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We have two iron (assume real-life stiffness) manhole covers resting on friction-less, perfectly smooth shims on flat ground. One is circular and the other square.

If a force F is applied vertically downwards at the center of each manhole cover, is the pressure on the manhole shim distributed uniformly in each case? i.e. is the downward force at any given cover-contact-point on the shim the same for both manholes.

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What is the idea of that shims? In a purely theoretical question like this one You do not need shims. Just perfectly flat ground and covers. And: of course the pressure of the covers rims are different, if the covers have more or less uniform thickness. Such things are outside mathematical analysis, You have to apply fined element calculation or better, stress analysis by polarized light. –  Georg Oct 12 '11 at 10:22
    
The shim concept highlights the fact that the covers are only supported by a uniform ledge around the edge. This may or may not be relevant. Covers have uniform thickness. This is outside the bounds of newtonian analysis? –  user11911 Oct 12 '11 at 10:30
    
I agree with Georg. Imagine you have a table with 4 identical legs, staying on an ideally flat ground. Are the normal forces applied on the legs the same? You can't now. A deviation in the length of a single leg changes the answer dramatically. This question is ambiguous in the context of an ideal table/leg/ground approximation. You have to consider the micro-deformations that produce the normal forces. –  valdo Oct 12 '11 at 11:05
    
Can this question be modified to enable us to get an answer? Can we say for the purposes of the question that the shim does not deform and that the iron manhole cover deforms "perfectly"? –  user11911 Oct 12 '11 at 11:12
    
No, you can't. If the shim doesn't deform than how it "knows" it should apply a pressure? You may however select a model that describes the pressure as a function of the curvature of the shim shape, with an an arbitrary elasticity coefficient. Solve (hopefully) your question. Then tend your elasticity coefficient to infinity, and see if the answer converges to something reasonable. –  valdo Oct 12 '11 at 11:21
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Not it is not uniform, because the linear stiffness varies by location (radially). If you were to take a stylus and press down with a unit force on each location along the manhole and measure the deflection you will see that it varies by location.

Also what varies by location is the "back up", or the supporting shims.

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