Calculates the maximum mid-span deflection of a simply supported beam under a uniformly distributed load — a standard structural engineering result.
How it works
The load, span length raised to the fourth power, elastic modulus, and moment of inertia combine in a single formula derived from beam bending theory.
What this does not include
This covers only the single most common case: simple supports with a uniform load. Other support conditions (cantilever, fixed-fixed) or loading types (point loads) use different formulas entirely.
How to use this calculator
- Enter the uniform load, beam span, elastic modulus of the material, and the beam’s moment of inertia.
A worked example
A 1,000 N load on a 4 m beam, modulus 200 GPa, moment of inertia 0.0001 m⁴: deflection ≈ 0.0001666667 m (about 0.17 mm).
A 500 N load on a 3 m beam with the same modulus, inertia 0.00005 m⁴: deflection ≈ 0.0000527344 m.
What the variables mean
| Variable | Meaning |
|---|---|
| Load | Force applied to the beam, in newtons |
| Length | Beam span, in meters |
| Modulus | Material’s stiffness (Young’s modulus) |
| Inertia | Moment of inertia of the beam’s cross-section, reflecting its shape’s resistance to bending |
Edge cases worth knowing
Deflection grows with the cube of the beam’s length — a longer beam sags dramatically more than a short one under the same load, which is why span length is such a critical factor in structural design.
A modulus of zero has no physical meaning — a real material always has some stiffness, so the calculator declines to show a result for that input.
Frequently asked questions
Why does span length have such a large effect on deflection?
Because deflection scales with the span raised to the fourth power — doubling the span increases deflection sixteen-fold, all else equal.
What is moment of inertia, in plain terms?
A measure of how a beam’s cross-sectional shape resists bending — a taller, deeper beam has a much higher moment of inertia than a shallow one of the same area.
Does this apply to any support condition?
No — this formula specifically applies to a beam simply supported at both ends with a uniform load; other conditions require different formulas.
