# Beam Deflection Calculator

> Computes the maximum deflection, bending moment and shear force of a prismatic beam for four standard cases (simply supported with a mid-span point load or a uniform load, cantilever with an end point load or a uniform load) from span, load, elastic modulus and second moment of area, using the Euler–Bernoulli closed-form formulas.

- Calculator id: `beam-deflection` · Category: Engineering & Automotive (`engineering`) · Tool name: `calculate_beam_deflection`
- Canonical page: https://tttkmbb.com/engineering/beam-deflection · This document: https://tttkmbb.com/engineering/beam-deflection.md · JSON definition: https://tttkmbb.com/engineering/beam-deflection.json

## Purpose

Computes the maximum deflection, bending moment and shear force of a prismatic beam for four standard cases (simply supported with a mid-span point load or a uniform load, cantilever with an end point load or a uniform load) from span, load, elastic modulus and second moment of area, using the Euler–Bernoulli closed-form formulas.

**Use when:** You need the mid-span or tip deflection of a steel, aluminium or timber beam under one point load or a uniformly distributed load, or want to check a deflection limit such as L/360.

**Do not use when:** The beam carries several or off-centre loads, has overhangs, fixed ends or continuous supports (superpose cases or use structural software), or you need bending stress (divide the moment by the section modulus).

## Input

| Parameter | Type | Unit | Required | Description |
| --- | --- | --- | --- | --- |
| `beam_case` | enum: simply_supported_point_load_center \| simply_supported_uniform_load \| cantilever_point_load_end \| cantilever_uniform_load |  | required | Support conditions and load type. |
| `span_m` | number | m | required | Length between the supports, or the cantilever length from the fixed end to the free end, in metres. (> 0, max 1000) |
| `load` | number | kN or kN/m | required | Point load P in kilonewtons for the point-load cases, or uniformly distributed load w in kN per metre of span for the uniform-load cases (1 kN ≈ 102 kgf ≈ 225 lbf; include self-weight in w if relevant). (> 0, max 10000000) |
| `elastic_modulus_gpa` | number | GPa | optional, default 200 | Young's modulus of the material: structural steel 200 (Eurocode uses 210), aluminium alloys 70, softwood timber about 11, concrete about 30. (> 0, max 2000) |
| `moment_of_inertia_cm4` | number | cm⁴ | required | Second moment of area (area moment of inertia) about the bending axis in cm⁴, from section tables (e.g. IPE 200: 1943 cm⁴) or b·h³/12 for a rectangle (1 in⁴ = 41.62 cm⁴). (> 0, max 10000000000) |

## Output

| Field | Type | Unit | Description |
| --- | --- | --- | --- |
| `max_deflection_mm` | number | mm | Largest vertical deflection: at mid-span for simply supported beams, at the free end for cantilevers. |
| `max_bending_moment_knm` | number | kN·m | Largest bending moment: at mid-span (simply supported) or at the fixed end (cantilever). |
| `max_shear_kn` | number | kN | Largest shear force, at the supports or at the fixed end. |
| `deflection_ratio` | string |  | Span divided by deflection written as L/n, the form used in deflection limits (e.g. L/360). |
| `span_over_deflection` | number |  | span_m divided by the maximum deflection, as a number. |
| `meets_l_over_360` | boolean |  | True when the deflection does not exceed span/360, the usual serviceability limit for floors carrying brittle finishes. |
| `deflection_limit_l_360_mm` | number | mm | Deflection allowed by the L/360 limit for this span (span_m × 1000 / 360). |
| `stiffness_ei_knm2` | number | kN·m² | E × I in kN·m² (elastic_modulus_gpa × 1e6 × moment_of_inertia_cm4 × 1e-8). |
| `formula_used` | string |  | Closed-form expressions for the chosen case (P = point load, w = uniform load, L = span). |

## Formula

`EI = elastic_modulus_gpa × 1e6 × moment_of_inertia_cm4 × 1e-8 (kN·m²), L = span_m, P or w = load; simply supported point load: δ = P·L³/(48·EI), M = P·L/4, V = P/2; simply supported uniform load: δ = 5·w·L⁴/(384·EI), M = w·L²/8, V = w·L/2; cantilever point load: δ = P·L³/(3·EI), M = P·L, V = P; cantilever uniform load: δ = w·L⁴/(8·EI), M = w·L²/2, V = w·L; max_deflection_mm = δ × 1000`

Euler–Bernoulli beam theory for a linear-elastic prismatic beam with small deflections; shear deformation is neglected, which is accurate when the span exceeds about ten times the section depth. Deflection limits (L/360 for floors, L/180–L/240 for roofs) come from building codes and may apply to live load only or to total load.

