Beam Span Calculator

Beam span calculator: solving the deflection formula for length

This beam span calculator answers one question: how long can a simply supported beam be before its sag reaches your deflection limit? It takes the standard uniform-load deflection formula, Δ = 5wL⁴ ÷ (384EI), sets Δ equal to L ÷ ratio, and solves for L. The result is L = ∛(384 × E × I ÷ (5 × ratio × w)), with w in pounds per inch and L in inches, then shown in feet.

What each input means

• E, modulus of elasticity (psi). How stiff the material is. Douglas fir-larch No. 2 lumber is listed at about 1,600,000 psi; structural steel is about 29,000,000 psi.
• I, moment of inertia (in⁴). How the cross-section resists bending. A deeper beam has a far larger I, because I grows with depth cubed.
• Uniform load (lb per ft). The load spread along the beam. For a joist, that is floor load in psf times joist spacing in feet.
• Deflection ratio. The limit as span ÷ number, such as 360 for floors.

The calculator divides the load by 12 to get pounds per inch, runs the cube root, then divides by 12 again to return feet.

Worked examples

1. The default beam

E = 1,600,000 psi, I = 300 in⁴, w = 40 lb/ft (3.33 lb/in), L/360.

• 384 × 1,600,000 × 300 = 184,320,000,000
• 5 × 360 × 3.333 = 6,000
• 184,320,000,000 ÷ 6,000 = 30,720,000; cube root = 313.2 in
• Span: 313.2 ÷ 12 = 26.10 ft

2. A 2x10 floor joist at 16 in on center

A 2x10 (1.5 × 9.25 in actual) has I ≈ 98.9 in⁴. A 40 psf live load on joists 16 in (1.333 ft) apart is 40 × 1.333 = 53.3 lb/ft. With E = 1,600,000 and L/360, the calculator returns 16.38 ft of deflection-limited span. Published span tables for this joist also weigh bending strength and dead load, so always check the table too.

Span sensitivity reference

Using the default E = 1,600,000 psi, I = 300 in⁴:

Load (lb/ft)	Ratio	Max span (ft)
40	L/360	26.10
40	L/240	29.88
80	L/360	20.71

Typical inputs:

Material or section	E (psi)	I (in⁴, strong axis)
2x8 sawn lumber	~1,600,000 (DF-L No. 2)	47.6
2x10 sawn lumber	~1,600,000 (DF-L No. 2)	98.9
2x12 sawn lumber	~1,600,000 (DF-L No. 2)	178
LVL	often 1,900,000–2,000,000 (check maker)	from maker’s tables
W8x10 steel	~29,000,000	30.8

Doubling depth raises I about eight times, which is why a deeper member gains span far faster than a wider one.

How to gather the inputs

1. Find E for your species and grade, or from the LVL or steel manufacturer.
2. Work out I with the moment of inertia calculator, using actual (not nominal) lumber dimensions.
3. Build the line load: area load (psf) × tributary width (ft). Add any wall or point loads separately; this tool assumes a uniform load only.
4. Pick the ratio your local code sets for the member.

Reading the result against real loads

Live load and dead load are usually checked separately. Floors commonly use a 40 psf live load in living areas, but the dead load of joists, subfloor, and finishes adds more weight that does not move. The usual practice is to check L/360 against live load alone, and a looser limit such as L/240 against total load. Run the calculator twice, once for each case, and use the shorter span.

A uniform-load formula assumes the beam rests on two simple supports. Continuous beams over three or more supports, cantilevers, and beams with fixed ends deflect differently and need their own formulas.

Common mistakes

• Using nominal sizes. A 2x10 is 9.25 in deep, not 10. Using 10 in overstates I by about 26%.
• Entering psf instead of lb/ft. Multiply by the spacing or tributary width first.
• Stopping at deflection. Run the beam load capacity calculator for bending and use the shorter span.
• Ignoring point loads. A post or girder bearing mid-span changes the formula entirely.
• Ignoring creep. Wood sags further under long-term dead load, which this simple check does not model.

To see how much a known span sags, use the beam deflection calculator.

This is a planning estimate using standard mechanics formulas, not engineering design. Have member sizes and code compliance confirmed by a licensed structural engineer.