Practice Question: Critical Slenderness Ratio for a Square Column
NCEES Civil PE Specification IV.E.: Axial
Problem
A 16-foot long square 18” x 18” column supports axial load P. The column is fixed at the base and unrestrained against any sort of movement at the top. Given the brace points pictured in each direction, determine the critical slenderness ratio to be considered for design of this column.
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Solution
Answer: C) 38
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:
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:
$$ L_y=8\ ft $$
