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 |  موضوع: كتاب Finite Element Analysis for Building Assessment - Advanced Use and Practical Recommendations الثلاثاء 06 أغسطس 2024, 8:55 pm | |
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أخواني في الله
أحضرت لكم كتاب
Finite Element Analysis for Building Assessment - Advanced Use and Practical Recommendations
Paulo B. Lourenço and Angelo Gaetani
و المحتوى كما يلي :
Contents
Foreword xi
Preface xiii
Authors xvii
1 Fundamentals of the finite element method 1
1.1 Equilibrium of a three-dimensional body in elasticity 1
1.2 Basics 4
1.2.1 Weak form of governing equations in elasticity 4
1.2.2 Example of an axially loaded column 8
1.3 Generalization 13
1.3.1 Convergence criteria on shape functions 15
1.3.2 Matrix formulation 18
1.3.3 Example of a beam subjected to bending 22
1.4 Shape functions and numerical integration 28
1.4.1 Hermitian beam elements 29
1.4.2 Isoparametric elements 33
1.4.3 Completeness of shape functions
for quadrilateral elements 42
1.4.4 Numerical integration, integration
points and integration schemes 44
1.4.5 Recovery process 51
1.5 Commonly used finite elements 55
Bibliography and further reading 62
2 Nonlinear structural analysis 63
2.1 Potential sources of nonlinearities 64
2.2 Geometrical nonlinearities and stability 67
2.2.1 Introduction to finite displacements:
the von Mises truss structure 68vi Contents
2.2.2 Geometric stiffness and linear buckling
analysis 74
2.2.3 Postbuckling analysis and geometrical
imperfection 78
2.2.4 Stability of compressed elements 82
2.3 Computational strategies 88
2.3.1 Incremental-iterative methods
with material nonlinearities 88
2.3.1.1 Forward Euler method 91
2.3.1.2 Newton–Raphson method
(regular, modified and linear) 94
2.3.1.3 Quasi-Newton methods 100
2.3.2 Advanced solution procedures 104
2.3.2.1 Arc-length control 106
2.3.2.2 Direct and indirect displacement
control 114
2.3.2.3 Line search 116
2.3.3 Convergence criteria 119
2.3.4 Example 122
Bibliography and further reading 125
3 Constitutive models 127
3.1 Material behaviour 128
3.1.1 Main constitutive models 128
3.1.2 Failure of concrete-like materials 133
3.1.3 Simple nonlinear problems 136
3.2 Elastoplasticity models 143
3.2.1 Elastic domain and perfect elastoplasticity 145
3.2.2 Yield functions for isotropic materials 149
3.2.3 Constitutive law with associated plastic
flow and perfect elastoplasticity 159
3.2.4 Constitutive law with associated
plastic flow and hardening 163
3.2.5 Euler Forward and Euler Backward incremental
methods with perfect elastoplasticity 168
3.3 Damage models 177
3.3.1 Uniaxial loading 178
3.3.2 Isotropic and anisotropic damage models 182
3.3.3 Damage deactivation 188
3.4 Smeared crack models 190
3.4.1 Main concept and strain decomposition 190Contents vii
3.4.2 Implementation of the fixed and
rotating smeared crack model 193
3.5 Strain localization and mesh sensitivity 196
3.6 Examples of constitutive models in
commercial software codes 199
3.6.1 Smeared crack model (in tension)
and plasticity in compression 199
3.6.2 Plasticity models in tension and
