Failure Behavior and Vulnerability of Containment Structures Subjected to Overpressure Loads Considering Different Failure Criteria

Abstract

This paper investigates the failure behavior and vulnerability of the containment structure (CS) under internal pressure according to different functional and structural failure criteria. Through nonlinear numerical analysis, the deformation of the structure, degree of concrete cracking, and the distribution of stresses and strains in the prestressing tendons and steel liner under different failure criteria are compared. Combined with probabilistic analysis methods, the vulnerability curves, HCLPF capacities, and total failure probabilities of CS determined by different failure criteria are systematically discussed. Results show that there are some differences in the failure behavior and pressure capacity of CS under different failure criteria for both functional and structural failure modes. Under the same failure mode, the variability of the pressure capacities obtained using different criteria is relatively small. Generally, the total failure probabilities of CS subjected to overpressure loads determined by different failure criteria exhibit significant differences. If it is considered that all the investigated failure criteria have a certain ability to predict containment failure, the probabilistic assessment results derived from the functional failure criterion based on the fracture strain of steel liners and the structural failure criterion based on the global strain are deemed to be more reasonable.

1. Introduction

Nuclear power provides an important opportunity to reduce humankind’s dependence on traditional fossil energy sources, which contributes to the diversification and sustainable development of the energy landscape. However, the utilization of nuclear power comes with inherent risks and challenges. Throughout history, catastrophic incidents such as the Chernobyl and Fukushima nuclear accidents have caused severe impacts on human health, socio-economics, and the ecological environment. Consequently, preventing nuclear leakage has emerged as a priority concern in the development of nuclear power. The containment structure (CS) in nuclear power plants serves as a crucial barrier against nuclear leakage, which can effectively mitigate the possible adverse effects of reactor accidents or external disasters (such as earthquakes, tsunamis, and missile impacts) [1,2,3]. Conducting a comprehensive analysis of the CS, encompassing its performance under various potential accident scenarios, holds paramount importance in ensuring the safe operation of nuclear power plants.

During abnormal accident scenarios, high steam pressures are often generated inside the CS. Therefore, the analysis and assessment of the CS subjected to overpressure loads is a major concern in nuclear engineering [4,5,6]. To date, a series of tests have been undertaken to explore the mechanical behavior and safety performance of the CS under internal pressure. The Central Electricity Generating Board (CEGB) of the U.K. conducted a 1:10 scale model test according to the CS of the Sizewell B nuclear power plant to verify the safety margin of the structure under overpressure conditions [7]. Sandia National Laboratories (SNL) of the U.S. carried out a 1:4 scale model test using the CS of Ohio Unit 3 as a prototype [8]. This test systematically elucidated the failure process, damage mechanism, and ultimate pressure capacity of the CS subjected to overpressure loads. The Metallurgical Corporation of China (MCC) constructed a 1:10 scale model of a typical CS of CNP1000 and tested the safety of the CS under overpressure conditions [9]. Electricité de France (EDF) performed a 1:3 scale model test to investigate the structural behavior, leak tightness, and aging effects of the CS under severe accident conditions [10,11]. The Bhabha Atomic Research Centre (BARC) of India conducted a 1:4 scale model test to study the leakage pattern of the CS under various internal pressures and assess its ultimate pressure capacity [12]. These valuable experimental data offer an important reference for the design, analysis, and evaluation of the CS.

Given the prolonged test cycles and substantial costs associated with the model tests, the finite element (FE) analysis method is extensively utilized to investigate the performance of the CS under internal pressure. Noh et al. [13] and Shokoohfar et al. [14] simulated the 1:4 scale model test carried out by SNL and further explored the mechanical behavior of the CS under thermal-pressure coupled loading. Hu et al. [15] examined the ultimate pressure capacity and failure mode of the CS under long-term prestress loss. Huang et al. [16] evaluated the long-term performance of the CS under internal pressure by numerical simulation taking into account the time-dependent aging factor. Yan et al. [5] investigated the structural behavior and damage mechanism of the CS under accident pressure based on a 3D FE model. Jim’enez et al. [17] explored the bearing capacity of the CS under beyond-design basis pressures with the consideration of material aging over the 40-year design life. Li et al. [18] systematically discussed the impart of steel liner corrosion on the leak tightness of the CS subjected to overpressure loads by combining a global model and a local refined sub-model. Yang et al. [19] analyzed the influence of concrete creep on the mechanical performance of the CS and evaluated its pressure capacity under long-term service conditions. Zheng et al. [20] explored the feasibility of using steel fibers to enhance the damage performance of the CS under internal pressure.

The aforementioned studies focused on analyzing the structural performance of the CS under internal pressure from a deterministic perspective. Given the importance of the CS, it is necessary to incorporate probabilistic analysis methods to provide a comprehensive assessment of the safety performance of the CS [21]. The following is an overview of the probabilistic performance evaluation of the CS. Prinja et al. [22] employed a first-order reliability method to predict the failure probability of the CS under internal pressure. Ren et al. [23] assessed the vulnerability of the CS subjected to overpressure loads (hereafter, overpressure vulnerability) using the probability density evolution method and conducted a reliability assessment of the CS under different failure modes. Ding et al. [24] evaluated the overpressure vulnerability and failure probability of the CS at different service lifetimes considering chloride-induced corrosion. Fan et al. [25] investigated the overpressure vulnerability of the CS using a simplified analytical method and discussed the effect of factors such as concrete strength, steel liner thickness, and reinforcement amount on failure probability. Li et al. [26] proposed a probabilistic analysis method accounting for spatial variability in steel liner properties and applied it to evaluate the leakage risk of the CS under internal pressure.

