Euler's column theory: For columns of equal material and length l, the crippling load for a column with one end fixed and the other end free is __________ compared to a similar column hinged at both ends. Choose the correct comparison.

Introduction / Context:Euler's buckling theory relates the critical (crippling) load to the boundary conditions via the effective length. Knowing how end restraint changes buckling capacity is essential for safe column design.

Given Data / Assumptions:

• Same column length l, material (E), and section (I).
• Case 1: One end fixed and the other free.
• Case 2: Hinged (pinned) at both ends.
• Elastic buckling (Euler region).

Concept / Approach:Euler load P_cr = (pi^2 * E * I) / (l_e^2), where l_e is the effective length. For pinned–pinned: l_e = l. For fixed–free: l_e = 2l. Larger l_e reduces P_cr quadratically.

Step-by-Step Solution:

Verification / Alternative check:Effective length factors K: K = 1.0 (pinned–pinned), K = 2.0 (fixed–free). Since P_cr ∝ 1/K^2, doubling K quarters the load, consistent with the result.

Why Other Options Are Wrong:

• Equal to/More than: contradict the effective-length dependence where a cantilevered column is the weakest restraint case.

Common Pitfalls:Confusing fixed–free with fixed–fixed (the latter has higher capacity, K = 0.5).

Final Answer:Less than