# What is the beam deflection?

Beam calculator: support reactions, maximum shear, maximum bending moment, bending stress and maximum deflection of a simply supported beam or a cantilever under a point load or a uniform load.

- Page: https://www.acalculator.org/construction/beam-calculator
- JSON spec: https://www.acalculator.org/construction/beam-calculator.json
- Version: ab9422a93ed4

## Default answer

Example with the default inputs (Support Simply supported, Load Point load, Span (L) 10 ft, Load position (a) 5 ft, Point load (P) 1,000 lbf, Material Structural steel, 200 GPa, Cross-section Rectangle, Width (b) 2 in, Depth (h) 6 in): The beam sags at most 0.0344738 in (span ÷ 3,480.91) with a maximum moment of 2,500 lbf·ft.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| sup | Support | A beam on a support at each end, or a cantilever fixed at one end and free at the other. |
| ld | Load | One point load, or a load spread evenly along the whole beam. |
| L | Span (L) | The length of the beam between supports, or from the fixed end to the free end. |
| a | Load position (a) | The distance from the left support to the point load. |
| P | Point load (P) | The point load. On a cantilever it acts at the free end. |
| w | Uniform load (w) | The load per unit length along the beam, in the unit chosen below. |
| wu | Uniform load unit | The unit of the uniform load. |
| mat | Material | The beam's material, which sets Young's modulus E. |
| E | Young's modulus (E) | The material's modulus of elasticity. |
| sec | Cross-section | A solid rectangle from its width and depth, or a second moment of area I from a table. |
| b | Width (b) | The width of the rectangular section. |
| h | Depth (h) | The depth (height) of the rectangular section, in the direction of the load. |
| I | Second moment of area (I) | The second moment of area about the bending axis, in the unit chosen below. |
| iu | I unit | The unit of the second moment of area. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| deflectionIn | Maximum deflection (in) | The largest sag of the beam, in inches. |
| deflectionMm | Maximum deflection (mm) | The largest sag of the beam, in millimetres. |
| ratio | Span ÷ deflection | L ÷ δ, to compare with limits such as L/360. |
| where | Deflection position (ft) | Where the largest sag is: from the left support, or from the fixed end of a cantilever. |
| momentFt | Maximum moment (lbf·ft) | The largest bending moment, in pound-force feet. |
| momentNm | Maximum moment (N·m) | The largest bending moment, in newton metres. |
| shear | Maximum shear | The largest shear force in the beam. |
| r1 | Left reaction (R₁) | The upward force at the left support, or at the fixed end of a cantilever. |
| r2 | Right reaction (R₂) | The upward force at the right support (simply supported only). |
| stressPsi | Bending stress (psi) | The largest bending stress M × (h/2) ÷ I of a rectangular section, in psi. |
| stressMpa | Bending stress (MPa) | The largest bending stress of a rectangular section, in MPa. |
| inertia | Second moment of area (in⁴) | I = b × h³ ÷ 12 for a rectangle, or the I you typed, in in⁴. |

## Method

Simply supported, point load P at a (b = L − a): R₁ = Pb/L, R₂ = Pa/L, M = Pab/L, δ = Pc(L² − c²)^(3/2) ÷ (9√3 LEI) with c = min(a, b). Uniform w: R = wL/2, M = wL²/8, δ = 5wL⁴/(384EI). Cantilever, end load: M = PL, δ = PL³/(3EI); uniform: M = wL²/2, δ = wL⁴/(8EI).

## Assumptions

- A straight, prismatic, linear-elastic beam with small deflections (Euler–Bernoulli); the beam’s own weight is not added unless you include it in w.
- Structural steel E = 200 GPa and aluminum E = 69 GPa (Engineering ToolBox). This is an estimate, not a design check: a structural engineer must size a real beam to the building code.

