This study evaluates the plastic responses of thick cylinders made of transversely isotropic materials under mechanical cyclic loads, using the kinematic hardening theory of plasticity. The Hill yield criterion is adapted to the kinematic hardening theory of plasticity. The constitutive equations of plastic strains are obtained using the adapted yield criterion. The flow rule based on the kinematic hardening theory of plasticity associated with the Hill yield criterion is represented to evaluate the cyclic behavior of transversely isotropic cylindrical vessels. A numerical method is proposed to calculate the stresses and plastic strains in this structure due to the cycling of pressure at its inside surface. The numerical solution is validated simplifying the results with those of isotropic materials. Using the proposed method, the effect of anisotropy on ratcheting and shakedown response of the vessel is evaluated. It has been shown that the ratcheting or shakedown response of the vessel and the rate of ratcheting are highly affected by the anisotropy ratio. The numerical results of this paper show that the yield strength ratio, which is affected by initial work hardening of the metal, may control the ratcheting behavior of the cylindrical vessels.