Beam Deflection Calculator
Compute the maximum mid-span deflection of a simply supported beam under a uniformly distributed load (UDL) using δ = 5wL⁴ / (384EI). The standard structural-design check for serviceability — verifies whether a beam stays within deflection limits like L/250 or L/360 under service loads.
Last updated: May 2026
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About this calculator
The maximum deflection of a simply supported beam carrying a uniformly distributed load (UDL) occurs at mid-span and is given by δ = 5wL⁴ / (384EI), where w is the load per unit length (N/m or kN/m), L is the span (m), E is the modulus of elasticity of the beam material (MPa or N/mm²), and I is the second moment of area of the cross-section about the bending axis (cm⁴ or mm⁴). The factor 5/384 is geometric — it arises from integrating the deflection differential equation across the span with the UDL applied. Note that deflection scales with the fourth power of span (L⁴), meaning doubling the span produces 16× the deflection at the same load. It scales linearly with load, inversely with E (stiffness of material), and inversely with I (geometric stiffness of cross-section). Variables: w = UDL in N/m, L = span in m, E = Young's modulus in MPa (typical values: structural steel 210,000; reinforced concrete 25,000–30,000; structural timber 8,000–15,000), I = second moment of area in cm⁴ from cross-section tables (a wide-flange IPE 200 has I_y = 1,943 cm⁴). Edge cases: this formula applies only to simply supported beams (pinned-roller supports) under UDL; cantilever beams, fixed-end beams, and point loads use different coefficients. For combined loadings, superpose deflections from each load case. The formula assumes elastic small-deflection behaviour; for very flexible beams (deflection > L/50) geometric non-linearity matters. Serviceability deflection limits per Eurocode are typically L/250 for general beams, L/360 for floors with sensitive finishes, and L/500 for cantilevers; ULS strength is a separate calculation.
How to use
Example 1 — Steel beam check. IPE 200 floor beam, span L = 5 m, UDL w = 10 kN/m, E = 210,000 MPa (structural steel), I = 1,943 cm⁴. Apply δ = 5wL⁴/(384EI) in consistent SI units. Convert I to m⁴: 1,943 cm⁴ = 1.943×10⁻⁵ m⁴. δ = 5 × 10,000 × 5⁴ / (384 × 2.1×10¹¹ × 1.943×10⁻⁵) = 3.125×10⁷ / 1.567×10⁹ = 0.01995 m ≈ 20 mm. ✓ Compare to deflection limit L/250 = 5,000/250 = 20 mm — the beam is right at the serviceability limit. A larger section (e.g., IPE 220 with I = 2,772 cm⁴) would reduce deflection by 1,943/2,772 ≈ 70% to ~14 mm, well within limits. Example 2 — Timber beam check. Pine beam (E = 11,000 MPa), span L = 4 m, UDL w = 3 kN/m, cross-section 100 × 200 mm (I = 100 × 200³ / 12 = 6.667×10⁷ mm⁴ = 6,667 cm⁴ = 6.667×10⁻⁵ m⁴). δ = 5 × 3,000 × 4⁴ / (384 × 1.1×10¹⁰ × 6.667×10⁻⁵) = 3.84×10⁶ / 2.816×10⁸ ≈ 0.01363 m ≈ 13.6 mm. ✓ Compare to L/300 = 13.3 mm (Eurocode 5 typical limit for visual finishes) — the beam just exceeds the serviceability limit. Either increase the section size or add a mid-span support.
Frequently asked questions
What is the maximum allowable deflection for a beam?
Serviceability deflection limits depend on the beam's role and the type of finishes attached. Eurocode 3 (steel) and BS 5950 typical limits are: L/200 to L/250 for general beams; L/360 for beams supporting brittle finishes (plaster ceilings, ceramic tiles); L/180 for cantilevers in general use; L/500 for cantilevers supporting brittle finishes. Eurocode 5 (timber) uses similar limits: L/300 for short-term deflection under service loads, L/250 for total long-term deflection including creep. ACI 318 (concrete) uses L/240 for floors not supporting brittle finishes, L/480 for those that do. These are minimum limits — for some applications (laboratory tables, machine bases, precision floors) tighter limits apply. Excessive deflection isn't a strength failure (the beam doesn't break), but it causes serviceability problems: visible sag, cracked finishes, malfunctioning doors, water ponding on flat roofs, and occupant complaints about 'springy' floors.
