Nonlinear Mathematics in Structural Engineering

Structural engineering is a discipline with a distinguished history in its own right with its landmark monuments and famous personalities from centuries past to the present [1, 2]. Moreover, it is also a discipline that relies on rich nonlinear mathematics as its basis. The aim of this article to show some of the interesting features and practical relevance of nonlinear mathematics in the behaviour of real structures. It is an area of research where the UK has led the way for many years. In the response of structures under loading there are many different sources of nonlinearities. However, for the purpose herein the various cases can be grouped into two distinct categories: (1) material and (2) geometric nonlinearities. The sources of nonlinear material behaviour can arise from the response where the constitutive law (relating stress to strain) in the elastic range is not linear—termed nonlinear elasticity. Materials such as mild structural steel have a linear elastic constitutive law, but other important structural materials such as concrete, aluminium, and alloys of iron such as stainless steel are all examples where the elastic constitutive law is nonlinear. Another route to nonlinearity in the material response can occur even in linear elastic materials when the stress exceeds the so-called yield stress; permanent deformation (plasticity) ensues and the constitutive law departs from the initial linear relationship (Fig. 1). For brittle materials, such as cast iron, fracture, rather than plasticity, follows the elastic response; a further example of material nonlinearities governing the mechanical response during failure. The main focus herein is, however, on geometric nonlinearities that govern structural behaviour when large and possibly sudden deflections are seen, often as a loss of stability when the phenomenon known as buckling is triggered. In structural engineering this is most likely elements in whole or in part compression such as columns and beams. Most rudimentary structural mechanics principles are based on the linear assumptions in that although structures deform, they do so slowly with small deflections. Linearization in this context can be typified by the familiar assumption when dealing with small angles: