{
  "id": "beam",
  "version": "ab9422a93ed4",
  "status": "published",
  "name": "Beam Calculator",
  "question": "What is the beam deflection?",
  "summary": "Beam calculator: support reactions, maximum shear, maximum bending moment, bending stress and maximum deflection of a simply supported beam or a cantilever under a point load or a uniform load.",
  "category": "construction",
  "subcategory": "materials",
  "url": "https://www.acalculator.org/construction/beam-calculator",
  "markdown": "https://www.acalculator.org/construction/beam-calculator.md",
  "kind": "function",
  "method": "Simply supported, point load P at a (b = L − a): R₁ = Pb/L, R₂ = Pa/L, M = Pab/L, δ = Pc(L² − c²)^(3/2) ÷ (9√3 LEI) with c = min(a, b). Uniform w: R = wL/2, M = wL²/8, δ = 5wL⁴/(384EI). Cantilever, end load: M = PL, δ = PL³/(3EI); uniform: M = wL²/2, δ = wL⁴/(8EI).",
  "assumptions": [
    "A straight, prismatic, linear-elastic beam with small deflections (Euler–Bernoulli); the beam’s own weight is not added unless you include it in w.",
    "Structural steel E = 200 GPa and aluminum E = 69 GPa (Engineering ToolBox). This is an estimate, not a design check: a structural engineer must size a real beam to the building code."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "sup": {
        "title": "Support",
        "description": "A beam on a support at each end, or a cantilever fixed at one end and free at the other.",
        "type": "string",
        "enum": [
          "simple",
          "cantilever"
        ]
      },
      "ld": {
        "title": "Load",
        "description": "One point load, or a load spread evenly along the whole beam.",
        "type": "string",
        "enum": [
          "point",
          "uniform"
        ]
      },
      "L": {
        "title": "Span (L)",
        "description": "The length of the beam between supports, or from the fixed end to the free end.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.001,
        "maximum": 1000
      },
      "a": {
        "title": "Load position (a)",
        "description": "The distance from the left support to the point load.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.001,
        "maximum": 1000
      },
      "P": {
        "title": "Point load (P)",
        "description": "The point load. On a cantilever it acts at the free end.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 0.001,
        "maximum": 1000000000
      },
      "w": {
        "title": "Uniform load (w)",
        "description": "The load per unit length along the beam, in the unit chosen below.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 1000000000
      },
      "wu": {
        "title": "Uniform load unit",
        "description": "The unit of the uniform load.",
        "type": "string",
        "enum": [
          "lbf-ft",
          "lbf-in",
          "N-m",
          "kN-m"
        ]
      },
      "mat": {
        "title": "Material",
        "description": "The beam's material, which sets Young's modulus E.",
        "type": "string",
        "enum": [
          "steel",
          "aluminum",
          "custom"
        ]
      },
      "E": {
        "title": "Young's modulus (E)",
        "description": "The material's modulus of elasticity.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "pressure",
        "minimum": 1000000,
        "maximum": 10000000000000
      },
      "sec": {
        "title": "Cross-section",
        "description": "A solid rectangle from its width and depth, or a second moment of area I from a table.",
        "type": "string",
        "enum": [
          "rect",
          "typed"
        ]
      },
      "b": {
        "title": "Width (b)",
        "description": "The width of the rectangular section.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 10
      },
      "h": {
        "title": "Depth (h)",
        "description": "The depth (height) of the rectangular section, in the direction of the load.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "length",
        "minimum": 0.0001,
        "maximum": 10
      },
      "I": {
        "title": "Second moment of area (I)",
        "description": "The second moment of area about the bending axis, in the unit chosen below.",
        "type": "number",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "iu": {
        "title": "I unit",