Experimental studies have demonstrated that functional failure (i.e., loss of leak-tight function) and structural failure (i.e., reaching the ultimate bearing capacity) are the typical failure modes of the CS subjected to overpressure loads [27]. In the FE analysis or probabilistic risk assessment combined with the FE analysis, a key issue is how to define the failure thresholds for both the functional and structural failures of the CS. However, owing to the complexity of the failure of the actual CS, establishing a recognized failure criterion for predicting the failure remains challenging. As a result, various failure criteria were employed in previous literature for the FE analysis and probabilistic assessment of the CS. Notably, the adoption of different failure criteria for the same CS often yields different analysis results, which may seriously affect the assessment of the actual failure behavior and safety margin of the CS. Therefore, the following two key issues need to be clarified: (1) What is the basis or source for the establishment of the different failure criteria? (2) What differences exist in predicting the failure of the same CS using different failure criteria? Addressing these questions is essential for the design of the CS and the assessment of its performance under extreme accident conditions. However, few studies have delved into the differences in the analysis and evaluation results arising from different failure criteria.

The main objective of this study is to examine the differences in the failure behavior and vulnerability of the CS under different failure criteria and to provide rational recommendations for the selection of failure criteria. To accomplish this objective, a refined 3D FE model of the CS is first established. Subsequently, functional and structural failure criteria for the CS, as reported in the literature, are thoroughly investigated. Based on these failure criteria, the failure behavior of the CS subjected to overpressure loads is compared in detail, including the deformation of the structure, degree of concrete cracking, and distribution of stresses and strains in the prestressing tendons and steel liner. Combined with probabilistic analysis methods, the vulnerability curves, HCLPF (High Confidence of Low Probability of Failure) capacities, and total failure probabilities of the CS under internal pressure determined by different failure criteria are systematically discussed. The findings in this work can provide a reference for the structural analysis and probabilistic performance evaluation of the CS under beyond-design basis pressures.

2. Modeling Process of the CS

2.1. Structure Configuration

The CS selected in this paper has similar dimensions to an in-service CS in China. The CS features a single-wall configuration, comprising a cylinder wall, four equally spaced buttresses, a dome, a ring beam, and a base slab. Furthermore, the inner surfaces of the cylinder wall and dome are lined with a steel plate (known as the steel liner) to provide the required leakage protection. Figure 1 depicts the structural composition and geometry of the prototype CS.

The cylinder wall of the CS is equipped with mutually perpendicular circumferential and longitudinal tendons, and U-shaped tendons with a uniform curvature radius are arranged in the dome, as shown in Figure 2. All prestressing tendons are placed at the mid-depth of the wall and are tensioned at both ends using post-tensioning techniques. The circumferential tendons in the cylinder wall are anchored between alternate buttresses, and the longitudinal tendons are anchored via a tendon gallery in the base slab and ring beam. The U-shaped tendons within the dome extend to the ring beam for anchorage. Besides the prestressing tendons, the cylinder wall and dome incorporate inner, middle, and outer layers of ordinary rebars, which are staggered along the longitudinal and circumferential directions of the wall. Moreover, additional layers of reinforcement are provided in the buttresses, ring beam, and thickened areas of the equipment hatch to ensure the integrity and stability of the local regions.

2.2. Material Model Selection

2.2.1. Concrete

The Concrete Plastic Damage (CDP) model developed by Lubliner et al. [28] has gained widespread utilization for simulating the nonlinear behavior of concrete. Built upon principles of plasticity and continuity, the CDP model can effectively describe the complex behaviors of the concrete material during the loading process, such as damage, plastic deformation, and the recovery of stiffness. Additionally, the CDP model integrates the two main failure mechanisms of tensile cracking and the compressive crushing of concrete materials by introducing two damage factors $d_{\mathrm{t}}$ and $d_{\mathrm{c}}$, as follows:

$$d_{\mathrm{t}}=1-\frac{{\sigma }_{\mathrm{t}}}{\left(1/D_{\mathrm{t}}-1\right)E_0{\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{pe}}+{\sigma }_{\mathrm{t}}}$$

(1)

$$d_{\mathrm{c}}=1-\frac{{\sigma }_{\mathrm{c}}}{\left(1/D_{\mathrm{c}}-1\right)E_0{\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pe}}+{\sigma }_{\mathrm{c}}}$$

(2)

where ${\sigma }_{\mathrm{t}}$ and ${\tilde{\epsilon }}_{\mathrm{t}}^{\mathrm{pe}}$ are the tensile stress and equivalent tensile plastic strain of concrete; $E_0$ is the initial modulus of elasticity; ${\sigma }_{\mathrm{c}}$ and ${\tilde{\epsilon }}_{\mathrm{c}}^{\mathrm{pe}}$ are compressive stress and equivalent compressive plastic strain; $D_{\mathrm{t}}$ and $D_{\mathrm{c}}$ are the damage parameters, which can be determined as $D_{\mathrm{t}}=0.1$ and $D_{\mathrm{c}}=0.7$ according to Birtel et al. [29].

