Stochastic Structural Dynamics

Dedication xi

Preface xiii

Acknowledgements xv

1. Introduction 1

1.1 Displacement Formulation Based Finite Element Method 2

1.2 Element Equations of Motion for Temporally and Spatially Stochastic Systems 13

1.3 Hybrid Stress Based Element Equations of Motion 14

1.4 Incremental Variational Principle and Mixed Formulation Based Nonlinear Element Matrices 18

1.5 Constitutive Relations and Updating of Configurations and Stresses 36

1.6 Concluding Remarks 48

References 49

2. Spectral Analysis and Response Statistics of Linear Structural Systems 53

2.1 Spectral Analysis 53

2.2 Evolutionary Spectral Analysis 56

2.3 Evolutionary Spectra of Engineering Structures 60

2.4 Modal Analysis and Time-Dependent Response Statistics 76

2.5 Response Statistics of Engineering Structures 79

References 94

3. Direct Integration Methods for Linear Structural Systems 97

3.1 Stochastic Central Difference Method 97

3.2 Stochastic Central Difference Method with Time Co-ordinate Transformation 100

3.3 Applications 102

3.4 Extended Stochastic Central Difference Method and Narrow-band Force Vector 114

3.5 Stochastic Newmark Family of Algorithms 122

References 128

4. Modal Analysis and Response Statistics of Quasi-linear Structural Systems 131

4.1 Modal Analysis of Temporally Stochastic Quasi-linear Systems 131

4.2 Response Analysis Based on Melosh-Zienkiewicz-Cheung Bending Plate Finite Element 141

4.3 Response Analysis Based on High Precision Triangular Plate Finite Element 156

4.4 Concluding Remarks 166

References 166

5. Direct Integration Methods for Response Statistics of Quasi-linear Structural Systems 169

5.1 Stochastic Central Difference Method for Quasi-linear Structural Systems 169

5.2 Recursive Covariance Matrix of Displacements of Cantilever Pipe Containing Turbulent Fluid 174

5.3 Quasi-linear Systems under Narrow-band Random Excitations 184

5.4 Concluding Remarks 188

References 190

6. Direct Integration Methods for Temporally Stochastic Nonlinear Structural Systems 191

6.1 Statistical Linearization Techniques 191

6.2 Symplectic Algorithms of Newmark Family of Integration Schemes 194

6.3 Stochastic Central Difference Method with Time Co-ordinate Transformation and Adaptive Time Schemes 199

6.4 Outline of steps in computer program 211

6.5 Large Deformations of Plate and Shell Structures 213

6.6 Concluding Remarks 224

References 226

7. Direct Integration Methods for Temporally and Spatially Stochastic Nonlinear Structural Systems 231

7.1 Perturbation Approximation Techniques and Stochastic Finite Element Methods 232

7.2 Stochastic Central Difference Methods for Temporally and Spatially Stochastic Nonlinear Systems 241

7.3 Finite Deformations of Spherical Shells with Large Spatially Stochastic Parameters 251

7.4 Closing Remarks 255

References 257

Appendices

1A Mass and Stiffness Matrices of Higher Order Tapered Beam Element 261

1B Consistent Stiffness Matrix of Lower Order Triangular Shell Element 267

1B.1 Inverse of Element Generalized Stiffness Matrix 267

1B.2 Element Leverage Matrices 268

1B.3 Element Component Stiffness Matrix Associated with Torsion 271

References 276

1C Consistent Mass Matrix of Lower Order Triangular Shell Element 277

Reference 280

2A Eigenvalue Solution 281

References 282

2B Derivation of Evolutionary Spectral Densities and Variances of Displacements 283

2B.1 Evolutionary Spectral Densities Due to Exponentially Decaying Random Excitations 283

2B.2 Evolutionary Spectral Densities Due to Uniformly Modulated Random Excitations 286

2B.3 Variances of Displacements 288

References 297

2C Time-dependent Covariances of Displacements 299

2D Covariances of Displacements and Velocities 311

2E Time-dependent Covariances of Velocities 317

2F Cylindrical Shell Element Matrices 323

3A Deterministic Newmark Family of Algorithms 327

Reference 331

Index 333