Abstract
Assessment of seismic resilience of critical infrastructure such as liquefied natural gas (LNG) storage tanks, is essential to ensure availability and security of services during and after occurrence of large earthquakes. In many projects, it is preferred to build energy storage facilities in coastal areas for the ease of sea transportation, where weak soils such as soft clay and loose sand with liquefaction potential may be present. In this study, threedimensional finite element model is implemented to examine the seismic response of a 160,000 m^{3} full containment LNG tank supported by 289 reinforced concrete piles constructed on liquefiable soil overlaying the soft clay deposit. The seismic soilstructure interaction analysis was conducted through direct method in the time domain subjected to the 1999 ChiChi and the 1968 Hachinohe earthquakes, scaled to Safe Shutdown Earthquake hazard level for design of LNG tanks. The analyses considered different thicknesses of the liquified soil deposit varying from zero (no liquefaction) to 15 m measured from the ground surface. The key design parameters inspected for the LNG tank include the acceleration profile for both inner and outer tanks, the axial, hoop and shear forces as well as the von Mises stresses in the inner tank wall containing the LNG, in addition to the pile response in terms of lateral displacements, shear forces and bending moments. The results show that the seismic forces generated in the superstructure decreased with increasing the liquefied soil depth. In particular, the von Mises stresses in the inner steel tank exceeded the yield stress for nonliquefied soil deposit, and the elastic–plastic buckling was initiated in the upper section of the tank where plastic deformations were detected as a result of excessive von Mises stresses. However, when soil liquefaction occurred, although von Mises stresses in the inner tank shell remained below the yield limit, localised stress concentrations were observed in the lower section of the tank near the base, increasing the risk of the elephant foot buckling. The lateral displacements, shear forces and bending moments in the piles increased with increasing depth of the liquefied soil. Indeed, increasing the pile lateral displacement amplified the bending moment at the pile head, thus resulting in increases in the pile bending moments especially when the liquefied soil depth exceeded one third of the entire soil deposit. In particular, the bending moment at the pile head exceeded the yield moment capacity of the pile and subsequent plastic hinges were formed. Moreover, when the thickness of the liquefied soil was more than half of the entire soil depth, the mobilised bending moments in the piles exceed the ultimate moment capacity of the pile and thus total failure of the piles were observed. In addition, in the absence of liquefied soil layer, the inertial interaction had a dominant impact on the pile response in this study. However, with increasing the thickness of the liquefied layer, further loads were developed in the piles due to amplified kinematic interaction, while the inertial interactioninduced loads decreased.
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1 Introduction
Liquid storage tanks have always been an important link in the distribution of water, chemical and refined petroleum products. In particular, during the past 30 years, the demand on the Liquefied Natural Gas (LNG) tanks has increase significantly (Animah and Shafiee 2020). As the LNG is cleaner and a cheaper fuel for power generation than oil and coal, it is considered as a transition and rather environmentally friendly energy fossil fuel source. While large water storage tanks located near cities and populated areas are in demand to ensure constant water supply, oil and liquefied natural gas storage tanks are generally located in refineries, terminals and ports to reduce the transportation cost and stay away from populated areas for safety reasons (Solakivi et al. 2019).
Indeed, natural gas is primarily made of methane, and to reduce the storage space, it is liquefied and stored at atmospheric pressures while operating at temperatures around − 166 °C. LNG tanks are usually built in areas near the shorelines to increase the flexibility of LNG transportation and reduce the traveling and storage cost. These tanks have capacities ranging from 160,000 to 225,000 m^{3} corresponding to diameters of 80 m to 100 m and heights from 30 to 50 m (Nagashima et al. 2011; Calderón et al. 2016). Thus, the foundation soil may not be capable of bearing the load of such heavy structure, and usually deep foundations are used to support these tanks built on weak soils near ports. Pile foundations are commonly used to support these types of extralarge tanks to carry the entire load and control the settlement (Hor et al. 2017). Indeed, when utilising the pile foundation, the LNG tank load is transferred to the piles through rigid reinforced slab at the base of the tank, which acts as a load distribution mat.
The LNG industry is growing quickly and many LNG tanks are constructed in seismically active coastal regions; hence, potential damage or leak due to cracking triggered by earthquake can result in the catastrophic environmental and safety threatening events (Zhang et al. 2011; Zhai et al. 2019). The LNG tanks should be designed wisely for safe and secure energy storage, and to meet the stringent requirements, the liquid inside the tank is usually contained within a steel container with high ductility material to ensure high performance subjected to low temperatures. Researchers have found that the inner container performance could be enhanced by using 9% nickel steel inner tank, surrounded by insulation materials (Chung et al. 2019). Moreover, for external protection and vapour containment, prestressed concrete outer tanks are usually used (Zhai et al. 2019). This type of tank, with inner steel tank and outer concrete tank with insulation material in between, is known as full containment tank, which is the most commonly used tank in practice to store LNG.
The seismic performance of LNG storage tanks has been a matter of special importance, beyond the economic value of the structure, due to the requirement to remain functional after a major earthquake event (Di Sarno 2020). Any potential danger associated with the failure of tanks containing highly flammable products can lead to extensive uncontrolled fires while any possible spillage of such content might cause extensive environmental damage and affect populated areas (Zhang et al. 2018). Therefore, the need for quite advanced experimental, analytical and numerical investigations to assess the seismic response of such structures, is quite evident.
Under earthquake loading, the dynamic behaviour of LNG tank is quite complicated due to SoilStructure Interaction (SSI) and Fluid–Structure Interaction (FSI) effects. Under dynamic loading, the LNG exerts additional hydrodynamic forces on to the tank walls. These hydrodynamic forces have two components; in the first component, portion of the liquid is accelerated with the tank walls and acts as added mass (or impulsive force), while in the second component, the upper portion of liquid sloshes generating surface waves (i.e. convective force), and the characteristics of these waves are affected by ground displacement rather than ground acceleration (Christovasilis and Whittaker 2008). Many researchers studied the FSI effects on liquid storage tanks under dynamic loading (e.g. Housner 1957; Veletsos and Tang 1987; Haroun and Housner 1981; Malhotra et al. 2000) and proposed a simple mechanical model to capture the FSI, where the two hydrodynamic forces can be represented by concentrated forces attached to the tank wall with appropriate links. Indeed, the simplified analytical models are adopted in several wellestablished design codes such as API 650 (2007) and NZSEE (2009). Later, Virella et al. (2006) enhanced the simplified mechanical models to capture FSI by replacing the hydrodynamic concentrated forces with distributed forces along the tank wall as adopted in Eurocode (2008).
Evidently, the SSI could greatly alert the dynamic behaviour of the superstructures and their supporting foundation system, and indeed the contribution of SSI is significantly increased when the superstructure is constructed on soft soil deposits in seismically active areas (Gazetas and Mylonakis 1998; Stewart et al. 2000; Mylonakis and Gazetas 2000; Trifunac 2000; Carbonari et al. 2008; Padrón et al. 2009; Gičev and Trifunac 2012; Hokmabadi and Fatahi 2016; Asimaki et al. 2019; Elwardany et al. 2019; Cavalieri et al. 2020; Brunelli et al. 2021). Medina et al. (2013) assessed the SSI effects on seismic characteristics of pile supported structures including period of vibration and damping implementing substructure analysis method. The results highlighted the importance of pile characteristics on the response of both high and short buildings. Zimmaro and Ausilio (2020) investigated the dynamic properties and seismic behaviour of earth fill dam foundations using modal and seismic hazard analyses, and showed that the substantial underestimation of the fundamental period of the dam could be observed when the SSI effects was ignored.
Similarly, since the SSI can significantly affect the seismic response of large capacity structures like LNG tanks, several researchers investigate SSI effects on the seismic response of LNG tanks (Willford et al. 2010; Ruiz and Gutiérrez 2015; Tajirian et al. 2019). For example, Sun and Cui (2015) studied the seismic response of the baseisolated LNG tank taking into account the SSI effect using a simplified mechanical model and elastic soil foundation. Their results illustrtaed the importance of SSI on selecting the best isolation system. Son and Kim (2019) highlighted the importance of SSI effects on the seismic response of LNG tank subjected to vertical earthquake component. Moreover, Hokmabadi et al. (2019) considered the impact of SSI on the seismic response of a large LNG tank sitting on the improved ground, and highlighted the importance of conducting rigorous numerical modelling to optimise the seismic design of LNG tanks.
One of the most dramatic causes of damage to structures during earthquakes is the occurrence of liquefaction in saturated sand deposits. Loose sand tends to contract under the cyclic loading imposed by earthquake shaking, which can cause increase in excess pore water pressure if the soil is saturated and unable to drain during earthquake. This results in reduction in soil effective stress and consequently substantial loss of soil strength and stiffness (Booth and Fenwick 1994; Bhattacharya and Madabhushi 2008). Several infrastructures and buildings suffered severe damages from soil liquefaction in the past earthquakes such as the 1964 Niigata, the 1971 San Fernando, the 1989 Loma Prieta and the 1995 Kobe earthquakes (Abdoun and Dobry 2002; Brandenberg et al. 2018; Jiménez et al. 2019; Zimmaro et al. 2020).
The liquefaction hazard should be evaluated in the seismic design of the projects. This includes undertaking liquefaction assessment to identify layers with liquefaction potential under the design earthquakes. The consequences of onset of potential liquefaction should be assessed and addressed in the design. From authors design experience, ground improvement techniques (e.g. deep cement mixing, jet grouting, stone columns) or pile foundations are common solutions to mitigate the liquefaction hazard in the design of large LNG tanks. While ground improvement techniques are considered to be more economical in many cases, their application depends on site ground conditions, extent of the liquefiable hazard, and project specific requirements. On the other hand, pile foundations offer a robust solution for the design of large LNG tanks and have been adopted in many projects.
