19.2 I. C. Taig, "Finite Element Analysis in Industry—Expertise or Proficiency?," in Accuracy, Reliability, and Training in FEM Technology (Proc. of Fourth World Congress), J. Robinson, ed., Robinson and Associates, Wimborne, England, 1984. 19.3 T. Sussman and K. J. Bathe, "Studies of Finite Element Procedures—Stress Band Plots and the Evaluation of Finite Element Meshes," Engineering Computations, Vol. 3, No. 3, 1986, pp. 178–191. 19.4 R. T. Haftka and J. C. Robinson, "Effect of Out-of-Planeness of Membrane Quadrilateral Finite Elements," AIAA Jnl., Vol. 11, No. 5, 1973, pp. 742–744. 19.5 R. H. MacNeal, ed., The NASTRAN Theoretical Manual (Level 16.0), NASA-SP-221(03), March 1976 (N79-27531, N.T.I.S.). 19.6 K. Christensen, “Writing Easy-to-Use Programs for Computers,” Mechanical Engineering, Sept. 1983, pp. 66–69. 19.7 H. H. Fong, "An Evaluation of Eight U.S. General Purpose Finite-Element Computer Programs," 23rd AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, 1982, pp. 145–160. 19.8 P. Naur, B. Randell, and J. N. Buxton, Software Engineering: Concepts and Techniques, Petrocelli-Charter, New York, 1976. 19.9 J. R. Rice, Numerical Methods, Software, and Analysis, McGraw-Hill, New York, 1983. 19.10 R. Evans, "Guidelines for the Selection of Analysis Software," Mechanical Engineering, March 1987, pp. 42–43. 19.11 J. A. Swanson, "The Development of General Purpose Software, or What is a Software Supplier?," in Structural Mechanics Computer Programs: Surveys, Assessments, and Availability, W. Pilkey et al., eds., University Press of Virginia, Charlottesville, 1974, pp. 687–702. 19.12 E. Schrem, "Status and Trends in Finite Element Software," in State-of-the-Art Surveys on Finite Element Technology, A. K. Noor and W. Pilkey, eds., ASME, New York, 1983, pp. 325–340. 19.13 K. Bell, “Some Thoughts on Design, Development and Maintenance of Engineering Software,” Advances in Engng. Software, Vol. 8, No. 2, 1986, pp. 66–72. 19.14 Anon., A Finite Element Primer, Dept of Trade and Industry, National Engineering Laboratory, Glasgow G75 OQU, U.K., 1986. 19.15 T. Slot and W. J. O'Donnell, "Effective Elastic Constants for Thick Perforated Plates with Square and Triangular Penetration Patterns," J. Eng. Industry, Vol. 93, No. 4, 1971, pp. 935–942. 19.16 C. Meyer, ed., Finite Element Idealization, Am. Soc. of Civil Engrs., New York, 1987. # APPENDIX A A.1 G. Strang, Linear Albegra and Its Applications, 3rd Ed., Harcourt-Brace-Jovanovich, San Diego, 1988. # APPENDIX B B.1 I. S. Duff, "A Survey of Sparse Matrix Research," Proceedings of the IEEE, Vol. 65, No. 4, 1977, pp. 500–535 (cites 604 references). B.2 W. F. Tinney and J. W. Walker, "Direct Solutions of Sparse Network Equations by Optimally Ordered Triangular Factorization," Proceedings of the IEEE, Vol. 55, No. 11, 1967, pp. 1801-1809. B.3 D. P. Mondkar and G. H. Powell, "Towards Optimal In-Core Equation Solving," Computers & Structures, Vol. 4, No. 3, 1974, pp. 531-548. B.4 E. L. Wilson and H. H. Dovey, "Solution or Reduction of Equilibrium Equations for Large Complex Structural Systems," Advances in Engng. Software, Vol. 1, No. 1, 1978, pp. 19–25. B.5 C. A. Felippa, "Solution of Linear Equations with Skyline-Stored Symmetric Matrix," Computers & Structures, Vol. 5, No. 1, 1975, pp. 13–29. B.6 E. Mendelssohn and M. Baruch, "Solution of Linear Equations with a Symmetrically Skyline-Stored Nonsymmetric Matrix," Computers & Structures, Vol. 18, No. 2, 1984, pp. 