The vertex solution theorem and its coupled framework for static analysis of structures with interval parameters

Abstract

SummaryThis work gives new statement of the vertex solution theorem for exact bounds of the solution to linear interval equations and its novel proof by virtue of the convex set theory. The core idea of the theorem is to transform linear interval equations into a series of equivalent deterministic linear equations. Then, the important theorem is extended to find the upper and lower bounds of static displacements of structures with interval parameters. Following discussions about the computational efforts, a coupled framework based on vertex method (VM) is established, which allows us to solve many large‐scale engineering problems with uncertainties using deterministic finite element software. Compared with the previous works, the contribution of this work is not only to obtain the exact bounds of static displacements but also lay the foundation for development of an easy‐to‐use interval finite element software. Numerical examples demonstrate the good accuracy of VM. Meanwhile, the implementation of VM and availability of the coupled framework are demonstrated by engineering example. Copyright © 2017 John Wiley & Sons, Ltd.