compression 201
3.6.3 Total strain crack model 203
3.6.4 Damaged plasticity 204
Bibliography and further reading 207
4 Recommended properties for advanced
numerical analysis 209
4.1 Masonry 210
4.1.1 Masonry morphologies 211
4.1.2 Introduction to numerical modelling 213
4.1.3 Mechanical properties of masonry
components 218
4.1.3.1 Unit 218
4.1.3.2 Mortar 223
4.1.4 Mechanical properties of interface 225
4.1.4.1 Elastic parameters 228
4.1.4.2 Mode I failure 229
4.1.4.3 Mode II failure 232
4.1.4.4 Shear–tension and shear–
compression interaction 239
4.1.4.5 Compressive failure 240
4.1.5 Mechanical properties of masonry
as a composite material 240
4.1.5.1 Axial behaviour (compression) 241
4.1.5.2 Axial behaviour (tension) 247
4.1.5.3 Biaxial behaviour 248
4.1.6 Structural behaviour of masonry as a
composite 249
4.1.6.1 Shear walls 251
4.1.6.2 Masonry Quality Index 253
4.1.7 Synthesis of the mechanical properties 261
4.1.7.1 Individual masonry components 262
4.1.7.2 Masonry as a composite material 262viii Contents
4.1.7.3 Data from Italian code 263
4.1.7.4 Data from American code 268
4.2 Timber 268
4.2.1 Strength grading 270
4.2.2 Mechanical properties 272
4.3 Reinforced concrete 273
4.3.1 Mechanical properties of concrete
and reinforcement 274
4.3.1.1 Elastic parameters 275
4.3.1.2 Compressive behaviour 276
4.3.1.3 Tensile behaviour and tension
stiffening 278
4.3.1.4 Biaxial behaviour 282
4.3.1.5 Reinforcement behaviour and
modelling 283
4.3.2 Nonlinear models of frame elements 284
4.4 Steel and iron 287
4.4.1 General behaviour and material properties 288
4.4.1.1 Tensile strength 289
4.4.1.2 Ductility 289
4.4.1.3 Code properties 290
4.4.2 Mechanical properties of various
types of iron and steel 290
4.5 Safety assessment 296
4.5.1 Eurocode 8 and Italian code 297
4.5.1.1 Knowledge levels KL and
confidence factors CF 298
4.5.1.2 Masonry 299
4.5.1.3 Timber 301
4.5.1.4 Concrete and steel 303
4.5.2 Italian guidelines for built heritage 307
4.5.3 Safety formats for nonlinear analysis
according to Model Code 2010 310
4.5.3.1 Basic principles 311
4.5.3.2 Probabilistic method 313
4.5.3.3 Global resistance factor
method (GRF) 314
4.5.3.4 Method of estimation of a coefficient
of variation of resistance (ECOV) 314
4.5.3.5 Partial safety factor method (PSF) 315
Bibliography and further reading 315Contents ix
5 Guidelines for practical use of nonlinear finite
element analysis 321
5.1 Overview 321
5.2 Preprocessing 324
5.2.1 Type and main aspects of analysis 325
5.2.2 Geometry 327
5.2.3 Type of element 332
5.2.4 Mesh 338
5.2.4.1 Mesh refinement 339
5.2.4.2 Mesh quality and controls 344
5.2.5 Examples of element types and mesh
refinement 349
5.2.6 Material properties 359
5.2.7 Constraints and loads 360
5.3 Solution process 363
5.4 Postprocessing 367
5.5 Structural analysis report 369
5.6 Examples 371
5.6.1 Steel plate 371
5.6.2 Concrete beam 377
5.6.3 Masonry shear wall 385
5.6.3.1 Macro-modelling 388
5.6.3.2 Micro-modelling 390
Bibliography and further reading 397
Index 399 xi
Index
Note: Bold page numbers refer to tables and italic page numbers refer to figures.
