I tried building a bow from a birch slab, following a ring along the outer edge of the slab. I put a rawhide backing on it that may have been too thick. I ran into a problem of compression failure cracks on the belly of the bow and it lost it's ability to return to straight when unstrung. The string follow stayed the way it was with the string on it, after I took the string off. I don't know why that happened, but I'm guessing that birch might not be as good under compression as some other woods.
I hope you have better luck with your bow!
Amber