Figure 3 illustrates the schematic diagram of the CDP model. To completely characterize the CDP model, the stress–strain relationships for uniaxial tension and the compression of concrete also need to be provided. In this paper, the uniaxial constitutive model of concrete recommended by the Chinese code [30] is adopted, which takes the following form:

$${\sigma }_{\mathrm{t}}=\left\{\begin{array}{ll}(1.2-0.2x_{\mathrm{t}}^5)\frac{f_{\mathrm{t}}}{{\epsilon }_{\mathrm{t}0}}{\epsilon }_{\mathrm{t}}& x_{\mathrm{t}}\le 1\\ \frac{f_{\mathrm{t}}}{\left[x_{\mathrm{t}}+{\alpha }_{\mathrm{t}}{\left(x_{\mathrm{t}}-1\right)}^{1.7}\right]{\epsilon }_{\mathrm{t}0}}{\epsilon }_{\mathrm{t}}& x_{\mathrm{t}}>1\end{array}\right.$$

(3)

$${\sigma }_{\mathrm{c}}=\left\{\begin{array}{ll}\frac{f_{\mathrm{c}}n}{(n-1+{x_{\mathrm{c}}}^n){\epsilon }_{\mathrm{c}0}}{\epsilon }_{\mathrm{c}}& x_{\mathrm{c}}\le 1\\ \frac{f_{\mathrm{c},\mathrm{r}}}{\left[x_{\mathrm{c}}+{\alpha }_{\mathrm{c}}{\left(x_{\mathrm{c}}-1\right)}^2\right]{\epsilon }_{\mathrm{c}0}}{\epsilon }_{\mathrm{c}}& x_{\mathrm{c}}>1\end{array}\right.$$

(4)

where $x_{\mathrm{t}}={\epsilon }_{\mathrm{t}}/{\epsilon }_{\mathrm{t}0}$, $x_{\mathrm{c}}={\epsilon }_{\mathrm{c}}/{\epsilon }_{\mathrm{c}0}$, and $n=E_{\mathrm{c}}{\epsilon }_{\mathrm{c}0}/(E_{\mathrm{c}}{\epsilon }_{\mathrm{c}0}-f_{\mathrm{c}})$; $E_{\mathrm{c}}$ is the elastic modulus of concrete. $f_{\mathrm{t}}$ and ${\epsilon }_{\mathrm{t}0}$ are the tensile strength and peak tensile strain of concrete; $f_{\mathrm{c}}$ and ${\epsilon }_{\mathrm{c}0}$ are the compressive strength and peak compressive strain of concrete; ${\alpha }_{\mathrm{t}}$ and ${\alpha }_{\mathrm{c}}$ are the parameters of the descending part of the tensile and compressive stress–strain curves.

2.2.2. Prestressing Tendons

The prestressing tendons utilized in the CS are fabricated from 1770-grade low-relaxation strands. In this paper, the constitutive model suggested by Tavakkoli et al. [4] is used to simulate the elastic–plastic behavior of prestressing tendons. The stress–strain curve of this model is defined by the modified Ramberg–Osgood function, characterized by the following expression:

$${\sigma }_{\mathrm{p}}=E_{\mathrm{p}}\left(0.025+\frac{1-0.0025}{{\left[1+{(118{\epsilon }_{\mathrm{p}})}^{10}\right]}^{1/10}}\right){\epsilon }_{\mathrm{p}}\le f_{\mathrm{pu}}$$

(5)

where ${\sigma }_{\mathrm{p}}$ and ${\epsilon }_{\mathrm{p}}$ represent the stress and strain of the prestressing tendons; $E_{\mathrm{p}}$ represents the elastic modulus; and $f_{\mathrm{pu}}$ represents the ultimate tensile strength.

To simulate the plastic behavior of prestressing tendons, the kinematic hardening model based on the von Mises criterion is employed. The elastic modulus of the prestressing tendons is taken as 195,000 MPa, and Poisson’s ratio is assumed to be 0.3. Figure 4a depicts the schematic diagram of the stress–strain relationship of the prestressing tendons.

2.2.3. Ordinary Rebar and Steel Liner

The ordinary rebar used in CS consists of both HRB335 and HRB400 steel. The steel liner is made of P265GH steel plate, which has excellent ductility. To enhance the computational efficiency, the ideal elastic–plastic constitutive relationship commonly used in engineering analysis is employed to simulate the mechanical behaviors of the ordinary rebar and steel liner, as shown in Figure 4b. Additionally, the elastic modulus for both the ordinary rebar and steel liner is taken as 200,000 MPa, and Poisson’s ratio is assumed to be 0.3.

2.3. FE Mesh of the Analytical Model

In accordance with the geometric properties of the CS, a refined 3D analytical model is developed using ABAQUS software (version 2020), and the detailed mesh information for each component of the CS is illustrated in Figure 5. Prestressing tendons and ordinary rebars in the CS are modeled separately, and they are arranged in the concrete by the “embedded” technique without considering the slip effect. A perfect bond between the steel liner and the inner surface of the concrete wall is assumed [5,26], and the steel liner shares nodes with the concrete elements through the “skin” technique. Following a mesh sensitivity analysis, the global mesh density of the analytical model is set to 800 mm to strike a balance between computational efficiency and accuracy. Moreover, finer meshes are employed in the singular areas (such as the penetrations, dome, and ring beam) of the CS.

