the restriction of small strains, the constitutive relations (3*55) are equivalent to the rate equations
T iJ= c iJkiDki (3.56)
where x* j is the Jaumann or corotational rate of Kirchhoff stress, and D kl is the rate of deformation tensor
Dki - v k,1+vi,k ) (3.57)
Here, v k>1 = 8vk/ax1 are instantaneous velocity gradients referred to the current configuration. These objective stress and strain measures are commonly employed in large strain elastoplastic analysis.
Further insight into the origin of the additional stress terms in (3.55), is gained by considering the deformation of an infinitesimal unit element loaded with a force P in direction 1. After deformation the side lengths of the element are l+e11# l+e22, l+e33. The Cauchy stress vector is {an ,0f 0} = {q21/ (l+e22 ) (l+e33), 0, 0>, where q la = P is the nominal stress. Since small elastic strains imply negligible volume change, q 21 and a xx are approximately related through
q n - a 11/(l + e 11) = crx x (l-e2 1+Q(e2 ) ) (3.58)
Consequently, for small incremental strains, one can write
qi 1 = = ^ u n “ ffn ) e ii = ^ u i i e ii (3*59)
86
which shows that the real and nominal tangent moduli! of the material differ by an amount equal to the Cauchy stress.
Writing C iJkl in the matrix form
C = C-C — — — <r ( 3 . 6 0 )
an explicit definition of for a beam is
Uor 2CTrs 2<*rtc = %— <r 0
sym. CTr
Similarly, for a shell,
2ar 0 ^rs CTrt 02 a s CTrs 0 <*st
C = — <r l4(crr+crs) %<*st JSCTrtsym. i$ar s
( 3 . 6 1 )
( 3 . 6 2 )
As noted by Biot [33 r the distinction between materialand geometric nonlinearity has no exact physicalinterpretation. Thus, if small changes in the assumedinterface between the two forms of nonlinearity are made ina consistent manner, the actual combined nonlinear responsewill be unaltered. Therefore, as an alternative to using themodified constitutive law C, it is also possible to treatthe correction matrix C as an initial stress effect. This- <rwas the approach adopted by McMeeking and Rice IUU], and leads to a modified geometric stiffness. For present purposes, the author has found it both convenient and efficient to base the beam and shell models directly on the modifified constitutive relations (3.60-3.62).
For finite element analysis, the equilibrium equations>can be derived from the principle of virtual work, and are usually written in a linearized incremental form suitable for solution by a predictor-corrector technique such as Newton-Raphson iteration. In the present instance, the
87
equations take the updated Lagrangian form
j k i®k l i s j dV = SAW ex-Jat j Set jdV (3.63)v v v
where SAWe is the external virtual work in AC, e id and )<1 j are the linear and nonlinear parts of the incremental Green strain tensor, and V is the current volume. Since this is an updated Lagrangian formulation, all quantities (other than
• the external virtual work) are referred to the current(known) configuration C. Note that the only non-standard aspect of these equations is the use of the modifiedconstitutive law C 1Jkl.
The Cauchy stress tensor figures predominantly on both sides of these equations (note that for hypoelasticmaterials, C iJkl depends on the history of Cauchy stress). However, whilst approximations to are admissible on theleft-hand side, it is clearly important that a ±3 in the last term on the right-hand side should be determined with high precision, since it is this term that controls the actual structural response.
For a continuum, the current Cauchy stresses can be found exactly by using equation (3.55) together with the accumulation rule
AXi j = crij+Tij (3.64)
and the transformation
= (aPn,/Pn' ) % , i AT1J4Xn,J (3.65)
where Apm /pm is the mass density ratio, and Tij are increments of second Piola-Kirchoff stress. However, this method is poorly suited to beam and shell models having rotational as well as translational freedoms. As explained in Section 3*2, the introduction of exact beam or shell kinematics into Green's strain tensor creates complications. Moreover, the transformation (3.65) is a computational overhead that should be avoided whenever possible.
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The corotational approach to stress accumulation offers an efficient and accurate alternative. To apply the method it is only necessary to write the basic relations (3’. 55) in the corotational form:
a i J = c i J k 1E k l
where
( 3 . 66 )
a i j = CTi J - a i JA „ „ _ i. r / A nK1 e k l “ E k l = u k ♦ 1 + U 1 . k ^ ~ ( u k , l + u l , k ) ]
(3.67)
For thin walled beam and shell models, objective definitions of the displacement gradients ukfl and u lfk have already been derived. Thus equations (3.66) and (3.67) provide a straightforward and high precision method of finding the current Cauchy stresses suitable for use on both the left- and right-hand sides of equations (3.63).
When the material is elastoplastic, the components of C i j k l are themselves a function of the current Cauchy stresses and , in this case, equation (3.66) is strictly valid only for infinitesimal strain increments ekl. This difficulty is easily overcome by writing
j - Jc iikidEki (3.68)
and applying the Euler forward integration method over a sufficient number, say n, of subincrements ekl/n. Note that when the incremental deformation results in a transition from elastic to elastoplastic behaviour, numerical integration is only required over the elastoplastic part of the total strain increment ekl.
The corotational method of stress accumulation has considerable physical and intuitive appeal. The stress and strain increments are measured in identical corotational frames which, in the case of thin walled structures, are practically coincident with the principal axes of strain. Since the corotational strain increments exclude the finite
89
rotation of the material, objectivity is self-evident. Furthermore, when observed in a corotational frame, incremental changes in the components of Cauchy stress can be directly accumulated using
AatJ = CTiJ+CTu (3.69)
A comparison of equations (3.5*0 and (3*66) suggests that the former is a special case of the latter in which rotations of the principal stretch directions are presumed to be small. To demonstrate that this is indeed true, consider the limiting situation in which these rotations vanish. Evidently, in this case, u t = u i and the second of equations (3.67) reduces to
Eki = J$(uk , x +ux t k ) (3.70)
But, in the absence of rigid rotation, this is the small strain tensor. Therefore ekl = ekl, and the assertion is verified.
3.7 On the definition of conservative moments
This investigation of finite rotations would not be complete without an examination of the corresponding work conjugate loads, i.e., applied concentrated moments. The potential energy associated with such loads must be included in the overall equilibrium equations of the structure, and the conditions under which these loads are conservative, requires detailed examination.
If all the loads acting on a structure are conservative (noncirculatory), then the equilibrium equations derived via the principle of virtual work will be symmetrical. This symmetry derives from the fact that conservative loads possess a unique (path-independant) potential function. In this case, the system as a whole is conservative in the classical sense of energy conservation, and loss of
90
stability will always take the form of static instability (divergence).
On the other hand, the presence of nonconservative (circulatory) loads will result in equilibrium equations that are, in general, nonsymmetric but can still, under special conditions, be symmetric [47,48]. In the context of finite element analysis, the path dependent nature of the external loading leads to an additional contribution to the tangent stiffness, known as the load correction or load stiffness matrix. Symmetry of the corrected stiffness matrix is then indicative of a system that behaves conservatively in a global sense even though individual external forces are nonconservative. Conversely, a nonsymmetrical stiffness is indicative of a system which is globally nonconservative in the classical sense. If the corrected stiffness is symmetric then instability will take the form of static instability (divergence), whereas, if the stiffness is nonsymmetric then loss of stability may either be of the static or dynamic (flutter) type.
Although the treatment of circulatory loads does not present any fundamental theoretical difficulties, it does require some radical changes to be made to any ’standard* (static) nonlinear finite element facility. Thus, for example, a nonsymmetric out-of-core equation solver must be used to replace the usual symmetric solver. Additionally, if the response in the post-flutter range is to be studied then a full nonlinear dynamic capability is required.
The present work is limited to the study of structures that are loaded conservatively. In this way, symmetry of the equilibrium equations is guaranteed under all boundary conditions, and instability is always of the static type.
As is well-known, concentrated forces, distributed pressures and body forces which act along fixed directions, and whose magnitude is Independent of the motion of the structure to which they are applied, are conservative. However, Ziegler [49] has shown that a concentrated moment acting about an axis fixed in space is nonconservative. As this result may appear surprising to some readers, Ziegler’s
91
explanation will be reproduced here. Consider a constant moment m acting about the fixed axis X lf as shown in Figure 3.5. Clearly, a rotation of n about X x , yields positive workW = mTt. However, two successive rotations of it about thefixed axes X 2» X 3 respectively, lead to the same finalposition of the body. but this time with zero work.Therefore, the work is path-dependent, and the moment is nonconservative.
X?, x3
m n
X3
W= 0
F I G U R E 3 . 5 The n o n c o n s e r v a t i v e n a t u r e o f a m o m e n t a c t i n g a b o u t a. f i x e d a x i s .
Now applying two successive rotations of it about the follower axes x 2, x 3 respectively, one finds once again that the work is path-dependent, and hence follower moments are also nonconservative.
These results lead naturally to the conclusion that, in
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order to be conservative, a moment must be applied in such a way that it partially follows the rotations of the structure at the point of application. Based on this idea, various alternative models of a conservative moment were introduced by Ziegler [50] as a prerequisite to his study of the torsional instability of shafts. Later, Argyris et al. [9] generalized Ziegler’s models to cover concentrated moments acting about an axis having an arbitrary inclination to the undisturbed structure. Unfortunately, Argyris et al. adopted the same terminology but altered the physical attributes of the earlier Ziegler models. Actually, the conflict is relatively unimportant, because, over the restricted rotation range for which they were designed, both the Ziegler and Argyris models are analytically equivalent.
As a prelude to the development of models that admit a conservative potential under arbitrary finite rotations, it will evidently be useful to look in greater detail at both the physical models and the corresponding moment vector transformations proposed by Ziegler and Argyris et al..
3-7-1 Ziegler's models
Three alternative methods of applying a conservative torque to a shaft were described by Ziegler [49.50]. The three methods, termed quasitangential, semitangential, and pseudotangential, are depicted in Figure 3.6.
To apply a quasitangential torque, a circular disc is rigidly connected to the shaft in a plane which is perpendicular to the shaft axis x x. For present purposes, it will be sufficient to assume that the axes Xj and X* coincide initially. Two strings wound around the disc are acted upon by equal and opposite forces of magnitude P, parallel to the global axis X 3. The strings are assumed to be of infinite length so that the direction of the forces P is unaffected by deformation of the shaft.
m = 2Pa
0 M = m {1, 0 , 0)
M= m { 1 , X i , 0 }
m = 4Pa
° M = m (1 , 0 , 0 }
M=m{1, JXi.fx;}
m = 2Pa
*M = m {1 , 0 . 0 )
M = m { cosoc, X i cos » + Xjsinoc, 0 }
(a) quasitangential (b) semitangential (c) pseudofangential
i«oG)i
FIGURE 3.6 Alternative models for the application of a conservative torque - after Ziegler [32]
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The addition of a second pair of forces P, applied by means of strings parallel to the global axis X 2, leads to a semitangential torque model.
Finally a pseudotangential torque corresponds to the application of two equal and opposite forces P acting at the ends of a rigid crossbar attached perpendicular to the shaft axis.
Ziegler's definitions of the moment vectors, which correspond to a given flexural deformation of the shaft in each of the three cases, are reproduced in Figure 3.6. The quantities X 2 and X 3 represent the slopes of the deformed axis of the shaft with respect to the fixed axes X 2 and X 3 respectively. In all cases, Ziegler assumed that the pre-buckling values of these slopes were small. Consequently, his moment vector expressions do not remain conservative in the presence of finite rotations.
Returning to the physical models themselves, it is easily verified that the pseudotangential model is in fact strictly conservative even in the presence of arbitrary spatial rotations. In contrast, the extent to which the remaining two models are conservative is not at all obvious, and a specific test must therefore be devised.
Suppose the quasitangential and semitangential models are subjected to two alternative finite rotation paths as follows (see Figure 3.7):
Path 1: rotate by l$n about x x and then by Jsit about -x3Path 2: rotate by i$Tl about x 2 and then by Vrt about x x
These paths are equivalent in the sense that they both lead to exactly the same final orientation of the associated follower axes x t. The calculation of the moment vectors and the corresponding work done for each path is straightforward and leads to the values shown in Table 3.2. The results provide a clear indication that, in the presence of finite rotations, both models are path-dependent and therefore nonconservative.
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Path 2
F I G U R E 3 . 7 Two e q u i v a l e n t f i n i t e r o t a t i o n p a t h s .
Model Path Moment vector M Work W
Quasitangential 1 m{1.0.0} Hmit2 m{0.-1,0 > 0
Semitangential 1 m{^, 0, -$£> WmTt2 m{0,-h,-%} mit
T A B L E 3 . 2 M o m e n t v e c t o r s a n d w o r k v a l u e s f o r t h e t w o f i n i t e r o t a t i o n p a t h s s h o w n in F i g u r e 3 . 7 ( °M - m i l , 0 , 0 } ) .
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3-7-2 Argyris’s models
The two models Introduced by Argyris [9], are shown in Figure 3.8, together with the corresponding moment vector definitions that are actually used in his finite element formulation. As mentioned earlier, the notation for these models conflicts with the notation introduced by Ziegler. However, the small rotation approximations used by Argyris to represent the moment vectors in the quasitangential and semitangential models respectively do infact agree with the corresponding definitions used by Ziegler (this follows because for small rotations = cx3 and x; = -oc2).Consequently, the critical torsional instability values for shafts under various boundary conditions, are the same for both the Ziegler and Argyris models.
(a) Quasitangential (b) Semitangential
F I G U R E 3 . 6 A l t e r n a t i v e m o d e l s f o r t h e a p p l i c a t i o n o f a c o n s e r v a t i v e t o r q u e — a f t e r A r g y r i s [9]
In Figure 3.8, the loads P are conservative and are applied at the ends of rigid crossbars which are rigidly attached perpendicular to the shaft. The conservative nature of these models is self-evident. However, it is equally
97
clear* that the approximate moment vector definitions used by Argyris cannot be exactly conservative unless the rotations are infinitesimal.
For the more general case of a moment m applied semitangentially about an inclined axis e , Argyris derives the following small rotation transformation
M = °M + 55(ax°M) (3.71)
where the initial moment vector °M is equal to me. In the quasitangential case the transformed moment depends not only on °M and a, but on the direction of the crossbar as well, so that a simple vector relation analogous to (3.71) does not exist.
3-7*3 Exact models
Within the context of an iterative self-correcting solution of the equilibrium equations (3.63), it is usual to specify that the external load vector corresponding to the external virtual work SAWex should be defined exactly, or at least to within a fixed precision. The deformation independent character of most conservative loads means that, usually, this condition is easily satisfied. However, it has been demonstrated above that conservative moment couples are exceptional in that they display a marked dependence on rotation. Therefore, in this section, moment vector-rotation relations are sought that remain exactly conservative in the presence of arbitrary rotations. (The approximate transformation (3.71) is not suitable for this purpose because it not valid for finite rotations).
Two alternative models of an exactly conservative moment vector are depicted in Figure 3.9* As before, the directions of the forces that are applied to the rigid crossbars, are assumed to be independent of any subsequent rotation of these bars. Thus, it is immediately apparent that, provided exact moment vector-rotation tranformations are used, the
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models will remain exactly conservative in the presence of finite rotations. For complete generality, the models are orientated with respect to an arbitrary set of orthogonal unit vectors i, d, k.
The first model, denoted GE for 'generalized exact *, can be thought of as the basic component from which any arbitrary method of applying a conservative moment can be built. Generality is assured by the fact that the crossbar and the force vector can initially lie along any pair of directions.
The second model, denoted STE for 'semitangential exact', is studied because it exhibits certain features that are not found in any other alternative method. The full significance of semitangential moments should become clear from the discussion that follows, both in this section, and the next.
°M = 2a(jxP) °M = lXa.P±
(a) Model GE (b) Model STE
F I G U R E 3 . 9 A l t e r n a t i v e c o n c e p t u a l m o d e l s f o r t h e a p p l i c a t i o n o f a n e x a c t l y c o n s e r v a t i v e m o m e n t v e c t o r .
The transformed moment vector M is obtained by applying a Euler rotation a to each model in turn. For model GE, thisA/ W
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results in,
M = 2a(j*xP) = 2a[R(a)jxP] (3-72)
where denotes the orientation taken up by the crossbar as a result of the rotation a. Similarly, for model STE,
M = 2aP(d*xk-k*x;J) = 2aP[R(a)dxk-R(a)kxd] (3-73)^ «v 'V /V /y • #V ^
At first sight, it would appear that the transformedmoment vector for both models is dependent on the initial orientation of the crossbar. It turns out however, that although it is impossible to remove this dependence from the model GE, the same is not true for model STE. In order to demonstrate this property, the rotation matrix R is first written in the expanded form
1*11 **12 **131*21 **22 **23r31 **32 ** 3 3
(3.7*0
Substitution of equation (3.7*0 into (3*73), leads, after considerable manipulation, to
M = 2aP2 2 -r -r
+ **3 3 "**2 1 "**3 11 2 ** 1 1 "*"**3 3 — ** 3 2 i (3.75)1 3 "**23 **ll+**22
However, since the initial moment vector takes the value
'M = 4aPi (3.76)
equation (3.7*1) can be written
** 2 2 + ** 3 3 -**21 — **3 1M = H -**12 ** 1 l+**33 — ** 3 2
"**13 — ** 2 3 r 11 +r’M = T ~ — s — (3.77)
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This result comfirms the fact that moment vector transformation corresponding to model STE is independent of the directions of the crossbars. It follows that (3.77) is also true for n crossbars arranged symmetrically at an angular spacing of Tt/n where n = 2, ll, 6, . . . , oo (°M = 2naPi). In the interesting limiting case of n = oo, the model can be visualized as a circular disc of radius a subjected to distributed tractions p which remain strictly tangential to the initial boundary of the disc. In this case the initial moment vector is given by
°M = 2Ttazpi (3-78)
For future reference, a linearised version of the transformation (3.77), corresponding to a small Eulerrotation a, will be needed. To this end, the first-order approximation (2.22) is appropriate, i.e.,
R(oc) = I+S(ot) (3.79)
leading to
M = (I+H§(a)]°M = °M+Js(ax°M) (3.80)
The significance of semitangential moments has been discussed in considerable detail by Argyris et al. (9*10,2/1]. They show that the semitangential model leads to incremental potential energy terms that are in exact correspondence with those arising from the internal stress field. This correspondence is a direct, and unique, consequence of using Euler rotation components as the basis for calculating both the internal and external potential and, providing the loading is conservative, leads to tangent stiffness matrices that are symmetric. Equivalently, load correction matrices are needed only in the case where external moments are not applied in a semitangential manner. The particular form of symmetric correction required in the case of model GE is derived in the next section.
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The space beam formulations presented In references C9.24] are contingent, either implicitly or explicitly, on a semitangential definition of the internal moment field. In Chapter Ut it will be demonstrated that an a-priori assumption concerning the nature of the internal moment field in a space beam is not strictly necessary. Provided the nonlinear kinematic relations of Section 3*^*3 are used as the basis for the virtual work derivation of the equilibrium equations, then the true nature of these moments is revealed automatically.
Model GE is undoubtedly the most useful when it comes to representing the external virtual work for finite element analysis. Infact, by adding together several models of the GE type at different orientations, practically all methods of generating a conservative couple are admitted, including mechanical linkages and couplings etc.. In the special case where the overall structure is partitioned into two or more substructures, then the stress resultants that are applied at the position of each cut must be of the STE type. In any event, for nonlinear finite element analysis, any externally applied conservative moment vector, be it of the GE or STE type, should be continuously updated in accordance with (3«72) or (3«77) respectively.
Conservative body forces, surface pressures, line loads, and concentrated loads, applied eccentrically to the reference line of a beam or midsurface of a shell, can all be statically reduced by standard techniques to a single equivalent force vector and moment vector at each individual node. The total moment vector at any point must then be understood to comprise the requisite number of GE-type models with orientations and lever arms corresponding to the individual components of the actual load system. Then, by applying the transformation (3.72) to each component in turn, it is possible to take exact account of the rotation-dependent changes in external potential energy that occur during the overall motion of the structure.
The ability to identify the method by which moments are transmitted is essential if an adequate prediction of
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critical buckling loads is to be made. Considerable errors can accrue from a failure to properly account for the actual method of application of a couple. Thus, for example, in a study of the buckling of a simple right-angle frame, Argyris et al. [9] found that when the free end was loaded by three alternative forms of conservative couple, the corresponding critical load levels were approximately in the ratio 0.5: 1.0: 3.4.
The relationships derived in this section have been expressed in terms of an Euler rotation a applied to the rigid crossbars of the various models. For finite element analysis, the crossbars can evidently be identified with either the convected axes r,s,t or the materially-attached axes *rfTnsfmt. Actually, although the latter alternative is formally correct, the error introduced by using the convected axes r,s,t as reference will be negligible because the difference between the two frames is due only to transverse shear strain. Therefore, the replacement
a =* 0 (3.81)
is recommended whilst noting that can be substituted for 0, if this is more convenient.
3-7-4 Load correction matrix for GE-type moment vectors
Before proceeding with a formal derivation of the load correction matrix, it is worth clarifying, still further, some of the issues involved. First is is necessary to draw a distinction between the terminology 1 path-dependence * and 'motion-dependence1. (Very often, the latter term is amgiguously called 'deformation-dependence'. In the present work, however, the word deformation refers exlusively to the small changes of shape that result from internal straining, whereas the word motion refers to the absolute movement of individual particles.)
For present purposes, it is sufficient to consider the
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most common source of motion-dependence, that Is, rotation-dependence. Within this category, concentrated moment vectors that are generated by applying conservative forces to the ends of rigid crossbars, exhibit the unusual property of being both rotation-dependent and conservative. In fact, a still stronger principle holds, namely that moment vectors generated in the aforementioned manner cannot remain conservative unless they are also rotation-dependent and vice-versa.
Thus, it can be seen that although path-independent load systems are evidently conservative, they are not necessarily motion-independent. Since load correction matrices arise out of motion- and not path-dependence, the fact that a load system is conservative does not necessarily mean that the correction is not needed. On the other hand, in cases where the load is both conservative and rotation-dependent, such as is the case for conservative moment vectors, the requisite correction matrix will be symmetric.
What is the physical origin of the load correction matrix? The uncorrected geometric stiffness contains terms that correspond to the incremental changes in internal moments that occur at a given point as a result of the incremental rotation of this point. Now consider the case where the point is loaded externally by a concentrated moment vector, so that both the internal and external load vectors experience identical incremental rotations. Under these conditions, correspondence between the internal and external load vectors at the end of the increment will occur only in the special case when both vectors exhibit identical rotation-dependence. (In practice, of course, an iterative solution to reach equilibrium is still required because of linearisation errors, inherent in the tangential stiffness.) Therefore, a necessary and sufficient condition for the original uncorrected geometric stiffness to be valid, is simply that the external moment vector should be of the STE-type. Equivalently, applied moment vectors of any other type will create a residual incremental moment vector that depends on the incremental rotation of the point. The local
ion
change of stiffness is then a reflection of the internal adjustment that must be made to accomodate these residual moments and to hence bring the point into equilibrium.
With these preliminaries, it is clear that thederivation of the load correction matrix associated with GE-type moment vectors actually entails the consideration of a new composite model, this being the difference between model GE and model STE. The first step is to match the moment vectors of each model in the current configuration C, but using the loads AP that correspond to the adjacent incremented configuration AC. This is achieved by writing
M = M = M = 2a(J x AP ) (3.82)~ S ~ g n, ~
where the suffices s and g refer to the STE and GE models respectively.
The incremental changes in the moment vector of the composite system now derive from
M s = M -M~g ~K ~ s
where
M = AM -m~g ~g ~M~ s = am -m~ S 'V'
Further, let
M = 2aAPA/ /V
So that, observing (3.82),
(3.83)
(3.8ft)
(3.85)
M = JxM ( 3 . 86 )
According to the Trefftz criterion, linearapproximations of the incremental moment vectors will suffice because these lead to quadratic potential energy
. Therefore, adopting the linear approximation (3-79)terms
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for R(a), one obtains
M = S(a)jxM'V-g — ~ ~ ~
M_ = H(axM)<V 5(3.87)
where a is the incremental Euler rotation of the loaded point between configurations C and AC. Expanding (3.87), and observing (3.86), now leads directly to the desired incremental moment-rotation relation
M' = k , a- l C - M (3 .88)
with the load correction matrix k, given by— 1 C
-J 3M 3-J 2M z ( hi d ) 1| hi. J xMg+jgMj )k = — l c 11 hi d2M 3-*-J3M 2 )
sym. 1 ~J 2^2 — d 3M 3
The finite element equilibrium equations for a structure subject to GE-type moments at one or more nodes, can now be written
R -R, ~ e x /v i n (3-90)
Here, K x, Kn , and K lc are the linear, geometric, and load correction stiffnesses respectively, and A is the vector of incremental nodal freedoms. R is the vector of equivalent^ 6 Xexternal nodal loads, and R, the vector of internal nodal' ~ i nforces in equilibrium with the current internal stressfield. When GE-type moments are applied to one or more nodes, the requisite components of R ex should be updated in accordance with the exact transformation (3.72).
Observe that K x is based on the updated geometry of configuration C, and that Kn1 derives from the nonlinear kinematic relations (3.25 or 3*^3) together with thequadratic terms of Green's strain expansion. The latter stiffness corresponds with the application of external
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semitangential moment couples in the sense described above. K, denotes the assembled load correction matrix: itconsists of one or more 3^3 sub-matrices, of the form (3.89). located on the main diagonal. The position of each sub-matrix corresponds with the rotational freedoms of each node subject to a GE-type moment. Evidently, the correction matrix is symmetric and highly sparse. This sparsity, together with the fact that equation (3.89) relates directly
• to the global axes, means that, in practice, the correctionto the tangential stiffness is relatively straightforward and efficient.
