This paper analyzes the effect of the polymer matrix non-viscoelastic behaviour in the mechanical behaviour of thick multilayered cylinders. The original contribution of this work is to provide novel approximate analytical solutions to compute the timedependent internal stress state throughout the pipe thickness within the  framework of nonlinear viscoelasticity theory. The structures considered are thick, multilayered anisotropic infinitive long cylinders subjected to axisymmetric mechanical loading .Under such conditions there is an exact elastic solution which naturally satisfies equilibrium, strain-displacement, compatibility and boundary  conditions for the stated constitutive equations and loading. Due to the continuous stress variations  throughout the cylinder thickness, the proposed nonlinear viscoelastic solution assumes the averaged stress state to calculate the nonlinear elastic and viscoelastic factors in each  layer. Furthermore the solution is obtained assuming that the creep strains, within each  layer, are constant through the thickness. The proposed algorithm converges to the exact solution when the number of layers is artificially increased. For the linear viscoelastic case the proposed solution proved to match the exact known solution for isotropic viscoelastic materials. Finally several invented cases are run to illustrate the importance of the viscoelasticity phenomenon on the internal stress field throughout thick laminated cylinders.