# Predicting elastic and plastic wheel deformation

1. Feb 10, 2009

### RayRoc

I am trying to predict elastic wheel deformation. I am not quite sure how to do it. How might one approach the problem?

Take a steel wheel: 6 in. diameter, 2 in. thick, with a contact force of 1000lbs

E = 30,000,000 psi
Poisson ratio: .27

Would one calculate this in finite layers? Considering sections at a time and applying shear modulus through the layers to find the finite stress of the steel?

I have seen the stress distribution of a steel wheel against a track, and it looked like a strait coulomb from the contact area to the axial. Can it be as simple as:

Contact pressure = (volume of wheel displaced/ volume of .5 the wheel cross section) * Poisson ratio * E

There are of course many effects not realized by such a simplistic view especially when one wants to find the conditions that will cause plastic deformation at the contact area; which is ultimately what I want to know.

2. Feb 11, 2009

### minger

Contact stresses for a cylinder on a cylinder are as follows. The area of contact is a narrow rectangle of width 2b, and length l, where l is the length of the cylinders, and
$$b = \sqrt{ \frac{2F}{\pi l} \frac{ (1-\nu^2_1)/E_1 + (1- \nu^2_2)/E_2}{1/d_1 + 1/d_2} }$$
With the maximum pressure being
$$p_{max} = \frac{2F}{\pi b l}$$
You can apply this to a cylinder in contact with a plane by making $$d_2 = \infty$$

edit: Just seen that you're looking for deformation, not stress. Let's see....stress in the z-direction, perpendicular to the contact line is:
$$\sigma_z = \frac{-p_{max}}{ \sqrt{1+ z^2/b^2}}$$
Perhaps you could integrate that from z = 0, to z = r to calculate the summation of strain?