Since the theory of continuous vibrations is based on classical elasticity theory, Chapter 5 begins with an introduction to elasticity with the concepts of stress and strain presented in terms of classical elasticity theory. First, the relationship between force and stress is presented and then the equations of motion are developed for a general three-dimensional system. The concept of plane stress is then presented. Following this, the relationship between deformation and strain are developed with the classical elastic stress–strain relations being presented. Special cases for plane stress and strain as well as uniaxial stress are then developed. This is followed by a discussion of anisotropic materials. The displacement equations for elastic bodies are developed along with relevant boundary conditions for various circumstances. The concepts of work and energy are presented with specific details for both strain energy and kinetic energy being presented. Various energy methods pertinent to vibrations are represented with discussions of the principal of virtual work, Hamilton's principle and the general energy theorem. Detailed discussions of Rayleigh's method, Ritz method, and Galerkin's method are also presented.