5.12. Energy Method for Deflections

Strain energy methods are frequently employed to analyze the deflections of beams and other structural elements. Of the many approaches available, Castigliano’s second theorem is one of the most widely used. In applying this theory, the strain energy must be represented as a function of loading. Detailed discussions of energy techniques are found in Chapter 10. In this section we limit ourselves to a simple example to illustrate how the strain energy in a beam is evaluated and how the deflection is obtained by the use of Castigliano’s theorem (Sec. 10.4).

The strain energy stored in a beam under bending stress σ_x only, substituting M = EI(d²v/dx²) into Eq. (2.63), is expressed in the form