accuracy 15, 120
additive strain decomposition 128
Almazán, F. J. A. 271
Amadio, C. 235
arc-length control 106, 106–113,
107–110, 112–113
Atkinson, R. H. 235
Augenti, N. 235
automatic loading 365, 366
Bauschinger effect 144, 289
beam elements 56–57
Bernoulli–Euler beam 16, 23, 56
bifurcation 77, 79
Binda, L. 214, 235, 242
Bisoffi-Sauve, M. 230
bond stone see diatone
bond-slip behaviour 226
Borri, A. 253, 260
boundary value problem 2, 4, 7
Brazilian test 221
Briceño, C. 231
buckling
buckling length 85
elastic and inelastic buckling 86
linear buckling analysis 74–77,
74, 75
nonlinear buckling analysis
86–87, 87
postbuckling analysis 78–82,
78–82
Bussell, M. N. 291
cast iron 290, 291
Collins, M. P. 277
commercial software codes
damaged plasticity 204–207,
205–207
plasticity models 201–203, 202, 203
smeared crack model 199–201,
199–201
total strain crack model
203–204, 204
compatibility (FEM) 15
composite material (masonry) 240–241
axial behaviour (compression)
241–247, 242–243,
245–247
axial behaviour (tension)
247–248, 248
biaxial behaviour 248–249,
249, 250
computational strategies
convergence criteria 119–122,
120, 121
incremental-iterative methods
88–91, 90 (see also
incremental-iterative methods)
solution procedure (see solution
procedure)
concentrated plasticity (concrete) 284
concrete-like materials 133–136,
133–135
confidence factor (CF) 297
constitutive models 128–132, 129–130,
131–132
contact elements 61, 61
convergence 15
criterion 120, 121
rate 119–120, 124–125400 Index
Crack Mouth Opening Displacement
control (CMOD) 116
crack-opening behaviour 226
damage 131
damage function 183
damage models 177–178, 176, 178
anisotropic 182–188, 184–188
constitutive laws 181
isotropic 182–188, 184–188
uniaxial loading 178–182, 179–181
damage surfaces 184
Degrees of Freedom (DOF) 4, 5
design working life 307
detailed micro-modelling 215
deviatoric plane 150
diatone 211
dilatancy 162, 237–238, 238
direct displacement control
114–116, 115
discretization 4, 18
displacement control 73, 104
distributed plasticity in concrete 284
Drucker’s stability 159
Drucker–Prager yield function 155
dry-joint masonry 210
Drysdale, R. G. 235
duplicate elements 347
duplicate nodes 346–347
elastic perfectly plastic 128
elastic stiffness 76
elasticity 128, 129–130
elasticity problems
closed-form solution 2
integral form solution 4
strong form of equilibrium 3
weak form of equilibrium 7
elastoplastic stiffness 168
elastoplasticity models
consistency condition 145
constitutive law
elastic domain evolution 165
perfectly plastic materials
159–163, 161, 162
plastic flow and hardening
163–168, 164–166
elastic domain 145–148, 147
Euler Backward method (see Euler
Backward method)
Euler Forward method (see Euler
Forward method)
hardening (see hardening)
load directions 147, 153
load history 144, 146
loading process 145
yield functions 145, 149–159
yield surface 145, 149
errors in FEM calculation, sources of 18
estimation of a coefficient of variation
(ECOV) 313, 314–315
Euler Backward method
(elastoplasticity) 168–177,
170, 171, 173, 174
Euler Forward method (elastoplasticity)
168–177, 170, 171, 173, 174
Euler’s critical load 83
Eurocode 8 see safety assessment
event-driven approach 139, 142
explicit method 174
external constrains 362–363
fibre model (concrete) 285, 285
finite element method (FEM) 5
elasticity 4–8
matrix formulation 18–22
shape functions 15–18
flat shell elements 59
flow rule 143; see also elastoplasticity
models
force control 73
Forward Euler method 91–94, 92,
92, 93
fracture energy 134
fracture modes 133
free edges 342, 373
free faces 346
free nodes 346
full integration scheme 50
fully probabilistic design method 313
