📏 Beam Deflection Calculator

The moment of inertia (I) — also called the second moment of area — measures a cross-section's resistance to bending. A larger I means less deflection for the same load. It depends on both the area and how that area is distributed relative to the neutral axis. Deep sections (like I-beams) have much higher I values than equivalent solid rectangular sections, which is why I-beams are structurally efficient. I is measured in m⁴ or cm⁴.

Excessive deflection causes: (1) cracking of brittle finishes like plaster or tiles; (2) doors and windows jamming; (3) ponding of water on flat roofs; (4) visual distress to occupants; (5) secondary structural effects (P-delta effects). Building codes set deflection limits to prevent these issues. Common limits: L/250 for general floors, L/360 when supporting brittle finishes, L/500 for facades, L/1000 for sensitive equipment floors.

Elastic modulus (E) measures a material's stiffness — its resistance to elastic deformation under stress. A higher E means the material is stiffer and deflects less under the same load. Steel (200 GPa) is about 3× stiffer than aluminum (69 GPa) and 6.5× stiffer than concrete (30 GPa). E is used in beam deflection formulas as EI (flexural rigidity), so increasing either E or I reduces deflection proportionally.

A simply supported beam rests on two supports (pinned and roller) and is free to rotate at both ends. The maximum deflection is at midspan. A cantilever beam is fixed at one end and free at the other. It deflects more than an equivalent simply supported beam — a cantilever with a point load deflects 16× more than a simply supported beam (PL³/3EI vs PL³/48EI). Cantilevers develop large hogging moments at the fixed support.

To reduce deflection: (1) Increase depth of section — deflection reduces with the cube of depth increase; (2) Use a stiffer material (higher E); (3) Reduce span — deflection increases with L³ or L⁴, so halving span reduces deflection by 8–16×; (4) Add intermediate supports to create a continuous beam; (5) Pre-camber the beam so it deflects to level under service load; (6) Increase moment of inertia by choosing a deeper or wider section.