A simply-supported beam under a uniformly distributed load bends into a parabolic shape. Deflection scales with load and the fourth power of span, but inversely with the material stiffness and the beam depth cubed (moment of inertia).
δ_max = 5wL⁴ / (384·E·I)
σ_max = M·c / I, M = wL²/8
I = b·h³ / 12 (rectangular section)
- Beam span — distance between supports; deflection grows with the 4th power of span.
- Beam depth — governs moment of inertia I; small increases dramatically stiffen the beam.
- Distributed load — the uniform load per metre pressing down on the beam (e.g. floor loading).
- Material — switches Young's modulus E and allowable stress between steel, timber, and concrete.
Engineers check both stress (won't break) and deflection (won't sag visibly, typically limited to span/360) — this is exactly the serviceability check performed when sizing real floor beams.