In this paper, we investigate the necessary and sufficient conditions on the network for non-faulty nodes to reach consensus using only local information in the presence of structured Byzantine (StrBYZ) faulty agents. After presenting a new fault model classification, we carry out the analysis not only on the original network but also on the expanded network built on copies of faulty nodes. It is shown that non-faulty nodes achieve consensus without using any fault tolerant algorithm if and only if they are in the same layer with at least one in-neighbor. The theoretical results are also illustrated with numerical results.