Engineering Reference

Section Modulus Calculator

Compute the elastic section modulus S = I/c for a rectangular or circular section, and the bending stress a given moment produces.

Page last changed 2026-10-19 · based on n/a — standard engineering relationships, no single governing revision

Quick Answer

Section modulus is S = I ÷ c, where c is the distance from the neutral axis to the extreme fibre. For a rectangle, S = b·h² ÷ 6; for a solid circle, S = π·d³ ÷ 32. Bending stress is then σ = M ÷ S.

Elastic Section Modulus

The Formulas Used

Elastic section modulus: S = I ÷ c
Rectangle: Sx = b·h² ÷ 6   Circle: S = π·d³ ÷ 32
Bending stress: σ = M ÷ S

Section Modulus vs Moment of Inertia

The two are related but answer different questions. The moment of inertia I governs deflection — how far a beam sags under load. The section modulus S governs stress — whether the beam yields or breaks. A design check needs both: a member can be stiff enough and still be overstressed, or strong enough and still sag too far to be usable.

Because S = I/c and c is half the depth for a symmetric section, S scales with the square of depth where I scales with the cube. That is why a shallow wide section can be strong enough while still deflecting excessively — it has less I for its S than a deep narrow one.

Only the elastic modulus is computed here, which is the right one for design against yield. Where a section is loaded past yield — plastic design of steel frames — the plastic modulus Z is used instead, and it is about 10 to 15% larger than S for an I-section.

Units and the M/S Check

The calculator returns S in in³ and the resulting stress in psi when the moment is entered in lb·in. Keep the units consistent: mixing a moment in lb·ft with a section modulus in in³ gives an answer twelve times too large, and it is a common mistake because drawings often give moments in kip·ft.

The stress figure is the extreme-fibre bending stress only. It does not include axial load, shear, torsion, stress concentrations at holes and notches, or the residual stresses from forming — all of which add to the governing stress in a real part.

Frequently Asked Questions

What is the section modulus of a rectangle?
S = b·h²/6, where h is the depth in the direction of bending. For a 2 × 4 section bending about the strong axis, S = 2 × 16 / 6 = 5.33 in³, which with a 1,000 lb·in moment gives a bending stress of 188 psi.
What is the difference between section modulus and moment of inertia?
Moment of inertia I governs deflection and has units of in⁴. Section modulus S = I/c governs stress and has units of in³. A beam can be adequate in one and not the other, so a design check normally looks at both.
How do I calculate bending stress from a moment?
σ = M/S. Divide the bending moment by the section modulus. Keep the units consistent — a moment in lb·in with a section modulus in in³ gives psi. A moment in kip·ft must be converted to lb·in first, multiplying by 12,000.
Does section modulus depend on which way the load is applied?
Yes. A rectangular section has two section moduli, one for bending about each axis. A 2 × 4 is 5.33 in³ about the strong axis and 1.33 in³ about the weak axis — four times weaker when it bends the flat way, which is why joists are set on edge.
What is the plastic section modulus?
The plastic modulus Z describes the section after it has fully yielded, when the entire section carries the yield stress. It is about 10–15% larger than the elastic modulus S for typical I-sections, and it is the basis of plastic design methods in structural steelwork.

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