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Static Loading Shaft or Axle Analysis Formula and Calculator

Strength and Mechanics of Materials

Shafts and Axles Design Analysis Formulas and Calculator - Maximum Shear Stress and Von Mises Stress

Most shafts are subject to combined bending and torsion, either of which may be steady or variable. Impact conditions, such as sudden starting and stopping, will cause momentary peak stresses greater than those related to the steady or variable portions of operation.

Design of shafts requires a theory of failure to express a stress in terms of loads and shaft dimensions, and an allowable stress as fixed by material strength and safety factor. Maximum shear theory of failure and distortion energy theory of failure are the two most commonly used in shaft design.

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Static Loading of a Shaft or Axle: The maximum shear stress and the von Mises stress may be calculated in terms of the loads from

Eq. 1 Maximum Shear Stress
τmax = 2 / ( π d3 ) [ ( 8 M + F d )2 + 8 T )2 ](1/2)

Eq. 2 Von Mises Stress
σ' = 4 / ( π d3 ) [ ( 8 M + F d )2 + 48 T )2 ](1/2)


M = bending moment
F = axial load
T = torque
d = diameter



  • 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