Cross-Scale Reliability Analysis Framework for LNG Storage Tanks Considering Concrete Material Uncertainty

Abstract

The reliability of liquefied natural gas (LNG) storage tanks is an important factor that must be considered in their structural design. Concrete is a core component of LNG storage tanks, and the geometric uncertainty of concrete aggregate material has a significant impact on their reliability. However, owing to the significant size difference between the concrete aggregate compared to the LNG storage tank, structural analysis using an accurate finite element model that includes all the geometric characteristics of the aggregate incurs significant analytical costs. In particular, for reliability analysis requiring a large number of samples, the computational costs incurred by finite element models are infeasible. Therefore, a dual acceleration strategy based on the asymptotic homogenization method and surrogate model technology is proposed to improve the efficiency of LNG storage tank reliability analysis. In the cross-scale analysis of a LNG storage tank based on asymptotic homogenization, order reduction of the LNG storage tank analysis model was realized. Based on this, a surrogate model construction method with the aggregate fraction and mass moment as inputs was proposed to further accelerate the reliability analysis of LNG storage tanks. Subsequently, a Monte Carlo method was used to perform a reliability analysis of the LNG storage tank considering the uncertainty of the concrete aggregate geometry and distribution under the action of liquid weight and wind load. The analysis showed that the wind load has a significant influence on the safety of the design of the roof of a LNG storage tank. The directionality of the wind load has a significant impact on the distribution of the sample point response for reliability analysis and the failure mode of the LNG storage tank. Owing to the directionality of the wind load, the response distributions of the maximum displacement and maximum stress of LNG were more concentrated, and the reliability of the LNG storage tank decreased after considering the wind load. In particular, the stress reliability of the tank decreased by 5.86%. When only the liquid load was considered, the maximum displacement and stress exhibited asynchronous failure, and the two almost never occurred simultaneously. When the wind load was considered, the failure mode of the LNG storage tank was dominated by the maximum stress. Moreover, the numerical example also demonstrated that the degree of freedom involved in structural analysis, as well as the time of structural analysis can be significantly reduced. So, the proposed cross-scale analysis framework can significantly improve the efficiency of reliability analysis. The conclusions established in this study provide theoretical and methodological guidance for the reliable design of LNG storage tanks.

1. Introduction

Liquefied natural gas (LNG), a low-carbon, clean, high-quality, green energy source, has become the third largest energy source in the world after oil and coal [1]. In the next five years, the global consumption of LNG is expected to increase to 26.5 trillion cubic feet per year [2]. Owing to the high-pressure, flammability, and explosive nature of LNG, its storage safety is a core factor to be considered during its use. As a key equipment for LNG storage, the design reliability of a LNG storage tank is directly related to the safety of LNG use. However, the production cost of LNG storage tanks is high and the construction period is long. Once a tank is damaged and fails, it causes economic losses and could lead to major accidents and disasters, such as explosions and leakages [3].

Based on structural forms, LNG storage tanks can be divided into single-, double-, and full-containment storage tanks [4]. Full-containment tanks (Figure 1) have the advantages of simple structure and strong safety and cost performance and have been widely used in LNG storage and transportation in recent years [5]. Owing to the good sealing properties of full-capacity LNG storage tanks, even if LNG leaks out of the inner tank, the outer tank can still provide a leak-proof barrier, thus ensuring the safety of the LNG storage. Therefore, this study considers a full-capacity LNG storage tank (hereinafter referred to as a LNG storage tank) as the research object. The main structure of the LNG storage tank is composed of an inner tank of low-temperature-resistant nickel steel, an outer tank of prestressed reinforced concrete, glass brick, elastic felt, and expanded perlite powder between the inner and outer tanks [6,7]. Concrete is a core material in LNG storage tanks, and its performance is directly related to the safety of the tanks [8]. However, the size and content of the aggregates in concrete materials are random, which leads to significant uncertainty in the performance of LNG storage tanks. Therefore, a reliability analysis of LNG storage tanks that considers the geometry and content of the aggregate is of great significance for improving the safety of LNG storage and transportation.

Many researchers have conducted studies on the mechanics of concrete materials. Considering the microstructural uncertainty, a micromechanical modeling framework was proposed by Nguyen et al. [9] to analyze the effects on the mechanical properties of the concrete material. It was found that the inclusion size and distribution, as well the volume fraction have a significant influence on the failure modes. Göbel et al. [10] proposed a probabilistic assessment methodology for a micromechanics-based model, and the effects of phase volume fraction uncertainty on Young’s modulus of the hierarchical multiscale concrete material were investigated. The analysis results showed that the effects of uncertainty will be magnified during the upscaling process from nanoscale, microscale, and mesoscale to macroscale. Tao and Chen [11] studied the uncertainty about the constitutive parameters of concrete over different hierarchies, i.e., structure, floor, component, and RVE (Representative Volume Element) levels. They found that only considering the uncertainty of concrete on the hierarchy of structure will seriously overestimate the reliability of a structure.

Up to now, for the reliability analysis of a LNG storage tank, the most focus has been on the structural level. Bursi et al. [12] performed a reliability analysis for a LNG storage tank to enable a seismic design, where a complete 3D model including the LNG tank, support structures, and pipework was utilized for structural analysis. Zhang and Wu [13] studied the dynamic responses and structural reliability of a LNG storage tank under deterministic and stochastic seismic actions. And to improve the computational efficiency, the mass–damper–spring model and mass–spring model were employed to describe the interactions of soil–pile and fluid–structure, respectively. It was found that the concrete wall–roof junction has a greater influence on the reliability of a LNG storage tank. Bhattacharyya et al. [14] presented a numerical approach-based methodology to analyze the reliability of four LNG storage tanks in blast scenarios. The analysis results showed that a single containment tank experienced more severe damage compared with double-, full-, and membrane-containment tanks. Simultaneously, cracks in the concrete wall due to tensile damage were observed, indicating that material plays an important role in the risk assessment of LNG storage tanks. This conclusion is also supported by Liu et al. [15]. They gave a review on evolution laws and the mechanism of concrete performance for a full-concrete LNG storage tank and found that the material parameters (e.g., sand ratio, aggregates, mineral admixture, and fiber) have significant effects on strength, elastic modulus, thermal conductivity, damage, and so on.

