jlight;
By design, an edge joint is like a notched weld joint specimen because you have two members along the same plane that are joined on one end - e.g., the tight gap of the two members that are fused in the weld deposit is like a pre-cracked weld joint. Typically, this type of weld joint is used in thin member welding, like sheet metal.
To determine joint efficiency you could weld a coupon simulating your material and wall thickness and send it to a mechanical test lab. By definition, the joint efficiency is the weld strength/strength of the parent metal. The weld coupon could be designed with both members having a 90 degree flat so that the welded edge joint can be pulled apart or separated with grips in a tensile machine.
The mechanical test would be inexpensive and would yield practical results. As far as stress concentration factor, I would use the inverse of the efficiency factor calculated from the tensile test results, unless somone has a better idea.