Shaft Analysis Calculator — Fatigue, Deflection & Critical Speed

At every section the tool combines the bending, axial and torsional stresses into the von Mises equivalent stress and reports the yield safety factor against that segment's material yield strength. Bending moments are evaluated in Macaulay (singularity-function) form using both bearing reactions, so overhung loads carry their true moment, and the section is checked at every kink in the moment diagram rather than on a fixed grid.

A rotating shaft sees fully reversed bending, so the bending stress is the alternating component and the steady torque supplies the mean. The alternating stress is amplified by the fatigue stress-concentration factor Kf at the governing notch, so a notched shaft is never shown infinite life on its nominal stress alone. Kt comes from the Peterson/Pilkey shoulder-fillet chart as a function of both r/d and the step ratio D/d, and is converted to Kf through the Neuber notch-sensitivity relation.

The endurance limit is the Marin-corrected value: Se = ka·kb·ke·Se′, with the surface factor ka = a·Su^b, the size factor kb from the shaft diameter, and the reliability factor ke = 0.814 at 99%. Se′ = 0.5·Su, capped at 700 MPa for ultimate strengths above 1400 MPa where that correlation stops holding.

Deflection is solved with a Timoshenko beam finite-element model of the stepped geometry exactly as entered: exact nodes are placed at every segment boundary, bearing and load, so bearing position is a live input rather than one snapped to a mesh grid, and the answer does not depend on how the shaft happens to be subdivided. Timoshenko elements include shear flexibility, which Euler–Bernoulli omits — negligible for a slender shaft but 26% of the total deflection at L/d = 2.5 and 58% at L/d = 1.25.

Deflection is reported relative to the line through the two bearings — what a dial indicator referenced to the journals would read. Supports are modelled as near-rigid: real bearing compliance is not included and would increase deflection and lower the critical speed.

The lateral critical speeds are the eigenvalues of the same beam model — det(K − ω²·M) = 0 — with consistent mass including rotary inertia. Because the mode shapes come from the actual stepped geometry, the second critical speed is computed rather than assumed: for a uniform simply-supported shaft the ratio f₂/f₁ approaches the classical 4, but for a typical stepped spindle with overhangs it is nearer 2.7–3.2.

The analysis is undamped and does not model gyroscopic stiffening, so it is a screening calculation. For a damped, gyroscopic Campbell analysis with bearing stiffness, use the Rotor Dynamics tool.

Torsional shear stress follows from the transmitted torque and the polar section modulus, with the angle of twist summed per segment from each segment's shear modulus and polar moment. A user-defined load spectrum is assessed with the Palmgren–Miner linear damage rule against a Wöhler line anchored on the endurance limit.