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)?

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:

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