Column End Conditions and Equivalent Length Concepts

Understanding the end conditions of a column is crucial in structural mechanics, particularly for calculating its buckling load using Euler's formula. The **equivalent length** (or effective length, \( L_e \)) represents the length of a theoretically pinned-pinned column that would have the same buckling load as the column with the given end conditions. It is calculated as \( L_e = K \times L \), where \( L \) is the actual length and \( K \) is the effective length factor, which depends on the end conditions.

This question asks us to match different column end conditions (List-I) with their corresponding equivalent lengths relative to a standard hinged-hinged column (List-II).

Matching Column End Conditions (List-I) with Equivalent Lengths (List-II)

Let's analyze each condition:

• a. Both ends hinged:

  Condition: The column is pinned at both ends, allowing rotation but preventing translation.

  Equivalent Length (List-II): 1. \( L \)

  Explanation: This is the basic reference case. For a column hinged at both ends, the effective length factor \( K \) is 1.0. Therefore, the equivalent length \( L_e \) is equal to the actual length \( L \). So, the match is a - 1.

• b. One end fixed and other end free:

  Condition: The column is rigidly fixed at one end (no rotation or translation) and completely free at the other end (like a cantilever).

  Equivalent Length (List-II): 4. \( 2L \)

  Explanation: For this configuration (a cantilever), the effective length factor \( K \) is 2.0. This means the equivalent length \( L_e \) is twice the actual length, \( L_e = 2 \times L = 2L \). So, the match is b - 4.

• c. One end fixed and pin-jointed:

  Condition: The column is rigidly fixed at one end and pinned (hinged) at the other end.

  Equivalent Length (List-II): 2. \( \frac{L}{\sqrt 2} \)

  Explanation: According to the provided correct answer, this end condition matches with option 2. This implies an effective length factor \( K = \frac{1}{\sqrt 2} \approx 0.707 \), making the equivalent length \( L_e = \frac{L}{\sqrt 2} \). (Note: Standard theoretical value is often \( \frac{2}{\sqrt 3} L \), but we adhere to the mapping given in the question's options and correct answer.) So, the match is c - 2.

• d. Both ends fixed:

  Condition: The column is rigidly fixed at both ends, preventing both rotation and translation.

  Equivalent Length (List-II): 3. \( L/2 \)

  Explanation: For a column fixed at both ends, the effective length factor \( K \) is 0.5. This means the equivalent length \( L_e \) is half the actual length, \( L_e = 0.5 \times L = \frac{L}{2} \). So, the match is d - 3.

Summary of Correct Matches

Based on the analysis and aligning with the provided correct answer, the complete matching is:

Therefore, the correct option is the one that represents this specific matching: a - 1, b - 4, c - 2, d - 3.