## 4 Case Study: The Grottalunga Viaduct
### Preliminary Discussion for Developing Modal Dynamic Analysis and POA in Longitudinal Direction
#### 4.3.1 Modal Linear Dynamic Analysis in the Longitudinal Direction  
Since the studied bridge is divided into two substructures by the constraints of the Gerber saddles, the results of standard Modal Dynamic Analysis (MDA) should be evaluated with special care. It has to be taken into account that since 1 st mode and 3 rd mode have similar periods, namely \(T_{1}=1.74\) s and \(T_{3}=1.67\) s, when the relative displacement between the two parts of the deck is evaluated, the Complete Quadratic Combination (CQC) rule instead of the standard SPSS rule should be used for the deck displacement at the abutment. By contrast, in order to evaluate the relative displacement at the RELCEIDIS, besides the SRSS and CQC rules, as well as the aforementioned DDC rule ([PERSON] et al., 2018), the combination coefficient proposed by [PERSON] et al. (2019) (NAD), and proposed by [PERSON] and [PERSON] ([PERSON] et al., 2019) M&J should be considered.  
In Figure 11b, the deformed shape due to the design seismic action (PGA\({}_{\mathrm{d}}\) = 0.333 g) acting in the longitudinal direction is shown, together with the zoom of the displacements at the left abutment (Figure 11a) and at the RELC-LEIDS (Figure 11c), where bounding or unseating are likely to occur. Table 1 gives the contribution of each mode in determining the displacement at the left and right abutment, and the relative displacement at the RELC-LEDIS, with the value obtained by the modal combination rules. It has to be emphasized that the relative displacement should be evaluated by the modal combination of the relative displacement of each mode rather than between the difference of multimodal estimation of the maximum displacement of each node. This result can be obtained either by using correlation coefficients derived from noise modeling of the seismic excitation for an elastic system; alternatively, a more reliable estimation can be obtained by using a correlation coefficient that takes into account the energy content at the different frequencies of the input ([PERSON] and [PERSON], 2019), or the expected non-linear behavior of the system ([PERSON] and [PERSON], 2019). In this regard, it is noteworthy that combination coefficients are estimated to be \(\rho_{\mathrm{wn}}\) = 0.960 when white noise approximation of the seismic input is assumed (Equation (2)) and \(\rho_{\mathrm{PSD}}\) = 0.965 when spectrum-compatible seismic input is assumed (Equation (3)). The coefficients suggested by [PERSON] assume the value \(\rho_{\mathrm{NAD}}\) = 0.307, while that proposed by [PERSON] and [PERSON] \(\rho_{\text{M}\&\text{Y}}=0.691\). Thus, the application of the SRSS provides an estimated displacement of 143 mm and 145 mm at the left and right abutments, respectively (Table 2). The relative displacement at the RELC-LEDIS is \(\Delta\text{d}_{\text{SRSS}}=207\) mm, while the use of CQC, leaving the estimate of the displacement at the abutments almost unchanged, provides an increased value of the relative displacement, \(\Delta\text{d}_{\text{CQC}}=281\) mm. Reduced values are provided by the modal combination rules derived on purpose for the bounding phenomenon: the DDC rule provides a very small value (\(\Delta\text{d}_{\text{DDC}}=79\) mm); a value close to that provided by the SRSS rule is obtained by using the Naderpour coefficient (\(\Delta\text{d}_{\text{NDP}}=172\) mm), and an intermediate value for the Miari and Jankowsky coefficient (\(\Delta\text{d}_{\text{NDP}}=115\) mm).  
Thus, while the modal dynamic analysis for the PGA\({}_{\text{D}}\) design value does not show a risk of bounding at the abutments, the risk of drop-in span unseating is estimated if the CQC rule is used for modal displacement superimposition.