Euler’s formula — end condition constant: In the form p = π^2 * E * I / (C * l^2), the value of C for a column with both ends fixed is

Introduction / Context:
Euler’s buckling load can be expressed using an end-condition constant C such that Pcr = π^2 * E * I / (C * l^2). The constant C encodes the support restraints via the effective length factor.

Given Data / Assumptions:

• Effective length Le = K * l.
• Pcr = π^2 * E * I / Le^2 = π^2 * E * I / (K^2 * l^2).
• Therefore, C = K^2.

Concept / Approach:
For both ends fixed, K = 0.5. Hence C = K^2 = 0.25 = 1/4. This gives the largest buckling load among common end conditions due to the shortest effective length.

Step-by-Step Solution:

Verification / Alternative check:
Check other cases: pinned–pinned K = 1 → C = 1; fixed–free K = 2 → C = 4; fixed–pinned K ≈ 0.7 → C ≈ 0.5. These confirm consistency.

Why Other Options Are Wrong:

• 4 corresponds to fixed–free, not fixed–fixed.
• 1 is for pinned–pinned; 1/2 is not a standard C; 2 is incorrect for any basic case.

Common Pitfalls:
Mixing effective length K with its square C when switching formula forms.

Final Answer:
1/4