Beam Deflection Calculator
Maximum deflection and bending moment of a simply supported beam under a centre point load or a UDL, with an L/ratio serviceability check.
When to use this calculator
Use for a quick serviceability check on a simply supported span — a pipe support beam, a monorail, a platform stringer, an equipment skid member — where you have the section properties and want to know whether it sags too much before running a full analysis. It reports both the deflection and the maximum bending moment, and compares the deflection against an allowable of L divided by a ratio you choose, which is how AISC 360, IBC 1604.3, and Eurocode 3 express serviceability limits.
Required inputs
- Load case — centre point load, or uniformly distributed load
- Load magnitude: P in N for a point load, w in N/mm for a UDL
- Span L between supports
- Modulus of elasticity E and second moment of area I
- Deflection limit ratio — 240, 360, or as your specification requires
Expected outputs
- Maximum deflection at mid-span
- Maximum bending moment
- Allowable deflection L/ratio, and a pass or fail against it
Formula overview
Consistent SI: load P in N or w in N/mm, span L in mm, E in MPa (= N/mm²), I in mm⁴ — giving deflection in mm and moment in N·mm. Watch the two easy slips: a UDL quoted in kN/m is the same number in N/mm, and a section table in cm⁴ needs ×10⁴ to reach mm⁴.
Simply supported, small-deflection Euler–Bernoulli theory.
Centre point load P:
δ_max = P·L³ / (48·E·I) M_max = P·L / 4
Uniformly distributed load w:
δ_max = 5·w·L⁴ / (384·E·I) M_max = w·L² / 8
Serviceability:
δ_allow = L / ratio pass when δ_max ≤ δ_allowWorked example
IPE 200 carrying a UDL, simply supported:
w = 10 N/mm (10 kN/m), L = 4000 mm, E = 210 000 MPa,
I = 1943 cm⁴ = 19.43 × 10⁶ mm⁴
δ_max = 5 × 10 × 4000⁴ / (384 × 210 000 × 19.43e6)
= 1.280e16 / 1.567e15 = 8.2 mm
M_max = 10 × 4000² / 8 = 2.0e7 N·mm = 20 kN·m
Serviceability at L/360: δ_allow = 4000/360 = 11.1 mm
8.2 mm ≤ 11.1 mm → pass
At the stricter L/500 the allowable is 8.0 mm and it just fails.Common mistakes
- Mixing unit families. E in MPa and I in mm⁴ demand the load in N and N/mm and the span in mm. Leaving I in cm⁴ under-predicts deflection by a factor of 10 000 and turns a failing beam into a passing one.
- Applying these cases to a beam that is not simply supported. Fixed or continuous ends are far stiffer — a fully fixed beam under a UDL deflects only a fifth as much — so using the simply supported result there is very conservative, while using it for a cantilever is dangerously unconservative.
- Treating a deflection pass as a strength check. Serviceability and capacity are separate criteria: the moment reported here still has to be checked against the section's bending capacity, along with shear, lateral-torsional buckling, and web crippling at the supports.
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