How beam deflection is calculated
For a straight, elastic beam, deflection depends on the load, the span, the material's stiffness E and the section's second moment of area I. Stress depends on the bending moment and how far the outer fibers sit from the neutral axis.
- Simply supported, uniform load:
δ = 5wL⁴/(384EI),M = wL²/8 - Cantilever, tip load:
δ = PL³/(3EI),M = PL - Bending stress:
σ = M·c/I
Worked example: W8×18 floor beam
A W8×18 steel beam spanning 4.5 m with 8 kN/m of uniform load has I ≈ 25.4×10⁶ mm⁴. Its maximum moment is 20.25 kN·m, giving a bending stress of about 83 MPa (a safety factor of about 3.0 on A36 yield) and a deflection of about 8.4 mm, or L/535, inside the L/360 limit for floors.
Good to know
- Long spans are usually governed by deflection rather than stress. Depth helps far more than width.
- Shear, lateral-torsional buckling and connections are not checked.
- Pro adds the beam's own weight and plots the deflected shape against your limit.