# Joint Layout Design: Finding the Strongest Connections within Segmental Masonry Arched Forms
## 1 Introduction
### Research Objectives  
The objective of this paper is to find the strongest joints of a curved structure, which can take on a wide range of shapes. Given the overall geometry of a structure and a loading scheme, this work investigates the joint shapes and locations (i.e., joint layout) providing the strongest connections between the blocks so that the load-bearing capacity of the structure is maximized. To this aim, limit analysis [51] and discontinuity layout optimization (DLO) [52] are adopted to develop two optimization methods that find the strongest joint layout within a masonry structure. These fall into the discontinuous analysis methods abstracting structural systems to solid and interface elements, which basically are the planes of weakness within a model [53]. It is worth highlighting that finite-element (FE) [54] and discrete-element (DE) [55] approaches are based on the deformability of solids, while interfaces between the solids make jumps in the possible stress-strain rate fields. On the contrary, the two considerably computational efficient and yet perfectly accurate methods of limit analysis and DLO set the solids to be fully rigid while the stresses are solely distributed on the potential failure interfaces. These two methods allow calculating the maximum load factor by which the internal forces on the interfaces are statically admissible (lower bound theorem or static approach) or the minimum load factor by which the flow rule at the interfaces is met (upper bound theorem or kinematic approach). The main difference between the static approach of the classic limit analysis (used in this paper) and DLO is that the former solves the equilibrium problem for every rigid element spanning the interfaces to obtain the stress state at the interfaces, while the latter solves the equilibrium problem for each individual interface (discontinuity) so that the forces at the endpoints of an interface equilibrate the shear and normal forces acting on that interface as well as the external normal and tangential forces applied to it [52]. Using the limit analysis approach, a continuum can be represented by a limited number of interfaces where the major cracks in an assemblage may occur, as proposed by macro-block [56; 57] and NURBS-based [58] limit analyses; as an alternative, the continuum can be represented by several interfaces considered as joints between the constituent units of an assemblage [59; 60] as well as by potential failure planes within a micro-block at which the unit may crack [48; 61]. Then, an assemblage of blocks with finite shear and tensile resistances can be represented by a set of so called \"internal interfaces\" within the constituent blocks (depicted in red dashed lines in Figure 2b), and \"peripheral interfaces\" at the boundary of those blocks (depicted in green continuous lines in Figure 2b). The peripheral interfaces are in fact joints between the blocks (Figure 2c), and from now on, only the former name is used in this paper. Assuming an overall geometry with a network of interfaces (depicted in blue dotted lines in Figure 2a), the designer can choose the peripheral ones between the constituent blocks, so that the rest of the interfaces are taken as the internal interfaces within the blocks. Different segmental models can then be presented as different sets of internal and peripheral interfaces with different load-bearing capacities. This paper in fact explores a set of internal and peripheral interfaces providing the maximum load bearing capacity as the strongest segmental model. Apparently, this approach is highly mesh-dependent; i.e., the final result is fully dependent on the initial prescribed network of interfaces. Therefore, by increasing the number of interfaces, the more accurate collapse load would be obtained. However, in classic limit analysis, the number of interfaces is rather limited since, as explained above, they are constrained to be at the boundaries of the prescribed rigid solid elements.