Engineering CalculatorsShaft Sizing Calculator
Analyse a rotating stepped shaft (solid or hollow) on two bearings under combined bending and torsion — two-plane moment diagram, static and fatigue safety at every shoulder (Shigley / ASME with a full Marin endurance limit and Peterson stress-concentration factors), deflection, angle of twist, and the first critical speed.
| Station | x | Ø | n_fat | n_yld | req Ø |
|---|---|---|---|---|---|
| Shoulder 30→40 mm | 60 | 30 | 2.34 | 4.06 | 28.5 |
| Shoulder 40→30 mm | 180 | 30 | 4.64 | 13.58 | 22.7 |
| Max bending moment | 120 | 40 | 7.35 | 22.2 | 25.9 |
The shaft is modelled as cylindrical (solid or hollow) sections on two simple bearing supports. Bearing reactions and the shear/bending-moment diagrams are resolved independently in the vertical and horizontal planes and combined to a resultant M(x) = √(M_y² + M_z²); the torque diagram is the running sum of the applied torques. Bending deflection comes from numerically integrating M/(E·I(x)) twice with the section-varying I, fixing zero deflection at the two bearings; the angle of twist is ∫T/(G·J) dx. Each shoulder and declared feature is checked for static first-cycle yield (peak distortion-energy stress vs Sy) and fatigue by the distortion-energy criteria (DE-Goodman / Gerber / ASME-elliptic / Soderberg, Shigley Ch. 7), assuming a rotating shaft (fully-reversed bending, steady torque). The endurance limit uses the Marin factors Se = ka·kb·ke·(0.5·Sut) with ka = a·Sut^b for the surface finish, the rotating-beam size factor kb, and the reliability factor ke; the load factor is carried by the von Mises combination. Fatigue stress-concentration factors are Kf = 1 + q(Kt − 1) with the Peterson/Pilkey shoulder-fillet Kt fits (or Shigley Table 7-1 estimates for keyseats/grooves) and Neuber notch sensitivity q. The first lateral critical speed is a Rayleigh estimate from the self-weight-plus-disk deflection. Not modelled: axial load, bearing/support flexibility, gyroscopic and multi-mode dynamics, residual/mean bending stress, and stress-concentration interaction — confirm against detailed FEA and the applicable standards before production.
Validated: the four distortion-energy diameter equations reproduce Shigley's Mechanical Engineering Design Prob. 7-1 (A = 338.4, B = 265.5 N·m, Se = 210, Sut = 700, Sy = 560 MPa → d = 27.27 mm Goodman, 25.77 mm elliptic, 27.70 mm Soderberg at n = 2); the Marin surface factor (machined, Sut = 560 → ka = 0.84) and Neuber notch sensitivity (r = 1.07 mm → q = 0.72 bending, 0.77 torsion) match Prob. 7-2; and a uniform simply-supported shaft reproduces the closed-form reactions (W/2), moment (WL/4), central deflection (WL³/48EI) and single-mass Rayleigh critical speed (ωc = √(g/δ)) to machine precision.