The pictures are from my run...we get very little snow here so I don't need to worry about the flat roof necessarily, though we are working on getting it slanted.
To keep it from bowing you could put in posts, like porch columns. I don’t think it should bow too much if they’re shorter pieces. But if you have a lot of snow/rain, you might want to seriously consider a slanted run. ‘Cause mine isn’t slanted, and all the water congregates on the roof and...