## Worked examples

1. sup = simple, ld = point, L = 3.048, a = 1.524, P = 4,448.221615, mat = steel, sec = rect, b = 0.0508, h = 0.1524 gives r1 = 2,224.110808, momentFt = 2,500, inertia = 36, stressPsi = 2,500, deflectionIn = 0.034474. Source: Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads (M = qL²/8, δ = 5qL⁴/(384EI); M = FL/4, δ = FL³/(48EI); R₁ = Fb/L, M = Fab/L), https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html (retrieved 2026-10-05).
2. sup = simple, ld = uniform, L = 4, w = 2, wu = kN-m, mat = custom, E = 10,000,000,000, sec = typed, I = 100,000,000, iu = mm4 gives r1 = 4,000, momentNm = 4,000, deflectionMm = 6.666667, ratio = 600. Source: Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads (M = qL²/8, δ = 5qL⁴/(384EI); M = FL/4, δ = FL³/(48EI); R₁ = Fb/L, M = Fab/L), https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html (retrieved 2026-10-05).
3. sup = cantilever, ld = point, L = 2, P = 1,000, mat = aluminum, sec = typed, I = 1,000, iu = cm4 gives momentNm = 2,000, r1 = 1,000, deflectionMm = 3.864734. Source: Wikipedia, Deflection (engineering) (cantilever: δ = FL³/(3EI), δ = qL⁴/(8EI); simply supported off-centre: δmax = Fa(L² − a²)^(3/2) ÷ (9√3 LEI), a to the nearer support), https://en.wikipedia.org/wiki/Deflection_(engineering) (retrieved 2026-10-05); MechaniCalc, Beam Deflection Tables (cantilever: M = FL, M = wL²/2), https://mechanicalc.com/reference/beam-deflection-tables (retrieved 2026-10-05).
4. sup = simple, ld = point, L = 6, a = 2, P = 9,000, mat = steel, sec = typed, I = 100,000,000, iu = mm4 gives r1 = 6,000, r2 = 3,000, momentNm = 12,000, deflectionMm = 1.741859, where = 8.969861. Source: Wikipedia, Deflection (engineering) (cantilever: δ = FL³/(3EI), δ = qL⁴/(8EI); simply supported off-centre: δmax = Fa(L² − a²)^(3/2) ÷ (9√3 LEI), a to the nearer support), https://en.wikipedia.org/wiki/Deflection_(engineering) (retrieved 2026-10-05); Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads (M = qL²/8, δ = 5qL⁴/(384EI); M = FL/4, δ = FL³/(48EI); R₁ = Fb/L, M = Fab/L), https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html (retrieved 2026-10-05).
5. sup = cantilever, ld = uniform, L = 3, w = 1, wu = kN-m, mat = steel, sec = typed, I = 100,000,000, iu = mm4 gives r1 = 3,000, momentNm = 4,500, deflectionMm = 0.50625. Source: Wikipedia, Deflection (engineering) (cantilever: δ = FL³/(3EI), δ = qL⁴/(8EI); simply supported off-centre: δmax = Fa(L² − a²)^(3/2) ÷ (9√3 LEI), a to the nearer support), https://en.wikipedia.org/wiki/Deflection_(engineering) (retrieved 2026-10-05); MechaniCalc, Beam Deflection Tables (cantilever: M = FL, M = wL²/2), https://mechanicalc.com/reference/beam-deflection-tables (retrieved 2026-10-05).

## FAQ

### How do you calculate beam deflection?

Use the formula for your supports and load. A simply supported beam with a point load P at mid-span sags δ = PL³ ÷ (48EI); with a uniform load w it sags δ = 5wL⁴ ÷ (384EI). E is the material’s stiffness and I the second moment of area of the section.

### What is the maximum bending moment of a simply supported beam?

PL ÷ 4 for a point load at the middle, Pab ÷ L for a point load at distance a from one end (b from the other), and wL² ÷ 8 for a uniform load. A 1,000 lbf load at the middle of a 10 ft span gives 2,500 lbf·ft.

### How do I find I for a rectangular beam?

I = b × h³ ÷ 12, with b the width and h the depth in the direction of the load. A 2 × 6 in section has I = 2 × 216 ÷ 12 = 36 in⁴. Turning the same board flat makes I = 6 × 8 ÷ 12 = 4 in⁴, so it sags 9 times as much.

### What does L/360 mean?

It is a deflection limit: the span divided by 360. A 10 ft (120 in) span limited to L/360 may sag at most 0.333 in. Compare the span ÷ deflection output with the limit your code or design guide sets.

### How is a cantilever different?

A cantilever is fixed at one end and free at the other. It carries all the load at the fixed end, where the moment is largest (PL for an end load, wL² ÷ 2 for a uniform load), and it sags most at the free end.

### Can I size a real beam with this calculator?

Use it to compare options and check rough numbers. A real beam must also meet the building code (load combinations, lumber grades, lateral support, connections), so have a structural engineer or the code’s span tables confirm it.

## Sources

- Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads: reactions, moments and deflections of simply supported beams, retrieved 2026-10-05. https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html
- Wikipedia, Deflection (engineering): cantilever and simply supported deflection formulas, including the off-centre point load, retrieved 2026-10-05. https://en.wikipedia.org/wiki/Deflection_(engineering)
- MechaniCalc, Beam Deflection Tables: cantilever moments FL and wL²/2, retrieved 2026-10-05. https://mechanicalc.com/reference/beam-deflection-tables
- Engineering ToolBox, Young's Modulus, Tensile Strength and Yield Strength Values: structural steel 200 GPa, aluminum 69 GPa, retrieved 2026-10-05. https://www.engineeringtoolbox.com/young-modulus-d_417.html
- NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 lbf = 4.4482216152605 N, retrieved 2026-10-05. https://www.nist.gov/pml/special-publication-811