Full Description
The last decades have witnessed the development of methods for solving struc tural reliability problems, which emerged from the efforts of numerous re searchers all over the world. For the specific and most common problem of determining the probability of failure of a structural system in which the limit state function g( x) = 0 is only implicitly known, the proposed methods can be grouped into two main categories: • Methods based on the Taylor expansion of the performance function g(x) about the most likely failure point (the design point), which is determined in the solution process. These methods are known as FORM and SORM (First- and Second Order Reliability Methods, respectively). • Monte Carlo methods, which require repeated calls of the numerical (nor mally finite element) solver of the structural model using a random real ization of the basic variable set x each time. In the first category of methods only SORM can be considered of a wide applicability. However, it requires the knowledge of the first and second deriva tives of the performance function, whose calculation in several dimensions either implies a high computational effort when faced with finite difference techniques or special programs when using perturbation techniques, which nevertheless require the use of large matrices in their computations. In or der to simplify this task, use has been proposed of techniques that can be regarded as variants of the Response Surface Method.
Contents
1 A Discussion on Structural Reliability Methods.- 1.1 Performance and Limit State Functions.- 1.2 Methods Based on the Limit State Function.- 1.3 Transformation of Basic Variables.- 1.4 FORM and SORM.- 1.5 Monte Carlo Methods.- 1.6 Solver Surrogate Methods.- 1.7 Regression and Classification.- 1.8 FORM and SORM Approximations with Statistical Learning Devices.- 1.9 Methods Based on the Performance Function.- 1.10 Summary.- 2 Fundamental Concepts of Statistical Learning.- 2.1 Introduction.- 2.2 The Basic Learning Problem.- 2.3 Cost and Risk Functions.- 2.4 The Regularization Principle.- 2.5 Complexity and Vapnik-Chervonenkis Dimension.- 2.6 Error Bounds and Structured Risk Minimization.- 2.7 Risk Bounds for Regression.- 2.8 Stringent and Adaptive Models.- 2.9 The Curse of Dimensionality.- 2.10 Dimensionality Increase.- 2.11 Sample Complexity.- 2.12 Selecting a Learning Method in Reliability Analysis.- 3 Dimension Reduction and Data Compression.- 3.1 Introduction.- 3.2 Principal Component Analysis.- 3.3 Kernel PCA.- 3.4 Karhunen-Loève Expansion.- 3.5 Discrete Wavelet Transform..- 3.6 Data Compression Techniques..- 4 Classification Methods I — Neural Networks.- 4.1 Introduction.- 4.2 Probabilistic and Euclidean methods.- 4.3 Multi-Layer Perceptrons..- 4.4 General Nonlinear Two-Layer Perceptrons.- 4.5 Radial Basis Function Networks.- 4.6 Elements of a General Training Algorithm.- 5 Classification Methods II — Support Vector Machines.- 5.1 Introduction.- 5.2 Support Vector Machines.- 5.3 A Remark on Polynomial Chaoses.- 5.4 Genetic Algorithm..- 5.5 Active Learning Algorithms.- 5.6 A Comparison with Neural Classifiers.- 5.7 Complexity, Dimensionality and Induction of SV Machines.- 5.8 Application Examples.- 5.9 An Application to Stochastic Stability.- 5.10 Other KernelClassification Algorithms.- 6 Regression Methods.- 6.1 Introduction.- 6.2 The Response Surface Method Revisited.- 6.3 Neural Networks.- 6.4 Support Vector Regression.- 6.5 Time-Dependent MLP for Random Vibrations.- 7 Classification Approaches to Reliability Indexation.- 7.1 Introduction.- 7.2 A Discussion on Reliability Indices.- 7.3 A Comparison of Hyperplane Approximations.- 7.4 Secant Hyperplane Reliability Index.- 7.5 Volumetric Reliability Index.- References.- Essential Symbols.
