GATE 2022 AE – Question 58
A simply supported Aluminium column of length 1 m and rectangular cross- section w× t with t≤w, is subjected to axial compressive loading. Young’s modulus is 70 GPa. Yield stress under uniaxial compression is 120 MPa. The value of t at which the failure load for yielding and buckling coincide is _______mm.
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Correct answer: 45.42 to 45.88
Explanation
The column fails either by yielding or by buckling, and we want the thickness at which both loads are equal.
**Yield load:** $P_y=\sigma_y\,A=\sigma_y\,w\,t$.
**Euler buckling load** of a pin-ended column, buckling about the weaker axis (the thickness direction, since $t\leq w$), with $I=\dfrac{wt^3}{12}$:
$$P_{cr}=\frac{\pi^2EI}{L^2}=\frac{\pi^2E\,w\,t^3}{12L^2}.$$
**Equate:**
$$\sigma_y\,w\,t=\frac{\pi^2E\,w\,t^3}{12L^2}\;\Rightarrow\;t^2=\frac{12\,\sigma_yL^2}{\pi^2E}.$$
**Numbers.** $\sigma_y=120\times10^6$ Pa, $E=70\times10^9$ Pa, $L=1$ m:
$$t=L\sqrt{\frac{12\sigma_y}{\pi^2E}}=\sqrt{\frac{12\times120\times10^6}{9.8696\times70\times10^9}}=\sqrt{2.0842\times10^{-3}}=0.04565\ \text{m}.$$
The thickness is **45.65 mm**. (The width $w$ cancels out.)