## 4 Results
### Influence of Low Joint Stiffness on the Simulation Results  
To decrease the global stiffness of the model, the interface stiffness is now reduced (Figure 4). As noted in Section 3, given the low stress acting in the experiments, block deformation is unlikely to affect the results significantly: therefore, blocks were kept rigid. On the contrary, it is known that low stresses lead to lower joint stiffness [33; 34; 46]. As an example, the formulas found by [33; 46] for other materials would give the following stiffness at the low vertical stress of the experiments (\(\sigma\) = 2.6 kPa at the bottom of the model):  
* \(\mathrm{k_{n}}\) = 17.52 \(\times\) 2.6 = 4.6 \(\times\) 10\({}^{7}\) Pa/m [46]
* \(\mathrm{k_{n}}\) = 8.8-30.4 \(\times\) 2.6 = 2.3 \(\times\) 10\({}^{7}\)-7.9 \(\times\) 10\({}^{7}\) Pa/m [33]  
Obviously, these values have been obtained for other materials, and one could argue that the fitting carried out in [33; 46] does not perfectly match the initial curve of the joint closure tests. However, at least it gives reference values for numerical purposes. In addition, block imperfections (non-planarity and large asperities) have been acknowledged to further decrease joint stiffness [32; 35]. As the contact area decreases, the local stress increases locally, thus the deformation increases [32]. More importantly, [35] brought to light that the actual local normal joint stiffness in a masonry wall is less than that identified through classical joint closure tests because of the misalignment of staggered blocks (height differences) and geometrical imperfections of the blocks themselves (non-parallelism of faces). This last comment totally applies to the considered experimental campaign [22]. From the results in [PERSON] et al. [35], the ratio between normal stiffness in the wall and normal stiffness of joint closure tests lies between 2 and 3. Finally, a few numerical studies focused on dry-stone masonry walls assembled with rubble stones. Given the highly imperfect geometries of stones, they used relatively low stiffness values (\(\mathrm{k_{n}}\) = 5 \(\times\) 10\({}^{7}\)-1.25 \(\times\) 10\({}^{8}\) Pa/m) to reproduce field experiments [47; 48; 49]. Figure 5: Collapse mechanism observed for WD 1. (**a**) Experimental [22]; (**b**) Limit Analysis (Non-Associative) [27]; (**c,d**) DEM simulations with \(\mathrm{k_{n}}\) = \(\mathrm{k_{s}}\) = 1 \(\times\) 10\({}^{9}\) Pa/m for different tilting angles \(\theta\). Note that both modelling strategies predict the failure of one brick of the last row, but, contrary to LA, DEM can further increase the tilting angle to get the global collapse. Imperfections in block geometry and block contacts seem to be easily linked with interface stiffness parameters. Based on this bibliographic information, a parametric study of joint stiffness parameters from \(\mathrm{k_{n}}\) = 1 \(\times\) 10\({}^{7}\) Pa/m until \(\mathrm{k_{n}}\) = 1 \(\times\) 10\({}^{10}\) Pa/m is carried out. From several simulations at \(\mathrm{k_{n}}\) = 1 \(\times\) 10\({}^{7}\), 4 \(\times\) 10\({}^{7}\), 7 \(\times\) 10\({}^{7}\), and 1 \(\times\) 10\({}^{8}\) Pa/m, the tangential stiffness has shown very little effect on the collapse tilting angle \(\theta\), except small changes (max 1.5\({}^{\circ}\)) when increasing \(\mathrm{k_{s}}\) from 1 \(\times\) 10\({}^{7}\) to 2 \(\times\) 10\({}^{7}\) Pa/m. For instance, when increasing the tangential stiffness from 4 \(\times\) 10\({}^{7}\) Pa/m to 1 \(\times\) 10\({}^{8}\) Pa/m, the collapse tilting angle is only slightly increased by maximum 0.3\({}^{\circ}\) for each WD, considering different levels of normal stiffness (1 \(\times\) 10\({}^{7}\), 4 \(\times\) 10\({}^{7}\), 7 \(\times\) 10\({}^{7}\), 1 \(\times\) 10\({}^{8}\) Pa/m). For this reason, tangential stiffness has always been considered equal to the normal stiffness \(\mathrm{k_{s}}\) = \(\mathrm{k_{n}}\), to simplify the parametric analysis, though the tangential stiffness may have a larger effect on other similar structures.