Beam Deflection Calculator
Estimate how much a beam sags under load. Enter the span and the beam's width and depth, pick the load case (simple or cantilever, point or uniform), the material and the deflection limit, and get the maximum deflection, the L/ratio achieved, a pass/fail against the limit, the bending moment and stress — with by-span and by-depth tables and a downloadable PDF.
| Span | Deflection | Ratio | vs limit |
|---|---|---|---|
| Enter a beam to compare | |||
| Depth | I (in⁴) | Deflection | Ratio |
|---|---|---|---|
| Enter a beam to compare | |||
How to use the beam deflection calculator
Four quick steps turn a beam’s size and load into its maximum deflection — with the L/ratio achieved, a pass/fail against your limit, the moment of inertia, bending moment and stress, by-span and by-depth tables, and a downloadable report.
Enter the beam
Span in feet, and the beam’s actual width and depth in inches.
Pick the load case
Simply supported or cantilever, with a point load or a uniform load.
Set material & limit
Choose the material (modulus E) and the deflection limit — L/360, L/240 or L/180.
Read results & save a PDF
Get the deflection, pass/fail, stress and section properties — then download a report.
The formulas behind the deflection
Deflection is a load factor times the load times the span cubed, over E times I. The factor depends on the load case; the section properties come from the width and depth.
▶ Worked example
A softwood beam 3.5 in wide × 9.25 in deep spans 12 ft (144 in), simply supported, carrying a uniform 100 lb/ft (1,200 lb total).
I is 3.5 × 9.25³ ÷ 12 ≈ 230.8 in⁴. Deflection is 5 × 1,200 × 144³ ÷ (384 × 1.6M × 230.8).
δ ≈ 0.126 in vs 0.40 in allowable (L/360) — passes, achieving L/1,140
The bending moment is W L ÷ 8 = 21,600 lb-in and the stress M ÷ S ≈ 433 psi.
Beam deflection quick reference
The deflection formula for each load case, the stiffness (E) of common materials, the usual deflection limits, and the section properties of common lumber sizes.
| Load case | Max deflection | Max moment |
|---|---|---|
| Simply supported, centre point | P L³ ÷ 48 E I | P L ÷ 4 |
| Simply supported, uniform | 5 W L³ ÷ 384 E I | W L ÷ 8 |
| Cantilever, end point | P L³ ÷ 3 E I | P L |
| Cantilever, uniform | W L³ ÷ 8 E I | W L ÷ 2 |
L is the span in inches, P a point load, W the total uniform load (load per foot × span).
| Material | Modulus E (psi) |
|---|---|
| Softwood (SPF / Douglas fir) | 1,600,000 |
| Hardwood | 1,800,000 |
| Aluminum | 10,000,000 |
| Steel | 29,000,000 |
A higher E deflects less. Within a species, use the published design value for your grade.
| Limit | Typical use | Allowable sag, 12 ft span |
|---|---|---|
| L/360 | Floors, finished ceilings (stiffest) | 0.40 in |
| L/240 | Roofs, general framing | 0.60 in |
| L/180 | Rougher / less critical members | 0.80 in |
| Nominal | Actual b × h | I (in⁴) | S (in³) |
|---|---|---|---|
| 2×6 | 1.5 × 5.5 in | 20.8 | 7.56 |
| 2×8 | 1.5 × 7.25 in | 47.6 | 13.14 |
| 2×10 | 1.5 × 9.25 in | 98.9 | 21.39 |
| 2×12 | 1.5 × 11.25 in | 178.0 | 31.64 |
I = b·h³÷12, S = b·h²÷6. Double the width (e.g. a built-up beam) to double I and S.
Everything the calculator gives you
One beam and load produce a full deflection profile, not just a single number.
Beam deflection reference numbers
Built for anyone sizing a beam
DIY builders, remodelers, and students all need a quick read on how much a beam will sag.
Checking whether a deck beam, door header or long shelf will sag before committing to a size. A quick deflection check turns a guess into a number.
- See if a joist or beam stays within L/360
- Compare a 2×10 against a 2×12 at a glance
- Decide whether to double up or go deeper
Sizing headers, floor joists and dropped beams across a remodel, and sanity-checking a span before pulling a span table or calling the engineer.
- Test a span and section against the right limit
- Try softwood, hardwood, aluminum or steel
- Hand the plan-check a clean PDF of the figures
Learning beam theory or running a fast first-pass check — deflection, moment, section properties — before a full design in the office software.
- See I, S, moment and stress worked out together
- Watch deflection scale with span cubed and depth cubed
- Compare all four classic load cases quickly
Get a reliable deflection check
Small habits make a beam deflection estimate far more trustworthy.
Add depth before width
Stiffness rises with depth cubed but only linearly with width. A deeper section beats a wider one.
Use actual dimensions
A 2×10 is 1.5 × 9.25 in, not 2 × 10. Enter net sizes or the result will be too optimistic.
Deflection is not strength
Passing the L/ratio check doesn’t mean the beam is strong enough. Bending and shear still need checking.
Pick the right E
Use the modulus for your actual species and grade, or the manufacturer’s value for LVL and glulam.
