Euler buckling of columns — determine the constant C for end conditions: According to Euler's column theory, the crippling load is P = π^2EI/(Cl^2). What is C for a column with both ends hinged (pinned-pinned)?

Introduction:
Euler buckling predicts the elastic critical load for slender columns. The end conditions enter via the effective length factor through C (or K).

Given Data / Assumptions:

• Slender, perfectly straight column; elastic material, small deflections up to buckling.
• Both ends hinged (pinned).

Concept / Approach:
For end conditions, the effective length Le modifies the denominator: Pcr = π^2EI/(Le^2). Writing Pcr = π^2EI/(Cl^2) implies C = (l/Le)^2.

Step-by-Step Solution:

Verification / Alternative check:
Known table: pinned-pinned K = 1 ⇒ Le = l ⇒ Pcr = π^2EI/l^2, matching C = 1.

Why Other Options Are Wrong:

• 1/4 or 1/2 or 2 or 4 correspond to other end conditions (e.g., fixed-free, fixed-pinned, fixed-fixed) and not pinned-pinned.

Common Pitfalls:
Mixing K (effective length factor) with C; forgetting which boundary conditions give which Le.

Final Answer: