Section Modulus — Solid Circular Section What is the section modulus Z about a centroidal axis for a solid circular section of diameter d?

Introduction:
Section modulus links bending moment to extreme-fiber stress: sigma_max = M / Z. For a solid circle, Z depends on its centroidal second moment of area and the outermost fiber distance.

Given Data / Assumptions:

• Solid circular cross-section of diameter d.
• Centroidal bending about any diametral axis.

Concept / Approach:
For a solid circle, I = (pi * d^4) / 64 about a centroidal axis. The extreme distance is y_max = d / 2. Hence Z = I / y_max.

Step-by-Step Solution:
I = (pi * d^4) / 64y_max = d / 2Z = I / y_max = ((pi * d^4) / 64) / (d / 2) = (pi * d^3) / 32

Verification / Alternative check:
Dimensional check: length^4 divided by length gives length^3, correct for Z.

Why Other Options Are Wrong:
pi * d^2 / 4 and pi * d^2 / 16: these correspond to areas, not section modulus.pi * d^3 / 16: off by a factor of 2; incorrect denominator.

Common Pitfalls:
Mixing up I with J (polar) or area A with Z; always confirm the correct formula.

Final Answer:
pi * d^3 / 32