## Data Sources

- Wikipedia – Deflection (engineering) — https://en.wikipedia.org/wiki/Deflection_(engineering) (reference, retrieved 2026-09-24)
- Gere JM, Timoshenko SP – Mechanics of Materials, Appendix: deflections and slopes of beams (PWS / Cengage) — https://www.worldcat.org/search?q=Gere+Timoshenko+Mechanics+of+Materials (textbook, retrieved 2026-09-24)
- Wikipedia – Euler–Bernoulli beam theory — https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory (reference, retrieved 2026-09-24)

Data freshness: `static`. Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output.

## API

- `GET https://tttkmbb.com/api/v1/calculate/beam-deflection?beam_case=…&span_m=…&load=…&moment_of_inertia_cm4=…`
- `POST https://tttkmbb.com/api/v1/calculate/beam-deflection` with JSON body `{"inputs": {…}}`
- Response: unified envelope (`success`, `request`, `result.values`, `result.units`, `sources`, `freshness`, `timestamp`, `next_actions`, `links`); see https://tttkmbb.com/docs/response-format.md
- Schema: https://tttkmbb.com/api/v1/calculators/beam-deflection · OpenAPI operationId `calculate_beam_deflection` in https://tttkmbb.com/openapi.json
- Authentication: none. Rate limit: fair use, see https://tttkmbb.com/docs/rate-limits.md.

## MCP

- Server: `https://tttkmbb.com/mcp` (Streamable HTTP, JSON-RPC 2.0, no auth)
- Tool:  `run_calculator` with `{"calculator_id": "beam-deflection", "inputs": {…}}`

## Example

- Simply supported, 10 kN at mid-span, 4 m, steel, I = 2000 cm⁴: inputs `{"beam_case":"simply_supported_point_load_center","span_m":4,"load":10,"elastic_modulus_gpa":200,"moment_of_inertia_cm4":2000}` → `{"max_deflection_mm":3.333,"max_bending_moment_knm":10,"max_shear_kn":5,"deflection_ratio":"L/1200","span_over_deflection":1200,"meets_l_over_360":true,"deflection_limit_l_360_mm":11.111,"stiffness_ei_knm2":4000}`
- Cantilever, 2 kN/m over 3 m, steel, I = 2000 cm⁴: inputs `{"beam_case":"cantilever_uniform_load","span_m":3,"load":2,"elastic_modulus_gpa":200,"moment_of_inertia_cm4":2000}` → `{"max_deflection_mm":5.063,"max_bending_moment_knm":9,"max_shear_kn":6,"deflection_ratio":"L/593","span_over_deflection":592.6,"meets_l_over_360":true}`

```
GET https://tttkmbb.com/api/v1/calculate/beam-deflection?beam_case=simply_supported_point_load_center&span_m=4&load=10&elastic_modulus_gpa=200&moment_of_inertia_cm4=2000
```

## Limitations

The beam carries several or off-centre loads, has overhangs, fixed ends or continuous supports (superpose cases or use structural software), or you need bending stress (divide the moment by the section modulus). Euler–Bernoulli beam theory for a linear-elastic prismatic beam with small deflections; shear deformation is neglected, which is accurate when the span exceeds about ten times the section depth. Deflection limits (L/360 for floors, L/180–L/240 for roofs) come from building codes and may apply to live load only or to total load. All values are computed from the formula above; no measurement or live data is involved.

## FAQ

**Is the load in kN or kN/m?**

For the two point-load cases enter the concentrated force P in kN; for the two uniform-load cases enter the load per metre w in kN/m (a total uniform load W spread over the span is w = W / span_m).

**Where do I get the second moment of area?**

From steel or timber section tables (I about the strong axis, usually Ix or Iy), or for a solid rectangle b·h³/12 with b and h in cm; a hollow section is the outer rectangle minus the inner one.

**Which deflection limit applies?**

Common serviceability limits are L/360 for floors and members supporting plaster or tile, L/240 for roofs and ceilings, and L/180 for members without finishes; check the governing code and whether it applies to live load or total load.

## Related

- [Hooke's Law Calculator](https://tttkmbb.com/physics/hookes-law.md) — The linear elastic force–displacement relation that deflection formulas rest on.
- [Force Converter](https://tttkmbb.com/conversion/force.md) — Convert the load between kN, kgf and lbf.
- [Pressure Calculator](https://tttkmbb.com/physics/pressure-force-area.md) — Turn a distributed pressure on an area into the line load w.