However, the reliability analysis of LNG storage tanks considering the geometry and distribution of concrete aggregates is challenging. This is because the challenge of a huge computational cost will be encountered. Since the concrete aggregate size is very small compared to the LNG storage tank size, a complete LNG tank model considering the concrete material geometry details for structural analysis will incur large computational costs. Advanced cross-scale analysis methods enable reliability analysis by considering the geometry and distribution of concrete. Commonly used analysis methods for lattice structures include the representative volume element (RVE) [16], the multiscale finite element method (MsFEM) [17,18], and asymptotic homogenization (AH) [19]. Among them, the asymptotic homogenization method is based on the asymptotic development theory. Through the complete decoupling of material analysis and structural analysis, the analysis problem is reduced, thus greatly improving the efficiency of the structural analysis [20]. In the process of implementing the traditional asymptotic homogenization method, it is necessary to derive the corresponding solution format for microanalysis according to the unit type of the specific microcell, and the established analysis code lacks generality. Thus, Cheng et al. [21] derived and established a novel implementation of asymptotic homogenization (NIAH) based on the asymptotic homogenization energy scheme proposed by Sigmund [22]. The NIAH method provides a multiscale equivalent reduction analysis method for generalized porous media, which significantly broadens the application range of the asymptotic homogenization method. Under a unified analytical framework, microscopic cells can freely choose the cell type and even allow multiple cell types to be mixed. Based on the NIAH method, the equivalent analysis theory of one-dimensional periodic beam structures [23] and periodic plate–shell structures [24,25] has been studied. Recently, this method has gradually expanded from linearity to the buckling problem [26] and viscoelastic composites [27].

The asymptotic homogenization method establishes an efficient bridge between the scales of the material and the structure to accelerate the analysis process. But conventional reliability analysis methods (such as the Monte Carlo methods [28]) require a large number of samples. Structural analyses for all the samples based on asymptotic homogenization are still time consuming. A surrogate model [29] (e.g., the Response Surface Method model [30], Radial Basis Function model [31], and Kriging model [32]) provides a good solution to reduce the scale of FEM calculation, and thereby improves the efficiency of a reliability analysis. For instance, Gu et al. [33] proposed a Kriging model based on adaptive adding point strategy to accelerate a reliability analysis. A reliability analysis for pipelines based on the Radial Basis Function model was carried out by Sousa et al. [34] to evaluate the effects of corrosion on structural failure. However, the choice of inputs has an important impact on the predictive accuracy of a surrogate model [35]. So, the surrogate model technology will be introduced to accelerate the reliability analysis of LNG storage tanks, and the input method is also determined.

In the present study, considering concrete material uncertainty, a cross-scale reliability analysis framework based on a dual acceleration strategy was proposed to enable an efficient reliability analysis of LNG storage tanks. Firstly, a cross-scale structural analysis method for LNG storage tanks based on asymptotic homogenization was developed to address the huge computational cost challenge in single FEM analysis. After that, a surrogate model acceleration strategy that reflects the uneven distribution of concrete aggregates was established to further enhance the efficiency. Based on the proposed framework, a reliability analysis of the LNG storage tank under the liquid weight and wind load was performed. The rest of the paper is organized as follows: In Section 2, the reliability analysis framework of a LNG storage tank, considering the geometric uncertainty of concrete aggregate, is developed. In Section 3, concrete material analysis of a LNG storage tank based on asymptotic homogenization and structural analysis under liquid weight and wind load are presented. In Section 4, an acceleration analysis method based on a surrogate model is established and a reliability analysis of the LNG storage tank is presented. Section 5 gives a numerical example to show the advantages of the proposed framework. Section 6 summarizes and concludes the study.

2. An Overview of the LNG Storage Tank Reliability Analysis Framework Based on Asymptotic Homogenization Method and Surrogate Model Technology

The main structure of a LNG storage tank is composed of concrete. Concrete is a typical multiphase composite material composed of aggregates, mortar, microcracks, and bubbles [34]. To simplify the research problem, this study ignored the effects of microcracks and bubbles and only considered concrete materials composed of aggregates and mortar, and their material properties are significantly different. Moreover, in the process of concrete pouring, the volume fraction, shape, spatial distribution, and orientation of aggregates are highly uncertain, which leads to randomness and heterogeneity in the performance of concrete materials. Thus, it has a significant impact on the performance evaluation and reliability of LNG storage tanks. Therefore, it is important to establish a mechanical property analysis method for concrete materials that considers random factors to accurately evaluate the influence of the mechanical behavior of LNG storage tanks.

The mean geometry of concrete differs by two to three orders of magnitude from the overall size of the LNG tank. In the process of structural analysis of a LNG storage tank, if a complete finite element model including specific characteristics of the aggregate is used to perform the structural performance analysis, huge analysis costs will be incurred. Therefore, this study establishes a concrete representative body element modeling method that considers material uncertainty, introduces a representative body element performance analysis method based on asymptotic homogenization, and decomposes the analysis process of a LNG storage tank into two scales: material and structure. As shown in Figure 2, the equivalent material properties of concrete materials with anisotropic characteristics can be obtained through material analysis based on asymptotic homogenization. In structural analysis, equivalent material properties can be used to solve the displacement and stress responses of LNG storage tanks without considering the detailed concrete composition, thereby significantly reducing the calculation cost of the structure.

Based on the above equivalent analysis, a reliability analysis of the LNG storage tank was conducted using the Monte Carlo method. Although an order reduction of LNG tank analysis can be achieved based on the asymptotic homogenization analysis method, a large number of sample point calculations are still required in the reliability analysis, and frequent finite element calls increase the calculation costs. Therefore, it is necessary to develop a reliability acceleration technology. The surrogate model technique is a common acceleration strategy used in reliability analysis. Implicit or explicit functional relationships between the aggregate and the structural parameters can be obtained by calculating a few sample points. In reliability analysis, the established surrogate model can be used to quickly solve the structural response, and there is no need to perform finite element analysis of the structure to accelerate the reliability analysis process. However, the selection of different aggregate parameters significantly affects the accuracy of the surrogate model. The selection of aggregate parameters must reflect the uneven distribution in the concrete material. Therefore, the construction method of the surrogate model was investigated in this study.

