Beam Deflection (Uniform Load)

Deflection of a simply supported beam

When a simply supported beam carries a uniformly distributed load (UDL), the deflection peaks at midspan and works out to δ = 5wL⁴ / (384·E·I). Here w is the load per unit length, L the span, E Young's modulus, and I the second moment of area. Swap the UDL for a single concentrated load P at midspan and you get δ = P·L³ / (48·E·I) instead. To see it in numbers: w = 500 N/m, L = 4 m, E = 200 GPa, I = 1000 cm⁴ gives δ ≈ 0.83 mm.

On the serviceability side, Brazilian code NBR 6118 (concrete) puts the general minimum at L/250, tightens to L/350 where fragile partitions or finishes rest on the slab, and allows L/300 for special cases. For structural steel, NBR 8800 recommends L/350 on floor beams under live load. One thing that trips people up: keep your units consistent (E in Pa, I in m⁴, L in m, w in N/m) and δ comes out in metres.

Applications

This shows up when designing concrete, steel and timber beams for floors, roofs and industrial mezzanines, when sizing joists and purlins, and when checking serviceability limit states (SLS) for cracked or uncracked sections. It also helps with preliminary studies of footings and tie beams. Think of the formula as where stiffness checks begin, before anyone reaches for an FEM model with multiple loads and supports.

FAQ

Why does L appear to the fourth power for UDL? Deflection comes out of integrating the moment diagram twice. Under a UDL the moment is quadratic in x, so two further integrations push it up to L⁴. A concentrated load gives a linear moment, which lands you at L³.

What value of E should I use for concrete? Go with the secant modulus, E_cs ≈ 0.85·E_ci, where E_ci depends on f_ck under NBR 6118 §8.2.8. C30 concrete lands around E_cs ≈ 26 GPa.

Does this account for creep? It does not. To estimate long-term deflection in concrete, take the instantaneous δ and multiply by (1 + φ), where φ is the creep coefficient (usually somewhere in the 2.0–2.5 range).

What if the load is not centred? Reach for the general formula δ_max = P·a·b·(L² − a² − b²)^(3/2) / (9√3·L·E·I), evaluated at x = √((L²−b²)/3). When in doubt, beam-deflection tables cover the common cases.