A Generalized Formulation of Time Integration Methods for Nonlinear Dynamic Analysis of Hysteretic Mechanical Systems

The chapter presents a generalized formulation of time integration methods that allow for the numerical solution of the nonlinear equilibrium equations characterizing mechanical systems having hysteretic behavior. Two families of time integration methods are derived from such a generalized formulation: the celebrated Newmark’s family of methods and Chang’s family of explicit methods. The former is presented since it represents one of the most employed families of conventional time integration methods available in the literature. On the contrary, the latter is illustrated since it is one of the most efficient families of recently developed structure-dependent time integration methods. For each family, the formulation, the expression for the evaluation of the unknown generalized displacement, velocity, and acceleration vectors, as well as the main numerical properties are first presented. Then, some instances as well as the implementation scheme of each family are illustrated. Finally, nonlinear time history analyses are performed on a rate-independent hysteretic mechanical system, characterized by a stiffening behavior and subjected to an external generalized random force, to illustrate the numerical performance, in terms of accuracy and computational efficiency, of some methods selected within the above-described families.