A method for stress analysis in cylindrical pressure vessels with ellipsoidal heads, based on the axisymmetric shell theory, was proposed. The starting point were the approximate solutions of the differential equation system that were used to get mathematical expressions for determining internal forces, moments and displacements in the vessel walls. Final expressions that can be applied were acquired by joining the membrane and moment theory and by settingand solving equations of boundary conditions. Diagrams that show distribution of internal forces and moments are in dimensionless form which enables their use regardless of dimensions and load. These expressions were used to develop a method for testing strength of pressure vessels with ellipsoidal heads in the design phase. Application of the method was shown on a selected numerical example, while a special computer programme was created for calculation purposes.