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 sampled at 61 stations, with the load position itself added to the sample set for an offset load so the peak moment is exact.

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. Shear stress follows from τ = V·Q / (I·b) and is checked against the shear yield strength σy/√3 per EN 1993-1-1 §6.2.6. The deflection is checked against a serviceability limit you select (L/200 to L/500 — EN 1993-1-1 §7.2.1 defers the number to the National Annex).

All three checks — bending, shear and deflection — are folded into a SINGLE governing verdict, so the on-screen hero, the header status pill, the sketch and the PDF report can never disagree with one another. A check that cannot be evaluated (for example the bending stress of a custom section whose extreme-fibre distance was not supplied) is reported as NOT CHECKED, never as a pass.

The first three natural frequencies of the beam in bending come from the exact closed-form eigenvalue expression for uniform beams under the three common boundary conditions (βL = π, 1.8751 and 4.7300 for simply supported, cantilever and clamped-clamped respectively). Any added lumped mass is then folded in with a Rayleigh effective-mass correction using the appropriate equivalent-mass fraction (0.5 for SS, 0.24 for cantilever, 0.3965 for fixed-fixed — the last from Blevins 1979, which avoids the 12% error of the naive 0.5 value). The resonance advice always quotes the frequency WITH the added mass, since that is the one the installed beam actually has.

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 uses Shigley's Se′ = 0.5·Sut estimate, capped at 700 MPa for Sut above 1400 MPa, corrected by a surface factor Csurf. A compressive mean stress receives no mean-stress benefit (the criterion reduces to σa ≤ Se), which is the conservative reading of the Goodman diagram.

The fatigue LIFE is read at the equivalent completely-reversed stress σrev = σa / (1 − σm/Sut) (Shigley 11e §6-14), not at the raw amplitude — so a large tensile mean stress properly shortens the life instead of being ignored. Above the endurance limit the Basquin power-law (Shigley 11e §6-7, high-cycle 10³–10⁶ fit) gives a finite N_f; at or below it the life is reported as infinite (run-out); below 10³ cycles the result is flagged as low-cycle and outside the fit's validity rather than extrapolated. Only the surface Marin factor is applied — size, loading, temperature and reliability factors are not, and the panel carries an EstimateBadge saying so.