The seismic response of large LNG tanks founded on potentially liquefiable soil needs a precise consideration of soilpilestructure interaction. This is not only required for the design of pile elements, but the impact of soil liquefaction on the overall response of the LNG tanks is crucial and should be well understood. Many studies had highlighted the potential failure of pile foundation in liquefiable soils due to buckling instability, bending, shear or settlement failures of piles during earthquakes (e.g. Tokimatsu et al. 1996; Dash et al. 2009; Haldar and Babu 2010; Zhuang et al. 2015). However, there is very limited research available in the literature on the response of LNG tanks founded on pile foundation in potentially liquefiable soil deposits. Thus, in this study, the effect of the depth of liquefied soil deposit on the seismic response of LNG tank supported by pile foundation will be examined using threedimensional finite element analysis. Results of this study can be used to assess the seismic efficiency of endbearing pile foundation to support extralarge LNG tanks on liquefiable soils without ground improvement.
2 Overview of adopted LNG tank and soil profile
In this study, a 160,000 m^{3} full containment LNG tank is used to perform the seismic analysis. The adopted LNG tank is an example of extralarge tank constructed in highly to moderately seismic regions. This tank consists of an open top 9% Ni steel inner tank and an outer reinforced concrete (RC) tank with fixed roof, sitting on endbearing pile raft foundation system. Figure 1 shows the overview of the structural and geometrical features of the adopted LNG tank. The total height of the outer RC tank is 51.20 m, the outer tank wall is 42.27 m high, the inner tank wall is 40.0 m high and the LNG design level in the inner tank is 36.0 m. The inner 9% Ni steel tank wall thickness is varying from 40.0 mm (bottom) to 10.0 mm (top), and the outer tank wall thickness is 800 mm. Figure 1b illustrates the layout of 289 reinforce concrete piles, while Fig. 1c shows the details of the piles supporting the raft foundation (i.e. tank slab).
Figure 2 shows the adopted soil profiles in this study to assess the impacts of depth of liquefiable soil on the seismic response of the LNG tanks. Figure 2a shows the adopted benchmark scenario in which the LNG tank sitting on soft clay soil deposit (nonliquefiable) with the average shear wave velocity V_{s} = 225 m/s for the top 15.0 m of soil deposit, and V_{s} = 270 m/s from 15.0 to 30.0 m depth. This benchmark subsoil profile (Scenario I) is classified as site Class C (Soft Soil Site) according to AS/NZS1170.4 (2004). It should be note that it is assumed that the soil deposit is underlain by Sandstone rock with shear wave velocity in excess of 760 m/s and the unconfined compressive strength greater than 50 MPa. To assess impacts of presence of liquefiable soil deposit on the seismic response of LNG tank system, three other scenarios with different depths of liquefiable soil are considered as in Fig. 2, namely Scenario II (5 m deep liquefiable soil), Scenario III (10 m deep liquefiable soil) and Scenario IV (15 m liquefiable soil). Further details about adopted properties of the liquefied soil deposits are provided in the next section.
3 Details of numerical modelling
In this study, the three dimensional finite element analysis was conducted using ABAQUS (2018) software, where the superstructure, pile foundation and the soil were simulated using fully nonlinear direct method (Kramer 1996), where the LNG tank, foundation and the soil were treated with the same rigour. Taking the advantage of the fact that the problem is symmetric about the line of symmetry parallel to the direction of the applied earthquake, only half of the model was simulated in this study as shown in Fig. 3.
3.1 Modelling of the inner and outer tanks and pile foundation
The inner 9% Ni steel and outer reinforced (RC) concrete tanks were modelled using S4R shell elements. Additionally, the steel reinforcements in the RC tank wall, roof and slab were modelled as uniaxial reinforcement layers embedded in the shell element, similar to the technique used by other researchers to simulate axial and circumferential steel rebars (e.g. Nateghi and Yakhchalian 2011; Hafez 2012; Zhai et al. 2019).
3.1.1 Modelling of outer reinforced concrete tank
The concrete damage plasticity (CDP) model was implemented in this study to model the outer concrete containment of LNG tank under the seismic loading. As explained by several researchers (Zhai et al. 2019; Hafez 2012; Dulinska and Jasinska 2014), this model is suitable to capture the concrete behaviour under dynamic loading particularly for thin wall structures such as concrete tanks and pipes. Indeed, this model adopts plasticitybased damage for the concrete, assuming two failure mechanisms, namely the tensile cracking and compressive crushing to represent the inelastic behaviour of concrete. The evolution of yield surfaces was controlled by two hardening variables, namely the tensile plastic strain (\({\overline{\varepsilon }}_{t}^{pl})\), and the compressive plastic strain (\({\overline{\varepsilon }}_{c}^{pl})\). where \({f}_{b0}\) and \({f}_{c0}\) are the biaxial and uniaxial compressive yield strengths, respectively. Moreover, \({\tilde{\sigma }}_{c}\) and \({\tilde{\sigma }}_{t}\) are the effective compressive and tensile stresses respectively (i.e. the stresses determined based on undamaged elastic stiffness). \({K}_{c}\) is the ratio of the second stress invariants on tensile and compressive meridians, which defines the shape of the yield surface. Since the CDP model adopts nonassociated flow rule proposed by Lee and Fenves (1998), the yield surface differs from the potential plastic flow, which utilises the DruckerPrager hyperbolic function to formulate the flow potential function Q as follows:
where \({\sigma }_{t0}\) is the tensile stress at failure (i.e. tensile strength), \(\epsilon\) is the eccentricity of plastic potential surface, and \(\psi\) is the dilation angle measured in \(\overline{q } \overline{p }\) space.
The degradation of concrete strength and stiffness in this adopted CDP model is captured via tension and compression damage parameters (\({d}_{t }\, \mathrm{and} \,{d}_{c})\), where these model parameters refer to the weakened concrete characteristics during unloading response as a result of cracking and crushing impacting the initial elastic stiffness (i.e. \({E}_{0})\). Indeed, the damage of the concrete under the tensile and the compressive stresses is characterized by damage plasticity theory developed by Lubliner et al. (1989) and Lee and Dale (1998), so the adopted stress–strain relation under uniaxial tension and compression loading can be presented as follows:
where the subscripts t and c refer to tension and compression, respectively, and \({E}_{0}\) refers to the initial stiffness of the concrete under compression and tension before any yielding occurs. The damage or degradation parameters \(({\text{i}}.{\text{e}}.{ }d_{t } \;{\text{and}} \;d_{c}\)) impacting the concrete stiffness can take a value from zero, where there is no loss in stiffness, to one, which represents the total damage state of the concrete.
Similar to many previous studies (e.g. Miglietta et al. 2016; Murray et al. 2018), the impacts that reinforcing bars would have on the concrete behaviour were modelled by introducing tension stiffening into the tension softening section of concrete damage plasticity model to simulate the load transfer across cracks through the rebar. As Wahalathantri et al. (2011) explained, by introducing tension stiffness, strain softening behaviour of the cracked concrete would be revised. The key parameter required to formulate the tension stiffening is the cracking strain \({\overline{\varepsilon }}_{t}^{cr}\), which is calculated by subtracting the elastic strain corresponding to the undamaged material from the total strain (\({\varepsilon }_{t}\)).
As discussed by Wahalathantri et al. (2011) and reported in ABAQUS (2018), the cracking strain (\({\overline{\varepsilon }}_{t}^{cr}\)) and plastic strain (\({\overline{\varepsilon }}_{t}^{pl}\)) are correlated as:
Nayal and Rasheed (2006) reviewed different tension stiffening models, then developed a model based on the homogenized stress–strain relationship established by Gilbert and Warner (1978), which accounts for tension stiffening, softening and local bond slip effects. This model captures the response caused by primary and secondary cracking phenomena on the stress–strain graph. As this model is applicable to both steel bar reinforced and fibre reinforced concrete with only minor changes, Wahalathantri et al. (2011) modified the model to be consistent with the general formulations on CDP model available in ABAQUS. Figure 4 represents the modified tension stiffening model implemented in this study. Indeed, the stress–strain relation is linear till reaching the tensile strength \({\sigma }_{t0}\) (MPa), where it can be determined according to ACI 31808 (2014) as below:
where \(f_{c}^{\prime }\) is the concrete compressive strength in (MPa). The axial strain corresponding to peak tensile strength \({\sigma }_{t0}\) is called critical tensile strain ε_{cr}. Then there is a rather abrupt drop in the state of stress to point (1.25 ε_{cr}, 0.77\({\sigma }_{t0}\)). The primary cracking stage ends at (4ε_{cr}, 0.45\({\sigma }_{t0}\)), while the secondary cracking stage stopes at (8.7ε_{cr}, 0.10 \({\sigma }_{t0}\)) as shown in Fig. 4a.
However, to capture the compressive stress–strain behaviour of the concrete following the initial elastic response, the compressive stress data are input in terms of the inelastic crushing strain \({\overline{\varepsilon }}_{c}^{in}\) which is defined as:
while the plastic strains \({\overline{\varepsilon }}_{c}^{pl}\) required for evolution of the yield surface were calculated based on the recommendation by Genikomsou and Polak (2015) as follows:
To establish the compressive stress–strain behaviour of concrete to be used for the calibration of the CDP model parameters, data and approach provided by Saenz (1964), shown in Fig. 4b, were utilised in this study. It should be noted that many other researchers (e.g. Asran et al. 2016; Tahnat et al. 2018; and Sakr et al. 2019) had validated and utilised the compressive stress–strain relationship developed by Saenz (1964) for various grades of concrete.
For the concrete C45 grade (i.e. \(f_{c}^{\prime }\) = 45 MPa) adopted in this study for the outer reinforced concrete containment, the modulus of elasticity of concrete (MPa) was determined based on ACI 31808 (2014) as E_{c} = 0.0043 * \(\rho_{c}^{1.5} \sqrt {f_{c}^{\prime } }\), where \({\rho }_{c}\) is concrete density (kg/m^{3}) taken as 2500 kg/m^{3} in this study. The basic parameters of grade C45 concrete corresponding to the calibrated CDP model are summarised in Table 1. In addition to these parameters, variations of the compressive yield stress and the stiffness degradation with inelastic strains (i.e. σ_{c}\({{\overline{\varepsilon }}_{c}^{in} \mathrm{and } \; d}_{c}{\overline{\varepsilon }}_{c}^{in})\), as well as variations of the tensile yield stresses and the stiffness degradation with cracking strains (i.e. σ_{t}\(\) \({\overline{\varepsilon }}_{t}^{cr}\) and \({d}_{t}{\overline{\varepsilon }}_{t}^{cr}\)), illustrated in Fig. 5, were adopted as the input parameters in this study. Indeed, Fig. 5 shows the comparison of the typical C45 concrete response obtained from Saenz (1964) and Wahalathantri et al. (2011) for compression and tension, respectively and calibrated CDP model predictions adopting model parameters reported in Table 1. A reasonable agreement is observed, which is confirming the suitability of adopted concrete model parameters in this study.
