## 2 Design and Modelling of Bridges  
To investigate the influence of using different methods of analysis and different procedures to determine the mean seismic demands using a number of ground motion records, two bridges with different column arrangements were considered. The bridges had four spans with equal lengths of 50 m, as shown in Figure 2, and the superstructure was continuous over the columns. The bridge columns were hinged at the top and fixed at the base. The superstructure consisted of a box girder with a uniform dead load of 200 kN/m. At the abutments the superstructure had no restraint in the longitudinal direction, but was restrained against transverse movements by means of shear keys at the abutments. It is noted that the roller abutment condition in the longitudinal direction can be considered due to the presence of gaps between the superstructure and the abutment back wall (e.g., Roller abutment model discussed by [PERSON] et al. (2013)). The effects of abutment modelling on the seismic response of bridges have been studied by [PERSON] and [PERSON] (2014) based on the abutment models defined by [PERSON] et al. (2013). However, in the present research, the effects of abutments are neglected in structural modeling because this paper is intended to focus on different subjects including directivity of records, different averaging methods and the use of 2D and 3D analysis. It is noted that similar abutment conditions have been used in many studies (e.g., (1; 35; 36; 37; 38; 39; 40)). All column diameters were 2.0 m. For the case of the regular bridges, all column heights were 7 m, while for the irregular bridge the column heights were 14, 7 and 21 m. The bridges were designed based on the 2006 Canadian Highway Bridge Design Code (CHBDC) provisions (41) except that the design spectrum given in the 2010 National Building Code of Canada (NBCC) (42) was adopted. The design spectra in the 2010 NBCC corresponds to 2% probability of exceedance in 50 years. The bridges were designed for Vancouver and site class C (i.e., Vs = 555 m/sec) with an importance factor, I of 1.0. The central columns of the regular and irregular bridges contained the longitudinal reinforcement ratios of 1.23% and 1.88%, respectively. All other bridge columns contained the minimum longitudinal reinforcement ratio of 0.8% based on the CHBDC. A confinement reinforcement ratio of 1.2% governed the design of the transverse reinforcement in the columns. The concrete  
Figure 2: Bridges properties: (**a**) regular bridge; (**b**) irregular bridge. compressive strength and the yield stress of the reinforcing bars were taken as 40 and 400 MPa, respectively. The displacement limit for the critical central columns of the bridges was determined as 370 mm based on the equations developed by [PERSON] [43] for the bar buckling damage state. In predicting the nonlinear response, the modified Takeda hysteresis model [44] was used in this study to model the behavior of the ductile RC columns using Ruaumoko software [45] using lumped plasticity method, as shown schematically in Figure 3a. This model has two main parameters, alpha and beta, which control the unloading and the reloading stiffness, respectively (see Figure 3b). The values of alpha = 0.5 and beta = 0 were adopted based on the recommendations by [PERSON] et al. [1]. The structural modelling considered in this study is similar to that used by [PERSON] et al. [1]. The superstructure was modelled using elastic beam elements and the superstructure mass was lumped at the nodal points as shown in Figure 3a. In addition, the mass of the columns was considered in the corresponding nodal points. Rigid elements were defined to model the superstructure depth. The heights of these elements were considered as half of the superstructure depth (see Figure 3a). For the 3D analyses, the coupling interaction effect of both directions has been considered through the yield interaction of moments with interaction factors of 2 for the circular columns in the Ruaumoko program [45].