Column effective length factor (buckling) A column that is effectively held in position (translation restrained) at both ends and restrained in direction (rotation fixed) at one end only has an effective length equal to:

Introduction / Context:
Effective length is a key concept in column buckling. It converts an actual member with given end restraints into an equivalent pin-ended member of length Le producing the same buckling load. Choosing the right Le ensures safe compressive design.

Given Data / Assumptions:

• The column is held in position (no translation) at both ends.
• One end is also restrained in direction (rotation fixed), the other end is not (rotation free → hinge-like).
• Euler-type buckling concept and standard effective length factors are applicable.

Concept / Approach:
Common end conditions and their factors K include: fixed–fixed (K ≈ 0.5), fixed–pinned (K ≈ 0.7), pinned–pinned (K = 1.0), and fixed–free (K = 2.0). The described condition corresponds to fixed–pinned behavior, for which K is commonly taken as about 0.67–0.70.

Step-by-Step Solution:
Identify boundary condition → fixed at one end, pinned at the other.Adopt K ≈ 0.67 for this condition.Effective length Le = K * L = 0.67 L.

Verification / Alternative check:
Many handbooks list K = 0.7 for fixed–pinned; several tables round to 2/3 (≈ 0.67). Either value leads to very similar design results; 0.67 L matches the option given.

Why Other Options Are Wrong:

• L: corresponds to pinned–pinned, not this case.
• 0.85 L: too large for fixed–pinned restraint.
• 1.5 L and 2 L: represent very poor restraint (approaching cantilever), not applicable.

Common Pitfalls:
Confusing “held in position” (translation restraint) with “direction restraint” (rotation fixity). Only one end is fixed in rotation here.

Final Answer:
0.67 L