## 2 SiDMACIB Formulation for the Multi-Lock Interface  
### Formulation of the Interlocking Interface  
According to the static formulation of limit analysis using the concavity approach, an assembly is modelled as a set of rigid elements with some discontinuity planes in between, where the model can deform and fail [13]. The equilibrium problem is posed to find the stress states at these discontinuities equilibrating the external forces, which are abstracted to internal forces at contact points. The typical internal forces at every contact point are a component normal to the face, which must always be in compression if the discontinuity has zero tensile resistance, and two components tangential to the discontinuity plane, which are perpendicular to each other. For a planar isotropic dry joint, each tangential component must be less than the joint frictional resistance, which equals the coefficient of friction multiplied by the normal force.  
SiDMACIB has extended this formulation to interlocking faces with orthotropic behaviour as follows [18]. In this model, two types of discontinuities are taken into account: one consists of strips between the blocks (dry faces) and the other consists of cohesive strips connecting the locks to the main body of the blocks (inner faces) (Figure 1a,b). It is assumed that both types of discontinuities have zero tensile resistance. Instead, the tangential behaviour of the dry and inner faces is different and is governed by Coulomb's law of friction and the shear strength of the material of which the block is made, respectively. The SiDMACIB model first merges each pair of dry and inner strips representing a lock and then distributes contact points on the centreline CL of each lock, as represented in Figure 1c.  
Several options for the number and location of the contact point have been considered and their validity has been reported in [31]. Similar to the conventional concave formulation, three components of internal forces are considered at each contact point \(i\) of CL \(j\) (Figure 1c). The normal component \(r_{ij}\)\({}^{n}\) should always be in compression; the tangential component \(\mu\) parallel to the locks \(r_{ij}\)\({}^{11}\) should be less than the frictional resistance (\(\mu\)\(r_{ij}\)\({}^{n}\)), where \(\mu\) is the friction coefficient; the second tangential component \(\mu_{ij}\)\({}^{12}\) must be less than the so-called shear resistance of the contact point, which is a portion \(p_{ij}\) of the shear resistance \(T_{0}\) of the whole discontinuity (see Figure 1d for the symbols). \(T_{0}\) equals \(\tau\,a\,b\), where a and b are the length and width of the lock, respectively (Figure 1a).  
The equilibrium problem then can be formulated as follows:  
\[\left\{\begin{aligned} & C_{eq}\overset{\rightarrow}{r}+\overset{ \rightarrow}{E}=0\\ & r_{ij}^{1}\leq 0\\ &\left|\begin{matrix}r_{ij}^{1}\\ r_{ij}^{2}\end{matrix}\right|\leq\mu\left|r_{ij}^{n}\right|&\forall i\in\text{ contact points on merged strip }j;\,\sum_{i=1}p_{i,ij}=1\\ &\left|r_{ij}^{2}\right|\leq p_{i,j}\,T_{0}\end{aligned}\right. \tag{1}\]  
where \(C_{eq}\) is the equilibrium coefficient matrix, \(\overset{\rightarrow}{r}\) is a vector of all normal and tangential components of internal forces, and \(\overset{\rightarrow}{E}\) is a vector of all external forces \(EF\) and torques \(ET\).  
As introduced above, this paper aims to implement the torsion-shear resistance of a multi-lock interface shared between two stacked blocks, assuming that all the locks of the interface are involved in the overall planar capacity. In the following section, how this condition is posed in SiDMACIB is presented.