The Finite Element Method – Volume 2 : Solid Mechanics

The Finite Element Method – Volume 2 : Solid Mechanics
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O.C. Zienkiewicz
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8 سبتمبر 2021
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The Finite Element Method – Volume 2 : Solid Mechanics
Fifth Edition
O.C. Zienkiewicz, CBE, FRS, FREng
UNESCO Professor of Numerical Methods in Engineering
International Centre for Numerical Methods in Engineering, Barcelona
Emeritus Professor of Civil Engineering and Director of the Institute for
Numerical Methods in Engineering, University of Wales, Swansea
R.L. Taylor
Professor in the Graduate School
Department of Civil and Environmental Engineering
University of California at Berkeley
Berkeley, CaliforniaPreface to Volume 2
General problems in solid mechanics and
non-linearity
Introduction
Small deformation non-linear solid mechanics
problems
Non-linear quasi-harmonic field problems
Some typical examples of transient
non-linear calculations
Concluding remarks
Solution of non-linear algebraic equations
Introduction
Iterative techniques
Inelastic and non-linear materials
Introduction
Viscoelasticity – history dependence of
deformation
Classical time-independent plasticity theory
Computation of stress increments
Isotropic plasticity models
Generalized plasticity – non-associative case
Some examples of plastic computation
Basic formulation of creep problems
Viscoplasticity – a generalization
Some special problems of brittle materials
Non-uniqueness and localization in
elasto-plastic deformations
Adaptive refinement and localization
(slip-line) capture
Non-linear quasi-harmonic field problemsPlate bending approximation: thin
(Kirchhoff) plates and C1 continuity
requirements
Introduction
The plate problem: thick and thin
formulations
Rectangular element with corner nodes (12
degrees of freedom)
Quadrilateral and parallelograpm elements
Triangular element with corner nodes (9
degrees of freedom)
Triangular element of the simplest form (6
degrees of freedom)
The patch test – an analytical requirement
Numerical examples
General remarks
Singular shape functions for the simple
triangular element
An 18 degree-of-freedom triangular element
with conforming shape functions
Compatible quadrilateral elements
Quasi-conforming elements
Hermitian rectangle shape function
The 21 and 18 degree-of-freedom triangle
Mixed formulations – general remarks
Hybrid plate elements
Discrete Kirchhoff constraints
Rotation-free elements
Inelastic material behaviour
Concluding remarks – which elements?
Thick’ Reissner – Mindlin plates – irreducible
and mixed formulations
Introduction
The irreducible formulation – reduced
integrationMixed formulation for thick plates
The patch test for plate bending elements
Elements with discrete collocation constraints
Elements with rotational bubble or enhanced
modes
Linked interpolation – an improvement of
accuracy
Discrete ’exact’ thin plate limit
Performance of various ’thick’ plate elements

  • limitations of twin plate theory
    Forms without rotation parameters
    Inelastic material behaviour
    Concluding remarks – adaptive refinement
    Shells as an assembly of flat elements
    Introduction
    Stiffness of a plane element in local
    coordinates
    Transformation to global coordinates and
    assembly of elements
    Local direction cosines
    ’Drilling’ rotational stiffness – 6
    degree-of-freedom assembly
    Elements with mid-side slope connections
    only
    Choice of element
    Practical examples
    Axisymmetric shells
    Introduction
    Straight element
    Curved elements
    Independent slope – displacement
    interpolation with penalty functions (thick or
    thin shell formulations)Shells as a special case of
    three-dimensional analysis –
    Reissner-Mindlin assumptions
    Introduction
    Shell element with displacement and rotation
    parameters
    Special case of axisymmetric, curved, thick
    shells
    Special case of thick plates
    Convergence
    Inelastic behaviour
    Some shell examples
    Concluding remarks
    Semi-analytical finite element processes –
    use of orthogonal functions and ’finite strip’
    methods
    Introduction
    Prismatic bar
    Thin membrane box structures
    Plates and boxes and flexure
    Axisymmetric solids with non-symmetrical
    load
    Axisymmetric shells with non-symmetrical
    load
    Finite strip method – incomplete decoupling
    Concluding remarks
    Geometrically non-linear problems – finite
    deformation
    Introduction
    Governing equations.
    Variational description for finitite deformation
    A three-field mixed finite deformation
    forumationA mixed-enhanced finite deformation
    forumation
    Forces dependent on deformation – pressure
    loads
    Material constitution for finite deformation
    Contact problems
    Numerical examples
    Concluding remarks
    Non-linear structural problems – large
    displacement and instability
    Introduction
    Large displacement theory of beams
    Elastic stability – energy interpretation
    Large displacement theory of thick plates
    Large displacement theory of thin plates
    Solution of large deflection problems
    Shells
    Concluding remarks
    Pseudo-rigid and rigid-flexible bodies
    Introduction
    Pseudo-rigid motions
    Rigid motions
    Connecting a rigid body to a flexible body
    Multibody coupling by joints
    Numerical examples
    Computer procedures for finite element
    analysis
    Introduction
    Description of additional program features
    Solution of non-linear problems
    Restart option
    Solution of example problems
    Concluding remarks
    Appendix AA1 Principal invariants
    A2 Moment invariants
    A3 derivatives of invariants
    Author index
    Subject indexAuthor index
    Page numbers in bold refer to the list of references at the end of each chapter.
    Abramowitz, M. 165, 172
    Adams, M. 35, 37
    Adini, A. 124, 169
    Agelet de Saracibar, C. 355, 363
    Ahmad, S. 173, 177, 188, 212; 217, 224, 241,
    242; 244, 264; 266, 270, 274, 275, 280, 284,
    Bathe, K.-J. 28, 36; 144, 169; 183, 185, 188, 189,
    191, 196, 213, 214; 286, 287; 347, 355, 362,
    363; 386, 390, 394; 415, 425, 426, 431
    Batoz, J.L. 31, 37; 144, 162, 169, 170, 171; 196,
    214; 231, 236, 242; 386, 393
    Bauchau, O.A. 396, 411
    Bauer, J. 257, 258, 259, 260, 261, 262, 265
    Bayless, A. 90, 109
    Bazant, Z.P. 90, 91, 92, 93, 109
    Bazeley, G.P. 123, 124, 139, 144, 148, 167
    Beisinger, Z.E. 217, 241
    Bell, K. 123, 154, 168
    Belytschko, T. 90, 91, 109; 217, 241; 286, 287;
    347, 362; 390, 394
    Benson, D.J. 347, 362; 396, 404, 407, 410
    Benson, P.R. 297, 310
    Bergan, P.G. 31, 32, 35, 37; 124, 132, 169; 226,
    242; 384, 388, 393
    Besseling, J.F. 53, 105
    Best, B. 88, 108
    Betsch, P. 210, 215; 286, 287; 386, 394; 404, 411
    Bicanic, N. 24, 36; 53, 70, 84, 90, 105, 106, 108,
    109; 179, 212
    Biezeno, C.B. 388, 394
    Bingham, E.C. 79, 107
    Birkhoff, G. 154, 170
    Bischoff, M. 210, 215; 218, 242; 286, 287, 288;
    332, 361; 386, 394
    Bittencourt, E. 347, 362
    Bjorck, A. 24, 36
    Blaauwcndraad, J. 84, 108
    Bogner, F.K. 123, 144, 168; 217, 240; 384, 393
    Bonet, J. 314, 317, 328, 333, 338, 345, 360; 398, 411
    Bonnes, G. 217, 240
    Booker, J.R. 89, 90, 109
    Boot, J.C. 156, 170
    Boresi, A.P. 3, 20
    Boroomand, B. 94, 110
    286
    Ahmed, H . 101, 110
    A1 Mikdad, M. 367, 392; 401, 411
    Alart, P. 347, 355, 363
    Albasiny, E.L. 234, 243
    Alfrey, T. 48, 104
    Allman, D.J. 157, 158, 171; 226, 229, 242
    Allwood, R.J. 157, 171
    Ambrosio, J.A.C. 396, 411
    Andelfmger, U. 286, 287; 332, 360
    Anderson, C.A. 15, 21; 102, 103, 110; 386, 392
    Anderson, R.G. 385, 393
    Ando, Y. 132, 169
    Argyris, J.H. 123, 124, 129, 154, 168; 217, 240;
    390, 395; 401, 411
