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Shaft Design

Stress, fatigue safety factor, deflection and critical speed of a stepped shaft on two bearings carrying gears, pulleys or couplings: the radial loads in two planes give the shear and bending-moment diagrams, the applied torques give the torque diagram, and at every step and every load station the rotating shaft's fully reversed bending and steady torsion are checked by the distortion-energy Goodman criterion with Marin's endurance-limit factors and the entered fatigue stress-concentration factors, and by the von Mises static criterion against yield. The deflection curve is integrated numerically through the varying stiffness, giving the slopes at the bearings for the bearing check and the first critical speed by Rayleigh's method from the station masses and the shaft's own mass. Charts of the diagrams and the deflection, and a drawing of the shaft.

  • Shear, bending-moment (two planes and resultant) and torque diagrams of a simply supported stepped shaft

  • Endurance limit by Marin's surface, size and reliability factors; DE-Goodman fatigue factor and von Mises yield factor at every section

  • Deflection and slope by numerical integration through the stepped stiffness; slopes at the bearings

  • First critical speed by Rayleigh's method with the station masses and the shaft mass; drawing of the shaft and charts of every diagram

Shaft Design

Unit System: /

Bearings and Speed (x along the shaft from bearing A; y up, z out of the page)



Shaft Sections (from the left; each runs to the next section's start)
1
2
3
4

Loads (gears, pulleys, couplings)
1
2

Material and Factors








Shaft Loads, Fatigue and Critical Speed | Method page
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Shaft Design

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 = Lb. Gears, pulleys and couplings load it with radial forces in two planes (Fy and Fz, 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 Kf (bending) and Kfs (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: Se = ka kb kc kd ke Se′ with the rotating-beam limit Se′ = 0.5 Sut up to 200 ksi and 100 ksi beyond, the surface factor a Sutb 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/nf = (16/(π d³))[2 Kf M / Se + √3 Kfs T / Sut], written here as the von Mises alternating stress over Se plus the von Mises mean stress over Sut. 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 ωc = √(g Σ wi yi / Σ wi yi²), 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.



References

Budynas, R. G., and Nisbett, J. K. (2015). Shigley's Mechanical Engineering Design, 10th ed. McGraw-Hill Education, New York.

Norton, R. L. (2020). Machine Design: An Integrated Approach, 6th ed. Pearson, Hoboken, NJ.

Marin, J. (1962). Mechanical Behavior of Engineering Materials. Prentice-Hall, Englewood Cliffs, NJ.

Rayleigh, J. W. S. (1894). The Theory of Sound, 2nd ed., Vol. 1. Macmillan, London.

Worked Examples

Published examples this tool reproduces; each page lists the inputs to enter and the printed answers.

All worked examples

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