In this paper, we deal with the integral domain D( S, r ):= D+( X 1, X 2,…, X r ) D S [ X 1, X 2,…, X r ], where D is an integral domain and S is a multiplicative set of D. The purpose is to pursue the study, initiated by Costa-Mott-Zafrullah in 1978, concerning the prime ideal structure of such domains. We characterize when D( S,r ) is a strong S-domain, a stably strong S-domain, a catenarian domain and a universally catenarian domain. As a consequence, we obtain a new class of non-Noetherian universally catenarian domains. Moreover, we give an explicit formula for the Krull dimension of D( S,r ) (depending on S and on the Krull dimensions of D and D S [ X 1, X 2,…, X r ]) and we compute its valuative dimension.