Understanding the Seismic Resilience of Metallic Cylindrical Tanks Through Parametric Analysis

Abstract

This research investigates the seismic behavior of rigid and flexible cylindrical steel tanks, focusing on tanks with an open top and fully anchored at the base. The primary objective is to evaluate the hydrodynamic pressures exerted by the fluid on the tank walls during seismic excitation. Three widely recognized design approaches—New Zealand NZSEE recommendations, European code UNI EN 1998-4:2006 (CEN, 2006), and American Water Works Association AWWA D100-05 standard (ANSI/AWWA, 2005)—were implemented and compared with high-definition finite element models and then validated against the experimental results. Nonlinear fluid–structure interaction (FSI) was modeled using an Arbitrary Lagrangian–Eulerian (ALE) formulation with the Navier–Stokes equations governing the fluid motion and material and geometric nonlinearities considered in the tank walls. Parametric analyses were conducted to investigate the impact of tank geometry, specifically height-to-radius and radius-to-thickness ratios, on seismic response, identifying a transition between rigid and flexible behavior. The study also examined the influence of seismic input using a set of ten displacement spectrum-compatible ground motions. The findings contribute to a better understanding of the seismic resilience of cylindrical steel tanks, offering valuable insights for improving design standards and safety in earthquake-prone regions where these systems may abound.

1. Introduction

The vulnerability of steel storage tanks and containment structures in industrial plants to earthquake-induced damage has been well documented across numerous seismic events as well as a wide variety of experimental and/or analytical/numerical studies available in the literature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Significant incidents include the 1960 earthquake in Chile [29], the 1964 Alaska earthquake, the 1964 Niigata earthquake, and several major earthquakes in California, such as the 1971 San Fernando, 1979 Imperial Valley, 1989 Loma Prieta, and 1994 Northridge events. Other noteworthy earthquakes include the 1995 Kobe earthquake and the 2006 Silakhor earthquake in western Iran [30]. In Italy, the 2009 L’Aquila earthquake and the 2012 Emilia earthquake [31] highlighted the vulnerabilities of industrial facilities, particularly those that house critical containment structures. The 2013 Sedan and Lake Grassmere earthquakes in New Zealand (with magnitudes of 6.5) and the 2016 Kaikōura earthquake (magnitude 7.8) [32] further demonstrated the severe impact of seismic events on industrial infrastructure, especially on storage tanks and containment facilities that store hazardous materials, chemicals, and fuel. These events underscored the necessity for improved seismic resilience in the design, construction, and retrofitting of such facilities to prevent catastrophic failures that could result in the loss of containment, fires, hazardous spills, and disruption of essential services critical for public safety and environmental protection.

The seismic performance of these tanks is critical not only due to the economic value of the structures and their contents but also because they often play a key role in the operation of emergency services following a seismic event. Disruptions in the supply of essential products such as oil and water can significantly hinder the effectiveness of critical facilities, such as hospitals and fire stations, which rely on these resources. For instance, a lack of water following an earthquake could trigger major fires, potentially causing more damage than the earthquake itself. The failure of tanks holding highly flammable petroleum products can lead to catastrophic, uncontrolled fires, which is similar to those observed after the Niigata and Alaska earthquakes in 1964. Furthermore, the release of hazardous materials, such as liquefied gases or toxic chemicals from containment structures damaged during an earthquake, can pose significant risks to both human health and the environment [26,33].

Past seismic events have revealed various forms of damage and failure in steel tanks [1]. The most common forms of structural damage include buckling of the tank shell, such as elastic–plastic buckling (often referred to as “elephant foot” buckling), which is typically seen in large oil containment structures, and elastic buckling (or “diamond buckling”), which occurs more frequently in thin-shell tanks, like those used in wine storage. Beyond these buckling issues, other common forms of damage have been identified, such as failures in the tank roofs, substantial settlements caused by soil liquefaction beneath the tank foundation, and anchor failures. Additionally, damage to the connecting piping systems is one of the leading causes of product loss during seismic events.

Design and analysis guidelines aimed at addressing these failure modes in steel tanks can be found in documents like Eurocode 8 Part 4 [34] and the recommendations from the NZSEE study group [33]. However, there are ongoing debates about the adequacy of current guidelines, particularly regarding the seismic loads acting on the tank walls, which can affect the necessary design parameters for earthquake-resistant structural systems. The design methods found in current international codes trace their origins back to early studies by Jacobsen [35] and Housner [36] on rigid tanks, which were later expanded by Haroun and Housner [37], Veletsos [38], and Malhotra [39] to include the flexibility of tanks. These methods use simplified mechanical analogies to represent the complex phenomena of fluid–structure interaction, based on solutions to the Laplace equation under specific boundary conditions, allowing for the calculation of hydrodynamic pressure distributions induced by seismic forces.

