Completely hierarchical two-dimensional curved beam element for dynamics

Abstract

This paper presents a completely hierarchical two-dimensional curved beam element formulation for linear dynamic analysis based on p-version. 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. Since the element displacement approximation is hierarchical, the resulting element stiffness matrix, the element mass matrix and the equivalent nodal load vectors are also hierarchical. The formulation ensures C° continuity. The element properties are derived using the principles of virtual work and the hierarchical element approximation. In the development of the element properties complete two-dimensional state of stress ( δ x, δ y, τ xy) and strain ( ϵ x, ϵ y, γ xy) is utilized, i.e. the stress normal to axis of t neglected (an assumption commonly used in the two-dimensional beam element formulations) in the present formulation. The formulation permits variable p-level control in axial and transverse directions of the element. Thus higher order transverse deformation coupled with higher-order longitudinal effects in deep and short beams can be easily and accurately simulated for dynamic applications. The formulation also permits nondifferentiable geometry (sharp corners) and the dynamic stress concentrations at such locations can be calculated accurately. 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 a variety of slender and deep 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 Timoshenko beam theory and other available analytical solutions.