Nonlinear analysis of structural frame systems by the state-space approach

This article presents the formulation and solution of the equations of moti on for distributed parameter nonlinear structural systems in state space. T he essence of the state-space approach (SSA) is to formulate the behavior o f nonlinear structural elements by differential equations involving a set o f variables that describe the state of each element and to solve them in ti me simultaneously with the global equations of motion. The global second-or der differential equations of dynamic equilibrium are reduced to first-orde r systems by using the generalized displacements and velocities of nodal de grees of freedom as global state variables. In this framework, the existenc e of a global stiffness matrix and its update in nonlinear behavior a corne rstone of the conventional analysis procedures, become unnecessary as means of representing the nodal restoring forces. The proposed formulation overc omes the limitations on the use of state-space models for both static and d ynamic systems with quasi-static degrees of freedom. The differential algeb raic equations (DAE) of the system are integrated by special methods that h ave become available in recent years. The nonlinear behavior of structural elements is formulated using a flexibility-based beam macro element with sp read plasticity developed in the framework of state-space solutions. The ma cro-element formulation is based on force-interpolation functions and an in trinsic time constitutive macro model. The integrated system including mult iple elements is assembled, and a numerical example is used to illustrate t he response of a simple structure subjected to quasi-static and dynamic-typ e excitations. The results offer convincing evidence of the potential of pe rforming nonlinear frame analyses using the state-space approach as an alte rnative to conventional methods.