        "description": "The unit of the second moment of area.",
        "type": "string",
        "enum": [
          "in4",
          "cm4",
          "mm4"
        ]
      }
    }
  },
  "outputs": {
    "deflectionIn": {
      "label": "Maximum deflection (in)",
      "description": "The largest sag of the beam, in inches.",
      "format": "number"
    },
    "deflectionMm": {
      "label": "Maximum deflection (mm)",
      "description": "The largest sag of the beam, in millimetres.",
      "format": "number"
    },
    "ratio": {
      "label": "Span ÷ deflection",
      "description": "L ÷ δ, to compare with limits such as L/360.",
      "format": "number"
    },
    "where": {
      "label": "Deflection position (ft)",
      "description": "Where the largest sag is: from the left support, or from the fixed end of a cantilever.",
      "format": "number"
    },
    "momentFt": {
      "label": "Maximum moment (lbf·ft)",
      "description": "The largest bending moment, in pound-force feet.",
      "format": "number"
    },
    "momentNm": {
      "label": "Maximum moment (N·m)",
      "description": "The largest bending moment, in newton metres.",
      "format": "number"
    },
    "shear": {
      "label": "Maximum shear",
      "description": "The largest shear force in the beam.",
      "format": "quantity",
      "unit": "lbf"
    },
    "r1": {
      "label": "Left reaction (R₁)",
      "description": "The upward force at the left support, or at the fixed end of a cantilever.",
      "format": "quantity",
      "unit": "lbf"
    },
    "r2": {
      "label": "Right reaction (R₂)",
      "description": "The upward force at the right support (simply supported only).",
      "format": "quantity",
      "unit": "lbf"
    },
    "stressPsi": {
      "label": "Bending stress (psi)",
      "description": "The largest bending stress M × (h/2) ÷ I of a rectangular section, in psi.",
      "format": "number"
    },
    "stressMpa": {
      "label": "Bending stress (MPa)",
      "description": "The largest bending stress of a rectangular section, in MPa.",
      "format": "number"
    },
    "inertia": {
      "label": "Second moment of area (in⁴)",
      "description": "I = b × h³ ÷ 12 for a rectangle, or the I you typed, in in⁴.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "sup": "simple",
      "ld": "point",
      "L": "10 ft",
      "a": "5 ft",
      "P": "1000 lbf",
      "w": 100,
      "wu": "lbf-ft",
      "mat": "steel",
      "E": "1600000 psi",
      "sec": "rect",
      "b": "2 in",
      "h": "6 in",
      "I": 36,
      "iu": "in4"
    },
    "outputs": {
      "deflectionIn": 0.03447378646584181,
      "deflectionMm": 0.8756341762323819,
      "ratio": 3480.905705525021,
      "where": 5,
      "momentFt": 2500,
      "momentNm": 3389.544870828501,
      "shear": 2224.11080763025,
      "r1": 2224.11080763025,
      "r2": 2224.11080763025,
      "stressPsi": 2500,
      "stressMpa": 17.236893232920902,
      "inertia": 36
    },
    "text": "The beam sags at most 0.0344738 in (span ÷ 3,480.91) with a maximum moment of 2,500 lbf·ft."
  },
  "examples": [
    {
      "given": {
        "sup": "simple",
        "ld": "point",
        "L": 3.048,
        "a": 1.524,
        "P": 4448.2216152605,
        "mat": "steel",
        "sec": "rect",
        "b": 0.0508,
        "h": 0.15239999999999998
      },
      "expect": {
        "r1": 2224.11080763025,
        "momentFt": 2500,
        "inertia": 36,
        "stressPsi": 2500,
        "deflectionIn": 0.03447378646584181
      },
      "source": "Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads (M = qL²/8, δ = 5qL⁴/(384EI); M = FL/4, δ = FL³/(48EI); R₁ = Fb/L, M = Fab/L), https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html (retrieved 2026-10-05); hand calculation in content.mdx: M = PL/4 = 1,000 × 10 ÷ 4 = 2,500 lbf·ft; I = 2 × 6³ ÷ 12 = 36 in⁴; σ = 30,000 lbf·in × 3 ÷ 36 = 2,500 psi; δ = PL³/(48EI)",
      "tolerance": 1e-9
    },
    {
      "given": {
        "sup": "simple",
        "ld": "uniform",
        "L": 4,
        "w": 2,
        "wu": "kN-m",
        "mat": "custom",
        "E": 10000000000,
        "sec": "typed",
        "I": 100000000,
        "iu": "mm4"
      },
      "expect": {
        "r1": 4000,
        "momentNm": 4000,
        "deflectionMm": 6.666666666666667,
