Large Deflection and Post-Buckling Analysis of Two and Three Dimensional Elastic Spatial Frames J. L. MEEK and FRY, H. S. TAN TA 1 .U4956 esearch Report No. CE49 December, 1983 T.4 I . t)41S{; \\\ \ 1\ I 1\ \1\ 11\ I I\\II I \1\1111�\1 I \ \ \ 5614 3 4067 03257 �.4-9 I Pte1£ � CIVIL ENGINEERING RESEARCH REPORTS This report is one of a continuing series of Research Reports published by the Department of Civil Engineering at the University of Queensland. This Department also publishes a continuing series of Bulletins. Lists of recently published titles in both of these series are provided inside the back cover of this report. Requests for copies of any of these documents should be addressed to the Departmental Secretary. The interpretations and opinions expressed herein are solely those of the author(s) . Considerable care has been taken to ensure the accuracy of the material presented. Nevertheless, responsibility for the use of this material rests with the user. Department of Civil Engineering, University of Queensland, St Lucia, Q 4067, Australia, [Te1:(07) 377-3342, Telex:UNIVQLD AA40315] LARGE DEFLECTION AND POST -BUCKLING ANALYSIS OF TWO AND THREE DIMENSIONAL ELASTIC SPATIAL FRAr·1ES by J. L. r1EEK, MS Calif. , BE, BSc, PhD, Associate Professor of Civil Engineering and H. S. TAN, BSc, ACGI, Post Graduate Student RESEARCH REPORT NO. CE 49 Department of Civil Engineering University of Queensland December, 1983 Synopsis The safe and economical design of curved spatial frames necessitates a geometrical nonlinear analysis of the structure. In this report, the nonlinear equations governing such structural behaviour are derived. Solution strategies which are capable of analysing the post buckling behaviour of structures are also critically reviewed. In addition, an extrapolation stiffness method that lends itself to a more efficient solution of the nonlinear equations than conventional Newton-Raphson methods is presented. The post-buckling responses of various spatial frames are analysed and the results found to be in good agreement with available published work. •.. " �- :: �.·. "\ " .<-." ;.J CONTENTS .a.. l�a.. &)"..�.. ....Q,. <,.J <._) _,. lC\�4 Fr '6e-r Page 1 1. INTRODUCTION 2. DERIVATION OF GOVERNING NONLINEAR EQUATIONS 3 2.1 GOVERNING EQUATIONS FOR PLANE FRAME 7 2.1.1 7 2.2 Member basic force-displacement relation GOVERNING EQUATIONS FOR SPACE FRAMES 12 2.2.1 Member basic force-displacement relation 12 2.2.2 Transformation from Member Basic Force/ Displacements to Member Nodal Force/ Displacement (Local Co-ordinates) 16 Transformation from Member Co-ordinates to Global Co-ordinates 22 Evaluation of Member Rotational Deformations from Joint Orientation Matrix 23 2.2.3 2.2.4 3. SOLUTION STRATEGIES FOR TRAVERSING LIMIT POINTS 25 4. CONSTANT ARC LENGTH STRATEGY OF CRISFIELD 33 5. EVALUATION OF POST-BIFURCATION EQUILlBRIU�1 PATH (SECONDARY PATH) 36 6. EXTRAPOLATED STIFFNESS PROCEDURE 40 7. NUMERICAL EXAMPLES 42 7.1 CANTILEVER BEAM 42 7.2 WILLIAMS' TOGGLE FRAME 42 7. 3 TWELVE MEMBER HEXAGONAL SPACE FRA�1E 45 7.4 TWENTY-FOUR MEMBER STAR-SHAPED DOME 50 8. CONCLUSION 50 APPENDIX A - NOMENCLATURE 52 APPENDIX B - REFERENCES 56 -1- 1. INTRODUCTION Curved structural space frames belong to a class of structures where a large proportion of the applied loading is resisted by the axial forces in the members. This together with the trend towards optimum light weight members, for reasons of economy and aesthetics, has brought to the fore the stability analysis of this type of structure, as it is inclined to fail through elastic instability before exhibiting any sig nificant nonlinear material response. The elastic buckling load has in the main been calculated using linear instability theory, through a linearised eigenvalue approach. For structures where the pre-buckling path is linear or close to being linear, this method will suffice. How ever, in the presence of initial imperfections and geometric nonlinearities, this procedure can grossly overestimate the actual buckling load. In addition, the information furnished by a linearised eigenvalue analysis is limited. The pre-buckling load-deflection path is not traced out and more importantly the nature of the bifurcation failure, whether the failure is catastrophic or there is an increase in post buckling stiff ness as the structure deforms further, is not known. These considerations lead us to the desirability of considering the geometrical nonlinearities when formulating the equilibrium equations, the result of which is a system of nonlinear equations. The formulation of the nonlinear equations governing the large deflection behaviour of framed structures has been extensively studied by various authors and is based on either the finite element method [1-5] or on the 'beam-column' approach [6-13]. In describing the motion of the element, a total Lagrangian or an updated Lagrangian description can be employed. The finite element formulation is generally used in con junction with a total Lagrangian material description. Analyses based -2- on this approach are however 1 imited by the magnitude of the joint rotations. To overcome this problem of large joint rotations, an up dated Lagrangian approach can be employed which separates the effects This is of pure member deformations from the joint displacements. achieved by introducing a local, convective reference system attached to the member. The nonlinear equations can be· solved by utilising the uncon strained minimisation algorithms of mathematical programming. All the geometrical nonlinearities are included in the potential energy function al which is then minimi.sed. Techniques such as the random search method [14] and Powell's method of conjugate direction [15] utilise the object function only; the method of steepest descent [14] and the conjugate gradient method [16], in addition makes use of the gradient (first derivative); while the quasi-Newton [17-19], secant-Newton [20] and Newton-Raphson [21] require the evaluation of the 2nd derivates as well. This approach has been adopted by Berke and Mallet [2] as early as 1969, and more recently by Papadrakakis [13] and Kamat [22] to trace the complete load-deflection path of spatial framed structures. An alternative to solving the nonlinear equations lies in using an incremental form of the equilibrium equations. The linearised incremental-iterative techniques appear to be more efficient than the minimisation methods for large scale problems. It however, breaks down at the first onset of instability due to the singularity of the incre mental stiffness matrix. Earlier work on the large deflection behaviour of framed structures employing an incremental iterative method has thus been limited to studying their prebuckling behaviour [8-10,23] and it is only very recently that robust and efficient techniques that circum vent this problem have been developed. Several good reviews of these -3- techniques for traversing the limit points exist in the literature [24-27]. These solution strategies have invariably been applied to the analysis of thin shell structures. To the authors' knowledge the only published work utilising these nonlinear solution strategies to spatial frames is that of Reference [4,28] on two-dimensional frames and of Reference [5] on three-dimensional frames; were presented. where only limited examples There is thus scope for further work in applying these newly developed nonlinear solution strategies to analysing the postbuckling response of spatial frames. In the linearised incremental-iterative technique, equilibrium iterations are performed within each increment. ( NR ) In the Newton-Raphson method, each iteration would entail the assembly and factorisation of the tangent stiffness matrix. This is, however, very expensive and in practise the tangent stiffness is kept constant throughout the incre ment or is changed only after a few iterative cycles. will be needed for the modified Newton-Raphson method More iterations ( mNR ) , but this may more than outweigh the cost of reforming and refactorising the incremental stiffness matrix at every cycle. The authors in Reference [29] have developed an extrapolated stiffness strategy which was found to be more efficient than the mNR or NR methods, when tested for plane frames. The applicability of the conclusions reached in Reference [29], to three-dimensional frames is also demonstrated in this report. 