Pressure vessels are wildly used in many fields, such as chemical, petroleum, military industries as well as in nuclear
power plants. Pressure vessels should be designed with great care because rupture of pressure vessel may cause
catastrophic accident. The common problem to the pressure vessel's designer is the accurate evaluation of stresses due
to the applied mechanical and /or thermal loads. The finite element method FEM is one of the numerical stress
analysis for many subjects. In this paper a complete stress analysis through the wall of pressure vessel under the effect
of constant and cyclic loading is presented. Hoop, radial, axial and effective stresses for cylindrical pressure vessels
have been evaluated analytically as well as using the program package ANSYS as a numerica l finite element method. A
comparison between the analytical and numerical solution is presented, and it is found a good agreement between them