Bending Stress Calculator

How is bending stress calculated?

When a beam bends, the fibres on one side stretch and those on the other side shorten. In between lies a layer whose length does not change: the neutral axis. In the elastic range the normal stress grows in proportion to the distance from that axis: σ = M × y / I. M is the bending moment at the section, y the distance of the fibre from the neutral axis and I the second moment of area.

The largest stress is at the extreme fibre. With that distance called c, σ_max = M × c / I = M / S. The section modulus S = I / c (written W in European practice and in this tool) is listed in section tables.

Common sections

• Rectangle: I = b × h³ / 12, W = b × h² / 6
• Solid circle: I = π × d⁴ / 64, W = π × d³ / 32
• Tube: I = π × (D⁴ − d_i⁴) / 64, W = 2 × I / D

For a rectangle the depth enters the section modulus squared, so the same piece of material carries far more moment on edge than flat.

Example

A rectangular beam 100 mm wide and 200 mm deep carries a moment of 10 kN·m. The section modulus is 100 × 200² / 6 = 666 667 mm³ and the stress is 10 000 000 / 666 667 = 15 MPa, compression at the top fibre and tension at the bottom.

Use in design

With a known allowable stress the required section modulus is W = M / σ_allow, and a section with a larger value is picked from the table. Conversely, the moment an existing section can carry is M = σ_allow × W.

Limits

The formula is for a straight beam, elastic material and bending about a principal axis. Shear stress, deflection and lateral torsional buckling are separate checks. For the second moment of area of I, T and channel sections use the section properties tool.