    Arlctt, P.L. 302, 310
    Armero, F. 91, 109; 332, 358, 360
    Armstrong, P.J. 53, 105
    Arnold, 186, 197, 198, 213
    Arulanandan, K. 20, 21
    Ashwell, D.G. 217, 241; 388, 394
    Atluri, S.N. 396, 411
    Auricchio, F. 53, 66, 71, 105; 185, 199, 200, 203,
    213, 214; 344, 361
    Babuska, I. 120, 145, 167; 183, 208, 213, 214
    Bagchi, D.K. 385, 393
    Bahrani, A.K. 302, 310
    Bakhrebah, S.A. 283, 287
    Baldwin, J.T. 162, 171
    Ballesteros, P. 139, 169
    Barut, A. 390, 394, 395438 Author index
    Borri, M. 396, 411
    Bosshard, W. 123, 154, 168
    Bottasso, C.L. 396, 411
    Bouzinov, P.A. 347, 362
    Braess, D. 332, 361; 386, 394
    Braun, M. 210, 215; 286, 287
    Brebbia, C.A. 217, 240; 373, 376, 383, 386, 390,
    Cilia, M.A. 286, 287
    Cimento, A.P. 28, 36
    Clark, H.T. 396, 411
    Clough, R.W. 123, 124, 144, 149, 167, 168, 169;
    179, 212; 217, 226, 232, 240, 242; 267, 280,
    286; 384, 388, 393
    Cohen, H. 396, 397, 410
    Cohen, M. 179, 182, 184, 185, 212; 286, 287
    Collatz, L. 24, 36; 415, 431
    Comini, G. 14, 21
    Connor, J. 217, 240, 241; 373, 376, 383, 386, 390,
    392
    Brekke, T. 88, 108
    Brczzi, F. 188, 213, 214; 226, 229, 242
    Brogan, F.A. 390, 394
    Bron, J. 156, 170
    Brown, T.G. 297, 310
    Broyden, C.G. 34, 37
    Bucalem, M.L. 188, 214
    Bucciarelly, L. I l l, 167
    Bucco, D. 308, 310
    Biichter, N. 210, 215; 286, 287
    Bushnell, D. 286, 287
    Butlin, G.A. 123, 154, 168
    392
    Cook, R.D. 157, 171; 179, 212; 226, 242; 284,
    286, 287
    Cormeau, I.C. 79, 81, 82, 91, 107, 108
    Cornes, G.M.M . 157, 171
    Cowper, G.R. 123, 154, 168; 217, 241
    Cozzarelli, F.A. 3, 20; 59, 106; 220, 242; 314, 317,
    328, 338, 339, 341, 360; 433, 436
    Cramer, H. 308, 311
    Creus, G.J. 347, 362
    Crisfield, M.A. 28, 29, 31, 34, 36, 37; 162, 171;
    200, 214; 359, 364; 390, 395
    Cross, D.K. 162, 171
    Cuomo, M. 347, 363
    Curnier, A. 347, 355, 362, 363
    Cusens, A.R. 297, 309, 310
    Cwertny, A.J.(Jr.) 308, 311
    Cyr, N.A. 78, 107
    Cyrus, N.J. 124, 169
    Cantin, G. 217, 240
    Cardona, A. 396, 410
    Carpenter, N. 217, 241
    Carr, A.J. 217, 226, 240
    Carson, W.G. 385, 393
    Cassel, E. 17, 21; 34, 37
    Cazzani, A. 396, 411
    Cervera, M. 91, 109
    Chaboche, J.L. 53, 105
    Chadwick, P . 314, 360
    Chan, A.H.C. 19, 20, 21; 70, 82, 106, 108; 124,
    128, 133, 169; 182, 183, 212
    Chan, A.S.L. 244, 257, 258, 259, 264
    Chan, H.C. 297, 308, 310
    Chan, S.K. 347, 361
    Chan, Y.K. 385, 393
    Chang, C.T. 53, 105
    Chari, M.V.K. 101, 110
    Chaterjee, A. 156, 170
    Chattapadhyay, A. 386, 393
    Chaudhary, A.B . 188, 189, 191, 213; 347, 355,
    362, 363
    Chawla, V. 347, 359, 362
    Chen, A.J. 407, 412
    Chen, W.F. 51, 53, 104
    Cheung, M.S. 297, 309
    Cheung, Y.K. 48, 77, 104, 107; 123, 124, 126,
    132, 139, 144, 148, 167, 168, 169; 217, 231,
    240; 297, 305, 306, 307, 308, 309, 310, 311
    Chiang, M.Y.M. 188, 213, 214
    Chong, K.P. 3, 20
    Chou, P.C. 3, 20
    Christensen, R.M. 39, 104
    Church, K. 157, 171
    Ciarlet, P.G. 216, 240
    da Deppo, D.A. 371, 392
    Dahlquist, G. 24, 36
    Damilano, G. 396, 411
    Darve, F. 68, 106
    Davidon, W.C. 28, 36
    Davies, J.D. 297, 310
    Dawe, D.J. 124, 128, 168; 217, 230, 231, 241; 308,
    310
    De Borst, R. 84, 91, 93, 108, 109
    DeAndrade, J.C. 308, 311
    Defalias, Y. 70, 106
    Del Guidice, S. 14, 21
    Delcourt, C. 308, 310
    Delpalc, R . 244, 252, 254, 258, 259, 260, 264
    Demmel, J. 24, 36; 413, 430
    Dennis, J.E. 28, 36
    Desai, C.S. 82, 108
    Dhalla, A.K. 72, 107
    Dhatt, G. 31, 37; 156, 162, 170, 171; 217, 240;
    386, 393
    Dhatt, G.S. 144, 170, 196, 214
    Dhillon, B.S. 200, 214
    Dill, E.H. 384, 393
    Dinno, K.S. 75, 107
    Doan, D.B. 396, 410Author index 439
    Doblare, M. 367, 392
    Dong, S.B . 200, 214; 308, 311
    Draper, J.K. 123, 167
    Drucker, C.D. 51, 62, 104, 106
    Duncan, W. 123, 168
    Dupuis, G. 217, 241
    Duvaut, G. 79, 80, 107
    Dvorkin, E.N . 188, 189, 191, 213; 286, 287
    Dworsky, N. Ill, 167
    Gonzalez, O. 359, 364
    Goodier, J.N. 3, 20; 302, 310
    Goodman, R.E. 88, 108
    Goudreau, G.L. 36; 39, 45, 46, 104; 347, 362
    Gould, P.L. 244, 255, 264
    Govindjee, S. 347, 355, 362; 430, 431
    Grafton, P.E. 244, 248, 250, 251, 264; 303, 304,
    310
    Gray, N.G. 286, 287
    Greenbaum, G.A. 77, 107
    Greene, B.E. 157, 171; 217, 240, 241
    Griemann, L.F. 200, 214
    Gross, B. 39, 104
    Gruttmann, F. 210, 215; 286, 287; 386, 394; 404,
    Edwards, G. 157, 171; 217, 241
    Elias, Z.M. 244, 265
    Elsawaf, A.F. 34, 37
    Ericksen, J.L. 433, 434, 436
    411
    Falk, R.S. 186, 197, 198, 213
    Felippa, C.A. 123, 124, 132, 144, 149, 154, 168,
    169, 170; 226, 242
    Feshbach, H. 292, 309
    Finnie, I. 81, 108
    Firmin, A. 244, 257, 258, 259, 264
    Fish, J. 90, 109
    Fletcher, R. 34, 37
    Fltigge, W. 216, 240
    Foo, O. 297, 306, 310
    Ford, R . 123, 154, 168
    Fox, D.D. 218, 241; 286, 287; 386, 393, 394
    Fox, R.L. 123, 144, 168; 217, 240; 384, 393
    Fraeijis de Veubeke, B. 123, 132, 149, 150, 156,
    167, 168, 170; 183, 200, 213; 330, 360; 373,
    Guex, L. 144, 157, 158, 170
    Gunderson, K. 144, 162, 170
    Guo, D.J. 308, 311
    Gurtin, M.E. 314, 317, 338, 360; 398, 411
    Haisler, W.E. 72, 107; 144, 162, 170; 255, 265
    Hallquist, J.O. 347, 362; 396, 404, 407, 410
    Hanssen, L 124, 132, 169
    Harou, M. 179, 212
    Harter, R.H. 226, 227, 242
    Hartz, BJ. 385, 393
    Harvey, J.W. 156, 157, 170
    Haug, E.J. 396, 410
    Heegaard, J.-H. 347, 351, 362
    Heege, A. 355, 363
    Hellan, K. 157, 170
    Heller, W.R. 81, 108
    Henshell, R.D. 124, 157, 169, 171
    Herrmann, L.R. 45, 104; 156, 157, 170; 217, 241
    Hestenes, M. 34, 37
    Hibbitt, H.D. 14, 21; 336, 361
    Hildebrand, F.B. 57, 105
    Hill, R. 51, 89, 104
    Hilton, H.H. 48, 104
    Hinton, E. 34, 37; 53, 57, 105; 179, 185, 188, 189,
    190, 191, 212, 213; 286, 287; 297, 310
    Hirschberg, M.H. 77, 107
    Ho, L.W. 144, 169; 185, 196, 213, 214
    Hobbs, R.E. 17, 21; 34, 37
    Hoffstetter, G. 84, 108
    Hogge, M. 28, 36
    Hrabok, M.M. 144, 169
    Hrudey, T.M. 144, 169
    Hsiao, K.-M 390, 394
    Huang, H.C. 93, 96, 109, 110; 188, 189, 190, 191,
    213; 286, 287
    Hughes, T.J.R . 11, 21; 49, 51, 53, 56, 59, 66, 68,
    79, 80, 89, 104, 107; 124. 169; 179, 180, 182,
    184, 185, 188, 200, 212, 213, 214; 226, 229,
    242; 261, 265; 286, 287; 320, 344, 346, 347,
    358, 360, 361, 362
    392
    Francavilla, A. 347, 361
    Frederick, C.O. 53, 105
    Frey, F. 217, 241
    Fried, I. 123, 154, 168
    Fulton, R.E. 124, 169
    Gallagher, R.H. 72, 82, 107, 108; 217, 240, 241;
    244, 265; 386, 393
    Galvanetto, U. 359, 364
    Garcia Orden, J.C. 398, 411
    Gelder, D. 101, 110
    Gellately, R.A. 386, 393
    Geradin, M. 28, 36; 396, 410
    Gere, J.M. 375, 392
    Ghali, A. 297, 309, 310
    Ghionis, P. 34, 37
    Giannini, M. 244, 255, 257, 258, 259, 264
    GiD, CIMNE 430, 431
    Gill, S.S. 75, 107
    Giroux, Y.M. 217, 240
    Glaser, S. 332, 360
    Goel, J.J. 154, 170; 217, 241
    Goicolea, J.M. 398, 411
    Goldstein, H. 404, 412
    Golub, G.H. 69, 106; 407, 412; 413, 430440 Author index
    Huhlhaus, H.B. 91, 93, 109
    Hulbert, G.M. 396, 411
    Humpheson, C. 82, 108
    Hung, H.-C 390, 394
    Huston, R.L. 396, 411
    Ladkany, S.G. 157, 171
    Landers, J.A. 347, 362
    Langhaar, H.L. 373, 392
    Lardeur, P. 196, 214
    Larsson, R. 93, 110
    Laug, E.J. 396, 410
    Laursen, T.A. 347, 355, 359, 362, 363
    Lawson, C.L. 154, 170
    Leckie, F.A. 81, 108
    Ledesma, A. 70, 106
    Lee, E.H. 344, 361
    Lee, S.-H 390, 394
    Lee, S.L. 139, 169
    Lees, M. 14, 21
    Lefebvre, D. 124, 156, 169; 182, 183, 186, 196,
    197, 198, 207, 210, 212
    Leroy, Y. 90, 93, 109
    Leung, K.H. 69, 106
    Lewis, R.W. 14, 21; 63, 82, 106, 108
