Bending Ovalization of Thin-Walled Circular Tubes

Abstract

This paper presents a new analytical model to predict the mechanical behaviour of an elastic straight thin-walled circular tube (pipe) subjected to bending. A thin-walled pipe cannot be modelled as a simple beam, inasmuch it presents a shell-like behaviour due to the ovalization (in-plane warping) of the cross-section when loaded. The section modification implies a non-linear trend between the applied moment and the axial curvature, i.e. a non-invariance of the section moment of inertia. Two analytical ways are proposed in the reference literature: a rigorous one due to Reissner use a stress-function approach, and an approximated one, first due to Brazier, that follows Ritz approach. The Reissner model is analytically unsolvable and difficult to face numerically due to some integral conditions to fulfill. On the contrary, the Ritz approach is easy to implement, but being Heuristic is limited to circular sections. To overcome the previous mentioned difficulties a new geometrically exact Pipe model in terms of displacements is proposed, which turns out to be a mix between a shell and a beam model. The present approach leads to a system of ODEs accompanied by Boundary Value Problems (not requiring any integral conditions) that can be solved using a direct collocation method. After solving it, we build-up the dimensionless (non-linear) moment-curvature diagram, valid for any straight pipe, i.e. for every diameter, thickness, and whatever linear elastic material. The results are compared with other literature solutions performed using the Ritz’s approach and with Finite Element Analysis.