Three-dimensional curved beam element based on p-Version for dynamics

Abstract

This paper presents a three-node curved three-dimensional beam element for linear dynamic analysis, where the element displacement approximation in the axial (ξ) and transverse directions (η and ζ) can be of arbitrary polynomial orders, p ξ , p η , and p ζ . 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 of these 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 are hierarchical. The formulation guarantees C 0 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 tensors are retained. 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 available analytical solutions 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. To demonstrate the effectiveness of this hierarchical formulation in dynamics, numerical results are presented for eigenvalue analysis. The frequencies and the mode shapes are presented for beams with various boundary conditions to demonstrate the accuracy, efficiency, modelling convenience and overall superiority of the present formulation. Numerical results obtained from the present formulation are also compared with those given by the Timoshenko beam theory and other available analytical solutions.