Numerical Investigation of Key Structural Parameters for Middle-Buried Rubber Waterstops

Abstract

Leakage at the lining joints of mountain tunnels is frequent. According to the waterproofing mechanism of waterstops, it is known that the deformation of middle-buried rubber waterstops under stress in typical operating conditions determines their waterproof performance. In addition to the deformation of the adjacent lining concrete, the structural parameters of waterstops are the main factors influencing their deformation under stress. This study combines the common structural components of middle-buried waterstops and considers the bond strength between waterstops and the concrete. A localized numerical model of the lining joint is constructed to explore the impact of geometric parameters, such as hole size, number and position of waterstop ribs, and length and thickness of wing plates on the stress-induced deformation and waterproof performance of the waterstops. The effective mechanisms of different components are revealed, and recommended structural parameters are proposed to further optimize the design of middle-buried rubber waterstops.

1. Introduction

Engineering practice shows that tunnel joint leakage problems are very serious [1]. As a waterproofing measure for joints in a tunnel, the waterproofing effect of the waterstop belt is not ideal. Water leakage will bring great difficulties and hidden dangers to tunnel construction, and it will also have adverse effects on the safety and durability of tunnel lining [2] and the life of facilities in the tunnel [3,4]. At the same time, water leakage will worsen the environment in the tunnel and seriously affect the traffic [5], which is one of the main factors leading to problems during the tunnel operation [6,7]. Therefore, it is very crucial to optimize the design of the existing waterstop belts, solve any tunnel leakage problems, and ensure the safety and long-term stability of tunnels.

Currently, there are various geometric configurations of waterstops available on the market, and different specifications provide differing requirements for waterstop structures. The Chinese national standard “Polymeric Waterproofing Materials Part 2: Waterstops” [8] and the “Technical Specification for Waterstops in Hydraulic Structures” [9] do not explicitly specify the specific dimensions of waterstops but provide corresponding geometric configurations. The railway standard “Waterproofing Materials for Railway Tunnels Part 2: Waterstops” [10] presents specific dimensions and corresponding geometric configurations of waterstops. However, the geometric configurations of waterstops are not explicitly defined in the highway tunnel specifications. Some geometric configurations of waterstops specified in various standards are shown in Figure 1.

Based on the current structural forms of waterstops, they can be divided into three parts: the center hole, the wing plate, and the rib (flat waterstops do not have a center hole), as shown in Figure 2.

Based on the above description, it can be concluded that the current specifications mostly specify the structural form of waterstops, but only the railway tunnel specification provides specific dimensions. In addition, the existing specifications do not provide the design principles for waterstops, making it difficult to determine whether the current geometric designs of waterstops are reasonable. Due to the frequent occurrence of tunnel leakage, and considering the close relationship between the geometric form of waterstops and their installation, waterproofing, and deformation behavior, many researchers have conducted studies on the geometric form of waterstops.

Some researchers [11,12] have studied the stress–strain characteristics of rubber waterstops in immersed tube tunnels and waveform waterstops in concrete dam panel joints using finite element simulation methods. Chen et al. [13] simulated the displacement, stretching, and compression of expansion joint waterstops and analyzed their deformation characteristics and stress distribution.

Lin et al. [14] simulated the deformation of expansion joint waterstops in utility tunnels and analyzed the stress characteristics of these waterstops under different conditions. Furthermore, they analyzed the deformation and stress characteristics of waterstops with different hole sizes and shapes and optimized the structure of existing waterstops.

Meng et al. and Li et al. [15,16] conducted simulations on the tensile and deformation behavior of rubber strips. They analyzed the stress characteristics of waterstops and further investigated the influence of waterstop thickness and the friction coefficient between rubber and concrete on stress distribution. They proposed that increasing the rubber thickness or the friction coefficient between rubber and concrete can effectively reduce the maximum stress in waterstops.

At present, using a numerical method is a popular way to calculate, but there is a need to pay attention to the accuracy of the numerical model [17,18,19]. The aforementioned analyses of rubber waterstops have certain limitations in terms of the contact between rubber and concrete. Scholars have only simply bound the waterstop under study to the concrete lining or applied friction between the waterstop and lining, without considering the adhesive force between the waterstop and concrete. Consequently, the analytical results may not accurately reflect the deformation and stress distribution of waterstops. Moreover, previous studies on waterstops have often focused on individual components and thus were lacking in depth, comprehensiveness, and systematic research.

In this study, we focused on the commonly used middle-buried rubber waterstops in engineering projects. Starting from the waterproofing mechanism of rubber waterstops, we considered the adhesion between waterstops and concrete in our finite element analysis. We conducted research on each component of waterstops and optimized the design based on the influence of the corresponding components on the deformation and stress distribution of waterstops. The aim was to overcome the traditional experience design, improve the rationality of the geometric structure of the waterstop belt, and enhance its waterproof performance.

