What deflection means
Every beam bends under load. Deflection is how far it sags at its weakest point, the middle of the span. A beam can be strong enough not to break and still sag too much: floors that bounce, cracked drywall ceilings and doors that stick are deflection problems. Codes therefore limit deflection to a fraction of the span, such as L/360 for floors. This calculator estimates the deflection of a simply supported rectangular beam under an evenly spread load, compares it with the limit you choose, and gives the bending stress.
The formulas
For a beam resting on two supports with a uniform load w along its span L:
- Maximum deflection = 5 × w × L⁴ ÷ (384 × E × I)
- Moment of inertia of a rectangle: I = width × depth³ ÷ 12
- Maximum bending moment = w × L² ÷ 8
- Maximum bending stress = moment ÷ (width × depth² ÷ 6)
E is the modulus of elasticity: about 1.6 million psi for Douglas fir-larch No. 2, 1.4 million for spruce-pine-fir, 2.0 million for LVL and 29 million for steel.
Worked example: a floor joist
A 2 × 10 Douglas fir joist (actual size 1.5 × 9.25 in) spans 12 ft and carries 70 lb per foot, roughly 40 psf of live load and 10 psf of dead load at 16 in on center.
- I = 1.5 × 9.25³ ÷ 12 = 98.9 in⁴.
- Deflection = 5 × 70 × 12⁴ ÷ (384 × E × I), with units converted: 0.21 in, or L/696.
- The L/360 limit for 12 ft is 144 ÷ 360 = 0.40 in, so the joist passes comfortably.
- Bending moment: 70 × 12² ÷ 8 = 1,260 lb·ft, giving a stress of about 707 psi, below the roughly 900 psi reference design value for No. 2 Douglas fir before code adjustments.
Why span and depth dominate
Deflection grows with the fourth power of the span. For the same joist and load:
| Span | Deflection | Ratio |
|---|---|---|
| 8 ft | 0.04 in | L/2,349 |
| 10 ft | 0.10 in | L/1,203 |
| 12 ft | 0.21 in | L/696 |
| 14 ft | 0.38 in | L/438 |
| 16 ft | 0.65 in | L/294 |
At 16 ft the joist fails L/360. Going to a 2 × 12 would fix it, because stiffness grows with the cube of the depth.
Limits of this calculator
This is a single-span, uniform-load check of a solid rectangular section. It ignores the beam’s own weight, point loads, cantilevers, shear, bearing length, notches, creep and the load duration and size factors that the NDS applies to wood. Use it to compare options and understand trade-offs; for structural changes, follow the span tables in the International Residential Code or have the design checked by an engineer, and get a permit.