04 // Interactive Formula Compendium // ENGINEERINGσ_b = (M · y) / I = M / S
Euler-Bernoulli Beam Flexure (Bending Stress) Formula
The flexure formula σ = My/I computes longitudinal bending stress at distance y from the neutral axis of a beam with second moment of area I subjected to bending moment M.
Formula Reference
Main Formula
σ_b = (M · y) / I = M / S
Rearranged Forms
S = I / c
1 / ρ = M / (E · I)
Variables & Units
| Symbol | Name | Unit |
|---|---|---|
| sigma | Maximum Bending Stress | Pa |
| M | Internal Bending Moment | N·m |
| y | Distance to Extreme Fiber c | m |
| I | Second Moment of Area | m⁴ |
Inputs & Parameters
[N·m]
[m]
[m⁴]
Calculated Result (Maximum Bending Stress)
sigma =1.56250e+8Pa
Formula Notes & Background
- Assumes plane cross-sections remain plane and perpendicular to the neutral axis (kinematic linearity ε_x = −y/ρ) combined with Hooke's law σ_x = Eε_x.
- Section modulus S = I/c consolidates cross-section geometry into a single design parameter in m³.
Step-by-Step Examples
#1: Steel I-beam (depth 250 mm ⇒ c = 0.125 m, I = 3.6 × 10⁻⁵ m⁴) carrying peak moment M = 45 kN·m.
M = 45,000 N·my = 0.125 mI = 3.6×10⁻⁵ m⁴
σ_max = (45,000 × 0.125) / 3.6×10⁻⁵ = 5,625 / 3.6×10⁻⁵
→ σ_max = 156.25 MPa
#2: Rectangular timber joist (b = 0.05 m, h = 0.20 m ⇒ S = bh²/6 = 3.333×10⁻⁴ m³) under M = 3,000 N·m.
M = 3,000 N·mS = 3.333×10⁻⁴ m³
σ_max = 3,000 / 3.333×10⁻⁴
→ σ_max = 9.00 MPa
Was this tool helpful?
Frequently Asked Questions
Where is bending stress zero inside a beam?+
Along the centroidal neutral axis (y = 0), longitudinal normal bending stress is zero while transverse shear stress VQ/(Ib) typically reaches its maximum.
Why do I-beams place most material in the top and bottom flanges?+
Because second moment of area I = ∫y² dA weights material by the square of distance from the neutral axis, maximizing flexural strength per kilogram.
What happens when moment exceeds the elastic yield moment M_y = σ_y·S?+
Outer fibers yield plastically inward toward the neutral axis until reaching the full plastic moment M_p = σ_y·Z.