    Li, W.Y. 307, 308, 310, 311
    Liepins, A. 257, 259, 265
    Lin, F.B. 90, 91, 92, 93, 109
    Lin, J.I. 390, 394
    Lindberg, G.M. 123, 154, 168; 217, 241
    Ling, W. 355, 363
    Lions, J.-L. 79, 80, 107
    Liu, D.T. 344, 361
    Liu, W.K. 185, 213; 286, 287
    Liu, Z.A. 244, 264
    Lo, K.S. 234, 243; 283, 286
    Loden, W.A. 217, 240
    Lohner, R. 34, 37
    Loo, Y.C. 297, 309, 310
    Lubliner, J.L. 49, 51, 53, 71, 104, 105
    Lyness, J.F. 101, 110
    Lynn, P.P. 200, 214
    Lyons, L.P.R. 162, 171
    Ibrahimbegovic, A. 226, 230, 242; 367, 392; 401,
    411
    Idelsohn, S. 28, 36
    Irons, B.M. 34, 37; 123, 124, 139, 144, 148, 149,
    150, 154, 158, 162, 167, 168, 171; 173, 177,
    179, 188, 212; 217, 224, 230, 241, 242; 244,
    264; 266, 274, 275, 280, 284, 286; 385, 386,
    393
    Isenberg, J. 84, 108
    Ishihera, K. 70, 107
    Jean, M. 355, 363
    Jetteur, P. 217, 226, 229, 241, 242
    Jirousek, J. 144, 157, 158, 170, 171
    Johnson, C.P. 157, 171; 217, 232, 240
    Johnson, K.W. 24, 36; 51, 53, 104
    Jones, R.E. 157, 171; 217, 241; 244, 264; 347, 362
    Joo, T. 396, 411
    Jordan, F.F. 257, 259, 265
    Jourdan, F. 355, 363
    Ju, J.-W. 355, 363
    Kabaila, A.P. 385, 393
    Kalker, J.J. 347, 361, 362
    Kanchi, M B. 78, 81, 107
    Kane, C. 347, 363
    Kang, D.S. 396, 411
    Kanoknukulchai, W. 179, 185, 212; 261, 265
    Kapania, R.K. 286, 288
    Kapur, K.K. 124, 136, 139, 169; 385, 393
    Kasper, E.P. 334, 336, 361
    Kawai, T. 384, 393
    Kelsey, S. 156, 157, 170
    Key, S.W. 217, 241
    Khachaturian, W. 84, 108
    Khojasteh-Bakht, M. 244, 255, 264
    Kikuchi, F. 132, 169
    Kikuchi, N. 347, 362
    King, I.P. 39, 56, 73, 84, 104, 105, 107, 108; 124,
    132, 169; 217, 226, 227, 232, 240
    Kirchoff, G. I l l, 167
    Klapka, I . 396, 411
    Klein, S. 244, 248, 264
    Koiter, W.T. 51, 56, 80, 104, 108
    Kosko, E. 123, 154, 168
    Kotanchik, J.J. 244, 264
    Krahl, N.W. 84, 108
    Kraus, H. 257, 260, 265
    Krieg, D.N 57, 105
    Krieg, R.D. 57, 105
    McHenry, D. 48, 104
    Mackerle, J. 347, 362
    McLay, R.M . 157, 171; 217, 241
    McMeeking, R.M. 72, 107
    MacNeal, R . H. 226, 227, 242
    Madenci, E. 390, 394, 395
    Maenchen, G. 58, 105
    Malkus, D.S. 182, 212
    Mallett, R.H. 384, 386, 393
    Malvern, L.E. 3, 20; 220, 242; 345, 361; 398, 411
    Mandel, J. 53, 89, 105, 109; 344, 361
    Mang, H.A. 84, 108
    Mansfield, L. 154, 170
    Manson, S.S. 77, 107
    Manzoli, O. 91, 109
    Marcal, P.V. 14, 21; 73, 78, 107; 336, 361; 384,
    390, 391, 395
    Marcotte, L. 162, 171Author index 441
    Needleman, A. 90, 93, 109
    Nelson, R .B . 308, 311
    Newton, R.E. 385, 393
    Nguyen, Q.A. 79, 107
    Noll, W. 345, 361
    Noor, A.K. 390, 395
    Norris, V.A. 63, 106
    Novozhilov, V.V. 244, 245, 252, 265; 303, 310
    Nygard, M.K. 124, 132, 169
    Marcus, H . 139, 169
    Marguerre, K. 373, 392
    Mark, R. 237, 243
    Marketos, E. 73, 107
    Marsden, J.E. 345, 347, 361, 363
    Martin, D.W. 234, 243
    Martin, H.C. 384, 385, 393
    Martin, J .B. 81, 108
    Martins, R.A.F. 162, 171
    Mason, W.E. 217, 241
    Masud, A. 286, 288
    Matsui, T. 386, 390, 393
    Matsuoka, O. 386, 390, 393
    Matte, Y. 162, 171
    Matthies, H. 28, 29, 31, 34, 36; 415, 416, 430
    Mawenya, A.S. 297, 310
    Mazars, J. 90, 109
    Mazumdax, J. 308, 310
    Meek, J.L. 390, 395
    Mellor, P.W. 51, 53, 104
    Melosh, R .J. 124, 126, 169; 384, 393
    Mendelson, A. 77, 107
    Mescall, J.F. 388, 394
    Meschke, G. 84, 108
    Meyer, C. 308, 311
    Miles, G.A. 244, 255, 258, 259, 264
    Milford, R.V. 286, 287
    Mindlin, R.D. I l l , 167
    Mira, R. 93, 110
    Moita, G.F. 390, 395
    Mooney, M. 341, 361
    More, J. 28, 36
    Morgan, K. 14, 21; 96, 110; 257, 258, 259, 260,
    261, 262, 265
    Morley, L.S.D. 124, 133, 134, 157, 168, 169; 207,
    Oancea, V.G. 355, 363
    Obrecht, H. 308, 311
    Oden, J.T. 162, 171; 336, 347, 361, 362
    Ogden, R.W. 341, 342, 343, 361
    Oliver, J. 91, 109
    Olson, M.D. 123, 154, 168; 217, 241
    Onate, E. 162, 163, 164, 172; 188, 191, 194,
    195, 213, 214; 257, 258, 259, 260, 261, 262,
    265
    Ortiz, M. 90, 93, 109; 347, 363
    Otter, J.H.R. 17, 21; 34, 37
    Ottoscn, N.S. 93, 110
    Owen, D.R.J. 54, 57, 78, 81, 84, 93, 101, 105,
    107, 108, 110; 162, 171
    Pacoste, C. 390, 395
    Paczelt, I. 347, 363
    Padlog, J. 386, 393
    Pagano, N.J. 3, 20
    Pamin, J. 91, 93, 109
    Pande, G.N. 63, 82, 88, 91, 106, 108, 109
    Papadopoulos, P. 188, 191, 195, 196, 200, 211,
    213, 214, 215; 347, 355, 362, 363; 396, 397,
    410
    Papadrakakis, M. 34, 37
    Parekh, CJ. 217, 226, 227, 231, 232, 240, 242
    Parisch, H. 286, 287; 347, 362; 386, 393, 394
    Parks, D.M. 72, 107
    Parlett, B.N. 373, 392; 426, 431
    Pastor, M. 19, 20, 21; 69, 70, 82, 93, 96, 106, 108,
    109, 110
    Paul, D.K. 70, 106
    Pawsey, S.F. 179, 212; 267, 280, 283, 286
    Peano, A. 154, 170
    Penzien, J. 244, 264
    Peraire, J. 93, 96, 110
    Percy J.H. 244, 248, 264
    Peric, D. 93, 110
    Perzyna, P. 79, 83, 107
    Peterson, F.E. 45, 104
    Petocz, E.G. 397, 411
    Phillips, D.V. 84, 108
    Pian, T.H.H. 31, 36; 157, 171; 244, 248, 255, 264;
    334, 361; 391, 395
    Pica, A. 34, 37
    Pietruszczak, S.T. 89, 90, 91, 109
    214
    Morse, P.M. 292, 309
    Mroz, Z. 53, 69, 89, 90, 105, 106, 109
    Mullen, R . 390, 394
    Muncaster, R.G. 396, 397, 410
    Murray, D.W. 384, 385, 393
    Muscat, M. 101, 110
    Naghdi, P.M. 262, 265
    Nagtegaal, J.C. 72, 107
    Nakazawa, S. 69, 106; 124, 169; 182, 183, 212
    Narasimhan, R. 183, 213
    Narayanaswami, R . 124, 168
    Nath, P . 84, 96, 108
    Navaratna, D.R. 244, 248, 255, 264
    Nay, R.A. 162, 163, 171
    Nayak, G.C. 51, 54, 57, 62, 104, 105; 388, 394;
    434, 436
    Naylor, D.J. 63, 106
    Neal, M.O. 347, 362
    Neale, B.K. 157, 171442 Author index
    Pietrzak, G. 355, 363
    Pijaudier-Cabot, G. 90, 91, 92, 93, 109
    Pister, K.S. 4, 21; 39, 45, 46, 104
    Pnezien, J. 244, 264
    Polak, E, 34, 37
    Popov, E.P. 70, 106; 244, 264
    Poster, K.S. 4, 21
    Prager, W. 51, 52, 53, 62, 104, 106
    Prakash, A. 54, 105
    Pramono, E. 90, 109
    Pratt, C. 217, 241
    Press, W.H. 413, 430; 433, 436
    Pugh, E.D.L. 179, 185, 188, 212
    Schmit, L A. 72, 107; 123, 144, 168; 217, 240;
    384, 393
    Schnabel, F. 308, 311
    Schnobrich, W.C. 283, 286, 287
    Schoenberg, IJ. 306, 310
    Schrefler, B.A. 19, 20, 21; 70, 82, 106, 108
    Schwarz, H . R. 24, 36
    Schweizerhof, K. 337, 347, 361, 362; 373, 392
    Scordelis, A.C. 234, 243; 283, 286; 308, 311
    Scott, R.F. 20, 21
    Seiler, C. 308, 311
    Seiss, C.P. 84, 108
    Sen, S.K. 244, 255, 264
    Setlur, A.V. 156, 170
    Shabana, A.A. 396, 407, 410
    Shames, I .H . 3, 20; 59, 106; 220, 242; 314, 317,
    328, 338, 339, 341, 360; 433, 436
    Shapiro, G.S. 166. 172
    Sharma, K.G. 88, 109
    Shewchuk, J. 430, 431
    Shi, G.-H . 397, 411
    Shi, J. 359, 364; 390, 395
    Shiomi, T. 70, 106
    Silvester, P. 101, 110
    Simo, J.C. 4, 11, 21; 25, 36; 49, 51, 53, 56, 58, 59,
    66, 68, 80, 89, 90, 91, 104, 105, 106, 109;
    124, 128, 133, 169; 182, 183, 212; 218, 226,
    241, 242; 286, 287; 320, 328, 332, 337, 338,
    341, 342, 343, 344, 346, 347, 355, 358, 359,
    360, 361, 362, 363, 364; 366, 367, 373, 386,
    392, 393, 394; 396, 399, 401, 404, 410, 411
    Sloan, S.W. 89, 90, 109
    Sluys, L.J. 91, 93, 109
    Smith, I.M. 123, 168
    Sokolnikoff, I.S. 3, 20; 276, 286; 299, 304, 310
    Solberg, J.M. 347, 362; 396, 397, 410
    Southwell, R.V. 124, 169
    Specht, B. 124, 132, 169
    Stebbins, F.J. 308, 311
    Stegun, I.A. 165, 172; 386, 393
    Stein, E. 210, 215; 286, 287; 355, 363; 386, 394;
    404, 411
    Steinmann, P. 93, 110
    Stiefel, E. 34, 37
    Stolarski, H. 217, 241; 286, 287; 390, 394
    Stolarski, H.K. 188, 213, 214; 355, 363
    Strang, G. 28, 29, 31, 34, 36; 415, 416, 430
