Research On Structural Reliability Analysis Methods Under Epistemic Uncertainties

Uncertainties associated with geometric tolerances, material properties, boundary conditions widely exist in practical engineering problems. Quantifying, propagating and managing the concerned uncertainty based on related uncertainty theory have become extremely important to improve the performances of products or structures. Uncertainty can be categorized into stochastic and epistemic types. Stochastic uncertainty quantification always requires a large amount of experimental samples to construct the precise probability distributions of uncertain parameters, and cann’t be reducible as the state of knowledge increases. While epistemic uncertainty arises from lack of knowledge without sufficient sample information to construct the corresponding precise probability distributions, and can be reducible as the state of knowledge increases. Actually, epistemic uncertainty widely exists during the design, manufacturing, service and aging stages of the whole life cycle of current products or structures. Structural analysis and design by only using stocastic modeling may not provide effective assessments of structural performances under epistemic uncertainties, or even will lead to unreliable designs.Comparatively, structural stochastic response analysis and reliability theory has exhibited full developments presently in support of a unified systematic framework of probability theory. In contrast, there co-exist several theories to characterize the epistemic uncertainty, which makes epistemic uncertainty quantification more complicated. Several progresses have been made in the field of structural response and reliability analysis theories and algorithms under epistemic uncertainty in recent years, however, it seems that it is still in its preliminary stages and several related important issues have not been well solved. Thus, this dissertation concentrates on two types of relatively convinent uncertainty modeling approaches, namely convex sets and evidence theory, and conducts systematic studies at structural response analysis algorithm based on evidence theory, structural response analysis under convex sets and probability hybrid uncertainty model, non-probability reliability analysis methods by using ellipsoid-type convex model, and evidence-theory-based structural reliability analysis. As a result, the following studies are carried out in this dissertation:(1) A structural uncertainty analysis method is developed to calculate the structural static and dynamic response under epistemic uncertainties by directly integrating evidence theory with finite element method. By introducing the moment concept in probability theory, the moments of evidence variables and associated functions are developed to describe their distributions. The static and dynamic response analysis method is developed by integrating the moment concept with static and dynamic finite element method to compute the respone moments. Besides, the interval structural analysis technique is used to efficiently calculate the approximate response for each focal element.(2) A structural uncertainty analysis method with interval and stochastic field hybrid model is proposed to conduct structural response analysis under stochastic field with interval correlation length. The correlation length of the stochastic correlation model is quantified by using interval due to lack of sufficient experimental data, and this proposed hybrid model can more reasonably describe the spatial variability characteristics of the uncertain parameters. The stochastic field with interval correlation length is discretized based on Karhunen-Loeve series expansion, and the mean element stiffness matrix and the interval weighted element stiffness matrix can be further obtained. The structural displacement response is approximated by using polynomial chaos expansion with interval weights. The Element-by-Element technique is used to assemble the corresponding global stiffness matrix to avoid the couping effects of the element stiffness matrices. An extended-order system of interval linear equations is formulated, and is further solved to obtain the corresponding interval weights, and the interval moments of the structural response can be achieved.(3) For ellipsoid-model-based non-probabilistic reliability problems with black-box limit-state equations, a sequential response surface method is developed to conduct reliability analysis by referring to the basic ideas of conventional response surface method in probability-based reliability analysis. The correlation analysis technique is used to construct the multi-dimensional ellipsoid model, which will be more convenient and economic to quantify uncertain parameters. A quadratic polynomial without cross terms is adopted to approximate the limit-state function, and an iterative strategy is employed to update sampling center, design space and metamodel until the convergence criteria is satisfied.(4) A local-densifying metamodeling approach is suggested to conduct ellipsoid-model-based non-probabilistic reliability analysis. A quadratic polynomial without cross terms is used to approximate the limit-state function in the first iterative step, and the approximate design point is located based on the created metamodel as the sampling center of the design space. In subsequent iterative steps, radial basis function is used to construct the metamodel of the limit-state function, and the obtained approximate design point is added into the previous sampling point sets to update the metamodel. The proposed method can gradually improve the approximation accuracy of the limit-state function at the local region nearby the design point, and hence much less samples are required comprared with the sequential response surface method, and whereby improve the reliability analysis solution efficiency.(5) For the extremely computational cost problems due to the discontinuous nature of epistemic uncertainty quantification based on evidence theory, three types of evidence-theory-based reliability analysis method are developed by intergating the quadratic polynomial without cross terms, radial basis function and high-dimensional model representation in combination with moving least square method to overcome the low-efficiency problem, respectively. Compared to probability-based and convex-model-based reliability analysis, evidence-theory-based reliability analysis requires relatively high approximation accuracy of the limit-state function across the frame of discernment. The proposed three metamodel-based approaches are systematically compared to test the applicability of different metamodeling techniques in evidence-theory-based reliability analysis under different sampling points, threshold setting, basic probability assignments, and different problem scale by introducing a statistical measure. Besides, their advantages and disadvantages are further investigated for low-failure-probability problems.(6) A structural system reliability analysis method is developed to deal with series and parallel system under epistemic uncertainty by extending evidence theory to system reliability problems. By introducing the concept of system reliabillity, the definitions of failure probability belief and plausibility for series and parallel systems are provided. By constructing two-level optimization formulations to locate the relative position between the focal element and system failure region, and whereby efficiently determine the focal element sets belonging to failure belief or plausibility. The first-order Taylor expansion and simplex method are employed to determine the relative positions for the max-min and min-max optimization problems, while the interval analysis method is used for min-min and max-max optimization problems.