The reliability analysis framework for LNG storage tanks established in this study, considering the geometric uncertainty of concrete aggregates, is shown in Figure 3. By establishing a dual acceleration strategy based on the asymptotic homogenization method and surrogate model technology, a large calculation cost in the reliability analysis is avoided, and the efficiency of the reliability analysis of LNG storage tanks is improved. The analysis framework is as follows:

(1)

In the cross-scale analysis of LNG storage tanks based on the asymptotic homogenization method, the characteristic displacement construction method in the analysis of concrete materials was explored. Using the cross-scale analysis method, the displacement and stress of the LNG storage tank under the liquid weight and wind loads were efficiently solved.

(2)

A surrogate model input method was established to reflect the aggregate inhomogeneity of concrete materials, and the cross-scale analysis method was used to solve the response of LNG storage tanks at each sample point (400 samples were used in this study). Based on the input and output of each sample point, a surrogate model for the structural analysis of the LNG storage tank was established to further improve the efficiency of the reliability analysis.

(3)

Reliability analysis sample points conforming to a normal distribution (10,000 in this study) were randomly generated and the surrogate model was used to perform sample point analysis. Thus, reliability analysis of LNG storage tanks considering the uncertainty of concrete aggregate geometry and distribution was realized based on the Monte Carlo method.

3. Concrete Material Modeling and Analysis Considering Material Uncertainty

3.1. Concrete Material Modeling Considering Aggregate Uncertainty

Spherical inclusions were used to approximate the aggregate model, and the secondary development of the ABAQUS 2023 commercial software based on the Python programming language was used to establish a concrete representative body element model. To facilitate the subsequent application of periodic boundary conditions in asymptotic homogenization in the modeling process, the concrete representative body elements are first divided into regular hexahedral grids and then on the grid according to the location and size parameters of the aggregate, and the unit that meets the conditions is selected as the concrete aggregate. The modeling process is shown in Figure 4. The detailed process is as follows:

(1)

In the given range, the aggregate number of the current concrete representative body element is randomly generated.

$$n_i\in \left[{\mathrm{N}}_{\mathrm{m}\mathrm{i}\mathrm{n}}, {\mathrm{N}}_{\mathrm{m}\mathrm{a}\mathrm{x}} \right]$$

(1)

where N is the number of the current concrete representative body element, N is the number of aggregates, and ${\mathrm{N}}_{\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{N}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are, respectively, the upper and lower limits of the number of aggregates.

(2)

In the given shaft length range, the aggregate radius is randomly generated.

$$r_{i,j}\in \left[{\mathrm{r}}_{\mathrm{m}\mathrm{i}\mathrm{n}}, {\mathrm{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$$

(2)

where $r_{i,j}\left(j\le n_i\right)$ is the radius of the jth sphere in the ith representative body element, and ${\mathrm{r}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\mathrm{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are the lower and upper limits of the radius values, respectively.

(3)

Within the size range of the single cell, the spatial coordinates ${\mathit{p}}_{i,j}=(x_{i,j},y_{i,j}, z_{i,j})$ of the center of the jth aggregate are randomly generated, and whether the current aggregate interferes with the aggregate that has been invested is judged. If the distance between the two aggregate centers is less than the sum of their radii, no interference occurs.

$$‖{\mathit{p}}_{i,j}-{\mathit{p}}_{i,k}‖\le r_i+r_k, 0\le k<j$$

(3)

where k represents the kth aggregate participating in the judgment. If the positions of the two aggregates do not meet the above criteria, the central coordinates of the aggregates are regenerated.

(4)

As shown in Figure 5, according to the radii and the center position of the aggregate that has been put in, judge whether each unit in the concrete representative body element belongs to each aggregate. The inclusion criteria are as follows:

$${\left(\frac{x_s-x_{i,j}}{r_{i,j}}\right)}^2+{\left(\frac{y_s-y_{i,j}}{r_{i,j}}\right)}^2+{\left(\frac{z_s-z_{i,j}}{r_{i,j}}\right)}^2\le 0, 0<s\le {\mathrm{N}}_{\mathrm{e}}$$

(4)

where $\left(x_s,y_s,z_s\right)$ are the central coordinates of the sth element and ${\mathrm{N}}_{\mathrm{e}}$ represents the total number of elements in the concrete material model. When the above equation is satisfied, the sth element belongs to the jth aggregate of the ith concrete material model.

3.2. Analysis of Concrete Material Properties Based on Asymptotic Homogenization

Based on the abovementioned modeling method, a structural analysis of the LNG storage tank can be performed. As shown in Figure 2, to overcome the high computational costs due to the significant difference between the aggregate size in concrete and the structural size of the LNG storage tank, a novel implementation of the asymptotic homogenization (NIAH) method was developed to obtain the equivalent properties of the concrete material model. According to the NIAH theory, the equivalent properties ${\mathbf{E}}^{\mathrm{H}}$ of the concrete RVE can be solved by

$${\mathbf{E}}^{\mathrm{H}}=\frac{1}{\left|\mathbf{Y}\right|}{\left({\mathit{\chi }}^0-{\mathit{\chi }}^{\ast }\right)}^{\mathrm{T}}({\mathit{f}}^0-{\mathit{f}}^{\ast })$$

(5)

where $\left|\mathbf{Y}\right|$ is the volume of the concrete representing the body element; ${\mathit{\chi }}^0$ represents the displacement field generated by unit strain; ${\mathit{f}}^0$ is the corresponding nodal force vector of ${\mathit{\chi }}^0$; ${\mathit{\chi }}^{\ast }$ is the characteristic displacement field, i.e., the displacement field caused by periodic boundary conditions; and ${\mathit{f}}^{\ast }$ is the characteristic load vector, i.e., the corresponding node force vector of ${\mathit{\chi }}^{\ast }$.

