Nonlinear Analysis of Structural Frame Systems by the State‐Space Approach

Abstract

This article presents the formulation and solution of the equations of motion for distributed parameter nonlinear structural systems in state space. The essence of the state‐space approach (SSA) is to formulate the behavior of nonlinear structural elements by differential equations involving a set of variables that describe the state of each element and to solve them in time simultaneously with the global equations of motion. The global second‐order differential equations of dynamic equilibrium are reduced to first‐order systems by using the generalized displacements and velocities of nodal degrees of freedom as global state variables. In this framework, the existence of a global stiffness matrix and its update in nonlinear behavior, a cornerstone of the conventional analysis procedures, become unnecessary as means of representing the nodal restoring forces. The proposed formulation overcomes the limitations on the use of state‐space models for both static and dynamic systems with quasi‐static degrees of freedom. The differential algebraic equations (DAE) of the system are integrated by special methods that have become available in recent years. The nonlinear behavior of structural elements is formulated using a flexibility‐based beam macro element with spread plasticity developed in the framework of state‐space solutions. The macro‐element formulation is based on force‐interpolation functions and an intrinsic time constitutive macro model. The integrated system including multiple elements is assembled, and a numerical example is used to illustrate the response of a simple structure subjected to quasi‐static and dynamic‐type excitations. The results offer convincing evidence of the potential of performing nonlinear frame analyses using the state‐space approach as an alternative to conventional methods.