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a-discretization 36, 44, 66, 75, 228 a-function set 45

random 81 a-level optimization 52, 1143, 11415, 157,213,228,248,273,284

problem 157 solution techniques 157

a-level representation 45,81 a-level set 22, 37

algebra Boolean set 55 Boolean a- 58 Borel a- 55 a-54

application of fuzzy cluster design 288 Fuzzy Finite Element Method 205 Fuzzy First-Order Reliability Method 253,301 Fuzzy Stochastic Finite Element Method 216 fuzzy structural analysis 11841, 266

approximate reasoning 18, 104 assessment criterion 98

Bayesian estimation 2 methods 13, 14 theorem 59, 306

biuniqueness 147, 11418, 187, 189,237 blasting processes 10 Boolean

a-algebra 58 set algebra 56

bunch parameter representation 413,44,741,811,87, 179,212 Cartesian product 27, 28

322 Index

centroid method 38 clusters

crisp 276 design 282, 287 fuzzy 276 nonpermissible 285, 290 numberof277,278,281,285,290, 296,298,299,304 permissible 285, 290 reduced permissible 286

cluster analysis 273, 276 cluster method 276

crisp 276 fuzzy 276, 279, 298, 304 k-medoid 277, 290

clustering 277 quality of 278, 281, 285, 290, 304

complement 25 complementary event 72 computational mode1135, 174,250,274

arbitrary nonlinear 174,180,184,209,216,232,274 confidence interval 117, 119 constraint distance 287, 297 continuity 147, 151 convex

fuzzy result set 29, 33 fuzzy result variable 143, 145 fuzzy set 20, 37, 45,143,145,180

credibility 59, 63 crisp

cluster 276 cluster methods 276 limit state surface 233, 256, 263 measure 54 n-dimensional subspace 37, 143, 158,161, 173, 180, 183,213,214,273 reliability index 270 set 19, 22, 54

defuzzification 38, 287 methods after Jain and Chen 39

design algorithm 284 constraints 273, 282, 283, 285, 289, 295, 304 fuzzy cluster 282, 288 parameters 273, 282, 289, 302

design structural 273 variants 274, 286, 287, 293, 297, 299, 305

design parameter values 282 nonpermissible 282 permissible 282

deterministic fundamental solution 135, 231 influence of 174, 200, 250, 269, 271, 272

dimensionality of spaces 147, 1418 discretization

u-36,44,66,75,228 point 49, 182, 2lO time 419, 182

dissimilarity 276 distribution

function 55, 56 fuzzy compound H4, 132 fuzzy probability 66 parameters 110 possibility 61 probability 57 standard normal 233, 236 type llO

dynamic analysis 185,201,288 loading 185,202,288

elementary event 55,57,66,80 empirical

fuzzy parameter values 117 fuzzy probability distribution function 117, 130, 222

envelope curve 145 estimation theory 12 estimation

Bayesian 2, 13, 14,306 interval 12, 13, 306 point 12,13 uncertain parametric 117 uncertain nonparametric 117

evolution strategy 159, 161 modified 161, 176, 248

existing structures 9 experience 91, 101

Index 323

324 Index

expert knowledge 91, 101 opinion of 101

extended mapping model 32 extension principle 28, 32, 137

field fuzzy 43, 210 fuzzy random 85, 210 linear fuzzy 206 real random 210 stationary fuzzy 206 stationary fuzzy random 211

First Order Reliability Method 228, 248 function

bounding 46 bunch of 45, 83,110,231 distribution 55, 56 empirical fuzzy probability distribution 117, 130, 222 fuzzy 41, 73, 231, 237 fuzzy approximation 52 fuzzy correlation 84,211,218 fuzzy covariance 84 fuzzy expected value 84 fuzzy probability density 74, 111,229,230 fuzzy probability distribution 66,73,82,83,110,114,117,120,126,129,132,211 fuzzy random 79, 209, 210 fuzzy transformation 237 fuzzy variance 84 fuzzy weighting 114 initial draft of the membership 91, 118 k-th fuzzy central moment 84 k-th fuzzy initial moment 83 membership 20, 91, 118 original 80, 82, 210 possibility density 61 possibiliy distribution 61 probability density 57 probability distribution 57, 73 random bounding 81 sample 117,121 set 54 stationary fuzzy 47 stationary fuzzy random 85

