Columns – Effective Length with One End Fixed and the Other Free A straight column of actual length L has one end fixed and the other end free. What is the Euler equivalent length Le used in buckling calculations?

Introduction / Context:
Euler buckling analysis uses an equivalent length to represent how end restraints affect the critical load. Different end conditions change effective length factors and thus the stability of columns.

Given Data / Assumptions:

• Column of actual length L.
• One end fixed, the other end free.
• Perfectly straight, centrally loaded, elastic behavior, pinned-plane assumption for flexure.

Concept / Approach:

Equivalent length Le is defined such that the Euler critical load Pcr = pi^2 E I / (Le^2) reproduces the correct buckling load for the given boundary conditions. For fixed–free, the effective length factor K equals 2.

Step-by-Step Solution:

Verification / Alternative check:

Mode shape for fixed–free is a quarter sine, matching the K = 2 value seen in standard tables and energy methods.

Why Other Options Are Wrong:

L corresponds to pinned–pinned. L / 2 and L / sqrt(2) are too short and overpredict stability. pi * L / 2 is not an effective length factor used for Euler K.

Common Pitfalls:

Using the wrong K for fixed–pinned or fixed–fixed, or confusing actual length with effective length when computing slenderness ratio.

Final Answer:

2 L