It looks like the arched member is 2 separate pieces of wood joined together at the central point with a long key set into vertical slots and cross pinned. There may be some half lap joint there; but without seeing both sides, I can't tell.
If that is the case, it isn't a candidate for beam calculations since it isn't a full length member. Visualize it without the vertical post there and ask yourself if 2 straight pieces joined like that in the middle would be a good beam.
To me it looks like 3/5ths of a good truss. With the addition of rafters and a joint at the top of the post that will resist tension it could be triangulated into a very rigid structure.
It reminds me of some window mullions I did structural testing on. The company wanted to reinforce the center of a 4 way joint with a key like you have rather than a full length stiffener. We framed it up and put a bit of a wind load on it in the lab, maybe 5-10psf. It ripped apart. The code requirement was 50psf wind load.