This research focuses on assessing the seismic response of both rigid and flexible cylindrical steel tanks, specifically above-ground tanks that are fully anchored at their base with an open top, which is commonly used for water or other liquid storage. The primary goal of the study is to evaluate the hydrodynamic pressures exerted by the fluid on the tank walls due to seismic excitation, as these pressures are crucial for determining the tank’s seismic response. The circumferential stresses in the tank shell are highly influenced by these hydrodynamic forces, and their accurate evaluation is essential. While mechanical models provided by international standards can calculate global response parameters such as shear and overturning moments at the base of the tank, they do not account for the circumferential stresses, which are vital for a comprehensive analysis of the tank’s structural integrity. Cylindrical tanks are typically preferred over rectangular ones because of their ability to withstand large circumferential stresses due to their geometry.

This study examines the effects of various parameters, such as the tank’s aspect ratio $H/R$ and wall flexibility, on seismic performance. To assess the accuracy of current code provisions for the seismic analysis and design of liquid-containing structures, the research compares numerical results with those proposed by existing design standards. Specifically, three international codes are analyzed and compared: the New Zealand NZSEE recommendations [33], the European code UNI EN 1998-4:2006 [34], and the American Water Works Association AWWA D100-05 standard [40]. However, before applying numerical techniques for design purposes, the accuracy of these methods must be validated against experimental data. For this purpose, a fully nonlinear fluid–structure interaction (FSI) model based on the Arbitrary Lagrangian–Eulerian (ALE) formulation of the liquid-structure finite element method is used. Two existing experimental studies of anchored tanks are utilized for model validation. Fluid motion is governed by the compressible Navier–Stokes equations, and both material and geometric nonlinearities are considered to accurately determine stress, strain, and pressure distributions throughout the tank. The study proposes a rigorous finite element model to capture the three-dimensional behavior of cylindrical tanks, including the sloshing behavior of the contained liquid.

Given that the methods used by the three standards to assess hydrodynamic pressures vary, this research aims to evaluate their reliability by comparing the results obtained from advanced finite element analyses. The findings from this study will contribute valuable insights into the actual behavior of above-ground cylindrical tanks under seismic motions and offer recommendations for improving the accuracy of seismic analysis and design for these structures.

2. Finite Element Modeling, Analysis and Verification

The analysis of the seismic behavior of steel tanks using the finite element method (FEM) involves addressing an FSI problem, which presents several challenges. One of the main difficulties in solving this problem is the need for specialized finite element (FE) formulations capable of handling the large distortions in computational meshes that arise during simulations of dynamic fluid–structure interactions. To address this issue, several FE formulations for FSI problems are discussed and experimentally validated for use in this research. Particular emphasis is placed on the ALE method, as it is widely regarded as the most effective approach for solving FSI problems in fluid–structure simulations. The ALE method, which allows for the handling of mesh deformations, was specifically chosen for the analyses conducted in this study due to its ability to accurately model the behavior of both the fluid and the tank structure under seismic loads. This method ensures that the evolving geometry of the system during seismic excitation is properly captured, allowing for more accurate predictions of the tank’s response to seismic events.

In this research, the general-purpose FE code LS-DYNA is employed to study the FSI problem of deformable steel tanks, which are completely filled with fluid and subjected to seismic loading at the base. The interactions between the fluid and the structure are modeled using the ALE method, which is well suited for handling large deformations in both the fluid and the structure. This method enables an accurate representation of the fluid’s behavior and the tank’s deformation under seismic excitation. The use of the LS-DYNA FE code allows for the incorporation of both material and geometric nonlinearities, providing a more realistic simulation of the tank’s dynamic response. Additionally, LS-DYNA performs dynamic analyses using an explicit time integration scheme based on the central difference method, which is particularly effective for simulating high-speed dynamic events like seismic loading. This combination of advanced modeling techniques and robust numerical methods ensures a comprehensive and accurate assessment of the seismic response of steel tanks under extreme conditions.

Numerical techniques for simulating tank–fluid interaction problems are highly effective tools for evaluating the seismic response of these systems, as they can significantly reduce the need for time-consuming, costly experimental tests, which are typically conducted under specific boundary and excitation conditions. Once validated against experimental data, numerical simulations can serve as reliable design tools to assess various system configurations subjected to different earthquake ground motions. In this study, experimental tests are used as reference solutions for validating the tank models then used in the parametric evaluation of the resilience.

Before developing the FEMs for the case study tanks, the chosen modeling approach is verified by comparing the results of FE analyses with experimental data obtained from two separate tests. The first test is the pioneering work by Clough [41], which involved a small-scale tank model. As a continuation of this research, Manos and Clough [42] conducted a series of experiments on the same model subjected to three different scaled earthquake motions with varying boundary and soil conditions. This study focuses solely on a model tank with an open top, anchored support, and a rigid foundation. The second experimental test considered is presented in Haroun [43], which describes a shake table test performed on an open-top cylindrical tank. Haroun’s work also includes numerical results, allowing for a direct comparison between the experimentally observed behavior and the predictions obtained from an LS-DYNA model developed according to the previously described modeling approach.

The comparison is based on several key parameters, including the maximum radial displacement at the top of the tank shell, the maximum meridional force observed in the shell, the maximum base shear, pressure distributions, and the sloshing wave height. By comparing these parameters, the study aims to assess the accuracy and reliability of the numerical model in replicating the observed behavior from the experimental tests, thus validating the chosen modeling approach for simulating the seismic response of steel tanks then used in the parametric evaluation.

