PSI - Issue 78

Samuele Faini et al. / Procedia Structural Integrity 78 (2026) 718–725

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5. NLTH analysis of the retrofitted bridge The effectiveness of the proposed retrofit layout is evaluated through NLTH analyses. Classical formulas for TMD s’ tuning allow to define the optimal frequency ( ) and damping ( ) solely basing on the mass ratio ( = ⁄ ) and assume that the structure (nearly) remains in its elastic range [Ayorinde et al., 1980]: = ∙ = √ 11−+ ⁄ 2 ∙ (5) =√ 4(1 + ( 1 )−( 1 −⁄4 ) ⁄2) (6) being: (i) = ⁄ ; (ii) the frequency of the mode shape to be mitigated; (iii) its participating mass. The bridge considered in this study is characterized by 1 = 2.09 and = 2491 (i.e., 71% of the bridge seismic weight 1 + 2 = 34400 ). Considering a low mass ratio ≅1.9% and the installation layout of Fig. 4-a, four TMDs of 12 tons (i.e., , = 48 ), were employed. The optimal damping =6.9% is calculated according to Eq. (6). Since, as will be shown later, it is not possible to keep the bridge in its elastic regime, the optimal TMDs’ frequency ( ) is identified by varying among these values: 0.80, 0.85, 0.875, 0.90, 0.95, 1.00, 1.05, 1.10. Unidirectional NLTH analyses (transversal excitation at the SLV limit state) are performed in Midas-Gen v2023, modeling the TMDs with “General link” elements [Midas -Support] characterized by axial stiffness = ∙ (2 ) 2 and damping coefficient = ∙ = ∙ (2√ ∙ ) . The reduction in peak transversal drift of the shortest spandrel columns (element n°1 in Fig. 1-a) is selected as performance index. As witnessed by Fig. 5-a, all tested values of improve the seismic response of the bridge with a peak benefit at =0.875 associated to 41.2% of reduction of the lateral drift. By comparing the displacement time-histories at mid-span (see Fig. 5-b), one can appreciate a drop from 35.5 mm , for the “as - built” layout (i.e., no TMD), to 26.5 mm (25.4% reduction), for the bridge equipped with optimized TMDs ( =0.875 ). Another benefit is recorded in terms of reduction of moment-curvature demand on short piers (see Fig. 5-c): for the “ as-built ” bridge (i.e., case “NO TMD”), peak bending moment and curvature are equal to 188 and 2.23 ∙ 10 −4 / , respectively; while, for the bridge with optimized TMDs, to 176 and 1.33 ∙ 10 −4 / (i.e., -40.4%). At sectional level, for the “ as-built ” bridge, the maximum elongation of the steel rebars is 7.7% while concrete fibers reach peak compression stress and strain of 59.9MPa and 0.228%, respectively. Therefore, the crushing-failure of the concrete is very close. Thanks to the introducti on of the optimized TMDs, the maximum rebars’ elongation is 4.4% (i.e., -42.9%) while concrete fibers are engaged at 50.2MPa (i.e., about 83% of their strength). These improvements are due to the counterphase dynamic forces generated by the TMDs. In this regard, Fig. 5-d shows the overall force-displacement loops of the four devices. A peak restoring force of about 440 kN is reached at (1) a downgraded FEM model, obtained neglecting rebars’ corrosion, is considered. This scenario offers initial insights into whether TMD tuning might require adjustments as the structure ages. It also reflects the type of simplified models often adopted by practitioners due to time constraints. Neglecting corrosion leads to a slightly stiffer structure (see Fig. 6-a): at around t = 9.5 s, the “ corroded model ” shows a peak deck displacement of 17.2 mm, compared to 13.4 mm of the “ non-corroded ” case (−22.1%). This suggests that minor adjusting of TMDs ’ tuning may be needed over the bridge ’s lifespan, which can be easily done by adjusting the modular stacked mass; (2) t he TMDs’ effectiveness is also tested under simultaneous application of all three components of the seismic input (i.e., longitudinal, transversal, and vertical). This is important because TMD s’ tuning is typically based on uni directional NLTH analyses, while real earthquakes are multi-directional. However, the additional components have minimal influence (see Fig. 6-b): the transverse deck displacements obtained from both 1D and 3D NLTH analyses are practically identical. However, it should be noted that in both cases, six longitudinal hysteretic dampers (200 kN each) are included to limit deck-abutment displacements (further detail can be found in [Gandelli et al., 2025]). Therefore, such findings are context-specific and broader conclusions would require a parametric study. 70 mm of displacement, with a not negligible energy dissipation (area within the loops). Lastly, the robustness of the optimized TMDs is evaluated in two specific conditions: