Elastic deflectionL/240 · L/360Free

Beam Deflection
Checker

Maximum elastic deflection for common beam load cases, checked against the usual serviceability limits. Work in consistent SI units (mm, N, MPa, mm⁴).

Who it's for: structural engineers checking elastic beam serviceability — you need span, load intensity, E, and moment of inertia; this is a code-level tool, not a rough estimate.

Closed-form deflection

Simply-supported UDL: δ = 5wL⁴/(384EI). SS point at midspan: δ = PL³/(48EI). Cantilever UDL: δ = wL⁴/(8EI). Cantilever point at tip: δ = PL³/(3EI). Allowable = L / limit. E in MPa (N/mm²), I in mm⁴, L in mm.

Deflection

δ vs allowable

How to use this calculator

Pick the load case

Select the boundary condition and load pattern that matches your beam: simply supported (pinned both ends) or cantilever, and whether the load is uniform (UDL) or a single point at midspan or tip.

Enter the span L

The full clear span (or cantilever length from the fixed support) in millimetres — 6,000 mm = 6 m.

Enter the load

For UDL cases enter w in N/mm (1 kN/m = 1 N/mm); for point-load cases enter P in N. Only the active load field is used — the other is ignored.

Select the elastic modulus E

Choose a material preset (Steel 200,000 MPa, Concrete 25,000 MPa, Timber 11,000 MPa, Aluminium 69,000 MPa) or select "Custom" and type a value.

Enter the moment of inertia I

The second moment of area of the beam cross-section about its bending axis, in mm⁴. Use the Moment of inertia calculator for standard sections if needed.

Choose the limit and read the check

L/360 for live load on floors with plastered or tiled finishes; L/240 for total deflection; L/180 for roof live load. The result shows max δ, the allowable, and the actual L/δ ratio with a pass/fail verdict.

The formula

Maximum elastic deflection scales with load and span (to the third or fourth power), and inversely with the bending stiffness EI. The four closed-form solutions are:

Simply supported — UDL: δ = 5wL⁴ / (384EI)
Simply supported — point at midspan: δ = PL³ / (48EI)
Cantilever — UDL: δ = wL⁴ / (8EI)
Cantilever — point at tip: δ = PL³ / (3EI)

δ = maximum elastic deflection (mm)
w = uniform distributed load (N/mm; 1 kN/m = 1 N/mm)
P = point load (N)
L = span or cantilever length (mm)
E = elastic modulus (MPa = N/mm²)
I = second moment of area (mm⁴)

Serviceability check: δ ≤ L / limit (e.g. 6,000 mm ÷ 360 = 16.7 mm allowable)

Worked example

Example

A simply-supported steel floor beam spans 6,000 mm with a UDL of w = 5 N/mm (5 kN/m), E = 200,000 MPa, and I = 84,000,000 mm⁴. Applying the formula: δ = 5 × 5 × 6,000⁴ ÷ (384 × 200,000 × 84,000,000) = 5.02 mm. The L/360 allowable = 6,000 ÷ 360 = 16.67 mm. Actual ratio = L/1,194 — the beam passes with a comfortable margin.

When this estimate will be off

Single load case per run — real beams carry multiple load patterns simultaneously. Add component deflections by superposition (δtotal = δ₁ + δ₂ + …).
Elastic, prismatic cross-section only — assumes constant EI along the full span. Tapered members, haunched beams, or composite sections with partial shear connection need a more detailed model.
No long-term effects — concrete beams require an additional multiplier for creep and shrinkage. ACI 318 §24.2 specifies a sustained-load factor that raises effective long-term deflection to 1.4–2.0× the elastic value.
Serviceability check only — a deflection pass does not confirm adequate bending or shear strength, or lateral-torsional buckling resistance. Preliminary screening, not a substitute for a stamped design.

Frequently asked questions

Why does doubling the span increase deflection by 16 times?

Span enters the UDL formula to the fourth power (L⁴). Double L → 2⁴ = 16× more deflection for the same load and stiffness EI. For a point load it scales as L³, so 8×. This is why a modest span increase demands a much stiffer section to maintain serviceability.

What is the difference between L/360, L/240, and L/180?

These are serviceability limits from IBC Table 1604.3. L/360 governs live-load deflection of floors supporting brittle finishes (plaster, ceramic tile). L/240 is the common total-load limit for floors. L/180 applies to roof members under live load. A smaller denominator means a more lenient limit — more deflection is permitted.

Does this calculator account for concrete creep?

No — it gives elastic, instantaneous deflection only. Concrete beams require a long-term multiplier applied to the sustained-load portion. ACI 318 §24.2 specifies a time-dependent factor that raises effective long-term deflection to 1.4–2.0× the elastic value, depending on compression reinforcement and load duration.

Can I use this for a continuous beam?

No — all four load cases assume a simply-supported span or a fixed-base cantilever. Continuous beams carry different moment distributions and deflect differently. Use superposition with fixed-end deflection coefficients from a beam table, or a structural-analysis program. Treating a continuous span as simply supported over-estimates deflection and is conservative.

Sources

AISC Steel Construction Manual Table 3-23 — Beam diagrams and formulas for common loading conditions
IBC 2021 Table 1604.3 — Allowable deflection limits for structural members (L/360, L/240, L/180)

This free Calculator is built and maintained by DataDrivenAEC, using the relevant codes and standards. It does not substitute for professional judgment.