2.4. Boundary Conditions and Loading Steps

Since the base slab has considerable stiffness, its restraining effect on the CS is simulated by fixing the degrees of freedom in all directions at the bottom of the analytical model [31]. During accident conditions, pressure generation within the CS often coincides with temperature loading. Relevant literature has examined the temperature effect on the global behavior and pressure capacity of the CS and found it to be limited [1,32,33]. Given the larger computational effort involved in conducting the vulnerability analysis, temperature effects are not considered within the scope of this study. The loading procedure for the analytical model of the CS consists of two steps. Initially, self-weight and prestressing loads are simultaneously applied to the entire structure. Subsequently, a uniformly rising pressure load is imposed on the inner surface of CS along the direction of the perpendicular steel liner. It should be noted that the soil–structure interaction is ignored in this study to simplify the analytical work. However, the effect of this factor on the CS needs to be scrutinized in future investigations.

Figure 6 shows the distribution pattern of the displacements of the analytical model following the application of gravity and prestressing loads. It can be seen that the displacement of the lower part of the CS is smaller than that of the upper part, which is basically consistent with the displacement distribution pattern monitored in the model test of the 1:4 scale prestressed concrete containment conducted by the SNL [27]. Furthermore, the measured displacement in the dome after tensioning the prestressing tendons in the model test is 0.6 cm. Considering the scale difference between the model test (with a total height of 16 m) and the prototype CS (with a total height of 64 m), the measured displacement tends to be four times larger, i.e., 2.4 cm. As can be seen from Figure 6, the maximum magnitude of the displacement in the dome of the analytical model is 2.12 cm. In addition, Figure 7 shows the failure patterns for both the analytical model and the 1:4 scale model test. When the CS reaches its limit state, the analytical model and the model test experience rupture at the side of the equipment hatch. Generally, similar results derived from the numerical simulation and model test preliminary validate the effectiveness of the analytical model developed in this paper.

3. Failure Criteria

As mentioned earlier, functional failure and structural failure are the typical failure modes of the CS subjected to overpressure loads. Previous studies have primarily employed two categories of failure criteria to assess the functional and structural failure of the CS: stress-based failure criteria and strain-based failure criteria. The stress-based failure criterion defines containment failure based on the stress threshold of its constituent materials (e.g., steel liner, ordinary rebar, or prestressing tendons up to their yield strength) [33,34], while the strain-based failure criterion defines failure in terms of the strain threshold of the material or the overall strain threshold of the structure. Due to the plastic behavior exhibited by the CS subjected to overpressure loads, it is difficult to use the stress-based criterion to accurately predict the failure state. Therefore, the commonly accepted method in nuclear engineering involves the utilization of strain-based failure criteria to predict the failure of the CS under internal pressure. In view of this, this study focuses solely on investigating the typical strain-based failure criteria.

3.1. Functional Failure Criteria

Functional failure is defined as the scenario where the leakage rate of the CS surpasses a specified threshold, resulting in the loss of the sealing function of the CS. Generally, such failure is characterized by the tearing of steel liners in localized areas. The three following functional failure criteria are extensively employed in the relevant literature [26,31,35,36,37,38,39,40,41]:

(1)

Functional failure criterion based on the global free-field strain of the steel liner

According to Regulatory Guide 1.26 from the U.S. Nuclear Regulatory Commission [42], for cylindrical prestressed concrete containments, localized tearing will occur when the strain of the steel liner away from the discontinuity of the CS, referred to as global free-field strain, reaches 0.4%. In this case, the CS is considered to have experienced functional failure. In this paper, this failure criterion is abbreviated to FFC-1.

(2)

Functional failure criterion based on the near-field strain of the steel liner

Dameron et al. [35] developed a criterion for predicting containment leakage by incorporating a series of strain-concentration factors, concluding that tearing will occur when the strain field in the vicinity of the discontinuity of the steel liner, referred to as near-field strain, reaches approximately 1–2%. In this paper, it is reasonable to assume that the functional failure of the CS occurs when the near-field strain of the steel liner reaches 1.5%. Similarly, this failure criterion is abbreviated to FFC-2.

(3)

Functional failure criterion based on the fracture strain of the steel liner

Tearing of the steel liner is mainly determined by its fracture strain. Therefore, Cherry et al. [40] proposed a tearing criterion based on the fracture strain of the steel liner. Similarly, this failure criterion is abbreviated to FFC-3. The FFC-3 criterion adjusts the strain of steel liners derived from FE analysis to its uniaxial equivalent strain through the incorporation of several factors. The tearing of steel liners occurs when the uniaxial equivalent strain surpasses the uniaxial fracture strain of steel liners. For the convenience of application, the tearing criterion was further improved by Spencer et al., and a modified version of the FFC-3 criterion has the following expression [41]:

$${\epsilon }_{\mathrm{p}}={\epsilon }_{\mathrm{pe},\mathrm{FEA}}{f}_{\mathrm{g}}{f}_{\mathrm{cor}}{f}_{\mathrm{ms}}$$

(6)

where ${\epsilon }_{\mathrm{p}}$ is the multiplied factor strain; ${\epsilon }_{\mathrm{pe},\mathrm{FE}}$ is the effective plastic strain of steel liners obtained from FE calculations; $f_{\mathrm{g}}$ is a factor to consider the sophistication of the FE model; $f_{\mathrm{cor}}$ is a factor to account for the effect of steel liner corrosion, and for uncorroded conditions, $f_{\mathrm{cor}}=1$; $f_{\mathrm{ms}}$ is a factor to consider the multi-axial stress state of steel liners, which can be calculated according to the following equation [43]:

$$f_{\mathrm{ms}}=\frac{1.0}{1.648{\mathrm{e}}^{-({\sigma }_1+{\sigma }_2+{\sigma }_3)/2{\sigma }_{\mathrm{vm}}}}$$

(7)

where ${\sigma }_i$ represents the principal stresses; ${\sigma }_{\mathrm{vm}}$ represents the von Mises stress.

