## Preliminary Numerical Modelling

In order to plan and design the experimental activity, several numerical analyses have been performed. The purpose has been to estimate either the minimum and maximum load to be applied and, on the other hand, the out of plane displacements of asymmetric specimens. These preliminary results will give precious information in order to organize in a proper manner the test setup, giving an esteem of the maximum load to be applied by the test machine. Indeed, the knowledge of ultimate displacements let to properly position the measurement devices (that will be LVDT in this case). Moreover, these initial numerical models are to be considered as the starting step of an essential theoretic activity finalized to deepen the knowledge of riveted lap joints' structural behaviour. Therefore, referring to Figure 12, four riveted specimens have been initially analyzed. In details, they are the specimens:

Numerical analyses have been carried out by means of the finite element program ABAQUS 6.5 The 8-node brick continuum element C3D8R, with 8 nodes per element, 3 degrees of freedom per node and a linear interpolation function was adopted for modelling both rivet and plates. All degree of freedom of the end portion of the plates were restrained in order to reproduce the actual boundary condition during the test. The material stress-strain relation-ships for the rivets and plates have been obtained starting from the relevant experimental test on materials (shown in Section 2.1.). In particular, the average experimental stressstrain curve has been converted into the true stress-true strain relationship, with a plateau corresponding to the ultimate strength ( fu). The plasticity behaviour was based on the Von Mises yield surface criterion. Large deformation effects have been considered.

In particular, it has been introduced the following contact conditions: (a) at interface between the plates (Figure 13a); (b) between the internal surfaces of the hole and rivet shank (Figure 13b); (c) between the rivet heads and the external plate surfaces (Figure 13c,d). After a preliminary sensitivity study of the mesh, in order to minimize numerical discrepancy due to the contact boundary conditions, it needed to thicken the mesh sizing, making refined partitions of plates and rivet as shown in Figures 14a,b.

Moreover, the Coulomb friction model (with a friction coefficient of 0.3) has been considered in all contact conditions. In addition, the rivet pre-stressing has been introduced by means of an imposed relative shortening of the rivet shank, so simulating the shank shrinking due to the cooling process after the Figure 13. Modelling assumptions: implemented contact conditions. Figure 14. Mesh thickening in the parts in contact.

riveting. The magnitude of the residual clamping force estimated assuming an equivalent shortening showed the attainment of a clamping force that approaches the yield load of rivets. Finally, the load pattern has been simulated by applying a relative displacement between the two opposite terminal ends of each connected plates, as shown in Figure 15.

The load versus in-plane deformation behaviour of lap joints is summarized per investigated models in Figure 16. This plot interestingly shows that in case of smaller specimens, the shear strength increase proportionally to the number of shear planes of the splice (in this case two times corresponding to two shear planes). In case of stronger specimens, the final shear capacity of both symmetric and asymmetric splices is almost the same. In fact, the failure seems to be governed by the yielding of steel plates.

In details, the predicted collapse mechanism of the specimen U16-10-1 is shown in Figure 17. As it can be observed, the shear failure concentrated in the rivet shank and the effect of the flexural deformation due to the eccentricity of the applied load is evident and Figure 15. Simulation of the experimental load pattern. Figure 16. Load vs. in-plane displacement of examined riveted specimens. Figure 17. Predicted collapse mechanism of U16-10-1 (amplified deformed shape). Figure 18. Predicted collapse mechanism of S16-10-1.

Figure 16. Load vs. in-plane displacement of examined riveted specimens.

Figure 17. Predicted collapse mechanism of U16-10-1 (amplified deformed shape).

Figure 18. Predicted collapse mechanism of S16-10-1.

confined to the regions where plate discontinuities occur.

Figure 18 summarizes the failure mechanism of the symmetric splice S16-10-1. Contrary to the previous case, the prevalent collapse mechanism is the heading of the inner plate, essentially due to the shear load concentration.

Figure 19 shows the failure mechanism of the symmetric splice U22-12-4. In this case it is evident the plastic engagement of the connected plates due to the out-of plate displacement induced by the secondary bending. In this case, the collapse is mainly due to the stress concentration in the zone of plates where the geometrical discontinuity occurs.

Finally, Figure 20 highlights the failure mechanism of the symmetric specimen S22-12-4. Contrary to the previous case, no secondary bending obviously occurs and, like the S16-10-1 specimen, the prevalent collapse mechanism is, again, the heading of the inner plate, essentially due to the shear load concentration.

Generally speaking, it can be affirmed that in case of lap joints unrestrained against out-of-plane displacements the joints showed considerable deformation due to the eccentricity of the load. It is evident that the effects of bending are mainly confined to the regions where plate discontinuities occur. Obviously, as the joint length increases, bending will become less pronounced, and the influence on the behaviour of the connection should decrease (similarly to that highlighted by Shoukry & Haisch, 1970). The influence of bending is most pronounced in a splice with only a single fastener in the direction of the applied load. In such a joint the fastener is not only subjected to Figure 21. Out-of vs. in-plane displacement of examined riveted specimens. Figure 19. Collapse mechanism of U22-12-4 (amplified deformed shape). Figure 20. Collapse mechanism of S22-12-4.

Figure 21. Out-of vs. in-plane displacement of examined riveted specimens.

single shear, but a secondary tensile component may be present as well. Furthermore, the plate material in the direct vicinity of the splice is subjected to high bending stresses due to the load eccentricity. Hence, the bending tended to decrease slightly the ultimate strength of short connections. The shear strength of longer asymmetric lap joints seems to be less affected by the effects of bending.

Figure 20 shows the predictive numerical response in term of out-of plane vs. relative in-plane displacement for each modelled unsymmetrical specimen. From these preliminary numerical analyses, it seems that the influence of bending is most pronounced in a splice with only a single rivet in the direction of the applied load. However, this has little influence on the load capacity, since the material will strain-harden and cause yielding on the gross area of the connected plate, as it can be observed comparing the joint capacity to the hand calculation. Moreover, it was found that the bending stress component varies significantly within the splice region. This may be attributed to the large stiffness variation caused by the rivets. 