$\Sigma M_{pin \,\, support} = 0$
$3P_{br} + 12P_{st} = 8(50\,000)$
$3P_{br} + 12P_{st} = 400\,000 \, \to \, $ Equation (1)
$\dfrac{\delta_{st}}{12} = \dfrac{\delta_{br}}{3}$
$\delta_{st} = 4 \delta_{br}$
$\left( \dfrac{PL}{AE} \right)_{st} = 4 \left( \dfrac{PL}{AE} \right)_{br}$
$\dfrac{P_{st} (10)}{0.5(29 \times 10^6)} = 4 \left[ \dfrac{P_{br} (3)}{2(12 \times 10^6)} \right]$
$P_{st} = 0.725P_{br}$
From equation (1)
$3P_{br} + 12(0.725P_{br}) = 400\,000$
$P_{br} = 34\,188.03 \, \text{lb}$
$\delta_{br} = \left( \dfrac{PL}{AE} \right)_{br} = \dfrac{34\,188.03(3 \times 12)}{2(12 \times 10^6)}$
$\delta_{br} = 0.0513 \, \text{in}$
$\dfrac{\delta_W}{8} = \dfrac{\delta_{br}}{3}$
$\delta_W = \frac{8}{3} \delta_{br}$
$\delta_W = \frac{8}{3} (0.0513)$
$\delta_W = 0.1368 \, \text{ in}$ answer
Check by δst:
$P_{st} = 0.725P_{br} = 0.725(34\,188.03)$
$P_{st} = 24\,786.32 \, \text{lb}$
$\delta_{st} = \left( \dfrac{PL}{AE} \right)_{st} = \dfrac{24\,786.32(10 \times 12)}{0.5(29 \times 10^6)}$
$\delta_{br} = 0.2051 \, \text{in}$
$\dfrac{\delta_W}{8} = \dfrac{\delta_{st}}{12}$
$\delta_W = \frac{2}{3} \delta_{st}$
$\delta_W = \frac{2}{3} (0.2051)$
$\delta_W = 0.1368 \, \text{ in}$ answer
