Using the x coordinate shown in Fig. 7.24(\mathrm{~b}), we write the equation for the bending moment in the beam as

M=-P x

The partial derivative of M with respect to P is given by

\frac{\partial M}{\partial P}=-x

The deflection at B can now be obtained by applying the expression of Castigliano’s second theorem, as given by Eq. (7.60), as follows:

\begin{aligned} \Delta &=\int_{0}^{L}\left(\frac{\partial M}{\partial P}\right)\left(\frac{M}{E I}\right) d x \\\end{aligned}      (7.60)

\begin{aligned} \Delta_{B} &=\int_{0}^{L}\left(\frac{\partial M}{\partial P}\right)\left(\frac{M}{E I}\right) d x \\ \Delta_{B} &=\int_{0}^{L}(-x)\left(-\frac{P x}{E I}\right) d x \\ &=\frac{P}{E I} \int_{0}^{L} x^{2} d x=\frac{P L^{3}}{3 E I} \\ \Delta_{B} &=\frac{P L^{3}}{3 E I} \downarrow \end{aligned}