Area Moment of Inertia – Rectangle about a Centroidal Axis Parallel to depth d For a rectangular section of width b and depth d, what is the second moment of area (area moment of inertia) about an axis that passes through the centroid and is parallel to the depth d (i.e., the vertical centroidal axis)?

Introduction / Context:
In strength of materials and structural analysis, standard formulas for second moments of area (also called area moments of inertia) are used to evaluate bending stiffness about different centroidal axes. For a rectangle, the values differ depending on whether the reference axis is parallel to the width or the depth.

Given Data / Assumptions:

• Plane figure: rectangle with width b (horizontal) and depth d (vertical).
• Axis: centroidal axis parallel to the depth d (vertical centroidal axis through the center).
• Homogeneous, thin area; standard definitions apply.

Concept / Approach:
The two principal centroidal area moments for a rectangle are: Ix (about the centroidal horizontal axis, parallel to width) Ix = (b * d^3) / 12 and Iy (about the centroidal vertical axis, parallel to depth) Iy = (d * b^3) / 12. The requested axis is parallel to d, so we need Iy.

Step-by-Step Solution:
Identify axis: centroidal vertical axis ⇒ use Iy. Write formula: Iy = (b^3 * d) / 12. Check dimensions: length^4 (b^3 * d) ⇒ correct for area moments.

Verification / Alternative check:
By symmetry, swapping b and d swaps the roles of Ix and Iy. If the axis were parallel to width, the answer would be (b * d^3) / 12 instead.

Why Other Options Are Wrong:
(b) corresponds to Ix, not the asked axis. (c) and (d) have the correct variables but wrong denominator (3 instead of 12). (e) has wrong power (dimensionally incorrect).

Common Pitfalls:
Confusing which axis (x or y) is parallel to width or depth.

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