A contact structure on a three manifoldM is a completely non-integrable tangent two-plane distribution. A contact structure is called overtwisted if it contains an embedded disk D such that is everywhere tangent to D along @D. Eliashberg [6] showed that the topologically interesting case to study is tight contact structures. He did this by showing that the classes of overtwisted contact structures correspond to homotopy classes of two-plane distributions on M. The purpose of this work is to classify tight contact structures on M = 2 I with a specied boundary condition. This is done by applying cut and paste contact topological techniques developed by Honda, Kazez and Matic [15, 18].