2.1. FEM of Manos and Clough Experimental Tank [42]

The tank model used in the experiments was a 1/3 scale model of a steel prototype with a radius of 1.83 m and a total height of 1.83 m. The tank was filled with water (density of 1000 kg/m³) to a height of 1.53 m. The tank was constructed from aluminum with a density of 2700 kg/m³, an elastic modulus of 71,000 MPa, and a yield stress of 100 MPa. The thickness of the aluminum bottom plate and the shell section near the bottom was 0.002 m, while the second shell course had a thickness of 0.0013 m. A steel wind girder, shaped in an L configuration, was placed on top of the second shell course. The geometry of the tank is shown in Figure 1a.

In the experiment, the shaking table motion was derived from the horizontal component of the El Centro 1940 earthquake with a peak acceleration of 0.50 g. The motion was scaled with respect to time by a factor of 1/√3 to meet similitude requirements. For the numerical verifications, the same tank model with an open top, anchored support, and resting on a rigid foundation was used. The numerical model retained the same dimensions and material properties as the experimental tank. During model development, the effect of element size and formulation, for both the tank and the fluid, was carefully investigated and adjusted to accurately capture the fluid–structure interaction. The tank was discretized using four-noded, fully integrated shell elements with two integration points through the thickness. The resulting finite element model consisted of 5793 nodes, 890 shell elements, and 4320 solid elements, as shown in Figure 1b. An explicit nonlinear transient analysis was performed to replicate the experimental test. The time histories of pressure (excluding hydrostatic pressure) at three specific locations (at $R$ = 1.83 m with heights of 0.05 m, 0.45 m, and 0.86 m above the tank base) during the experiment were presented in Figure 2. These results were used to verify the numerical model and compare the simulated pressures with the experimentally observed values. Pressure is a key parameter in the study of storage tanks because it plays a crucial role in evaluating the dynamic response of the tank to seismic loading. Figure 2 presents a comparison of the pressure distribution at three different levels along the height of the tank: at the base, one-third of the height, and approximately half the height. This comparison is crucial, as it captures the pressure variation along the tank wall during seismic loading, offering valuable insights into the fluid–structure interaction over time. By showing the overlap between the experimental results and the numerically evaluated data, the figure clearly demonstrates that the numerical model successfully replicates the actual behavior of the fluid inside the tank. The pressure profiles at these three heights highlight the nonlinear behavior of the fluid’s displacement. The significant variation in pressure over time, especially at higher fluid levels, emphasizes the complex dynamics involved in fluid motion during seismic events. The agreement between the experimental and numerical results confirms that the model effectively captures the real-world fluid behavior. This validation is key to understanding the dynamic response of the tank under seismic loading and supports the accuracy of the proposed numerical methods. In summary, Figure 2 summarizes these findings, illustrating the alignment between experimental and numerical data across different fluid heights, and reinforces the robustness of the methodology used in this research.

2.2. FEM of Haroun Experimental Tank [43]

The cylindrical tank tested by Haroun [43] was an open-top, anchored support tank filled with water. The shell was made of aluminum with a Young’s modulus of E = 68,940 MPa and a density of 2600 kg/m³. The tank had a radius of 1.18 m and the height of the aluminum shell was 4.57 m, while the height of the water’s free surface was 3.96 m. The shell thickness varied with height: up to 3 m, the thickness was 2.3 mm, and above this, the thickness was reduced to 1.6 mm. The shake table test was conducted using a record from the 1994 El Centro earthquake. To account for the scaling of the model, the record was sped up by a factor of 1.73 and scaled by 1.43 to achieve a peak ground acceleration (PGA) of 0.5 g. The FEM was developed using the previously outlined approach. The tank was discretized with 1236 shell elements for the base plate and the wall, while the fluid was modeled using 6084 solid elements with the ALE formulation. The final model, depicted in Figure 3a, consists of 8009 nodes. The discretization was chosen to balance the accuracy of the results with computational efficiency.

An explicit nonlinear transient analysis was performed to replicate the experimental test. The results from the LS-DYNA FE model were compared with the experimental data from Haroun [43]. The comparison of the results is presented in

Table 1. Comparison of experimental and numerical results.

Parameter	Experimental	FEM	[%]
Max radial disp at shell top [mm]	3.32	3.40	2.41
Max meridional compression [N/m]	63,396	62,205	1.88

, and the deformed shape of the model at different times during the transient analysis is shown in Figure 3b. The maximum meridional compression, measured 20 cm above the tank’s base plate, was found to be nearly identical in both the experimental test and the FE model with a difference of less than 2%. The maximum radial displacement and base shear were also closely matched with differences between the numerical and experimental results consistently below 3%. These findings suggest that the modeling approach used in this study is sufficiently reliable for simulating the seismic behavior of cylindrical tanks under similar conditions.

