This paper contains the derivation of the basic differential equations which govern the elastic-plastic behaviours of rotationally symmetric shells. Geometrical nonlinearity is considered in the same degree as in the nonlinear theory of thin elastic shells derived by E. Reissner. Using the assumption of plane stress, the Euler-Bernoulli hypothesis, and the Prandtl-Reuss equations for stress-strain relations in the plastic region, the basic differential equations are expressed in terms of the increments of the variables. These differential equations are approximated in finite difference forms and they are solved by an elimination method. These solutions relating to the increments are integrated numerically, and the final states of shells are obtained. As an example of the numerical analysis a U-shaped bellows loaded by an axial force is discussed, and it is ascertained that our results coincide well with the solutions obtained by another method for the elastic problem.