2. DERIVATION OF GOVERNING NONLINEAR EQUATIONS The 'beam-column' approach has been used here in deriving the governing nonlinear equations. Here in the members are assumed to be of constant cross-section with the external loads applied only at the joints. Only conservative or displacement independent loading is considered. -4- Shear deformations are neglected and the matedal is assumed to be linearly elastic. For the three-dimensional frames, the members' cross sections are doubly symmetric, thus excluding coupltng of the torsiona 1 stiffness to that of the bending and axial stiffness. Warping effects are also neglected. In developing the nonlinear equations, two types of co-ordinate systems are employed; a fixed global set of co-ordinates and a local convective system which rotates and translates with the member. The member deformations are thus separated from the joint displacements. The basic member force-deformation relations are derived in the local convective co-ordinates with member deformations assumed to be small relative to it, through the principle of minimum potential energy. This formulation of the equilibrium equation through an updated Lagrangian approach is thus applicable to analysing structures exhibiting large rotation small strain behaviour. The member forces and displacements in the various co-ordinate systems are illustrated i.n Figures 1 and 2 for the two and three-dimensional frames respectively. The rotational degrees of freedom are described by Euler angles for the case of the two-dimensional frames. Due to the occurence of large rotations in post-buckling analysis, this description by Euler angles is unsatisfactory for the analysis of three-dimensional space frames. This stems from the fact that finite rotations in three dimensional space are non-vectori.al quantities in that they do no comply with the rules of vector transformation. To overcome this difficulty, the concept of a joint orientation matrix comprising of a triad of orthogonal unit vectors as presented in References [12 and 30] is employed to describe the arbitrary large nodal rotations. -5- / Mz 1-P-x1 / (a) Member basic displacements and forces M,' ,, T+\ :__- L __ t- ; �o x, i inal pos tion sf (b) Member· intermediate displacements and forces f '{6 us J XL' L-r�---- "' '•( i'_, - f4 I u, x, (c) Member nodal forces and displacements (global co·ordmates) Figure 1. Member deformations and associated forces (2-dimensional frame) -6- Figure 2. Member deformations and associated forces (3-dimensional frame) -7- 2.1 GOVERNitlG EQUATIONS FOR PLANE FRM1ES 2.1.1 Member basic force-displacement relation On assuming a cubic lateral deflection curve and that the axial deformation, e is small compared to the original member length, � (1) and the axial lengthening due to bowing is (2) = � 30 ( 282 1 - 88 I 2 + 282 2 } The axial load, P is thus (3) where EA Neglecting the shear strain energy, the axial rigidity. = potential energy due to lateral deflection along the prinicipal axis is '¥ = -21 � I 0 EI [ l2 � d2 � 2 dx dx 1 - t1 8 - �1 8 1 1 2 2 + I [ 1 ,1 P -21 0 I d I� dx 2 dx 1 (4) 1 Through the principle of minimum potential energy, the member forces are thus, M (5) 1 [ill ) � _ P� 8 30 1 + [lli � + )2 4P� 8 30 - where EI 8 - flexural rigidity The basic incremental stiffness matr1x is obtained by partial differentiating Equations (3) and (5} with respect to the member basic deformations. Hence EA EA dP 30 T (48, k dM dr1 J EA - 82 30 22 symmetric ( - 8 + 1 I< 23 k 33 48 ) 2 de de de 2 where k 22 k 23 2EI --� _ 30 300 ( - 282 + l 68 e 1 2 _ ) 282 2 (6) [kJ {1'- v} {lis} 2.1.2 EAe + EA� Transformation from member basic force displacement to member intermediate force/displacement Considering gross deformation, e =I'(�+ u')2 + v'2 - � 8 8 (7) -9 - On partial differentiating, de {} [ c�+u')2 +v'2 ]� 2(�+u'l } = r[ c.�+ +<1 [2 de de UI 2 ) + VI 2 r� 2V I 1 ;_> ) du' dv' + (0) de' + (0) de' 2 l �+u' v' __ du' ---- dv' + (1) de'I + (0) de' 2 (�+e)> c�+e)2 __ = I v' 2 (� + e)2 du' - � dv' + (O) de' + (1) de' I 2 (�+ e)2 Thu s � +u' de de de � v' I (�+e)2 2 (�+ e)2 v' {Liv} [�] v' T+e 0 �+u' 0 (� + e)2 �+u' (� + e) 2 {Liu'} 0 0 du' dv' de' de' 2 (8) By the contragredient principle { f' } where and (9) -10 - { s} T {P, �, ' M } 1 2 (9) On differentiating Equation {M'} [A]T ' {flu } [�] {t.s} + (10) where d 0 0 d d 0 0 0 0 0 0 0 0 0 0 d 11 21 [�] 12 22 and 1 d 11 d 12 (� + e)' 21 2.1.3 (� - v' 2(� + u') T + [ -v'3 --(� d (2(� + u')2 (� + u')3 u'l (�; u')2 + 2v'2 T- + e)" [(� + ( + v'2 ] ) v•s + u')2 v's s] + e)' d 22 [ ) T- (� ] Transformation from member intermediate force/displacement to nodal global force/displacement The member intermediate displacements are related to the nodal displacements by - ' u -cosa v' sina e' 1 e' 2 11 - sina 0 cosa sina 0 u -cosa 0 -sina cosa 0 0 u 0 1 0 0 0 0 u 0 0 0 0 - u u u 1 2 3 4 5 • (11) and {!.1�'} = [!] (12) '{!.1�} By the contragredient principle, [!]T ([�]T [�] [�] + [�]I [!J {!.I�} (15) -12- 2.2 GOVERNING EQUATIONS FOR SPACE FRA"1ES 2.2.1 Nember basic force-displacement relation In deriving the member basic force-deformation relation, it is i. assumed that the lateral deflection curve is cubic in each of the two principal directions, and that e is small relative to X 2 X 3 Therefore and e I (16) 2 with the axial lengthening due to bowing being ,, = i 1 [:::r ( 2e _! 30 2 13 ''· • - e e 13 23 i 1 [:::r ''. ) ( + 2 28 23 + _! 30 2 28 12 - e e 12 22 + 282 ) (17) 22 The potential energy functional due to lateral deflection along the two principle axis, neglecting the shear strain, is 'I'= lJi 2 - �1 EI e 3 ( ] 22 22 d 2 \ dx 1 2 dx + .!._ 2 1J 0 EI [ ] d2x 2 3 dx I 2 dx 1 - �l 13 e 13 - �1 8 23 23 - M e 12 12 -132EI = Jl. 3 8 __ 2 + 13 2EI + + 2 2 -8 Jl. 22 [ P!l. jiT 2EJ 3 8 Jl. _ __ 8 + 13 23 - M 8 13 13 t·l - 2El 3 Jl. __ e 2.3 23 2Et 2EI 2 + � e2 + ____g. e 8 8 12 Jl. -� ]2 22 -23 _ - (282 - 8 8 + 282 l + 12 J2 2.3 23 ,. 8 12 J2 {_282 �� 8 22 22. - - 8 12 2 8 + 28 J2 23 22 1] (181 Through the principle of minimum potential energy, this leads us to M ( 4E I I3 I2 [ [ 22 ( 2EI �� 3 _ _ 23 �� 4P Jl. 4P!l. EA � Jl. + 2EI 3 Jl. ) ) 30 8 ( + + 8 30 4P!l. ) 3 -!1. -+ 3lr 2EI 2 Jl. 8 12 I2 ] P!l. _ [ + e13 P!l. _ where EI and EI 2 respectively. I3 ( + 4EI 3TIJ 4EI 2EI 2 Jl. 8 PJl.) - 2 !1.+ � '' p ) !l. - 3 !1. + 3lr [ 4EI ) P!l. _ 30 4P!l. 23 8 23 a 22 ) 2 -!1.+ � 8 22 (19) are the flexural rigidities about the x and x axis 3 2 The torsional moment and the axial force is given by _l_ 30 2 (28 13 8 + 282 ) + 1__ 8 13 23 23 30 ( 282 12 - 8 8 + 2 82 ) 12 22 22 (20) where EA = axial rigidity and GJ torsional rigidity. Partial differentiating Equations (19) and (2.D) wi.th respect to the basic member deformations will lead us to the incremental stiffness matrix in the convected co-ordinate system. -1�.- Thus 61� I K K 3 .I J K 6�1 6�1 )3 K· K K 3 K K K 33 K K K K 12 K 2 22 K )2 14 1 5 25· 24 K 22 K .. Symmetry 66 26 3 35 34 68 16. 45 55 23 68 G. 12 K 66 K 68 K ile 46 K )3 22 t 56 66 where K 11 4EI -T EA� 4EAe + 30 + 3oo 12 EAe EA� 2EI T- 30- 3oo (28�3 13 900 K 14 900 K 0.0 K K 15 K EA� (168 8 )3 EA� (- 48 8 EA 16 12 30 13 12 (48 13 - 8 3 ) 2 48 8 (a8�3 68 38 48 8 13 22 + 168 8 + 13 23 13 22 1 23 48 8 + 8 8 23 23 28 ) 8 e 2 + 12 23 3 82 ) 23 + 12 23 22 ) 48 8 ) 23 22 -154Ef 4EAe -.EM 8 3oo. c3 3 K 22 T+ � + K 23 EM (- 48 8 + e 13 e 22 9TIO 12 �3 K 24 EM ( e 900 13 e 12 48 0.0 K EA K -yL+�+ 900 26 33 ( -e 30 . 