Multi-Code 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
Moment of inertia is in cm⁴ — the same unit our moment of inertia calculator reports and the cantilever and Euler-buckling calculators accept, so a section property carries straight across without conversion.
| Maximum bending moment | — |
| Flexural rigidity EI | — |
| Code reference | IBC Table 1604.3, live load: L/360 floors and plastered roofs · L/240 roofs with a non-plaster ceiling |
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 are the four single-span cases tabulated in every mechanics-of-materials appendix; continuous, fixed-ended and partially loaded beams are not modelled.
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: the section property, in cm⁴. Reference values: a W8×18 steel beam is ~2,580 cm⁴; a dressed 2×10 timber joist (38 × 235 mm) is ~4,120 cm⁴; a 200 mm concrete slab taken as a 1 m wide strip is ~66,700 cm⁴. Our moment of inertia calculator reports cm⁴ directly, so its answer can be pasted straight in.
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. IBC Table 1604.3 sets L/360 (live load) for floor members and for roof members supporting a plaster ceiling, L/240 for roof members with a non-plaster ceiling and L/180 with no ceiling; Singapore's National Annex to SS EN 1993-1-1 uses L/360 for beams carrying plaster or other brittle finishes and L/200 for other beams. Stricter targets such as L/480 come from project specifications or finish manufacturers, not from the code tables. A larger number after the slash means a 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 and Daniel Bernoulli; Stephen Timoshenko's 1921 refinement added the shear deformation that the classical theory leaves out. 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 cap deflection as a fraction of span. In the International Building Code (Table 1604.3) floor members are limited to L/360 under live load, roof members to L/360 with a plaster ceiling, L/240 with a non-plaster ceiling and L/180 with no ceiling. Eurocode leaves the numbers to each country's National Annex: Singapore's annex to SS EN 1993-1-1 uses L/360 for beams carrying plaster or other brittle finishes and L/200 for other beams, the same values as the UK annex it follows. A 3 m span at L/240 is 12.5 mm; at L/360 it is 8.3 mm. Stricter targets such as L/480 or L/600 for sensitive equipment or stone finishes come from the project specification or the finish manufacturer rather than the code table. This calculator bands its result at L/240 and L/360 as reference values, not as a code check — a real beam must satisfy serviceability (deflection, vibration, cracking) and strength together.
ASEAN structural practice
Singapore adopted the structural Eurocodes as the SS EN series in 2013 and withdrew the British Standards they replaced in 2015; Malaysia publishes them as MS EN and Indonesia writes its own SNI structural codes. The beam theory behind this page is the same everywhere — what changes by country is the deflection limit in the National Annex and the partial factors on load and material. Structural steel is specified to EN 10025 grades such as S275 and S355 (the number is the yield strength in MPa). For reinforced-concrete beams in a hot, humid climate, long-term deflection under sustained load is governed by the creep and shrinkage provisions of EN 1992-1-1 rather than by the elastic formula here; for timber and bamboo, use the species and grade design values in the local code, not a European default.
Span cubed, flexural rigidity and the L/360 limit — what drives 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.
IBC Table 1604.3: L/360 floors and plastered roofs, L/240 other roof ceilings; SS EN 1993-1-1 Singapore NA: L/360 brittle finishes, L/200 otherwise.
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. Our own moment of inertia calculator handles rectangles, circles, tubes, box sections and I-beams and reports I in cm⁴ — the unit this page expects, so the number transfers without conversion. Watch the units on any figure from elsewhere: section tables quote I in cm⁴, in⁴ or mm⁴ depending on the source, and 1 in⁴ = 41.62 cm⁴ while 1 cm⁴ = 10,000 mm⁴. Entering mm⁴ where cm⁴ is expected overstates stiffness by 10,000× and will tell you a badly undersized beam is fine.
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.
It depends on what the beam carries. IBC Table 1604.3 gives L/360 (live load) for floor members and for roof members supporting a plaster ceiling, L/240 for roof members with a non-plaster ceiling and L/180 with no ceiling; Singapore's National Annex to SS EN 1993-1-1 gives L/360 where plaster or another brittle finish would crack and L/200 for other beams. Tighter limits such as L/480 or L/600 are project or manufacturer requirements for sensitive equipment or stone and tile finishes, not code minima. This page bands at L/240 and L/360 as reference values only.
