This paper studies a stress-strain state of the U-shaped expansion bellows under an internal-pressure induced load. The bellows is considered as a corrugated shell of revolution under axisymmetric load. The governing equations have been derived and the numerical calculations of the stress-strain state were carried out. A variant of the classical shell theory based upon Lagrangian mechanics was used. The finite-difference method was applied to solve the obtained system of ordinary differential equations. The ultimate internal pressure resulting in plastic deformations was determined. A simulation of the loss of equilibrium stability of the expansion bellows was performed. The ANSYS code was used for Finite-Element Method (FEM) in order to calculate the stress-strain state in the bellows.