Book about a post-nuclear war, domed, underground city where criminals were banished

I am looking for a book that I read in the mid 90’s. The book was written in a post nuclear war setting. It was about a group of people that lived in an underground city with a glass dome at the top part of the city.

If anyone committed a serious crime their punishment was to be banished outside the dome. In the book the main character get banished for a crime he didn’t commit and finds the outside world is safe to live in.

He bangs on the glass dome to try and tell everyone, but everyone thinks he is going insane from radiation poisoning.