    Strickland, G.E. 217, 240
    Stricklin, J.A. 72, 107; 144, 162, 170; 244, 255,
    264, 265; 308, 311
    Strome, D.R. 157, 171; 217, 240; 244, 248, 250,
    251, 264; 303, 304, 310
    Sture, S. 90, 109
    Suarez, B. 188, 194, 214
    Sumihara, K. 334, 361
    Suneda, D.K . 101, 110
    Qu, S. 69, 106; 124, 169; 182, 183, 212
    Ralston, A. 24, 29, 36
    Ramm, E. 31, 37; 210, 215; 218, 242; 286, 287,
    288; 332, 337, 360, 361; 386, 393, 394
    Rankin, C.C. 390, 394
    Rao, R.S. 188, 214
    Razzaque, A. 124, 144, 150, 159, 162, 168, 170,
    171; 231, 242; 384, 393
    Reeves, C.M . 34, 37
    Reinsch, C. 415, 425, 430
    Reissner, E. 111, 167; 229, 242; 366, 388, 392,
    394
    Repetto, E.A. 347, 363
    Rice, J.R. 72, 89, 90, 107, 109; 336, 361
    Riesa, J.D. 237, 243
    Rifai, M.S. 218, 241; 286, 287; 386, 394
    Riks, E. 31, 36
    Ristic, S. 390, 395
    Roberts, T.M. 388, 394
    Robichaud, L.P.A. 217, 240
    Rockey, K.C. 385, 393
    Roehl, D. 210, 215; 286, 287; 332, 360
    Rots, J.G. 84, 108
    Rubinstein, M.F. 77, 107
    Rubio, C. 93, 110
    Rudnicki, J.W. 89, 90, 109
    Runcsson, K. 93, 110
    Russell, H.G. 48, 104
    Rybicki, E.F. 72, 107
    Sabir, A. 217, 241
    Sackman, J.L. 347, 362
    Sacks, S. 58, 105
    Saigal, S. 286, 288
    Sakurai, T. 56, 105
    Samuelsson, A. 124, 132, 169
    Sander, G. 123, 149, 157, 167
    Sanders, J.L.(Jr.) 257, 259, 265
    Scapolla, T. 120, 145, 167; 208, 214
    Scharpf, D.W. 123, 154, 168; 401, 411
    Schmidt, R. 371, 392Author index 443
    Sved, G. 308, 310
    Swaddiwudhipong, S. 308, 310
    Szabo, B.A. 347, 363
    Szabo, T. 347, 363
    Szilard, R . 376, 392
    von Mises, R. 51, 104
    von Reisman, W. 72, 107
    Vu Van, T. 355, 363
    Vu-Quoc, L. 25, 36; 367, 392
    Walters, D. 124, 169
    Walz, J.E. 124, 169
    Warburton, G.B. 124, 169
    Wasfy, T.M. 390, 395
    Washizu, K. 4, 21; 244, 245, 262, 265
    Watson, M. 39, 48, 104
    Webster, JJ. 217, 241; 244, 256, 264
    Wehage, R.A. 396, 410
    Weikel, R.C. 217, 240
    Wempner, G.A. 162, 171
    Westergaard, H.M. 433, 436
    White, J.L. 39, 104
    Wilkins, M.L. 58, 105
    Wilkinson, J.H. 415, 425, 430
    Will, G. 217, 241
    Wiliam, K.J. 90, 93, 109, 110
    Wilson, E.L. 149, 170; 203, 214; 226, 242; 249,
    265; 302, 310; 336, 361; 384, 385, 393; 415,
    425, 431
    Winnicki, L.A. 63, 106; 308, 311
    Winslow, A.M. 101, 102, 110
    Withum, D. 154, 170
    Witmer, E.A. 244, 264
    Wittaker, E.T. 401, 411
    Woinowski-Krieger, S. I l l, 113, 121, 167; 216,
    240; 244, 245, 252, 257, 265
    Wong, K. 396, 399, 401, 410
    Wood, R.D. 124, 134, 168; 314, 317, 320, 328,
    333, 338, 339, 345, 360; 372, 384, 388, 392;
    398, 411
    Wriggers, P. 332, 337, 338, 347, 350, 351, 355,
    360, 361, 362, 363; 373, 392
    Wu, J. 96, 110
    Wunderlich, W. 308, 311
    Tabbara, M. 91, 109
    Tadros, G.S. 297, 310
    Tahar, M.B. 144, 170; 231, 236, 242
    Tarnow, N. 359, 363, 364; 367, 386, 392, 394
    Tatsueka, T. 70, 107
    Taylor, R.L. 3, 4, 20, 21; 39, 45, 46, 53, 58, 59,
    66, 68, 69, 71, 73, 79, 88, 104, 105, 106, 107,
    108; 124, 128, 133, 169; 179, 182, 183, 185,
    188, 191, 194, 195, 196, 199, 200, 203, 211,
    212, 213, 214, 215; 226, 229, 242; 261, 265;
    267, 270, 274, 278, 280, 283, 286; 328, 329,
    332, 334, 336, 337, 338, 341, 342, 344, 347,
    355, 360, 361, 362, 363; 373, 392; 413, 430;
    434, 436
    Tessler, A. 200, 214; 390, 395
    Teter, R.D. 78, 107
    Tezduyar, T. 188, 213
    Tham, L.G. 307, 308, 310, 311
    Theocaris, P.S. 73, 107
    Theron, NJ. 396, 411
    Thomas, G.R. 217, 241
    Timoshenko, S.P. 3, 20; 111, 113, 121, 167; 216,
    240; 244, 245, 252, 257, 265; 302, 310; 375,
    392
    Tisdale, P. 144, 162, 170
    Tocher, J.L. 123, 124, 136, 139, 144, 167, 169;
    217, 240
    Tong, P. 31, 36; 157, 171; 391, 395
    Tong, R. 157, 171
    Too, J . 179, 185, 212; 267, 270, 274, 278, 280,
    283, 286; 295, 308, 309
    Torbe, I. 157, 171
    Toyoshima, S. 182, 212
    Truesdell, C. 345, 361
    Tsay, C.-S. 390, 394
    Tu, T. 198, 214
    Tuba, I.S. 347, 361
    Turner, M.J. 384, 393
    Xie, Y.M. 70, 106
    Xu, Z. 185, 186, 197, 198, 200, 206, 213, 214
    Yamada, Y. 56, 105
    Yang, H.T.Y. 286, 288; 386, 393
    Yang, T.Y. 388, 394
    Yishimura, N. 56, 105
    Yoshimura, N. 384, 393
    Yu, J. 93, 110
    Utku, S. 162, 163, 171; 217, 240
    Vahdati, M. 96, 110
    Valliappan, S. 56, 73, 84, 86, 105, 108
    Van Loan, C.F. 69, 106; 407, 412; 413, 430
    van Randen, Y. 347, 361
    Ventura, G. 347, 363
    Vilotte, J.P. 182, 212
    Visser, W. 123, 154, 156, 168, 170
    Vlachoutsis, S. 286, 287
    Volker, R.E. 101, 110
    Zanisek, A . 154, 170
    Zarate, F. 162, 163, 164, 172
    Zarka, J. 79, 107
    Zavarise, G. 332, 351, 360, 363
    Zeng, L.F. 185, 198, 200, 206, 213
    Zhong, H.G. 390, 395444 Author index
    Zhong, W.X. 183, 213
    Zhong, Z. 347, 362
    Zhu, D.S. 308, 311
    Zhu, J.Z. 94, 110; 211, 215
    Ziegler, H. 52, 104
    Zienkiewicz, O.C. 3, 4, 14, 15, 19, 20, 20, 21; 31,
    34, 36, 37; 39, 48, 51, 53, 56, 57, 62, 63, 69,
    70, 73, 77, 78, 79, 81, 82, 84, 88, 91, 93, 94,
    96, 101, 102, 103, 104, 105, 106, 107, 108,
    109, 110; 123, 124, 126, 128, 132, 133, 139,
    144, 148, 156, 167, 168, 169, 170; 173, 177,
    179, 182, 183, 185, 186, 188, 191, 194, 195,
    196, 197, 198, 200, 207, 210, 211, 212, 213,
    214
    , 215; 217, 224, 226, 227, 231, 232, 240,
    242; 244, 257, 258, 259, 260, 261, 262, 264,
    265; 266, 267, 270, 274, 275, 278, 280, 283,
    284, 286; 295, 297, 302, 308, 309, 310, 311;
    347, 361; 371, 385, 386, 388, 392, 393, 394;
    413, 430; 434, 436Subject index
    Acceleration of convergence 30
    Accuracy, axisymmetric shells 244
    Adaptive analysis, plastic flow deformation 97
    Adaptive refinement 93, 211
    based on energy norm error estimates 93
    Additional enhanced mode 248
    Additional plastic potentials 88
    Aerospace structures 244
    Ageing effects 42
    Algebraic non-linear solutions, use of 8
    Algorithms:
    BFGS 414, 419
    GN22 13, 419, 424
    line search computer programs 414
    line-search 414
    Newton-type 415, 416, 417
    return map 58
    return map implicit 80
    Almansi strain tensor 316
    Alternative refinement using error indicators 95
    Analogies:
    McHenry-Alfrey 48
    solution by 48
    Angular momentum 399
    balance of 4
    Approximations:
    curved shells, polygonal shape 249
    displacement approximation, plates 121
    finite element 163, 322
    finite element virtual displacements 5
    finite elements displacements 5
    Galerkin 156
    Galerkin weighting 156
    Herrmann 156
    pressure 11
    quadratic displacement 162
    shells, flat element 217
    thick plates 180
    volumetric strain 11
    Arc-length controls 31
    Arch, clamped-hinged 371
    Arch dam , analysis example 231
    Associative plasticity 52, 60
    Assumptions:
    Kirchhoff’-Love 245
    Reissner-Mindlin 266
    thick and thin plates 114, 115
    Asymptotic rates 34
    Augmented Lagrangian form 351
    Axisymmetric body, torsion problem 300
    Axisymmetric shells 217, 244, 387
    curved , thick 275
    with non-symmetrical load 303
    Axisymmetric solids, with non-symmetrical load
    297
    Barrel vaults:
    analysis example 234
    problem 230
    Beams:
    with attached mass, numerical example 409
    beam functions 153
    definition 1
    deformed configuration 367
    large displacement theory of 365
    one-dimensional example 159
    stress resultants 113
    two-dimensional theory of 366
    Behaviour:
    deformation of solids 137
    elasto-plastic 343
    geometric non-linearity 1
    inelastic material 164, 210
    inelastic shells 279
    laminates 88
    limit loads 74
    material non-linearity 1
    non-linear elastic 48
    path dependent 35
    plastic 48446 Subject index
    Behaviour – emit.
