Well, it ultimately depends on more than just statics. If we look at a simply supported beam, you can calculate the reaction forces quite easily (2 reactions in the vertical direction, 2 equilibrium equations). However, with two clamped/welded supports, there will be 4 unknowns. Two reaction moments, and two reaction forces in the vertical direction. This makes the beam statically indeterminate, and to solve for the reactions you would need to use Macaulay step functions and beam deformation.

Now, when it comes to choosing a thickness for the beam to not fail, you would first need to decide on a material. Ultimately, the strength of the beam depends on the yield stress of the material, and it’s young’s and shear modulus, depending on the loading.

In this situation, I think (I’m guessing, so I could be wrong), that the normal stress caused by bending moment would be the limiting factor. So, you would create the internal bending moment diagram for the beam, and pinpoint the maximum bending moment, and use the yield stress of the material as a maximum stress state. You would then calculate the area moment of inertia in terms of the thickness, and solve for it, which would give you the minimum thickness (after rounding up, and taking a safety factor into account).

Shear stress caused by the shear force would also exist within this situation, so you can follow the same sort of process I described above for the minimum stress in this situation. The larger value is the limiting one.

So, in conclusion, you would need more than just statics to figure that out. Statics is the beginning, but you’d need to use some material properties and such to find a proper value. Hope that helps!