Beam Deflection Calculator

Beam Deflection Calculator that finds the sag of a simple, cantilever, fixed or propped beam, checks it against L/360 or L/240, and sizes the lightest lumber, LVL or steel W-shape.

What the Beam Deflection Calculator Checks

Pick a beam type and a load type, then enter the span, the load, the modulus of elasticity E and the moment of inertia I. The beam deflection calculator returns the maximum deflection, where along the span it happens and the span ratio, such as L/659.

The math runs on a stiffness model that splits the beam into about 200 short elements. That model handles off-center loads, end moments and a point load combined with a uniform load. A closed-form textbook formula runs beside it as a check, and the step table shows both results.

Two results go further than most deflection tools. One card gives the minimum I that keeps your loads inside the limit, the smallest lumber, LVL or W-shape that passes, and the longest span the current beam can reach. The diagram panel draws the deflected shape with live shear and moment diagrams.

Under Section Shape you can pick sawn lumber, LVL plies, a steel W-shape, a solid rectangle, a round bar, a round tube or a rectangular tube. The beam deflection calculator then works out I and S from the dimensions and can add the beam's own weight to the load. For loads alone, the Beam Load Calculator gives reactions and moment, and the Bending Stress Calculator checks M/S against the allowable stress for your grade.

Beam Deflection Formulas

Every beam deflection formula has the same form. Load times a power of the span, divided by the flexural rigidity EI. Span is cubed for a point load and raised to the fourth power for a spread load, so doubling the span of a uniformly loaded beam makes it sag 16 times more.

For a simply supported beam with a point load P at midspan, the maximum deflection is

$${\delta }_{max}=\frac{P{L}^3}{48EI}$$

For the same beam carrying a uniform load w along the full span, the simply supported beam deflection formula becomes

$${\delta }_{max}=\frac{5w{L}^4}{384EI}$$

A cantilever with a point load at its free end follows the formula below. It is 16 times the simply supported midspan value, which is why short cantilevers can still sag more than long simple spans.

$${\delta }_{max}=\frac{P{L}^3}{3EI}$$

The table lists the closed-form cases the beam deflection calculator uses for its check row. Here b is the shorter distance from the point load to a support, and every location is measured along the span.

Maximum deflection sits at midspan only when the load and the supports are symmetric. Off-center loads, a propped end or a varying load all move the peak, and the calculator reports that spot as a distance from the left support or the fixed end.

Worked Example: 2x12 Over 12 ft

The default inputs model a Douglas Fir-Larch No.2 2x12 spanning 12 ft (144 in) with a 1,000 lb point load at center. E is 1,600,000 psi from the NDS Supplement, and I for the actual 1.5 x 11.25 in section is 1.5 × 11.25³ / 12 = 178 in⁴.

$${\delta }_{max}=\frac{1000\times 144^3}{48\times 1600000\times 178}=0.218\text{in}$$

The span ratio is 144 / 0.218, or L/659. The L/360 limit allows 144 / 360 = 0.400 in, so the beam uses 55% of it with 0.182 in to spare. Peak moment is PL/4 = 3,000 lb·ft, and bending stress is 36,000 lb·in over S = 31.6 in³, about 1,138 psi.

The sizing card shows a minimum I of 97.2 in⁴ for L/360, so a single 2x10 at 98.9 in⁴ would also pass this load. Kept as a 2x12, the same beam could stretch to 16.2 ft before the 1,000 lb load pushed it past L/360.

Deflection Limits: L/360, L/240 and More

A deflection limit is the span divided by a number. L/240 means the beam may sag no more than its span divided by 240, so a 12 ft beam may move 144 / 240 = 0.60 in. A bigger divisor is a stricter limit, and a ratio like L/659 passes every limit with a smaller divisor.

In the US these limits come from IBC Table 1604.3. Floor members are held to L/360 under live load and L/240 under dead plus live. Roof members get L/360 under a plaster ceiling, L/240 under a non-plaster ceiling and L/180 with no ceiling at all.

For a cantilever, the IBC table notes take the length as twice the cantilever, so a 4 ft cantilever is checked like an 8 ft span. The beam deflection calculator applies that rule on its own. Beams that carry unreinforced masonry use L/600 under dead plus live in TMS 402, and natural stone tile floors are usually held to L/720 under the TCNA Handbook.

