Reanalysis of Structures: A Unified Approach for Linear, Nonlinear, Static and Dynamic Systems

Reanalysis of Structures - ... Linear, Nonlinear, Static and Dynamic Systems, 2008......Page 3
Contents......Page 10
1 Static Analysis......Page 14
1.1 Linear Analysis of Framed Structures......Page 16
1.2 Linear Analysis of Continuum Structures......Page 19
1.3.1 Matrix Factorization......Page 26
1.3.2 Iterative Solution Procedures......Page 28
1.4 Nonlinear Analysis......Page 33
1.4.1 Geometric Nonlinearity......Page 34
1.4.2 Material Nonlinearity......Page 39
1.4.3 Time Varying Loads......Page 42
1.4.4 Buckling Analysis......Page 43
1.4.5 Solution of the Nonlinear Equations......Page 45
References......Page 48
2.1 Free Vibration......Page 50
2.1.1 Properties of the Eigenproblem......Page 52
2.1.2 The Standard Eigenproblem and the Rayleigh Quotient......Page 53
2.2.1 Static Condensation......Page 55
2.2.2 Solution Methods......Page 56
2.3.1 Inverse Vector Iteration......Page 58
2.3.2 Vector Iteration with Shifts......Page 61
2.3.3 Matrix Deflation and Gram-Schmidt Orthogonalization......Page 63
2.4 Transformation Methods......Page 64
2.5 Polynomial Iterations......Page 65
2.6.1 Approximate Eigenproblem Solution......Page 67
2.6.2 Load-Dependent Ritz Vectors......Page 68
2.7 The Lanczos Method......Page 71
2.8 Subspace Iteration......Page 73
References......Page 76
3.1 Linear Dynamic Analysis......Page 78
3.1.1 Direct Integration......Page 79
3.1.2 Mode Superposition......Page 85
3.1.3 Special Analysis Procedures......Page 91
3.2 Reduced Basis......Page 94
3.2.1 Static Analysis......Page 95
3.2.2 Dynamic Analysis......Page 97
3.3.1 Implicit Integration......Page 99
3.3.2 Mode Superposition......Page 100
References......Page 104
4 Reanalysis of Structures......Page 106
4.1 Design Variables......Page 108
4.2.1 Linear Static Reanalysis......Page 110
4.2.2 Nonlinear Static Reanalysis......Page 112
4.3.1 Eigenproblem Reanalysis......Page 115
4.3.2 Iterative Procedures......Page 116
4.4.1 Linear Dynamic Reanalysis......Page 118
4.4.2 Nonlinear Dynamic Reanalysis......Page 120
4.5 Reanalysis Methods......Page 122
4.5.2 Approximate Methods......Page 123
4.5.3 The Combined Approximations Approach......Page 125
References......Page 129
5. Static Reanalysis......Page 134
5.1.1 The Binomial Series......Page 135
5.1.2 Calculation of the Basis Vectors......Page 136
5.2.1 Coupled Basis Vectors......Page 138
5.2.2 Uncoupled Basis Vectors......Page 144
5.3.1 Number of DOF is Decreased......Page 152
5.3.2 Number of DOF is Increased......Page 155
5.4.1 Problem Formulation......Page 159
5.4.2 Solution by Combined Approximations......Page 161
5.4.3 Procedures for Analysis and Reanalysis......Page 164
References......Page 171
6 Vibration Reanalysis......Page 172
6.1.1 Problem Formulation......Page 173
6.1.2 Determination of the Basis Vectors......Page 174
6.2.1 Gram-Schmidt Orthogonalizations of the Modes......Page 176
6.2.2 Shifts of the Basis Vectors......Page 177
6.3 General Solution Procedure......Page 179
6.4 Numerical Examples......Page 181
6.5 Reanalysis by Iterative Procedures......Page 188
References......Page 196
7.1 Linear Dynamic Reanalysis......Page 198
7.1.1 Solution by Direct Integration......Page 199
7.1.2 Solution by Mode Superposition......Page 203
7.2.2 Solution by Mode Superposition......Page 210
References......Page 217
8 Direct Reanalysis......Page 220
8.1.1 A Single Rank-One Change......Page 221
8.1.2 Multiple Rank-One Changes......Page 222
8.1.3 General Procedure......Page 225
8.2 Direct Solutions by Combined Approximations......Page 228
8.2.1 Multiple Rank-One Changes......Page 229
8.2.2 Solution Procedure......Page 230
8.3.1 Topological Changes......Page 234
8.3.2 Geometrical Changes......Page 236
References......Page 237
9 Repeated Sensitivity Analysis......Page 240
9.1 Finite-Difference Derivatives......Page 241
9.2.1 Analytical Derivatives......Page 243
9.2.2 Semi-Analytical Derivatives......Page 244
9.2.3 Repeated Analytical Derivatives......Page 246
9.2.4 Repeated Finite-Difference Derivatives......Page 250
9.2.5 Errors Due to Rigid Body Motions......Page 253
9.3.1 Analytical Derivatives......Page 256
9.3.2 Repeated Finite-Difference Derivatives......Page 257
9.3.3 Improved Basis Vectors using Central-Differences......Page 260
9.4.1 Modal Analysis Equations......Page 262
9.4.2 Analytical Derivatives......Page 263
9.4.3 Reducing the Number of Differential Equations......Page 264
9.5.1 Modal Analysis Equations......Page 268
9.5.2 Nonlinear-Dynamic Sensitivity Analysis......Page 269
9.5.3 Efficient Solution Procedures......Page 271
References......Page 275
10 Computational Considerations......Page 278
10.1.1 Static Reanalysis......Page 279
10.1.2 Dynamic Reanalysis......Page 282
10.1.3 Dynamic Sensitivity Analysis......Page 283
10.2.1 Linearly Dependent Basis Vectors......Page 286
10.2.2 Scaled and Nearly Scaled Designs......Page 289
10.3 Equivalence of the PCG and CA methods......Page 291
10.4.1 Static Reanalysis......Page 294
10.4.2 Vibration Reanalysis......Page 297
10.5 Accuracy of Forces and Stresses......Page 302
References......Page 304
Index......Page 306