# Joint Layout Design: Finding the Strongest Connections within Segmental Masonry Arched Forms
## 3 Implementation and Validation  
In this section, and to assess the applicability of the segmentation method proposed in Section 2 to curved structural forms, a simple semi-circular arch with four radial neutral interfaces is considered, with angles of \(26.67^{\circ}\), \(60^{\circ}\), \(120^{\circ}\), and \(160^{\circ}\) with respect to the horizontal line (Figure 11). Considering all the interfaces to be peripheral, the model is identical to the first example provided by [PERSON] et al. [64]. The segments below interfaces 1 and 4 are fully fixed, while the arch is subjected to its weight and a constant lateral acceleration (in terms of load factor). The material density is also set to be 400 kg/m\({}^{3}\). [PERSON] et al. [64] used a kinematic collapse load calculator (KCLC) previously implemented as a MATLAB-CAD limit analysis tool, by which the designer can first choose a mechanism (with given location of hinges and/or slipping joints) among seven kinematically admissible ones for a dry stacked masonry arch subjected to its own weight and a constant lateral acceleration. Then, the tool checks the feasibility of the chosen mechanism configuration and (if it is feasible) computes the related load factor. In this paper, the analysis is carried out for three different friction angles, while the peripheral interfaces (that are in fact dry joints) are governed by the cohesionless Coulomb's law of friction. How the segmentation method of Section 2 is extended to the [PERSON]'s friction law is presented in the following. It is worth noting that both the current paper and [64] adopt an associated sliding rule. Finding the optimal joint layout for this simple example basically proves that the proposed method is perfectly applicable to the more complex problem of, e.g., Figure 12 with the denser network of neutral interfaces that, apparently, allows segmental arches to be generated with planar and/or non-planar interlocking joints. Modelling segmental arched forms having joints with complex geometries will be comprehensively investigated in future work. Figure 11: Semi-circular arch as originally presented by [PERSON] et al. [64]. For the model illustrated in Figure 11, when a neutral interface is chosen as an internal interface, the shear behaviour of the interface is independent of the normal force (i.e., it is a cohesive face with friction angle = 0). On the other hand, when a neutral interface is selected as a peripheral interface, the frictional behaviour of the interface is governed by the cohesionless [PERSON]'s law of friction. To this aim, Equation (2) in Section 2.1 based on the lower bound theorem of limit analysis is updated as follows:  
\[\left\{ \begin{array}{l}\max\lambda\\ \textit{subjected to }:\\ \lambda\cdot L+A\cdot f\ =\ -D\\ N_{k}\ \leq 0\\ |M_{k}|\leq 0.5\ N_{k}\cdot l_{k}\\ \forall\ k\ \in\textit{peripheral interfaces}\ \rightarrow|T_{k}|\leq\tan\varphi\cdot N_{k}\\ \forall\ k\in\textit{internal interfaces}\ \rightarrow|T_{k}|\leq s_{i}\cdot l_{k} \end{array} \right. \tag{8}\]  
where \(\varphi\) is the friction angle at the peripheral joints, \(s_{i}\) is the masonry shear strength at the internal interfaces that here is set to be \(3\times 10^{4}\) N/m\({}^{2}\), and \(k\) refers to all interfaces, except the supports depicted by blue continuous lines in Figure 11.  
Considering all the neutral interfaces to be peripheral, the first row of Table 2 reports the load factors \(\lambda\) found by Equation (8) for two friction angles (\(35^{\circ}\) and \(17.6^{\circ}\)), while the second row shows the computed load factors \(\lambda\) reported by [PERSON] et al. [64]. In particular, the first load factor reported in the second row of Table 2 (\(\varphi=35^{\circ}\)) is related to the standard (hinging) mechanism with four hinges when the frictional resistances are considered large enough, while the second one is related to the mechanism in which hinge 1 of the standard mechanism is exchanged with a slipping joint, when the friction angle is set to be \(17.6^{\circ}\).