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
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A4
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B1
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C1/2
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D1/4
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E2
Answer
Correct Answer: 1/4
Explanation
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:
Identify K for fixed–fixed = 0.5.Compute C = K^2 = (0.5)^2 = 0.25.Thus C = 1/4.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