Beam Deflection Calculator
Calculate maximum deflection, moment and stress for a loaded beam.
Please note: These are idealised textbook models using representative material data. They are for study, sizing and sanity-checking only — never for final design. Real engineering requires code-compliant analysis, verified material certificates and a qualified engineer’s sign-off.
What the Beam Deflection Calculator does
A loaded beam bends by an amount set by the load, the span, the material’s stiffness and the shape of its cross-section. Deflection grows with span far faster than stress does, which is why long spans are usually governed by how much they sag rather than by whether they break.
Formula
Simply supported, UDL: δ = 5wL⁴ ÷ (384 E I)Simply supported, central point load: δ = PL³ ÷ (48 E I)Cantilever, point load: δ = PL³ ÷ (3 E I)Bending stress σ = M ÷ S
Inputs explained
| Input | Unit | Required | Notes |
|---|---|---|---|
| Support and loading | one of 5 options | Yes | — |
| Span / length | number | Yes | — |
| Span / length unit | one of 9 options | Yes | — |
| Point load | number | In some modes | Shown in 10 of the 25 modes. |
| Point load unit | one of 5 options | In some modes | Shown in 10 of the 25 modes. |
| Distributed load | kN/m | In some modes | Accepts more than 0. Shown in 15 of the 25 modes. |
| Material | one of 13 options | Yes | — |
| Cross-section | one of 5 options | Yes | — |
| Width | number | In some modes | Shown in 15 of the 25 modes. |
| Width unit | one of 9 options | In some modes | Shown in 15 of the 25 modes. |
| Height / depth | number | In some modes | Shown in 15 of the 25 modes. |
| Height / depth unit | one of 9 options | In some modes | Shown in 15 of the 25 modes. |
| Outer diameter | number | In some modes | Shown in 10 of the 25 modes. |
| Outer diameter unit | one of 9 options | In some modes | Shown in 10 of the 25 modes. |
| Inner diameter | number | In some modes | Shown in 5 of the 25 modes. |
| Inner diameter unit | one of 9 options | In some modes | Shown in 5 of the 25 modes. |
| Wall thickness | number | In some modes | Shown in 5 of the 25 modes. |
| Wall thickness unit | one of 9 options | In some modes | Shown in 5 of the 25 modes. |
| Flange thickness | number | In some modes | Shown in 5 of the 25 modes. |
| Flange thickness unit | one of 9 options | In some modes | Shown in 5 of the 25 modes. |
| Web thickness | number | In some modes | Shown in 5 of the 25 modes. |
| Web thickness unit | one of 9 options | In some modes | Shown in 5 of the 25 modes. |
| Deflection limit | number | Optional | Span divided by this. L/360 is typical for floors, L/240 for roofs. Accepts 1 or more. |
How to use it
- Choose Support and loading and Span / length unit.
- Enter Span / length.
- Fill in the remaining inputs the form shows for your choice.
- Optionally add Deflection limit.
- Select Calculate.
Worked example
A 3 m simply supported steel beam, 100 × 200 mm solid rectangle, carrying 5 kN/m.
- Case
- Simply supported, UDL
- Span
- 3 m
- Load
- 5 kN/m
- Section
- 100 × 200 mm
I = 66.67 × 10⁶ mm⁴. δ = 5 × 5000 × 3⁴ ÷ (384 × 200 GPa × I) = 0.396 mm, or L/7,585. Moment 5.63 kN·m gives 8.44 MPa — 3.4% of yield.
Frequently asked questions
What deflection limit should I use?
L/360 for floors carrying plaster or brittle finishes, L/240 for roofs, and L/180 where appearance is the only concern. Codes set the binding value.
Why does depth matter so much more than width?
Second moment of area is width times depth cubed. Doubling the depth gives eight times the stiffness; doubling the width only doubles it.
Does this include the beam’s own weight?
No. Add self-weight to the distributed load — for a steel beam it is usually small, but for concrete it can dominate.
Method and sources
Method. Standard elastic beam deflection formulas for the support and loading case selected — for a simply supported beam under uniform load, δ = 5wL⁴ ÷ 384EI.
Assumptions
- Linear-elastic material, small deflections, and a prismatic beam of constant section.
- Idealised supports: a pin and roller genuinely free to rotate, or a fully fixed end.
Limitations
- Real connections are neither perfectly pinned nor perfectly fixed, and the difference changes deflection by a large factor — the two cases differ by five times under the same load.
- Shear deformation is ignored, which is acceptable for slender beams and understates deflection for deep, short ones.
- Deflection is a serviceability check, not a strength check, and neither is a design. Codes govern permitted deflection limits and this applies none.
Sources
- The design code governing the structure, and standard elastic beam theory — Varies by jurisdiction and material. The deflection limits that make a result acceptable or not. The formulas themselves are classical Euler-Bernoulli beam theory.