Beam Deflection Calculator
Compute how much a simply supported beam sags. Choose a center point load or a uniformly distributed load, enter the load in pounds, span in feet, modulus of elasticity E in psi, and moment of inertia I in in⁴, and get deflection in inches with an L/360 check.
Example: with Load case Point load at center: PL³/48EI · Load P or total W (lb) 1000 · Span (feet) 10 · Modulus of elasticity E (psi) — steel 29,000,000, lumber ≈ 1,600,000 29000000 · Moment of inertia I (in⁴) — 2x10 ≈ 98.9, W8x18 = 61.9 100 → Max deflection (at midspan): 0.0124 in.
- Span-to-deflection ratioL/9,667
- L/360 serviceability checkOK — under the L/360 floor limit of 0.333 in
Computed by the calculator below using its default values. Change any input to see your own numbers.
δ = PL³/48EI for a centered point load, 5WL³/384EI for a uniform load (W = total). Span enters cubed — doubling span means 8× the sag.
The two classic load cases
For a beam resting on supports at both ends, the midspan deflection under a centered point load is δ = PL³/48EI, and under a uniformly distributed load it is δ = 5WL³/384EI, where W is the total distributed load (w per foot × span). Everything must be in consistent units — this calculator converts your span to inches so the answer comes out in inches. The striking feature is the cubed span: stretching a beam from 10 to 12 feet increases sag 73% before you change anything else.
E describes the material's stiffness (29,000,000 psi for steel, roughly 1,400,000–1,900,000 for structural lumber, 10,000,000 for aluminum), while I describes the shape — a 2x10 on edge has I ≈ 98.9 in⁴, and a W8x18 steel beam 61.9 in⁴. Depth dominates I, which is why joists stand on edge.
How much deflection is acceptable
Building codes limit deflection for comfort and finishes, not strength: a floor can be safe yet bouncy. The standard serviceability limits are L/360 for floor live load (a 12-foot span may sag 0.4 inches) and L/240 for total load, with L/180 tolerated on some roofs without ceilings. This calculator flags your result against L/360. A beam can pass deflection and still fail in bending or shear — deflection is one check among several, so have an engineer size anything structural.
How it’s calculated
Simply supported beam formulas: δ = P·L³ ÷ (48·E·I) for a midspan point load; δ = 5·W·L³ ÷ (384·E·I) for a uniform load with W = total load (equivalent to 5wL⁴/384EI with w = W/L). Span converts to inches (×12); E in psi, I in in⁴, load in lb, deflection in inches. Serviceability compared against L/360.
Assumes an ideal simply supported, single-span, elastic beam — fixed ends, cantilevers, continuous spans, and shear deformation all change the answer; this is an educational check, not a structural design. Have a licensed engineer size real load-bearing members.
Common code deflection limits
| Member | Limit | Sag allowed on a 12 ft span |
|---|---|---|
| Floor, live load | L/360 | 0.40 in |
| Floor or ceiling, total load | L/240 | 0.60 in |
| Roof with plaster ceiling | L/360 | 0.40 in |
| Roof, no ceiling | L/180 | 0.80 in |
Typical limits per IBC Table 1604.3; allowed sag computed as 144 in ÷ limit denominator.
Common mistakes
- Leaving the span in feet inside the formula — L must be in inches, or the answer is off by 1,728×.
- Mixing up w (load per foot) and W (total load): the 5/384 formula here wants the total.
- Using point-load math for a fixed or continuous beam — fixed ends deflect only a quarter as much, so the formulas are not interchangeable.
- Forgetting the beam's own weight, which matters on long spans and heavy members.
Frequently asked questions
What is the beam deflection formula?
For a simply supported beam: δ = PL³/48EI with a centered point load, and δ = 5WL³/384EI with a uniform load (W = total). L in inches, E in psi, I in in⁴ gives δ in inches.
What are E and I?
E, the modulus of elasticity, is material stiffness — 29,000,000 psi for steel, about 1,600,000 for common structural lumber. I, the moment of inertia, is shape stiffness in in⁴ — for a rectangle, width × depth³ ÷ 12, which is why depth matters so much.
How much deflection is OK for a floor beam?
The usual code serviceability limit is L/360 under live load: span in inches divided by 360. For a 14-foot span that is 0.47 inches. Stiffer targets like L/480 are common under tile or stone finishes.
Can I use this to size a beam for my house?
Use it to understand and sanity-check, not to build from. Real design also checks bending stress, shear, bearing, and lateral bracing under code load combinations — that is a licensed structural engineer's job.