4. Numerical example The latter idea has been applied to the FE structure printed on the right side of figure 2. The structure consists of two 0.5mm iron sheets which are connected with four spot welds. The sheets are outlined in blue and orange color and the positions of the spot welds are indicated by green ‘spiders’. The joint is defined by the overlapping area of the two sheets. The dimension of the joint can be given as 155mm x 75mm. Both ends of the structure are considered as rigid, which is indicated by a black color in figure 2. One end is constrained and the other end is loaded by a single force which is indicated by the arrow in orange color. Figure 2: FE structure and reference gap The reference computation has been done with all nodal FE DOF and with the non penetrations condition (6) for all 512 master surface FE nodes in the joint. The selection of the master surface is unimportant because the FE nodes in the joint coincide in unstressed reference position. All computations have been performed using the ‘qpsolve’ routine of the software Scilab 5.2.2 [11]. The gap distribution of this reference computation can be seen on the left hand side of figure 2. The tip deflection of this computation is 5.3 mm. A series of mode based computations according to the equations (16) to (18) have been performed. Each computation varies either in the number of considered JIM or/and in the number of constraints (=subareas). The considered JIM are used as extension to 10 modes which have been computed based on an eigenvalue problem of the stiffness matrix K. Four different relative errors in percent have been evaluated. The first one is the relative error in the tip deflection which is computed according to § · ¨ ¸ © ¹ red T T T T x x e 100 x (19) where red Tx is the tip deflection of the load application point in y direction computed with the modal approach. The quantity Tx is the according reference value based on the reference computation done with all DOF and all available joint constraints. Three other errors are based on the different norms of the vector g. Formally these errors can be written as § · ¨ ¸ ¨ ¸ © ¹ red i i i i e 100 g g g and f i=1,2, . (20) If i = 1 the error of the Manhattan norm, if i = 2 the error of the Euclidian norm and if i = f the error of the maximum norm is computed. The computations have been performed for 0, 10, 20, 30, 40 and 50 considered JIM and the number of constraints (=subareas) have been 1, 2, 4, 8, 16, 32, 64, 128, 256 and 512. The results are plotted as contour plots and can be seen in figure 3. The four plots indicate that there is an optimum number of considered constraints (=subareas) with respect to the used number of JIM. Both, too less 33
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