215-246. # APPENDIX C C.1 D. S. Malkus and M. E. Plesha, "Zero and Negative Masses in Finite Element Vibration and Transient Analysis," Comp. Meth. Appl. Mech. Engng., Vol. 59, No. 3, 1986, pp. 281–306. C.2 D. S. Malkus and X. Qiu, "Divisor Structure of Finite Element Eigenproblems Arising from Negative and Zero Masses," Comp. Meth. Appl. Mech. Engng., Vol. 66, No. 3, 1988, pp. 365–368. C.3 D. S. Malkus, M. E. Plesha, and M-R. Liu, "Reversed Stability Conditions in Transient Finite Element Analysis," Comp. Meth. Appl. Mech. Engng., Vol. 68, No. 1, 1988, pp. 97–114. No. 1, 1988, pp. 97-114. C.4 K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1982. C.5 T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1987. C.6 H. Kardestuncer and D. H. Norrie, eds., The Finite Element Handbook, McGraw-Hill, New York, 1987. C.7 G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986. C.8 J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, England, 1965. C.9 J. R. Rice. Numerical Methods, Software, and Analysis, IMSL Reference Edition, McGraw-Hill, New York, 1983. C.10 B. T. Smith et al., Matrix Eigensystem Routines—EISPACK Guide, Lecture Notes in Computer Science No. 6, 2nd Ed., Springer-Verlag, New York, 1976. C.11 NAg Library Manual, Numerical Algorithms Group, Downers Group, C.12 IMSL Library Manual, International Mathematical and Statistical Library, Houston, TX 77036-5085. # INDEX a-basis formulation, 112 Accuracy: anisotropy, 23, 583 arch elements, 346, 347, 348 axial symmetry, 297 bar element, 92–93, 553–556 constant strain triangle, 156–157 digits lost, 547 dynamic analysis, 397, 404–405, 413–416 element geometry, 196-199 fluid-structure interaction, 491, 494 higher-order elements, 246 isoparametric elements, 187–188 mass condensation, 388–390 near-mechanisms, 192-193 numerical integration, 172, 188–189 plate elements, 325–328, 333 Rayleigh-Ritz solution, 82–83 singularity elements, 250 stress calculation, 81, 133–134, 195 substructuring, 259 transformations, 209 user-defined elements, 245 see also Bounds; Convergence; Errors: Modeling advice Acoustic modes, 490, 494 Admissible configuration, 71 Alpha-method, 409 Amplitude error, 413–416 Analogies, 486 Anisotropic material, 20–22, 23, 294, 583 Antisymmetry, 260–261, 298 Arches, 343–350 Arrested instability, 523 Artificial viscosity, 407–409 Assembly of elements: algorithms, 43, 44, 47, 51 discussed, 12, 38–44, 49, 59 in Galerkin method, 463-464 Asymmetric Loads: heat conduction, 480 shell of revolution, 353, 357 solid of revolution, 301–307 Attachment d.o.f., 258, 392, 393 Average acceleration method, 405 Axial symmetry, see Symmetry Banded matrices: bandwidth computation, 45, 46, 53 equation-solver, 595 storage format, 46 Bar elements: in space, 215 nodal loads, 40–41 plane, 8, 36–38, 113 stress calculation, 55–56 Basis functions, see Shape functions Basis vectors, 384 Beam, curved, 308 Beam-column, 429–432 Beam elements: direct method, 9–11 Mindlin, 278–280, 283, 285 shape functions for, 101 shear deformation, 114, 279 standard (Euler), 113-114 Beating, 404 BFGS method, 507 Bifurcation analysis, 432, 441–444 Bilinear elements: axially symmetric, 295–297 plane, 98–99, 115–116, 166–170, 173–176 Blocking, 258 Body forces: defined, 17 stress calculation, 133 Bounds: Gerschgorin, 401, 548 hybrid elements, 241 incompatible elements, 234 on buckling load, 445 on condition number, 548, 549 on Rayleigh quotient, 379 on stiffness, 82–83 on vibration frequency, 375, 