Model tests for the CS have demonstrated that the tearing of steel liners typically occurs in the welded region. Therefore, it is necessary to account for the impact of welding in the tearing criterion of Equation (7). Based on the tensile testing of welded specimens, Dameron et al. [44] concluded that welding primarily diminishes the ductility of steel liners, while its impact on the strength of steel liners is negligible. Therefore, a simplified approach involves approximating the welding effect by discounting the uniaxial fracture strain of steel liners in the tearing criterion. Referring to [36,44], a 40% reduction in uniaxial fracture strain in the vicinity of the welds is assumed.

3.2. Structural Failure Criteria

Structural failure means that the CS reached its ultimate bearing capacity. In this case, the main load-resisting materials within the CS (i.e., prestressing tendons and ordinary rebars) fracture, and the structure undergoes rapid depressurization. The three following failure criteria are widely employed to define the structural failure of the CS:

(1)

Structural failure criterion based on free-field strain of the circumferential rebar.

The first structural failure criterion is based on the test results of the 1:4 scale model test of the prestressed concrete containment conducted by SNL. Before the catastrophic rupture of the containment model, the maximum free-field strains of the circumferential and longitudinal rebar strains were 1.4% and 0.3%, respectively [8,45]. In this paper, this failure criterion is abbreviated as SFC-1.

(2)

Structural failure criterion based on the fracture strain of the prestressing tendons

The second structural failure criterion is based on the fracture strain of the load-resisting materials in the CS. As for the prestressed concrete containment subjected to overpressure loads, structural failure will occur when the strain of the prestressing tendons reaches their fracture strain. The fracture strain of the prestressing tendons used in the CS ranged from 2.0 to 7.7%, with a typical value of 4% [8]. However, the 1:4 scale model test conducted by SNL found that the measured strains of the prestressing tendons at the time of structural failure of containment often fell below the fracture strains obtained from the laboratory. Referring to the relevant literature [46,47], a threshold value of 3% is adopted in this paper to define the fracture of prestressing tendons. Correspondingly, this failure criterion is abbreviated as SFC-2.

(3)

Structural failure criterion based on global strain

The third structural failure criterion is based on the global strain of the CS [8,23,24]. The global strain is defined as follows:

$${\epsilon }_{\mathrm{global}}=\Delta v/R$$

(8)

where $\Delta v$ is the radial displacement of the CS, and R is the inner radius of the cylinder wall.

Experimental studies have shown that, under quasi-static pressurization, a fracture of the load-resisting material within the CS occurs rapidly when the global strain reaches 1–2% [27]. Additionally, Ren et al. [23] recommended that a threshold value of 1.5% for the global strain could be used to predict the structural failure of prestressed concrete containments. In this paper, this failure criterion is abbreviated to SFC-3.

4. Failure Behavior of the CS under Various Failure Criteria

4.1. Functional Failure Mode

4.1.1. Load–Displacement Curves of the CS under Functional Failure Mode

Figure 8 shows the relationship between internal pressure and radial displacement at the apex of the dome (point A) of the CS, along with the corresponding deformation distributions in the dome for different functional failure criteria. When the CS experiences functional failure under internal pressure, the dome tends to expand outward. The apex center of the dome exhibits maximum deformation, which gradually decreases from the apex to the ring beam. Additionally, the deformation of the dome corresponding to the three functional failure criteria exhibits notable differences. Specifically, the FFC-1 criterion shows the largest deformation with a radial displacement of 4.5 cm, whereas the FFC-2 criterion exhibits the smallest deformation with a radial displacement of 2.1 cm. The radial displacement of the dome for the FFC-3 criterion is 3.7 cm, which lies between the FFC-1 criterion and the FFC-2 criterion. Moreover, the largest and smallest displacements differ by more than 50%.

4.1.2. Principal Tensile Stress of Concrete and von Mises Stress of Prestressing Tendons under Functional Failure Mode

Figure 9 illustrates the principal tensile stress of the concrete corresponding to different functional failure criteria. The gray area indicates that the principal tensile stress of the concrete exceeds its ultimate tensile strength, leading to cracking. It can be found that, when the CS undergoes functional failure subjected to overpressure loads, the severe cracking of the concrete occurs in the equipment hatch. Meanwhile, varying degrees of cracking are observed in the concrete at the airlock, the bottom of the dome, and the top of the ring beam. The cracking patterns of the concrete corresponding to the FFC-1 and FFC-3 criteria are similar, whereas the degree of concrete cracking is more severe for these criteria compared to the FFC-2 criterion. In addition, Figure 10 depicts the von Mises stress of prestressing tendons corresponding to different functional failure criteria. Under the failure mode defined by the FFC-2 criterion, the circumferential tendons in the vicinity of the equipment hatch reach their yield strength. Comparatively, under the FFC-1 and FFC-3 criteria, there are more circumferential tendons yielding near the equipment hatch, and some of the longitudinal tendons in this area have also reached their yield strength.

