Beams must be strong enough not to break and stiff enough not to sag visibly or crack finishes. Building codes limit live-load deflection to L/360 for floors and L/240 for roofs, so the span divided by the deflection should be at least that number. Deflection depends on the load, the span (to the third or fourth power), the material stiffness E and the cross-section’s moment of inertia I.
How to use the Beam Deflection Calculator
- Beam and load
- Span
- Point load
- Uniform load
- Material
- Custom modulus of elasticity
- Moment of inertia
- Every result field is editable: type into Maximum deflection and Moment of inertia plus every other field update to match.
Results update as you type, and every field accepts the units shown in its menu.
How it works
Choose the support and load case, enter the span, the load, the material (or E) and the moment of inertia of the section. Type a target deflection to find the moment of inertia you need.
Inputs explained
| Field | What to enter |
|---|---|
| Beam and load | Choose one: Simply supported with point load at centre, Simply supported with uniform load, Cantilever with point load at free end, Cantilever with uniform load |
| Span | Units: m, ft, cm, in, yd |
| Point load | Units: N, kN, lbf, kip, kgf |
| Uniform load | Units: kN/m, N/m, lbf/ft, lbf/in |
| Material | Choose one: Structural steel (200 GPa), Aluminium (69 GPa), Softwood timber (11 GPa), Hardwood timber (13 GPa) … |
| Custom modulus of elasticity | Units: GPa, MPa, ksi, psi |
| Moment of inertia | Units: cm⁴, mm⁴, in⁴, m⁴ |
Example values
How the main result changes when Moment of inertia changes and everything else stays at the example values above:
| Moment of inertia (cm⁴) | Maximum deflection (mm) |
|---|---|
| 4,178 | 1.995 |
| 6,267 | 1.33 |
| 8,356 | 0.997 |
| 10,445 | 0.798 |
| 12,534 | 0.665 |
| 16,712 | 0.499 |
Formula
Simply supported, centre point load: δ = PL³ ÷ 48EI, M = PL ÷ 4
Simply supported, UDL: δ = 5wL⁴ ÷ 384EI, M = wL² ÷ 8
Cantilever, end load: δ = PL³ ÷ 3EI, M = PL
Cantilever, UDL: δ = wL⁴ ÷ 8EI, M = wL² ÷ 2
Worked example
Inputs: Beam and load: Simply supported – uniform load; Span: 4 m; Uniform load: 5 kN/m; Material: Structural steel (200 GPa); Moment of inertia: 8,356 cm⁴.
Results: Maximum deflection = 0.997 mm; Maximum bending moment = 10,000 N·m; Maximum shear = 10,000 N; Span ÷ deflection (L/δ) = 4,011.
Key terms
- E
- Young's modulus, the stiffness of the material.
- I
- Second moment of area of the cross-section.
Frequently asked questions
What is an acceptable beam deflection?
Typically L/360 for floors under live load, L/240 for roofs and L/180 for cantilevers. Check your code.
What is the moment of inertia?
A property of the cross-section shape (I = b h³/12 for a rectangle) that measures resistance to bending; steel tables list it for every section.
Why does doubling the span matter so much?
Deflection grows with L³ for point loads and L⁴ for distributed loads, so doubling the span makes a uniformly loaded beam sag 16 times as much.