Second Moment of Area — Rectangle about Centroidal Axis Parallel to Width For a rectangular section of width b and depth d, what is the moment of inertia about a centroidal axis parallel to the width (i.e., bending about the axis along b)?

Introduction / Context:
The second moment of area (also called the area moment of inertia) is crucial for deflection and bending stress calculations. For rectangles, two principal centroidal axes exist: one parallel to the width b and one parallel to the depth d.

Given Data / Assumptions:

• Rectangle with width b (horizontal) and depth d (vertical).
• Axis passes through the centroid and is parallel to the width b (i.e., bending about the horizontal axis).

Concept / Approach:
Standard formulae: about the centroidal axis parallel to width b (horizontal), I = (b d^3) / 12; about the centroidal axis parallel to depth d (vertical), I = (d b^3) / 12. The larger cubic power belongs to the dimension perpendicular to the neutral axis.

Step-by-Step Reasoning:

Verification / Alternative check:
Dimensional check: units of I are length^4; b d^3 has correct dimension. Symmetry check: swapping axes swaps b and d accordingly.

Why Other Options Are Wrong:
Using b^3 d or factors of 1/3 corresponds to wrong axis or to a non-centroidal formula; they over/under-estimate stiffness.

Common Pitfalls:
Mixing up which dimension appears cubed; forgetting that the axis orientation determines which side contributes the cube.

Final Answer:
I = (b * d^3) / 12.