Roark's book, Formulas for Stress and Strain

Ditto.  Is this the rectangular structure problem you mention above?  If so, the edges aren't "fixed"- they're "semi-fixed", so there's not a lot of point in getting too accurate on the solution if you have a gross approximation in the boundary conditions.

Yes it is the same problem. The reason I wanted to use this method is because I need to know the internal forces throughout the flat slab. The solution to the biharmonic equation would give me deflection and from that I would be able to determine the internal moments.

I actually found this situation in roark's book but the solution given is max stress and max deflection.

What's the problem, where are you getting stuck?  You should have the same approach with different boundary conditions.  

UcfSE

I am a little bit confused of the solution to the biharmonic equation because it seems to me that if I use the Navier solution to the equation I will get the same answer no matter which boundary conditions I choose (simply supported or fixed edges). For example my solution to the boundary conditions of the simply supported case also satisfies those of the fixed edges case.

Take a look at a theory book (e.g. Timoshenko's Theory of Plates and Shells); you'll see that there is no exact solution for the fixed edges condition.
In the site below you might find the answer to your questions: rectangular plates are solved by using the energy method.

prex

http://www.xcalcs.com
Online tools for structural design

Correct me if I am wrong, but I didn't think that Naviers solution was valid for a plate with fixed boundries.  I would think that a finite difference approach would be the best way to approach this problem which would give you deflections, shears, and moments at grid-line points in the slab.  Personally, and I don't mean to ruffle any feathers, but for design why are you approaching this from a finite element standpoint.  For concrete its not like you are going to be trimming out places with low stress.  Seems to me that it would be easier to just design it as a one or two-way strip and move on with life.  Any deflections you are getting in the fea solution wouldn't be accurate anyway if the concrete cracks so why go to the extra effort to do it this way?  It seems to me that assuming the slab is reasonably square the two way strip method would be a good approach.  If the slab is long, then a one-way method would be most satisfactory.