Beam Deflection Calculator
Find deflection under center point load or uniform distributed load.
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Formulas
δ = FL³ / 48EISimply supported beam, center point load
δ = 5wL⁴ / 384EISimply supported beam, uniform distributed load
Beam Deflection
For simply supported beams. E values: Steel 200 GPa, Aluminum 69 GPa, Wood 8-14 GPa. Design limit: typically L/360 for floor beams, L/240 for roof beams.
How Beam Deflection Is Calculated
When a load is applied to a beam, it bends. The amount it sags at its worst point is the deflection. For a simply supported beam (resting on a support at each end), the two most common load cases have well-established formulas:
δ = FL³ / 48EICenter point load: F is the load, L the span, E the modulus of elasticity, and I the second moment of area.
δ = 5wL⁴ / 384EIUniformly distributed load: w is the load per unit length along the beam.
Deflection grows with the cube (point load) or fourth power (distributed load) of the span, so length is by far the dominant factor. Doubling the span of a point-loaded beam increases deflection eightfold.
Worked Example
A 3 m simply supported steel beam carries a 5 kN point load at midspan. Steel E = 200 GPa = 200×10⁹ Pa, and the section has I = 8×10⁻⁶ m⁴:
δ = (5000 × 3³) / (48 × 200×10⁹ × 8×10⁻⁶)
δ = 135000 / 76800000 = 0.00176 m = 1.76 mm
Checking against the L/360 limit: 3000 mm / 360 = 8.3 mm allowed. At 1.76 mm the beam is comfortably within limits.
Deflection Limits
Structural codes cap deflection to protect finishes, prevent visible sag, and keep occupants comfortable. Limits are expressed as a fraction of span L.
| Application | Typical limit |
|---|---|
| Floor beams (live load) | L / 360 |
| Roof beams | L / 240 |
| Cantilevers | L / 180 |
| Beams supporting brittle finishes | L / 480 |
A beam can be strong enough to avoid breaking yet still fail a deflection check, so both strength and stiffness must be verified.
Modulus of Elasticity Reference
| Material | E (GPa) |
|---|---|
| Steel | 200 |
| Aluminium | 69 |
| Concrete | 30 (varies) |
| Timber | 8 – 14 |
A higher E means a stiffer material that deflects less under the same load. Steel's high modulus is why steel beams can span far with modest depth.
Frequently Asked Questions
What is I (second moment of area)?
It is a geometric property describing how a cross-section resists bending. Deep sections (like I-beams oriented tall) have a large I and deflect little, which is why beam orientation matters so much.
Why does a distributed load use a different formula?
A uniform load spreads weight along the whole span rather than concentrating it at the center, producing less midspan deflection for the same total load — hence the 5/384 coefficient versus 1/48.
Does this work for fixed or cantilever beams?
No. These formulas are specific to simply supported beams. Fixed-end and cantilever beams have different coefficients because their support conditions change how the beam bends.
Strength or stiffness — which governs?
It depends on the span. Short beams are usually limited by strength (stress), while long beams are typically limited by deflection (stiffness). Always check both.