# Verification & Validation — Lesson 8

This section contains a few formulas, which made the listed assumptions, found in the Pre-Analysis & Start-Up page.

The analytical formula for computing the radius of contact zone (a) is given as follows: The following command for the computation of the contact area can be downloaded here.

• This command was generously provided by Mr. Sean Harvey. (Lead Technical Services Engineer at Ansys.)

Theoretical Numerical Relative Error(%)
Contact radius, a [mm] 1.00964 1.02517 1.538

Using this value of contact radius, we can also compute the normal pressured induced at the contact zone. Theoretically, the maximum pressure (pmax) is induced along the y-axis, as expected, and is given by the following formula: Theoretical Numerical Relative Error(%)
Max Pressure, Pmax [MPa] 88.290 81.094 8.151

Furthermore, we can derive the following formula for the normal stresses σz and σr = σθ along the z-axis. Here we note that the principal normal stresses σ1 = σ2 = σr = σθ since the out-of-plane shear stresses, τrz = τθz = 0 and σ3 = σz. And we can deduce that τmax = |τ1|=|τ2|=|(σ12) / 2|. The effective stress (using the von-Mises criterion) along the y-axis can be computed as the following: Lastly, we also confirm that the applied load at the top vertex of the sphere matches our numerical contact pressure, integrated along the interface.

Mesh Size [m] 2.00E-04 1.00E-04 9.00E-05 Theoretical
Force Reaction [N] 187.95 188.32 188.52 188.50
Relative Error [%] 0.29 0.09 0.01 0.00  