Keywords

1 Introduction

As an important force to achieve China’s “double carbon” goal, the consumption of natural gas is increasing year by year, and the construction of LNG receiving stations and storage tanks is developing rapidly [1,2,3]. At present, China has several LNG receiving stations and storage tanks under construction or have been put into operation.

The loss of property and life caused by earthquake [4, 5] disasters cannot be measured, especially when the gas storage tank and liquid storage tank in the energy industry are damaged and leaked in the earthquake disaster, which causes huge damage. Therefore, the study of earthquake response has important engineering significance for the safety design of storage tank. In 1969, Edwards used finite element method for the first time to conduct numerical simulation research on the coupled seismic response of tank and liquid [6]. W.a.ash and Haroun et al., from the United States, have conducted various researches on cylindrical liquid storage tanks by applying the finite element theory [7,8,9,10]. They apply the theory of potential flow to discrete the fluid into ring elements with rectangular sections without the free sloshing of the fluid surface, and transform the circumferential displacement into a one-dimensional problem assumed to be linear elastic deformation. The Sanders shell theory is used as a thin elastic shell on the wall of the tank and is discrete into ring elements. Finally, the coupling problem of tank and liquid is simplified into an empty tank vibration problem with mass. The dynamic response of empty tank and liquid storage depth, the influence of soil contact and geometrical initial defects were studied. It is reasonable in theory to use finite element theory to calculate the natural characteristics and dynamic response, and it can be used to simulate the vibration situation of storage tank. However, the finite element method is more suitable for solving the natural characteristics of storage tank because of its large calculation amount and the difficulty in compiling the calculation program. This simulation study is of great significance to the research of dynamic characteristics and overall seismic performance of the storage tank.

2 Storage Tank Geometry

Taking 160,000 m3 LNG storage tank as an example, the tank Liquefied Natural Gas is 51.00 m tall, and the largest diameter is 87.10 m. As shown in Fig. 1, it is a sealed double-wall structure with an inner tank made of 9% nickel steel and supported by anchoring bands, and an outer tank consisting of prestressed reinforced concrete walls and a combined reinforced concrete and steel plate dome structure.

Fig. 1.
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Prototype structure of full containment LNG storage tank

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2.1 Structural Unit Analysis Model

The basic components of the overall structure of LNG storage tanks can be divided into two categories: beam and column components and wall panel components. In this simulation, fiber beam element and layered shell element are used to simulate the composite beam-column component and wall panel component respectively. Figure 2 shows the fiber beam element model.

Fig. 2.
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The fiber beam element model.

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Figure 3 shows the model of the layered shell element. For reinforced concrete wall panels, the in-plane deformation and out-of-plane bending are described by plane stress element and layered shell element respectively. For the plane stress element (sometimes called the membrane element in structural engineering), the element model commonly used in the traditional finite element method can be used. The strain in the plane of the steel bar and the concrete is consistent, and the internal force of the plane element is the sum of the internal forces of the steel bar and the concrete.

Fig. 3.
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Layered shell element model

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2.2 Structural Analysis Model

Based on the finite element method, the spatial discretization of the continuous structure can form the following structural dynamic equation:

$$ M\ddot{u} + C\dot{u} + Ku = f^{ext} $$
(1)

where M, C and K are the mass matrix, damping matrix and stiffness matrix respectively. Is the external force vector; u is the displacement vector. The solution of the above dynamic equation is generally based on the Newmark method. Here, the three-stage Newmark method expression is adopted.

2.3 Additive Mass Model

The additional mass method [11] is an approximate method to calculate the liquid-solid coupling problem. The basic idea is to equate the impact dynamic pressure of the liquid on a certain point of the tank wall with the inertia force of the additional mass moving together at the point, and simulate the impact effect of the liquid by applying the additional mass to the inner tank wall. This calculation model makes the calculation decouple, thus reducing the calculation amount.

2.4 Constitutive Relation of Concrete Damage

In order to simulate the whole process of cracking, crushing, damage and destruction of concrete after being stressed, the concrete damage model [12,13,14] is mainly adopted. According to the Code for Design of Concrete Structures (GB50010-2015), the concrete damage model is introduced as follows:

By introducing the tensile damage variable, the uniaxial tensile stress-strain curve of concrete is expressed as:

$$ \begin{gathered} \sigma = \left( {1 - d_{t} } \right){\text{E}}_{{\text{c}}} \varepsilon \\ d_{t} = \left\{ {\begin{array}{*{20}c} {1 - \rho_{t} \left[ {1.2 - 0.5x^{5} } \right]x \le 1} \\ {1 - \frac{{\rho_{t} }}{{\alpha_{t} \left( {x - 1} \right)^{1.7} + x}}x > 1} \\ \end{array} } \right. \\ x = \frac{\varepsilon }{{\varepsilon_{t,r} }} \\ \rho_{t} = \frac{{f_{t,r} }}{{E_{c} \varepsilon_{t,r} }} \\ \end{gathered} $$
(2)

where: \({{\upalpha }}_{\text{t}}\) is the uniaxial tension pressure-strain curve.

