Non-probabilistic Uncertainty Convex Model And Reliability Analysis Method Incorporating Correlation

Uncertainty widely exists in practical engineering problems, which are commonly related to various factors, such as material properties, loads, boundary conditions, etc. While uncertainty arises from lack of knowledge without sufficient sample information to construct the corresponding precise probability distributions, the approximation to the bounds on the uncertain information can be obtained more easily, it is an effective approach to adopt the convex model to depict the uncertainty under such circumstances. However, the research advance of convex model was becalmed slightly in recent years, the weak mathematical foundation, especially the immature scheme on solving correlation analysis between different uncertain-but-bounded parameters, has stunted the application of convex model. In addition, the existing non-probabilistic methods based on convex model are lack of acceptability for engineers, which call for more intuitionistic new non-probabilistic methods to offer valuable information for engineering design. Thus, this dissertation concentrates on two types of relatively convenient convex model for uncertainty modeling and reliability analysis, namely the interval model and ellipsoidal model, and conduct systematic studies at measure character definitions and corresponding operation rule, algorithm for the construction of the convex model, mechanical response analysis under different convex sets, non-probability reliability analysis methods in principle of indifference. As a result, the following studies are carried out in this dissertation:(1) Based on the comprehensive analysis on morphology of the interval model and the ellipsoid model, some measure characters are defined, including the midpoint, variance, covariance, full matrix, and some corresponding operation rules are derived; In principle of basic nonlinear programming, the minimum volume enclosing ellipsoid problem is transformed into a semi-definite programming problem. Under the circumstances of complete information and incomplete information, algorithms for the construction of multi-dimensional ellipsoid model and interval model with high efficiency and stability respectively.(2) The spatial distribution of sample data exhibit no strongly characteristic of edges and corners in most engineering cases, it is reasonable to adopt the ellipsoid model to depict the uncertain-but-bounded parameters preferentially. Based on the theorem revealing the projection regularity of multi-dimensional ellipsoid in sub-space, the relationship of the ellipsoid and its axis-alighed box is derived. The unified form for the uncertainty evolution based on convex sets is given, and comparisons between the multi-dimensional ellipsoid model and its axis-alighed box are performed from the mathematical proofs. The fluctuation ranges of the two mentioned convex model are illustrated in the uncertain buckling analysis of piezoelectric functionally graded cylindrical shells.(3) In the opinion of the traditional non-probabilistic reliability analysis method, the system is reliable if it allowed the accuracy of uncertainty to fluctuate within a certain range. However, the approach did not provide a specific non-probabilistic safety measure. A new non-probabilistic safety measure for structural reliability is proposed in view of the principle of indifference, it turns out that the non-probabilistic reliability exactly equals the ratio of the safe performance volume in the uncertain parameters domain to the total volume of their variation. To meet the requirement of the present non-probabilistic measure, a Monte Carlo simulation is formulated. Besides, two kinds of first order approximation method(FOAM) are proposed to solve the reliability model based on a linear approximation of the failure surface for ellipsoid model.(4) Since a linear approximation is used in the first order approximation method developed for the ellipsoid model, the curvature information of the limit-state function can not be reflected in FOAM. Generally, FOAM can only provide a sufficiently accurate solution for limit-state functions with relatively small nonlinearity at the design point. To improve the analysis precision of the non-probabilistic reliability, a second order approximation method(SOAM) is further formulated for limit-state functions with relatively strong nonlinearity. Besides, the response surface method is introduced to analysis the complicated engineering problems with black-box type limit-state function.