PSI - Issue 78

Gabriele Fiorentino et al. / Procedia Structural Integrity 78 (2026) 245–252

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3. Numerical Model Given the isostatic nature of the structural system, the first step in understanding the dynamic behaviour of the bridge is to model two of the cantilever piers (Pier N.5 and Pier N. 4-6). The open-source finite element (FE) platform OpenSees (McKenna et al., 2011) was utilised, along with the pre- and post-processor software STKO (Petracca et al., 2020). Fig.2 shows the numerical model of the two piers. Shell elements ( ASDShellQ4 ) were used to model the deck and piers. The geometry has been defined to be as accurate as possible and consistent with the real structure, with a few main exceptions: (a) the slope of the road (around 2%) was neglected, as it has a negligible influence on the results; (b) the slope of the deck in the transversal direction was also neglected, to obtain two symmetrical objects (Figure 2); (c) to define the shell elements, the midlines of the external walls were assumed to be in the same position, while in reality they are slightly misaligned.

Fig. 2. Front and 3D view of the numerical models in OpenSees/STKO: piers (a) 4-6 and (b) 5.

The 3D mesh has 2772 and 3500 Nodes and 4842 and 5890 elements for Piers 4-6 and Pier 5, respectively. An equalDOF condition was assigned to the deck/pier interface, ensuring a monolithic condition between the two parts of the model. As a first assumption, the concrete materials for the deck and pier were defined as Elastic Isotropic materials, having Young's moduli of 28.9 and 37.1 GPa, respectively, obtained from in-situ measurements. First, the fixed-based model was examined. A modal analysis was performed, allowing for the determination of natural vibration periods of 1.093 and 2.116 s in the longitudinal direction, and of 1.072 and 1.710 s in the transverse direction, for Piers 4-6 and Pier 5, respectively. To investigate the effect of the SSI, soil impedances were defined at the base of the model. To account for the SSI, the equations by Gazetas (1991) and Mylonakis et al. (2006) for fully embedded foundations were employed to estimate the impedances for the six DOFs. A ZeroLengthElement was assigned between two nodes at the base of the model, one fixed at the base for all DOFs and the other connected to the base of the pier with an equalDOF constraint. Elastic uniaxial materials were defined at the ZeroLengthElement for each DOF. The modal analysis resulted in natural vibration periods of the 1st mode of 1.095 s for Pier 4-6 and 2.118 for Pier 5 (See Table 1). Thus, in this case, the influence of SSI appears to be limited in terms of inertial interaction when using a simplified impedance model. This is also confirmed by the value of 1/ σ = h/(Vs*T), where 1/ σ is the so -called wave parameter (Stewart et al 1999, Veletsos and Meek 1974). The estimated value is 0.06 for Pier 5 and 0.09 for Pier 4-6, assuming Vs=550 m/s and the height of the equivalent SDOF (height of the center of mass) h equal to 52.2 and 69.4 for Piers 4-6 and 5, respectively.