Civil Engineering

Beam Calculator

Simply supported beam analysis: deflection, max moment, and shear

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N/m
m
Pa
m⁴

How it works

Analyses a simply supported beam carrying a uniformly distributed load, returning the mid-span deflection along with peak bending moment and shear. Deflection scales with the fourth power of span, which is why doubling the length makes a beam sixteen times more flexible under the same load per metre.

Formula
δ_max = 5wL⁴ / (384·E·I)
M_max = wL² / 8
V_max = wL / 2
(simply supported beam, uniformly distributed load)
Variables
  • wuniformly distributed load, N/m
  • Lclear span between supports, m
  • Emodulus of elasticity of the material, Pa
  • Isecond moment of area of the cross-section, m⁴
  • δ_maxmaximum deflection, which occurs at mid-span for this case
Worked example
Inputs: w = 1,000 N/m over a 6 m span; steel with E = 200 GPa and I = 1×10⁻⁴ m⁴
  1. L⁴ = 6⁴ = 1,296
  2. Numerator: 5 × 1,000 × 1,296 = 6,480,000
  3. Denominator: 384 × 200×10⁹ × 1×10⁻⁴ = 7.68×10⁹
  4. δ = 6,480,000 / 7.68×10⁹ = 8.4375×10⁻⁴ m
  5. M_max = 1,000 × 6² / 8 = 4,500 N·m
  6. V_max = 1,000 × 6 / 2 = 3,000 N at each support
Result: 0.84 mm deflection, 4,500 N·m moment, 3,000 N shear
Notes
  • These coefficients belong to one load case only. A central point load on the same span gives PL³/48EI, a cantilever under uniform load gives wL⁴/8EI (9.6 times the simply supported value), and fixed ends reduce it to wL⁴/384EI, one fifth.
  • On longer spans serviceability governs before strength. A common deflection limit is L/360 for floors carrying brittle finishes, which caps the 6 m span above at about 16.7 mm regardless of how much moment capacity the section has.
  • Keep E, I, w and L in one consistent unit system. Mixing a section property quoted in mm⁴ with a span in metres is the error that produces answers wrong by a factor of 10¹².

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