This research paper applies the finite element method as a methodology to evaluate the structural performance of nonlinear viscoelastic solids. A finite element algorithm was built and developed to simulate the mathematical nonlinear viscoelastic material behavior based on incremental constitutive equations. The derived Equation of the incremental constitutive included the complete strain and stress histories. The Schapery’s nonlinear viscoelastic material model was integrated within the displacement-based finite element environment to perform the analysis. A modified Newton-Raphson technique was used to solve the nonlinear part in the resultant equations. In this work, the deviatoric and volumetric strain–stress relations were decoupled, and the hereditary strains were updated at the end of each time increment. It is worth mentioning that the developed algorithm can be effectively employed for all the permissible values of Poisson’s ratio by using a selective integration procedure. The algorithm was tested for a number of applications, and the results were compared with some previously published experimental results. A small percentage error of (1%) was observed comparing the published experimental results. The developed algorithm can be considered a promising numerical tool that overcomes convergence issues, enhancing equilibrium with high-accuracy results.