Kirsch Design-Oriented Analysis of Structures

A Unified Approach
1. Auflage 2006
ISBN: 978-0-306-48631-9
Verlag: Springer Netherlands
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 95, 255 Seiten

Reihe: Solid Mechanics and Its Applications

This book deals with problems of multiple repeated analyses (reanalysis) of structures. It introduces various concepts and methods, and presents them in a unified approach. This should prove useful to students, researchers, consultants, and practising engineers involved in analysis and design of structures. Reanalysis is common to numerous analysis and design tasks, and it is needed in such areas as structural optimisation, damage analysis, non-linear analysis, and probabilistic analysis.

The material presented in the text is related to a wide range of applications in such fields as aerospace engineering, civil engineering, mechanical engineering, and naval architecture. The book discusses various analysis models, including linear and non-linear analysis, static and dynamic analysis, and design sensitivity analysis.

It presents direct as well as approximate methods, and demonstrates how various concepts and methods are integrated to achieve effective solution procedures. Previous books on structural analysis do not cover most of the material presented in the book. To clarify the presentation, many illustrative examples and numerical results are demonstrated.

Autoren/Hrsg.

Kirsch, Uri

Inhaltsverzeichnis

1;Table of Contents;7
2;Preface;11
3;Acknowledgements;15
4;PART ONE CONCEPTS AND METHODS;16
4.1;1. Introduction;18
4.1.1;1.1 ANALYSIS AND REANALYSIS;18
4.1.1.1;1.1.1 Structural Analysis;18
4.1.1.2;1.1.2 Design Variables;20
4.1.1.3;1.1.3 Changes in the Structural Model;22
4.1.1.4;1.1.4 Reanalysis of Structures;26
4.1.2;1.2 SCOPE OF TEXT;27
4.1.3;1.3 REFERENCES;30
4.2;2 Structural Analysis;32
4.2.1;2.1 LINEAR ANALYSIS OF FRAMED STRUCTURES;33
4.2.1.1;2.1.1 Basic Relations;33
4.2.1.2;2.1.2 Solution by the Displacement Method;35
4.2.2;2.2 CONTINUUM STRUCTURES;38
4.2.3;2.3 NONLINEAR ANALYSIS;45
4.2.3.1;2.3.1 Geometrical Non-linearity;46
4.2.3.2;2.3.2 Material Non-linearity;48
4.2.4;2.4 DYNAMIC ANALYSIS;48
4.2.4.1;2.4.1 The Eigenproblem;49
4.2.5;2.5 COLLAPSE AND BUCKLING ANALYSIS;53
4.2.6;2.6 REFERENCES;55
4.3;3 Reanalsis of Structures;56
4.3.1;3.1 FORMULATION OF REANALYSIS PROBLEMS;56
4.3.1.1;3.1.1 Linear Reanalysis;56
4.3.1.2;3.1.2 Nonlinear Reanalysis;57
4.3.1.3;3.1.3 Vibration Reanalysis;59
4.3.2;3.2 REANALYSIS METHODS;61
4.3.2.1;3.2.1 Direct Methods;62
4.3.2.2;3.2.2 Approximate Methods;62
4.3.3;3.3 REFERENCES;66
4.4;4 Direct Methods;70
4.4.1;4.1 A SINGLE RANK-ONE CHANGE;70
4.4.2;4.2 MULTIPLE RANK-ONE CHANGES;72
4.4.3;4.3 GENERAL PROCEDURE;75
4.4.4;4.4 REFERENCES;78
4.5;5 Local Approximations;80
4.5.1;5.1 SERIES EXPANSION;80
4.5.1.1;5.1.1 The Taylor Series;80
4.5.1.2;5.1.2 The Binomial Series;81
4.5.1.3;5.1.3 Homogeneous Functions;83
4.5.2;5.2 INTERMEDIATE VARIABLES;85
4.5.2.1;5.2.1 Conservative and Convex Approximations;86
4.5.2.2;5.2.2 Intermediate Response Functions;88
4.5.3;5.3 IMPROVED SERIES APPROXIMATIONS;90
