A slender uniform column of length $L$ and flexural rigidity $EI$ is pinned at both ends. It rests on a continuous linear elastic foundation of Winkler stiffness modulus $k$ (restoring force per unit length per unit lateral deflection). The critical Euler buckling load $P_{cr}$ for the fundamental buckling mode ($n = 1$) is given by:

(A) $P_{cr} = \frac{\pi^2 EI}{L^2} + \frac{k L^2}{\pi^2}$

(B) $P_{cr} = \frac{\pi^2 EI}{L^2} + k L$

(C) $P_{cr} = \frac{\pi^2 EI}{L^2} + \frac{k}{\pi^2}$

(D) $P_{cr} = \sqrt{\frac{\pi^2 EI}{L^2} \cdot k}$

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### 1. Strategy & Theoretical Principles The governing differential equation for lateral deflection $w(x)$ of an axially loaded column on a continuous elastic foundation is: $$EI \frac{d^4 w}{dx^4} + P \frac{d^2 w}{dx^2} + k w = 0$$ For pinned-pinned boundary conditions ($w(0) = w(L) = 0$ and $w''(0) = w''(L) = 0$), the deflection mode shape is: $$w(x) = C_n \sin\left(\frac{n\pi x}{L}\right)$$ ### 2. Analytical Formulation Substitute $w(x) = C_n \sin\left(\frac{n\pi x}{L}\right)$ into the differential equation: $$EI \left(\frac{n\pi}{L}\right)^4 \sin\left(\frac{n\pi x}{L}\right) - P \left(\frac{n\pi}{L}\right)^2 \sin\left(\frac{n\pi x}{L}\right) + k \sin\left(\frac{n\pi x}{L}\right) = 0$$ Canceling $\sin\left(\frac{n\pi x}{L}\right)$: $$EI \left(\frac{n\pi}{L}\right)^4 - P \left(\frac{n\pi}{L}\right)^2 + k = 0$$ ### 3. Step-by-Step Calculation Solving for buckling load $P$: $$P = EI \left(\frac{n\pi}{L}\right)^2 + \frac{k}{\left(\frac{n\pi}{L}\right)^2} = \frac{n^2 \pi^2 EI}{L^2} + \frac{k L^2}{n^2 \pi^2}$$ For the fundamental mode $n = 1$: $$P_{cr} = \frac{\pi^2 EI}{L^2} + \frac{k L^2}{\pi^2}$$ ### 4. Distractor Forensics - **Option A:** Correct analytic solution. - **Option B:** Dimensionally inconsistent ($k L$ does not have dimensions of force). - **Option C:** Missing $L^2$ in the foundation stiffness term. ### 5. Exam Trap & Standard Reference - **Trap Alert:** Always verify dimensional homogeneity: $[EI/L^2] = \text{N}$, $[k L^2] = (\text{N/m}^2) \cdot \text{m}^2 = \text{N}$. - **Standard Reference:** Timoshenko, S.P. & Gere, J.M., *Theory of Elastic Stability*, 2nd Edition, McGraw-Hill, Chapter 2.

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