On the use of spring models to analyse the lateral-torsional buckling behaviour of cracked beams

Abstract

This paper is focused on the lateral-torsional buckling of cracked or weakened elastic beams. The crack is modelled with a generalised elastic connection law, whose equivalent stiffness parameters can be derived from fracture mechanics considerations. The same type of generalised spring model can be used for beams with semi-rigid connections, typically in the fields of steel and timber engineering. As the basis for the present investigation, we consider a strip beam with fork end supports and exhibiting a single vertical edge crack, subjected to uniform bending in the plane of greatest flexural rigidity. We adopt both direct and variational approaches. In the former, conducted within the framework of the Kirchhoff–Clebsch theory, the effect of prebuckling curvature is taken into consideration. This effect is subsequently neglected in the variationally based analyses. First, the three-dimensional elastic connection law adopted is a direct extension of the planar case, which leads to a paradoxical conclusion: the critical moment is not affected by the presence of the crack, regardless of its location. It is shown that the above paradox is due to the non-conservative nature of the connection model adopted (in the sense that the connection law cannot be derived from a potential). Simple alternatives to this cracked-section constitutive law are proposed, based on conservative moment-rotation laws (quasi-tangential and semi-tangential) and consistent variational arguments. Only one of the proposed alternatives leads to a critical moment that depends on both the torsional and flexural stiffnesses of the spring modelling the cracked cross-section. The analyses are always conducted analytically, yielding closed-form characteristic equations for the buckling moments.