In this thesis, the stress analysis of the shell steam boiler in the normal operating load case and in the testing load case was performed using the finite element method. Model of the boiler was created in Autodesk Inventor and then analysed in Inventor Nastran. The analysis was made according to the existing version of shell boiler designed according to HRN EN 12953-3:2016. Before carrying out the numerical analysis, the applicability of using the software was confirmed by verifying all used finite elements on a simplified example of a thin-walled cylindrical pressure vessel with a flat bottom with internal pressure load. Comparison of the results obtained by the finite element method and analytical calculation according to the referenced literature was carried out. Numerical analysis of the boiler includes the implementation of linear steady-state heat transfer analysis, linear static analysis, definition of boundary conditions of constant internal pressure, hydrostatic pressure, weight and temperature load, and boundary conditions of connection with the environment. The issue of connecting parabolic solid finite elements (tetrahedrons) with linear line elements (beams) by means of rigid body connectors (RBE2) is also addressed. For strenght calculation, design by analysis approach defined by Annex C of the HRN EN 13445-3:2021, called the method based on stress categories, was applied. To achieve certainty in the obtained results, the existing design by formulae strenght calculation carried out according to HRN EN 12953-3:2016 and available in the project documentation of the boiler was used. In this paper, a calculation methodology was developed according to the specified standard, harmonized with the options and output data available in the used finite element analysis software. Stress linearisation was performed at critical locations in the boiler, and finally stress evaluation of the entire structure according to the given annex.