What's the name for a bar connecting two parts for reinforcement?
Another term for this sort of feature is gusset.



Often enough, a gusset is a flat plate engaging both parts at one edge, but it appears that when used with "shelf" the brace reference is translated to gusset. Both terms appear to work well.



The image above shows double braces held in place by gussets, although if there were no gussets, the term apparently still applies. Brace is better terminology, though.