400-402 Boundary conditions: acoustic modes, 490 hear conduction, 477–482 in Galerkin method, 461 inclined support, 545–546 on displacement, 48–53 on stress, 18–19 plate bending, 332–334 solid of revolution, 297, 303 types of, 83–84, 478 with a fluid, 487–489 with derivative d.o.f., 246 Boundary element method, 255 Box beam, 322 Brick element, 180–181 Broyden methods, 506–508 Bubble function, 178, 230 Buckling: analysis restrictions, 446–447 columns, 431 condensation of d.o.f., 444 Buckling (Continued) defined, 429 eigenvalue problem, 442 shells, 445, 447 with nonlinearity, 441, 443 Bulk modulus, 280 $C^0$ elements, 96-99 $C^1$ elements, 99-101 $C^m$ continuity, 95 Cables, 446, 533 Calculus of variations, 83–87 Cantin-Clough element, 348 Central-difference method, 397–405, 411–412, 415–416, 525–527 CFL condition, 401 Change of phase, 533 Characteristic matrix, defined, 7 Checking of model and results, 24, 579, 583-584 Circulation mode, 493 Collapse analysis, 447 Collocation method, 456, 459 Compatibility, 124–126, 234, 319, 320 Compatibility condition, 18 Complementary energy, 239–240 Completeness: isoparametric elements, 186–188 of trial series, 82 Component mode synthesis, 391–395 Computer program steps, 4, 57 Condensation: algorithm for, 231 and constraints, 273–274 in buckling analysis, 444 in dynamics, 387–391 static, 228–229 Condition number, 546–550 Conditional stability, 398, 399, 485, 523 Conforming, see Compatibility Connections, modeling of, 577 Conservative system, 70 Consistent penalty method, 287 Constraints: and quadrature points, 282 counting, 283–285 for joining of elements, 218–222 Lagrange multipliers, 275–276 modes, 392–393 multipoint, 272 naturally arising, 278–282, 325–328 penalty functions, 276–278 ratio, 284 rigid elements, 220–222 single-point, 272 transformations, 272–274 see also Boundary conditions Constant-strain triangle, 114–115, 155–157 Contacts, moving, 533 Continuity: degree of, 95 integration by parts, 461 least squares methods, 458 Convergence: and path test, 129 general discussion, 542–566 monotonic, 560 nonlinear problems, 502–510, 522, 529, 530 requirements for, 82, 126–128 with lumped loads, 124 Coordinates (natural or intrinsic): for area, 149–150, 152 for line, 147–148, 164 for quadrilateral, 166–167 for tetrahedra, 151–152 Core element, 297 Costs: and dimensionality, 574 in nonlinear problems, 501 program writing and running, 586–587 Courant number, 402 Cracks, 247–250, 533, 557 Craig-Bampton method, 393 Crank-Nicolson method, 405, 485 Creep, 533 Critical condition (buckling), 429, 432, 441-444 Critical time step: explicit methods, 398–404, 408–409, 411–412 thermal transients, 485 with material nonlinearity, 523 Curvatures: arch, 344 beam, 113 plate, 317, 318, 320, 324 shell, 354 Cyclic symmetry, 262–263 Damping: numerical, 407–409, 510 physical, 376–377 Dead loads, see Body forces Degrees of freedom, defined, 2, 31, 69 Derivatives, as nodal d.o.f., 246 Determinant, calculation of, 593–594, 596 Diagnostics, see Errors Diagonal decay test, 550–552 Differential stiffness matrix, 429 Differentiation of matrix forms, 590–591 Direct method of forming [k], 7–11, 32–33, 36–37 Direct integration (transients), 395–418, 484–485, 522–529 Direct substitution, 502–504 Dirichlet boundary condition, 478 Discrete element, 31 Discrete Kirchhoff elements, 321, 328–332 Discretization error, 543, 553–563 Discretization into finite elements, 1-2 Dissection, 258 Dissipation, numerical, 407–409, 