The Linear Theory of Beams

Abstract

The equations describing the mechanics of a three-dimensional continuum are formidable to solve even for a simple constitutive model like isotropic hyper-elasticity. Even in the age of computers and the finite element method, it is still not feasible to treat every solid body as a three-dimensional continuum. Bodies with certain geometric features are amenable to a reduction from three dimensions to fewer dimensions, from the perspective of the governing differential equations. These bodies are usually called beams (one dimension), plates (two dimensions, flat), and shells (two dimensions, curved). These reduced theories comprise a subset of solid mechanics generally referred to as structural mechanics. Among the theories of structural mechanics, beam theory is the simplest.