Moment of Inertia for Common Sections

The second moment of area I decides how much a section resists bending. Bending stiffness is EI; stress capacity is M/Z with Z = I/y. Two shapes with the same area can differ by an order of magnitude in I — which is why steel is rolled into I-sections instead of squares.

The base formulas

Section	I (strong axis)	y (to extreme fibre)
Rectangle b × h	bh³ / 12	h/2
Circle Ød	πd⁴ / 64	d/2
Hollow rect. B×H minus b×h	(BH³ − bh³) / 12	H/2
Hollow circle D minus d	π(D⁴ − d⁴) / 64	D/2
I-section (see below)	outer minus cutouts, /12	H/2

I-section

For flange width B, total height H, web thickness t_w, flange thickness t_f:

I = [B·H³ − (B − t_w)·(H − 2t_f)³] / 12

Outer rectangle minus the two side cutouts between the flanges. The flanges sit far from the neutral axis, so they carry most of I with most of the area.

Built-up sections: parallel axis theorem

For a shape made of parts, the total is

I_total = Σ (I_part + A_part · d²)

where d is the distance from the part’s own centroid to the group neutral axis. The A·d² term is why moving material away from the axis works: it grows with the square of the distance.

Worked example — plate girder: 500×10 web with two 200×20 flanges.

• Flange area A = 4000 mm², distance d = 260 mm
• Flange contribution each: 200·20³/12 + 4000·260² = 1.33e5 + 2.704e8 ≈ 2.705e8 mm⁴
• Web: 10·500³/12 = 1.042e8 mm⁴
• Total: 2·2.705e8 + 1.042e8 ≈ 6.45e8 mm⁴

The flanges — 44% of the area — contribute 84% of I.

Quick sanity checks

• Doubling depth multiplies I by 8 (cubed) — the cheapest stiffness you can buy.
• Doubling width doubles I — linear, expensive.
• A hollow circle’s I depends on wall thickness cubed-ish: thin walls are efficient in bending, but check local buckling.

Compute any of these interactively with the cross-section calculator.