Beam Deflection Formulas
Beam deflection is a measure of the displacement of a structural element under a load. The calculation varies depending on the beam's supports and the type of load applied.
Frequently Asked Questions
The maximum deflection occurs at the center and is calculated as: delta = (P * L^3) / (48 * E * I).
Beam deflection is the vertical displacement of a structural beam from its unloaded position under applied loads. Excessive deflection causes serviceability issues (cracking in finishes, door binding) even if the beam is structurally adequate. Most codes limit deflection to L/360 for live loads and L/240 for total loads, where L is the span length.
Beam deflection depends on: applied load type and magnitude; span length (deflection increases with the cube of span — doubling span increases deflection 8×); moment of inertia (I) of the cross-section (larger I = less deflection); modulus of elasticity (E) of the material (steel deflects less than timber); and support conditions (cantilever deflects 4× more than simply supported for the same loading).
For a cantilever beam with a point load P at the free end: maximum deflection δ = PL³ / (3EI), occurring at the free end. For a uniformly distributed load w (N/m): δ = wL⁴ / (8EI). Cantilever beams deflect 4 to 5 times more than simply supported beams of the same span and loading — this is a critical design consideration.
Modulus of elasticity (Young's Modulus): Structural Steel — 200 GPa; Aluminium alloy — 69 GPa; Concrete — 20–30 GPa (varies with grade); Timber along grain — 8–15 GPa (species dependent); Carbon fibre composite — 70–300 GPa. Steel is 3–10× stiffer than timber, which is why steel beams deflect far less for the same cross-section.
The second moment of area (I), or moment of inertia, measures a cross-section's resistance to bending. For a rectangular section: I = b×h³/12, where b is width and h is depth. Doubling the depth increases I by 8×, reducing deflection 8×. This is why I-beams (wide flange sections) are efficient — most material is placed far from the neutral axis where it contributes most to stiffness.