The wood dead plus live check is easy to misread. The American Wood Council explains that the L/240 limit applies to live load deflection plus the creep part of dead load deflection. For wood that is dry when installed, that creep part can be estimated as the deflection from half the dead load.

Wood, LVL and Steel Values

E depends on the material and the grade. Douglas Fir-Larch No.2 sawn lumber carries E = 1,600,000 psi and Hem-Fir No.2 carries 1,300,000 psi in the NDS Supplement. LVL sold as 2.0E has an E of 2,000,000 psi, and structural steel uses 29,000,000 psi, about 18 times the stiffness of DF-L.

LVL weighs more than sawn lumber of the same size. The LP LVL technical guide lists a 1-3/4 x 11-7/8 in ply at 5.9 lb/ft, and the calculator uses that weight when self-weight is on.

That stiffness gap is why a wood beam deflection calculator and a steel beam deflection calculator land on very different sizes for the same span. Steel W-shape values in the tool, such as Ix = 61.9 in⁴ and Sx = 15.2 in³ for a W8x18, match the AISC shapes database.

Wood also creeps under loads that stay on for years. NDS Section 3.5.2 raises the long-term load deflection by 1.5 for seasoned lumber and by 2.0 for unseasoned or wet lumber, as the AWC creep guidance describes. The calculator shows short-term deflection, so add that creep share yourself when dead load is large.

Beam Size for a Span

Questions like what size beam spans 12, 16, 20 or 30 ft all come down to the required moment of inertia. Rearranging the uniform load formula for a limit of L/n gives the I a beam needs, and any section with at least that I will pass.

$$I_{req}=\frac{5w{L}^{3}n}{384E}$$

The table assumes an 8 ft tributary width at 40 psf floor live load, which is w = 320 lb/ft on a simply supported beam checked at L/360. It covers deflection only, so bending strength, shear, dead load and bearing still need their own checks.

LP lists 1-3/4 in LVL only up to 18 in deep, so past about 24 ft the LVL column needs four or more plies while a single light W-shape still passes. A 5 m span is about 16.4 ft, so it sits just past the 16 ft row. For other loads, enter your own w and pick LVL or steel under Section Shape, and the beam deflection calculator lists the smallest passing size with its own weight counted. The Wood Beam Span Calculator and Floor Joist Calculator cover lumber spans from the strength side.

Once a size is chosen, the support reactions become post and footing loads. The Bearing Pressure Calculator checks a footing against soil capacity, and the Ground Pressure Calculator converts a load and contact area into pressure.

Cantilever, Fixed and Propped Beams

The five common beam types are simply supported, cantilever, fixed at both ends, propped cantilever and continuous or overhanging. The first four are built in. Simply supported is the default, so the tool opens as a simply supported beam deflection calculator with a pin at the left and a roller at the right.

Switch to Cantilever and it works as a cantilever beam deflection calculator, fixed at the left and free at the right. It then reports the free-end slope and the sag at midspan, which matter for railings, finishes and doors hung near the tip, and it checks the limit at twice the length.

End fixity makes a large difference. Under a uniform load, a beam fixed at both ends deflects one fifth as much as a simply supported beam, since 1/384 is one fifth of 5/384. A propped cantilever sits between them at wL⁴/185EI and develops negative moment at the fixed end, so the moment card reports its inflection points for splice locations.

Excel and Hand Checks

The step table prints an Excel or Google Sheets formula filled with your numbers, such as =1000*144^3/(48*1600000*178). Paste it into a cell to build your own beam deflection calculator in Excel or to check the result by hand. Keep every input in one unit set, such as inches, pounds, psi and in⁴.

The most common unit slip is a uniform load left in lb/ft while the span is in inches. Divide lb/ft by 12 to get lb/in first, or the deflection comes out 12 times too large. A 320 lb/ft load is 26.7 lb/in in the formula.

Adding the peak deflection of each load, called superposition, is exact only when every load peaks at the same point. A centered point load plus a uniform load on a simple span qualifies. An off-center point load or a propped end does not, so the sum overstates the peak, and the beam deflection calculator reports the true combined value from its stiffness model.