2. Waterproofing Mechanism of Middle-Buried Rubber Waterstops

Based on previous research [20], it is known that the waterproofing ability of a middle-buried waterstop mainly depends on its water resistance and flow path. As shown in Figure 3, water resistance involves the contact pressure and adhesive force between the waterstop and concrete. Since the contact pressure is significantly smaller than the adhesive force, its influence on the deformation and stress of the waterstop can be disregarded during simulations. The flow path can dissipate hydraulic head pressure and increase the difficulty of groundwater seeping through tunnel joints.

Meanwhile, due to the uncertainty of rock and soil, the tunnel lining will be deformed [21,22,23]. At this time, a waterstop is subjected to uneven settlement [24] or expansion due to the deformation of the lining. The maximum settlement can reach 30 mm, the maximum elongation is 20 mm, and the maximum compression is 10 mm. When deformation is significant, the stress on the waterstop may exceed the critical stress level. Prolonged exposure to high-stress levels can accelerate the deterioration of various mechanical properties of the waterstop and reduce its service life, leading to waterproofing failure before the expected service life is reached.

From this, it can be concluded that there are two main causes of failure for an embedded rubber waterstop: (1) water resistance that is lower than the groundwater pressure, and (2) excessive deformation stress on the waterstop. Therefore, to prevent waterproofing failure, it is necessary to ensure that the deformation stress on a waterstop is kept within a reasonable range while maintaining sufficient water resistance. According to relevant research reports, the stress level for synthetic rubber and other polymer materials should not exceed 20% of their tensile strength at fracture [25], which can result in a service life of over 100 years. Based on current specifications, the tensile strength of rubber waterstops is not less than 10 MPa, and a maximum allowable deformation stress of 2 MPa can be selected to ensure safety.

3. Numerical Model Setups

The real waterstop and concrete used in this study are shown in Figure 4. Based on this, a plane–strain finite element model of a waterstop in concrete lining was established according to the ABAQUS/Standard. Figure 5 shows a mesh diagram of the base model, along with the basic dimensions of the model.

3.1. Constitutive Models of Materials

3.1.1. Constitutive Model of Concrete

The concrete lining strength is considered to be C30, with an elastic modulus (E_c) of 30 GPa and a Poisson’s ratio (ν_c) of 0.2. The concrete damage plasticity model in ABAQUS is used to simulate the failure of the lining. For the damage plasticity parameters, the eccentricity (e_f), the ratio of biaxial to uniaxial compressive strength (f_b0/f_c0), and the coefficient (k) are set to the default values of 0.1, 1.16, and 0.667, respectively. The viscosity parameter (μ_c) is set to 0.005 for implicit solving in ABAQUS. The dilation angle (ψ) is taken as 30°. The stress–strain curves for uniaxial compression and the tension of concrete are referenced from the concrete design code [26], as shown in

Table 1. Uniaxial stress–strain relationship of concrete.

σc	${\mathsf{\epsilon }}_{\mathbf{c}}^{\mathbf{in}}$	σt	${\mathsf{\epsilon }}_{\mathbf{t}}^{\mathbf{in}}$
14.070	0	2.030	0
20.100	0.000802	2.010	0.000028
14.637	0.002456	1.232	0.000149
10.073	0.004080	0.849	0.000257
7.501	0.005638	0.661	0.000359
5.931	0.007162	0.548	0.000458
4.890	0.008668	0.473	0.000556
4.153	0.010165	0.419	0.000653
3.607	0.011655	0.378	0.000749
3.186	0.013141	0.346	0.000846

. Here, σ_c and σ_t represent compressive and tensile stresses, respectively, and ${\mathsf{\epsilon }}_{\mathrm{c}}^{\mathrm{in}}$ and ${\mathsf{\epsilon }}_{\mathrm{t}}^{\mathrm{in}}$ represent non-elastic strains and cracking strains, respectively.

3.1.2. Constitutive Model of Rubber Waterstop

Rubber materials belong to hyperelastic materials, which exhibit significant deformations under certain loads and show dual nonlinearity in terms of material and geometry [27]. For numerical calculation, we need to determine the constitutive equation of the model used [28,29]. The constitutive modeling of rubber materials can be simulated using strain energy functions, including commonly used models such as the Mooney–Rivlin model, the Ogden model, and the Yeoh model [30,31,32] based on continuum mechanics, and the Arruda–Boyce model, Gent model, and Neo Hookean model [33,34,35] based on thermodynamic statistical theory. These models generally require the determination of parameters and model selection based on rubber testing data.

