Second moment of area (area moment of inertia) — symmetry of a hollow rectangle: For a hollow rectangular section, the second moment of area about the centroidal Y–Y axis is generally not equal to that about the centroidal X–X axis (unless it is square and concentrically holed). Is the given statement correct?

Introduction / Context:
Designers compare Ixx and Iyy (area moments of inertia) to assess bending stiffness about orthogonal axes. For rectangles, different side lengths produce different stiffness about X–X and Y–Y, and hollowing preserves this asymmetry unless special symmetry exists.

Given Data / Assumptions:

• Hollow rectangular cross-section with unequal outer and inner dimensions.
• Centroid located at the geometric center (concentric hole).

Concept / Approach:
The second moment of area depends on breadth and depth to the third power along the bending axis. For a rectangle, Ixx ∝ bh^3/12 and Iyy ∝ hb^3/12 (outer minus inner for hollow). Unless b equals h (square), these values differ. Hollowing (subtracting inner rectangle) preserves the general inequality unless the section is perfectly square and similarly proportioned.

Step-by-Step Solution:
Compute Ixx = (b* h^3 − b_i* h_i^3)/12.Compute Iyy = (h* b^3 − h_i* b_i^3)/12.For b ≠ h or b_i ≠ h_i, typically Ixx ≠ Iyy.Therefore, the statement that Iyy is not the same as Ixx (in general) is correct.

Verification / Alternative check:
Set a square case: b = h and b_i = h_i, then Ixx = Iyy by symmetry. This exception proves the general rule stated in the stem (which says “not the same” in general).

Why Other Options Are Wrong:

• No: would imply equality regardless of proportions, which is false except for the special square case.

Common Pitfalls:

• Confusing centroid location with equality of moments; centroid at center does not guarantee Ixx = Iyy.
• Ignoring the cubic dependence on the dimension perpendicular to the bending axis.

Final Answer:
Yes