Match the load case
A cantilever sags far more than a simply supported beam. Model how the beam is really held.
Choose the limit by use
L/360 for finished floors, L/240 for roofs, L/180 for rough work. The limit changes the verdict.
Confirm structural members
For anything load-bearing, have a licensed engineer or your building department confirm the design.
Frequently asked questions
Common questions about beam deflection, deflection limits, and section properties.
Beam deflection — how much a beam sags under load — depends on the load, the span, the material’s stiffness and the beam’s cross-section. The general form is a load factor times the load times the span cubed, divided by the modulus of elasticity (E) times the moment of inertia (I).
For a simply supported beam with a uniform load, the maximum deflection is 5 times the total load times the span cubed, divided by 384 times E times I. For a central point load it is the load times the span cubed divided by 48 E I.
Work in consistent units — this calculator uses inches, pounds and psi, with the span converted to inches. Enter the span, the width and depth, the load case and the material, and it returns the deflection, the L/ratio achieved and a pass or fail against your limit.
Deflection limits are written as the span (L) divided by a number, so the allowable sag is a fraction of the span. The common limits are L/360, L/240 and L/180.
L/360 is the usual limit for floors with finished or plastered ceilings — the stiffest — L/240 is typical for roofs and general framing, and L/180 is used for rougher or less critical members. On a 12-foot (144-inch) span these work out to 0.40, 0.60 and 0.80 inches.
These are serviceability limits meant to prevent bouncy floors, cracked finishes and ponding, not strength limits. This calculator checks your beam’s deflection against whichever limit you choose and tells you the ratio it actually achieves.
Depth matters far more. The moment of inertia of a rectangular beam is its width times its depth cubed, divided by 12, so stiffness rises with the cube of the depth but only in direct proportion to the width.
Doubling the width doubles the stiffness, but doubling the depth multiplies it by eight. That is why floor joists are tall and narrow rather than wide and flat — a 2×12 is dramatically stiffer than a 2×6, far more than its extra material suggests.
So to stiffen a beam, add depth before width wherever the framing allows. This calculator includes a by-depth table that shows how the moment of inertia and deflection change as you make the section deeper.
A simply supported beam rests on a support at each end, like a joist spanning between two walls. A cantilever is fixed rigidly at one end and free at the other, like a balcony or a diving board.
For the same length, load and section, a cantilever deflects far more. A cantilever with a point load at its free end deflects sixteen times as much as the same beam simply supported with a central point load, because the load has no second support to share it.
Pick the case that matches how your beam is actually held. This calculator covers simply supported and cantilever beams with either a central or end point load or a uniform load spread along the span.
The modulus of elasticity (E) measures a material’s stiffness, in pounds per square inch. Common values are about 1,600,000 psi for softwood framing lumber like SPF or Douglas fir, 1,800,000 for hardwoods, 10,000,000 for aluminum and 29,000,000 for steel.
A higher E means less deflection for the same load and section — steel deflects far less than wood. Within a species, the published E varies by grade, so a design value from a span table or supplier is more reliable than a round number.
This calculator offers softwood, hardwood, aluminum and steel. For engineered products like LVL or glulam, use the manufacturer’s published E, which is often printed on the beam or in its span tables.
No — this calculator checks deflection only. It tells you how far the beam sags and whether that is within your chosen L/ratio limit, but passing that check does not mean the beam is strong enough.
A full structural design also has to check bending strength, shear, bearing at the supports, lateral stability, the connections, and the effect of load duration and combined loads. A beam can be stiff enough to pass deflection yet still be overstressed in bending or shear.
Use this tool for a quick serviceability estimate and for comparing sections, but have a licensed engineer or your building department confirm any structural member before you build.
Because deflection depends on load, span cubed, and E times I, you can reduce it by changing any of those. The most effective moves, in rough order, are to shorten the span, make the beam deeper, use a stiffer material, or reduce the load.
Span has a cubed effect, so even a small reduction helps a lot; depth is also cubed through the moment of inertia, so a deeper section is far stiffer than a wider one. Adding a mid-span support effectively halves the span and cuts deflection dramatically.
This calculator lets you try each change and see the new deflection and L/ratio at once, with by-span and by-depth tables to compare options side by side.
The moment of inertia (I) describes how a cross-section resists bending. For a rectangular beam it is the width times the depth cubed, divided by 12 — written b times h cubed over 12 — and it is measured in inches to the fourth power.
Because the depth is cubed, it dominates: a 1.5-inch by 9.25-inch joist has a moment of inertia of about 98.9 in to the fourth, while the same width at 5.5 inches deep has only about 20.8. The related section modulus, used for bending stress, is width times depth squared divided by 6.
This calculator works out the moment of inertia and section modulus from the width and depth you enter, then uses them in the deflection and stress formulas.
Disclaimer: US units (feet, inches, pounds, psi). Elastic deflection for a rectangular section, I = b × h³ ÷ 12. This checks deflection against the chosen L/ratio only — it does not check bending strength, shear, bearing, lateral stability, connections, duration of load or combined loads. Use net (actual) dimensions and the correct E for your grade; real framing must follow the building code and span tables. For any structural member, have a licensed engineer or your building department confirm the design.