The selection of an appropriate constitutive model for hyperelastic materials is a fundamental issue. Scholars have compared [36] and modified existing models [37,38,39] based on experiments, theoretical analyses, and numerical simulations in order to obtain more accurate models. However, modified models are often specific and complex, making them difficult to apply widely. Some researchers argue that higher-order strain energy functions have limited value since the reproducibility of rubber-like materials is low, making it challenging to estimate a large number of parameters accurately [40]. Among the various constitutive models mentioned above, the Mooney–Rivlin model is the most widely used.

The Mooney–Rivlin model [41] is relatively simple compared to other models. It assumes that rubber materials, under a short period of time and constant temperature, behave as isotropic incompressible materials. The model parameters can be determined based on empirical formulas using rubber hardness [42]. It is widely accepted in the industry that the Mooney–Rivlin model can accurately simulate the mechanical properties of rubber and other hyperelastic materials when the strain does not exceed 150%, providing reasonable approximations compared to actual conditions.

In this study, the rubber waterstop was simulated using the dual-coefficient Mooney–Rivlin model in ABAQUS, and the strain energy function is defined as follows [43]:

W = C₁₀(l₁ − 3) + C₀₁(l₂ − 3)

(1)

In the equation, W represents the strain energy density, l₁ and l₂ are the strain tensor invariants, and C₁₀ and C₀₁ are the material mechanical property constants. Considering the incompressibility of rubber (Poisson’s ratio μ = 0.5), the relationship between the shear modulus G, the elastic modulus E₀, and the material constants can be derived as follows:

G = 2(C₁₀ + C₀₁)

(2)

E₀ = 3G

(3)

E₀ = 6(C₁₀ + C₀₁)

(4)

The elastic modulus E₀ of rubber can be determined based on the rubber hardness H_A, and the relationship is as follows:

E₀ = (15.75 + 2.15H_A)/(100 − H_A)

(5)

According to current specification, the hardness of rubber waterstops for tunnel joints is 55~65. For this simulation, the intermediate value of 60 was chosen for calculation. Based on empirical data, a value of C₁₀/C₀₁ = 0.25 can be used, which gives C₁₀ = 0.484 MPa and C₀₁ = 0.121 MPa.

3.2. Contact Model

For the interface issues of composite materials, it is a crucial problem to choose an appropriate constitutive model and utilize numerical computation methods to effectively predict the nonlinear behavior of the interface [44]. Currently, there is limited research on the bonding strength of rubber–concrete interfaces, both domestically and internationally. However, insights can be gained from studies on interface damage and cracking in other composite materials [45,46]. Among them, cohesive force models based on damage mechanics have significant advantages in addressing interface problems in composite materials, and the techniques for simulating interface damage and crack propagation under static loads are relatively well-developed.

Cohesive force represents the interaction forces between material molecules, and a cohesive force model is a simplified model used to describe the mechanical behavior of interfaces. By appropriately selecting parameters and defining the evolution of damage processes, it is possible to simulate interface cracking accurately. Interfaces in composite materials are held together by cohesive forces, which are related to the relative displacement between the interfaces. Currently, there are various cohesive force constitutive models available for simulation, as shown in Figure 6. Alfano G. conducted comparative calculations on multiple models and ultimately concluded that a bilinear model can meet the requirements of calculation accuracy and efficiency.

Generally, the relative displacement of an interface increases with an increase in external load. At this point, the cohesive force also increases. When the displacement reaches ${\mathsf{\delta }}_{\mathrm{n}}^0$, the cohesive force reaches its maximum value σ₀. After reaching the maximum value, the cohesive force gradually decreases as the displacement continues to increase. When it decreases to zero, the relative displacement reaches its maximum value ${\mathsf{\delta }}_{\mathrm{n}}^{\max }$, indicating the failure of the bond at the interface of the composite material.

In order to determine the initiation displacement and failure displacement of damage, it is necessary to introduce damage initiation criteria and damage evolution criteria for research. Currently, there are four commonly used damage initiation criteria, which are as follows:

(1)

Maximum nominal stress criterion:

$$\max \left\{{<\mathsf{\sigma }}_{\mathrm{n}}{>/\mathsf{\sigma }}_{\mathrm{n}}^0{,\mathsf{\sigma }}_{\mathrm{s}}{/\mathsf{\sigma }}_{\mathrm{s}}^0{,\mathsf{\sigma }}_{\mathrm{t}}{/\mathsf{\sigma }}_{\mathrm{t}}^0\right\}=1$$

(6)

(2)

Maximum nominal strain criterion:

$$\max \left\{{<\mathsf{\delta }}_{\mathrm{n}}{>/\mathsf{\delta }}_{\mathrm{n}}^0{,\mathsf{\delta }}_{\mathrm{s}}{/\mathsf{\delta }}_{\mathrm{s}}^0{,\mathsf{\delta }}_{\mathrm{t}}{/\mathsf{\delta }}_{\mathrm{t}}^0\right\}=1$$

