## 3 Numerical Examples
### Buttressed Arches  
In this section, two types of buttressed arches are investigated, as shown in Figure 8. The figure is drawn based on [PERSON] and [PERSON] [17]. The width of the buttress is \(B\), the height from \(O\) to the base is \(H_{1}\), the height of the buttress is \(H\), the height from the top of the buttress to the arch crown is \(h\) and other parameters are similar to the arches shown in Figure 3.  
Similar to that in Section 3.1, each arch is composed of 11 voussoirs. Geometric parameters of the investigated buttressed arches are: (i) circular buttressed arch, \(R_{c}\) = 1.0 m, \(H_{1}/R_{c}\) = 3, \(B/R_{c}\) = 0.8 and \(t/R_{c}\) = 0.2; (ii) basket-handle buttressed arch, \(R_{c}\) = 1.0 m, \(H_{1}/R_{c}\) = 3, \(B/R_{c}\) = 0.8, \(d/R_{c}\) = 0.5, \(r/R_{c}\) = 0.5 and \(t/R_{c}\) = 0.2. The mesh configurations of the buttressed arches are shown in Figure 9, and the total element numbers are: (i) circular buttressed arches, \(N\) = 50 for \(\beta\) = 120\({}^{\circ}\) and 150\({}^{\circ}\), \(N\) = 61 for \(\beta\) = 180\({}^{\circ}\); (ii) basket-handle buttressed arches, \(N\) = 50 for \(\beta\) = 120\({}^{\circ}\), \(N\) = 58 for \(\beta\) = 150\({}^{\circ}\) and \(N\) = 63 for \(\beta\) = 180\({}^{\circ}\). The material properties are as same as in Table 1. The time step of the simulation was set to 1.0 \(\times\) 10\({}^{-6}\) s, and a large friction coefficient of 0.9 was adopted to avoid potential sliding. Similarly, failure of all buttressed arches followed the four-hinge mechanism, too. Herein, the buttressed basket-handle arch with \(\beta\) = 180\({}^{\circ}\) was chosen and presented representatively, and the failure process with \(\lambda\) = 0.18 is shown in Figure 10. At the first instance (Figure 10a), four hinges initiated quickly. Three upper hinges (one extrados hinge and two intrados hinges) appeared within the arch, and a lower hinge formed at the bottom of the right buttress; therefore, the entire structure behaved as a linked mechanism. The hinges continued to develop till the collapse of the structure. The seismic capacity factors \(\lambda^{*}\) of the buttressed basket-handle arches are presented in Table 3, including the results from [PERSON] and [PERSON] [17]. Apparently, the FDEM simulation results agree well with the results based on the limit analysis and the UDEC simulation. The failure modes of buttressed basket-handle arches are illustrated in Figures 11 and 12. All the structures failed due to the instability of the four-hinge mechanism, and the simulation results by the FDEM are in excellent agreement with those by UDEC, especially the hinges and their locations. It is worth mentioning that the locations of hinges are not as same as the corresponding single arches in Section 3.1, i.e., one hinge formed at the bottom of the right buttress. With the ground acceleration at the base, the overturning moments of the right buttress induced by the reaction of the right arch springer and the inertial force of itself are all clockwise. When they surpassed the anticlockwise moment of self-weight, the lower hinge appeared at the exterior corner of the bottom of the buttress. Figure 8: Geometry of buttressed arches (after [17]): (a) circular arch; (b) basket-handle arch. Figure 10: Failure process of the buttressed basket-handle arch with \(\beta=180^{\circ}\): (**a**) t = 1.0 s; (**b**) t = 1.3 s; (**c**) t = 1.5 s. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Embrace Angle \(\beta\) (\({}^{\circ}\))** & \multicolumn{3}{c}{\(\lambda\) *} \\ \cline{3-5}  & & **FDEM/\(e\) (\%)** & **Limit Analysis**[17] & **UDEC [17]/\(e\) (\%)** \\ \hline \multirow{3}{*}{Circular} & 120 & 0.166/3.75 & 0.16 & 0.17/6.25 \\  & 150 & 0.178/1.11 & 0.18 & 0.18/0.00 \\  & 180 & 0.188/1.05 & 0.19 & 0.21/10.53 \\ \hline \multirow{3}{*}{Basket-handle} & 120 & 0.150/7.14 & 0.14 & 0.15/7.14 \\  & 150 & 0.160/6.67 & 0.15 & 0.16/6.67 \\ \cline{1-1}  & 180 & 0.174/8.75 & 0.16 & 0.17/6.25 \\ \hline \hline \end{tabular}
\end{table}