Beam Deflection Example: Simply Supported Steel Beam
A beam can be strong enough and still sag too much: floors bounce, ceilings crack and doors stick. So beams are also checked against a deflection limit, usually a fraction of the span such as L/360. This example works out the deflection of a simply supported steel beam by hand, checks it, and picks a beam that passes.
The formulas
For a simply supported beam of span L, stiffness E and moment of inertia I, the largest deflection is at midspan:
- Uniform load w along the span: Δ = 5wL⁴ / (384EI)
- Point load P at midspan: Δ = PL³ / (48EI)
Deflection is linear in the load, so when both act, add them (superposition). Use consistent units: here kips and inches, with E = 29,000 ksi for steel.
The problem
A W12x26 steel beam spans 20 ft between simple supports and carries, as live load:
- a uniform load of 1.0 kip/ft, and
- a 6 kip point load at midspan, from a post above.
The building code limit for live load deflection of a floor beam is L/360 (International Building Code Table 1604.3). Does the beam pass?
Values: L = 20 ft = 240 in, w = 1.0 kip/ft = 0.0833 kip/in, Ix = 204 in⁴ for a W12x26 (see the moment of inertia example for where that comes from).
Step by step
Step 1: deflection from the uniform load
Δw = 5 × 0.0833 × 240⁴ / (384 × 29,000 × 204) = 0.61 in
Step 2: deflection from the point load
ΔP = 6 × 240³ / (48 × 29,000 × 204) = 0.29 in
Step 3: total and limit
Δ = 0.61 + 0.29 = 0.90 in against L/360 = 240 / 360 = 0.67 in
0.90 in is more than 0.67 in, so the W12x26 fails the deflection check. It deflects L/266.
Step 4: the moment of inertia needed
Deflection is inversely proportional to I, so the beam needs
Irequired = 204 × 0.90 / 0.667 = 276 in⁴
The lightest W shape with Ix of at least 276 in⁴ is a W16x26 at 301 in⁴. It deflects 0.90 × 204 / 301 = 0.61 in, under 0.67 in, so it passes, and weighs the same 26 lb/ft as the W12x26. A deeper beam is stiffer for the same steel, because the flanges sit farther from the middle.
Deflection is only one check: the beam also has to be strong enough in bending and shear. The steel beam design tutorial goes through those.
Common deflection limits
| Member | Live load | Total load |
|---|---|---|
| Floor beams and joists | L/360 | L/240 |
| Roof members supporting a plaster ceiling | L/360 | L/240 |
| Roof members supporting a non-plaster ceiling | L/240 | L/180 |
| Roof members with no ceiling | L/180 | L/120 |
From IBC Table 1604.3. Roof rows use roof live, snow or wind load in the "Live load" column. Check the code adopted where you build.
Check it with the calculators
- The beam calculator gives deflection and slope for 17 load cases, with the equations, so you can check each step above.
- The shear and moment diagram calculator handles several loads at once and continuous beams.
- The beam designer runs the deflection, bending and shear checks together and suggests shapes that pass.
Frequently asked questions
What is the deflection formula for a simply supported beam?
5wL⁴/(384EI) for a uniform load and PL³/(48EI) for a point load at midspan. Add them when both act.
What does L/360 mean?
The beam may deflect no more than its span divided by 360. For a 20 ft (240 in) span, that's 0.67 in.
Do I check deflection with factored loads?
No. Deflection is a serviceability check, so it uses the loads as they are, without load factors.