(7)

(3)

Secondary nominal stress criterion:

$${\left\{{<\mathsf{\sigma }}_{\mathrm{n}}{>/\mathsf{\sigma }}_{\mathrm{n}}^0\right\}}^2+{\left\{{\mathsf{\sigma }}_{\mathrm{s}}{/\mathsf{\sigma }}_{\mathrm{s}}^0\right\}}^2+{\left\{{\mathsf{\sigma }}_{\mathrm{t}}{/\mathsf{\sigma }}_{\mathrm{t}}^0\right\}}^2=1$$

(8)

(4)

Secondary nominal strain criterion:

$${\left\{{<\mathsf{\delta }}_{\mathrm{n}}{>/\mathsf{\delta }}_{\mathrm{n}}^0\right\}}^2+{\left\{{\mathsf{\delta }}_{\mathrm{s}}{/\mathsf{\delta }}_{\mathrm{s}}^0\right\}}^2+{\left\{{\mathsf{\delta }}_{\mathrm{t}}{/\mathsf{\delta }}_{\mathrm{t}}^0\right\}}^2=1$$

(9)

In the above equations, σ_n, σ_s, and σ_t represent the stresses in the normal and the two tangential directions; ${\mathsf{\sigma }}_{\mathrm{n}}^0$, ${\mathsf{\sigma }}_{\mathrm{s}}^0$, and ${\mathsf{\sigma }}_{\mathrm{t}}^0$ are the critical onset stresses for damage initiation in each direction; δ_n, δ_s, and δ_t are the strains in each direction; and ${\mathsf{\delta }}_{\mathrm{n}}^0$, ${\mathsf{\delta }}_{\mathrm{s}}^0$, and ${\mathsf{\delta }}_{\mathrm{t}}^0$ are the critical onset strains for damage initiation in each direction.

The maximum nominal stress criterion and the maximum nominal strain criterion both indicate that the interface bond starts to undergo damage when the stress or strain in a particular direction reaches the critical onset value. On the other hand, the secondary nominal stress criterion and the secondary nominal strain criterion indicate that the interface bond starts to undergo damage when a certain value is reached by coupling the stresses or strains in all three directions.

As for damage evolution, it refers to the failure process of the interface bond after damage initiation. It is achieved by introducing a stiffness degradation parameter SDEG, which ranges from 0 to 1. When SDEG is 0, it indicates that the interface bond is intact, while a value of 1 indicates complete failure of the interface bond. For a bilinear constitutive model, the stiffness of the interface bond after damage initiation can be expressed as follows:

$$\mathrm{SDEG}=\left[{\mathsf{\delta }}_{\mathrm{m}}^{\mathrm{f}}\left({\mathsf{\delta }}_{\mathrm{m}}^{\max }{-\mathsf{\delta }}_{\mathrm{m}}^0\right)\left]/\right[{\mathsf{\delta }}_{\mathrm{m}}^{\max }\left({\mathsf{\delta }}_{\mathrm{m}}^{\mathrm{f}}{-\mathsf{\delta }}_{\mathrm{m}}^0\right)\right]$$

(10)

$${\mathrm{Ki}=\mathrm{K}}_1^0\left(1-\mathrm{SDEG}\right)$$

(11)

In Equation (10), ${\mathsf{\delta }}_{\mathrm{m}}^0$ represents the displacement value at the node when the interface bond starts to undergo damage, ${\mathsf{\delta }}_{\mathrm{m}}^{\mathrm{f}}$ represents the displacement value at the node when the interface bond completely fails, ${\mathrm{K}}_{\mathrm{i}}^0$ is the stiffness of the interface bond when it is intact, and K_i is the stiffness of the interface bond after damage initiation. Furthermore, to determine the value of the stiffness degradation parameter SDEG, it is necessary to introduce a damage evolution criterion. Commonly used damage evolution modes include those based on failure displacement or failure energy. Energy-based damage evolution criteria include the power criterion and B-K criterion, all of which can be directly selected in the ABAQUS 2021 finite element analysis software.

3.3. Element Types and Boundary Conditions

The concrete was simulated using CPE4 elements, while the rubber waterstop belt was simulated using CPE4RH elements. Coupling was applied to the concrete lining on both sides using the reference points. During simulation, deformations were applied to the reference points, with the other two directions being fixed when a single-direction deformation was applied.

