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<p>Dedication xi</p> <p>Preface xiii</p> <p>Acknowledgements xv</p> <p><b>1. Introduction 1</b></p> <p>1.1 Displacement Formulation Based Finite Element Method 2</p> <p>1.2 Element Equations of Motion for Temporally and Spatially Stochastic Systems 13</p> <p>1.3 Hybrid Stress Based Element Equations of Motion 14</p> <p>1.4 Incremental Variational Principle and Mixed Formulation Based Nonlinear Element Matrices 18</p> <p>1.5 Constitutive Relations and Updating of Configurations and Stresses 36</p> <p>1.6 Concluding Remarks 48</p> <p>References 49</p> <p><b>2. Spectral Analysis and Response Statistics of Linear Structural Systems 53</b></p> <p>2.1 Spectral Analysis 53</p> <p>2.2 Evolutionary Spectral Analysis 56</p> <p>2.3 Evolutionary Spectra of Engineering Structures 60</p> <p>2.4 Modal Analysis and Time-Dependent Response Statistics 76</p> <p>2.5 Response Statistics of Engineering Structures 79</p> <p>References 94</p> <p><b>3. Direct Integration Methods for Linear Structural Systems 97</b></p> <p>3.1 Stochastic Central Difference Method 97</p> <p>3.2 Stochastic Central Difference Method with Time Co-ordinate Transformation 100</p> <p>3.3 Applications 102</p> <p>3.4 Extended Stochastic Central Difference Method and Narrow-band Force Vector 114</p> <p>3.5 Stochastic Newmark Family of Algorithms 122</p> <p>References 128</p> <p><b>4. Modal Analysis and Response Statistics of Quasi-linear Structural Systems 131</b></p> <p>4.1 Modal Analysis of Temporally Stochastic Quasi-linear Systems 131</p> <p>4.2 Response Analysis Based on Melosh-Zienkiewicz-Cheung Bending Plate Finite Element 141</p> <p>4.3 Response Analysis Based on High Precision Triangular Plate Finite Element 156</p> <p>4.4 Concluding Remarks 166</p> <p>References 166</p> <p><b>5. Direct Integration Methods for Response Statistics of Quasi-linear Structural Systems 169</b></p> <p>5.1 Stochastic Central Difference Method for Quasi-linear Structural Systems 169</p> <p>5.2 Recursive Covariance Matrix of Displacements of Cantilever Pipe Containing Turbulent Fluid 174</p> <p>5.3 Quasi-linear Systems under Narrow-band Random Excitations 184</p> <p>5.4 Concluding Remarks 188</p> <p>References 190</p> <p><b>6. Direct Integration Methods for Temporally Stochastic Nonlinear Structural Systems 191</b></p> <p>6.1 Statistical Linearization Techniques 191</p> <p>6.2 Symplectic Algorithms of Newmark Family of Integration Schemes 194</p> <p>6.3 Stochastic Central Difference Method with Time Co-ordinate Transformation and Adaptive Time Schemes 199</p> <p>6.4 Outline of steps in computer program 211</p> <p>6.5 Large Deformations of Plate and Shell Structures 213</p> <p>6.6 Concluding Remarks 224</p> <p>References 226</p> <p><b>7. Direct Integration Methods for Temporally and Spatially Stochastic Nonlinear Structural Systems 231</b></p> <p>7.1 Perturbation Approximation Techniques and Stochastic Finite Element Methods 232</p> <p>7.2 Stochastic Central Difference Methods for Temporally and Spatially Stochastic Nonlinear Systems 241</p> <p>7.3 Finite Deformations of Spherical Shells with Large Spatially Stochastic Parameters 251</p> <p>7.4 Closing Remarks 255</p> <p>References 257</p> <p><b>Appendices</b></p> <p>1A Mass and Stiffness Matrices of Higher Order Tapered Beam Element 261</p> <p>1B Consistent Stiffness Matrix of Lower Order Triangular Shell Element 267</p> <p>1B.1 Inverse of Element Generalized Stiffness Matrix 267</p> <p>1B.2 Element Leverage Matrices 268</p> <p>1B.3 Element Component Stiffness Matrix Associated with Torsion 271</p> <p><b>References 276</b></p> <p>1C Consistent Mass Matrix of Lower Order Triangular Shell Element 277</p> <p><b>Reference 280</b></p> <p>2A Eigenvalue Solution 281</p> <p><b>References 282</b></p> <p>2B Derivation of Evolutionary Spectral Densities and Variances of Displacements 283</p> <p>2B.1 Evolutionary Spectral Densities Due to Exponentially Decaying Random Excitations 283</p> <p>2B.2 Evolutionary Spectral Densities Due to Uniformly Modulated Random Excitations 286</p> <p>2B.3 Variances of Displacements 288</p> <p><b>References 297</b></p> <p>2C Time-dependent Covariances of Displacements 299</p> <p>2D Covariances of Displacements and Velocities 311</p> <p>2E Time-dependent Covariances of Velocities 317</p> <p>2F Cylindrical Shell Element Matrices 323</p> <p>3A Deterministic Newmark Family of Algorithms 327</p> <p><b>Reference 331</b></p> <p><b>Index 333</b></p>

<p><b>Cho Wing Solomon To is Professor of Mechanical Engineering at the University of Nebraska-Lincoln, USA.</b> He gained his Ph.D from the University of Southampton, UK, in 1980. Prior to joining UNL, he was a Professor at the University of Western Ontario and an Associate Professor at the University of Calgary. He was a Research Fellow of the Natural Sciences and Engineering Research Council of Canada from 1982-1992, and a Research Fellow at the Institute of Sound and Vibration Research (ISVR), University of Southampton. He is a member of the American Society of Mechanical Engineers (ASME), the American Academy of Mechanics (AAM), the Society of Industrial and Applied Mathematics (SIAM), and a founder Fellow of the Institution of Diagnostics Engineers, U.K. He served as chair of the ASME Finite Element Techniques and Computational Technologies Technical Committee. Dr To’s research interests include Sound and Vibration Studies, Solid and Computational Mechanics, and System Dynamics and Controls.</p>

<p>The parallel developments of the Finite Element Methods (FEM) and the engineering applications of stochastic processes in the mid-20^th century provided a combined numerical analysis tool for the studies of dynamics of structures and structural systems under random loadings. In the open literature, there are books on statistical dynamics of structures and books on structural dynamics with chapters dealing with random response analysis. However, a systematic treatment of stochastic structural dynamics applying the finite element methods seems to be lacking, and this book redresses the balance.</p> <p>Aimed at advanced and specialist levels, the author presents and illustrates closed form solutions and the stochastic central difference methods for the computation of response statistics of linear and quasi-linear structures to nonstationary random loads. In addition to the linear and quasi-linear problems, the text also addresses highly nonlinear systems with uncertain parameters of large variations. The systems considered in the text include beam, plate and shell structures which are represented by the FEM.</p> <p>Key features:</p> <ul> <li>Fully illustrated throughout and aimed at advanced and specialist levels, the focus is on closed form solutions and computational approaches</li> <li>Emphasizes results of time-dependent response statistics of beam, plate and shell structures represented by the FEM</li> <li>Presents and illustrates computed results of highly nonlinear deformations of finite rotations and finite strains with large uncertain variations</li> <li>Presents efficient methods for large-scale computation of time-dependent linear and nonlinear response statistics of structural systems approximated by the FEM</li> </ul>

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