The steel reinforcement bars in the outer concrete tank were Grade 60 (\({f}_{y}=\) 420 MPa) referring to ASTM A615 (2018), and were embedded in concrete in both vertical and circumferential directions with reinforcement content ratio 1% in both directions. The rebars were modelled as an equivalent smeared layer with a constant thickness determined based on the rebar size and spacing, while adopting rebar layered shell option available in ABAQUS, particularly customised for reinforced concrete simulation as mention earlier. In this study, it is assumed that the rebar layers were completely bonded to the concrete and the linear kinematic model explained earlier was used to model the steel reinforcement bars, with adopted model parameters summarised in Table 2. It should be noted that the outer reinforced concrete tank has a fixed reinforced concrete roof, and in this study the roof material is assumed to be same as the tank wall with same reinforcement ratios.
3.1.2 Modelling of reinforced concrete piles
For the simulation of concrete piles, concrete grade C40 (\(f_{c}^{\prime } = 40{\text{ MPa}}\)) was adopted in this study. The hybrid modelling technique was used in this study to capture the elastic–plastic response of the piles by implementing the moment–curvature relationship for the reinforced concrete pile section, where a 2node linear beam element (B31) was circumscribed by solid eightnode brick elements (C3D8R) with a minimal flexural rigidity (i.e. 10^{6} times less than the beam elements of the pile) as used by several other researchers (e.g. Anastasopoulos et al. 2013; Rasouli and Fatahi 2021). Considering the pile geometrical characteristics and reinforcement content, the bending moment–curvature relationship of the piles was established as in Fig. 6 using SAP2000 software, and assigned to the beam element in the centre of concrete pile. It should be noted that other general beam section details including the Young’s modulus of 30.1 GPa, the Poisson’s ratio of 0.3, and the mass density of 2500 kg/m^{3} were assigned to the pile element.
3.1.3 Modelling of inner 9% Ni steel tank
The linear kinematic hardening model was adopted in this study to model the inelastic behaviour of the inner steel tank made of 9% Ni steel, recognized by the ASTM A353 (2014) for cryogenic service temperatures, while considering linear elastic behaviour when the stress state is within the yield surface. The linear kinematic hardening model was used in several previous research studies for seismic analysis of steel structures (e.g. Zakavi and Nourbakhsh 2014; Mizuno et al. 2014). The adopted constitutive model to simulate the metal inelastic behaviour under cyclic loading, adopts Von Mises yield surface as defined in Eq. (9). In this model, the equivalent Mises stress (i.e. \(f\left(\sigma \alpha \right)\)) is defined with respect to kinematic shift stress (also known as the backstress α) which describes the translation of the yield surface in the stress space. It should be noted that in Eq. (9), the yield stress \({\sigma }_{y}\) defines the size of yield surface which separates elastic and elastoplastic responses.
where \({\sigma }_{y}\) is the yield stress and \(f\left(\sigma \alpha \right)\) is the equivalent Mises stress with respect to the back stress, α, which determines the kinematic evolution of the yield surface in the stress space. The adopted linear kinematic hardening model in this study assumes associated plastic flow rule which is acceptable for metals subjected to cyclic loading (Chun et al. 2002; Hashiguchi and Ueno 2017; Koo et al. 2019). The evolution of the hardening law adopted in this study follows the linear Ziegler (1959) hardening law as formulated below:
where \({\dot{\overline{\varepsilon }}}^{pl}\) is the equivalent plastic strain rate, \({C}_{s}\) is the kinematic hardening modulus for steel which remains constant in this model, and \({\sigma }_{y}\) is the equivalent stress defining the size of the yield surface which remains constant and equal to yield stress at zero plastic strain (\({\sigma }_{0}\)). In this model the kinematic hardening component is defined to be a purely kinematic following the linear Ziegler (1959) hardening law, and the relaxation term (or the recall term \({\gamma }_{k}\alpha {\dot{\overline{\varepsilon }}}^{pl}\) mentioned in Eq. 23) is omitted since \({\gamma }_{k}\) is assumed to be equal to zero.
As explained by Driver et al. (1998), the linear kinematic hardening model adopted here takes the Bauschinger effect into consideration. In other words, the model allows the straining in one direction (e.g. tension of compression) decrease the yield stress in the opposite direction (i.e. compression and tension correspondingly). Figure 7 illustrates the adopted stress–strain behaviour considering the linear kinematic hardening model with hysteresis effects to simulate response of 9% Ni steel. Indeed, the stress–strain relation follows linear elastic behaviour with initial stiffness \({E}_{s}\) until the yield condition at point (\({\sigma }_{y},{\varepsilon }_{y}\)) is reached. Then the stress–strain relationship continues with a postyield modulus \({E}_{t}\). As shown in Figs. 7 and 8, it should be noted that the post yield modulus is different from the plastic modulus; the post yield modulus refers to the slope of the total strain and stress, while the plastic modulus (i.e. C_{s}) is the slop of the plastic strain versus stress. Furthermore, the inelastic material properties must be input into ABAQUS as Cauchy stress \({\sigma }_{true}\) and true logarithmic strain \({\varepsilon }_{true}\) values, which can be calculated from the nominal stress strain values (\({\sigma }_{eng},{\varepsilon }_{eng})\) using Lubliner (1990) equations:
The inelastic input parameters for this model into ABAQUS are only two data pairs, the yield stress at zero plastic strain \({\sigma }_{0},\) and the yield stress \({\sigma }_{y}\) at finite plastic strain \({\varepsilon }^{pl}\); which are used to determine the linear kinematic hardening modulus (i.e. plastic modulus) from the following relation:
The linear kinematic hardening model parameters for 9% Ni steel used in this study is summarized in Table 3.
3.2 Modelling of the fluid–structure interaction
The hydrodynamic forces exerted from the accelerated fluid to the tank walls are commonly considered in two main modes of vibration; one is due to the rigid movement of the fluid which is called the impulsive force, while the other mode is due to surface waves and sloshing of the liquid, called convective force. In practical analysis and design, these forces are represented using simplified mechanical springmass models (Houser 1957), where these two forces are modelled as concentrated equivalent masses attached to the tank wall via link or spring elements with appropriate stiffness. Further developments were done by Virella et al. (2006) by adopting distributed masses, also recognized by Eurocode8 (2006), which is known as a rigorous and a sound alternative model to Houser (1957) simplified mechanical model. In Virella et al. (2006) model, instead of using one concentrated mass to represent the impulsive force, the mass is distributed along the tank wall in several segments as added masses. Indeed, usually these distributed masses are attached to the wall nodes via rigid springs. In general, the Virella et al. (2006) replaced the hydrodynamic concentrated forces of Houser (1957) mechanical model with equivalent distributed pressure along the tank wall.
In this study, the FSI was considered by combining the Houser (1957) springmass model and the distributed pressure method of Virella et al. (2006). Indeed, the concentrated convective mass was used to model the convective force based on Houser (1957) and API650 (2007), while the impulsive mass model based on Virella et al. (2006) and Eurocode8 (2006) pressure distribution method was used to simulate the impulsive force. The adopted techniques deem suitable since the convective mode is generally uncoupled and independent from the impulsive mode, and can have less impact on the overall response of the liquid storage tank under horizontal loading, and many previous studies ignored the convective mode of liquid when adopting the addmass approach for the sake of simplicity (e.g. Virella et al. 2006; Buratti and Tavano 2014).
As mentioned above, the convective force was modelled using spring—mass model, including the first convective mode of vibration only for the single convective mass (i.e. M_{con}) attached to the inner tank at a height of H_{con} (the centre of action or effective height of the convective mass) via springs with a stiffness equal to K_{con,} where in this study the springs were used to connect the convective mass to the tank wall in the horizontal direction (i.e. earthquake acceleration direction). The required parameters to capture the convective hydrodynamic effect were determined using the following equations (API650 2007):
where \({M}_{con}\) is convective mass, \({M}_{total}\) is the total mass of liquid in the tank, \({T}_{con}\) is the convective period, \({C}_{con}\) is sloshing period coefficient, \({K}_{con}\) is the effective stiffness for the convective mode (N/m), \({H}_{con}\) is the convective mass height, and H and D are the liquid height and tank radius, respectively.
For the impulsive mass modelling, Eurocode8 (2006) was used, in which the spatial–temporal distribution of the rigid impulsive component of the hydrodynamic pressure (\({p}_{imp}\)) were determined by the following expressions to obtain the distributed impulsive masses \({m}_{imp}\):
where ξ = r/R_{tank} and ς = z/H_{L} are the nondimensional coordinates, R_{tank} is the radius of the tank, r, θ, z are components of cylindrical coordinates with origin at the centre of the tank while the z axis is vertical, t is time, \({C}_{imp}\) is impulsive pressure distribution coefficient, ν_{n} = ((2n + 1)/1) π, γl = H_{L}/R, \({I}_{1}\), \({\stackrel{`}{I}}_{1}\) are the modified Bessel functions of order 1 and its derivatives, respectively, \({A}_{g}\left(t\right)\) is the horizontal ground acceleration time history in the freefield condition, and ρ is the mass density of the fluid. Indeed, Eq. (20) was used to calculate the equivalent mass at each segment of tank wall, which was 1.0 m segment along the tank wall in this study. As recommended by Virella et al. (2006), a convenient alternative to avoid attaching lumped masses via rigid link elements, is including these extra masses by increasing the wall density in each segment, which was used in this study to reduce the computational time.
3.3 Modelling of the soil deposit
3.3.1 Modelling of clay soil deposit
The nonlinear kinematic hardening model, was implemented in this study to model the cyclic behaviour of clay (nonliquefiable soil) under seismic loading. This pressureindependent plasticity model based on Von Mises yield criteria, and associated flow rule, can be considered as appropriate model for nonlinear cyclic behaviour of pressure independent material, such as clay under undrained condition (Zhang and Tang 2007). As explained by George et al. (2016) and Zhu (2019), the key parameters of this model are Young’s modulus, Poisson’s ratio, and stress–strain curve of stabilised loading cycle obtained from the triaxial shear test. Based on the modification work done by Armstrong and Frederick (1966) on the original model which was proposed by Lemaitre and Chaboche (1994), the yield surface of the adopted nonlinear kinematic hardening constitutive model is defined according to Eq. (9), while the equivalent Mises stress can be defined by the following relation:
