Beam Deflection Calculator
A beam deflection calculator uses 5wL⁴ ÷ 384EI for a simply supported beam: 200 lb/ft over 12 ft with E = 1.6M psi and I = 100 in⁴ sags 0.583 in.
How much will my beam sag?
This beam deflection calculator finds the mid-span sag of a simply supported beam carrying an even load, using the standard formula deflection = 5 × w × L⁴ ÷ (384 × E × I). With 200 lb per foot on a 12 ft span, E of 1,600,000 psi, and I of 100 in⁴, the beam sags 0.583 inches. The calculator also shows the L/360 limit for that span, which here is 0.400 inches, so this beam is too flexible.
Enter the uniform load in pounds per foot, the span in feet, the modulus of elasticity E in psi, and the moment of inertia I in in⁴.
The formula in plain words
The calculator converts everything to inches and pounds first:
- w = load per foot ÷ 12, giving pounds per inch.
- L = span in feet × 12, giving inches.
- Deflection = 5 × w × L⁴ ÷ (384 × E × I).
- Limit = L ÷ 360.
Two things in that formula matter most. Span is raised to the fourth power, so a beam that spans twice as far sags 16 times as much. E and I sit together underneath the line, so doubling either one halves the sag.
Worked examples
- Default inputs: 200 lb/ft, 12 ft span, E = 1,600,000 psi, I = 100 in⁴
- w = 200 ÷ 12 = 16.67 lb/in
- L = 12 × 12 = 144 in
- Deflection = 5 × 16.67 × 144⁴ ÷ (384 × 1,600,000 × 100) = 0.583 in
- L/360 limit = 144 ÷ 360 = 0.400 in, so the beam fails the check
- Same load and span, double 2x12 (I = 2 × 178.0 = 356.0 in⁴)
- Deflection = 0.583 × 100 ÷ 356.0 = 0.164 in
- Limit is still 0.400 in, so the deeper beam passes with room to spare
Going from I = 100 to I = 356 cut the sag by more than two-thirds without changing the load, span, or wood.
Moment of inertia for common lumber
Calculated as width × depth³ ÷ 12, using actual dressed sizes, for a single ply bending about its strong axis.
| Nominal size | Actual size (in) | I (in⁴) | Two plies | Three plies |
|---|---|---|---|---|
| 2x6 | 1.5 × 5.5 | 20.8 | 41.6 | 62.4 |
| 2x8 | 1.5 × 7.25 | 47.6 | 95.3 | 142.9 |
| 2x10 | 1.5 × 9.25 | 98.9 | 197.9 | 296.8 |
| 2x12 | 1.5 × 11.25 | 178.0 | 356.0 | 533.9 |
For steel shapes, read I (often listed as Ix) from the manufacturer’s or AISC shape tables. For other shapes, use the moment of inertia calculator.
Typical E values
| Material | Typical E (psi) |
|---|---|
| Spruce-pine-fir No. 2 | about 1,400,000 |
| Douglas fir-larch No. 2 | about 1,600,000 |
| LVL beam | about 1,900,000–2,000,000 |
| Structural steel | 29,000,000 |
Wood values vary by species, grade, and size. Use the published design value for your exact lumber.
Deflection limits you will see
| Member | Common limit | Limit on a 12 ft span |
|---|---|---|
| Floors, live load | L/360 | 0.40 in |
| Ceilings with flexible finishes | L/240 | 0.60 in |
| Rafters with no ceiling attached | L/180 | 0.80 in |
These follow common US residential practice. Check local code for the limit that applies to your member and load case.
How to gather the inputs
- Span: measure the clear distance between supports in feet.
- Load per foot: multiply the load in psf by the width of floor the beam carries. A beam holding up 5 ft of floor on each side at 40 psf carries 10 × 40 = 400 lb/ft. The beam load calculator does this step.
- E and I: from the tables above or the product’s spec sheet.
Common mistakes
- Mixing units. Span goes in feet and load in lb per foot; the calculator handles the inch conversion.
- Using nominal sizes for I. A 2x10 is 9.25 in deep, not 10.
- Checking total load against a live-load limit. L/360 usually applies to live load only.
- Treating a cantilever as simply supported. An overhang bends very differently. Use the cantilever beam deflection calculator.
This is a planning estimate for stiffness only. Beam size, connections, and bearing are structural decisions. Have a licensed structural engineer or your building department confirm the design.
Frequently asked questions
What is beam deflection?
Deflection is how far a beam sags under load, measured at the point of greatest movement. For a beam resting on two supports with an even load, that point is mid-span. A beam can be strong enough not to break and still sag enough to cause bouncy floors, cracked drywall, or sticking doors.
What is the L/360 deflection limit?
It means the beam may sag no more than its span divided by 360. A 12 ft span is 144 inches, so the limit is 144 ÷ 360 = 0.40 in. L/360 is the usual limit for floor live load in US residential codes; other members, such as rafters, often use L/240 or L/180. Check local code for your member.
What are E and I in the deflection formula?
E is the modulus of elasticity, the material's stiffness in psi: about 1,600,000 psi for Douglas fir-larch No. 2 and 29,000,000 psi for structural steel. I is the moment of inertia of the cross-section in in⁴; a rectangular board is width × depth³ ÷ 12, so a 2x10 (1.5 × 9.25 in) has I ≈ 98.9 in⁴.
How do I reduce excessive beam deflection?
Make the beam deeper, use a stiffer material, or shorten the span. Deflection falls in direct proportion to I, and I grows with depth cubed, so going from a 2x10 to a 2x12 raises I from 98.9 to 178 in⁴ and cuts deflection by about 44%. Halving the span cuts deflection to one-sixteenth.
Does this calculator check beam strength?
No. It checks stiffness only. A beam also needs a bending-stress and shear check, and the supports need enough bearing. Use the beam load capacity tools for a first look, and have a licensed engineer or your building department confirm the final size.
Estimates only. Verify quantities with your supplier before purchasing.