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Beam Bending Stress Calculator

Bending, shear and von Mises stress with a yield safety factor.

Calculate the maximum bending stress and shear stress in a beam section from bending moment and shear force. Section properties for I-beams, rectangular tubes, round tubes, round bars and flats are computed for you, including the first moment of area Q for shear.

  • Live results
  • Full step-by-step working
  • PASS / FAIL checks
  • Free PDF report

Inputs

Results

Bending stress σ
100.5 MPa
Shear stress τ
30.37 MPa
Von Mises σvm
113.4 MPa
Safety factor
2.2
PASS
Step-by-step calculation (7 steps)
  1. 1Second moment of area
    FormulaI = [B·H³ − (B−tw)(H−2tf)³] / 12
    SolutionI = [80 × 150³ − 75 × 134³] / 12
    ResultI = 7461850 mm⁴ (746.2 cm⁴)
  2. 2Section modulus
    FormulaZ = I / c
    SolutionZ = 7461850 / 75
    ResultZ = 99491 mm³
  3. 3Maximum bending stress
    Formulaσ = M·c / I
    Solutionσ = 1×10⁷ × 75 / 7461850
    Resultσ = 100.5 MPa
  4. 4First moment of area at neutral axis
    FormulaQ = B·tf·(H−tf)/2 + tw·((H−2tf)/2)²/2
    SolutionQ = 80 × 8 × 142/2 + 5 × 67²/2
    ResultQ = 56663 mm³, b = 5 mm
  5. 5Maximum shear stress
    Formulaτ = V·Q / (I·b)
    Solutionτ = 20000 × 56663 / (7461850 × 5)
    Resultτ = 30.37 MPa
  6. 6Von Mises equivalent stress (conservative)
    Formulaσvm = √(σ² + 3τ²)
    Solutionσvm = √(100.5² + 3 × 30.37²)
    Resultσvm = 113.4 MPa
  7. 7Safety factor on yield
    Formulan = Sy / σvm
    Solutionn = 250 / 113.4
    Resultn = 2.2 ≥ 1.5PASS

Engineering takeaways

  • Combining peak σ (outer fibre) and peak τ (neutral axis) is conservative: the two maxima occur at different points. For I-beams also check the web-flange junction.
  • Under fatigue (cyclic) loads use the endurance limit and weld detail category (EN 1993-1-9), not yield strength.
  • Open sections in bending need a lateral-torsional buckling check when the compression flange is not restrained.

I-section 150 × 80, tw 5, tf 8 mm (no root radii)

The PDF report contains every input, formula, substitution and check shown above, in LTM calculation-sheet format.

Worked Example

The calculator combines the two into a conservative von Mises equivalent stress and compares it with the yield strength of E250, S355, stainless 304, aluminium 6061-T6 or your own material.

The calculator above opens with this example. Change any input to run your own case.

Inputs

Bending moment M10000 N·m
Shear force V20000 N
Section profileI-beam / H-beam
Width / flange width B80 mm
Height H150 mm
Web thickness tw5 mm
Flange thickness tf8 mm
MaterialStructural steel IS 2062 E250 / S235 (Sy 250 MPa)
Required safety factor1.5

Results

Bending stress σ100.5 MPa
Shear stress τ30.37 MPa
Von Mises σvm113.4 MPa
Safety factor2.2 PASS

Bending Stress Formulas Used

Bending stress

σ=M cI=MZ\sigma = \frac{M\,c}{I} = \frac{M}{Z}
  • Mbending moment
  • cdistance to extreme fibre
  • Isecond moment of area

Shear stress

τ=V QI b\tau = \frac{V\,Q}{I\,b}
  • Vshear force
  • Qfirst moment of area above neutral axis
  • bwidth at neutral axis

Von Mises stress

σvm=σ2+3τ2\sigma_{vm} = \sqrt{\sigma^2 + 3\tau^2}

Safety factor

n=Syσvmn = \frac{S_y}{\sigma_{vm}}

Basis: Euler-Bernoulli bending; Jourawski shear formula; von Mises criterion. Results are for preliminary design and should be confirmed against supplier data and the governing standard.

Bending Stress Calculator FAQ

How do I calculate bending stress in a beam?

Multiply the bending moment by the distance from the neutral axis to the outer fibre, and divide by the second moment of area: σ = M·c/I. With M in N·mm and dimensions in mm, the result is in MPa.

Where is shear stress highest?

At the neutral axis, where bending stress is zero. In an I-beam almost all shear is carried by the web, so τ ≈ V / (web height × web thickness) is a good quick check.

Why is combining peak values conservative?

Peak bending stress is at the outer fibre and peak shear at the neutral axis, so they do not occur at the same point. Adding them in von Mises gives an upper bound. In I-beams, check the web-flange junction where both are significant.

What safety factor should I use?

For static loads on ductile steel, 1.5 on yield is common in machine design. Use higher factors for uncertain loads, and use fatigue methods with weld detail categories for cyclic loads.

Which yield strength is E250?

IS 2062 E250 has a minimum yield of 250 MPa for thickness up to 20 mm, similar to S235 in EN 10025. E350 / S355 has 350 to 355 MPa.

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