function support 45 trend 45

fuzzification 12, 90, 91, 98, 99, 101 of parameters U7, 126 of the probability distribution 126, 129, 131

fuzziness 7, 19,77, 135,245 separation of 126, 129,240

fuzzy acceleration 202 approximation function 52 approximation polynomials 52 arrangement of the reinforcement steel 197, 208 bending moment 204 bifurcation load 195 boundary conditions 199,200 bunch parameters 43,75,81, 1lO, 206, 208, 2lO, 212, 213 central moment 79 central moment function 84 cluster 276 cluster design 282, 288 cluster methods 276, 279, 298, 304 compound distribution n4, 132 concrete compressive strength 206, 207 correlation function 84,211,218 correlation length 212, 218, 224 covariance 211 covariance function 84 damping parameters 185 design point 233, 240, 243, 259, 263 differential equations l78, 187 displacement 189, 195, 200, 289 displacement-time dependency 189,203,204 distance 289 distribution parameters no, 1l7, 120, 126 distribution type 114, 1l7, 120 element stiffness matrix 177 event 58 expected value 79 expected value function 84 failure load 198, 266 failure probability 228, 229, 240,259,265,268 failure region 229, 231,232, 239 field 43, 206, 2lO

Index 325

326 fudex

fuzzy function 41, 73, 231 functional parameters 117, 120 Gaussian process 83, 85 hyperplane 242 initial moment 78 initial moment function 83 initial value problem 185, 189 initial velocity 186 input set 28, 33, 37 input variable 137 interpolation nodes 52,206 interval 21 limit state surface 229,231,234,238,241,257,265,267 load-displacement dependency 198,200,207,208 loading 199, 206, 208 mass 289 material behavior 206 number 21 parameters 78, 110, 117 parameter vector 79 process 43 realization 66, 120 reliability index 229, 233, 240,243, 245, 259, 265, 267, 302 result set 29, 33 result variable 143, 145 rotational restraint 194 rotational spring stiffness 194, 196, 199,201,202,266 sample mean 121 set 19 sensitivity factors 242 standard deviation 79, 121 surface force 178 survival probability 228 survival region 229, 231, 232, 239 transformation function 237 trapezoidal interval 21 triangular number 21 variable 12, 20 variance 79, 211 variance function 84 vector 20, 210 velocity 186, 189 weighting factors 242

fuzzy weighting function 114 Young's modulus 189

Fuzzy Finite Element Method 136, 177 application of 205

Fuzzy First Order Reliability Method 228, 248, 301 application of 253, 301 numerical realization 248

fuzzy model 103 for crack formation 104 for a concrete material law 105 for the arrangement of reinforcement 107 for geometrical imperfections 107 parameters 90, 103, 106, 107, 108, 137, 180,231,235,266

fuzzy probabilistic basic variables 228,229,238,301 safety assessment 228, 270, 274

fuzzy probabilistics 18 fuzzy probability 58, 69, 70, 72, 228

density function 74, 111,229,230

Index 327

distribution function 66,73,82,83,110,114,117,120,126,129,132,211 fuzzy random

concrete strength 218,222,224 crack state 220 displacement 222, 225 field 86, 210, 211 function 79,85,209,210 load 253,263,266,301 loading process 220,222 process 85,88 rotational spring stiffness 263,301 variable 15, 67, Jl09, 228, 236, 306 vector 67,210,228 yield stress 255

fuzzy randomness 7, 66, 109, 135 Fuzzy Stochastic Finite Element Method 137,209, 2J10

