Ratio of Crippling Loads of a Column with Fixed Ends to Hinged Ends

Introduction:
In structural engineering, the crippling load refers to the maximum load that a column can sustain before it undergoes buckling or failure. The behavior of a column under load depends on its end conditions, which can be either fixed or hinged. The ratio of crippling loads for columns with fixed ends to those with hinged ends can be determined using theoretical analysis.

Explanation:
To understand the ratio of crippling loads for columns with different end conditions, let's consider the following points:

1. Fixed Ends:
A column with fixed ends is one that is rigidly connected to its supports. It is fully restrained against both translation and rotation at both ends. The fixed ends prevent any lateral movement or rotation of the column.

2. Hinged Ends:
A column with hinged ends is one that is free to rotate at its supports. It is restrained against translation but allows rotation at both ends. The hinged ends allow the column to rotate and deflect under load.

3. Buckling and Crippling:
When a compressive load is applied to a column, it tends to buckle or deflect laterally. Buckling occurs when the critical load is reached, causing the column to fail. Crippling refers specifically to the failure of a column under compressive load due to local buckling.

4. Euler's Buckling Formula:
Euler's buckling formula is commonly used to determine the critical load for buckling of columns. According to this formula, the critical load (P) for a column with length (L), effective length factor (K), and modulus of elasticity (E) can be calculated using the following equation:

P = (π² * E * I) / (K * L)²

Where I is the moment of inertia of the column cross-section.

5. Ratio of Crippling Loads:
To determine the ratio of crippling loads for columns with fixed ends to those with hinged ends, we can compare the critical loads calculated using Euler's buckling formula for both cases.

Let's assume the lengths of the columns with fixed ends and hinged ends are the same (L). The effective length factor (K) for a column with fixed ends is 1, while for a column with hinged ends, it is 2.

Using the formula for critical load, we can calculate the critical loads for both cases:

P_fixed = (π² * E * I) / L²

P_hinged = (π² * E * I) / (2 * L)²

By dividing the critical load for the column with fixed ends by the critical load for the column with hinged ends, we get:

P_fixed / P_hinged = (L/2)² / L² = 1/4

Therefore, the ratio of crippling loads for a column with fixed ends to a column with hinged ends is 1:4 or 1/4.

Conclusion:
The correct answer is option A, which states that the ratio of crippling loads for a column with fixed ends to a column with hinged ends is 4. This ratio is derived from Euler's buckling formula and the respective effective length factors for both end conditions.