Recently, Andrews proved two congruences conjectured by Beck on Beck’s partition statistic. Motivated by Andrews’ work, Chan, Mao and Osburn, and Kim established several Andrews-Beck type congruences for overpartitions. In this paper, we prove some new Andrews-Beck type congruences on \(\overline{NT}(r,m,n)\) , where \(\overline{NT}(r,m,n)\) denotes the total number of parts in the overpartitions of n with rank congruent to r modulo m by using some identities proved by Gu and Su, and Kim. In particular, we present characterizations of congruences modulo 4 and 8 for \(\sum _{r=1}^{m-1} r\overline{NT}(r,m,n)\) with \(m=4 \) or 8. As a corollary, we prove that the arithmetic density of the set of integers such that \(\sum _{r=1}^{7} \overline{NT}(r,8,n)\equiv 0 \pmod {8}\) is 1.
