# Influence of Multiple-Support Excitation on Seismic Response of Reinforced Concrete Arch Bridges
## 3 Methodology
### Seismic Analysis Accounting for Multiple-Support Excitation Effects
The response spectrum method for multiple-support excitation is labelled as MSRS (analyses exclude the bridges of static system F). The cases of simplified method according to Eurocode 8 are labelled as EC8-SRSS for the case in which the combination of the most unfavorable dynamic response and the most unfavorable effects obtained by quasi-static analysis is calculated using the SRSS rule, while the approach in which this combination is obtained by summing absolute values of the dynamic and quasi-static response is labelled EC8-S.  
Figure 11: Ratios \(\rho_{1}\) of all analyzed bridges grouped by static system: (**a**) horizontal displacements; (**b**) vertical displacements. Figure 12. Ratios \(\rho_{2}\) at arch quarter span for: (**a**) axial forces; (**b**) bending moments; (**c**) vertical displacements *. * **note:** for clarity of the figure (**c**), the large \(\rho_{2}\) of the bridge Upp100 amounting to 5.4 obtained using the MSRS method was disregarded. Figure 13: Ratios \(\rho_{2}\) at arch crown for: (**a**) axial forces; (**b**) bending moments; (**c**) vertical displacements *. * **note**: for clarity of the figure (**c**), the large \(\rho_{2}\) of the bridge Upp100 amounting to 3.65 obtained using the MSRS method was disregarded  
Results show that simplified procedure proposed by EC8 generally underestimates the response of analyzed arch bridges to MSE, regardless of span. At quarter span, the ratios \(\rho_{2}\) of the internal forces are in the range of 0.67 (0.72 for EC8-S) to 0.94 for EC8-SRSS (1.28 for EC8-S) for the bending moment of the bridge Upp100. The vertical displacements are in relatively good agreement with the reference values (in the range of 0.82 to 0.94 for EC8-SRSS and 0.97 to 1.14 for EC8-S). At the arch crown, the values obtained by both EC8 cases are far below the reference values, with the lowest ratio amounting to 0.32 occurring for bending moments in the bridge U100. Vertical displacements are in the range of 0.62 to 0.87 for EC8-SRSS and 0.81 to 1.01 for EC8-S for all bridges. Results are not presented for the arch springing as they follow the same trend as those commented above. Regarding the records used in the time-history method, it may be observed that all values obtained using records from NYCDOT are significantly lower than all other values, which has been expected as the elastic response spectrum prescribed by NYCDOT is more favorable for analyzed bridges than that of EC8. The most detrimental values are calculated using the record generated using unconditional simulation, what has also been anticipated accounting for the characteristics of the signal, which cannot be considered realistic. The MSRS method mostly gives conservative values for MSE. At quarter span, the values of the internal forces are in relatively good agreement (up to 1.16), except for the bridge Upp100 where the ratio for axial force amounts to 1.49 and 3.73 for the bending moment. Vertical displacements are significantly overestimated for stiff bridges, especially for the bridge Upp100 where the ratio amounts to 5.4. At arch crown, the values obtained by MSRS are very different than the reference values, ranging from 0.5 to 3.65. It is interesting to compare the pseudo-static and dynamic component of the response, in MSRS method, for two bridges of the same span but different stiffness, U100 and Upp 100 (Figure 14a,b). It may be observed that for the displacement of the bridge U100 the pseudo-static component prevails, while the dynamic one is more important for the internal forces. For very stiff bridge Upp100, pseudo-static component is more important for all response values and it is the main reason for very unfavorable response of this bridge to MSE. These components are also presented for a flexible bridge U400, where the dynamic component dominantly affects all the response values (Figure 14c). It is repeated herein that the dynamic component of the response to MSE differs from the one to uniform excitation.