## 2 Modelling of Base-Isolated Buildings with DFP Bearings  
### Bidirectional Behaviour  
The bidirectional behaviour of the DFP bearing having the schematic and cross-section shown in Figure 1 can be obtained using a series model developed by [PERSON] and [PERSON] [7], as shown in Figure 2. The model in each direction consists of two single friction bearing elements in series, where each element is of a mass \(m_{bi}\) (\(i=1\) for the bottom part and \(i=2\) for the top one), a spring with stiffness \(k_{bi}\), a dashpot with damping coefficient \(c_{bi}\), a rigid-plastic friction element with a friction coefficient \(\mu_{ie}\) that represents the friction behaviour between the bearing surfaces and a gap that represents the displacement limit \(d_{i}\)The spring stiffness is calculated based on the effective radius \(R_{effi}\) of the spherical surface that represents the linear restoring behaviour (i.e. restoring force), \(k_{bi}=W/R_{effi}\), with \(R_{effi}=R_{i}-h_{i}\), where \(W\) is the total weight acting on the bearing, \(R_{i}\) is the radius of curvature \(i\), and \(h_{i}\) is the height from the centre of the pendulum to the surface of curvature \(i\) (see Figure 1). The friction coefficients \(\mu_{ie}\) have magnitudes depending on the sliding velocity and surface pressure and are determined as  
\[\mu_{ie}=\mu_{max}-(\mu_{max}-\mu_{min})e^{-a|i_{i}|}, \tag{1}\]  
where \(\mu_{max}\) and \(\mu_{min}\) are friction coefficients corresponding to maximum and minimum shear velocities, \(\alpha\) is a constant depending on the surface pressure, and \(\dot{u}_{i}=\sqrt{\dot{u}_{ix}^{2}+\dot{u}_{iy}^{2}}\) is the relative radial sliding velocity of element \(i\). In this friction model, the heat flux and temperature increase at each sliding surface are ignored. A more complex model that takes into account the instantaneous velocity and temperature effects on friction at each sliding  
Figure 1: Schematic of the double friction pendulum bearing and its cross-section. Figure 2: Series model of the DFP bearing and multi-degree of freedom model of the superstructure. surface can be found in the recent work of [PERSON] and [PERSON] [16]. However, this temperature effect is not very significant; the authors suggested that their complex model can be only used in some special cases. The model of the superstructure is also presented in Figure 2, in which a multi-degree of freedom linear elastic model is assumed; this is reasonable since the purpose of implementing base isolation is to reduce the earthquake forces on the structure and limit the structural behaviour to an elastic range. As presented in Figure 2, \(m_{i}\), \(k_{i}\) and \(c_{i}\) are the equivalent lumped mass, stiffness, and damping coefficient of floor \(i\) of the building, respectively. The system of differential equations of motion in each direction includes \((n+2)\) equations (\(n\) is the number of floors of the building) can be obtained based on the D'Alembert's principle, given as  
\[\left\{ \begin{array}{l}m_{b1}\big{(}\ddot{u}_{b1}+\ddot{u}_{g}\big{)}+k_{b1 }u_{b1}+F_{f1}+F_{r1}+k_{b2}(u_{b1}-u_{b2})-F_{f2}-F_{r2}=0\\ m_{b2}\big{(}\ddot{u}_{b2}+\ddot{u}_{g}\big{)}+k_{b2}(u_{b2}-u_{b1})+F_{f2}+F_{ 2}+k_{1}(u_{b2}-u_{1})+c_{1}\big{(}\dot{u}_{b2}-\dot{u}_{1}\big{)}=0\\ m_{1}\big{(}\ddot{u}_{1}+\ddot{u}_{g}\big{)}+k_{1}(u_{1}-u_{b2})+c_{1}\big{(} \dot{u}_{1}-\dot{u}_{b2}\big{)}+k_{2}(u_{1}-u_{2})+c_{2}\big{(}\dot{u}_{1}-\dot {u}_{2}\big{)}=0,\\...\\ m_{n}\big{(}\ddot{u}_{n}+\ddot{u}_{g}\big{)}+k_{n}(u_{n}-u_{n-1})+c_{n}\big{(} \dot{u}_{n}-\dot{u}_{n-1}\big{)}=0\end{array} \right. \tag{2}\]  
where \(u_{b1}\), \(u_{b2}\) and their 1 st derivatives are the displacements and velocities of the bottom and top bearing masses, respectively, and \(u_{1}\),..., \(u_{n}\) and their 1 st, 2 nd derivatives are the displacements, velocities, and accelerations of the floor masses. The friction force components on the concave surfaces are represented by the Bouc-Wen hysteretic model,  
\[\left\{