The static displacement and the stress analysis of structures with bounded uncertainties using the vertex solution theorem Original Research Article

In practical engineering, on account of physical imperfections, model inaccuracies, and system complexities, almost all structures have physical and geometrical uncertainties in some degrees. In this paper, we focus on the static response problem and the stress distribution of structures with bounded uncertainties. In terms of the vertex notation in interval analysis, a new mathematical proof of the vertex solution theorem which is used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties is given. Then, we extend this theorem for calculating the interval stress and the deformation distributions, which was only used to determine the supremum and the infimum of the static responses of structures with bounded uncertainties. The basic idea of the vertex solution theorem is to convert the interval linear equations into systems of equivalent deterministic linear equations. The resulting deterministic linear equations are then solved by using the familiar techniques like the Gauss elimination method or the Gauss–Seidel iteration method. Another advantage of this method is that the increase in number of the uncertain parameters in a structure element will not result in the increase in number of the uncertain elements in the stiffness matrix. For practical engineering problem with bounded uncertainties, the chief thing is to avoid the disadvantages of large calculation and much runtime of the vertex solution. In this paper, we present the parallel arithmetic of the vertex solution theorem, which can be used in large-scale computations in practical engineering to avoid much runtime. On the other hand, we can use the symmetrical characteristic of the stiffness matrix and the substructure method to reduce the computational effort in practical engineering. The numerical results of the typical example presented by Deif calculated by the proposed vertex solution theorem show that the width between the supremum and the infimum on the static response yielded by the vertex solution theorem is tighter than those produced by the KCN’s method presented by Köylüoğlu, Cakmak, and Nielson. The numerical examples of the two-dimensional shell with an edge fixed and the two-dimensional shell with surrounding fixed are used to illustrate the computational feasibility of the vertex solution theorem.