Scheme dependence and equivalence of sensitivity for nonlinear problems

Abstract

In static nonlinear structural analysis, there are several schemes widely employed to calculate the responses of structures. Examples are the Newton-Raphson method, displacement control method, arc length method, and work control method. It is known that the responses obtained are independent of the schemes chosen for the analysis problem. It is shown in this paper that if sensitivities of response variables are computed, different schemes do give rise, in general, to different results. However, these sensitivities are all equivalent. A systematic approach is derived in order to show this equivalence. This approach is derived in incremental form using the analytical differentiation approach. Numerical examples for validating the present approach include a nonlinear elastic spring model and geometric nonlinear analysis of trusses. The finite element method is used in the truss examples. Interestingly, it is also found that the finite difference method performs poorly in obtaining the sensitivities corresponding to the arc length and work control methods, especially for structures with limit point response.