510 Distortion, effect of, 187–188, 196–197, 578 DKT element, 328–332 D.o.f., see Degrees of freedom Drift, nonlinear solutions, 505–506, 513, 521 Drilling d.o.f., 236–238, 242–244, 351–352 Dynamic response analysis: accuracy, 381, 407, 413–418 alpha method, 409 basic equations, 368–370 beating, 404, 405 choice of procedure, 407, 417, 418 direct integration, 395–418 efficiency, 397, 400, 401 explicit methods, 395–405, 408–409 implicit methods, 396, 405–407, 407–410 mixed methods, 410 modal methods, 381–387 multi-step methods, 396 Newmark methods, 408–409 nonlinearities, 387, 398, 417–418, 522–529 numerical noise, 404, 418 operator splitting, 410 Ritz vectors, 385–386 single-step methods, 396 spurious oscillations, 404 stability, 398, 399, 404, 408–409, 410–413 Dynamic storage allocation, 584–585 Eigenproblems: algorithms and programs, 603–604 general discussion, 378, 598–604 in heat conduction, 482, 484 Rayleigh quotient. 379, 601 Eigenvalue economization, 387 Eigenvalue test for elements. 563–565 Elastic kernel, 244–245 Elastic support, 576 Elasticity, theory of, 15-20 Elastic-plastic problems: dynamic, 522–529 elastic to plastic transition, 512, 520 general formulation, 515–519 initial-stiffness method, 513–515, 522 material property matrix, 517, 521 one dimensional, 510–515 tangent stiffness method, 513, 520–522 Energy and energy principles: arches and shells, 344, 349, 355 bars and beams, 77, 279 complementary energy, 239–240 general expressions, 75, 109–110 minimum and stationary, 71–73, 86–87 nonlinear terms, 433, 436, 438 plates, 320, 324, 330 Energy balance check, 523–524 Envelope, 46, 596 Equation solving, 48, 53–54, 592–597 Equilibrium: and constraints, 274–275 between and within elements. 124–125 check of, 553 differential equations, 17–19 in hybrid elements, 239, 242 Errors: and load lumping, 123–124 and mass matrix, 375–376, 415 buckling analysis, 441, 447–448 causes of, 24–25, 209, 566, 573, 579–580 diagonal decay test, 550–552 dynamic analysis, 381, 383, 413–416, 492, 523–524, 527 in cyclic symmetry, 263 nonlinear problems, 505, 513, 521, 523–524, 527 numerical, analysis and discussion. 542–566 of discretization, 553–563 penalty formulations, 278, 283 reduced by extrapolation, 559–562 residual tests, 383, 527, 552–553 stress computation, 81, 132–135, 194–195, 297, 580–581 terminology, 542–543 use of symmetry, 261, 293, 381, 583 various elements, 156, 223, 297, 326–327 very stiff elements, 221–222, 544–546 with singularities, 557 see also Accuracy; Bounds; Convergence; Modeling advice Euler equation, 84–85 Euler integration, 485, 505 Extra nodal d.o.f., 246 Extrapolation: from multiple meshes, 559–562 of Gauss point stresses, 195–196 Factorization, see Equation solving Fills, defined, 47 Film coefficient, 474–475 Finite difference method, 256 Finite element method: advantages and disadvantages. 4–5 defined, 4, 90 Finite prism method, 307 Finite strip method, 307, 322 Flexural rigidity, defined, 317 Flow rule (plasticity), 515 Fluid flow, 287, 486–488 Fluid-structure interaction, 491–494 Folded plate, 322, 362 Forces, internal and external. 