    softening behaviour 31
    soil 83
    solids, non-linear:
    geometric 1
    material non-linearity 1
    small deformation problems 3
    symmetric stress 164
    Bending strains, plates 117
    Bergan mcthod/proccdurc 32
    BFGS computer program algorithm 414, 419
    BFGS update 28
    Bifurcation 22
    instability 384
    Bilinear elements 179
    Blending function mesh generation 358
    Boundary conditions 291
    constrained 184
    displacement 318
    fixed 119
    mixed 120
    plates 116
    simply supported prism 295
    Boundary constraints 187
    Boundary integral terms 132
    Boundary layers 250
    Boundary traction 120
    Bounding surface plasticity models 70
    Box structures, thin membrane 295, 297
    Boxes, with flexure 296
    Branching shell 260
    Bridge, railway 138
    Bridge with beams, skewed, curved 137
    Brittle materials, special problems of 84
    Broyden method 34
    Bubble modes 187. 196, 198
    Buckling 1
    Euler 374
    with large deflections 385
    relation to discrete Kirchhoff formulation 159,
    194
    for triangular elements 194
    Components, skew-symmetric stress 229
    Composites, plates 118
    Computed central deflection, square plates 136
    Computer contour plot 139
    Computer procedures, finite element analysis 413
    Computer programs:
    FEAPpv 413, 418, 425, 428, 429
    GID 430
    Concentrated discontinuities 91
    Condensation, static 249
    Condensation process 203
    Conical frustra 246, 247
    Conservation of linear momentum 399
    Consistency tests, irreducible forms 186
    Consistent tangent matrix 66
    Constant isotropic hardening modulus 52
    Constant strain 132
    Constitutive law 84
    Constitutive models 9, 166, 333
    Constitutive relations, plates 118
    Constrained boundary conditions 184
    Constrained elements 155
    Constraints:
    boundary 187
    discrete collocation 187, 192
    discrete Kirchhoff 162
    Kirchhoff 158
    Kuhn —Tucker form 51
    Lagrange multiplier 405
    patch count and discrete collocation 188
    rotational 406, 407
    translation 405
    triangles/triangular elements 194
    see also Collocation constraints: Joints
    Contact:
    frictionless 352
    Hertzian 348
    node-node 348
    Contact problems 313, 347
    Continuity:
    Cu requirements, plates 156, 159, 177, 182, 188,
    262
    Ci requirements, plates 111, 122, 145, 261
    cross derivatives 123
    difficulty avoidance 155
    patch tests 135
    Continuum problems 314
    Continuum rate formulation 55
    Convective heat transfer coefficient 14
    Convergence:
    criteria for 8, 33
    with Newton-Raphson method 25
    non-conforming elements 124
    Cantilever, propped, Euler buckling 374
    Cartesian and curvilinear coordinates 269
    Cartesian tensor form 3, 4
    Cauchy stress 316, 325, 329, 339, 342, 345
    Cauchy-Green deformation tensor 316, 338, 342
    Circular bar, necking of 357
    Circular plate, bending 262
    Clamped edges, spherical dome 257
    Clamped hinged-arch 371
    Clamped plates 136, 139, 175
    Classical time-dependent plasticity theory 48
    Closed loops, and joints 407
    Clough and Toucher triangles 148
    Collocation constraints:
    element matrices 192
    quadrilaterals 187Subject index 447
    and plate and shell problems 278
    rate of 145
    study 263
    Cooling tower, analysis example 233
    Coordinates:
    cartesian and curvilinear relation 269
    curvilinear 128, 244, 268
    local and global 220
    use of 3
    Corner nodes 124
    Comer supported square plate 136
    Cosines:
    direction 221, 224
    normal direction 225
    Crash analysis 18
    Creep, in metals 80
    Creep compliance 40
    Creep law, Norton-Soderberg 81
    Creep problems:
    basic formulation of 75
    fully explicit solutions 77
    Creep strain 78
    Criterion, constant strain 132
    Cross derivatives 123
    Cubic axisymmetric shell elements 277
    Cubic elements 179
    Cubic expansion 129
    Cubic functions 130
    Cubic Hermitian interpolations 296
    Cubic interpolation 161
    Cubic lagrangian interpolation 188, 286
    Current configuration 312
    Current configuration formulation 324
    Current deformed configuration 314
    Current stiffness parameter 35
    Curved dams 283
    Curved elements 244, 251
    properties 254
    shape functions for 252
    strain expressions for 254
    Curved, isoparametric hexahedra 267
    Curved shells 267
    polygonal shape approximation 249
    Curvilinear coordinates 128, 244, 268
    relation with cartesian 269
    Curvilinear elements 262
    Cylinder:
    edge loaded 282
    thick-walled, subject to internal pressure 46
    Cylindrical billet, upsetting of 356
    Cylindrical shells 250, 281
    Cylindrical vault 282
    Definitions, strains and stresses 271
    Deformation:
    behaviour of solids 137
    deformation analysis 349
    deformation gradient 315, 316, 321, 332. 344
    polar decomposition of 398
    deformation tensors 316, 328
    Cauchy-Green 338
    equations for 314
    finite 312
    process of 319
    thin case axisymmetric shell 303
    thin plates 112
    Deformed configuration 312
    Degrees of freedom see DOF
    Delpak elements 260
    Delta weighting, Dirac 189
    Deviatoric term 42
    Differential equations:
    forth order 112
    model 41
    Dirac delta weighting 189
    Direct elimination 159
    Direct iteration 29
    Direct iterative methods 34
    Direction cosines 221, 224
    Discontinuity capture 95
    Discontinuity and slip lines 95
    Discontinuous deformation analysis (DDA) 397
    Discontinuous S interpolation 182
    Discrete collocation constraints 1S7, 192
    Discrete exact thin plate limit 202
    Discrete Kirchhoff constraints 158
    Discrete Kirchhoff theory ( DKT) elements 161,
    188, 194. 196
    Discretization 4, 71, 121, 176, 202
    error of 33, 216
    Discretized equilibrium equation 77
    Displacement approximation equations, plates
    121
    Displacement boundary conditions 318
    Displacement boundary restraints 73
    Displacement control 31
    Displacement field 270
    Displacement formulation:
    plates 122
    use of 9
    Displacement functions 124, 305
    multiplicative type 305
    Displacement increment 166
    Displacement-based finite element models 327
    Displacements:
    finite element approximations to 5
    global 276
    global nodal 254
    tangential 256
    D’Alembert principle 323
    Damping effects, with GN22 algorithm 424
    Dams, curved 283448 Subject index
    Distributed loads 294
    DKT (discrete KirchholT theory) elements 161,
    188, 194, 196
    Delpak 260
    discrete collocation constraints 187, 192
    discrete KirchholT triangular plate element 226
    eight-node serendipity 286
    enhanced mode 196
    equilibrium 157
    failure of 183
    finite 251, 329
    matrix notation 334
    method for shells 217
    geometric definition of 268
    heterosis 180
    Hinton-Huang 191
    hybrid 157
    integration 184
    isoparametric quadratic 75
    isoparametric three dimensional 174
    joints 86
    Kelvin 40
    linear beam and plate 261
    linear triangular 198
    with mid-side slope connections only 230
    mixed 155
    use of 72
    parabolic-type 297
    parallelogram 128
    Pian-Sumihara 199, 334
    plane element stiffness 218
    plate 157
    plate bending 183
    quadratic fully integrated (QS) 179
    quadratic of Huang and Hinton 189
    quadratic T6S3B3 196, 369
    quadrilateral 128, 148, 201
    quadrilateral mixed 185
    quartic and quintic 154
    quasi-conforming 150
    rectangular with collocational constraints 192
    rectangular quadrilateral 209
    rotation-free 162
    rotational bubble modes 196
    serendipity eight-node 286
    serendipity quadratic 179
    shell properties and transformations 272
    shells with displacement and rotation 266
    smoothed 150
    strain and virtual strain in 10
    superparametric 271
    thick plate theory limitations 203
    transformation to global coordinates 219
    Trefftz-hybrid 158
    see also Finite clement; Triangles/triangular
    elements
    ELMTnn computer element module 413
    Elongation strain 91
    Embankment under gravity 83
    DOF:
    6 133, 225
    9 128, 144
    12 124, 144
    16 144
    18 148, 153
    21 124, 144, 153
    Dome spherical 257
    under uniform pressure 281
    Drilling rotational stiffness 225
    Drucker-Prager yield conditions 62
    Dynamic buckling, computer programs for 428
    Dynamic relaxation methods 34
    Earthquake response, soil structures 19
    Edge loaded cylinder 282
    Eigenfunctions 306
    Eigenproblem 342
    Eigensolutions, computer programs for 425
    Eight-node serendipity element 286
    Elastic bulk modulus 41
    Elastic calculations 87
    Elastic constitutive equations, rate form 345
    Elastic deformation in joints 407
    Elastic material, isotropic 300
    Elastic moduli, isotropic material 340
    Elastic modulus matrix 61
    Elastic shear modulus 42
    Elastic stability, energy interpretation 373
    Elasticity:
    isotropic, and finite deformation 338
    isotropic for plates 118
    three-dimensional 155
    as a variational problem 320
    Elasticity matrix 55, 245, 272
    Elasto-plastic behaviour 343
    Elasto-plastic deformations:
    localization in 88
    non-uniqueness in 88
    Elasto-plastic matrix 55
    Elasto-plastic modulus matrix 56
    Elasto-plastic slope 99
    Elasto-plastic solutions 249
    Elasto-viscoplastic material 78
    Elements:
    assembly of 219
    axisymmetric shell 246
    bilinear 179
    choice of 230
    constrained 155
    cubic 179
    curved 244, 251
    curvilinear 262Subject index 449
    Energy:
    constrained potential principle 193
    minimum potential 162
    transverse shear 189
    Energy function, stored, Saint-Venant-Kirchhoff
