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Beam Deflection Calculator

Computes the maximum deflection, bending moment and shear force of a prismatic beam for four standard cases (simply supported with a mid-span point load or a uniform load, cantilever with an end point load or a uniform load) from span, load, elastic modulus and second moment of area, using the Euler–Bernoulli closed-form formulas.

When to use

You need the mid-span or tip deflection of a steel, aluminium or timber beam under one point load or a uniformly distributed load, or want to check a deflection limit such as L/360.

Do not use when: The beam carries several or off-centre loads, has overhangs, fixed ends or continuous supports (superpose cases or use structural software), or you need bending stress (divide the moment by the section modulus).

Formula

EI = elastic_modulus_gpa × 1e6 × moment_of_inertia_cm4 × 1e-8 (kN·m²), L = span_m, P or w = load; simply supported point load: δ = P·L³/(48·EI), M = P·L/4, V = P/2; simply supported uniform load: δ = 5·w·L⁴/(384·EI), M = w·L²/8, V = w·L/2; cantilever point load: δ = P·L³/(3·EI), M = P·L, V = P; cantilever uniform load: δ = w·L⁴/(8·EI), M = w·L²/2, V = w·L; max_deflection_mm = δ × 1000

Euler–Bernoulli beam theory for a linear-elastic prismatic beam with small deflections; shear deformation is neglected, which is accurate when the span exceeds about ten times the section depth. Deflection limits (L/360 for floors, L/180–L/240 for roofs) come from building codes and may apply to live load only or to total load.

Inputs

ParameterTypeUnitRequiredDescription
beam_caseenum: simply_supported_point_load_center | simply_supported_uniform_load | cantilever_point_load_end | cantilever_uniform_loadyesSupport conditions and load type.
span_mnumbermyesLength between the supports, or the cantilever length from the fixed end to the free end, in metres. Range: > 0, ≤ 1000
loadnumberkN or kN/myesPoint load P in kilonewtons for the point-load cases, or uniformly distributed load w in kN per metre of span for the uniform-load cases (1 kN ≈ 102 kgf ≈ 225 lbf; include self-weight in w if relevant). Range: > 0, ≤ 10000000
elastic_modulus_gpanumberGPadefault 200Young's modulus of the material: structural steel 200 (Eurocode uses 210), aluminium alloys 70, softwood timber about 11, concrete about 30. Range: > 0, ≤ 2000
moment_of_inertia_cm4numbercm⁴yesSecond moment of area (area moment of inertia) about the bending axis in cm⁴, from section tables (e.g. IPE 200: 1943 cm⁴) or b·h³/12 for a rectangle (1 in⁴ = 41.62 cm⁴). Range: > 0, ≤ 10000000000

Outputs

OutputTypeUnitDescription
max_deflection_mmnumbermmLargest vertical deflection: at mid-span for simply supported beams, at the free end for cantilevers.
max_bending_moment_knmnumberkN·mLargest bending moment: at mid-span (simply supported) or at the fixed end (cantilever).
max_shear_knnumberkNLargest shear force, at the supports or at the fixed end.
deflection_ratiostringSpan divided by deflection written as L/n, the form used in deflection limits (e.g. L/360).
span_over_deflectionnumberspan_m divided by the maximum deflection, as a number.
meets_l_over_360booleanTrue when the deflection does not exceed span/360, the usual serviceability limit for floors carrying brittle finishes.
deflection_limit_l_360_mmnumbermmDeflection allowed by the L/360 limit for this span (span_m × 1000 / 360).
stiffness_ei_knm2numberkN·m²E × I in kN·m² (elastic_modulus_gpa × 1e6 × moment_of_inertia_cm4 × 1e-8).
formula_usedstringClosed-form expressions for the chosen case (P = point load, w = uniform load, L = span).

Example

Simply supported, 10 kN at mid-span, 4 m, steel, I = 2000 cm⁴: {"beam_case":"simply_supported_point_load_center","span_m":4,"load":10,"elastic_modulus_gpa":200,"moment_of_inertia_cm4":2000} → {"max_deflection_mm":3.333,"max_bending_moment_knm":10,"max_shear_kn":5,"deflection_ratio":"L/1200","span_over_deflection":1200,"meets_l_over_360":true,"deflection_limit_l_360_mm":11.111,"stiffness_ei_knm2":4000}

Cantilever, 2 kN/m over 3 m, steel, I = 2000 cm⁴: {"beam_case":"cantilever_uniform_load","span_m":3,"load":2,"elastic_modulus_gpa":200,"moment_of_inertia_cm4":2000} → {"max_deflection_mm":5.063,"max_bending_moment_knm":9,"max_shear_kn":6,"deflection_ratio":"L/593","span_over_deflection":592.6,"meets_l_over_360":true}

GET https://tttkmbb.com/api/v1/calculate/beam-deflection?beam_case=simply_supported_point_load_center&span_m=4&load=10&elastic_modulus_gpa=200&moment_of_inertia_cm4=2000

Machine access

Sources

FAQ

Is the load in kN or kN/m?

For the two point-load cases enter the concentrated force P in kN; for the two uniform-load cases enter the load per metre w in kN/m (a total uniform load W spread over the span is w = W / span_m).

Where do I get the second moment of area?

From steel or timber section tables (I about the strong axis, usually Ix or Iy), or for a solid rectangle b·h³/12 with b and h in cm; a hollow section is the outer rectangle minus the inner one.

Which deflection limit applies?

Common serviceability limits are L/360 for floors and members supporting plaster or tile, L/240 for roofs and ceilings, and L/180 for members without finishes; check the governing code and whether it applies to live load or total load.

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