Section modulus of a rectangular cross-section about its centroidal axis\n\nFor a rectangular section of breadth b and depth d, determine the section modulus about the centroidal axis parallel to b (i.e., the neutral axis through the centroid).

Introduction / Context:
Section modulus is a geometry property used to relate bending moment to maximum bending stress. For a given bending moment, increasing the section modulus reduces the bending stress. It is essential in sizing beams and machine elements.

Given Data / Assumptions:

• Rectangular cross-section of breadth b (horizontal) and depth d (vertical).
• Neutral axis through the centroid, parallel to the breadth b.
• Linear elastic bending; plane sections remain plane.

Concept / Approach:
The section modulus Z is defined as Z = I / c, where I is the second moment of area about the neutral axis and c is the distance from the neutral axis to the extreme fiber.

Step-by-Step Solution:
For a rectangle about its centroidal axis: I = b d^3 / 12.Extreme fiber distance: c = d / 2.Therefore Z = I / c = (b d^3 / 12) / (d / 2) = b d^2 / 6.

Verification / Alternative check:
Dimensional check: I has units of length^4; dividing by c (length) gives length^3, consistent with Z.

Why Other Options Are Wrong:
b/2 and d/2 are lengths, not section modulus. b d^2 / 2 overestimates by a factor of 3. 2 b d^2 is dimensionally correct but numerically wrong.

Common Pitfalls:
Using I = b d^3 / 3 (incorrect) or mixing up axes; forgetting c = d/2 for the extreme fiber.

Final Answer:
b d^2 / 6