The contact torsion problem of two cylindrical rods is investigated by considering the sliding contact. When the two semi-infinite rods are contacted at each end, pressure is divided at the corner due to the stress singularity. Thus, the analysis proceeds in consideration of this stress singularity by using the series expansion method. It is considered that the shear stress is linear to the pressure in the sliding region. The relationships between the sliding region and the torque or the friction coefficient are obtained by numerical calculation.