3. Case Studies, Material, Geometry and FEM

The case study structures analyzed in this research are vertical cylindrical tanks with both flexible and rigid walls, which were all perfectly anchored at their base. To ensure a comprehensive study, ten different geometric configurations have been considered. The slenderness ratios ($H/R$) for the case study steel tanks were selected to represent both squat tanks (with 0.5 < $H/R$ < 1.5) and slender tanks (with 1.5 < $H/R$ < 6), allowing for an analysis of a wide range of tank geometries. The radius-to-thickness ($R/t$) ratio was also varied to investigate the effects of both rigid walls (with $R/t$ close to 100) and flexible walls (with $R/t$ close to 2000).

Table 2. Case studies: geometrical configurations in terms of $H/R$ and $R/t$ ratios.

$\mathit{H}/\mathit{R}$	0.50	0.75	1.00	1.50	2.00	2.50	3.00	4.00	5.00	6.00
$\mathit{R}/\mathit{t}$	100	125	150	250	400	600	800	1000	1500	2000

provides a summary of all the geometric configurations in terms of the $H/R$ and $R/t$ ratios considered for the FE analysis.

The ten case study tanks were designed to contain water with a fixed radius ($R$) of 1.83 m. The height ($H$) of the tanks ranged from 0.91 m to 10.97 m, covering the full range of slenderness ratios (0.5 < $H/R$ < 6.0). The tanks were made of aluminum with a density of 2700 kg/m³, an elastic modulus of 71,000 MPa, and a yield stress of 100 MPa. The shell thickness ($t$) of the tanks varied between 0.91 mm and 18.29 mm and was kept constant along the entire height of the tank. To avoid unrealistic geometrical configurations in terms of the $R/t$ ratio, preliminary designs were performed, and only significant $R/t$ ratios were selected for each $H/R$ case study tank (see

Table 3. Main design properties and modelling dimensions of the case studies tanks.

Thickness Wall, t [m]
	H/R	0.50	0.75	1.00	1.50	2.00	2.50	3.00	4.00	5.00	6.00
R/t	100	-	-	-	-	-	-	-	-	-	0.018288	Rigid
	125	-	-	-	-	-	-	-	-	0.014630	0.014630	----------------------->
	150	-	-	-	-	-	-	-	0.012192	0.012192	0.012192
	250	-	-	-	-	-	0.007315	0.007315	0.007315	0.007315	0.007315
	400	-	-	0.004572	0.004572	0.004572	0.004572	0.004572	0.004572	0.004572	0.004572
	600	0.003048	0.003048	-	0.003048	0.003048	0.003048	0.003048	0.003048	0.003048	-
	800	0.002286	0.002286	0.002286	0.002286	0.002286	0.002286	0.002286	0.002286	-	-
	1000	0.001829	0.001829	0.001829	0.001829	0.001829	0.001829	0.001829	-	-	-
	1500	0.001219	0.001219	0.001219	0.001219	0.001219	0.001219	0.001219	-	-	-
	2000	0.000914	0.000914	0.000914	0.000914	0.000914	-	-	-	-	-	Flex.
Water height, H [m]		0.9144	1.3716	1.8288	2.7432	3.6576	4.5720	5.4864	7.3152	9.1440	10.9728
N° of shells		16	18	20	24	30	36	40	50	60	72
Height of shell, l [m]		0.057150	0.076200	0.091440	0.114300	0.121920	0.127000	0.137160	0.146304	0.152400	0.152400
Wall length, L [m]		1.20015	1.75260	2.28600	3.31470	4.26720	5.20700	6.17220	8.04672	9.90600	11.73480
Modulus of stiffening ring, Z [cm3]		0.7196	1.0794	1.4392	2.1587	2.8783	3.5979	4.3175	5.7566	7.1958	8.6350
Thickness ring, s [m]		0.009	0.009	0.010	0.011	0.012	0.013	0.014	0.015	0.017	0.018

). Figure 4 illustrates the FE models used for the geometrically and materially nonlinear analyses, which include fluid–structure interaction capabilities in LS-DYNA. These models were designed to simulate the full range of seismic behaviors and provide insight into the dynamic response of the tanks under earthquake loading. In the proposed FE approach, the motion of the fluid was described using compressible Navier–Stokes equations, which were implemented through a mesh of eight-noded, one-point integration ALE solid elements. For modeling the tank wall, four-noded, two-point integration shell elements were selected, as described in [44]. This type of shell element provides a computationally efficient solution, as it is based on a combined co-rotational and velocity–strain formulation. To model the wall’s material behavior, a bilinear elastic–plastic constitutive law (MAT_003) was employed, incorporating both isotropic and kinematic hardening. This approach was used to capture the cyclic permanent deformations exhibited by the tank wall under seismic excitation. In this study, null material was used to simulate the fluid, which is conservatively assumed to be water. Consequently, the bulk modulus of the fluid was assumed to be 2.25 × 10⁹ N/m². The resulting FE models, based on different tank geometries, vary in complexity, with the number of nodes ranging from 6162 to 25,034, depending on the specific tank configuration being analyzed. This range reflects the model’s ability to handle different tank sizes and geometries while maintaining computational efficiency.

