What this calculator does
Beam deflection is how far a beam bends or sags under load, measured as the distance the loaded point moves away from its unloaded position. It is a standard check in structural and mechanical design, since a beam can be strong enough not to break while still deflecting more than is acceptable for the job it is doing.
This calculator covers the two most common textbook cases: a cantilever beam, fixed at one end and loaded at the free end, and a simply-supported beam, resting on two supports with a point load at its centre. Both use fixed, standard engineering formulas built from the load, the beam length, the material's elastic modulus and the beam cross-section's second moment of area.
The formula
For a cantilever with a point load at the free end, maximum deflection is the load times the length cubed, divided by three times the elastic modulus times the second moment of area. For a simply-supported beam with a centred point load, the same load and length cubed is divided by 48 times the elastic modulus times the second moment of area, reflecting how the load is shared between two supports instead of resisted by a single fixed end.
| Term | Meaning |
|---|---|
| P | The point load applied to the beam. |
| L | The beam length, either from the fixed end to the load (cantilever) or between the two supports (simply supported). |
| E | Elastic modulus (Young's modulus), a property of the beam material describing its stiffness. |
| I | Second moment of area, a property of the beam's cross-sectional shape describing its resistance to bending. |
| δ | Deflection, the distance the loaded point moves under the applied load. |
The inputs explained
| Field | What to enter |
|---|---|
| Beam type | Whether the beam is a cantilever, fixed at one end with the load at the free end, or simply supported on two ends with the load at the centre. |
| Load (P) (N) | The point load applied to the beam. |
| Beam length (L) (m) | The beam length: from the fixed support to the load for a cantilever, or the full span between supports for a simply-supported beam. |
| Elastic modulus (E) (GPa) | The elastic modulus of the beam material, commonly around 200 GPa for structural steel or roughly 10 to 13 GPa for typical structural timber. |
| Second moment of area (I) (cm⁴) | The second moment of area of the beam's cross-section, which depends on its shape and dimensions, not just its material. |
When to use it
Checking a shelf or bracket bracket under load
A shelf bracket or cantilevered support fixed at one wall end and loaded at the free tip is the classic cantilever case, and this calculator gives how far the tip will sag under a known load.
Checking a floor joist or beam spanning two supports
A joist or beam resting on two supports with a load at its midpoint, such as a person standing at the centre of a span, is the simply-supported case, and excessive deflection here is a common sign a beam is undersized.
Comparing two beam materials or sizes
Holding the load and length fixed and changing only the elastic modulus or second moment of area shows directly how much stiffer a different material or a deeper section would make the beam.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How cantilever deflection changes with load
A fixed 2 m cantilever in a 200 GPa material with a second moment of area of 8,000 cm⁴, across a range of point loads.
| Point load | Maximum deflection | Maximum deflection (m) |
|---|---|---|
| 500 N | 0.0833 mm | 0.00008333 m |
| 1000 N | 0.1667 mm | 0.000167 m |
| 2000 N | 0.3333 mm | 0.000333 m |
| 3000 N | 0.5000 mm | 0.000500 m |
| 4000 N | 0.6667 mm | 0.000667 m |
| 5000 N | 0.8333 mm | 0.000833 m |
How simply-supported deflection changes with span length
A fixed 2,000 N centred load in a 200 GPa material with a second moment of area of 8,000 cm⁴, across a range of span lengths.
| Span length | Maximum deflection | Maximum deflection (m) |
|---|---|---|
| 1 m | 0.0026 mm | 0.00000260 m |
| 1.5 m | 0.0088 mm | 0.00000879 m |
| 2 m | 0.0208 mm | 0.00002083 m |
| 2.5 m | 0.0407 mm | 0.00004069 m |
| 3 m | 0.0703 mm | 0.00007031 m |
| 4 m | 0.1667 mm | 0.000167 m |
Questions
Why does deflection depend on length cubed?
Deflection accumulates along the beam from both the increasing bending moment further from a support and the increasing distance over which that curvature is integrated, and both effects grow with length, compounding into a cubic relationship overall.
What is second moment of area, in plain terms?
It is a measure of how a beam's cross-sectional shape resists bending, independent of the material. A deeper beam section has a much larger second moment of area than a shallow one of the same cross-sectional area, which is why standing a plank on its edge is far stiffer than laying it flat.
Does a stiffer material always mean less deflection?
Yes, for the same load, length and cross-section, since elastic modulus sits in the denominator of both formulas here: a higher elastic modulus directly reduces the calculated deflection.
What other loading cases exist besides these two?
Beams can also carry distributed loads, multiple point loads, or loads at other positions along the span, each with its own deflection formula. This calculator sticks to the two most common textbook cases; more complex or unusual loading needs a formula specific to that case.
For the maximum bending stress in the same beam under load, see the bending stress calculator.