The body consists of two substructures with rectangular shaped cross section (a ¼10 mm, b ¼20 mm) and an overall length of l ¼110 mm. The substructures are connected by a contact interface and a single bolt. All structural components are modeled using a linear elastic material with Young’s Modulus E ¼210,000 N/mm2, Poisson ratioν ¼0.29 and density ρ ¼7820 kg/m3. The discretization bΩ of the full model comprises 1834 elements and 2780 nodes. The CHEXA, CBEAM, RBE2 and RBE3 element formulations of the commercial FEM software package MSC Nastran are used to generate the system matrizes within Eq. (15.10). The contact interface bΓjc is discretized with an 4 (8) node zero thickness element formulation which allows to obtain the contact stress equivalent nodal force vector within Eq. (15.13). A convergence study regarding the meshsize of the full model is not within the scope of this contribution. It is assumed that the accuracy of the contact stresses obtained from this numeric model are sufficiently accurate for the purpose of this model. The purpose of the reduced order model is to reproduce these contact stresses with sufficiently low error but significantly higher computational efficiency. 15.4.2 Sticking Friction Definition During generation of the system matrices using a commercial FEM software package there is no connection of the two contacting substructures except the beam element based bolt model. The resulting Eq. (15.13) is utilized for a contact simulation where the structure is constrained at bΓd and the bolt is incrementally pretensioned to a nominal value. The resulting contact stresses are depicted on the left hand side of Fig. 15.3. All contact node pairs possessing a contact pressure over a certain threshold are assumed to stick together in tangential direction during all subsequent simulations. The node pairs selected this way are marked on the right hand side of Fig. 15.3. The sticking friction condition within this area is approximated utilizing a penalty approach. All subsequent steps consider these additional penalty stiffness related entries in the stiffness matrix. 15.4.3 Reduction Basis The computation of static and dynamic loadcases to generate snapshots for obtaining a reduction basis was carried out according to Sect. 15.3.4. The static loadcases represent load combinations of bolt pretension, vertical tip load and torsional moment at the free end of the cantilever. The dynamic loadcases, computed with a nominal pretension of the bolt, are limited to combinations of vertical load and torsional moment at the free end of the cantilever. For easier computation the equation of motion comprises an additional Rayleigh approach based damping matrix. The step responses are obtained from the equation of motion, which is transformed to a system of first order differential equations, utilizing the implizit Euler integration scheme. Fig. 15.3 Contact stresses in surface normal direction resulting from a nominal pretension of the bolted joint (left). Locations with sticking friction are identified based on a contact stress threshold. To suppress the tangential relative movement of the adjoining contact node pairs a penalty approach is utilized at the marked node pairs (right) 15 Numerical Studies on the Reduced Order Modeling of Frictionless Joint Contact Interfaces 175
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