Every beam bends under load. How much it bends — its deflection — determines whether a floor feels bouncy, whether a shelf sags visibly, or whether drywall below it cracks. This calculator applies the Euler–Bernoulli beam equations to the four most common cases: simply supported or cantilever beams carrying a single point load or a uniform load along their length. It returns the maximum deflection and where it occurs, the bending moment, shear and reactions, the bending stress when the section is known, and a span-to-deflection ratio you can compare with building-code limits.
How to use the beam deflection calculator
- Choose Simply supported (resting on a support at each end) or Cantilever (fixed at the left end, free at the right).
- Choose a uniform load w (per foot or meter) or a point load P. For a point load, give its position from the left end, or leave it blank for midspan (simple beams) or the free end (cantilevers).
- Enter the span and the load with its units.
- Pick a material to set the elastic modulus E, or choose Custom and enter it.
- Describe the cross-section: a rectangle (b × h), a solid round bar, a round tube, or a moment of inertia from a table. Add the section depth to get bending stress for tabulated shapes.
Beam deflection formulas
| Case | Maximum deflection | Maximum moment |
|---|---|---|
| Simple, uniform load | 5wL4 ÷ 384EI | wL2 ÷ 8 |
| Simple, center point load | PL3 ÷ 48EI | PL ÷ 4 |
| Cantilever, uniform load | wL4 ÷ 8EI | wL2 ÷ 2 |
| Cantilever, end point load | PL3 ÷ 3EI | PL |
For an off-center point load on a simple beam, the calculator uses the general form δ = Pb(L2 − b2)3/2 ÷ (9√3 EIL), where b is the shorter distance from the load to a support. The product EI, called flexural rigidity, captures everything about the beam’s stiffness.
Worked examples
Douglas fir-larch No. 2 2×10, 12 ft span, 60 lb/ft uniform load (the default)
I = 1.5 × 9.25³ ÷ 12 = 98.93 in⁴, and E = 1,600,000 psi.
δ = 5 × 5 lb/in × 144⁴ ÷ (384 × 1,600,000 × 98.93) = 0.177 in, or L/814 — well inside L/360 (0.4 in).
Moment = 60 × 12² ÷ 8 = 1,080 lb·ft, and bending stress = 12,960 lb·in × 4.625 in ÷ 98.93 in⁴ = 606 psi.
A steel beam with a center load. A W8×18 (I = 61.9 in⁴, depth 8.14 in) spanning 20 ft with a 10 kip point load at midspan deflects 10,000 × 240³ ÷ (48 × 29,000,000 × 61.9) = 1.60 in, or L/150. The bending stress is 39,450 psi, below the 50 ksi yield of A992 steel, but the deflection is far too large for a floor — a deeper section is needed.
Deflection limits
Building codes limit deflection to keep floors comfortable and finishes intact. Common limits from IBC Table 1604.3 are L/360 for floor members under live load, L/240 under total load, and L/180 for some roof members. For cantilevers, the code measures L as twice the cantilever length, which the calculator applies. Manufacturers of engineered lumber and steel decking often recommend stiffer limits, such as L/480, for tile floors.
Assumptions and limits
These are linear-elastic Euler–Bernoulli results. They assume small deflections, a prismatic beam, loads applied in the plane of bending and no lateral-torsional buckling. The beam’s own weight is not included — add it to w if it matters. Shear deformation, which becomes significant in short, deep beams, is ignored. Wood also creeps under long-term load, so sustained deflection can grow by half or more over time.
For the stress and stretch of a bar in tension or compression, use the stress, strain and Young’s modulus calculator. Lumber quantities for a framing project come from the board foot calculator, and spring-like stiffness is covered by Hooke’s law.
Results are for estimation and learning. Structural members must be designed or checked by a qualified engineer under the applicable building code, including load combinations, connections and stability.
Frequently asked questions
What is the deflection formula for a simply supported beam with a uniform load?
δmax = 5wL⁴ ÷ (384EI) at midspan, where w is the load per unit length, L the span, E the elastic modulus and I the moment of inertia. A 12 ft Douglas fir 2×10 carrying 60 lb/ft deflects about 0.18 in.
What is the deflection of a cantilever with a load at the end?
δmax = PL³ ÷ (3EI) at the free end. For a uniform load along a cantilever it is wL⁴ ÷ (8EI). Cantilevers deflect far more than simply supported beams of the same span: an end load produces 16 times the deflection of the same load at midspan of a simple beam.
What does L/360 mean?
It is a deflection limit expressed as a fraction of the span: a 12 ft (144 in) floor joist limited to L/360 may deflect no more than 0.4 in under live load. The International Building Code (Table 1604.3) uses L/360 for floor live load and L/240 for total load on many members.
How do I find the moment of inertia of a beam?
For a solid rectangle, I = bh³ ÷ 12, using the actual dimensions — a 2×10 is 1.5 × 9.25 in, giving 98.9 in⁴. For a solid round bar, I = πd⁴ ÷ 64. For steel shapes such as W8×18, take I from the manufacturer's or AISC tables and use the strong-axis value.
Why does a deeper beam deflect so much less?
Moment of inertia grows with the cube of depth, and deflection is inversely proportional to it. Doubling a rectangular beam's depth makes it eight times stiffer, while doubling its width only doubles the stiffness.