## 2 Strategies to Model Behaviour of FPS and of Masonry Infills in RC Framed Structures
### Principles of Behaviour of FPS with Single Concavity  
The FP isolators are devices able to improve the seismic performance of RC-framed buildings in areas with high seismicity [17; 19; 26]. Particularly, the FP devices allow for disconnecting the superstructure from the foundation level and are able to accommodate most of the seismic demand for displacement. In addition, these devices are able to provide high energy dissipation through friction developed between the sliding surfaces [12]. These devices are realized through a slider device that can move on a surface having a concave shape, which is characterized by a specific curvature radius \(R\)[29] and friction coefficient \(\mu_{d}\)[30; 31]. The adoption of single-concave FP leads to the main advantage of having the first natural period of the base-isolated structure \(T_{is}\) dependent only on the curvature radius \(R\)[12]. The above-mentioned dependence can be expressed as follows:  
\[T_{is}=2\pi\sqrt{R/g} \tag{1}\]  
where the acceleation of gravity is denoted by \(g\).  
The basic response in terms of the dynamic equilibrium of the FPS device is represented in Figure 1a by means of the free-body diagram of the slider and the related equilibrium equation with respect to translation along the tangent direction to the sliding surface. After mathematical manipulation of the equilibrium equation shown in Figure 1a and assuming that the discrepancy between the normal direction to the sliding surface and the vertical one is negligible according to [11; 12] (i.e., tangent direction to the sliding surface coincides, reasonably, with the horizontal one), the restoring force \(f_{b}(t)\) of the FP bearing device can be determined as follows:  
\[f_{b}(t)=\frac{Mg}{R}u_{b}(t)\textbf{+}u_{d}(t)Mg\text{sgn}\big{(}i_{b}(t) \big{)} \tag{2}\]  
where \(M\) is the mass of the portion of structure pertaining to the specific FPS device, \(u_{b}(t)\) denotes the projection on the horizontal plane of the slider displacement relative to the ground, \(t\) is the time, \(\mu_{d}(t)\) represents the coefficient of friction in a dynamic regime, and \(sgn\big{(}i_{b}(t)\big{)}\) determines the sign of the velocity of the slider \(i_{b}(t)\) during the motion.  
Mechanical behavior of single-concave FPS devices may be idealized according to the non-linear hysteretic model proposed by [29], characterized by the definition of  
Figure 1: Representation of the free-body diagram of the FPS under imposed lateral force and related equilibrium equation to translation in horizontal direction of the slider. **(a)** Theoretical non-linear hysteretic response of the FPS device **(b)**.  
3 parameters: the characteristic strength \(Q_{d}=\mu_{d}W\) with \(W=Mg\); the stiffness after the elastic behavior \(K_{2}=W/R\); the stiffness in the elastic regime \(K_{1}\), assumed equal to \(51\cdot K_{2}\). The representation of the theoretical hysteretic model for the FP device is reported in Figure 1b. Regarding friction coefficient in a dynamic regime, investigations of [31] suggest that \(\mu_{b}(t)\) can be expressed as being dependent on the sliding velocity \(\dot{u}_{b}(t)\) as follows:  
\[\mu_{d}(t)=\mu_{fast}-(\mu_{fast}-\mu_{slow})\exp(-\alpha\big{|}\dot{u}_{b}(t) \big{|}) \tag{3}\]  
where the terms \(\mu_{fast}\) and \(\mu_{slow}\) represents the values of the coefficient of friction in regimes of high and low, respectively, velocity of sliding \(\dot{u}_{b}(t)\); \(\alpha\) represents a constant that governs the variation of the coefficient of friction \(\mu_{b}(t)\) between minimum and maximum values.