## 3 Numerical Examples
### Arches without Buttresses  
[PERSON] and [PERSON] [17] proposed an analytical model to evaluate the seismic capacity of masonry arches and buttressed arches based on the static theory of limit  
Figure 2: Contact forces in the 2D FDEM. analysis, and their predictions of the model were also compared and verified with results from the DEM program UDEC [17]. In this section, two types of arches, i.e., the circular arch and the basket-handle arch, were investigated. Their geometric configurations are shown in Figure 3. The circular arch (Figure 3a) is simply characterised by the embrace angle \(\beta\), the thickness \(t\) and the centreline radius \(R_{c}\). Figure 3b shows the geometry of a basket-handle arch, which is composed of three circular arches: a larger arch with the centre \(O_{1}\), two smaller arches with the centres \(O_{2}\) and \(O_{3}\) and the radius \(r\). \(d\) is the distance between \(O_{1}\) and \(O\), where \(O\) is the centre of the reference circle (red dash line). The seismic capacities of two circular arches and two basket-handle arches with the embrace angles \(\beta\) = 150\({}^{\circ}\) and \(\beta\) = 180\({}^{\circ}\) were investigated. Each arch is composed of 11 voussoirs. The geometric parameters are: (i) circular arches, \(R_{c}\) = 1.0 m and \(t/R_{c}\) = 0.2; (ii) basket-handle arches, \(R_{c}\) = 1.0 m, \(t/R_{c}\) = 0.2, \(d/R_{c}\) = 0.5 and \(r/R_{c}\) = 0.5. The geometric parameters were chosen such that direct comparison could be made with results in the literature [17]. The material properties are shown in Table 1. Mesh configurations of the arches used in the FDEM simulations are illustrated in Figure 4, in which the arches are yellow, the bases are blue and \(N\) is the total number of elements. The time step of the simulation was set to 1.0 \(\times\) 10\({}^{-6}\) s, and a friction coefficient of 0.6 was defined. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Young’s Modulus (GPa)** & **Poisson’s Ratio** & **Density (kg/m\({}^{3}\))** \\ \hline
40.0 & 0.2 & 2000.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties of arches. Figure 4: Mesh configurations: (**a**) circular arch with \(\beta\) = 150\({}^{\circ}\) (\(N\) = 36); (**b**) circular arch with \(\beta\) = 180\({}^{\circ}\) (\(N\) = 34); (**c**) basket-handle arch with \(\beta\) = 150\({}^{\circ}\) (\(N\) = 106); (**d**) basket-handle arch with \(\beta\) = 180\({}^{\circ}\) (\(N\) = 46). Figure 3: Geometry of arches: (**a**) circular arch; (**b**) basket-handle arch. The \(\lambda^{*}\) was tested using every single value. For each \(\lambda\), the simulation was performed for a period of time that was long enough so that the arch would collapse if the assigned \(\lambda g\) was larger than its seismic capacity. Should the arch be still standing with little noticeable deformation at the end of the simulation, the structure was considered stable. Otherwise, it was unstable with the corresponding \(\lambda g\). Besides the current approach, a criterion based on the unbalanced force ratio/max velocity could have been a better way and shall be considered in future research. The explicit time integration in the FDEM was automatically terminated once the computed steps reached the predefined maximum step, or it could be terminated manually as long as the arch had collapsed under certain \(\lambda g\). According to [PERSON] [28], viscous damping with a value of \(2\lambda h\sqrt{E\rho}\) was considered, where \(\Delta h\) is the characteristic size of the smallest element, \(E\) is the Young's modulus and \(\rho\) is the material density. For \(\lambda>\lambda^{*}\), all of the four arches failed following the hinge mechanism. Take the instance of the circular arch with \(\beta=180^{\circ}\); the entire failure process with \(\lambda=0.3\) was simulated by the FDEM and is presented in Figure 5. Initially, there were five hinges (see Figure 5a). Subsequently, one hinge closed, and the motion of the arch was dominated by the remaining four hinges until collapse, as shown in Figure 5b-d.