With the advancement of technology, usage of materials taken on a new form. Functionally graded materials is a composite material with a gradual change in two sides has been caused many capabilities to develop new products. The use of FGMs in thick-walled tanks, can be a fundamental solution to the passage of the challenges faced by the nuclear and chemical industries. Based on my search has been among the studies about the behaviour of elastic-plastic FG materials perfectly plastlc is intended. Exact and comprehensive study to consider the effect of work hardening not been done. In addition to increasing the accuracy of the analysis with regard to the work hardening, after all cylinder Become plastic, the solution can be continued. In this thesis study Analytical and numerical of Elastic-Plastic deformation of a FG cylinder under Internal Pressure, considering linear work hardening have been investigated. The behaviour of cylinder is assumed axisymmetric. Analysis of the theory of small deformations and plane-strain is given. To generalized results and better curve fitting material properties, Dimensionless variables is used. Changes in Young's modulus, yield stress and strain hardening exponent as an exponentiation function of the radial coordinate are assumed. To describe the behaviour of the material in the plastic region, the flow rule associated with Tresca yield function is used, assuming a linear work hardening. Previous research has been categorized into three types to start yield, in the thesis yield to the five different states referenced. Yield initiation can occur from internal radius or outer radius, or inner and outer radius simultaneously, or in the interval between the inner and outer radius occur, or in particular condition, in the area between the inner and outer radius at the same time. Due to the flow rule associated with Tresca yield function, an equation of radial stress in the plastic region is achieved. Then method of extracting constants of differential equations in elastic and plastic region has been explained. Analytical solutions with the assumption of equality exponent of the Young's modulus and work hardening functions is provided. Also, strain and displacement are obtained. To evaluate and validate the analysis, results with the relationship of the homogeneous Elastic-Plastic cylinder assuming the work hardening and Elastic-Plastic FG cylinder assuming perfectly plastlc compared. Matching numerical with analytical solution has been investigated in two examples of initial and boundary value. In addition, by providing five example of different states of developed plastic zone has been studied. These include the expansion plastic from the inside, outside, middle, both inside and outside and around the area between the inner and outer radius. Finally, by expression another example, the effect of work hardening is shown in problem solving. The results show that the work hardening assumption can have a significant effect on the distribution of stresses. Finally, Programs is used in extraction yield initiation pattern and numerical solution have been brought. Keywords Functionally Graded Material, Cylinder Under Internal Pressure, Elastic-Plastic Deformation, Plasticity Associated The Tresca Criterion, Work Hardening.