GATE 2025 AE – Question 58
A prismatic vertical column of cross-section $a \times 0.5a$ and length $l$ is rigidly fixed at the bottom and free at the top. A compressive force P is applied along the centroidal axis at the top surface. The Young's modulus of the material is 200 GPa and the uniaxial yield stress is 400 MPa. If the critical value of P for yielding and for buckling of the column are equal, the value of $\left(\frac{l}{a}\right)$ is _____ (rounded off to one decimal place).
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Correct answer: 5.07 to 5.13
Explanation
Yielding needs $P_y = \sigma_y A = 400 \times 10^6 \times 0.5a^2$. A column fixed at one end and free at the other has an effective length $2l$, so $P_{cr} = \frac{\pi^2EI}{(2l)^2}$ with the smaller moment of inertia $I = \frac{a(0.5a)^3}{12} = \frac{a^4}{96}$. Equating, $\frac{l^2}{a^2} = \frac{\pi^2 E}{96 \times 4 \times 0.5\,\sigma_y} = \frac{\pi^2 \times 200 \times 10^9}{192 \times 400 \times 10^6} = 25.7$, so $\frac{l}{a} = 5.07 \approx 5.1$.