4EI + EA� J UU � 2EI 13 4EAe EA� 2 (8e 12 - 4e 12e 22 EAe EM 0.0 K EA (46 - 6 l 3D_ - 12 22 - K 4EI EA2 4EAe -yL + � + 9TIIT (Be�, K 0.0 44 45 - 4e e l 23 22 + 16e 23 822 ) e 12 23 + 88223 l + 3e222 ) K 36 4e (86 � 3 -yL- 35 + 16823 e 12 + 46 23 ) K 34 2 8 + 88 ) 13 23 23 e - 4e e 13 22 23 12 K 25 48 � 30- 9oo c2e�3 e 2 8 + 26 1 13 23 23 46 j 3 e 2 3 + 86 2 r 23 - 46 EA (e 30. 12 55 GJ T K K K + 46 22 16 - 1 0.0 56 EA T K 66 {6$} 2..2..2 (211 { [�] liV} Transformation from Mem5er Basic Force/Disolacements to ��ember Nodal Force/Displacement (Local Co-ordinatesl From considerations of geometry and equilibrium, F F' 2 F3 F 0 0 1 � � 0 0 Q Q 0 ""1 I 0 M 23 0. 0 11 0 -1 0 M 0 0 Q M p 0 -I � 0 0 0 0 F 1 0 0 0 0 0 F 0 0 0 0 0 1 0 0 Q Q 1 � 0 Q � 0 F' 5 1 F 8 � � Fs 0 0 F1 o 0 0 0 0 0 0 - F ]] - F I 2 0 0 3 0 0 0 4 �1 0 0 0 0 Q 0 12 22 t - [B l 17 - (22) {S} Using the contragredient principle, (23} . {liV} By differentiating Equation [BJ {liS} (22), its incremental form is obtained, + (24} [ll�]{�} The matrix[�] is the change in [�] resulting from {ll�}. To evaluate [ll�) let us consider the member moving from position (I} to (II} in Figure 3, which is equivalent to rotating the translational forces by p X 2 = - (t.u - 9 � (t.u t.u ) 3 p 3 = - 8 � t.u ) 2 2 >---- x1 Figure 3. Rotational transformation of member nodal force (in local co-ordinates) (25) -18- The nodal forces in local co-ordinates are about the member axis. thus F' 1 -P 1 F' p 0 0 0 0 0 0 0 0 0 F 0 0 Q 0 0 0 0 0 0 0 F 3 2 F' p 3 -P a 1 0 0 0 0 0 0 Q 0 0 F 0 0 0 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 0 0 0 0 0 F 0 0 Q 0 Q 0 0 0 Q Q a 0 F 0 0 0 0 a 0 1 p Q 0 0 f 0 0 0 0 f 0 l 0 0 0 f 3 F' 4 F' 5 F' 6 F' - P 7 F' 3 0 0 0 0 0 0 8 F' p 3 0 0 0 0 0 0 -P 9 F' 2 0 0 0 0 0 0 0 0 a 0 0 0 F 0 0 0 0 0 0 0 0 Q Q 0 0 F 0 0 0 0 0 0 0 0 0 0 0 0 F 10 F' 11 F' 12 1 0 11 1 2 (26) Equation (26) when used to evaluate the matrix [��] will result in an unsymmetric co-efficient matrix when the term{[��] {�}) is expressed in terms of the vector {�u}. To restore symmetry; in evaluating [��] we take into account the change in length of the member and that the moments producing shear are constant rather than the transverse nodal forces. If S 0 is the initial transverse force; and �S0 its variation, ignoring 2nd order terms -19- v1here t>u 0 - t>u 7 1 {�'>�u} 1�hich is the change in the vector {E} res·ulting frof'l the variation {t>u} is thus {C,fu} t>F 0 t>F P -o/9., 1 2 t>F 2 4 t>F 5 t>F 6 t>F t>F 8 t>F t>F t>F 10 11 12 3 p 0 0 0 0 0 0 0 0 Q F 0 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 0 0 F 2 0 -0/9., 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 -P p 0 0 0 r: 0 0 0 0 0 0 p -0/9., 0 0 0 0 F o - o / 9., 0 0 0 F 3 0 t>F -P 3 -P t>F {f} {F'} 0 0 0 0 0 -p 2 0 0 0 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 0 0 0 0 0 F 0 0 0 0 0 0 0 0 0 0 0 0 F 10 11 12 -20- "p �P -P .,p .. 11 3 T 3 T 2 -�- 2. T 0 c :i .1',2 -6 .1',2 0 0 0 -P 0 0 6 .1',2 6 .1',2 0 0 0 0 0 0 0 "' 0 0 0 0 0 0 •·.1 c 0 0 0 0 0 p p 3 2 3 2 2 2 2 2 0 0 6 .1',2 6 .1',2 0 0 0 p 3 0 0 6 .1',2 -6 .1',2 0 -p 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 [6B] {S} p p - p p 3 2 1 3 r� 23 � 12 22 •t -21- 0 M +M � 1a M +M _ ..ll...-..U. 1a p I p ;: 0 0 0 0 0 M +M 0 -....!.!....-U ,. 0 0 0 M +M � ,. 0 0 0 p -;: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ... 0 0 0 ... 0 0 0 6U, 0 0 0 0 0 0 0 0 0 0 0 p -;: � -� M +M M +M p ;: p ;: S�try ... ' . . ... . ... " (27) Substituting Equations (21), (23) and (27) into (25), the incremental equilibrium equation in member co-ordinates are {llF} [BJ {K} [ll� l ([�) [KJ [BJT + + [NJ {llii} [�)) {llii} (28) -222.2.3 Transformation from Member Co-ordinates to Global Co-Ordinates The member orientation matrix, which is the direction cosines of the member axes with respect to the global co-ordinates X1, X2 and X3 is denoted by lrl. [R] Let [r] 0 0 0 0 [r] 0 0 0 0 [r] 0 0 0 0 [r] We thus have U.F} (29) and by contragredience, T [R] {t>u} (30) Substituting Equation {28) into (29) {t�F} r�J r�J {t�u} (31) which is the incremental equilibrium eouation in global co-ordinates. -232.2.4 Evaluation of Member Rotational Deformations from Joint Orientation Matrix The 'joint orientation matrix', parallel to the global axes. [�]is initially assumed to be The incremental rotation matrix for a joint in global co-ordinates is -l>O [l>O] 3 0 (32) where and l>O l>O = I 2 6u , 60 4 L'lu = I 0 , l>O 2 = L'lu , 60 5 = L'lu I I , 60 = 3 = 3 6u 6u 6 I 2 for joint (1) for joint (2) This results in an increment in the joint orientation matrix which is (33) [60] [a] and thus the updated jo.int orientation matrix, [a] + [L'la] can be evaluated. The two end sections of the member will in general be not parallel to each other. 'End section orientation matrices' are thus defined to describe them with the matrices section (1) and (2) respectively. [P(t)] and [PC2l] for end They are related to the joint orientation matrices by [pl (34) -24- where [t::(o}] = member orientation matrix in undeformed configuration. The first column of the member orientation matrix, [�] repre sents the vector of direction cosines of the member chord (the line joining the end sections). This can be obtained from the global trans lational displacements of the nodes. Denoting the first column of [�] by [r ], the following relation applies for the case of small relative -· rotations of the member. {r } -· (35) and where { p (i) j } = j column of the end orientation matrix of joint i. To evaluate the second and third columns of [r] , which repre- 1 sents the vector of direction cosines of the 2 member principal directions, let us define the following incremental rotation matrix for the two end sections. -25- [: (!)] [ [ [:(zJ] 8 -8 13 8 12 23 8 22 -8 1 2 0 0 8 -8 13 23 -8 22 0 0 l l (36) Due to the member relative rotations, the new end section orientation 2 2 matrices will be [p(1l] [e( 1l] and [p( J] [e( l), The member x2 and x3 axes could then be defined as the principal direction of an average cross-section, hence (37) [�] SOLUTION STRATEGIES FOR TRAVERSING LIMIT POINTS 3. One of the earliest techniques developed was to introduce spring coefficients into the incremental stiffness matrix [31,32]. This has the effect of augmenting the stiffness matrix so that it remains positive definite throughout the entire range of the analysis. This augmentation is achieved by adding to the structural stiffness matrix [K] by the following matrix of rank one; !._ {Q} {Q} T 82 - where k (38) - qiven stiffness of a fictitious elastic spring. {Q} reference load vector. 8 the norm of the matrix [K]. -26The augmented equilibrium equations are thus {P} where {P} = (39) total external load of augmented system, and the load reduction factor, A applied to {P} to obtain the actual external load is 1 _1.._ 62 which can be computed T {Q} {r} - (40) - once Equation (39) is solved for {�}. case of simple load and one spring, as illustrated in Figure method is easy to apply. For the 4, this However, this technique cannot be easily justified from a mathematical viewpoint when multiple springs are added. In addition, for a multi degree of freedom system, the coupling of the artificial stiffness may destroy the banded nature of the stiffness matrix. Also the reference stiffness of the applied springs have to be obtained by trial and it should not be used for structures with local buckling or a tendency to bifurcate. Due to the singularity of the incremental stiffness matrix at the extremum points, equilibrium iterations easily break down. To overcome this problem which has plagued earlier researchers employing an incremental-iterative solution strategy, Bergan [24,25,33] the concept of a current stiffness parameter, S . With reference to P Figure 5, S p introduced is defined as [�J i r�t (41) -27- Augmented structure Figure 4. Actual structure Spring Constant Method - 28 - Load,P Deflection, Figure 5. 