Simplified Solution for Stresses and Deformations

A shortcut to the classical Hertzian solution for local stress and deformation of two elastic bodies in contact is presented. The shortcut is accomplished by using simplified forms for the ellipticity and for the complete elliptic integrals of the first and second kinds as a function of the geometry. Thus the interdependence of these variables can be uncoupled, and the resulting transcendental equation, which must be solved through use of the computer or design charts, avoided. Simplified formulas that make the elastic deformation at the center of contact easy to calculate have been previously reported by the authors. However, the range of applicability was limited to ellipticities greater than or equal to 1. This paper extends the range of validity to include ellipticities less than 1, that is, where the semimajor axis in the elliptical contact lies in a direction parallel to the rolling direction rather than being perpendicular as in previous studies. Furthermore an auxiliary shear stress parameter is expressed in simplified form as a function of the geometry. This enables a shortcut calculation to be made for the location and magnitude of the maximum subsurface shear stress.