Gardner, L. 296
Gauss quadrature 47
generalized strain 28, 29
generalized stress 29
geometric stiffness 75
geometrical nonlinearities
linear kinematics 82, 82
stability 72
stability of compressed elements
82–87, 82, 84–87
stiffness 74–77, 74, 75
von Mises truss structure 68–74,
68–70, 72, 73Index 401
Giuffrè, A. 211
global resistance factor (GRF) 313–314
h-refinement method 340, 356
hanging nodes 348
hardening, anisotropic 144
hardening, kinematic 144, 164
hardening, isotropic 143, 164
hardening modulus 166, 167
hardening parameter 165
hardening process 131, 143
Hegemeier, G. A. 235
Hermite interpolation polynomial
function 30
Hermitian beam elements 29–33
Hilsdorf, H. K. 241
historical-critical analysis 298
Hoffmann, G. 242
Hofmann, P. 235
homogeneous materials 128
hourglassing 335, 336, 337
hp-refinement method 340
hydrostatic stresses 149
ideal plasticity 128
ill-posed problem 80
implicit method (elastoplasticity) 174
incremental-iterative methods
Forward Euler method 91–94, 92,
92, 93
Newton–Raphson method 94–100,
96, 99, 99
Quasi-Newton methods 100–104,
101, 103, 103, 104
indirect displacement control method
106, 114–116, 115
initial elastic domain 164
integration points 45
interface (masonry) 225–228, 227, 228
compressive failure 240
elastic parameters 228–229
Mode I failure 229–232, 230,
231, 232
Mode II failure 232–239, 233, 234,
235, 236, 237, 238
shear-tension and shearcompression interaction
239–240, 239
interface element 59, 60
inverse parameter identification 324
isoparametric elements
bar element 34
completeness and compatibility 42
definition 33
four-node quadrilateral element
36, 37
Jacobian operator 38–39
local reference system 42
shape function matrix 40
stiffness matrix 35
strain-displacement transformation
matrix 35
isotropy 149
Italian code see safety assessment
iterative solution algorithm 88
Jacobian operator 38–39
joints (masonry) 214
Khelifa, M. 269
knowledge level (KL) 297, 298
Kržan, M. 224, 224, 231
leaf 212
limit point 74, 104
line search algorithm 116–119, 118
linear elastic stiffness 72
load/force control method 104
load increments 73
load path–following methods 74
load steps 73
Lobatto quadrature 48–49
local coordinate system 34, 34
Lourenço, P. B. 230, 236, 249, 263
lumped plasticity (concrete) 286
macro-modelling 216
Magenes, G. 235, 385, 385
Magonette, G. 385, 385
Mann, W. 233
masonry 210–211
compressive behaviour 214
compressive strength 245, 244, 245,
247, 246, 247
in-plane action 212
mechanical properties 218
composite material (see
composite material)
interface (see interface)
mortar (see mortar)
nonlinear static analysis 217
units 218–223, 219, 220,
221, 223
modelling strategies 215402 Index
masonry (cont.)
morphologies 211–213
numerical modelling 213–218
structural behaviour 249–251, 250
MQI 253–261, 252, 254,
255–257, 258, 259, 260, 261
shear walls 251, 253, 251
synthesis of mechanical properties
261–268
Masonry Quality Index (MQI)
253–261, 252, 254, 255–257,
258, 259, 260, 261
master/slave nodes 62
material behaviour
concrete-like materials 133,
133–136, 134, 135
constitutive models 128–132,
129–130, 131, 132
nonlinear problems 136–143, 137,
138, 138, 140, 142
matrix formulation (FEM) 18–22
membrane elements 57
mesh; see also preprocessing
dependency 194, 196–198, 197
sensitivity 196–198, 197, 351
meshing blocks 344
Mindlin–Reissner plate 58
vonMises, R. 68
Model Code 2010 220, 222, 223, 273,
275, 277, 278, 280
model calibration 323
model optimization 322, 322
Modulus Of Elasticity (MOE) 271
Mohr–Coulomb yield surface
151, 152, 155, 156,
151, 153
monotonic loads 144
mortar 223–225, 224, 226
Müller, H. 233
Muñoz, G. R. 271