The above variables have the following relation:

$${\mathit{f}}^0=\mathbf{K}{\mathit{\chi }}^0$$

(6)

$${\overline{\mathit{f}}}^0=\overline{\mathbf{K}}{\overline{\mathit{\chi }}}^{*}$$

(7)

$${\mathit{f}}^{\ast }=\mathbf{K}{\mathit{\chi }}^{\ast }$$

(8)

where $\mathbf{K}$ is the stiffness matrix of the concrete RVE and $\overline{\left[∎\right]}$ represents the matrix or vector by applied periodic boundary conditions, i.e.,

$${\mathit{\chi }}^{\ast }=\mathbf{P}{\overline{\mathit{\chi }}}^{*},\hspace{1em}\hspace{1em}{\overline{\mathit{f}}}^0={\mathbf{P}}^{\mathrm{T}}{\mathit{f}}^0,\hspace{1em}\hspace{1em}\overline{\mathbf{K}}={\mathbf{P}}^{\mathrm{T}}\mathbf{K}\mathbf{P}$$

(9)

where $\mathbf{P}$ is a multi-degrees-of-freedom (multi-DOF) constraint matrix. The construction method for the multi-DOF constraint matrix is described in [35].

From Equations (5)–(9), the key to solving the equivalent property ${\mathbf{E}}^{\mathrm{H}}$ of the concrete RVE is to use the deformation mode assumption under unit strain. The displacement modes of the concrete RVE used in this study are listed in

Table 1. Unit strain and corresponding characteristic displacement field of RVE.

Load Case	Strain ${\mathsf{\epsilon }}^0={\left[{\mathit{\epsilon }}_{11}^0{\mathit{\epsilon }}_{22}^0{\mathit{\gamma }}_{12}^0{\mathit{\gamma }}_{23}^0{\mathit{\gamma }}_{31}^0\right]}^{\mathbf{T}}$	Assumed Displacement Field ${\mathsf{\chi }}^0={\left[{\mathit{\chi }}_{11}^0{\mathit{\chi }}_{22}^0{\mathit{\chi }}_{33}^0{\mathit{\chi }}_{12}^0{\mathit{\chi }}_{23}^0{\mathit{\chi }}_{31}^0\right]}^{\mathbf{T}}$
1	${\mathsf{\epsilon }}_1^0=[1,0,0,0,\mathrm{0,0}{]}^{\mathrm{T}}$	${\mathsf{\chi }}_1^0=[x,0,0,0,0,0{]}^{\mathrm{T}}$
2	${\mathsf{\epsilon }}_2^0=[0,1,0,0,0{,0]}^{\mathrm{T}}$	${\mathsf{\chi }}_2^0=[0,y,0,0,0,0{]}^{\mathrm{T}}$
3	${\mathsf{\epsilon }}_3^0=[0,0,1,0,0{,0]}^{\mathrm{T}}$	${\mathsf{\chi }}_3^0=[0,0,z,0,0,0{]}^{\mathrm{T}}$
4	${\mathsf{\epsilon }}_4^0=[0,0,0,1,0{,0]}^{\mathrm{T}}$	${\mathsf{\chi }}_4^0=[y/2,x/2,0,0,0,0{]}^{\mathrm{T}}$
5	${\mathsf{\epsilon }}_5^0=[0,0,0,0,1{,0]}^{\mathrm{T}}$	${\mathsf{\chi }}_5^0=[0,z/2,y/2,0,0,0{]}^{\mathrm{T}}$
6	${\mathsf{\epsilon }}_6^0=[0,0,0,0,0{,1]}^{\mathrm{T}}$	${\mathsf{\chi }}_6^0=[z/\mathrm{2,0},x/2,0,0,0{]}^{\mathrm{T}}$

. The unit strain field is not unique to the corresponding displacement mode. To obtain the equivalent elastic properties of the concrete representative volume elements, six conditions must be applied, namely, stretching along the x, y, and z directions and shearing in the x-y, y-z, and x-z planes.

Periodic boundary conditions of the three-dimensional concrete representative body elements are applied such that they can be described by the following formula:

$${\mathsf{\chi }}^{*\left({\mathrm{A}}^{-}\right)}={\mathsf{\chi }}^{*\left({\mathrm{A}}^{+}\right)},\hspace{1em}\hspace{1em}{\mathsf{\chi }}^{*\left({\mathbf{B}}^{-}\right)}={\mathsf{\chi }}^{*\left({\mathrm{B}}^{+}\right)},\hspace{1em}\hspace{1em}{\mathsf{\chi }}^{*\left({\mathbf{C}}^{-}\right)}={\mathsf{\chi }}^{*\left({\mathrm{C}}^{+}\right)}$$

(10)

where $\left(A^{+},A^{-}\right)$, $\left({\mathrm{B}}^{+},{\mathrm{B}}^{-}\right),$ and $\left({\mathrm{C}}^{+},{\mathrm{C}}^{-}\right)$ are the node pairs of the three opposite surfaces in the RVE, as shown in Figure 6. In the detailed implementation process, it is necessary to constrain the displacement of the 8 corner points to be equal, the displacement of all nodes except the corner points on the 12 edges to be equal, and all nodes except the 12 edges on the 6 surfaces to be equal.

3.3. Structural Analysis of LNG Storage Tank under Liquid Weight and Wind Load

The equivalent elastic properties of the concrete RVE obtained by the NIAH method above were used to conduct a structural analysis of the LNG storage tanks. In the structural analysis, two typical loads, liquid weight and wind load, were considered. The LNG liquid load acts on the inner wall and bottom of the tank. The designed specific gravity of the LNG after liquefaction was 0.48, and the maximum liquid height was 38.51 m. The liquid weight borne by the inner tank wall increased linearly from top to bottom. This indicated that the hydraulic pressures at the top and bottom of the liquid were 0 and 181.34 kPa, respectively. The wind load $P_{\mathrm{w}}$ can be calculated as follows:

$$P_w=p_0·{\mu }_z·{\mu }_s·{\beta }_Z$$

(11)

where $p_0$ is the basic wind pressure of the area where the storage tank is located, set to $p_0=0.45 \mathrm{k}\mathrm{P}\mathrm{a}$ in this study. ${\mu }_{\mathrm{z}}$ is the wind pressure height variation coefficient; its value is shown in

Table 2. ${\mu }_{\mathrm{z}}$ as a function of height $H$.

