Beam deflection is 5 × w × L⁴ ÷ (384 × E × I) for a uniform load on a simple span: two 2x10s of Douglas fir-larch No. 2 carrying 100 plf over 12 ft sag 0.15 in, about L/977. Pick the supports and the load, a material and a section above, and the calculator gives the sag, the L/ ratio, the largest moment and shear, and the deflected shape.
How beam deflection is calculated
Deflection depends on four things: the load, the span, the stiffness of the material (its modulus of elasticity E, in psi) and the stiffness of the cross section (its moment of inertia I, in in⁴). The formulas below are from the American Wood Council's Design Aid No. 6, and they hold for any material as long as the beam stays elastic.
Simple span, uniform load: Δ = 5 × w × L⁴ ÷ (384 × E × I). Simple span, center load: Δ = P × L³ ÷ (48 × E × I). w in lb per inch, P in lb, L in inches.
The section stiffness of a solid rectangle is I = width × depth³ ÷ 12. That cube is why depth matters so much: a 2x12 (11¼ in deep) has 1.8 times the I of a 2x10 (9¼ in) for the same width.
Worked example: two 2x10s over 12 ft
Two 2x10 plies are 3 in wide and 9¼ in deep, so I = 3 × 9.25³ ÷ 12 = 197.9 in⁴. Douglas fir-larch No. 2 has E = 1,600,000 psi. A uniform load of 100 plf is 8.33 lb per inch, and the span is 144 in.
Δ = 5 × 8.33 × 144⁴ ÷ (384 × 1,600,000 × 197.9) = 0.147 in, which is L/977. That passes every common limit. The largest moment is w × L² ÷ 8 = 21,600 lb-in (1,800 lb-ft) and the largest shear 600 lb.
Beam deflection formulas for every case
For the same uniform load, the supports change the sag a lot. Compared with a simple span of the same length, load and section:
| Supports, uniform load | Maximum deflection | Compared with a simple span |
|---|---|---|
| Simple span | 5 w L⁴ ÷ 384 E I | 1.00 |
| Cantilever | w L⁴ ÷ 8 E I | 9.6 |
| Fixed at both ends | w L⁴ ÷ 384 E I | 0.20 |
| Fixed at one end, supported at the other | w L⁴ ÷ 185 E I | 0.42 |
With a point load: simple span, center P L³ ÷ 48 E I; simple span with two equal loads at the third points P a (3 L² − 4 a²) ÷ 24 E I with a = L ÷ 3; cantilever, load at the free end P L³ ÷ 3 E I; fixed at both ends, center P L³ ÷ 192 E I; fixed at one end and supported at the other, center 0.009317 P L³ ÷ E I. A 2x10 (one ply, 1.6 million psi) with a 200 lb load at the end of a 4 ft cantilever sags 0.047 in.
Modulus of elasticity of common materials
Values in psi from published design tables:
| Material | E, psi | Source |
|---|---|---|
| Structural steel | 29,000,000 | AISC 360-16 |
| 2.0E LVL | 2,000,000 | Weyerhaeuser TJ-9000 |
| Douglas fir-larch No. 2 | 1,600,000 | AWC design values |
| Southern pine No. 2 | 1,400,000 | AWC design values |
| Spruce-pine-fir No. 2 | 1,400,000 | AWC design values |
| Hem-fir No. 2 | 1,300,000 | AWC design values |
| MDF grade 130 | 313,000 | Composite Panel Association |
The calculator also carries sugar maple, red and white oak, walnut, poplar and cherry from the USDA Wood Handbook, and lets you enter your own E from a data sheet. Steel is about 18 times stiffer than Douglas fir at the same section, but no one uses the same section: use the steel beam calculator to size one.
How much sag is too much
The IRC span tables limit floor joists to L/360, ceiling joists to L/240 and rafters without a finished ceiling to L/180. The calculator checks your sag against each and against L/480, a stricter limit for brittle finishes. A 12 ft floor beam allowed L/360 may sag 0.40 in. Long-term creep in wood adds to the sag shown here, so keep some margin.
Next questions
- A wood or steel beam for a real load: the LVL beam calculator and the steel beam calculator.
- Sag of a shelf: the shelf sag calculator.
- Sawn-lumber girders and headers from the code tables: the beam span calculator and the floor joist span calculator.
- Steel section properties: the I-beam size chart.
- All span tools: framing and lumber calculators.
Deflection is one check of three; bending strength and shear also have to pass, and shear deflection is not included here. Have an engineer verify any beam that holds up a floor, roof or wall.