Index

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||u||E 507

||f||0 505

||u||m 506

(p, u) 505

(p, ϕi) 507

(p,w) 505

C0-elements 338

C1-elements 424

Gj 81

Hm(Ω) 506

L2 scalar product 505

Ph 28

S 22

V 22

V

_

559

Vh 6

V

_

h 559

V +

h 185

δ0 69

δi 69

div 503

div2M 426

tr, trace 395

∇ 503

⊗ 504

∂i w 536

C[ ], elasticity tensor 395

E 503

S 503

uG 134

a(u, ˆu) 504

a(w, ˆ w) 35

a(w,w) 10

p(ϕi) 507

a posteriori error indicators 147

accuracy 80

actio = reactio? 177

adaptive mesh refinement 383

adaptive methods 147

adaptivity, dimension 154

adaptivity, model 154

adding a member 323

approximations for stiffness matrices

298

asymptotic error estimates 560

Aubin–Nitsche trick 540, 564

Babuˇska–Brezzi condition 399

backward error analysis 201

Bathe–Dvorkin element 439

beam girders 462

beam models 459

beam-like elements 356

Bessel’s inequality 10

bilinear elements 340

bilinear form 504

boolean matrix 329

boundary layer effect 436

Boussinesq 256

brachistochrone 175

bulk modulus 393

cables 305

capacity 56

Castigliano’s Theorem 50

Cauchy-Schwarz inequality 506

change in an elastic support 316

changes in a structure 313

circular arches 491

circular slabs 461

columns 443

composition 22

concentrated forces 44

condition number 230

continuity 221

594 Index

corner force 421

coupling degrees of freedom 224

cracks in a beam 316

critical angles, plane problem 336

critical load 302

CST element 338

cubic elements 338

cycloid 175

d-adaptivity 154

deformation space 6

degenerate shell elements 490

dichotomy 149

dimension argument 45

dipole 97

Dirac delta 69

Dirac energy 110, 111

direct design method 476

direct product 504

discontinuities 389

displaced point supports 354

displacement methods 393

displacement model of slabs 421

displacements 40

distance 37

DKT element 440

drilling degrees of freedom 344

drop panels and column capitals 451

DST element 441

dual load 69

dual quantities 70

dual weighted residual error estimate

525

duality 40

duality techniques 156, 160, 520

effective length factor 302

effective width 467

efficiency of estimator 567

eigenvalue 230

eigenvector 230

eigenwork 17

Element Free Galerkin method 215

element stresses 357

elements 220

elements, plane problems 338

elements, slabs 422

ellipticity 21

energy method 4

energy norm 151

equilibrium 86, 184, 420

equilibrium of resultant forces 378

equilibrium point 20

error estimators 566

error in energy 151

error indicator 566

essential boundary conditions 570

Euler equation 25, 40

Euler-beam I 193

explicit error estimators 567

finite element 197

first fundamental form 486

flat elements 493

flexibility matrix 198

folded plates 463

forces 40

foundation slabs 469

fundamental solution 241

Galerkin method 193

Galerkin orthogonality 11

Gateaux derivative 402

Gaussian curvature 488

Generalized Finite Element method

213

geometric matrix 301

global equilibrium 185

goal-oriented recovery 160, 520

goal-oriented refinement 412

Green’s first identity 508

Green’s function 51, 57, 516

Green’s identities 508

Green-Lagrangian strain tensor 401

h-methods 154

half-space model 256, 470

hard support 436

hat functions 197

Heaviside function 218

Hellinger–Reissner principle 396

Hermite Polynomials 201

hierarchical elements 309

higher-order elements 344

homogeneous deflection curve 289

hourglass modes 230

How to predict changes 113

hp-method 154

Index 595

HSM element 425

hybrid elements 425

hyperelastic material 402

implicit error estimators 567

incompressible material 393

infinite energy 208

influence functions 42, 69

influence functions for mixed formulations

399

influence of a single element 126

integral operators 82

integration by parts 508

interpolation 197

isotropy 221

kernel of a matrix 231

kernel of an influence function 58