natural coordinate system 34, 34
Newton–Cotes method 45, 46, 47
Newton–Raphson method 94–100,
96, 99, 99, 108, 110,
111, 175
nonassociated plastic flow 162
nonlinear constitutive relations 18
nonlinearities, sources of 64
numerical integration 44–45
Oudjene, M. 269
p-refinement method 340, 352
Parisi, F. 235
partial safety factor method (PSF)
312, 315
Pereira, J. M. 263
performance-based design 310
plane strain elements 58
plane stress elements 57, 57
plastic flow 143, 159–163, 161, 162;
see also elastoplasticity
models
plastic potential 168
plate bending element 59, 59
polynomial interpolation 43
postbuckling analysis 78, 78–82, 79,
81, 81; see also buckling
postprocessing 317, 367–369, 368, 369
preprocessing 322, 324–325
constraints 360–363, 361, 362
geometry 327–332, 328–331
material properties 359–360
mesh (see also mesh)
automatic 338
convergence study 339
quality and controls 344–349,
345, 347–349
refinement 15, 339–344,
342, 343
type of analysis 325–327
type of element 332–337, 333, 334,
335, 337, 349–359, 352,
350–355, 356, 357, 358
pressure-dependent behaviour 277
Principle of Virtual Displacements 3
Principle of Virtual Work 3
probabilistic method 312
proportional loading 144
quadratic convergence 124
quadrilateral elements 42–44
Quasi-Newton methods 100–104, 101,
103, 103, 104, 110, 111
quasi-static problems 1
Rajgelj, S. 235
Ramberg–Osgood model 296
Rankine yield function 156, 157
Rankine–Hill anisotropic model 203
recovery process 53–54
reduced integration scheme 50
Regula–Falsi method 118
reinforced concreteIndex 403
biaxial behaviour 282–283
compressive behaviour 276,
276–277
elastic parameters 275–276
nonlinear effects 275
nonlinear models 284–287,
285–286
reinforcement behaviour and
modelling 283, 283–284, 284
tensile behaviour 278–282,
278–282
report (contents) 320, 369–371,
370–371
residual stresses 139
reversible strain 128
Ridley-Ellis, D. 270
Riks–Wempner method 111, 112
Roberti, G. M. 235
Roca, P. 307, 310
Roman opus quadratum 211, 212
Rots, J. G. 234
rules of the art (masonry) 211
safety assessment
Eurocode 8/Italian code
concrete and steel 303–307,
305, 306
geometric survey 301
KL/CF 298
masonry 299–301
mechanical characterization 299
timber 301–303, 301
Italian guidelines for built heritage
307–310, 309
Model Code 2010
ECOV 314–315
GRF 314
safety formats 310–313
probabilistic method 313
PSF 315
Saint-Venant’s principle 17, 342
Schubert, P. 242
secant elastic modulus 131, 181
Secant methods see Quasi-Newton
methods
Selby, R. G. 285
semi-probabilistic method 312
shear locking 335
shear retention factor 194
shell normals 347
Shim, W. 377, 379
simplified micro-modelling 215–216
Simpson rule 48–49
skewness 345
smeared crack models
concept 190–193, 190–192
fixed multi-directional crack
concept 194
fixed smeared crack models 191
rotating smeared crack models 191
strain decomposition 190–193,
190–192
total strain crack model 195
snap-back instability 106, 107
snap-through instability 74, 106, 107
softening 128, 132
solid elements 59, 60
solution procedure 105, 107
arc-length control 107, 106–113,
107, 110, 112, 113
displacement control 114–116, 115
line search 116–119, 118
solution process 363–367
spherical arc-length method 111
spring elements 61
stainless steel 294
steel and iron
basic behaviour 288–289, 288
material properties 288–289, 288
code properties 290, 291
ductility 289–290
tensile strength 289, 290
mechanical properties 291,
290–296, 292, 293, 294,
295, 295
stiffness matrix 6
Stöckl, S. 235
strain localization 196–198, 197
strain singularities 55
stress stiffening 75
superconvergent points 51
tangent modulus 128, 166
tangent stiffness 72
target reliability level 311, 312
