## 3 Numerical Examples
### A Multi-Span Pointed Arch Church  
[PERSON] and [PERSON] [47] investigated the seismic capacity of a multi-span pointed arch church, as shown in Figure 14a, using the approach of limit analysis. It is a three-span buttressed pointed masonry arch structure. Similar to the arch structures in Sections 3.1 and 3.2, it was subjected to the gravitational acceleration \(g\) and a constant horizontal acceleration \(\lambda g\). Various potential failure sections were studied by [PERSON] and [PERSON] [47], e.g., 'horizontal' stereotomy in the buttresses and 'polar' stereotomy in the arches, as shown in Figure 14a. Accordingly, the FDEM discretisation is illustrated in Figure 14b, denoted by Scheme A. The buttresses were discretised by horizontal blocks with a height of 0.26 m. The pointed arches were discretised by polar voussoirs with an embrace angle of \(\beta=2^{\circ}\), and a special block was defined at the crown of each arch because of its pointed shape. As a comparison, another discretisation scheme with monolithic buttresses and the same polar voussoirs was also modelled, as shown in Figure 14c, denoted by Scheme B. Similar to the arch structures in Sections 3.1 and 3.2, only the dry-joint contact and friction between adjacent blocks or voussoirs were considered in the FDEM simulations. The total number of elements is 786 for Scheme A, and 342 for Scheme B. Material properties are as same as in Table 1. In Nodargi and Bisegna [47], 'a sufficiently large friction angle is assumed to be available', and the friction coefficient was set to 0.6 in the FDEM simulation, as it is large enough to avoid sliding. The time steps for Scheme A and B were 1.0 \(\times\) 10\({}^{-7}\) s and 1.0 \(\times\) 10\({}^{-6}\) s, respectively. Figure 13: Influences of [PERSON]’s modulus of masonry blocks on the seismic capacity. Figure 15 shows the failure modes of the multi-span pointed arch church under gravity and a constant horizontal acceleration from both the FDEM simulations and the limit analysis [46]. It can be observed that they are in excellent agreement. Seven hinges were predicted by both the FDEM simulations and the limit analysis [47], and the locations of these hinges are the same. As shown in Figure 15, the right part of the church inclined to the right, while the left part still stood vertically without noticeable deformation. Three hinges appeared within the arch at the middle span, i.e., one extrados hinge and two intrados hinges; therefore, the arch turned into a three-hinge mechanism. Another two hinges formed in the arch at the right span, i.e., one extrados hinge and one intrados hinge, due to the deformation of the structure. The other two hinges formed at the bottom of two right buttresses due to their rotations under the horizontal acceleration \(\lambda g\). Figure 14: The multi-span pointed arch church: (**a**) geometry (unit: m); (**b**) FDEM discretisation of Scheme A; (**c**) FDEM discretisation of Scheme B. Table 4 presents the seismic capacity factor \(\lambda^{\star}\) of the multi-span pointed arch church from the FDEM simulations and the limit analysis [47]. Apparently, the \(\lambda^{\star}\) predicted by the FDEM (both Scheme A and B) is as same as that from the limit analysis, and the difference is only 0.1%. To further examine the differences between Scheme A and B, an extrados hinge, denoted as hinge II in Figure 15b,c, was selected. The time histories of hinge II in the two schemes with \(\lambda=0.0884\) are shown in Figure 16. Each curve terminates at the instant when hinge II disappears since the two adjacent voussoirs at the hinge separated completely in the final and fell down. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{2}{c}{**FDEM**} & \multirow{2}{*}{**Limit Analysis [47]**} \\ \cline{2-3}  & **Scheme A** & & **Scheme B** \\ \hline \(\lambda^{\star}/\text{e}\) (\%) & 0.0883/0.1 & 0.0883/0.1 & 0.0882 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Seismic capacity of the multi-span pointed arch structure.