acalculator

What is the beam deflection?

Pick the supports and the load, then type the span and the load. The beam calculator gives the support reactions, the maximum shear and bending moment, and, from the material and cross-section, the bending stress and the maximum deflection.

Your numbers

Units
Support
Load
More options
Cross-section
Maximum deflection (in)
0.0344738

The beam sags at most 0.0344738 in (span ÷ 3,480.91) with a maximum moment of 2,500 lbf·ft.

Maximum deflection (mm)
0.875634
Span ÷ deflection
3,480.91
Deflection position (ft)
5
Maximum moment (lbf·ft)
2,500
Maximum moment (N·m)
3,389.54
Maximum shear
500 lbf
Left reaction (R₁)
500 lbf
Right reaction (R₂)
500 lbf
Bending stress (psi)
2,500
Bending stress (MPa)
17.2369
Second moment of area (in⁴)
36

Maximum deflection (in): 0.0344738. The beam sags at most 0.0344738 in (span ÷ 3,480.91) with a maximum moment of 2,500 lbf·ft.

How to calculate

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.

Example with the default inputs (Support Simply supported, Load Point load, Span (L) 10 ft, Load position (a) 5 ft, Point load (P) 1,000 lbf, Material Structural steel, 200 GPa, Cross-section Rectangle, Width (b) 2 in, Depth (h) 6 in): The beam sags at most 0.0344738 in (span ÷ 3,480.91) with a maximum moment of 2,500 lbf·ft.

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).

  • 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.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Support Simply supported, Load Point load, Span (L) 10 ft, Load position (a) 5 ft, Point load (P) 1,000 lbf, Material Structural steel, 200 GPa, Cross-section Rectangle, Width (b) 2 in, Depth (h) 6 in gives Left reaction (R₁) 500 lbf, Maximum moment (lbf·ft) 2,500, Second moment of area (in⁴) 36, Bending stress (psi) 2,500, Maximum deflection (in) 0.034474.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)
  2. Support Simply supported, Load Uniform load, Span (L) 13.12 ft, Uniform load (w) 2, Uniform load unit kN/m, Material Other: type E, Young's modulus (E) 1,450,377 psi, Cross-section Type I, Second moment of area (I) 100,000,000, I unit mm⁴ gives Left reaction (R₁) 899.236 lbf, Maximum moment (N·m) 4,000, Maximum deflection (mm) 6.666667, Span ÷ deflection 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)
  3. Support Cantilever, Load Point load, Span (L) 6.562 ft, Point load (P) 224.8 lbf, Material Aluminum, 69 GPa, Cross-section Type I, Second moment of area (I) 1,000, I unit cm⁴ gives Maximum moment (N·m) 2,000, Left reaction (R₁) 224.809 lbf, Maximum deflection (mm) 3.864734.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)
  4. Support Simply supported, Load Point load, Span (L) 19.69 ft, Load position (a) 6.562 ft, Point load (P) 2,023 lbf, Material Structural steel, 200 GPa, Cross-section Type I, Second moment of area (I) 100,000,000, I unit mm⁴ gives Left reaction (R₁) 1,348.85 lbf, Right reaction (R₂) 674.427 lbf, Maximum moment (N·m) 12,000, Maximum deflection (mm) 1.741859, Deflection position (ft) 8.969861.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)
  5. Support Cantilever, Load Uniform load, Span (L) 9.843 ft, Uniform load (w) 1, Uniform load unit kN/m, Material Structural steel, 200 GPa, Cross-section Type I, Second moment of area (I) 100,000,000, I unit mm⁴ gives Left reaction (R₁) 674.427 lbf, Maximum moment (N·m) 4,500, Maximum deflection (mm) 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)

How it works

L is the span, P a point load, w a uniform load per unit length, E Young's modulus and I the second moment of area.

Simply supported, point load P at distance a from the left support (b = L − a, and c = the smaller of a and b):

  • Reactions R₁ = P × b ÷ L (left), R₂ = P × a ÷ L (right); maximum shear = the larger reaction.
  • Maximum moment M = P × a × b ÷ L, under the load.
  • Maximum deflection δ = P × c × (L² − c²)^(3/2) ÷ (9√3 × L × E × I), at √((L² − c²) ÷ 3) from the support farther from the load. At mid-span this is P × L³ ÷ (48EI).

Simply supported, uniform load w: R₁ = R₂ = w × L ÷ 2 (the maximum shear); M = w × L² ÷ 8 at mid-span; δ = 5 × w × L⁴ ÷ (384 × E × I) at mid-span.

Cantilever, point load P at the free end: R₁ = P (the maximum shear); M = P × L at the fixed end; δ = P × L³ ÷ (3 × E × I) at the free end.

Cantilever, uniform load w: R₁ = w × L (the maximum shear); M = w × L² ÷ 2 at the fixed end; δ = w × L⁴ ÷ (8 × E × I) at the free end.

Section and stress. A rectangle of width b and depth h has I = b × h³ ÷ 12, and its largest bending stress is σ = M × (h ÷ 2) ÷ I. With a typed I there is no stress output, because the depth is not known.

Materials. Structural steel E = 200 GPa; aluminum E = 69 GPa (Engineering ToolBox); or type E in psi or MPa. The deflection position is measured from the left support, or from the fixed end of a cantilever.

