138 I. Zare et al. This advancement prompted Jewell et al. [3] to apply this technique to detailed finite element models that included nonlinear contact between the bolted interfaces using a commercial software package. While their results showed that such an analysis is feasible, they struggled to obtain accurate results and noted that, once the structure had been meshed with adequate fidelity to capture micro-slip, the computational cost was very significant even to perform a single static analysis. This work explores a more computationally efficient alternative that follows the work of Ahn and Barber [4, 5]. Specifically, static reduction is used to eliminate all of the DOF except those at the contact interface, and then a Gauss-Seidel algorithm is used to solve the nonlinear contact problem in Matlab. A variant on the Gauss-Seidel algorithm has been developed, based on the Block-Gauss Seidel approach and will be elaborated in the presentation, that seems to improve the computational efficiency and the resulting algorithm is evaluated by comparing it with the commercial software package, Abaqus ®. 11.2 Sample Results The proposed algorithm was tested by applying it to the structure shown in Fig. 11.1, which represents a very simple structure with a bolted joint. The normal load, P, represents the clamping load from a bolt and the lateral load, Q, provides a loading to the structure that induces micro-slip. A finite element model was created of this structure, with 300 nodes along the contact interface. Then the stiffness matrices for the top and bottom blocks was exported to Matlab and statically reduced using the approach in [5]. The proposed algorithm was then to find the contact forces at the interface, and they were compared to a solution of the full order FEM in Abaqus. The results were indistinguishable, and so only the result from the proposed algorithm is shown. The difference between the solutions, expressed as a percentage by dividing it by the largest force over the contact surface, was computed and is shown below. There are small differences at the edges of the contact, presumably due to a slightly different number of nodes being in contact in the two solutions, but the forces are within 1% of each other (Fig. 11.2). While this simple model could be solved very quickly in Abaqus, the solution above took only 30 s, the proposed algorithm was even faster, solving in uncompiled code in less than 0.15 s. This difference is expected to be very important when going to three-dimensional models, where solve times have been in excess of 24 h even for relatively simple models [3]. Furthermore, commercial contact algorithms are known to favor robustness over accuracy, which perhaps explains the challenges encountered in [3] in obtaining physically reasonable results using a commercial code. In the future this algorithm will be used to compare predictions of the stiffness and damping of joints to experimental measurements. P P Q Q 2h 2h h h 4h Fig. 11.1 Beam structure with a clamping load representing a bolted joint and a shear load which exercises the joint
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