# Seismic Risk Analysis of Existing Link Slab Bridges Using Novel Fragility Functions
## Appendix A Details on Seismic Hazard and Ground Motion Model  
The seismic scenario is described by a source model characterized by two main random seismological parameters, namely the moment magnitude \(M\) and the epicentral distance \(R\). A Gutenberg-Richter recurrence law is used to describe the magnitude-frequency relationship of the seismic source:  
\[\
u_{M}(m)=10^{(a-bm)} \tag{1}\]  
in which the parameter \(a\) accounts for the mean number of earthquakes expected from the seismic source, while the parameter \(b\) is a regional seismicity factor governing the proportion of small to large earthquakes. The assumed recurrence law, bounded within the range of magnitudes of interest \([m_{0},m_{max}]\), leads to the following probability density function (PDF) of the moment magnitude:  
\[f_{M}(m)=\beta\frac{e^{-\beta(m-m_{0})}}{1-e^{-\beta(m_{max}-m_{0})}} \tag{2}\]  
being \(\beta=b^{*}\) log\({}_{e}\)(10), \(m_{0}\) the magnitude value below which non-significant effects are expected on the structures, and \(m_{max}\) the physical upper bound of the magnitudes expected from the source. In this application, it is assumed \(m_{0}=5.5\), \(m_{max}=8\), \(a=4.35\) and \(b=0.9\), in order to provide a seismic hazard representative of Italian medium-high seismicity zones. The epicentral distance is modeled according to the following PDF:  
\[f_{R}(r)=\left\{\begin{array}{ll}\frac{2r}{r_{max}}&if\ r<r_{max}\\ 0&otherwise\end{array}\right. \tag{3}\]  
which is obtained under the hypothesis that the source produces random earthquakes with equal likelihood anywhere within a distance from the site \(r_{max}=100\) km, beyond which the seismic effects are assumed to become negligible. The soil condition is described by a deterministic value of the shear-wave velocity parameter \(V_{530}=255\) m/s, representative of deformable soil condition.  
The [PERSON] (1958) source-based ground motion model is used to describe the attenuation from the source to the building site. This model, combined with the stochastic point source simulation method of ([PERSON], 2017) is employed to generate synthetic accelerograms consistent with the random variables characterizing the seismic scenario, i.e., mainly magnitude \(M\) and source-to-site distance \(R\).  
For the simulation of two horizontal ground motion components, the stochastic model of ([PERSON], 2017) is modified as suggested by ([PERSON], 2017), according to which the radiation spectra can be considered to have the same shape for the two horizontal orthogonal directions, but the intensities are scaled by two different scaling random parameters (random scaling disturbance \(\varepsilon_{1}\), \(\varepsilon_{2}\)) generated as samples of a multivariate lognormal distribution with zeromean, standard deviation \(\sigma=0.523\) and correlation \(\rho=0.8\). This way, the target radiation spectra along directions 1 and 2 will be equal to \(A_{1}(\omega)=\epsilon_{1}A(\omega)\) and \(A_{2}(\omega)=\epsilon_{2}A(\omega)\).  
The random scaling disturbance \((\epsilon_{1},\epsilon_{2})\), together with the Gaussian white noise process ensures the ground motions record-to-record variability to be accounted for. In particular, for each earthquake sample a Gaussian white noise signal is generated (Figure 11a) and, after being windowed through the envelope functions \(e(t)\) (Figure 11b), its normalized frequency spectrum is applied to the target radiation spectrum (Figure 11c), thus providing the variability of the energy content within the frequency domain. Such variability is further amplified by the lognormally distributed multiplicative factors \((\epsilon_{1},\epsilon_{2})\) of the radiation spectra. The resulting sample of acceleration time series is finally generated by back-transforming the spectrum from the frequency to the time domain (Figure 11d).