4. Adhesive Force Test and Numerical Model Verification

4.1. Adhesive Force Test

In summary, in this study, a bilinear cohesive model was used to simulate the bonding interface between rubber and concrete. To obtain the required parameters for the model, rubber–concrete bond strength tests were conducted. The performance of concrete varies greatly with different mix ratio [48,49]. Due to the small size of the test specimens and the main bonding between concrete and rubber being the bonding between cement paste and rubber, cement blocks were used as a substitute for concrete to simulate the bond between concrete and rubber. Ordinary Portland cement with a compressive strength of 42.5 MPa (28 d) was used. The tensile bond strength and shear bond strength of rubber–cement were taken as the evaluation criteria. The rubber was cut into blocks with dimensions of 100 mm × 50 mm × 20 mm, and the cement blocks had the same dimensions as the rubber blocks. Good end surfaces were selected for pouring. After pouring, the specimens were cured in an environment with a temperature of 20 ± 2 °C and a relative humidity of over 95% for 28 days. The specimens were divided into two groups: tension and shear, with three specimens in each group. The test areas for tensile and shear bond strength were rectangular with dimensions of 50 mm × 20 mm and square with dimensions of 50 mm × 50 mm, respectively. The tensile and shear specimens are shown in Figure 7.

The test equipment is shown in Figure 8. The tension machine consisted of four parts: a displacement measuring device, a loading apparatus, a tensiometer, and fixing installation. The measuring range of the tensiometer was 0~1000 N, and the accuracy was 1 N. The test loading took the displacement as the variable, and the accuracy of the loading device was 0.1 mm. Because the shape of the specimen used was rectangular, the adhesive force between rubber and concrete was small, and the bonding interface was brittle; the tensile speed of the selected loading device was 5 mm/min.

The maximum stress sustained by the rubber–cement specimens during the bonding failure caused by external forces is referred to as the tensile or shear strength. The average strength value of the three specimens in each group was taken as the measurement result, as shown in

Table 2. The bonding strength of rubber to concrete.

Conditions	1/MPa	2/MPa	3/MPa	Bonding Strength/MPa
Tensile specimen	0.414	0.387	0.356	0.386
Shear specimen	0.162	0.141	0.149	0.151

. Additionally, the deformation of the specimens during the testing process was recorded. The average deformation for the tensile specimens from start to finish was 7.7 mm, while the average deformation for the shear specimens was 5.9 mm.

Therefore, the contact between rubber and concrete could be set as normal hard contact, while the tangential contact was implemented using the penalty function method with a friction coefficient of 0.3. Additionally, the bonding behavior was modeled as cohesive contact, with a normal stiffness of 3860 MPa/mm, a tangential stiffness of 1510 MPa/mm, a normal cohesive strength of 0.386 MPa, a tangential cohesive strength of 0.151 MPa, and a plastic displacement of 0.0009 mm.

4.2. Numerical Model Validation

To validate the rationality of the Mooney–Rivlin rubber constitutive model and the bilinear cohesive force model, a finite element model of rubber–concrete tension and shear was established using ABAQUS/Standard. The model is shown in Figure 9. The reference points were used to couple the concrete and rubber, with the concrete being fixed and displacement applied to the rubber reference points, while the reaction forces were recorded.

The calculated results are shown in Figure 10, indicating that the bond strength and deformation obtained from the finite element analysis were slightly smaller than the measured values. The calculated tensile strength was 0.339 MPa, and the shear strength was 0.141 MPa, with differences of 12.2% and 6.6% from the measured values, respectively. The calculated tensile deformation was 7.45 mm, and the shear deformation was 5.2 mm, with differences of 3.2% and 11.9% from the measured values, respectively. The discrepancies may be attributed to the deformation of the rubber during simulation, but they were relatively small, and the simulated values were lower than the measured values. Therefore, it can be concluded that using the Mooney–Rivlin rubber constitutive model and the bilinear cohesive model to simulate the bond between rubber and concrete is reasonable.

5. Numerical Analysis of Key Structural Parameters

5.1. Different Parameters of Waterstop Center Hole

5.1.1. Fixed Outer Diameter of the Center Hole

By keeping the outer diameter of the center hole of the waterstop constant (r = 15), different inner diameters of the center hole were used to establish a 2D strain finite element model of buried waterstops in the concrete lining using ABAQUS/Standard, as shown in Figure 11. Tension, compression, and settlement deformations were applied with deformation magnitudes of 20 mm, 10 mm, and 30 mm, respectively.

The calculation results indicated that there was no damage to the concrete lining. The deformation stress of the waterstops is shown in Figure 12, revealing the following trends regarding the influence of the inner hole size:

(1)

Under tensile deformation, the deformation stress initially increases and then decreases with an increase in the aperture size. The maximum stress occurs when the aperture size is R = 3 mm, possibly due to the aperture size being too small, thus resulting in stress concentration during deformation. The deformation stress in all scenarios remains below the critical stress value.

(2)

During compressive deformation, the overall deformation stress decreases with an increase in the aperture size (though stress concentration may occur when 0 < R ≤ 3). However, when the aperture size is less than 5 mm, the deformation stress of the waterstop exceeds the critical stress value.

