% Students Correctly Answered:
30.9%
Solution
Answer: C) 40
The slenderness ratio is computed is using following formula:
$$ KL/r $$
where:
$$ K\ =\ effective\ length\ coefficient $$
$$ L\ =\ unbraced\ length $$
$$ r\ =\ radius\ of\ gyration $$
The critical slenderness ratio shall be the largest of the ratio in either direction:
$$ SR_{cr}=\ max\ \left(\frac{K_x\ast L_x}{r_x},\frac{K_y\ast L_y}{r_y}\ \right) $$
The radius of gyration is based on the section’s area (A) and moment of inertia (I). Since the column is square, radii of gyration in each direction are equal:
$$ r_x=r_y=\ \sqrt{\frac{I}{A}}=\ \sqrt{\frac{\frac{1}{12}\ast\left(18\ in\right){\ast\left(18\ in\right)}^3}{\left(18\ in\right)\ast\left(18\ in\right)}}=5.2\ in $$
The effective length coefficient value, K, is considered based on a fixed base and free headed support:
$$ K=2 $$
Given that:
$$ K_x=K_y $$
$$ r_x=r_y $$
The critical slenderness ratio will be controlled by the maximum unbraced length. In this case, that is in the y-direction:
$$ L_y=8\ ft $$
$$ SR_{cr}=\frac{2\ast8\ ft\ast\frac{12\ in}{1\ ft}\ }{5.2\ in}\ \approx40 $$