        "ratio": 600
      },
      "source": "Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads (M = qL²/8, δ = 5qL⁴/(384EI); M = FL/4, δ = FL³/(48EI); R₁ = Fb/L, M = Fab/L), https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html (retrieved 2026-10-05); hand calculation in content.mdx: R = wL/2 = 4 kN; M = wL²/8 = 4 kN·m; δ = 5 × 2,000 × 4⁴ ÷ (384 × 10¹⁰ × 10⁻⁴) = 6.667 mm",
      "tolerance": 1e-12
    },
    {
      "given": {
        "sup": "cantilever",
        "ld": "point",
        "L": 2,
        "P": 1000,
        "mat": "aluminum",
        "sec": "typed",
        "I": 1000,
        "iu": "cm4"
      },
      "expect": {
        "momentNm": 2000,
        "r1": 1000,
        "deflectionMm": 3.864734299516908
      },
      "source": "Wikipedia, Deflection (engineering) (cantilever: δ = FL³/(3EI), δ = qL⁴/(8EI); simply supported off-centre: δmax = Fa(L² − a²)^(3/2) ÷ (9√3 LEI), a to the nearer support), https://en.wikipedia.org/wiki/Deflection_(engineering) (retrieved 2026-10-05); MechaniCalc, Beam Deflection Tables (cantilever: M = FL, M = wL²/2), https://mechanicalc.com/reference/beam-deflection-tables (retrieved 2026-10-05); hand calculation in content.mdx: M = PL = 2,000 N·m; δ = 1,000 × 8 ÷ (3 × 69 × 10⁹ × 10⁻⁵) = 3.865 mm",
      "tolerance": 1e-12
    },
    {
      "given": {
        "sup": "simple",
        "ld": "point",
        "L": 6,
        "a": 2,
        "P": 9000,
        "mat": "steel",
        "sec": "typed",
        "I": 100000000,
        "iu": "mm4"
      },
      "expect": {
        "r1": 6000,
        "r2": 3000,
        "momentNm": 12000,
        "deflectionMm": 1.741859372645816,
        "where": 8.96986114268076
      },
      "source": "Wikipedia, Deflection (engineering) (cantilever: δ = FL³/(3EI), δ = qL⁴/(8EI); simply supported off-centre: δmax = Fa(L² − a²)^(3/2) ÷ (9√3 LEI), a to the nearer support), https://en.wikipedia.org/wiki/Deflection_(engineering) (retrieved 2026-10-05); Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads (M = qL²/8, δ = 5qL⁴/(384EI); M = FL/4, δ = FL³/(48EI); R₁ = Fb/L, M = Fab/L), https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html (retrieved 2026-10-05); hand calculation in content.mdx: R₁ = 9 × 4 ÷ 6 = 6 kN; M = 9 × 2 × 4 ÷ 6 = 12 kN·m; δ = 9,000 × 2 × (36 − 4)^(3/2) ÷ (9√3 × 6 × 2 × 10⁷)",
      "tolerance": 1e-12
    },
    {
      "given": {
        "sup": "cantilever",
        "ld": "uniform",
        "L": 3,
        "w": 1,
        "wu": "kN-m",
        "mat": "steel",
        "sec": "typed",
        "I": 100000000,
        "iu": "mm4"
      },
      "expect": {
        "r1": 3000,
        "momentNm": 4500,
        "deflectionMm": 0.50625
      },
      "source": "Wikipedia, Deflection (engineering) (cantilever: δ = FL³/(3EI), δ = qL⁴/(8EI); simply supported off-centre: δmax = Fa(L² − a²)^(3/2) ÷ (9√3 LEI), a to the nearer support), https://en.wikipedia.org/wiki/Deflection_(engineering) (retrieved 2026-10-05); MechaniCalc, Beam Deflection Tables (cantilever: M = FL, M = wL²/2), https://mechanicalc.com/reference/beam-deflection-tables (retrieved 2026-10-05); hand calculation in content.mdx: R = wL = 3 kN; M = wL²/2 = 4.5 kN·m; δ = 1,000 × 3⁴ ÷ (8 × 2 × 10⁷) = 0.50625 mm",
      "tolerance": 1e-12
    }
  ],
  "sources": [
    "Engineering ToolBox, Beams - Supported at Both Ends - Continuous and Point Loads: reactions, moments and deflections of simply supported beams, retrieved 2026-10-05. https://www.engineeringtoolbox.com/beam-stress-deflection-d_1312.html",
    "Wikipedia, Deflection (engineering): cantilever and simply supported deflection formulas, including the off-centre point load, retrieved 2026-10-05. https://en.wikipedia.org/wiki/Deflection_(engineering)",
    "MechaniCalc, Beam Deflection Tables: cantilever moments FL and wL²/2, retrieved 2026-10-05. https://mechanicalc.com/reference/beam-deflection-tables",
    "Engineering ToolBox, Young's Modulus, Tensile Strength and Yield Strength Values: structural steel 200 GPa, aluminum 69 GPa, retrieved 2026-10-05. https://www.engineeringtoolbox.com/young-modulus-d_417.html",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 ft = 0.3048 m, 1 in = 0.0254 m, 1 lbf = 4.4482216152605 N, retrieved 2026-10-05. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "rafter",
    "framing",
    "moment-of-inertia"
  ],
  "changelog": []
}