# Energy Methods

• Anthony Bedford
• Kenneth M. Liechti
Chapter

## Abstract

When an archer draws a bow and shoots an arrow, it demonstrates that energy can be stored in an object when work is done to deform it relative to a reference state. Many advanced techniques used in mechanics of materials, including finite elements, are based on energy methods. This chapter discusses some of the fundamental concepts underlying these methods. The work done by a force or couple is first defined. By equating the work done by stress components in deforming a cubic element, the strain energy per unit volume of an isotropic linear elastic material subjected to a general state of stress is shown to be $$u=\frac{1}{2}\left({\sigma}_x{\varepsilon}_x+{\sigma}_y{\varepsilon}_y+{\sigma}_z{\varepsilon}_z+{\tau}_{xy}{\gamma}_{xy}+{\tau}_{yz}{\gamma}_{yz}+{\tau}_{xz}{\gamma}_{xz}\right).$$ With this result the strain energy of simple structural elements subjected to loads can be derived. For example, the strain energy of a prismatic bar of length L and cross-sectional area A that is subjected to axial loads P is U = P2L/2EA. By subjecting objects or structures to a force at a point and calculating the work done by the force, the displacement of the point in the direction of the force can be determined. Strain energy can be applied to a much broader class of problems using Castigliano’s second theorem. Consider an object of elastic material that is subjected to N external forces F1, F2,   … ,  FN. Let ui be the component of the deflection of the point of application of the ith force Fi in the direction of Fi. Then ui = ∂U/∂Fi, where U is the strain energy resulting from the application of the N forces. If couples also act on an object, the angle of rotation of the point of application of the ith couple Mi in the direction of Mi is given by θi = ∂U/∂Mi. Examples of applications to trusses and beams are presented.

## Keywords

Castigliano’s second theorem Deflection Energy Energy methods Strain energy Work

## Supplementary material

© Springer Nature Switzerland AG 2020

## Authors and Affiliations

• Anthony Bedford
• 1
• Kenneth M. Liechti
• 1
1. 1.University of TexasAustinUSA