(3)

During settlement deformation, the deformation stress initially decreases and then increases with an increase in the aperture size. The minimum stress occurs when the aperture size is R = 5 mm, and only in this case is the deformation stress below the critical stress.

In summary, under the condition of a constant outer diameter of the waterstop aperture, increasing the inner diameter of the waterstop aperture primarily improves the stress distribution during compressive deformation. However, care should be taken to avoid having an excessively thin wall for the aperture, which could lead to tearing or puncturing of the waterstop. When designing a waterstop, an inner aperture diameter of R = 5 mm can be chosen.

Taking the tensile deformation of the simulated waterstops as an example, Figure 13 shows the region of bond failure between each waterstop and concrete after deformation for different inner hole diameters while keeping the outer hole diameter constant. Figure 14 shows the adhesive failure range of the waterstop belts with different inner diameters (single-side failure range of waterstops). It can be observed that the bond failure range of the waterstops is approximately the same for different inner hole diameters (the bond failure range for compression deformation and settlement deformation of the waterstops is similar). Therefore, it can be concluded that the inner hole diameter has little effect on the bond between a waterstop and concrete when the outer diameter of the waterstop’s center hole is constant.

5.1.2. Fixed Inner Hole Diameter

Based on the above analysis, to avoid stress concentration in the inner hole of a waterstop, the inner hole diameter should be maintained at a certain size. By keeping the inner hole diameter of the waterstop constant (R = 7 mm), different outer hole diameters were used to establish a plane–strain finite element model of buried waterstops in the concrete lining using ABAQUS/Standard, as shown in Figure 15. The same deformations as mentioned earlier were applied.

According to the calculation results, no damage was observed in the deformation of the concrete lining. The deformation stress of the waterstops is shown in Figure 16. With a constant inner hole diameter of the waterstops, the following could be observed:

(1)

During tensile deformation, the deformation stress generally increases with an increase in the outer diameter. When the outer diameter of the aperture r > 21 mm, the deformation stress exceeds the critical stress threshold.

(2)

During compressive deformation, the deformation stress initially increases and then decreases with an increase in the outer diameter of the aperture. The deformation stress in all scenarios remains below the critical stress level.

(3)

During settlement deformation, the deformation stress shows a notable change with variations in the outer diameter of the aperture. It decreases as the outer diameter of the waterstop aperture increases. When the outer diameter r > 15 mm, the deformation stress is below the critical stress threshold.

In summary, under a constant inner diameter of the waterstop aperture, increasing the outer diameter of the aperture primarily serves to improve the stress distribution during settlement deformation. However, it is also important to prevent excessive tensile deformation stress. When designing a waterstop, an outer aperture diameter of r = 18 mm is recommended.

Similarly to the case of a constant outer hole diameter, when the inner hole diameter of the waterstop is fixed, the size of the outer hole has a minimal impact on the range of adhesive failure after the deformation of the waterstop.

5.1.3. Fixed Thickness of the Center Hole

The thickness of the center hole in the waterstop was kept constant at 8 mm. Different sizes of center holes were selected, and a 2D finite element model of the embedded waterstops in the concrete lining was established using ABAQUS/Standard, as shown in Figure 17. The simulation parameters were the same as described earlier.

The calculation results are shown in Figure 18. It can be observed from the results that, with a constant waterstop aperture wall thickness of 8 mm, the aperture size has a relatively minor effect on tensile deformation stress but a significant impact on compression and settlement deformation stress:

(1)

During tensile deformation, the deformation stresses in all scenarios are below the critical stress value.

(2)

During compression deformation, the deformation stress decreases with an increase in the aperture size. When the aperture outer diameter r ≥ 15 mm, the deformation stress is below the critical stress value (at r = 9 mm, the deformation stress is excessively high, and the complete curve is not shown in the graph).

(3)

During settlement deformation, the deformation stress decreases with an increase in the aperture size. When the aperture outer diameter r > 15 mm, the deformation stress is below the critical stress value.

In conclusion, under the condition of a constant waterstop aperture wall thickness, enlarging the waterstop aperture can effectively improve the stress distribution during waterstop deformation. It is recommended to choose an aperture size with r > 15 mm when designing a waterstop.

Similarly to the previous calculations, when the wall thickness of a waterstop’s hole is constant, the size of the center hole has a minor influence on the range of bond failure after deformation.

5.2. Different Parameters of Waterstop Ribs

5.2.1. Number and Placement of Waterstop Ribs

To analyze the effect of the number and placement of waterstop ribs on the deformation and stress of waterstops, and taking the previous waterstop as an example, the distances from the waterstop ribs to the waterstop orifice and the spacing between the waterstop ribs are shown in Figure 19. The number of ribs on the waterstop was selectively reduced based on their positions, resulting in the creation of eight different waterstop models, as illustrated in Figure 20. ABAQUS/Standard was used to establish the plane–strain finite element models of the embedded waterstop in concrete lining, using the same simulation parameters as mentioned earlier.

