The problem of synthesizing the reduced-order dynamic models for continuous elastic systems in a geometrically nonlinear formulation (primarily thin-walled structures) based on the finite element method is considered. The approaches under consideration are based on the idea of identifying the nonlinear (quadratic-cubic) stiffness characteristic of an elastic system in its modal coordinates, followed by the application of the theory of nonlinear normal modes and Poincaré normal forms to construct an invariant manifold tangent to the selected modal subspace. The developed algorithm is used to construct a nonlinear model of coupled longitudinal-bending vibrations of a clamped-clamped beam and its verification based on an approximate analytical solution by the Galerkin method. The features of the software implementation of the presented method based on the ABAQUS finite element analysis software system are discussed.