PSI - Issue 78

Stefano Bracchi et al. / Procedia Structural Integrity 78 (2026) 745–752

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varying seismic IMLs. The dynamic impedance functions required for calibration are computed using the DynaMat code, which adopts a three-dimensional hybrid approach. This method combines finite element modeling in the near field with a semi-analytical representation of the far field, allowing for the accurate evaluation of frequency-dependent stiffness and damping while accounting for wave propagation, material damping, radiation effects, and layered soil profiles. Soil properties used in the analysis correspond to those mobilized under free-field seismic wave propagation. For each pier and IML, translational and rotational impedance functions are derived based on the average profiles of shear modulus and damping ratio mobilized during seismic shaking (shaded lines of Fig. 3).

Fig. 3. Translational and rotational impedance function (shaded lines) and calibration of translational and rotational impedance functions (solid lines) for the foundations of the panels with length b = 2.25 m (a, c) and b = 6.5 m (b, d).

These impedance functions serve as the basis for calibrating the LPMs. The real part is approximated using a parabolic function, while a linear function is adopted for the imaginary part, as demonstrated for IML 2. The general form of the impedance function K ij (ω) is defined as: K ij (ω) = k ij + i ω c ij − ω 2 m ij (1) where k ij , c ij , and m ij represent the frequency-independent stiffness, damping, and inertial coefficients, respectively. These parameters are identified through a least-squares fit in the frequency range 0–5 Hz (solid lines of Fig. 3). The subscripts i and j may refer to the same degree of freedom (e.g., vertical or rocking motion) or different ones in the case of coupling, although coupling terms are neglected in the present model. The calibration procedure is repeated systematically for each pier and IML to capture both the spatial variability of the SFSI response and the nonlinear degradation of soil stiffness with increasing seismic intensity. The resulting sets of calibrated LPM coefficients are provided in RINTC Workgroup (2024). It is worth noting that the damping coefficient c ij , primarily associated with radiation damping, shows limited sensitivity to changes in seismic intensity across all vibration modes considered. 4. Nonlinear analyses of the case study Nonlinear analyses are performed by means of the equivalent-frame modelling strategy, implemented in the TREMURI software (Lagomarsino et al. 2013), adopting advanced mechanics-based macroelements for piers