kinematic constraints 347

Kirchhoff plates 416

Kirchhoff shear 420

Kronecker delta 199

Lagrange element 200, 343

Lagrange multipliers 226

Lagrange polynomials 200

least action 20

least squares 33

linear elements 338

linear strain triangle 341

lintels 356

load case δ 17

load case p 17

loads 255

local equilibrium 185

local solution 136

locking 437

loss of a frame member 320

loss of a support 317

LST element 341

m-adaptivity 154

mapped polynomials 203

master degree of freedom 226

Maxwell’s theorem 550

membrane locking 491, 492

membrane stresses 487

membranes 498

Mindlin shell elements 490

missing member 325

mixed methods 393

Mohr’s integral 240, 535

natural boundary conditions 570

Nitsche trick 540

nodal influence functions 77

nodal stresses 357

nonlinear problems 123, 161, 167, 401,

526

normal equations 34

order of the singularity 46

order of the strain energy 46

output 181

p-methods 154

particular deflection curve 289

partition of unity 211

Partition of Unity method 213

Pascal’s triangle 200

patch test 344

peaks 102

piecewise polynomials 197

piled raft foundations 258

plane elements 338

plane strain 222, 334

plane stress 222, 334

point supports 353, 477

Poisson’s ratio 416, 419

polynomials 199

potential energy 6

primal problem 157

primary load case 386

principal stresses 336

principle of minimum potential energy

7

principle of virtual displacements 25

principle of virtual forces 240

projection 515

projection method 2

pulley 111

Pythagorean theorem 573

Q4 340

Q4 + 2 341

Q8 343

quadratic elements 338

596 Index

Rayleigh quotient 195

recovery based error estimators 567

reduced global stiffness matrix 228

reduced integration 228

reduced stiffness matrix 185, 224

Reissner–Mindlin plates 431

remeshing 154

residual 236

residual forces 15

restriction 1

retrofitting structures 130

Riesz’ representation theorem 521,

559

rigid-body motions 221

rope 6

rotational invariance 221, 338

rounding errors 230

safety of structures 326

scalar product 33, 40, 503

scalar product of matrices 504

second fundamental form 486

second Piola-Kirchhoff stress tensor

401

self-equilibrated stresses 386

sensitivity analysis 313

serendipity elements 200, 343

set theory 16

settlement of supports 187

settlements 387

shape functions 337

shear forces 451

shear locking 437

shear modulus 336, 393

shearing strain 432

shell equations 485

shells 485

shells of revolution 488

shift 82

Shifted Green’s Function Theorem

547

shifted Green’s functions 541

singular matrices 227

singularities 175

skew projection 13, 22

slabs, Reissner–Mindlin 431

slave degree of freedom 226

smoothing process 362

Sobolev’s Embedding Theorem 46

soft support 436

spurious modes 211

St. Venant’s principle 172

stability problems 193

stiffness matrices 221

stiffness matrices for 1-D elements 292

stiffness matrices, slabs 424

stiffness matrices, three properties 223

stiffness matrix, 2-D 339

strain energy product 31, 35, 221, 339,

395

stream model 330

substitute load case 2

superconvergence 206

support conditions 183

support reactions 99, 353

supports 441

symmetries 354

T beams 226

tangential supports 354

Taylor series 199

temperature 187

tension 6

tension stiffening 288

test range 36

test space 22

three-moment equation 198

Timoshenko beam 296

Timoshenko beam element 301

Tottenham’s equation 64

translations 212

trial space 6, 22

truss models 363

values at a point 62

variable thickness 452

virtual displacements 24

volume elements 490

weak convergence 40

weak equal sign 4

weak form of influence functions 535

weak solution 40

weighted least squares 35

weighted residual method 194

wheel loads 460

r-adaptivity 154 Sobolev norms 12

Index 597

Winkler model 255, 469

X-FEM 215

yielding of a rigid support 319

zero-energy modes 230

Wilson’s Element 341