H/m	5	10	15	20	30	40	50	60
${\mu }_z$	1.09	1.23	1.42	1.52	1.67	1.79	1.89	1.97

. The shape coefficient ${\mu }_s$ of the outer surface of the tank is shown in

Table 3. ${\mu }_{\mathrm{s}}$ as a function of angle $\theta$.

$\theta$	0	15	30	45	60	75	90	105	120	135	150	165	180
${\mu }_s$	1	0.8	0.1	0.7	1.2	1.5	1.7	1.2	0.7	0.5	0.4	0.4	0.4

. The ${\mu }_s$ of the top of the tank can be obtained by ${\mu }_s=-{\cos }^2\Phi$. The wind vibration coefficient ${\beta }_z$ is calculated as follows:

$${\beta }_z= 1 + 2g·I_{10}·B_z\sqrt{1+R^2}$$

(12)

where $g=2.5$ is the peak factor; $I_{10}=0.12$ is the nominal turbulence intensity at height $H=10 \mathrm{m}$; $B_z$ is the background component factor of the pulsating wind load; and $R$ is the resonance component factor of the pulsating wind load. $R$ and $B_z$ are calculated as follows:

$$R =\sqrt{\frac{\pi }{6{\zeta }_1}\frac{x^2}{{\left(1+x^2\right)}^{4/3}}}$$

(13)

$$x=\frac{30f_1}{\sqrt{k_w{p}_0}}, x>5$$

(14)

$$B_z=k{H_0}^{\alpha }{\rho }_x{\rho }_z\frac{{\Phi }_1\left(z\right)}{{\mu }_z}$$

(15)

$${\rho }_x=\frac{10\sqrt{B+50e^{-B/50 }-50}}{B}$$

(16)

$${\rho }_z=\frac{10\sqrt{H+60e^{-H/60 }-60}}{H}$$

(17)

where $f_1$ is the first natural vibration frequency of the structure, approximated as $f_1=3.57 \mathrm{H}\mathrm{z}$; $k_w=1.28$ is the ground roughness correction coefficient; ${\zeta }_1=0.05$ is the damping ratio of the structure; $k=0.944$ and $\alpha =0.155$ are constant coefficients; ${\Phi }_1\left(z\right)$ is the first mode coefficients of the structure, which are related to the relative height coefficient $z$ of the structure, i.e., $z=H/H_0$, and its values are shown in

Table 4. ${\Phi }_1\left(z\right)$ as a function of height $H$.

z	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
${\Phi }_1\left(z\right)$	0.02	0.08	0.17	0.27	0.38	0.45	0.67	0.74	0.86	1.00

; $H$ is the height variable of the LNG tank storage structure, and $H_0$ is the total height; and ${\rho }_x$ and ${\rho }_y$ are the correlation coefficient of the horizontal and vertical directions of the pulsating wind load, respectively. The wind load distribution is shown in Figure 7, and its direction is assumed to be along the normal plane of a LNG storage tank.

4. Reliability Analysis of LNG Storage Tank

4.1. Establishing the Surrogate Model between RVE Parameters and Structural Responses

By using the proposed cross-scale analysis method above, the responses (e.g., displacement and stress) of the LNG storage tank can be calculated. However, a time-consuming structural analysis is also needed for each sample point during the reliability analysis. In general, lots of sample points are often required to evaluate the reliability of LNG storage tanks accurately. When the sample point size is large, there is a large computational cost owing to the frequent concrete representative element material analysis and LNG tank analysis. To improve the efficiency of the reliability analysis of a LNG storage tank, this study built a surrogate model between the concrete representative element aggregate information and the structural response of the LNG storage tank based on the above random sample points. The radial basis function (RBF) is a common method for constructing surrogate models and has a high fitting accuracy for problems with high nonlinearity. When the RBF surrogate model is established, the subsequent sample analysis can be based on the RBF surrogate model to achieve a rapid solution for the LNG storage tank. Therefore, the cost of structural analysis with large sample sizes can be effectively reduced, and the efficiency of the reliability analysis of LNG storage tanks can be improved.

When the displacement and size parameters of the aggregate are used to construct the surrogate model, the complexity of the surrogate model increases because there are too many input parameters, which makes it difficult to obtain a high-precision surrogate model. This increases the number of sample points required in the construction of the surrogate model, resulting in a large calculation cost. Therefore, the key to constructing a RBF surrogate model with a high fitting accuracy is to select the appropriate concrete representative parameters. In addition, the parameter should reflect the uneven distribution of the representative volume elements of the aggregate while characterizing the aggregate content. In view of this, four parameters, including the bulk fraction of the aggregate and the mean mass moment of each aggregate mass in the representative volume element to the x-, y-, and z-axes, were selected as the inputs of the RBF model, and a surrogate model was established between the representative volume element information of concrete and the maximum Mises stress and maximum displacement of the LNG storage tank. The mean mass moment reflects the unevenness of the aggregate distribution in the representative volume element and is expressed as follows:

$$I_x=\sqrt{\left(\frac{\sum _{i=1}^{N_{\mathrm{I}}}{m}_i{l}_{x,i}}{N_{\mathrm{I}}}-\overline{m}{x}_0\right)}$$

(18)

$$I_y=\sqrt{\left(\frac{\sum _{i=1}^{N_{\mathrm{I}}}{m}_i{l}_{y,i}}{N_{\mathrm{I}}}-\overline{m}{y}_0\right)}$$

(19)

$$I_z=\sqrt{\left(\frac{\sum _{i=1}^{N_{\mathrm{I}}}{m}_i{l}_{z,i}}{N_{\mathrm{I}}}-\overline{m}{z}_0\right)}$$

(20)

where $I_x$, $I_y$, and $I_z$ are the average mass moments of the aggregate along the x-, y-, and z-axes, respectively; the larger the mass moments, the more uneven the distribution of the aggregate. $m_i$ represents the mass of the ith aggregate in the RVE. $l_{x,i}$, $l_{y,i}$, and $l_{z,i}$ are the distances from the aggregate to the center point $\left(x_0,y_0,z_0\right)$ of the RVE. $\overline{m}$ is the average mass of the aggregates in the RVE.

