Beam Formulas — Cheat Sheet
Reactions, max moment and deflection for the common support/load cases.
Cantilever (fixed–free)
| End load P | M_max = PL, δ_max = PL³ / 3EI |
| Load P at a | M_max = Pa, δ_tip = Pa²(3L−a) / 6EI |
| UDL w | M_max = wL²/2, δ_max = wL⁴ / 8EI |
Simply supported
| Central P | R = P/2, M_max = PL/4, δ_max = PL³ / 48EI |
| P at a (b = L−a) | R₁ = Pb/L, M_max = Pab/L, δ_max = Pb(L²−b²)^1.5 / (9√3·LEI) |
| UDL w | R = wL/2, M_max = wL²/8, δ_max = 5wL⁴ / 384EI |
Fixed–fixed
| Central P | M_end = PL/8, δ_max = PL³ / 192EI |
| UDL w | M_end = wL²/12, M_mid = wL²/24, δ_max = wL⁴ / 384EI |
Propped cantilever
| UDL w | R_fix = 5wL/8, R_prop = 3wL/8, M_fix = wL²/8, δ_max ≈ wL⁴ / 185EI |
| Central P | M_fix = 3PL/16, δ_max = 7PL³ / 768EI |
Notes
| σ from M | σ = M / Z |
| Governing check | Deflection often governs before stress |
| Interactive | Run any case in the Beam Calculator tool |
Formula reference — verify against the governing standard. Not a substitute for engineering judgment.