Beam Deflection Calculator
Beam deflection calculator. Simply supported + cantilever beams under point load or UDL. δ = PL³/(48EI) classic structural textbook formulas with L/240, L/360 deflection limit checks.
Beam Deflection
| Maximum bending moment | — |
| Flexural rigidity EI | — |
| Common limits | L/240 general · L/360 plaster · L/480 sensitive |
How to use the beam deflection calculator
Pick the beam case
Simply supported: pinned at both ends. Cantilever: fixed at one end, free at the other. Point load: concentrated force at one location. UDL: uniformly distributed (per unit length). These 4 combinations cover ~80% of basic structural beam problems.
Enter geometry
Span L in meters. Young's modulus E: 200 GPa steel, 70 GPa aluminum, 12 GPa wood (parallel to grain), 30 GPa concrete. Moment of inertia I: section property in m⁴ × 10⁻⁶. Common sections: W8x18 ~30×10⁻⁶ m⁴; 2x10 lumber ~125×10⁻⁶; 8" deep concrete ~430×10⁻⁶.
Enter loads
Point load P in Newtons (1 kN = 1000 N; 1 kg ≈ 9.81 N). Distributed load w in N/m (kN/m × 1000). Only one type matters depending on the beam case picked. Both inputs always shown for flexibility.
Read deflection + ratio
Maximum deflection in mm. The L/δ ratio (e.g. L/240) is the standard engineering format. Building codes require: L/240 general, L/360 with plaster ceilings, L/480 for sensitive finishes (some hospitals + labs). Smaller ratio = stiffer beam.
Verify with structural standards
These are first-order linear elastic formulas. Real beams need: load combinations (live + dead + wind), buckling check, shear check, lateral-torsional buckling, vibration check, fatigue (cyclic loads). Use AISC, Eurocode, IBC for real design.
Beam deflection — the structural engineering staple
Beam deflection — the bending displacement of a structural member under load — is one of the foundational calculations in structural engineering. The classical Euler-Bernoulli beam theory dates to the 18th century work of Leonhard Euler + Daniel Bernoulli, refined by 19th century engineers including Stephen Timoshenko. The fundamental formula δ = PL³/(48EI) for a simply supported beam with center point load encapsulates the key insight: deflection scales with the cube of span L (so length is the dominant factor), inversely with flexural rigidity EI (where E is material stiffness + I is section geometry). Doubling the span = 8× deflection. Doubling EI = halving deflection. Engineers use these dependencies to size beams.
The four classic cases
Simply supported + center point load: δ = PL³/(48EI), max moment PL/4 at center. Simply supported + UDL: δ = 5wL⁴/(384EI), max moment wL²/8. Cantilever + end point load: δ = PL³/(3EI) — much larger than SS (factor of 16). Cantilever + UDL: δ = wL⁴/(8EI). Cantilevers are inherently more flexible than simply supported beams of the same dimensions because there\'s no reaction at the free end. The numerator coefficient (3, 5, 48, 384, 8) comes from integrating the curvature equation EI·d²y/dx² = M(x) with the appropriate boundary conditions.
Beam deflection scales with span cubed. Quadrupling the floor span doesn\'t make a beam 4× weaker — it makes it 64× more flexible. The 4³ = 64 dependency is why span lengths are the dominant structural design constraint.
L/240, L/360, L/480 — what those mean
Building codes specify maximum allowable deflection as a fraction of span. L/240: general structural (most residential floors, roofs). L/360: with brittle finishes like plaster ceilings, drywall over deep deflection. L/480: hospitals + labs with sensitive equipment, occupant comfort. A 3 m span at L/240 = 12.5 mm max deflection; at L/480 = 6.25 mm. Strength-only design might pass with a 25 mm deflection, but real beams need to satisfy serviceability (deflection, vibration, cracking) PLUS strength. Modern building codes (IBC, Eurocode, BCA) embed these limits.
ASEAN structural practice
ASEAN construction uses Eurocode-influenced standards (Singapore SS EN, Malaysia MS 1462, Indonesia SNI). Same Euler-Bernoulli beam theory; same L/240 / L/360 deflection limits. Common structural steels: ASEAN equivalent of S275 (275 MPa yield), S355 (355 MPa yield). Concrete grades: typically C25, C30 for general, C40 for high-rise. Hot tropical climates require thermal expansion design + creep coefficients for concrete beams under sustained load. Wood + bamboo (Indonesia, Vietnam): local species E + design values different from European.
10 Things to Know About Beam Deflection
Simply supported + point load: δ = PL³/(48EI). The classic formula.
Deflection ∝ L³. Doubling span = 8× deflection. Span dominates design.
Cantilever same load: 16× more deflection than simply supported.
UDL on SS: δ = 5wL⁴/(384EI). Max moment wL²/8 at center.
Building codes: L/240 general, L/360 plaster, L/480 sensitive.
Steel E ≈ 200 GPa. Aluminum ~70. Wood ~12. Concrete ~30 (effective).
Moment of inertia I = bh³/12 for rectangle. Depth matters cubically.
Euler-Bernoulli theory: valid for slender beams (L/h > 10). Shorter beams need Timoshenko theory.
Real design adds: shear deflection, lateral-torsional buckling, vibration, creep for concrete.
Maximum bending moment determines stress + strength check; max deflection determines serviceability.
Frequently asked questions
For standard sections: AISC steel manual (W shapes, channels), wood manuals (2x4, 2x10, glulam). For custom shapes: I = ∫y²dA (numerical integration). For rectangle b×h: I = bh³/12. Online calculators for standard sections — bring the result back to this calculator.
SS has supports at both ends — moment + shear go to zero at each end. Cantilever has one fixed end (carrying full moment + shear) and one free end. For the same load + span, cantilever deflects ~16× more. Cantilevers are inherently flexible.
Depends on what\'s on/under the beam. L/240: general floors, roofs. L/360: ceilings with brittle finishes (plaster, gypsum) — too much deflection cracks the finish. L/480: occupant-comfort floors. L/600: sensitive equipment (precision labs). Modern building codes mandate per use category.
No — you must include self-weight in the UDL input or as part of total load. Steel beam weight ≈ 7850 × cross-section kg/m. Wood ~500-800 kg/m³ density. For accurate design, total load = dead (self + finishes) + live (occupants + furniture).
Euler-Bernoulli ignores shear deflection — valid for slender beams (L/h > 10). Short, deep beams (L/h < 10) have substantial shear contribution; use Timoshenko theory which adds shear correction. Common L/h ratios for typical floor beams: 15-25.
Wood E varies by species + grain orientation. Common values (parallel to grain): Southern Pine 12-14 GPa, Douglas Fir 11-13, Spruce-Pine-Fir 10-12. Hardwoods (oak, maple) 12-14 GPa. Wood has higher creep than steel — long-term deflection can be 2× short-term. Wood design codes (NDS) include creep factors.
No. All inputs stay in your browser.
Continuous beams (multiple supports) have smaller deflections than simply supported because adjacent spans share load. Coefficients differ — use 3-moment theorem or modern matrix methods. This calculator is single-span only.
Same formula, but concrete has additional complications: cracked-section EI ≠ uncracked. Short-term vs long-term (creep) effective E. Reinforcement transforms the section. Building codes (ACI 318, Eurocode 2) provide effective stiffness equations + creep coefficients.
Hibbeler "Mechanics of Materials" Ch.12. Beer + Johnston "Mechanics of Materials". Timoshenko "Theory of Elasticity". AISC Steel Construction Manual. ACI 318 for concrete. Eurocode 3 (steel) + Eurocode 2 (concrete).
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