tearing mode 133
tension stiffening 280
through stone see diatone
timber 268, 269–270, 269
mechanical properties 268,
272–273, 273, 274
strength grading 270–271, 271
Tomaževič, M. 245
Total Lagrangian formulation 326404 Index
Tresca yield function 149, 150, 166
truss elements 56, 56
uncertainties 325
unloading–reloading process 128
Updated Lagrangian formulation 326
Van der Pluijm, R. 230, 233, 233, 234,
237, 237, 238
Vasconcelos, G. 233, 234, 235,
251, 251
Vecchio, F. J. 273, 281, 377, 379
verificationprocess 323
volumetric locking 335, 336
von Mises yield function 154, 170
well-posed problem 80
wrought iron 291, 292
wythe see leaf
yield stress 128
yield surface 145, 149
Young’s modulus 64
Yun, X. 29400 Index
Crack Mouth Opening Displacement
control (CMOD) 116
crack-opening behaviour 226
damage 131
damage function 183
damage models 177–178, 176, 178
anisotropic 182–188, 184–188
constitutive laws 181
isotropic 182–188, 184–188
uniaxial loading 178–182, 179–181
damage surfaces 184
Degrees of Freedom (DOF) 4, 5
design working life 307
detailed micro-modelling 215
deviatoric plane 150
diatone 211
dilatancy 162, 237–238, 238
direct displacement control
114–116, 115
discretization 4, 18
displacement control 73, 104
distributed plasticity in concrete 284
Drucker’s stability 159
Drucker–Prager yield function 155
dry-joint masonry 210
Drysdale, R. G. 235
duplicate elements 347
duplicate nodes 346–347
elastic perfectly plastic 128
elastic stiffness 76
elasticity 128, 129–130
elasticity problems
closed-form solution 2
integral form solution 4
strong form of equilibrium 3
weak form of equilibrium 7
elastoplastic stiffness 168
elastoplasticity models
consistency condition 145
constitutive law
elastic domain evolution 165
perfectly plastic materials
159–163, 161, 162
plastic flow and hardening
163–168, 164–166
elastic domain 145–148, 147
Euler Backward method (see Euler
Backward method)
Euler Forward method (see Euler
Forward method)
hardening (see hardening)
load directions 147, 153
load history 144, 146
loading process 145
yield functions 145, 149–159
yield surface 145, 149
errors in FEM calculation, sources of 18
estimation of a coefficient of variation
(ECOV) 313, 314–315
Euler Backward method
(elastoplasticity) 168–177,
170, 171, 173, 174
Euler Forward method (elastoplasticity)
168–177, 170, 171, 173, 174
Euler’s critical load 83
Eurocode 8 see safety assessment
event-driven approach 139, 142
explicit method 174
external constrains 362–363
fibre model (concrete) 285, 285
finite element method (FEM) 5
elasticity 4–8
matrix formulation 18–22
shape functions 15–18
flat shell elements 59
flow rule 143; see also elastoplasticity
models
force control 73
Forward Euler method 91–94, 92,
92, 93
fracture energy 134
fracture modes 133
free edges 342, 373
free faces 346
free nodes 346
full integration scheme 50
fully probabilistic design method 313
Gardner, L. 296
Gauss quadrature 47
generalized strain 28, 29
generalized stress 29
geometric stiffness 75
geometrical nonlinearities
linear kinematics 82, 82
stability 72
stability of compressed elements
82–87, 82, 84–87
stiffness 74–77, 74, 75
von Mises truss structure 68–74,
68–70, 72, 73Index 401
Giuffrè, A. 211
global resistance factor (GRF) 313–314
h-refinement method 340, 356
hanging nodes 348
hardening, anisotropic 144
hardening, kinematic 144, 164
hardening, isotropic 143, 164
hardening modulus 166, 167
hardening parameter 165
hardening process 131, 143
Hegemeier, G. A. 235
Hermite interpolation polynomial
function 30
Hermitian beam elements 29–33
Hilsdorf, H. K. 241
historical-critical analysis 298
Hoffmann, G. 242
Hofmann, P. 235