'Current Stiffness Parameter' Method liP; with � �c characteristic load, which is chosen to be the reference load = , T 1\r R - -c vector, R ref. Now, II�; incremental external load LIP s ; R ref. llp . llp • 1 1 p ! llr liR. -1 -1 llr T IIR � llp llp 0 0 r -29- [liP;l·2 llp o llr T -0 K -0 llr -0 (42) t�r "!" K. llr. _, _, - 1 The current stiffness parameter has an initial value of 1.0 and is zero at a singular point. This parameter can thus sense in advance the ap proach of an instability, and by suppressing equilibrium iterations when the magnitude of this parameter is less than a threshold value, the solution is prevented from breaking down. The solution, by pure load incrementation, is then made to 'jump' over the limit point and advance along its path. On the negative definite portion of the solution path, the 'current stiffness parameter' is negative and when this occurs the algorithm reverses the load and displacement increments. The solution then proceeds with negative load increments until the next extremum point is reached. Due to the low stiffness at the limit points, the load increment can produce uncontrolled displacements which may cause the solution to diverge. Small load increments are thus necessary, and this also helps to prevent drifting away from the equilibrium path. with small increments, the displacement can still be too large; Even and a simple method to prevent this is to calculate a norm of the displacement increment and scale back both load and displacement increments according to how much a specified maximum value has been exceeded. A popular method for avoiding the singularities is to increment the load parameter to the limit point, solving for displacements, then on beyond this point by incrementing a characteristic displacement to evaluate the corresponding load parameter. Argyris [34], but unfortunately his approach led to a system of non symmetric equations. Tong [35], This was first described by Yamada [36] Alternative approaches were presented by Pian and and Zienkiewicz [37] which circumvented the non- symmetric matrix problem, but were applicable only for a step-by-step -�- procedure without iterating to equilibrium within the displacement increment. These also required a modification to the incremental stiff- ness matrix. A scheme suggested by Haisler and Stricklin [38], which makes use of an initial value formulation, allows one to iterate to convergence. The proposed computational procedure is, however, cumber some and a simpler procedure as described by Batoz and Dhatt [39] exists. In the algorithm presented by Batoz and Dhatt, the tangent stiffness matrix K is used to complete 6u which is the incremental . -r displacement due to the residual forces, � and also 6� e which is that due to an arbitrary external load increment P i.e. (43) The actual external load applied is 6AP and this factor is obtained by specifying that the displacement increment at degree of freedom, n, satisfies a displacement, o, where (44) The actual incremental displacement vector, 6u is then given by (45) 6u The above procedure enables one to obtain the increment at the beginning of the iteration sequence. Subsequent iterations are performed in exactly the same manner except that o is now specified to be zero; iterating to convergence is achieved through keeping the selected nodal -31displacement constant. This particular strategy, in a more generalised form was also described by Powell and Simons [40]. From the writers' experience with the above strategy, it has proved to be both stable and efficient. However, it has the distinct disadvantage that the displacement component at the degree of freedom chosen. to control the incrementation has to be monotically increasing. If the controlling displacement snaps back from one load level to another, the strategy will fail to converge. A proper choice of the controlling displacement is thus essential and.for some structures this is not obvious and some experimentation will be necessary. It has been observed that instead of using a single component of the displacement vector, it is possible to use some measure of the complete vector as the controlling parameter. by Riks [41) and Wempner [42] This is the path followed where the load step, �A, is limited by the following constraint equation. (46) In the equation, ��is a prescribed scalar which fixes the length of the increment. This constraint equation was originally added to the incre mental stiffness expression but this unfortunately destroys the banded ness and symmetry of the stiffness matrix. By adopting a two step technique used by Batoz and Dhatt, as described above, this problem can be· overcome. This modified 'constant arc length' strategy of Riks and Wempner is presented by Ramm in Reference [26]; where both iteration in a 'plane' normal to the tangent and iteration in a 'sphere' are described. [43], This is illustrated in Figure 6. In the work of Crisfield instead of applying the constraint equation of of the following constraint. (46), use was made -32- ,---I-- nitial solution .----Normal to tangent ·< r-----r-Final solution a:: UJ 1UJ ::E <( a:: L_ Spherical path ct. Cl <( 0 ....J DISPLACEMENT, u Figure 6. Rik's method with iteration about a 'sphere' or iteration about a 'normal plane' (47) which from his experience appears to be numerically better. The 'constant arc length' method, by generalising the constraint equation from that for a single component of the displacement vector to that for the scalar product of the vector, is thus able to overcome the problem of the choice of a monotonically increasing displacement component associated with the single displacement component control techniques. Equilibrium iterations near a singularity do not pose any particular problems, as is the case with the 'current stiffness parameter' of Bergan. By iterating about a constant arc length or a constant displacement component as in the Batoz and Dhatt strategy, we are iterating in the externa1 1 oad space as we 11 as in the d isp1 acement space, and this is more efficient than conventional Newton-type strategies at constant load. This point is illustrated by the work of Crisfield in Reference [43]. From this brief review, the writers conclude that in terms of generality, -33- effectiveness and robustness, the 'constant arc length' method is superior to the other described strategies. :I I 4. CONSTANT ARC LENGTH STRATEGY OF CRISFIELD J At the start of an arc mental displacement vector, t>:\ ( o) length increment, 6u ( o) is the incre the corresponding load multiplier and 6� the prescribed generalised arc length. Application of the arc length constraint results in (48) with (49) K = the incremental stiffness matrix, and vector. Solving Crisfield [43) (48) and and Ramm (49) [26), � = an arbitrary external load 0 will then yield t>:\ • In the paper of it was suggested that the sign of 6:\ 0 should follow that of the previous increment unless the sign of the determinant of the incremental stiffness matrix has changed. l t From the examples presented by them, this criterion appears to be.satisfactory. However, in the case of structures exhibiting multiple negative eigen values behaviour, this simple criterion does not always work and it appears better to base the sign change on that of the sign of the incre mental work done, t>W where 6W (50) -34If the sign of sign of n"A ( o ) W has changed from that of the previous increment, the is reversed from that of its previous value. After the initial increment, the displacement is then iterated to convergence. nu ( -e At the i th iteration i) K -� (5 1 ) p and nu ( -r i K � R ( i-1 ) ) (52) (i ) where nu displacement increment due to the arbitrary externa 1 1 oad, -e i-l } i nu ( ) displacement increment due to the residual force vector, R ( -r = = at the end of the matrix � ( i-1 ) th iteration. In the Newton-Raphson method the is reformed after every iteration, while in the