## 2 Design and Modelling of Bridges
The effects of the variation of the axial forces on the seismic response were found to be small for the single-column bridges considered [39, 46] and hence were neglected. However, it is noted that although the use of fiber models could more precisely include the interaction effects, such models have difficulty in including realistic shear deformations of columns. Therefore, the use of the modelling adopted in this paper, including the interaction effects, is deemed to provide sufficient accuracy for the sake of this research. The plastic hinge lengths and the strain penetration depths were computed using the equations given by [PERSON] et al. [1]. To consider the deformations due to bond slippage, the strain penetration depth, Lsp, was considered in modelling, as recommended by [PERSON] et al. [1, 47]. Although this modelling procedure (i.e., using lumped plasticity model) has been widely used in research and practice for design and evaluation of bridges and is recommended in many seismic codes and guidelines such as Caltrans [2] and AASHTO Guide Specifications [3], to further validate the modeling assumptions the numerical results were compared to some experimental data. The data from the RC column tested by [PERSON] and [PERSON] [48] (data were available in the PEER structural database [49]; details are available in [50]) as well as the bridge model labelled as B213C in the experimental study by [PERSON] et al. [51] were considered to validate the predictions from the numerical models. This large-scale (1:2.5) bridge model was tested in pseudo-dynamic fashion. The experimental data from this test was provided by the European Laboratory for Structural Assessment (ELSA). A comparison of the results from numerical modelling with experimental results (examples are provided in Figure 4) indicates that the recommendations made by [PERSON] et al. [1] for modelling bridge structures provided sufficient accuracy in predicting the maximum displacement demands [50]. Therefore, these modelling assumptions were adopted for the  
Figure 3: Structural modelling of the bridges: (**a**) Schematic presentation of the structural modelling of the bridge in the RUAUMOKO program (adapted from [34]) (**b**) Modified Takeda hysteresis loop (adapted from [45]). seismic analysis and evaluation of the bridges in this research. More details concerning the modelling of the bridges and the verification of computer models are available in [34; 40; 50]. The modal properties of the bridges including the fundamental transverse mode, \(\rm T_{T}\), second transverse mode, \(\rm T_{T2}\), fundamental longitudinal period, \(\rm T_{L}\), geometric mean period, \(\rm T_{G}\), (defined as \(\rm T_{G}=(T_{L}\times T_{T})^{0.5}\)) and the effective modal mass ratios are presented in Table 1. It is noted that in the longitudinal direction due to the rigidity of superstructure (i.e., large axial stiffness of the bridge deck) the bridges act similar to a single degree of freedom system and as a result the effective modal mass ratio in the longitudinal direction was almost 100%. The periods are computed using the effective stiffness of the columns (i.e., \(\rm K_{e}=M_{y}/\theta_{y}\) where \(\rm M_{y}\) and \(\theta_{y}\) are the moment and rotation at yield, respectively). It is noted that, in this research, regular and irregular four-span bridges were considered only as example cases to investigate the problems discussed in this research, because these types of bridges were the subject of previous studies by the authors (e.g., [34; 40; 52; 53]). Also it should be noted that this study is limited to 2 bridges to demonstrate different methods and present the proposed methodology for considering the variability due to directivity effects for different averaging methods. More research is needed on this subject for a larger number and different types of bridges.