Beam Deflection Calculator

How is beam deflection calculated?

Deflection is the displacement of the beam axis under load. In the elastic range it is proportional to the load and inversely proportional to the bending stiffness E × I. E is the elastic modulus of the material (210 GPa for steel) and I is the second moment of area about the bending axis. Span enters to the third or fourth power: doubling the span makes the deflection under a point load eight times larger.

Basic formulas

• Simply supported, point load at midspan: δ = P × L³ / (48 × E × I), M = P × L / 4
• Simply supported, uniformly distributed load: δ = 5 × w × L⁴ / (384 × E × I), M = w × L² / 8
• Cantilever, point load at the tip: δ = P × L³ / (3 × E × I), M = P × L
• Cantilever, uniformly distributed load: δ = w × L⁴ / (8 × E × I), M = w × L² / 2

When a point load is off centre, the maximum deflection is not under the load but at a point closer to midspan. The calculator reports that location as well.

Example

A steel beam spanning 6 metres with a second moment of area of 8356 cm⁴ carries 10 kN at midspan. The deflection is 10 000 × 6³ / (48 × 210 × 10⁹ × 8356 × 10⁻⁸) = 2.56 mm, about one part in 2340 of the span.

Limits

The calculation covers a single span beam of constant section with one load. For several loads the results can be found separately and added (superposition), but the maxima may occur at different places. Add the beam's self weight separately as a distributed load. Deflection limits (such as L / 300) and resistance checks come from the relevant standard.