## 1 Introduction  
Liquid storage tanks are critical components in modern infrastructure, serving a wide range of functions, from water distribution and firefighting to storing oil and hazardous materials in industrial facilities. These tanks are indispensable for the extraction, processing, and distribution of vital resources. However, given the global prevalence of storage tanks, many are inevitably situated in seismically active regions. This has driven significant interest in developing earthquake-resistant designs and conducting seismic fragility analyses, with numerous experimental and analytical studies aimed at mitigating earthquake risks and evaluating tank vulnerability under seismic loads [1; 2; 3; 4; 5; 6; 7; 8; 9]. The failure of these tanks during earthquakes can result in catastrophic losses, environmental contamination, and disruption to essential services, as evidenced during recent seismic events in Italy (2012) [10] and Chile (2010), with buckling of the tank shell and failure of the anchor or support legs as primary causes. Similarly, [PERSON] and [PERSON] [11] highlight the damage to larger on-grade storage tanks during earthquakes in the United States, Turkey, and Japan. In these cases, foundation instability and the sloshing of liquid inside the tanks also contributed to damage, with sloshing waves not only harming the upper portions of the tanks but also causing liquid overspill. Over the past five decades, substantial research has been conducted to understand the seismic behavior of liquid storage tanks. Foundational studies by [PERSON] [12] and [PERSON] [13], followed by subsequent research [14; 15; 16; 17; 18; 19], laid the groundwork by introducing the concepts of decoupling the tank-liquid system motion into two distinct components: the impulsive component, representing the dynamic behavior of the tank's lower part, and the convective component, which describes the movement of surface waves in the liquid. These studies have been widely adopted in seismic design codes, such as API 650 [20], ASCE/SEI 7-10 [21], CEN [22], and NZSEE [23], which adopt simplified models to estimate dynamic properties for idealized tank geometries. More recently, fragility curves have emerged as a powerful tool for assessing seismic vulnerability, with significant research focused on developing fragility models for buildings and bridges [24; 25; 26] and, to a lesser extent, for liquid storage tanks. Notable early work by ALA [27; 28] provided empirical fragility curves for on-ground supported liquid containers. [PERSON] [29] later categorized tank damages into five distinct states and proposed corresponding fragility curves. Building on this, [PERSON] et al. [30] developed fragility curves for unanchored liquid storage tanks using a surface response approach that considered a variety of tank geometries. [PERSON] and [PERSON] [31] used Bayesian methods to develop fragility curves. [PERSON] et al. [32] incorporated uncertainty factors that influence their seismic performance and, using Gaussian process regression, developed fragility models reflecting these uncertainties [2]. Additionally, [PERSON] et al. [7; 33] expanded on this by developing fragility functions for liquid tanks supported by special concentrically braced frames. In related work, [PERSON] et al. [34] carried out dynamic analyses to optimize the freeboard height accounting for record-to-record variability in seismic excitation. Despite these advances, most studies assumed an idealized, undeformed geometry for tanks, neglecting the potential influence of pre-existing deformations caused by prior seismic events. Such deformations, including shell denting and localized thinning, can substantially alter the dynamic behavior of tanks. The resulting changes in stiffness, mass distribution, and natural periods impact both impulsive and convective responses, rendering the assumptions in current design codes insufficient. This gap in the literature limits our ability to accurately assess the seismic performance of deformed tanks.