How does span length affect deflection in this formula?
Deflection scales with the fourth power of span: δ ∝ L⁴. This is one of the most dramatic scaling laws in structural engineering. Doubling the span produces 2⁴ = 16× more deflection at the same load and section. Halving the span reduces deflection to 1/16. Adding a single intermediate support (turning a single-span beam into two half-spans) cuts the maximum deflection by roughly a factor of ~40, depending on continuity conditions — a striking reduction. This is why structural engineers add intermediate supports whenever deflection is the limiting design criterion: it's far more effective than upsizing the beam section. Conversely, increasing span by 20% increases deflection by 1.2⁴ ≈ 2.07 — more than doubling — which is why long-span designs typically use higher-grade materials, optimised cross-sections (deep I-sections, hollow box sections, prestressed concrete), or composite construction to manage deflection.
What is the second moment of area (I) and how do I find it?
The second moment of area (also called moment of inertia, though more strictly the area moment of inertia) measures a cross-section's geometric resistance to bending. For a rectangular cross-section of width b and depth h: I = b·h³/12. For a circular cross-section of diameter d: I = π·d⁴/64. For wide-flange (I-shape) and hollow steel sections, standard manufacturer's tables list I directly (the IPE 200 has I_y = 1,943 cm⁴ from the EN steel-section tables). I has units of length⁴ (cm⁴, mm⁴, or in⁴). Larger I means stiffer in bending — depth contributes the most because of the cubic h term: doubling depth gives 8× I, while doubling width only gives 2× I. This is why structural beams are deep and narrow rather than square or wide. Composite sections (steel-concrete, timber-concrete) need transformed-section analysis to compute an effective I.
What are the most common mistakes engineers make with deflection calculations?
The first is unit inconsistency: mixing MPa with mm or N/m with m without unit conversion gives results off by powers of 10. Always work in consistent SI units. The second is forgetting that deflection limits are for service loads, not factored ULS loads — Eurocode service loads are typically 65–80% of ULS. The third is using the wrong support condition formula: a simply supported beam has the (5/384)·w·L⁴/EI formula; a fixed-end beam is (1/384)·w·L⁴/EI (much smaller); a cantilever is w·L⁴/(8EI) (much larger). The fourth is forgetting that deflection is non-linear in span (L⁴ scaling); engineers used to rules of thumb for short spans get caught out at long spans. The fifth is using nominal cross-section dimensions instead of the actual cross-section (which can be reduced by holes, slots, or section reductions). The sixth is ignoring creep in concrete and timber, where long-term deflection can be 1.5–3× the elastic short-term deflection due to material creep over years. And the seventh is using elastic modulus values from datasheets without verifying they apply to the loading direction and grain orientation (especially for timber).
When should I not use this calculator?
Skip it for any support condition other than simply supported with UDL — cantilevers, fixed-end beams, continuous multi-span beams, propped cantilevers, and beams with point loads all have different deflection formulas. Avoid it for short, deep beams where shear deflection becomes significant alongside bending deflection (typically when L/h < 10–15); use Timoshenko beam theory or include the shear deflection term. It is the wrong tool for composite sections where the simple I value doesn't capture the cross-section stiffness; use transformed-section analysis or FEA. Do not use it for beams in the inelastic range where the linear-elastic assumption breaks down — past yield, deflections grow non-linearly and require plastic analysis. Skip it for dynamic loading (vibration, impact) where natural frequency and dynamic amplification matter more than static deflection. And for any final design submitted to a regulator or used for construction, use proper engineering software or hand-check against full Eurocode/AISC/ACI formulas; a calculator gives a quick estimate, not a defensible engineering design.