Formula
Deflection here is elastic bending for a prismatic member. Simply supported, center load: δ = P L³ / (48 E I), M = P L / 4, V = P / 2. Simply supported, uniform load: δ = 5 w L⁴ / (384 E I), M = w L² / 8, V = w L / 2. Cantilever, end load: δ = P L³ / (3 E I), M = P L, V = P. Cantilever, uniform: δ = w L⁴ / (8 E I), M = w L² / 2, V = w L. A rectangle uses I = b d³ / 12 and S = b d² / 6. Stress is M / S and is shown as a number only.
δ = P L³ / (48 E I)
I = b d³ / 12
Units in the arithmetic are inches, pounds, and psi, so a foot of span is 12 inches and a pound per foot is divided by 12. L/180, L/240, L/360, and L/480 are serviceability comparisons people use. They are not strength limits. This tool does not cover species and grade allowable stress, bracing, load combinations, shear deflection, composite action, bearing, or connections. Published span tables and an engineer exist for that.
Worked examples
10 ft simple span, 1,000 lb at center, 1.5 × 7.25 in, E 1.2×10⁶ psi. I is 47.63 in⁴. Deflection is about 0.630 in, near L/190. Moment 2,500 ft-lb. Shear 500 lb. Stress about 2,283 psi. L/360 is 0.333 in, and the computed deflection is about 0.297 in past that comparison.
Same beam, 100 lb/ft. Deflection about 0.394 in. Moment 1,250 ft-lb. Cantilever 6 ft, 200 lb. About 0.435 in. I = 100 in⁴, E = 29×10⁶ psi, 12 ft, 2,000 lb. About 0.0429 in.
Reference
Example dressed sizes are dry sizes from PS 20. They are not a recommendation.
| Nominal | Dressed b × d |
|---|---|
| 2x8 | 1.5 × 7.25 in |
| 2x10 | 1.5 × 9.25 in |
| 2x12 | 1.5 × 11.25 in |
| 4x8 | 3.5 × 7.25 in |
| 4x10 | 3.5 × 9.25 in |
| 4x12 | 3.5 × 11.25 in |
Questions
How do I calculate beam deflection?
For a simply supported beam with a load at midspan, deflection is P L³ divided by 48 E I. The other three cases on this page have their own formulas. The result is a number of inches for the inputs you typed.
What is the beam deflection formula for a simply supported beam?
Center load: P L³ / (48 E I). Uniform load: 5 w L⁴ / (384 E I). Moment is P L / 4 or w L² / 8. This page keeps E in psi, I in in⁴, and L in inches.
What is L/360?
It is a serviceability comparison: the span divided by 360. The page shows how the computed deflection sits next to that length. It is not a code check and not a pass or fail.
How do I calculate the moment of inertia of a rectangular beam?
For a rectangle, I is b d³ / 12, and the section modulus is b d² / 6. Use the dressed size, not the nominal name.
Can I use this to size a header or a floor beam?
No. This page does not recommend a size. Span tables and allowable stresses live in the IRC, AWC, and AISC documents, and a licensed structural engineer applies them.
How does wood E compare with steel?
You type E. The worked example uses 1,200,000 psi as an example number, not a grade value. A textbook mild-steel modulus is about 29,000,000 psi. Verify the number for the species, grade, or alloy you have.
What loads should I use?
This page does not select design loads. Ask an engineer or use the loads the code prescribes.
Sources
- The four deflection, moment, and shear formulas are the standard elastic cases: simply supported center load, simply supported uniform load, cantilever end load, and cantilever uniform load. E, I, and L are kept in psi, in⁴, and inches.
- NIST Voluntary Product Standard PS 20, dressed dry sizes. A nominal 2×8 is 1.5 × 7.25 in, a 2×10 is 1.5 × 9.25, a 2×12 is 1.5 × 11.25, and a nominal 4-inch thickness is 3.5 in with the same width series. These are example sections, not a span table.
- 1 in⁴ = 416,231 mm⁴. 1 psi = 6.894757 kPa. 1 lb = 4.448222 N. IRC, AWC, and AISC are where a designer looks. None of their span tables are copied here.