Functionally-graded materials (FGM) have recently been widely used. Furthermore, FGM are widely recommended in cylinder design. This study represents a mathematical analysis of the stresses and strains of an FGM cylinder. The paper is interested in introducing a stress analysis to an FGM cylinder where its properties vary exponentially in the r-direction. Firstly, a benchwork has been done and compared with recent works in the same field. The radial stress, the hoop stress, and the longitudinal stress on a cylinder under mechanical and thermal loading have been theoretically derived. In addition, a finite element analysis has been investigated through ANSYS software. Finally, the values of stresses obtained from the derived equations and the corresponding values of stresses obtained from ANSYS have been compared. The obtained results established the efficiency of the proposed stress distribution and the optimization model in this research, which would be helpful for understanding well the technical design of FGM vessels or related constructions.