Simply Supported Beam Deflection

Deflection of a simply supported beam

Put a uniformly distributed load on a simply supported beam and the midspan deflection peaks at δ = 5wL⁴ / (384·E·I). A concentrated load P at midspan gives δ = PL³ / (48·E·I) instead. Try it: w = 10 kN/m, L = 5 m, E = 200 GPa, I = 8000 cm⁴ lands at δ ≈ 5.1 mm. Just keep your units consistent (E in Pa, I in m⁴, L in m, w in N/m) and δ comes out in metres.

For serviceability, Brazilian code NBR 6118 (concrete) caps deflection at L/300 for general slabs and L/350 where brittle partitions rest on the beam. NBR 8800 (steel) suggests L/350 for floor beams under live load, while NBR 7190 (timber) usually sits at L/300. Check the instantaneous deflection and the long-term one (with creep factor 1+φ), not just one of them.

Applications

Use it to size concrete, steel and timber beams in floors, roofs and mezzanines, and to verify serviceability limit states under NBR 6118, NBR 8800 and NBR 7190. It also handles preliminary stiffness checks on joists, purlins and tie beams before any FEM analysis, and it lets you weigh up profiles (I, W, H, channel) by their moment of inertia I.

FAQ

Why L⁴ for UDL but L³ for point load? You integrate the moment diagram twice to get deflection. A UDL produces a quadratic moment, which ends up as L⁴; a point load produces a linear one, which ends up as L³.

Which E for concrete? The secant modulus, E_cs ≈ 0.85·E_ci, as defined in NBR 6118 §8.2.8. That works out to roughly 26 GPa for C30 concrete.

Does the result include creep? No, it doesn't. For long-term deflection in concrete, multiply the instantaneous δ by (1 + φ), where φ usually falls between 2.0 and 2.5.

What if the load is off-centre? Switch to the general formula δ_max = P·a·b·(L² − a² − b²)^(3/2) / (9√3·L·E·I), or just look it up in beam-deflection tables.