where S is the deviatoric stress tensor, and α^{dev} is the deviatoric part of back stress tensor. In the adopted model, an associated flow rule is adopted to obtain the required kinematic hardening flow rule as reported by Zhu (2019). The hardening rule of this model is a mixed nonlinear isotropickinematic hardening making it suitable for simulating cyclic behaviour of soils as explained by Elia and Rouainia (2016) and Mucciacciaro and Sica (2018). Indeed, the nonlinear kinematic hardening component describes the translation of the yield surface in stress space through the back stress (α), as it is defined by the superposition of pure kinematic and relaxation (source of the nonlinear behaviour) terms, while the isotropic hardening component describes the change of the equivalent stress defining the size of the yield surface (\({\sigma }_{y}\)) as a function of plastic deformation. The isotropic hardening behaviour adopted in this study follows a simple exponential law as reported by ABAQUS (2018) and Zhu (2019) and presented in Eq. (22), and data fitting can be utilised to obtain the model parameters by directly introducing data points for the yield surface size and plastic strains.
where \({\sigma }_{0}\) is the yield stress at zero plastic strain and \(Q\infty\) and b are material parameters, corresponding to the maximum change in the size of the yield surface, and the parameter defining the rate at which the size of the yield surface changes as plastic straining develops, respectively. It should be noted that when the equivalent stress defining the size of the yield surface remains unchanged (\(i.e.{\sigma }_{y}={\sigma }_{0}\)), the general combined nonlinear isotropickinematic hardening model reduces to a nonlinear kinematic hardening model, which is adopted in this study.
The evolution of the kinematic component for the yield surface on the stress space based on Ziegler (1959) kinematic hardening law is presented below:
where (\({\alpha }_{k}\)) is the backstress tensor, \({C}_{k}\) and \({\gamma }_{k}\) are material parameters for each back stress\({\alpha }_{k}\), which are obtained through calibration using cyclic test data. Indeed, \({C}_{k}\) is the initial kinematic hardening moduli, and \({\gamma }_{k}\) determines the rate at which the kinematic hardening moduli decreases with the plastic strains for each back stress \({\alpha }_{k}\). It should be noted that the recall term \({\gamma }_{k}\alpha {\dot{\overline{\varepsilon }}}^{pl}\) introduces the nonlinearity to the evolution law proposed by Chaboche and Lemaitre (1990).
Moreover, each back stress is covering a different range of equivalent plastic strains, so the overall back stress is computed by summation of back stresses:
The kinematic and isotropic hardening components are illustrated in Fig. 9a for uniaxial loading, while Fig. 9b illustrates the same for more generalised stresses for multiaxial loading. It is evident that the kinematic hardening component implies that the back stress is contained within a cylinder of radius \(\sqrt{\frac{2}{3}} {\alpha }^{s}\), where \({\alpha }^{s}\) is the magnitude of α at large plastic strains (known as saturation condition), while the stress points lie within a cylinder with radius \(\sqrt{\frac{2}{3}} {\sigma }_{max}\) (represented in Fig. 9) since the yield surface remains bounded.
As explained above, the adopted nonlinear kinematic hardening model requires input parameters \({C}_{k}\) and \({\gamma }_{k}\), or directly input triaxial shear test data corresponding to the stress–strain data points for stabilised cyclic curve. In this study, the soil properties, corresponding to the site class C and the density and shear wave velocity taken from insitu tests are ρ_{clay} = 1650 kg/m^{3} and 1950 kg/m^{3}, with V_{s} = 225 m/s and 270 m/s, for the top 15 m and the second 15.0 m, respectively. The maximum shear modulus (\({G}_{max})\) of the adopted soil was obtained from the following relationship:
Sun et al. (1988) recommended backbone curves for practical use in seismic site response evaluations for cohesive soils as reported in Fig. 10. The backbone curves adopted in this study capture the variations of modulus reduction ratio (G/G_{max}) and damping ratio (ξ) with cyclic shear strain (γ) for cohesive soils obtained from cyclic simple shear and resonant column test results. To obtained an analytical formulation fitting the backbone laboratory measurements, Hardin and Drnevich (1972) model, as shown below, was adopted in this study.
where \(\tau\) is the shear stress, ξ is the damping ratio, G is secant shear modulus, γ is the cyclic shear strain of the soil, and γ_{ref} is Hardin and Drnevich (1972) constant to get a best fit for backbone curves for modulus reduction and damping ratio. The calibrated value of γ_{ref} = 0.234 resulted in the best match for backbone curves with coefficient of determination R^{2} equal to 0.88 and 0.80 for the modulus reduction curve and damping ratio, respectively, as evident in Fig. 10.
Since the kinematic hardening model adopted in this study for cyclic behaviours of soils required the stabilised stress–strain cycle, Masing (1926) rule presented in Eq. (28) was used to obtain the cyclic shear stress–strain behaviour. In this study, it is assumed that the stabilised cyclic loading–unloading curve is reached at cyclic shear strain value of 2% beyond which the shear modulus remains unchanged.
where \({\tau }_{rev}\) and \({\gamma }_{rev}\) are the shear stress and cyclic shear strain at the reversal point.
The stabilised loading cyclic was used to obtain the kinematic hardening parameters by inputting the shear stress and plastic shear strain data points from the reyielding point to the reversal point. Indeed, using an ABAQUS subroutine, a curve fit analysis was conducted to determine the combined hardening constants based on the shear–stress strain data points. Referring to Eq. (24), by adopting several back stresses and calibrating the parameters, a larger strain range can be covered resulting in more accurate predictions. Thus, in this study two back stresses were adopted which resulted in more reliable predictions. Table 4 summarises soil properties and the calibrated kinematic hardening model parameters adopted in this study.
Figure 11 illustrates the comparison of the calibrated kinematic hardening model adopted in this study versus the laboratory measurements obtained from Sun et al. (1988) experiments while adopting Masing (1926) rule to obtain closed loading–unloading cyclic loop. As evident, the predictions adopting kinematic hardening model are in good agreement with the experimental data, confirming the suitability of the adopted nonlinear kinematic hardening model and calibrated parameters to simulate the cyclic response of adopted soft clay deposits.
3.3.2 Modelling of liquefied soil deposit
During an earthquake, the rapid shaking of the ground can generate large pore water pressures in saturated loose granular soils under undrained condition which results in a reduction in effective stresses and subsequently a substantial loss in strength and stiffness. The variables that influence the onset of liquefaction mainly include the presence of groundwater, the particle size distribution of the soil, the insitu relative density of the soil, the effective confining stress and the amplitude and duration of ground motions.
The current stateofthepractice approach commonly adopted in projects has been used to obtain the postliquefaction properties of the liquifiable layers for modelling. The adopted approach is in accordance with Idriss and Boulanger (2008), where the residual shear resistance of liquefied sand, Sr, were estimated based on back analysis of several field case histories. Idriss and Boulanger (2008) correlated the residual shear strength with the ground insitu penetration test measurements (SPT or CPT). In this study, the correlation (Eq. 29) based on equivalent clean sand corrected SPT value, (N1)60cs − Sr, is adopted for condition in which the effects of void redistribution is considered negligible:
where S_{r}/σ'_{vc} is residual shear strength ratio, σ′′_{vc} is effective vertical stress preliquefaction, and φ’ is effective friction angle of the liquefiable layer (before onset of liquefaction). The idea of correlating residual strength with (N1)60cs − Sr was initially proposed by Seed (1987) and considered logical on the basis of criticalstate concepts and established correlations between the overburden correlated penetration resistance and insitu relative density (Idriss and Boulanger 2008).
Table 5 summarizes the postliquefaction soil properties used in this study. The damping behaviour of the liquefied soil was captured using Rayleigh damping formulation via Rayleigh damping coefficients \({\alpha }_{damping}\) and \({\beta }_{damping}\), calculated based on the first and second mode natural frequency of the liquefied soil deposit and 20% damping ratio for the liquefied soil referring to Poulos (2017), Boulanger et al. (2014) and Lombardi and Bhattacharya (2014). Rayleigh damping coefficients adopted in this study for different liquefied soil thicknesses are presented in Table 5.
3.4 Modelling of interfaces, boundary conditions and the adopted earthquakes
To include the SSI effect and capturing the possible separation or sliding between the subsoil and the piles, and between the subsoil and the foundation raft (i.e. outer tank slab), interfaces were defined between the mentioned surfaces. For the normal response of interacting surfaces, the hard contact algorithm was used by defining the relationship between the contact pressure (p) and the overclosure (h) between the master and slave surfaces. Indeed, the pressure was transferred between interacting surfaces when the overclosure between them was zero (i.e. h = 0). In this adopted hard contact model, penalty algorithm, based on stiff approximation of contacting surfaces, was implemented to avoid overconstraining issue in the modelling, and thus a small degree of penetration for the interacting surfaces, was allowed. Moreover, for the tangential interaction between the interacting surfaces, the Coulomb frictional model via penalty formulation was implemented, where the relative movement between the contacting surfaces was controlled by defining a critical shear stress between them \({({\varvec{\tau}}}_{{\varvec{c}}{\varvec{r}}{\varvec{i}}{\varvec{t}}{\varvec{i}}{\varvec{c}}{\varvec{a}}{\varvec{l}}}\) = μ. P), which is proportional to the contact pressure through the friction coefficient (μ), where the relative movement can occur when the shear stress between the contacting surfaces exceeds the critical shear stress (i.e. the frictional resistance between the surfaces). The friction coefficients (μ) between the clay subsoil and the raft surfaces, and between the subsoil and pile surfaces were considered to be 0.4 and 0.35, respectively (Rasouli and Fatahi 2021; Agalianos et al. 2020). It should be noted that frictionless interfaces were used between the liquefied soil and foundation elements. Moreover, the rigid connection between the pile head and the raft foundation was imposed via the tie constraint condition available in ABAQUS.
