Reliability-based structural design

As modern structures require more critical and complex designs, the need for accurate ways to assess uncertainties in loads, geometry, material properties, manufacturing processes and operational environments has increased. Reliability assessment techniques help to develop safe designs and identify where contributors of uncertainty occur in structural systems.

This book provides readers with an understanding of the fundamentals and applications of structural reliability, stochastic finite element method, reliability analysis via stochastic expansion, and optimization under uncertainty. Probability theory, statistic methods, and reliability analysis methods are discussed. In addition, the use of stochastic expansions for the reliability analysis of practical engineering problems is also examined throught the use of examples of practical engineering applications.

This book will be of value to graduates and post graduates studying in this field as well as engineers, researchers, and technical managers.

Author Notes

Dr Seung-Kyum Choi earned his PhD in mechanical and materials engineering at Wright State University, OH, USA. His research interests include structural reliability and probabilistic mechanics, statistical approaches to design of mechanical systems, and multidisciplinary design optimization.

Dr Ramana Grandhi is the distinguished professor of mechanical and materials engineering at Wright State University, OH, USA. His research interests are in multidisciplinary analysis and optimization, probabilistic mechanics, and metal forming. Dr Grandhi has conducted sponsored research for the US Air Force, US Navy, NSF, NASA, DARPA, GE, GM and Caterpillar. He is a fellow of the ASME and an associate fellow of the AIAA.

Dr Robert A. Canfield is an associate professor of aerospace engineering in the Department of Aeronautics and Astronautics at the Air Force Institute of Technology (AFIT), OH. USA. His research interests include structural optimization, multidisciplinary analysis and design methods, uncertainty quantification, structural dynamics and control, and aeroelasticity. He retired as a Lieutenant Colonel in the US Air Force, where he was the project engineer for the Automated Structural Optimization System (ASTROS), an AFIT instructor, the program manager for basic research in computational mathematics, the chief of plans and budget, and then the director of policy and integration at the Air Force Office of Scientific Research. He is an Associate Fellow of the AIAA, and he chaired the AIAA Multidisciplinary Design Optimization (MDO) Technical Committee for two years.

