What is the moment of inertia of a cross-section?

The moment of inertia (second moment of area) I measures how the cross-section's area is distributed about the neutral axis. A larger I means more resistance to bending. Units are mm⁴ or cm⁴.

What is section modulus?

Section modulus W (or S) equals I/c, where c is the distance from the neutral axis to the extreme fiber. It directly relates bending moment to stress: σ = M/W. A larger W means lower bending stress.

What is the radius of gyration?

The radius of gyration r = √(I/A) is a measure of how far the area is distributed from the neutral axis. It is used in column buckling calculations to determine the slenderness ratio λ = L/r.

How do I calculate I for an I-beam?

For a symmetric I-beam: I = [B×H³ − (B−tw)×(H−2tf)³] / 12, where B is flange width, H is total height, tw is web thickness, and tf is flange thickness. This subtracts the hollow portions from the bounding rectangle.

What is the difference between Ix and Iy?

Ix is the moment of inertia about the horizontal (x) axis, and Iy is about the vertical (y) axis. For bending in the vertical plane (most common), Ix is used. For lateral bending, Iy applies. This calculator computes the primary axis I.

Free Engineering Tool

Section Moment of Inertia Calculator

Calculate cross-section properties — moment of inertia I, section modulus W, area A, and radius of gyration r — for common shapes.

CircleRectangleI-Beam#165

Cross-Section Shape

Diameter d (mm)

Outer Diameter D (mm)

Inner Diameter d (mm)

Width b (mm)

Height h (mm)

Total Height H (mm)

Flange Width B (mm)

Web Thickness tw (mm)

Flange Thickness tf (mm)

Total Height H (mm)

Flange Width B (mm)

Web Thickness tw (mm)

Flange Thickness tf (mm)

Quick presets

Results

Moment of Inertia I

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Section Modulus W

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Cross-Section Area A

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Radius of Gyration r

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Distance to Extreme Fiber c

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Solid Circle

Hollow Circle (Tube)

Rectangle

I-Beam (Symmetric)

General Relationships

ℹ️ Section modulus W = I/c relates bending moment to stress: σ = M/W. Radius of gyration r = √(I/A) is used in buckling analysis.

Section Moment of Inertia Calculator | Free Online Tool

Published by Nikolai Shelkovenko on February 15, 2026 March 5, 2026