Stochastic Structural Dynamics

Cho Wing Solomon To is Professor of Mechanical Engineering at the University of Nebraska-Lincoln, USA. 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.

• Dedication xiPreface xiiiAcknowledgements xv1. Introduction 11.1 Displacement Formulation Based Finite Element Method 21.2 Element Equations of Motion for Temporally and Spatially Stochastic Systems 131.3 Hybrid Stress Based Element Equations of Motion 141.4 Incremental Variational Principle and Mixed Formulation Based Nonlinear Element Matrices 181.5 Constitutive Relations and Updating of Configurations and Stresses 361.6 Concluding Remarks 48References 492. Spectral Analysis and Response Statistics of Linear Structural Systems 532.1 Spectral Analysis 532.2 Evolutionary Spectral Analysis 562.3 Evolutionary Spectra of Engineering Structures 602.4 Modal Analysis and Time-Dependent Response Statistics 762.5 Response Statistics of Engineering Structures 79References 943. Direct Integration Methods for Linear Structural Systems 973.1 Stochastic Central Difference Method 973.2 Stochastic Central Difference Method with Time Co-ordinate Transformation 1003.3 Applications 1023.4 Extended Stochastic Central Difference Method and Narrow-band Force Vector 1143.5 Stochastic Newmark Family of Algorithms 122References 1284. Modal Analysis and Response Statistics of Quasi-linear Structural Systems 1314.1 Modal Analysis of Temporally Stochastic Quasi-linear Systems 1314.2 Response Analysis Based on Melosh-Zienkiewicz-Cheung Bending Plate Finite Element 1414.3 Response Analysis Based on High Precision Triangular Plate Finite Element 1564.4 Concluding Remarks 166References 1665. Direct Integration Methods for Response Statistics of Quasi-linear Structural Systems 1695.1 Stochastic Central Difference Method for Quasi-linear Structural Systems 1695.2 Recursive Covariance Matrix of Displacements of Cantilever Pipe Containing Turbulent Fluid 1745.3 Quasi-linear Systems under Narrow-band Random Excitations 1845.4 Concluding Remarks 188References 1906. Direct Integration Methods for Temporally Stochastic Nonlinear Structural Systems 1916.1 Statistical Linearization Techniques 1916.2 Symplectic Algorithms of Newmark Family of Integration Schemes 1946.3 Stochastic Central Difference Method with Time Co-ordinate Transformation and Adaptive Time Schemes 1996.4 Outline of steps in computer program 2116.5 Large Deformations of Plate and Shell Structures 2136.6 Concluding Remarks 224References 2267. Direct Integration Methods for Temporally and Spatially Stochastic Nonlinear Structural Systems 2317.1 Perturbation Approximation Techniques and Stochastic Finite Element Methods 2327.2 Stochastic Central Difference Methods for Temporally and Spatially Stochastic Nonlinear Systems 2417.3 Finite Deformations of Spherical Shells with Large Spatially Stochastic Parameters 2517.4 Closing Remarks 255References 257Appendices1A Mass and Stiffness Matrices of Higher Order Tapered Beam Element 2611B Consistent Stiffness Matrix of Lower Order Triangular Shell Element 2671B.1 Inverse of Element Generalized Stiffness Matrix 2671B.2 Element Leverage Matrices 2681B.3 Element Component Stiffness Matrix Associated with Torsion 271References 2761C Consistent Mass Matrix of Lower Order Triangular Shell Element 277Reference 2802A Eigenvalue Solution 281References 2822B Derivation of Evolutionary Spectral Densities and Variances of Displacements 2832B.1 Evolutionary Spectral Densities Due to Exponentially Decaying Random Excitations 2832B.2 Evolutionary Spectral Densities Due to Uniformly Modulated Random Excitations 2862B.3 Variances of Displacements 288References 2972C Time-dependent Covariances of Displacements 2992D Covariances of Displacements and Velocities 3112E Time-dependent Covariances of Velocities 3172F Cylindrical Shell Element Matrices 3233A Deterministic Newmark Family of Algorithms 327Reference 331Index 333