Before conducting a series of case study analyses, the acceleration history of the input motion used in this research was the same as that employed by Manos and Clough [42] for their experimental test. The acceleration history and response spectrum of the input motion, used in both the experimental tests and the numerical simulations of the case studies, are provided in Figure 5a and Figure 5b, respectively. These figures illustrate the time history of the seismic input and its corresponding response spectrum, which serve as the foundation for analyzing the seismic behavior of the tanks in the subsequent case study simulations.

4. Discussions on FE Results and Parametric Sensibility

To study the effect of rigid and flexible wall tanks, the time history responses of the tank models were calculated, and a comparison was made between the obtained results. All the FEMs described earlier have been used to perform explicit nonlinear analyses with the seismic excitation applied in the form of the acceleration time history that was used for the experimental test by Manos and Clough [42]. The resulting hydrodynamic pressure profiles for the ten different tank configurations, characterized by various $H/R$ slenderness ratios, are presented and were thoroughly analyzed. The envelopes of these pressure profiles provided a detailed understanding of how different geometric configurations influence the seismic response of the tanks particularly in terms of the distribution and magnitude of hydrodynamic pressures during the earthquake simulation. This analysis will help to assess the performance of both rigid and flexible wall tanks under seismic loading and highlight key differences between the two types of tank behavior.

We completed the analysis but decided to present the results for slenderness ratios of 0.5, 0.75, and 1.5, as 1.5 represents a transition between squat and slender tanks. We excluded 1.0 because it follows the trend of 0.75. Additionally, we focused on 3.0 and 6.0 while avoiding the intermediate ratios of 2.0, 2.5, 4.0, and 5.0. This approach provides a clear representation of the transition and extreme behaviors of squat and slender tanks, with both flexible and rigid walls, covering a wide range of thickness ratios.

4.1. Tank H/R = 0.5

To study the effect of both rigid and flexible wall tanks, the trend of pressures exerted by the fluid on the tank walls has been thoroughly investigated. The seismic response, expressed in terms of total wall pressure, of a tank with an $H/R$ ratio of 0.5, obtained from numerical analyses, is compared with the design predictions of the AWWA [40], EC8-4 [34], and NZSEE [33] standards. These standards use the input motion response spectrum from Figure 5 and apply the combination rule for impulsive and convective pressure components specified in each respective code. Figure 6a shows the pressure profiles normalized by the tank height ($z/H$) for five different configurations of the $R/t$ ratio, while Figure 6b compares the predictive expressions for impulsive and convective pressure distributions with the results of the FE simulations. As observed in Figure 6a, the pressure profiles exhibit a similar trend for all five different $R/t$ ratios investigated. Pressure peaks occurred at the top of the wall due to the convective waves both for rigid and flexible wall tanks. These peaks were generally similar in magnitude, except for the rigid tank ($R/t$ = 600), where the peak was smaller, occurring at approximately 0.9 normalized height ($z/H$). In Figure 6b, a comparison between the numerical results and the standard codes is presented. Generally, the results obtained from the FE analyses align well with the analytical predictions in the lower part of the tank wall. As expected from the preliminary analyses performed to calibrate the FE model, the impact of convective waves on the tank shell caused a pressure peak at the top of the wall, which was not captured by the design approaches proposed by the three regulations considered in this study. Under the assumption of flexible circular tanks, EC8-4 predicted the highest values for the pressure, as it recommended directly summing the impulsive and convective effects. On the other hand, when assuming rigid circular tanks, EC8-4 underestimated the pressure profiles compared with the FE simulations. The NZSEE standard provided more or less the same pressure values along the tank wall for both rigid and flexible tanks. However, the AWWA code tended to underestimate the pressure profiles compared to the FE results.

These observations highlight the limitations of current design codes in accurately predicting the pressure distribution for tanks subjected to seismic loading, especially at the top of the wall where convective waves play a significant role.

4.2. Tank H/R = 0.75

As observed in Figure 7a, the pressure profiles exhibit a similar trend across the five different $R/t$ ratios investigated. Significant peaks of pressure occurred at the top of the wall due to convective waves both for rigid and flexible wall tanks. The largest peak occurred for the rigid tank ($R/t$ = 600), at approximately 0.9 normalized height ($z/H$), and this peak decreased as the flexibility of the wall increased, reaching its lowest value for the flexible tank ($R/t$ = 2000). In the lower part of the tank, the pressure profiles remained relatively uniform, showing no significant differences across the different $R/t$ ratios. Figure 7b presents a comparison between the numerical results and the pressure distribution recommendations from the various standards along the wall. In general, the results obtained from the FE analyses aligned well with the analytical predictions in the lower part of the tank wall. However, as in previous cases, the impact of convective waves on the tank shell caused a pressure peak at the top of the wall that cannot be predicted by the design codes. Under the assumption of flexible circular tanks, EC8-4 predicted the highest pressure values, while for rigid circular tanks, EC8-4 underestimated the pressure profiles compared to the FE simulations. The NZSEE standard resulted in similar pressure values along the wall for both rigid and flexible tanks. In contrast, the AWWA code consistently underestimated the pressure profiles when compared to the FE results. These findings further emphasize the limitations of existing design codes, particularly in capturing the effect of convective waves at the top of the tank wall, which is critical for accurately predicting the seismic response of tanks.

