The prediction of the nonlinear response of structures exhibiting softening behaviour is sometimes too computationally demanding due to the need for an extremely small incremental step, and it often suffers from convergence failure of the nonlinear solution procedure in many real problems. To improve the efficiency and convergence of such analyses, different solution procedures are proposed in this work for nonlinear static and dynamic structural analysis, with particular emphasis on softening structures.
A new solution procedure for tracing the static equilibrium path is developed, which is based on controlling the work done by the out-of-balance forces over the iterative displacement corrections, referred to as the residual-power control method. A supplementary root selection strategy is developed to ensure the correct root is selected for the incremental estimate and to prevent elastic unloading and back-tracing of the equilibrium path. This strategy works well in combination with the residual-power control method for handling a wide spectrum of practical structural engineering problems.
A notable discovery is made in this work that nonlinear static analysis of softening structures can unexpectedly terminate due to discontinuity in the equilibrium path at a specific state, called the terminal equilibrium state. Even though the constituent materials and structural elements exhibit a continuous response, it is shown in this work that the equilibrium path is discontinuous at the terminal equilibrium state, which leads to convergence failure with gradient-based solution procedures.
Finally, enhanced line search methods are developed which aim to improve the overall convergence performance of nonlinear solution procedures, with particular focus in the present study on dynamic problems. The proposed line search methods seek the local minimum solution of the nonlinear system of discrete equilibrium equations by searching along alternative directions other than the search direction of the standard line search method.