# Seismic Risk Analysis of Existing Link Slab Bridges Using Novel Fragility Functions
## 3 Case Studies
### OpenSees Numerical Model  
All the case studies were modeled in OpenSees [49] following the strategy described below and schematically represented in Figure 7.  
Bridge models are 3D and account for both mechanical and geometrical nonlinearity. Deck girders are not expected to show nonlinear responses and are thus modeled through elastic frames. The kinematic links at the joints are explicitly modeled to reproduce the scheme shown in Figure 6, with a central bare portion of the DYWIDAG bar and two lateral parts embedded within the concrete slabs. This detail is realized in OpenSees adopting a straight layer of fibers with _Steel02_ uniaxial material for the DYWIDAG bars and the _Concrete01_ uniaxial material for the slab in the side portions. A bilinear contact element (see [50]) is adopted to account for the potential bounding between adjacent spans, assuming a gap distance in compression equal to 40 mm; in particular, such a phenomenon is expected after the link bar failure, which is explicitly accounted by the model thanks to the implementation of a _MinMax_ material, i.e., a fictitious material wrapping the bars constitutive law that deactivates the DYWIDAG elements once attained their ultimate strain capacity. An analog contact material is inserted to catch the bounding between the deck and the abutments. _Zero-length_ elements with _Steel01_ uniaxial material are used to model the neoprene bearings [51], assuming an elastic stiffness \(k=\frac{\mathrm{G}A}{\mathrm{I}}\) and a maximum admissible strength equal to the lower value between the friction resistance \(\mu_{0}w\) (where \(w\) is the vertical load \begin{table}
\begin{tabular}{l l l l l l l l l l l} \hline \hline
**Joint Number** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline DYWIDAG bars: & L/B = 5 & 2\(\updownarrow\)40 & 2\(\updownarrow\)40 & 3\(\updownarrow\)40 & 4\(\updownarrow\)40 & 5\(\updownarrow\)40 & - & - & - & - & - \\ \cline{2-11} number/diameter (mm) & L/B = 10 & 2\(\updownarrow\)40 & 2\(\updownarrow\)40 & 3\(\updownarrow\)40 & 4\(\updownarrow\)40 & 5\(\updownarrow\)40 & 6\(\updownarrow\)40 & 7\(\updownarrow\)40 & 8\(\updownarrow\)40 & 9\(\updownarrow\)40 & 10\(\updownarrow\)40 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Joint design properties (number and diameter of DYWIDAG bars) for different H/L ratios. \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **H/D = 3** & **H/D = 5** & **H/D = 7** \\ \hline Pier height H (m) & 6 & 10 & 14 \\ Long. Rebars: number/diameter (mm) & 30\(\updownarrow\)26 & 59\(\updownarrow\)26 & 66\(\updownarrow\)30 \\ Long. Rebar geometric ratio (\%) & 0.51 & 1.00 & 1.49 \\ Stirrups: diameter (mm)/spacing (mm) & \(\updownarrow\)14/250 & \(\updownarrow\)14/250 & \(\updownarrow\)14/250 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Piers design properties for different H/D ratios. Figure 7: Numerical model details. transmitted on the bearing, and \(\mu_{0}=0.25\) is the friction coefficient between concrete and neoprene surfaces), and the shear strength \(\gamma_{u}GA\) (assuming \(\gamma_{u}=1.0\) is the ultimate shear strain of the neoprene pads). Distributed plasticity with fiber sections is used for the piers, with the _Concrete01_ material set according to [PERSON] [52] in order to simulate both confined and unconfined portions (i.e., parts within and outside the hoops, respectively). Pier steel reinforcements are also modeled using fibers exploiting the _Steel02_ uniaxial material. Pier caps are modeled as rigid. Regarding the abutments, a classic seat type resting on piles is considered. The interaction between abutments and soil is considered following validated strategies available in the literature, briefly recalled below. Along the longitudinal direction, bridge motions inducing compression of the soil behind the abutment backwall produce, as the main effect, the development of a passive resistance within the soil, and this is simulated using the hyperbolic model proposed by [PERSON] et al. [53] for a granular soil type.