{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"beam-deflection"},"result":{"entity_type":"calculator","id":"beam-deflection","calculator_id":"beam-deflection","canonical_url":"https://tttkmbb.com/engineering/beam-deflection","name":"Beam Deflection Calculator","title":"Beam Deflection Calculator – Maximum Deflection, Bending Moment and Shear for Simply Supported and Cantilever Beams","category":"engineering","category_name":"Engineering & Automotive","tool_name":"calculate_beam_deflection","featured_mcp_tool":false,"description":"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.","use_when":"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).","inputs":[{"name":"beam_case","label":"Support and load case","type":"enum","required":true,"values":[{"value":"simply_supported_point_load_center","label":"Simply supported, point load at mid-span","aliases":["simply_supported_point","ss_point"]},{"value":"simply_supported_uniform_load","label":"Simply supported, uniformly distributed load","aliases":["simply_supported_udl","ss_udl"]},{"value":"cantilever_point_load_end","label":"Cantilever, point load at the free end","aliases":["cantilever_point"]},{"value":"cantilever_uniform_load","label":"Cantilever, uniformly distributed load","aliases":["cantilever_udl"]}],"description":"Support conditions and load type.","example":"simply_supported_point_load_center"},{"name":"span_m","label":"Span","type":"number","unit":"m","required":true,"max":1000,"exclusive_min":0,"description":"Length between the supports, or the cantilever length from the fixed end to the free end, in metres.","example":4},{"name":"load","label":"Load","type":"number","unit":"kN or kN/m","required":true,"max":10000000,"exclusive_min":0,"description":"Point 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).","example":10},{"name":"elastic_modulus_gpa","label":"Elastic modulus E","type":"number","unit":"GPa","required":false,"default":200,"max":2000,"exclusive_min":0,"description":"Young's modulus of the material: structural steel 200 (Eurocode uses 210), aluminium alloys 70, softwood timber about 11, concrete about 30.","example":200},{"name":"moment_of_inertia_cm4","label":"Second moment of area I","type":"number","unit":"cm⁴","required":true,"max":10000000000,"exclusive_min":0,"description":"Second 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).","example":2000}],"outputs":[{"name":"max_deflection_mm","label":"Maximum deflection","type":"number","unit":"mm","decimals":3,"description":"Largest vertical deflection: at mid-span for simply supported beams, at the free end for cantilevers."},{"name":"max_bending_moment_knm","label":"Maximum bending moment","type":"number","unit":"kN·m","decimals":3,"description":"Largest bending moment: at mid-span (simply supported) or at the fixed end (cantilever)."},{"name":"max_shear_kn","label":"Maximum shear force","type":"number","unit":"kN","decimals":3,"description":"Largest shear force, at the supports or at the fixed end."},{"name":"deflection_ratio","label":"Deflection ratio","type":"string","decimals":4,"description":"Span divided by deflection written as L/n, the form used in deflection limits (e.g. L/360)."},{"name":"span_over_deflection","label":"Span / deflection","type":"number","decimals":1,"description":"span_m divided by the maximum deflection, as a number."},{"name":"meets_l_over_360","label":"Meets L/360","type":"boolean","decimals":4,"description":"True when the deflection does not exceed span/360, the usual serviceability limit for floors carrying brittle finishes."},{"name":"deflection_limit_l_360_mm","label":"L/360 limit","type":"number","unit":"mm","decimals":3,"description":"Deflection allowed by the L/360 limit for this span (span_m × 1000 / 360)."},{"name":"stiffness_ei_knm2","label":"Flexural stiffness EI","type":"number","unit":"kN·m²","decimals":3,"description":"E × I in kN·m² (elastic_modulus_gpa × 1e6 × moment_of_inertia_cm4 × 1e-8)."},{"name":"formula_used","label":"Formulas used","type":"string","decimals":4,"description":"Closed-form expressions for the chosen case (P = point load, w = uniform load, L = span)."