## 2 Three-Dimensional Finite Element Modelling and Validation
### Constitutive Model and Parameters  
This study employs the Concrete Damage Plasticity (CDP) constitutive model, initially proposed by [PERSON] [27], to simulate the mechanical behavior of concrete. This model simulates the formation and propagation of microcracks in concrete under load by introducing  
Figure 2: Finite element model. (**a**) Full ring; (**b**) segment.  
a damage variable, thus enabling a more accurate prediction of the mechanical properties and failure process of concrete. Specifically, the Concrete Damage Plasticity (CDP) model divides the stress-strain relationship of concrete into elastic, plastic, and damage phases. In the elastic phase, concrete exhibits linear elastic behavior. As stress increases, concrete enters the plastic phase, beginning to undergo irreversible deformations. Upon reaching a certain stress level, concrete moves into the damage phase, where microcracks start to form and propagate, leading to a gradual reduction in the concrete's strength and stiffness. The choice of the CDP model to simulate the mechanical behavior of concrete was primarily due to its ability to accurately capture the nonlinear behavior and damage evolution of concrete under complex stress states. This is of significant importance for our study of the failure mechanisms of concrete structures, optimization of design, and enhancement of structural safety. Widely used in reinforced concrete structure research, the CDP model comprehensively covers both the tension and compression behavior of concrete, including its damage factors. Figure 3 shows the compressive damage and tensile damage behaviorof the concrete C50. The concrete's stress-strain relationship is determined in line with the Code for Design of Concrete Structures (GB50010-2010) [28]. Additionally, the concrete damage plasticity model's damage factor is calculated based on Sidoroff's energy equivalence principle [29]. During the full-scale model experiments, [PERSON] and [PERSON] [12; 13] tested the compressive strength of concrete, determining it to be 50 MPa. However, the tensile strength of concrete was not assessed. Thus, this study utilized the formula provided by the standards for calculating the tensile strength of concrete from its compressive strength. According to the currently utilized Building Code Requirements for Structural Concrete and Commentary (ACI 318-99) [30] in the United States, the concrete tensile strength (\(f_{\text{t}}\)) can be calculated from the compressive strength (\(f_{\text{c}}\)) using the formula (\(f_{\text{t}}=0.56f_{\text{c}}^{0.5}\)), resulting in \(f_{\text{t}}=3.96\) MPa. Table 1 lists the parameters of CDP model.  
In this model, both the reinforcement and bolts exhibit elastic-plastic bi-polyline behavior, whereas the bolt washers and sleeves are modeled with linear elasticity. The elastic modulus and Poisson's ratio for the reinforcement, bolts, and washers are uniformly 206,000 MPa and 0.3. In contrast, the bolt sleeves have an elastic modulus of 2000 MPa and a Poisson's ratio of 0.35. According to the Code for the Design of Concrete Structures (GB 50010-2010), the standard value of the yield strength and the ultimate strength of HPB400 reinforcement are 400 MPa and 500 MPa, respectively. The grade of bolts is 8.8, of which the yield strength and ultimate strength are 640 MPa and 800 MPa, respectively.