Second moment of area — hollow rectangular section about centroidal X–X For a rectangular section of outer breadth B and outer depth H with a concentric rectangular void of breadth b and depth h, what is the second moment of area about the centroidal X–X axis (parallel to breadth)?
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A(B * H^3 − b * h^3) / 12
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B(B^3 * H − b^3 * h) / 12
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C(B * H^3 + b * h^3) / 12
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D(B^3 − b^3) / 12
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E(H^3 − h^3) / 12
Answer
Correct Answer: (B * H^3 − b * h^3) / 12
Explanation
Introduction / Context:The second moment of area (area moment of inertia) of a hollow rectangle is widely used for deflection and bending stress calculations. Engineers often need a general formula independent of a specific diagram.
Given Data / Assumptions:
- Outer rectangle: breadth B, depth H.
- Inner void (hole): breadth b, depth h, concentric and axis-aligned.
- Neutral axis X–X is horizontal through the centroid (mid-depth).
Concept / Approach:For composite areas, subtract the inner void’s second moment of area from that of the outer area about the same axis. For a solid rectangle about its own centroidal horizontal axis, I_xx = (b * h^3) / 12 (breadth times depth cubed over 12).
Step-by-Step Solution:
Outer I_xx(out) = (B * H^3) / 12.Inner I_xx(in) = (b * h^3) / 12.Hollow I_xx = I_xx(out) − I_xx(in).Therefore, I_xx = (B * H^3 − b * h^3) / 12.Verification / Alternative check:When b = 0 and h = 0 (no hole), the formula reduces to (B * H^3)/12 for a solid rectangle. When b = B and h = H (no material left), the result becomes zero, as expected.
Why Other Options Are Wrong:
- (B^3 * H − b^3 * h)/12: Swaps breadth and depth exponents; correct for I_yy, not I_xx.
- Sum form: Should be the difference (subtract the void).
- (B^3 − b^3)/12, (H^3 − h^3)/12: Incomplete; ignore the other dimension and units.
Common Pitfalls:Mixing up I_xx and I_yy; forgetting to keep axes through the common centroid (no parallel-axis shift needed if concentric).
Final Answer:(B * H^3 − b * h^3) / 12