Questions: Moment of Inertia about Centroidal Axes

Question 1

A rectangular cross-section has centroidal moment of inertia I_c = 100 cm⁴ and area A = 20 cm². Its centroid is 6 cm from a parallel axis. What is the moment of inertia about that parallel axis?

• A. 100 cm⁴ — the parallel-axis theorem doesn't apply when moving away from the centroid
• B. 220 cm⁴ — using I = I_c + A·d = 100 + 20·6 = 220
• C. 820 cm⁴ — using I = I_c + A·d² = 100 + 20·36
• D. 620 cm⁴ — subtracting A·d² because the reference axis passes through the shape

Answer: (C)

The parallel-axis theorem states I = I_c + A·d², where d is the perpendicular distance between axes. Here: I = 100 + 20·(6²) = 100 + 720 = 820 cm⁴. The term A·d² is always added (never subtracted), confirming I_c is the minimum moment of inertia for any parallel axis. Option B is a common error: using d instead of d² (forgetting to square the distance). Always square the distance in the parallel-axis theorem.

Question 2

When computing the moment of inertia of a composite L-section, a student applies the parallel-axis theorem to each component before locating the composite centroid. What is wrong with this approach?

• A. Nothing — the parallel-axis theorem can be applied to each component independently at any stage
• B. The parallel-axis theorem requires knowing the composite centroid to compute d_i for each component correctly
• C. The parallel-axis theorem only applies to simple shapes, not composite sections
• D. The student should use mass moment of inertia formulas for composite cross-sections

Answer: (B)

The distance d_i in I_i = I_c,i + A_i·d_i² must be measured from each component's own centroid to the *composite* centroidal axis. If the composite centroid hasn't been found yet, d_i is unknown. The composite centroid is found by area-weighted averaging of component centroids first — only then can d_i be computed for each component. Applying the parallel-axis theorem prematurely produces an incorrect total I because the reference axis is wrong.

Question 3

Among all axes parallel to a given centroidal axis, the centroidal axis has the smallest moment of inertia.

• T. True
• F. False

Answer: True

This follows directly from the parallel-axis theorem: I = I_c + A·d². Since A > 0 and d² ≥ 0, any non-centroidal parallel axis adds a positive term A·d² to I_c. The extra term is zero only when d = 0 — i.e., at the centroidal axis itself. This minimum property is why standard tables tabulate I_c: it is the most compact, reference-independent description of a cross-section's bending geometry.

Question 4

The parallel-axis theorem can be applied in reverse — subtracting A·d² — to find a moment of inertia about an axis closer to the centroid than the given reference axis.

• T. True
• F. False

Answer: False

The parallel-axis theorem is I = I_c + A·d², where I_c is *specifically the centroidal* moment of inertia. You can rearrange it to I_c = I − A·d² to find I_c from a known non-centroidal I — but you cannot chain this to transfer between two arbitrary non-centroidal axes directly. The correct procedure is always: go through the centroidal axis. Transfer from a known axis to the centroid (subtract A·d²), then from the centroid to the target axis (add A·d'²). Directly subtracting to jump between non-centroidal axes is incorrect.

Question 5

For a composite cross-section, why must you locate the composite centroid *before* applying the parallel-axis theorem to each component?

Model answer: The parallel-axis theorem transfers each component's I from its own centroidal axis to the composite centroidal axis. The distance d_i in I_i = I_c,i + A_i·d_i² is measured from the component's centroid to the composite centroid. Without knowing where the composite centroid is, d_i cannot be computed. The composite centroid is found first via area-weighted averaging of the component centroids, and only then can d_i be determined for each component.

The structural reason this matters: in bending, stresses are computed about the neutral axis of the full cross-section, which coincides with the composite centroidal axis (for symmetric sections under pure bending). A wrong d_i produces a wrong total I, which produces wrong bending stress predictions via σ = M·y/I. The three-step order — (1) find composite centroid, (2) compute d_i for each component, (3) sum I_c,i + A_i·d_i² — is rigid and cannot be reordered.