Two nodes cusp geometry beam element by using condensed IGA

Abstract

A cusp is a curve which is made by projecting a smooth curve in the 3D Euclidean space on a plane. Such a projection results in a curve whose singularities are self-crossing points or ordinary cusps. Self-crossing points created when two different points of the curves have the same projection at a point. Ordinary cusps created when the tangent to the curve is parallel to the direction of projection on a single point. The study of a cusp geometry beam is more complex than that of a straight beam because the structural deformations of the cusp geometry beam depend also on the coupled tangential displacement caused by the singular geometry. The Isogeometric Approach (IGA) is a computational geometry based on a series of polynomial basis functions used to represent the exact geometry. In IGA, the cusp geometry of the beam element can be modeled exactly. A thick cusp geometry beam element can be developed based on the Timoshenko beam theory, which allows the vertical shear deformation and rotatory inertia effects. The shape of the beam geometry and the shape functions formulation of the element can be obtained from IGA. However, in IGA, the number of equations will increase according to the number of degree of freedom (DOF) at the control points. A new condensation method is adopted to reduce the number of equations at the control points so that it becomes a standard two-node 6-DOF beam element. This paper highlights the application of IGA of a cusp geometry Timoshenko beam element in the context of finite element analysis and proposes a new condensation method to eliminate the drawbacks elevated by the conventional IGA. Examples are given to verify the effectiveness of the condensation method in static and free vibration problems.