Section Modulus Calculator

Find the section modulus and moment of inertia of a rectangle, circle, hollow circle, or I-beam cross-section, plus the bending stress it produces under an applied moment.

Cross-Section

Shape

Width (b), mm Cross-section width, in millimetres.

Height (h), mm Cross-section height (depth), in millimetres.

Bending moment (M), kN·m (optional) Applied bending moment. Leave blank to skip the bending-stress calculation.

Result

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Enter the cross-section dimensions above to compute.

4 min read 3 steps 7 terms 3 examples 6 FAQs S = I / c

Section modulus tells you how efficiently a beam's cross-section resists bending, independent of the material it's made from.

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Walk-through

How to Use This Calculator

3 steps

Pick a cross-section shape

Choose Rectangle, Circle, Hollow circle, or I-beam from the shape dropdown. Each shape swaps in the dimension fields it needs — width and height for a rectangle, a single diameter for a solid circle, inner and outer diameter for a tube, and flange/web dimensions for an I-beam.

Enter the cross-section dimensions

Fill in every dimension field in millimetres. The calculator updates instantly and computes the moment of inertia (I) and section modulus (S = I/c) for the selected shape.

Add a bending moment for stress (optional)

Enter an applied bending moment in kN·m to see the resulting bending stress (σ = M/S) in megapascals. Leave it blank if you only need the section properties.

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Reference

Formula & Methodology

2 formulas

Section modulus

S = I / c

The section modulus S is the moment of inertia I divided by c, the distance from the neutral (centroidal) axis to the extreme fiber of the cross-section. It has units of length cubed (mm³) and measures how efficiently a cross-section resists bending — a larger S means a lower bending stress for the same applied moment.

Bending stress

σ = M / S

For a beam in pure bending, the maximum bending stress σ at the extreme fiber equals the applied bending moment M divided by the section modulus S. With M in kN·m and S in mm³, the calculator converts internally so σ comes out in megapascals (MPa = N/mm²).

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Glossary

Key Terms Explained

7 terms

Section modulus (S) A geometric property of a cross-section, S = I/c, that relates the applied bending moment to the maximum bending stress. Larger section modulus means the shape resists bending stress more efficiently for its size.

Moment of inertia (I) Also called the second moment of area. A measure of how a cross-section's area is distributed relative to its neutral axis — larger I means the shape resists bending deflection more, independent of material stiffness.

Neutral axis The line through a beam's cross-section where bending stress is zero — everything on one side is in tension, the other side in compression, for a beam bent by a moment about that axis.

Bending stress The internal stress produced by a bending moment, maximum at the extreme fiber (farthest point from the neutral axis) and zero at the neutral axis itself. Calculated as σ = M/S.

Extreme fiber The point in a cross-section farthest from the neutral axis — where bending stress reaches its maximum magnitude. The distance from the neutral axis to the extreme fiber is denoted c.

I-beam A structural steel section shaped like the letter I, with two horizontal flanges connected by a vertical web. Most of its material sits far from the neutral axis in the flanges, giving it a high section modulus relative to its weight — an efficient shape for resisting bending.

Elastic (bending) Bending behavior where stress is proportional to strain and the beam returns to its original shape once the load is removed. The S = I/c and σ = M/S formulas assume elastic bending with stress staying below the material's yield point.

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Scenarios

Real-World Examples

3 worked examples

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Structural designer

Rectangular timber beam

Shape Rectangle Width (b) 200 mm Height (h) 400 mm

For a rectangle, S = b·h²/6 = 200 × 400² / 6 ≈ 5,333,333 mm³, and I = b·h³/12 ≈ 1,066,666,667 mm⁴. Doubling the height would multiply S by 4 (it scales with h²) — height is by far the most powerful lever for a rectangular section's bending resistance.

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Machine designer

Solid round shaft

Shape Circle (solid) Diameter (d) 150 mm

For a solid circle, S = π·d³/32 = π × 150³ / 32 ≈ 331,340 mm³. A round shaft has a smaller section modulus than a rectangle of similar overall size because its material is distributed closer to the center rather than pushed to the extremes.

