Consider a simply supported beam PQ of length L, carrying a point load F at the center. What will be the deflection at the center of the beam?

Here, the reactions at both ends of the beam are equal, and each is half of the central load.

Consider the segment PC and choose any point at a distance x from the end P. At this point, the moment due to the reaction at point P is the load at point P times the distance. A partial differentiation of the moment equation with respect to the load at end P is half of x. 

A similar analysis can be conducted for the segment QC of the beam.

Using Castigliano's theorem, the deflection at point C is determined by the partial derivatives of the strain energy due to the applied load.

This equation can be simplified by considering the two segments of beam PC and QC.

Performing the integration over half of the length of the beam gives an expression for the deflection at the center of the beam.