Deflection

Deflection is how far a beam moves under load. Learn the elastic curve EI·y″ = M, standard formulas for cantilevers and simply supported beams, superposition, and serviceability limits.

A beam can be strong enough and still unusable: floors that bounce, shelves that sag, machine frames that lose accuracy. Deflection checks — often L/250 to L/360 for buildings, much tighter for machines — frequently decide the size of a beam.

Deflection depends on load, span (cubed or to the fourth power!), E and I.

An analogy

A diving board: a longer board droops much more under the same diver. Span is the most sensitive variable.

What controls deflection

Deflection grows with load and dramatically with span (L³ or L⁴), and shrinks with stiffness EI. Doubling the span of a uniformly loaded beam makes it sag 16 times more.

Standard cases and superposition

Handbook formulas cover common cases. Because behaviour is linear, deflections from several loads can simply be added (superposition) — e.g. a beam's self-weight UDL plus a point load.

Serviceability limits

Typical limits: floors L/360 (live load), roofs L/250, crane girders L/600–L/1000, precision machine beams a few micrometres. Vibration (natural frequency) of floors is related: f ≈ 18/√δ(mm) Hz.

Integration, moment-area and energy methods

Double integration of EIy″ = M with boundary conditions gives the full curve; Macaulay brackets handle point loads. Moment-area and Castigliano's theorem are quicker for single values; FEM handles varying sections and indeterminate supports.

Formulas

• Elastic curve: EI · d²y/dx² = M(x)
• Simply supported, central point load: δ_max = P L³ / (48 E I)
• Simply supported, UDL: δ_max = 5 w L⁴ / (384 E I)
• Cantilever, end load: δ_max = P L³ / (3 E I)
• Cantilever, UDL: δ_max = w L⁴ / (8 E I)

Worked example: Simply supported beam

Given: L = 4 m, P = 10 kN at centre; E = 200 GPa, I = 20 × 10⁶ mm⁴

Find: Maximum deflection and check L/360.

• δ = 10 000 × 4000³/(48 × 200 000 × 20 × 10⁶) = 3.33 mm
• L/360 = 11.1 mm → OK

Answer: 3.33 mm ✓

Worked example: Cantilever

Given: L = 1.5 m, P = 2 kN at tip; Steel 50 × 100 mm (on edge)

Find: Tip deflection.

• I = 50 × 100³/12 = 4.17 × 10⁶ mm⁴
• δ = 2000 × 1500³/(3 × 200 000 × 4.17 × 10⁶) = 2.7 mm

Answer: 2.7 mm

Common mistakes

• Wrong: Mixing units in EI (GPa with mm⁴ and kN). — Right: Use N, mm, MPa throughout: E in N/mm², I in mm⁴, L in mm.
• Wrong: Checking only strength. — Right: Always check deflection too — it often governs.

Applications

• Floors & bridges: Deflection and vibration limits keep structures comfortable and prevent cracking of finishes.
• Machine tools & gantries: Micrometre-level deflection budgets drive very stiff, deep sections.
• Shafts: Shaft slope at bearings and gear mesh deflection affect bearing life and gear noise.

Key takeaways

• EI·y″ = M; stiffness EI resists deflection.
• Span dominates: L³ (point) or L⁴ (UDL).
• Superpose standard cases.
• Check serviceability limits like L/360.

Going further

• Macaulay's method: Singularity functions ⟨x − a⟩ write one M(x) expression valid along the whole beam for multiple point loads.
• Statically indeterminate beams: Compatibility (e.g. zero deflection at a prop) plus deflection formulas solves propped cantilevers and continuous beams.
• Creep deflection: Timber and concrete deflect further over time; codes apply long-term multipliers (≈ 2–3).