Moment of inertia of I, T, channel and angle sections

Plate sections: find the centroid, then Ix′ and Iy′ with the parallel-axis theorem.

Educational practice only. Not for designing or checking real structures.

CentroidsMoment of inertia

Set

Method, signs and how to type answers
  1. Each section is two or three rectangular plates. Use symmetry first: an axis of symmetry is a centroidal axis.
  2. Find the centroid along any axis that is not one of symmetry: A, x̄ or ȳ, and A·x̄ or A·ȳ for each plate.
  3. For each plate, Ī = bh³/12 about its own centroid, where h is the side across the axis.
  4. Add A·d² for each plate, with d from the section’s centroid to the plate’s. A plate centred on the axis has d = 0.
  5. Ix′ = ΣĪx + ΣA·dy² and Iy′ = ΣĪy + ΣA·dx².

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      Questions

      How do you find the moment of inertia of an I-section?

      Split it into two flanges and a web. Each plate’s own Ī is bh³/12. The flanges are far from the neutral axis, so their A·d² terms carry most of I_x′. Add every plate’s Ī and A·d².

      How do you find the centroid of a T-section?

      The vertical axis of symmetry holds the centroid, so only Ȳ is needed: Ȳ = (A₁ȳ₁ + A₂ȳ₂) ÷ (A₁ + A₂) with ȳ measured from one edge, usually the bottom of the stem.

      Why is I_y′ of an I-section just the sum of Ī_y?

      Every plate is centred on the vertical axis of symmetry, so each d_x is zero and the parallel-axis terms vanish. The same holds for a T-section.

      Which side is cubed in bh³/12?

      The side across the axis. For I_x′ of a flange 120 wide and 20 thick, b = 120 and h = 20; about the y axis the 120 is cubed instead.

      Are these real steel sections?

      No. They are invented plates with square corners and no fillets, for exam practice. Real rolled sections have published properties. Educational practice only: not for designing or checking real structures.

      How do I give my class the same problems?

      Press Share with class and send the link: everyone gets the same set code and sections. Print makes up to 30 different copies, each with its own answer key.

      Sources, checked 2 November 2026: Baker and Haynes, Engineering Statics: Open and Interactive, 7.5 Centroids using composite parts (holes as negative areas), Table C.1.1 Centroids of common shapes, Table C.2.1 Moments of inertia of common shapes and 10.4 Moment of inertia of composite shapes. The same formulas are on the inside covers of Hibbeler and of Beer and Johnston.