This will depend greatly on the stiffness of the remaining structure (the supports indicated around the $D_1$ span, whatever those rest on, etc).
Assuming that the remaining structure can be considered relatively stiff, but that the plate's connection to its supports (around $D_1$) is hinged, then the structure is isostatic and can be trivially solved.
Here's the bending moment, for example (this obviously depends on the actual values of $D$ and $D_1$) (ignore the units, they're irrelevant).

On the other hand, if the supports can be considered fixed (not allowing rotations, then the bending moment diagram is as follows:

As it happens, regardless of the chosen condition, the maximum bending moment on the plate in this example is 150 kNm (or whatever unit you desire). The maximum stress applied on the plate will be equal to
$$\sigma = \dfrac{My}{I}$$
where $M$ is the moment, $y$ is the distance to the fiber farthest from the centroid (in this case, $y = \frac{t}{2}$, where $t$ is the thickness of the plate) and $I$ is the section's inertia (in this case, $I = \dfrac{bt^3}{12}$).
Adding appropriate safety factors to both the applied load and the maximum allowable stress in your steel, you can therefore reverse-engineer the necessary thickness of your plate. This ignores the bending moment due to the structure's self-weight, which is obviously a function of the plate's thickness, but it works as a very decent first approximation (and, actually, in this layout, the plate's thickness shouldn't change the bending moment anyway).
There are other verifications such as for shear strength, but these follow a similar concept as shown for bending moment. You must also check the connection itself, but that requires more details regarding the connection.
D1
distance apart? $\endgroup$ – John Alexiou May 15 '16 at 14:47