4.1.3. Strain of Steel Liner under Functional Failure Mode

Figure 11 depicts the development of strains of steel liners corresponding to various functional failure criteria. When the CS is subjected to internal pressure, significant stress concentration occurs in the steel liner of the equipment hatch, with the maximum strain located near the upper and lower edges of the hole. Furthermore, the strain distributions of steel liners corresponding to the three functional failure criteria exhibit similarity, but the strain levels have significant differences. Specifically, the strain level of steel liners corresponding to the FFC-1 criterion is higher than that of the FFC-2 criterion, while the strain level of steel liners corresponding to the FFC-3 criterion falls between the other two criteria.

4.2. Structural Failure Mode

4.2.1. Load–Displacement Curves of the CS under Structural Failure Mode

Figure 12 shows the curves of internal pressure versus radial displacement at the apex of the dome (point A) of the CS and the corresponding deformation distributions of the dome for different structural failure criteria. The deformation pattern of the dome under structural failure mode is similar to that of the deformation pattern under functional failure mode, but the magnitude of deformation increases further. The radial displacements of the apex of the dome corresponding to the SFC-1, SFC-2, and SFC-3 criteria are 9 cm, 13 cm, and 11.3 cm, respectively. In general, significant differences in dome deformation are observed when applying the three structural failure criteria, and the largest and smallest displacements determined using these criteria differ by approximately 30%.

4.2.2. Principal Tensile Stress of Concrete and von Mises Stress of Prestressing Tendons under Structural Failure Mode

Figure 13 illustrates the principal tensile stresses of the concrete corresponding to the two structural failure criteria. The extensive cracking of the concrete occurs when the CS reaches its ultimate bearing capacity. The cracking patterns observed at the bottom of the dome, the junction area between the ring beam and the cylinder wall, the equipment hatch, the airlock, and the gusset are similar under the SFC-2 and SFC-3 criteria. In contrast, the degree of concrete cracking corresponding to the SFC-1 criterion is slightly less severe. Additionally, Figure 14 depicts the von Mises stress of prestressing tendons corresponding to different structural failure criteria. When the CS undergoes structural failure, most of the prestressing tendons (both circumferential and longitudinal) in the cylinder wall reach their yield strengths. Compared to the failure scenarios corresponding to the SFC-1 and SFC-3 criteria, there are more prestressing tendons yielding within the cylinder wall and dome under the SFC-2 criterion.

5. Overpressure Vulnerability of the CS under Various Failure Criteria

5.1. Structural Vulnerability

5.1.1. Basic Concept of Overpressure Vulnerability

Structural vulnerability denotes the conditional failure probability of an engineering structure upon reaching its limit state under a certain load, which is typically characterized by the vulnerability curve. Within the PSA framework of nuclear power plants, the overpressure vulnerability of the CS assumes paramount importance [21]. It enables the quantitative characterization of the risk associated with the CS reaching a specific limit state from a probabilistic standpoint. Specifically, the overpressure vulnerability analysis aims to quantify the probability of functional failure or structural failure for the CS subjected to different overpressure loads considering the associated randomness and uncertainty.

In the overpressure vulnerability analysis of the CS, the internal pressure load serves as the variable, while the stochastic nature of the mechanical properties of materials used in the CS and the uncertainty of the analytical model constitute the primary factors affecting the results of the vulnerability analysis. The overpressure vulnerability curve is typically described using a lognormal distribution [21,31,47], and the failure probability of the CS at a given internal pressure can be expressed as follows:

$$F_R\left(p\right)=\mathrm{\Phi }\left(\frac{\ln p-\ln P_{\mathrm{m}}}{\sqrt{{\beta }_R^2+{\beta }_M^2}}\right)$$

(9)

where $p$ is the internal pressure; $P_{\mathrm{m}}$ is the mean value of the pressure capacity of the CS taking into account the randomness of the material properties; ${\beta }_R$ represents the logarithmic standard deviation of the pressure capacity of the CS caused by the randomness of the material properties; and ${\beta }_M$ represents the modeling uncertainty parameter, reflecting the discrepancy between the analytical model and the actual CS. According to Li et al. [26], ${\beta }_M$ can be taken as 0.12 for prestressed concrete containments.

5.1.2. Analysis Method for Overpressure Vulnerability of the CS

Due to the complexity of the CS, the overpressure vulnerability analysis of the CS is usually carried out by combining nonlinear FE analysis and Monte Carlo simulation. Specifically, n random variables associated with the material properties of the CS are first sampled to generate k random samples characterizing the component parts of the CS. Nonlinear FE analyses are carried out for each random sample, and the pressure capacity of the CS for the corresponding sample is determined according to the failure criterion, denoted by $P_1,P_2,\dots ,P_k$. Then, statistical analysis is performed on these pressure capacities to determine the statistical parameters of the pressure capacity of the CS, as shown in Equations (10) and (11). Finally, the overpressure vulnerability curve of the CS is determined according to Equation (9).

$$P_{\mathrm{m}}=\exp \left(\frac{1}{k}\sum _{i=1}^k\ln P_i\right)$$

(10)

$${\beta }_R=\sqrt{\frac{1}{k-1}\sum _{i=1}^k{\left(\ln P_i-\ln P_{\mathrm{m}}\right)}^2}$$

(11)

5.1.3. Statistical Characteristics of Material Properties

To account for the effect of the randomness of material properties, the typical mechanical parameters of the material are treated as random variables. Through reviewing the relevant codes and published literature [26,30,31,48,49], the statistical parameters for the mechanical properties of these materials are obtained, as shown in

Table 1. Statistical parameters of the material properties.

