Alexander Revisited: Upper- and Lower-Bound Approaches for Axial Crushing of a Circular Tube

Abstract In the classical Alexander model for a rigid, perfectly plastic circular tube under axial progressive crushing, an expression for the average crushing force over one complete folding cycle was given in terms of the folding length while the value of the folding length was determined by minimising the average force. This approach is often misunderstood as a correct upper-bound method and has been widely used for analysing large plastic deformation of structures. In this paper, it is highlighted that Alexander approach is strictly not an upper-bound analysis and alternative analyses are presented. A new initial collapse mechanism involving folding both inwards and outwards of the tube wall is proposed for an upper-bound analysis, which leads to the determination of the folding characteristics such as the folding length. Subsequently, a repeatable mechanism is idealised for the periodic folding process. The upper-bound analysis is performed by minimizing the instant force required for maintaining the folding process, which is equivalent to adopting the energy balance for the incremental plastic deformation rather than the total deformation of one complete folding cycle. Instead of merely obtaining the average force as Alexander did, the present theoretical analysis gives a complete force-displacement curve as well as plastic folding length, etc., which agrees well with the existing knowledge. Furthermore, an equilibrium approach is presented involving detail stress state for a deforming configuration of the tube. The lower-bound results re-produce the variation of the force as obtained from the upper-bound analysis when the folding configurations are the same as those in the assumed collapse mechanisms. The present study conceptually reconciles the upper-bound and low-bound theories in structural plasticity for large deformation analysis.