4.3. Tank H/R = 1.5

As observed in Figure 8a, the pressure profiles for the five different $R/t$ ratios show a similar trend. The rigid tank displayed a very small peak of pressure at the top of the wall, around 0.9 normalized height ($z/H$), while the flexible tanks exhibited a much smoother profile with no noticeable pressure peaks. The results from the FE analyses are in good agreement with the analytical predictions in both the bottom and upper parts of the wall. In this specific case, compared to previous configurations where pressure peaks were absent in the upper part of the tank, the pressure profiles were very well predicted by the three design standard codes. Only the AWWA and EC8-4 standards, when applied to the rigid tank assumption, slightly underestimated the FE results, but the discrepancy was not significant. Looking at Figure 8b, it is immediately apparent that the pressure peaks observed in earlier analyses at the top of the shell were not present for this case study. This is because the convective contribution was considerably smaller in this case, which was due to the different slenderness ratio of the containment structure. The tank with an $H/R$ = 1.5 ratio represents the transition point between squat tanks and slender tanks. It is important to note that despite the filling height of the tank being limited to 80% of the shell height, the FE analyses recorded pressures above the filling height as well. This is due to the free movement of the fluid inside the tank. When fluid moves vertically as part of the sloshing motion, it exerts horizontal pressures above the hydrostatic free surface.

This phenomenon highlights the complexity of fluid behavior in seismic conditions particularly when sloshing effects contribute to pressure distributions above the typical water level in the tank. The results underscore the importance of accurately modeling fluid dynamics, especially for tanks with different slenderness ratios, to capture the full range of pressures exerted during seismic excitation.

4.4. Tank H/R = 3.0

As observed in Figure 9a, the pressure profiles for the five different $R/t$ ratios show a similar trend. As in previous cases, significant pressure peaks occur at the top of the wall due to convective waves both for rigid and flexible wall tanks. The largest peak occurred for the flexible tank ($R/t$ = 1000) at approximately 0.9 normalized height ($z/H$), and this peak decreased as the rigidity of the wall increased, reaching its lowest point for the rigid tank ($R/t$ = 250—Figure 9b). In the lower part of the tank, the pressure profiles remained relatively consistent, except for the most flexible wall ($R/t$ = 1000), which exhibited very large peaks due to the low resistance of the shell to horizontal excitations. It is evident that the $R/t$ = 1000 tank configuration failed through the classical elastic–plastic elephant foot buckling mechanism just above the base of the wall, highlighting the vulnerability of highly flexible tanks under seismic loading. The predictions made using the international regulations (EC8-4, NZSEE, and AWWA) consistently underestimate the envelope of pressures observed from the FE analyses. While the small discrepancies in the lower part of the tank can be accounted for in design, the significant underestimation in the upper portion of the tank, caused by the convective pressure component, may lead to an unsafe design, particularly if a substantial reduction in shell thickness is implemented along the tank height. Finally, it is important to note that the contribution of sloshing motion increased clearly as the radius-to-thickness ratio ($R/t$) rose. However, current analytical proposals are not fully capable of accounting for this behavior, which is observed to become more pronounced as the flexibility of the tank increases. This underscores the need for a more accurate prediction of fluid–structure interaction in tanks particularly in cases involving highly flexible structures that are more susceptible to large convective forces.

4.5. Tank H/R = 6.0

As observed in Figure 10a, the pressure profiles for the five different $R/t$ ratios show a similar trend. For the more slender tanks, large pressure peaks occurred at the top of the wall due to convective motion waves both for rigid and flexible wall tanks. The largest peak occurred for the flexible tank ($R/t$ = 250) at approximately 0.9 normalized height ($z/H$), and this peak decreased as the rigidity of the wall increased, reaching its lowest point at ($R/t$ = 125). In the lower part of the tank, the pressure profiles remained relatively regular, except for the most flexible wall ($R/t$ = 400), which exhibited very large peaks due to the low resistance of the shell to horizontal excitations. It is clear that the $R/t$ = 400 configuration tank failed through the classical elastic–plastic elephant foot buckling mechanism just above the base of the wall.

This case behaves similarly to the $H/R$ = 5 case study, and the comparison between the numerical results and standard codes is shown in Figure 10b. As in previous analyses, the predictions made using international regulations underestimated the envelope of pressures observed in the FE analyses. These results further demonstrate the limitations of current design codes in accurately predicting the pressure distribution especially in cases where large convective waves and buckling mechanisms significantly affect the tank’s behavior under seismic loading.

5. Numerical Results Considering a Set of Ten Scaled Recorded Accelerograms

To better understand how pressure trends may vary, a series of explicit dynamic nonlinear analyses were performed using a set of different accelerograms as input ground shaking motions. The dynamic response of the case study tanks was assessed by considering a set of ten scaled recorded accelerograms. These records were selected from the PEER NGA database, with magnitudes ranging from 6.2 to 7.6, ensuring compatibility with the EC8 design response spectrum. The reference response spectrum was defined with a peak ground acceleration (PGA) of 0.4 g and a soil category C, according to the European classification. Figure 11 shows the response spectra for the ten accelerograms used in the dynamic response analysis of the case study tanks, and it is complemented by

Table 4. Characteristics of ground motion records for nonlinear dynamic analyses.

