What is the second moment of area?

The second moment of area (I), also called area moment of inertia, measures a cross-section's resistance to bending. A higher value means the beam is stiffer and deflects less under the same load.

What units does this calculator use?

Dimensions are entered in millimeters (mm) and results are given in mm⁴ for I, mm³ for section modulus (Z), and mm for radius of gyration (r).

What is the section modulus?

Section modulus Z = I / y, where y is the distance from the neutral axis to the outermost fiber. It is used to find the maximum bending stress: σ = M / Z.

What is the radius of gyration?

Radius of gyration r = √(I / A) relates the moment of inertia to the cross-sectional area. It is used in column buckling calculations (Euler's formula).

Can I use this for circular hollow sections (CHS)?

Select "Hollow Rectangular" and enter equal width and height to approximate a square hollow section. For true circular hollow sections, use the circular formula and subtract the inner circle, which is not yet a separate shape here.

Moment of Inertia Beam Calculator

Select a beam cross-section shape, enter dimensions, and instantly get the second moment of area (I) about the neutral axis.

Cross-Section Inputs

Cross-Section Shape

Results

Second Moment of Area

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I (mm⁴)

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mm³

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mm²

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Summary

This tool computes the second moment of area (I) for five common beam cross-sections used in structural and mechanical engineering. Enter dimensions in millimeters and get I, section modulus (Z), cross-sectional area (A), and radius of gyration (r) instantly — all calculated client-side with no data sent to any server.

How it works

Each formula is the standard closed-form solution for that cross-section shape.
For I-beams and T-beams the parallel axis theorem is applied to each sub-rectangle.
Area is computed from dimensions and used to derive the radius of gyration.

Use cases

Determine beam stiffness for structural analysis.
Calculate deflection under load for steel or timber beams.
Select appropriate beam profiles for floor joist design.
Verify I-beam properties against manufacturer data sheets.
Compare hollow vs solid sections for weight-to-stiffness ratio.
Academic structural mechanics coursework and exam prep.

Frequently Asked Questions

Last updated: 2026-06-04 · Reviewed by Nham Vu