}],"input_schema":{"type":"object","properties":{"beam_case":{"description":"Support conditions and load type.","type":"string","enum":["simply_supported_point_load_center","simply_supported_uniform_load","cantilever_point_load_end","cantilever_uniform_load"],"examples":["simply_supported_point_load_center"]},"span_m":{"description":"Length between the supports, or the cantilever length from the fixed end to the free end, in metres. Unit: m.","type":"number","maximum":1000,"exclusiveMinimum":0,"examples":[4],"x-unit":"m"},"load":{"description":"Point 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). Unit: kN or kN/m.","type":"number","maximum":10000000,"exclusiveMinimum":0,"examples":[10],"x-unit":"kN or kN/m"},"elastic_modulus_gpa":{"description":"Young's modulus of the material: structural steel 200 (Eurocode uses 210), aluminium alloys 70, softwood timber about 11, concrete about 30. Unit: GPa.","type":"number","maximum":2000,"exclusiveMinimum":0,"default":200,"examples":[200],"x-unit":"GPa"},"moment_of_inertia_cm4":{"description":"Second 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). Unit: cm⁴.","type":"number","maximum":10000000000,"exclusiveMinimum":0,"examples":[2000],"x-unit":"cm⁴"}},"additionalProperties":false,"required":["beam_case","span_m","load","moment_of_inertia_cm4"]},"output_schema":{"type":"object","properties":{"max_deflection_mm":{"description":"Largest vertical deflection: at mid-span for simply supported beams, at the free end for cantilevers. Unit: mm.","type":"number","x-unit":"mm"},"max_bending_moment_knm":{"description":"Largest bending moment: at mid-span (simply supported) or at the fixed end (cantilever). Unit: kN·m.","type":"number","x-unit":"kN·m"},"max_shear_kn":{"description":"Largest shear force, at the supports or at the fixed end. Unit: kN.","type":"number","x-unit":"kN"},"deflection_ratio":{"description":"Span divided by deflection written as L/n, the form used in deflection limits (e.g. L/360).","type":"string"},"span_over_deflection":{"description":"span_m divided by the maximum deflection, as a number.","type":"number"},"meets_l_over_360":{"description":"True when the deflection does not exceed span/360, the usual serviceability limit for floors carrying brittle finishes.","type":"boolean"},"deflection_limit_l_360_mm":{"description":"Deflection allowed by the L/360 limit for this span (span_m × 1000 / 360). Unit: mm.","type":"number","x-unit":"mm"},"stiffness_ei_knm2":{"description":"E × I in kN·m² (elastic_modulus_gpa × 1e6 × moment_of_inertia_cm4 × 1e-8). Unit: kN·m².","type":"number","x-unit":"kN·m²"},"formula_used":{"description":"Closed-form expressions for the chosen case (P = point load, w = uniform load, L = span).","type":"string"}}},"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","method":"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.","sources":[{"name":"Wikipedia – Deflection (engineering)","url":"https://en.wikipedia.org/wiki/Deflection_(engineering)","type":"reference","retrieved_at":"2026-09-24"},{"name":"Gere JM, Timoshenko SP – Mechanics of Materials, Appendix: deflections and slopes of beams (PWS / Cengage)","url":"https://www.worldcat.org/search?q=Gere+Timoshenko+Mechanics+of+Materials","type":"textbook","retrieved_at":"2026-09-24"},{"name":"Wikipedia – Euler–Bernoulli beam theory","url":"https://en.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"Simply supported, 10 kN at mid-span, 4 m, steel, I = 2000 cm⁴","inputs":{"beam_case":"simply_supported_point_load_center","span_m":4,"load":10,"elastic_modulus_gpa":200,"moment_of_inertia_cm4":2000},"expected":{"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},"url":"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"},{"name":"Cantilever, 2 kN/m over 3 m, steel, I = 2000 cm⁴","inputs":{"beam_case":"cantilever_uniform_load","span_m":3,"load":2,"elastic_modulus_gpa":200,"moment_of_inertia_cm4":2000},"expected":{"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},"url":"https://tttkmbb.com/api/v1/calculate/beam-deflection?beam_case=cantilever_uniform_load&span_m=3&load=2&elastic_modulus_gpa=200&moment_of_inertia_cm4=2000"}],"faq":[{"q":"Is the load in kN or kN/m?","a":"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)."},{"q":"Where do I get the second moment of area?","a":"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."},{"q":"Which deflection limit applies?","a":"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."}],"tags":["beam deflection","cantilever deflection","simply supported beam","bending moment","beam calculator","l/360"],"related":[{"calculator_id":"hookes-law","reason":"The linear elastic force–displacement relation that deflection formulas rest on."},{"calculator_id":"force","reason":"Convert the load between kN, kgf and lbf."},{"calculator_id":"pressure-force-area","reason":"Turn a distributed pressure on an area into the line load w."}],"links":{"html":"https://tttkmbb.com/engineering/beam-deflection","markdown":"https://tttkmbb.com/engineering/beam-deflection.md","json":"https://tttkmbb.com/engineering/beam-deflection.json","api":"https://tttkmbb.com/api/v1/calculate/beam-deflection","schema":"https://tttkmbb.com/api/v1/calculators/beam-deflection","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T03:44:11Z"}