A Quadratic Hermite Triangular Finite Element Method For Thin Plate Analysis

The4th order thin plate governing equation necessitates a C1numerical approximation within the Galerkin weak formulation. Compared with the widely used C0finite element approximation, it is not trivial to construct completely conforming C1finite element shape functions even for simple triangular elements. In this thesis, the C1elements are summarized and their shape functions are discussed in detail. Subsequently the Powell-Sabin-6quadratic Hermite triangular element is introduced into the thin plate analysis. This three-node triangular element ideally has one deflectional and two rotational degrees of freedom and is completely C1continuous over the problem domain. Meanwhile, this element has a constant curvature representation and thus is highly simple and efficient from the computational point of view. The accuracy and convergence of this element is systematically demonstrated through a series of benchmark thin plate examples.On the other hand, similar to the plane constant strain triangles, the constant curvature Hermite triangular elements also share the advantages of simplicity and efficiency, but their accuracy is not very attractive. In order to further improve the accuracy, a smoothed finite element method is developed based on the quadratic Hermite triangular elements within the curvature smoothing framework. In this method, the sub-triangle for the PS-6shape function construction is used as the curvature smoothing domain. The resulting smoothed curvature is employed to evaluate the bending energy which leads to the smoothed quadratic Hermite finite element method. Both static and free vibration analysis of sample thin plate problems prove that the smoothed finite element formulation significantly improve the numerical accuracy of the quadratic triangular Hermite finite element method.