Keywords: uncertainty, structural reliability, optimization.

This paper is a state-of-the-art review of the handling of uncertainty in structural analysis and optimization. As computational resources available to the average designer continue to increase exponentially, so does the level of complexity of the structural systems being designed in the modern era. Such complexity is typically associated with a corresponding decrease in knowledge of the safety characteristics of the structures. With little or no previous data available regarding the structural behaviour and potential failure modes, and little experience in designing such complex systems, quantifying the risk associated with structural failure becomes challenging. As a result, modern structural design methods must invariably include a non-deterministic component, that is, one that allows the reliability of a component or a structural system to be quantified with the help of numerical simulations. Quantifying the risk of structural failure using deterministic or stochastic simulation models is the subject of this review paper.

Four critical aspects of uncertainty in structural applications are identified, including uncertainty modelling, structural system reliability assessment, reliability-based structural optimization formulations, and use of surrogate models for computational efficiency. Each aspect is investigated from a theoretical perspective, the current state-of-the-art methods and algorithms are described, followed by potential limitations, and future directions. The choice of an appropriate uncertainty modelling technique is perhaps the most critical component of the structural reliability assessment process. Depending on the depth of knowledge of the system and the available computational resources, one can choose between traditional probability theory, evidence theory, or fuzzy-set based possibility theory. We discuss the use of these theories from a structural reliability viewpoint.

Probability theory is perhaps the most widely used approach for modelling uncertainty and its propagation in structural systems. First and second-order reliability methods (FORM and SORM) are commonly used techniques for evaluating the reliability of a structural element, and can be extended to evaluate the overall reliability of the structural system. Sampling methods, such as Monte Carlo simulation, are accurate methods to evaluate structural system reliability, but they can be computationally impractical, especially if the limit state function evaluation requires expensive finite element analysis. We review the basic concepts of these reliability assessment techniques, and discuss applications in structural design.

The non-deterministic analysis module can be easily integrated into an optimal design framework. The advantage of an optimization framework is that it allows us to efficiently explore the design space to identify a location with favorable properties, both from an objective function and a probabilistic viewpoint. We review some efficient structural optimization formulations that include probabilistic constraints. Iterative methods such as optimization can significantly increase the overall computational cost, primarily because the non-deterministic component needs to be invoked iteratively. Response surface methods or surrogate modelling methods can help alleviate this additional computational burden by constructing computationally benign approximations of the limit state and objective functions. A review of some of the recent work in response surfaces for structural reliability assessment is presented.

Finally, we discuss some of the recent work in stochastic finite element methods and stochastic cellular automata, that is, approaches that can significantly improve the computational efficiency of reliability assessment. This efficiency improvement is realised through simultaneous calculation of the limit state function value and its gradient in a single finite element analysis. Stochastic cellular automata also provide the possibility of efficient parallelization, leading to considerable reduction in wall-clock time. These methods can also be used to model spatial distribution of random properties.

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