What beam deflection is and why it usually governs
Deflection is how far a loaded beam sags below its unloaded line. You calculate it because a beam can be nowhere near breaking and still be unacceptable: the drywall above it cracks, the tile grout lines open, the door at mid-span stops latching, and the floor bounces when someone walks across it. In wood framing the deflection check controls the member size far more often than the bending-stress check does, and it is the reason a 2x10 that is theoretically strong enough at 16 feet is still the wrong beam.
The number you get here is elastic deflection under service loads — the loads as they actually occur, not factored loads. That matters, because deflection limits in the building code are serviceability criteria and are written against unfactored load combinations. It is also instantaneous deflection: wood creeps under sustained load, and codes handle that separately by amplifying the dead-load portion.
Three quantities come out of the same analysis and you need all three. Deflection tells you whether the beam is stiff enough. The bending moment feeds a stress check against the section modulus, which is where the section modulus calculator takes over. The end reaction tells you what the post, jack stud or foundation below has to carry, and how much bearing length the beam needs.
Reading the formula: why span is raised to the fourth power
For a simply supported beam under a uniform load, the maximum deflection is Δ = 5wL⁴ / (384·E·I), and every symbol in it earns its place.
w is load per unit length. Deflection is directly proportional to it: double the load, double the sag. This is the only term in the formula with a linear effect, which is why chasing a deflection problem by reducing load is usually futile.
L is the span, and it enters to the fourth power. Extend a beam from 10 feet to 12 feet, a 20% increase, and deflection rises by 1.2⁴ = 2.07 — it more than doubles. Meanwhile the allowable deflection L/360 only grows by 20%. That mismatch is the single most important fact in beam sizing, and it is why adding one intermediate post transforms a marginal beam into a comfortable one.
E is the modulus of elasticity, the material's stiffness in pounds per square inch. It is a property of the species and grade, not of the shape. Douglas Fir-Larch No. 2 is 1,600,000 psi, Spruce-Pine-Fir No. 2 is 1,400,000 psi, common laminated veneer lumber is 1,900,000 to 2,000,000 psi, and structural steel is 29,000,000 psi. Deflection is inversely proportional to E, so swapping SPF for DF-L at the same size buys you 12.5% less sag and nothing more.
I is the moment of inertia, purely geometric, and for a solid rectangle it is b·d³/12. Depth is cubed. Going from a 2x10 (I = 98.9 in⁴) to a 2x12 (I = 178.0 in⁴) at the same width cuts deflection by 44%. Adding a second ply of the same depth only halves it. Depth beats plies, every time, which is why engineers reach for a deeper member before a wider one.
The 5/384 is a shape constant that comes from integrating the curvature of that particular load and support arrangement twice. Change the arrangement and only the constant changes: a centre point load gives 1/48, a cantilever under uniform load gives 1/8, and fully fixed ends give 1/384.
Worked example: a 2x10 spanning 12 feet at 100 lb/ft
Take a single 2x10 of Douglas Fir-Larch No. 2, simply supported over a 12 ft span, carrying 100 pounds per foot of service load, checked against L/360.
- Convert the span to inches. L = 12 × 12 = 144 in.
- Convert the load to pounds per inch. w = 100 ÷ 12 = 8.3333 lb/in.
- Find the moment of inertia. A dressed 2x10 is 1.5 in × 9.25 in, so I = 1.5 × 9.25³ ÷ 12 = 1.5 × 791.453 ÷ 12 = 98.932 in⁴. The NDS Supplement lists exactly this value.
- Raise the span to the fourth power. 144⁴ = 429,981,696 in⁴.
- Build the numerator. 5 × 8.3333 × 429,981,696 = 17,915,904,000.
- Build the denominator. 384 × 1,600,000 × 98.932 = 614,400,000 × 98.932 = 60,783,820,800.
- Divide. Δ = 17,915,904,000 ÷ 60,783,820,800 = 0.2947 in.
- Express it as a ratio. 144 ÷ 0.2947 = 489, so the beam runs at L/489.
- Compare with the limit. L/360 allows 144 ÷ 360 = 0.4000 in. The beam uses 0.2947 ÷ 0.4000 = 73.7% of its deflection allowance and passes.
The companion numbers follow from the same load. The moment is M = wL²/8 = 100 × 12² ÷ 8 = 1,800 lb·ft, and each end reaction is wL/2 = 100 × 12 ÷ 2 = 600 lb. To just reach L/360 you would need I = 98.932 × 0.2947 ÷ 0.400 = 72.9 in⁴, so a 2x8 (I = 47.6 in⁴) would fail and a 2x10 has room to spare.
How to read the result
Start with the utilisation percentage. Below 100% the beam satisfies the limit you chose; above 100% it does not. Because deflection is inversely proportional to I, the ratio of the required I to the actual I is the same as the utilisation, which is why the calculator reports both — if you are at 150%, you need a section half again as stiff.
Then look at the L/x figure. Framers and engineers speak in these terms because the number is scale-free: L/480 feels solid under foot at any span, L/360 is the code floor for a floor, L/240 is what IRC Table R301.7 requires of most other structural members, including a roof beam or a garage door header, and L/180 is the loosest limit the table contains. The code sets floors at L/360 rather than at the strength limit precisely because a beam can be nowhere near failure and still perform badly under foot.
Treat the deflection limit as a serviceability criterion, not a safety criterion. Passing L/360 does not mean the beam is strong enough — you still owe it a bending check against allowable stress, a shear check, and a bearing check at the supports. Failing L/360 does not mean it will collapse; it means the finishes above it will not survive.
Two limits deserve a note. Floor systems under hard finishes are designed stiffer than the structural minimum because thin-set and grout crack long before gypsum does: the TCNA Handbook calls for L/360 under ceramic tile and L/720 under natural stone, and it applies those ratios to the total load rather than to the live load alone, which makes even the tile figure more demanding than the IRC's L/360. And floor vibration, the springy feeling in a long open span, is not captured by any static deflection limit at all; a joist bay can pass L/360 comfortably and still bounce, which is why the stiffer denominators in the selector are a hedge rather than a vibration check.
Deflection and moment for the standard beam cases
| Case | Maximum deflection | Where it occurs | Maximum moment |
|---|---|---|---|
| Simple span, uniform load | 5wL⁴ / 384EI | Mid-span | wL² / 8 |
| Simple span, point load at centre | PL³ / 48EI | Mid-span | PL / 4 |
| Simple span, two equal loads at third points | 23PL³ / 648EI | Mid-span | PL / 3 |
| Both ends fixed, uniform load | wL⁴ / 384EI | Mid-span | wL² / 12 at the supports |
| Cantilever, uniform load | wL⁴ / 8EI | Free end | wL² / 2 at the fixed end |
| Cantilever, point load at the free end | PL³ / 3EI | Free end | PL |
Compare the first and fourth rows: fixing both ends cuts deflection to one fifth of the pinned case. Real timber connections are nowhere near fixed, which is why residential beams are analysed as simply supported.
Code deflection limits and what they allow
| Span | L/180 | L/240 | L/360 | L/480 |
|---|---|---|---|---|
| 8 ft (96 in) | 0.53 in | 0.40 in | 0.27 in | 0.20 in |
| 12 ft (144 in) | 0.80 in | 0.60 in | 0.40 in | 0.30 in |
| 16 ft (192 in) | 1.07 in | 0.80 in | 0.53 in | 0.40 in |
| 20 ft (240 in) | 1.33 in | 1.00 in | 0.67 in | 0.50 in |
IRC Table R301.7 assigns L/360 to floors and plastered ceilings, L/240 to most other structural members, and L/180 to rafters with slopes over 3:12 and no finished ceiling attached. For a cantilevered member, IBC Table 1604.3 directs you to take the reference length as twice the projection, and this calculator follows that convention.
Which standards this follows
The deflection expressions are classical Euler–Bernoulli beam theory and appear unchanged in every strength-of-materials text and in Roark's tables. The L/180, L/240 and L/360 entries in the selector are the limits of the 2021 IRC Table R301.7, which agrees with IBC Table 1604.3 for the same member types; L/480 and L/600 are stricter-than-code choices you may impose yourself, and L/720 is the TCNA Handbook requirement under natural stone. The cantilever convention — apply the limit to twice the projection — comes from IBC Table 1604.3. Material stiffness values for sawn lumber come from the AWC NDS Supplement, and the dressed section dimensions used for the b × d option come from the same document. If you are designing to Eurocode or CSA O86 the arithmetic is identical; only the limits and the load combinations change.
Mistakes that make a deflection check wrong
- Mixing feet and inches. The most common error by a wide margin. If L is in inches, w must be in pounds per inch and E in psi. A load of 100 lb/ft is 8.333 lb/in, not 100.
- Using nominal instead of dressed dimensions. A 2x10 is 1.5 in by 9.25 in. Using 2 × 10 gives I = 2 × 10³ ÷ 12 = 166.7 in⁴ against the true 98.9 in⁴, an overstatement of 68%, and it makes an under-strength beam look comfortable.
- Forgetting that plies add width, not depth. Three 1.75 in LVL plies of the same depth triple I. One deeper ply usually does better for less money.
- Checking total load where the code checks live load. The L/360 floor limit is a live-load check; IBC Table 1604.3 pairs it with a separate, looser L/240 check on dead plus live load. Entering the full dead-plus-live load and comparing it to L/360 is conservative, and it can push you a size larger than the code demands.
- Ignoring long-term creep. Wood under sustained load keeps deflecting. The NDS handles this by adding a creep factor to the dead-load deflection — 1.5 for seasoned lumber and glulam, 2.0 for unseasoned members. This calculator gives instantaneous deflection only.
- Assuming a fixed end that is not fixed. Nailed and bolted timber connections rotate. Analyse residential beams as simply supported unless a moment connection has actually been detailed.
- Skipping the strength checks. Deflection says nothing about bending stress, horizontal shear near the supports, or crushing at the bearing. All three still have to pass.
Where this sits among the other checks
A complete beam design runs four checks in sequence, and this calculator handles the first and supplies the inputs for the rest. Deflection comes first for wood because it usually governs. Bending stress comes second: take the moment reported here, divide by the section modulus, and compare with the allowable bending stress adjusted for load duration — that is exactly the workflow in the section modulus and moment of inertia calculator. Horizontal shear comes third and matters on short heavily loaded spans. Bearing comes last: the end reaction shown here, divided by the compression-perpendicular-to-grain value of the plate below, gives the bearing area you need.
For prescriptive residential work you often do not need to run any of it. Joist span tables and the IRC header tables already embed these calculations for standard load cases, and the header sizing calculator works the tributary loads over an opening for you. Use the general beam analysis here when the load case is not standard: a girder truss landing mid-span, a beam picking up a point load from a post above, a cantilevered balcony, or a flitch beam.
The loads themselves come from elsewhere. Roof loads depend on ground snow, which the snow load calculator converts to a flat-roof design load, and on the roof geometry that the rafter length calculator resolves. What lands underneath matters too — the reaction figure here is the starting point for footing sizing when the beam bears on a pad rather than a wall.
Key terms
- Moment of inertia (I)
- The second moment of area of a cross-section about the bending axis, in in⁴. It measures how the material is distributed away from the neutral axis and controls stiffness. Purely geometric — species does not change it.
- Modulus of elasticity (E)
- The ratio of stress to strain in the elastic range, in psi. A material property published by grade for lumber and by grade of steel for shapes.
- Flexural rigidity (EI)
- The product of E and I. Deflection is inversely proportional to it, so a beam twice as rigid sags half as much under the same load and span.
- Serviceability limit state
- A performance criterion about comfort, appearance and the survival of finishes rather than about collapse. Deflection limits are serviceability criteria and use unfactored service loads.
- Simply supported
- A beam free to rotate at both bearing points and restrained only against vertical movement. It carries no end moment, which makes it the conservative assumption for timber framing.