Question 11.14: A load P is supported at B by two rods of the same material ...

We apply a dummy horizontal load $\mathbf{Q}$ at $B$ (Fig. 11.45). From Castigliano’s theorem we have

$x_{B}=\frac{\partial U}{\partial Q} \quad y_{B}=\frac{\partial U}{\partial P}$

Recalling from Sec. 11.4 the expression (11.14) for the strain energy of a rod, we write

$U=\frac{F_{B C}^{2}(B C)}{2 A E}+\frac{F_{B D}^{2}(B D)}{2 A E}$

where $F_{B C}$ and $F_{B D}$ represent the forces in $B C$ and $B D$, respectively. We have, therefore,

$x_{B}=\frac{\partial U}{\partial Q}=\frac{F_{B C}(B C)}{A E} \frac{\partial F_{B C}}{\partial Q}+\frac{F_{B D}(B D)}{A E} \frac{\partial F_{B D}}{\partial Q}$      (11.83)

and

$y_{B}=\frac{\partial U}{\partial P}=\frac{F_{B C}(B C)}{A E} \frac{\partial F_{B C}}{\partial P}+\frac{F_{B D}(B D)}{A E} \frac{\partial F_{B D}}{\partial P}$     (11.84)

From the free-body diagram of pin $B$ (Fig. 11.46), we obtain

$F_{B C}=0.6 P+0.8 Q \quad F_{B D}=-0.8 P+0.6 Q$      (11.85)

Differentiating these expressions with respect to $Q$ and $P$, we write

$\begin{array}{rlrl} \frac{\partial F_{B C}}{\partial Q} =0.8 & \frac{\partial F_{B D}}{\partial Q}=0.6 \\ \frac{\partial F_{B C}}{\partial P}=0.6 & \frac{\partial F_{B D}}{\partial P}=-0.8 \end{array}$       (11.86)

Substituting from (11.85) and (11.86) into both (11.83) and (11.84), making $Q=0$, and noting that $B C=0.6 l$ and $B D=0.8 l$, we obtain the horizontal and vertical deflections of point $B$ under the given load $\mathbf{P}$ :

$\begin{aligned} x_{B} & =\frac{(0.6 P)(0.6 l)}{A E}(0.8)+\frac{(-0.8 P)(0.8 l)}{A E}(0.6) \\ & =-0.096 \frac{P l}{A E} \\ y_{B} & =\frac{(0.6 P)(0.6 l)}{A E}(0.6)+\frac{(-0.8 P)(0.8 l)}{A E}(-0.8) \\ & =+0.728 \frac{P l}{A E} \end{aligned}$

Referring to the directions of the loads $\mathbf{Q}$ and $\mathbf{P}$, we conclude that

$x_{B}=0.096 \frac{P l}{A E} \leftarrow \quad y_{B}=0.728 \frac{P l}{A E} \downarrow$

We check that the expression obtained for the vertical deflection of $B$ is the same that was found in Example 11.09.