Structural parts as quadrics -- elasticity ellipses revisited

TL;DR

This paper generalizes the concept of elasticity ellipses in structural analysis to higher dimensions and degenerate cases, exploring how projective transformations preserve system equilibrium and compatibility.

Contribution

It introduces a generalized framework for elasticity quadrics, extending classical elasticity ellipses to broader geometric and dimensional contexts, including transformations.

Findings

Generalization of elasticity ellipses to higher dimensions and degenerate quadrics

Analysis of projective transformations preserving mechanical equilibrium

Enhanced understanding of geometric properties in structural elasticity

Abstract

Elasticity ellipses or central ellipses have been long used in graphic statics to capture the elastic behaviour of structural elements. The paper gives a generalisation the concept both in dimensions and in the possibility of degenerate conics / quadrics. The effect of projective transformations of these quadrics is also given, such that the entire mechanical system can be transformed preserving equilibrium and compatibility between its elements.