Reference · Euler-Bernoulli

Beam deflection formulas

All cases assume linear-elastic, prismatic beams and small deflections. P = point load, w = load per length, L = span, E = modulus, I = inertia. Downward loads and deflection positive; sagging moment positive (cantilever hogging negative).

Simply supported

Center P: δ_max = PL³/48EI @ L/2 · M = PL/4
Off-center P at a: δ = Pb(L²−b²)^3/2/9√3LEI · M = Pab/L @ a
Full UDL: δ_max = 5wL⁴/384EI @ L/2 · M = wL²/8
End moment M₀: δ = M₀L²/16EI near mid · M(0⁺) = M₀
Two P at third points: δ_max = 23PL³/648EI @ L/2

Cantilever (fixed wall)

Tip P: δ_max = PL³/3EI @ tip · M_wall = −PL
P at a from wall: δ_P = Pa³/3EI · δ_tip = Pa²(3L−a)/6EI
Full UDL: δ_max = wL⁴/8EI @ tip · M_wall = −wL²/2
Tip moment M₀: δ_tip = M₀L²/2EI

Fixed-fixed and propped

Fixed center P: δ_max = PL³/192EI · M_ends = M_mid = PL/8
Fixed UDL: δ_max = wL⁴/384EI · M_ends = wL²/12, M_mid = wL²/24
Propped UDL: δ_max ≈ wL⁴/185EI · M_wall = −wL²/8
Propped center P: δ_max ≈ 7PL³/768EI

Stiffness ranking (same P, L, EI)

Fixed-fixed PL³/192EI (baseline, stiffest) → simply supported 4× softer → propped ≈ 6.8× → cantilever 64× softer. For UDL: fixed wL⁴/384EI → simply supported 5× → propped ≈ 2.1× → cantilever 48×. Boundary conditions dominate: doubling span multiplies point-load deflection 8× (L³) and UDL deflection 16× (L⁴).

Superposition

With P and UDL together, δ_total(x) = δ_P(x) + δ_w(x). Exact for linear-elastic small deflections — the calculator samples 121 stations and adds both fields, so peaks land correctly even off-center.

Try the calculator →Worked examples →