Second moment of area of a rectangle about centroidal axis (parallel to width): For a rectangular cross-section of width B and depth D, find the moment of inertia about the centroidal axis that is parallel to the width (i.e., the horizontal centroidal axis through the section).

Introduction / Context:Design against bending requires the second moment of area (also called area moment of inertia) about the relevant centroidal axis. For a rectangle, the standard formulas are frequently used in beam deflection and bending stress calculations (sigma = M * y / I).

Given Data / Assumptions:

• Rectangular section with width B and depth D.
• Axis passes through centroid and is parallel to the width, i.e., a horizontal centroidal axis.
• Linear elastic behavior; small deflections; standard geometric properties apply.

Concept / Approach:

The second moment of area about a centroidal axis parallel to the width depends on the dimension perpendicular to that axis. When the axis is parallel to the width (horizontal), the dimension contributing with a power of 3 is the depth D, leading to a D^3 dependence.

Step-by-Step Solution:

Verification / Alternative check:

Swapping axis orientation swaps which dimension is cubed. About a vertical centroidal axis (parallel to depth), the formula becomes (D * B^3)/12, confirming orientation sensitivity.

Why Other Options Are Wrong:

• (B^3 * D)/12 corresponds to the vertical centroidal axis, not the horizontal one.
• (B * D^3)/3 and (B * D^3)/36 have incorrect constants; 1/12 is standard for centroidal axis.
• (B^3 * D)/3 has both wrong axis and constant.

Common Pitfalls:

• Confusing which dimension is cubed—always the dimension perpendicular to the axis.
• Using base about base axis (B * D^3 / 3) instead of centroidal axis (B * D^3 / 12).

Final Answer:

I = (B * D^3) / 12.