2. Introduction and Motivation The presence of joints, like bolted joints, spot welded seems and others, is a computational challenge in assembled structures. From a mechanical point of view, a joint leads to a contact and friction problem with a possibly time invariant contact area. It is well documented that these local nonlinearities can significantly influence the structures local and global response, see exemplarily [1], [2] and [3]. For the accurate consideration of contact phenomenon’s inside complex jointed structures typically the Finite Element Method (FEM, [4] and [5]) has been used. The FEM is characterized by accurate results in case of a satisfying discretization of the domain of interest. This, on the other hand, leads to a high number of degrees of freedom (DOF) which disqualifies the method for the time integration of jointed structures. Recently developed Joint Interface Modes (JIM) overcome that problem. Joint Interface Modes (see [6]) are a problem oriented Ritz vector type extension to existing, well proven mode bases like the one of Craig and Bampton [7]. The basic idea of JIM is the generalization and reduction of the joint DOF. Based on a proper mode base which is enriched with JIM, an efficient and accurate time integration of jointed structure is possible even the nonlinear contact problem is considered, see [6] and [8]. The publications [6] and [8] have in common that the contact forces are realized via a penalty approach. The penalty approach works like a nonlinear spring which connects both contact surfaces. This ‘virtual’ spring has zero stiffness in case of gaping and a nonzero stiffness in case of penetration. The penalty approach is quiet straight forward to implement. The contact forces are computed as a function of the gap state and finally projected into the subspace for the time integration. This approach does not influence the number of degrees of freedom of the equation of motion. Another approach for computational contact mechanics is the use of non penetration constraints which modify the structure of the equation of motion because a set of active constraint equations has to be considered as well. Normally the number of nodal joint DOF of the FE model is higher as the number of considered Ritz vectors. The straight forward approach to impose a non penetration constraint onto all nodal joint DOF will lead to an over constraint system and is therefore not applicable. One possibility to overcome that problem will be introduced in this contribution. Instead of a FE node oriented application of the non penetration constraints, an area wise application of the latter constraint is suggested and investigated. The joint area will be subdivided into several subareas and the non penetration constraint is applied in an averaged form. After the idea is formally introduced in the next section a static example will be given. The contribution ends with a discussion of the results and some conclusions. 3. Theory A FE structure with n nodal DOF which contains a joint is outlined in figure 1. Note that for the reasons of simplicity the FE structure as no rigid body DOF. The external forces Bf are acting on the structure exclusively via the interface B and the according DOF are collected in the (nB x 1) vector Bx . The nodal DOF of Bx are outlined in figure 1 by white dots. The nodal DOF which are involved in the non-linear joint contact are collected in the (nIJ x 1) vector IJ x and are outlined in figure 1 by grey dots. Note, that the problem under consideration can be characterized as self contact. The remaining (n - nB - nIJ) DOF are denoted as Rx . The nodal DOF of Rx are outlined in figure 1 by black dots. According to that scheme, the vector of DOF can be written as ª º ¬ ¼ T T T T B IJ R x x x x and (1) the vector of external forces Bf takes on the form ª º ¬ ¼ T B T T T B f f 0 0 . (2) 30
RkJQdWJsaXNoZXIy MTMzNzEzMQ==