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, grade and grain orientation — take it from the design-value tables of the code you are working to (the NDS Supplement in North America, EN 338 strength classes in Europe), not from a single "wood" number; the 12 GPa default here is only a placeholder for softwood parallel to grain. Wood also creeps: the National Design Specification (NDS 2018, §3.5.2) multiplies the long-term component of deflection by a creep factor of 1.5 for seasoned lumber, glulam and structural composites and 2.0 for unseasoned or wet-service lumber. This page reports the immediate elastic deflection only.
It is the span divided by the maximum deflection, the form in which every code table states its limit. A 3 m span deflecting 10 mm is L/300. The page compares that number with L/240 and L/360: below L/240 it warns, between the two it notes that a plastered ceiling would be at risk, above L/360 it passes both reference values. Because the check uses the load you entered, a result under the point load alone says nothing about the beam under its full dead-plus-live load combination — enter the governing combination to get a meaningful ratio.
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.
Gere and Goodno, Mechanics of Materials (9th ed., 2018), Appendix G tabulates the deflection and slope formulas for these and many more single-span cases; Hibbeler, Mechanics of Materials (10th ed., 2017), Chapter 12 derives them by double integration. For the limits, see IBC 2021 Table 1604.3 and the National Annex to SS EN 1993-1-1 (or your own country's annex). For design rather than analysis: ANSI/AISC 360 for steel, ACI 318 or EN 1992-1-1 for concrete, the NDS or EN 1995-1-1 for timber.
Method & sources
How it computes
Euler–Bernoulli closed forms for four single-span cases — simply supported δ = PL³/48EI (centre point load) and 5wL⁴/384EI (uniform load), cantilever δ = PL³/3EI (end load) and wL⁴/8EI (uniform load) — with E in GPa, I in cm⁴ (1 cm⁴ = 10⁻⁸ m⁴) and L in metres; also reports the maximum bending moment (PL/4, wL²/8, PL, wL²/2) and EI. Immediate elastic deflection only; the span/deflection ratio is banded against L/240 and L/360 as reference values, not a code check.
What this tool implements
- Euler–Bernoulli (thin-beam) theory: shear deformation, partial-length loads, continuity over supports and creep are excluded; valid for slender members, conventionally L/h greater than about 10, beyond which Timoshenko beam theory is needed.
- Second moment of area is entered in cm⁴, the unit the moment-of-inertia calculator on this site returns (1 cm⁴ = 10⁻⁸ m⁴; 1 in⁴ = 41.62 cm⁴).
- Reference serviceability bands L/240 and L/360 follow IBC Table 1604.3 (L/360 live-load limit for floor members and roof members supporting a plaster ceiling; L/240 for roof members with a non-plaster ceiling; L/180 with none); the Singapore National Annex to SS EN 1993-1-1 uses L/360 for brittle finishes and L/200 for other beams. The band is informational — it is applied to whatever load was entered, not to a code load combination.
- Default stiffnesses are nominal: steel E = 200 GPa (AISC 360 uses 29 000 ksi ≈ 200 GPa); aluminium ≈ 70 GPa; the softwood (≈ 12 GPa) and concrete (≈ 30 GPa) defaults are placeholders to be replaced by the code design value — concrete needs a cracked-section, creep-adjusted stiffness (EN 1992-1-1 / ACI 318) and timber a species-and-grade E plus the NDS creep factor.
Sources
- Gere JM, Goodno BJ. Mechanics of Materials. 9th ed. Boston: Cengage Learning; 2018. Appendix G, Deflections and slopes of beams — the four closed-form cases and their maximum-moment expressions.
- Hibbeler RC. Mechanics of Materials. 10th ed. Hoboken: Pearson; 2017. Chapter 12, Deflection of beams and shafts (double-integration derivation); Appendix C, Slopes and deflections of beams.
- International Code Council. 2021 International Building Code. Section 1604.3 and Table 1604.3, Deflection limits. https://codes.iccsafe.org/content/IBC2021P1/chapter-16-structural…
- American Wood Council. National Design Specification (NDS) for Wood Construction, 2018 edition. Section 3.5.2, Long-term loading — creep factor Kcr = 1.5 (seasoned lumber, glulam, structural composite lumber) and 2.0 (unseasoned or wet-service lumber). https://awc.org/publications/2018-nds/
What can make this go out of date
- None at runtime — the formulas are fixed. The L/240 and L/360 reference bands quote IBC Table 1604.3 and the SS EN 1993-1-1 National Annex, which change only on a code cycle (the IBC is revised every three years); check the edition in force.
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