Section Modulus — Solid Circular Section What is the section modulus Z about a centroidal axis for a solid circular section of diameter d?
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Api * d^2 / 4
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Bpi * d^2 / 16
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Cpi * d^3 / 16
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Dpi * d^3 / 32
Answer
Correct Answer: pi * d^3 / 32
Explanation
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