Beam Deflection Formulas: A Working Table

These are the closed-form maximum deflections every structural engineer keeps close. All assume a prismatic beam (constant EI), small deflections, and linear elastic material.

Simply supported, span L

Load case	Max deflection	Where
Central point load P	PL³ / 48EI	midspan
Point load P at distance a (a ≥ L/2)	P·a·(L² − a²)^1.5 / (9√3·L·EI)	√(L²−a²)/3 from far support
Full-span uniform load w	5·w·L⁴ / 384EI	midspan
Uniform load on half span	5·w·L⁴ / 768EI	near midspan
Central applied moment pair M₀	M₀·L² / 8EI	midspan

Cantilever, length L

Load case	Max deflection	Where
End point load P	PL³ / 3EI	free end
Full-span uniform load w	wL⁴ / 8EI	free end
End applied moment M₀	M₀·L² / 2EI	free end

Fixed–fixed, span L

Load case	Max deflection	Where
Central point load P	PL³ / 192EI	midspan
Full-span uniform load w	wL⁴ / 384EI	midspan

Note the fixed-end uniform case is 1/5 of the simply supported value — fixity is expensive to build but cheap on deflection.

Using the table

1. Compute I for your section — the cross-section calculator does it for common shapes.
2. Pick E for the material: steel 200 GPa (29,000 ksi), aluminium ≈ 69 GPa (10,000 ksi), timber along grain ≈ 10–12 GPa (1.4–1.7 Msi).
3. Watch the units. With N and mm, E for steel is 200,000 MPa and the result comes out in mm. Mixing m and mm is the single most common deflection-formula error — see SI vs imperial pitfalls.

Serviceability limits

Deflection is usually a serviceability check, not a strength one. Common governing limits: L/360 for live load, L/240 for total load on roof members, tighter for finishes-sensitive floors. A beam can pass stress and fail deflection — check both.

For anything off-table — partial loads, varying loads, multiple load cases — use the beam calculator, which integrates the actual moment diagram rather than a special-case formula.