Column buckling – effective length: For a compression member held in position and restrained in direction at both ends (i.e., effectively fixed at both ends), the effective length is approximately equal to which of the following?

Introduction / Context:
Effective length, Le, is a key concept for column buckling. It converts actual member length and end restraints into an equivalent pin-ended length for use in Euler-type stability checks or code-based slenderness limits. Knowing standard Le factors for typical end conditions speeds preliminary design and checking.

Given Data / Assumptions:

• Prismatic column, uniform cross-section.
• End conditions: held in position and restrained in rotation at both ends (fixed–fixed idealization).
• Elastic buckling concept underlying effective length.

Concept / Approach:
The relation is Le = K * L. For typical end conditions: pinned–pinned K ≈ 1.0; fixed–free K ≈ 2.0; fixed–pinned K ≈ 0.7–0.8; fixed–fixed commonly ≈ 0.65–0.70. Many examination problems adopt 0.67 L as a practical rounded value for fixed–fixed restraint to account for real-world semi-rigidity.

Step-by-Step Solution:
Recognize end restraint: both ends held and restrained in direction.Adopt K ≈ 0.67 for fixed–fixed boundary.Compute Le = 0.67 L → select the listed option 0.67 L.

Verification / Alternative check:
Comparative capacity: Buckling load is proportional to 1 / Le^2. Fixed–fixed with Le ≈ 0.67 L provides about (1 / 0.67^2) ≈ 2.2 times the Euler load of a similar pinned–pinned column, matching engineering expectations.

Why Other Options Are Wrong:

• L: corresponds to pinned–pinned.
• 0.85 L: closer to fixed–pinned or partial restraint.
• 1.5 L or 2 L: represent weak-end or cantilever conditions, not both ends restrained.

Common Pitfalls:

• Assuming perfect fixity; real joints may justify slightly higher K values.
• Confusing “held in position” (translation fixity) with “restrained in direction” (rotation fixity).

Final Answer:
0.67 L