homogeneous materials 128
hourglassing 335, 336, 337
hp-refinement method 340
hydrostatic stresses 149
ideal plasticity 128
ill-posed problem 80
implicit method (elastoplasticity) 174
incremental-iterative methods
Forward Euler method 91–94, 92,
92, 93
Newton–Raphson method 94–100,
96, 99, 99
Quasi-Newton methods 100–104,
101, 103, 103, 104
indirect displacement control method
106, 114–116, 115
initial elastic domain 164
integration points 45
interface (masonry) 225–228, 227, 228
compressive failure 240
elastic parameters 228–229
Mode I failure 229–232, 230,
231, 232
Mode II failure 232–239, 233, 234,
235, 236, 237, 238
shear-tension and shearcompression interaction
239–240, 239
interface element 59, 60
inverse parameter identification 324
isoparametric elements
bar element 34
completeness and compatibility 42
definition 33
four-node quadrilateral element
36, 37
Jacobian operator 38–39
local reference system 42
shape function matrix 40
stiffness matrix 35
strain-displacement transformation
matrix 35
isotropy 149
Italian code see safety assessment
iterative solution algorithm 88
Jacobian operator 38–39
joints (masonry) 214
Khelifa, M. 269
knowledge level (KL) 297, 298
Kržan, M. 224, 224, 231
leaf 212
limit point 74, 104
line search algorithm 116–119, 118
linear elastic stiffness 72
load/force control method 104
load increments 73
load path–following methods 74
load steps 73
Lobatto quadrature 48–49
local coordinate system 34, 34
Lourenço, P. B. 230, 236, 249, 263
lumped plasticity (concrete) 286
macro-modelling 216
Magenes, G. 235, 385, 385
Magonette, G. 385, 385
Mann, W. 233
masonry 210–211
compressive behaviour 214
compressive strength 245, 244, 245,
247, 246, 247
in-plane action 212
mechanical properties 218
composite material (see
composite material)
interface (see interface)
mortar (see mortar)
nonlinear static analysis 217
units 218–223, 219, 220,
221, 223
modelling strategies 215402 Index
masonry (cont.)
morphologies 211–213
numerical modelling 213–218
structural behaviour 249–251, 250
MQI 253–261, 252, 254,
255–257, 258, 259, 260, 261
shear walls 251, 253, 251
synthesis of mechanical properties
261–268
Masonry Quality Index (MQI)
253–261, 252, 254, 255–257,
258, 259, 260, 261
master/slave nodes 62
material behaviour
concrete-like materials 133,
133–136, 134, 135
constitutive models 128–132,
129–130, 131, 132
nonlinear problems 136–143, 137,
138, 138, 140, 142
matrix formulation (FEM) 18–22
membrane elements 57
mesh; see also preprocessing
dependency 194, 196–198, 197
sensitivity 196–198, 197, 351
meshing blocks 344
Mindlin–Reissner plate 58
vonMises, R. 68
Model Code 2010 220, 222, 223, 273,
275, 277, 278, 280
model calibration 323
model optimization 322, 322
Modulus Of Elasticity (MOE) 271
Mohr–Coulomb yield surface
151, 152, 155, 156,
151, 153
monotonic loads 144
mortar 223–225, 224, 226
Müller, H. 233
Muñoz, G. R. 271
natural coordinate system 34, 34
Newton–Cotes method 45, 46, 47
Newton–Raphson method 94–100,
96, 99, 99, 108, 110,
111, 175
nonassociated plastic flow 162
nonlinear constitutive relations 18
nonlinearities, sources of 64
numerical integration 44–45
Oudjene, M. 269
p-refinement method 340, 352
Parisi, F. 235
partial safety factor method (PSF)
312, 315
Pereira, J. M. 263
performance-based design 310
plane strain elements 58
plane stress elements 57, 57
plastic flow 143, 159–163, 161, 162;
see also elastoplasticity
models
plastic potential 168
plate bending element 59, 59
polynomial interpolation 43
postbuckling analysis 78, 78–82, 79,
81, 81; see also buckling
postprocessing 317, 367–369, 368, 369
preprocessing 322, 324–325
constraints 360–363, 361, 362
geometry 327–332, 328–331