modified Newton Raphson iterative scheme it remains constant or is reformed after a number of iterations. i If K is constant, then so also is nu ( 1 throughout -e the increment and there i s thus no need t o recalculate it at every iteration. At the i th iteration, the iterative displacement, nu ( i) , is (53) and the incremental displacement up to the th iteration, ou ( i ) is (54) This incremental displacement is made to satisfy the arc length con straint of Equation (48) and thus -35- (55) which leads to the following quadratic where a a a 1 2 (56) 3 i Two values of �A( ) are possible and to void 'doubling-back' the solution i path, the correct choice of �A( ) is that in which the scalar product of i the incremental displacement vector, o�( -l) is a positive quantity. In i the event that both choices of �A( ) yield a positive result, the correct root is the one nearest to the linear solution. i 6A( ) a /a 3 2 (57) -36- The success of this nonlinear solution strategy depends on the quadratic of Equation (56) yielding real roots. Imaginary roots will occur if a2 - 4a a I 2 3 < 0 and this will indeed be so if a norm measure of i than that of llu( ). -e (58) liU( i ) is much 1 arger -r This will happen if the stiffness in the direction of the degree of freedoms .not acted on by the external force vector become very small relative to that of the degree of freedoms carrying the external loads. In the majority of cases, this is unlikely to occur, and from the writers' experience this will probably occur only in structures exhibiting multiple instability directions at a point. A structure showing this type of behaviour is described in the numerical examples. EVALUATION OF POST-BIFURCATION EQUILIBRIUM PATH (SECONDARY PATH) 5. To investigate the stability of an equilibrium configuration it is necessary to formulate the quadratic form (59) where V V potential energy of the system. q generalised co-ordinates ij coefficient matrix of the quadratic form which is equivalent to the incremental stiffness matrix, � defined above. -37- According to the theory of stability of cons-ervative systems, the stalltltty of the equilibrium state ts· ens-ured if the above quadratic form is positive definite. E"quation (59) is indefinite for an un stable configuration while the transition between a stable and an unstable point of the equilibrium path is denoted by a positive semi definite Equation (59). The critical equilibrium state is in general either a limit point or a bifurcati.on point and they are illustrated in Figure 7. A necessary but not sufficient condition for the existence of a critical state is the vanishing of the determinent of the incremental stiffness matrix. det l�l = (601 0 This can be easily obtained during the factorisation of the tangent stiffness matrix in the incremental-iterative solution strategy, since it is the product of all terms of the diagonal matr.ix in the L D L decomposition. The precise definition of a critical equilibrium T state is given by the solution of the following eigenvalue problem (61) where a(k ) denote the eigenvector and the critical states. n(k) the eigenvalues which define A limit point is distinguished from a bifurcation point in that it is characterised by a vanishing load increase. The incremental-iterative solution strategy described in the sections above is capable of continuing the computation of the fund amental (basic) equilibrium path beyond the critical points. However, to follow a secondary path after a bifurcation point, will require a modification to the described solution strategy at the arc length increment immediately after the bifurcation point. The vector �F - 38 - load Parameter load Parameter Deflection (a) limit point Deflection (b) Figure 7. Bifurcation point Critical points tangential to the fundamental path at the bifurcation point, B can be calculated approximately as �F where 6gm = = 1 2 ( 6g m m+1 + 6g ) {62) incremental generalised displacement vector at point m. The unit vector ! ' colinearwith � is thus F F �F According to References [41 and 44] the vector (63) !s secondary post bifurcation path at B is given by tangential to the -39- t a{� -s + �!F} (64) where �= eigenvector of the incremental stiffness matrix, � and the coefficient � is a function of the third derivative of the potential energy [41]. This evaluation of the third derivative is difficult and thus an approximation to gonal to !F is used. !s such that the approximation, � ! i) is ortho Therefore (65) The displacement increment for the first iteration of the arc length increment after the critical point, 9c can be assumed to be (66) where��= prescribed generalised arc length. Iteration to equilibrium is then performed as described in Section 4, with the modification that the incremental stiffness matrix used is �(9c g8) + to prevent the solution from coming back to the fundamental equilibrium path. This described technique of superimposing a fraction of the eigenmode on the displacement field at the bifurcation point can be called the 'perfect approach'. A simpler alternative without the need to compute the eigen vectors wi11 be to impose either a sma 11 perturbation in 1 oad or geometry to the structure. This 'imperfect approach' will yield a load-deflection path which approaches the perfect post-bifurcation path with increased - 40 - deformation of the structure (_see Figure 7). 6. EXTRAPOLATED STIFFNESS PROCEDURE Let us consider the efficiency of the linearised incremental., iterative arc length technique. It is apparent from Figure 8 that for the one degree of freedom cases, the point (a) which is the result after the first iteration, is a poor guess of the final solution. If instead of using the gradient to the solution path at the beginning of the arc length increment, it is possible to use the gradient of the point half arc length ahead, it would appear that the result after the first iter ation will give a much better estimate to the final solution. illustrated in Figure 8 where m is midway along the arc a-b. This is For a multi degree of freedom system, by using the tangent stiffness half an arc length ahead instead of the value at the beginning of the increment, the same conclusion should apply. The pertinent question now is; how to obtain the tangent stiffness half an arc length ahead since the nodal displacement associ ated with the point are not known at the beginning of the increment. For the strategy proposed herein, this is achieved by extrapolating forward the tangent stiffness from previous arc length increments. 3 A point Lagrange interpolation polynominal [45] will be. used and thus in addition to the tangent stiffness at the beginning of the arc length increment, its value at the previous two increments must be retained. The tangent stiffness matrices are stored in 'skyline' linear arrays. Let k., k. J J-1 and k. 1 represent and element of the tangent stiffness at J- the beginni.ng of the j, (j-1) and U-2) increment respectively. k j is the corresponding element of the extrapolated tangent stiffness half an arc length ahead at the j th increment and it is given by -41- k' j (2.5 - 1) (2.5 - 2) k j-2 (O - 1} (O - 2) (2.5 - 0) (2.5 - 2) + ------- (1 - 0} (1 - 2) (2.5 - 0) (2.5 - 1) + ------- (2 - 0} ( 2 - 1) 0.375 k. J-2 1.25 k. J-1 + 1.875 k. (67) J The above extrapolation is repeated for each element of the tangent stiffness to fonn the complete extrapolated stiffness matrix. This matrix is then factorised and used throughout the increment until convergence to the final solution. Since the extrapolation procedure described above required the tangent stiffness for the previous two increments, this technique can only be applied after the second arc length increment. For the initial two increments either the M.N.R. or the N.R. procedure will have to be used to iterate to convergence. The usefulness of this extrapolated stiffness strategy rests on the assumptions that (i ) for each element of the tangent stiffness, its value along the solution path does indeed have a relationship than can be modelled closely enough by a 3 point Lagrange interpolation polynomial. ( ii ) the extension from the single d.o.f. case to the multi d.o.f. system that by using the gradient ( tangent stiff ness ) at half an arc length increment ahead gives a much closer estimate to the final solution; and - ( iii ) 42 - this better initial estimate will iterated more efficient ly to the final solution than the current Newton-Raphson methods. These assumptions have been validated in Reference [29) for two dimen sional frames. The increased efficiency of these extrapolated stiffness strategy over the mNR method is demonstrated in the following numerical example for three dimensional space frames. 