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Fatigue and Maximum Shear Stress Theory Equations and Calculator

Strength and Mechanics of Materials

Maximum Shear Stress Theory Fatigue of a Shaft or Axle Formula and Calculator

Material strengths can be estimated from any one of several analytic representations of combined-load fatigue test data, starting from the linear (Soderberg, modified Goodman) which tend to give conservative designs to the nonlinear (Gerber parabolic, quadratic, Kececioglu, Bagci) which tend to give less conservative designs.

Preview Premium Access Only Maximum Shear Stress Theory Fatigue of a Shaft or Axle Calculator

Fatigue Loading Shaft or Axle using the maximum shear stress theory combined with the Soderberg line for fatigue, the diameter and safety factor are related by

Eq. 1
π d3 / 32 = n [ ( Mm / Sy + Kf Ma / Se )2 + ( Tm / Sy + Kfs Ta / Se )2 ](1/2)


Eq. 2
Shaft or axle diameter required
d = { ε n / π [ ( Mm / Sy + Kf Ma / Se )2 + ( Tm / Sy + Kfs Ta / Se )2 ](1/2) } (1/3)


ε = 32 (maximum shear theory)
= diameter, in;
n = safety factor, %;
Ma = alternating moment, in-lbs;
Mm = mean moment, in-lbs;
Ta = alternating torque, in-lbs;
Tm = mean torque, in-lbs;
Se = fatigue limit, cycles;
Sy = yield strength, lbs/in2
Kf = fatigue strength reduction factor;
Kfs = fatigue strength reduction factor for shear.

A shaft is a rotating member, usually of circular cross section, used to transmit power or motion. It provides the axis of rotation, or oscillation, of elements such as gears, pulleys, flywheels, cranks, sprockets, and the like and controls the geometry of their motion. An axle is a nonrotating member that carries no torque and is used to support rotating wheels, pulleys, and the like. The automotive axle is not a true axle; the term is a carryover from the horse-and-buggy era, when the wheels rotated on nonrotating members. A nonrotating axle can readily be designed and analyzed as a static beam, and will not warrant the special attention given in this chapter to the rotating shafts which are subject to fatigue loading.

There is really nothing unique about a shaft that requires any special treatment beyond the basic methods already developed in previous chapters. However, because of the ubiquity of the shaft in so many machine design applications, there is some advantage in giving the shaft and its design a closer inspection. A complete shaft design has much interdependence on the design of the components. The design of the machine itself will dictate that certain gears, pulleys, bearings, and other elements will have at least been partially analyzed and their size and spacing tentatively determined. The details of the shaft itself that should be examined, include the following:

  • Material selection
  • Geometric layout
  • Stress and strength
    • Static strength
    • Fatigue strength
  • Deflection and rigidity
    • Bending deflection
    • Torsional deflection
    • Slope at bearings and shaft-supported elements
    • Shear deflection due to transverse loading of short shafts
  • Vibration due to natural frequency

In deciding on an approach to shaft sizing, it is necessary to realize that a stress analysis at a specific point on a shaft can be made using only the shaft geometry in the vicinity of that point. Thus the geometry of the entire shaft is not needed. In design it is usually possible to locate the critical areas, size these to meet the strength requirements, and then size the rest of the shaft to meet the requirements of the shaft-supported elements. The deflection and slope analyses cannot be made until the geometry of the entire shaft has been defined. Thus deflection is a function of the geometry everywhere, whereas the stress at a section of interest is a function of local geometry. For this reason, shaft design allows a consideration of stress first. Then, after tentative values for the shaft dimensions have been established, the determination of the deflections and slopes can be made.


  • Budynas, Richard G., and J. Keith Nisbett, Shigley's Mechanical Engineering Design, 8th ed., New York: McGraw-Hill, 2008.
  • Marks Standard Handbook for Mechanical Engineers