The numerical analyses for the tanks—foundation—soil system was performed in two steps; the initial step, which was static analysis, considered the gravity loads for the entire system, followed by the second step which was application of earthquake acceleration at the model base in Xdirection only adopting dynamic implicit stepping technique. During the static analysis, the bottom of the soil deposit was fixed in all directions, and the vertical movements were allowed on the side boundaries, while the displacements normal to the side boundaries were disallowed. During the dynamic analysis, the infinite boundaries were introduced on the side boundaries replacing the initial roller boundaries. The adopted one way 8 nodes infinite elements CIN3D8 (see Fig. 3) are appropriate solid elements representing the farfield soil, capturing the initial static equilibrium conditions (Zienkiewicz et al. 1983) and absorbing the energy from the unbounded soil area under dynamic loading (Lysmer and Kuhlemeyer 1969). During the dynamic analysis the response of these elements are isotropic linear elastic and represent the infinite model where the normal waves are not allowed to be reflected and the damping of the normal velocity of these boundaries are governing by the following equations:
where c_{p} and c_{sv} are the compressive and shear wave velocities of the soil, \(\rho\) is the mass density of the material, and \(\lambda\) and \(G\) are Lamé's constants. It should be noted that the adopted infinite elements eliminate energy transmission for plane waves crossing the boundary, while the wave propagation upward will not be distorted.
For the input seismic excitations, one nearfield earthquake 1999 ChiChi earthquake and one farfield earthquake, 1968 Hachinohe earthquake, were used to study the dynamic response of the LNG tank. In general, the seismic design of LNG tanks follows the same scenario of nuclear power plants by applying the two earthquake levels, namely the Operating Basis Earthquake (OBE) and Safe Shutdown Earthquake (SSE). Under OBE, the LNG storage system needs to remain fully operational with no damage, while subjected to SSE, no leakage of LNG should occur. In this study, the response of LNG tank system subject to SSE was assessed.
To minimise the scatter in the response of earthquake engineering demand parameters, the seismic input motions are suggested to be scaled using spectral matching method (Guzel 2019). The spectral matching which adjusts the frequency content of an accelerogram till its response spectrum almost matches a target response spectrum with the minimal alteration of the velocity and displacement histories of the record was conducted using SeismoMatch software (SEISMOSOFT 2016).
To simulate earthquake wave transmitted from bedrock (i.e. high shear wave velocity) via assessed ground condition to the LNG tank, the response spectra of input motions were scaled to the target response spectrum representing Site Class A (i.e. strong rock site) from AS/NZS1170.5 (2004) as shown in Fig. 12. The shape of target spectrum is defined by the response spectrum acceleration factor C(T) using Eq. (32).
where \({C}_{h}\left(T\right)\) is spectral shape factor which depends on the site subsoil class and structure period (T), Z is the hazard factor equal to 0.4 for Wellington city in New Zealand based on the available site hazard map. R is the return period factor taken as 1.8 (i.e. annual probability of exceedance (APE) of 1/2500) for the SSE earthquake level for tank containing hazardous liquid based on importance level of 4 and design working life of 100 years taken from AS/NZS1170.5 (2004). \({N}_{max}\) and D_{k} are factors related to near fault properties, where \({N}_{max}\) is the maximum near fault factor and taken to be 1 in this study; and D_{k} is the shortest distance between the site and the nearest fault which was considered to be 2 in this study.
A specified period range, the minimum and maximum periods of 0.4 s and 1.5 s, respectively which covers the period range of soilstructure system under seismic load, was defined to perform spectral matching. Figure 13 shows the original (PEER 2014) and scaled accelerogram. It should be noted that baseline correction was conducted after the selected accelerogram being spectral matched. The scaled accelerogram was applied at the base of the soilstructure model to perform dynamic analysis.
4 Results and discussions
In this section, the results of free vibration and time history analyses are presented to assess the dynamic response of LNG tank system while capturing effect of soil liquefaction on the seismic response of the pile foundation and the superstructure. Initially, the free vibration analysis results are presented, and then numerical predictions from the nonlinear time history analysis under the 1999 ChiChi and the 1968 Hachinohe earthquakes are presented and discussed.
4.1 Free vibration analysis
In this study, the modal analysis was performed for the LNG tank using Block Lanczos algorithm. Firstly, the modal analysis on the fixed base tank (no foundation movement) was conducted as a reference to highlight importance of SSI when compared with other cases capturing the soil and foundation movements. Since the impulsive and convective hydrodynamic forces were modelled using added mass and springmass methods, respectively, the first impulsive and convective modes of the LNG were captured in the numerical model and the corresponding predicted natural periods were compared with the analytical formulations available in API650 (2007) as below:
where \({C}_{con}\) is sloshing period coefficient (explained earlier in Eq. (23)), \({C}_{l}\) is the coefficient for determining the impulsive period of tank system, which depends on the H/D ratio, and could be obtained from API650 (2007), t_{u} is the equivalent uniform thickness of the tank wall in mm, H_{L} \({\mathrm{H}}_{\mathrm{L}}\) \({\mathrm{H}}_{\mathrm{L}}\) is the fluid design level in meter, D is the diameter of the tank in meter, Es is the modulus of elasticity of the inner tank in MPa, and ρ is the fluid density in kg/m^{3}, which was assumed to be 480 kg/m^{3} for LNG.
Table 6 summarise the FEA predictions and the analytical calculations for the natural period of inner LNG tank. Moreover, Fig. 14a and b show the vibration mode shapes for both impulsive and convective first modes. It can be noticed that the first impulsive mode shape is the beam type mode, where the liquid and the tank wall vibrated similar to a cantilever beam as a result of rigid movement of the impulsive LNG mass attached to the tank wall. In addition, the convective mass vibrated in the horizontal direction since it was tied to the tank wall via springs controlling the stiffness of the vibrating mass. Figure 14c represents the first mode shape for outer concrete tank, and it is evident that the first mode shape corresponded to tank roof. Moreover, Table 6 verifies the suitability of the proposed mechanical model to simulate fluid–structure interaction (FSI) in the threedimensional finite element model used in this study, since a good agreement was observed between the natural periods obtained from the 3D numerical and the analytical formulations in API650 (2007).
The effects of the soil liquefaction on the dynamic properties of LNG tank supported by endbearing pile foundation namely frequency and damping are also discussed. Table 7 summarise the results of the modal analysis conducted to obtain the natural frequency of the LNG tank for different depths of liquefied soil deposit varying from zero (no liquefaction) to 15 m. In general, the vibration characteristics of pile foundation impact the response of the entire LNG tank system. Indeed, Table 7 show that the natural period of the LNG tank system increased significantly when the soil deposit liquefied; for example, the fundamental period of the LNG tank excluding soil liquefaction was 0.46 s, while when the top 5 m of soil liquefied, the fundamental natural period increased by 69% to 0.78 s. This is due to the fact that loss of soil stiffness around the piles supporting the LNG tank as a result of liquefaction increased the overall structural flexibility and thus altered the dynamic characteristics of the LNG tank system significantly. Further increasing the depth of liquefied soil to 10 m and 15 m, increased the natural period of the LNG tank system by 104% (to 0.94 s) and 154% (to 1.17 s) comparing to the LNG tank on nonliquefiable soil, respectively. Figure 15 shows the corresponding fundamental vibration mode shapes for LNG tank. When the soil deposit liquefied, the mode shape was governed by the vibration of the section of the pile embedded in the liquefied soil layer which deform more compared with the pile in the nonliquefied soil deposit. In general, when the soil liquefied and the liquefaction extended deeper, the dynamic properties of the LNG tank alerted significantly, which could trigger different responses for different parts of the LNG tank system. However, for the convective mass mode, it can be notice that increasing the depth of the liquefied soil deposit slightly decrease the natural period since the convective mass system was already more flexible due to the presence of springs attaching the mass to the wall, in contrast to the rigid connection between the impulsive mass and the tank wall.
4.2 Nonlinear time history results
The seismic analyses for the LNG tank on the different sub soil scenarios as shown in Fig. 2 were conducted in time domain for two spectral matched earthquake records, namely, the 1999 ChiChi and the 1968 Hachinohe earthquakes. The seismic results were extracted and compared in terms of the response spectrum, tanks maximum acceleration profile, inner tank wall maximum structural response, and the pile seismic response (i.e. lateral displacements, shear forces and bending moments developed along the pile).
4.2.1 Response spectrum
Figure 16 displays the acceleration response spectra at the ground surface (i.e. the raft level of LNG tank) for four soil deposit scenarios with different depth of liquefied soil. The induced motion in the soil gets modified as it propagates through the soil deposit from the bedrock level to ground surface, which is known as the site effect. The extent of the site effect depends on the geometrical and mechanical characteristics of the soil deposit and the applied earthquake. Indeed, Fig. 16a shows the amplification of the 1999 ChiChi earthquake input motion from the bedrock to the ground surface for the no liquefaction scenario, where the peak ground acceleration (PGA) for the input motion at the bedrock level was 1.85 g and increased to 3.5 g on the ground surface (i.e. almost twice amplification). Similar observations could be made in Fig. 16b for the 1968 Hachinohe earthquake where the PGA amplified from 1.95 g (bedrock level) to 4.1 g (ground surface level). However, the acceleration response spectra decreased when the soil liquefaction occurred, as a result of energy dissipation within the liquefied soil layer. As also evident in Fig. 16, increasing the thickness of the liquefied soil layer from 5.0 to 15.0 m incrementally resulted in continuous increase in the dissipation of the seismic wave as evident in the corresponding reduction in the PGA measured on the ground surface. However, the presence of liquefied soil deposit amplified the spectral acceleration in the long period rang and shifted the peak response spectrum towards the long period rang. These observations are comparable with the results reported by Youd and Carter (2005) and Gingery et al. (2015). Indeed, Youd and Carter (2005) studied five real liquefied sites and found that the liquefaction induced softening reduced the spectral acceleration in the short period (i.e. period range less than 1.0 s), while in long period range (i.e. period range more than 1.0 s), the amplification of the spectral acceleration was observed due to the ground oscillations in this range. Referring to Fig. 16, it can be observed when the thickness of the liquefied soil increased, the extent of soil softening was more, alerting the vibration characteristics of the soil deposit, filtering the high frequency components of the input motions, and delaying the transition of the seismic motion to the ground surface.
