Beam Deflection Calculator — Deflection, Moment, Shear & Fatigue (EN 1993-1-1)

The calculator solves the Euler-Bernoulli beam differential equation EI·d²y/dx² = M(x) in closed form for each of the six supported load configurations. This produces exact analytical expressions for deflection, bending moment and shear at every section along the beam, with no numerical integration error. The peak deflection and peak bending moment — and the section where each occurs — are identified from the full distribution across N = 100 points.

Bending stress follows from the flexure formula σ = M·c / I, where c is the distance from the neutral axis to the outermost fibre and I is the second moment of area of the cross-section. The yield safety factor SF = Sy / σ_max is computed and flagged against pass/warn/fail thresholds that match the in-report check, so there is no ambiguity between the on-screen status and the PDF.

The first three natural frequencies of the beam in bending are found from the Rayleigh eigenvalue expression, which gives the exact closed-form frequencies for uniform beams under the three common boundary conditions. The effective mass accounts for the distributed beam mass plus any added lumped mass at the appropriate equivalent-mass fraction (0.5 for SS, 0.24 for cantilever, 0.397 for fixed-fixed — the last from Blevins 1979, which avoids the 12% error of the naive 0.5 value). The result is the undamped natural frequency in Hz, which should sit at least 20% clear of any excitation frequency to avoid resonance.

For beams under cyclic loading, the calculator applies the Goodman mean-stress criterion to the alternating and mean bending stresses derived from the maximum and minimum load amplitudes. The modified endurance limit is corrected by a surface factor Csurf. Above the endurance limit the Basquin power-law (Shigley 11e §6-7, using the high-cycle 10³–10⁶ fit) gives a finite fatigue life estimate N_f; at or below the limit the life is reported as infinite (run-out).