2.5 Main Material Parameters

The main mechanical parameters of concrete, steel and thermal insulation materials involved in the scale model of LNG storage tank are listed in Table 1 and Table 2.

Table 1. Concrete material parameters.
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Table 2. Other material parameters
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3 LNG Tank Structure ABAQUS Finite Element Model

The finite element software ABAQUS is used to analyze the seismic response of the LNG storage tank model. The basic idea of numerical simulation is as follows: firstly, the whole model is established, including prestressed tendons, ordinary rebar, concrete cap, concrete outer tank, steel dome, steel inner tank and thermal insulation layer. According to Housner theory [15, 16], the equivalent additional mass of hydraulic pressure is obtained, and the additional mass is distributed on the wall of the tank. The corresponding structural analysis of the seismic response results of the storage tank is carried out to evaluate its seismic performance.

3.1 ABAQUS Numerical Model for LNG Storage Tank

Outer and Inner Tank Models. The ABAQUS finite element analysis model of the LNG storage tank model structure was established using the aforementioned unit model. The geometric model and mesh division of the inner and outer tanks are shown in Fig. 4 respectively.

Fig. 4.
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Geometric model of inner and outer tanks

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Reinforcement and Prestressing.

T3D2Truss unit is used for both common reinforcement bars and prestressed tendons of the storage tank, and the geometric model is shown in Fig. 5. In ABAQUS, a cooling method is used to apply prestressing force to the steel bars.

Fig. 5.
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ABAQUS Model of Steel Fabric

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Fig. 6.
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ABAQUS model of insulation layer

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Geometric Model of Insulating Layer.

The insulating layer on the side wall of the storage tank is 46mm thick expanded perlite and 24mm thick glass fiber felt. ghABAQUS uses C3D8 solid unit to build the insulating layer, as shown in Fig. 6. The insulating layer is processed in different layers, and each layer is assigned with different material properties. The bottom insulating layer is modeled in the same way as the side wall insulating layer.

Additional Mass Distribution.

As shown in Fig. 7 and Fig. 8, ABAQUS adopts the additional mass method to carry out dynamic response numerical simulation analysis of the inner tank. A layer of user-defined units is laid at the interface between the storage tank and the liquid and attached to the wall of the storage tank. One is tank wall element mesh, and the two types of finite element mesh share nodes.

Fig. 7.
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Finite Element Model of Storage Tank

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Fig. 8.
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Additional Mass Model

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4 ANSYS Builds the Finite Element Model

The seismic response analysis of LNG storage tank model is carried out with ANSYS finite element software. The basic idea of numerical simulation is as follows: firstly, the whole model is established, including prestressed tendons, concrete cap, concrete outer tank, steel dome, steel inner tank and thermal insulation layer. Then, according to Housner theory, the equivalent additional mass of hydraulic pressure is obtained, and the additional mass is distributed on the wall of the tank. ANSYS-APDL is used to analyze the seismic response results of the tank and evaluate its seismic performance [17,18,19,20].

4.1 Establishment of LNG Storage Tank Model

Outer Tank Model. The external tank structure includes the external wall and the roof of the tank, as shown in Fig. 9. The external wall and roof structure adopts solid SOLID65 unit, which is a three-dimensional solid unit specially designed in ANSYS software to face concrete materials and can simulate the unique mechanical phenomena of concrete materials.

Fig. 9.
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Finite Element Model of Outer Tank

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Steel Bar Model.

As shown in Fig. 10, the prestressed tendon of the storage tank adopts Link180 unit. In the shaking table test, the prestressed steel bar imposes prestressed constraint on the external wall of concrete through the anchor hole installed on the external wall. For the convenience of modeling, the prestressed steel bundle is directly attached to the surface of the external wall, and shared with the external wall concrete.

Fig. 10.
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Model of prestressed reinforcement and insulation layer for storage tanks

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According to Housner’s theory, when a storage tank is subjected to a horizontal acceleration \({a}_{1}\)(t) from the bottom of the tank, the liquid impact pressure acting on any point on the tank wall (θ, y) is shown in formula (3).

$$ P_{R} = a_{1} \rho h\left( {{y \mathord{\left/ {\vphantom {y h}} \right. \kern-0pt} h} - \frac{1}{2}\left( {{y \mathord{\left/ {\vphantom {y h}} \right. \kern-0pt} h}} \right)^{2} } \right)\sqrt 3 \tan h\left( {\sqrt 3 \frac{R}{h}\cos \sigma } \right) $$
(3)

In the formula:

\({a}_{1}(t)\)—horizontal acceleration (m/\({s}^{2}\)).

\(\rho \)—Liquid density in the storage tank (kg/\({\text{m}}^{3}\)).

r—radius of the storage tank (m).

h—Liquid level height (m).

\(\sigma \)—Azimuth angle of any point along the circumference (rad).

y—Height from the point to the bottom plate (m).