Although not necessary here, extension of the techniques introduced in this section to include nonconservative moment vectors, such as fixed and follower moments, does not present any difficulties. Indeed all that is necessary is to replace (3.72) and (3.89) by new relations that reflect the rotation dependency of these forms of moment. Note however, that in this case, the requisite load correction matrices will be nonsymmetric.
3-7-5 On the application of concentrated moments to shells
For space beams, equations (3.72) and (3.77) can be applied in conjunction with any arbitrary moment vector °M. However, smooth shell formulations cannot usually support moment couples which act about the midsurface normal. Therefore, in this case, the vectors °M and M must both be tangential to the midsurface, i.e.,
°M-°t = M •t = 0 (3.91)A/ -"V A/ A#
where °t and t denote the initial and current orientations of the shell normal. In general, equation (3*91) is evidently not satisfied for applied moments of the semitangential type, unless the rotation at the point of application is infinitesimal. However, a GE-type moment,
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applied using a rigid lever initially orientated along the normal °t, is admissible. Clearly, equation (3.91) will then be satisfied provided the resulting vector is updated using the exact transformation (3.72).
Nonlinear theories which admit the application of arbitrary moment vectors to smooth shells have been developed - notable examples are given in references [51-5^]. To the author's knowledge, however, such theories have not yet been successively implemented into the numerical formulations of thin walled shells. Indeed, the development of robust general purpose shell elements capable of modelling both smooth and discontinuous shells, and of resisting moment couples turning about the midsurface normal, must surely stand as an important unresolved challenge.
3«8 Concluding remarks
The relationships derived sound basis for the finite el structures in the large displac These finite elements will features that are rarely formulations. These are:
in this chapter provide a ement analysis of thin walled ement, small strain range,
exhibit several important available in alternative
1
2
3
Euler rotations are used as generalized rotation coordinates throughout the derivation of the equilibrium equations.The internal force field that is limited only by the the shape functions that These forces are a unique displacement parameters and presence of arbitrary rigid External moment couples are renders them truly path-
will exhibit an accuracy kinematic assumpions and are used for the model, function of the nodal
vanish identically in the body motion.applied in a manner which
independent (conservative)even in the presence of arbitrary finite rotations
108
l l . U n d e r c o n s e r v a t i v e e x t e r n a l l o a d i n g t h e t a n g e n t s t i f f n e s s m a t r i x d e r i v e s f r o m a p o t e n t i a l f u n c t i o n t h a t is u n i q u e , s y m m e t r i c a n d s a t i s f i e s t h e p r e r e q u i s i t e s o f t h e t h e T r e f f t z b u c k l i n g c r i t e r i o n .
5. The equilibrium path pertaining to the incremental/iterative solution of a geometrically nonlinear problem will be independent of the selected incremental step size. Provided certain conditions are met, then it is possible to also achieve step size independence (to within an appropriate tolerence) in the presence of material nonlinearity.
It is o f c o u r s e p o s s i b l e to m o d e l t h e k i n e m a t i c a l b e h a v i o u r o f t h i n w a l l e d s t r u c t u r e s u s i n g g e n e r a l i z e d f r e e d o m s o t h e r t h a n E u l e r r o t a t i o n s . H o w e v e r , o t h e r c h o i c e s , s u c h as E u l e r o r m o d i f i e d E u l e r a n g l e s , w i l l l e a d to e q u i l i b r i u m e q u a t i o n s w h i c h d o n o t f u l l y s a t i s f y t h e n e c e s s a r y c o n d i t i o n s of s y m m e t r y a n d u n i q u e n e s s . T h e r e a s o n l i e s e s s e n t i a l l y in t h e n o n - v e c t o r i a l c h a r a c t e r of t h e s e f o r m s o f r o t a t i o n w h i c h r e n d e r t h e m n o n - c o m m u t a t i v e a n d d e p e n d e n t o n t h e s e q u e n c e o f a p p l i c a t i o n . In c o n t r a s t , t h e v e c t o r i a l c h a r a c t e r of an E u l e r r o t a t i o n m e a n s t h a t its c o m p o n e n t s m u s t be c o n s i d e r e d to o c c u r s i m u l t a n e o u s l y , a n d t h e r e f o r e t h e q u e s t i o n of s e q u e n c e b e c o m e s m e a n i n g l e s s .
A v a r i a n t o f t h e s t a n d a r d E u l e r r o t a t i o n t h e o r y , d e r i v e s f r o m the u s e o f f o u r g e n e r a l i z e d r o t a t i o n c o o r d i n a t e s , k n o w n as E u l e r p a r a m e t e r s . T h e s e a r e d e f i n e d as
ip = c o s a / 20tp = e s i n a / 21 itp = e s i n a / 22 2ip = e , s i n a / 23 d
(3.92)
w h e r e t h e f o l l o w i n g c o n s t r a i n t c o n d i t i o n m u s t be used:
<Po+<Pi+<P2+<Pi = 1 (3-93)
If t h e s e p a r a m e t e r s a r e u s e d in c o n j u n c t i o n w i t h the b a s i c
109
transformation (2.15), they lead to a rotation matrix RC^j) whose components are quadratic functions of <1 (i=0,l,2,3).This matrix exhibits precisely the same kind of symmetry as the Euler rotation form (2.21). It follows that Euler rotations and Euler parameters are simply two alternative algebraic representations of Euler’s rigid body rotation theorem. Properly formulated, both representations should lead to identical results.
Details of the way in which the Euler parameter method can be applied to the finite element analysis of structural systems, can be found in the recent papers by Besseling [55] and Geradin et al. [56]. However, a study of these papers reveals several significant disadvantages that do not arise in the proposed Euler rotation approach. These are:
1. The tangent stiffness matrix for the structure is not positive definite in any configuration. This arises because only three of the four Euler parameters are linearly independent.
2. In order to implement the constraint condition, special methods such as Lagrange multipliers, lambda elements or penalty functions are needed.
3. The number of generalized rotation coordinates associated with each free node of a finite element mesh is increased from 3 to 4. Still more equations are used if the system matrix is augmented by the Lagrange constraints
It should be pointed out, however, that if complex constraints (links, couplings, Hooke Joints, mechanisms, etc.) constitute a basic requirement of the finite element program, then the Lagrange multiplier method may already be available. Also, according to Besseling [55], the presence of zeros on the leading diagonal of the system matrix does not lead to any special problems with the decomposition. Thus, the biggest disadvantage of the method appears to lie in the apparently unecessary use of additional freedoms. For finite element analysis in general, and especially for
nonlinear analysis where efficiency is at a high premium, there appears to be little Justification for introducing these additional freedoms.
References
[1] MALVERN, L.E., "Introduction to the mechanics of a continuous medium", Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969.
[2] BELYTSCHKO, T. and HSIEH, B.J., "Non-linear transient finite element analysis with convected coordinates", Int. J. Num. Meth. Engrg., Vol. 7, 1973, PP. 255-271.
[3] BIOT, M.A., "Mechanics of incremental deformation”, John Wiley & Sons, New York, 1965.
[4] WEMPNER, G., "Finite elements, finite rotations and small strains of flexible shells", Int. J. Solids and Struct., Vol. 5, 1969, PP. 117-153.
[5] BATHE, K.-J. and BOLOURCHI, S., "Large displacement analysis of three-dimensional beam structures", Int. J. Num. Meth. Engrg. , Vol. 14, 1979, PP* 961-986.
[6] MURRAY, D.W. and WILSON, E.L., "Finite-element large deflection analysis of plates", Proc. ASCE, Engrg. Mech. Div., 1969, PP. 143-165.
[7] MATSUI, T. and MATSUOKA, O., "A new finite elementscheme for instability analysis of thin shells", Int. J. Num. Meth. Engrg., Vol. 10, 1976, pp. 145-170.
[8] BELYTSCHKO, T. and GLAUM, L., "Applications of higher-order corotational formulations for nonlinear finite element analysis". Comp, and Struct., Vol. 10, 1979, PP. 175-182.
[9] ARGYRIS, J.H., DUNNE, P.C., and SCHARPF, D.W., "Onlarge displacement-small strain analysis of structures with rotational degrees of freedom", Comp. Meth. Appl. Mech. Engrg., Vol. 14, 1978, PP. 401-451; Vol. 15,1978, pp. 99-135.
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[10] ARGYRIS, J.H., BALMER, H. , DOLTSINIS, J. St., DUNNE,P.C., HAASE. M. , KLEIBER, M., MALEJANNAKIS, G.A.,MLEJNEK, H.-P., MULLER, M . , and SCHARPF, D.W., "Finite element method - the natural approach", Comp. Meth. Appl. Mech. Engrg., Vols. 17/18, 1979, PP. 1-106.
[11] YOSHIDA, Y. and NOMURA, T., "A formulation and solutionprocedure for post-buckling of thin-walled structures", Comp. Meth. Appl. Mech. Engrg., Vol. 32, 1982, pp.285-309.
[12] MARTINS, R.A.F. and OWEN, D.R.J., "Elastoplastic andgeometrically nonlinear thin shell analysis by the semiloof element", Comp, and Struct., Vol. 13. 1981,PP. 505-513.
[13] B A T H E , K.-J. and HO, L.W., " A simple and effectiveelement for analysis of general shell structures", Comp, and Struct., Vol. 13, 1981, pp. 673-681
[14] HORRIGMOE, G. and BERGAN, P.G., "Instability analysisof free form shells by flat finite elements", Comp. Meth. Appl. Mech. Engrg., Vol. 16, 1978, pp. 11-35-
[15] RANKIN, C.C. and BROGAN, F.A., "An element independentcorotational procedure for the treatment of large rotations", in: Collapse Analysis of Structures, (L.H.Sobel and K. Thomas, eds.), ASME, New York, 1984, pp. 85-100.
[16] BELYTSCHKO, T., SCHWER, L., and KLEIN, M.J., "Largedisplacement transient analysis of space frames", Int. J. Num. Meth. Engrg., Vol. 11, 1977, PP. 64-84.
[17] KARAMANLIDIS, D., HONECKER, A., and KNOTHE, K., "Largedeflection finite element analysis of pre- andpost-critical reponse of thin elastic frames", in: Nonlinear Finite Element Analysis in StructuralMechanics (W. Wunderlich, E. Stein, and K.-J. Bathe,eds.), Springer-Verlag, 1981.
[18] RAMM, E. and OSTERRIEDER, P., "Ultimate load analysis of three-dimensional beam structures with thin-walled cross sections using finite elements", Int. Conf. on Stability of Metal Structures, Paris, Nov. 16-17, 1983.
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[19] TREFFTZ, E., "Zur theorie der stabllitat der elastichen gleichgewichts”, Zeit. Angew. Math. Mech., Vol. 13, 1933, PP. 160-165.
[20] TIMOSHENKO, S.P. and GERE, J.M., "Mechanics ofmaterials”, Van Nostrand Relnhold Co., London, 1972.
[21] HIBBIT, H.D., MARCAL, P.V., and RICE, J.R., ”A finite element formulation for problems of large strain and large displacement”, Int, J. Solids and Struct., Vol.6, 1970, pp. 1069-1086.
[22] NOVOZHILOV, V.V., "Foundations of the nonlinear theory of elasticity”, Graylock Press, New York, 1953.
[ 2 3 ] HUGHES, T.J.R. and LIU, W.K., "Nonlinear finite elementanalysis of shells: Part I. Three-dimensional shells”,Comp. Meth. Appl. Mech. Engrg., Vol. 26, 1981, pp.331-362.
[24] ARGRYRIS, J.H., HILPERT, O., MALEJANNAKIS, G.A., andSCHARPF, D.W., ”On the geometrical stiffness of a beam in space - a consistant V.W. approach”, Comp. Meth. Appl. Mech. Engrg., Vol. 20, 1979, PP. 105-131.
[25] SANDERS, J.L., "Nonlinear theories for thin shells”, Quarterly of Applied Mathematics”, Vol. 21, 1963, PP. 21 - 3 6 .
[26] KOITER, W.T., "On the nonlinear theory of thin elastic shells”, Proc., Kon. Ned. Ak. Wet., Series B, Vol. 69, 1966, pp. 1-54.
[27] BUDIANSKY, B., "Notes on nonlinear shell theory", J.Appl. Mech., ASME, Vol. 35, 1968, pp. 393-401.
[28] RAMM, E., ”A plate/shell element for large deflectionsand rotations", in: Formulations and ComputationalAlgorithms in Finite Element Analysis, (K.-J. Bathe, J.T. Oden and W. Wunderlich, eds.), M.I.T. Press, Cambridge, Mass., 1977.
[29] PARISCH, H., "Geometrical nonlinear analysis ofshells”. Comp. Meth. Appl. Mech. Engrg., Vol. 14, 1978,PP. 159-178.
[30] PARISCH, H., "Large displacements of shells includingmaterial nonlinearities”, Comp. Meth. Appl. Mech. Engrg., Vol. 27, 1981, PP. 183-214.
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[31] SURANA, K.S., "Geometrically nonlinear formulation for curved shell elements", Int. J. Num. Meth. Engrg., Vol. 19, 1983, PP. 581-615.
[32] OLIVER, J. and ONATE, E., "A total Lagrangianformulation for the geometrically nonlinear analysis of structures using finite elements, Part I.Two-dimensional problems: shell and plate structures",Int. J. Num. Meth. Engrg., Vol. 20, 1984, pp.2253-2 2 8 1 .
[33] KRAKELAND, B., "Large displacement analysis of shellsconsidering elastic-plastic and elastic-viscoplastic materials", Report No. 77-6, Division of Structural Mechanics, The Norwegian Institute of Technology,Trondheim, Dec. 1977.
[34] BATHE, K.-J. and BOLOURCHI, S.. "A geometric andmaterial nonlinear plate and shell element", Comp, and Struct., Vol. 11, 1980, pp. 23-48.
[35] DVORKIN, E.N. and BATHE, K.-J., "A continuum mechanicsbased four-node shell element for general non-linearanalysis", Engrg. Comput., Vol. 1, 1984, pp. 77-88.
[36] COHEN, J.W., "The inadequacy of classical stress-strain relations for the right helicoidal shell", Proc. IUTAM Symp. on Theory of Thin Elastic Shells, North-Holland Publishing Co., Amsterdam, i960, pp. 415-433-
[37] REISSNER, E., "On the form of variationally derived shell equations", J. Appl. Mech., ASME, Vol. 31, No. 2, 1964, p. 233.
[38] KANOK-NUKULCHAI, W., A simple and efficient finiteelement for general shell analysis", Int. J. Num. Meth. Engrg., Vol. 14, 1979, PP. 179-200.
[39] FRIED, I., "Shear in C° and C 1 bending finite elements", Int. J. Solids and Struct., Vol. 9, 1973, pp. 449-460.
[40] FRIED, I., "Residual energy balancing technique in thegeneration of plate bending finite elements", Comp, and Struct., Vol. 4, 1974, pp. 771-778.
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[41] TESSLER, A. and HUGHES, T.J.R., "An improved treatment of transverse shear in the Mindlin-type four-node quadrilateral element", Comp. Meth. Appl. Mech. Engrg., Vol. 39, 1983, PP. 311-335.
[42] T E S S L E R , A. and H U G H E S , T . J . R . , "A three-node Mindlinplate element with improved transverse shear", Comp. Meth. Appl. Mech. Engrg., Vol. 50, 1985, PP. 71-101.
[4-3] EIDSHEIM, O.M. and LARSEN, P.K., "Nonlinear analysis of elasto-plastic shells by hybrid finite elements", Comp. Meth. Appl. Mech. Engrg., Vol. 34, 1982, pp. 989-IOI8.
[44] McMEEKING, R.M. and RICE, J.R., "Finite elementformulations for problems of large elastic-plastic deformation", Int. J. Solids and Struct., Vol. 11,1975. PP. 601-616.
[45] KLEIBER. M., KONIG, J.A., and SAWCZUK, A., "Studies onplastic structures: stability, anisotropic hardening,cyclic loads", Comp. Meth. Appl. Mech. Engrg., Vol. 33, 1982, pp. 487-556.
[46] SAMUELSSON, A. and FROIER, R., "Numerical methods inelasto-plasticity - a comparative study", in: Nonlinear Finite Element Analysis in Structural Mechanics, (W. Wunderlich, E. Stein, and K.-J. Bathe, eds.),Springer-Verlag, Berlin, 1981, pp. 274-289.
[47] ARGYRIS, J.H. and SYMEONIDES, Sp., "Nonlinear finite element analysis of elastic systems under nonconservative loading - natural formulation. Part I. Quasistatic problems", Comp. Meth. Appl. Mech. Engrg., Vol. 26, 1981, pp. 75-123.
[48] RAMM, E. and STEGMULLER, H., "The displacement finite element method in nonlinear buckling analysis", in: Buckling of Shells, (E. Ramm, ed.), Springer-Verlag, Berlin, 1982.
[49] ZIEGLER, H., "Principles of structural stability", Blaisdell Publishing Co., 1968.ZIEGLER, H., "Knickung gerader stabe unter torsion", ZAMP 3, Vol. 96, 1952, pp. 96-119.
[50]
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SIMMONDS, J.G. and DANIELSON, D.A. theory with a finite rotation vector” Ak. Wet., Series B, Vol. 73, 1973, PP SIMMONDS, J.G. and DANIELSON, D.A. theory with finite rotation and vectors”, J. Appl. Mech., ASME,
, "Nonlinear shell , Proc. Kon. Ned. . 460-478., "Nonlinear shell
stress-function Vol. 39, 1972, PP.
1085 - 1090 .
[53] REISSNER, E., "On finite symmetrical deflections ofthin shells of revolution", J. Appl. Mech., ASME, Vol. 36, 1969, PP. 267-270.
[54] REISSNER, E., "On the equations of nonlinear shallow shell theory", Studies in Applied Mathematics, Vol. 48,1969, PP. 171-175.
[55] BESSELING, J.F., "Large rotations in problems of structural mechanics", in: Finite Element Methods for Nonlinear Problems, Europe-US Symp., The University of Trondheim, Norway, Aug. 12-16, 1985.
[56] GERADIN, M., ROBERT, G., and BUCHET, P., "Kinematic anddynamic analysis of mechanisms - a finite element approach based on Euler parameters", in: Finite ElementMethods for Nonlinear Problems, Europe-US Symp., The University of Trondheim, Norway, Aug. 12-16, 1985.
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C H A P T E R 4
N O N L I N E A R THEORY OF C U R V E D THIN W A L L E D B E A N S
Introduction
In this chapter, the mechanical behaviour of curved thin walled beams of arbitrary cross sectional shape will be considered in some detail. As far as possible, the presentation is not pre-disposed towards any particular form of numerical model. Indeed, the explicit and detailed treatment of many aspects of thin walled beam theory, should make this chapter an invaluable reference to anyone planning a numerical model, regardless of its disposition. However, in recognition of the importance of the finite element method, details of the various matrices that arise when the equations of equilibrium are represented in discretized form, have been included (see Section (J..7).
The theory derives from the classical semi-inverse approach in which the three-dimensional field equations ofcontinuum mechanics are reduced to a one-dimensional formexpressed in terms of a single longitudinal reference line.With the advent of finite element modelling, andisoparametric elements in particular, the technique has generally been referred to simply as 'degeneration'. In the present case, the process depends on two primary assumptions
A1. Stress components that correspond to deformation of the cross section in its own plane may be neglected.
A2. Points that originally lie in planes that are normal to the longitudinal reference line, can move out of these planes during the deformation. The relative motion parallel to the reference line comprises two uncoupled comonents:- a component due to transverse shear which leaves
the section plane, but not normal to the reference line
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- a component due to torsion which warps the section in accordance with a predefined torsional warping function
With these assumptions it is possible to think of the proposed model as deriving from a curved Bernoulli-Euler beam in two distinct stages. In the first stage, the average effects of transverse shear deformation are accounted for by adding a Reissner-Mindlin type shear model. This leads to what is generally referred to as a Timoshenko beam [1], In the second stage, the torsional behaviour is accounted for by using the theories originally developed by St. Venant [2] (uniform torsion), and Vlasov [3] (nonuniform torsion).
Although the degeneration process leads to a more complex kinematical description, the number of generalized coordinates is considerably reduced. Since equation solving is usually the most time consuming component of a numerical analysis, degenerated models can lead to significant computational advantage. However, a price must be paid, and the use of simplifying assumptions must inevitably restrict the range of application of the model. Specifically, in the present case, the formulation cannot account for the effects of cross sectional distortion or differential warping (shear lag).
It is important to note that, because distortional effects are not included, the formulation cannot model local buckling modes that involve significant changes in the shape of the beam cross section. In practice, this limitation is actually less restrictive than it might seem. The dimensions of open section bars will usually be selected so that local buckling is preceded by a longitudinal buckling mode. In such cases, cross sectional distortion in the pre-buckling range will evidently be negligible. On the other hand, local buckling in thin closed sections, is usually precluded by the use of transverse stiffening frames or diaphragms. Moreover, distortional effects are most significant in unstiffened closed section beams that are relatively short and thick walled [fl.], and such beams fall outside the scope
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of t h i s study.One way to account for cross sectional distortion, is to
add the appropriate distortional modes to the basic one-dimensional model. This approach was used by Zhang and Lyons [5] to model distortional effects in symmetric multibox bridge sections, and by Rajasekaran [6,7] to model local buckling of I-section beams. The disadvantages of the method are that it cannot easily be generalized to the case of arbitrary cross sectional shapes, and that it results in a significant increase in the number of generalized freedoms. In fact, the complete removal of all dependency on sectional shape, leads inevitably to a model that has approximately the same number of generalized freedoms as a full folded plate or multi-shell idealization. One of the main objectives of the present work is to provide a viable alternative to such idealizations without imposing any restrictions, other than the requirement that the walls should be reasonably thin, on the disposition of the cross section. Consequently, no attempt is made to include distortional effects in the theory that follows.
B y r e f e r r i n g t h e b e a m c o o r d i n a t e s to a l o n g i t u d i n a l r e f e r e n c e l i n e w h o s e p o s i t i o n w i t h r e s p e c t to t h e c r o s s s e c t i o n is a r b i t r a r y , t h e f o r m u l a t i o n c a n be u s e d e i t h e r in s t a n d - a l o n e f o r m o r as a s t i f f e n e r . In t h e l a t t e r case, t h e b e a m s h a r e s t h e s a m e d e g r e e s o f f r e e d o m as t h e u n d e r l y i n g p l a t e o r she l l . In t h i s way, c o m p l e x s t i f f e n e d s h e l l p r o b l e m s c a n b e a n a l y s e d at a c o n s i d e r a b l y r e d u c e d c o s t w h e n c o m p a r e d to a f u l l s h e l l - s h e l l m o d e l . O n c e aga i n , h o w e v e r , it s h o u l d b e e m p h a s i z e d t h a t t h e s h e l l - b e a m m o d e l p r e c l u d e s l o c a l b u c k l i n g . If l o c a l b u c k l i n g is k n o w n to b e i m p o r t a n t , t h e n it m a y be n e c e s s a r y to u n d e r t a k e a s e c o n d - s t a g e a n a l y s i s i n w h i c h t h e a l t e r n a t i v e s h e l l - s h e l l i d e a l i z a t i o n is u s e d in t h e z o n e ( s ) o f i n t e r e s t .
T h e i n f l u e n c e of w a r p i n g on t h e b e h a v i o u r o f b e a m s u b j e c t e d to n o n u n i f o r m t o r s i o n is w e l l e s t a b l i s h e d . T h e w a r p i n g f u n c t i o n a d o p t e d f o r t h e p r e s e n t s t u d y is s u i t a b l e f o r open, c l o s e d , a n d m i x e d c r o s s s e c t i o n s . T h e l a s t c a t e g o r y r e p r e s e n t s t h e g e n e r a l c a s e of a s e c t i o n w i t h o n e
119
or more closed branches as well as open branches. In the special case of an open section beam whose walls are very thin, the warping function reduces to the classic Vlasov definition.
Warping can be important in stiffened plate and shell structures as well as in stand-alone bars. Indeed, the torsion of a stiffener will always be nonuniform because of the distributed warping restraint that occurs at the stiffener-shell Junction. In certain sensitive cases, such as the buckling of longitudinally stiffened cylinders, numerical studies have shown that even the very low warping rigidity afforded by a thin flat stiffener, can significantly affect the equilibrium path of the structure. Obviously, in cases where the torsional-warping rigidity of the stiffeners is enhanced by outstands, bulbs etc., the errors associated with a no-warping formulation are increased. This can lead to an underestimation of the load carrying capacity of the structure, and, in some instances, may obscure the correct post-buckling mode.
4-2 Geometry and kinematics
4-2*1 Cross sectional geometry and assumptions
A thin walled beam section is depicted in Figure 4.1. The point P lies on the longitudinal reference line of the beam and serves as the origin of the local section coordinates s and t. The location of the cross section with respect to P, and the orientation of the axes s, t are arbitrary. Generic points Q and Q are located in the wall of the section with Q at the contour (midsurface) and Q at a distance n from the contour. The thickness of the wall is denoted by h so that n lies in the range -h/2 n ^ h/2.
