# Joint Layout Design: Finding the Strongest Connections within Segmental Masonry Arched Forms
## 2 Joint Layout Optimization
### Joint Layout Optimization Based on DLO  
As briefly explained in Section 1, using the DLO approach, a continuum is modelled as several nodes properly distributed on the continuum, and the optimization algorithm finds the maximum load factor by which each individual interface connecting two arbitrary nodes are in equilibrium, and the normal and tangential forces on the interfaces meet the related constraints. DLO was initially developed based on the Tresca and Mohr-Coulomb failure criteria for 2D problems only involving shear failure [52] and later was extended to include the rocking failure [63]. However, for the sake of an easy explanation of the segmentation optimization for 2D problems, the Tresca model is herein adopted, and the rocking failure is excluded from the optimization so that interfaces can only undergo shear displacements (no rotation). According to the static approach of DLO, shear \(T_{k}\) and normal \(N_{k}\) forces acting on interface \(k\) equilibrate the applied shear and normal dead loads (\(D^{T}{}_{k}\) and \(D^{N}{}_{k}\)) and live loads (\(L^{T}{}_{k}\) and \(L^{N}{}_{k}\)) on the interface as well as the horizontal (\(v^{x}{}_{i}\) and \(v^{x}{}_{i+1}\)) and vertical (\(v^{y}{}_{i}\) and \(v^{y}{}_{i+1}\)) forces at the nodes \(i\) and \(i\)+1 placed at the endpoints of the interface. Figure 8 illustrates an equilibrated interface with no applied external load, while the general equilibrium equation is [52]:  
\[\left[ \begin{array}{ccc}\cos\theta&\sin\theta&-\cos\theta&-\sin\theta\\ -\sin\theta&\cos\theta&\sin\theta&-\cos\theta\end{array} \right]\left[ \begin{array} []{c}v^{x}_{i}\\ v^{x}_{i+1}\\ v^{y}_{i+1}\end{array} \right]+\lambda\left[ \begin{array}{c}L^{T}_{k}\\ L^{N}_{k}\end{array} \right]-\left[ \begin{array}{c}T_{k}\\ N_{k}\end{array} \right]\ =\ -\left[ \begin{array}{c}D^{T}_{k}\\ D^{N}_{k}\end{array} \right] \tag{5}\]  
The yield constraints on \(T_{k}\) and \(N_{k}\) can also be written as:  
\[\left[ \begin{array}{cc}-1&0\\ 1&0\end{array} \right]\left[ \begin{array}{c}T_{k}\\ N_{k}\end{array} \right]\leq\left[ \begin{array}{c}s_{k}\cdot l_{k}\\ s_{k}\cdot l_{k}\end{array} \right] \tag{6}\]  
where \(s_{k}\) is the shear strength at interface \(k\). Then, to find the strongest set of internal and peripheral interfaces, an optimization problem can be formulated to maximize the load factor \(\lambda\) by finding the optimal shear strengths \(s_{1}\) to \(s_{m}\) for the neutral interfaces 1 to \(m\) (all the interfaces except those that are located at supports), which can be of either peripheral \(s_{p}\) or internal \(s_{i}\) type. Figure 7: **(a,b)** Two subsets of initial input interfaces for complex geometries; **(c)** a set of internal and peripheral interfaces generating **(d)** invalid and **(e)** valid segments. Same as the optimization problem formulated in Section 2.2, to encourage the algorithm to choose \(s_{p}\) over \(s_{i}\), minimization of \(s_{1}\) to \(s_{m}\) is added to the objective function, while maximization of \(\lambda\) is weighted as the main objective of this function:  
\[\left\{ \begin{array}{ll}\max(\lambda-\ 10^{-c}\cdot S)&\qquad c\in\mathbb{N}\\ \text{\emph{subjected to }}:\\ \lambda\cdot L+A\cdot v-q\ =\ -D&\\ \forall\ k\
otin\text{\emph{supports}}&\rightarrow|T_{k}|\ \leq\ s_{k}\cdot l_{k}& \qquad s_{k}\in\{s_{p}\,\ s_{i}\}\\ \forall\ k\in\text{\emph{supports}}&\rightarrow|T_{k}|\ \leq\ s_{b}\cdot l_{k}& \end{array} \right. \tag{7}\]  
where \(v\) is the vector of horizontal and vertical loads on the nodes, \(A\) is the related equilibrium coefficient matrix, and \(q\) is the vector of tangential and normal forces on the interfaces. Figure 9 depicts a continuum with a network of neutral interfaces, subjected to the loading scheme similar to the example provided in Sections 2.1 and 2.2, taking into account that, using DLO, loads can only be applied to interfaces. The variables involved in the optimization are illustrated in Figure 9-d.