Abstract: The solution to problem of an orthotropic long cylinder subjected to torsional loading is first obtained by means of separation valuables. The cylinder is twisted by two lateral shear tractions and the ends of the cylinder surface of the cylinder are stress-free. First, the domain under consideration is weakened by an axisymmetric rotational Somigliana ring dislocation. The dislocation solution is employed to derive a set of Cauchy singular integral equations for the analysis of multiple axisymmetric planner cracks. The numerical solution to these integral equations is used to determine the stress intensity factors (SIFs) for the tips of the concentric planar cracks A preliminary comparison between results of this study and those available in the literature is performed to confirm the validity of the proposed technique. Several examples of multiple concentric planner cracks are solved and displayed graphically. Furthermore, Configuration of the cracks and the interaction between cracks is studied.