EngineeringFormulaCivil & Structural Engineering
Bending Stress Calculator
Calculate elastic bending stress, bending moment, section modulus, moment of inertia, or extreme-fiber distance using the classic flexure formulas.
Bending stress tells you how intensely a beam or cross-section is being stressed by an applied bending moment. In simple elastic bending, the maximum normal stress at the extreme fiber follows either of two equivalent forms: σ = M / S when section modulus is known, or σ = M c / I when you want to work directly with moment of inertia and extreme-fiber distance. This calculator supports both approaches so you can move between applied moment, stress, and required section properties without repeating hand conversions.
M · Applied internal or external bending moment at the section being checked.
S · Elastic section modulus about the bending axis, equal to I / c.
This calculator uses positive magnitudes for maximum bending stress checks. One extreme fiber may be in tension and the opposite one in compression, but the reported value here is the stress magnitude.
Solution
Enter the required values to calculate bending stress.
σ = M / S
Formula Sheet
- σBending Stress
- MBending Moment
- SSection Modulus
- IMoment of Inertia
- cExtreme-Fiber Distance
Variables & Units
| Symbol | Variable | Description | Common Units |
|---|---|---|---|
| σ | Bending Stress | Maximum normal stress magnitude at the extreme fiber under the applied bending moment. | Pa, kPa, MPa, GPa, psi, ksi |
| M | Bending Moment | Applied internal or external bending moment at the section being checked. | N·m, N·mm, kN·m, lbf·in, lbf·ft, kip·in, kip·ft |
| S | Section Modulus | Elastic section modulus about the bending axis, equal to I / c. | mm³, cm³, m³, in³, ft³ |
| I | Moment of Inertia | Second moment of area about the bending axis. | mm⁴, cm⁴, m⁴, in⁴, ft⁴ |
| c | Extreme-Fiber Distance | Distance from the neutral axis to the outermost fiber where bending stress reaches its maximum magnitude. | mm, cm, m, km, in, ft |
How to Use This Calculator
- 01Choose the calculation method first. Use Section Modulus when you already know S for the cross-section. Use Moment of Inertia when you want to work directly with I and the extreme-fiber distance c.
- 02Choose which variable to solve for. The calculator then shows only the required inputs for that formula form.
- 03Enter positive magnitudes for moment, stress, section modulus, inertia, and extreme-fiber distance. This page reports maximum stress magnitude, not compression-versus-tension sign.
- 04Select Calculate to see the answer, the active formula, and a substitution line with internally coherent units so the dimensional relationship is easy to verify. If you are checking a real member, compare the result with the allowable or design stress from the governing material standard.
How the Formula Works
For straight beams in simple elastic bending, normal stress varies linearly through the depth of the section and reaches its maximum at the farthest fiber from the neutral axis. That gives the familiar flexure relation σ = M c / I, where M is the bending moment, c is the distance from the neutral axis to the extreme fiber, and I is the second moment of area about the bending axis.
Because section modulus is defined as S = I / c, the same stress relation can be rewritten as σ = M / S. This is often the more convenient structural-design form because rolled shapes, HSS, and many timber or masonry checks are tabulated directly by section modulus. Larger I or S reduces bending stress for the same applied moment, while larger c increases stress if I does not increase proportionally.
Worked Example 01
Stress from bending moment and section modulus
Known
- Bending Moment (M): 50 kN·m
- Section Modulus (S): 800 cm³
Formula
σ = M / S
Substitution
Convert first: 50 kN·m = 50,000 N·m and 800 cm³ = 0.0008 m³, then σ = 50,000 / 0.0008
Result
σ = 62.5 MPa
A section with 800 cm³ of elastic section modulus carrying a 50 kN·m moment develops 62.5 MPa of maximum elastic bending stress.
Worked Example 02
Moment from allowable stress and section modulus
Known
- Bending Stress (σ): 165 MPa
- Section Modulus (S): 1,200 cm³
Formula
M = σ S
Substitution
Convert first: 165 MPa = 165,000,000 Pa and 1,200 cm³ = 0.0012 m³, then M = 165,000,000 × 0.0012
Result
M = 198 kN·m
If the allowable elastic bending stress is 165 MPa and the section modulus is 1,200 cm³, the corresponding bending moment is 198 kN·m.
