Surface cracks commonly propagate in rotating machines due to cyclic loads. The propagation of these cracks in shafts is an important problem that can lead to catastrophic failure and put lives at risk. Although the majority of the works on cracked shafts consider that the shape of the front of the surface cracks is straight or almond, also sickle-shaped cracks can appear in shafts. In this study, we propose an expression that provides the value of the Stress Intensity Factor (SW) of an open crack with sickle shape, located at the central section of a shaft, as a function of the relative depth of the crack, the shape factor and the relative position on the crack front. To this end, the SW along the crack front has been determined through a 3D numerical analysis considering different elliptical geometries of the crack. To validate the expression, we have compared the obtained results with other solutions obtained by numerical approaches of the literature. It has been found that they are in good agreement. Finally, the sickle crack propagation has been analyzed using the expression for sickle cracks proposed and a developed algorithm based on the Paris Law.