The theoretical analyses for stresses induced in axial direction in the buried pipelines are reviewed. The influences of the axially directed stresses on the surface elliptical crack are studied in detail and thus some engineering technical informations are provided to use reliability assessment of buried pipelines. The change in temperature, the effect of inner pressure and soil friction in the buried pipeline constrained in axial direction are included to determine the axial stresses in the buried pipeline. Furthermore, the stress induced by the pipeline bending are also considered. The stress intensity factors calculated by two models such as a simple plane crack and an elliptical surface crack for a circumferential surface elliptical crack are compared.