Topics in Structural Reliability

Course Outline

The subject of structural reliability offers a rational framework to quantify uncertainties mathematically. The subject combines theories of probability, random variables and random processes with principles of structural mechanics and forms the basis on which modern structural design codes are developed. The present course aims to introduce the basics of the structural reliability analysis procedures; Formulation of reliability for structural components and systems. Exact solutions, first- and second-order reliability methods. Reliability indices. Simulation based methods. Reliability sensitivity measures.

Goals

This course offers a comprehensive review of structural reliability assessment methods and their applications to engineering problems in general. Topics include formulation of structural reliability, first-order and second-order reliability methods(FORM and SORM), system reliability analysis, structural reliability analysis under model or statistical uncertainties, simulation methods and uncertainty quantification methods. Students will apply the methods to example problems using available computer codes. As a final term project, each student will review, apply or develop reliability methods for an engineering/science application he/she chooses.

Syllabus

I. Introduction: Uncertainty & risk; Structural/system reliability

II. Basic Theory of Probability and Statistics (“Primer”) Events and probability – set theory Mathematics of probability: axioms, theorems and rules Random variables: probability functions and partial descriptors Normal and Lognormal distributions Distributions related to Bernoulli sequence/Poisson process Multiple random variables; Joint probability functions; Correlation Mathematical expectations of functions of random variables (linear/nonlinear) Distribution of functions of random variables

III. Structural Reliability – Component Joint probability distribution models Elementary reliability analysis & indices Reliability index by Mean-Value First-Order Second-Moment Method (MVFOSM) Hasofer-Lind reliability index (HL/FOSM) Generalized reliability index Reliability “index” & “methods” First-Order Reliability Method (FORM) FORM examples and issues Second-Order Reliability Methods (SORM) FORM importance vectors FORM sensitivity measures

IV. Structural Reliability - System Definition of “system”; “Structural” system reliability analysis Inclusion-exclusion formula; FORM approximation Bounding methods: theoretical bounding formulas; bounds by linear programming Matrix-based System Reliability (MSR) method and its applications

V. Reliability under Model & Statistical Uncertainties; Midterm Exam Bayesian parameter estimation; Reliability under epistemic uncertainties

VI. Simulation Methods Monte Carlo simulations; Importance sampling; adaptive sampling, etc. Latin Hypercube Sampling; Markov Chain Monte Carlo methods VII. Random Field; Term Project Abstract Mathematical model of random fields Discrete representation of random fields: K-L expansion, etc.

VIII. Response Surface Basic formulations and approaches to construct RS models; UQ methods

IX. Finite Element Reliability and Sensitivity Uncertainty/Reliability quantification using FE analysis; Sensitivity calculations

X. Reliability-Based Design Reliability-based design codes; Reliability-based design/topology optimization