Non-Linear Strain Description for Two-Dimensional Shear Deformable Beam Element Based on the Absolute Nodal Coordinate Formulation

In the absolute nodal coordinate formulation, the transverse shear deformation can be accounted for by using a fully parametrized element or, alternatively, by replacing longitudinal slope coordinates by a vector that describes the orientation of the cross-section. The use of a fully parametrized element allows the description of cross-section deformations in the case of beams and, correspondingly, fiber deformations in the case of plates and shells. It is noteworthy, however, that cross-section or fiber deformations are usually associated with high natural frequencies complicating the time integration of a fully parametrized element. A procedure to replace longitudinal slope coordinates by the vector that describes cross-section orientation was recently applied to a two-dimensional beam element based on the absolute nodal coordinate formulation. In this study, the procedure to account for shear deformation using the vector that describes cross-section orientation is extended to account for the nonlinear strain-displacement relationship in the definition of the elastic forces of the beam element. To accomplish this, the exact displacement field is used in the description of element kinematical and strain measures. This makes it straightforward to implement the non-linear strain-displacement relationship in the description of the elastic forces. Numerical results demonstrate that the enhanced beam element yields accurate results in eigenfrequency analysis. Results obtained in large deformation cases are in line with previously proposed elements based on the absolute nodal coordinate formulation.