## 3 Numerical Discrete Element Method (DEM)
 \tag{2}\]  
Here, \(\text{u}_{\text{i}}\), \(\text{v}_{\text{i}}\), \(\text{a}_{\text{i}}\), \(\text{m}_{\text{i}}\) and \(\text{F}_{\text{i}}\) are the position, velocity, acceleration, mass and applied forces of block i. New block positions generate increment in the contact forces (Equation (1)). At this stage, the constitutive law of the joint is applied to update the contact force. In the case of the studied masonry specimens, a classical Mohr-Coulomb behaviour with the experimental friction coefficient \(\mu\), no tension, no cohesion, and no dilatancy has been considered [38]. It reads:  
\[ \begin{split}\sigma(\text{t}+\Delta\text{t})=\sigma(\text{t})+ \Delta\sigma,\,\sigma<0\text{ (only compression)}\\ \tau(\text{t}+\Delta\text{t})=\tau(\text{t})+\Delta\tau,\,|\, \tau\,|<\tau_{\text{max}}=-\sigma\cdot\mu.\end{split}  \tag{3}\]  
Note that the convention used in Equation (3) leads to compression stresses displayed as negative numbers. Stresses are then multiplied by the associated contact areas, and new contact forces are derived and are inputted into Equation (2) for the next timestep. Such an integration scheme requires that the timestep \(\Delta\)t is chosen to be sufficiently small, related to the maximum ratio between local stiffness and local mass, and is automatically computed in 3 DEC. In addition, DEM simulations are often artificially damped for static analysis to avoid oscillations around the equilibrium position and to speed-up the analysis. This work considers the classical local damping of \(\xi,=0.8\)[4; 37], which reduces the acceleration \(\text{a}_{\text{i}}\), replacing the first equality shown in Equation (2) by:  
\[\text{m}_{\text{i}}\text{a}_{\text{i}}(\text{t})=\Sigma\text{F}_{\text{i}}( \text{t})-\,\xi,\,\cdot\,|\,\Sigma\text{F}_{\text{i}}(\text{t})\,|\,\cdot \,\text{sign}(\text{v}_{\text{i}}(\text{t}-\Delta\text{t}/2)). \tag{4}\]  
From Equation (4), one can note that the damping force is always opposed to the velocity of the computational node. For each wall design (WD), the model has been directly imported from the Rhinoceros3D environment through the VRML format (\"block generate from-vrml\" command) [37; 39]. Then, an additional block, called \"base\", is added to support the whole model: its six degrees of freedom are fixed (Figure 2). The contact interface between the base block and the first course of the masonry model is similar to the interface between blocks, as no information is provided in [22] to support other modelling strategies. In addition, no sliding of the first course of blocks has been noticed during both the experiments [22] and the simulations (Section 4), validating the accuracy of the modelling strategy. All blocks were also fixed in the Y-direction as no movement was allowed in this direction during the experiments [22]. Finally, all the material properties (density and interface parameters) are applied to the model. As a first stage, static loading with only vertical gravity is activated, and the model runs until stability is achieved. This is assumed when the maximum velocity in the model is below a threshold of \(1\times 10^{-13}\) m/s. Applying vertical gravity with 100 progressive steps or a single one was not affecting the results (i.e., the collapse tilting angle \(\theta\)), given the low nonlinear effects in gravity loading for the different designs. Therefore, the most time-efficient strategy (single-step) was used. In a second stage, gravity is progressively inclined by steps of 0.1\({}^{\circ}\), according to the formula:  
\[\theta\ =\ \theta\ +\ 0.1^{\circ} \tag{5}\]  
Here, g = 9.81 m/s\({}^{2}\) is the intensity of gravity and the inclination of gravity \(\theta\) corresponds to the inclination of the tilting plane in the experiments. For each tilting increment, the system waits until stabilisation before moving to the next step, using the threshold defined in the first stage. The simulation runs until the horizontal displacement of the top block reaches a limit (Figure 2), here chosen as 10 cm (i.e., half the model length).