Material	Grade	Variable	Distribution	Mean (MPa)	COV
Concrete	C50	fc	Normal	42.9	0.149
		ft	Normal	3.38	0.149
		Ec	Normal	34,500	0.08
Prestressing tendons	1770	fp	Normal	1845.9	0.025
		Ep	Normal	195,000	0.033
Ordinary rebars	HRB400	f400	Normal	435.9	0.05
		E400	Normal	200,000	0.03
	HRB335	f335	Normal	378.6	0.05
		E335	Normal	200,000	0.03
Steel liner	P265GH	fs	Normal	325	0.07
		Es	Normal	200,000	0.03

Note: $f_{400}$, $f_{335}$, and $f_{\mathrm{s}}$ are the yield strength of the rebar and steel liner, respectively. $E_{400}$, $E_{335}$, and $E_{\mathrm{s}}$ are the elastic modulus of the rebar and steel liner, respectively.

. It should be noted that the mean values and coefficients of variation for the concrete are taken from the Chinese code GB50010-2010 [30]. For the sake of simplicity, it is assumed that the mechanical parameters of all materials follow a normal distribution, and the variability of the density and Poisson’s ratio of all materials is ignored. In addition, the Latin hypercube sampling technique is employed in this paper to generate 30 groups of random samples for vulnerability analysis.

5.2. Vulnerability Analysis Results

The pressure capacity of each analyzed sample can be determined based on the failure criteria outlined in Section 3. Figure 15 shows the box charts of the pressure capacities of the CS under different failure criteria. Meanwhile, the statistical parameters of the pressure capacities of the CS are listed in

Table 2. Statistical parameters for the vulnerability analysis results.

Failure Criteria	FFC-1	FFC-2	FFC-3	SFC-1	SFC-2	SFC-3
$P_{\mathrm{m}}$ (MPa)	1.347	1.222	1.314	1.516	1.614	1.574
${\beta }_R$	0.0353	0.0372	0.0346	0.0298	0.0279	0.0272

. Under the functional failure mode, the mean values of the pressure capacities determined by the FFC-1 and FFC-3 criteria are close to each other, while the mean values of the pressure capacities determined by the FFC-2 criterion show a large difference from these two criteria. As for the structural failure mode, the mean values of the pressure capacities determined using the different structural failure criteria also exhibit some differences. Specifically, the pressure capacity determined by the SFC-2 criterion is the largest, and the pressure capacity determined by the SFC-1 criterion is the smallest. The pressure capacity determined by the SFC-3 criterion lies between these two criteria. Furthermore, differences in the logarithmic standard deviation of the pressure capacity obtained under the same failure mode with different criteria are relatively small.

Figure 16 illustrates the overpressure vulnerability curves of the CS under different failure modes. In the case of the functional failure mode, the vulnerability curves determined by the FFC-2 criteria exhibit significant deviation from those determined by the FFC-1 and FFC-3 criteria, and the FFC-2 curve is positioned closer to the left of the coordinate axis. This indicates that the functional failure probability of the CS corresponding to the FFC-2 criterion is the highest with the identical internal pressure. For the structural failure mode, the deviation from the vulnerability curves determined by the three failure criteria is relatively small. Under the same internal pressure, the structural failure probability corresponding to the SFC-1 criterion is the largest.

5.3. HCLPF Capacity of the CS

The HCLPF capacity refers to the internal pressure that corresponds to a 5% failure probability on the vulnerability curve at a 95% confidence level [48]. This indicator is commonly employed to assess the safety margin of the CS against beyond-design basis pressures. The overpressure vulnerability curves for various confidence levels can be obtained with the following equation:

$$F_R(p,\psi )=\mathrm{\Phi }\left(\frac{{\beta }_M\cdot {\mathrm{\Phi }}^{-1}(\psi )+\ln \left(p/P_{\mathrm{m}}\right)}{{\beta }_R}\right)$$

(12)

where ${\mathrm{\Phi }}^{-1}$ is the inverse function of the standard normal distribution function, and $\psi$ is the confidence level.

According to Equation (12), the internal pressure on the overpressure vulnerability curve corresponding to a 5% failure probability can be calculated as follows:

$$\mathrm{HCLPF}=P_{\mathrm{m}}\cdot \exp [-1.645({\beta }_R+{\beta }_M)]$$

(13)

The pressure capacities of CS under various confidence levels and failure criteria are shown in

Table 3. Pressure capacities of the CS under different confidence levels and failure criteria.

Failure Criteria	${\mathit{P}}_{\mathbf{m},5\mathbf{\%}}$ (MPa)	${\mathit{P}}_{\mathbf{m},50\mathbf{\%}}$ (MPa)	${\mathit{P}}_{\mathbf{m},95\mathbf{\%}}$ (MPa)	HCLPF (MPa)
FFC-1	1.106	1.347	1.641	1.043
FFC-2	1.003	1.222	1.489	0.944
FFC-3	1.078	1.314	1.6	1.018
SFC-1	1.244	1.515	1.846	1.184
SFC-2	1.325	1.614	1.967	1.266
SFC-3	1.292	1.574	1.917	1.235

Note: $P_{\mathrm{m},5\%}$, $P_{\mathrm{m},50\%}$, and $P_{\mathrm{m},95\%}$ represent the mean values of the pressure capacities that corresponds to the vulnerability curves with 5%, 50%, and 95% confidence levels, respectively.

