Beam Deflection Formulas
Quick reference for beam deflection equations. Use the calculator below for instant results, or scroll down for the complete formula reference with all common beam configurations.
Beam Deflection Calculator
Understanding Beam Deflection
When a beam is loaded, it bends and deflects from its original position. The amount of deflection depends on four factors:
- Load magnitude - More load means more deflection
- Span length - Deflection increases dramatically with span (to the 3rd or 4th power)
- Material stiffness (E) - Higher modulus means less deflection
- Section stiffness (I) - Higher moment of inertia means less deflection
The product EI is called the flexural rigidity and appears in the denominator of all deflection formulas. To reduce deflection, you can increase E (use a stiffer material) or increase I (use a deeper or wider section).
Simply Supported Beam Formulas
Uniformly Distributed Load (UDL)
Maximum deflection occurs at midspan. This is the most common case for floor beams supporting uniform floor loads. The coefficient 5/384 ≈ 0.013.
Point Load at Center
Maximum deflection at midspan under the load. The coefficient 1/48 ≈ 0.021, which is larger than the UDL case-a concentrated load causes more deflection than the same total load distributed.
Point Load at Any Position
δmax = Pa(L² − a²)3/2 / (9√3 · EIL) (true maximum, when a > b)
where a + b = L
When the load is not at center, the deflection at the load point is Pa²b²/(3EIL). The true maximum deflection occurs slightly off-center and is given by the second formula. For loads near midspan, both give similar results.
Cantilever Beam Formulas
UDL Along Full Length
Maximum deflection at the free end. The coefficient 1/8 = 0.125 is much larger than simply supported beams-cantilevers deflect about 10× more for the same load and span.
Point Load at Free End
Maximum deflection at the free end under the load. The coefficient 1/3 ≈ 0.333 shows why cantilevers need careful deflection checks.
Fixed-End Beam Formulas
UDL Along Full Length
Maximum deflection at midspan. The coefficient 1/384 ≈ 0.0026 is 5× smaller than the simply supported case-fixed ends dramatically reduce deflection.
Point Load at Center
Maximum deflection at midspan. The coefficient 1/192 ≈ 0.0052 is 4× smaller than the simply supported case.
Deflection Limits
Building codes limit deflection to prevent cracking, damage to finishes, and occupant discomfort. Common limits include:
- L/360 - Floor beams supporting brittle finishes (plaster)
- L/240 - Floor beams with flexible finishes
- L/180 - Roof beams (less critical)
- L/500 - Beams supporting sensitive equipment
These limits apply to live load deflection only-dead load deflection can be pre-cambered. Always check your applicable building code for specific requirements.
Units and Consistency
The formulas work with any consistent unit system. Common combinations:
| System | Span (L) | Material (E) | Section (I) | Result (δ) |
|---|---|---|---|---|
| Metric (mm) | mm | MPa | mm⁴ | mm |
| Imperial (in) | in | psi (or ksi/1000) | in⁴ | in |
Related Calculators
Section Properties
Area, inertia, section modulus, and torsion constant
Moment of Inertia
Second moment of area for common shapes
Beam Load Capacity
Maximum allowable load governed by stress or deflection
Unit Converter
Convert force, stress, mass, and more
Young's Modulus
Calculate E from stress-strain data
K-Factor (Effective Length)
Column effective length from alignment chart
Embodied Carbon
Estimate kgCO₂e from steel tonnage
Steel Weight
Calculate weight from section mass and length