MultiPhysicsVault/.raw/ConceptsApplicationsFiniteElementAnalysis/ConceptsApplicationsFiniteElementAnalysis_065.md

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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.

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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.

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# 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

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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

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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

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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

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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

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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

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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

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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