# Joint Layout Design: Finding the Strongest Connections within Segmental Masonry Arched Forms
## 3 Implementation and Validation
As shown in the table, the results obtained by Equation (8) and [PERSON] et al. [64] are in good agreement with each other. Figure 12: (a) A dense network of neutral interfaces, allowing the (b) modelling of arch-shaped structures having joints with complex shapes that can generate (c) interlocking concave segments. Furthermore, the lines of thrust obtained by Equation (8) in Figure 13 evidently confirm that the arch with the high friction coefficient (\(\varphi\) = 35\({}^{\circ}\)) fails due to the standard four-hinge mechanism, while the expected collapse mechanism for an arch with a lower friction coefficient (\(\varphi\) = 17.6\({}^{\circ}\)) is similar to the second mechanism introduced in [64]. The thrust line is modelled through a spline whose points are the application points of the internal forces on the interfaces, while the curve is tangential to these forces. After validating Equation (8) for the semi-circular arch with five segments (as elaborated above), this equation is used to find the load-bearing capacities of segmental arches modelled with different combinations of peripheral and internal interfaces. First, it has been observed that for any set of peripheral and internal interfaces, the load factor becomes 0.8279 when the friction angle is 35\({}^{\circ}\), and four hinges are generated at the interfaces 1 to 4 (standard failure mechanism). The results simply show that when the frictional resistance at the peripheral interfaces is large enough, all the segmental arches have the same load-bearing capacity as the unsegmented model, since both the internal and peripheral interfaces are resistant enough to tolerate the related shear forces. On the other hand, Table 3 presents the computed load factors, thrust lines, and failure mechanisms of different segmental arches for two lower friction angles, i.e., 27.28\({}^{\circ}\) (second column) and 17.6\({}^{\circ}\) (third column). \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\varphi\) = 35\({}^{\circ}\) & \(\varphi\) = 17.6\({}^{\circ}\) \\ \hline \(\lambda\) obtained by Equation (8) & 0.8279 & 0.483 \\ \(\lambda\) obtained by [PERSON] et al. [64] & 0.8118 & 0.4878 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results obtained by Equation (8) and [PERSON] et al. [64] for the simple example of Figure 11 when all the interfaces are peripheral. Figure 13: Internal forces (in Newtons) and moments (in Newton-meters) as well as the thrust line obtained by Equation (8) for the simple example of Figure 11 with peripheral joints, when the friction angle is: (**a**) 35\({}^{\circ}\) and (**b**) 17.6\({}^{\circ}\). As shown in the table, when \(\varphi=27.28^{\circ}\), two different collapse mechanisms can be generated for different sets of peripheral and internal forces. First, it can be observed that interfaces 2 to 4 are strong enough against shear forces, being either of internal or peripheral type. On the contrary, interface 1 slips if acting as a peripheral interface and rotates when set to be an internal interface. In fact, for \(\mathrm{E}^{c}{}_{i}\) with \(i=1\), 3, 5, 8, 10, 11, 12, 13, the four-hinge mechanism is always obtained with \(\lambda=0.8279\), while the obtained mechanism for \(\mathrm{E}^{c}{}_{i}\) with \(i=2\), 4, 6, 7, 9, 15, 16 comprises a slipping joint on interface 1 and three hinges on interfaces 2 to 4 with the load factor reduced to 0.8084. Given the lowest friction angle of \(17.6^{\circ}\), three different collapse mechanisms can be generated for different sets of peripheral and internal forces. It can be noted that two interfaces 2 and 3 are strong enough against shear forces, being either peripheral or internal interfaces. Therefore, the obtained mechanisms for \(\mathrm{E}^{c}{}_{i}\) with \(i=1\), 3, 10, 11 simply have four hinges with the highest load factor 0.8279. On the other hand, interface 1 acts as a slipping joint when it is peripheral and as a hinge when internal.