Solution to Problem 256 Statically Indeterminate

$\cos 25^\circ = \dfrac{2.75}{L_{br}}$

$L_{br} = 3.03 \, \text{ m }$
 

 

$\Sigma F_V = 0$

$2P_{br} \cos 25^\circ + P_{st} = 7.5(1000)$

$P_{st} = 7500 - 1.8126P_{br}$

$\sigma_{st} \, A_{st} = 7500 - 1.8126 \sigma_{br} \, A_{br}$
$\sigma_{st} (250) = 7500 - 1.8126 \, [ \, \sigma_{br} (250) \, ]$

$\sigma_{st} = 30 - 1.8126 \sigma_{br}$       → Equation (1)
 

$\cos 25^\circ = \dfrac{\delta_{br}}{\delta_{st}}$

$\delta_{br} = 0.9063 \delta_{st}$

$\left( \dfrac{\sigma L}{E} \right)_{br} = 0.9063 \left( \dfrac{\sigma L}{E} \right)_{st}$

$\dfrac{\sigma_{br} (3.03)}{83} = 0.0963 \, \left[ \dfrac{\sigma_{st} (2.75)}{200} \right]$

$\sigma_{br} = 0.3414 \sigma_{st}$       → Equation (2)
 

From Equation (1)
$\sigma_{st} = 30 - 1.8126(0.3414 \sigma_{st})$

$\sigma_{st} = 18.53 \, \text{ MPa}$           answer
 

From Equation (2)
$\sigma_{br} = 0.3414(18.53)$

$\sigma_{br} = 6.33 \, \text{ MPa}$           answer