Local coordinates m, n are associated with the generic point Q, with n normal and m tangential to the contour. These coordinates are rigidly attached to the contour and
120
F I G U R E 4.1 G e o m e t r y a n d c o o r d i n a t e s o f t h e b e a m c r o s s s e c t i o n .
therefore follow completely the motion of the particle Q. Although the plane m-n can evidently move out of the reference plane s-t during the deformation, for small strains, the angular deviation between the two planes will be small. Consequently, the basic transformation
— - — — ”■m -sa ca s = R sn ca sa t — OC t
_ — _ — — _ —
where sa = sina and cot = cosa, will be adopted for both the deformed and undeformed configurations.
The point M serves as an arbitrary origin for the contour coordinate m. In the context of the Vlasov theory of thin walled open section bars, it is sometimes referred to
121
as the sectorial origin.In addition to the two primary assumptions (A1 and A2)
already stated in Section 4.1, we add the following:
A3. The beam is reasonably slender (L/B, L/D > 10, where B and D represent typical lateral dimensions of the section, and L is the length of of the reference line).
A4. The walls of the beam are reasonably thin(B/h, D/h > 10, where h is a typical wall thickness).
A5. The beam may be both curved and twisted in theinitial undeformed state.
A6. Longitudinal fibres lying on the external surfaces of the untwisted beam are approximately parallel to the reference line (dB/dL, dD/dL < 0.04).
A7. The end sections of the beam are initially planeand are approximately perpendicular to thereference line.
A8. The warping displacement is small relative to thelateral dimensions of the beam.
A9. The warping function is valid for both elastic and elastoplastic conditions.
A10. Deformation of the cross section in the s-t plane has a negligible influence on the remaining strain components.
All. The strains remain small throughout the deformation (e < 0.04).
The values given in assumptions A3 and A4 are intended only as an approximate guide. In reality, the appropriate minimum slenderness and wall thickness ratios, depend both on the type of problem, and on the degree of precision required. For most purposes, however, the quoted limitations are conservative. Note that because the present theory allows for both transverse shear deformation and local warping of the beam cross section, the geometrical restrictions can be somewhat relaxed in comparison with
122
theories that exclude these effects.According to Assumption A5, the reference line of the
beam is not restricted to be a plane curve, i.e., the beam may have a nonvanishing twist in the initial undeformed configuration. This allows the formulation to include the complex axial-flexural-torsional interaction effects that are known to occur in pretwisted beams (e.g., helicalsprings, turbine blades etc.).
Assumption A6 evidently permits a degree of variation in the cross sectional dimensions and/or shape along the length of the beam. Since the effect of pretwist on fibreorientation is explicitly included in the formulation, the quoted maximum fibre slopes must be measured with respect to the untwisted beam. Observe that, in accordance withassumption A10, the maximum fibre strain is 0.04. It follows that by also limiting the fibre slope to a maximum of 0.04, the lateral component of the axial strain in any fibre will be 0(e2), which can be safely ignored.
The assumption made in A7 that the end sections of the bar are plane, is the natural and, in pactice, the most useful choice. Obviously, this limitation implies that, when the end sections deviate significantly from planarity, the actual geometry can ony be modelled in an average sense.
It has sometimes been argued that full geometriccompatibility is important when modelling the Junction between two shells or beams. The term 'full geometric compatibility' refers here to the use of inclined end sections that are exactly matched so that there is no gap or overlap at the Junction. In fact there is very little rational basis behind such an assertion. The reason is simply that the finite elements that comprise the Joint are based on degenerate one- and two-dimensional theories which although ideal for representing the behaviour in regions remote from the Joint, are clearly incapable of modelling the local three-dimensional behaviour of the particles that make up the Joint itself.
There is nothing beam- or shell-like about the stress distributions arising within the Joint, and therefore no
123
reason to believe that inclined (matching) end sections will give a better representation of the behaviour of the joint region. On the other hand, the fundamental justification for using the conventional nonmatching idealisation, is that in accordance with the St. Venant principle, the specialized local behaviour of the joint will not substantially alter the behaviour of the rest of the structure. A somewhat improved estimate of the stresses in the vicinity of the joint can be found by interfacing the basic beam-beam or shell-shell model with specially designed transition elements [8,9].
Assumption A8 is needed because the adopted warping function is not capable of an accurate description of warping displacements whose magnitude is a significant fraction of the lateral dimensions of the bar. In practice, even for slender bars, warping displacements of this magnitude will usually be accompanied by strains that exceed the limit set in assumption All. Consequently, in the theory that follows, the angular deviation between the longitudinal fibres passing through points P and Q respectively, is assumed to be small.
The degree to which assumption A9 is satisfied is difficult to judge because experimental evidence which might support or refute the proposition is difficult to find. Neverthless, for slender thin walled beams and small warping displacements it is reasonable. The assumption is used implicitly in references [7,10-13].
Assumption A10 implies that the conditions
E s — E t ~ E s t = 0
E m = E n = E m n = 0U . 2 )
can be used when deriving the strain-displacement relations of the beam. It is important to emphasize that although these strain components can be ignored when determining the extension and relative rotation of material fibres, they should not be set to zero in the constitutive relations. To do so would impose plane strain conditions that are
124
appropriate only in the special case of sections stiffened by very closely spaced diaphragms. Note also that, except in the special case when Poisson's ratio is zero, the stress components corresponding to equations (4.2) will not vanish. However, in respect of the internal strain energy of the bar, they are evidently redundent.
Assumption All reiterates the basic limitation on the magnitude of strains discussed in Chapter 3. The nominal limit of 0.04 is taken from reference [9].
4-2-2 Cross sectional warping
The warping function is taken as
io(s,t) = co(m,n) = J(cp/h+n-pn )dm+npm (4.3)c
where C denotes the section of contour between M and Q, and Pm, Pn are the local rotated coordinates shown in Figure 4.1.
The parameter cp, usually referred to as the torsional function, is independent of material properties and external loads, and is defined by the equation
$ = GcpXr (4.4)
where G is the shear modulus and $ is the average shear flow over the wall thickness h. The function cp gives the contour distribution of $ per unit change of twist Xr . It vanishes along open branches and is m times statically indeterminate with respect to the remaining m closed branches. Methods of determining the value of cp for each closed branch can be found in most 'Strength of Materials* texts. Consequently, for present purposes, it will be assumed that the cp * s are known.
The two terms on the right-hand side of equation (4.3) evidently correspond to the warping of the contour and the warping of the wall relative to the contour. Following
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Gdelsvik [4], these components will be called the contour warping and thickness warping respectively.
For application to thin walled sections, the thickness warping is frequently discounted [3t7,11-16]. This approach may be justified for the elastic analysis of thin walled open section bars. However, as pointed out in [U], the effect of thicknss warping on the extreme fibre stresses of a closed section can be significant even though the effect on torsional rigidity is negligible. Thus, for example, in the case of a circular tube with a radius to thickness ratio of five, thickness warping alters the maximum torsional shear stress by approximately 20% even though the torsional rigidity is increased by only 1%. Clearly, in the case of elastoplastic analysis of thin walled bars of arbitrary cross sectional shape, the retention of thickness warping can be important.
For open section beams, there is an additional reason why the full warping definition (U.3) should be used. If the thickness warping is ignored, then the St. Venant torque associated with uniform torsion cannot be derived from kinematical considerations alone, and recourse must be made instead to stress function theory (Prandtl’s membrane analogy or St. Venant’s stress function). On the other hand, for certain section profiles (i.e., I, L, T, -4-), locating the reference point at the shear centre and dropping the thickness warping, will cause both the warping displacement and warping rigidity to vanish. In this case, not only is stress function theory required to determine the torsional rigidity, but also uncoupling of torsion and flexure via the use of the shear centre becomes mandatory. Actually, under elastoplastic conditions, the latter process of uncoupling the equations of equilibrium is no longer possible because the associated concept of a single unique shear centre axis no longer exists. Therefore, the unified approach afforded by the use of the full warping function with arbitrary origin P, offers considerable advantages.
Note that, in general, discontinuities may occur in the thickness warping whenever there is a sudden change in
126
direction of the contour, such as at Junctions. However, this is a local phenomenon associated with the stress concentrations that occur at such points, and has little substantive affect on the overall theory [4].
Equation (4.3) may be differentiated with respect to m and n respectively to give the warping gradients
to m = <p/h+2n-pn’ (4.5)“,n = Pm
These warping gradients are needed to define the distribution of torsional shear strain over the cross section.
The warping displacement of point Q parallel to the reference line is given by
ur = a)Xr (4.6)
In accordance with assumption A2, the total motion of point Q will comprise three additive components. These are the rigid body motion of the reference point P, and the two P-relative motions associated with flexure and torsional warping (equation (4.6)) respectively. Note that because the rigid body motion is referred to the point P, this point does not move when the reference line is subjected to a pure extension. By implication, the P-relative flexural-torsional motion of the eccentric point Q applies to the stretched rather than the unstretched beam.
4-2-3 Geometrical description of motion
Figure 4.2 shows the beam in both the initial and deformed configurations. The position vector of point Q in the initial configuration is evidently
q = ° p + s °V+t°V/V A/ ^ S ^ t
o (4.7)
127
(a) Initial state - °C
(b) Deformed state - C
F I G U R E 4 m2 G e o m e t r y o f a c u r v e d b e a m in c o n f i g u r a t i o n s °C a n d C r e s p e c t i v e l y .
128
Differentiation with respect to °r, °s, °t repectively leadsto the covariant base vector definitions
°G =~ l 3°q/8 °r = °v +s°v'+t°v'~ s ^ t°G = ~ 2 a°q/a°s = 0 V ~ s (4.8)°G = ~ 3 a°q/a°tr>j = °Vt^ t
where °V*A/ f denotes a°v /a0r ' etc..rIn accordance with assumption Al, the initial
longitudinal axis °V may be both curved and twisted.rConsequently, the vector derivatives °Vp etc. can be derived using the Frenet-Serret formulae for a space curve, i.e.,
°v'~~ r° V ' _~ s°v;~ t
-k0
-kV~ sXt
(tt.9)
where the initial curvatures kr, k s, kt are defined as
kr - k- =ki- =
V . • ° v *
>o>o1II
~ X ~ s ~ s ~ tv ■ ° v !
>o•>o1II~ r X ~x rV . ° v *~ s ~ r = — ° v . ° V 'A/ S
(Zl. 10)
Hence equations (4.8) become
0G = (1 + tk - skJ°V -tk °V +sk °V,~ i s t ~ r r ~ s r ~ t°Gz = °Vs (4.11)°G = ° V~ 3 ~ t
This result shows that, in the presence of an initial twistkr, the longitudinal fibres can significantly deviate fromthe reference direction °V .~r
In the deformed state, the relation analogous to (4.7) is
Q p+sv s f tv tf ■G)XrVr (4.12)
where the last term accounts for the small warping
129
displacement parallel to the reference axis (see equation (4.6)). Since the difference between the convected frame (V , V , V .) and the deformed fibre frame (V , V _, V ) corresponds to a small pure deformation, the two systems must be related by
V „ 1+E r 0 0 r “ V~ r f L ~ rx „ = ^ r s 1+ES ^ S t Xs (4.13)X,e r t ^ s t 1+ E t Xt
where er etc. are the strains at the reference line s = t = 0. But, in keeping with (4.2),
= Et = Est = 0
which in turn implies
(4.14)
V~ s f V~ s + E V~ rXtf = V.J-E-.V ~ t r T~r
(4.15)
Therefore, equation (4.12) can be rewritten
q = p+(sers+tErt+uXr)Vr+sVs+tVt (4.16)
Note that the difference between the strain matrix used in (4.13) and the symmetric strain tensor of continuum mechanics, arises because the convected axes r, s, t corotate with the material in the restricted sense discussed in Section 3.2.
The derivatives of P with respect to r, s, t are evidently
P = P = V~ *r /VPA' »s = P++ tt = 2
(4.17)
Thus the covariant base vectors in the deformed state are given by
130
G. = aq/3r = (l + S£ls+tErt+(0Xr)Vw^ j v 4 ** * w * ^ |7
G ~ 2 = d q /a s =+ (sirs*tErt-K0Xr ) V ^ S V ^ t V ^ <EPs*“ . sXr >yr-*-Vs
(4.18)
G 3 = aq/at = (irt+«>.tXr)yr+ Y t
The vector derivatives that appear In the first of equations (4.18) can be expressed In terms of the curvatures of the
• deformed reference line in the manner of (4.9)» i.e.,
K 10K
(4.19)
where the current curvatures Kr, K s, Kt are given by
(4.20)
Substitution of equations (4.19) into (4.18) now leads to
G a = [l + t(Ks + Ert )—s(Kt — EpS ) +toXp ] Vp - [tKr-(sers+tert+coXr)Kt] V s+ [sKr-(sErs+tErt+dJXr )KS ] V t (4.21)
5 2 = ( Er s+t0, s*r ) Y r+J£sg 3 = (EPt+« , tx r )vr+v t
4-2-4 Stress and strain transformations
Transformations of stress and strain components from the rectangular V-system to the nonorthogonal G-system and vice-versa, represent an important ingredient in the technical formulation that follows. Exact Jacobians can be found, but for present purposes, transformation matrices with a truncation error of O( e ) will provide adequate precision.
131
Recalling the discussion of Section 3.6, two alternative work congugate stress-strain systems are needed to define the virtual work equations. These are the 2nd Piola-Kirchhoff stresses congugate to Green strains, and Cauchy stresses congugate to corotational (convected Eulerian) strains, respectively. Therefore, the required transformations take the form
T VJJ
and
( f t . 2 2 )
E = J E~ a — G ~ Va
A/ T= J T CT~ V — a(^.23)
where the suffices V and G refer to the two frames formed by the V vectors and the G vectors respectively. In the present case, the stress and strain components read
E G = { e g , E G . E~ V r r se g = { E c . E G » E~ G l 1 2T = { T . T . T~ V r r sT~ G = < T x . T . T 1 2
sr t s1 3 r t i 3
>>>>
(ft. 2ft)
with similar notation applying to the four remaining vectorsE » E >~ V G a v» a n d ?G-
The Jacobian matrix J of— G (ft. 22) is defined as
G • V ~ l ~ r G • V ~ s G • V,l tJ- G = G • V - 2 ~ r G • V~ 2 ** S G • V + ~ 2 t (ft.25)
G • V ~ 3 ~ r G • V~ 3 — S G • V.~3 ~ t
where the scalar products can be found directly from (ft.2 1 ).After discarding terms of O(e) and higher, this leads to the following approximation:
132
J G
0 -tKr sKr 0 1 00 0 1
where
0 = l+tKs-sKt+uXr
(4. 26)
(4. 27 )
To within terms of O( e ), the Jacobian matrix J of the“ Gconvected system (4.23)* is given by
J- G (4.28)
and therefore one gets immediately
J G
1 —tKr sKr0 1 00 0 1
(4.29)
Note the definitions (4.26) and (4.29) satisfy the usual condition that the virtual work should be invariant with respect to change of frame, that is,
x • S e G = t •Se®~ G ~ G ~ V ~ V ( 4 . 3 0 )a • Se~ G ~ G = <7 • Se~ V ~ V
To complete this section, the transformations involved when using the local contour attached frame (l,m,n) are considered. As mentioned earlier, this frame serves as the basis for describing the material behaviour of the beam walls. In keeping with the notation used above, the contour frame is referred to by the suffix C. The C- and G-frames differ by a rigid rotation a about the 1-axis. Thus, analogous to (4.22) and (4.23)> one can write
= J e— c ~ v= J Tx- c ~ c
(4.31)
133
and
8 = J 8~ c - c ~ v= J 1 a - c~ c
(4.32)
where, in correspondence with (4.26) and (4.29), the new J-matrices are
8 — tKr sKr0 -sot ca0 cot sex
and
1 - tKr sKr0 -sex ca0 ca sa
(4.33)
(4.34)
4-2-5 Differential fibre length ratios
In the case of a beam whose reference axis is curved and/or twisted, the arc lengths of longitudinal fibres will obviously vary over the cross sectional plane s-t. In the undeformed configuration, the fibre length variation is described analytically by
°Q = d ° x 1/d°r = (°G • °G )** (4.35)~ i ~ i
Using the first of equations (4.11), one finds
°Q = [ (l + t k s-skt ) 2 + ( pkr ) 2 (4.36)
where
p = (s2+t2)** (4.37)
Similarly, for the deformed configuration
134
Q = = [(l+tKs-sKt+uX')z + (pKr)2]^ (4.38)
Unlike (4.36), the expression on the right-hand side of (4.38) is not exact. However, for present purposes, it has adequate precision.
The influence of fibre length variation on the overall behaviour of a beam or shell will be referred to here simply as ’the curvature effect*. It is well-known that under small strain conditions the curvature effect in thin shells can be neglected. However, there are two reasons why to draw the same conclusion for beams could be dangerous. Firstly, the thickness/radius ratio for a beam may typically be an order of magnitude higher than that of a curved shell. This isparticularly true for thin walled open section bars andapplies both to stand-alone and stiffener usage. Secondly, equation (4.38) is a function not iust of K s and K t, but of Kr as well. The twist Kr results in complicated stress interactions that have no comparable counterpart in shell theory, but can be important particulary in relation tocertain forms of buckling. Thus, for example, the retention of the Kr term in (4.38) is pivotal to the solution of torsional instability in a slender bar. Therefore, even though the curvature effect frequently results in quite small changes in the distribution of strain, it isnevertheless important that it should be retained in the kinematical model.
Another important role of the parameter Ci concerns the definition of the elemental volume in the deformed state. This can be written
dV = dXidXjdXg = dx2dsdt = Qdrdsdt = QdAdr (4.39)
Note that in view of (4.14), the elemental arc lengths dx2 and dx3 are not affected by the deformation (i.e., d°x2 = dx2 = d°s = ds, etc.).
135
4-2-6 Curvatures
In equations (/l. 10) and (4.20), the reference line curavatures corresponding to the initial and deformed configurations, have been given in terms of the vector derivatives “Vp, °Vg, °Mt and Vri ,Ys» v't respectively. The curvatures kSf kt and Ks, K t are more conveniently caste in terms of second-order derivatives of the reference line position vectors °p and p respectively. That is
where
o ' ' _ J 2 / J 0 2 / o , , 0 „ 0 _ ip = d / d r i x , y, z y p" = d2/dr2{x,y,z}
(4.40)
(4.41)
For finite element models in which the reference line axis is described parametrically using shape functions, the curvatures (4.40) are evidently determinate.
Since the beam model is essentially one-dimensional, it is clear that any initial twist (referred to here as pretwist) of the reference line must be known a-priori. In practice this is achieved by direct specification of a pretwist angle at each node. If the pretwist angle at any point along the reference line is denoted by °0r, then the initial curvature kr is simply
kr = °0r (4.42)
Just as it is impossible to determine kr from °p alone,A/
it is also impossible to determine Kr from p alone. This difficulty is actually peculiar to generalised one-dimensional beam theories - it does not arise in classical shell theories because the metric tensor of the
136
midsurface provides a complete definition of the requisite vector derivatives.
As a preliminary to finding the current twist Kr, it is necessary to emphasize the fact that the reference line curvatures are objective and frame indifferent quantities. This means that equations (4.20) remain valid irrespective of the orientation of the global frame x, y, z depicted in Figure 4.2. On this basis, suppose the frame x, y, z is rotated into coincidence with °V , °V , °V4. Then, inkeeping with (4.20), one can write
Kr A/ «V~ s -V ~ s ft (a.4 3 )
where V s, Vt are base vectors parallel to s, t respectively, but are measured with respect to the new global frame, say x, y, z. Thus, by definition, V and V a r e equal to the second and third columns of the rigid rotation matrix R(§) that carries the local frame at P from °V , °V , 0V_ to Vr * Vs» V f Since explicit expressions for the coefficients of R(^) have already been derived in equation (2.21), an exact determination of Kr is now possible. In practice, however, the presence of the trigonometric functions f(^) and g('§') in the coefficients of the rotation matrix, lead to a result that is extremely cumbersome.
An alternative, and much more elegant, method of finding Kr, is to utilize the polar decomposition theorem. Under present conditions the change of twist Xr = Kr-kr, is a pure deformation. Therefore, in accordance with the theorem, Xr is independent of the rigid body component of the overall motion. Furthermore, because Xr is frame indifferent, it is possible to work with local, rather than global, Euler rotation components. Consequently, Xr can be found by operating with the second and third columns of the residual (corotational) rotation matrix R($ ), where
g L = {0r’0s’0t> (4.44)
In the vicinity of the reference point P, 0l is O ( e ), s o
137
that, under the assumption of small strains, a linear approximation of R(gL) will suffice. Thus, following (2.22), R(0l) can be written
1 - 6 tA
0 sA A
0 t l - 0 rA A
0 s 0 r 1(4. 45)
Since the right-hand side of (4.45) is only approximately orthogonal, for maximum precision, the following symmetric definition of Kr is appropriate:
Kr = kr+Xr = kr+^(3*2'-2-3') (4.46)
where 2 and 3 denote the scond and third columns of R(0 )^ ^ Lrespectively. Using (4.45) and (4.46) now leads to the desired result, i.e.,
Kr = kr + 0r+J*(0tes-es0t)+°( E* E * ) (4.47)
Since the underlined term of (4.47) is O(ee'), under small strain but finite rotation conditions, it can be neglected. Note that although the corotational component §r vanishes identically at P, it must be retained in (4.45) until after the differentiation with respect to r has been carried out.
The above analysis is suggestive of the following generalization:
---1 X ►i __1 « ( 3*2 -2•3
= l4 t 1 * 3 ’-3 * 1'1 «2*1 -1*2A# /V A/ A/
A I , , A I A0 ,+*6(0,x 0 , )~ L ~ L L iu.ua)
Here, X is a symmetric 'change of curvature' tensor analogous to that used in the technical theory of shells. As before, the underlined term on the right-hand side of (4.48) can be neglected under conditions of small strain but arbitrary rotation. Comparing (3-34) and (4.48), it is immediately apparent that an exact correspondence exists
138
between the Incremental and corotatlonal forms of the change of curvature tensor. With hindsight, this correspondence is to be expected, nevertheless, bearing in mind the disparity in their individual derivations (the former result derives from Green’s strain expansion), this is a significant and satisfying result. Moreover, the validity conditions that pertain to the use of a simple linear relation X = 0' are/v /v L
consistent with the ideas of Section 3.3.Finally, the reader is warned that, notwithstanding the
credentials of both (4.47) and (4.48), the latter result does not mean that a generalization of the former, obtained by appending two similar equations for Kr and K t, is valid. The reason is simply that X s and Xt include the effect of transverse shear deformation, so that the relations X j = (Ks-ks) and X t = (Kt-kt) are not exactly true. Therefore, as suggested at the outset, K s and Kt should be determined using the exact definitions (4.40).
4-3 Strains
The convected (Eulerian) strain components at the generic point Q can be written
r sr t
H(q •q -°q • °q )~ 1 • 1 ~ t 1 ~ • 1q • q -°q •° q~ , Z ~ , 2 2 1q • q - q~ , 3 3 , 3 3 , 3
(4.49)
where, to within terms in e2,
°3, i = (a°q/ax1) = (a°q/a°r)(d°r/dxx) = Q*3., = (a°q/ax2) = (a°q/a°r) (d°r/dx2 ) = °<°3.3 = (a°q/ax3) = (a°q/a°r) (d°r/dx3 ) = °«3.i = (aq/axx) = (aq/3r)(dr/dxx) = 1G i3.i = (aq/ax2 ) = (aq/ar)(dr/dx,) = G ~ ~ 23. 3 = (aq/ax3 ) = (aq/ar)(dr/dx3 ) = G->/ a A. 3
- 1
(4.50)
Hence
139
Er = J$0“2 [G^ G i-(l-2Er ) °Gi • °Gi ]ers = Q - 1 ( G ^ G z - ° G i • °Gz ) (U. 51)Ert - 0-1(Gi.G3-°Gi .°G3)
As before, engineering measures of shear strain have been used. These are exactly twice the magnitude of the corresponding tensor components. The bracketed terms in (4.51) can now be expanded using the base vector definitions (4.11) and (4.21). To a precision that is adequate for present purposes, this leads to
E r = Q~z [er + tXs-sXt +o)Xp 2 (Kr+kr)Xr]E r s = Q-x(Ers+4>sXr) (4.52)Ert = Q-1(Ert-t-0>tXr )
where
Xr = Kr-krxs = K s —k s + E r t
= Kt—k t + Erj= 0O)f s-1+o)Kt= 0(iit t + s-a)Ks
0 = l+tKs-sKt+
(4.53)
Note that when deriving these results, terms involving nonlinear products of the curvatures X and the lateral coordinates of the beam, have generally been neglected. On this basis, it could be argued that the last term in the equation for er could also be neglected. The reason for retaining this term is that it plays an important role in coupling the axial and torsional forces in a twisted beam. As is well-known, such force interactions have a direct bearing on the classical problem of lateral-torsional instability.
In order to take proper account of the local behaviour of the beam walls, it is necessary to work with strains in the nonorthogonal contour attached frame (1, m, n) rather than the rectangular frame (r, s, t). The transformation can
i4o
be a c h i e v e d u s ing
e~ c J- c E~ V (4.54)
where an explicit definition of the Jacobian J c has already been given in equation (4.34). Substitution of (4.34) and (4.52) into (4.54) now leads to
Ex = Q_1[£r+t(Xs-Krers)-s(Xt-Krert)+wXp +^P2 (Kr + kr )Xr]
E im = ^ ( E r te a — E r s sot+^in^r )ein = Q_1 ( ersca+Er tsa+0>nXr )
(4.55)
where
4>m = 4>tca-^ssa <|>n = M>sca+iptsa (4.56)
To arrive at this result, use has been made of theapproximation
(3/0)§r i §r (4.57)
In accordance with (4.38), the truncation error in (4.57) is 0(<t2K2Er ), where K is the resultant curvature of thereference line, and <t is the (corresponding) radius of gyration of the cross section.
