Hierarchical three dimensional curved beam element based on p-version

Abstract

This paper presents a three node curved three dimensional beam element for linear static analysis where the element displacement approximation in the axial (ξ) and transverse directions (η and ζ) can be of arbitrary polynomial orders pξ, pη and pζ. This is accomplished by, first constructing one dimensional hierarchical approximation functions and the corresponding nodal variable operators in ξ, η and ζ directions using Lagrange interpolating polynomials and then taking the products (also called tensor product) of these hierarchical one dimensional approximation functions and the corresponding nodal variable operators. The resulting approximation functions and the corresponding nodal variables for the three dimensional beam element were hierarchical. The formulation guarantees C0 continuity. The element properties are established using the principle of virtual work. In formulating the properties of the element all six components of the stress and strain tensor are ratained. The geometry of the beam element is defined by the coordinates of the nodes located at the axis of the beam and node point vectors representing the nodal cross-sections. The results obtained from the present formulation are compared with analytical solutions (when available) and the h-models using isoparametric three dimensional solid elements. The formulation is equally effective for very slender as well as deep beams since no assumptions are made regarding such conditions during the formulation.