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

Studies of contact problem have been widely executed by researchers with variable scopes, methods and definitions. A common problem 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]. The common problem that occurs at an element boundary is a stable convergence result is hard to achieve [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. Authors have been investigated geometrically non-linear analysis with extremely large displacements by using Tangent Stiffness Method (TSM) [5], a robust non-linear analysis method to execute analysis and produce results with high accuracy. In this study, authors propose the modification of the 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 Method
The 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.

=
(1) 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.

Comparison of Euler-Bernoulli and Timoshenko beam theory for extremely large loading increment
With the aforementioned method, TSM could solve any geometrically non-linear problem, even for extremely large deformation. Therefore, in this section, author will provide a comparison for extremely large loading for both Euler-Bernoulli beam and Timoshenko beam with a common plane frame structure.  Fig. (1) shows a simply supported beam with a roller support at one end and a pinned support at the other. An extremely large bending moment is applied in a single incremental step at the roller support until the beam deformed to a circular shape. In addition, as shown in Fig. 3, the beam

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International Integrated Engineering Summit 2014 meshes are set from 6 to 200 meshes, and for this case, a stable convergence result for Euler-Bernoulli beam is until 52 meshes, while for Timoshenko beam, stable convergence result has been achieved even until 200 meshes.

Comparison of Euler-Bernoulli and Timoshenko beam theory for frictionless contact problem
In this section, the element force equation for Euler-Bernoulli and Timoshenko beam is introduced. Fig. (4) shows an equilibrium condition of an elastic and homogeneous simply supported beam which is subjected by an axial force N, edge moments M i and M j , and contact force Y c . The element force equation for Euler-Bernoulli beam in contact case is shown in Eq. (6) and for Timoshenko beam is shown in Eqs. (7), (8) and (9). The difference between these two theories are in Timoshenko beam, shear deformation (γ) is considered even for the large deformational case.   Fig. (7), it is significantly clear that by the consideration of shear deformation in Timoshenko beam, a stable yet converged solution have been successfully achieved at the edge of the segment which ranges from 99.499% to 99.933%. For Euler beam, the percentage ranges from 87.408 % to 92.251% and for cantilever coordinate, it ranges from 96.135% to 97.836%. Figure 6 The cantilever beam deformation due to the contact node Figure 7 The relation between percentage l i /l and the distance of the contact node

Summary
For the Euler-Bernoulli beam theory, unbalanced force will either diverge or no convergence result could be achieved beyond the ranges. On the other hand, for Timoshenko beam, unbalanced force is steadily converged around the tip of the segment, and the contact node is able to slide through to the next segment.