Linear elastic mechanics: Integration technique needed

Factual Questions
1

Disclaimer: This is not homework, and although it is work related, no one is paying me for this work. This is for my own personal education.

I’ve got two solutions for change in stress due to vertical loading beneath the surface of a semi-infinite homogeneous linear-elastic mass. The first is for a concentrated load acting at a point. The second is for a load distributed over a rectangular area. The surface of the semi-infinite mass is defined by z=0 and z increases as depth beneath the surface increases. x and y are in orthogonal horizontal directions.

For the point load solution, the load acts at one point and I can calculate the change in vertical stress at another (different) point.

For the distributed load solution, the load acts on a rectangle parallel to the surface at depth z=h, and I can calculate the change in the vertical stress at another point, but the point must be beneath one corner of the rectangle. For example, the length of the recangle is 0 <= y <= b, the width is 0 <= x <= a, and the point for evaluation is x=0, y=0, and z=whatever.

Now I should be able to get the distributed load solution by integrating the point load solution over the rectangle, i.e. if “P” is the point load and “p” is the distributed load, I substitute pdxdy for P in the point load solution, integrate that once with respect to x from x=0 to a and again with respect to y from y=0 to b, and I should get the distributed load solution. However, it’s not working for me. The original solutions date from the 1930s to the early 1960s, and were therefore probably done by hand. I’ve tried integrating just the first term of the point load solution using Sage (www.sagemath.org) and Integrator (integrals.wolfram.com) and I can get the first integral, but for the second integral the Integrator times me out and Sage returns the input; it seems like an analytical solution does not exist. But I know it exists–I already have it.

I’m guessing that some transform was applied that made an analytical solution possible. My question is: What transformation was applied or what other technique was used?

Since the equations are complex, allow me to place links to them (.pdf documents):
Point load solution (eq. 2.4c)
Distributed load solution (eq. 4.1)

In case you’re wondering why I care if I already have the solution: I would like to obtain the solution for a distributed load over other shapes besides rectangles, and I suspect I need the same trickery to get the integration to work.

2

So it turns out, at least for the first term, that my integration terms were “simplifying” the result of the first integration so that the second integration appeared impossible. (The “simplified” form was not really the simplest, at least to my eyes.) So now I can move on to the second term (or third term since the second is almost the same as the first).

Thanks to those who at least took a look at this thread.

3

Just a random tidbit I recall running across one time.

It was a scenario like you described, but I think it was a 2 layer medium and it had no exact solution for “normal” loads.

However, there was an exact solution if the load was distributed like a Bessel function.

Just thought I would through that out there in case it helps.