    model 320
    Energy interpretation , elastic stability 373
    Energy norm 144
    Energy norm error 95
    estimates and adaptive refinement 93
    Enhanced modes 196
    Enhanced strain methods 332
    Enhanced strain mode 248
    Enthalpy 14
    Equations:
    deformation 314
    differential 41, 112
    discretized equilibrium 77
    displacement approximation, plates 121
    elastic constitutive, rate form 345
    equilibrium 3, 6, 163, 225, 317, 371
    Euler, for static equilibrium 321
    finite element convergence 60
    finite elements, plates 121
    fourth-order differential 112
    Galerkin (variational) 10
    inelastic constitutive 12
    integral model 43
    kinematics and deformation 314
    non-linear algebraic 22
    plate equilibrium 115
    plate problem 113
    Prandtl-Reuss 63
    of rigid body motion 399
    singular systems 182
    strain-displacement, mixed form 11
    stress measures 316
    tangent system 32
    traction measures 317
    virtual work 5
    Equilibrating stress field 157
    Equilibrium, weak form for 4
    Equilibrium elements 157
    Equilibrium equation 3, 163. 225, 317, 371
    residual form 6
    Equivalence, discontinuous S interpolation 182
    Erratic yielding 249
    Error indicators, alternative refinement using
    FEAPpv computer program 413, 418
    and eigensolutions 425
    example problem solutions 429
    restart option 428
    and static and steady-state problems 418
    Field, displacement 270
    Field problems, non-linear quasi-harmonic 12
    Fine element solution 377
    Finite angle solution 373
    Finite deformation 1, 312
    material constitution for 338
    mixed-enhanced 332
    plasticity model 358
    problems 313, 319
    three-field mixed 328
    variational description 319
    Finite elasticity, variational theorem for 320
    Finite element analysis, computer procedures
    413
    Finite element approximations 163, 322
    Finite element contact 348
    Finite element discretization 4
    Finite element equation convergence 60
    Finite element equations, plates 121
    Finite element evaluation , strain-displacement
    matrices 378
    Finite element forms, use of 9
    Finite element formulation 326
    Finite element method 22
    Finite element model, construction from 400
    Finite element processes 289
    Finite elements in hemispherical shells 251
    Finite precision arithmetic 33
    Finite strip method 289, 297
    incomplete coupling 305
    semi-analytical 305
    spline finite strip 305
    First order problem solution using GN11 420
    Fixed boundary conditions 119
    Flexible bodies, connected to a rigid body 402
    Flow rule ( normality principle ) 51
    Fluid dynamics, high speed 95
    Folded plate structure, analysis example 237
    Forces:
    external 127
    in-plane 244
    membrane 244
    nodal 127, 337
    pressure load 336
    Forces dependent on deformation-pressure loads
    95
    Errors, from incomplete solutions 33
    Euler buckling, propped cantilever 374
    Euler column 1
    Euler equations, for static equilibrium 321
    Euler ( explicit) method 79
    Euler implicit schemes 36
    Evaluation, tangent matrix 379
    336
    Formulae:
    Gauss-Lobatto-type 165
    Gaussian quadrature 274
    recursion 46
    Sherman-Morrison-Woodbury 69450 Subject index
    Formulations:
    continuum rate 55
    current configuration 324
    displacement, plates 9, 122
    finite element 326
    Saint-Venant-Kirchhoff model 320
    stress 302
    substitute shape 150
    viscoelastic relaxation 46
    yield 50
    free of locking 330
    irreducible 112, 173, 176
    isoparametric 255
    isotropic elasticity 338
    Kirchhoff 194
    large displacement with small rotations
    Galerkin form 319, 353
    Galerkin method 321
    Galerkin procedures 13, 181
    Galerkin Process 176
    Galerkin statement, variational 137
    Galerkin (variational) equations 10
    Galerkin weighting approximations 156
    Gauss points 183, 194, 204, 262, 278
    quadrature 160, 248, 274
    Gauss-Lobatto-type formulae 165
    Gaussian integration 180
    General constitutive models 9
    Generalized Maxwell model 41, 43, 44
    Generalized Maxwell solid 39
    Generalized plasticity:
    forms 69
    J2 model 71
    non-associative case 68
    Geometric definition of elements 268
    Geometric matrix 383
    Geometric modelling 348
    Geometric non-linearity behaviour 1
    Geometric stiffness 3, 328
    Geometric tangent 332, 371
    Geometric vector algebra 223
    Geometrically exact specialization 367
    Geometrically non-linear problems 312
    GID computer program 430
    Global displacements 276
    Global and local coordinates 220
    Global nodal displacements 254
    Global strains 276
    GN 11 and Newmark (GN22 ) algorithms 13, 419
    GN22 method 6, 13, 419, 424
    Green strain matrices 321
    Green strain tensor 316
    Green-Lagrange strain 368, 369, 376, 381, 387
    non-zero 370
    370
    mixed 155, 173, 180, 332
    non-associated 89
    non-linear 6, 379
    plates displacement 122
    reference configuration 320
    reference configuration for 320
    small rotations, large displacement 370
    steady-state non-linear problems 6
    thick plates 180
    three-field mixed finite deformation 328
    use of 9
    Foundation with eccentric loading 100
    Fourier components/series 290, 292, 294
    Fourier expansion 297
    Frames, definition 1
    Freezing moist soil 14
    Frictional materials, non-associative case 68
    Frictionless conditions 348
    Frictionless contact 352
    Frustra, conical 246, 247
    Functions:
    beam 153
    blending mesh generation 358
    conforming shape 148
    cubic 130
    displacement 124, 305
    eigenfunctions 306
    energy, stored, Saint-Venant-Kirchhoff model
    320
    Hermitian rectangular shape 151
    Huber—von Mises yield 62, 67
    internal shape 256
    least square substitute shape 151
    linking, derivation of 200
    mesh generation, blending 358
    non-conforming shape 124
    orthogonal 289
    penalty 261, 350
    serendipity 201
    shape 112, 124, 129, 145, 208, 290
    shape for curved elements 252
    shear relaxation modulus 43
    spline 306
    stored energy 340
    Hardening:
    hardening/softening rules 52
    isotropic 51, 52
    kinematic 51, 52, 55
    non-associative 53
    plastic material 48
    Heat conduction, transient 14
    Hellinger-Reissner principle 156
    Hemispherical shells 251
    Hermitian interpolations 247, 253
    Hermitian polynomials 296Subject index 451
    Hermitian rectangular shape function 151
    Herrmann, approximations 156
    Hertzian contact 348
    High-speed fluid dynamics 95
    Hinton-Huang elements 191
    History dependent paths 48
    Homogeneous fields 157
    Homogeneous motion 397
    Homogeneous plates 158
    Hoop stresses 284
    Hu-Washizu functional 162
    Huber-von Mises flow rule 72
    Huber-von Mises yield conditions/function 62, 67
    Huber-von Mises-type materials 166
    Hybrid elements 157
    Hyperelastic material 313
    Hyperelasticity 338
    linked 199, 229
    membrane 227
    Invariants:
    first derivatives of 434
    moment 433
    second derivatives of 435
    Invariants of second-order tensors 432
    Irreducible forms:
    consistency tests for 186
    use of 9
    Irreducible formulations 112, 173, 176
    Irreducible systems 175
    Isoparametric concepts 208
    Isoparametric formulation 255
    Isoparametric hexahedra, curved 267
    Isoparametric mapping 203
    Isoparametric quadratic elements 75
    Isoparametric quadrilaterals 81
    Isoparametric shell elements 285
    Isoparametric three dimensional elements 174
    Isoparametric transformations 262
    Isotropic elasticity:
    elastic material 300
    formulation in invariants 338
    formulation in principal stretches 338
    plates 118
    Isotropic elasto-plastic material forms 313
    Isotropic hardening 51, 52, 54. 63
    Isotropic homogeneous plate 380
    Isotropic materials 61, 179, 272
    Isotropic models 40
    Isotropic plasticity models 61
    Isotropic yield surfaces 61
    Iteration:
    direct (or Picard) 29
    Newton-Raphson 166, 327, 351
    use of 8
    Iteration number 24
    Iterative correction 24
    Iterative processes, use of 7
    Iterative solution scheme
    , viscoelasticity 80
    Iterative techniques 23
    Identity matrix 28
    Impenetrability condition 347
    In-plane forces 244
    In-plane stress action 218
    Increment size 35
    Incremental return map form 65
    Incremental-secant method 26
    Independent slope-displacement interpolation 261
    Induction time 15
    Inelastic behaviour, shells 279
    Inelastic constitutive equations 12
    Inelastic material behaviour 164, 210
    Inelastic materials 38
    Initial conditions 319
    Initial stress matrix for plate bending 381
    Initial stress method 26
    Initial stress process 84
    Initial stress stiffness 3
    Instability, bifurcation 384
    Integral equation model 43
    Integration:
    Gaussian 180
    numerical 274
    reduced (selective) 182
    through-thickness 165
    Integration elements 184
    Integration order 278
    Internal shape functions 256
    Interpolations:
    boundary-type 189
    cubic 161
    cubic Hermitian 296
    cubic lagrangian 188, 286
    four-term 199
    Hermitian 247, 253
    independent slope-displacement 261
    Lagrangian 160
    linear 247
    Ji model:
    with isotropic and kinematic hardening 63
    and volumetric plastic strain 68
    J2 plane stress 66
    Jacobian matrix 26, 273
    computer program for 416
    Jacobian transformations 199
    Jaumann-Zaremba stress rate form 346
    Joint elements 86
    Joints:
    and closed loops 407
    coupling multibodies 404
    revolute 407452 Subject index
    Joints – cont.
