Formulas
| Case | Max deflection | Max moment | Max shear |
|---|---|---|---|
| Simple span, center point load | PL³ / 48EI | PL / 4 | P / 2 |
| Simple span, uniform load | 5wL⁴ / 384EI | wL² / 8 | wL / 2 |
| Cantilever, end point load | PL³ / 3EI | PL | P |
| Cantilever, uniform load | wL⁴ / 8EI | wL² / 2 | wL |
Units are converted internally: span to inches, E to psi and uniform load to lb/in, so deflection comes out in inches.
Deflection limits
Building codes limit deflection as a fraction of the span. Common limits from IBC Table 1604.3:
- L/360: floors under live load, and members supporting plaster.
- L/240: floors under total load, and roofs with a non-plaster ceiling.
- L/180: roofs without a ceiling.
Equipment supports, crane runways and conveyor frames usually have tighter limits from the equipment maker, often L/600 to L/1000.
Worked example
A W8×18 steel beam (I = 61.9 in⁴) spanning 12 ft with a 1,000 lb load at midspan: δ = 1,000 × 144³ ÷ (48 × 29,000,000 × 61.9) = 0.035 in, about L/4,150. The moment is 3,000 ft·lbf.
Frequently asked questions
- Where do I find the moment of inertia?
- For steel shapes, the AISC Steel Construction Manual or the manufacturer's tables. For a solid rectangle, I = b·h³ ÷ 12 with h as the depth in the direction of bending. A 2×10 lumber joist (1.5 × 9.25 in) is 98.9 in⁴.
- Does this include the beam's own weight?
- No. Add self-weight to the uniform load. A W8×18 weighs 18 lb/ft.
- Is deflection the only check?
- No. You also need to check bending stress, shear, lateral-torsional buckling, bearing and connections. Deflection often governs long, lightly loaded spans.
Results are estimates for planning and education. Verify against the current code edition your jurisdiction has adopted, manufacturer data and a licensed professional before you build or install.