On the Concept of the Virtual Structure to Provide an Inextricable Link between Upper Bound Calculations and the Virtual Power Principle

Abstract

On encountering the upper bound theorem of plasticity, students tend to be both fascinated and slightly mystified. They are favourably impressed by both the mathematical elegance of, and the significance of the output from, the application of the theorem, but they also perceive the origins of the theorem to be shrouded in mystery. Specifically, student confusion arises over the received wisdom that while virtual work arguments can be used to elegantly prove the upper bound theorem, upper bound calculations cannot, strictly speaking, be generally regarded as virtual work calculations on the original structure. As a result, the potential for innovation in structural analysis and design which stems from a firm grounding in the theorem may remain untapped in the students when they enter the design office upon graduation. The present paper tries to resolve this student enigma by arguing that if the concept of a virtual structure with special properties is introduced, then upper bound calculations on several frames can be shown to be unmitigated applications of the virtual work principle. It is then shown that reference to virtual power is more appropriate than that to virtual work, and that the hinges used in upper bound calculations are not meant to represent real plastic hinges, but rather are mathematical abstractions required for the work calculations; indeed, it is shown that the concept of the hinge applies with equal rigour to both plastic and elastic analyses. Finally a graph employing the idea of gradually varying section topology is presented to help students gain an intuitive feel for the concept of the shape factor as used for plastic analysis of structural sections. It is concluded that these ideas may well enhance students' appreciation of some key concepts of plastic theory.