4.5.3.1;5.3.1 Scaling of the Initial Design;91
4.5.3.2;5.3.2 Scaling of Displacements;97
4.5.4;5.4 REFERENCES;99
4.6;6 Global Approximations;100
4.6.1;6.1 POLYNOMIAL FITTING AND RESPONSE SURFACE;101
4.6.1.1;6.1.1 Polynomial Fitting;101
4.6.1.2;6.1.2 Least-Square Solutions;105
4.6.2;6.2 REDUCED BASIS;107
4.6.2.1;6.2.1 Static Analysis;108
4.6.2.2;6.2.2 Dynamic Analysis;110
4.6.3;6.3 THE CONJUGATE GRADIENT METHOD;116
4.6.3.1;6.3.1 Solution Procedure;116
4.6.3.2;6.3.2 Preconditioned Conjugate Gradient;118
4.6.4;6.4 REFERENCES;120
5;PART TWO A UNIFIED APPROACH;122
5.1;7 Combined Approximations (CA);124
5.1.1;7.1 COUPLED BASIS VECTORS;125
5.1.1.1;7.1.1 Determining the Basis Vectors;125
5.1.1.2;7.1.2 Solution Procedure;127
5.1.2;7.2 UNCOUPLED BASIS VECTORS;130
5.1.2.1;7.2.1 Determining the Basis Vectors;130
5.1.2.2;7.2.2 Solution Procedure;133
5.1.3;7.3 ACCURATE SOLUTIONS;134
5.1.3.1;7.3.1 Linearly Dependent Basis Vectors;134
5.1.3.2;7.3.2 Equivalence of the CA Method and the PCG Method;135
5.1.3.3;7.3.3 Error Evaluation;137
5.1.3.4;7.3.4 Scaled and Nearly Scaled Designs;139
5.1.3.5;7.3.5 High-Order Approximations;147
5.1.4;7.4 REFERENCES;150
5.2;8 Simplified Solution Procedures;152
5.2.1;8.1 LOW-ORDER APPROXIMATIONS;152
5.2.1.1;8.1.1 Structural Optimization;154
5.2.1.2;8.1.2 Reanalysis of Damaged Structures;159
5.2.1.3;8.1.3 Efficiency of the Calculations;163
5.2.1.4;8.1.4 Limitations on Design Changes;165
5.2.2;8.2 EXACT SOLUTIONS;166
5.2.2.1;8.2.1 Multiple Rank-One Changes;167
5.2.2.2;8.2.2 Equivalence of the CA Method and the S-M Formula;168
5.2.2.3;8.2.3 Equivalence of the CA Method and the Woodbury Formula;169
5.2.2.4;8.2.4 Solution Procedure;169
5.2.3;8.3 REFERENCES;173
5.3;9 Topological and Geometrical Changes;176
5.3.1;9.1 TOPOLOGICAL CHANGES;177
5.3.1.1;9.1.1 Number of DOF is Unchanged;178
5.3.1.2;9.1.2 Number of DOF is Decreased;184
5.3.1.3;9.1.3 Number of DOF is Increased;187
5.3.2;9.2 GEOMETRICAL CHANGES;191
5.3.2.1;9.2.1 Accurate Solutions;192
5.3.2.2;9.2.2 Exact solutions;199
5.3.3;9.3 REFERENCES;200
5.4;10 Design Sensitivity Analysis;202
5.4.1;10.1 EXACT ANALYTICAL DERIVATIVES;203
5.4.1.1;10.1.1 Direct Method;204
5.4.2;10.2 APPROXIMATE FIRST-ORDER DERIVATIVES;205
5.4.2.1;10.2.1 Direct Approximations (DA);206
5.4.2.2;10.2.2 Adjoint-Variable Approximations (AVA);207
5.4.2.3;10.2.3 Finite Difference Approximations (FDA);208
5.4.3;10.3 COMPARISON OF RESULTS;211
5.4.3.1;10.3.1 Accuracy of the Calculations;211
5.4.3.2;10.3.2 Computational Efficiency;211
5.4.3.3;10.3.3 Ease-of-Implementation;212
5.4.4;10.4 SECOND-ORDER DERIVATIVES;212
5.4.5;10.5 COMPUTATIONAL PROCEDURE;213
5.4.6;10.6 REFERENCES;219
5.5;11 Nonlinear Reanalysis;222
5.5.1;11.1 GEOMETRIC NONLINEAR ANALYSIS;222
5.5.2;11.2 NONLINEAR ANALYSIS BY THE CA METHOD;223
5.5.3;11.3 NONLINEAR REANALYSIS BY THE CA METHOD;225
5.5.4;11.4 REFERENCES;235
5.6;12 Vibration Reanalysis;236
5.6.1;12.1 VIBRATIONANALYSIS;236
5.6.2;12.2 FORMULATION OF EIGENPROBLEM REANALYSIS;238
5.6.3;12.3 REANALYSIS BY THE CA METHOD;239
5.6.4;12.4 EVALUATION OF MODIFIED EIGENVALUES;245
5.6.5;12.5 REFERENCES;246
6;Subject Index;248
7;More eBooks at www.ciando.com;0

