DOF Condensation of Thick Curved Beam Element Formulated by Isogeometric Approach

Abstract

The study of a thick in-plane curved beam is more complex than that of the straight beam because the structural deformations of the curved beam depend not only on the rotation and transverse displacement but also on the coupled tangential displacement caused by the curvature of the structure. The Isogeometric approach is a computational geometry based on a piecewise ratio function (Non-Uniform Rational B-Spline (NURBS)) used to represent the exact geometry. In the Isogeometric approach, the free curvature geometry of the beam element can be represented exactly. A thick two-node curved beam element can be developed by using the Isogeometric approach based on Timoshenko beam theory, which allows the transverse shear deformation and rotatory inertia effects. The natural shape of the beam curvature and the shape functions formulation of the element can be formulated by using the Isogeometric approach. However, in the Isogeometric approach, the number of equations will increase according to the number of degree of the polynomial and its control points. A novel technique is been proposed to condense the number of equations of the DOFs at control points so that it is equal to the standard two-node six DOFs beam element. This paper highlights the application of the NURBS for a curved Timoshenko beam element in the context of finite element analysis and proposes a new condensation method to eliminate the drawbacks raised from the Isogeometric approach. Examples are given to verify the effectiveness of the condensation method in static and free vibration problems.