The parameters ipm and <l>n can be expanded using (4.1) and the fourth and fifth of equations (4.53). This gives
= 0<*\m + pn-o>Kn *>n = 0 W tn-Pm+wKm
(4.58)
Introducing the warping gradient definitions (4.5), equations (4.58) can also be written
4>m = 3 (cp/h+2n-pn ) + pn-a)K 0»n = (0-l)pm+a)Km (4.59)
l4l
Evidently, equations (4.55) define the strain field at any point on the cross section in terms of the seven generalized strain components
r (4.60)
Note that each component of r is a function of r only, and is associated with an easily identified and independent mode of deformation. Also observe that the strain field is applicable to beams of arbitrary cross section. Specialization to thin walled sections occurs via the use of the 4>-factor definitions (4.59).
The components of eim and Ein associated with flexural action do not yet reflect the well-known requirement that the resultant shear strain in most thin walled sections is practically tangential to the contour. Moreover, these components should be adjusted to allow for the average flexurally induced warping of the section. These effects are accounted for here by introducing two new parameters, and Hn , and rewriting equations (4.55) in the form
Ex = Q “ 1 [er+t(Xs-KrErs )-s(Xt-KrErt)+wXr +>SP2 (Kr+kr )Xr]
E im = ^ (v rn e r t cot~ E r s sot)+v^ m ^ r i ein = Q " 1 [un ( Ersca+Irtsa)+4>nX r ]
(4.61)
or, alternatively,
e = Hr (4.62)~ Q
where H is the 3x7 transformation matrix
1 — tKr r 1| sKr !| t 11 -s %P2(Kr+kr ) (0H = Q“1 0 -HmSoe 1I Umcoc I1 o | 0
|| 0 (4.63)
0 Unccx 11 Mnsot l
1 0 0 0>n 0
Explicit definitions of the flexural shear factors um and un are derived later in Section 4.3.1.
1/12
The last term ±n the equation for er can be Identified with the so-called Weber effect [17]. When a beam of arbitrary cross section is twisted about any longitudinal axis, the fibres become helically shaped and initially take up a tensile strain equal to the above term. However, the final distribution of axial stress must satisfy the equilibrium condition
= J ovdA = 0 (/l. 6/1)
Consequently, provided kr = 0, the axis of rotation will always become shorter when the beam is twisted. Moreover, from the form of the Weber effect term, it is easy to see that the associated axial strain vanishes for Xr = -2kr as well as for Xr = 0, and is a minimum when Xr = -kr. Consideration of the behaviour of a single helical fibre on the surface of a pretwisted cylindrical bar, provides physical confirmation of these observations.
The Wagner effect [18], whereby the torsional rigidity is increased by a nett tensile distribution of axial stress, and vice-versa, is a manifestation of the same pattern of axial deformation. In this case, however, the effect arises because, a x is not, in general, parallel to the reference axis r. In contrast to the Weber effect, which has only a second-order influence on the axial distribution of strain, fibre orientation has a first-order influence on the transverse components of the internal stress field. Consequently, Wagner’s is by far the most significant of the two effects. In fact, the classical problem of torsional, or lateral-torsional, instability in a beam-column cannot be resolved unless the Wagner torque terms are included in the differential equations of equilibrium.
For a straight thin walled beam subjected to twist only, we have Km = Kn = 0 and Q = @ = l, so that the tl>-factors of (/l. 59) reduce to
0
143
<l>m = (<p/h+2n)( a . 65)
«l>n = 0
The corresponding shear strains are therefore
6 un = (<P/h + 2n)Xr (4. 66)E i n = 0
which agrees with the predictions of St. Venant's theory of uniform torsion and Prandtl's membrane analogy. If the section is open, then tp = 0, so that, in this case, the appearance of eim as an active strain component, is directly attributable to the thickness warping term of equation (4.3).
4-3-1 Flexural warping and shear attenuation
The parameters and un in the second and third of equations (4.61), are intended to allow for two higher-order flexural effects. These effects will be termed ’flexural warping* amd ’shear attenuation* respectively. They are independently accounted for by writing
Urn = UWUn = uw
(4.67)
The flexural warping parameter uw plays the familiar role of the ’shear correction factor*. For ’solid* section bars whose outer surfaces are stress free, the appropriate value can be determined analytically. In the particular case of a rectangular section, Reissner [19] gives the static value 75/6. Alternatively, for dynamic flexural motions, Mindlin [20] derives the value 7t//T2. For an arbitrary section, uw always lies in the range
Tt/yi2 uw ^ 1 (4.68)
i u u
In practice, because transverse shear deformation is a second-order effect, it is usually sufficient to use a single fixed value regardless of section shape. Thus, for present purposes, the more familiar Reissner value is adopted
uw = 75/6 (U .69)
Note that the conventional method of accounting for flexural warping, is to apply a factor of 5/6 to the diagonal tranverse shear term(s) of the constitutive matrix. In the case of beams, however, material nonlinearity can give rise to coupling between the incremental shear stress t 1Tn and axial stress Tj. This implies that, in addition to the aforementioned correction, the appropriate off-diagonal components of the constitutive matrix should be multiplied by 75/6. More seriously, the correction factors should operate only on those parts of the incremental stress and strain fields that are attributable to flexure, and not on the total values. The proposed method, in which only the flexural components of the transverse shear field are modified, overcomes these difficulties, and has a proper mechanical basis. This approach is consistent with the recommendations put forward recently by Hughes and Liu [21] in relation to shell analysis.
The term ’shear attenuation* has been introduced by the author to describe the process involved in bridging the gap that exists between the conventional first-order theories of solid and thin walled beams respectively. The reason for introducing new terminology was simply that a formalized description of the process could not be found in the available litrature.
In a first-order, solid section shear theory (Timoshenko, Mlndlln, Reissner), the entire section rotates into an average location defined locally by the plane
r = uw (sErs+tErt) (U.70)
145
It is obvious that, in general, there are no directions on this plane that are orthogonal to the reference line r, and therefore no directions with respect to which the resultant shear strain vanishes. Consequently, if one now imagines that a thin walled section is embedded in this plane, it is immediately apparent that the resultant shear strain will not, in general, be tangential to the contour. In other words, the first-order solid section theory is not sufficiently general to comply with one of the basic assumptions of the first-order thin walled section theory. Conversely, and more seriously, if the tangential shear strain requirement is enforced by setting Ern = 0, then special cases of practical importance, such as a thin rectangular section, cannot be classified as thin walled because the n-direction shear resultant generated in such sections will vanish or, at least, be unrealistically small.
The introduction of the shear attenuation factor uQ eliminates these conflicts by linking the two theories together. When m q = 1, the walls of the bar lie in the shear plane defined by (4.70), and the direction of the resulltant shear strain is entirely independent of the contour direction m. On the other hand, when |ia = 0, the section contour lies in the same shear plane, but material fibres normal to the contour actually remain parallel to the reference plane s-t. Thus, ern vanishes, and the resultant shear strain is tangential to the contour in accord with thin walled section theory. Clearly, the intermediate situation, 0 < uQ < 1, is characterized by sections in which the resultant shear strain is partially dependent on the direction of the contour. Thus, shear attenuation describes the degree of directional dependence that exists between the resultant shear strain and the cross sectional contour.
Table 3.1 provides a useful summary of the role of shear attenuation in terms of the three categories of thin walled section corresponding to ua = 0, 0 < ua < 1, and na = 1 respectively.
146
Section type Typical shapes Resultant shear TheoryTypical thin H * r n □ O 0 tangential to thinwalled contour walledShallow thin /S f 1—1 , 1— , E3 > 0 partiallywalled < 1 follows contourThin _ 1 independent of >1r
rectangular contour solid
T A B L E Q - I The role o f s h e a r a t t e n u a t i o n i n r e s p e c t o f v a r i o u s c a t e g o r i e s o f t h i n w a l l e d s e c t i o n m
Evidently, if uQ can be related to some characteristic geometrical property of the section, then all three categories shown in Table 4.1 fall within the scope of the proposed unifying theory. It turns out that this is indeed possible provided the normal shear stress o in is assumed to remain elastic and uncoupled in the constitutive law. In this context, it is obviously not necessary to consider the effect of curvature on shear attenuation. Therefore, setting Q = 1 and <|>n = 0 in the third of equations (4.61) and observing (4.67)» leads to the following definition of the elastic shear stress a in:
CTin = ^n®Ein = ® ern (4.71)
Note that the assumption of an elastic shear response in the n-direction does not significantly detract from the precision of the model. Moreover, it leads to an appropriate consistency with the commonly adopted treatment of tranverse shear in the nonlinear numerical analysis of thin plates and shells.
An approximate definition of uQ can now be found by considering the static equilibrium conditions for an elemental length of an initially straight beam. Figure 4.3 shows the shell forces that are relevant to satisfying equilibrium in the n-direction. These forces are the shear qn, moment mm , and twisting moment m 1Tn acting on a unit length of the contour. The requisite equilibrium equation reads
- 147 -
m ^ - a /a m ( m l m )+qn = O (4.72)
But, in accordance with thin shell theory.
mm Eh312(1-v z )
h312Im (Mtca-MsSa) (4.73)
and
a8m ( m i m ) Gh3
6 -f- (Xr ) 3m r 0 (4.74)
F I G U R E 4 . 3 S h e l l f o r c e s a s s o c i a t e d w i t h eqtiili l i b r i u m a l o n g t h e n o r m a l n o f a u n i t l e n g t h o f t h e c o n t o u r .
In equation (4.73)> Im is the moment of inertia of the beam cross section with respect to an axis through P and parallel to m. The approximation sign occurs because the influence of the product of inertia Imn has been neglected. This is Justified because the shell moment mm is only significant in sections whose profiles are nearly flat, in such cases, Imn « Im .
Substituting (4.73) and (4.74) into (4.72) leads to
and
148
(4.75)
But
Ms = Ft = M$GAErt M t = — F s = —U^GAErs (4.76)
where A is the area of the cross section. Therefore, (4.75) can be written
where <tm is the radius of eyration corresponding to Im . Note that, as expected, for a rectangular section, equation(4.78) reduces identically to Ma = 1, whereas, for a typical non-shallow thin walled section, 4m » h and uQ -*► 0. Furthermore, |iQ exhibits the proper invariance with respect to the orientation of the local reference axes s and t.
4-3-2 Generalized strain-displacement relations
In accordance with the arguments of Section 3.2, the components of the generalized strain vector r can be expressed directly in terms of the local convected
(4.77)
so that, observing (4.71), one finds
(4.78)
(corotational) displacements of the reference line. Thus for small strain but arbitrary rotations, one can write
- 149 -
1
u110
1_______
A »
u r
E rs 4J
«D1
E rtA
9s*s =
A 1
0s
X r A 1
0r_xr J A fl
_ ®r _
(4.79)
Despite their very simple form, the truncation errors for X, and Xp are 0 ( e 2 ), O ( e e '), and 0 ( e e ") respectively.It is interesting to note that r is a function of only
four corotational displacement components, that is, ur, 0r, 0S, and 0t. This arises because the displacement
A | A |derivatives us and u t vanish at the reference point P.
4-3-3 Incremental Green strain field
As is well-known, the incremental Green strain tensor can be decomposed into linear and nonlinear parts. Thus, in the present case,
e g ~ v = e ■ X = ~ v e + x rs r se r t +5<rt
(4.80)
where e and x are linear and nonlinear functions~ V ^ V
respectively of the generalized displacement gradients, referred to configuration C.
Since the linear and nonlinear components of the incremental strain field make independent contributions to the internal virtual work of the system, it is not essential that they are measured in the same frame. In fact, different frames are admissible provided the virtual work is the same in both. Here, it is convenient to refer the linear components of the incremental strain to the contour related frame (l,m,n), but to refer the nonlinear components to the
150
basic frame (r,s,t). This is valid because
t Se = ~c ~ c x Se ~ v ~ v (4.82)
The transformation equations (4.31) and (4.32) of Section4.2.4 satisfy this requirement identically.
An explicit expansion of e in terms of the incrementalcdisplacements of the reference line, can be found by writing equations (4.62) in the incremental form
HE— (4.83)
where E comprises the linear incremental reference strains- — “ —fire r s®rt a i + e s
*s = • «©s©i
X r ©rx; _ ©^
(4.84)
and H is the transformation matrix of equation (4.63).The nonlinear incremental strain components *cv are best
found directly from the Green strain expansion
EEe
0 ~ 2 Cur , r +H ( u £ L r-Mj|t r + u | , r )]Q~X tur , A , r -*-( r , sfrr . r + Us, s^s, r +ut , r>3O " 1 Cur , t ^ t tr+ ( U r , t U r , r +U s tt U s tr ^ t , t U t , r ) l
(4.85)
where, on account of assumptions A9 and A10, the underlined terms can be neglected.
Equations (3.25) relating the incremental motion of the points P and Q, may now be employed to express (4.85) interms of the generalized displacements ur, u s etc. of thereference line. In the present case, it is of coursenecessary to add the axial displacement due to warping.Hence, observing (4.6),
151
u r = Q r + t ( 0 s+^0r0 1 )-s ( 9 t- % Q r Q s ) +0)0p u s = 6 s-t 0 r u t = 6 t+s0r
(4. 86)
Substituting (4.86) in (4.85), and discarding terms that are quadratic functions of s and t, one finds
In arriving at this result, use has been made of the approximations
which, in accordance with (4.84), are evidently true to within terms in e2.
Note that, as discussed in Section 3.4.3, both thelinear and nonlinear components of (4.85) contribute to k .vBecause the conventional approach erroneously ascribes a linear relation to the relative motion of the two points P and Q, it fails to pick up the former contribution, and leads to an incomplete definition of the internal potential energy of the structure.
4-3-4 Convected incremental strain field
The convected incremental strains at the generic point Q are defined as the change in corotational strain corresponding to the incremental motion that carries C to AC. Thus, taking the contour related frame (l,m,n) as reference, the requisite relation is
^rt
+sier0s-0se')+(pe')2j 1 (-Qr0t+Oi0r+^0r8s- t 0 t0 s )
Q_ 1 ( O r 0s- U50r+)£er0t- s0s0 't )
(4.87)
(265—0t )0p = 0t0p(2Ci| + 0s )0p = - 0 s0r
(4.88)
ec (4.89)
152
Here, In order to admit numerical evaluation, theconfigurations C and AC must be viewed retrospectively. Inthe general case of an incremental-iterative solution of theequilibrium equations, this means that AC is taken as thelatest iterative estimate of the equilibrium configurationthat is currently being sought, and C is taken as the lastknown equilibrium configuration (i.e., configuration at theend of the previous step). The strain increments 8 must be^ cdetermined with high precision because they have a direct influence on the internal stress field and on the response of the structure.
In practice, corotational strain increments are needed only for nonlinear materials, since it is only in this case that the constitutive relations need to be cast in an incremental form. But, nonlinear material response is dependent on the history of Cauchy stress at each individual material point, so that, the direct evaluation of the new stress state at any point, requires a knowledge of both the previous stress state crc and the previous strain state ec. For a large problem, such as say a ring stiffened cylinder with several hundred elements, the volume of information involved in such a process can be very large, and inevitably entails the use of an external storage device. Since the storage and recovery of data from an external device is a relatively inefficient process, it follows that any reduction in the amount of imformation that must be transferred in this way, will improve the efficiency of the computer model.
A simple, and fairly obvious, way to achieve the foregoing objective, is to express the strains at point Q in terms of the generalized strain components of the reference line. In other words, (4.89) is written in the form
e = AH Ar - H r = * H r + H r ( 4 . 9 0 )
where
153
H = AH-Hr = Ar-r (4.9 1 )
Here, AH and H take the same form as (4.63)» but are written In terms of parameters defined in configurations AC and C respectively.
Since H is O( e ), the second term on the right-hand side of (4.90) is O(e e ). Therefore, in practice, although the evaluation of (4.90) does not present any special difficulties, it is worth noting that the approximation
e _ = AHr+0(ee) (4.92)
is quite acceptable. The advantage of (4.92) is that no information, other than the seven generalized strain components f , is required for the preceding configuration.
4-4 Constitutive relations
Following the discussion of Section 3.6, the constitutive relations for the material can be written either in the unconvected form
dT = (C-C )de® = Cde® (4.93)~ c — — <r ~c — c
or in the convected (corotational) form
da = (C-C )de = Cde (4.94)~ c — — ff ~ c — ~c
where t c are 2nd Piola-Kirchhoff stresses conjugate to Greenstrains e®, and a c are Cauchy stresses conjugate to thecorotational strains e . The correction matrix C takes the~ c — orsame form as that given previously in equation (3.6l), but,for present purposes, the Cauchy stress components crrs andcrrt are replaced by CTim and a in respectively. Recalling thedecision to treat a in as elastic and uncoupled, and notingthat the coefficients of C are negligible in comparison— <r
154
with the elastic moduli! E and G, an adequate approximationfor C is — o
Ca4 CTl 2crim 0
o-i 0sym 0
(4.95)
In the general case of an elastoplastic material, the modulus matrix can be split into elastic and plastic parts in the usual way, that is,
C = C - 5(C + C ) (4.96)— — e — p — <t
where Ce is the elastic modulus matrix
C e = |~E, G , gJ (4.97)
C is the plastic modulus matrix, and the multiplier 5 takes— p
the values 0 or 1 according to whether the material is elastic or plastic respectively.
The plastic modulus matrix Cp governs the nonlinear response of the material, and is a function of the current levels of Cauchy stress. For present purposes, two alternative nonlinear material models will be considered. These are referred to in the literature as the J2 flow and J2 deformation theories respectively.
The reason for considering two alternative constitutive models is that certain aspects of plasticity theory remain controversial. One of the main difficulties is centred on whether or not yield surfaces can develop corners at the loaded point. Paradoxically, it turns out that a yield surface with a corner at the active point implies a much wider range of validity to the older total-strain (deformation) theories than had previously been thought possible [22]. A number of theories incorporating a corner at the loaded point have been proposed - notable examples are Koiter [23], Sanders [24], Budiansky [25], and Kliushnikov [26]. However, for a restricted range of
155
deformations, such theories are essentially equivalent to the J2 deformation theory [27-29].
Unfortunately, despite considerable effort, experimental evidence for the existence of yield surface corners remains inconclusive. Nevertheless, the most recent studies [27-29]> do appear to support the disconcerting notion that J 2 flow theory may be overestimating the load carrying capacity of imperfection sensitive structures, especially in cases where there is a significant departure from proportional loading. It appears therefore that the situation is still some way from being resolved. Futhermore, should definite confirmation of yield surface corners be forthcoming, then it is quite possible that the lower-bound J 2 deformation theory may also need to be abandoned in favour of the more elaborate corner theories mentioned above.
In the following sections the constitutive relations for a beam are derived assuming that the material obeys the von Mlses yield condition and exhibits isotropic hardening. The corresponding relationship between the effective (uniaxial) stress and strain, takes the bilinear form shown in Figure 4. a.
0
F I G U R E 4.4 E f f e c t i v e uniax i a l s t r e s s — s t r a i n r e l a t i o n s h i p for a v on M i s e s material with i s o t r o p i c h a r d e n i n g .
156
4>4-l J2 flow theory
This theory is based on the well-known Prandtl-Reuss equations. In the three-dimensional case, the modified elastoplastic constitutive tensor for a material of the type depicted in Figure 4.4, can be written
c i J k i - c i J k i “ 5 ( P S i j S k i + c f J k i ) (4.98)
In keeping with (4.96), the three terms on the right-hand side of (4.98) correspond to the elastic, plastic and correction tensors respectively. The plastic tensor comprises the deviatoric stress components,
8 i J - a i j ” 3 a i i © i j
a plastic hardening parameter,
3G
(4.99)
P = ct2 [l-*-H/(3G)j (4.100)
and a material state parameter,
5 = 0 for - CT < CTct > ct
5 = 1 for a > ct
y o - elasticy o and si 3de i j < 0 - unloadingy o and 8 i j dE i j 0 - plastic
where a is the effective stress
a = (SsjjStj)^ = (3J2)** (4.102)
and H is the hardening modulus
E„H = dade. l-ED/E (4.103)
In accordance with assumption Al, specialization to the present case is achieved by setting
do 22 - da33 = da23 - 0 (4.104)
157
The corresponding strain components can then be eliminated by statical condensation. However, following [30], it ‘ appears reasonable to neglect the in-plane distortional shear strain as well. Thus
de2 3 = 0 (4. 105)
leaving only de2z and d£33 to be eliminated. After a considerable amount of algebraic manipulation, the following explicit definition for the plastic modulus matrix emerges:
2E ( 1+v ) or2 j 09 G ° l m 1i 0
° ! 0 1i 0
where v is Poisson’s ratio and
a = 3a2 [1+H/(3G)]-of(l-2v)ct = (<yl+3olm + 3 o l n )*
(4.106)
(4.107)
4-4-2 J2 deformation theory
In this case, the constitutive tensor C iJkl of equation (4.98) takes the more complicated form [30],
c iikl = Pitc iJkl'5^P23 iJskl"P3^iJ^kl+Ci J k l (4.108)
where
P a = (l+5g)-1P2 = P(ayo/a)p3 = -|Eg/(l-2v) (4.109)2 _ 3 Q fg y o Q~-<?y o ) / g p \ _ ±* \ a I E
and the remaining symbols have the same meaning as those of the previous section.
158
Following exactly the same procedure as before, the plastic modulus matrix can be written
C- p= a
2E(l+v) (bcrJ+g/3) 3Ebaxa im
0
3Ebaxa im I 0r , 1T T £ (9bca?m+ae) I 0I |i 0 , 0
(4. no)
where
a = c-b<rx (l-2v )
b = -3 CTy°---ct3(3+H/G) c = l+|g(l+v)
(4.111)
4-4-3 Stress accumulation
Following the discussion in Section 3.6, the Cauchy stress components in the matrices C and C can be found— p - O’using the accumulation rule
^ w*<7 = ct + f Cde~ c ~ c J — ~ c (4.112)
where e „ are finite increments of corotational strain. Anexplicit definition of these strain increments has beengiven in Section 4.3.4. Recall that the datum for measuringec, is the previous equilibrium configuration. Thesignificance of this datum, is that it allows the integralterm in (4.112) to be independent of any intermediateapproximations to AC that occur when equilibrium iterationsare used in the overall solution process. In other words, itadmits path independent stress accumulation. The alternativepath dependent scheme corresponds to the case in which e is~ ctaken as the the difference between the corotational strains in two successive iterations. A serious drawback of the latter scheme is that the converged solution depends on intermediate configurations that are not in equilibrium [31*32]. Also, according to [32], the scheme can lead to
159
In practice, because ec is finite, the integral term in (4.112) must be evaluated numerically. For this purpose, the relations are conveniently recast in the form
s p u rious o s c i l l a t i o n s between e l a stic and plastic states.
~cepa = a +qa ~ c ~ c ct + CdE ~ c e J - ~ (4.113)
where q lies in the range 0 4: q 4: 1. Here, the general caseof an elastic-elastoplastic transition is taken into accountby dividing the strain increment s into elastic and~ celastoplastic parts
c e =E = (1-q)E~cep ^ c
(4.114)
Evidently, the second and third terms of (4.113) account for the elastic and elastoplastic parts of the transition respectively. Furthermore, since the stress (crc + qg-ce) must lie on the current yield surface, q can be found from the condition
(a1+qaie)2+3 C(c7lm+qCTin,e)2 + (CTin+qaine)2] = a2 (4.115)
where cxy is the current yield stress (allowing for isotropic expansion). Equation (4.115) has only one root in the range 0 4: q ^ 1, that is,
q = a"1 [-b+(b2-ac)Js] (4.116)
where
a = a?e+3(CT^me+CTjne)b = <JX <JX e + 3 ( °’xInariTne+CTi nCTi n e ) cHHct
c = cr2+3(aJm+CT2n )-o2
Numerical evaluation of the last term of (4.112) is nowstraightforward. The usual procedure is to divide e „ into a^ c
160
suitable number of subincrements, and then to apply an Euler forward integration scheme. Note that in the special case of an elastic-ideally plastic material (H = Ep = 0), drift from the yield surface can be eliminated by using a simple ’radial return* at the end of each subincremental step.
4«5 Stress resultants and rigidities
The internal stress resultants acting at the reference point P, play an important role on both sides of the virtual work based equations of equilibrium. On the predictor side they lead to a set of nodal forces that are equivalent to the current internal stress field, and on the corrector side they are needed to define the initial stress effect (geometric stiffness). The rigidities are the symmetric coefficients connecting the incremental stress resultants with the generalized strains. They are referred to in the litrature as tangent rigidities.
4-5-1 Total stress resultants
Figure 4.5 depicts the cross section of the beam in the deformed configuration C. Three local force vectors act on the section. These are
The quantity, Mu is called the bimoment by Vlasov [3] or warping moment by Timoshenko [14]. It is represented in the figure by a three-headed arrow tangential to the reference line. In correspondence with the generalized strains of equation (4.60), the generalized stresses are written
FL <Fr,Fs,Ft>
{Mu ,0,0}(4.118)
S = {Fr,Fs,FtfM slM t,Mr,Mu } (4.119)
161
F I G U R E Q .5 S t r e s s resultants a c t i n g at the r e f e r e n c e point.
