A thin rectangular aluminum plate of width $b$, length $a$, and thickness $t$ is simply supported along all four edges and subjected to uniform uniaxial compressive stress $\sigma_x$. The elastic critical buckling stress is given by $\sigma_{cr} = \frac{k_c \pi^2 E}{12(1 - \nu^2)} \left(\frac{t}{b}\right)^2$. If the plate aspect ratio is an integer $a/b = 2$, the theoretical buckling coefficient $k_c$ for the critical buckling mode is:

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### 1. Strategy & Theoretical Principles From Bryan's plate buckling theory, for a simply supported rectangular plate under uniaxial compression with aspect ratio $\alpha = a/b$: $$k_c = \left(m \frac{b}{a} + \frac{1}{m} \frac{a}{b}\right)^2 = \left(\frac{m}{\alpha} + \frac{\alpha}{m}\right)^2$$ where $m$ is the number of half-sine waves along the length $a$. ### 2. Analytical Formulation & Calculation When the aspect ratio $\alpha = a/b$ is an exact integer, the minimum value of $k_c$ occurs when the plate buckles into $m = \alpha$ square half-waves ($m = 2$): $$k_c = \left(\frac{2}{2} + \frac{2}{2}\right)^2 = (1 + 1)^2 = 2^2 = 4.0$$ Indeed, for any plate of integer aspect ratio ($a/b = 1, 2, 3, \dots$), the minimum critical buckling coefficient is identically $k_{c,\min} = 4.0$. ### 3. Distractor Forensics - Candidate testing $m=1$: $k_c = (1/2 + 2/1)^2 = (2.5)^2 = 6.25 > 4.0$. - Minimum occurs at $m=2$, yielding $k_c = 4.0$. ### 4. Exam Trap & Standard Reference - **Trap Alert:** Always minimize over all possible integer half-wave numbers $m$. For integer $a/b$, $k_c = 4.0$ exactly. - **Standard Reference:** Megson, T.H.G., *Aircraft Structures for Engineering Students*, Chapter 9 (Buckling of Thin Plates).

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