369 Form factor, 279 Forming processes, 533 Forward reduction, see Equation solving Foundation, elastic, 250–252 Fourier equation, 465, 475 Fourier series, 298–300 Fox-Goodwin method, 409 Fracture mechanics, 247–248 Frame element, 114, 216, 345–346 Frequency, terms for, 378 Frontal method, 48, 596–597 Full integration, 188, 282 Functional: acoustical modes, 490 defined, 69, 83 element matrices from, 90–94 for potential energy, 109 heat conduction, 93, 476, 479 possible unavailability, 455 wave equation, 489 Galerkin method, 87, 457–458, 460–466, 468–469 Guass elimination, 53–54, 593–596 Gauss quadrature, 170–173, 183–185 Generalized coordinates, 79, 382, 384–386, 484 Geometric stiffness matrix, 429 Gerschgorin bound, 401, 548 Global, 12 Gravity load, see Body forces Green-Lagrange strain, 437 Guyan reduction, 387–391 Hardening rule, plasticity, 511, 516-518 Hardening structure, 502 Harmonic functions, 468 Hat functions, 463 Heat conduction: boundary conditions, 477–482 Fourier equation, 465, 475 functional for, 93, 476, 479 general element expressions, 481 multidimensional, 85, 93–94, 477–480 nonlinear, 475, 482, 485 one-dimensional, 465–466, 475–477 quantities and units, 474 Heterosis plate element, 326–328 Higher-order element, 246 Hilbert matrix, 542 Hilber-Hughes-Taylor method, 409 Hinge, 124, 230 History of finite elements, 14 Houbolt method, 407 Hourglass mode, 190–194, 238 see also Instabilities Hybrid elements, 235, 239–244 Ill-conditioning, 52, 343, 358, 543–550 Imperfections, and buckling, 432, 447 Incompatible elements, 125, 233–236, 320 Incompressibility: and ill-conditioning, 550, 580 constraint counting, 285 elastic-plastic problems, 521 fluid elements, 492 penalty constraints, 280–282, 285–287 Indeterminacy, static. 75 Inertia, rotary, 371, 378, 602 see also Mass matrices Inertia forces, see Body forces Inextensibility condition, 345 Infinite elements and media, 252–255 Influence lines, 120 Initial stress and strain: bar element, 22–23, 40–41, 56, 118 in general, 56, 110, 170 in plates, 317 stress calculation, 56, 132–134, 196 Initial stress stiffness matrix, 429 Initial value problem, 381 Instabilities, element and numerical: mechanisms, element or mesh, 130, 190- 193, 238, 381, 492, 564 numerical, in dynamics, 398, 404, 485, 523 stabilization matrix, 193, 238, 352, 493 see also Buckling Integration (spatial): and instabilities, 188–193 by parts, 86, 87, 461, 466–467 full, reduced, selective, 188, 189, 282, 326 Gauss quadrature, 170–173, 183–185 in natural coordinates, 149, 151, 152 special quadrature rules, 189, 374 see also Dynamic response analysis; Transients (thermal) Internal d.o.f., 177–178, 228–230, 233, 258, 326,348 Interpolation: defined, 95 Hermitian, 100 Lagrange's formula, 97–99 modified, 325, 356–357 Interpolation functions, see Shape functions Invariance, geometric, 128–129 Inversion, of matrix, 54–55 Isoparametric elements: defined, 164 for shells, 358–362 general discussion. 163–199 incompatible modes, 232–236 validity of, 186–188 Isotropy: geometric (spacial), 128–129 of material, 21–22 Iterative improvement, 134–135, 552 Jacobian and Jacobian matrix: for area, 168–169 for line, 165 for solid, 180 other terminology, 507 Jointed media, 533 Joints, modeling of, 577 Kernel, elastic, 244–245 Kickoff forces, 435 Kinematic mode, 190–194, 238 see also Instabilities Kirchhoff plate theory, 315, 316, 319–320 Lagrange element, 99, 177–180, 197 Lagrange's interpolation formula, 97–99 Laplace's equation, 468, 478, 487 Laplacian mesh, 559 Least squares methods, 457, 458, 460 Limit point, 446–447, 508, 510 Linear-strain triangle, 156–159, 183, 197–198 Load vector: bar element, 40-41, 118, 120 beam elements, 123 body of revolution, 306 consistent, 91, 110, 119 for arches, 346 for plates, 322–323 in Galerkin method, 464 inconsistent (lumped), 119, 