The force vectors F l and M l , and warpine moment M w , of equations (4.118) are defined as
F = fa dA ^ l J ~ vA
M = [p xa dA~ L J Z L ~ VA
= Jcrra>dA
where
PL * {0,s , t >£ v — ^CTr»CTrs»a rt^
Here, in accordance with (4.32) and (4.3/1), of ct can be written
r J V
(4.120)
(4.121)
the components
162
CTr - crx- crx nca-CTxmsa-tKr<Tx (4.122)
CTr t = CTxmca+CTxnsa+sKrCTx
Thus, expanding (4.120), one finds
Fr = JardA = JcrxdAA A
F s = JcrrsdA ± / (ainca- a imsa-tKra 1 )dAA A
F t = J a rtdA =*= J( a linca+alnsa+sKra 1 )dAA A
M r = J (crrts-crrst )dA ± J( PnCTim“Pmcrin + PlKrCTi >dA (4.123)A A
M s = J o rtdA ± J"axtdAA A
M t = - JCTrsdA i - Jctx sdAA A
M w = jo’j.udA i Jcrxa)dAA A
The foregoing equations are valid for ’solid* type sections of arbitrary cross section. Specialization to thin walled sections is achieved by introducing the u- and ip-f actors. Recall that the former account for flexural warping and shear attenuation whilst the latter account for torsional warping. After accomodating these effects, the stress resultants become
Fr — J a xdAA
Fs = J(PnCTxnca-Pina iTnSa-tKrCTxA
)dA
Fs = J P,ncrimca+Pn<7i nsa+sKro,xA
) dA
M r = J C4>mcrxm+a)nCT1n + PZKra x )dAA
M s = Jctx tdAA
M t = -|ax sdAA
Mu = Ja x tod AA
Note that, strictly speaking, the u-factors appearing in the equations for the transverse shear forces should operate
163
only on those parts of <7lm and a in that are attributable to flexure. However, because the tranverse shear stresses associated with torsion make no contribution to Fs and Ft» and because shear attenuation and flexural warping are only second-order effects, the accuracy of the given expressions is adequate.
The second, third and fourth of equations (4.124) can be written in the alternative form
Fs = Q s-KrM sFt = Q t-KrM t (4.125)M r = Tr+KrY
where
Qs = /(nnCTinca-umc7imsa)dAA
Qs = J ( ^ i m C « ^ n ^ in8a)dAi (a.126)Tr = J ( jn<7im+ ,n<Ti n
A
Y = Jar1pzdAA
Equations (4.125) clearly demonstrate the nonlinear influence of the twist Kr. The two components of M r can be identified with the first- and second-order St. Venant torques respectively. The latter term accounts for the Wagner effect, and the associated quantity y is known as the Wagner coefficient.
4-5-2 Generalized stress—strain relations
The next objective is to establish the generalized stress-strain relations, that is, the linearized relationship between the incremental stress resultants and generalized incremental strains
2 = DErs/ “• rv (4.127)
16 4
where D is a symmetric 7*7 matrix referred to as the tangent rigidity matrix. Recall that the incremental reference line strains E are based on the linear components of Green’s strain expansion. Consequenty, the components of E are not exactly equal to the physical changes in the components of S. In other words,
S = AS-S ?t g (4. 128)
This inequality arises because S and E are conjugate to the convected and unconvected strains (r and E) respectively.
An explicit definition for E isrsj
E = [h tt OdA (4.129)/v J — “ CA
where the operator matrix H, relating ec to E, is given by (4.63)* But, in keeping with (4.93),
t c = Cec = CHE (4.130)
so that
E = }HTCHQdAE (4.131)~ A
and therefore, comparing (4.127) and (4.131),
D = jHTCHQdA (4.132)A
Using the transformation (4.63)# and denoting the nonzero coefficients of C by c llf c 12 (= c21), c22, and c33(= G), the explicit rigidity expansions shown in Table 4.2 are obtained. These rigidity coefficients provide an extremely comprehensive and versatile model for the behaviour of thin walled beams. Apart from elastoplastic coupling and the influence of Cg., the rigidity matrix of Table 4.2 accounts for the effects of differential fibre length (Q), shear attenuation (uQ), flexural warping (uw ), and current curvature (Kr, Km, Kn ).
C i i 1 Q. d 2 c 41t - c aas d* C aa(0
G c J a + c „ s 5a +
( d a+ c aas a ) t K r
( G - c 22) s a c a -
( das + c 12t c a ) K P- d at d as
“ d,Pni s a - d ^ t Kr -
G p „ ^ „ c a- d a0)
G s J a + c „ c J a +
( d2+ c a2ccr ) s K rd , t “ d , s
d 3pra coc + dvs K r +d 2co
a - c ais t di»t C 111 co
C 11 s 2 - d * s - c aisco
sym .d ,i | i„ + G i|) ;+
4 d „ p ’ ( K r +!<,.)d vw
C ^ O J 2
d a = c „ s a + c ait K r d 3 = „ p’ CKr + k r ) c ia “ C a2p m G = G p2
CL H II c a2c a + c ais K r dw = c aai|jin+ 4 c 11 P2 ( K P + k r ) C 71 ~ C 2-2 P ID
TABLE 4-. 2 Tangent rigidity matrix for a curvilinear thin mailed beam.
165
166
In the special case of an elastic beam with no pretwist, the transverse shear forces and St. Venant torque become uncoupled, and the rigidity matrix takes the much simpler form:
E y 0 ~| 0 J~Et I -Es Ir s t
P = Jo~
“Ii__ - _____ 1_______•_____i_______
Et | -C.B l
0 0
-EstEs
1EG)00
0 EtG)
0 I - E s g )
sym. E g)-
d A (4.133)
where
G s = G(M£c2a+M2s2a)G t = G ( M ^ c 2a + n J s 2a). , (4.134)Gst = G(Mn-u„)saca
Note that, in the case of an elastic bar that is initially straight as well as untwisted, the curvature factor O is l+0(e ), and can therefore be dropped from equations (4.133)*
It is of interest to compare the foregoing stress-strain relations with those derived by Timoshenko [33] for the restricted case of a plane curved bar in the elastic small displacement range. Timoshenko assumed that the centroidal axis of the bar was a plane curve and that the cross section had an axis of symmetry in this plane. For present purposes it is sufficient to take a rectangular bar curved in the r-t plane, as shown in Figure 4.6.
In accordance with equations (4.133) and (4.134), the generalized stress-strain relations for the bar depicted in Figure 4.6 can be written
F r br 1 t E r= E h [ Q " 1 d t
_M r_ Ja t t 2 *s(4.135)
167
t
f, m
R= kj1
F I G U R E 4.6 Elemental length of a c u rved r e c t a n g u l a r Ti-moshenko b a r .
where a = -d/2, b = d/2, and
Qt = (5/6)GA e rt (4.136)
The expression for Q t can be recognized as the standard definition of tranverse shear in a Timoshenko beam, and therefore need not be considered further. But, according to equation (4.36),
°Q = l + tks (4.137)
Therefore, explicit integration of equations (4.135) is possible. Carrying out the integration, one obtains
Fr EA 1 -© rM s_ ~ l-©ks -© e/ks *s
(4.138)
where 0 is an auxiliary parameter defined as
168
e ■ ir-los"(f^afe)
[l+( l / 1 5 ) d 2kl
Inversion of (4.138) leads to
(4.139)
E r / Tl ] k s F r= 1 / E A 1*s [_KS | K s /© M s
(4.140)
Apart from notational differences, this result coincides with that given by Timoshenko. Note that in accordance with the series expansion on the right-hand side of (4.139),
lim(ks/©)k = 12/d 2 (4.141)
Therefore, for a straight beam, the familiar uncoupled relations
E r 1 / E A 0 F r*s 0 1 / E I S M s
are recovered.
(4.142)
Virtual work equilibrium equations
Following the discussion of Section 3.7.4, the incremental virtual work equations (3*63) can be specialized to the present case by writing
ff(r • Se +ct -8* )GdAdr-SWlc = SAW ex-ffcr . 8e QdAdr (4.143) JJ ~ a ~a 1C ex J J-vc ~ g 'LA LA
Here, the appearance of the curvature parameter Q is a direct consequence of the elemental volume definition (4.39). The new term on the left-hand side of (4.143) represents the motion dependent increment of virtual work that arises in the presence of externally applied GE-type moment vectors.
Taking into account the invariance of virtual work with
169
respect to change of frame (see Section *1.3.3), the equilibrium equations may now be cast in the alternative form
ff(T • 8e +ct • 8* )QdAdr-8Wlc = S^W.v-ffcx • 8e QdAdr (4.1 44) J J ~ c ~v ~ v AC ex JJ^C ~cLA LA
Integrating over the cross section and utilizing equations (4.63), (4.83), (4.87), (4.124), (4.126), (4.127), and(4.129), one finds
J d e •&Edr+SWnl- S W lc = 8AW ex-Js • SEdr (4.145)L L
where
| = {FrlFslFt,Ms,Mt,MrfM u } Fr = Fr-KsM s-KtM t (4.146)
and the nonlinear increment of the internal virtual work, &Wn l * is civen by
SWn 1 j"[Fr (8ds6s + SOtQt)+F s8(-Dr0t+^t®r','^®r®s)L + F t 8 ( Up0 S-Qj 0r+i$0r0 t ) +)gMr8 (0t®s~®s®t )+)$MS 8 ( 0r 0t-0t0r ) + H M t 8(0s0p-9r0s) +^80^0^ 3 dr (4.147)
The form of the St. Venant torque term in this equation requires some elaboration. The term derives from the virtual work component
SW =
However,
M r =
- f J [crrst8(0t0s )+CTrtsS(0s0t ) ] dAdrL A
for sections of arbitrary shape.
J(CTr t S - C T r s t )dAA
(4.148)
(4.149)
where, according to elasticity theory [34], each term contributes exactly half of the total St. Venant torque M r, that is,
170
Jar t sdA = -JarstdA = **Mr (4.150)A A
For present purposes, it is reasonable to assume that this result is also approximately valid under elastoplastic conditions. Substitution of (4.150) into (4.148) now leads to the requisite term in (4.147).
An important feature of equation (4.147) can be demonstrated by considering the terms that are a function of the internal moment vector M l = (Mr,Ms,Mt >. It is easily verified that the associated increment of virtual work can be written in the compact form
8W = )$8 Jm l* ( 0^x0 l )dr = J ( 0 l xM l ) • 0 dr (4.151)L L
which implies that the moment vector M transforms asA* L
aM l = M l+*s(0l xM l ) (4.152)
This transformation coincides with the linearized definition of a semitangential moment vector introduced by Argyris et al. [35] and derived independently in (3.80). In other wordsthe internal moment vector is semitangential!
An alternative method of deducing the same result, is to start from the virtual work expressions given by Ziegler [36]. For the restricted case of an axial semitangential torque, Ziegler gives
5W = J$|Mr8 ( u^Us-UjU t )dr (4.153)L
But since Ziegler did not consider transverse shear deformation,
Uj = 01 t u| = — 0 s (4.154)
Consequently, (4.153) can be written
= )$|MrS( 0t0s-0s0i )drL
aw (4.155)
- 171 -
which agrees with the corresponding term In (4.147). The previous assertion that the Internal moment vector Is semltangentlal, is now easily verified on the grounds of symmetry alone.
Explicit expansions of the remaining virtual work terms 8AW ex and 8Wlc, corresponding to specific forms of externally applied loading, are given below.
4-6-1 Virtual work terms 8AWex and 8Mlc
In order to evaluate the external virtual work 8AW ex, the two eccentric load systems shown in Figure 4.7 will be considered. Because cross sections of the beam are assumed to be effectively ’rigid*, surface loads and body forces can be statically reduced to distributed line loads acting at specific eccentricities. Consequently, the proposed loading types have considerable generality, and actually cover all the most common situations encountered in practice.
(a) D i s t r i b u t e d l i n e l o a d (b) C o n c e n t r a t e d l o a d s
F I G U R E 4 . 7 G e n e r a l i z e d c o n s e r v a t i v e load s y s t e m s f o r b e a m a n a l y s i s .
172
acting along the eccentric curvilinear line £. Both the intensity and direction of w may vary with c, but theintensity and direction of any given elemental component, wdc, is assumed to be entirely independent of the overall motion of the beam. Certain restrictions will, however, be made in respect of the lever arm 17. These are
A*
1) The lever arm r\ is perpendicular, or approximately perpendicular, to the reference line r.
2) The variation of r\ with respect to r is small.A/
Load system (b) comprises a number of eccentricconcentrated loads Pjt acting on the reference line atlocations i, together with a number of discretesemitangential moment vectors M sk, acting on the reference line at locations k. As before, the direction and magnitude of P 3 and M sk are assumed to be independent of the overall motion of the beam. In this case, however, no restrictions are placed on either the direction or magnitude of the lever arms 3 .
Provided the lever arms 1? and are visualized as rigid^ v Jlevers that follow completely the motion of the reference line, then both of the proposed load systems are evidently conservative. Furthermore, the moment vectors associated with the eccentric load components w and P., will be of the GE-type, and will therefore exhibit the rotation dependence discussed in Section 3.7.4. This rotation dependence leads to the virtual work term 8W lc, and the associated stiffness (load correction) will take a form similar to that already derived in equation (3.89). In contrast, the semitangential couples M s are exactly balanced by the corresponding internal virtual work terms, and therefore contribute nothing to 8W lc.
Recall that, in accordance with the discussion of Section 3.7.4, SWlc actually reflects the difference in the rotation dependence of GE- and STE- type moment vectors. In contrast, the external virtual work 8AW ex must always take
Load s y s t e m (a) c o n s i s t s of a d i s t r i b u t e d line load w/ V
173
account of the absolute rotation dependence of any applied moment vector, regardless of its disposition. In practice, this is achieved by employing the exact transformations (3.72) or (3-77) for GE- and STE-type moment vectors respectively.
In keeping with restrictions 1) and 2) listed above, the differential length d? associated with dw can be found from equation (4.35), i.e.,
dC = °Qwd°r (4.156)
with
°Ow = [ (l + twk s-swk t ) 2 + (*)kr w ) 2 (4.157)
and
krw = dV/d°r , n = (s£+t£)*$ (4.158)
Here, sw and tw are the local coordinates of the lever arm g with respect to the reference line. The new curvature, krw, is introduced to allow for the fact that the twist of the C line does not necessarily match the reference line twist kr. Thus, for example, if a straight beam is pretwisted, but one wishes to apply the load w along a straight line parallel to r, then krw = 0 even though kr * 0. In practice, however, assuming that 17 does not greatly exceed p, then restriction 2) implies
(*?krw)2 « 1 (4.159)
Therefore, the underlined term in (4.157) can be safely ignored giving
°GW * l+twk s-swkt (4.160)
Note that the deliberate use of the undeformed configuration in these equations ensures that, apart from a
17/1
r i g i d t r a n s l a t i o n , t h e l i n e l o a d r e m a i n s c o m p l e t e l y i n d e p e n d e n t o f t h e s t r u c t u r a l r e s p o n s e .
W i t h t h e s e p r e l i m i n a r i e s , t h e e x t e r n a l v i r t u a l w o r k c a n be w r i t t e n
bS AW e x = J ( Aw • Su + A x Aw • S 8 ) °Qwdr
a
*Z(4? 1- ss J+43 i * 4iei‘8Sj>
sO (a.161)k
Here, t h e c u r r e n t f o r c e s a n d l e v e r a r m s a r e r e l a t e d to t h e i r i n i t i a l v a l u e s by
Aw = ^w°w
> i*
II R ( 0 ) %
- X i ° £ i= R ( 0 . )°t|.— j <%> j
*J5 « = X T (0 k - s ~ k
(/1.162)
In equations (/l. 162), X w , X j , and X k are scalar load factors corresponding to the specified load history of eachindividual load component, R is the standard orthogonal rotation matrix of equation (2.21), and T g is thetransformation matrix of equation (3.77). The use of individual load factors allows for the possibility of nonproportional loading. Within the quoted restrictions, the foregoing expansion of the external virtual work is exact. It is therefore valid for reference line motions of arbitrary magnitude.
Following the method detailed in Section 3.7»^> the virtual work term SWlc can be written
bSWlc = f [S(e) °>7x aw -)40x ( °r)xAw) ] °Qwdr • 50
a
)] • U.163)Y ^ J /V j A/ J /Vj ^ J ^ J A/ J
or
175
S W lcb
se .~ j (a.16ft)
Here, the matrices kw and take a form similar to equation (3.89), i.e.,
O A oTJ2aW,-°T) W,
sym.
°T)xAW y + °rJyAw x ) | - 0,>X*WX- 0T)Z*WZ J
%( °’?xAW z + °r)zAwx ) *S( °T}y AW 2 + Ail2 Aw y )
O „ A. , 0 _ A. ,^ y w y ^z w z(4.165)
and
is,
O „ A n O „ A ■’JzJ P Z J " ^y J y Jsym.
*5(°’1xJi‘P y J +0 ’lyi'‘Pxi> | - t’ix/rxj-',')zj1PI j |
^ 0 , l x J 6 p z J + 0 l 1 z J 4 p x J ) ^ ( OTlyJApzJ + 4,lzJ4pyJ ) - N y J flpyJ-0l}zJ4pzi
(4.166)
where the suffices x, y, z signify vector components parallel to the global x, y, z axes in the usual way.
Equations (4.165) and (4.166) are true to within terms in 0 3. Consequently, when used in conjunction with (4.164), they provide a definition of 8 W lc that complies with the Trefftz buckling criterion.
4-7 Finite element equilibrium equations
Transformation of the incremental virtual work equations of Sections 4.6 and 4.6.1 to a form suitable for finite element analysis, follows the standard procedures of spatial discretization and Boolean assembly.
176
T h e o v e r a l l s t r u c t u r a l d o m a i n is d i s c r e t i z e d i n t o a n u m b e r o f n - n o d e d b e a m e l e m e n t s . W i t h i n e a c h e l e m e n t , d i s p l a c e m e n t s h a p e f u n c t i o n s a r e i n t r o d u c e d so t h a t t h e g e n e r a l i s e d d i s p l a c e m e n t s o f t h e r e f e r e n c e l i n e at a n y p o i n t P c a n b e w r i t t e n in t e r m s o f t h e g e n e r a l i z e d d i s p l a c e m e n t s of t h e n o d e s . T h a t is,
U = N A (4.167)
where
A = (u ,U . . u 4~ l f ~ 2 ~ iU 4 = (u , 0 , e ' . >~ i ~ i ~ i r iu , = « , u , u ,— i X i y i * 2 i0 = {0 ., 0 * ® .~ i X i y i z 1
•Hn>(11.168)
F o r p r e s e n t p u r p o s e s it is a l s o n e c e s s a r y t o i d e n t i f y the i n d i v i d u a l s h a p e f u n c t i o n m a t r i c e s a s s o c i a t e d w i t h t h e t r a n s l a t i o n a l a n d r o t a t i o n a l c o m o n e n t s o f t h e r e f e r e n c e l i n e f r e e d o m s U. T h i s is a c h i e v e d b y w r i t i n g
u0 N A — 0~
(^.169)
A d d i t i o n a l l y , l i n e a r a n d n o n l i n e a r s t r a i n - d i s p l a c e m e n t m a t r i c e s a r e i n t r o d u c e d v i a t h e r e l a t i o n s
E = B 1AG = B A ~ - n l ~
(4.170)
where the new vector G, lists the nine incremental displacement derivatives that appear in equation (4.147), in any convenient order.
Substituting (4.168), (4.169), and (4.170) into the virtual work equations (4.145), (4.146), and (4.147), one finds
6 A T (K. + K - K )A ~ — l — n l — lc - 8^ T(5 ex- 5 1n> (11.171)
177
where
= Jb Jd b ,drL
K , — n 1 = Tb ^ Q B .dr J — n 1 - - n 1 Lb
K,— 1 c = fNTk N d r + Y (Nt .k.N 4) J-e-w-e v -eJ-i-eJ a j
~in _ f§TSdrLb
R~e x = f [NTAw + N T ( Ar?x Aw) ] °Qwdr a
(4.172)
+£[NT 4 AP + N T 4 , x AP 4 ) ] +7(Nt aM „ )V -uJ ~ i — e l ~ l ' •' l 4-> — ©k « s k
Since equations (4.171) are true for arbitrary values of the virtual displacements 8A, they can also be written in the standard form
( K ^ K m - K ^ J A = R -R,~ex ~ in (4.173)
Note that the equilibrium equations (4.173) are valid either for a single element or for an arbitrary assemblage of elements. In the latter case, however, it is of course necessary to apply the standard process of Boolean assembly to the nodal stiffnesses and equivalent nodal loads.
4-7-1 Shape functions and transformation matrices
In order to obtain explicit definitions of the strain-displacement matrices B, and B ,, it is necessary to specify the form of displacement interpolation in more detail. For n-noded isoparametric beam elements, the mapping relations adopted by the author are
178
u
0'V
9r
n£ N «5«i = i 1^ in _ _Z (N .0 , + J N . 0 * )l = i 1 r 1 1 r i
(4.17^)
where N it N lt and N t are assumed to be known functions of a natural longitudinal coordinate, say 5, where -1 1. The parameter J in the third of equations (4.174) is the one-dimensional Jacobian
J = (dr/d*) = lPfCl = (xj €+yf f+z2 C )** (4.175)
The shape functions are designed to provide C° continuity to u and 0, and C1 continuity to thetorsional-warping freedoms 0r and 0p. This is achieved by using polynomial expansions of order n-1 for N 4, and order 2n-l for N 1 and N t. Derivation of the coefficients of these expansions follows standard techniques and will not be repeated here. However, coefficients for the particular case n = 3 are reproduced in Table 4.3 (for further details of this element see Appendix B).
Shape CoefficientsFunctions 1 K 53
Ni 0 -'LIZ LIZ 0 0 0n 2 0 LIZ LIZ 0 0 0n 3 1 0 -1 0 0 0N i 0 0 1 -5/4 -1/2 3/4n 2 0 0 1 5/4 -1/2 -3/4n 3 1 0 -2 0 1 0Nx 0 0 1/4 -1/4 -1/4 1/4n 2 0 0 -1/4 -1/4 1/4 1/4n 3 0 1 0 -2 0 1
TABLE 4 .3 L a g r a n g i a n shape f u n c t i o n s f or 3 — n o d e d bean elenent (see A p p e n d i x B) .
179
An explicit definition of the 7*7n matrix can now be found by using the Lagrangian mapping functions (4.174) in conjunction with the incremental strain-displacement relations (4.84). After transformation to the global frame, the result is
N ' V T i ~ r 0 11
0
n !v tI n S - N , V li ~ t 1 0
n N! V *i ~ t N V T1~ s1 0
I 0 n ; y ;1l 0
l = i a _ 10 » Vi ~ t 1 0
0 1 n 'v tl ^ r11 j n ;
0 1 N > tl ** rl1
(4.176)
Similary, if the components of G are listed in the order W
G (4.177)
then the 9*7n matrix B , is given by— n l
n 'v t 1l ~ r 0 I1 0
i 0 1I
0
" i x ; i 0 11
0
n 0 !n , v t1 ~ s 1 0
£i = io n ! V T i~ s
11
0
0 11 N . V l i~ t 1 0
0 ,1 n 'v *1 ~ t11
0
0 | N V T 1~ r 1 J N i
0 !N' V t 1 ~ r 1
j n ;
(4.178)
The shape function derivatives appearing in the matricesB x and Bnl can easily be found using
Hi = N i. r = J _ l N i .c
Mi = M l . r = J ’ ^ l . c
n ; = M l . r == M l . r r =
s i = M l . r r = J " 2 N lf
rr^irr^i
Ce
(4.179)
180
where
5 , rr (4.180)
Note that these formulae are exact. However, for elements with small or vanishing curvature, the last term in the fourth and fifth of equations (4.179) can be safely ignored.
A simplified six degree of freedom model that excludes warping can easily be derived by deleting the seventh column of the transformation matrices B, and B ,, and replacing Nj by Nj. In this case, however, it becomes mandatory to refer the local freedoms u s, ut, and 0r to the elastic shear centre axis. Assuming that the shear centre has local coordinates s0, tQ relative to the reference point P, then the B-matrices for the simplified model are
n(4.181)
and
n(4.182)
in the introductory chapter, many existingelement models fail to provide an exact
As noted curved finite representation of constant strain states even under small
181
(linearized) motions. In contrast, the linear transformations B x given here do admit an exact representation of linearized constant strain states, including the all important zero strain state associated with rigid body motion (recall that is the corotational method that allows these states to be maintained under motions of arbitrary magnitude).
The use of B-matrices analogous to (4.176) and (4.181) in the context of isoparametric beam and shell models, is apparently quite rare. Indeed, the only publisized exceptions known to the author are the linear curved beam model of Jirousek [37], and the resultant stress degenerated shell model of Liu et al. [38]. Although both these papers introduce transformation matrices that are analogous to those proposed here, neither offers much insight into the underlying rationale, or includes any comments concerning the provision of constant strain states.
The only matrix still to be specified is the 9*9 initial stress matrix Q. Making use of equations (4.147) and (4.177), the components of Q are easily deduced by inspection. The result is
Q
^ I r 0 1 0 1 0 1
— , — t-------------1 .
1 °
1| --------1.
