# Influence of Multiple-Support Excitation on Seismic Response of Reinforced Concrete Arch Bridges
## 3 Methodology
### Seismic Analysis Accounting for Multiple-Support Excitation Effects  
Results are presented for the arch as the main load-bearing element. Arch sections in which axial force (N), bending moment (M), shear force (T) and displacements (u\({}_{\text{x}}\) and u\({}_{\text{z}}\)) were analyzed include arch springings (p), quarter span (f) and arch crown (c). Firstly, the results of seismic analysis using the time-history method will be presented. The ratio of absolute values of peak response for the multiple-support and uniform excitations was determined for each analyzed arch section and for each set of time-histories (20 in total, without NYCDOT motions) according to the expression:  
\[\rho_{1}=\frac{\text{peak response to the multi - support excitation (C, WP or WPC)}}{\text{peak response to the uniform excitation (U)}} \tag{3}\]  
For ratios \(\rho_{1}>1\), multiple-support excitation (MSE) has a detrimental effect on the seismic response in relation to the uniform excitation. The arithmetic mean of all the ratios for each set of time-histories and for each case of multiple-support excitation was calculated, and the standard deviation was \(\sigma<1/3\)\(\rho_{1}\) in all analyzed cases. The average ratios \(\rho_{1}\) for all analyzed cases of MSE are shown in Figures 10 and 11, for all analyzed bridges grouped by static system. It may be observed that the most detrimental effect of MSE appears for the crown bending moment of the bridge U100 for the WPC case, with the ratio of 3.63. Regarding the displacements, the most unfavorable response occurs due to WPC case for the crown vertical displacement of the bridge Upp100, amounting to 1.86. Such a detrimental values occurring for MSE in the arch crown have been anticipated, because both symmetric and antisymmetric modes are excited. For uniform excitation symmetrical mode, shapes are not excited, so the crown of the symmetrical arch bridge does not undergo a vertical displacement (Figure 2) due to seismic action. Bending moments in the springings occurring for uniform excitation are acceptable for all bridges, except the very stiff ones (Upp) where the largest ratio amounts to 1.35 for the WPC case. However, the values at quarter span due to uniform excitation are unacceptable and the ratios are up to 1.45 for the WP case of the bridge Upp300. Figure 9: Elastic response spectra EC8 with modal mass participating ratios for all analyzed bridges. Further unfavorable responses may be observed for axial forces in all analyzed sections of all the bridges, with the crown being the most critical one with ratios up to 1.68 of the bridge U200. It may be observed that in almost all analyzed cases the most unfavorable is the WP case. The unfavorable response of axial forces obtained is in accordance with the results presented in [29; 30; 33]. Regarding the shear forces, the most detrimental response of MSE may be observed at the quarter span of the stiff bridges amounting to 1.42 for the WPC case of the bridge Upp100. For the shear force in the arch crown the uniform excitation is on the safe side. All longitudinal displacements are in order of \(\pm 10\%\), so disregarding MSE is acceptable. Figure 10: Ratios \(\rho_{1}\) of all analyzed bridges grouped by static system: (**a**) axial forces; (**b**) shear forces; (**c**) bending moments. Following the results of time-histories, the ratio \(\rho_{2}\) of the absolute values of peak response for the multiple-support cases calculated using different methods and the most detrimental case obtained by the time-history method using all generated records (UK) was determined for each analyzed arch section. Therefore, the average ratios \(\rho_{2}\) are presented in Figures 12 and 13 in order to compare results of different analysis methods. Results of time-history method are given for records generated using different methods (denoted by G for unconditional simulation, P for conditional simulation, N for NYCDOT motions and UK for all generated motions excluding NYCDOT).