Beam Deflection Calculator
Calculate the maximum deflection of a simply supported beam under a point load, uniformly distributed load (UDL), or end load using standard Euler-Bernoulli beam theory. Use it to size beams in structural design or to verify deflection stays within serviceability limits (typically L/250 or L/360).
Last updated: May 2026
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About this calculator
Beam deflection is the displacement of a beam under load, governed by Euler-Bernoulli beam theory. Three load cases are supported: (1) Simply-supported with a central point load: δ = P·L³ / (48·E·I); (2) Simply-supported with uniformly distributed load (UDL): δ = 5·w·L⁴ / (384·E·I); (3) End-loaded option (third formula, included for special cases). Variables: P is the point load in newtons (N); w is distributed load in N/m; L is span length in meters (m); E is Young's modulus / elastic modulus in pascals (Pa) — about 200 GPa for steel, 25–30 GPa for concrete, 10–14 GPa for timber; I is the area moment of inertia about the neutral axis in m⁴ — depends on cross-section shape (for a rectangular b×h beam, I = b·h³/12). Edge cases: deflection scales with L⁴ for UDL and L³ for point load, so doubling span increases deflection 16× or 8× — span is by far the most influential variable. The Euler-Bernoulli theory assumes small deflections (linear elastic behaviour), uniform cross-section, isotropic homogeneous material, and no shear deformation. For deep beams (L/h < 5) shear deformation becomes significant and Timoshenko beam theory is required. Serviceability limits in design codes: residential floor live load deflection ≤ L/360, roof ≤ L/240, cantilever ≤ L/180. Always check both strength and serviceability.
How to use
Example 1 — Steel beam, point load. Simply supported steel I-beam with span L = 4 m, central point load P = 10,000 N (10 kN), E = 200 GPa (200×10⁹ Pa), I = 1×10⁻⁵ m⁴ (typical W6×15 wide-flange). Step 1: numerator = 10,000 × 4³ = 640,000. Step 2: denominator = 48 × 200×10⁹ × 1×10⁻⁵ = 96,000,000. Step 3: δ = 640,000 / 96,000,000 = 0.00667 m = 6.67 mm. Verify ✓. Serviceability check: L/360 = 4,000/360 = 11.1 mm — the 6.67 mm deflection passes by a comfortable margin. Example 2 — UDL on a timber joist. Wood beam with span L = 3 m, UDL w = 2,000 N/m (typical floor live load on 600mm spacing), E = 12 GPa, I = 4×10⁻⁶ m⁴ (a 50×200mm joist). Step 1: numerator = 5 × 2,000 × 3⁴ = 810,000. Step 2: denominator = 384 × 12×10⁹ × 4×10⁻⁶ = 18,432,000. Step 3: δ = 810,000 / 18,432,000 ≈ 0.0440 m = 44 mm. Verify ✓. Serviceability check: L/360 = 8.3 mm — this beam fails badly. Either upsize the joist (taller depth has huge effect since I scales with h³), reduce span, or add a beam below to halve the span.
Frequently asked questions
How is deflection different from bending stress?
Deflection (δ) is the displacement of the beam under load — measured in millimetres or inches. Bending stress (σ) is the internal force per unit area at the most-stressed fibre of the beam — measured in MPa or psi. They are related but distinct: σ = M·c/I where M is bending moment, c is the distance from neutral axis to extreme fibre, and I is the moment of inertia. A beam can be strong enough (bending stress below yield) but still deflect unacceptably (serviceability failure causing bouncy floors, cracked drywall, ponding on roofs). Conversely, a stiff but undersized beam might pass deflection but fail strength. Structural design always checks both — strength (governed by code yield-stress limits with safety factors) AND serviceability (governed by deflection limits like L/360). The two checks rarely give the same answer; you size for the more restrictive of the two.
What does L/360, L/240, and L/180 mean as deflection limits?
These are span-to-deflection ratios used by building codes (IBC in the US, Eurocode in Europe) for serviceability. L/360 means the deflection should not exceed 1/360 of the span length: a 6m beam would have a 16.7mm limit. L/240 allows twice the deflection (more permissive); L/180 allows 4× more (most permissive). Typical applications: L/360 for floors carrying brittle finishes (plaster, ceramic tile, drywall — which crack at excessive deflection); L/240 for ceilings and roofs; L/180 for cantilevers and short-term loads. These limits prevent visible sag, bouncy floors, jamming doors, cracked finishes, and ponding water on flat roofs (which itself causes additional load creating a dangerous feedback loop). Modern codes apply different limits to long-term (creep) vs short-term loads — creep deflection is typically limited to L/240 for ceilings to prevent cumulative damage over decades.
What are the most common mistakes in beam deflection calculations?
The biggest is mixing unit systems — combining E in psi with L in mm and getting nonsense. Always convert everything to one consistent system (SI: Pa, m, m⁴, N) before computing. The second is using the gross moment of inertia for reinforced concrete; in cracked concrete the effective moment of inertia (Ieff) can be 30–60% of the gross value, and the actual deflection is correspondingly larger. The third is forgetting end conditions — a fixed-end (clamped) beam deflects only 1/4 as much as a simply supported one at the same load; a cantilever deflects 16× more at the free end than a simply supported beam at midspan for the same length and load. Always use the formula that matches your actual support conditions. The fourth is ignoring creep in timber and concrete; long-term deflection can be 2–3× the initial elastic deflection. The fifth is using nominal cross-section properties instead of effective ones after accounting for openings (drilled holes, notches, services).
When should I NOT use simple beam deflection formulas?
Skip these formulas for short deep beams (L/h < 5) where Euler-Bernoulli theory breaks down and shear deformation contributes meaningfully to deflection — use Timoshenko beam theory or finite element analysis instead. Avoid them for beams with axial loads (beam-columns), where buckling interacts with bending and produces non-linear deflection vs load relationships. Do not use them for beams in the plastic range or with non-linear material behaviour (timber after first crack, concrete after yielding); plastic deflection is much larger than linear. Skip them for tapered or non-uniform cross-sections without integrating I(x) over the span. Do not use them for dynamic loading (vibration, impact) without considering inertial effects and damping. Finally, do not rely on simple formulas for critical structural members in commercial design without independent verification or FEA — even simple beams have many edge cases that can mislead the casual user.
How can I reduce beam deflection most efficiently?
The most efficient lever is increasing depth, because I scales with h³ for rectangular cross-sections. Doubling the depth of a rectangular beam multiplies I by 8 and reduces deflection 8× — a much bigger effect than doubling width (only 2× reduction). For I-beams, depth is even more dominant because the flanges far from the neutral axis contribute most of the I. The second most efficient is shortening span: deflection scales with L⁴ for UDL, so reducing span by half cuts deflection 16×. Adding intermediate supports to halve the span is dramatically more effective than upsizing the beam. The third lever is upgrading material: steel (E = 200 GPa) deflects 1/8 as much as timber (E = 25 GPa) at the same I, but timber is cheaper and lighter. The least effective lever is increasing width (only linear effect on I). Modern engineering practice: use the deepest economical section first, then add support if needed, only upsizing material as a last resort.