Beam Deflection.
Maximum deflection, slope, bending stress, and end reactions for the four most common loading cases. Enter your beam length, load, material, and cross-section.
Beam case
Point load at free end
UDL (w · L)
Point load at centre
UDL (w · L)
Beam Deflection and Stress
Beam deflection analysis predicts how much a loaded beam will sag and what stress it will experience. Get this wrong and structures fail (or pass code with no margin).
Core Formulas
Cantilever, end load: δ = PL³/(3EI)
Simply supported, center load: δ = PL³/(48EI)
Bending stress: σ = Mc/I
Where δ = deflection, P = load, L = span, E = elastic modulus, I = moment of inertia, M = bending moment, c = distance to extreme fiber.
Acceptable Deflection Limits
- Floor beams: L/360 (live load), L/240 (total load)
- Roof beams: L/240
- Cantilevers: L/180
- Machinery frames: L/500 to L/1000 (vibration concerns)
- Precision instruments: L/2000 or tighter
Worked Example
Steel I-beam (W200×27), 6m simply-supported span, 5 kN center load.
- I (W200×27) ≈ 25.8 × 10⁶ mm⁴, E_steel = 200 GPa
- δ = (5000 × 6000³) / (48 × 200000 × 25.8e6) = 4.4 mm
- L/δ = 6000/4.4 = 1364 — exceeds L/360 by wide margin (good)
- M_max = PL/4 = 7.5 kN·m, σ_max = 7.5e6 × 100 / 25.8e6 = 29 MPa
- Steel yield 250 MPa → SF = 8.6 (very safe)
How to Increase Stiffness
- Increase depth — stiffness goes with depth³. Doubling depth = 8× stiffer. Cheap.
- I-beam shape — 5-10× stiffer than rectangular bar of same area.
- Reduce span — δ goes with L³. Halving span = 8× less deflection.
- Add intermediate supports — converts simply-supported to continuous (much stiffer).
Related Tools
For pressed components, see Press-Fit Calculator. For bolted connections, Bolt Torque. For dimensional verification, Tolerance Stack-Up.