    single rotational constraint 407
    single translational constraint 407
    spherical 407
    with rotational constraints 406
    with translation constraints 405
    Linear viscoelasticity 39
    model for 45
    Linearization 327, 350, 354, 380
    Linked interpolation 199, 229
    Linking function, derivation of 200
    load bifurcations 1
    Load correction matrix 336
    Load matrix 127, 133, 134
    Loading, axisymmetric 245
    Loading factor expansion 291
    Local direction cosines 221
    Local and global coordinates 220
    Localization , in elasto-plastic deformations 88
    Localization (slip line) capture 93
    Kelvin element/model/solid 40
    Kinematic boundary conditions 321
    Kinematic hardening 51, 52, 55, 63
    Kinematic hardening back stress 67
    Kinematic hardening relations 66
    Kinematics equations 314
    Kirchhoff constraints 158
    Kirchhoffplates 111
    Kirchhoff stress 316, 317, 325, 329, 342, 343, 344,
    345, 358
    Kirchhoff triangular plate bending element 226
    Kirchhoff—Love assumption 245
    Koitr treatment for multi-surface plasticity 80
    Kronecker delta quantities 315
    Kuhn-Tucker constraint form 51
    McHenry-Alfrey analogies 48
    Magnetic response, and non-linear quasiharmonic field problems 101
    Mapping, isoparametric 203
    Martices, strain displacement 378
    Master nodes 348, 351
    Material:
    behaviour of 1, 164.210
    brittle 84
    constitution for finite deformation 338
    elasto-viscoplastic 78
    finite deformation 338
    frictional 68
    Huber-von Mises-type 166
    hyperelastic 313, 320
    inelastic 38, 164, 210
    isotropic 61, 179, 272, 300, 313, 340
    laminar 86
    Maxwell 43
    neo-Hookean 341, 357
    no-tension 84
    non-linear 1 , 38
    plastic, hardening/softening of 48
    Matrices:
    elastic modulus 61
    elasticity 55, 245, 272
    elasto-plastic modulus 56
    element for discrete collocation constraints 192
    finite elements notation 334
    geometric 383
    Green strain 321
    identity 28
    and index form 5
    Jacobian 26, 273, 416
    linear stiffness 6
    load 127, 133, 134
    load correction 336
    Newton method 30. 31
    orthogonal 366
    Piola-Kirchhoff stress 321
    plate bendinginitial stress 381
    Lagrange multiplier constraint 405
    Lagrange multiplier form 350, 353
    perturb of 406
    Lagrange multipliers 398, 402
    Lagrangian form, augmented 351
    Lagrangian interpolation 160, 188, 286
    Lagrangian quadratics (QL ) 179
    Lagrangian variables 113
    Laminar behaviour 88
    Laminar materials 86
    Laminar structures 308
    Laplace transform theory 40
    Large deflection problems, solution of 383
    Large displacement formulation with small
    rotations 370
    Large displacement theories:
    beams 365
    thick plates 375
    thin plates 381
    Least square substitute shape functions 151
    Legendre polynomial 256
    Limit load behaviour 74
    Limit plastic state 71
    Line search computer programs 416
    algorithm for 414
    Line search procedures 30
    Linear axisymmetric shell elements 277
    Linear elasticity 6
    Linear interpolations 247
    Linear momentum 399
    balance of 4
    Linear stiffness matrix 6
    Linear triangular element (T3S1B1 ) 198, 200Subject index 453
    stiffness 127, 128, 131, 133, 134, 219, 221, 248,
    272, 293, 300
    strain-displacement 5, 182, 378
    substitute shear strain 194
    tangent 66, 337, 379, 383, 416
    velocity 399
    Matrix form, relation with index form 5
    Maximum plastic dissipation principle 53
    Maxwell material 43
    Maxwell model 39
    generalized 41, 43, 44
    Membrane box structures 295
    Membrane forces 244
    Membrane interpolations 227
    Membrane locking 286
    Membrane stresses 250
    Mesh density 140, 142, 143
    Mesh generation, blending function 358
    Metals, creep of 80
    Mid-side nodes 146
    Minimum potential energy 162
    Mixed elements 155
    use of 72
    Mixed forms, use of 9
    Mixed formulations 155, 173
    Mixed patch test 230
    Mixed variational principle 162
    Mixed-enhanced finite deformation formulation
    Multiaxial stress 49
    Multibodies, coupled by joints 404
    Multiple branches 22
    Multiple solutions 23
    Multipliers, penalty 261
    Necessary requirements 184
    Necking of a circular bar 357
    Neo-Hookean models/materials 341, 357
    Neutral equilibrium, and stability 374
    Newmark procedure 6, 419
    Newton matrix method 30, 31
    Newton-Raphson iteration 35, 59, 102, 166, 327,
    351
    Newton-Raphson method 24, 79
    modified 26, 84
    Newton-Raphson process 59, 336, 379, 392
    negative features of 25
    Newton-Raphson scheme 78, 331, 335, 346, 359
    Newton-Raphson solution 313, 324, 326, 339,
    345, 352, 354, 369, 403
    Newton-type computer algorithms 415, 416, 417
    Newton-type solution 46
    strategy for 42
    Newton’s method see Newton-Raphson method
    No-tension material 84
    Nodal circles 246
    Nodal displacement 218
    Nodal forces 127
    and tangent matrix terms 337
    Nodal load vector 127
    Nodal parameters 123, 192
    Nodal rotation 255
    Nodal values 188
    Nodal variables 134
    Node transforms 220
    Node-node contact 348
    Node-surface contact/treatment 351
    between discs 355
    Nodeless variables 255
    Nodes:
    master 348, 351
    slave 348, 351
    Non-associated formulations 89
    Non-associated plasticity 91
    Non-associative case:
    frictional materials 68
    generalized plasticity 68
    Non-associative hardening 53
    Non-associative plasticity 52, 60
    Non-linear algebraic equations 22
    Non-linear behaviour of solids:
    geometric non-linearity 1
    material non-linearity 1
    small deformation problems 3
    Non-linear elastic behaviour 48
    332
    Modelling, geometric 348
    Models:
    neo-Hookean 341
    resultant constitutive 166
    Modes:
    additional enhanced 248
    bubble 187, 196, 198
    enhanced 196
    enhanced strain 248
    rigid body 130
    vibration 428
    Modified modulus term 12
    Modified Newton-Raphson method 26, 34, 84
    Moduli:
    direct elastic 114
    shear elastic 114
    Young’s 272
    Mohr-Coulomb surface 56, 89
    Mohr-Coulomb yield conditions 62
    Moist soil, freezing 14
    Moment invariants 433
    Momentum, linear and angular 399
    Motions:
    pseudo-rigid 396
    rigid 398
    Multi-degree-of-freedom systems 30
    Multi-surface plasticity, Koiter treatment 80454 Subject index
    Non-linear formulation, transient and steadystate problems 6
    Non-linear materials 38
    Non-linear problems, computer program
    solutions 415
    Non-linear quasi-harmonic field problems 12, 101
    Non-linear structural problems 365
    Non-local approach 92
    Non-uniqueness 23, 90
    in elasto-plastic deformations 88
    Normal direction cosines 225
    Normal slopes 146
    Normality principle (flow rule) 51
    Norton-Soderberg creep law 81
    Numerical integration 165, 274
    Numerical patch tests 186
    Picard iteration 29
    Piola-Kirchhoff stress 317, 320, 328, 333, 339,
    345, 368, 369
    Piola-Kirchhoff stress matrix 321
    Pipe penetration 284
    Plane element stiffness 218
    Plane strain solutions, perforated plate 73
    Plane stress 165
    J2 66
    Plane stress solutions, perforated plate 72
    Plastic behaviour 48
    Plastic computation , examples of 71
    Plastic correction 58
    Plastic deformation, sustained 51
    Plastic deviatoric strain rates 64
    Plastic flow rule potential 51
    Plastic localization calculation 96
    Plastic material, hardening/softening of 48
    Plastic mechanisms 93
    Plastic potentials, additional 88
    Plastic strain rate 51, 54
    Plastic stress-strain relations 54
    Plastic yield surfaces 87
    Plasticity:
    associative 52, 60
    constitutive model construction 54
    non-associated 91
    non-associative 52, 60
    rate form equations 64
    time-independent theory 48
    upper and lower bound theorems 52
    Plasticity models 343
    finite deformation 358
    isotropic 61
    Plate bending, initial stress matrix for 381
    Plate elements 157
    Plates:
    bending 111
    bending elements 183
    bending stiffness 114
    bending strains 117
    bending triangles 151
    boundary conditions 116
    circular, bending 262
    clamped 175, 384
    composites 118
    cylindrical bending 113, 114
    definition 1
    discrete exact thin limit 202
    displacement formulation 122
    equilibrium equations 115
    with flexure 296
    governing equations 113
    homogeneous 158
    hybrid 157
    in-plate plate stiffness 114
    One-dimensional beam example 159
    Orthogonal functions 289
    Orthogonal matrix 366
    Orthogonal transformations 220
    Overlay models 54
    Parabolic axisymmetric shell elements 277
    Parabolic-type elements 297
    Parallelogram elements 128
    Parameters:
    forms without rotational 208
    nodal 123. 192
    rotational 208
    Pascal triangle 154
    Patch count, and discrete collocation constraints
    188
    Patch tests 131
    an analytical requirement 134
    and collocation constraints 195
    continuity condition 135
    curvilinear coordinates 128
    numerical 186
    plate bending elements 183
    quadrilateral mixed elements 185
    simple count 187
    thick plate elements 203
    triangular mixed elements 186
    Patches:
    four-element 190
    single-element 190
    Path dependent behaviour 35
    Penalty functions 261, 350
    Penalty multiplier 261
    Penalty parameters 175
    Perforated plate:
    plane strain solutions 73
    plane stress solutions 72
    Perturbing a Lagrange multiplier form 406
    Pian-Sumihara elements 199, 334Subject index 455
    internal/external virtual work 121
    isotropic elasticity 118
    isotropic homogeneous 380
    large deflection problems 383
    non-homogeneous 158
    non-linear formulation 379
    point loads 176
    rectangular 125
    square with clamped edges 136, 139
    square simply supported 278
    strain components in 165
    thick 173. 180, 203
    large displacement theory 375
    Reissner-Mindlin 173
    and shells 277
    thin 111, 112
    irreducible approximation 120
    transverse shear group strains 117
    twisting moments 118
    Point collocation of nodes 193
    Point loads on plates 176
    Point numbering 222
    Poisson ratio operator 48
    Poisson’s ratio 67, 73, 114. 118, 272
    Polar decomposition of the deformation gradient
    static condensation 203, 336
    Prony series 43
    Properties, curved elements 254
    Propped cantilever, Euler buckling 374
    Pseudo-rigid motions 396
    Push forward transformation 340
    Quadratic displacement approximations 162
    Quadratic fully integrated elements (QS) 179
    Quadratic interpolation 194. 198
    Quadratic triangles 135
    Quadratics, lagrangian (QL ) 179, 190
    Quadrature formulae 248, 274
    Quadrilaterals:
    and discrete collocation constraints 187
    elements 128, 149, 183
    linear 200
    mixed elements, patch tests 185
    Quartic and quintic elements 154
    Quasi-conforming elements 150
    Quasi-harmonic field problems, non-linear 12, 101
    Quasi-harmonic problem 13
    Quasi-Newton method 26, 31, 34
    Quintic polynomials 153
    Railway bridge 138
    Rate constitutive models 345
    Rates of convergence 145
    Rayleigh quotient form 374
    Rectangular element (12 DOF) 124
    Rectangular quadrilateral elements 209
    Recursion formula 46
    Reduced (selective) integration 182
    Reference configuration 312, 314
    Reference configuration formulation 320
    Regula falsi procedure 30
    Reinforced concrete:
    cracking of 86
    as a no tension material 84
    Reissner-Mindlin assumptions 266
    Reissner-Mindlin postulates 111
    Reissner-Mindlin thick plates 173
    Relaxation moduli 39
    Relaxation modulus function 43
    Relaxation times 39, 41
    REP methods 94
