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Page last changed 2026-11-08 · based on n/a — standard engineering relationships, no single governing revision
Quick Answer
Beam Deflection
The Formulas Used
Simply supported, central point load: δ = P·L³ ÷ (48·E·I)
Simply supported, uniform load w: δ = 5·w·L⁴ ÷ (384·E·I)
Cantilever, end point load: δ = P·L³ ÷ (3·E·I) Cantilever, uniform: δ = w·L⁴ ÷ (8·E·I)
Maximum moment, simply supported central point load: M = P·L ÷ 4
The L-Cubed Rule Is the Whole Story
Every one of these formulas contains L³. Double the span and the deflection goes up by a factor of eight, at the same load and the same section. This is why long spans drive section size so hard, and why a floor that feels solid at 12 ft can feel bouncy at 16 ft with nothing else changed.
The corollary is that adding depth is the cheapest way to control deflection — I goes with the cube of depth, so a modest increase in section depth buys a large reduction in sag. Adding width helps only linearly.
The cantilever case is worth noting separately: at the same load and span, a cantilever deflects 16 times as much as a simply supported beam (a factor of 48/3). Ends of beams and overhangs are always the softest part of a structure, which is why they feel springy.
Deflection Limits and What This Does Not Check
Deflection limits are usually expressed as a fraction of span — L/360 for floors under live load, L/240 or L/180 for roofs, L/500 or tighter for precision machinery and for finishes that will crack. The calculator returns the ratio so you can compare directly against the governing limit.
This is an elastic, small-deflection calculation with a constant section. It does not cover large deflections, where the geometry changes enough that the linear formulas no longer hold; shear deflection, which matters for short deep beams; continuous or multi-span beams; or the effect of connections, which in timber and cold-formed steel often governs. For a structural design, use the span tables or the governing code rather than a formula.
Frequently Asked Questions
- What is the formula for beam deflection?
- For a simply supported beam with a central point load, δ = P·L³/(48·E·I). For a cantilever with an end load it is P·L³/(3·E·I). Both are elastic formulas valid for small deflections and constant section.
- Does doubling the span double the deflection?
- No — it multiplies it by eight, because every beam deflection formula contains L cubed. That is why span drives section size so strongly, and why a beam that is adequate at one span may be completely inadequate at a slightly longer one.
- How much should a floor beam be allowed to deflect?
- The common rule is L/360 under live load for floors, and L/240 or L/180 for roofs, though the governing code and the finish materials set the actual requirement. Tile and plaster demand tighter limits — often L/480 or L/600 — because they crack before the structure is in any danger.
- Why does a cantilever deflect so much more?
- Because the load acts at the free end with no support on the other side of the load, so the full moment is carried to the root. At equal load and span a cantilever deflects 16 times as much as a simply supported beam, which is why cantilevers are always the springiest part of a structure.
- Does the deflection formula account for the material?
- Only through E, the modulus of elasticity. Steel is about 29,000,000 psi and aluminium about 10,000,000, so an aluminium beam deflects nearly three times as much as a steel one of the same section — even though the two may have similar strength.
Related
Value Sources
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| # | Source | Type | Revision / method |
|---|---|---|---|
| [1] | Values computed in your browser | derived | Evaluated locally from the formulas shown on the page. No data leaves the device. |
Data Sources
| Standard | Revision | What it covers on this page |
|---|---|---|
| Formulas as shown on this page | n/a — standard engineering relationships, no single governing revision | every value this calculator produces |
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