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.

Beam Deflection Calculator

Deflection and bending stress for a simply supported or cantilever beam under a central point load or a uniform load.

Data verified 2026-09-29 · based on n/a — standard engineering relationships, no single governing revision

Quick Answer

Simply supported, central point load: δ = P·L³ ÷ (48·E·I). Cantilever, end point load: δ = P·L³ ÷ (3·E·I) — a cantilever deflects 16 times as much as the same beam simply supported.

Beam Deflection

Load, P (lb)

Span, L (in)

Modulus, E (psi)

Moment of inertia, I (in⁴)

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.

Value Sources

Each data column on this page is tied to the source it came from. The numbers in square brackets correspond to the table headers above.

#	Source	Type	Revision / method
[1]	ASME B1.1 — Unified Inch Screw Threads	standard	ASME B1.1-2019 — source
[2]	ASTM A615 — Deformed steel bars for concrete reinforcement	standard	ASTM A615/A615M-20 — source
[3]	ASTM E140 — Hardness Conversion Tables	standard	ASTM E140-12b — source
[4]	Values computed in your browser	derived	Evaluated locally from the formulas shown on the page. No data leaves the device.
[5]	ISO 4287 — Surface texture: Profile method	standard	ISO 4287:1997 — source
[6]	ISO 68-1 — Basic profile	standard	ISO 68-1:2023 — source
[7]	NFPA 70 NEC Table 310.16	standard	NEC 2023 (NFPA 70-2023) — source

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

Cross-checked against:

Formulas verified against the corresponding reference chart on this site, so the calculator and the published tables agree

Derived values — the following values on this page are calculated, not taken directly from the standard:

Value	How it is derived
All outputs	Computed in the browser from the formulas above. No data leaves the device.

Outputs are computed from the formulas shown. Verify against the governing standard for design or acceptance work.

Accuracy and use. The values on this page are compiled from the published standards and cross-checked sources listed above. Where values are derived, the derivation is stated. No warranty, express or implied, is made as to the accuracy or completeness of this information, and no liability is accepted for any loss or damage arising from its use. Engineering reference data is provided for guidance in preliminary work — before a value is used for design, fabrication or acceptance testing, verify it against the current revision of the governing standard and against your own inspection. The user assumes all risk and responsibility in connection with the use of this information.

Cite This Page

Every value on this page is traceable to the sources listed above. If you use the data in a document, paper or report, cite it as:

Plain text

Engineering Reference. "Beam Deflection Calculator." https://refmanual.com/calculators/beam-deflection/ (accessed 2026-09-29). Data from Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision.

BibTeX

@misc{beam-deflection,
  title        = {Beam Deflection Calculator},
  author       = {{Engineering Reference}},
  howpublished = {\url{https://refmanual.com/calculators/beam-deflection/}},
  note         = {Accessed: 2026-09-29},
  source       = {Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision}
}

RIS (EndNote, Zotero, Mendeley)

TY  - ELEC
TI  - Beam Deflection Calculator
AU  - Engineering Reference
PB  - Engineering Reference
UR  - https://refmanual.com/calculators/beam-deflection/
PY  - 2026
DA  - 2026-09-29
N1  - Source: Formulas as shown on this page, n/a — standard engineering relationships, no single governing revision
ER  -

Each row in the tables above also has a permanent link — hover a row and use the # link to cite a single value rather than the whole page.