Beam Deflection Fundamentals
Deflection is how much a beam bends under load. Keeping deflection within limits (usually L/360 for floors) prevents cracking, discomfort, and structural failure.
Simply Supported Beam — Point Load at Centre
δ_max = PL³ / (48EI)
P = applied load (N)
L = span (m)
E = Young's modulus (Pa)
I = second moment of area (m⁴)Simply Supported — Uniform Distributed Load (UDL)
δ_max = 5wL⁴ / (384EI)
w = load per unit length (N/m)Cantilever — Point Load at Free End
δ_max = PL³ / (3EI)
M_max = P × L (at fixed support)Second Moment of Area (Rectangular Section)
I = bh³ / 12
b = width, h = height (same units)
For 200×300mm beam: I = 0.2×0.3³/12 = 4.5×10⁻⁴ m⁴Typical E Values
- Structural steel: 200 GPa
- Aluminium: 69 GPa
- Concrete: 25-35 GPa
- Timber (Douglas fir): ~13 GPa
Calculate beam deflection: Free Beam Deflection Calculator
Common Deflection Formulas
- Simply supported beam, central point load P: δ_max = PL³/(48EI)
- Simply supported beam, UDL w: δ_max = 5wL⁴/(384EI)
- Cantilever, tip point load P: δ_max = PL³/(3EI)
- Cantilever, UDL w: δ_max = wL⁴/(8EI)
E = Young's modulus (Pa), I = second moment of area (m⁴), L = span, P = load, w = load per unit length.
Second Moment of Area
I quantifies a cross-section's resistance to bending. For a rectangle of width b and depth d: I = bd³/12. For a circle of diameter d: I = πd⁴/64. I-sections (steel UB/UC profiles) concentrate material at the flanges — furthest from the neutral axis — maximising I for a given weight. Doubling the depth of a rectangular beam increases I eightfold (d³ term) and halves deflection by a factor of 8, which is why deeper beams are far more efficient than wider ones for bending resistance.
Frequently Asked Questions
What is the permitted deflection limit for floors?
Most building codes limit floor deflection to L/360 under imposed load (L = span length), to prevent cracking in plaster or brittle finishes. Pre-cambering (building in an upward bow) is used in longer spans so deflection under load results in a near-level floor. For roof beams, L/200 or L/250 are common limits where cracking of finishes is not an issue.
What increases beam stiffness most effectively?
Increasing depth (d) is by far the most effective: I ∝ d³, so doubling depth reduces deflection by a factor of 8. Increasing width (b) is less efficient: I ∝ b, so doubling width only halves deflection. Reducing span (L) is very powerful: δ ∝ L³ or L⁴, so halving the span reduces deflection by a factor of 8 or 16. Using stiffer material (higher E) reduces deflection proportionally.
What is the difference between deflection and stress in a beam?
Deflection (serviceability) and stress (strength) are two separate design checks. A beam can be strong enough not to yield but still deflect excessively. Conversely, a very stiff deep beam may be over-stressed at its flanges if not sized correctly for strength. Both checks must be satisfied in structural design; neither alone is sufficient.