A compression member of actual length L is held in position and restrained in direction (fixed) at one end, and at the other end it is effectively restrained in direction but not held in position. What effective length should be used for buckling design?

Introduction / Context:
Effective length captures the influence of boundary restraints on the buckling strength of columns. Real-world columns seldom have identical end restraints, so design tables present K-factors for various combinations of translation and rotation restraints at ends.

Given Data / Assumptions:

• Member length = L.
• End A: held in position (no lateral translation) and restrained in direction (no rotation) → fixed.
• End B: effectively restrained in direction (no rotation) but not held in position (translation free) → rotationally fixed but laterally free (sway permitted).
• Elastic buckling; concentric axial compression.

Concept / Approach:
For the mixed case where one end is fully fixed and the other end cannot rotate but is free to translate (sway), standard alignment charts give a relatively large K due to the permitted sway mode. The widely used value for this end pair is K = 1.5, hence L_eff = 1.5 L. This reflects a much lower buckling capacity than fixed–fixed and even lower than pin–pin in many alignment conditions because sway increases the effective length.

Step-by-Step Solution:
Identify end A (fixed) and end B (rotation restrained, translation free).Select corresponding K-factor from end-condition table: K ≈ 1.5.Compute effective length: L_eff = K * L = 1.5 L.Use in design checks: λ = L_eff / r and P_cr = (pi^2 * E * I) / (L_eff^2).

Verification / Alternative check:
Compare with common cases: fixed–fixed (0.5L), fixed–pin (≈0.7L), pin–pin (L), fixed–free (2L). The given arrangement reasonably lies between pin–pin and cantilever, thus 1.5L is consistent.

Why Other Options Are Wrong:

• L, 0.85L, 0.67L: correspond to stiffer end-restraint combinations (no sway or partial fixity).
• 2L: is for a cantilever (fixed–free) and is too large here.

Common Pitfalls:

• Misreading “held in position” (translation restraint) vs “restrained in direction” (rotation restraint).
• Assuming the same K for non-sway and sway frames.

Final Answer:
1.5 L