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

This article presents the formulation and solu- tion 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 dif- ferential equations of dynamic equilibrium are reduced to rst-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 stiff- ness matrix and its update in nonlinear behavior, a cor- nerstone of the conventional analysis procedures, become unnecessary as means of representing the nodal restor- ing forces. The proposed formulation overcomes the limita- tions 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 ele- ments is formulated using a exibility-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 exam- ple is used to illustrate the response of a simple struc- ture 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. The formulation and solution of nonlinear inelastic struc- tures subjected to static and dynamic loading present great difculties and take a high toll on computational resources. This article presents a formulation and the solution of non- linear structural analysis in state-space form applied to ini- tial boundary value problems. The proposed solution strat- egy overcomes the limitations on the use of state-space models for both static and dynamic systems with quasi- static degrees of freedom. The state variables of a discrete structural system consist of global and local element quan- tities. The global state variables are the generalized dis- placements and velocities of certain nodal degrees of free- dom (DOF) of the system. The local state variables are the iendi restoring forces and forces and deformations at intermediate sections, which describe the local nonlineari- ties. Each state variable is associated with a state equation, which characterizes its evolution in time in relation to other state variables or kinematic constraints. These are usually rst-order differential or algebraic equations. Global state equations describe the motion or equilibrium of the entire system, whereas local state equations describe the evolution of the state variables needed for determining the conditions of all nonlinear elements. There is a single independent variable, represented by the ireal timei in dynamic analy- sis and iintrinsic timei in quasi-static analysis. This article suggests a formulation of the structural elements using a exibility approach with spread plasticity in terms of local and global state variables, which in turn assemble differen- tial algebraic equations solved by iterative solvers. Two alternative schemes can be considered for solving the equations of equilibrium of discrete nonlinear dynamic or quasi-static systems. First, the most commonly used solution strategy is based on implicit time-stepping meth-