## 6 Numerical Investigation and Validation
### Double-Lap Test  
The double-lap test is a standard experiment used to assess the bolt connection between structural elements. The full details of the experimental campaign and test setup are fully described in the work of [57], in which it is possible to determine the applied load using impose displacements. Some of the material properties adopted for the numerical test are shown in Table 5, and these were obtained in the previously referenced work while others were estimated. Due to some findings' fluctuations, the fracture energy was calculated using the same values as in the earlier example in Table 2. For this numerical test, the bolt edge distances (cover) of 15 mm and 70 mm were simulated since these clearly show a difference between a shear-out and bearing collapse, respectively, as shown in Figure 15.  
A full 3D non-linear analysis was performed using the ABAQUS standard with C3D8 and C3D8R solid finite elements using full and reduced integration and hourglass control to evaluate any mesh sensitivity due to the high variability of the stress field. The time step restrictions are the same as the previous numerical models. For the GFRP plate, in order to reduce the computational cost, part of the plate near the opening was modelled with the UMAT, but for the other part, an elastic orthotropic material from the ABAQUS library was used. For the steel bolt, an isotropic elastoplastic material from the ABAQUS library was used (\(E_{s}=195~{}GPa\), \(\
u=0.3,f_{y}=300~{}MPa\), \(f_{u}=691~{}MPa\)). Again, to reduce the computational cost, only one-quarter of the structure was simulated, which represents one-half of the GFRP plate and one steel plate, as shown in Figure 16, in which the imposed vertical displacement is inserted into the bottom of the steel plate. Only the ITE algorithm was employed in this example, with the primary goal of numerically evaluating the simulation of the bearing collapse in the contact between the steel bolt and the GFRP plate. Figure 15: Test setup for the loading conditions and impose displacement for the double-lap test, and used bolt edge distance (cover) for the circular opening [57]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\text{E}_{1}\) [MPa] & \(\text{E}_{2}\) [MPa] & \(\text{E}_{3}\) [MPa] & \(\text{G}_{12}\) [MPa] & \(\text{G}_{13}\) [MPa] & \(\text{G}_{23}\) [MPa] \\ \hline
21,300 & 2900 & 2900 & 3000 & 2500 & 2500 \\ \hline \(\text{v}_{12}\) [\(-\)] & \(\text{v}_{13}\) [\(-\)] & \(\text{v}_{23}\) [\(-\)] & \(\text{X}_{\text{t}}\) [MPa] & \(\text{X}_{\text{c}}\) [MPa] & \(\text{Y}_{\text{t}}\) [MPa] \\ \hline
0.28 & 0.28 & 0.30 & 334.0 & 316.0 & 29.0 \\ \hline \(\text{Y}_{\text{c}}\) [MPa] & \(\text{Z}_{\text{t}}\) [MPa] & \(\text{Z}_{\text{c}}\) [MPa] & \(\text{S}_{\text{L}}\) [MPa] & \(\text{S}_{\text{T}}\) [MPa] & \(\text{n}\) [\(-\)] \\ \hline
51.9 & 29.0 & 51.9 & 52.4 & 33.8 & \(10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Material properties for the pultruded GFRP material used in the double-lap tests. To correctly simulate the bearing, the residual compressive stress after the maximum peak must also be simulated. Two values of compressive residual stress were considered, namely, 15%, as estimated from the work of [60], and 80%, which is approximately the relation between the bearing resistance and compressive resistance according to [64], when using the experimental values form [57]. In addition, to improve convergence, the tensile residual stress was 1% and 10%, respectively. A sensitivity test showed that residual tensile stress below 15% had no influence on the maximum numerical resistance and its softening branch. The structural response of the numerical models and its comparison with the experimental campaign is exhibited in Figure 17 for both 15 mm and 70 mm cover, with 15% and 80% residual compressive strength, for both reduced (RED) and full (FULL) integration.