Figure 17 shows the effect of the liquified soil layer thickness on the frequency content of the seismic input motion. By comparing the frequency corresponding to the maximum Fourier amplitude (f_{d}), it can be seen that when the thickness of the liquified soil layer increased, f_{d} decreased. For the 1999 ChiChi earthquake, for the liquified soil thickness of 0, 5 m, 10 m, and 15 m, f_{d} values were 0.224 Hz, 0.204 Hz, 0.189 Hz and 0.181 Hz, respectively, while the corresponding values from the 1968 Hachinohe earthquake were 0.169 Hz, 0.159, 0.131 Hz and 0.125 Hz.
4.2.2 Maximum tank acceleration profile
The profiles of the seismic acceleration developed along the inner and outer tank walls are reported in Figs. 19 and 20, respectively, where the acceleration profile was extracted along the tank wall when the maximum acceleration occurred at the top of the tank walls in X direction (i.e. at polar coordinate where \(\theta\) = 0\(^\circ\) and parallel to the direction of applied earthquake). Figure 18 shows the horizontal acceleration time history record of the outer tank roof for different soil conditions. It is evident that the maximum horizontal acceleration occurred at different times for each soil condition scenario under same earthquake; for example, for nonliquefied soil deposit, the maximum horizontal acceleration was observed at t = 6.0 s and t = 7.4 s under the 1999 ChiChi and the 1968 Hachinohe earthquakes, respectively. However, the maximum horizontal accelerations were observed at t = 6.2 s, 7.5 s and 7.8 s when the liquefied soil deposit increased to 5.0 m, 10 and 15.0 m under the 1999 ChiChi earthquake (Fig. 18a). The corresponding values under the 1968 Hachinohe earthquake were t = 7.8 s, 8.3 s, and 10.8 s, respectively (Fig. 18b). This can be explained by referring to the Fig. 16 where the liquefied soil layer changed the amplitude and frequency content of the seismic load. Referring to Fig. 19a and as expected, the maximum acceleration profile was observed in nonliquefied soil case (i.e. Scenario I), where the maximum recorded acceleration at the top of the outer tank wall was 2.3 g, and decreases gradually to 1.62 g, 0.98 g and 0.92 g in the presence of 5.0 m (Scenario II), 10.0 m (Scenario III) and 15.0 m (Scenario IV) thick liquefied soil deposit, respectively. Similarly, Fig. 19b shows the maximum acceleration of the outer tank wall under the effect of the 1968 Hachinohe earthquake, where the maximum acceleration values of 1.83 g, 1.78 g, 0.82 g, and 0.52 g were observed for Scenarios I to IV, respectively.
Figure 20 shows the maximum acceleration for the inner steel tank wall under the 1999 ChiChi (Fig. 20a) and the 1968 Hachinohe (Fig. 20b) earthquakes. As evident, the acceleration along the inner tank wall experienced significant fluctuations in Z direction (i.e. tank wall elevation highlighting the impacts of seismic fluid – structure interaction (FSI) as a result of impulsive and convective hydrodynamic forces applied on the tank wall. For both earthquakes, presence of thicker liquefied soil deposit reduced the seismic acceleration induced at the tank base, as well as along the tank wall. Besides, it can be seen that at the location where the convective mass was connected to the wall (i.e. H = 20.0 m), the peak acceleration for the LNG tank under Scenarios III and IV behaved quite differently to Scenario I and II, as the horizontal acceleration increased at convective mass level when the soil liquefied, while an opposite trend was observed for the nonliquefied soil deposit. Indeed, as the liquefied soil thickness increased, the frequency of the seismic load experience by the inner tank decreased, contributing to the increase in the convective mass acceleration, similar to observations made by Kianoush and Ghaemmaghami (2011).
4.2.3 The structural response of the LNG tanks
The maximum structural response of the inner steel tank under the applied earthquakes in terms of generated forces and displacements along the tank wall are reported and discussed in this section. Assessing the impacts of the soil liquefaction surrounding the piles on the resilience of the steel tank against different failure modes is very crucial. Indeed, the hydrodynamic forces applied to the inner tank as a result of earthquake result in amplified hoop and axial forces in the steel tank wall and potentially cause the inner tank shell buckling. In fact, the shell buckling is the main failure mode for ground supported steel tanks (Brunesi et al. 2015; Dogangun et al. 2009). The failure of steel tanks subjected to the hydrodynamic forces, can be due to elastic or elastoplastic buckling. The elastoplastic buckling occurs when the axial compression and the circumferential hoop stresses (due to the selfweight and hydrostatic and hydrodynamic pressures) exceed the yield limit, often near the based on the tank, and result in an outward bulge appearing which is known as elephant’s foot buckling mode. The diamondshaped buckling mode, which is an elastic buckling often happens at small values of hoop stresses, where inward hydrodynamic suction at the base level of the tank wall or at upper level of the tank wall (corresponding to secondary diamond shape buckling mode) exceed the outward hydrostatic pressure. Indeed, the distribution of the hydrodynamic forces along the tank wall plays a critical role in formation of diamondshaped buckling, even if stresses remain in the elastic range. Thus, the axial force (F_{11}) and the hoop force (F_{22}) for circumferential unit width of the tank and radial displacements of the inner tank wall are presented and discussed for various foundation conditions.
Figure 21 shows the time history of the horizontal displacement of the inner tank top, indicating that the maximum lateral displacements of the inner tank were observed at different times for different soil deposit scenarios. Figure 21a indicates that the maximum horizontal displacement of the inner tank under 1999 ChiChi earthquake occurred at t = 9.4 s, 9.3 s, 10.0 s and 10.4 s, for Scenarios IV, III, II, and I, respectively. The corresponding value under the 1968 Hachinohe earthquake were t = 11.0 s, 10.1 s, 10.0 s and 13.0 s, respectively (Fig. 21b). It can be noticed that increasing the thickness of the liquefiable soil deposit increased the lateral displacement amplitude and the maximum lateral deformation of the superstructure.
Figure 22 shows the distribution of the hoop forces (F_{22}) within the inner tank shell under 1999 ChiChi and 1968 Hachinohe earthquakes when the maximum deformation of the tank was recorded. The distribution of the hoop forces along the inner steel tank wall is in a line with the horizontal accelerations recorded for the inner steel tank wall (see Fig. 20). The maximum hoop forces in the upper portion of the tank wall for the soil deposit Scenarios I and II, exceeded those of Scenarios III and IV. Indeed, since the impulsive mode periods for Scenarios I and II (i.e. 0.51 s and 0.78 s as in Table 7), were located in the shorter period range of the acceleration response spectrum (Fig. 16), the amplification of seismic forces was observed. However, the lengthened impulsive mode periods for Scenarios III and IV (i.e. 0.94 s and 1.17 s as in Table 7) were located in longer period range with decreased spectral accelerate range, which resulted in reduced the seismic forces due to impulsive mass. It is noticed that the maximum hoop forces for the tank built on nonliquefiable soil deposit subjected to the 1999 ChiChi earthquake (refer to Fig. 22a) and the 1968 Hachinohe earthquake (Fig. 22b) were 1.3 MN/m and 2.87 MN/m, respectively.
On the other hand, Fig. 23 shows the maximum axial forces (F_{11}) along the inner tank wall at θ = 0˚ for different soil deposits considered (i.e. Scenarios I to IV) when the maximum horizontal displacement of the tank was observed. It is evident that the axial forces in the inner tank generally increased from the top to the bottom along the inner steel tank wall with the maximum observed at Z/H = 0.10–0.375 (or Z/H_{L} = 0.12–0.44). In addition, referring to Fig. 23, the maximum axial forces in the inner tank wall decreased with the increasing depth of the liquefied soil. For example, under 1999 ChiChi earthquake, when the depth of liquefied soil was increased from zero to 5.0 m, the axial force reduced by 21%, and increasing the depth of liquefied soil layer to 10.0 m and 15.0 m reduced the axial force by 42% and 50%, respectively (Fig. 23a).
Figure 24 captures the distribution of the shear forces along the inner steel tank wall when the maximum lateral displacement was observed (as indicated in Fig. 21). It is evident that for both earthquakes, the maximum shear forces were observed at the base of the inner tank wall where the maximum axial forces were reported. Moreover, a second local peak of the shear force was recorded near the middle of the tank wall (i.e. Z/H = 0.45 or Z/H_{L} = 0.52 in the vicinity of the location where the convective mass was attached to the wall, at which significant variation of the hoop forces were also reported. As evident in Fig. 24, among the soil deposit scenarios considered, the nonliquefiable soil deposit (Scenario I) resulted in the maximum mobilised shear forces in the inner tank under both earthquakes.
Figure 25 shows the radial displacements along the inner tank wall when the maximum lateral displacement was reported (as indicated in Fig. 21). It is evident that the most significant radial displacement gradient (or section rotation) was observed in vicinity of the midheight of the inner tank, where the second peak of the shear forces were observed. Indeed, the convective hydrodynamic force applied at the mid height of the inner tank wall could alert the structural response of the inner tank. As discussed earlier, increasing the liquefied soil depth resulted in changing the seismic waves, mainly by reducing the frequency, resulting in the reduction in the impulsive forces, and in contrary amplification of the convective forces. This observation highlights the importance of including the convective mass in the seismic analysis of the LNG tanks, especially, where the presence of the liquefiable soil deposit can increase the flexibility and natural period of the soilstructure system significantly. This can result in amplified radial displacement of the LNG tank system which can in turn introduce more convective hydrodynamic forces to the system.
Figure 26 presents the maximum Von Mises stresses generated in the inner tank wall under the 1999 ChiChi (Fig. 26a) and the 1968 Hachinohe (Fig. 26b) earthquakes. The results show that the maximum induced Von Mises stresses corresponded to the LNG tank built on the nonliquefiable soil deposit (i.e. Scenario I) for both earthquakes, where the maximum induced plastic strains reached 0.29% and 0.89% for the 1999 ChiChi earthquake (Fig. 27a) and the 1968 Hachinohe (Fig. 27b) earthquakes, respectively. The above observations show that for the LNG tank on the nonliquefiable soil deposit, elastic–plastic buckling may happen in the upper section of the tank where plastic deformations are observed as a result of the von Mises stresses exceeding the yield stress. However, when soil liquefaction occurs, due to period lengthening and significant soil damping, stresses in the inner tank may reduce below the yield limit, while more concentrated stresses may be observed in the lower section of the tank near the base, where potential elephant foot buckling failure may occur.
Figures 28 and 29 present the maximum von Mises stresses generated in the outer tank wall under the 1999 ChiChi and the 1968 Hachinohe earthquakes, respectively. It should be mentioned here that since no failure was observed in the inner tank under any analysed scenarios, no leakage of LNG from the inner tank was considered, and thus LNG was not in the direct contact with the outer tank wall in this study. It is evident that the nonliquefied soil deposit scenario (Scenario I) led to the maximum generated von Mises stresses in the outer tank wall. According to Fig. 28, the maximum von Mises stress decreased from 25.9 MPa to 19.9 MPa when the top 5.0 m of soil deposit had liquefied under 1999 Chi Chi earthquake, while when the liquefied soil layer extended to 10.0 m and 15.0 m, the maximum von Mises stresses reduced to 18.4 MPa and 10.9 MPa respectively. It can be seen that the maximum stresses in the outer tank were generated at the connection between the tank wall and the roof for both earthquakes. Moreover, referring to Fig. 29, the maximum von Mises stresses decreased by 5%, 4% and 19% when the liquefied soil depth increased from zero to 5.0 m, 10.0 m and 15.0 m respectively. These results are in a line with the maximum acceleration reported for the outer tank as in Figs. 18 and 19.
4.2.4 Seismic response of pile foundation
To investigate the effects of liquefiable soil deposit on the response of piles supporting the LNG tank, the pile in the outermost ring of piles in Y direction as shown in Fig. 30 was selected. This pile was selected because it is under the outer tank wall and resists the highest shear forces, lateral displacement and overturning effects (Tajirian et al. 2019). Referring to Figs. 30, 31 and 32, the lateral displacement, shear force, and bending moment profiles along the pile length were reported when the maximum response (i.e. displacement) at the pile head was observed.
Referring to Fig. 30, as the liquefied soil depth increased, the lateral pile displacement also increased. Indeed, since the liquefied soil layer lost the stiffness and shear strength significantly, the ground displaced more laterally and piles experienced larger deflections. According to Fig. 30a, the maximum lateral deflection of the pile head increased from 72 to 150 mm, 272 mm and 330 mm when the liquefiable soil depth increased from 0 (nonliquefiable soils deposit Scenario I) to 5.0 m (Scenario II), 10.0 m (Scenario III), and 15.0 m (Scenario IV) under the 1999 ChiChi earthquake, respectively. The corresponding lateral displacements for the 1968 Hachinohe earthquake were 80 mm (Scenario I), 155 mm (Scenario II), 243 mm (Scenario III) and 410 mm (Scenario IV), respectively (see Fig. 30b).
Figures 31 and 32 show the shear force and bending moment envelopes along the pile length. In general, once the seismic wave affected the superstructure, the inertial forces transferred from the superstructure to the pile heads and ultimately to the soil deposit. After the liquefaction occurred, more lateral displacement was developed in the vicinity of the ground surface which induced more bending moments near the top of the piles. Therefore, the piles behaved like rather unsupported column in that section. As the liquefied soil depth increased, the pile head displacement increased and subsequently the shear forces increased as in Fig. 31.
Similar to shear forces, the bending moments generated at the pile head increased as the depth of liquefied soil increased. As Fig. 32 shows, in Scenario I, where the soil deposit was nonliquefiable, the maximum bending moment was observed at the pile head, while for other scenarios where soil liquefaction occurred, local amplified bending moments were observed well below the ground surface. In addition, the higher values of bending moment and shear forces were observed at the boundary between the liquefied and nonliquefied soil layers where abrupt the soil stiffness change was observed. These results are comparable with observations made by Rostami et al. (2017) and Dobry et al. (2003) where plastic hinges were generated in the piles at the interfaces between the liquefied and nonliquefied soil layers. However, for heavy superstructure such as LNG tanks, the maximum bending moments are expected to occur at the pile head due to the large inertial forces, which could exceed the bending moments observed at the boundary between the liquefied soil layer and nonliquefied soil layer. In this study when over one third of the entire soil was liquefied soil (i.e. Scenarios III and IV), the bending moments in the pile exceeded the yield stress. For example, as shown in Fig. 32, the generated bending moments in the piles for Scenario III were 6.6 MN.m (1999 ChiChi earthquake) and 6.5 MN.m (1968 Hachinohe earthquake) which exceeded the yield moment of pile M_{yield} = 5.8 MN.m, resulting in formation of plastic hinges in the piles. Similarly, the corresponding bending moment generated at the pile head for Scenario IV for both earthquakes reached the ultimate moment capacity of the pile M_{ultimate} = 7.5 MN.m resulting in the bending failure of the piles.
The variations of soil stiffness and strength with depth can impact the inertial and kinematic interactions influencing the seismic response of piles (Gazetas and Dobry 1984; Nikolaou et al. 2001) and resulted reported in Figs. 30, 31 and 32 illustrate these effects for pile foundation used in this study. The observed inertial interaction stemmed from the movement of the superstructure generating large shear forces and bending moments in the top part of the piles due to the rigid connection between piles and pile cap, and the forces decayed rapidly with depth in the case of nonliquefiable soil (Scenario I) as evident in Figs. 31 and 32.
When the soil deposit liquefied (i.e. Scenarios II to IV), although the impact of inertial interaction on piles reduced (i.e. less base shear experienced by the superstructure as shown in Fig. 24), the response of piles in terms of shear forces and bending moments increased in the top part of the piles (see Figs. 31 and 32). This is due to the fact that the kinematic interaction between the piles and the surrounding soil became more significant. Indeed, kinematic interaction is caused by the inability of a foundation to match the deformation of the surrounding soil (Kramer 1996). The presence of liquefied soil layer (Scenarios II to IV) introduced larger difference between the stiffnesses of the piles and the surrounding soil compared to Scenario I. Therefore, the inability of the piles to match the deformation of the surrounding soil was more significant resulting in more kinematic interaction induced loads (i.e. shear forces and bending moments). It can be concluded that when the liquefaction occurred in the shallow depth only, the kinematic interaction had a dominant impact on the pile response over inertial interaction. Indeed, with the increase in the thickness of the liquefied layer, the loads developed in the piles due to kinematic interaction increased and extended deeper, while the loads due to inertial interaction decreased.
It should be mentioned that the soil constitutive model used for the clay deposit was the nonlinear kinematic hardening model, which is capable of capturing the variation of soil stiffness with strain under cyclic loading and the required parameters were calibrated against the exiting data in the literature as in Sun et al. (1988) and reported in Fig. 11. While it is preferred to use noneassociated flow rule for the response of the soil subjected to cyclic loading, associated flow rule was adopted as a simplifying assumption for the clay soil in undrained conditions, where soil resistance is pressureindependent (Mucciacciaro and Sica 2018). For the liquefied soil deposit, the elastic—perfectly plastic Mohr–Coulomb soil model with Rayleigh damping coefficients was implemented to capture the post liquefaction response of the sand. However, more advanced nonlinear soil models can be implemented in future studies for more accurate predictions, such as Yang (2000) constitutive soil model, which can capture nonlinear hardening behaviour with multi yield surfaces as implemented by Zhuang et al. (2015).
In addition, due to the complexity of the model and lack of experimental results available for the fullscale system capturing all components, attempts were made in this study to evaluate suitability of different individual components of the system. However, consideration of different utilities such as pipelines attached to the system, more rigorous simulation of response of liquified soil and fluid–structure interaction with more accurate simulation of slushing are examples of improvement recommended for future studies.
5 Conclusions
This paper presented the results of seismic analysis of a 160,000 m^{3} full containment LNG tank considering soil structure interaction (SSI) and the fluid–structure interaction (FSI) while assessing the impacts of the depth of soil liquefaction on performance of different components of the system. Four different soil deposit scenarios were investigated, namely, nonliquefied soil deposit (Scenario I), and liquefied soil deposit with their liquefied layer depth increasing from 5.0 m (Scenario II) to 10.0 m (Scenario III) and 15.0 m (Scenario IV). A threedimensional numerical model of the entire tanks, piles and soil system was developed, and the free vibration and time history analyses were conducted adopting fully nonlinear analysis in the time domain. Indeed, direct method of analysis was adopted in which the LNG tank, foundation and soil were simulated and analysed in one step.
The free vibration analyses were performed for the LNG tank using Block Lanczos algorithm, and the results showed that the presence of the liquefied soil significantly alerted the dynamic characteristics of the LNG tank, impacting the seismic performance of the tank system. The ground response analysis showed the liquefied soil layer reduced the spectral acceleration in short period range and amplified the spectral accelerations in the long period range, while increasing the liquefied soil depth lengthened the natural period of the LNG tank.
The nonlinear time history results showed that the seismic forces developed in the inner and outer tank systems reduced as the depth of liquefaction increased. Indeed, increased soil damping and increased structural flexibility directed less seismic forces to LNG tank, and thus the case with nonliquifiable soil deposit resulted in the maximum forces and stresses in the superstructure. The observed potential failure of the inner steel tank was the elastic–plastic buckling mode at the midheight of the inner tank wall, as the von Mises stresses exceed the yield stress for Scenario I. Since, increasing the depth of the liquefied soil, reduced the seismic response of the superstructure, no damage or failure was observed in the inner steel tank for Scenarios II to IV, while localised Von Mises stress were observed near the base of the tank. However, when the liquefied soil depth increase, the impulsive forces along the tank wall decreased, while amplification of convective forces particularly for Scenarios III and IV were observed, since the dominant frequency of the seismic wave was reduced which in turn induced increased convective forces.
On the other hand, soil liquefaction and its depth impacted the seismic response of the pile foundation adversely, due to the observed amplified shear forces, bending moments and deflections in the piles. Moreover, presence of liquefied soil layer exacerbated the inability of the piles to match the deformation of the surrounding soil, and thus resulted in more kinematic interaction induced loads (i.e. shear forces and bending moments). Indeed, with the increase in the thickness of the liquefied layer, although the loads due to inertial interaction decreased, the loads developed in the piles due to kinematic interaction increased and extended deeper, and eventually additional localised plastic deformations in the piles were observed at depths well below the pile head.
Abbreviations
 \(\left\langle \cdot \right\rangle\) :