123 initial strain and stress, 40, 56, 58–59, 110, 118 isoparametric elements, 170, 185–186 non-nodal point loads, 119 quadratic elements, 121–122, 185 sign of, 13 work-equivalence, 119 Loads: body of revolution, 297, 301–307 corrective, 521, 529, 530 deformation dependent, 447, 533 fictitious (pseudoload), 387, 417, 503 internal and external, 369 on plates, 318–319 types of, 76 Lobatto quadrature rule, 374 Locking: arch elements, 346–350 bilinear element, 233 constraint counting, 283–285 elastic-plastic problems, 521 fluid elements, 492 incompressible material, 281 Mindlin elements, 280, 325–327 penalty constraints, 278, 280–283 reduced integration, 282 Lumping: error of, 123-124 of foundation stiffness. 252 of loads, 119 of masses, 370–376, 378 of thermal matrices, 482 Macroelements, 228, 242, 258 Mass condensation, 387 Mass matrices: and eigenproblem, 378, 379, 601–602 choice of (accuracy), 375–376, 415 choice of (efficiency), 376, 398, 401, 407 consistent, 370–372 diagonal (lumped), 370–376 Master and slave d.o.f., 220, 259, 387-390 Material coordinates, 438 Material properties: heat conduction, 474, 477–478 temperature dependent, 23, 482 see also Stress-strain relations Matrix manipulations, 589–591 Mechanisms, 190–194, 238 see also Instabilities Mechanisms (linkages), 446 Membrane forces: calculation of, 432 defined, 340 in buckling, 429 Membranes, 446, 533 Meridian, defined, 340 Mesh: and ill-conditioning, 549–550 discretization errors, 553–563 layout (modeling), 573–584 local refinement, 582 revision of, 256–257, 575–576 Mindlin elements: arches, 348–350 beam, 278–280 plates, 321, 323–328 shells, 355–362 Mock-fluid elements, 491-494 Modal methods, 381–387, 484 Modal synthesis, 391–395 Modeling advice, 24–25, 573–584 Modification of structure, 256–257, 575–576 Moment, bending vs. nodal, 114 Moment-curvature relations, 114, 316–318 Negative area or Jacobian, 150, 152, 170, 198–199 Neumann boundary condition, 478 Newmark methods, 408–409 Newton-Raphson methods, 504–505, 509 Node numbering schemes, 44–48 Nodeless d.o.f., 229–230, 233, 348 Noise, numerical, 404, 418, 526 Nonconforming elements, 125, 233–236, 320 Nonlinear problems: solution methods, 502–510 Nonlinearities: discussion and methods, 501–533 effect on buckling, 441, 447 geometric, 439–441, 443, 529–533 in dynamic problems, 387, 398, 417–418, 522–529 in heat conduction, 475, 482, 485 in plate bending, 319 in strain-displacement relation, 437 Nonlinearities (Continued) incremental methods, 440–441, 505, 513, 520–522 initial stiffness method, 513–515, 522, 527–529 modeling advice, 582 types of, 501 Offsets, 220-221 Operator-splitting methods, 410 Orthogonality of eigenvectors, 381 Parasitic shear: in beam and plate elements, 280, 327 in plane elements, 194, 197, 232–233 Patch test, 129–131, 297, 334 Penalty constraints: choice of penalty number, 283 discussed, 276–278 for boundary conditions, 50–52 in transient analysis, 401, 482 Period error, 413-416 Pipe elbow, 308 Plane stress and plane strain, 21, 23 Plasticity, see Elastic-plastic problems Plates and plate bending: boundary conditions, 332–333 discussed, 314–335 sandwich plate, 318 test cases, 331, 334–335 Point sources, 122, 481, 482 Positive definite matrix, 75, 590 Potential flow, 486–488 Potential function (fluids), 486 Prandtl-Reuss relation, 518 Pressure calculation, 286–287 Prestress, see Initial stress and strain Profile, 46, 596 Programming and programs: commercial, 587–588, 593, 