김경종
29 KiB
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