Euler’s buckling (crippling) load forms: Identify the correct Euler critical load expression P_cr for a prismatic column (modulus E, second moment I, actual length L) under different end conditions, using the effective length concept.

Introduction / Context:
Euler’s column theory gives the elastic buckling (critical) load for slender, perfectly straight columns with ideal end conditions. The formula uses an effective length L_e that depends on end restraints. Greater end fixity reduces L_e and increases the buckling load.

Given Data / Assumptions:

• P_cr = π² E I / L_e².
• Idealized end conditions; column remains elastic up to buckling.
• No initial imperfections or eccentricity (theoretical model).

Concept / Approach:

Effective length L_e embodies end restraint: hinged-hinged L_e = L; fixed-fixed L_e = L/2; fixed-free (cantilever) L_e = 2 L; fixed-hinged L_e = L/√2. Substituting each L_e into P_cr = π² E I / L_e² yields the respective expressions.

Step-by-Step Solution:

Verification / Alternative check:

Ordering by restraint: fixed-fixed has 4× the hinged-hinged capacity; fixed-hinged has 2×; fixed-free has 1/4×, consistent with classical tables.

Why Other Options Are Wrong:

Each individual option is correct for its stated end condition; only the combined choice confirms all correctly.

Common Pitfalls:

Mixing L with L_e; forgetting that the cantilever (fixed-free) is the weakest case (largest effective length).

Final Answer:

All of the above (each matches its end condition)