Measuring Deflection on Site

To test a beam in place, stretch a tight string line or set a laser level between the bottom edges at the two supports. Measure the gap at midspan, where a simple beam sags most, and compare it with the calculator's result for the same loads.

For load tests, engineers use a dial indicator or an LVDT sensor under the beam to read deflection to a thousandth of an inch. When measuring the beam itself, use actual sizes rather than nominal ones, since a 2x12 is 1.5 x 11.25 in, and take the span between the centers of the bearings.

Long-term sag in an old floor includes creep, so a measured value can run well above the short-term figure the tool gives. Steel members hold their shape over time, and their weight is easy to find with the Steel Weight Calculator.

Beam Deflection FAQs

These answers cover the beam deflection questions people search most, with numbers you can check in the calculator above.

How do I calculate beam deflection?

Identify the beam type and load case, then pick the matching formula, such as PL³/48EI for a centered point load on a simple span. Find E from the material grade and I from the section, using bh³/12 for a rectangle. Convert all inputs to one unit set and solve.

For a 2x10 DF-L No.2 beam spanning 10 ft with a 300 lb/ft uniform load, w = 25 lb/in, L = 120 in, E = 1,600,000 psi and I = 98.9 in⁴. The deflection is 5 × 25 × 120⁴ / (384 × 1,600,000 × 98.9), or about 0.43 in, which is L/281.

What is acceptable deflection in a beam?

Acceptable deflection depends on what the beam supports. Floors are commonly held to L/360 under live load, so a 16 ft floor beam may deflect 0.53 in. Members under brittle finishes need tighter limits, such as L/600 under unreinforced masonry and L/720 under natural stone tile.

Roofs with no ceiling can go as loose as L/180. Beyond the code value, many builders choose L/480 or tighter for floors that feel bouncy, since a beam can pass L/360 and still feel lively underfoot.

Is L/800 a deflection limit?

Yes, but for bridges rather than buildings. The older AASHTO Standard Specifications suggest L/800 for live load deflection on bridges without sidewalks and L/1000 with sidewalks, as a New Hampshire DOT notice summarizes. The current AASHTO LRFD specifications make these limits optional.

IBC Table 1604.3 has no L/800 row for buildings. You can still use it as a self-imposed target by setting the limit in your own check, since an L/800 limit on a 20 ft span allows only 0.30 in.

What is the deflection ratio?

The deflection ratio is the span divided by the actual deflection, written as L/x. A 12 ft beam that sags 0.218 in has a ratio of 144 / 0.218, or L/659. A higher number means a stiffer beam.

The beam passes a limit when its ratio number is larger than the limit's, so L/659 passes L/360 but would fail L/720. For a cantilever the tool reports 2L/x, matching the IBC rule for cantilever length.

How heavy is a 20 ft steel beam?

Multiply the nominal weight in pounds per foot by the length. The number after the x in a W-shape name is that weight, so a 20 ft W8x18 weighs 18 × 20 = 360 lb and a 20 ft W12x26 weighs 520 lb.

When you pick a W-shape and leave self-weight on, the calculator adds that weight as a uniform load. It is small next to most floor loads, yet it adds deflection on long spans.

How do you calculate I-beam size?

Work out the required Ix from the deflection formula and your limit, then choose the lightest W-shape with at least that Ix. Check its section modulus Sx for bending stress as a separate step. For the 20 ft row in the sizing table, the required Ix is 59.6 in⁴, and a W12x14 at 88.6 in⁴ is the lightest pass.

The beam deflection calculator does this search for you when steel is selected. It lists the lightest passing W-shape from its built-in set, sorted by weight, with the beam's own weight counted in the deflection.

How do you calculate slab deflection?

Concrete slabs follow ACI 318 rather than the beam formulas alone. Once a slab cracks, its stiffness drops, so engineers use an effective moment of inertia between the gross and cracked values. Long-term deflection is then multiplied by a factor of up to 2.0 for sustained loads over five years or more.

ACI 318 limits floor slabs to L/360 under live load, with tighter values where partitions could crack. The calculator here models beams with a constant I, so it suits steel, wood and LVL better than cracked concrete.