4.2. Reliability Analysis of LNG Storage Tank Based on Monte Carlo Method

Monte Carlo simulation is a common method used for reliability analysis. It is not limited by the nonlinearity and distribution of random variables of the limit state function and obtains the approximate solution of the original problem by means of continuous approximation. Therefore, Monte Carlo methods can accurately reflect the actual situation of uncertainty problems and are often used to verify the accuracy of other reliability analysis methods. Monte Carlo simulation is a computer simulation based on statistics. When a large number of $N$ samples are randomly selected by the probability density function, the limit state function is used to determine whether the structure of the sample is invalid. According to the law of large numbers, an unbiased estimate of the failure probability can be obtained using the statistical sample failure frequency $n_f$ as follows:

$${\widehat{P}}_f=\frac{n_f}{N}$$

(21)

In this study, two failure modes were considered: maximum displacement and yield failure. The maximum displacement failure criterion ensures that the LNG tank can have a good resistance for load bearing, such as liquid weight and wind loads. And the stress failure criterion is used to ensure the strength safety, because if the maximum stress exceeds the allowable stress, damage may occur. Their state functions can be expressed as

$$Z_u=g_u\left(\mathit{X}\right)=u_0-\max \left\{u_i| i=\mathrm{1,2},\dots ,n_d\right\}$$

(22)

$$Z_{\sigma }=g_{\sigma }\left(\mathit{X}\right)={\sigma }_0-\max \left\{{\sigma }_i| i=\mathrm{1,2},\dots ,n_e\right\}$$

(23)

where $\mathit{X}$ is the space of design variables and $Z_u$ and $Z_{\sigma }$ are the state function for maximum displacement failure and yield failure, respectively. $u_0$ and ${\sigma }_0$ are the maximum displacement and maximum Mises stress allowed in the design of the LNG storage tank, respectively. $n_d$ and $n_e$ are the total number of degrees of freedom and the total number of elements over the structure, respectively.

The displacement failure probability $P_u$ and yield failure probability $P_{\sigma }$ can be expressed as

$$P_u=P\left[g_u\left(\mathit{X}\right)<0\right]=\underset{{\Omega }_f}{\int }{f}_u\left(\mathit{X}\right)dx$$

(24)

$$P_{\sigma }=P\left[g_{\sigma }\left(\mathit{X}\right)<0\right]=\underset{{\Omega }_f}{\int }{f}_{\sigma }\left(\mathit{X}\right)dx$$

(25)

where ${\Omega }_f$ is the feasible domain and $f_u\left(x\right)$ and $f_{\sigma }\left(x\right)$ are the joint probability density function when displacement failure and stress failure occur, respectively. By introducing the characteristic functions ${\delta }_u\left(\mathit{X}\right)$ and ${\delta }_{\sigma }\left(\mathit{X}\right)$ in Equations (24) and (25), we obtain

$${\delta }_u\left(\mathit{X}\right)=\left\{\begin{array}{c}1,x\in {\Omega }_u\\ 0,x\notin {\Omega }_u\end{array}\right.,\hspace{1em}\hspace{1em}{\delta }_{\sigma }\left(\mathit{X}\right)=\left\{\begin{array}{c}1,x\in {\Omega }_{\sigma }\\ 0,x\notin {\Omega }_{\sigma }\end{array}\right.$$

(26)

$$P_u={\int }_{-\infty }^{+\infty }\delta \left[g_u\left(\mathit{X}\right)\right]f_u\left(\mathit{X}\right)dx=E_u\left\{{\delta }_u\left[g_u\left(\mathit{X}\right)\right]\right\}$$

(27)

$$P_{\sigma }={\int }_{-\infty }^{+\infty }\delta \left[g_{\sigma }\left(\mathit{X}\right)\right]f_{\sigma }\left(\mathit{X}\right)dx=E_{\sigma }\left\{I_{\sigma }\left[g_{\sigma }\left(\mathit{X}\right)\right]\right\}$$

(28)

where ${\Omega }_u$ and ${\Omega }_{\sigma }$ are the feasible space when the displacement and stress are safe, respectively, and $E_u$ and $E_{\sigma }$ are the expected failure probabilities of the displacement and stress, respectively.

When a Monte Carlo method is used to determine the reliability of a LNG storage tank, the displacement and stress failure probabilities corresponding to Equation (21) can be written as

$${\widehat{P}}_{f,u}=\frac{1}{N}\sum _{i=1}^N{\delta }_u\left[g_u{\left(\widehat{\mathit{X}}\right)}_i\right]$$

(29)

$${\widehat{P}}_{f,\sigma }=\frac{1}{N}\sum _{i=1}^N{\delta }_{\sigma } \left[g_{\sigma }{\left(\widehat{\mathit{X}}\right)}_i\right]$$

(30)

where $\widehat{\mathit{X}}$ is the sample space of the Monte Carlo method.

5. Numerical Example

A reliability analysis of the LNG storage tank, as shown in Figure 8, will be utilized to demonstrate the advantage of the proposed framework. The outer diameter of the LNG storage tank was 87 m, the inner diameter was 85 m, and the height of the outer tank was 46 m. The structure was discretized by 68,181 C3D8R elements.

Table 5. Material properties of aggregates and mortar in concrete.

	Modulus of Elasticity/GPa	Poisson’s Ratio
Mortar	30	0.2
Aggregate	52	0.25

gives the properties of the aggregates and mortar in the concrete material [36].

