# Influence of Multiple-Support Excitation on Seismic Response of Reinforced Concrete Arch Bridges
## 3 Methodology
### Analysis Methods and Multiple-Support Analysis Cases  
As already mentioned in the Introduction, the methods used in this study are in accordance with [12], but the most important features will be summarized herein. Analysis methods include response spectrum method according to Der Kiureghian and Neuenhofer (using Multiple Support Response Spectrum (MSRS) fortran code) [55; 56], time-history method using generated records and simplified procedure for consideration of spatially variable seismic excitation proposed in Eurocode 8 [57]. The fundamental information on these methods is presented in Figure 5, while the methods of generating records for time-history method will be briefly elaborated below. Ground-motion records needed for time-history method were generated using two procedures. In the first one, conditional simulation using spectral representation method proposed by [PERSON] [58] was applied, but with his extension to adjust the records to be compatible with the target spectra. In this study, the procedure was coded in Wolfram Mathematica according to the scheme presented in Figure 6.  
The second procedure is unconditional simulation proposed by [PERSON] et al. [59] in which a recorded ground motion is applied for generation of ground motions at other supports based on a spatial variability model. Basic input consisted of location of the supports, the spectral density function (SDF), the frequency-dependent spatial correlation function and the known ground motion at support. The procedure was performed in the program SIMQKE-II [60]. In addition to records generated by conditional and unconditional method, time-histories provided by seismic guidelines of New York City Department of Transportation (NYCDOT) were also applied [61]. These are readily available spatially-variable time histories sets for 1000-year, 1500-year and 2500-year earthquake return periods. Each one of these three sets incorporates the effects of spatial variation along 21 hypothetical piers on very hard rock spaced at 100 m and extended over a straight line having a total length of 2 km. The formulation used for generation is according to [PERSON] [50]. NYCDOT motions were used for checks as they are not compatible with Eurocode spectrum. In total, 13 ground motion sets were generated using the unconditional simulation, 7 using conditional simulation and 3 sets of prescribed motions were taken from the NYCDOT standards. Figure 5: Analysis methods applied in this study. Elastic response spectra of recorded motions at the left abutment (for conditional simulation method) and spectra of generated motions using unconditional method and of prescribed motions by NYCDOT are presented in the left part of Figure 7. On the right side of the same figure, one example set of displacement records at bridge supports is shown. Figure 6: Scheme of iteration procedure for generating seismic motions compatible with target response spectrum. It is important to point out that, generally, the response of structures to MSE cannot be analyzed using the recorded accelerograms only, as a large number of simultaneously recorded time-histories at different distances, in different soils, of different magnitudes, etc., should be available. As such seismic database does not exist, a procedure for generation of records must be applied. For example, Eurocode 8 [57] in its informative annex D related to MSE does not even mention the use of recorded motions in time-history analysis, but only the generated ones. Their number should be such as to yield stable estimates of the mean of the maximum responses of interest. Therefore, in this study the analysis was made with the 20 generated motions, excluding the NYCDOT ones. Furthermore, in the part of the standard not directly related to MSE, where the time-history representation is discussed, it is suggested that the appropriate modified/simulated records are used when the recorded ones with location-compatible characteristics are not available.