Based on the calculation results, no damage was observed in the deformed concrete lining. The deformation stresses of the waterstop are shown in Figure 21. The following could be observed:

(1)

Under tensile deformation, the stress values in all conditions are below the critical stress. Conditions 2-1, 2-2, 2-3, and 2-5 exhibit nearly identical deformation stresses, while conditions 2-4 and 2-6 also exhibit similar deformation stresses. The common factor among these pairs of waterstops with similar deformation stresses is that the position of the first waterstop rib on both sides is the same, with the difference being the number of ribs or the positions of the remaining ribs. Furthermore, conditions 2-5, 2-6, 2-7, and 2-8 demonstrate that as the distance between the first rib on both sides and the waterstop orifice increases, the tensile deformation stress of the waterstop decreases continuously. The difference between the maximum and minimum tensile deformation stresses can reach 233.5%.

(2)

Under compressive deformation, the deformation stresses in all conditions are below the critical stress. Similarly to tensile deformation, the deformation stress of the waterstop is also primarily influenced by the distance between the first rib on both sides and the waterstop orifice.

(3)

Under settlement deformation, when the distance between the first rib on both sides and the waterstop orifice exceeds 60 mm, the deformation stress is below the critical stress. The deformation stress of the waterstop under settlement deformation follows a similar pattern as described above.

Based on the above analysis, it can be concluded that the deformation stress of the waterstop is independent of the number of ribs and is only related to the distance between the first rib on both sides of the waterstop and the hole. The farther the first rib is from the hole, the smaller the deformation stress of the waterstop. When the ribs of the waterstop are closer to the hole, the deformation stress may reach the critical stress. Therefore, when optimizing the design of a waterstop, it is advisable to increase the distance between the first rib on both sides of the waterstop and the hole.

Taking tensile deformation of the waterstop as an example, Figure 22 illustrates the zone of adhesive failure after deformation for different models with varying rib numbers and positions. Figure 23 shows the adhesive failure range of the waterstop belt under different conditions (single-side failure range of the waterstop). It can be observed that the adhesive failure range of the waterstop in each condition is only related to the position of the first rib on both sides of the waterstop. The farther the first rib is from the hole, the larger the range of adhesive failure after deformation. When there are no ribs on the waterstop, the adhesive failure area already includes almost the entire waterstop, indicating a nearly complete loss of adhesion between the waterstop and the concrete due to tensile deformation.

Therefore, it can be concluded that the ribs of the waterstop can effectively prevent the adhesive failure between the concrete and the waterstop, and the failure range slightly exceeds the position of the first rib on both sides. To enhance the anchorage between the waterstop and the concrete and to ensure effective waterproofing, it is recommended to have at least one rib on the waterstop.

5.2.2. Spacing between Waterstop Ribs

When setting the spacing between waterstop ribs, the influence of coarse aggregate size in concrete should be considered. As shown in Figure 24, if the spacing between waterstop ribs is too small and it becomes difficult for coarse aggregates to penetrate, the quality of the concrete between waterstop ribs may be poor, resulting in lower strength and the formation of areas with quality defects. These areas can become weak points in waterproofing.

According to the concrete mix design specifications, for pumped concrete, the minimum size of coarse aggregates should be greater than 5 mm. The maximum aggregate size, when pumping height is less than 50 m, should be less than one-third of the diameter of the conveying pipe, which is generally less than 40 mm. This indicates that the coarse aggregate size for pumped concrete should be between 5 mm and 40 mm. In tunnel construction, the typical gradation of coarse aggregates used is 5–10 mm and 10–25 mm. Therefore, to meet the requirements for concrete quality, it is recommended to set the distance between the ribs of the waterstop to the waterstop or the spacing between the ribs of the waterstop to be greater than 25 mm.

5.3. Different Parameters of Waterstop Wing

5.3.1. Wing Thickness

To investigate the influence of wing thickness on the performance of waterstops, while keeping other parameters constant, different wing thicknesses, denoted as t, were selected. As shown in Figure 25, finite element models of the embedded waterstop in concrete lining were established using ABAQUS/Standard, with the simulation parameters consistent with the previous analysis.

The calculation results are shown in Figure 26. It can be seen that the deformation stress of the waterstop belt increases with an increase in the thickness of the wing plate:

(1)

During tensile deformation, when the thickness of the wing plate is greater than or equal to 14 mm, the tensile deformation stress is greater than the critical stress level.