In ANSYS, a cooling method is used to apply prestress to steel bars. The temperature difference T between the initial and the temperature field that reaches the effective stress value is calculated by formula (4).

$$ \sigma_{pe} = \lambda \cdot \Delta T \cdot E $$
(4)

In the formula:

\({\upsigma }_{\text{pe}}\)—effective tensile stress;

\(\uplambda \)—Linear expansion coefficient of steel bars, taken as 1.2 × 10–5;

E—Elastic modulus of steel bars, taken as 2.3269 × 1011 N/m2;

When the initial temperature is 0 °C, the effective stress value can be reached by applying − 55.64 °C to the prestressed steel bar.

5 Comparative Analysis of Two Finite Element Simulation Results

5.1 Seismic Wave Input

In order to study the seismic performance of LNG storage tank under earthquakes with different spectrum and amplitude acceleration, two artificial seismic waves were synthesized by El Centro wave (El), and the site-specific seismic response spectrum. Artificial waves are denoting RG-1 respectively. The acceleration time history are shown in Fig. 11.

Fig. 11.
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Acceleration time history of two seismic waves

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5.2 Analysis of Test Results

As shown in Fig. 12 and Fig. 13, for seismic wave input, the amplitude of the acceleration dynamic amplification coefficient for the outer tank of the empty tank model is between 0.85 and 1.50, and the acceleration dynamic amplification coefficient for the inner tank is between 1.00 and 2.00; Under the same PGA and seismic wave conditions, the dynamic amplification coefficient amplitude of the half tank water outer tank model is slightly lower than that of the empty tank outer tank model, ranging from 0.90 to 1.60, while the dynamic amplification coefficient amplitude of the half tank water inner tank model is larger than that of the empty tank inner tank model, ranging from 1.30 to 2.10.

For different seismic wave inputs, the distribution of the acceleration dynamic amplification coefficient of the model along the height has a similar variation pattern, with the following characteristics:

  1. (1)

    There is a significant turning point in the outer tank, and for different operating conditions and whether the interior is filled with water, there is a significant turning point that occurs at the transition between the outer tank sidewall and the dome.

  2. (2)

    The acceleration dynamic amplification coefficient at the end of the model increases more significantly from above the outer tank ring beam to the observation opening of the dome compared to other tank side parts of the outer tank.

  3. (3)

    For different working conditions and whether the interior is filled with water, the acceleration amplification coefficient of the inner tank increases with height.

Fig. 12.
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Comparison of Acceleration Amplification Coefficients of Outer Tank under El Centro Wave Seismic Action

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Fig. 13.
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Comparison of Acceleration Amplification Coefficients of Outer Tank under RG-1 Wave Seismic Action

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As shown in Fig. 14 and Fig. 15, in the maximum relative displacement of the outer tank, for the horizontal relative displacement within the height range, it can be observed that the values of the outer wall and dome increase as the height increases.

For the acceleration amplification coefficient in the height range, when the design seismic wave is small, the lower part of the exterior wall has a good barrier effect on the seismic wave, resulting in a smaller acceleration value. However, when the design seismic wave is small, the acceleration amplification coefficient is larger, while when the design seismic wave value is large, the acceleration amplification coefficient is smaller. Within the height range, as the height increases, the overall amplification coefficient shows an increasing trend.

The relative displacement at the variable cross-section of the outer tank is relatively large, and the maximum interlayer displacement angle is significantly greater than other positions. Due to the large bending moment and shear force at the bottom of the structural model under earthquake action, it often has large relative displacement and interlayer displacement angle.

Below the middle of the outer tank, the maximum relative displacement increases with height, and decreases from the middle to below the ring beam. For vibrations dominated by the first mode of vibration, if the deformation mechanism is mainly bending, the relative displacement will increase with the increase of height, while for structures dominated by shear deformation, the relative displacement will decrease with the increase of height.

Fig.14.
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Maximum relative displacement of outer tank along height under El Centro wave seismic action

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Fig. 15.
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Maximum relative displacement of outer tank along height under RG-1 wave seismic action

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6 Conclusion

This article establishes a finite element analysis model for the vibration table test model of large LNG storage tanks using ABABQUS and ANSYS finite element calculation software. The dynamic time history response analysis of the structure with empty and half tank water was conducted through a finite element model, and the results of dynamic time history analysis such as acceleration, displacement, and stress were obtained. The main conclusions are as follows:

  1. (1)

    Under the same peak ground motion (PGA) condition, the acceleration amplification coefficient of the model along the height under the action of artificial waves is generally larger than that under the action of natural waves. Overall, there is no trend of decreasing the dynamic acceleration amplification coefficient with the increase of earthquake amplitude under the same seismic wave action.

  2. (2)

    Under the action of seismic waves with the same waveform, as the peak ground motion increases, the relative displacement response of the model increases. At the same time, the sudden change in size at the variable cross-section of the outer tank leads to stress concentration, making the stiffness at the variable cross-section less than other positions, resulting in the maximum displacement angle between the variable cross-section position of the outer tank and the root of the outer tank being greater than other positions; Due to the shaking of water during vibration, the relative displacement below the middle of the outer tank increases. As the distance from the water surface increases, it can be roughly assumed that the impact of water vibration on relative displacement is decreasing.