    Restart option, FEAPpv computer program 428
    Resultant constitutive models 166
    Retaining wall, earthquake excitation of 19
    Retardation time parameters 40
    Return map algorithm 58, 80
    Rcvolutc joint 407
    Rigid bodies:
    connected to a flexible body 402
    equations of motion 399
    modes 130
    398
    Polygonal shape approximation, to a curved shell
    249
    Polynomials:
    area coordinate 130
    complete linear 129
    Hermitian 152, 296
    Legendre 256
    quintic 153
    Postulates, Reissner-Mindlin 111
    Potential energy principle 193
    Power station , underground 84
    Prandtl-Reuss equations 63
    Prandtl-Reuss-strain relations 65
    Pressure approximations 11
    Pressure loads, and deformation dependent
    forces 336
    Principal stretches, logarithmic form 342
    Principal tensile stresses, elimination of 84
    Principles:
    constrained potential energy 193
    d’Alembert 323
    Hellinger— Reissner 156
    Prismatic bar 292
    Procedures:
    discretization 176
    Galerkin 181
    Processes:
    Galerkin 176
    Newton-Raphson 336
    semi-analytical finite element 289456 Subject index
    Rigid footing, failure of 98
    Rigid motions 398
    Rods, definition of 1
    Rotating disc, numerical example 409
    Rotation, nodal 255
    Rotation-free elements 162
    Rotational constraints, on joints/couplings 406,
    407
    Rotational parameters 208
    Rotational stiffness, 6 DOF 225
    Rotational treatment 227
    Rotations 202
    and displacement fields 270
    Runge-Kutta method/process/procedure 36. 57
    flat element approximation for 217
    general theories 218
    hemispherical 251
    in-plane force resultants 216
    inelastic behaviour 279
    non-linear response and stability 386
    problem treatment methods 216
    and rigid bodies 404
    shallow, co-rotational forms 388
    shallow shell theory 217
    spherical shell problem 227
    stability of 390
    stress representation 274
    thick, axisymmetric, curved 275
    and thick plates 277
    three-dimensional analysis 266
    toroidal, under pressure 258
    Sherman—Morrison—Woodbury formula 69
    Single rotational constraint joints 407
    Single translational constraint joints 407
    Skew curved bridge 137
    Skew-symmetric stress component 229
    Slave nodes 348, 351
    Slip line (localization) capture 93
    Slope-displacement interpolation 261
    Slopes, normal 146
    Small deformation non-linear solid mechanics
    problems 3
    Small rotations, large displacement formulation
    Saint-Venant-Kirchhoff material model 409
    Saint-Venant-Kirchhoff relation 397
    Saint-Venant-Kirchhoff stored energy model
    320
    Secant method 28
    Secant procedure 30
    Secant update method 29
    Second-order system solution using GN22 423
    Second-order tensors, invariants of 432
    Seepage flow 101
    Selective integration 182
    Semi-analytical finite strips 305
    Semi-analytical finite element processes 289
    Serendipity element, eight-node 286
    Serendipity functions 201
    Serendipity quadratic elements 179, 183
    Shallow shells 388
    Shape functions 124, 129, 133, 145, 208, 290
    and continuity requirements 122
    simple triangular element 145
    substitute 150
    thin plates 112
    Shear deformation, plates 114
    Shear locking 369
    Shear relaxation modulus function 43
    Shear rigidity 189
    Shearing stresses, and laminar material 87
    Shells:
    as an assembly of flat elements 216
    axisymmetric 217, 244, 387
    axisymmetric, curved , thick 275
    axisymmetric with non-symmetrical load 303
    bending 111
    branching 260
    classical treatment of 216
    curved 249, 267
    cylindrical 250, 281
    definition 1
    with displacement and rotation parameters 266
    divided into triangular elements 223
    finite element method 217
    370
    Smoothed elements 150
    Softening behaviour 31
    Softening of plastic material 48
    Softening/hardening rules 52
    Soil mechanics/behaviour:
    associated and non-associated issue 83
    viscoplastic models for 82
    Solid mechanics:
    general problems 1
    small deformation 3
    Solution, fine element 377
    Solution by analogies 48
    Solutions:
    elasto-plastic 249
    large deflection problems 383
    Newton-Raphson 324, 326, 339
    Spherical cap 284
    Spherical dome 257, 279
    Spherical dome under uniform pressure 281
    Spherical path controls 31
    Spherical points 407
    Spherical shell problem 227
    Spherical test problem 226
    Spinning constrained disc 405
    Spline finite strip method 305, 308
    Spline functions 306Subject index 457
    Spontaneous ignition 15, 102
    SPR methods 94
    Spring-dashpot model 39
    Square plates, clamped edges 136, 139
    Stability:
    of equilibrium 373
    and large plate deflections 385
    and neutral equilibrium 374
    shells 390
    Standard barrel vault problem 230
    Static condensation 203, 249, 336
    Static equilibrium, Euler equations for 321
    Static problems, computer program solutions
    415
    Steady-state problems:
    computer program solutions 415
    non-linear formulation 6
    Steel pressure vessel, plastic computation with 75
    Step-size control, computer programs 416
    Stiffness:
    geometric 3, 328
    initial stress 3
    rotational 225
    Stiffness matrix 127, 128, 131, 133, 134, 219, 221,
    248, 272, 293
    isotropic elastic material 300
    Stored energy function 340
    Saint-Venant —Kirchhoff model 320
    Strain:
    enhanced strain methods 332
    global 276
    Green 322
    Green-Lagrange 376, 387
    mixed form expression 10
    recovery procedures 94
    Strain components 245
    Strain-displacement equations, mixed form 11
    Strain-displacement matrices:
    evaluation of 182
    finite element evaluation 378
    Strain-displacement matrix 5
    Strain-displacement operator 174
    Strain-displacement relations 247
    Strain-driven form 42
    Strain expressions for curved elements 254
    Strain rates:
    plastic 51
    plastic deviatoric 64
    Strain softening 89, 97
    Strain tensor, Green-Lagrange 368
    Strain tensors 316
    Strains and stresses, definitions 271, 303
    Stress:
    Cauchy 316, 318, 325, 329, 339, 342, 345, 359
    Kirchhoff 316, 317, 325, 329, 342, 343, 344,
    345, 358
    membrane 250
    mixed form expression 10
    multiaxial 49
    Piola Kirchhoff 317. 320, 321, 328, 339, 345,
    368
    recovery procedures 94
    stress function 302
    uniaxial 49
    Stress deviator 41
    Stress divergence 6
    Stress field, equilibrating 157
    Stress force 6
    Stress hardening relations 66
    Stress increment computation 56
    explicit methods 57
    implicit methods 57
    Stress measures 316
    Stress rate form, Jaumann-Zaremba 346
    Stress representation, shells 274
    Stress resultants 113, 245
    Stress tensor, Piola-Kirchhoff 334
    Stress transfer method 26
    Stresses and strains, definitions 271
    Stretching ratio 96
    Structural dynamics 15
    Structural problems, non-linear 365
    Subincrementation 57
    Sublayer models 54
    Subparametric derivation 252
    Substitute shape functions 150
    Substitute shear strain matrices 194
    Superparametric elements 271
    Surface plasticity models 70
    Symmetric stress behaviour 164
    Systems, irreducible 175
    Tangent computation 12
    Tangent equation system 32
    Tangent matrix 66, 77. 337, 379, 383, 416
    computer program for 416
    evaluation of 379, 383
    use of 8
    Tangent matrix terms 371
    and nodal forces 337
    Tangent moduli 42
    Tangential displacements 256
    Tensor form, relation with index relation 5
    Tensors:
    Almansi strain 316
    Cauchy-Green deformation 316, 338, 342
    Green strain 316
    Green-Lagrange strain 368
    mixed deformation 328
    nine component computations 50
    second-order, invariants of 432
    two-point 315458 Subject index
    non-linear 6
    typical examples 14
    Transient solutions 401
    Translation constraints on couplings 405
    Translational constraint joints 407
    Transverse shear energy 189
    Transverse shear group strains, plates 117
    Trefftz-hybrid elements 158
    Tresca surface expressions 89
    Tresca yield conditions 62
    Triangles/triangular elements
    6 DOF 133
    9 DOF 128, 144
    12 DOF 144
    15 DOF 148
    18 DOF 153
    21 DOF 153
    arbitrarily oriented in space 223
    Clough and Toucher 148
    and collocation constraints 194
    compatible 148
    linear 200
    mixed 186
    Pascal 154
    plate bending 151
    quadratic 135
    simple 145
    Twisting moments, plates 118
    Two-point tensor 315
    Tests:
    mixed patch 230
    spherical shell problem 227
    spherical test problem 226
    Theory:
    large displacement of beams 365
    large displacement of thick plates 375
    large displacement of thin plates 381
    shallow shell 217
    two-dimensional of beams 366
    Thick box bridge prism 294
    Thick plate elements, theory limitations 203
    Thick plates 111
    large displacement theory 375
    mixed formulation for 180
    numerical solution for 173
    and shells 277
    Thick shells, axisymmetric, curved 275
    Thick-walled cylinder subject to internal pressure
    46
    Thin case, axisymmetric shell:
    no shear deformation 303
    shear deformation 304
    Thin membrane box structures 295
    Thin plates 111
    irreducible approximation 120
    large displacement theory 381
    theory of 112
    thin plate limit 203
    Three-dimensional analysis, shells 266
    Three-dimensional elasticity 155
    Three-field mixed finite deformation formulation
    UMESHn computer input routine 413
    Underground power station, and no-tension
    issues 84
    Uniaxial stress 49
    Unsymmetric solver 313
    Unsymmetrical loads 244
    328
    Three-field mixed method for general constitutive
    models 9
    Time-independent plasticity theory, classical 48
    Tolerance limits 33
    Toroidal shell, under pressure 258
    Torsion, variable section circular bar 300
    Total increment, use of 7
    Tower, axisymmetric, under non-symmetric load
    Variables:
    lagrangian 113
    nodeless 255
    Variational description for finite deformation
    301 319
    Traction boundary 120
    Transformation matrix form 4
    Transformations:
    isoparametric 262
    Jacobian 199, 273
    orthogonal 220
    push forward 340
    and rotational stiffness 226
    and shell element properties 272
    to global coordinates 219
    Transforms, node 220
    Transient heat conduction 14
    Transient problems:
    computer procedures for 419
    Variational Galerkin statement 137
    Variational principle 162
    Variational theorem for finite elasticity 320
    Vault, cylindrical 282
    Vector, nodal 127
    Velocity matrices 399
    Vibration modes, computer programs for
    428
    Virtual displacements, finite element
    approximations to 5
    Virtual work 134
    plates 121
    Virtual work equation 5
    Virtual work statement 132Subject index 459
    Viscoelastic relaxation function 46
    Viscoelasticity 39
    iterative solution scheme 80
    linear models for 39
    Viscoplastic laws 80
    Viscoplastic model 79
    Viscoplastic (or creep ) strain rate 78
    Viscoplasticity 88
    Viscoplasticity/viscoplastic models 78
    Volumetric strain approximations 11
    Volumetric stress effects 11
    Yield:
    and multiaxial stress 49
    surface and normality criterion 50
    Yield conditions:
    Drucker-Prager 62
    Huber-von Mises 62
    Mohr-Coulomb 62
    Tresca 62
    Yield functions 50
    Yield stress 48
    Yield surfaces, isotropic 61
    Yielding, erratic 249
    Weak form for equilibrium 4 Young’s modulus 272, 369
    Weighted residual Galerkin form 156
    Weighting, Dirac delta 189
    Work, internal/external virtual, plates 121
    Zero stress increment 60
    Zone of attraction 26

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