Area Moment of Inertia – Circle About Any Diameter For a circular cross-section of diameter D, what is the second moment of area about any centroidal diameter in its plane?

Introduction / Context:
The second moment of area, also called the area moment of inertia, measures a section resistance to bending. For symmetric shapes like circles, closed-form expressions are standard and widely tabulated.

Given Data / Assumptions:

• Section is a solid circle of diameter D.
• Axis is any centroidal diameter within the section plane.
• Homogeneous, prismatic, small deflection elastic bending context.

Concept / Approach:

For a circle, the polar second moment is J = pi D^4 / 32 about the centroid. By the perpendicular axis theorem, J equals the sum of the two equal in-plane centroidal moments about any pair of orthogonal diameters, so each in-plane moment is J/2.

Step-by-Step Solution:

Verification / Alternative check:

Equivalent expression in radius R = D / 2 is I = pi R^4 / 4. Substituting R reproduces pi D^4 / 64.

Why Other Options Are Wrong:

pi D^4 / 32 is the polar moment, not the planar diameter moment. pi D^3 / 64 and pi D^5 / 64 have wrong dimensions. pi D^4 / 128 is off by a factor of 2.

Common Pitfalls:

Confusing polar and planar moments or mixing radius and diameter forms without proper conversion.

Final Answer:

pi D^4 / 64