5.1. Sample Points Generation and Their Cross-Scale Analysis Based on Asymptotic Homogenization

In the present study, a surrogate model was introduced to improve the reliability analysis efficiency of LNG tanks. And 400 sample points, i.e., concrete material models in total, were generated to establish the surrogate model. For each sample point, a cross-scale analysis was performed. Specifically, the equivalent property of the concrete material model was calculated based on the asymptotic homogenization method, where the properties of aggregates and mortar shown in Table 1 were used. Then, the obtained equivalent material property will be used to calculate the structural responses of the LNG storage tank.

In the sample generation process, the size of the concrete material model was $100\times 100\times 100$, and 1,030,301 8-node C3D4R elements were used to discretize the concrete material model. The lower and upper limits of the aggregate radius were set to ${\mathrm{r}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=4$, ${\mathrm{r}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=7$. A total of 400 concrete material models were generated, and the upper and lower limits of the number of aggregates were set to ${\mathrm{N}}_{\mathrm{m}\mathrm{i}\mathrm{n}}=20$, ${\mathrm{N}}_{\mathrm{m}\mathrm{a}\mathrm{x}}=32$. Figure 9 shows the typical concrete material model and its aggregate distribution. The size distribution of the aggregates in the 400-group concrete material model is shown in Figure 10. In the random process, both the location and size of the aggregates satisfy the statistical law of the Gaussian distribution.

The equivalent properties of each sample point were calculated based on the asymptotic homogenization method, as shown in Section 3.2. And the accuracy of the asymptotic homogenization method was vilified by comparison with the estimated results of the classical Mori–Tanaka model [37]. For a two-phase concrete material, the equivalent stiffness tensor $\mathbb{C}$ can be calculated by

$$\mathbb{C}={\mathbb{C}}_{\mathrm{m}}+v_a\left({\mathbb{C}}_{\mathrm{a}}-{\mathbb{C}}_{\mathrm{m}}\right):{\mathbb{A}}^{\mathrm{M}\mathrm{T}}$$

(31)

where ${\mathbb{C}}_{\mathrm{m}}$ and ${\mathbb{C}}_{\mathrm{a}}$ are the stiffness tensor of mortar and aggregate in concrete material, respectively; $v_a$ is the volume fraction of aggregates; and ${\mathbb{A}}^{\mathrm{M}\mathrm{T}}$ is the localization tensor about average strain of aggregates in the Mori–Tanaka model, calculated by

$${\mathbb{A}}^{\mathrm{M}\mathrm{T}}={\mathbb{A}}^{\mathrm{E}\mathrm{s}\mathrm{h}}{\left[v_{\mathrm{m}}\mathbb{I}+v_{\mathrm{a}}{\mathbb{A}}^{\mathrm{E}\mathrm{s}\mathrm{h}}\right]}^{-1}$$

(32)

where $\mathbb{I}$ is the identity tensor; $v_{\mathrm{a}}$ is the volume fraction of mortar in concrete; and ${\mathbb{A}}^{\mathrm{E}\mathrm{s}\mathrm{h}}$ is the Eshelby’s localization tensor, i.e.,

$${\mathbb{A}}^{\mathrm{E}\mathrm{s}\mathrm{h}}={\left[\mathbb{I}+\mathbb{S}:{\mathbb{C}}_{\mathrm{m}}^{-1} :\left({\mathbb{C}}_{\mathrm{a}}-{\mathbb{C}}_{\mathrm{m}}\right)\right]}^{-1}$$

(33)

where $\mathbb{S}$ is the Ashelby tensor [38].

The relative error between the estimated concrete material properties of the asymptotic homogenization and Mori–Tanaka model is given in Figure 11. From the figure, it can be found that for all sample points the estimation error for tension and shear modulus is less than 8% and 3%, respectively. It demonstrates that the established method based on the asymptotic homogenization method can give a relatively accurate estimation for the equivalent material properties. In the Mori–Tanaka model, it is assumed that the aggregates are distributed homogeneously over the concrete material. But for a real concrete material, a heterogeneity is always presented, which will lead to the estimation error.

Figure 12 shows the distribution of the displacement and stress at each sample point under the liquid weight load, wind load, and combined loads. The small amplitude of the wind load had little influence on the minimum and maximum displacements and Mises stresses of the LNG storage tank structure. However, from Figure 7b, the wind load gradually increased from the bottom to the top of the tank, whereas the influence of the liquid weight load on the tank was mainly concentrated at the bottom. Therefore, the wind load had a greater impact on the top of the tank. Figure 13 shows the structural displacement and stress cloud map corresponding to a typical sample point. By comparing Figure 13a-II with Figure 13c-II, it can be observed that the displacement of the top of the structure under the combined action of the liquid weight and wind load was significantly increased compared with the displacement map of a single liquid load. Therefore, although the amplitude of the wind load is small, it has a certain impact on the reliability of LNG storage tanks.

Moreover, the number of DOF (degrees of freedom) involved in the structural analysis can reflect the computational cost, i.e., the larger number of DOF indicates higher time consumption. For the current LNG storage tank model, when a complete model contains the detailed concrete material configuration, the total DOFs $N_c$ can be estimated by

$$N_{\mathrm{c}}\approx N_{\mathrm{m}}\times N_{\mathrm{s}}$$

(34)

where $N_{\mathrm{m}}=\mathrm{304,896}$ and $N_{\mathrm{s}}=\mathrm{3,090,903}$ are the number of DOF of the concrete material and the LNG storage tank structure. It means that $N_c\approx 9.4\times {10}^{11}$ DOF will be included in a complete model. However, when the cross-scale analysis based on asymptotic homogenization is employed, the total number of the involved DOF $N_{\mathrm{a}\mathrm{h}}$ can be estimated by

$$N_{\mathrm{a}\mathrm{h}}\approx N_{\mathrm{m}}+N_{\mathrm{s}}$$

(35)

and $N_{\mathrm{a}\mathrm{h}}\approx 3.4\times {10}^6$. It is obvious that the cross-scale analysis method based on asymptotic homogenization can improve the efficiency of LNG storage tank analysis.

