Nonlinear inelastic analysis of space frames

Abstract

In this paper, a fiber beam–column element which considers both geometric and material nonlinearities is presented. The geometric nonlinearities are captured using stability functions obtained from the exact stability solution of a beam–column subjected to axial force and bending moments. The material nonlinearities are included by tracing the uniaxial stress–strain relationship of each fiber on the cross sections. The nonlinear equilibrium equations are solved using an incremental iterative scheme based on the generalized displacement control method. Using only one element per member in structure modeling, the nonlinear responses predicted by the proposed element compare well with those given by commercial finite element packages and other available results. Numerical examples are presented to verify the accuracy and efficiency of the proposed element.

Introduction

In the past few decades, there have been numerous studies to improve the accuracy of the beam–column element for the nonlinear analysis of steel frames. In general, the nonlinear response of steel frames can be predicted by using either the finite element method or the beam–column approach. The finite element approach is often based on a stiffness or displacement formulation in which cubic and linear interpolation functions are used for the transverse and axial displacements, respectively 1., 2., 3., 4., 5.. Since this method is based commonly on an assumed cubic polynomial variation of transverse displacement along the element length, it is unable to capture accurately the effect of axial force acting through the lateral displacement of the element ($P\text{–}\delta$ effect) when one element per member is used [6]. Hence, it overestimates the strength of a member under significant axial force. Although the accuracy of this method can be improved by using several elements per member in the modeling, it is generally recognized to be computationally intensive because of a very refined discretisation of the structures. The beam–column approach is based on the stability functions which are derived from the exact stability function of a beam–column subjected to axial force and bending moments 7., 8., 9., 10., 11., 12.. This approach can capture accurately the $P\text{–}\delta$ effect of a beam–column member by using only one or two elements per member in the modeling, hence, to save computational time.

In parallel with the above developments, different beam–column models have been proposed to represent inelastic material behavior. These models can be grouped into two categories: lumped plasticity 9., 10., 13. model and distributed plasticity model 5., 14., 15., 16., 17., 18.. In the lumped plasticity model, the inelastic behavior of material is assumed to be concentrated at point hinges that are usually located at the ends of the member. The force–deformation relation at these hinges is based on force resultants. The advantage of this model is that it is simple in formulation as well as implementation. However, the disadvantage of this model is that the force–deformation relation at the hinges is not always available and accurate for every section. In the distributed plasticity model, the inelastic behavior of material is distributed along the member length since the element behavior is monitored through numerical integration of constitutive behavior at a finite number of control sections. The nonlinear constitutive behavior at these sections is derived using one of the following methods: (1) moment–curvature relations; (2) force and deformation resultants; and (3) uniaxial stress–strain relations of fibers on the cross sections. Although fiber model is the most computationally intensive among others, it represents the inelastic behavior of material more accurately and rationally than concentrated plasticity model.

This paper proposes a fiber beam–column element for the nonlinear inelastic analysis of space steel frames. The spread of plasticity over the cross section and along the member length is captured by tracing the uniaxial stress–strain relations of each fiber on the cross sections located at the selected integration points along the member length. The Gauss–Lobatto integration rule is adopted herein for evaluating numerically element stiffness matrix instead of the classical Gauss integration rule because it always includes the end sections of the integration field. Since inelastic behavior in beam elements often concentrates at the ends of the member, monitoring the end sections of the element results in improved accuracy and numerical stability [19]. Although the fiber model is included in DRAIN-3DX [20] and OpenSees [21] programs to represent the material nonlinearity, the geometric nonlinearity caused by the interaction between the axial force and bending moments ($P\text{–}\delta$ effect) was not considered. Therefore, these methods overestimate the strength of a member subjected to significant axial force if only one or few elements per member are used in the modeling. In this research, the stability functions obtained from the closed-form solution of a beam–column subjected to end forces are used to accurately capture the $P\text{–}\delta$ effect. Numerical examples are presented to verify the accuracy and efficiency of the proposed element in predicting nonlinear inelastic response of space steel frames.