Large deformation analysis for anisotropic and inhomogeneous beams using exact linear static solutions

Abstract

The existing statically exact beam finite element (FE) based on the first order shear deformation theory (FSDT) is used to study the geometric nonlinear effects on static and dynamic responses in isotropic, composite and functionally graded material (FGM) beams. In this beam element, the exact solution of the static part of the governing differential equations is used to construct the interpolating polynomials for the element stiffness and mass matrix formulation. The rotary inertia is also taken into account while formulating the mass matrix. Consequently, the stiffness matrix is statically exact and the mass matrix is much more accurate compared to any other existing FEs. These two aspects make the element devoid of shear locking and an efficient instrument for analysing wave propagation problems, wherein refined mesh is essential. A total Lagrangian formulation of the linear beam element is employed for large displacement and large rotation analysis. The superconvergent property of the beam element is shown by comparing the convergence rate of this beam element to the reduced integrated FSDT beam element formed using the linear shape functions. Numerical examples deal with the static and wave propagation problems to highlight the nonlinear effects. In the static case, both Von-Karman strain tensor and Green’s strain tensor is used, whereas, for the wave propagation studies only the Von-Karman strains are used. Power-law variations of the material property distribution is used for the FGM beam. Both high and low frequency pulse loading is employed to bring out the effect of nonlinearity on the transient response.