!*
! ! - F s !i
0 i------------- 1_ 0
I F r 1 0 1------------
0 1 0 I 0 1 0 1-------------4- - - : F i l
0
*1 1 0 1 01 . ! ? ■0 1 F s | 0
i 0 I 0 1 0 1 - ) $ M r i * F S |
1 0L_
) * M r l. --------------- 1-
□ - * M t | 00 | i $ F t | - J $ M S
s y m . 1L
0 1-----------------1-
1^ M s ,
0 100i 0 1 0i-------1------
I v
(4.183)
where, for the six degree of freedom no-warping model, theterms HM and % should be replaced by J$(M -F t ) and s t s r oJ$(M +F s ) respectively, t r o
The conventional approach to formulating curved beam elements of the degenerate serendipity type, exactly parallels that used for shells. Typical examples are to be
182
found In [9,39,40]. a straightforward application of the shell-type methology leads to a three-dimensional rectangular beam formulation. However, as pointed out in [9»40], such elements do not lead to the accurate representation of torsion. Removal of the restriction to rectangular sections can be achieved by carrying out the integration with respect to s and t explicitly. Additionally, in order to recover an accurate torsional response, it becomes necessary to introduce an appropriate (classical) model of the torsional-warping behaviour (e.g., St. Venant or Vlasov).
Unfortunately, the application of explicit integration to degenerate continuum elements can effect their ability to undergo strain free rigid body motion. Specifically spurious bending strains occur when thin walled elements of nonzero Gaussian curvature are subjected to a small rigid rotation. This fact has been noted independently by Milford and Schnobrich [41], and Crisfield [42]. As pointed out in [42], it was also a defect recognized by Irons and was undoubtedly one of the main motivations for developing the semiloof element [43-45]. It should be emphasized, however, that the strain-displacement transformations introduced by Irons are quite different from those proposed here. They are also much more involved.
4-7-2 Numerical integration with respect to r
Explicit evaluation of the integral terms of equation (4.172) is, in general, practically impossible. Therefore, numerical integration is required. This of course necessitates the replacement of differential length dr by Jd£. The adopted ordering of generalized components in E and G, allows the associated transformation matrices and equivalent load vectors to be easily partitioned into axial-flexural-shear and torsional-warping parts. Because the shape functions for the latter are of higher order than those those of the former, a selective form of Gauss
183
quadrature Is needed. For the 3-noded element developed by the author, the 2- and 4-point Gauss rules are recommended. The former rule is of the ’reduced integration’ type, and is motivated by the need to prevent membrane locking [46,48]. (Unlike its shell counterpart, the 3-noded beam element does not suffer from shear locking). The simplified six degree of freedom model, does not require selective integration because, in this case, only the standard shape functions are employed.
For maximum computational efficiency, it is possible to devise a technique whereby the total number of passes through the integration loop is only equal to the number of sampling stations required in the higher-order rule. Thus, for example, in the case of the 3-noded beam, the number of sampling stations can be reduced from six to four. This is achieved by linear extrapolation of the requisite function values from the 2- to the 4-point sampling positions. The process is a familiar one since it is similar to that used when exprapolating nodal stresses from the values at Gauss points [49]. Note that the proposed technique is exact in the sense that the values of the integrated functions are identical to those that would have been obtained in a standard 2-point quadrature,
4-7-3 Numerical integration with respect to s and t
Evaluation of the stress resultants S and rigidities D necessitates integration over the cross section. For an elastic beam of arbitrary cross section, exact analytic integration is possible. When the beam is curved, the presence of the irrational function Q-1 makes suchintegration extremely cumbersome. However, a simplified form of explicit integration is still possible provided Q-1 is replaced by a suitable polynomial approximation. On the other hand, if regions of plasticity develop, then analytic integration, although not impossible, becomes completely intractable. In this case, numerical integration should be
is 4
used, but only within the plastic regions of the cross section.
The foregoing method has the important advantage of maximizing the proportion of analytic to numerical integration. This leads to near optimum results in the sense that both accuracy and efficiency are maintained at a high level. The technique relies on the fact that the total stress state at any point can be decomposed into elastic and plastic parts, i.e.,
ar» C a -5a ~ c e ~ c p (4.184)
where a„ denotes the stresses attributable to plastic flow.«cpHence, the corresponding stress resultants are
S/v S -S ~ e ~ p D r-s— e ~ ~ P (4.185)
where S are the stress resultants for a section that is * eentirely elastic, and S are the resultants corresponding to~ pthe distribution of plastic stress components a over^cpplastic regions of the section. The latter resultants take the same form as (4.124), but with the total stresses a replaced by the plastic stresses £ cp» in this way, the evaluation of S can be reduced to an analytic integration to find De, followed by numerical integration over plastic zones.
In a similar manner, and in keeping with (4.96), the calculation of tangent rigidities can be reduced to an analytic integration to find De, followed by numerical integration (over plastic zones) to find the modified plastic rigidities (D +D ). Expressions governing the— p — a -
coefficients of the latter rigidities are the same as those given in Table 4.2, but with C replaced by (Cp+C^). Explicit definitions for the integrated coefficients of De can be found by setting
185
C i i — Ec 12 = 0 (4.186)c2 2 = G
In Table 4.2, together with the linear approximation
Q-i * p-i * i-tKs + sKt+0(<tK) 2 (4.187)
When using (4.187), terms in De involving products of the curvatures are neglected. The corresponding truncation errors in the generalised stresses can then be found approximately from the relations
S « D r[i + o(<tK)2]7 e . (4.188)£ = D ECl + O U K ) 2]r-sj e
Observing the strain limit of assumption All, equations (4.188) imply that target truncation errors of 0 (eD r) ande ++0(eD E), in S and £ respectively, are achieved provided
(<tK) “ 1 = (R/$) > 5 (4.189)
where R is the radius of curvature of the reference line at P, and <t is the radius of gyration of the cross section about P. Both R and <t are measured along any common direction in the cross sectional plane s-t. (Normally, it is sufficient to apply the condition in the s and t directions only.) Note that in most cases of practical interest, the constraint (4.189) is easily satisfied. However, when this is not the case, it may well be necessary to undertake a full three-dimensional continuum analysis.
To further simplify the results, the effect of curvature on the tranverse shear-St Venant torsion interaction terms ( i.e., d 26, d 62, d36 and d 63) can be discounted. This isJustified because fibre length variation over the wall thickness h has only a minor effect on the magnitude of such interactions. Also, because of the idealisations implicit in the transverse shear modelling, it is somewhat doubtful that
186
higher* order interaction terms can be correlated to real physical behaviour.
Applying the foregoing approximations, the rigiditiesare found to be functions of a total of thirty two sectionconstants. A full specification of these constants is givenin Table 4.4 while Table 4.5 provides an explicit definitionof the corresponding elastic rigidity matrix D . The— estrength of the method lies in the fact that all thirty two values can be determined on a once-and-for-all basis prior to the analysis proper. In a typical nonlinear run, the tangential stiffness of the structure may need to be reformed a hundred times or more. The ability to construct a refined elastic stiffness, including the effects of curvature, from a set of predefined constants therefore constitutes a major computational advantage. Observe also that, once the constants are known, the operation count involved in calculating the rigidities is completely independent of the complexity of the section.
I t = j > i < J AA
X i l i * 1 l i * i l i
1 X1 I * C 2 S l i t s 2 1 . 3 to)
1 Zt I 1 o C 3 1 1 a s 21 l a s SCO
1 3s I n C 3 t 1 1 0 s 3 1 * 7 t 2 CO
c i Iis c 3s Iso c * I*B S t O )
I s c i t I 1 3 t 2 l a s C 3 I 2 9 S 2 0)
Is c is l u S t ^ 2 2 I 3 0 oi2
I* C 2 I l S t 3 1 . 3 c 60) I 3 l tO)2
l a c 2t I l S S t 2 l a s 01 X 3 Z sc*)2
C 1 = n£c 2a + u ^ s 2a = ^ m c 6s aC 2 = ) s a c a °3 = ^ m c 6c aC 3 = n£s 2a + u £ c 2oc C 6 = <p/h + 2n
TABLE 4.4 D e f i n i t i o n of s e c t i o n p r o p e r t i e s 1 to I c u r v i l i n e a r thin w a l l e d bean. 1 3 for a
EIi,2.3 0 0 E I 2.13,l»t E I3 .m- . 1 7 E( Ii3+ Ii7)Kr E I2it.2 s.2 B
G K ,5,6 G I 7 .0 . 9 -E J iGK r E l ^ K r - G I 2a ” E l25
GI 1 0 , n, 1 2 E IiifKr -EI17Kr E I 21 EI2e
EId3,lS,16 “ E lm .,10.10 E ( I is 110 ) r E I25,27.2B
sy m . E Il7.10.19 “ E ( Iie + Il9^ ” E I2G,2B,29
G ( l2 2 “ 2 l-23Kn) E ( 27 + 29 ) K r
E I3 0 ,31.32
Il .M.N “ I L ” 2 ( K p -*-|<r )
TABLE 4-. 5 Elastic rigidity matrix for a curvilinear thin walled beam.
187
188
The explicit evaluation of the thirty two section properties in any given case is a tedious, but nevertheless straightforward, task involving analytic integration of known functions over a known domain. From the point of view of developing general purpose software, it makes good sense to * build* the section out of standard segments. In the first Instance, a facility to construct the section using simple rectangular segments is recommended. Then, if necessary, circular and variable thickness segments can be implemented at a later stage. Obviously, it is only necessary to define the thirty two constants for a typical segment. Summation over all such segments then gives the overall section properties. Explicit definitions of I x to I32 for a rectangular segment are listed in Appendix C.
As mentioned earlier, evaluation of S and (D +D ) must be carried out numerically. Alternative methods, commonly recommended for this kind of application, are
1. Mid-ordinate rule2. Trapezoidal rule3. Newton-Cotes quadrature4. Gaussian quadrature
Because the yield surface is convex, the mid-ordinate and trapezoidal rules lead, over regions in which the functional is smooth, to upper and lower bounds respectively [50]. The Newton-Cotes formulae are based on a least-squares polynomial fit, and, although accurate for smooth functions, they are generally less reliable than the mid-ordinate or trapezoidal rules for piecewise (discontinuous) functions. Gauss quadrature suffers from the same disadvantage as the Newton-Cotes method and, in addition, it has the drawback of being based on lrregulary spaced sampling stations none of which lie at the extreme fibres.
Against this background, the author has developed an alternative method that is more accurate than the mid-ordinate or trapezoidal rules for discontinuous functions, and yet can maintain adequate precision for
189
smooth functions. It would seem unlikely that the technique Is original. Nevertheless, as far as the author is aware, it has apparently not previously been applied in the structural mechanics context.
The basic idea is to factorise the functional into two parts, 1.e. ,
b bI = J$(x)dx = Jf(x)g(x)dx (4.190)
a a
where f(x) is a continuous or piecewise function of order one, and g(x) is a continuous function of order m-1 (where m is the order of the basis function «£). Now apply Simpson’s rule over the range of collocation b-a. However, Instead of the usual n-intervals, use 2n intervals, and denote the ordinates of f and g
f(0),f(Js),f(l).... f (n-J$), f (n)(4.191)e(o),g(J$),g(D....e(n-H), a(n)
The result is
I = ^^[f(0)g(0)+4f(H)e(H)+2f(l)g(l)+....o2f (n-1 )g(n-l)+4f (n-H)s(n-J$)+f (n)g(n) ] (4.192)
Next eliminate intermediate terms by substitution of the average values
f(i-fc) = J4[f (i-l) + f (i)] i = 1,2,--- n (4.193)e(i-H) = isCcd-l )+ff(i) ]
After some manipulation, this leads to
I = (b-a){Jsf (0) [g(0)+Ag1]+f(l)g(l) +....n
hf (n) [g(n)-Agn] +£}sf (i) [Agi-Agi-1 ] > (4.194 )i = i
where
190
Since g(x) Is continuous, the underlined term vanishes identically in the limit n -*oo. Therefore, (4.195) becomes
I = (b-a){%f(0)[g(0)+Ag!]+f(l)g(l)+....f(n-1)s(n-l)+H?(n)[g(n)-Agn]> (4.196)
Comparing this result with the standard trapezoidal rule, it is easy to see that the only change consists of a ’slope* correction to the first and last ordinates of s(x). If $(x) is a continuous function of order three or less, then the proposed rule, with n = 2, suffices to give exact results. The technique also yields exact results when $(x) is a piecewise function each part of which is of order two or less, provided the discontinuities are located exactly at station positions.
Table 4.6 compares the accuracy of the proposed rule (TREC) with the standard trapezoidal rule (TRAP). They are applied to a quadratic and a cubic function respectively, using in each case n = 1,2,4, 6,8 over the range 1 4: x 5.
Asi = [ g ( i ) - e ( i - l ) ]/3 (4.195)
Function n TRAP TREC Exact
1 0 325 2 24 32J (13~3x)xdx 4 30 32 321 6 31.11 32
8 31.50 321 -36 92
3 2 36 60J(9-2x)x*dx 4 54 59 601 6 57.33 59.41
8 58.50 59. 63
TABLE 4 .6 C o m p a r i s o n of a l t e r n a t i v e q u a d r a t u r e rules for f u n c t i o n s of o r d e r two an d three r e s p e c t i v e l y .
191
It Is clear from the Table that, for a given value of n, the proposed rule is by the far the most accurate. Observe that, for functions of third order or higher, the convergence of the new rule is not monotonic. This is of course a direct consequence of enforcing an exact result for both a quadratic function with n > 1 and for a cubic function with n = 2.
Apart from its accuracy and ease of application, there are two further reasons why the new rule is ideally suited to the numerical evaluation of elastoplastic rigidities and stress resultants. Firstly, the predominant terms that arise in such cases are of order two or less. Cubic terms arise only from plastic flow within the material, and from curvature effects. Secondly, because the position of plastic discontinuities are not, in general, known in advance, for actual numerical evaluation, they will automatically be relocated at the nearest sampling station. This immediately implies that, in order to minimize the associated error, a relatively small interval is required, and this in turn points to the use of a low-order rule.
References
[l]
[2 ]C3]
C 5 3
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[6] RAJ ASEKARAN , S. and MURRAY, D.W. , "Coupled local buckling in wide flange beam columns", Proc. ASCE, Vol. 99, No. ST6, 1973, PP. 1003-1023.
[7] RAJASEKARAN, S., "Finite element method for plasticbeam-columns", in: Theory of Beam Columns, Vol. 2 -Space Behaviour and Design (W.F. Chen and T. Atsuta, eds.), McGraw-Hill Book Co., New York, 1976, Ch. 12.
[8] BATHE, K.-J. and HO, L.W., "Some results in theanalysis of thin shell structures", in: NonlinearFinite Element Analysis in Structural Mechanics, (W. Wunderlich, E. Stein, and K.-J. Bathe, eds.),Springer-Verlag, Berlin, 1981.
[9] BATHE, K.-J., "Finite element procedures in engineering analysis", Prentice-Hall, Inc., Englewood Cliffs, N.J., 1982.
[10] HARSTEAD, G.A., BIRNSTIEL, C., and LEU, K.C.."Inelastic H-columns under biaxial loading", Proc. ASCE, Vol. 9Uf No. ST10, 1968, pp. 6173-6398.
[11] RAJASEKARAN, S. and MURRAY, D.W., "Finite elementsolution of inelastic beam equations", Proc. ASCE, Vol. 99, No. ST6, 1973, PP. 102/1-10/12.
[12] CHEN, W.F. and ATSUTA, T., "Inelastic response of column segments under biaxial loads", Proc. ASCE, Vol. 99, No. EM/l, 1973, PP. 685-701.
[13] RAMM, E. and OSTERRIEDER, P.. "Ultimate load analysisof three-dimensional beam structures with thin-walled cross sections using finite elements", Int. Conf. on Stability of Metal Structures, Paris, Nov. 16-17, 1983.
[14] TIMOSHENKO, S.P. and GERE. J.M., " Theory of elasticstability", 2nd. ed., McGraw-Hill Book Co., New York,1 9 6 1 .
[15] ROBERTS, T.M., "Second order strains and instability ofthin walled bars of open cross-section", Int. J. Mech. Sci., Vol. 23. 1981, pp. 297-306.
[16] GUNNLAUGSSON, G.A. and PEDERSON, P.T.,"A finite element formulation for beams with thin walled cross-sections". Comp, and Struct., Vol. 15. No. 6, 1982, pp. 691-699.
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[17] WEBER, C., "Die lehre der drehungsfestigkeit",Forsharb. Ing. Wes., Vol. 249, 1921, pp. 60-62.
[18] WAGNER, H. and PRETSCHNER, W., "Verdrehung und knickungvon offenen profilen", Luftfahrt-Forschung, Vol. 11, 1934, translated in NACA Tech. Mem. 784, 1936.
[19] REISSNER, E., "The effect of transverse sheardeformation on the bending of elastic plates", J. Appl. Mech., ASME, Vol. 12, 1945, A69-A72.
[20] MINDLIN, R.D., "Influence of rotatory inertia and shearon flexural motions of isotropic elastic plates", J. Appl. Mech., ASME, Vol. 18, 1951, PP. 31-38.
[21] HUGHES, T.J.R. and LIU, W.K., "Nonlinear finite elementanalysis of shells: Part I. Three-dimensional shells", Comp. Meth. Appl. Mech. Engrg., Vol. 36, 1981, pp.331-362.
[22] MALVERN, L.E., "Introduction to the mechanics of acontinuous medium", Prentice-Hall, Inc., EnglewoodCliffs, N.J., 1969, p. 372.
[23] KOITER, W.T., "Stress-strain relations, uniqueness and variational theorems for elastic plastic materials with a singular yield surface", Q. Appl. Math., Vol. 11, 1953, PP. 350-354.
[24] SANDERS, J.L., "Plastic stress-strain relations basedon H e a r loading functions", Proc. 2nd U.S. Natl.Congr. Appl. Mech., ASME, New York, 1954, pp. 455-460.
[25] BUDIANSKY, B., "A reassessment of deformation theoriesof plasticity", J. Appl. Mech., Vol. 26, 1959* PP.259-264.
[26] KLIUSHNIKOV, V.D., "On a possible method ofestablishing the plasticity relations", TPMM(translation of Prikl. Mat. i. Mekh.), Vol. 23, 1959, pp. 405-418.
[27] HUTCHINSON, J.W., "Plastic buckling", in: Advances inApplied Mechanics, (C.S. Yih, ed.), Vol. 14, Academic Press, New York, 1974, pp. 67-146.
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[28] HUTCHINSON, J.W. and BUDIANSKY, "Analytical andnumerical studies of the effects of initialimperfections on the inelastic buckling of a cruciform section", in: Buckling of Structures, Proc. IUTAMSymp., Cambridge, (B. Budiansky, ed.), Springer-Verlag, Berlin, 1976, pp. 98-105.
[29] NEEDLEMAN, A. and TVERGAARD, V., "Aspects of plastic post-buckling behaviour", in: Mechanics of Solids - The Rodney Hill 60th Aniversary Volume, (H.G. Hopkins and M.J. Sewell, eds.), Pergamon Press, Oxford, 1981.
[30] WUNDERLICH, H.O. and SCHRODTER, V., "Nonlinear analysisand elastic-plastic load-carrying behaviour of thin-walled spatial beam structures with warping constraints", Int. J. Num. Meth. Engrg., Vol. 22, 1986,pp. 671-695.
[31] POWELL, G. and SIMONS, J., "Improved iteration strategy for nonlinear structures", Int. J. Num. Meth. Engrg., Vol. 17, 1981, pp. 1455-1467.
[32] TRUEB, U., "Stability problems of elasto-plastic plates and shells", Ph.D. Thesis, University of London, 1983,p. 111.
[33] TIMOSHENKO, S.P., "Strength of materials. Part I", 3rd. ed., Van Nostrand Reinhold Co., New York, 1955, Ch. 12.
[34] TIMOSHENKO, S.P., "Theory of elasticity", 3rd. ed., McGraw-Hill Book Co., New York, 1970, p. 296.
[35] ARGYRIS, J.H., DUNNE, P.C., and SCHARPF, D.W., "Onlarge displacement - small strain analysis ofstructures with rotational degrees of freedom", Comp. Meth. Appl. Mech. Engrg.. Vol. 14, 1978, pp. 401-451.
[36] ZIEGLER, H., "Knickung gerader stabe unter torsion",ZAMP 3, Vol. 96, 1952, pp. 96-119.
[37] JIROUSEK, J., "A family of variable section curved beamand thick-shell or membrane-stiffening isoparametric elements", Int. J. Num. Meth. Engrg., Vol. 17, 1981,pp. 171-186.
[38] LIU, W.K., LAM, E.S., and BELYTSCHKO, T., "Resultantstress degenerated-shell element", Comp. Meth. Appl.Mech. Engrg., Vol. 55, 1986, pp. 259-300.
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[39] BURAGOHAIN, D.N., AGRAWAL, S.B., and AYYAR, R.S., "Amatching superparametric beam element for shell-beam systems", Comp, and Struct., Vol. 9* 1978, pp. 175-182.
[40] FERGUSON, G.H. and CLARK, R.D., "A variable thickness, curved beam and shell stiffening element with shear deformation”, Int. J. Num. Meth. Engrg., Vol. 14, 1979* PP. 581-592.
[41] MILFORD, R.V. and SCHNOBRICH, W.C., "Degeneratedisoparametric finite elements using explicit integration", Int. J. Num. Meth. Engrg., Vol. 23. 1986,PP. 133-154.
[42] CRISFIELD, M.A., "Explicit integration and the isoparametric arch and shell elements", Comm. Appl. Num. Meth., Vol. 2, 1986, pp. 181-187.
[43] IRONS, B.M., "The semiloof shell element", internal report, University of Calgary, 1973*
[44] IRONS, B.M., "The semiloof shell element", in: FiniteElements for Thin Shells and Curved Members, (R.H. Gallagher and D.G. Ashwell, eds.), John Wiley & Sons, New York, 1976, Ch. 11.
[45] IRONS, B.M., "Techniques of Finite Elements", Ellis Horwood, Chichester, 1980.
[46] RAMM, E. and STEGMULLER, H., "The displacement finiteelement method in nonlinear buckling analysis of shells", Buckling of Shells - A State-of-the-Art Colloquium, Institut fur Baustatik UniversitatStuttgart, May 6-7* 1982.
[47] STOLASKI, H. and BELYTSCHKO, T., "Membrane locking andreduced integration for curved elements", J. Appl. Mech., ASME, Vol. 49* 1982, pp. 172-176.
[48] STOLASKI, H. and BELYTSCHKO, T., "Shear and membrane locking in curved C° elements", Comp. Meth. Appl. Mech. Engrg., Vol. 41, 1983, PP. 279-296.
[49] HINTON, E., SCOTT, F.C., and RICKETTS, R.E., "Localleast squares smoothing for parabolic isoparametric elements", Int. J. Num. Meth. Engrg., Vol. 9, 1975* PP.235-256.
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[50] CORMEAU, I., "Elastoplastic thick shell analysis by viscoplastic solid finite elements", Int. J. Num. Meth. Engrg., Vol. 12. 1978, pp. 203-227.
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C H A P T E R 5
S U M M A R Y A ND C O N C L U D I N G R E M A R K S
5-1 Summary
In this work techniques have been developed for modelling the three-dimensional quasistatic behaviour of thin walled structures undergoing large deformations. Since the structures were presumed thin, and thickness updating procedures have not been used, the focus has mainly been on problems of the small strain but finite rotation type. Particulary notable in this category is the pernicious problem of elastic and plastic buckling of slender beams and shells. Although attention has been restricted to conservative systems, extension to cover circulatory forces as well as instabilities of the flutter type istheoretically straightforward (see Section 5.2).
5-1-1 Theory of rigid body rotation
A comprehensive study of the theory of finite rigid body rotation was undertaken in Chapter 2. It was concluded that the classical theory due to Euler forms an excellent basis for the continuous description of the spatial orientation of a rigid body rotating in a three-dimensional space. The theory is shown to lead to a definition of the rotation tensor that is symmetric with respect to three linearly independent rotation coordinates, termed herein, Euler rotation components. These coordinates are the natural three-dimensional generalization of the usual engineering notion of angular rotation in a plane, and are therefore ideally suited to numerical models based on say finite elements. Euler rotations were shown to have many of the attributes of a vector. In particular, they are sequence independent, and can be transformed from one frame to
198
another In the standard manner. This contrasts with Euler angles which, although frequently recommended for finite rotation modelling, do not possess the same symmetry and are highly sequence dependent.
5-1-2 Kinematics of thin walled structures
The role of finite rotations in the context of the pre- and post-buckling behaviour of thin walled structures, was investigated in Chapter 3. Degeneration of full continuum formulations inevitably leads to the introduction of rotation coordinates, and it was shown that direct assimulation of the finite rotation tensor leads to very complicated strain-displacement relations involving trigonometric functions. Greatly simplified relations of high precision can, however, be obtained by utilizing the polar decomposition theorem in order to discard the rigid body component of the overall motion. In effect, this enables the strains to be observed in a frame that translates and rotates with the point in question. Such formulations are known in the literature as corotational.
Apart from very high precision, an important advantage of the corotational method, is the fact that spurious straining under finite rigid body rotation is directly eliminated from the calculation of the internal stress field. Explicit definitions of the corotational freedoms have been derived, along with exact methods of updating the local referential frame and of accumulating finite (noncommutative) increments of Euler rotation.
Working from the Trefftz buckling criterion, it was argued that the incremental form of the virtual work based equations of equilibrium, must contain a complete quadratic expansion of the generalized incremental strain-displacement relations. In degenerate beam and shell models, it is necessary to introduce kinematical relations defining the relative motion of points lying off and on the reference line or surface. Usually the distance between such points is
199
assumed, a-priori, to be invariant. Consequently, in order to satisfy the Trefftz criterion, it is is essential that the the associated incremental motion should be based on a consistent second-order approximation of the rotation tensor. This leads to quadratic terms that are additional to those arising from Green’s strain expansion, and results in a geometric stiffness that contains a complete, and geometrically consistent, set of critical buckling eigenvalues and eigenvectors. In contrast, the conventional approach derives from the usual skew-symmetric first-order approximation of rotation increments, and therefore leads to a geometric stiffness that is incomplete in the aforementioned sense.
