Exact bounds for the static response set of structures with uncertain-but-bounded parameters

The numerical estimation of the static displacement bounds of structures with uncertain-but-bounded parameters is considered in this paper. By representing each uncertain-but-bounded parameter as an interval number or vector, a static response analysis problem for the structure can be expressed in the form of a system of linear interval equations, in which the coefficient matrix and the right-hand side term are, respectively, the interval matrix and the interval vector. In this study, we present two new simple mathematical proofs of the vertex solution theorem using Cramer’s rule for solving linear interval equations, different from the other proof methods, to find the upper and lower bounds on the set of solutions. The first takes advantage of optimization theory, while the second is based on interval extension. By means of a typical example considered first by Hansen, it can be seen that the result calculated by the vertex solution theorem is the same as one predicted by the Oettli–Prager criterion. Examples of a three-stepped beam and a 10-bar truss are presented to illustrate the computational aspects of the vertex solution theorem in comparison with the interval perturbation method.