Macaulay bracket
 \(\Delta \varvec{h}\) :

Small height segment of tank wall
 α :

Back stress
 α ^{dev} :

Devatoric part of back stress tensor
 α ^{s} :

Back stress at large plastic strains
 α _{ damping } :

Rayleigh coefficients viscous damping
 A _{g}(t):

Horizontal ground acceleration
 β _{ damping } :

Rayleigh coefficients viscous damping
 γ _{ k } :

Hardening modulus decreasing rate
 γ _{ ref } :

Refernce shear strain
 γ :

Cyclic shear strain
 γl:

Nondiamntional factor
 ρ :

Fluid density
 ρ _{soil} :

Soil density
 ρ _{ c } :

Concrete density
 ρ _{ r } :

Steel rebar density
 ρ _{ steel } :

Ni steel desity
 C _{ con } :

Sloshing period coefficient
 C _{h(T)} :

Spectral shape factor
 C _{ imp } :

Impulsive pressure distribution coefficient
 C _{ k } :

Initial kinematic hardening modulus
 C _{ l } :

Coefficient of impulsive period
 c _{ p } :

Dilatational (pressure) speed
 CS :

Kinematic hardening modulus for steel
 c _{ sv } :

Shear wave speeds
 C _{ u } :

Undrained shear strength
 D :

Tank diameter
 D _{ k } :

Building damping ratio
 dt :

Tensile damage parameter
 dc :

Compressive damage parameter
 E _{0} :

Intial elastic stiffness
 Ec :

Young’s modulus of elasticity of concreter
 Er :

Young’s modulus of elasticity of rebar
 Es :

Young’s modulus of elasticity of 9% Ni steel
 h :

Overclosure
 H :

Tank height
 H _{ con } :

Convective mass height
 H _{ L } :

Fluid design level
 F :

Yield surface function
 \(f\left( {\sigma  \alpha } \right)\) :

Equivelant Mises stress
 f _{ bo } :

Biaxial compressive yield strengths
 f _{ co } :

Uniaxial compressive yield strengths
 f _{ ult } :

Ultimate tensile strength
 f _{ y } :

Tensile yield stress
 \(f_{c}^{\prime }\) :

Concrete compressive strength
 \(G/G_{{max}}\) :

Modulus reduction ratio
 G _{ max } :

Maximum shear modulus
 G _{ sec } :

Secant shear modulus
 \(I^{\prime}_{1} ,I_{1}\) :

Modified Bessel functions of order 1 and its derivative
 I _{ g } :

Reduction factor modulus
 K _{ c } :

Tensile to the compressive meridian ratio
 I _{ con } :

Convective mode spring stifness
 N _{ max } :

Coffiecient for fualt distance
 \(\left( {N_{1} } \right)_{{60cs  Sr}}\) :

Equivalent SPT value
 M _{ con } :

Covective mass
 m _{ imp } :

Implusive mass
 P :

Contact pressure
 \(p_{{imp}}\) :

Hydrodynamic pressure
 \(\bar{p}\) :

Hydrostatic stress
 \(\bar{q}\) :

Von Mises effective stress
 Q :

Flow potential function
 R :

Return period factor
 R ^{2} :

Coefficient of determination
 R _{tank} :

Tank radiuos
 r :

Cylindrical coordinates of tank
 S :

Deviatoric stress tensor
 S _{ r } :

Residual shear strength
 t :

Time
 T :

Structure period
 T _{ imp } :

Impulsive natural period
 T _{ con } :

Convective natural period
 t _{ u } :

Equivalent uniform thickness of the tank wall
 V _{ s } :

Shear wave velocity
 Z :

Hazard factor
 z :

Cylindrical coordinates of tank
 δ :

Elongation
 \(\dot{\bar{\varepsilon }}^{{pl}}\) :

Equivalent plastic strain rate
 \(\bar{\varepsilon }_{c}^{{in}}\) :

Inelastic crushing strain
 \(\bar{\varepsilon }_{c}^{{pl}}\) :

Compressive plastic strain
 \(\bar{\varepsilon }_{r}^{{cr}}\) :

Cracking strain
 \(\bar{\varepsilon }_{t}^{{pl}}\) :

Tensile plastic strain
 \(\dot{\varepsilon }^{{pl}}\) :

Plastic flow rate
 \({\varepsilon }_{t}^{{pl}}\) :

Tensile plastic strain
 \(\in\) :

Eccentricity of plastic potential surface
 ε _{ c } :

Compressive strain
 ε _{ eng } :

Nominal strain values
 ε _{ t } :

Tensile strain
 ε _{ true } :

True (logarithmic) strain
 v :

Poisson's Ratio
 ξ :

Nondimensional coordinates
 ξ _{ damping } :

Soil damping
 ξ _{ max } :

Maximum damping ratio
 ς :

Nondimensional coordinates
 \(\bar{\sigma }_{{max}}\) :

Maximum principal effective stress
 \(\bar{\sigma }_{{c}}\) :

Effective compressive stress
 \(\bar{\sigma }_{{t}}\) :

Effective tension stress
 \({\sigma }_{{c}}\) :

Compression stress
 \({\sigma }_{{eng}}\) :

Nominal stress strain values
 \({\sigma }_{{t}}\) :

Tensile stress
 \({\sigma }_{{t0}}\) :

Tensile stress at failure
 σ _{ true } :

Cauchy stress
 σ _{ ult } :

Ultimate stress
 \(\mathop \sigma \limits^{\prime }\) _{ vc } :

Effective vertical stress
 σ _{ y } :

Yield stress
 \(\sigma _{0}\) :

Initial yield stress
 \(\tau\) :

The shear stress
 \(\tau _{{critical}}\) :

Critical contact shear stress
 \(\tau _{{rev}}\) :

Shear stress at the reversal point
 ψ :

The dilation angle
 \(\mathop \varphi \limits^{\prime }\) :

Soil friction angle
 μ :

Friction coefficient
 θ :

Cylindrical coordinates of tank
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The first author would like to acknowledge the support provided by AlHussein Bin Talal University during the course of this research.
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Sharari, N., Fatahi, B., Hokmabadi, A. et al. Seismic resilience of extralarge LNG tank built on liquefiable soil deposit capturing soilpilestructure interaction. Bull Earthquake Eng 20, 3385–3441 (2022). https://doi.org/10.1007/s10518022013841
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DOI: https://doi.org/10.1007/s10518022013841