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Mechanics · Stress and deformation

Bending stress

Calculate maximum elastic bending stress with σ = Mc/I = M/W, where W = I/c.

Created v1.0

By Thibaut Grzelak, Mechanical Analysis Engineer

Stress and deformation
01Calculation inputsσmax = |M| / W = |M|c / I
M neutral axis
Input identificationM acts about the studied axis; W is the section modulus associated with that axis.
Enter W manually, derive it from a verified standard profile, or enter I and c directly.
Value entered manually or derived from the selected profile, depending on the selected mode.

Results are up to date.

03

Results

Metric
Maximum stress
50MPa

Absolute value at the extreme fibre.

Utilization
31.25%

Ratio σmax/σallow.

Safety factor
3.2—

Ratio σallow/σmax.

View calculation detailsFormulas, SI conversion, numerical substitution and resultsOpen ↓

Calculation note

Full calculation details

Simple bending in the linear elastic range. Constant section and Navier–Bernoulli assumption. W must match the studied axis and extreme fibre. The calculation does not replace code checks for stability, fatigue or interaction.

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IPE beam: a complete worked example

Major axis: 22.8 MPa and 1.81 mm. Minor axis: 156 MPa and 24.8 mm. These are model results, not an approval of the beam.

01Formulas and symbols

Formulas used

σmax|M| / WMaximum normal stress
ησmax / σadmUtilization
Sσadm / σmaxSimplified safety factor
02Assumptions and limits

Scope of validity

  • Simple bending in the linear elastic range.
  • Constant section and Navier–Bernoulli assumption.
  • W must match the studied axis and extreme fibre.
  • The calculation does not replace code checks for stability, fatigue or interaction.
03Validation example

Reference numerical case

  1. M = 12 kN·m and W = 240 cm³.
  2. σmax = 50 MPa.
  3. With σallow = 160 MPa: utilization = 31.25%.
  4. Simplified safety factor = 3.2.
04Frequently asked questions

Questions about the calculation

Which section modulus should be used?

Use Wx for bending about x and Wy about y. For asymmetric sections, distinguish top and bottom extreme fibres when required.

Why use the absolute moment?

The sign identifies the tension and compression side. The absolute value measures stress magnitude for comparison with an allowable limit.

Does this provide a complete strength check?

No. It does not automatically cover lateral buckling, shear, fatigue, stress concentrations or code load combinations.

05References

Technical references

  1. Gere & Goodno, Mechanics of Materials, elastic bending theory.
  2. Roark’s Formulas for Stress and Strain, 9th edition.

Calculation path and sources

A strength check should connect loads, section properties, stresses and relevant failure modes.

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External technical sources

Elastic bending

Bending-stress formula: M/W or Mc/I

Both forms are equivalent because W = I/c. Moment, inertia and extreme-fibre distance must refer to the same axis.

Direct calculation with W

Divide maximum bending moment by elastic section modulus.

σmax = |M| / W

Calculation with I

When I and c are known, use Mc/I. Do not replace W with I without including c.

σmax = |M| · c / I