Moment of Inertia and Section Properties Calculator

What is the second moment of area?

The second moment of area, often called the area moment of inertia, is the geometric property that shows how strongly a section resists bending. It does not depend on the material, only on the shape and size of the section. Its unit is length to the fourth power: mm⁴ or cm⁴. Beam deflection and bending stress are inversely proportional to it.

For a rectangle I_x = b × h³ / 12. Because height enters cubed, the same piece of material is many times stiffer on edge than laid flat. For a solid circle I = π × d⁴ / 64.

How are built up sections calculated?

Sections such as I, T, channel and angle are split into rectangles. First find the centroid: y_c = Σ(A_i × y_i) / ΣA_i. Then add to each part's own second moment its area times the square of its distance from the centroid. This is the parallel axis theorem: I_x = Σ(I_i + A_i × d_i²).

Section modulus and radius of gyration

The section modulus is W = I / e, where e is the distance from the centroid to the extreme fibre. Bending stress is then σ = M / W. The radius of gyration i = √(I / A) sets the slenderness in buckling checks.

Example

An I section 100 mm wide and 200 mm deep with a 5.6 mm web and 8.5 mm flanges has an area of 27.25 cm², I_x of 1845.6 cm⁴ and W_x of 184.6 cm³.

Limits

The calculator works with sharp cornered sections. Rolled profiles have root radii, so table values are a few percent higher: for standard profiles rely on the manufacturer's table.