Explicit definitions of the incremental strain-displacement relations have been derived for degenerate beam and shell models. In the former case, comparison of initial stability problems with results published by Argryris et al. [1], has indicated that an equivalence exists between the geometric stiffnesses based on virtual work (in its ’complete* form) and on the natural approach. Although Argyris et al. [2] recognize the fact that the conventional virtual work equations are incomplete, their explanation of the deficiency is unsatisfactory in that it fails to identify the true underlying cause. Consequently, the consistent assimilation of finite rotations into the virtual work equations, is undoubtedly one of the major achievements of the present work.
Critical stability (initial eigenvalue) levels for a series of simple beam and frame problems involving general out-of-plane buckling, are given in reference [1]. Comparison with conventional virtual work based models, indicated that errors as high as 60% in the critical buckling loads could accrue as a result of the missing terms in the geometric stiffness. These results have been independently confirmed by the present author. In addition, when using the conventional equations, it was found impossible to investigate the post-buckling behaviour of such structures. Indeed, in some cases, convergence of the
200
incremental-iterative solution was lost well before the instability could be reached even when using the full Newton-Raphson method.
An interesting observation concerning the proposed incremental strain-displacement relations for shells, is that the additional quadratic terms are functions of the in-plane rotation (i.e., rotation about the shell normal). Much has been written about the role of the in-plane rotation on the nonlinear behaviour of shells. Although it is generally accepted that this role can be safely ignored in the majority of situations, exceptions do exist [3,/lJ, and this degree of freedom should not be discarded too lightly. Indeed, in the context of the incremental-iterative numerical solution of the nonlinear equations of equilibrium, it is not sufficient to think in terms of the absolute effect of approximations on the the control (corrector) side of the equations. In addition, the correlation between the predictor and corrector sides of the equations must be taken into account, since it is this correlation that controls the convergence characteristics of the solution. Buckling problems in general, and plastic buckling in particular, frequently lead to conditions in which the convergence rate is very sensitive to quite minor changes in the topology of the total potential surface. Against this background, it has been the author's experience that retention of the additional stiffness terms on the predictor side of the equations can often improve convergence significantly, in some cases leading to solutions where previously convergence was seemingly impossible.
To understand the basis for the five degree of freedom element proposed in Section 3.5.6, it is essential to appreciate, that the motion of a pair of orthogonal vectors in a three-dimensional space cannot, in general, be described using less than six displacement freedoms. This means that special conditions pertain to the convected frame with respect to which the two local rotational freedoms are measured. Assume that the frame axes are labelled r, s, t
201
and that Initially r and s are tangential to the reference surface of the shell. Then, at each stage of an Incremental analysis, the frame must be updated by applying the corresponding Incremental Euler rotation to an axis that lies strictly in the current r-s plane. This frame is of course unique in the sense that it is the only frame with respect to which the in-plane rotation vanishes identically. Needless to say, conventional five freedom shell elements have completely ignored this point. In other words, they assume a-priori that the in-plane rotation is irrelevant. Even if one accepts the loss of precision, a clear danger of this approach is that convergence difficulties are likely to be encountered in certain buckling problems, particulary in the post-buckling range.
An important corollary to the foregoing observations is that because the in-plane rotation vanishes identically, no additional terms arise in the geometric stiffness of the proposed five freedom element. Equivalently, in this case, the conventional form of the incremental strain-displacement equations, based on linear kinematic relations, is actually sufficient to satisfy the Trefftz criterion.
The practical need for six degree of freedom shell elements was discussed along with a technique for dealing with the troublesome case when several flat elements meet in a single plane. It may be worth pointing out that an alternative to the proposed ad-hoc procedure is to introduce the special five degree of freedom model (discussed above) over smooth regions of the shell, the six freedom element being used only to accommodate stiffeners and shell Junctions.
Nonlinear constitutive relations were introduced, with particular emphasis being paid to the problem of maintaining objectivity in the presence of finite rotation increments. Based on the recommendations of references [5-7]» a modified definition of the material law has been used. This modification ensures that the material modelling is consistent with large strain theories. The additional terms are functions of the current Cauchy stresses, and can be
202
significant in materials with low or vanishing hardening rates. Objectivity was assured by using a corotational method of stress accumulation.
The final objective of Chapter 3 was to examine the effects of applying concentrated moment vectors to a thin walled structure. As pointed out by Ziegler [8], concentrated moments applied to a flexible structure via an axis that remains fixed in space, are nonconservative. Argyris [1,9] has shown that such moments lead to equilibrium equations that are nonsymmetric. Therefore, in order to meet the stated objectives of the present work, moment vector transformations that admit exact path independent work in the presence of finite rotations are introduced.
Two alternative methods of applying an exactly conservative moment vector to a flexible rotating body have been discussed. The first method, termed semitangential after its introduction by Ziegler [10], has been shown by Argryis [1,9] to be unique in the sense that it is the only method of applying a conservative moment that does not affect the tangent stiffness of the structure. In other words, no load-stiffness correction terms are required. The second method, termed here GE (generalized exact), is needed because it corresponds to the common situation in which a conservative load is applied eccentrically to the end of a rigid lever.
In contrast to semitangential moment vectors, the application of GE-type moment vectors to beams and shells does introduce load-stiffness correction terms. It is worth noting that such terms are analogous to, and of the same order as, the effects of geometric imperfections and initial stresses, and can have a very significant effect on the buckling characteristics of the structure. Once again, the reader is warned that it is not sufficient to continuously update the external potential on the corrector side of the equilibrium equations. The aforementioned stiffness correction terms must also be added to the predictor side.
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5-1-3 Nonlinear theory of curved thin walled beams
A comprehensive technical theory of curved thin walled beams of arbitrary cross sectional shape was developed in Chapter U. The theory is based on degeneration of the general three-dimensional equations of continuum mechanics to a one-dimensional form expressed in terms of a single reference line. The resulting generalized freedoms are work conjugate with the generalized stresses (stress resultants) obtained by integrating over the cross section. A method of including warping, that is suitable for open, closed and mixed open/closed sections, and includes the effects of differential warping through the wall thickness, was presented. The theory embraces all the more pertinent features discussed in earlier chapters, as well as a number of additional features and refinements that are specifically beam orientated.
The incremental equations of equilibrium were derived using virtual work. It was shown that an increment of internal virtual work involves rotation terms that are work conjugate with respect to a semitangential definition of internal moment. This finding shows, unequivocally, that objections to virtual work based derivations of the geometric stiffness, such as those raised by Argyris et al. [2], were ill-concieved. Potential energy terms associated with the application of several forms of eccentrically applied load have been included in the equations. To ensure that these loads remain conservative, they are applied to the ends of rigid levers that are, in turn, rigidly attached to the reference line of the beam.
Details of the way in which the equations of equilibrium can be applied in a form suitable for finite element discretization, were presented. Explicit definitions of the various matrices that arise in this process have been given, including the load-stiffness correction matrices associated with eccentrically applied forces. The problem of defining, at the shape function level, strain-displacement transformations that admit genuine strain free rigid body
2 .0 li
motion in curved degenerate elements, was also discussed.Some useful procedures for the efficient evaluation of
stress resultants and tangent rigidities, were elucidated. The basic idea was to minimize the amount of numerical integration with respect to the transverse directions s and t, by restricting numerical integration to those parts of the cross section that are currently plastic. Analytic integration under elastic conditions was made possible by applying a series expansion to the curvature effect, and truncating terms that are quadratic and higher in the curvature components. The resulting elastic rigidity matrix is a function of thirty two analytically integrated constants. Explicit expressions for the thirty two constants for a single rectangular segment have been given in AppendixC. This allows many of the commonly occurring section shapes to be built up using a series of such segments. The elastic rigidities are then obtained by a straightforward summation over the entire section.
An accurate and easily applied method of numerical quadrature was proposed for evaluating the reduction in stress resultants and rigidities associated with plastification of discrete areas of the cross section. This method is essentially a modified composite trapezoidal rule which combines Simpson’s rule accuracy with the ability to cope with discontinuities in the functional.
5«2 Concluding remarks and suggestions for further work
Inevitably there are a number of aspects of the present work where further study may be fruitful and extensions can be made. In particular, the models can be extended to include nonconservative loads, dynamics, large strains, and local buckling. Also, much work is still required in the development of nonlinear finite element formulations, that can accurately reproduce the equilibrium paths of slender structures undergoing plastic buckling.
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Although the present thesis has been limited to conservatively loaded systems, extension to cover circulatory forces is straightforward. Note however that circulatory loads lead to a nonsymmetric tangent stiffness, so that a nonsymmetric equation solver is needed. Moreover, if the loads produce instabilities of the flutter type, then in order to model the behaviour in the post-flutter range a full transient time-stepping procedure is needed. Although these enhancements clearly involve fairly radical changes to the basic quasistatic conservative formulation, they do not present any special theoretical problems, and the techniques involved are firmly established. It is as well to bear in mind, however, that the transference of theoretical concepts into working numerical models nearly always throws up new and unexpected problems!
In theory, extension to cover large strains is also straightforward. This is true because the main ingredients that are needed to cover large strains, are already present in the proposed method of incremental stress evaluation and stress accumulation. Thus, for example, work congugate measures of stress and strain are employed, and objectivity is assured by referring the increments of stress and strain to identical corotational frames. The main missing ingredient concerns the ’thickness* updating process. That is, the effect of lateral strains on the length and orientation of tranverse fibres should be accounted for.
In the author’s opinion, however, the practical Justification for including such refinements withindegenerate models of beams and shells is quite problematical. This viewpoint is largely based on the common sense notion that any system is only as good as its weakest link. Unfortunately, the precision available in the best finite element beam and shell models currently available, is simply not good enough in the regime in which finite strains operate. Errors accrue from a number of sources - geometric discretization, shape function mapping, numericalquadrature, Mindlin/Reissner assumptions, convergencetolerance, and so on. Such errors are bound to have a
206
masking effect on any refinement that produces changes to the model that theoretically fall below a certain average error threshold.
In the case of the proposed beam formulation, there are two additional reasons why the implementation of large strains would be inappropriate. Firstly, the validity of using an idealized Vlasov type warping model in the large strain range is problematical, and secondly the exclusion of local buckling in the presence of large strains creates an obvious conflict.
As discussed in Chapter 4, the modelling of local buckling involves the use of additional out-of-plane flexural freedoms. One method of incorporating these freedoms within the basic one-dimensional model was pioneered by Rajasekaran [11,12]. Unfortunately, the shape functions associated with the new freedoms are dependent on the particular cross sectional shape that is required (for example, Rajasekaran works exclusively with an I-section). In order to achieve greater generality, it may well be necessary to build the section out of individual shell elements. Comparative numerical studies using alternative models constructed from the author’s beam and shell elements respectively, would provide useful imformation. From the results of such a study, it should be possible to quantify the role of cross sectional distortion both in the pre- and post-buckling regime.
Although they already include all the main features, some work remains to bring the beam and shell elements developed by the author fully into line with the theoretical proposals of Chapters 3 and 4. In addition, development of the proposed five degree of freedom shell element would provide a useful alternative to the six freedom model, at least within those regions of the shell that are free from discontinuities. The five freedom model has the advantage that it obviates the use of stabilization techniques that are needed in the six freedom model to control the in-plane rotation of the reference surface.
Another challenging, and as yet unresolved, difficulty
207
relates to the fact that existing six freedom shell models (including the author’s) do not, In general, have the ability to resist concentrated moment vectors turning about the midsurface normal. Basically the problem is how to incorporate a realistic drilling freedom into a plane stress situation which will nevertheless become totally uncoupled in situations where no moments are actually applied to the nodes. One way of achieving this goal is by using a drilling freedom whose order is one degree higher than that of the in-plane freedoms. Then, by suitable selection of the shape functions, exact C 1 compatibility can be enforced between the drilling freedom and the in-plane rotation defined by the right-hand side of equation (3*53). However, a fairly obvious drawback of this scheme is that reduced integration, which is needed to control membrane and shear locking in curved elements, will not be sufficient to capture the higher order in-plane ’bending* modes of the new model (for a quadratic 8- or 9-noded element, these modes will be cubic).
One possible solution to this dilemma would be to devise an independent method of controlling membrane locking that does not require reduced integration. Then, the foregoing proposals could be applied to a 9-noded Lagrange shell element with full 3*3 integration. This scheme has two important advantages. Firstly, it eliminates the troublesome spurious modes that appear when using reduced integration. Secondly, because shear locking in the fully integrated Lagrange element is low, it is easily controlled by using the residual energy balancing technique first introduced by Fried [13-15]. This actually imparts a third advantage because it allows the flexural stiffness, which will probably tend to be too high, to be ’fine-tuned*.
Note that the 8-noded element detailed herein is less amenable to the above approach because, under full integration, the shear locking is high. In this case, although the aforementioned locking control method still works, its effectiveness is significantly impaired. Moreover, the spurious hour-glass modes that appear in the
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underintegrated 8-noded element, are largely irrelevant, because only in pathological cases can they propogate through a properly supported mesh of two or more elements.
Finally, the author hopes to be able to carry out an in-depth comparative study with the recent work of Wunderlich et al. [16]. As discussed in the introductory chapter, the formulation presented in reference [16], possibly represents the only published curved beam model that is genuinely capable of modelling the plastic buckling behaviour of thin walled open section bars. Furthermore, a number of extremely valuable numerical results, involving the elastoplastic buckling of simple imperfection sensitive structures, are presented. These include studies of Roorda's two-bar frame and of a cruciform section beam-column.
The latter structure exhibits a particularly strong relationship between the material description and the buckling characteristics, and has recieved the attention of a number of eminent researchers [17-20]. The results of these studies, as well as those of Wunderlich et al., show a remarkable discrepency between the buckling load obtained when using J2 flow and J 2 deformation theory respectively. According to reference [16], this has only served to heighten the controversy over whether or not yield surfaces develop corners. It appears likely that, in the absence of unambiguous experimental evidence, the issue may remain unresolved for some time to come.
References
[1] ARGYRIS, J.H., DUNNE, P.C., and SCHARPF, D.W., "On large displacement-small strain analysis of structures with rotational degrees of freedom". Comp. Meth. Appl. Mech. Engrg., Vol. 1U, 1978, pp. U01-U5H Vol. 15,1978, pp. 99-135.
[2] ARGYRIS, J . H . , HILPERT, O. , MALEJANNAKIS, G.A., and SCHARPF, D.W. , "On the geometrical stiffness of a beam in space - a consistant V.W. approach”, Comp. Meth. Appl. Mech. Engrg., Vol. 20, 1979, pp. 105-131.
[3] COHEN, J.W., ”The inadequacy of classical stress-strain relations for a right helicoidal shell", Proc. IUTAM Symp., I960, p. 415.
[4] REISSNER, E., "On the form of variationally derivedshel equations", J. Appl. Mech., ASME, Vol. 31, No. 2, 196/1. p. 233.
[5] McMEEKING, R.M. and RICE, J.R., "Finite elementformulations for problems of large elastic-plasticdeformation", Int. J. Solids and Struct., Vol. 11,1975, PP. 601-616.
[6] KLIEBER, M. , KONIG, J.A., and SAWCZUK, A., "Studies onplastic structures: stability, anisotropic hardening,cyclic loads", Comp. Meth. Appl. Mech. Engrg., Vol. 33, 1982, pp. 487-556.
[7] SAMUELSSON, A. and FROIER, R., "Numerical methods inelasto-plasticity - a comparative study", in: NonlinearFinite Element Analysis in Structural Mechanics, (W. Wunderlich, E. Stein, and K.-J. Bathe, eds.),Springer-Verlag, Berlin, 1981, pp. 274-289.
[8] ZIEGLER, H., "Principles of structural stability",Blaisdell Publishing Co., 1968.
[9] ARGYRIS, J.H., BALMER, H., DOLTSINIS, J. St., DUNNE,P.C., HAASE, M., KLEIBER, M., MALEJANNAKIS, G.A.,MLEJNEK, H.-P., MULLER, M . , and SCHARPF, D.W., "Finite element method - the natural approach" Comp. Meth. Appl. Mech. Engrg., Vols. 17/18, 1979* PP. 1-106.
[10] ZIEGLER, H., "Knickung gerader stabe unter torsion",ZAMP 3, Vol. 96, 1952, pp. 96-119.
[11] RAJASEKARAN, S. and MURRAY, D.W., "Coupled localbuckling in wide flange beam columns", 99, No. ST6, 1973, PP. 1003-1023.
Proc. ASCE, Vol.
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[12] RAJASEKARAN, S., "Finite element method for plasticbeam-columns", in: Theory of Beam Columns, Vol. 2 -Space Behaviour and Design (W.F. Chen and T. Atsuta, eds.), McGraw-Hill Book Co., New York, 1976, Ch. 12.
[13] FRIED, I., "Shear in C° and C 1 bending finite elements", Int. J. Solids and Struct., Vol. 9, 1973, pp. HU9 -U6 O .
[1*1] FRIED, I., "Residual energy balancing technique in the generation of plate bending finite elements", Comp, and Struct., Vol. *1, 1974. pp. 771-778.
[15] TESSLER, A. and HUGHES, T.J.R., "An improved treatment of transverse shear in the Mindlin-type four-node quadrilateral element", Comp. Meth. Appl. Mech. Engrg. , Vol. 39, 1983, PP. 311-335.
[16] WUNDERLICH, W., OBRECHT, H., and SCHRODTER, V. , "Nonlinear analysis and elastic-plastic load carrying behaviour of thin-walled spatial beam structures with warping constraints", Int. J. Num. Meth. Engrg., Vol. 22, 1986, pp. 671-695.
[17] ONAT, E.T. and DRUCKER, C., "Inelastic instability and incremental theories of plasticity", J. Aeron. Sci., Vol. 20, 1953, PP. 181-186.
[18] HUTCHINSON, J.W., "Plastic buckling", in: Advances inApplied Mechanics", Vol. 1*1, (C.S. Yih, ed.), AcademicPress, New York, 197**, PP. 67-1*16.
[19] HUTCHINSON, J.W. and BUDIANSKY, B., "Analytical andnumerical studies of the effects of initial imperfections on the inelastic buckling of a cruciform column", in: Buckling of Structures, Proc. IUTAM Symp., Cambridge, 197*1, (B. Budiansky, ed.), Springer-Verlag,Berlin, 1976, pp. 98-105.
[20] NEEDLEMAN, A. and TVERGAARD, V. , "Aspects of plastic post-buckling behaviour", in: Mechanics of Solids - The Rodney Hill 60th Anniversary Volume, (H.G. Hopkins and M.J. Sewell, eds.), Pergamon Press, Oxford, 1981.
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A P P E N D I X A
SHELL F I N I T E E L E M E N T
Many curved shell element formulations have been based on the so-called ’degenerate* models (serendipity elements) first introduced by Ahmed et al. in the early seventies [1]. Whilst such elements are cabable of dealing satisfactorily with thick plate and shell problems, their performance deteriorates rapidly as the plate or shell becomes thin. This phenomenon is called shear locking and is known to be related to the inability of the element to approach the limiting condition of zero transverse shear strain at the appropriate quadratic rate. The problem is alleviated by the use of reduced integration (2x2 Gauss quadrature over the midsurface of the shell) but, contrary to the claims made in reference [2], it is by no means eliminated.
When curved serendipity elements are used to model shells they exhibit a tendency to produce spurious membrane effects under pure bending [3]. This phenomenon is known as membrane locking and, as before, the effects are alleviated but not eliminated by reduced integration.
Although largely unreported in the literature, numerical tests reveal the following additional defects:
1. The element stiffness is not, in general invarariant with respect to the choice of local coordinates. Loss of invariance occurs whenever the local coordinate directions vary over the midsurface (i.e., for curved or distorted elements).
2. Thin curved elements in which integration through the thickness is carried out analytically produce spurious strains when subjected to rigid rotation.
3. Curved elements cannot exactly reproduce a state of constant uniaxial or biaxial bending irrespective of whether the integration through the thickness is analytic or numeric.
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Evidently, these latter* defects are closely related to the basic requirements for convergence of finite element models, and therefore are of fundamental significance. Fortunately, the conditions for convergence are not actually violated since in practice it is sufficient for the rigid body and constant strain states to be represented exactly only for plane rectangular geometries (c.f., convergence of piecewise flat (facet) models of curved shells). However, unless these defects can be eliminated, the advantages of using curved elements to model curved shells must surely be brought into question (because the spurious strain may well be of the same order as the geometric discretisation errors associated with the alternative facet model). Quite apart from questions of robustness and precision, in the demanding context of nonlinear finite element analysis, inadequate representation of rigid body and/or constant strain states in curved elements will inevitably lead to problems (loss of precision, loss of symmetry, poor convergence, etc.).
The element to be described here is designed to overcome all the problems discussed above whilst retaining the simple 8-noded geometry and isoparametric shape functions of the quadratic serendipity element. The formulation also retains the Mindlin-Reissner assumptions. Thus, normal sections of the shell are assumed to remain plane and unextended, but may deviate from the normal in the presence of transverse shear. In addition, it is assumed that the strains remain small (< 0.04), transverse normal stresses are negligible, and transverse shear stresses can be ignored in relation to plastic straining of the material.
The incomplete representation of rigid body and constant strain states can be shown to derive from truncation errors that arise in the interpolated values of the shell coordinates and displacements. To eliminate these errors, the two local rotations are replaced by three global values, and new relationships governing the interpolation of the coordinates and displacements are introduced. To incorporate the sixth degree of freedom, the rotation about the midsurface normal at any point is linked to the average
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in-plane rotation of the midsurface at this point. This link is achieved by the penalty function method.
These expedients ensure that two important objectives are satisfied. Firstly, rigid body motion and constant strain states are represented exactly even for curved and distorted shapes. Secondly, by increasing the number of degrees of freedom from five to six the element can be used to model stiffened plates and shells, folded plates, shell Junctions, box-sections etc..
A further advantage of the new shell model is that membrane locking vanishes identically for elements of arbitrary shape provided 2x2 integration is used over the midsurface. Therefore 2x2 integration is adopted as the standard (mandatory) rule for this element. However, this rule still fails to prevent shear locking and therefore a special energy balancing technique is needed to allow elements of arbitrarily high aspect ratios to respond accurately to flexural deformations. The technique was first introduced by Fried [il,5] • More recently, it has been applied to C° plate bending elements by Tessler and Hughes[6]. An innovative aspect of the present implementation is to relate the correction factors which control thetransverse shear strain energy to the components of the shell metric, i.e., to the covariant base vector lengths at a Gauss integration station.
Geometric nonlinearity is based on a special form of the updated Lagrangian (Eulerian) method in which the displacement field is referred to a set of local convected or corotational coordinates at each Gauss integration station [7]• The corotational method derives from the polar decomposition theorem of continuum mechanics which asserts that any motion can be decomposed into a rigid body motion followed by a pure deformation [8]. For thin shells, the strains will remain small even when preceded by large rigid body rotations - consequently, by explicitly discarding the rigid body component of the overall motion, all restrictions on the magnitude of displacements and rotations are removed.
The noncommutative nature of finite spatial rotations is
2 1 4
accounted for in an elegant manner via the use of Euler’s theorem of rigid body motion. In contrast to the conventional method, this leads to new geometric stiffness terms that are associated with in-plane rotation of the shell about the midsurface normal. Numerical studies appear to suggest, however, that these terms are generally of less significance than their beam counterparts.
By removing one or more of the midside nodes, various • nodal configurations are admitted. However, the full 8-noded
configuration is by far the most accurate and should therefore be used whenever possible. In this context, it is worth noting that the geometric discretisation error associated with the use of flat (linear) elements to model curved shells, can lead to unacceptable errors even for fine meshes. Therefore, elements with from 4 to 7 nodes are recommended only for restricted situations (e.g., thetransition from one mesh size to another).
Provided 2x2 integration is specified, then no spurious modes are possible in a properly supported mesh of at least two 4- to 8-noded elements. Note, however, that if the 4-noded element is used to model a shell then, to avoid membrane locking, a 1-point integration rule is necessary. Unfortunately, this leads to the possibility of hour-glass mechanisms. Since no hour-glass control techniques are provided, the 4-noded element should be treated with caution.
In common with classical shell theories, the formulation presented here is a two-dimensional generalized theory cast in terms of eight generalized midsurface strains and eight stress resultants, i.e.,
ml *1 E s , E r s * X r , x s , X r s » E r t * E stNr. Ns, N r s » M r, M s , Mrs. Qr » Q S
For thin shells, the effect of curvature on the through thickness distribution of strain, is known to be small. Consequently, the usual generalised constitutive law relating the stress resultants to the generalised midsurface
2 1 5
strains Is applicable, and numerical Integration through the shell wall Is required only when modelling nonlinear material behaviour.
Plasticity is handled by applying the von Mises yield condition and Prandl-Reuss flow rule to discrete points through the shell wall. The active stress components are assumed to be crr, crs» ars that is, the plane stress components at any level t. In contrast to the conventional multi-layer approach in which the integration stations are located at the centre of each layer, the present method is based on equi-spaced stations with the outermost points at the extreme fibres. In this way, first yielding of the shell wall is accurately predicted even when a low number of stations are used. Numerical integration of the constitutive relations is based on a special composite end corrected trapezoidal rule.
