This paper deals with flexural vibrations of a continuous slender shaft with a crack located at a distance l n from the left end of the shaft. The mathematical model of this problem is formulated by means of the large finite element method (LFEM). The crack effect is modelled by a switching crack [1] (Jun et al. 1992 Journal of Sound and Vibration 155, 273-290). The increase in crack depth causes decrease in bending stiffness, whereas the non-linearity is related to opening and closing of the crack faces in the process of flexural vibrations (the so called crack ''breathing''). These are generated by the rotating shaft unbalance and by deflection due to shaft own weight. As the zero approximation of the solution a linearized model is used, in which a permanently opened crack is assumed. Based on this simplified model a condition is given discriminating whether the crack remains permanently open/closed during the shaft rotation or it ''breathes''. For the first approximation of the solution of the non-linear mathematical model the averaging method, based on the small parameter theory, is used. The theoretical results are illustrated by calculation of the amplitudes and phases of the first, second and third harmonics of the forced shaft flexural vibrations.