Abstract: Shen（1991） and Li（1998） presented the strong collapsibility of additive interaction in I x J x K tables.. However, tile concept is not suitable for an ordinal background variable since it is meaningless to pool disjoint levels in this case. In this paper, we introduce the consecutive collapsibility of additive interactions over an ordinal background and present necessary and sufficient conditions for consecutive collapsibility of additive interactions. The conditions are important for guiding the possible merging of consecutive levels of tile background.