Beam Deflection (Center Load)

Beam deflection: δ = 5·w·L⁴ / (384·E·I)

Take a beam supported at both ends carrying a uniformly distributed load w across the whole span L. Its largest deflection happens at mid-span and comes out to δ = 5·w·L⁴ / (384·E·I), with E being Young's modulus and I the second moment of area. Steel runs around E ≈ 200 GPa. Concrete bends much more easily at E ≈ 25 GPa, and aluminium falls in between near 70 GPa. Watch that L⁴ term closely. Double the span and the deflection jumps by a factor of 16, which is exactly why long spans tend to fail serviceability checks long before they run out of strength. Most codes cap deflection at δ < L/300, loosening to L/250 for roofs and tightening to L/500 where partitions are sensitive. To see it in numbers: L = 4 m, w = 5 kN/m, E = 200 GPa, I = 2·10⁷ mm⁴ gives δ ≈ 8 mm, which works out to L/500 and stays inside the limit.

Applications

It shows up when you size reinforced-concrete beams and slabs (NBR 6118), steel rafters and joists for industrial roofing (NBR 8800), or timber floors (NBR 7190). Engineers also lean on it to check vibration comfort, to size crane runways and bridges, and to verify serviceability limit states (SLS).

FAQ

Why is the limit L/300 and not based on stress? A stress check tells you whether the beam will actually break. A deflection check is about whether people can live with it: cracked plaster, doors that stick, a floor that bounces underfoot. You need to pass both.

What if the load is concentrated, not distributed? For a single point load at mid-span, switch to δ = P·L³/(48·E·I). The exponent on L drops from 4 to 3, though everything else works the same way.

Does the formula handle cantilevers? No. A cantilever loaded at its free end follows δ = P·L³/(3·E·I), which makes it roughly 16 times more flexible than the same beam supported at both ends.