The problem of whipping of an eccentric, long circular shaft under linearly varying tension rotating in a fluid medium is investigated within linear theory. In view of the great lengths under consideration, the bending stiffness is assumed negligible. Internal damping is neglected and the external damping is assumed viscous. The mass eccentricity is assumed to be a deterministic function of the axial coordinate. Only steady‐state conditions are considered. The deflection of the shaft is obtained by finite Hankel transforms, and the bending stresses are subsequently determined approximately from the curvature of the deflection curve. Numerical results are given in graphical form for a shaft with a pulse‐type eccentricity function in one plane for several values of speed of rotation and tension at the lower end.