A Comparison for the Simulation of Frictionless Contact Problem with Large Displacement

Abstract

Abstract. Studies of contact problem have been widely executed by researchers with variable scopes, methods and definitions. A common problem that occurs while handling contact phenomena is sliding through element boundary [1], due to the discontinuity of the local coordinate between elements and a contact point [2] [3]. This common problem that occurs at an element boundary is such that a stable convergence result is not easily obtained. [4], thus inspires authors to make a comparison of two beam methods which are Euler-Bernoulli beam theory and Timoshenko beam theory for frictionless contact problem. The authors have been investigate geometrically non-linear analysis with extremely large displacement by the aid of Tangent Stiffness Method (TSM) [5], a robust non-linear analysis method to execute analysis and produce results with high accuracy. In this study, the authors propose the modification of beam elements with three nodes by considering the adaptation of shear deformation by Timoshenko beam theory. The modification enables the contact point to slide through the element edge smoothly and some numerical examples are provided in this study. Tangent Stiffness MethodThe TSM was solely idealized to overcome numerical cases exhibiting significant nonlinearity. The superiority of this method is that it converges the unbalanced force with high accuracy by defining element behavior using a simple form of the element force equation. This theory requires the element edge forces to be treated separately and independently of each other. In addition, strict compatibility and an equilibrium equation are disseminated in the iteration configuration to converge the unbalanced force.Here, let an element constituted by two edges with its element edge forces and the force vector for both edges is assumed as S. Let the external force vector as U, in a plane coordinate system with J as the equilibrium matrix, and the equilibrium condition could be expressed as the following equation.With the differentiation of Eq. (1), the tangent stiffness equation could be expressed as;Here, the differentiation of Eq. (1) simultaneously extract δS and δJ makes it possible to express a linear function of displacement vector, δd in the local coordinate system. Meanwhile, in Eq. (2), K O represents the element stiffness matrix which also simulates the element behavior, correspondent to the element stiffness in the coordinate system while K G , represents the element displacement originated by the tangent geometrical stiffness.