Eq. ID	PEER ID	Event	Component	MW [-]	Dr [km]	ttot [s]	Vs [m/s]
01	1233	Chi-Chi, Taiwan	E	7.62	36	90	194
02	1153	Kocaeli	090	7.51	127	102	275
03	851	Landers	000	7.28	157	70	272
04	1810	Hector	090	7.13	92	60	345
05	1629	St Elias, Alaska	279	7.54	80	83	275
06	777	Loma Prieta	090	6.93	28	39	199
07	1043	Northridge-01	090	6.69	52	48	309
08	728	Superstition Hills-02	180	6.54	13	40	194
09	172	Imperial Valley-06	140	6.53	22	39	237
10	2615	Chi-Chi, Taiwan-03	N	6.20	40	107	273

, which provides details of the seismic input for nonlinear dynamic simulations. More specifically, PEER ID, event, component, moment magnitude (M_W), closest distance from recording site to fault area (D_r), duration (t_tot), and shear wave velocity over the first 30 m (V_s) are shown in Table 4. The matching between the mean response spectrum of the ten unscaled records and the EC8 design spectrum was verified for periods between 0.15 s and 4 s. This range is quite large because sloshing modes and tank impulsive modes have significantly different fundamental periods [45,46].

For the analysis, three case studies were selected to represent squat and slender tanks, specifically with $H/R$ = 0.75, 1.5, and 4 slenderness ratios. Figure 12a shows the pressure distributions for the squat tank ($H/R$ = 0.75 and $R/t$ = 2000) obtained from the FE simulations. A comparison between the results using shaking table excitations and those using the average of the accelerograms indicates that the pressure distributions for the average earthquakes overestimated the trends of pressures observed from the shaking table excitations. The discrepancies were small and are primarily observed at the bottom and upper parts of the tank, which were influenced by the severity of the selected accelerograms, especially for the “LC9” earthquake.

The same accelerograms were applied to the $H/R$ = 1.5 and $R/t$ = 400 case study tank. Figure 12b shows the comparison of the total pressure profiles obtained from the shaking table excitations and the average of the ten accelerograms. Again, the pressures from the average earthquakes overestimated the trend of pressures obtained from the shaking table with the impact of the sloshing wave being mostly negligible. The “LC9” earthquake, however, had the most significant effect due to its pronounced sloshing wave impact.

For the slender tank case study ($H/R$ = 4 and $R/t$ = 400), the comparison of pressure profiles is shown in Figure 12c. In this case, the total pressure profile from the average earthquakes closely matched the profile obtained using the accelerogram from the shaking table. For slender tanks, it is important to note that the effect of sloshing wave impact is well captured by both input motions, whether using the accelerogram from the shaking table or the average of the set of accelerograms.

These findings demonstrate that for squat tanks, the average accelerograms tend to overestimate the pressures observed from shaking table excitations, while for slender tanks, both input motions accurately capture the sloshing wave effects, resulting in almost identical pressure profiles [47].

6. Conclusions

To determine the transition zone between rigid and flexible tank idealizations, the trend of hydrodynamic pressures exerted by the fluid on the tank wall during an earthquake has been extensively studied. Specifically, the seismic response of open-top steel tanks, both with rigid and deformable walls, fully anchored at the base, is analyzed. Understanding hydrodynamic pressures is critical because they directly influence the circumferential stresses in the steel shell, which can lead to shell buckling. The importance of selecting a rigid or flexible tank idealization lies in the potential for under- or over-estimating hydrodynamic pressures compared to actual values, which impacts the accuracy of the seismic analysis.

After calibrating the model with two experimental tests, 50 transient explicit analyses were performed using LS-DYNA software ver. 9 to generate the envelopes of hydrodynamic pressure profiles for ten different configurations of $H/R$ slenderness ratios and five configurations of $R/t$ ratios. The seismic responses obtained from the numerical analyses were then compared with the design predictions from the AWWA, EC8-4, and NZSEE standards. Given the differences in assumptions regarding rigid or flexible tanks and the complexity of the design methodologies in the international standards, the goal of this research was to investigate the prediction of hydrodynamic pressures to provide more accurate seismic analysis and design recommendations for above-ground cylindrical tanks. Key conclusions from the study are outlined below:

