Beam Deflection Calculator
Calculate maximum beam deflection for simply supported, cantilever, or fixed beams under point or distributed load.
Handles E in GPa and I in cm⁴ for clean inputs.
Beam deflection is how far a beam bends under load. It’s the practical answer to “will this floor feel springy?” or “will this beam clear the ceiling joist below it when loaded?” Building codes limit deflection because excessive bending cracks finishes, causes vibration, and looks structurally unsound even when it’s still safe.
The maximum-deflection formulas by support and load type:
| Support | Load | Maximum δ | Where it occurs |
|---|---|---|---|
| Simply supported | Point load P at center | PL³ / (48 EI) | At midspan |
| Simply supported | Uniform load w (N/m) | 5 w L⁴ / (384 EI) | At midspan |
| Cantilever | Point load P at free end | PL³ / (3 EI) | At free end |
| Cantilever | Uniform load w | wL⁴ / (8 EI) | At free end |
| Fixed both ends | Point load P at center | PL³ / (192 EI) | At midspan |
| Fixed both ends | Uniform load w | wL⁴ / (384 EI) | At midspan |
What each variable means
- L is the span (m). Deflection scales as L³ for point loads and L⁴ for distributed loads. Doubling the span makes the same beam under the same load deflect 8 times more under a point load, 16 times more under uniform load.
- E is Young’s modulus (Pa, usually quoted in GPa). It’s a material property: how stiff the material is in tension/compression.
- I is the second moment of area of the cross-section (m⁴, usually quoted in cm⁴). It’s a geometry property: how stiff the shape is against bending.
- EI together is flexural rigidity, the beam’s full resistance to bending.
Typical Young’s modulus values
| Material | E (GPa) |
|---|---|
| Steel (carbon) | 200 |
| Steel (stainless) | 195 |
| Aluminum alloy | 69 |
| Cast iron | 110 |
| Concrete (normal weight) | 25 to 30 |
| Hardwood (oak, parallel to grain) | 11 to 14 |
| Softwood (pine, parallel to grain) | 8 to 12 |
| Glass | 70 |
| Carbon fiber composite | 70 to 230 |
Typical second moment of area for common shapes
For a rectangle of width b and height h (height = vertical dimension under load): I = b × h³ / 12
| Section | Dimensions | I (cm⁴) |
|---|---|---|
| 2 × 4 lumber (38 × 89 mm actual) | b=38, h=89 mm | 223 |
| 2 × 6 lumber (38 × 140 mm actual) | b=38, h=140 mm | 870 |
| 2 × 10 lumber (38 × 235 mm actual) | b=38, h=235 mm | 4111 |
| W8×10 steel I-beam | per AISC tables | 1660 |
| W12×26 steel I-beam | per AISC tables | 8400 |
| 100 mm steel pipe (OD 100, ID 90 mm) | I = π(OD⁴ − ID⁴)/64 | 169 |
I-beams concentrate material at the top and bottom flanges where bending stress is highest. Same mass of steel arranged as an I-beam can have 5 to 10 times the I of a solid rectangle.
Worked example: floor joist check
A 2×10 wood floor joist spans 4 m and supports a uniformly distributed load of 2 kN/m (typical residential live load). E for the wood is 11 GPa, I = 4111 cm⁴.
δ = 5 × 2000 × 4⁴ ÷ (384 × 11×10⁹ × 4111×10⁻⁸) δ = 5 × 2000 × 256 ÷ (384 × 11×10⁹ × 4.111×10⁻⁵) δ = 2,560,000 ÷ 173,648 δ ≈ 0.0147 m = 14.7 mm
The code limit (L/360 for live load) is 4000/360 ≈ 11.1 mm. This joist exceeds the deflection limit. You’d upsize to a 2×12 or add a sister joist.
Building code deflection limits
| Limit | Where used |
|---|---|
| L / 360 | Floor live load (most residential codes) |
| L / 240 | Floor total load (live + dead) |
| L / 180 | Roof rafters under snow |
| L / 480 | Bare floors supporting brittle finishes (tile, plaster) |
If you need stricter than L/360, build with deeper joists or shorter spans. Going to a higher-grade species (Douglas fir vs SPF) helps modestly through E; going from 2×8 to 2×10 helps dramatically through I.
A few things worth knowing
- Cantilever vs simply supported: a cantilever under end load deflects 16× more than a simply supported beam (denominator 3 vs 48). Cantilever balconies need much stiffer members than equivalent enclosed-span floors.
- Fixed-end beams are stiffer than simply supported by a factor of 4 to 5 (denominator 192 vs 48). Welded steel frames take advantage of this. Wood and bolted joints rarely achieve true fixity in practice.
- The formulas assume linear elastic behavior. They are accurate to a few percent for normal service loads on engineered beams. Near yield, the beam starts to flow plastically and these formulas underpredict the real deflection.
- Real-world loads are rarely uniform or perfectly centered. The calculator gives the textbook maximum; field cases need finite-element analysis or load-combination engineering judgment.