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.

Data verified 2026-09-29 · 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.

Related

Value Sources

Each data column on this page is tied to the source it came from. The numbers in square brackets correspond to the table headers above.

#SourceTypeRevision / method
[1]ASME B1.1 — Unified Inch Screw ThreadsstandardASME B1.1-2019 — source
[2]ASTM A615 — Deformed steel bars for concrete reinforcementstandardASTM A615/A615M-20 — source
[3]ASTM E140 — Hardness Conversion TablesstandardASTM E140-12b — source
[4]Values computed in your browserderivedEvaluated locally from the formulas shown on the page. No data leaves the device.
[5]ISO 4287 — Surface texture: Profile methodstandardISO 4287:1997 — source
[6]ISO 68-1 — Basic profilestandardISO 68-1:2023 — source
[7]NFPA 70 NEC Table 310.16standardNEC 2023 (NFPA 70-2023) — source

Data Sources

StandardRevisionWhat it covers on this page
Formulas as shown on this pagen/a — standard engineering relationships, no single governing revisionevery value this calculator produces

Cross-checked against:

Derived values — the following values on this page are calculated, not taken directly from the standard:

ValueHow it is derived
All outputsComputed in the browser from the formulas above. No data leaves the device.

Outputs are computed from the formulas shown. Verify against the governing standard for design or acceptance work.

Accuracy and use. The values on this page are compiled from the published standards and cross-checked sources listed above. Where values are derived, the derivation is stated. No warranty, express or implied, is made as to the accuracy or completeness of this information, and no liability is accepted for any loss or damage arising from its use. Engineering reference data is provided for guidance in preliminary work — before a value is used for design, fabrication or acceptance testing, verify it against the current revision of the governing standard and against your own inspection. The user assumes all risk and responsibility in connection with the use of this information.

Cite This Page

Every value on this page is traceable to the sources listed above. If you use the data in a document, paper or report, cite it as:

Each row in the tables above also has a permanent link — hover a row and use the # link to cite a single value rather than the whole page.