Shaft Design: Method

Shaft Loads, Fatigue and Critical Speed

Loads and diagrams

The shaft is a beam on two simple supports, the bearings A at x = 0 and B at x = L_b. Gears, pulleys and couplings load it with radial forces in two planes (F_y and F_z, which for a gear are the tangential and radial tooth forces) and with torques that enter at one station and leave at another, so the torque diagram is a cumulative sum from the left. Shear and bending moment are computed in each plane and combined into the resultant moment, since a rotating shaft sees the resultant as a fully reversed stress whatever its direction. Stations outside the bearings are overhung and handled by the same statics.

Sections and stress concentrations

The shaft is entered as stepped sections, each with its diameter and the fatigue stress-concentration factors K_f (bending) and K_fs (torsion) at the shoulder where it begins. These come from Peterson's charts for the fillet radius and diameter ratio, reduced by the notch sensitivity (the Notch Sensitivity tool on this site does that step); typical values for a well-filleted shoulder are 1.5 to 1.7 in bending and 1.2 to 1.5 in torsion, and a keyseat about 2.1 and 3.0. Sections are checked at every shoulder with its factors, at every load station and at the point of maximum moment within each section with factors of 1.

Endurance limit

Shigley's form of Marin's equation: S_e = k_a k_b k_c k_d k_e S_e′ with the rotating-beam limit S_e′ = 0.5 S_ut up to 200 ksi and 100 ksi beyond, the surface factor a S_ut^b from the finish table, the size factor 0.879 d^−0.107 (inches, 0.11 to 2 in) or 0.91 d^−0.157 (2 to 10 in), the reliability factor from the normal-distribution table, and the load and temperature factors taken as 1.

Fatigue and static criteria

With fully reversed bending and steady torsion the DE-Goodman criterion gives 1/n_f = (16/(π d³))[2 K_f M / S_e + √3 K_fs T / S_ut], written here as the von Mises alternating stress over S_e plus the von Mises mean stress over S_ut. It is the usual conservative choice for shafts; Gerber and ASME-elliptic give higher factors. The static check compares the von Mises stress at the first cycle with yield. Sections whose fatigue factor falls below the entered design factor are flagged.

Deflection and critical speed

The deflection is found by integrating M/EI twice along the shaft through the stepped stiffness with zero deflection at the bearings; the slopes at the bearings are compared with the limits bearing makers quote. The first critical speed is Rayleigh's ω_c = √(g Σ w_i y_i / Σ w_i y_i²), with the station masses and the shaft's own mass as the weights and the static deflection under those weights as the shape; it is an upper bound usually within a few percent of the true first critical.

Not included

Axial loads, press-fit and keyway stresses beyond the entered factors, bearing stiffness, gyroscopic effects of heavy disks, torsional vibration and the higher critical speeds.