📊 Beam formulas (moment & deflection)
Classic cases for a beam of span L, uniform load w and point load P (EI = stiffness):
| Case |
Max bending moment |
Max deflection |
| Simply supported + w | wL²/8 | 5wL⁴/384EI |
| Simply supported + P | PL/4 | PL³/48EI |
| Cantilever (fixed) + w | wL²/2 | wL⁴/8EI |
| Cantilever (fixed) + P | PL | PL³/3EI |
📐 Section & stiffness
Rectangular section: I = b·h³/12. Stiffness EI uses the modulus E (steel ≈ 200 GPa, concrete ≈ 25, aluminium ≈ 69, wood ≈ 11). A common serviceability limit for deflection is L/300 to L/500.
🌍 Naming around the world
Beam = viga = Träger / Balken · Span = luz = Spannweite · Deflection = flecha = Durchbiegung · Bending moment = momento flector = Biegemoment · Cantilever = voladizo = Kragträger · Moment of inertia = inercia = Trägheitsmoment.
❓ Frequently Asked Questions
How do you calculate the maximum bending moment of a beam?
For a simply-supported beam, a uniform load gives M = wL²/8 (at mid-span) and a central point load gives M = PL/4. For a cantilever, a uniform load gives M = wL²/2 and a tip point load M = PL, both at the fixed end. With several loads you add the moments by superposition.
How is the deflection of a beam calculated?
The deflection depends on the stiffness EI (E = modulus, I = moment of inertia). Simply-supported: δ = 5wL⁴/384EI (uniform) or PL³/48EI (central point). Cantilever: δ = wL⁴/8EI (uniform) or PL³/3EI (tip). For a rectangular section I = b·h³/12.
Can I use this to design a structural beam?
No. This is an educational tool with the classic statics formulas for a single beam. A real design checks the resistance and deflection limits of the code, load combinations, buckling and connections, and must be calculated and signed off by an engineer.