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## Q. 7.16

Use Castigliano’s second theorem to determine the deflection at point $B$ of the beam shown in Fig. $7.24(\mathrm{a})$.

## Verified Solution

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}