# Learning from the Past: Parametric Analysis of Cob Walls
## 2 Materials and Methods
### Macro-Element Kinematic Limit Analysis  
A second method of analysis was implemented to compare the results obtained with the simplified FEM analysis. This method is based on the abacus of collapse mechanisms reported by the new integrated-knowledge-based approaches to the protection of cultural heritage from the earthquake-induced risk (NIKER) European project [40]. The abacus of collapse mechanisms basically classifies the collapse mechanisms into two categories: in-plane and out-of-plane mechanisms. The identified in-plane damage caused by horizontal loads consist mainly in rocking, sliding shear failure, and diagonal cracking. On the other hand, the out-of-plane damage is mainly characterized by the development of vertical cracks. Vertical cracks may either start from the bottom or the top of the wall and some run  
Figure 4: FEM model setup for (**a**) in-plane analysis and (**b**) out-of-plane analysis. through the entire height of the wall. This causes the separation of important segments of the wall and can lead to the overturning of the element. The macro-element kinematic limit analysis consists basically in finding the equilibrium condition under which, for a certain value of the \(\alpha\) coefficient, the studied collapse mechanism will form [41; 42; 43; 44]. Thus, \(\alpha\) is defined as the seismic mass multiplier that leads to the formation of the collapse mechanism assumed and eventually to the element failure [45]. \(\alpha\) is computed as the ratio between \(a\), the ground acceleration, and \(g\), the acceleration due to gravity. The out-of-plane mechanism considered in this parametric analysis is presented in Figure 5. The thickness of the wall is represented by \(T_{wall}\), its height by \(H_{wall}\), and its length by \(L_{wall}\). It is assumed that cob has zero tensile strength and that no internal sliding would happen, in other words, the walls behave as rigid bodies and rotate while supported in two idealized pined supports. For the case shown in Figure 5a, it was assumed that cob has infinite compressive strength and the bottom support is located at the edge of the wall. On the other hand, for the case presented in Figure 5b, the compressive strength of the material, \(f_{c}\), is taken into account and the bottom support is located at a distance \(t\) from the edge of the wall (this is a conservative assumption, a more realistic one would consider the location of the support at a distance of \(t/3\) from the edge of the wall). The force \(P\) represents the self-weight of the wall and the force \(N\) the external loads transferred from the roof. To compute \(N\), a thatch root with a thickness of 305 mm (including battens) and a self-weight of 450 N/m\({}^{2}\)[36] has been considered with a tributary span of 3 m giving a value of 1350 N/m for every meter of wall length. The equilibrium condition of the rigid body for the case is presented in Figure 5a, from where the expression to compute the \(\alpha\) coefficient is obtained (see Equation (2)). It is worth noting that for this parameter analysis, the distance \(d\) was assumed to be half of the wall thickness, thus, placing the external loads over the centroid of the wall. \[\alpha=\frac{P(T_{wall}/2)+Nd}{P(H_{wall}/2)+NH_{wall}}, \tag{2}\]  
To find the equilibrium condition of the rigid body shown in Figure 5b, it is necessary to find the value of \(t\) at which the support is located from the edge of the wall. This distance is obtained by equating the vertical forces acting on the wall with the force developed at the support as a function of the material's compressive strength, \(f_{c}\). Thus, for a unitary segment of wall (1 m length):  
\[t=\frac{2(P+N)}{f_{c}*1}, \tag{3}\]  
Figure 5: Out-of-plane mechanisms (**a**) assuming infinite compressive strength, and (**b**) accounting for the compressive strength of the material, \(f_{c}\). The seismic mass multiplier is calculated using Equation (4).