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 — for the steels, stainless, aluminium and titanium the delivery standard's guaranteed minimum for the diameter the material was heat-treated at (by default the bar it is turned from: its largest diameter plus an assumed machining stock), not a small-bar or typical value; case-hardening steels use DIN 743-3's size-scaled core strength and grey cast iron a typical value. 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 Peterson 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. Wrought aluminium has no endurance limit: for it Se′ is an estimated 5×10⁸-cycle fatigue strength of 0.35·Su, and a result inside the Goodman line means beyond 5×10⁸ cycles, not infinite life. Grey cast iron uses an estimated 0.4·Su.

The proof is built on the 2012-12 editions of DIN 743-1, -2 and -3. Part 3 (the material strengths) is now DIN 743-3:2024-07, with an updated material table and strength values, and this implementation has not been checked against it or against the 2014 correction to Part 3.

At every shoulder, keyway, retaining-ring groove and at each segment's weakest plain section, the tool runs the DIN 743 proof on nominal stresses at the notch root, taken from the same bending moment as the rest of the calculation and from the same section torque and axial force as the static check (the thrust may be located at either bearing or by a coupling at either end; torque signs are used only when the user asks for them). The smooth-specimen fatigue strength is reduced to the component's by the technological size factor K1 (from the diameter that governed heat treatment), the geometric size factor K2, the roughness factor K_F and the notch factor β — at a shoulder β = α/n, the form factor over the support factor from the stress gradient; at a keyway or ring groove, DIN 743-2's measured notch factor scaled to the actual diameter. No credit is taken for surface strengthening (K_V = 1).

The mean stress enters through the material's mean-stress sensitivity ψ, in load case 2 (mean and amplitude rise together — the default, and DIN 743's advice when in doubt) or load case 1 (mean held constant), with the yield line taking over beyond each case's limit. An application factor K_A scales every stress to its peak — rotating bending fully reversed at that peak, the torque and axial force fluctuating by (K_A − 1) times their nominal — and sets the peak loads for the static proof. A keyway, ring groove or plain section is proven at every candidate place along its segment (its ends and every load and bearing) and reported where it is weakest, because a feature's exact position along the segment is not an input.

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 the shear-deformable element's own consistent mass and rotary inertia, at the same shear parameter as its stiffness. 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 2.7–3.3.

The mass of a gear, pulley or hub is included only where it is entered at its load station (as a point mass); a station left blank is a BARE station, the result says so, and above 35 % of that bare critical the speed check reads REVIEW until the masses are entered — attached masses can only lower the critical speed.

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.