Certain engineering materials exhibit different elastic behaviour in tension and compression. Such materials are commonly modeled as bimodular, characterized by different Young’s moduli in tension and compression. As a result, the neutral axis of a bimodular beam shifts from the geometric centroid of the cross-section, lying above or below it depending on the curvature sign. Consequently, the governing equation for flexural vibrations is formulated using an effective two-layer laminate model with a discontinuous neutral axis. In this framework, the neutral axis position in each bending state depends not only on the material elastic properties but also on the cross-sectional characteristics, where the cross-section varies along the beam axis. Depending on these properties, a bimodular beam may exhibit either linear or nonlinear dynamic response. When the effective bending stiffness differs between upward and downward bending, a single closed-form solution valid over the entire motion domain cannot be derived. Therefore, an isogeometric finite element discretization is employed to compute approximate solutions, focusing on the nonlinear vibrations of a bimodular beam. Stiffness and mass orthogonal load-dependent Ritz vectors are used to achieve higher accuracy with fewer vectors than approaches based on exact eigenvectors. Numerical studies of bimodular tapered beams with different cross-sectional properties demonstrate the significant influence of bimodular material behavior on the structural response, especially in relation to cross-sectional properties.