Overview

The problem of modelling straight beams as an idealized 1D solid is further considered here with the aim of presenting a few distinct topics of theoretical and practical importance, namely:

1. Beam models with deformed cross-sections

2. Bending in the presence of axial loads and buckling

3. Concentrated quantities described in terms of singular distributions

4. Properties of symmetry of stiffness and mass operators

5. Finite element models.

It will be shown that Hamilton s principle provides a suitable theoretical framework to deal with these distinct subjects in a clear and unified manner.

3.1. Introduction

The variational approach introduced in Chapter 1 in the case of three-dimensional solids can be applied to beams without any difficulty. The global strains or stresses have to be substituted for local ones and the three-dimensional domain of integration has to be adapted to one-dimensional beam geometry. One major interest of the variational approach is its convenience for improving the basic models established in Chapter 2. Based on a global energy balance and making use of Hamilton s principle, corrective terms can be suitably defined, which take into account the local deformations of the beam cross-sections. With this aim in mind, it is found useful to split the transverse displacement field into two distinct components; that marked by the subscript s is related to transverse shear and the other, marked by the subscript b , is related to bending. Whatever the material law may be, by using the results given in subsections 2.1.3...