Stress
Beam Deflection Calculator — Simply Supported Beam
Simply supported beam with a central point load: deflection, bending stress and safety factor per Euler–Bernoulli beam theory.
Parameters
Results
- Max deflection (midspan)(δ_max)
- 1.896 mm
- Deflection at probe position(δ(x))
- 1.896 mm
- Span-to-deflection ratio(L/δ)
- 527
- Support reaction (each)(R)
- 500.0 N
- Max bending moment(M_max)
- 250.0 Nm
- Max bending stress(σ_max)
- 93.8 MPa
- Safety factor(n)
- 2.24
Design checks
- [PASS] Safety factor n = 2.24 ≥ 1.5strength_check
- [PASS] δ_max 1.9 mm ≤ L/250 = {spanLength/250} mmdeflection_check
Formulas
δ_max = F·L³ / (48·E·I)= 1.89625
Expression: load * spanLength^3 / (48 * elasticModulus * momentOfInertia)
δ(x) = F·x·(3L² − 4x²)/(48·E·I) for x ≤ L/2 (mirrored beyond midspan)= 1.89625
Expression: load * min(probePosition, spanLength/2) * (3*spanLength^2 - 4*min(probePosition, spanLength/2)^2) / (48 * elasticModulus * momentOfInertia)
L/δ — serviceability index (aim for ≥ 250)= 527.357
Expression: spanLength / maxDeflection
R = F/2 at each support= 500.000
Expression: load / 2
M_max = F·L/4 (N·mm)= 250000
Expression: load * spanLength / 4
4 — simply supported, single central point load
σ = M·c / I (N·mm × mm / mm⁴ = MPa)= 93.7506
Expression: maxBendingMoment * outerFiberDistance / momentOfInertia
n = σ_allow / σ_max= 2.23999
Expression: allowableStress / bendingStress
Results are for preliminary design. Verify before manufacturing.
Export
STEP export runs entirely in your browser; the geometry kernel downloads once on first export.
How the beam deflection calculator works
The model is an Euler–Bernoulli simply supported beam of length L with a single point load F at midspan. The midspan deflection is δ_max = F·L³/(48·E·I), and the deflection at any position x ≤ L/2 along the span follows δ(x) = F·x·(3L² − 4x²)/(48·E·I) — the probe value is mirrored symmetrically beyond midspan. Support reactions are F/2 each, and the maximum bending moment M_max = F·L/4 occurs at midspan, giving the extreme-fibre bending stress σ = M·c/I where c is the distance from the neutral axis to the outer fibre. For a rectangular section, I = b·h³/12 and c = h/2.
Two serviceability and strength checks run on every change: the safety factor n = σ_allow/σ must be at least 1.5, and the deflection is compared against the common L/250 span-to-deflection serviceability guideline (EN 1990 advisory). Typical material moduli: structural steel 206,000 MPa, aluminium 69,000 MPa. Both metric and imperial display units are supported — angles and counts are never converted, and the round-trip unit switch error stays below 0.1%.
Need drivetrain geometry instead? Head to the Spur gear calculator with STEP export.