The part-through circumferential crack is a typical crack form in pipe structure, which often lead to the structure failure. Firstly, the local flexibility equations of a pipe with a part-through circumferential crack are derived by using linearly elastic fracture mechanics. Secondly, an adaptive Simpson quadrature-revisted was used to numerically calculate the local flexibility coefficients. Finally, the model test were performed to validate the local flexibility for a part-through circumferentially cracked pipe. The results show that the natural frequencies of cracked pipe obtained by the model test agree well with the theoretical values from the proposed equations of the local flexibility. It is demonstrated the feasibility of the proposed method for vibtaion analysis and damage identification of a cracked pipe.