# Uncertainties in the Seismic Assessment of Historical Masonry Buildings
## 2 Case Studies
### Modelling Approach  
The buildings were modelled using the equivalent frame model (EFM) approach. EFM works by discretizing facades into piers, spandrels and rigid nodes using beams or macroelements [14]. For this work in particular, a macroelement newly developed by [PERSON] et al. [12] was used that builds on the work of [PERSON] et al. [15] to precisely model the behaviour of masonry panels in-plane. The macroelement is formulated as  
Figure 3: Lausanne Malley. (**a**) Main facade. (**b**) Numerical model with the coordinate system.  
Figure 2: Holsteiner Hof. (**a**) Main facade. (**b**) Numerical model with the coordinate system.  
a one-dimensional element defined by three nodes in three-dimensional space. Three non-linear sections account for axial deformation, and the central section accounts for the non-linear shear response. The out-of-plane response is coupled with the in-plane response. Nonlinear geometrical effects are captured through a second-order Taylor series expansion of the compatibility equations, obtaining the \(\mathrm{P}-\Delta\) formulation. To correctly simulate the out-of-plane behaviour of a masonry building, it is not sufficient to choose a macroelement formulation that captures the out-of-plane behaviour, if walls are restrained in the numerical model by rigid wall-to-wall and floor-to-wall connections. Therefore, it is crucial that the model allows for relative displacements between the nodes of the intersection walls and between the nodes of the floor and the wall [16]. For this purpose, interface elements and material models for wall-to-wall and wall-to-floor connections were implemented in OpenSEES [17].  
All piers and spandrels were assigned the same material properties, while the nodal regions between them were modelled as rigid. Timber floors were modelled as an orthotropic elastic membrane, with a higher stiffness in the direction of the beam span and a lower stiffness in the orthogonal direction. The shear stiffness of the membrane was defined by the shear modulus, meaning the membrane was defined by the two moduli of elasticity in two orthogonal directions, as well as the shear modulus and the thickness of the diaphragm. The floors were modelled as linear elastic, but the connection between the floor and the wall was modelled to account for a nonlinear behaviour and a possible connection failure, which can result in the out-of-plane failure of a pier. The nodes of the floor were modelled separately from the nodes of the walls to allow the possible relative displacement. This connection was modelled using a zero-length element to which a frictional model was assigned. The sliding occurred in the direction perpendicular to the wall, and bounding occurred in the opposite direction. Although it is possible to model slip parallel to the wall, for this study, the connection parallel with the wall was assumed to be linear elastic. Therefore, the properties of the zero-length element were defined by the friction coefficient characterising the interface between floor beams and walls, the modulus of elasticity and the shear modulus of this connection [16]. Another set of zero-length elements was used to model the connection between orthogonal walls. This connection simulated the potential formation of a vertical crack and separation of the orthogonal walls due to poor interlocking, which could lead to the out-of-plane failure of a wall. The one-dimensional material for the interface was defined as a damage tension law with exponential softening and a linear elastic model in compression [16]. The material was defined by the elastic modulus, the tensile strength and the Mode I fracture energy.