material properties 359–360
mesh (see also mesh)
automatic 338
convergence study 339
quality and controls 344–349,
345, 347–349
refinement 15, 339–344,
342, 343
type of analysis 325–327
type of element 332–337, 333, 334,
335, 337, 349–359, 352,
350–355, 356, 357, 358
pressure-dependent behaviour 277
Principle of Virtual Displacements 3
Principle of Virtual Work 3
probabilistic method 312
proportional loading 144
quadratic convergence 124
quadrilateral elements 42–44
Quasi-Newton methods 100–104, 101,
103, 103, 104, 110, 111
quasi-static problems 1
Rajgelj, S. 235
Ramberg–Osgood model 296
Rankine yield function 156, 157
Rankine–Hill anisotropic model 203
recovery process 53–54
reduced integration scheme 50
Regula–Falsi method 118
reinforced concreteIndex 403
biaxial behaviour 282–283
compressive behaviour 276,
276–277
elastic parameters 275–276
nonlinear effects 275
nonlinear models 284–287,
285–286
reinforcement behaviour and
modelling 283, 283–284, 284
tensile behaviour 278–282,
278–282
report (contents) 320, 369–371,
370–371
residual stresses 139
reversible strain 128
Ridley-Ellis, D. 270
Riks–Wempner method 111, 112
Roberti, G. M. 235
Roca, P. 307, 310
Roman opus quadratum 211, 212
Rots, J. G. 234
rules of the art (masonry) 211
safety assessment
Eurocode 8/Italian code
concrete and steel 303–307,
305, 306
geometric survey 301
KL/CF 298
masonry 299–301
mechanical characterization 299
timber 301–303, 301
Italian guidelines for built heritage
307–310, 309
Model Code 2010
ECOV 314–315
GRF 314
safety formats 310–313
probabilistic method 313
PSF 315
Saint-Venant’s principle 17, 342
Schubert, P. 242
secant elastic modulus 131, 181
Secant methods see Quasi-Newton
methods
Selby, R. G. 285
semi-probabilistic method 312
shear locking 335
shear retention factor 194
shell normals 347
Shim, W. 377, 379
simplified micro-modelling 215–216
Simpson rule 48–49
skewness 345
smeared crack models
concept 190–193, 190–192
fixed multi-directional crack
concept 194
fixed smeared crack models 191
rotating smeared crack models 191
strain decomposition 190–193,
190–192
total strain crack model 195
snap-back instability 106, 107
snap-through instability 74, 106, 107
softening 128, 132
solid elements 59, 60
solution procedure 105, 107
arc-length control 107, 106–113,
107, 110, 112, 113
displacement control 114–116, 115
line search 116–119, 118
solution process 363–367
spherical arc-length method 111
spring elements 61
stainless steel 294
steel and iron
basic behaviour 288–289, 288
material properties 288–289, 288
code properties 290, 291
ductility 289–290
tensile strength 289, 290
mechanical properties 291,
290–296, 292, 293, 294,
295, 295
stiffness matrix 6
Stöckl, S. 235
strain localization 196–198, 197
strain singularities 55
stress stiffening 75
superconvergent points 51
tangent modulus 128, 166
tangent stiffness 72
target reliability level 311, 312
tearing mode 133
tension stiffening 280
through stone see diatone
timber 268, 269–270, 269
mechanical properties 268,
272–273, 273, 274
strength grading 270–271, 271
Tomaževič, M. 245
Total Lagrangian formulation 326404 Index
Tresca yield function 149, 150, 166
truss elements 56, 56
uncertainties 325
unloading–reloading process 128
Updated Lagrangian formulation 326
Van der Pluijm, R. 230, 233, 233, 234,
237, 237, 238
Vasconcelos, G. 233, 234, 235,
251, 251
Vecchio, F. J. 273, 281, 377, 379
verificationprocess 323
volumetric locking 335, 336
von Mises yield function 154, 170
well-posed problem 80
wrought iron 291, 292
wythe see leaf
yield stress 128
yield surface 145, 149
Young’s modulus 64
Yun, X. 296
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