Beam Deflection Calculator
Maximum deflection of a beam under a central or end point load.
That is 0.00008333 m for a simply supported beam. Stiffer E or I means less deflection.
Maximum beam deflection under a point load is δ = F·L³ ÷ (48·E·I) for a simply supported beam and δ = F·L³ ÷ (3·E·I) for a cantilever. A 1 kN load at the centre of a 2 m steel beam (E 200 GPa, I 1×10⁻⁵ m⁴) deflects 0.0833 mm.
What beam deflection is
Deflection is how far a beam bends out of its straight line when a load is applied. For a single point load, the largest deflection sits under the load on a simply supported beam and at the free end of a cantilever. It grows with the cube of the length, so doubling the span makes a beam bend eight times as much — and it shrinks as the material stiffness E or the section stiffness I rises.
Simply supported, central point load · F = load (N), L = span (m), E = Young’s modulus (Pa), I = second moment of area (m⁴)
Worked example
A simply supported steel beam spans 2 m and carries a 1 kN load at midspan. Steel has E = 200 GPa = 200×10⁹ Pa, and the section has I = 1×10⁻⁵ m⁴.
- 1 Pick the load case. Central point load on a simply supported beam uses the divisor 48; an end load on a cantilever uses 3.
- 2 Put everything in SI units. F in newtons, L in metres, E in pascals (200 GPa = 200×10⁹ Pa), and I in m⁴.
- 3 Substitute into δ = F·L³ ÷ (48·E·I). δ = 1000 × 2³ ÷ (48 × 200×10⁹ × 1×10⁻⁵) = 8000 ÷ 9.6×10⁷.
- 4 Read the deflection. δ = 8.33×10⁻⁵ m = 0.0833 mm. The same beam as a cantilever would deflect 8000 ÷ 6×10⁶ = 1.333 mm.
Point-load deflection formulas
Both give the maximum deflection δ. A stiffer beam — larger E or I — deflects less, so raising I is the most effective fix.
| Load case | Maximum deflection | Where it occurs |
|---|---|---|
| Simply supported, central load | δ = F·L³ ÷ (48·E·I) | Under the load, at midspan |
| Cantilever, end load | δ = F·L³ ÷ (3·E·I) | At the free end |
Assumptions and limits
These formulas assume linear-elastic behaviour (the material obeys Hooke’s law and springs back), small deflections relative to the span, and a uniform, prismatic cross-section along the length. They cover a single concentrated point load only — distributed or multiple loads use different constants.
The section stiffness I depends entirely on the cross-section shape: a tall rectangle beam has I = b·h³ ÷ 12, so orienting the section with its depth vertical dramatically reduces deflection. Beyond the elastic limit, or once deflections get large, these results no longer hold and a full structural analysis is needed.