Prediction of elastic—plastic displacements of tubular joints under combined loading using an energy-based approach

Abstract

A series of 24 geometrically and materially non-linear finite element analyses of a simply supported YT tubular joint, with axial loads on the T- and Y-brace ends, was carried out to collapse, using solid three-dimensional element models. The analyses all have proportional and monotonic loading histories (i.e. radial load paths) and each analysis has a different T-brace to Y-brace load ratio so that the series ranges over all four quadrants of the two-dimensional load space. The results of the analyses are processed and combined by means of an energy approach, specifically a generalization of Castigliano's theorem on displacements using the concept of complementary work. A prediction method is thus developed to determine the non-linear, elastic-plastic, force-displacement responses of the joint for arbitrary (in-plane) radial load paths in the two-dimensional load space of the structure. One immediate application envisaged is the development of an elastic-plastic ‘joint finite element’ for use in the collapse prediction of tubular structures. The method is potentially applicable to a range of structures and loading scenarios, including other tubular joint types, both uniplanar and multiplanar, with bending moments and/or torques as well as transverse and axial loads.

On the prediction of elastic-plastic generalized load-displacement responses for tubular joints

“…Since the left-hand sides of equations (12) and (13) are known (see above) and the components of r are also known (from the direction of the prediction load path), we have two equations in the eight unknown backward and forward estimates of @W c =@ M n Y , @W c =@ M n T , @W c =@ F n Y and @W c =@ F n T . However, since the gradient vector of a scalar function at a point is normal to the level surface of the function through the point and since, in the present case, the complementary work W c is a scalar function of four independent variables, it follows that the normal vector to the W c2 level surface is parallel to the gradient vector of W c , the components of which are @W c =@ M n Y , @W c =@ M n T , @W c =@ F n Y and @W c =@ F n T .…”

“…These six relationships permit solution of equations (12) and (13) for the forward and backward estimates of each of…”

“…The ranges were chosen, for validation purposes only, because they were expected to give reasonably smooth level surfaces based on previous investigations of the F Y ± F T and M Y ± M T loading modes [12,13]. A critical aspect of the prediction procedure was the choice of an appropriate set of ten sample loading paths to establish the W c level surfaces.…”

“…proportional loading and no unloading) could be predicted from a set of evolving level curves of total complementary work [12,13]. The method was shown to be applicable to the simply supported or encastre YTjoint for combined loading modes involving two loads, such as T-and Y-brace axial forces [12] (i.e. F Y and F T in Fig.…”

“…to find the generalized displacements ö Y , ö T , u Y and u T for the radial load path. The procedure developed is an extension of the procedures presented for the (F Y :F T ), (M Y :M T ) and F Y :M T ) loading modes (see references [12], [13] and [14] respectively), which are based on the method of complementary work W c and the determination of the required partial derivatives of W c with respect to the applied loads via the use of W c level curves in the load space of applied loads. However, for the case of four applied loads the problem is significantly more complex since the M Y ± M T ± F Y ± F T load space is now four dimensional and visualization of the W c level hypersurfaces is no longer possible.…”