Higher-Order Non-Linear Analysis of Steel Structures Part I : Elastic Second-Order Formulation

Abstract

This paper presents a higher-order beam-column formulation that can capture the geometrically non-linear behaviour of steel framed structures which contain a multiplicity of slender members. Despite advances in computational structural frame software, analyses of large frames can still be problematic from a numerical standpoint, with efficacious and reliable convergence not always being ensured. To this end, the intent of this paper is to fulfil a need for versatile, reliable and efficient non-linear analysis of general steel framed structures with a large number of members suitable for engineering practice. Following a comprehensive review of numerical frame analysis techniques, a fourth-order element is derived, in which the crucial member bowing effect involved in the equilibrium equation is captured, and implemented in an updated Lagrangian formulation. Because of this, it is able to predict flexural buckling, snap-through buckling and the large displacement post-buckling behaviour of typical structures whose responses have been reported by independent researchers. The present approach with its efficacious and reliable convergence is shown in comparison studies to be applicable to selected applications which are prone to several forms of geometric non-linearity.Keywords: Beam-column, Frames, Geometric non-linearity, Higher-order element, Steel, Updated lagrangian INTRODUCTIONDisplacement-based finite element techniques have been the most favoured for frame structures, mainly because of their accuracy, versatility, fully-established mathematical (variational) basis, and their suitability for computer implementation. The finite element method is based on the general principle that equilibrium of the structure under applied loading is achieved at displacements which correspond to the potential energy of the structure being stationary. In the description of the kinematic deformations of framed structures is favourably based on an updated Lagrangian system. proposed a second-order analysis which allows for the geometric non-linear behaviour using one element. Their formulation makes use of stability functions for the transverse displacements, and it also considers the elastic coupling of axial, flexural and torsional deformations. Later, Liew et al. [8,9] developed their numerical non-linear analysis of a structure subjected to fire and explosion. Similarly, Iu et al. [10,11] presented the non-linear fire analysis of steel structure including heating and cooling behaviour.

Higher-Order Non-Linear Analysis of Steel Structures Part Ii : Refined Plastic Hinge Formulation

“…It depends on both the shape factor for the cross-section  and the residual stresses on the cross-section. The initial yield surface  y , or initial yield interaction equation, may be defined from the bending actions about the major principal axis M x , minor principal axis M y and axial force P collected in the vector f = {P, M x , M y } T and related by   (1) in which the numbers 0.8 and 0.9 in the denominator account for residual stresses, P y is the axial force capacity of the cross-section, and M px and M py are its full plastic moments about the major and minor principal axes respectively which have respective shape factors of  x and  y . When  y (f) < 1, the cross-section is taken as elastic.…”

“…The spring formulation is therefore able to capture the non-linear material behaviour, including its elastic domain, gradual or partial yielding, full plasticity, strain hardening as well as residual stresses, in the load-deformation relationship for the quartic beam-column finite element. These hinges may be incorporated into the second-order elastic stiffness formulation of the companion paper [1] using the procedures described in [7,16]. In reference to [15], it is also worth mentioning that the plastic hinges in Eqs.…”

“…is equivalent to the incremental secant stiffness in the second-order elastic stiffness formulation in the companion paper [1]. For axial actions, the incremental force equilibrium equation is written separately as …”

“…This paper is concerned with an innovative extension of the geometrically non-linear elastic finite element formulation with a higher-order element, described in the companion paper [1], to include steel yielding so as to produce a robust and highly efficient technique for analysing frames with a multiplicity of component structural members. The material yielding of an infinitesimal steel element or particle across the element section has been investigated numerically through the plastic hinge approach in the efficacious manner.…”

“…In the companion paper [1], the geometric non-linearities associated with a second-order analysis of an elastic framed structure were discussed. Commonly, material yielding is an important consideration for steel structures at their strength or ultimate limit state, and so generalised numerical non-linear analysis of steel framed structures at their strength limit state necessitates the accurate modelling of both geometric and material non-linearities.…”