1 Introduction	p. 1
1.1 Motivations	p. 1
1.2 Uncertainty and Its Analysis	p. 2
1.3 Reliability and Its Importance	p. 4
1.4 Outline of Chapters	p. 6
1.5 References	p. 7
2 Preliminaries	p. 9
2.1 Basic Probabilistic Description	p. 9
2.1.1 Characteristics of Probability Distribution	p. 9
Random Variable	p. 9
Probability Density and Cumulative Distribution Function	p. 10
Joint Density and Distribution Functions	p. 12
Central Measures	p. 13
Dispersion Measures	p. 14
Measures of Correlation	p. 15
Other Measures	p. 17
2.1.2 Common Probability Distributions	p. 20
Gaussian Distribution	p. 20
Lognormal Distribution	p. 25
Gamma Distribution	p. 28
Extreme Value Distribution	p. 29
Weibull Distribution	p. 31
Exponential Distribution	p. 34
2.2 Random Field	p. 36
2.2.1 Random Field and Its Discretization	p. 36
2.2.2 Covariance Function	p. 41
Exponential Model	p. 42
Gaussian Model	p. 42
Nugget-effect Model	p. 42
2.3 Fitting Regression Models	p. 43
2.3.1 Linear Regression Procedure	p. 44
2.3.2 Linear Regression with Polynomial Fit	p. 45
2.3.3 ANOVA and Other Statistical Tests	p. 46
2.4 References	p. 50
3 Probabilistic Analysis	p. 51
3.1 Solution Techniques for Structural Reliability	p. 51
3.1.1 Structural Reliability Assessment	p. 51
3.1.2 Historical Developments of Probabilistic Analysis	p. 56
First- and Second-order Reliability Method	p. 56
Stochastic Expansions	p. 58
3.2 Sampling Methods	p. 60
3.2.1 Monte Carlo Simulation (MCS)	p. 60
Generation of Random Variables	p. 62
Calculation of the Probability of Failure	p. 65
3.2.2 Importance Sampling	p. 68
3.2.3 Latin Hypercube Sampling (LHS)	p. 70
3.3 Stochastic Finite Element Method (SFEM)	p. 72
3.3.1 Background	p. 73
3.3.2 Perturbation Method	p. 73
Basic Formulations	p. 74
3.3.3 Neumann Expansion Method	p. 75
Basic Procedure	p. 75
3.3.4 Weighted Integral Method	p. 77
Formulation of Weighted Integral Method	p. 77
3.3.5 Spectral Stochastic Finite Element Method	p. 79
3.4 References	p. 79
4 Methods of Structural Reliability	p. 81
4.1 First-order Reliability Method (FORM)	p. 81
4.1.1 First-order Second Moment (FOSM) Method	p. 81
4.1.2 Hasofer and Lind (HL) Safety-index	p. 86
4.1.3 Hasofer and Lind Iteration Method	p. 88
4.1.4 Sensitivity Factors	p. 97
4.1.5 Hasofer Lind - Rackwitz Fiessler (HL-RF) Method	p. 99
4.1.6 FORM with Adaptive Approximations	p. 110
TANA	p. 111
TANA2	p. 111
4.2 Second-order Reliability Method (SORM)	p. 124
4.2.1 First- and Second-order Approximation of Limit-state Function	p. 125
Orthogonal Transformations	p. 125
First-order Approximation	p. 126
Second-order Approximation	p. 128
4.2.2 Breitung's Formulation	p. 130
4.2.3 Tvedt's Formulation	p. 133
4.2.4 SORM with Adaptive Approximations	p. 136
4.3 Engineering Applications	p. 138
4.3.1 Ten-bar Truss	p. 138
4.3.2 Fatigue Crack Growth	p. 142
4.3.3 Disk Burst Margin	p. 144
4.3.4 Two-member Frame	p. 146
4.4 References	p. 150
5 Reliability-based Structural Optimization	p. 153
5.1 Multidisciplinary Optimization	p. 153
5.2 Mathematical Problem Statement and Algorithms	p. 155
5.3 Mathematical Optimization Process	p. 157
5.3.1 Feasible Directions Algorithm	p. 157
5.3.2 Penalty Function Methods	p. 160
Interior Penalty Function Method	p. 160
Exterior and Quadratic Extended Interior Penalty Functions	p. 162
Quadratic Extended Interior Penalty Functions Method	p. 163
5.4 Sensitivity Analysis	p. 178
5.4.1 Sensitivity with Respect to Means	p. 181
5.4.2 Sensitivity with Respect to Standard Deviations	p. 182
5.4.3 Failure Probability Sensitivity in Terms of [beta]	p. 183
5.5 Practical Aspects of Structural Optimization	p. 197
5.5.1 Design Variable Linking	p. 197
5.5.2 Reduction of Number of Constraints	p. 198
5.5.3 Approximation Concepts	p. 198
5.5.4 Move Limits	p. 198
5.6 Convergence to Local Optimum	p. 200
5.7 Reliability-based Design Optimization	p. 200
5.8 References	p. 201
6 Stochastic Expansion for Probabilistic Analysis	p. 203
6.1 Polynomial Chaos Expansion (PCE)	p. 203
6.1.1 Fundamentals of PCE	p. 203
6.1.2 Stochastic Approximation	p. 209
6.1.3 Non-Gaussian Random Variate Generation	p. 211
Generalized Polynomial Chaos Expansion	p. 212
Transformation Technique	p. 212
6.1.4 Hermite Polynomials and Gram-Charlier Series	p. 213
6.2 Karhunen-Loeve (KL) Transform	p. 218
6.2.1 Historical Developments of KL Transform	p. 219
6.2.2 KL Transform for Random Fields	p. 220
6.2.3 KL Expansion to Solve Eigenvalue Problems	p. 226
6.3 Spectral Stochastic Finite Element Method (SSFEM)	p. 229
6.3.1 Role of KL Expansion in SSFEM	p. 230
6.3.2 Role of PCE in SSFEM	p. 231
6.4 References	p. 233
7 Probabilistic Analysis Examples via Stochastic Expansion	p. 237
7.1 Gaussian and Non-Gaussian Distributions	p. 237
7.1.1 Stochastic Analysis Procedure	p. 237
7.1.2 Gaussian Distribution Examples	p. 239
Demonstration Examples	p. 239
Joined-wing Example	p. 244
7.1.3 Non-Gaussian Distribution Examples	p. 248
Pin-connected Three-bar Truss Structure	p. 248
Joined-wing Example	p. 251
7.2 Random Field	p. 252
7.2.1 Simulation Procedure of Random Field	p. 253
7.2.2 Cantilever Plate Example	p. 253
7.2.3 Supercavitating Torpedo Example	p. 256
7.3 Stochastic Optimization	p. 260
7.3.1 Overview of Stochastic Optimization	p. 261
7.3.2 Implementation of Stochastic Optimization	p. 261
7.3.3 Three-bar Truss Structure	p. 264
7.3.4 Joined-wing SensorCraft Structure	p. 267
7.4 References	p. 270
8 Summary	p. 273
Appendices	p. 275
A Function Approximation Tools	p. 275
A.1 Use of Approximations and Advantages	p. 276
A.2 One-point Approximations	p. 277
A.2.1 Linear Approximation	p. 278
A.2.2 Reciprocal Approximation	p. 278
A.2.3 Conservative Approximation	p. 279
A.3 Two-point Adaptive Nonlinear Approximations	p. 280
A.3.1 Two-point Adaptive Nonlinear Approximation	p. 280
A.3.2 TANA1	p. 281
A.3.3 TANA2	p. 283
A.4 References	p. 289
B Asymptotic of Multinormal Integrals	p. 291
B.1 References	p. 293
C Cumulative Standard Normal Distribution Table	p. 295
D F Distribution Table	p. 297
Index	p. 301

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