Universal Grey System Theory for Analysis of Uncertain Structural Systems

The uncertainty present in many structural systems is commonly modeled using probabilistic, fuzzy, or interval approaches depending on the nature of uncertainty present. The probabilistic approach is based on the probability distributions of the uncertain parameters, which are not known for most practical systems. The fuzzy analysis is applicable to systems whose parameters are described in terms of linguistic and imprecise or vague statements. In most practical systems, particularly structural systems, the parameters such as the geometry (sizes of members), material properties, and loads are available as ranges or intervals. For such systems, the use of interval analysis, coupled with the finite element analysis, appears to be appropriate for predicting the ranges of the response quantities such as nodal displacements and element stresses. However, the accuracy of the results given by the interval analysis suffers from the so-called dependency problem, which can lead to undesirable expansions of the ranges of the computed results. Depending on the nature of expressions involved in the computation and the width of the ranges (intervals) of the uncertain parameters, the computed response quantities may sometimes violate the physical laws. Although several approaches, such as numerical truncation during interval arithmetic operations, parameterization of intervals, and the use of subintervals within an interval, have been suggested to limit the growth of the intervals of the results predicted, there has not been a simple approach that can improve the accuracy of the basic interval analysis. This work presents a new methodology, called universal grey number (or system) theory, for the analysis of systems whose parameters are described as intervals or ranges. The computational feasibility and improved accuracy (compared to the interval analysis) of the methodology is demonstrated by considering three examples: the stress analysis of a stepped bar, the stress analysis of a planar 10-bar truss, and the rigid-body (vertical) response of an airplane taxiing on a wavy runway based on a single-degree-of-freedom model. This work demonstrates that the universal grey system approach is a viable methodology for the accurate analysis of structural and other engineering problems involving uncertain parameters that are described in terms of ranges or intervals.