## 4 Computational and Mathematical Aspects
### Structural Modeling Techniques for Masonry Structures
#### 4.2.2 Modeling Masonry as a Two-Phase Material (Simplified Micro-Modeling Approach)  
In this approach, masonry units/blocks are represented by fictitious expanded units in size (Phase I) to keep the geometry consistent, however, mortar is not modeled explicitly. The fictitious expanded unit dimensions are of the same size as the original dimensions plus the real joint thickness, as shown in Figure 3c. The interface's stiffness is represented numerically by the stiffness of the real joint. The elastic or inelastic, isotropic model is used for the behavior of the masonry units to simulate properly their possible crushing and cracking propagation patterns. The interaction between the expanded units is represented by an interface element (Phase II). To simulate the behavior of mortar joints, the properties of the mortar and the interface elements are lumped into the zero-thickness common interface elements. This approach leads to the reduction in computational intensiveness and yields a model that is applicable to a wider range of structures. In fact, the drawback of the large computational effort required by the detailed micro-modeling is partially overcome by the simplified micro-modeling strategy. Cracking in the masonry units can also be simulated by assigning potential vertical zero thickness interfaces (unit to unit) at the unit's center lines ([PERSON] 1996 [100]). Another important factor influencing the accuracy of the simplified micro-modeling strategy is the masonry units' aspect ratio that should be taken into account. According to [PERSON] and [PERSON] (2004) [102], aspect ratio of stone masonry units (stone shape or the ratio of stone unit length to height lst/hst) has an effect on the joint failure and load-carrying capacity of masonry walls. For example, as shown in Figure 4, the load-carrying capacity of a masonry wall increases as a result of the decreasing stone rotation with an increasing ratio of lst/hst. Furthermore, it can be observed in this figure that an increase in the stone unit dimensions in each direction (expanding stone units), would increase the ultimate load to some extent depending on the ratio of lst/hst. Therefore, it can be concluded that expanding masonry units in the simplified micro-modeling strategy may alter the accuracy of the analysis depending on the mortar joint relative thickness. Finally, it should be kept in mind that, although this procedure is preferred to the abovementioned detailed micro-modeling approach, it is still time consuming as the masonry structure size increases [103]. This modeling strategy has been used successfully in the literature to investigate nonlinear static and dynamic behavior of masonry wall elements [104; 105; 106; 107; 108; 109], masonry buildings [110; 111], ancient structures [112; 113; 114; 115; 116; 117], stone masonry arches and aqueducts [118; 119; 120; 121; 122; 123] and masonry infilled frames [124].  
Recently, [PERSON] et al. (2018) [124] used this strategy to investigate the nonlinear dynamic behavior of an ancient dry-joint stone masonry tower subjected to earthquake excitations. The numerical model developed using the distinct element method software 3 DEC [125], in which the stone blocks are assumed to behave as rigid elements connected by zero-thickness interfaces. The zero thickness interfaces between adjacent blocks were modeled using the Mohr-Coulomb slip model. Since the tower has been constructed with dry joints, both the cohesion and tensile strength at the interfaces are assumed to be zero. During the dynamic analysis, no viscous damping is assumed, the only dissipation being due to frictional sliding on the joints. This conservative assumption is often used in simulating stone masonry structures containing dry joints ([PERSON] et al., 2002, [126]). This study shows that the tower sustains considerably large lateral drifts in each direction without collapse. This large lateral drift may be attributed to the governing failure mode of the tower, which is a sliding shear failure mode throughout the tower height.