Most of the existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Classical theories based on simplifying assumptions that are not applicable for nonlinear large deformation problems are often used. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this thesis, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation (ANCF). This formulation relaxes the assumptions of the Euler-Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. In this thesis, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These nonlinear constitutive models are based on the Neo-Hookean constitutive law for compressible materials, the Neo-Hookean constitutive law for incompressible materials, and the Mooney-Rivlin constitutive law in which the material is assumed to be incompressible. Several beam and plate ANCF finite elements based on linear and nonlinear constitutive models are applied to several applications including rotor blades, belt drives and tracked vehicles with rubber chains. In the case of belt drives, two large deformation three-dimensional ANCF finite elements are used to develop two different belt drive models that have different numbers of degrees of freedom and different modes of deformation. The results obtained using the two finite element formulations are compared with a simplified analytical solution, and the convergence of the finite element solution is examined. In this thesis, the effect of the centrifugal forces on the flap and lag modes of a rotating beam is also examined. The results obtained using the ANCF and the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation, are compared. The fundamental problem of modeling slope discontinuities is also addressed in this thesis. A new procedure that can be used to model structures with slope discontinuities in the ANCF is investigated. This procedure can be applied to model slope discontinuities in the case of fully parameterized elements as well as in the case of gradient deficient elements that are used for modeling thin beam and plate structures.