Aly, After cutting your first shape, the remainder can be patterned with the circular pattern tool. The cuts will only go around the part, it will not make another version below the first so a staggered pattern is obtained. You'll need to make the second cut, and then pattern it as well.
You may have some trouble with your shape cutting through the internal spokes. The spokes could be added last, or you could design the part with multiple solid bodies so the spokes remain unaffected by the cuts happening to the ring.
If you upload your model, it will be easier to see and recreate just what you are trying to do.
I added the file