Direct stiffness analysis of a composite beam-column element with partial interaction

Abstract

This paper presents a stiffness formulation for the analysis of composite steel–concrete beam-columns with partial shear interaction (PI). This formulation is based on the direct stiffness method (DSM). The advantage of the proposed method is that no approximated displacement and/or force fields are introduced in the element derivation, unlike other modelling techniques available in the literature. Some simple structural systems, such as simply supported beams and propped cantilevers, subjected to a point load and to a uniformly distributed load are then considered to validate the accuracy of the results obtained using the proposed formulation against results derived based on closed form solutions; for continuous beams, the results have been validated against those calculated using highly refined mesh of high order finite elements. This has been carried out for different levels of shear connection stiffness to highlight the ability of the proposed method to overcome the curvature locking problems observed in some conventional displacement formulations. The generic applicability of this technique to the analysis of continuous beams is then illustrated, in particular, highlighting its ability to account for material nonlinearities at both service and ultimate conditions.

Introduction

Over the last few decades, several researchers have investigated the behaviour of composite steel–concrete beams with partial shear interaction (PI). The seminal work by Newmark et al. [1] represents one of the earliest contributions in the understanding of the partial interaction behaviour of composite beams, and their model is usually referred to as Newmark’s model. Since then, several modelling techniques have been presented which usually require some sort of discretisation in the spatial domain (i.e. along the beam length) to be introduced, such as that for finite element methods and for finite difference methods. It is beyond the scope of this paper to provide a lengthy discourse of the current state of the art, and for this purpose reference should be made, among the others, to Spacone and El-Tawil [2] and to Leon and Viest [3].

Despite the fact that the partial interaction behaviour of composite beams has been studied over the last 50 years, there have been some recent interesting numerical contributions still focussing on their linear-elastic behaviour. Despite the problem appearing quite simple at face value, the analysis of steel–concrete composite members with PI is quite complicated. Worthy of mention is the contribution by Faella et al. [4] who, in 2002, presented a stiffness element with 6dof, viz. the vertical displacement, the rotation and the slip at both element ends, where the governing differential equation of the PI problem was expressed and solved with respect to the curvature, while the slip expression was defined in terms of the hyperbolic functions. Ranzi et al. [5] presented another formulation based on the direct stiffness approach for an element with 6dof, i.e. the vertical displacement, the rotation and the slip at both ends and observed that numerical instabilities occur in the calculation of some stiffness coefficients for low values of the dimensionless stiffness parameter αL, as defined by Girhammar and Pan [6], when these coefficients are derived using the exponential functions (or the hyperbolic functions) in the expression for the slip, and they proposed a modelling procedure to avoid such instabilities.

The modelling technique proposed in this paper intends to derive an 8dof stiffness element which represents an extension of the 6dof element mentioned previously, in which the freedoms consist of the axial displacement at the level of the reference axis, the vertical displacement, the rotation and the slip at both element ends, and these are depicted in Fig. 1. The main advantage of this technique is that no interpolation in the displacement fields and/or discretisations are introduced along the element length. Hitherto, the inclusion of axial force to produce a robust algorithm incorporating PI has not been reported.

Applications of this technique are then demonstrated for simply supported beams and for propped cantilevers subject to an uniformly distributed load and to an axial loading. These cases are also used to validate the accuracy of the proposed stiffness formulation against closed form solutions derived by the authors. This has been carried out for different levels of shear connection stiffness. By doing so, it will be demonstrated how the proposed element is capable of overcoming the curvature locking problems that has been observed in some conventional displacement formulations for high connection stiffness, which have been reported by Dall’Asta and Zona [7] to occur for values of the dimensionless stiffness parameter $\alpha L>10\text{.}$

A 2-span continuous composite beam is analysed to demonstrate the ease of use of this method and its results are validated against those obtained by means of a highly refined finite element with 16dof proposed by Dall’Asta and Zona [7].

Finally, the suitability of the proposed technique to describe the behaviour of indeterminate and of continuous beams is highlighted; in particular, it is shown how material nonlinearities can be easily implemented in the modelling taking advantage of the highly refined functions at the basis of the proposed stiffness element.