• NZSEE Method: This method is relatively complex as it requires the use of advanced mathematical tools such as Bessel functions and the solution of “heavy” equations. While initially developed for rigid tanks, the NZSEE method was extended to flexible tanks with minimal modification, resulting in discrepancies between predicted and observed responses.
• EC8-4 Method: This method is the only one that accounts for the influence of tank wall deformability. However, it is highly complex, as it requires an iterative procedure, and FE analyses are always necessary for accurate results.
• AWWA Method: This is the simplest of the three methods but does not account for the flexibility of the tank wall. Additionally, it is based on an admissible stress design approach, and the choice of the force reduction factor used in the calculations remains debatable, as no unanimous consensus can be gathered on it.
• The comparison of the methods proposed in the three standards reveals that differences between the results can be quite large when considering the various contributions. However, these differences become less significant when evaluating the total pressures. This is due to the fact that the amplitude of hydrodynamic pressures is generally lower than that of the hydrostatic pressures. Specifically, for slender tanks, the influence of hydrodynamic pressures is sufficiently diminished. While variations of around 10% were observed between the total pressure profiles computed using the different methods, variations in the order of 25–30% were found in the impulsive contributions.
• The previous comments, along with the subsequent considerations regarding the parametric sensitivities and the various ratios involved in the analyses, such as radius, height, thickness, as well as rigid vs. flexible and slender vs. squat configurations, lead to the most important result summarized in Figure 13. This figure provides guidance on the selection of flexible versus rigid tanks, depending on the $H/R$ (height-to-radius) and $R/t$ (radius-to-thickness) ratios. The figure offers valuable insights into how the tank’s geometric properties influence the decision to adopt a flexible or rigid model for seismic design, helping to optimize safety and performance based on the tank’s specific characteristics.
• Explicit dynamic nonlinear analyses were conducted to accurately reproduce the FSI problem. These analyses were performed using the LS-DYNA software, treating the FSI problem according to the ALE formulation. The modeling approach was verified by comparing the results from a shake table test performed by other researchers with the numerical results obtained from the model developed in this study. The comparison demonstrated that the chosen modeling approach provides a sufficiently accurate prediction of the tank’s seismic response, making it a reliable reference for future research in this area.
• Nonlinear transient analyses were performed to assess the seismic response of ten case study tanks, which were designed to represent both squat and slender configurations. Each tank was analyzed using the seismic excitation corresponding to the acceleration time history from the experimental test by Manos and Clough [42]. The pressure profiles obtained from these analyses exhibited similar trends across the five different $R/t$ ratios investigated. A noticeable peak in pressures occurred at the top of the wall for both rigid and flexible wall tanks: around 0.9 normalized height ($z/H$). However, the $H/R$ = 1.5 case study showed a smaller peak due to a significantly reduced convective contribution, which is attributed to the different slenderness of the containment structure. This peak has been shown to be related to the sloshing motion, specifically the impact of the convective wave on the tank wall, which generates the pressure peak. The $H/R$ = 1.5 case represents the transition point between squat and slender tanks. Based on the analysis of all the cases studied, the following conclusions can be drawn:
  • For squat tanks (0.5 < $H/R$ < 1.5), the largest pressure peak occurred in rigid tanks and decreased as the wall flexibility increased.
  • For slender tanks (1.5 < $H/R$ < 6), the largest pressure peak occurred in flexible tanks and decreased as the rigidity of the wall increased.
  These findings highlight the significant influence of tank flexibility and slenderness ratio on the seismic pressure distribution, emphasizing the need for careful consideration of these factors in the design and analysis of cylindrical tanks subjected to seismic excitation.
• The pressure profiles obtained from the FE analyses were compared with the predictions from the three standard approaches (EC8-4, NZSEE, and AWWA) using the input motion response spectrum and the combination rule for impulsive and convective pressure components specified in each corresponding code. Generally, a good agreement was observed in the lower part of the wall. However, the impact of the convective waves on the tank shell causes a peak in pressure at the top of the wall, which cannot be predicted by the design approaches proposed by the three regulations, except for the $H/R$ = 1.5 case study. It is important to note that these peaks could lead to significant problems during seismic events and should be taken into account during the design of steel tanks. Although small discrepancies at the base can be easily accounted for in design, the considerable underestimation of pressures in the upper portion of the tank could lead to an unsafe and non-conservative design, particularly if there is a large reduction in thickness along the tank height. Comparing the pressure response values obtained from the FE simulations with those predicted by the standard codes, the following conclusions can be drawn:
  • AWWA underestimated the FE results for all studied cases, including both squat and slender tanks.
  • EC8-4 for rigid tanks underestimated the FE results across all cases: both squat and slender tanks. EC8-4 for flexible tanks overestimated the FE results only for squat tanks (0.5 < $H/R$ < 1.5) but underestimated the results for slender tanks (1.5 < $H/R$ < 6).
  • NZSEE recommendations, both for rigid and flexible tanks, led to similar pressure values for squat tanks (0.5 < $H/R$ < 1.5) but underestimated the FE results for slender tanks (1.5 < $H/R$ < 6).
  These results highlight the limitations of the current design codes especially in predicting the impact of convective wave-induced pressure peaks at the top of the tank wall. The findings suggest a need for more accurate methods to capture the full seismic behavior of tanks—particularly those with slender designs.
• The trend of pressures was investigated using a set of ten scaled recorded accelerograms, which were all compatible with the EC8 design response spectrum. It can be concluded that the pressure profiles obtained from these accelerograms were almost identical to those obtained using the seismic excitation from the shaking table test used in the experimental study by Manos and Clough [42]. This similarity indicates that the selected accelerograms provide a reliable representation of the seismic response and produce pressure profiles comparable to those observed in the experimental setup.