What is Bending Stress?
When a beam carries loads — including its own weight — it bends, usually downward. During bending, normal stresses develop across the cross-section: compressive stress on the side closer to the load and tensile stress on the opposite side. These internal normal stresses are collectively called bending stress.
The upper and lower faces of the beam experience the greatest stress intensity — especially whichever face is farther from the beam's neutral axis (the imaginary line where bending stress is zero).
How to Calculate Bending Stress
The maximum bending stress formula is:
σ = M × c / I
- σ (sigma) — Bending stress at the extreme fiber (Pa = N/m², MPa, or psi)
- M — Applied bending moment (N·m, kN·m, lb·ft, kip·ft)
- c — Perpendicular distance from the neutral axis to the extreme fiber (mm or in)
- I — Area moment of inertia of the cross-section (mm⁴ or in⁴)
Understanding the Bending Stress Equation
The bending moment M is caused by loads acting perpendicular to the beam's long axis. For example, a 10 N point load at the center of a 3 m simply-supported beam produces a peak bending moment of M = 10 N × 3 m / 4 = 7.5 N·m at mid-span.
The distance c is measured from the neutral axis (located at the centroid of the cross-section) to the outermost fiber. For a symmetric section (rectangle, circle, I-beam) the neutral axis is at half the total height, so c = h / 2.
The area moment of inertia I (also called second moment of area) describes how the cross-sectional area is distributed relative to the neutral axis. A higher I means a stiffer beam that resists bending with less stress.
Cross-Section Formulas Supported by This Calculator
| Cross-Section | I (area moment of inertia) | c (extreme fiber distance) |
|---|---|---|
| Solid Rectangle (b × h) | b·h³ / 12 | h / 2 |
| Solid Circle (diameter d) | π·d⁴ / 64 | d / 2 |
| Hollow Rectangle (B×H outer, b×h inner) | (B·H³ − b·h³) / 12 | H / 2 |
| Hollow Circle / Pipe (D outer, d inner) | π·(D⁴ − d⁴) / 64 | D / 2 |
| I-Beam (bf, tf, hw, tw) | (bf·H³ − (bf−tw)·hw³) / 12 | H / 2 where H = hw + 2·tf |
How to Use This Bending Stress Calculator
- Select the unit system — Metric (SI: mm, N·m, MPa) or Imperial (US: in, lb·ft, psi).
- Choose your cross-section type — Rectangle, Circle, Hollow Rectangle, Hollow Circle (pipe), or I-Beam.
- Enter the bending moment and select the moment unit (N·m, kN·m, lb·ft, kip·ft, etc.).
- Enter the cross-section dimensions in the fields that appear for your chosen section type.
- Click "Calculate" to instantly see the maximum bending stress (σ), area moment of inertia (I), distance to extreme fiber (c), and section modulus (Z = I/c).
Frequently Asked Questions
What is the neutral axis?
The neutral axis is the line within the cross-section where bending stress is zero. For geometrically symmetric sections (rectangles, circles, I-beams with equal flanges) it lies exactly at the centroid — the geometric center of the cross-section. Bending stress increases linearly with distance from the neutral axis.
What is section modulus (Z)?
The section modulus Z = I / c is a property of the cross-section shape that directly relates bending moment to maximum stress: σ = M / Z. A larger Z means the section can carry a higher moment for the same allowable stress, making it a useful measure of bending efficiency when comparing beam shapes.
What units does the calculator output?
- Metric: stress in MPa (megapascals = N/mm²), inertia in mm⁴, section modulus in mm³.
- Imperial: stress in psi (pounds per square inch), inertia in in⁴, section modulus in in³.
What is the difference between bending stress and shear stress?
Bending stress (σ) is a normal stress — it acts perpendicular to the cross-section and is maximum at the top and bottom faces. Shear stress (τ) acts parallel to the cross-section and is usually maximum at the neutral axis. For most structural beams the bending stress governs design, but for short, deep beams shear stress can be the critical factor.
Is this calculator for American (Imperial) or Metric units?
Both. Switch the Unit System selector to use either the SI metric system (mm, N·m, MPa) or the US customary / Imperial system (inches, lb·ft, psi). The calculator automatically adjusts all inputs and outputs accordingly.