How To Calculate Slenderness Ratio

The slenderness ratio, which at first glance suggests an obscure metric in the fashion world, is an engineering concept that is used to estimate the strength of a weight-bearing column, usually one made of steel. In particular, the slenderness ratio is a measure of a column's tendency to be crushed (that is, give way vertically) or buckle (that is, give way both horizontally and vertically). As you might expect, columns that are short in relation to their radius are more susceptible to crushing, whereas longer, thinner ones have a higher risk of buckling.

A higher slenderness ratio implies a greater tendency to fail, all else being equal. Usually, values greater than 200 are considered unsafe.

The general formula for slenderness ratio is

\(\frac{KL}{r}\)

Calculate the Slenderness Ratio for Any Problem

1\. Find the effective length constant K.

K is ​the effective length constant.​ These values can be found in the American Institute of Steel Construction (AISC) manual and typically range between 0.5 and 2.0. The value of K depends on how the ends of the column are supported in a given structure, that is, on the nature of their attachments.

2\. Calculate the effective length.

This is done by simply multiplying the actual measured length of the column by K to derive the complete numerator in the algebraic expression of interest (K)(L).

3. Calculate the radius r.

If you are not explicitly given the minimum radius (assuming the column does not have constant width), derive it from the cross-sectional area of the column using the formula:

\(A=\pi r^2\text{ or }r=\bigg(\frac{A}{\pi}\bigg)^{1/2}\)

4\. Solve for the slenderness ratio.

Use the values from steps . and 3. to derive the ratio.

5. Interpret the results.

The larger the ratio, the less strength the column possesses. The AISC recommendation is a ratio less than or equal to 200.