Worked Example 03
Required section modulus
Known
- Bending Moment (M): 90 kN·m
- Allowable Bending Stress (σ): 150 MPa
Formula
S = M / σ
Substitution
Convert first: 90 kN·m = 90,000 N·m and 150 MPa = 150,000,000 Pa, then S = 90,000 / 150,000,000
Result
S = 600 cm³
A 90 kN·m moment limited to 150 MPa requires at least 600 cm³ of elastic section modulus.
Worked Example 04
Stress from moment, inertia, and extreme-fiber distance
Known
- Bending Moment (M): 25 kN·m
- Moment of Inertia (I): 8,000 cm⁴
- Extreme-Fiber Distance (c): 150 mm
Formula
σ = M c / I
Substitution
Convert first: 25 kN·m = 25,000 N·m, 8,000 cm⁴ = 0.00008 m⁴, and 150 mm = 0.15 m, then σ = (25,000 × 0.15) / 0.00008
Result
σ = 46.875 MPa
Working directly with I and c gives the same elastic flexure stress that you would get by first converting the section to S = I / c.
Worked Example 05
Required moment of inertia
Known
- Bending Moment (M): 30 kN·m
- Extreme-Fiber Distance (c): 100 mm
- Allowable Bending Stress (σ): 120 MPa
Formula
I = M c / σ
Substitution
Convert first: 30 kN·m = 30,000 N·m, 100 mm = 0.1 m, and 120 MPa = 120,000,000 Pa, then I = (30,000 × 0.1) / 120,000,000
Result
I = 2,500 cm⁴
At 100 mm to the extreme fiber, keeping stress to 120 MPa under 30 kN·m requires 2,500 cm⁴ of moment of inertia.
Applications
- 01Checking beam or shelf bending stress from a known bending moment and section property
- 02Estimating the required section modulus for a target allowable bending stress
- 03Back-solving moment of inertia or extreme-fiber distance during hand checks and preliminary design
Assumptions
- 01The formulas assume straight-beam, small-deflection, simple elastic bending.
- 02Plane sections remain plane, and the neutral axis passes through the centroid as in classic flexure theory.
- 03The page reports bending stress magnitude only; it does not classify one face as tension and the opposite face as compression.
- 04Section properties I, S, and c are taken about the same bending axis as the applied moment.
Where This Model Stops
- 01This is not a full member-design check. Final design may still require code-based capacity checks, stability checks, local buckling checks, combined-stress checks, and serviceability review.
- 02Does not include plastic stress redistribution, inelastic section behavior, curved-beam theory, or shear-stress effects.
- 03If your moment varies along the beam, this page evaluates only the section where the stated bending moment applies.
References
- [1]4.2: Stresses in Beams
Engineering LibreTexts
Derives the beam flexure relation σ = M y / I from elastic bending theory.
- [2]7.2.2: Structural design of shelf angles
Engineering LibreTexts
Shows the section-modulus form of elastic bending stress as f_b = M / S_x.
- [3]NIST Guide to the SI, Appendix B.9
National Institute of Standards and Technology
Reference for unit conversions used in the length-derived and stress-related engineering quantities on this page.
- [4]AISC Shapes Database v16.0
American Institute of Steel Construction
Useful published source for section properties when you want to obtain section modulus and moment of inertia for standard steel shapes.
Frequently Asked Questions
What is the difference between σ = M / S and σ = M c / I?
They are the same elastic bending relationship written two ways. Since section modulus is defined as S = I / c, substituting that into σ = M c / I gives σ = M / S.
Does this calculator tell me whether the stress is tension or compression?
No. This page reports the maximum stress magnitude only. In a bent section, one extreme fiber is typically in compression while the opposite extreme fiber is in tension, depending on the moment direction and sign convention.
Can I use this as a full beam design check?
Not by itself. Bending stress is only one part of design. Final structural checks may also need shear, deflection, lateral-torsional buckling, local buckling, connection behavior, and code-specific strength or allowable-stress rules.
Where do I get section modulus or moment of inertia?
Use a published shape table, manufacturer data, or this site's Section Modulus and Area Moment of Inertia calculators for simple sections. Do not guess S or I from only beam depth; cross-section shape controls both values.