Beam Deflection Calculator
Use this calculator to estimate maximum deflection for common beam and cantilever loading cases using linear elastic beam theory.
What this deflection calculator does
This tool estimates maximum beam deflection for four of the most common design scenarios: simply supported beams and cantilevers, each with either a point load or a uniformly distributed load (UDL). The calculation uses Euler-Bernoulli beam equations and assumes linear elastic behavior.
Built-in equations
| Case | Maximum Deflection Formula |
|---|---|
| Simply supported + center point load | δmax = P L³ / (48 E I) |
| Simply supported + full-span UDL | δmax = 5 w L⁴ / (384 E I) |
| Cantilever + point load at free end | δmax = P L³ / (3 E I) |
| Cantilever + full-length UDL | δmax = w L⁴ / (8 E I) |
Input guide
1) Span length (L)
Enter beam length in meters. Deflection is very sensitive to length, especially when the formula includes L⁴.
2) Elastic modulus (E)
Enter modulus in GPa. This reflects material stiffness. Higher E means less deflection. For example, steel typically deflects less than timber for the same beam shape and load.
3) Second moment of area (I)
Enter section inertia in mm⁴. This is the geometric stiffness of the cross-section. Increasing beam depth usually increases I significantly and can reduce deflection dramatically.
4) Load value
- Point load cases use P in kN.
- UDL cases use w in kN/m.
Worked example
Suppose you have a simply supported steel beam with:
- L = 5.0 m
- E = 200 GPa
- I = 9,000,000 mm⁴
- Center point load P = 10 kN
Plugging these values into δmax = P L³ / (48 E I) gives a maximum deflection in millimeters, along with useful comparison limits such as L/360 and L/240 for a quick serviceability check.
Common serviceability limits
In many projects, deflection limits are expressed as span ratios:
- L/360 for floors or members sensitive to finishes.
- L/240 for less sensitive elements.
Exact limits vary by building code, occupancy, and structural system. Always use the code and project specification that apply to your location.
Assumptions and limitations
- Small deflection, linear elastic behavior.
- Prismatic beam section (constant E and I along span).
- Idealized supports and load positions.
- No shear deformation or nonlinear effects included.
- No creep, cracking, or long-term behavior included.
Quick section property reminders
If you need rough values for I, these formulas are often used:
- Rectangle: I = b h³ / 12
- Solid circle: I = π d⁴ / 64
- Hollow circle: I = π (D⁴ - d⁴) / 64
Be consistent with units when deriving I from dimensions.