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:

$$ L_y=8\ ft $$

$$ SR_{cr}=\frac{2\ast8\ ft\ast\frac{12\ in}{1\ ft}\ }{5.2\ in}\ \approx40 $$