604 costs of, 586–587 documentation, 585–586 Pseudoloads, 387, 417, 503 Q6 or QM6 elements, 233–236, 297 Quadratic forms, 590 Quadrature, 170–173, 183–185 Quality tests for elements, 563–565 Quarter-point elements, 248–250 Quasiharmonic equation, 468 Quasistatic, defined, 367 Quasi-Newton methods, 506–508 Rank-deficiency, 190, 590 see also Singularity; Instabilities Rayleigh quotient, 379, 391, 601 Rayleigh-Ritz method: and Galerkin method, 458 classical form, 78–83 finite element form, 90–93 properties of solution, 82–83 Reactions, support, 35–36, 56–57 Reanalysis methods, 256–257 Recovery of condensed d.o.f., 229, 231, 388 Reduced basis, 384–386, 391–395 Reduction (in dynamics), 228, 387–391 see also Equation solving Refinement of mesh: and extrapolation, 559–562 $h$ and $p$ versions, 563 modeling strategy, 581–582 Reflected waves, 490 Reinforcing beam, offset, 220–221 Release of d.o.f., 230, 246 Repetition of substructures, 259, 262–263 Residual: as error test, 383, 527, 552–553 in dynamics, 383, 507, 527 weighted residual methods, 456 Richardson extrapolation, 560 Rigid body motion: and convergence, 127–129 and stress stiffening, 435, 438, 440, 445 arch elements, 344, 348, 349, 350 in dynamics, 375, 380, 602 in user-defined element, 245 yields no forces, 34–35, 564 Rigid elements, 220–222, 272 Ritz vectors, see Reduced basis Rotational d.o.f. in plane elements, 236–238, 242–244 Rotational periodicity, 262–263 Rounding error, 543 Scalar element, 252 Secant methods, 506–508 Sectorial symmetry, 262–263 Semianalytical method, 307, 322 Semidiscretization, defined, 369 Separation of variables method, 307, 322 Serendipity elements, 177–180, 197, 326–328, 373–375 Shape functions: axial elements, 89, 90, 96–98, 165 beam elements, 101, 280 bubble, 178, 230 derivation and properties, 89–90, 187 plane quadrilaterals, 99, 167, 177, 178, 255 plane triangles, 154 solid elements, 117, 159, 181 with drilling d.o.f., 238 with incompatible modes, 229, 233, 234 Shear deformation: beams and arches, 114, 278–280, 348 modified shear interpolation, 325, 356–357 plates and shells, 316, 355–356, 358 Shells: buckling, 445, 447 general discussion, 340–362 geometry of, 340, 353 thick vs. thin, 341 Simplex, 152 Single-element test, 565 Singularity: crack-tip elements, 248–250 error analysis, 557–558 infinite element, 254 of stiffness matrix, 13, 34–35 plate bending, 334 see also Instabilities Sink, 122, 481, 482 Skew support, 216-218, 545-546 Skyline, 46, 596 Slave d.o.f., 220, 259, 387-390 Sloshing of fluid, 488, 490, 493–494 Snapping instability, 510 Softening structures, 502, 509 Software, see Programming and programs Sound propagation, 488, 490 Source, point, 122, 481, 482 Sparsity: and constraints, 275, 278 of matrix, 44–48, 592, 596, 603–604 Spectral matrix, 382, 484 Spectral stability, 411–413 Spurious modes or oscillations, see Instabilities Stability coefficient matrix, 429 Stability, see Buckling; Instabilities Stabilization matrix, 193, 238, 352, 493 Static equivalence, 119 Statics check, 553 Stationary principles, 83–87 see also Functional Stick model, 574 Stiffener, offset, 220-221 Stiffness matrices (conventional): arch element (straight), 345 bar elements, 8, 37–38, 90, 264 beam or frame elements, 9–11, 114, 216 bilinear element, 116, 169 change of coordinates, 213–218 formulation procedures, 7–8, 32–33, 90, 110, 461–465 Fortran subroutines, 174–175, 332 general formula, 110 hybrid element, 240 physical meaning of, 9, 34 