Enrichment · STEM Explorations · Engineering

How far the shelf
sags before the
books go on it.

Choose how the beam is held and loaded, its material and its size. The calculator gives the sag at its worst point, checks it against a common limit, and shows how much a deeper beam would help.

Check a beamWhat you get

The beam deflection calculator

The worked example is a 19 mm pine shelf spanning 0.9 m with 30 kg of books spread along it. Nothing you type leaves this browser.

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What you get

Enrichment · STEM ExplorationsFree

What you finish with
The maximum sag and whether it is within span ÷ 360, 240 or 180, the bending moment and stress, and the sag at other depths. It prints as a summary and downloads as a spreadsheet.
Who it is for
Physics, engineering and technology students, woodworking classes, and students building model bridges or shelves.
Format
Browser calculator for a phone, tablet or computer, by touch or keyboard. Prints a summary; downloads a spreadsheet (.csv); export and import a file.
Access
Free. No account, no sign-in and nothing to install. Everything you type stays in this browser; nothing is sent to Caveat.
Where it lives
The STEM Explorations space in Caveat Enrichment.
Support
Questions, a correction or a request: team@caveatacademic.com.

How to use it

  1. Choose how the beam is supported and loaded.
  2. Choose the material, or type its modulus.
  3. Type the span, the total load in newtons, and the width and depth.
  4. Read the sag and the check; try a deeper section in the table.
  5. Print the summary for a lab report or a build log.

Check a beam

How the sag is worked out

Standard beam formulas for four cases.

The worked example: a pine shelfI = bh³ ÷ 12142,896 mm⁴250 × 19³ ÷ 125WL³ ÷ 384EI2.34 mm300 N, 0.9 m, 8.5 GPaAllowed2.50 mm900 mm ÷ 360Twice as thick0.29 mmone eighth of the sagThe worked example: a pine shelfI = bh³ ÷ 12142,896 mm⁴250 × 19³ ÷ 125WL³ ÷ 384EI2.34 mm300 N, 0.9 m, 8.5 GPaAllowed2.50 mm900 mm ÷ 360Twice as thick0.29 mmone eighth of the sag
Stiffness comes from the material (E) and the shape (I). Depth appears cubed in I, which is why joists stand on edge.

The formulas

For a solid rectangle, the second moment of area I = width × depth³ ÷ 12. With total load W, span L and modulus E, the maximum sag is 5WL³ ÷ 384EI for a simply supported beam with the load spread evenly, WL³ ÷ 48EI with the load at the middle, WL³ ÷ 8EI for a cantilever with a spread load and WL³ ÷ 3EI with the load at its tip. The maximum moment is WL ÷ 8, WL ÷ 4, WL ÷ 2 and WL; stress is moment × half the depth ÷ I.

Limits of the model

These formulas assume small sags, a uniform material and a beam that does not twist or buckle. Wood varies from board to board and creeps under long loads; its handbook modulus is for clear, dry wood. The beam's own weight is left out. Span ÷ 360 is a common limit for floors; shelves and models can accept more. Anything people stand under needs design values from the building code.

Sources

  1. U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. Wood Handbook: Wood as an Engineering Material (FPL-GTR-282), chapter 5, mechanical properties of clear wood.
  2. American Institute of Steel Construction. Steel Construction Manual, beam diagrams and formulas, and modulus of elasticity of structural steel.
  3. American Wood Council. Beam Design Formulas with Shear and Moment Diagrams (Design Aid No. 6) (awc.org).
  4. Aluminum Association. Aluminum Design Manual: properties of 6061-T6 (aluminum.org).

The worked example is made up. This is a learning tool, not a structural design; anything people rely on for safety needs an engineer and the building code. Your figures stay in this browser. Support: team@caveatacademic.com.