A slender uniform aerospace strut of length $L$ and flexural rigidity $EI$ has critical Euler buckling load $P_{cr,1}$ when its ends are pinned-pinned ($L_e = L$). If both ends are instead rigidly clamped (fixed-fixed, $L_e = 0.5 L$), what is the ratio of the new critical buckling load to the original load ($P_{cr,2} / P_{cr,1}$)?

Answer Key:

4.0

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Euler Buckling Formulation:** $P_{cr} = \frac{\pi^2 EI}{L_e^2}$, where $L_e$ is the column effective length. 2. **Stage 2: Pinned-Pinned Configuration:** * $L_{e,1} = 1.0 L \implies P_{cr,1} = \frac{\pi^2 EI}{L^2}$. 3. **Stage 3: Fixed-Fixed Configuration:** * The zero-slope boundary conditions introduce inflection points at $x = L/4$ and $x = 3L/4$. * The effective length is halved: $L_{e,2} = 0.5 L$. * $P_{cr,2} = \frac{\pi^2 EI}{(0.5 L)^2} = \frac{\pi^2 EI}{0.25 L^2} = 4 \frac{\pi^2 EI}{L^2} = 4 P_{cr,1}$. 4. **Stage 4: Ratio:** $P_{cr,2} / P_{cr,1} = 4.0$. 5. **Stage 5: Official Reference:** S. Timoshenko and J.M. Gere, *Theory of Elastic Stability*, Chapter 2.

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