7. NUMERICAL EXAMPLES 7.1 CANTILEVER BEAM Figure 9 depicts a cantilever beam loaded at its free end with a concentrated load. The direction of the load is kept constant in a vertical direction throughout the deformation. is used to model the beam. A single frame element An analytic solution to this problem, obtain ed through the use of elliptic integrals, is given by Frisch-Fary [46]. It is observed that the results of the numerical analysis using the two dimensional frame program is indistinguishable from the analytical solution. 7.2 WILLIAMS' TOGGLE FRAME This problem illustrated in Figure 10, has been solved analytically as well as experimentally tested by Williams [6]. In his analytical treatment of the frame, Williams took into consideration the finite change of geometry as 11ell as the effects of the axial forces on the flexural stiffness and the flexural shortening of the members. -43- Result from first iteration Loading parameter,>.. Final solution Deflection, u Figure 8. Constant arc length procedure combined with extrapolated stiffness strategy p l I. 12 L= Area= M.l.: E= 400cm 42cm2 6482 cm4 0,2x105kN/cm2 L - .....,.- = present analysis analytiCal method \ 10 _.8 w ;:;-. - a.b 4 0 Figure 9. 0,2 0,4 o.a · 10.0 Load deflection curve for cantilever beam -44- � 1 �k P,v 25, 87 2 . -=:c:_=-=-.._ - EA = EI = -----�-"'+ 6 1,85Sx10 lb 3lb/in2 9,27 x 10 70 Present analysis • a.. 60 + - Cl < 0 _J • II Analy tical _ Experimental } Williams[6] Wood & Z ienki ewi c z [4] Papadrakakis[131 50 40 QL---�----_J----�-----L----�-----�----� 0,2 0,3 0,4 0,6 0,7 0,5 0,1 DEFLECTION, v {in.) Figure 10. Load deflection curves for Williams toggle frame -45- Wood and Zienkiewicz [4] have also investigated this problem employing an assumed desplacement finite element approach with five elements per member. Papadrakakis [13] used a 'beam-column' approach to derive the nonlinear equilibrium equations, which were solved through a vector iteration approach. The authors results, utilising one plane frame element per member, are in very close agreement with the analytical solution of Williams and the finite element solution of Wood and Zienkiewicz. 7.3 TWELVE MEMBER HEXAGONAL SPACE FRAME This hexagonal frame has been experimentally studied by Griggs !47]. The prebuckling behaviour has also been solved by Chu and Rempetsreiter [10] while Papadrakakis [13] traced the solution into the post buckling range. From Figure 11, it is observed that the present analysis is in agreement with that of Chu and Rempetsreiter in the pre buckling range. Papadrakakis' curve appears to be in exact agreement with that of the experimental result, which yielded a snap through load 4% lower than that of the writers. The post-buckling region of the writers' solution path does not correspond exactly with that presented by Papadrakakis. In his formulation, he treated the rotations as a vector and it appears to the writers, therefore, that the validity of his results in the presence of genuinely large rotations, which do occur in the post-buckling range, is doubtful. Iteration to convergence was achieved using the mNR method and the proposed extrapolated stiffness method. ations The average number of iter per arc length was 3.5 for the mNR method and 2.3 for the -46- 250 E = 439800 lb/in2 159000 lb/in2 0,494 in2 12 :0,02 in 2 G = Area = 13 J = = O.D2 in2 0,0331 in4 200 150 Present analysis ..c • c < Papadrakakis 191 II Chu- Rampetsreiter + Experimental results [34) [5) Present analysis, with the 6 0 � bo undary nodes res t rained against translational movement 100 i =no. of iterations for mNR j = no. of iterations for extrapolated stiff ness (4,2) so Figure 11. Load-deflection curve for hexagonal frame -47- extrapolated stiffness strategy. This increased efficiency is further illustrated by the cpu times required for the two runs ( implemented on the VAX-11/780 computer ) , 22.5 sees for the mNR and 17.8 sees for the extrapolated stiffness method. Next, the structure with the six boundary nodes now restrained against translational movement in all directions, was reanalysed. As shown in Figure 11, this has the effect of yielding a larger snap-through load. In addition, it was observed at the snap-through load that two diagonal elements of the triangularised stiffness matrix changed sign ( from positive to negative ) at essentially the same load level. resulted in a This positive determinant and would indicate that the matrix is still positive definite if the determinant is used as the criterion for determining it. The incremental stiffness matrix is, however, negative definite at this stage, as is indicated by considering the sign of the incremental work done. At the lowest point of the post-buckling region, the solution strategy broke down as no real roots could be found for the quadratic of Equation ( 56 ) . On examination, it was found that at that load level, one of the rotational degree of freedoms exhibited a singularity before it occurred for the vertical displacement of the loaded node. This resulted in a norm mea:;ure of the displacement increment due to the residual forces becoming very much larger than that due to the external applied load which yielded the condition of Equation (58 ) . To prevent the applied solution strategy from breaking down, wherever the condition of Equation (58 ) occurs, iteration is suppressed and only pure incrementation is applied. -48- E = Area = G I + � r 25 t i- x 3, w = 5 3,03x 10 N/cm2 5 1, 096 x 10 N/cm2 3,17 cm 2 x2,v 25 1 B,216L� �� � k Figure 12(a). 43,3 . JI" 43,3 � -----+ � Geometry of 24 member star-shaped shallow dome x,,u - 49 - 1000 500 / z.a / I / 1 t.o I I _, ._/ // "' •1,0 -2.0 19l!!ll ·3,0 •4,0 ·5.0 •6,0 DEFlECTION, v • tO lc•l 11110 ·• ·• DEflECTION, u •1011cmJ ·6 ·4 ·2 12!!!..L Figure 12(b). Load-deflection curves ·7.0 •1,0 -50- TWENTY-FOUR MEMBER STAR-SHAPED DOME 7.4 The structure depicted in Figure 12 ( a) has been analysed by various authors [13,48,49] as a space truss to trace its load-deflection behaviour into the post-buckling range. Here, the structure is analysed as a space frame and its post-buckling path for the loading condition of Figure 12(b) presented. The supports of the dome are assumed to be pin ned and restrained against translational motion. The average number of iterations per arc length increment is 3.5 using the mNR method and 2.3 utilising the extrapolated stiffness method. The corresponding cpu times used are 46.3 sees for the mNR and 38.0 sees for the extrapolated stiffness technique. 