Solution to Problem 252 Statically Indeterminate

$\Sigma M_B = 0$

$4P_{al} + 2P_{st} = 4(20\,000)$

$4(\sigma_{al} \, A_{al}) + 2 \sigma_{st} \, A_{st} = 80\,000$

$4 \, [ \, \sigma_{al} (0.75) \, ] + 2 \, [ \, \sigma_{st} (0.5) \, ] = 80\,000$

$3\sigma_{al} + \sigma_{st} = 80\,000$       → equation (1)
 

 

$\dfrac{\delta_{st}}{2} = \dfrac{\delta_{al}}{4}$

$\delta_{st} = 0.5 \delta_{al}$

$\left( \dfrac{\sigma L}{E} \right)_{st} = 0.5 \left( \dfrac{\sigma L}{E} \right)_{al}$

$\dfrac{\sigma_{st} (3)}{29 \times 10^6} = 0.5 \left[ \dfrac{\sigma_{al} (4)}{10 \times 10^6} \right]$

$\sigma_{st} = \frac{29}{15} \sigma_{al}$       → equation (2)
 

From equation (1)
$3\sigma_{al} + \frac{29}{15} \sigma_{al} = 80\,000$

$\sigma_{al} = 16\,216.22 \, \text{psi}$

$\sigma_{al} = 16.22 \, \text{ ksi}$           answer
 

From equation (2)
$\sigma_{st} = \frac{29}{15}(16.22)$

$\sigma_{st} = 31.35 \, \text{ ksi}$           answer