Beam Deflection Formulas (Quick Reference)
Maximum elastic deflection δmax for common beam and loading cases. Symbols: E = modulus of elasticity, I = moment of inertia, L = span or cantilever length, P = concentrated (point) load, w = uniformly distributed load (force per unit length). Use consistent units; the calculator above handles unit conversion and combined loads for you.
| Beam & loading | Max deflection δmax | Location |
|---|---|---|
| Simply supported — central point load P | P·L³ / (48·E·I) | Midspan |
| Simply supported — uniform load w | 5·w·L⁴ / (384·E·I) | Midspan |
| Simply supported — two equal loads P at distance a from each end | P·a·(3L² − 4a²) / (24·E·I) | Midspan |
| Cantilever — point load P at free end | P·L³ / (3·E·I) | Free end |
| Cantilever — uniform load w | w·L⁴ / (8·E·I) | Free end |
| Cantilever — point load P at distance a from support | P·a²·(3L − a) / (6·E·I) | Free end |
| Fixed–fixed — central point load P | P·L³ / (192·E·I) | Midspan |
| Fixed–fixed — uniform load w | w·L⁴ / (384·E·I) | Midspan |
Deflection limits (serviceability)
A beam that is strong enough can still fail serviceability if it sags too much. Compare δmax against the governing span/limit ratio:
- L/360 — live-load deflection for floors (typical IBC/most-critical check).
- L/240 — total (dead + live) deflection for floors and members supporting brittle finishes.
- L/180 — roof members and less deflection-sensitive cases.
Example: a 20 ft (240 in) floor beam under live load should deflect no more than 240/360 = 0.67 in. Enter your span, supports, and loads in the calculator above to get shear, moment, and deflection instantly and check them against these limits.
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