application of 216 fuzzy stochastic structural analysis 135, 137, 209

application of 216 fuzzy structural analysis 17, 136, 143, Jl74, 273

application of Jl81/;, 288 numerical solution 174

328 fudex

genetic algorithms 104 geotechnical engineering 8 gradient method 159, 161

histogram 91,94, 126 homogeneous increments 47 hydraulic engineering 11 hypercuboid

n-dimensional 37

i.i.d. paradigm 3 inclusion 25 information

a priori 12, 13 type I 91 type II 91, 98 type III 91, 99 type IV 91, 101

integrating the fuzzy probability distribution 76 interaction 33, 51, 112, 120, 122, 127, 133, 145, 166, 173, 180,230,243,289

a priori 33 within the mapping 34

intersection 26

k-medoid cluster method 277,290

least squares method of 92

level rank method 39 limit state surface 6, 229

crisp 233,256,263 fUzzy 229,231,234,238,241,257,265,267 linear 256

linguistic assessment 62, 91, 98 variable 23, 98

loading dynamic 185,202,288 fuzzy 199,206,208 fuzzy random 220,222,253,263,266,301

loading process 197 random 223,225,269

mapping 28 biunique 147, 148, 187, 189,237 continuous 147, 151 monotonic 147, 153, 187, 195, 196 nonlinear 195 nonmonotonic 32, 122, 155, 187, 189 properties of the 147 uncertain 42 unique 147

mapping model 32, 114, 122, 135, 137, 177,273 extended 32 properties of the 147 uncertain 137

max-min operator 29, 143 measure 53

crisp 54 A-uncertain 60 Lebesgue 53 modified Shannon's entropy 65,245,269,288 objective uncertain 57 of uncertainty 641, 245 quality of clustering 277, 278, 281, 285, 290, 298, 304 Riemann 53 Shannon's entropy 64 Shannon's uncertainty 65 space 54 uncertain 53, 54, 55, 59

membership function 20,91 initial draft of 91, 118

mesh search techniques 159, 161 model

extended mapping 32 mapping 32, 114, 122, 135, 137, 177,273 properties of the mapping 147 uncertain mapping 137

model parameters 103 fuzzy 90,103,106,107,108,137,180,231,235,266

modified evolution strategy 161, 176, 248 control parameters of 168, 169, 170 postcomputation of 171

moment k-th fuzzy central 79 k-th fuzzy initial 78

Index 329

330 Index

moment function k-th fuzzy central 84 k-th fuzzy initial 83

monotonicity 147,153,187,189,195,196,237 Monte Carlo simulation 159, 161,216 multistory frame 185

necessity 59 neural networks 104 nodal-point method 210

objects 276 representative 277

operator maximum 247 max-min 29, 143 min 63 sup 63

opinion of experts 10 1 optimization

a-level 52, 143,145,157,213,228,248,273,284 problem 157 solution techniques 157

original 67, 73, 109, 230 function 80, 82, 210

original space of fuzzy probabilistic basic variables 229

partial safety factors 233 partition coefficient 281

normalized 281, 304 pavement engineering 11 permissible failure probability 245 plausibility 59, 63 possibility 59, 60

density function 61 distribution function 61 theory 17

postcomputation 171 probabilistic

fundamental solution 248 safety assessment 16, 262, 269 structural analysis 16

probability 56, 57 conditional 59 density function 57 distribution function 57, 73 distribution shadow 73 shadow 71 space 56,69

process fuzzy 43 fuzzy Gaussian 83, 85 fuzzy random 85,88 stationary fuzzy 48

product space 27 prototype 279,281

quality measure 277,278,281,285,290,298,304 of a cluster configuration 277, 278,281,285,290,298,304

random a-function set 81 bounding function 81 concrete strength 221 field 210 load 223,225,269 rotational spring stiffness 269 variable 12, 57,228,235 vector 57, 210

randomness 7, 77, 135 separation of 126, 129, 240

reinforced -concrete frame 196, 262, 301 prestressed frame 201 folded-plate structure 205, 223 plate 219 textile 216

reliability index crisp 270 fuzzy 229,233,240,241,243,245,259,265,267,302 required 245

representation a-level 45, 81 bunch parameter 43, 74, 81, 87, 179,212 of fuzzy random fields 210

