In this paper, the nonlinear behavior of beams with one-dimensional (1D) Functionally Graded Materials (FGM) is investigated by Carrera Unified Formulation (CUF). In the study, kinematic variants of CUF are extended with Lagrange Extension (LE), thus adopting Layer-Wise (LW) approach. In the CUF formulation combined with geometric nonlinear equations, the Lagrangian approximation and Newton-Raphson linearization scheme are used with the method based on arc length constraint. It is assumed that the variation of material properties in the thickness direction follows an exponential grading. The displacement and stress distributions of the Functionally Graded (FG) cantilever beam under transverse/axial loading are investigated for different compositional gradient exponents. The three-dimensional (3D) stress distributions of FG beams are investigated with the 1D CUF model and the results are confirmed by the literature. The effect of the compositional gradient exponent on the mechanical behavior is expressed and it is emphasized that the combined formulation is time-efficient and highly sensitive in determining the nonlinear behavior.