Critical Buckling Load for a Slender Column

A slender column is a long, narrow column with a uniform cross-section. When a slender column is subjected to an axial load, it may buckle under the compressive force. The critical buckling load is the maximum load that the column can withstand before buckling occurs.

There are different boundary conditions that can be applied to the ends of a slender column, which affect its buckling behavior. Two common boundary conditions are:

1. Both Ends Clamped: In this case, both ends of the column are fixed, meaning that they cannot move or rotate.

2. Both Ends Hinged: In this case, both ends of the column are allowed to rotate freely, but they cannot move in the axial direction.

Comparison of Critical Buckling Loads

The question asks for the ratio of the critical buckling load for the case with both ends clamped to the case with both ends hinged.

To understand this, let's consider a simple example. Assume we have a slender column with length L and a uniform cross-section. The critical buckling load for the case with both ends clamped is given by the Euler's formula:

P_critical_clamped = (π^2 * E * I) / L^2

Where:
- P_critical_clamped is the critical buckling load for the case with both ends clamped.
- E is the modulus of elasticity of the material.
- I is the moment of inertia of the cross-section.
- L is the length of the column.

Now, let's consider the case with both ends hinged. In this case, the critical buckling load is given by:

P_critical_hinged = (4 * π^2 * E * I) / L^2

Ratio of Critical Buckling Loads

To find the ratio of the critical buckling load for the case with both ends clamped to the case with both ends hinged, we can divide the two equations:

P_critical_clamped / P_critical_hinged = (π^2 * E * I) / (4 * π^2 * E * I)

Simplifying the equation, we get:

P_critical_clamped / P_critical_hinged = 1 / 4

Therefore, the ratio of the critical buckling load for the case with both ends clamped to the case with both ends hinged is 1:4, which corresponds to option C.

This ratio indicates that the critical buckling load for a column with both ends clamped is four times higher than the critical buckling load for a column with both ends hinged.