# Effectiveness of LRB in Curved Bridge Isolation:
## 2 Modeling
\[Q_{d}=f_{py}\;\times\;A_{p} \tag{1}\]  
For specified design displacement, \(D\), Post-yield stiffness, \(k_{2}\), the effective stiffness, \(K_{eff}\) of LRB is given by  
\[K_{eff}=k_{2}+\frac{Q_{d}}{D} \tag{2}\]  
The effective damping \(\beta_{eff}\) of Bilinear system for the yield displacement, \(D_{y}\), is given by  
\[\beta_{eff}=\frac{4\big{(}D-D_{y}\big{)}Q_{d}}{2\pi K_{eff}D^{2}} \tag{3}\]  
The coupled force-deformation relationship is given by  
\[\text{F}_{1} =\text{R}_{1}\text{K}_{\text{e1}}\text{u}_{1}+(1-\text{R}_{1}) \text{F}_{\text{y1}}z_{1} \tag{4}\] \[\text{F}_{2} =\text{R}_{2}\text{K}_{\text{e2}}\text{u}_{2}+(1-\text{R}_{2}) \text{F}_{\text{y2}}z_{2} \tag{5}\]  
where, \(\text{K}_{\text{e1}}\) and \(\text{K}_{\text{e2}}\) are the elastic stiffnesses in two orthogonal horizontal directions, \(\text{F}_{\text{y1}}\) and \(\text{F}_{\text{y2}}\) are the yield forces and \(\text{u}_{1}\) and \(\text{u}_{2}\) are displacements in the two directions, \(\text{R}_{1}\) and \(\text{R}_{2}\) are the ratios of post-yield stiffnesses to elastic stiffnesses and \(Z_{1}\) and \(Z_{2}\) are internal hysteretic variables which have a range of \(\pm 1\), with the yield surface as represented by \(\sqrt{Z_{1}^{2}+Z_{2}^{2}}=1\). As per the [PERSON] et al. model [34; 35], the hysteretic dimensionless quantity \(Z\) is governed by the following Equation (6). \[ZU^{y}=\Big{\{}A-|Z|^{\eta}\Big{(}\gamma Sign\Big{(}U\dot{U}Z\Big{)}+\beta \Big{)}\Big{\}}\dot{U} \tag{6}\]  
where \(\gamma\), \(\beta\), \(\eta\), and \(A\) are dimensionless quantities that control the shape of the hysteresis loop, \(\dot{U}\) stands for the velocity. In the model, \(Z_{x}\) and \(Z_{y}\) are the hysteretic dimensionless quantities, which are governed by the following coupled differential equations that account for the interaction between the \(X\) and \(Y\) directions. The paper represents \(Z_{x}\) and \(Z_{y}\) by \(Z_{1}\) and \(Z_{2}\), respectively. \[ \begin{Bmatrix}Z_{x}U_{x}^{y}\\ Z_{y}U_{y}^{y}\end{Bmatrix} =\text{AI} \begin{Bmatrix}\dot{U}_{x}\\ \dot{U}_{y}\end{Bmatrix} - \begin{Bmatrix}Z_{x}^{2}\Big{(}\gamma Sign\Big{(}U \dot{U}_{x}Z_{x}\Big{)}+\beta\Big{)}&Z_{x}Z_{y}\Big{(}\gamma Sign\Big{(}U \dot{U}_{y}Z_{y}\Big{)}+\beta\Big{)}\\ Z_{x}Z_{y}\Big{(}\gamma Sign\Big{(}U\dot{U}_{x}Z_{x}\Big{)}+\beta\Big{)}&Z_{y}^{2 }\Big{(}\gamma Sign\Big{(}U\dot{U}_{y}Z_{y}\Big{)}+\beta\Big{)}\end{Bmatrix}  \begin{pmatrix}\dot{U}_{x}\\ \dot{U}_{y}\end{Bmatrix} \tag{7}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Location**} & \multicolumn{2}{c}{**Span/Height**} & \multicolumn{2}{c}{**Nos. of Segments**} & \multicolumn{1}{c}{**Length of Each Element**} \\  & **(m)** & **(Elements) Considered** & **(m)** \\ \hline \multirow{2}{*}{Superstructure} & 20 & 20 & 1 \\  & 25 & 25 & 1 \\ \hline Pier & 11 & 11 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Details of the number of elements considered in bridge modeling. \(\dot{U}_{x}\) and \(\dot{U}_{y}\) are the velocity components in \(X\) and \(Y\) directions, respectively. Additional models, typically adopted for LRBs, are the ones described in [36], obtained on the basis of the uniaxial model proposed in [37], and a more recent one [38] obtained by extending a previous efficient model [39; 40].