# Lateral Capacity of URM Walls: A Parametric Study Using Macro and Micro Limit Analysis Predictions
## 2 Limit Analysis Formulation
### Macro LA Formulation  
Several authors have used macro-block LA formulations to assess the structural behaviour of unreinforced masonry structures [50; 53; 54; 55]. Typically, the failure mechanism is pre-defined (Figure 1), and the equilibrium equation is formulated using the virtual work principle, where the horizontal load multiplier is the only unknown variable. Regarding the in-plane sliding-rocking mechanism, the internal work is derived from the frictional resistance at the contact interfaces, whereas the external virtual work involves both the overturning and the stabilising works performed by the inertial forces:  
\[\delta W_{ext} =\lambda\cdot W_{OBC}\cdot\delta_{O,OBC}-W_{OBC}\cdot\delta_{S, OBC} \tag{1}\] \[\delta W_{\text{int}} =F_{real}\cdot\delta_{S,f}\]  
where \(W_{OBC}\) is the inertial force arising from the self-weight of the macro-block OBC, \(\delta_{O,OBC}\) and \(\delta_{S,OBC}\) are the virtual overturning and stabilising displacements of the centre of gravity of the macro-block, and \(F_{real}\) is the frictional resistance generated by the wall. It is worth highlighting that the failure often includes mix- sliding-rocking failure mode, which might cause uplifting of the units and subsequent reduction of the contact surfaces. In order to consider this phenomenon, the solution proposed in [PERSON] et al. [50] is adopted, where the frictional force is defined as a function of the crack inclination angle \(\alpha_{c}\). This is given by:  
\[F_{real}=W_{OAB}\cdot\mu\cdot\left(1-\frac{\alpha_{c}}{\alpha_{b}}\right)= \frac{\left(\overline{H}-Z_{o}\right)^{2}}{2}\cdot\tan(\alpha_{b})\cdot t_{w} \cdot\gamma\cdot\mu\cdot\left(1-\frac{\alpha_{c}}{\alpha_{b}}\right) \tag{2}\]  
where \(t_{w}\) is the thickness of the in-plane wall, \(\mu\) is the friction coefficient, \(\gamma\) is the specific weight of the masonry, \(\alpha_{c}\) is the actual crack inclination, and \(\alpha_{b}\) is the crack inclination upper threshold, which in the case of Running bond type is the function of the block aspect ratio:  
\[\tan(\alpha_{b})=\frac{\
u}{h} \tag{3}\]  
Here, \(\
u\) and \(h\) are half-width and height of the units, respectively.  
Since Equation (3) may only be adopted for Running patterns, an alternative solution to define \(\alpha_{b}\) for different quasi-periodic bond types is introduced next (Figure 1).  
As proposed in [51], in order to compute the crack inclination upper thresholds, one can refer to a representative masonry pattern window (RMPW) and calculate \(\alpha_{b}\) according to (Figure 1):  
\[\tan(\alpha_{b})=\frac{\sum\limits_{i=1}^{n_{c}}v_{i}}{(n_{c}+1)\cdot h} \tag{4}\]  
It is worth remarking that, in this case, \(n_{c}\) refers to the number of courses inside the RMPW, and \(v_{i}\) is the \(i\)-th horizontal segments of the structured path UP-RIGHT-UP-RIGHT.  
According to the analytical formulation proposed in [50], once the crack inclination upper threshold is defined, it is possible to apply the macro-block formulation, which allows the computation of the horizontal load multiplier and the geometry of the failure mechanism. In particular, the horizontal load multiplier can be obtained by equating the external and internal virtual works and solving for the load multiplier \(\lambda\). Hence,the horizontal load multiplier can be computed through the solution of a constrained minimisation problem where geometrical parameters characterising the failure mechanism, i.e., \(\alpha_{c}\) and \(Z_{O}\), are adopted as variables:  
\[\begin{array}{ll}\text{minimise}:&\lambda\\ \text{subject to}:&Z_{O}\leq\overline{H}\\ &\alpha_{c}\leq\alpha_{b}\end{array} \tag{5}\]  
where \(Z_{O}\) is the height position of the pivot point and \(\overline{H}\) is the total height of the wall.