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Steel detailer

I-beam under a bending moment

Shape I-beam Flange width (B) 150 mm Total height (H) 300 mm Flange thickness (tf) 12 mm Web thickness (tw) 8 mm Bending moment (M) 50 kN·m

With those dimensions the beam's section modulus works out to roughly 940,000 mm³. Bending stress is σ = M/S = (50 kN·m × 1,000,000) / 940,000 mm³ ≈ 53 MPa — comfortably below typical structural steel yield strengths of 250–350 MPa, so the section has margin left before yielding.

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Reference

Cite This Calculator

APA & MLA

Use either format to cite this calculator in a paper, report, or resource list.

APA

MLA

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Deep Dive

Understanding Section Modulus and Bending Stress

3 chapters · 4 min read

Section modulus tells you how efficiently a beam's cross-section resists bending, independent of the material it's made from. Engineers and hobbyist builders alike use it to compare cross-section shapes and to check whether a beam under a known load will stay within a safe bending-stress limit.

How the Section Modulus Calculator works

The calculator first computes the moment of inertia (I) for the selected shape about its centroidal (neutral) axis, then divides by c — the distance from that axis to the extreme fiber — to get the section modulus, S = I/c. Rectangle and solid-circle formulas are exact closed forms (bh²/6 and πd³/32 respectively). The hollow circle subtracts the inner circle's moment of inertia from the outer circle's before dividing by c. The I-beam treats the section as a full B×H rectangle with two side rectangles removed to leave the web and flanges, ignoring the small fillets real rolled sections have at the web-flange junction — a standard simplification that stays accurate to within a percent or two for typical proportions.

Inputs and what they mean

All dimensions are entered in millimetres, matching common structural drafting practice. Section modulus (S) comes out in mm³ and moment of inertia (I) in mm⁴ — both large numbers for realistic beam sizes, which is expected. The optional bending moment is entered in kilonewton-metres (kN·m), a standard structural-engineering unit, and the resulting bending stress is reported in megapascals (MPa = N/mm²) so it can be compared directly against a material's published yield strength. Height is the input with the biggest effect on both I and S for rectangular and I-beam sections, since it enters the formulas raised to the second or third power — small increases in height produce outsized gains in bending resistance.

Limits and edge cases

This calculator handles pure elastic bending about the horizontal centroidal axis only — it does not account for lateral-torsional buckling, shear stress, combined axial-and-bending loading, or unsymmetrical bending about a skewed axis. The I-beam formula ignores fillet radii and assumes a doubly-symmetric section (equal top and bottom flanges), so it will not exactly match every manufacturer's published catalog value for a specific rolled shape — use it for estimating and comparing custom or built-up sections, and cross-check against a steel manufacturer's tables for a specific standard profile. For any load-bearing design, verify results against your local structural code and, where required, have a licensed engineer review the calculation.

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Questions

Frequently Asked Questions

6 questions

What is the formula for section modulus?+

Section modulus is S = I/c, where I is the moment of inertia of the cross-section about its neutral (centroidal) axis and c is the distance from that axis to the extreme fiber — the farthest point of the section from the axis.

What is the section modulus formula for a rectangle?+

For a rectangle with width b and height h, S = b·h²/6. The height matters far more than the width because it's squared in the formula — doubling height quadruples the section modulus, while doubling width only doubles it.

How do I calculate bending stress from section modulus?+

Bending stress is σ = M/S, where M is the applied bending moment and S is the section modulus. This gives the maximum stress at the extreme fiber of the cross-section for a beam in pure bending.

Why does a bigger section modulus matter?+

A larger section modulus means a beam produces less bending stress for the same applied moment, so the beam can carry a bigger load — or the same load more safely — before reaching its material's yield stress. Increasing S is a way to make a beam stiffer against bending failure without necessarily using more material, by moving material farther from the neutral axis.

What units does this calculator use?+

Cross-section dimensions are entered in millimetres. The calculator reports moment of inertia in mm⁴ and section modulus in mm³. The optional bending moment is entered in kilonewton-metres (kN·m), and the resulting bending stress is reported in megapascals (MPa, equivalent to N/mm²).

Why is an I-beam shape so efficient for bending?+

An I-beam concentrates most of its cross-sectional area in the flanges, far from the neutral axis, where material contributes the most to moment of inertia and section modulus. This gives an I-beam a much higher section modulus per unit of material than a solid rectangle or circle of the same area — which is why structural steel beams are shaped this way instead of being solid bars.

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