Area Moments — Hollow Circular Section For a hollow circular section, the polar moment of inertia about an axis perpendicular to the section (through its centroid) is how many times the centroidal moment of inertia about the X–X axis (in-plane principal axis)?

Introduction / Context: For axisymmetric circular sections (solid or hollow), in-plane centroidal moments about X–X and Y–Y are equal. The polar moment about the perpendicular centroidal axis equals the sum of these two equal in-plane moments.

Given Data / Assumptions:

• Hollow circular section with centroid at the geometric center.
• Principal in-plane axes X–X and Y–Y through the centroid.
• Polar axis Z perpendicular to the section through the centroid.

Concept / Approach: By definition, J_z (polar) = I_x + I_y. For any circular (axisymmetric) section, I_x = I_y. Hence J_z = 2 I_x. Therefore, the polar moment is two times either in-plane centroidal moment.

Step-by-Step Solution:

Verification / Alternative check: For a thin ring of radius R and area A, I_x = I_y = (A R^2)/2 and J_z = A R^2. Indeed, J_z = 2 * I_x holds exactly.

Why Other Options Are Wrong: “Same” ignores J_z = I_x + I_y; “Half” and “Four times” contradict the exact identity for axisymmetric sections.

Common Pitfalls: Confusing mass polar moment with area polar moment; misapplying the relation J = I_x + I_y to non-centroidal axes.

Final Answer: Two times.