Unit sizes (exact): 1 lbf/ft = 4.4482216152605 ÷ 0.3048 N/m; 1 lbf/in = 4.4482216152605 ÷ 0.0254 N/m; 1 in⁴ = 0.0254⁴ m⁴; 1 cm⁴ = 10⁻⁸ m⁴; 1 mm⁴ = 10⁻¹² m⁴; 1 psi = 4.4482216152605 ÷ 0.0254² Pa.

Rules

  • Span from 1 mm to 1,000 m; the load position a from 1 mm to less than the span (a load at or past the right support has no answer); P from 0.001 to 10⁹ in its unit; w from 0.001 to 10⁹ in its unit; a typed E from 10⁶ to 10¹³ Pa; b and h from 0.1 mm to 10 m; a typed I from 10⁻⁶ to 10¹² in its unit.
  • A deflection too small or too large to show as a number has no answer.

Output format

Deflection in inches (US headline) and millimetres (Metric headline), span ÷ deflection, the deflection position in feet, moments in lbf·ft and N·m, stress in psi and MPa, and I in in⁴, each to 6 significant figures; forces to 6 significant figures in lbf (US) or N (Metric). Typed values are read back as exact decimals and the arithmetic is exact except the square root in the off-centre point load, which is exact only when (L² − c²) ÷ 3 is the square of a decimal. Each value is rounded once.

Worked examples by hand

A steel bar at mid-span. L = 10 ft = 120 in, P = 1,000 lbf at 5 ft, 2 × 6 in steel: R₁ = R₂ = 500 lbf; M = 1,000 × 10 ÷ 4 = 2,500 lbf·ft = 30,000 lbf·in; I = 2 × 6³ ÷ 12 = 36 in⁴; σ = 30,000 × 3 ÷ 36 = 2,500 psi. E = 200 GPa = 29,007,547.5 psi, so δ = 1,000 × 120³ ÷ (48 × 29,007,547.5 × 36) = 0.0344738 in, or L/3,481.

A uniform load. L = 4 m, w = 2 kN/m, E = 10 GPa, I = 10⁸ mm⁴ = 10⁻⁴ m⁴: R = 2,000 × 4 ÷ 2 = 4,000 N; M = 2,000 × 16 ÷ 8 = 4,000 N·m; δ = 5 × 2,000 × 256 ÷ (384 × 10¹⁰ × 10⁻⁴) = 0.0066667 m = 6.66667 mm, or L/600.

An aluminum cantilever. L = 2 m, P = 1,000 N at the free end, I = 1,000 cm⁴ = 10⁻⁵ m⁴: M = 2,000 N·m; δ = 1,000 × 8 ÷ (3 × 69 × 10⁹ × 10⁻⁵) = 3.86473 mm.

An off-centre load. L = 6 m, P = 9,000 N at a = 2 m (b = 4 m), steel, I = 10⁸ mm⁴: R₁ = 9,000 × 4 ÷ 6 = 6,000 N, R₂ = 3,000 N; M = 9,000 × 2 × 4 ÷ 6 = 12,000 N·m; c = 2, L² − c² = 32, δ = 9,000 × 2 × 32^(3/2) ÷ (9√3 × 6 × 200 × 10⁹ × 10⁻⁴) = 1.74186 mm, at √(32 ÷ 3) = 3.26599 m from the right support (2.73401 m from the left).

A cantilever with a uniform load. L = 3 m, w = 1 kN/m, steel, I = 10⁸ mm⁴: R = 3,000 N; M = 1,000 × 9 ÷ 2 = 4,500 N·m; δ = 1,000 × 81 ÷ (8 × 200 × 10⁹ × 10⁻⁴) = 0.50625 mm.

Other questions people ask

How do you calculate beam deflection?

Use the formula for your supports and load. A simply supported beam with a point load P at mid-span sags δ = PL³ ÷ (48EI); with a uniform load w it sags δ = 5wL⁴ ÷ (384EI). E is the material’s stiffness and I the second moment of area of the section.

What is the maximum bending moment of a simply supported beam?

PL ÷ 4 for a point load at the middle, Pab ÷ L for a point load at distance a from one end (b from the other), and wL² ÷ 8 for a uniform load. A 1,000 lbf load at the middle of a 10 ft span gives 2,500 lbf·ft.

How do I find I for a rectangular beam?

I = b × h³ ÷ 12, with b the width and h the depth in the direction of the load. A 2 × 6 in section has I = 2 × 216 ÷ 12 = 36 in⁴. Turning the same board flat makes I = 6 × 8 ÷ 12 = 4 in⁴, so it sags 9 times as much.

What does L/360 mean?

It is a deflection limit: the span divided by 360. A 10 ft (120 in) span limited to L/360 may sag at most 0.333 in. Compare the span ÷ deflection output with the limit your code or design guide sets.

How is a cantilever different?

A cantilever is fixed at one end and free at the other. It carries all the load at the fixed end, where the moment is largest (PL for an end load, wL² ÷ 2 for a uniform load), and it sags most at the free end.

Can I size a real beam with this calculator?

Use it to compare options and check rough numbers. A real beam must also meet the building code (load combinations, lumber grades, lateral support, connections), so have a structural engineer or the code’s span tables confirm it.