Beam deflection formula
Beam deflection is how far a loaded beam sags. For a simply supported beam with a central point load it depends on the load, the span cubed, and the beam stiffness, so span is by far the biggest factor.
Variables
| δ | Maximum deflection | in or m |
| F | Central point load | lb or N |
| L | Span length | in or m |
| E | Modulus of elasticity | psi or Pa |
| I | Area moment of inertia | in⁴ or m⁴ |
Rearranged
Cantilever, end load: δ = F × L³ / (3 × E × I)
Worked example
A simply supported steel beam spans 2 m, carries 10,000 N at center, with E = 200 GPa and I = 8 × 10⁻⁶ m⁴.
Apply the formula: 10,000 × 2³ / (48 × 200×10⁹ × 8×10⁻⁶) = 1.04 mm.
Deflection scales with the cube of the span, so doubling the length increases sag eightfold. It falls with the product of modulus and moment of inertia, the beam stiffness. The constant in the denominator depends on how the beam is supported and loaded.
Need stiffness inputs?
See the Modulus of Elasticity Table and the Area Moment of Inertia Formula.
Why span dominates deflection
The length term is cubed, so it overwhelms the other factors. A beam twice as long deflects eight times as much under the same load and section. This is why long spans need disproportionately deeper beams, and why shortening a span or adding a support is the most effective way to cut sag.
Different beam cases
The formula shown is for a simply supported beam with a central point load, where the constant is 48. A cantilever with an end load uses 3, giving far more deflection. Uniformly distributed loads use 384 over 5 for a simply supported beam and 8 for a cantilever. The variables are the same; only the constant changes with support and load type.
FAQ
What is the beam deflection formula?
For a simply supported beam with a central point load, deflection equals F times span cubed divided by 48 times E times I.
Why does deflection depend on span cubed?
The bending and rotation accumulate along the length, so the sag grows with the cube of the span. Doubling the length multiplies deflection by eight.
How do I reduce beam deflection?
Shorten the span, add a support, or increase E times I by using a deeper section or a stiffer material. Span has the largest effect because it is cubed.