## 3 Distinct Element Model of the Multi-Lock Interface  
As mentioned, 3 DEC 7.0 [30], which is a numerical modelling code based on the distinct element method (DEM), is herein used to validate the torsion-shear capacities of interlocking blocks obtained by the SiDMACIB model. The DE model is made of discrete particles (bodies) that can move independently including large displacements and rotations. Once particles are moved relative to each other, some contacts between the particles can be lost or newly generated [32]. DEM was initially developed for the simulation of soil modelled as an assembly of rigid and fine particles. The method was later adopted to model masonry assemblies in which masonry blocks are modelled as either rigid or deformable. The most common method to model a deformable body, also used in the 3 DEC software, is to subdivide it into finite elements, usually into uniform-strain tetrahedral elements [33; 34]. By adopting the assumption of rigid bodies in this paper, each interlocking block can be considered as made of several rigid bodies representing the locks and the main part of the block, as shown in Figure 4a for the three- and five-lock interfaces. The subdivision into smaller parts sketched in this figure is based on the observation that if each block is divided into the main body and the locks, 3 DEC displays an error. Considering the abstraction in Figure 1, these bodies are linked by horizontal cohesive interfaces (inner faces coloured red in Figure 4b) with finite shear and tensile strengths, while the horizontal interfaces shared between the blocks are dry faces with zero tensile strength and finite frictional resistance (blue faces in Figure 4b). Note that the provisional yellow parts over the upper blocks are considered fully fixed to avoid potential rocking failure. To avoid the potential sliding interaction between the fixed yellow part and the upper interlocking block, the friction coefficient and the interface between these two entities are set to zero. Later on, it has been observed that the rocking failure does not happen in any of the studied cases. The vertical faces, instead, are automatically generated when the main bodies of the interlocking blocks are divided. These faces are also categorised as 'inner' type, depicted in red. Figure 4: (**a**) 3 DEC model of the three and five-lock interface; (**b**) types of interfaces (dry faces in blue, inner faces in red), all discretized by 0.005 m mesh size. On the other hand, a simple triangulation method has been selected among others in 3 DEC to divide each shared face into 2D sub-contacts, using two different mesh sizes of 0.02 and 0.005 m. Both types of horizontal contact faces (i.e., dry faces between two interlocking blocks and inner faces between the rigid bodies of one interlocking block) are triangulated similarly. Table 1 shows the material properties of the blocks and the inner and dry faces, already used to model the mentioned experimental tests [27; 28]. Note that the properties of the inner interfaces, including their shear and tensile strengths, correspond to the block material properties. To simulate the torsion-shear interaction at the interlocking interface, the main part of the lower interlocking block is considered to be fixed (depicted in green in Figure 5b), while the top cannot rotate (yellow part in Figure 4a). A non-uniform and linearly distributed force is applied to the whole lateral face of the upper block shown in Figure 5a, involving both the side locks and the main body of this interlocking block. This distributed force is equivalent to a pointed load \(V_{y}\) applied to the block face with the eccentricity ex from the centroid of the face (with \(e_{y}\) = 0). This is equal to the lateral force \(V_{y}\) and torque \(M_{y}=V_{y}\cdot e_{x}\) applied to the face centroid.