Leseproben

6 Global Approximations (p. 85-86)

Global (multipoint) approximations are obtained by analyzing the structure at a number of design points, and they are valid for the whole design space (or, at least, large regions of it). This type of approximation may require much computational effort, particularly in problems with large numbers of design variables. This difficulty can be alleviated by the approach presented in Chapter 7.

Polynomial fitting and response surface methods are introduced in Section
6.1. In response surface methods, the response functions are replaced by simple functions (polynomials), which are fitted to data computed at a set of selected design points. So far in practice, the use of these methods has been limited to problems with a few design variables.

Reduced-basis methods are presented in Section 6.2. Using this approach, we approximate the response of a large system, which is originally described by many degrees of freedom, by a linear combination of a few pre-selected basis vectors. The problem is then stated in terms of a small number of unknown coefficients of the basis vectors. This approach is most effective in cases where highly accurate approximations can be achieved by the reduced system of equations. A basic question in using reduced basis methods relates to the choice of an appropriate set of the basis vectors. Response vectors of previously analyzed designs could be used, but an ad hoc or intuitive choice of these vectors may not lead to satisfactory approximations.

In addition, calculation of the basis vectors requires several exact analyses of the structure for the basis design points, which might involve extensive computational effort. A method for selecting the basis vectors that provides efficient and accurate results is presented in Section 7.1.1.

The conjugate gradient method described in Section 6.3 is an iterative method for solving a set of linear equations. The problem can be stated equivalently as the minimization of a quadratic function. The method generates a set of conjugate vectors such that the solution requires little storage and computation. If the quadratic function is minimized sequentially in an n dimensional space, once along each of a set of n conjugate directions, the minimum will be found at or before the nth step, regardless of the starting point.

A preconditioned conjugate gradient method, intended to accelerate convergence of ill-conditioned problems by transformation of the set of linear equations, is then developed.

6.1 POLYNOMIAL FITTING AND RESPONSE SURFACE

When the number of design variables is small it might be practical to analyze the structure at a number of design points, and use the response at those points to construct a polynomial approximation to the response at other points. Polynomial approximations obtained by analyzing the structure at a number of design points are global approximations. Obtaining such approximations can be quite expensive for problems with large numbers of design variables. For example, if the object is to fit the structural response by a quadratic polynomial, it is necessary to analyze the structure for at least n(n+1)/2 design points (typically, many more are required to ensure a robust approximation), where n is the number of design variables [1]. The most common global approximation is the response surface approach. Using this approach, we compute the response functions at a number of points, and then fit an analytical response surface, such as a polynomial, to the data.