Standard Methods for Aerospace Stress Analysis

4Energy Methods

4.1 Introduction to Energy Methods

Chapter 4 explores the application of energy methods for the determination of structural behavior of determinate and indeterminate structures. The strain energy concept is examined for the determination of displacements and rotations in solid elastic bodies.

4.2 Strain Energy

Strain energy is usually defined as the energy stored within an elastic body when a solid body is deformed by a specified applied loading. Thus, the strain energy is equivalent to the work done by the external forces acting on a solid body. The general expression defining the strain energy is shown as follows:

(4.1)

For any bar of the length, L, under only the axial loading of F, the expression above reduces to the following expression:

(4.2)

where the volume change is dV = Adx, the area times the change in length (L).

Substitute the terms for the axial stress and strain, and one would have

(4.3)

This expression is solved and reduced as

(4.4)

Which, for a series of “n” bars joined together, the strain energy takes the series form

(4.5)

Likewise, for ...