GATE 2023 CE (CE1) – Question 55

Solid Mechanics · Simple bending theory, flexural and shear stresses, shear centre · 2 marks · Numerical answer

The cross-section of a girder is shown in the figure (not to scale). The section is symmetric about a vertical axis (Y-Y). The moment of inertia of the section about the horizontal axis (X-X) passing through the centroid is _____ cm$^4$ (round off to nearest integer).

a T-shaped section. The top flange is 40 cm wide and 10 cm thick. The web below is 20 cm wide and 50 cm deep.

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Correct answer: 468810

Explanation

Measure from the bottom of the web. The web ($20 \times 50 = 1000$ cm²) has its centroid at 25 cm and the flange ($40 \times 10 = 400$ cm²) at 55 cm. The centroid is at $\bar{y} = \frac{1000 \times 25 + 400 \times 55}{1400} = 33.57$ cm. By the parallel axis theorem, $I = \frac{20 \times 50^3}{12} + 1000 \times (33.57 - 25)^2 + \frac{40 \times 10^3}{12} + 400 \times (55 - 33.57)^2 = 208333 + 73469 + 3333 + 183673 = 468809.5 \approx 468810$ cm⁴.