. As the confidence level increases, the pressure capacity of the CS decreases for both failure modes. Under the functional failure mode, the HCLPF capacities of the CS determined by the FFC-1, FFC-2, and FFC-3 criteria are 1.043 MPa, 0.944 MPa, and 1.018 MPa, respectively. Taking the HCLPF capacity determined by the FFC-1 criterion as the benchmark, the relative difference between the HCLPF capacity determined by the FFC-1 and FFC-2 criteria is 9.5%, while the HCLPF capacities corresponding to the FFC-1 and FFC-3 criteria are closely aligned. In the case of the structural failure mode, the HCLPF capacities determined by the SFC-1, SFC-2, and SFC-3 criteria are 1.184 MPa, 1.266 MPa, and 1.235 MPa, respectively. Similarly, the relative differences between the HCLPF capacities corresponding to the SFC-1 and SFC-2 criteria are 6.07%, and the relative differences between the HCLPF capacities determined by the SFC-1 and SFC-3 criteria are 4.13%.

5.4. Total Failure Probability of the CS

To comprehensively assess the distinctions among the different failure criteria, the total failure probability (TFP) of the CS subjected to overpressure loads is further compared. The aforementioned vulnerability analysis is conducted with the internal pressure serving as the independent variable, from which the conditional failure probability of the CS under a given internal pressure can be obtained. In fact, the internal pressure of the CS also involves uncertainty, the degree of which is related to factors such as the structural configuration of the CS and the severity of the reactor accident.

The internal pressure of the CS is usually assumed to follow a lognormal distribution. According to the structural reliability theory, the TFP of the CS subjected to overpressure loads can be calculated by the convolution of the probability density function ($f_P(p)$) of the internal pressure with the corresponding overpressure vulnerability curve:

$$\mathrm{TFP}={\displaystyle \underset{R<P}{\iint }{f}_R(r)\cdot f_P(p)drdp}={\displaystyle {\int }_0^{+\infty }{F}_R(p)\cdot f_P(p)dp}$$

(14)

Referring to [26], the mean value and logarithmic standard deviation of the internal pressure for the investigated CS are 0.663 MPa and 0.3, respectively. The TFP of the CS subjected to overpressure loads under different failure modes is shown in

Table 4. TFP of the CS subjected to overpressure loads based on different failure criteria.

Failure Criteria	FFC-1	FFC-2	FFC-3	SFC-1	SFC-2	SFC-3
TFP	0.0147	0.0303	0.0178	0.0055	0.0031	0.0039

. Generally, the TFP of the CS calculated according to the different failure criteria is all less than 0.1, which meets the probabilistic performance objective of the CS in the PSA framework [21]. However, there is a significant distinction in the TFP of the CS as determined using different failure criteria for both failure modes. For example, the relative difference in TFP determined by the FFC-1 and FFC-2 criteria is up to 51.49%, and the relative difference in TFP determined by the SFC-1 and SFC-2 criteria is up to 43.64%.

Combining the aforementioned analysis results, it is evident that the probabilistic assessment results based on the FFC-2 and SFC-1 criteria are the most conservative. However, employing these two failure criteria may lead to an underestimation of the probabilistic safety performance of the CS. For example, the HCLPF capacity, vulnerability curve, and TFP determined by the FFC-2 criterion significantly deviate from those determined using the other two functional failure criteria. If all the investigated failure criteria are considered to have a certain ability to predict the failure of the CS, then it is more reasonable to adopt a moderate failure criterion. After comprehensive comparison, adopting the functional failure criterion based on the fracture strain of steel liners (FFC-3) and the structural failure criterion based on the global strain (SFC-3) for assessing the probabilistic performance of CS subjected to overpressure loads is deemed more reasonable.

6. Conclusions

In this paper, a refined analytical model of the CS is established. The failure behavior of the CS subjected to overpressure loads is compared based on various functional failure criteria and structural failure criteria. Combining nonlinear numerical simulation and probabilistic analysis methods, the vulnerability curves, HCLPF capacities, and total failure probabilities of the CS under internal pressure as determined by different failure criteria are systematically discussed. The key conclusions are as follows:

(1)

When functional failure occurs in the CS, the magnitude of deformation, degree of concrete cracking, and stress and strain levels of the prestressing tendons and steel liner corresponding to the FFC-1 criterion are higher compared to those of the failure scenarios corresponding to the FFC-2 criterion. Meanwhile, the degree of damage associated with the FFC-3 criterion lies between these two criteria. In the case of structural failure, the failure scenarios corresponding to the SFC-2 and SFC-3 criteria exhibit similar characteristics. In contrast, the degree of damage corresponding to the SFC-1 criterion is less severe.

(2)

Vulnerability analysis results indicate variations in the mean value of the pressure capacities of the CS determined by different failure criteria, while differences in the logarithmic standard deviation of the pressure capacities obtained under the same failure mode with different criteria are relatively small.

(3)

Under the functional failure mode, the HCLPF capacity determined by the FFC-1 criterion is the largest, while that determined by the FFC-2 criterion is the smallest, with a relative difference of 9.5%. As for the structural failure mode, the HCLPF capacity determined by the SFC-2 criterion is the largest, while that determined by the SFC-1 criterion is the smallest, with a relative difference of 6.07%.

(4)

In general, there is a significant difference in the TFP of the CS as determined using different failure criteria for both failure modes. If it is considered that all the investigated failure criteria have a certain ability to predict containment failure, then adopting the FFC-3 and SFC-3 for assessing the probabilistic performance of the CS subjected to overpressure loads is deemed more reasonable.