The incremental strains for any given load step are defined as the difference between the current corotational strains and the corotational strains that existed in the previous equilibrium configuration (i.e.f at the end of the previous step). Within the context of small strain plasticity, these strain increments can be converted directly to increments of Cauchy stress without recourse to expensive transformations.
Accuracy is maintained by dividing the increments of elastoplastic strain into a suitable number of subincrements and accumulating the changes in Cauchy stress that occur within each subincrement. During this process, which can be viewed as an Euler forward integration, drift from the yield surface is eliminated by using a simple ’radial return* at the end of each step. Because the stress and strain changes are referred to the previous equilibrium configuration, the forward integration is said to be ’path independent’. The alternative ’path dedendent* approach, in which the stress and strain changes are referred to intermediate configurations which are not in equilibrium, is less robust, and can lead to spurious oscillations between plastic and elastic states [9,10].
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References
[1] AHMAD, S., IRONS, B .M., and ZIENKIEWICZ, O.C.,"Analysis of thick and thin shell structures by curved finite elements", Int. J. Num. Meth. Engrg., Vol. 2, 1970, pp. 419-451.
[2] ZIENKIEWICZ, O.C., TAYLOR, R.L., and TOO, J.M, "Reducedintegration technique in general analysis of plates and shells", Int. J. Num. Meth. Engrg., Vol. 3, 1971, pp.275-290.
[3] STOLARSKI, H. and BELYTSCHKO, T., "Membrane locking andreduced integration for curved elements", J. Appl. Mech., ASME, Vol. 49, 1982, pp. 172-176.
[4] FRIED, I., "Shear in C° and C 1 bending finiteelements", Int. J. Solids and Struct., Vol. 9, 1973,pp. 44 9-460.
[5] FRIED, I., "Residual energy balancing technique in thegeneration of plate bending finite elements", Comp, and Struct., Vol. 4. 1974, pp. 771-778.
[6] TESSLER, A. and HUGHES, T.J.R., "An improved treatment of transverse shear in the Mindlin-type four-node quadrilateral element", Comp. Meth. Appl. Mech. Engrg., Vol. 39. 1983. PP. 311-335.
[7] ODEN, J.T., "Finite elements of nonlinear continua", McGraw-Hill Book Co., New York, 1972.
[8] MALVERN, L.E., "Introduction to the mechanics of continuous medium", Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969.
[9] POWELL, G., and SIMONS, J., "Improved iteration strategy for nonlinear structures", Int. J. Num. Meth. Engrg.. Vol. 17, 1981, pp. 1455-1467.
[10] TRUEB, U., "Stability problems of elasto-plastic plates and shells", Ph.D. Thesis, University of London, 1984.
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C o n f i g u r a t i o n s
Variable number of nodes allowing any combination of straight and curved edges. By removing one or more of the midside nodes (5 to 8) the element can have any number of nodes from ll (corner nodes only) to 8.
Node numbering
Anti-clockwise, corner nodes first followed by midside nodes (Figure A.1) .
Local axes
The midsurface is represented in parametric form by two curvilinear coordinates % and v At any pointPC 5 »i? ), a right-handed set of orthogonal cartesian axes, denoted by r, s, t are used to define the internal stresses and strains. The axes r, s are tangential to the midsurface at P, and are arranged symmetrically with respect to 5 and
The t axis is therefore normal to the midsurface at P (Figure A.1 ) .
Nodal coordinates
Global cartesian coordinates x, y, z.
Degrees of freedom
uxl, uyi. u2l, 0xl» 0yi» 0zl at each node i (Figure A.l)
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Shape functions
Standard Isoparametric functions are used for the Interpolation of coordinates, translations and rotations.
Generalized strains
Generalized stresses
Nr, Ns, Nrs, M r, M s, M rs, Qr, Q s (Figure A.2)
Material properties
E - Young’s modulus v - Poisson’s ratiocryo - yield stress in simple tension H - strain hardening parameter (Ep/(1-Ep/E)) pm - mass density
Geometric properties
n - number of nodesh t - wall thickness at each node i
Note that if the wall thickness for a given element is constant only one h value is required.
Formulations
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Small displ., elastic material Small displ., nonlinear material Updated Lagrangian (corotational)
(NLOPT (NLOPT (NLOPT
0)1)3)
Material laws
Linear elastic, isotropic (MTMOD = 1) von Mises plasticity (MTMOD = k)
Load types
Uniformly distributed normal pressure Uniformly distributed global tractions Nodal loads
Note that the applied loads are assumed to be conservative, i.e., they must remain independent of the motion of the structure. Furthermore, the shell cannot resist moment vectors that have a component about the midsurface normal. Thus, any applied moment vector must be tangential to the midsurface in the initial configuration.
Mass modelling
Lumped diagonal mass Consistently diagonal mass Consistent distributed mass
The consistent distributed mass option is the most accurateand robust
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Numerical integration
The following Gauss quadrature rules are recommended:
5-8 noded: ngr = ngs = 2U noded: ngr = ngs = 1 (shells)k noded: ngr = ngs = 2 (plates)
• Note that when the 4-noded element is used with a 1-pointrule, an hour-glass type mechanism is possible.
For elastoplastic analysis the parameter ngt is used to specify the number of integration station through the shell wall (1 ^ ngt 10). The recommended value is ngt = 7.
Stress output
The eight generalized stress resultants listed above are printed out at each Gauss point in the sequence A to D (Figure A.2). In addition, for elastoplastic analysis, the values of the plane stress components ar, cts , and crrs together with the effective stress and plastic stateindicator, can be output at each station through the wall.
Sign convention
The positive directions of the stress resultants and stresses are shown in Figures A.2 and A.3 respectively.
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Z,Uz,0Z
F I G U R E A.1 tti-dsu.rfa.ce g e o m e t r y , local c o o r d i n a t e s , and nodal d e grees of f r e edomm
222
F I G U R E A .2 S t r e s s r esultants and Gauss point l o c a t i o n s .
F I G U R E A . 3 Str e s s e s a c t i n g on an element of the s h e l l -
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A P P E N D I X B
B EAN F I N I T E E L E M E N T
This is a general purpose beam element designed for use either in stand-alone form or as a stiffener element. In the latter case, the element is specifically designed to operate
• in conjunction with the doubly curved shell elementdescribed in Appendix A. The element can be employed to model straight, curved and pretwisted beams whose cross sections are of the thin walled open type. Currently, however, the section properties are assumed to be constant over the length of each element. The beam has three nodes lying along a curvilinear reference line and each node has seven degrees of freedom. Applications range from the elastic small deflection analysis of plane or space frames through to the elastoplastic buckling analysis of stiffened plates and shells.
The longitudinal fibres of the beam are not restricted to lie along plane curves. Consequently, the complex axial-flexural-torsional interaction effects that are known to occur in pretwisted bars, can be modelled. Applications of this facility include the analysis of helical springs, stiffener imperfections, and turbine blades.
The theoretical basis for the element derives from the classical process of ’degeneration' in which the three-dimensional field equations of continuum mechanics are reduced to a one-dimensional form expressed in terms of a single longitudinal reference line. The process depends on two main assumptions:
1. Stress components that correspond to deformation of the cross section in its own plane, may be neglected.
2. Points that originally lie in planes that are normal to the reference line, can move out of these planes during the deformation. The relative motion parallel
224
to the reference line comprises two uncoupled components:- a component due to transverse shear which leaves
the section plane, but not necessarily normal to the reference line.
- a component due to non-uniform torsion which warps the section in accordance with a predefined warping function.
These assumptions lead to a theoretical model that can be thought of as a curved Timoshenko beam [1] with a warping function added to account for the effects of uniform (St. Venant) and non-uniform (Vlasov) torsion [2]. Additional restrictions are that the bar is reasonably slender, the longitudinal warping displacements are small compared to the lateral dimensions of the section, and the strains are small (< 0.04).
Although the degeneration process leads to a more complex kinematical description, the number of generalised freedoms is considerably reduced. Since equation solving is usually the most time consuming component of a finite element analysis, degenerated models offer significant computational advantages. However, a price must be paid, and the use of simplifying assumptions must inevitably restrict the range of application of the model. Thus, in this case, the model cannot account for the effects of cross-sectional distortions or, perhaps more importantly, local buckling. (Local buckling is used here to mean a buckling mode that involves significant changes in the shape of the bar cross section).
In practice, the inability to model local buckling is actually much less restrictive than it might seem. The dimensions of the section will usually be selected so that local buckling is preceded by a longitudinal buckling mode and, in such cases, any change in shape of the section prior to buckling will normally be negligible. Nevertheless, when local buckling is known to be important, the various components of the section should be modelled using shell
225
elements.The Influence of warping on the behaviour of open
section bars subjected to non-uniform torsion is well established. Warping can be important in stiffened structures as well as in stand-alone beams. Indeed, the torsion of an eccentric stiffener will always be non-uniform because of the warping restraint that occurs along the stiffener-shell Junction. In certain sensitive cases (e.g., the buckling of longitudinally stiffened cylinders) numerical studies indicate that, even the very low warping resistance afforded by a thin flat stiffener, can significantly affect the response.
The adopted warping function is suitable for the modelling of both uniform and non-uniform torsion. This is achieved by using a Vlasov type function for the warping of the midsurface contour, but adding an additional contribution to allow for the warping of the wall relative to the contour. The latter contribution, termed ’thickness warping*, is important because it leads to the kinematical description of St. Venant (uniform) torsion, and ensures that the warping rigidity of the section never vanishes even under elastoplastic conditions. This contrasts with the conventional approach in which the vanishing warping rigidity for certain cross-sectional shapes (i.e., T, L, -f-, etc.), necessitates the decoupling of flexure and torsion via the use of the shear centre. Furthermore, recourse to stress function theory, such as Prandtl’s membrane analogy, becomes unecessary.
The cross section can be of any thin walled open shape. In the current version of the element, however, this shape must be built up using a series of constant thickness segments. All the pertinent integrated properties of such a section will be generated automatically from the specified geometric data, and are included as part of the output. For generality, the reference line passing through the beam nodes can have any position relative to the specified secton. Consequently, when modelling stiffened shells, the beam and shell will share the same nodes. This expedient
226
automatically satisfies the compatibility conditions at the stiffener-shell interface, and is extremely economical when compared with the alternative shell-shell idealisation.
For elastic analysis, it is possible to avoid the restriction that the beam cross section should be of the thin walled open type. However, to achieve this, the user must supply a set of integrated section properties as part of the input data.
• In order to model warping, the element comprises sevennodal degrees of freedom rather than the usual six. The warping freedom is defined as the 'rate of rotation* about the longitudinal reference axis, i.e., 0^ = d8r/dr. An optional 'warping excluded' formulation is available through the use of the flag wf. In effect, this option ignores the strain energy contribution associated with warping, and it follows that the associated freedoms must then be suppressed. Failure to comply with this rule will result in a singular system matrix.
Unlike the other freedoms, which are represented by standard quadratic shape functions, the torsional-warping freedom 8r requires a fifth-order function in order to maintain C 1 continuity (i.e., both 0r and 0^ are continuous across inter-element boundaries). Consequently, a mixed quadrature rule is appropriate for the integration of the equilibrium equations. In the case of the three-noded element, this takes the form of a 2-point Gauss rule for the axial, shear and flexural components and a 4-point Gauss rule for the torsional-warping components. With these quadrature rules, there are no spurious energy modes, and the element is free from either membrane or shear locking.
Removal of the midside node leads to a straight 2-noded element. However, the use of straight elements often introduces geometric discretisation errors which exhibit a high degree of problem dependency and may be slow to respond to mesh refinement. Therefore, the 3-noded element should be selected whenever possible.
Geometric nonlinearity is based on a special form of the updated Lagrangian method in which the displacement field is
227
referred to a set of convected or corotational coordinates defined at each Gauss point. The corotational method derives from the polar decomposition theorem of continuum mechanics which asserts that any motion can be decomposed into a rigid body motion followed by a pure deformation [3]. For slender bars, the strains will remain small even when preceded by large rigid body rotations - consequently by explicitly discarding the rigid body component of the overall motion, all restrictions on the magnitude of displacements and rotations are removed.
The noncommutative nature of finite spatial rotations is accounted for in an elegant manner via the use of Euler's theorem of rigid body rotation. In contrast to the conventional approach, this leads to new geometric stiffness terms that play an important role in defining the lateral-torsional buckling loads of beams and frames. Without these additional terms, errors as high as 60% in critical instability levels can occur (4]. The element is thus ideally suited to the high precision analysis of beams and frames in both the pre- and post-buckling range.
The evaluation of the internal stress field includes a number of refinements. Second-order terms that derive from both the Wagner [5] and Weber [6] effects are included. In addition, curvature (fibre length variation over cross section), thickness warping (warping of wall relative to contour) [7], shear attenuation (deviation of resultant shear strain from the midsurface tangent m), and tranverse shear deformation, are all taken into account. This allows the element to operate with high precision over a wide range of geometries and deformations.
Material nonlinearity is accounted for by applying the von Mises yield condition and associated flow rule at discrete points located both around the section profile and through the thickness of the wall. The active stress components are assumed to be the longitudinal stress <7r and tangential shear stress cxrTn. In this way, the spread of plasticity and the associated coupling between the axial, flexural and torsional stress resultants, is accurately
228
modelled. It is admissible to specify only one station through the wall thickness. In this case, however, the component of crrm that derives from St. Venant (uniform) torsion will remain elastic because it vanishes at the midsurface - consequently, the plastic coupling between torsion and flexure will not be picked up.
In contrast to the conventional multi-layer approach in which the integration stations are located at the centre of
• each layer, the present method is based on equi-spacedstations with the outermost points at the extreme fibres. In this way, first yielding of the bar walls is accurately predicted even when a low number of stations are used. Numerical integration of the constitutive relations is based on a special composite end-corrected trapezoidal rule.
The incremental strains for any given load step are defined as the difference between the current corotational strains and the corotational strains in the previous equilibrium state (i.e., at the end of the previous step). Within the context of small strain plasticity, these strain increments can be converted directly to increments of Cauchy stress without recourse to expensive transformations.
Accuracy is maintained by dividing the increments of elastoplastic strain into a suitable number of subincrements and accumulating the changes in Cauchy stress that occur within each subincrement. During this process, which can be viewed as an Euler forward integration, drift from the yield surface is eliminated by using a simple ’radial return* at the end of each step. Because the stress and strain changes are referred to the previous equilibrium configuration, the forward integration is said to be ’path independent*. The alternative ’path dependent* approach, in which stress and strain changes are referred to intermediate configurations which are not in equilibrium, is less robust, and, according to [8,9], can lead to spurious oscillations between plastic and elastic states.
229
References
[1] TIMOSHENKO, S. P. , "Strength of materials", Part I, 3rd ed., Van Nostrand Reinhold Co., New York, 1955.
[2] V L A Z O V , V . V . , "Thin walled elastic beams", 2nd ed., English translation by Israel program for scientific translations, Jerusalem, Israel, 1961.
[3] MALVERN, L.E., "Introduction to the mechanics of continuous medium", Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969.
[4] ARGYRIS, J.H., DUNNE, P.C. and SCHARPF, D.W., "On largedisplacement - small strain analysis of structures with rotational degrees of freedom”, Comp. Meth. Appl. Mech. Engrg., Vol. 14, 1978, pp. 401-451; Vol. 15, 1978, pp.99-135.
[5] WAGNER, H. and PRETSCHER, W . , "Verdrehung und knickungvon offenen profilen", Luftfahrt-Forschung, Vol. 11, 1934; translated in NACA Tech. Mem. 784, 1936.
[6] WEBER, C., "Die lehre der drehungsfestigkeit",Forscharb. Ing. Wes., Vol. 249, 1921, PP. 60-62.
[7] GJELSVIK, A., "The theory of thin walled bars", John Wiley & Sons, Inc., New York, 1981.
[8] POWELL, G. and SIMONS, J., "Improved iteration strategy for nonlinear structures", Int. J. Num. Meth. Engrg., Vol. 17, 1981, pp. 1455-1467.
[9] TRUEB, U., "Stability problems of elasto-plastic plates and shells", Ph.D. Thesis, University of London, 1983.
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C o n f i g u r a t i o n s
Straight 2-noded or curved 3-noded.
Node numbering
End nodes first followed by midside node (Figure B.l).
Local axes
r axis: covarient base vector tangential to curvilinearreference line at any point P (Figure B.l). Locations along the reference line are defined by a non-dimensional coordinate where -1 K 4. 1.
s,t axes: linear coordinates defined so that r f sf t form anorthogonal right-handed set at any point P. An explicit definition is shown in Figure B.l. In the figure, a 'curved ' beam is nominally defined as a beam which subtends an angle of ^ 0.01. Note that in the curved case, the attitude point A is used only in the auxiliary role of controlling the sense of s, t. Thus when -ti/2 < Q < 7t/2, the positive directions are as shown but when Tt/2 < 0 < 371/2, they will be reversed. This expedient prevents the s, t directions from being reversed at points of contraflexure.
Nodal coordinates
Global cartesian coordinates x, y, z.
Degrees of freedom (Figure B.3)
uxi> uyi, uzi, 0xi» 0yi» 021 * 0ri» at each node i
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S h a p e functions
Isoparametric for geometry and global d.o.f. Higher-order functions are used for the warping freedom 0^*
Generalized strains
Generalized stresses
Fr t Qs. Qtt M s, M t, T r »
Fr - axial forceQs. Qt - transverse shearsM s, Mt - flexural momentsTr - St. Venant torqueMw - warping moment
(Figure B.3)
Note that the total moment couple about the r axis is M r = Tr+Ttl)-*-Twg, where Tu = -M^ is known as the warping torque and Twg is known as the Wagner torque. The Wagner torque is a geometrically nonlinear effect associated with warping.
Material properties
E - Young’s modulusV - Poisson’s ratioCTy o - yield stress in simple tensionH - strain hardening parameter (Ep/(1-Ep/E))P m - mass density
G e o m e t r i c p r o p e r t i e s (Figures B. 2 and B.3)
n X A Y A ZA wf nc ns so rm al a2
Then either:
nc cp sc tc
— if ns * 0
ns -- is ds hs nsm nsn
or:
if ns = 0 and MTMOD = 1 2
where:
es et A As At Is It KSs St Ast 1st lui Sto Isa) Itto W1 W2 W3 W4
n - number of nodesX - X coordinate of attitude pointY - Y coordinate of attitude pointZ - Z coordinate of attitude pointwf - warping flag (0 = no-warping, 1 = warping)nc - number of contour points used to define cross sectionns - number of segments used to define cross section 3so - sectorial origin (rigidly linked with P) 4rm - section rigidity multiplier 5al - pretwist angle at node 1 (degs.)a2 - pretwist angle at node 2 (degs.)
6cp - contour point numbersc - 8 coordinate of contour pointtc - t coordinate of contour point
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isds
hsnsmnsn
start contour point numberfinish contour point number
8wall thicknessnumber of integration stations along segment length number of integration stations through segment wall
- 7
- 9
esetAAsAtIsItKSsStAst1stIu)Sq>ISO)ItO)W1W2W3WU
s coordinate of shear centre t coordinate of shear centre total cross sectional area shear area in s direction shear area in t directionJ t2 dA| s2dA
not used if wf 1
St. Venant torsion constant
Notes:
1) Provided the beam cross section is defined in segmental form, the program will generate and print out all the section properties automatically.
2) The direct specification of integrated rigidities allows cross sections of arbitrary shape to be analysed under the assumption that the material remains elastic (MTMOD = 1).
- 234 -
3) The maximum number of segments allowed is twenty.4) In the no-warping case (wf = 0), the sectorial origin is
not used, but, when warping is included (wf = 1), the sectorial origin must be placed at one of the contour points. For stand-alone beams any convenient point may be selected. If the beam is employed as a stiffener, then, in order to achieve proper compatibility, the sectorial origin and edge contour point should both be placed at the actual stiffener/shell interface. However, for thin shells, the ’overlap* error that occurs if the sectorial origin and edge contour point are made to coincide with the midsurface at P, will usually be negligible.
5) This is normally set to 1.0. Alternatively it can be used for halfing a section along a line of symmetry (rm = 0.5) or for removing an element altogether (rm = 0.0).
6) The contour points must be listed consecutively starting from one. The manner in which the contour numbers are assigned to the section profile is, however, arbitrary.
7) The first segment can be selected arbitrarily. Subsequent segments must, however, always be connected to one end of a previously defined segment.
8) If zero thickness is specified for a segment, then this segment will make no contribution to the section rigidities, but will be treated instead as a rigid link. Such a link enables the displacements to be transmitted across the ’missing* part of the section.
9) The integration parameters nsm, nsn are not used for elastic analysis (MTMOD = 1). For elastoplastic analysis (MTMOD = 4) the minimum permissible values of nsm and nsn are 2 and 1 respectively.
10) In the elastoplastic analysis of beams of rectangular cross section, the n-direction transverse shear rigidity is assumed to remain elastic.
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Formulations
Small displ., elastic material (NLOPT = 0) Small displ., nonlinear material (NLOPT = 1) Updated Lagrangian (corotational) (NLOPT = 3)
Constitutive laws
Linear elastic, isotropic (MTMOD = 1) von Mises plasticity (MTMOD = k)
Load types
Conservative nodal loads
Mass modelling
Lumped diagonal mass Consistently diagonal mass Consistent distributed mass
The consistent distributed mass option is the most accurate and robust
Numerical integration
For integration with respect to r, the following Gauss quadrature rules are mandatory:
3-noded (wf = 1): ngr = 42-noded (wf = 1): ngr = 33-noded (wf = 0): ngr = 22-noded OII3sy ngr = 1
236
These integration stations are numbered in the positive r direction.
For elastic materials the integration with respect to s, t is carried out analytically, so that ngs, ngt are not used. For elastoplastic analysis, ngs is not used but ngt should be set to the total number of integration stations in the cross section, i.e.,
nsngt = £ (nsmxnsn)k = 1
The layout of these cross sectional integration stations is shown in Figure B.U.
Stress output
The seven generalized stress resultants listed above are printed at the Gauss points 5 5 = 1//3 for the3-noded beam, or at the Gauss point 5 = 0 for the 2-noded beam (these positions are not affected by the warping flag wf). In addition, the values of the active stress components ar and arm together with the effective stress and plastic state indicator, can be output at each cross sectional integration station.
Sign convention
The positive directions of the stress resultants stresses are shown in Figures B.3 and B.5 respectively.
and
237
F I G U R E B.l G e o m e t r y of r e f e r e n c e line and d e f i n i t i o n of local axes r , s , t.
j s ( S C j . t C j )
Typical segment
/
F I G U R E B m2 D e f i n i t i o n of c r o s s - sectional g e o m e t r y .
%
238
Z , Uzj, 9 zj
F I G U R E B .3 S t ress res u l t a n t s a nd nodal d e g rees of f r e e d o m *
*
n
m
nsm = 5 , nsn = 4
1 --------------------- T----------------------r
5 4 3 2
4321
F I G U R E B .4 Typical i n t e g r a t i o n s t a tion layout f or a segment.
F I G U R E B .5 Local stress comp o n e n t s <rr an d crr m .
%
239
A P P E N D I X C
G E O M E T R I C C O N S T A N T S F OR A R E C T A N G U L A R S E G M E N T
F I G U R E Cm! Geom e t r i c a l p a r a m e t e r s for a r e c t a n g u l a r segment.
The generalized coordinates s, t, u at any point m, n of the segment shown In Figure c . 1 , can be written
s = s c + ( m / L ) A s + n c at = t c + ( m / L ) A t + n s a (C.l)to = toc + (m/L) Ato+nm
where
240
sc = ^ISj+Sj )tc = tx+t j )wc = HUOi+Wj ) (C.2)As = (Sj-Si)At = (•tj-ti)Au> = (COj-lOi) = Sjtj-Sitj
Substituting equations (C.l) and (C.2) into the basis ♦ functions given in Table 4.4, and integrating over the
domain -L/2 ^ m L/2, -h/2 n 4: h/2, leads to thefollowing definitions for Ix to I32:
II = A= At c
*3 = AS c= A(fcza+sza)
I 5 11 H * ct n
*6 - I ^ s c
) 1 7 = A(f-l)saca*8 = 17 1 c*9 = X 7 s c* 1 0 = A(fs2a+c2a)1 1 1 = 1 1 01* cI 1 2 = I 1 O 3 c
I 1 3 = A[t2+(1/12)(At2+h2sza)] (C« 3)I n = A[sctc+ (1/12)(AsAt+h saca)]I l S ~ I 1 3 cI 1 6 = X 13sc
•lie
= A[s £+(1/12)(As2+h2c2a)]= x 2 7 t c
I 1 9 = X 1 7 S cI 2 0 = (5/6)Ltpsa* 2 1 = (5/6)L<pcaI 2 2 = L[ (<p2/h) + (h3/3) ]I 23 = Lcpa)cI 2 4 = AtocI 2 5 = X2a)c + (A/12)AtAa>* 2 6 = 130)c-»-( A/12 ) AsAca ....
*
Z U l
X 2 7 = Ii 3a)c+2A[tcAtAa)+(l/lZlZl)LhzAt J I2a = Ii*coc+A(scAt + tcA s )Aw 12 o = Ix 7coc+2A[scAsAoj+(l/l/l/i )Lh2As] I30 = A[o)J + tl/lZl/U (A(x)2+A2 ) ]^ 3 1 = ^ 3 0 ^ C
^•32 = ^ 3 o a C
where
A = Lh A = (5/6 ) A f = [1+12(pnc/h)* J-1 Pnc = scca+tcsot
(C. U)