Geometrically nonlinear arc length sparse finite element analysis and optimal design of truss structures

A technique for the optimization of stability-constrained geometrically nonlinear shallow trusses with snap-through behavior is demonstrated using the arc length method and a strain energy density approach within a discrete finite element formulation. The optimization method uses an iterative scheme that evaluates the performance of the design variables and then updates them according to a recursive formula that is controlled by the arc length method. A minimum weight design is achieved when a uniform nonlinear strain energy density is found in all members. This minimal condition places the design load just below the critical-limit load that causes snap-through of the structure. The optimization scheme is programmed into a nonlinear finite element algorithm to find the large strain energy at critical-limit loads. Examples of highly nonlinear trusses that are found in the literature are presented to verify the method.