A uniform slender aerospace pushrod of length $L$ and flexural rigidity $EI$ has pinned-pinned (hinged) end conditions. What is the critical Euler buckling load $P_{cr}$ of the column?

(A) \frac{\pi^2 E I}{L^2}

(B) \frac{4 \pi^2 E I}{L^2}

(C) \frac{\pi^2 E I}{4 L^2}

(D) \frac{2 \pi^2 E I}{L^2}

Answer Key:

A

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Governing Eigenvalue Problem:** $E I \frac{d^4 w}{dx^4} + P \frac{d^2 w}{dx^2} = 0$. 2. **Stage 2: Pinned Boundary Conditions:** $w(0)=0, w(L)=0, w''(0)=0, w''(L)=0$. 3. **Stage 3: Fundamental Mode:** $w(x) = C \sin\left(\frac{\pi x}{L}\right) \implies P_{cr} = \frac{\pi^2 E I}{L^2}$. 4. **Stage 4: Selection:** Option A is correct.

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