What you are trying to do is called 'distribution analysis'. In the old days, before we had computers bridge engineers used series solutions of the plate biharmonic such as the methods authored by Maurice & Little, or Hendry & Jeager. Nowadays they use commercial software employing finite elements. Back then, in about 1964, I had such a project but I was keen to use finite differences to solve the plate equation and did so programming the solution in ALGOL to run on an ICL mainframe. Over the years I have re-written this in various forms of BASIC and recently I have a version in Mathcad 14, a pdf of a typical runs is attached. The abnormal vehicle load in Britain (HB load) consist of two bogies of two axles each with four wheels per axle, usually 45 units or 37.5 units, the units are Ton/axle. On a two-lane carriageway this would be in combination with 1/3rd the standard highway loading (HA load) in the other lane.
The attached worksheet distributes the HA loads and the HB wheel loads to the rectangular finite difference grid, assembles a stiffness matrix from the finite difference form of the plate equation, adds the effects of any beams on any of the grid lines and the support stiffnesses. The equations are solved to produce the deflexions at the nodes and the stress resulants are calculated from the relevant FD equations. So it can be done with a Mathcad worksheet, but not quickly.
Hope this helps, regards, Bill Wadsworth