(2)

During compression deformation, when the flange thickness is greater than or equal to 14 mm, the compression deformation leads to tensile damage in the concrete lining (only partial curves are shown for the cases of t = 14 mm and t = 18 mm). The damage is due to the fact that when the other parameters of the waterstop belt remain unchanged and the thickness of the wing plate increases, the outer contour of the hole in the waterstop belt decreases, the concrete lining joint narrows, and the compression deformation of the hole in the waterstop belt becomes increasingly intense with an increase in the thickness of the wing plate.

(3)

During settlement deformation, the deformation stress for all cases exceeds the critical stress value.

In summary, to reduce the deformation stress of a waterstop and avoid damage to the concrete lining, it is necessary to control the thickness of the waterstop’s flange. However, when the wing thickness is too small, there is a risk of tearing or puncturing the waterstop. According to the specifications for waterstops, the minimum thickness of waterstops should be 4 mm. From the perspectives of safety and waterproofing, it is recommended to design a waterstop with a wing thickness of 4–10 mm.

5.3.2. Wing Length

From the previous simulations, it can be observed that the deformation stress in a waterstop is mainly concentrated between the first pair of waterstop ribs on both sides. This indicates that increasing the length of the wing itself has a minimal impact on its deformation and stress state.

Considering the waterproofing mechanism of waterstops, it is known that a longer flow path can enhance the waterproofing reliability of the joint. The flow path length of the simulated embedded waterstop is determined by the length of the wing and the length of the waterstop ribs, as shown in Figure 27. Therefore, it can be inferred that increasing the length of the wing can result in a longer flow path, thereby improving the waterproofing reliability of the tunnel joint.

However, the embedded waterstop exhibits a cutting effect where the wing plates cut into the concrete lining. The cutting depth is equal to the length of the wing plates, resulting in a reduced integrity of the concrete lining. This creates a defect area that cannot be resolved through the construction process. The defect area is illustrated in Figure 28, showing a significant deficiency in the controllability of concrete quality in the arch area, with a risk of detachment at the ends. Particularly, when the positioning of the embedded waterstop is inaccurate or when deformation occurs, the uneven thickness of the cut concrete layers increases vulnerability to damage and detachment on the thinner side, thus compromising the reliability of the embedded connection between the waterstop and the concrete and leading to waterproofing failure. Moreover, longer wing plates exacerbate the cutting effect on the concrete lining. From this perspective, it is necessary to reduce the length of waterstop wing plates.

In summary, it is important to control the length of a waterstop’s wing plates to minimize the impact of cutting on the concrete lining while maximizing the waterstop’s flow path. However, the existing structure of the embedded waterstop makes it challenging to reconcile the contradiction between the flow path and the cutting effect. To address this issue, it may be necessary to explore alternative waterstop structures.

6. Conclusions

This paper is based on the waterproofing mechanism of an embedded rubber waterstop. It focuses on the structural characteristics of the waterstop and uses numerical methods to analyze the influence of the central hole, waterstop ribs, and wing on the stress and deformation characteristics of the waterstop. The effective mechanisms and the impact of key geometric parameters on the waterproofing performance are revealed, and recommended parameters for each component are proposed. Adopting the recommended parameters in the design of waterstops can reduce deformation stress, extend the service life of waterstops, and reduce the risk of waterproofing failure of waterstops. The main work and conclusions are as follows:

(1)

The bond strength between rubber and concrete was tested, and the feasibility of using the Mooney–Rivlin model and bilinear cohesive force model to simulate the bonding between rubber and rubber–concrete interface was verified.

(2)

The center hole of the waterstop primarily serves to accommodate the deformation of the joint and reduce the deformation stress. In the design of the center hole, increasing the outer diameter and reducing the wall thickness of the center hole can help mitigate excessive deformation stress.

(3)

The waterstop ribs primarily enhance the anchorage between the waterstop and the concrete. As the deformation stress is mainly borne by the first ribs on both sides of the waterstop, the deformation stress of the waterstop is independent of the number of ribs but depends on the distance between the first ribs and the center hole. A greater distance between the first ribs and the center hole results in lower deformation stress. To reduce the deformation stress of the waterstop, it is recommended to increase the distance between the ribs and the center hole, for example, increasing the spacing from 30 mm to over 35 mm. Additionally, to ensure the quality of the concrete between the ribs, it is suggested to have a rib spacing greater than 25 mm.

(4)

The wing of the waterstop serves as a part that is embedded in the concrete to provide fixation. The thicker the wing, the greater the deformation stress. To avoid excessive deformation stress and reduce the risk of tearing or puncturing the waterstop, it is recommended to have a wing thickness of 4–10 mm. Moreover, the wing contributes to the leakage pathway, and a longer length of the wing enhances the waterproofing reliability of the waterstop. However, the wing cuts into the concrete, and the longer the flanges, the more severe the cutting effect on the concrete. The existing waterstop structure has difficulty reconciling the contradiction between the leakage pathway and the cutting effect, thus requiring the proposal of new waterstop structures to address this issue.