About the Beam Deflection Calculator
This beam deflection calculator gives the maximum deflection, bending moment, shear force and bending stress for the four textbook load cases: a simply supported beam with a central point load or a uniformly distributed load (UDL), and a cantilever with an end point load or a UDL. It also plots the deflected shape along the span.
It is useful for students checking homework, engineers doing a quick hand-check, and builders sanity-checking a joist, lintel or shelf bracket. Choose a material to fill in Young’s modulus, then enter the second moment of area (I) and elastic section modulus (S) from a steel section table or from b·h³/12 and b·h²/6 for a rectangle.
Results assume linear-elastic, small-deflection Euler–Bernoulli beam theory, a prismatic beam and no self-weight unless you include it in the load. Shear deformation, lateral-torsional buckling and connection details are not checked, so use it for preliminary sizing — a qualified engineer should verify anything structural.
With the default inputs, the maximum deflection is 5.39 mm. Change any value above to recalculate instantly.
How to use the beam deflection calculator
- 1Pick the support type and whether the load is a point load or spread evenly.
- 2Enter the span in metres and the load in kN (point) or kN/m (UDL).
- 3Choose a material or enter a custom Young’s modulus.
- 4Enter I and S for the section from a table or from b·h³/12 and b·h²/6.
- 5Compare the maximum deflection with the L/n limit and review moment and stress.
Formula and method
The formulas come from integrating the Euler–Bernoulli beam equation EI·y″ = M(x) with the boundary conditions of each support type. Deflection grows with the cube (point load) or fourth power (UDL) of the span, so doubling the span of a uniformly loaded beam makes it sag sixteen times as much — span is usually the dominant factor.
Stiffness EI combines the material (Young’s modulus E) and the cross-section shape (second moment of area I). Maximum bending stress is the peak moment divided by the elastic section modulus S. Units are converted internally to newtons, metres and pascals; deflection is reported in millimetres and stress in MPa (N/mm²). Self-weight is not added automatically.
- δ
- Maximum deflection
- P
- Point load (kN)
- w
- Uniformly distributed load (kN/m)
- L
- Span length (m)
- E
- Young’s modulus of the material (GPa)
- I
- Second moment of area of the section (cm⁴)
- S
- Elastic section modulus (cm³)
- M
- Maximum bending moment (kN·m)
Worked examples
6 m steel beam (IPE 300), 20 kN at mid-span
δ = 20,000 × 6³ ÷ (48 × 200×10⁹ × 8,356×10⁻⁸) ≈ 5.39 mm, or L/1114 — well inside L/360 (16.7 mm). The peak moment is 20 × 6 ÷ 4 = 30 kN·m, giving 30,000 ÷ 557×10⁻⁶ ≈ 53.9 MPa.
Same beam with a 10 kN/m uniform load
A 10 kN/m UDL over 6 m totals 60 kN. δ = 5 × 10,000 × 6⁴ ÷ (384 × EI) ≈ 10.1 mm (L/594). The moment is 10 × 6² ÷ 8 = 45 kN·m and each support carries 30 kN.
2 m steel cantilever (IPE 200), 5 kN at the tip
δ = 5,000 × 2³ ÷ (3 × 200×10⁹ × 1,943×10⁻⁸) ≈ 3.43 mm. The fixed end resists 5 × 2 = 10 kN·m, which on a 194 cm³ section is about 51.5 MPa.
3 m timber cantilever (45×195 mm), 2 kN/m
For a 45×195 mm joist, I = 0.045 × 0.195³ ÷ 12 ≈ 2,781 cm⁴. With E = 11 GPa, δ = 2,000 × 3⁴ ÷ (8 × EI) ≈ 66.2 mm, far beyond the 16.7 mm L/180 limit — this cantilever needs a deeper section or a back-span.
Frequently asked questions
What is the formula for beam deflection?+
For a simply supported beam with a central point load, δ = PL³ ÷ 48EI; with a uniform load, δ = 5wL⁴ ÷ 384EI. A cantilever deflects PL³ ÷ 3EI under an end load and wL⁴ ÷ 8EI under a uniform load.
What is an acceptable beam deflection?+
Building codes commonly limit live-load deflection to L/360 for floors with brittle finishes, L/240 for total load on floors and roofs, and L/180 for some roofs and cantilevers. Check the code and finishes that apply to your project.
How do I find the moment of inertia of a beam?+
For a solid rectangle, I = b·h³ ÷ 12 with h the depth in the bending direction. For steel I-beams, channels and hollow sections, use the Ix value from the manufacturer’s section table.
Why does a cantilever deflect so much more than a simply supported beam?+
A cantilever has only one support, so the whole span acts like a lever from the fixed end. For the same span and point load it deflects 16 times as much as a simply supported beam loaded at mid-span.
Does this include the beam’s own weight?+
No. Add the self-weight to the uniform load yourself — for example, an IPE 300 weighs about 42 kg/m, or roughly 0.41 kN/m.