ABSTRACT
This chapter presents the derivation of the system of differential equations governing the behaviour of beams. The mathematical model is known as the Euler-Bernoulli beam model and is suitable for the analysis of statically determinate and indeterminate beams. The derivation of the governing differential equations is performed combining three sets of equations, namely, the equilibrium, constitutive and kinematic equations. The equilibrium equations have already been discussed in previous chapters and state the relationship between internal and external actions. The constitutive equations depend on the properties of the materials from which the beam is constructed and describe the relationship between uniaxial stress and strain. The kinematic equations provide a representation of how the deformations undergone by parts of the structure relate to the displacements of the whole structure. These are derived under the assumptions of what is known as the Euler-Bernoulli beam theory (see Chapter 13 for more details), in which plane sections perpendicular to the member axis remain plane and perpendicular to the member axis after deformation. Individual cross-section are therefore assumed not to deform in their plane. We will also assume that the displacements of the structure are small in comparison with the dimensions of the structure.
