How much surface area does a cylinder have to the ground?

For an infinitely rigid cylinder on an infinitely rigid ground, the answer is zero. As in, the contact looks like nothing more than an infinitesimally thin line with a length equal to the length of the cylinder.

For all REAL cylinders on the REAL ground, it depends on the structural properties of the cylinder and the ground (those being Young's modulus and Poisson's ratio), the radius of the cylinder, and the amount of compressive load supported by the contact (i.e. the cylinder weight, unless you are adding to it).

Contact mechanics can be a real challenge, and isn't really a topic of discussion until Junior year of an engineering course.

Now that I did some looking up, you can read about the equations here:

http://en.wikipedia.org/wiki/Contact_mechanics

Take a look at the formula for E*, and take a look at the section on "Contact between two cylinders with parallel axes". Remember, treat the ground as a second cylinder with "infinite radius".

Let me know if these equations scare you, or if you are still interested. They certainly scared me when I first saw them.

Now that I take a look at your goal to compact the soil to 116 psf (not pcf, psf, as in pounds per square foot)...

You do need to realize that the pressure is by no means uniform. It is what is known as the Hertzian pressure distribution. Greatest in the center, and zero at the edges. The graph of pressure as a function of position takes the shape of a half-ellipse.

Greatest pressure is 50% greater than the average pressure.

So, what I still need to know:

1. Elastic modulus of the local ground, or I can use a standard soil from this list

http://www.geotechnicalinfo.com/youngs_modulus.htm...

2. Elastic modulus of the roller. Is it steel, is it aluminum, is it zinc?

3. Poisson's ratios of the materials. Maybe it might be one of these:

http://en.wikipedia.org/wiki/Poisson's_ratio#Poiss...

4. LENGTH of the roller. Not known? then tell the weight per unit length.

This depends on the ground.

In theory, an ideal cylinder on an ideal surface would rest upon a single molecule thick line, the length of the cylinder. This assumes surfaces with no give. You say it is on it's sie, so the the size of the cylinder would have no bearing.

Picture it as an intersection of a line with a circle.

Are you sure you don't mean that it is sitting on end? That would be the easier Pi, r, squared, or about 452 square inches.

Assuming there is no compression of the cylinder or the surface the answer is 0.

The cylinder is resting on a 1 dimensional line equal to it's length.

element of one base is pir^2 the rest is purely a rectangle wrapped around, top * diameter of circle 2pirh A = pir^2 + 2pirh = pi*5^2 + 2*pi*5*11 = 25pi + 110pi = 135pi = 424.1m^2 ===