Reanalysis of Structures

Contents 1 Static Analysis 1.1 Linear Analysis of Framed Structures 1.2 Linear Analysis of Continuum Structures 1.3 Solution of the Linear Equilibrium Equations 1.3.1 Matrix Factorization 1.3.2 Iterative Solution Procedures 1.4 Nonlinear Analysis 1.4.1 Geometric Nonlinearity 1.4.2 Material Nonlinearity 1.4.3 Time Varying Loads 1.4.4 Buckling Analysis 1.4.5 Solution of the Nonlinear Equations References 2 Vibration Analysis 2.1 Free Vibration 2.1.1 Properties of the Eigenproblem 2.1.2 The Standard Eigenproblem and the Rayleigh Quotient 2.2 Solution of the Eigenproblem 2.2.1 Static Condensation 2.2.2 Solution Methods 2.3 Vector Iteration Methods 2.3.1 Inverse Vector Iteration 2.3.2 Vector Iteration with Shifts 2.3.3 Matrix Deflation and Gram-Schmidt Orthogonalization 2.4 Transformation Methods 2.5 Polynomial Iterations 2.6 Rayleigh-Ritz Analysis 2.6.1 Approximate Eigenproblem Solution 2.6.2 Load-Dependent Ritz Vectors 2.7 The Lanczos Method 2.8 Subspace Iteration References 3 Dynamic Analysis 3.1 Linear Dynamic Analysis 3.1.1 Direct Integration 3.1.2 Mode Superposition 3.1.3 Special Analysis Procedures 3.2 Reduced Basis 3.2.1 Static Analysis 3.2.2 Dynamic Analysis 3.3 Nonlinear Dynamic Analysis 3.3.1 Implicit Integration 3.3.2 Mode Superposition References 4 Reanalysis of Structures 4.1 Design Variables 4.2 Formulation of Static Reanalysis 4.2.1 Linear Static Reanalysis 4.2.2 Nonlinear Static Reanalysis 4.3 Formulation of Vibration Reanalysis 4.3.1 Eigenproblem Reanalysis 4.3.2 Iterative Procedures 4.4 Formulation of Dynamic Reanalysis 4.4.1 Linear Dynamic Reanalysis 4.4.2 Nonlinear Dynamic Reanalysis 4.5 Reanalysis Methods 4.5.1 Direct Methods 4.5.2 Approximate Methods 4.5.3 The Combined Approximations Approach References 5 Static Reanalysis 5.1 Determination of the Basis Vectors 5.1.1 The Binomial Series 5.1.2 Calculation of the Basis Vectors 5.1.3 Convergence of the Series 5.2 Linear Reanalysis 5.2.1 Coupled Basis Vectors 5.2.2 Uncoupled Basis Vectors 5.3 Topological Changes 5.3.1 Number of DOF is Decreased 5.3.2 Number of DOF is Increased 5.4 Nonlinear Analysis and Reanalysis 5.4.1 Problem Formulation 5.4.2 Solution by Combined Approximations 5.4.3 Procedures for Analysis and Reanalysis References 6 Vibration Reanalysis 6.1 The Reduced Eigenproblem 6.1.1 Problem Formulation 6.1.2 Determination of the Basis Vectors 6.2 Improved Basis Vectors 6.2.1 Gram-Schmidt Orthogonalizations of the Modes 6.2.2 Shifts of the Basis Vectors 6.2.3 Gram-Schmidt Orthogonalizations of the Basis Vectors 6.3 General Solution Procedure 6.4 Numerical Examples 6.5 Reanalysis by Iterative Procedures References 7 Dynamic Reanalysis 7.1 Linear Dynamic Reanalysis 7.1.1 Solution by Direct Integration 7.1.2 Solution by Mode Superposition 7.2 Nonlinear Dynamic Reanalysis 7.2.1 Solution by Implicit Integration 7.2.2 Solution by Mode Superposition References 8 Direct Reanalysis 8.1 Direct Methods 8.1.1 A Single Rank-One Change 8.1.2 Multiple Rank-One Changes 8.1.3 General Procedure 8.2 Direct Solutions by Combined Approximations. 8.2.1 Multiple Rank-One Changes 8.2.2 Solution Procedure 8.3 Topological and Geometrical Changes 8.3.1 Topological Changes 8.3.2 Geometrical Changes References 9 Repeated Sensitivity Analysis 9.1 Finite-Difference Derivatives 9.2 Static Problems 9.2.1 Analytical Derivatives 9.2.2 Semi-Analytical Derivatives 9.2.3 Repeated Analytical Derivatives 9.2.4 Repeated Finite-Difference Derivatives 9.2.5 Errors Due to Rigid Body Motions 9.3 Vibration Problems 9.3.1 Analytical Derivatives 9.3.2 Repeated Finite-Difference Derivatives 9.3.3 Improved Basis Vectors using Central-Differences 9.4 Linear Dynamic Problems 9.4.1 Modal Analysis Equations 9.4.2 Analytical Derivatives 9.4.3 Reducing the Number of Differential Equations 9.5 Nonlinear Dynamic Problems 9.5.1 Modal Analysis Equations 9.5.2 Nonlinear-Dynamic Sensitivity Analysis 9.5.3 Efficient Solution Procedures References 10 Computational Considerations 10.1 Efficiency of the Calculations 10.1.1 Static Reanalysis 10.1.2 Dynamic Reanalysis 10.1.3 Dynamic Sensitivity Analysis 10.2 Accuracy Considerations 10.2.1 Linearly Dependent Basis Vectors 10.2.2 Scaled and Nearly Scaled Designs 10.3 Equivalence of the PCG and CA Methods 10.4 Error Evaluation 10.4.1 Static Reanalysis 10.4.2 Vibration Reanalysis 10.5 Accuracy of Forces and Stresses References Index