# Shelf Sag & Deflection Calculator

> Check how much a shelf sags under books or a point load, with creep and a front edge strip. Calculate the mid-span sag of a shelf from its span, depth, thickness and material under a uniform and a point load, simply supported or fixed. Includes self-weight, long-term creep and a glued front strip or face-frame rail, and suggests a thickness, strip height or support.

Part of usefulweb.app (https://usefulweb.app/). Part of [Kerfwise](https://kerfwise.usefulweb.app/) (ownerApp: kerfwise). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://kerfwiseshelfsag.usefulweb.app/ (interactive tool at https://kerfwiseshelfsag.usefulweb.app/tool)
- JSON API: POST https://kerfwiseshelfsag.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://kerfwiseshelfsag.usefulweb.app/api/v1/tool)
- OpenAPI: https://kerfwiseshelfsag.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://kerfwiseshelfsag.usefulweb.app/mcp - tool `kerfwise_shelf_sag`

## Inputs
- span (number): Span between supports
- depth (number): Shelf depth (front to back)
- thickness (number): Shelf thickness
- material (select): Shelf material
- customE (number): Custom modulus of elasticity (optional) - Replaces the material value, for example from the board maker’s data sheet.
- support (select): Supports
- uniformLoad (number): Uniform load along the shelf - A full row of books is about 30–40 kg per metre on a 300 mm deep shelf.
- pointLoad (number): Point load at mid-span
- selfWeight (boolean): Include the shelf’s own weight
- longTerm (boolean): Long-term load (include creep)
- limit (select): Sag limit
- stripHeight (number): Front strip height (optional) - A solid edge strip or face-frame rail glued to the front edge, top flush with the shelf. Empty = none.
- stripThickness (number): Front strip thickness
- stripMaterial (select): Front strip material

## Method
- Euler–Bernoulli beam, maximum at mid-span: simply supported δ = 5wL⁴/(384EI) + PL³/(48EI); both ends fixed δ = wL⁴/(384EI) + PL³/(192EI). w = uniform load plus self-weight (density × g × section area), P = point load × g (g = 9.80665 m/s²).
- Composite section by the transformed-section method: a front strip with its top flush with the shelf is scaled by n = E_strip / E_shelf; the neutral axis and I follow from the parallel-axis theorem.
- Long-term (creep) deflection follows EN 1995-1-1 §2.3.2.2 for a permanent load: every part uses E / (1 + k_def) with k_def for service class 1 (solid wood 0.6, plywood 0.8, OSB/3 and P6 1.5, P4 particleboard and MDF 2.25).
- Moduli are typical: solid wood from the USDA Wood Handbook (clear wood, 12 % moisture content), panels from EN 12369-1 mean values, P2 furniture board from the EN 312 minimum requirement and birch plywood from the Handbook of Finnish Plywood. Use the custom modulus for your board.
- Suggestions: the thickness or strip height that just meets the limit (bisection, rounded up to 1 mm), the longest span that passes and the sag with a centre support (two separate half spans, conservative).

## Limits
- A serviceability estimate, not a strength check: bending stress, pin and joint strength, knots, moisture and shelf-pin rocking are not included. Real boards vary; a test load tells the truth.