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 the axis change section modulus?

Yes. An actual 2 × 4 in rectangle has Sx = 5.3333 in³ and Sy = 2.6667 in³, a ratio of 2. The two bending stresses must use the matching moments and axes.

Is plastic section modulus a fixed percentage above elastic modulus?

No. The ratio depends on section geometry. This tool calculates elastic section modulus only.

Section Modulus Calculator

Identified content errors were corrected on 2026-10-25. Sources and scope for this page: Formulas as shown on this page (standard engineering relationships, no single governing revision), covering every value this calculator produces. No separate source check is recorded for the correction; the figures rest on the sources listed and on the working shown on the page. Full sources, values and limits on this page.

Page last changed 2026-11-08 · 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

Section width, b (in)

Section height, h (in)

Circle diameter, d (in)

Bending moment (lb·in)

The Formulas Used

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

Axes and a reproducible rectangle example

For an actual 2 × 4 in rectangle, Sx = 2 × 4²/6 = 5.3333 in³ and Sy = 4 × 2²/6 = 2.6667 in³. The ratio is 2. A 1000 lb·in moment gives 187.5 psi about x or 375 psi about y. These are actual geometric dimensions, not nominal lumber dimensions.

I describes area distribution about the chosen axis; elastic S = I/c connects the moment to extreme-fibre stress in the applicable elastic bending model. Plastic section modulus requires a different stress distribution and must not be substituted here.

Keep moment units consistent

Multiply lb·ft by 12 to obtain lb·in, or kip·ft by 12000. Entering a numerical lb·ft value as lb·in makes the stress 12 times too small.

The reported stress is bending stress only, excluding axial force, shear, torsion, stress concentrations and other design checks.

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 the axis change section modulus?: Yes. An actual 2 × 4 in rectangle has Sx = 5.3333 in³ and Sy = 2.6667 in³, a ratio of 2. The two bending stresses must use the matching moments and axes.
Is plastic section modulus a fixed percentage above elastic modulus?: No. The ratio depends on section geometry. This tool calculates elastic section modulus only.

Value Sources

References and calculation methods used for this page are listed below. Numbered markers connect table columns where present. A reference listing does not itself establish that every value has been independently checked.

#	Source	Type	Revision / method
[1]	Values computed in your browser	derived	Evaluated locally from the formulas shown on the page. No data leaves the device.

Data Sources

Standard	Revision	What it covers on this page
Formulas as shown on this page	n/a — standard engineering relationships, no single governing revision	every value this calculator produces

Verification scope

No page-specific independent source review is recorded here. The listed references and formulas explain the basis of the compilation; automated consistency checks do not verify the original sources or establish product conformity.

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

Value	How it is derived
All outputs	Computed 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. Use the source notes and recorded verification scope above to assess each value. Calculated values follow the stated derivations; compilation values may lack a confirmed original document or product condition. 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

Read the source notes, calculation methods and verification scope above before reusing a value. To cite this page:

Plain text

Engineering Reference. "Section Modulus Calculator." https://refmanual.com/calculators/section-modulus/ (accessed 2026-11-09). Data from Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision.

BibTeX

@misc{section-modulus,
  title        = {Section Modulus Calculator},
  author       = {{Engineering Reference}},
  howpublished = {\url{https://refmanual.com/calculators/section-modulus/}},
  note         = {Accessed: 2026-11-09},
  source       = {Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision}
}

RIS (EndNote, Zotero, Mendeley)

TY  - ELEC
TI  - Section Modulus Calculator
AU  - Engineering Reference
PB  - Engineering Reference
UR  - https://refmanual.com/calculators/section-modulus/
PY  - 2026
DA  - 2026-11-09
N1  - Source: Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision
ER  -