GATE 2021 AE – Question 62
A 3 m ×1 m signboard is supported by a hollow square pole fixed to the ground. Its outer side is 250 mm, and material yield strength 240 MPa. The sign bottom is 5.5 m above ground. Under wind pressure 7.5 kPa, find inner dimension d in mm (nearest integer). Neglect transverse shear and wind loading on the pole.
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Correct answer: 235
Explanation
**Wind load.** The signboard is 3 m wide and 1 m high, so the wind force is $F=p\times A=7500\times3=22\,500$ N. The bottom of the sign is 5.5 m above the ground, so its centre is at 6 m, and
$$M=22\,500\times6=135\,000\text{ N m}.$$
**Bending stress at the base of the hollow square pole** (outer side $b=250$ mm, inner side $d$):
$$\sigma=\frac{M\,c}{I},\qquad c=\frac b2=125\text{ mm},\qquad I=\frac{b^4-d^4}{12}.$$
Set the maximum stress equal to the yield strength, 240 MPa:
$$I=\frac{M\,c}{\sigma_y}=\frac{135\,000\times10^3\times125}{240}=7.03125\times10^7\text{ mm}^4 .$$
**Solve for $d$:**
$$250^4-d^4=12\times7.03125\times10^7=8.4375\times10^8$$
$$d^4=3.90625\times10^9-8.4375\times10^8=3.0625\times10^9\;\Rightarrow\;d=235.2\approx\mathbf{235\ mm}.$$