Straight bar injuries aren't uncommon. Try to work around it instead of using an E-Z curl bar, which doesn't work the bis nearly as well. Try different grip widths, keep rests between sets to 1 minute at most to keep the muscles warm, keep the reps moderate 8-12, try dumbbells, try cable curls instead. Cables are widely under-rated, are great size builders as good or better than barbells, and won't hurt the joints as much.
Another thing that might help is to work another related muscle first, to warm up the area better-tris first, or lats, etc.