Beam Deflection & Bending Moment Calculator

Calculate beam deflection, bending moment, and shear force diagrams using Euler-Bernoulli theory for simply supported, cantilever, and fixed beams.

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▸Formula · assumptions · example(Euler-Bernoulli Beam Theory · AISC Steel Construction Manual · EN 1993-1-1)

Formula

\delta_{\max} = \frac{5 w L^4}{384 E I} \quad (\text{Simply supported, UDL}), \quad \sigma_{\max} = \frac{M_{\max} \cdot y}{I}

Euler-Bernoulli beam bending theory for deflection, shear force, bending moments, and outer fiber flexural stress.

Assumptions

  • · Linear elastic, isotropic material behavior with small displacements (δ << L)
  • · Cross-sections remain planar and perpendicular to the neutral axis during flexure (shear deformation neglected)
  • · Deflection limits checked against standard serviceability criteria (e.g. L/300 or L/500)
Beam deflection guide →

Worked example

Worked example — IPE 200 Steel Beam (L = 4.0 m, UDL w = 15 kN/m, E = 210 GPa)

Calculates maximum bending moment Mmax = 30.0 kN·m, bending stress σmax = 136.4 MPa, and maximum midspan deflection δmax = 5.82 mm (L/687).