Index 331

332 Index

representation of real random fields 210 discrete 210

reproduction conditions 66 known, nonconstant 126 unknown, nonconstant 120

required reliability index 245 restricted parameters 273, 283

space of 283 robustness 245,288,298

a-algebra 54 Boolean 58 Borel 56

safety assessment 16 fuzzy probabilistic 228,253, 270, 274 probabilistic 16,262,269

safety measure 53 safety verification 199, 245,270

fulfilled 246, 265 not fulfilled 246 partially fulfilled 246,247,261,268,271

sample function 117, 121 mean 121 size 306 small 91, 117 variance 121

separation

set

average 278, 292 degree 281, 304 of clusters 278, 281 of fuzziness and randomness 126, 129, 240

a-level 22, 37 a-function 45 convex fuzzy 20, 37,45, 143, 145 crisp 19, 22, 54 family of 54 function 54 fundainental19,27 fuzzy 19 fuzzy input 28, 33, 37

set fuzzy result 29, 33 of input points 273 of result points 273 power 54

shadow probability 71 probability distribution 73

Shannon's entropy measure 64 uncertainty measure 65

silhouette coefficient 278, 292 similarity 276 space

fuzzy probability 69 measure 54 of design parameters 282 of fuzzy bunch parameters 210, 213 of fuzzy probabilistic basic variables 228, 229 of restricted parameters 283 probability 56, 69 product 27 standard normal 233, 236, 238, 240 uncertain measure 55 x- 138,229 y- 233,238 z- 138

spatial coordinate 43,79,85 standard normal

distribution 233, 236 space 233,236,238,240

stationary fuzzy field 206 fuzzy function 47 fuzzy process 48 fuzzy random field 211 fuzzy random function 85 increments 47

statistics 121 steel

frame 193 girder 253, 288

Stochastic Finite Element Method 213 strengthening of structures 10

Index 333

334 Index

structural analysis dynamic 185,201,288 fuzzy 17, 136, 137,143,174,184,273 fuzzy stochastic 136, 209 geometrically and physically nonlinear 174,185,196,201,250,262,301 geometrically nonlinear 193 linear dynamic 203, 204, 288 nonlinear dynamic 185,201 physically nonlinear 205,219,223,253 statical 196 uncertain 135

structural design 273 algorithm of 284 for time-dependent load 288 fuzzy cluster 282, 288 robust 288, 298 variants 274, 286, 287, 293, 297, 299, 305 with Fuzzy First Order Reliability Method 301

structural parameters 90, 109 uncertain 90, 109

structural safety 53 subjective

aspects 101 assessment 268 decision 246 factors 90

subspace n-dimensional crisp 37, 143, 147, 158,161, 173, 180, 183,213,273

support 20, 273 function 45

textile-reinforced concrete 216 time

coordinate 43, 79, 85 dependent problems 181 discretization 49, 182 independent problems 178

trajectory 43,46,73,80,206,214,215,230 transformation of

fuzzy probabilistic basic variables 238 fuzzy random variables 236

uncertain mapping 42

uncertain mapping model 137 measure 53, 54,55,57,59 measure space 55 measurement 91,99 nonparametric estimation 117 parametric estimation 117 position 99 proposition 91 structural analysis 135 structural parameters 90, 109 transition 99

uncertainty 1 data 3, 4, 91, 228, 235, 263, 266 characteristic 7, 77 global 64 informal 7,90,117,120,135 lexical 7, 90, 135 local 64 measure of 641,245 model 3,4, 6, 103, 136, 229, 233, 234, 266 modeling of 11, 90, 109 nonstochastic 15 quantification of 12 type of 7 stochastic 7, 135

union 26 uniqueness 147 urban engineering 10

vertex method 17 virtual external fuzzy work 178 virtual internal fuzzy work 177

x-space 138, 229

y-space 233, 238

z-space 138

Index 335