# Hertz contact pressure formula for cone on a plane

Can anyone please point me to some literature that contains formula for Hertz contact pressure for conical body on a plane? I know that there are general solutions that probably cover also this, but I'm looking for this in particular because of practical reasons. If anyone can also confrim if this problem is covered enough exact with equations from simpler problems, that is also welcome.

I'm designing a familiy of linear guides of different sizes with a steel wheel with 90° groove on a steel tube and would like to do some local stress calculation when evaluating it.

## 1 Answer

Your case is a special case of Hertz contact forces between two cylinders. Except one has a radius of infinity. Say in the figure $$D_2= \infty$$. source.

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The equations are:

$${ b = \sqrt{\frac{2F}{{\pi}l} \frac{(1-v_1^2)/E_1 + (1-v_2^2)/E_2}{1/d_1 + 1/d_2}} }$$ $$\displaystyle { P_{max} = \frac{2F}{{\pi}bl} }$$

Your calculations assigning the load to each face seem correct. There is shear stresses that can be hard to calculate analytically!

• I thought about that too, but what is the radius of cone body in this example? Cone has two radiuses at the limits of contact - one smaller and one bigger and everything in between. If we are working on a plane that is oriented perpendiculary to the surface of one tube side, then I can't be sure which radius of cone body is correct for the solution you provided. Commented Aug 18, 2023 at 9:10
• @Grega, You can use the radius of average of the two radiuses. In fact because the cone and the angle surface don't mate, the cone is always skidding about somewhere near the average radius. The half that is farther from the center skids faster and the other half slower. Surfaces the roll on a contact face have to be designed like gears. Commented Aug 18, 2023 at 17:44