5.2. Establishment of RBF Surrogate Model

The RBF surrogate model between the concrete material model parameters (volume fraction of aggregates and the mass moments) and LNG storage tank responses (displacement and stress) was established. Figure 14 and Figure 15 show the relationship between the maximum displacement and maximum stress predicted by the RBF surrogate model and the input, considering the working condition under the combined action of the liquid weight and wind load as an example. And the prediction errors of the surrogate model for maximum displacement and stress are shown in Figure 15. The relative error of the maximum displacement surrogate model was found to be within 5%, although a nonlinear relationship between the input parameters and maximum displacement is exhibited (Figure 15). So, the established RBF model can predict the maximum displacement with high accuracy (Figure 16a). However, because the stress of the structure shows strong local and nonlinear characteristics (Figure 16), the prediction accuracy of individual samples is slightly lower than 5%; however, some points are still within 5% (Figure 16b). It indicates that the established surrogate model can be used to estimate the displacement and stress response straightforwardly while no more structural FEM analysis is needed.

Moreover, a greater volume fraction indicates that more aggregates are included within the concrete material, and the aggregate has a larger stiffness compared with mortar. So, the stiffness of a concrete material increases as the volume fraction of aggregates increases, and the maximum displacement and strain will decrease (Figure 14). Since the material properties of mortar and aggregates are constant, the maximum stress can thereby be decreased (Figure 15). Simultaneously, from Equations (18)–(20), the larger volume fraction will generally result in a larger average mass moment $I_y$, $I_y$, and $I_z$. So, from Figure 14 and Figure 15, it can be observed that all the maximum displacement and stress plots descend along their diagonal directions.

5.3. Reliability Analysis of LNG Storage Tank

Based on the RBF surrogate model, a Monte Carlo method was used to analyze the reliability of the LNG storage tank structure. The number of samples is set to N = 10,000 and the maximum allowable displacement $u_0$ and maximum Mises ${\sigma }_0$ stress to $u_0=6.8741 \mathrm{m}\mathrm{m}$ and ${\sigma }_0=4\times {10}^6 \mathrm{P}\mathrm{a}$, respectively. Importantly, by using the surrogate model, this reliability analysis can be completed in 18 s, and the efficiency of reliability analysis can be improved significantly.

The reliability values obtained under different operating conditions are listed in

Table 6. Reliability of LNG storage tank under different load cases.

	Displacement Reliability	Stress Reliability
Liquid weight load	0.9522	0.9749
Wind Load	0.9799	0.9416
Combined action of liquid weight and wind load	0.9518	0.9178

. The displacement was small owing to the small amplitude of the wind load. Therefore, when considering the effect of wind load, the LNG storage tank had a high reliability, and its reliability reached 0.9799. Because the amplitude of wind load was small, it mainly affected the top of the LNG storage tank (Figure 13b-I). Compared to cylindrical tank walls, the top is a weak point in the design of LNG tanks. Therefore, the stress reliability under the action of the wind load was lower than that under the action of the liquid weight. For the tank design method adopted in this study, the design of the top of the tank should be strengthened to enhance its reliability under the action of a wind load. In addition, from Table 6, when the combined action of the liquid weight and wind load is considered, the reliability of the structure decreases. Specifically, stress reliability decreased by 5.86%.

Figure 17, Figure 18 and Figure 19 show the distribution of displacement and stress failure sample points of the LNG storage tank under the action of liquid weight, wind load and action, and liquid weight and wind load, respectively. When only the liquid weight load was considered, the distribution of the sample points in space was relatively dispersed (Figure 17). Owing to the directional nature of the wind load, the distribution of the sample points was relatively concentrated (Figure 18). Compared with Figure 17, the sample points under the combined action of liquid weight and wind load were more concentrated (Figure 19). There was almost no intersection between the displacement failure points and stress failure points when considering only the liquid weight, which means that different concrete aggregate compositions mainly lead to the asynchronous failure of displacement and stress, but do not lead to the simultaneous failure of both. Therefore, it is necessary to pay attention to the reliability of both displacement and stress when considering only the liquid weight. However, when the wind load was considered, the structure was more prone to stress failure, as shown in Figure 18 and Figure 19. Displacement failure of the LNG storage tank was accompanied by stress failure; however, a stress failure may not lead to displacement failure. The spatial distribution of the displacement and stress points, and the subsequent failure mode of the LNG storage tank, depends on the type of load. Thus, detailed attention should be paid to stress reliability in the design of LNG storage tanks.

6. Conclusions

In this study, a dual analysis acceleration strategy based on asymptotic homogenization and surrogate modeling was established to accelerate the efficiency of LNG tank reliability analysis. It aimed to overcome the large computational cost caused by the significant size differences between material and structural scales, and the large number of sample analyses used in the Monte Carlo method. By establishing a cross-scale analysis method based on asymptotic homogenization, the material and structural analyses of the LNG storage tank were decoupled. A material analysis considering the geometry and distribution of concrete aggregates was performed to obtain the equivalent material property. And the calculated equivalent material property was then employed in the structural analysis without considering the detailed material configuration. So, the efficiency of LNG storage tank structural analysis can be significantly improved. After that, a surrogate model construction method was established with the aggregate fraction and mass moment as input. Via small-scale sample analysis, the mapping relationship between the input and output (i.e., the displacement and stress) was established. And the structural response can be estimated using the established surrogate model, and no more structural analysis is needed. So, the surrogate model technology can avoid the large computational cost in reliability analysis based on the Monte Carlo method. The numerical example showed that the proposed framework can give a good estimation for the structural response and improve the efficiency of the reliability analysis of LNG storage tanks.

A cross-scale structural reliability analysis considering material uncertainty was performed for a LNG storage tank under the liquid weight and wind loads. If only the liquid weight load was considered, the distribution of the failure sample points was relatively dispersed, and the failures of stress and displacement would not occur simultaneously. However, when the liquid weight and wind loads were both considered, the failure points distribution and failure modes were changed. Although the amplitude of the wind load was small, owing to its directionality, it led to an increased risk of failure at the top of the tank. Simultaneously, stress failure became the dominant failure mode. So, to improve the reliability of a LNG storage tank, the influences of a weak and directional load (e.g., wind load) applied on the top of the tank need to be considered. This study provides numerical methods and guidance for the design and reliability analysis of LNG storage tanks considering material uncertainty.