plate elements, 320, 324, 325, 330, 332 properties of, 34, 74 shell elements, 351, 352, 362 solid (brick) element, 116–117 triangular elements, 115, 155, 158–159 zero coefficients, 39–40, 48–49 Strain, effective plastic, 518 Strain-displacement matrix: bar and beam elements, 90, 112–114, 166, 249 bilinear element, 116, 169 body of revolution, 295–296, 305 from shape functions, 110 Mindlin plate, 324 triangular elements, 155, 158 Strain-displacement relations: arches, 344 bodies of revolution; 295, 302, 354, 356 general, 15–17 plates, 315, 316 Strain-hardening parameter, 511 Stream function, 487 Stresses calculated from elements: arches and shells, 346, 362 best locations for, 93, 194–195, 297, 328, 350 extrapolation from Gauss points, 195–196 iterative improvement, 134–135 oversmoothed contours, 132, 580–581 "strain gage," 132–133 stress concentration, 135–136 superconvergence, 195 with distributed loads, 133 with initial stress, 56, 59, 132–134, 196 with internal d.o.f., 232, 236 Stress intensity factor, 248–250 Stress stiffness matrices: bars, beams, and plates, 434–437 defined, 429 field compatibility, 444–445 general expression, 437–439 rigid body motion, 435, 445 various elements, 445 Stresses: accuracy of, 81 at boundaries, 19, 125 deviatoric and dilatational, 285–286, 517 plates and shells, 314, 340–341 Stress-strain relations: axial symmetry, 294, 301, 355 change of coordinates, 213, 318, 361, 478 in general, 20-23 plates and shells, 316–318, 355, 361 Strong form, 69, 84, 455 Structural dynamics, defined, 367 "Structure size," expansion to, 38, 41-44 Subdomain method, 456, 460 Subparametric elements, 164, 188 Substructuring: dynamic, 391–395 static, 257–260, 262–263 Superelement, 258 Superparametric elements, 164, 188 Support reactions, 35–36, 56–57 Supports, modeling of, 216–218, 545–546, 576 see also Boundary conditions Surface tractions, 19 Symbolic processing, 185 Symmetry: axial, 293–297, 354–355 Symmetry (Continued) cyclic (sectorial), 262–263 discussed, 260–261, 574 Fourier series terms, 298 possible misuse, 261, 293, 381 System, defined, 70 Tangent modulus or stiffness, 504, 511 Tetrahedron, 152, 159 Thermal stress, 22–23, 58–59, 132–134, 196 see also Initial stress and strain Time-history analysis, defined, 367–368 see also Dynamic response analysis; Transients (thermal) Torsion of shafts, 303, 307 Total-Lagrangian approach, 438 Transformation: and constraints, 272–274 isoparametric, 168 material properties; 213, 318, 478 possible errors, 209 various applications, 209–222 Transients (thermal), 482, 484–485 see also Dynamic response analysis Tranezoidal rule, 405–406, 412–413, 415–416, 485,528 Triangular elements: linear, 144-115, 153-157 quadratic, 156–159, 183, 197–198 Triangular coordinates, 152 Trilinear element, 116–117, 181 Truss element, see Bar elements Twisted ribbon, 335 Unbounded regions, 252–255, 490 Underrelaxation, 503 Unsymmetric, see Asymmetric loads User-defined element, 244–245 Variational methods, 85–87 Vibrations: nonlinear, 533 of fluid, 488–494 of structures, 367, 378–381 with stress stiffening, 446 Virtual work principle, 72, 87 Viscous relaxation, 510 von Mises criterion, 517–518 Warped elements, 583 Wave equation, 489 Wave propagation, 367, 401–405, 417–418, 490 Wavefront method, 48, 596–597 Weak form, 69, 84, 455 Weighted residual methods, 87, 455–470 Winkler foundation, 251 Work, see Energy and energy principles Yield criterion, 515 Zero-energy mode, see Instabilities