# Joint Layout Design: Finding the Strongest Connections within Segmental Masonry Arched Forms
## 2 Joint Layout Optimization  
### Load Bearing Capacity of a Segmental Model  
In this section, the concept of the proposed segmentation method is presented for a simple 2D problem, for which the joint layout optimization is then developed in Sections 2.2 and 2.3. For the sake of an easy explanation of the segmentation concept, this example is governed by the Tresca yield criterion (weightless cohesive material), involving shear resistances independent of normal forces [62]. The method, however, is easily extendable to include the Coulomb's yield criterion, as presented in Section 3, to segment a semi-circular arch. To find the best location and shape for its joints, the given unsegmented geometry of a structure is abstracted to a number of interfaces (segmenting lines), considered as potential joints. Two sets of interfaces are introduced, i.e., internal and peripheral interfaces, and internal forces and bending moments are applied to their centroid. These interfaces are spanned by rigid elements whose centroids are subjected to external forces (dead and live loads). All the segmenting lines are made of the same no-tension materials with finite shear. Adopting the Tresca model, the shear resistance of interface \(k\) is \(s\),..., \(l_{k}\), where \(s\) is the shear strength of the interface and \(l_{k}\) is length of the interface, whether internal or peripheral. The shear strength of an internal interface (within a segment) is equal to the segment material shear strength \(s_{i}\), while the shear strength of a peripheral interface (e.g., mortared joint) is equal to the mortar shear strength \(s_{p}\). It is worth noting that in this paper,'resistance' refers to the resultant force/moment on the overall interface, while'strength' refers to the interfacial material property per unit length. Based on the choice of internal or peripheral interface for each of them, different configurations of joints (represented by peripheral interfaces) can be assumed. The load bearing capacity of each configuration is assessed using the lower bound approach of limit analysis and the convex contact formulation [51], and the best configuration will be that providing the closest solution to the unsegmented structure. Figure 3 demonstrates a simple example of a rectangular continuum with four blue dotted segmenting lines, while the segments can be convex or concave. As introduced above, when the model is discretised along a segmenting line, this line acts as a peripheral interface shared between two segments (green continuous lines in Figure 3). Otherwise, the segmenting line can simply be taken as an internal interface within a segment with zero tensile and finite shear resistance (red dashed lines in Figure 3). Applying the lower bound theorem of limit analysis and the convex contact formulation [51], the internal forces and moments at the centroid of the internal and peripheral interfaces for each assumed configuration can be found by solving the problem formulated as:  
\[\left\{ \begin{array}{l}\max\lambda\\ \textit{subjected to}:\\ \lambda\cdot L+A\cdot f\ =\ \ -D\\ N_{k}\ \leq 0\\ |M_{k}|\leq 0.5\ N_{k}\cdot l_{k}\\ \forall\ k\ \in peripheral\ interfaces\ \rightarrow|T_{k}|\ \leq\ s_{p}\cdot l_{k}\\ \forall\ k\in internal\ interfaces\ \rightarrow|T_{k}|\ \leq\ s_{i}\cdot l_{k}\\ \forall\ k\in supports\ \rightarrow|T_{k}|\ \leq\ s_{b}\cdot l_{k}\end{array} \right. \tag{1}\]  
where \(\lambda\) is the collapse load factor; \(L\) and \(D\), respectively, are the vectors of live and dead loads, applied to the centroids of the solid elements among the segmenting lines; \(f\) is the vector of internal forces and moments; \(A\) is the equilibrium coefficient matrix; for each interface \(k\), \(N_{k}\), \(T_{k}\), and \(M_{k}\) are normal and tangential forces and bending moment at the interface centroid, respectively; and \(s_{p}\), \(s_{i}\), and \(s_{b}\) are the shear strengths of the peripheral, internal, and supporting interfaces, respectively.