Bending Stress MCQ with answer and solution

Concept

For a semicircle, MOI about the diameter and about the symmetry axis through the centre are both half of the full circle value.

Formula

I_{full circle} = πD⁴/64; I_semicircle = πD⁴/128 = πR⁴/8

Step-by-step solution

1. For a full circle, MOI about any diameter = πD⁴/64. 2. A semicircle is half of the circle, and both axes pass through the centre of the diameter. 3. So I_y = (1/2) × πD⁴/64 = πD⁴/128. 4. In terms of R: πR⁴/8 ≈ 0.393 R⁴. 5. Answer: πD⁴/128.

Explanation

The y axis passes through the centre of the diameter, so the semicircle is exactly half of a circle about this axis. Half of πD⁴/64 gives πD⁴/128. The value 0.10976 R⁴ is used only for the centroidal axis parallel to the diameter.

Why the other options are wrong

B: R³ has the wrong unit for MOI. C: πD⁴/16 is too large and is not a standard result for a semicircle. D: 0.10976 R⁴ is the MOI about the centroidal axis parallel to the diameter, not about the y axis.

Shortcut

For a semicircle, MOI about the diameter or the symmetry axis through the centre = half of the circle value = πD⁴/128.

Common mistake

Choosing 0.10976 R⁴, which is the centroidal value parallel to the diameter, not the value about the given axis.