P-Version hierarchical two dimensional curved beam element for elastostatics

Abstract

This paper presents a completely hierarchical two dimensional curved beam element formulation where the element displacement field can be of arbitrary polynomial orders p ξ and p η in the axial and the transverse directions of the element. The approximation functions and the corresponding nodal variables for the beam element are derived by first constructing the hierarchical one dimensional approximation functions of orders p ξ and p η , and the corresponding hierarchical nodal variable operators for each of the two directions ξ and η and then taking their product. This procedure yields approximation functions and nodal variables for the curved beam element that correspond to polynomial orders p ξ and p η in ξ and η directions. The element approximation is hierarchical, i.e. the approximation functions and the nodal variables are both hierarchical. Thus, the element matrix and the load vectors corresponding to the polynomial orders p ξ and p η are a subset of those corresponding to the polynomial orders ( p ξ + 1) and ( p η + 1). The element formulation ensures C 0 continuity. The element properties are derived using the principle of virtual work and the hierarchical element displacement approximation. The element geometry is constructed using the coordinates of the nodes located on the elastic axis of the element and the node point vectors indicating nodal depths and the element width at the nodes. The orders of approximation along the length of the element as well as in the transverse direction can be chosen independently to obtain optimum (maximum) rate of convergence. Numerical examples are presented to demonstrate the accuracy, simplicity of modeling, effectiveness, faster rate of convergence and overall superiority of the present formulation. Results obtained from the present formulation are also compared with h-approximation models and available analytical solutions.