sach4u1...you are new to the forums, but yes, all members have the right and expectation to tell others what they can post...it's part of the process of participation in the forums. It's also one of the reasons the Red Flag link is available. I chose not to use that for your post.
I did not consider your post to be a homework problem. From your wording of the post, it appeared like so many others we've seen in the forums....just give me the answer and I'll go away. My apologies if I misunderstood your intent.
As I think you understand, the forums are for professional interaction...an opportunity to share information, to bounce ideas around, to validate assumptions, to agree, to disagree, and the ultimate end result is to grow professionally and enhance our profession.
Now that I understand your intent a bit better, I'll offer my opinion on the approach to solving your problem....
If you do a lot of steel design, you probably have a finite element analysis program available to you. You can model the mat as a concrete plate element. You can model the soil as springs. Another piece of information you'll need that I don't believe was shown in your sketch is the modulus of subgrade reaction (K).
Another method you can use is to do your analysis upside down...your piers or columns become supports for a concrete beam, uniformly loaded by the soil and the other imposed loads. This will be good to use as a check for any other method you might employ.
Since you are in the US, use ACI 318 as your basis for the factors required in the analysis. It is an LRFD method, similar to what you've used in steel design.
Many foundation design texts offer reasonably good treatment and examples for design application. (Check "Principles of Foundation Engineering" by Braja Das, for example).
Mat foundation analysis can be detailed and difficult or treated simplistically and conservatively (not always right to do this!). In any case, I would suggest that once you've done your analysis and think that you have your design established, that you run it by a more experienced engineer, locally, to check the things that we as a group would not have the time nor the information to do properly.
Good luck and welcome to the forums.