What is the section modulus?

The elastic section modulus S = I / c relates the applied bending moment M to the maximum bending stress σ via σ = M / S. A higher S means lower stress for the same moment, so a larger section modulus indicates a stronger beam in bending.

What is the difference between elastic and plastic section modulus?

The elastic section modulus S = I/c is used in elastic (linear) analysis and assumes stress varies linearly across the section. The plastic section modulus Z accounts for full yielding of the cross-section and is used in plastic design. This tool calculates the elastic section modulus.

Does the unit matter?

Yes — you must use consistent units. If dimensions are in millimeters, I is in mm⁴ and S is in mm³. If dimensions are in inches, S is in in³. The tool labels results clearly; just ensure all inputs use the same unit.

What is c in S = I/c?

c is the perpendicular distance from the neutral axis to the outermost fiber of the cross-section (the point farthest from the neutral axis). For a symmetric section, c equals half the total height.

How do I use S to check a beam?

Compute the maximum bending moment M in the beam (in N·mm or lb·in), then calculate the required section modulus S_req = M / σ_allow, where σ_allow is the allowable bending stress of the material. If your beam's S ≥ S_req, the section is adequate.

Why does the hollow rectangle show a much higher section modulus for less material?

Removing material near the neutral axis — where bending stresses are low — barely reduces the section modulus while significantly reducing weight. This is the structural advantage of hollow sections and I-beams.

Section Modulus Calculator

Compute the elastic section modulus S = I/c for rectangular, circular, hollow rectangular, and I-beam cross-sections.

Cross-Section Inputs

Cross-Section Shape

Width b

mm / in

Height h

mm / in

Bending about the horizontal (x–x) neutral axis. c = h/2.

Enter cross-section dimensions and click Calculate

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Summary

The section modulus S = I/c is a geometric property of a beam cross-section that determines its resistance to bending stress. A larger section modulus means the beam can carry more bending moment before the extreme fiber reaches the allowable stress. This tool computes the moment of inertia I, the distance to the extreme fiber c, and the elastic section modulus S for four common structural shapes.

How it works

Select the cross-section shape (rectangle, circle, hollow rectangle, or I-beam).
Enter the required dimensions in millimeters or inches — units are consistent within each calculation.
The tool calculates the moment of inertia I about the neutral axis.
It determines c, the distance from the neutral axis to the extreme fiber.
Section modulus S = I / c is computed and displayed alongside all intermediate values.
Use S to find the maximum bending stress with σ = M / S, where M is the applied moment.

Use cases

Verify beam sizing against allowable bending stress in structural design.
Compare section efficiency between solid and hollow cross-sections.
Perform quick hand-check calculations for steel, timber, or aluminum beams.
Calculate section modulus for I-beams (wide-flange or standard) during preliminary design.
Determine if an existing beam cross-section satisfies a given bending moment requirement.
Support homework and study exercises in mechanics of materials courses.
Check section modulus for custom fabricated box beams or plate girders.
Quickly evaluate the effect of wall thickness on hollow section stiffness.

Frequently Asked Questions

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

Section Modulus Calculator

Cross-Section Inputs

Section Modulus Result

Formula Breakdown

Bending Stress Check (optional)

Summary

How it works

Use cases

Frequently Asked Questions

Section Modulus Calculator

Use the Section Modulus Calculator

Cross-Section Inputs

Section Modulus Result

Formula Breakdown

Bending Stress Check (optional)

Summary

How it works

Use cases

Frequently Asked Questions