## 4 Results
### Simulations with Classical Joint Stiffness Values  
First, simulations have been carried out using classical values for dry-joint stiffness. In the literature, joint stiffness is often calibrated to account for the deformability of masonry units and mortar [4; 12; 41], even when modelling mortarless masonry structures [4] to account for geometric irregularities. In other cases, the joint stiffness is based only on the  
Figure 3: Effect of the density of contact points used in the numerical DEM simulations on the collapse tilting angle \(\theta\). deformability of blocks [42; 43]. Finally, some studies take arbitrary, though reasonable, values [44; 45]. According to the references listed above, acceptable values for dry-joint behaviour range from 5 \(\times\) 10\({}^{8}\) to 1 \(\times\) 10\({}^{11}\) Pa/m. The adopted ratio between k\({}_{\rm s}\) and k\({}_{\rm n}\) varies between 0.4 and 1.0. Initially, two simulations were run for each model with the following parameters:  
* k\({}_{\rm n}\) = k\({}_{\rm s}\) = 1 \(\times\) 10\({}^{9}\) Pa/m
* k\({}_{\rm n}\) = k\({}_{\rm s}\) = 1 \(\times\) 10\({}^{10}\) Pa/m. The ratio k\({}_{\rm s}\)/k\({}_{\rm n}\) is taken as equal to 1.0, though different ratios do not seem to influence the results significantly (see Section 4.3). The collapse tilting angles \(\theta\) obtained for each simulation are gathered in Table 2. The original results of the Limit Analysis model are also shown [27], including associative and non-associative solutions. One can observe that the results provided by the different approaches are rather close (larger deviation in case of WD 1), even if quite different from the experimental results. The reason for the differences found is that limit analysis solutions do not consider joint (or block) deformation under applied stresses and progressive changes in the structure due to load incrementation. For example, limit analysis fails to reproduce the typical progressive bending behaviour observed for slender masonry piers (Figure 4). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Exp [22]** & **LA-A [27]** & **LA-NA [27]** & **DEM (1 \(\times\) 10\({}^{9}\))** & **DEM (1 \(\times\) 10\({}^{10}\))** \\ \hline WD 1 (Newcastle) & 18.3\({}^{\circ}\) & 23.9\({}^{\circ}\) & 23.5\({}^{\circ}\) & 25.3\({}^{\circ}\) & 26.5\({}^{\circ}\) \\ \hline WD 2 (Leeds) & 7.8\({}^{\circ}\) & 17.0\({}^{\circ}\) & 16.0\({}^{\circ}\) & 15.5\({}^{\circ}\) & 16.9\({}^{\circ}\) \\ \hline WD 3 (Lyon) & 19.8\({}^{\circ}\) & 26.5\({}^{\circ}\) & 24.5\({}^{\circ}\) & 25.6\({}^{\circ}\) & 26.3\({}^{\circ}\) \\ \hline WD 4 (Pavia) & 16.8\({}^{\circ}\) & 24.6\({}^{\circ}\) & 23.5\({}^{\circ}\) & 23.8\({}^{\circ}\) & 24.9\({}^{\circ}\) \\ \hline WD 5 (Leuven) & 7.8\({}^{\circ}\) & 17.5\({}^{\circ}\) & 17.4\({}^{\circ}\) & 16.4\({}^{\circ}\) & 18.0\({}^{\circ}\) \\ \hline WD 6 (Yildiz) & 7.8\({}^{\circ}\) & 18.3\({}^{\circ}\) & 17.1\({}^{\circ}\) & 15.9\({}^{\circ}\) & 17.4\({}^{\circ}\) \\ \hline WD 7 (Munich) & 18.6\({}^{\circ}\) & 26.6\({}^{\circ}\) & 26.6\({}^{\circ}\) & 26.5\({}^{\circ}\) & 27.0\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Collapse tilting angle \(\theta\) (\({}^{\circ}\)) obtained for different modelling strategies. The results of the Limit Analysis with associative (LA-A) and non-associative (LA-NA) flow rules [27] and DEM simulations for two different joint stiffness k\({}_{\rm n}\) = k\({}_{\rm s}\) = 1 \(\times\) 10\({}^{9}\) and 1 \(\times\) 10\({}^{10}\) Pa/m are presented. The experimental tilting capacities are also indicated for reference. Figure 4: Collapse mechanisms obtained for different modelling strategies on two WDs: (**a**) Experimental failure [22]; (**b**) Limit Analysis with associative behaviour [27]; (**c**) DEM simulations with k\({}_{\rm n}\) = k\({}_{\rm s}\) = 1 \(\times\) 10\({}^{9}\) Pa/m; (**d**) DEM simulations with k\({}_{\rm n}\) = k\({}_{\rm s}\) = 3 \(\times\) 10\({}^{7}\) Pa/m. The reader is specially referred to the left masonry pier bending.