Evidence-theory-based Structural Reliability Analysis Methods

Uncertainties associated with material properties, external loads, geometric sizes and boundary conditions widely exist in practical engineering structures. When uncertainties are propagated to system responses, they tend to result in fluctuations of structural performance and even structural failures. Therefore, quantifying and controlling uncertainties are becoming critical for the reliability and safety design of practical engineering structrues. According to the generation mechanisms and physical meanings, uncertainties can be categorized into three distinct types: aleatory uncertainty, fuzzy uncertainty and epistemic uncertainty. Epistemic uncertainty has become the research hot spot in structural reliability analysis field. Until now, a collection of theories have been developed, which include evidence theory, interval analysis theory, possibility theory etc. Compared with other theories, evidence theory employs a more flexible framework to represent uncertainty with respect to the body of evidence and its measures. It utilizes the discrete basic probability assignment function to quantify the evidence of input variables and the probability interval composed of belief measure and plausibility measure to represent the reliability of system responses. Under different situations, evidence theory can be equivalent to probability theory, fuzzy theory, interval analysis theory, etc. It has recently been introduced in structural reliability analysis area and some progressive achievements have been made. However, research of evidence-theory-based reliability analysis is still in the preliminary stage. A series of scientific issues are required to be further explored. Especially, the high computational cost severely hinders the applicability of evidence theory in practical engineering problems. The following studies are carried out in this dissertation:1) An efficient evidence-theory-based reliability analysis method using focal element reduction technique is proposed. Evidence-theory-based reliability analysis has to conduct extreme analysis over every focal element, the computational cost of which rises rapidly with the increase of the dimension of the problem and the number of the focal elements. To deal with this issue, an optimization problem is constructed to compute the non-probabilistic reliability index defined for evidence theory and simutaleously the design point is obtained. Subsequently, an assistant area is constructed using the design point. The focal elements located in this area are excluded from extreme analysis, by which the computational cost of evidence-theory-based reliability analysis can be significantly reduced. For the focal elements located outside the assistant area, an interval analysis method is employed to conduct extreme analysis. Finally, the belief measure and plausibility measure of the response are computed. The effectiveness of the proposed method is demonstrated using two numerical examples, and it is further applied in the reliability analysis of a vehicle side impact problem.2) An evidence-theory-based reliability analysis method using response surface technique is developed. Due to the complexity of practical engineering structures, the limit-state function between input variables and output responses is hard to be explicitly expressed, which generally behaves as an implicit “black box”. Time-consuming numerical analysis models, such as finite element analysis and computational fluid dynamics, are required to compute the “black box” response in this case, which will result in high computational cost. Aiming at this type of issues, the intersection points of the limit-state surface and the uncertainty domain are searched, which capture the key information of the limit-state surface. Here, these points are defined as control points. Subsequently, sample points are allocated in the uncertainty domain using the important control points, based on which the high-accuracy radial basis function is constructed. Finally, the explicit radial basis function is utilized to quickly compute the belief measure and plausibility measure. The effectiveness of the proposed method is demonstrated using three numerical examples, and it is further applied in the reliability analysis of a five-degrees-of-freedom vehicle single track model.3) The concept of most probable focal element is proposed, based on which a novel evidence-theory-based reliability analysis method is further developed. Similar as the most probable point in probabilistic reliability analysis, there exists a focal element contributing most to the calculation of the belief measure and plausibility measure in evidence-theory-based reliability analysis, which is defined as the most probable focal element. A uniformity approach is used to deal with the evidence variables, through which the original evidence-theory-based reliability problem is transformed to a traditional reliability problem with only random uncertainty. It is then solved using a sequential response-surface-based reliability analysis method, based on which the most probable focal element is identified. With the most probable focal element, a quadratic polynomial response surface which more precisely approximates the actual limit-state surface is established. Finally, the belief measure and plausibility measure of the original evidence-theory-based reliability analysis problem are efficiently computed. Several numerical examples are investigated to demonstrate the significance of the proposed most probable focal element. Furthermore, satisfied computational efficiency and accuracy of the proposed method is demonstrated by comparing its results with those of existing evidence-theory-based reliability analysis methods.4) The first and second order approximate evidence-theory-based reliability analysis methods are developed. Refering the idea of first order reliability method and second order reliability method in probabilistic reliability theory, the linear and quadratic Taylor series are expanded around the most probable focal element, based on which the belief measure and plausibility measure are efficiently calculated. The developed methods include several main steps. First, a uniformity approach is used to transform the original evidence-theory-based reliability problem into a probabilistic reliability problem. It is solved by the first-order reliability method and the most probable focal element is obtained. Subsequently, the linear and quadratic Taylor series are established for the actual limit-state function based on the important most probable focal element. Finally, the belief and plausibility measures are efficiently computed using the Taylor series. Two numerical examples are investigated to demonstrate the effectiveness of the present method, and it is further applied in the reliability analysis of a vehicle front impact problem.