## 5 Conclusions  
This study introduces a novel methodology for the seismic vulnerability assessment of ground-supported liquid storage tanks, with a particular focus on tanks exhibiting deformed shells. A key contribution of this work is the development of a simplified two-degree-of-freedom (2 DoF) system, which enables efficient IDA simulations for probabilistic analysis. The 2 DoF model, calibrated using detailed nonlinear finite element (NL FEM) simulations, offers a balance between computational efficiency and accuracy, allowing for rapid evaluation of seismic fragility. The NL FEM model plays a critical role in capturing the complex fluid-structure interaction and calibrating the stiffness and damping properties of the 2 DoF system, thereby bridging the gap between computational rigor and practical application.  
Another distinguishing feature of this work is the explicit consideration of tank deformations, which are often overlooked in traditional seismic assessments. This inclusion highlights the significant influence of geometric imperfections on tank behavior, particularly in amplifying vulnerability to local buckling and other damage mechanisms. By integrating deformations into the analysis, the proposed framework provides a more realistic and comprehensive evaluation of tank performance under seismic loads.  
The application of this methodology to a representative tank validated its ability to capture critical failure mechanisms and generate fragility curves associated with different  
Figure 10: Fragility curves for the critical damage states of the liquid storage tank.  
damage states. The results confirmed the efficacy of the 2 DoF system in reproducing seismic responses with sufficient accuracy while significantly reducing computational demands compared to full NL FEM simulations. The findings also underscored the importance of considering deformed geometries, which can exacerbate vulnerabilities in certain damage states.  
This study represents a contribution to the seismic risk assessment of tanks by providing a practical yet rigorous framework that can be adapted to various tank configurations and seismic conditions. The proposed methodology offers engineers and researchers an effective tool for evaluating seismic performance, developing design improvements, and prioritizing retrofitting efforts through its simplified representation using a 2-DOF system. However, it is important to note that certain nonlinear effects, such as plastic deformation, buckling, or complex fluid-structure interactions (e.g., sloshing and impulsive forces), may be underrepresented in this approach. These limitations could affect the accuracy of the assessment, particularly under high-intensity seismic events or for tanks with complex geometries or boundary conditions. Future research should aim to incorporate these nonlinearities for a more comprehensive analysis.  
Conceptualization, R.R.R. and G.M.; methodology, R.R.R. and G.M.; software, R.R.R.; validation, R.R.R. and G.M.; writing--original draft preparation, R.R.R.; writing--review and editing, R.R.R. and G.M.; supervision, G.M. All authors have read and agreed to the published version of the manuscript. This research was funded within the project DPC-ReLUIS 2024-26.  
Not applicable.  
Data Availability Statement: Data are contained within the article.  
The authors declare no conflicts of interest.