8. CONCLUSION An updated Lagrangian large rotation formulation of the non linear equilibrium equations governing the large deflection of plane and space frames, within the confines of 'beam-column' theory has been presented. A comprehensive survey of the most recent solution strategies available for analysing the post-buckling behaviour of structures was also undertaken. From this review, it was concluded that the arc length strategy due to Crisfield [43] is most robust and efficient, and thus this incremental-iterative method was used in conjunction with the derived system of nonlinear equations to study the post-buckling behaviour of spatial framed structures. The results of the analyses are in good agree ment with previously published experimental and analytical work. Modifi cations to the described strategy to enable it to trace the secondary post-bifurcation path are also discussed. -51- The solution of the nonlinear equations is expensive in comput ing time and this is especially so for large scale problems. An extra polated stiffness method which would reduce the computing time needed to arrive at a particular solution was presented. For the space frames analysed above, it was found to be more efficient than using the conven tional mNR method. For most problems, what is required is the snap-through load without the need to trace the· post-buckling path. However, in some cases, for example in studying the effects of a concentrated load on a restricted part of the structure, it is important to obtain information on the nature of the load shedding after the occurrence of a local instability, in order to assess the behaviour of the whole structure. Due to the lack of robust and efficient techniques for critical points, not much attention has been focused on the problem of analysing the post-buckling behaviour of spatial frames. It is hoped that this report will go some way in helping to remedy this situation. -52- APPENDIX A - NOMENCLATURE A area of cross-section (k} a eigenvector [A] displacement transformation matrix [B I l member rotation matrix [B] transformation matrix relating local member forces to member basic forces [�] {flu} [LIB] change in [�] stiffness modification matrix to account for the effect resulting from of member forces (two-dimensional frame) .9E_ generalised gradient of load-displacement curve dr E modulus of elasticity e axial displacement of member {FJ, {fl} intermediate force vector {FI l rotated member local force vector {LIF}, {llf} incremented joint force vector � variation of the vector {ll ul displacement {LIF}, {llf I } {F} resulting from incremental {llg} incremental joint force vector GJ torsional rigidity I, I m moment of inertia [�], [k] member basic incremented stiffness matrix lKl member incremental stiffness matrix in local co-ordinates K, K. tangent stiffness matrix [ �Al augmented stiffness matrix [�Gl' [�Gl global incremental stiffness matrix - -1 -53incremental stiffne.ss matrix correspondi.ng to displacement vector {gc + g6} j th e 1 ement of tti.e tangent stiffness matrix stored in a 'skyline' array stiffness of fictitious· spring k undeformed length parameter generalised arc length parameter bending moment torsional moment stiffness modification matrix to account for the effect of member force (three dimensional frame) p axial force P, {P} external load vector {.1P} incremental external load vector [ i p( ) ] p ll i end section orientation matrix for end section incremental load multiplier {g } reference load vector q generalised co-ordinates t>g i [R] incremental generalised displacement vector at point transformation matrix for local convective co-ordinates to global co-ordinates R - �c residual force vector characteristic load vector �ref reference load vector [�] member orientation matrix llr incremental displacement vector {S}, {s} member force vector so sP transverse force current stiffness parameter -54- {LS}, {Ls} member incremental force vector LS 0 incremental transvers-e force [!l disp 1 acement transformation matrix {u} joint displacement vector {u'} intermediate displacement vector u' relative displacement of member ends measured parallel to undeformed member axis {Lu} incremental joint displacement vector LU incremental displacement vector {Lu'}, {Lii} intermediate incremental displacement vector L incremental displacement vector due to arbitrary external �e load increment incremental displacement vector due to residual forces Lu -r i .Su( ) incremental displacement from first iterative cycle to i th iterative cycle potential energy of system v coefficient matrix of the quadratic form v .. lJ relative displacement of member ends measured perpendicular v' to undeformed member axis {LV}, {Lv} member incremental displacement vector LW incremental work X ' X ' X I X I 2 ' X 2 ' X 3 3 global coordinate axes member coordinate axes a initial inclination of member to global axes [a] joint orientation matrix [La] incremental joint orientation matrix .s .. lJ LA nk kronecker delta incremental load multiplier eigenvalue - 55 - [11�] incremental joint rotation matrix e norm of stiffness matrix e , 8 m mn member relative end rotation e• joint rotation of node e angle of twist of member m t <P potential energy A load reduction factor m -56- APPENDIX B 1. - REFERENCES MALLETT, R.H. and MARCAL, P.V. 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Wunderlich, W., Stein, E. and Bathe, K.J., MEEK, J.L. and TAN, H.S. 217-235. "A Stiffness Extrapolation Strategy for Nonlinear Analysis", to be published in Computer Methods Mechanics and Engineering. in Applied -60- 30. (_19771 BELYTSCHKO, T. and SCHWER, L. "Large Dfsplacement, Transient Analysis of Space Frames", lnternation.al Journal of Numerical Methods pp 65-84. SHARIF!, P. and POPOV, E.P. (1971) in·Engir'leertng, Vol. 31. 11, "Nonlinear Buckling Analysis of Sandwich Arches", Journal of Engineering Mechanics, ASCE, Vol. pp 32. 97, 1397-1411. SHARIF!, P. and POPOV, E.P. (1973) "Nonlinear Finite Element Analysis of Sandwich Shells of Revolution", American Institute of Aeronautics 33. and Astronautics Journal, Vol. 11, BERGAN, P.G. and SOREIDE, T .H. (1978) pp 715-722. "Solution of Large Displacement and Instability Problems using the Current Stiffness Parameter", in: Bergan, P.G., Finite Elements in Nonlinear Mechanics, Tapir, Trondheim, pp 34. 647-66g. ARGYRIS, J . H . (1965) "Continua and Discontinua",. Proceedings of 1st Conference on Matrix Methods in Structural . Mechanics, Wright-Patterson AFB, Ohio, pp 35. 11-189. PIAN, T.H.H. and TONG, P. (1971) Displacements Analysis", in: 36. "Variational Formulation of Finite De Venbeke, B.F., ( ed. ) , High Speed Computing of Elastic Structures, Liege, Belgium, pp 43-63. YAMADA, Y., IWATA, K., KAKIMI, T., and HOSOMURA, T. (1974) Large Deflection and Critical Loads Analysis of Framed Structures", in: Oden, J.T., ( ed. ) , Computational Methods in Nonlinear Mechanics, University of Texas, Texas, pp 819-829. CURRENT CIVIL ENGINEERING BULLETINS 4 Brittle Fracture of Steel - Perform 11 steels: C. O'Connor 5 blems (1964) Buckling in Steel Structures-1. The 12 column: O'Connor C. 13 (1965) design the 14 Land use prediction in transportation C. O'Connor (1965) Generated Wave observations Currents (1965) Brittle Fracture of Steel-2. Theoret 16 non-uniform, material: C. O'Connor 17. frame Meek and grid 18 J.L. structures: Methods: Traffic Lucas C. and (1974) and R. Owen Planning and of Evaluation K.B. Davidson, et a/. J.L. 19 Meek of Evaluation Quantitative a High Speed Brisbane-Gold Coast Rail Link: (1974) Brisbane Airport Development Flood way Studies: C.J. Ape/t Force Analysis of Fixed Support Rigid (1968) ·(1973) (1966) (1967) Frames: (1971) Wave Climate at Moffat Beach: M.R. K.B. Davidson polycrystalline Two with a free sur (1971) Assignment Analysis by Computer-Programmes for Methods Transportation Gravity Models: A. T.C. Gourlay ical stress distributions in a partially yielded, Element Philbrick - Some made in fixed bed hy Finite face: L. T. Isaacs 15 draulic models: M. R. Gourlay 10 (1969) dimensional seepage plane determinate of trusses against buckling in their plane: 9 (1969) Buckling in Steel Structures- 2. The in 8 pro methods: planning: S. Golding and K.B. David use of a characteristic imperfect shape 7 element Ground Water Hydrology: J.R. Watkins son 6 finite (1969) and its application to the buckling of isolated by J.L. Meek and G. Carey use of a characteristic imperfect shape an Computer - Axisy by Analysis metric solution of elasto-plastic ance of NO 1B and SAA A1 structural 20 (1911) Numbers of Engineering Graduates in Queensland: C. O'Connor (1977) -62- 44. WASZCZYSZYN, Z. "Numerical Problems of Nonlinear Stability (1983} Analysis of Elastic Structures", Computer arid Structures, Vol. pp 45. 17, 13-24. ISAACSON, E. and KELLER, H. B. John Wiley, New York, pp 46. FRISCH-FAY, R. 47. GRIGGS, H.P. (1962) (1966) (1966) Anal,YS is of Numerica 1 r�ethods, 187-189. Flexible·sars, Butterworths, London, pp 35-40. "Experimental Study of Instability in Elements of Shallow Space Frames", Research Report, Department of Civil Engineering, MIT, Cambridge, Mass. 48. PARADISO, M. and TEMPESTA, G. (1980} "Member Buckling Effects in Nonlinear Analysis of Space Frames�·, in: Owen, D. R.J. , ( eds. ) , Numeri ca 1 Methods for Non 1 inear Prob 1 ems, Pineridge Press, Swansea, Wales, pp 49. Taylor, C., Hinton, E. and WATSON, L.G. and HOLZER, S.M. (1983) 395-405. "Quadratic Convergence of Crisfield's Method", Computer and Structures, Vol. 17, pp 69-72. CIVIL EtlGitlEERING RESEARCH REPORTS CE No. Title Flood Frequency Analysis: Logistic Method Author(s) Date BRADY, D.K. Februar) 1979 for Incorporating Probable Maximum Flood 2 Adjustment of Ph r eatic Line in Seepage ISAACS, L.T. March, 1979 Analysis by Finite Element Method 3 Creep Buckling of Reinforced Concrete Coluinns 4 5 6 BEHAN, J.E. & O'CONNOR, Buckling Properties of Monosymmetric KITIPORNCHAI, !-Beams & TRAHAIR, Elasto-Plastic Analysis of Cable Net MEEK, Structures BROWN, A Critical State Soil Model for Cyclic CARTER, J.P., BOOKER, J.R. & Loading Resistance to Flow in Irregular Channels An Appraisal of the Ontario Equivalent 10 11 12 Decembet 1979 C.P. KAZEMIPOUR, A.K. O'CONNOR, C. Februar) 1980 A.K. APELT, C.J. Shape Effects on Resistance to Flow in KAZEMIPOUR, April, Smooth Rectangular Channels & 1980 April, The Analysis of Thermal Stress Involving BEER, Non-Linear Material Behaviour MEEK, J.L. G. & 1980 Buckling Approximations for Laterally DUX, April, Continuous Elastic !-Beams KITIPORNCHAI, S. 1980 A Second Generation Frontal Solution BEER, May, P.F. & G. 1980 Combined Stiffness for Beam and Column O'CONNOR, May, C. 1980 Braces 14 Beaches:- Profiles, Processes and GOURLAY, June, M.R. 1980 Permeability 15 16 Buckling of Plates and Shells Using MEEK, Sub-Space Iteration TRANBERG, The Solution of Forced Vibration Problems July, J.L. & W.F.C. Numerical Solution of a Special Seepage 1980 ISAACS, SeptembE L.T. 1980 Infiltration Problem 18 19 Shape Effects on Resistance to Flow in KAZEMIPOUR, Smooth & APELT, Semi-circular Channels The Design of Single Angle Struts 1980 August, SWANNELL, P. by the Finite Integral Method 17 Februar)l 1980 C.J. Program 13 1979 1979 Base Length 9 May, Novembet P.L.D. & APELT, 8 S. N.S. J.L. & WROTH, 7 April , 19 79 C. WOOLCOCK, A. K. S.T. KITIPORNCHAI, Novembet 1980 C.J. & S. Decembet 1980 CIVIL ENGINEERING RESEARCH REPORTS CE No. 20 21 Title Date Author(s) CARTER, J.P. Subjected to Non Axi-s�metric Loading BOOKER, J.R. Truck Suspension Models KUNJAMBOO, Consolidation of Axi-symmetric Bodies 23 C. Elastic Consolidation Around a Deep CARTER, J.P. & Harch, Circular Tunnel BOOKER, J.R. 1981 An Experimental Study of Blockage WEST, G.S. April, Inelastic Beam Buckling Experiments 1981 DUX, 11ay, P.F. & KITIPORNCHAI, 25 February, 1981 Effects on Some Bluff Profiles 24 January, 1981 K.K. & O'CONNOR, 22 & Critical Assessment of the ·International KORETSKY, Estimates for Relaxation Losses in PRITCHARD, s. A.V. & R.W. 1981 June, 1981 Prestressing Strands 26 Some Predications of the Non-homogenous CARTER, July, J.P. 1981 Behaviour of Clay in the Triaxial Test 27 The Finite Integral Method in Dynamic Analysis : 28 August, SWANNELL, P. 1981 A Reappraisal Effects of Laminar Boundary Layer on a ISAACS, September, L.T. 1981 Model Broad-Crested Weir 29 30 31 Blockage and Aspect Ratio Effects on WEST, Flow Past a Circular Cylinder for 104 < R < 10 5 APELT, Time Dependent Deformation in Prestressed SOKAL, Y.J. & November, Concrete Girder: TYRER, 1981 Measurement and Prediction Non-uniform Alongshore Currents and Sediment Transport - a One Dimensional G.S. & October, C.J. 1981 P. GOURLAY, M.R. January, 1982 Approach 32 33 A Theoretical Study of Pore Water Pressures ISAACS, L.T. & Developed in Hydraulic fill in Mine Stapes February, CARTER, J.P. 1982 Residential Location Choice Modelling: GRIGG, Gaussian Distributed Stochastic Utility July, T.J. 1982 Functions 34 The Dynamic Characteristics of Some Low WEST, August, G.S. 1982 Pressure Transducers r, 35 Spatial Choice ModellinG with Mutually Denendent Alternatives: GRIGG, September, 1982 T.J. Legit Distributed St � chastic Utility Functions 36 Buckling Approximations Beams for Inelastic DUX, I'.F. October, & KITIPORNCHAI, S. 1932 CIVIL ENGINEERING RESEARCH REPORTS CE No. Ti t l e Author(s) Date 37 Parameters of the Retail Trade Model: A Utility Based Interpretation GRIGG, T.J. October, 1982 38 Seepage Flow across a Discontinuity in Hydraulic Conductivity ISAACS, L.T. December, 1982 Probabilistic Versions of the Short-Run GRIGG, T.J. December, 1982 KOE, C.C.L. & BRADY, D.K. January, SWANNELL, P. March, 39 Herbert-Stevens Model 40 41 Quantification of Sewage Odours The Behaviour of Cylindrical Guyed Stacks 1983 1983 Subjected to Pseudo-Static Wind Loads 42 Buckling and Bracing of Cantilevers KITIPORNCHAI, s. DUX, P.F. & RICHTER, N.J. April, 1983 43 Experimentally Determined Distribution of Stress Around a Horizontally Loaded Model Pile in Dense Sand WILLIAMS,,D.J. & August, PARRY, R.H.G. 1983 44 Groundwater Model for an Island Aquif er : Bribie Island Groundwater Study ISAACS, L.T. & September, WALKER, F.D. 1983 45 Dynamic Salt-Fresh Interface in an Unconfined Aquifer: Bribie Islartd Groundwater Study ISAACS, L.T. September, 1983 46 An Overview of the Effects of Creep in Concrete Structures SOKAL, Y.J. October, 1983 47 Quasi-Steady Models for Dynamic Salt Fresh Interface Analysis ISAACS, L.T. November, 1983 48 49 Laboratory and Field Strength of Mine WILLIAMS, D.J. & Waste Rock WALKER, L.K. Large Deflection and Post-Buckling Analysis of Two and Three Dimensional Elastic Spatial Frames MEEK, J.L. TAN, H.S. & .November, 198 3 December, 1983 CURRENT CIVIL ENGINEERING BULLETINS 4 Brittle Fracture of Steel - Perform 11 steels: C. O'Connor 5 blems (1964) use of a characteristic imperfect shape 12 column: O'Connor C. 13 (1965) design the 14 Land use prediction in transportation C. O'Connor (1965) Generated Wave observations Currents (1965) Brittle Fracture of Steel-2. Theoret 16 non-uniform, material: C. O'Connor 17. frame Meek and grid 18 J.L. structures: Methods: Traffic Lucas C. and (1974) and R. Owen Planning and of Evaluation K.B. Davidson, et a/. J.L. 19 Meek of Evaluation Quantitative a High Speed Brisbane-Gold Coast Rail Link: (1974) Brisbane Airport Development Flood way Studies: C.J. Ape/t Force Analysis of Fixed Support Rigid (1968) ·(1973) (1966) (1967) Frames: (1971) Wave Climate at Moffat Beach: M.R. K.B. Davidson polycrystalline Two with a free sur (1971) Assignment Analysis by Computer-Programmes for Methods Transportation Gravity Models: A. T.C. Gourlay ical stress distributions in a partially yielded, Element Philbrick - Some made in fixed bed hy Finite face: L. T. Isaacs 15 draulic models: M. R. Gourlay 10 (1969) dimensional seepage plane determinate of trusses against buckling in their plane: 9 (1969) Buckling in Steel Structures- 2. The in 8 pro methods: planning: S. Golding and K.B. David use of a characteristic imperfect shape 7 element Ground Water Hydrology: J.R. Watkins son 6 finite (1969) and its application to the buckling of isolated by J.L. Meek and G. Carey Buckling in Steel Structures-1. The an Computer - Axisy by Analysis metric solution of elasto-plastic ance of NO 1B and SAA A1 structural 20 (1911) Numbers of Engineering Graduates in Queensland: C. O'Connor (1977)

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