Finite Element Methods for Structures With Large Stochastic Variations

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ISBN: 9780198526315 | 0198526318
Cover: Hardcover
Copyright: 9/4/2003

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The finite element method (FEM) can be successfully applied to various field problems in solid mechanics, fluid mechanics and electrical engineering. FEM is a numerical method widely used in computer aided engineering, which allows for modelling and analysis of engineering and mathematicalproblems, e.g. safety testing. This reference text is the first to discuss finite element methods for structures with large stochastic variations. Stochastic variations are those that follow a random probability distribution or pattern and whose behaviour may be analysed statistically but not predicted precisely. Graduate students, lecturers, and researchers in mathematics, engineering, and scientific computation will find this a very usefulreference. This book is the latest in the Oxford Texts in Applied and Engineering Mathematics series, which includes texts based on taught courses that explain the mathematical or computational techniques required for the resolution of fundamental applied problems, from the undergraduate through to thegraduate level. Other books in the series include: Jordan and Smith: Nonlinear Ordinary Differential Equations: an introduction to dynamical systems; Sobey: Introduction to Interactive Boundary Layer Theory; Scott: Nonlinear Science: emergence and dynamics of coherent structures; Tayler:Mathematical Models in Applied Mechanics; Ram-Mohan: Finite Element and Boundary Element Applications in Quantum Mechanics.

Prologue

(8)

1 Fundamentals of the finite element method

1.1 Finite element formulation for beam bending analysis

(4)

1.2 Finite element formulation for plane stress/strain analysis

(3)

2 The FEM for stochastic structures: A review and improvement

2.1 Finite element formulation by the perturbation technique

(2)

2.2 Finite element formulation by series expansion

(1)

2.3 Finite element formulation by homogeneous chaos

(3)

2.4 Improved first-order perturbation finite element formulation

(2)

2.5 Numerical results for a two-bar truss with stochastic Young's moduli

(5)

2.5.1 Improved FEMSS solutions

(1)

2.5.2 The conventional perturbation FEMSS solution

(1)

2.5.3 Exact solution

(4)

2.6 Lessons pertaining to the FEM for stochastic problems, learned from the simplest example (Brzakala and Elishakoff 2001)

(18)

2.6.1 Analysis

(1)

2.6.2 The perturbation method

(2)

2.6.3 Improved first-order perturbation

(1)

2.6.4 The Legendre polynomial chaos

(2)

2.6.5 The projection method and the Legendre polynomial chaos

(3)

2.6.6 Approximations within the Legendre polynomial chaos

(1)

2.6.7 The response surface method

(2)

2.6.8 Results and discussion

(4)

3 The FEM for stochastic structures based on the exact inverse of the stiffness matrix

3.1 The FEM based on the direct exact inverse of the stiffness matrix

(8)

3.1.1 Exact inverse of the stiffness matrix for a fixed-free bar extension

(2)

3.1.2 Solutions for a fixed-free bar under an end force

(2)

3.1.3 Solutions for a fixed-free bar under a uniform distributed load

(2)

3.1.4 Exact inverse of the stiffness matrix for a fixed-fixed bar extension

(1)

3.1.5 Numerical results for a Bar with a stochastic Young's modulus

(1)

3.2 The FEM through the generalized Fuchs method

(39)

3.2.1 New formulation of the finite element stiffness matrix

(3)

3.2.2 Imposition of displacement constraints

(2)

3.2.3 Mean and autocorrelation functions

(2)

3.2.4 Statically determinate beams

(15)

3.2.4.1 Clamped-free beam

(6)

3.2.4.2 Beam simply supported at both ends

(5)

3.2.5 Statically indeterminate beams

(23)

3.2.5.1 Clamped-simply supported beam

(9)

3.2.5.2 Beam clamped at both ends

4 Exact solutions of stochastic shear and Bernoulli-Euler beams as benchmark problems

4.1 Exact solutions for stochastic shear beams under deterministic loading

(5)

4.2 Exact solutions for stochastic shear beams under random loading

(2)

4.3 Exact solutions for stochastic Bernoulli-Euler hams

101

(16)

4.3.1 Deterministic beam under a random load represented as a random field

101

(6)

4.3.2 Beam with random flexibility represented by a Random variable, and subjected to a random field load

107

(3)

4.3.3 Beam with random field flexibility, and subjected to a load depending on a random variable

110

(3)

4.3.4 Beam with uncorrelated random flexibility and a random load

113

(4)

5 Variational principles-based FEM for stochastic beams

5.1 Stochastic variational principles for a shear beacon

117

(2)

5.2 Stochastic version of the Rayleigh-Ritz method

119

(1)

5.3 Basic equations

120

(2)

5.4 Variational principles

122

(2)

5.5 Stochastic version of the Bubnov-Galerku method

124

(2)

5.6 Stochastic version of the Rayleigh-Ritz method

126

(1)

5.7 The finite element formulation

127

(2)

5.7.1 Formulation for the mean displacement

127

(1)

5.7.2 Formulation for the covariance function

128

(1)

5.8 Numerical examples

129

(2)

5.8.1 Cantilever beam under uniform pressure

129

(2)

5.9 Simply supported beam under uniform pressure

131

(6)

5.9.1 Simply supported beam under uniform pressure-the FEM solution

133

(4)

6 Element-level flexibility-based FEM for stochastic structures

6.1 Formulation

137

(5)

6.2 Formulation for plane stress and strain analysis

142

(2)

6.3 Application of the simulation technique in the FEM for stochastic structures

144

(1)

6.4 Examples

145

(11)

6.4.1 A clamped beam subjected to a uniform load q

145

(3)

6.4.2 A square in a plane stress state

148

(3)

6.4.3 A rectangle under a concentrated load

151

(1)

6.4.4 A rectangle under a concentrated force by the simulation technique

152

(4)

7 A comparison of stochastic and interval finite elements

7.1 Introduction

156

(1)

7.2 Formulation of the element and global stiffness matrices

157

(6)

7.2.1 Stochastic model

158

(3)

7.2.2 Interval model

161

(2)

7.3 Numerical examples

163

(12)

7.3.1 Stochastic model

164

(2)

7.3.2 Interval model

166

(3)

Epilogue

169

(6)

Appendices

A Elements of the nodal displacement correlation matrix for a clamped-free beam

175

(1)

B Elements of the nodal displacement correlation matrix for a simply supported beam

176

(2)

C Elements of the nodal displacement correlation matrix for a clamped-simply supported beam

178

(3)

D Correlation matrix of elements of the nodal displacement correlation matrix for a clamped-clamped beam

181

(1)

E Stochastic clamped-free beam under stochastic loading

182

(2)

F Proof of the boundary conditions in egns (5.32)

184

(2)

G Proof of the functional r2 in eqn (5.35)

186

(4)

H Proof of the variational principle in eqn (5.36)

190

(1)

I Exact solutions for the cantilever beam in Example 5.1

191

(2)

J Exact solution for the simply supported beam in example 5.2

193

(2)

K Addition and subtraction of two intervals

195

(1)

Bibliography

196

(50)

Author Index

246

(11)

Subject Index

257

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Finite Element Methods for Structures With Large Stochastic Variations

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