Question

For a circular column having its ends hinged, the slenderness ratio is 160. The \( l/d \) ratio of the column is ............

• A. 80
• B. 40
• C. 57
• D. 20

Hint:

For a circular column, the slenderness ratio is related to the effective length and the radius of gyration, which depends on the column’s cross-sectional dimensions.

Answer 1

The Correct Option is B

The slenderness ratio \( \lambda \) for a column is given by the formula: \[ \lambda = \frac{l}{r} \] where:
- \( l \) is the effective length of the column,
- \( r \) is the radius of gyration of the column.
For a circular column, the radius of gyration \( r \) is given by: \[ r = \sqrt{\frac{I}{A}} \] where: - \( I \) is the moment of inertia of the cross-section, - \( A \) is the cross-sectional area. For a circular column of diameter \( d \), the moment of inertia \( I \) and area \( A \) are given by: \[ I = \frac{\pi d^4}{64} \quad \text{and} \quad A = \frac{\pi d^2}{4} \] Substituting for \( r \): \[ r = \sqrt{\frac{\frac{\pi d^4}{64}}{\frac{\pi d^2}{4}}} = \sqrt{\frac{d^2}{16}} = \frac{d}{4} \] Now, using the given slenderness ratio \( \lambda = 160 \), we have: \[ \lambda = \frac{l}{r} = \frac{l}{\frac{d}{4}} = \frac{4l}{d} \] Thus, the ratio \( \frac{l}{d} \) is: \[ \frac{l}{d} = \frac{\lambda}{4} = \frac{160}{4} = 40 \] Therefore, the \( l/d \) ratio of the column is 40.