A method for finding distinct solutions of geometrically nonlinear structures

Abstract

Geometrically nonlinear structures are extensively used as load-carrying systems or energy harvesting devices. The main characteristics of their behaviour include complex and/or multi-equilibrium paths. In this paper, a new method is proposed for analysing these structures wherein distinct deformation states of a structure under applied loads (forces or displacements) are directly obtained. Using the concepts of finite element approximations, the proposed method first establishes the nonlinear equilibrium equations for a structure undergoing large deformations. Then, contrary to the well-established incremental-iterative path-following methods that appeal to linearization concepts and use an arc-length constraint to solve the nonlinear equilibrium equations, the proposed method tries to directly find the solutions to these equations. To this end, a likelihood function is introduced based on the nonlinear equilibrium equations. This function is defined such that it reaches its maximum value at the solution(s) of the equilibrium equations. To maximize this likelihood function, the nested sampling algorithm is exploited. This algorithm explores the solution phase space thoroughly and finds the deformation state(s) that satisfy the nonlinear equilibrium equations. An application of the proposed method to analysing geometrically nonlinear truss structures is given and three truss structures are solved to demonstrate the capabilities of this method.