marmon: Correction: I had forgotten that you don't have to check the shear stress on the base metal adjacent to the weld leg (for steel welds), and the first couple of references I had handy at that moment didn't mention it.
Therefore, the allowable strength of the 0.3125 inch fillet weld you mentioned is 813 N/mm (4640 lbf/inch) or 0.40(Sty)(t), whichever is less, where Sty = plate tensile yield strength (250 MPa for your plate), and t = plate thickness.
For simplicity, if the load is perpendicular to your weld, you could assume the weld strength is still 4640 lbf/inch. So if your load were parallel to the plate and perpendicular to the weld, that would be, in your case, 4640(2 welds)(8) = 74240 lbf. (And obviously the plate tensile strength wouldn't govern here.) If the load is instead parallel to your weld, the shear strength of the plate through its thickness is 0.40(36000)(1.125)(8) = 129600 lbf (which doesn't govern) and the weldment strength is 4640(2 welds)(8) = 74240 lbf.
But I now see your perpendicular load is perpendicular to the plate and perpendicular to the weld. So I agree with your equation, except change the "D+a" to just "D". Hence the weld stress due to bending would be sigma_y = P*L/(b*D*a), where b = plate length = 8 inch. But you also have weld stress perpendicular to the plate, which would be sigma_z = P/(2*b*a). So I think that would be sigma = (sigma_y^2 + sigma_z^2)^0.5. Therefore, ensure sigma*b is less than the weld allowable strength.