What is the moment of inertia of a section?

The moment of inertia (second moment of area) I = integral of y^2 dA measures how a cross-section's area is distributed about an axis. A larger I means more material far from the axis, giving greater resistance to bending and buckling. Units are length to the fourth power (mm^4).

What is the parallel axis theorem?

The parallel axis theorem states I = I_c + A d^2, where I_c is the moment of inertia about the centroidal axis, A is the area, and d is the distance between the centroidal axis and the parallel axis. It lets you shift moments of inertia to a common axis to add them for composite sections.

How do you find the centroid of a composite section?

Split the section into simple shapes, find each shape's area and the centroid distance from a reference line, then use y_bar = sum(A_i y_i) / sum(A_i). The result is the distance of the whole section's centroid from the reference.

Centroid and Moment of Inertia of Sections — Parallel Axis Theorem with Examples

Almost every beam and column calculation begins with two geometric properties: the centroid (where the neutral axis sits) and the moment of inertia (how stiff the section is in bending). Master the parallel axis theorem and you can compute these for any built-up section. This article shows exactly how.

Centroid of a Composite Section

The centroid is the area-weighted average position. Divide the section into simple shapes (rectangles, triangles, circles), then:

ȳ = Σ(Aᵢ ȳᵢ) / Σ(Aᵢ)     (distance from a chosen reference line)

For symmetric sections, the centroid lies on the axis of symmetry, halving the work.

Moment of Inertia and the Parallel Axis Theorem

The second moment of area about a centroidal axis for common shapes:

Shape	I about own centroid	Area
Rectangle (b × d)	bd³/12	bd
Triangle (base b, height h)	bh³/36	bh/2
Circle (dia D)	πD⁴/64	πD²/4

To combine shapes about the section's centroid, shift each shape's I using the parallel axis theorem:

I = I_c + A d²

where d is the distance from the shape's own centroid to the section centroid. Then sum the shifted values.

Radius of Gyration

r = √(I / A)

It represents the distance at which the whole area could be concentrated to give the same I. It is the key parameter in column buckling (slenderness ratio L_e/r).

Worked Example 1 — T-Section

A T-section has a flange 100 mm wide × 20 mm deep on top of a web 20 mm wide × 80 mm tall. Find the centroid and moment of inertia about the horizontal centroidal axis.

Step 1 — Areas and centroids (measured from the bottom of the web):

Web: A₁ = 20 × 80 = 1600 mm²; ȳ₁ = 40 mm
Flange: A₂ = 100 × 20 = 2000 mm²; ȳ₂ = 80 + 10 = 90 mm
Total area A = 3600 mm²

Step 2 — Centroid:

ȳ = (1600×40 + 2000×90)/3600 = (64 000 + 180 000)/3600 = 67.78 mm from bottom

Step 3 — Moment of inertia about the centroid (parallel axis theorem):

Web: I_c = 20×80³/12 = 853 333; d = 67.78 − 40 = 27.78; A d² = 1600 × 771.7 = 1 234 700 → 2 088 000 mm⁴
Flange: I_c = 100×20³/12 = 66 667; d = 90 − 67.78 = 22.22; A d² = 2000 × 493.7 = 987 400 → 1 054 100 mm⁴
I_total = 2 088 000 + 1 054 100 ≈ 3.14 × 10⁶ mm⁴

Worked Example 2 — Hollow Rectangular (Box) Section

An outer rectangle 120 × 200 mm has a 80 × 160 mm rectangular hole (concentric). Find I about the centroidal axis parallel to the 120 mm side.

I_outer = 120 × 200³ / 12 = 8.0 × 10⁷ mm⁴
I_hole = 80 × 160³ / 12 = 2.731 × 10⁷ mm⁴
I = I_outer − I_hole = 8.0×10⁷ − 2.731×10⁷ = 5.27 × 10⁷ mm⁴

Because the section is symmetric, the centroids coincide and we simply subtract — no parallel-axis shift needed.

Why This Matters

The neutral axis in bending passes through the centroid — so the centroid locates where stress is zero.
I controls bending stress (σ = My/I), deflection (∝ 1/I) and buckling (P_cr ∝ I).
Maximising I for a given area is the whole logic behind I-beams, channels and hollow sections.

Common Mistakes

Forgetting the A·d² term — simply adding the I_c values gives a badly wrong (too low) result.
Measuring d from the wrong axis — it is the distance between the shape centroid and the section centroid.
Mixing axes — only add moments of inertia taken about the same axis.