Bending Stress Calculator — Beam Flexural Stress
Given a bending moment and a beam cross-section, calculate the maximum bending (flexural) stress using the standard σ = M/S formula — for rectangular or solid circular sections, with step-by-step working.
N·mm
Cross-section shape
mm
mm
Stress at the extreme fibre of the cross-section (N/mm²)
Moment of inertia (rectangle)
Distance to extreme fibre
Section modulus
Bending stress
- 1
Moment of inertia (I, mm⁴)
50 × 100³ ÷ 12 = 4,166,666.67 - 2
Distance to extreme fibre (c, mm)
100 ÷ 2 = 50 - 3
Section modulus (S, mm³)
4,166,666.67 ÷ 50 = 83,333.33 - 4
Bending stress (σ, MPa)
1,000 ÷ 83,333.33 = 0.012
How does this calculator work?
Maximum bending stress σ = M / S (MPa), where M is the bending moment (N·mm) and S = I / c is the section modulus. For a rectangle: S = bh²/6. For a circle: S = πd³/32. The result is the peak stress at the outermost fibre; compare it to your material's allowable stress.
Formula
How this is calculated
When a beam is subjected to a bending moment M (in N·mm), internal stresses develop across the cross-section. The outermost fibres carry the highest stress — tension on one side and compression on the other — and the neutral axis at the mid-plane carries zero stress.
The maximum bending stress σ (in MPa = N/mm²) is given by **σ = M / S**, where S is the section modulus (mm³). The section modulus combines the shape's resistance to bending: S = I / c, where I is the second moment of area (moment of inertia, mm⁴) about the neutral axis and c is the distance from that axis to the extreme (outermost) fibre.
For a **rectangular section** b × h (width × height, with bending about the horizontal axis): I = bh³/12 and c = h/2, giving S = bh²/6. For a **solid circular section** of diameter d: I = πd⁴/64 and c = d/2, giving S = πd³/32. This calculator assumes a homogeneous, elastic material and pure bending with no shear or axial loads. For combined loading (shear + bending, eccentric loading, composite sections), use the appropriate analysis for those cases.
Frequently asked questions
This calculator uses N·mm for the bending moment and mm for dimensions, producing stress in MPa (N/mm²). To convert from kN·m to N·mm, multiply by 10⁶. For example, 1 kN·m = 1,000,000 N·mm.
For a simply supported beam of span L with a central point load P, the maximum bending moment at mid-span is M = P × L / 4. For a uniformly distributed load w per unit length: M = w × L² / 8. Use a beam bending moment diagram calculator for other load configurations.
Typical values: mild steel ~165–250 MPa, structural steel grade S275 ~275 MPa, aluminium alloy 6061-T6 ~240 MPa, wood varies from ~10–40 MPa depending on species and grade. Always divide allowable values by an appropriate safety factor per your design code.
Also known as
TG we-Calculate Editorial Team. (2026). Bending Stress Calculator — Beam Flexural Stress [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/bending-stress-calculator
TG we-Calculate Editorial Team. "Bending Stress Calculator — Beam Flexural Stress." TG we-Calculate. 2026. https://we-calculate.com/calculator/bending-stress-calculator.
TG we-Calculate Editorial Team, "Bending Stress Calculator — Beam Flexural Stress," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/bending-stress-calculator
@misc{wecalculate_bending_stress_calculator, title = {Bending Stress Calculator — Beam Flexural Stress}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/bending-stress-calculator}}, year = {2026}, note = {TG we-Calculate} }
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