Application to Beam Problems:

In beams, the strain energy stored is mostly because of bending with shear contributing less & this contribution is usually neglected. Therefore, the strain energy U might be written as

                        (for very small curvatures)

 The bending moment shall be a function of the loads. After that, by Castigliano's theorem the deflection under one of the loads, P, in its direction shall be

If we need to find the deflection at a point where there is no load, a fictitious load Q ought to be introduced at that point in the direction in which the deflection is required and U is computed. After that, ∂U/∂Q is calculated. If in this expression Q is built equal to zero, the result shall be the deflection at the pint of application of Q.

From the similarities along the truss problems, it might easily be recognised that ∂M /∂Q shall be the moment at a section because of a unit load at the point of application of Q with no other loads present on the beam.