# Seismic Risk Analysis of Existing Link Slab Bridges Using Novel Fragility Functions
## 3 Case Studies
### Design Properties  
Each case study is conceived to be representative of existing Italian bridges constructed between 1970 and 1980, i.e., in the interval prior to the enforcement of proper specific seismic design codes. To this aim, a simulated design was carried out preliminarily on the whole set of bridges considering both gravity and seismic loads according to old Italian codes [46; 47].  
A simple cantilever static scheme is assumed for the seismic design of piers, with an equivalent punctual seismic force (7% of the gravity loads according to [46; 47]) applied on the top. Gravity loads derive from the deck self-weight and superstructure permanent loads (approximately 50 kN/m).  
Consistent with the bridge typology, spans are longitudinally connected, in correspondence with the joints and at the slab level, through kinematic links realized by means of DYWIDAG bars. These are designed considering the cumulative contributions of the seismic forces transmitted by the spans: abutment \(B\) is assumed as fixed, thus with a link slab designed to withstand the seismic mass of the whole kinematic chain (made of all the spans of the bridge), therefore at this joint, the number of link bars is the highest; on the contrary, an expansion joint is assumed in correspondence of abutment \(A\), where there are no links; all the intermediate joints (on piers) have a number of link bars which is scaled  
Figure 5: Graphical representation of the convolution integral performed numerically (rectangle approximate method) to estimate annual failure rates using (**a**) monotonic and (**b**) non-monotonic fragility functions.  
depending on the specific location within the chain, i.e., bars number reduces moving from a subtment B towards abutment A.  
Material properties (Table 2) are assumed as follows: concrete grade R\({}_{\text{ck}}\)300 (i.e., cubic characteristic strength 30 MPa), stirrups and longitudinal rebars steel grade FeB44k (i.e., yielding strength 435 MPa), link DYWIDAG bars made of harmonic steel with peak strength 950 MPa.  
Features stemming from the simulated design process are summarized in Table 3 (piers design properties) and Table 4 (DYWIDAG link bars design properties at every structural joint).  
Neoprene bearings are placed between each beam and the pier cap (or abutment); related properties are as follows [48]: bearing area \(A\) = 157.5 cm\({}^{2}\), thickness \(t\) = 52 mm and shear modulus \(G\) = 1 MPa.  
Schematic bridges representation (longitudinal prospective view, planar view and cross sections) are shown in Figure 6. Note that structural details of link slabs are taken from the design documents of a real existing bridge built in the 1980s [45].  
\begin{table}
\begin{tabular}{c c c c c c} \hline  & **E (MPa)** & **f\({}_{\text{yk}}\) (MPa)** & **f\({}_{\text{uk}}\) (MPa)** & \(\varepsilon_{\text{y}}\) & \(\varepsilon_{\text{u}}\) \\ \hline DYWIDAG & 206,000 & 950 & 1050 & 0.0046 & 0.04 \\ \hline Steel & 206,000 & 435 & 540 & 0.00207 & 0.12000 \\ \hline Confined & 30,000 & 26.75 & 18.57 & 0.0027 & 0.0114 \\ \hline Unconfined & 30,000 & 25.00 & 3.58 & 0.002 & 0.0060 \\ \hline \end{tabular}
\end{table}
Table 2: Steel and concrete mechanical properties.  
Figure 6: Long and short bridge case studies show the link bar arrangement at the slab level and a representative transverse cross-section of the deck.