When two bodies with curved surfaces are in contact under a force, point or line contact between these bodies changes to area contact, and  3 dimensional stresses are developed  These stresses are contact stresses. A knowledge of contact stresses is important in calculating strength of bearings, gear and worm drives, ball and cylindrical rollers, and cam mechanisms. Typical failures are seen as cracks, pits, or flaking in the surface material.

The contact stress calculator has been designed to calculate contact pressure and contact stress for spherical and cylindrical contact. Maximum shear stresses are drawn with respect to the depth from contact surface for Object-1 and Object-2. The formulas used for the calculations are given in the "List of Equations" section.

Contact configurations which can be calculated by "Hertzian contact stress calculator" are shown below.

Herzian contact configurations

Note: For more information on contact stresses, spherical contact and cylindrical contact, please refer to Chapter 3.19 (Contact Stresses) of Shigley's Mechanical Engineering Design .

Hertzian Contact Stress Calculator:

Parameter Object-1 Object-2 Unit
Object shape  
Poisson's ratio [v1,v2]  
Elastic modulus [E1,E2]
Diameter of object [d1,d2]
Force [F]

Parameter Obj-1 Obj-2 Unit
Maximum Hertzian contact pressure [pmax] ---
Max shear stress [τmax] --- ---
Depth of max shear stress [z] --- ---
Circular contact area diameter  [2a ] ---

Note: Use dot "." as decimal separator.




Cylindrical contact: The contact of two cylindrical parts where the contact line turns to rectangle area with the presence of a contact force. The area of contact is a narrow rectangle of width 2b and cylinder length l.

Modulus of elasticity (Young’s modulus): The rate of change of unit tensile or compressive stress with respect to unit tensile or compressive strain for the condition of uniaxial stress within the proportional limit. Typical values: Aluminum: 69 GPa, Steel: 200GPa.

Poisson’s ratio: The ratio of lateral unit strain to longitudinal unit strain under the condition of uniform and uniaxial longitudinal stress within the proportional limit.

Proportional Limit: The largest value of stress up to which a linear relation still exist between stress and strain (Hooke’s Law).

Spherical contact: The contact of two spherical parts where contact point turns to a circular area of radius a under the presence of contact force. A contact pressure pmax is occurred on this contact area.

Shear stress: A form of a stress acts parallel to the surface (cross section) which has a cutting nature.

Stress: Average force per unit area which results strain of material.

List of Equations:

List of equations and calculation steps for Hertzian contact stress calculations


An additional calculation tool for Contact Calculations of Sphere in Circular Race has been developed to calculate contact stresses in ball bearings.

Contact Calculations of Sphere in Circular Race