System Reliability Analysis of Slope Stability of Earth Rock Dams Based on Finite Element Strength Reduction Method

Abstract

To overcome the limitations of rigid body limit equilibrium methods in earth rock dam slope stability analysis, this study develops a system reliability framework using the finite element strength reduction method (FEM-SRM). An elastoplastic finite element model simulates dam construction and impoundment, identifying potential slip pathways. Each pathway, treated as a parallel system of shear-failed elements, is analyzed via the response surface method to derive explicit limit state functions. Reliability indices are computed using an improved first-order second-moment method, while interdependencies are assessed through stepwise equivalent linearization. System reliability is determined using Ditlevsen’s narrow bound method. Applied to a 314 m earth rockfill dam, three critical slip pathways were identified: upstream shallow (reliability index is 6.94), upstream deep (reliability index is 6.87), and downstream deep (reliability index is 7.44), with correlation coefficients between 0.26 and 0.89. The system reliability index (6.81) significantly exceeds the code target (4.2), highlighting the method’s ability to integrate material randomness, stress-strain nonlinearity, and multi-slip interactions. This framework provides a robust probabilistic approach for high earth rock dam stability assessment, enhancing engineering safety evaluations.

1. Introduction

Slope stability analysis and evaluation is one of the key issues in earth rock dam design. Currently, the analysis is predominantly based on the single safety factor method derived from the rigid body limit equilibrium approach. This method determines the minimum safety factor using fixed material parameters; if the computed value exceeds the permissible safety factor specified by design codes, the slope stability is deemed acceptable. However, due to the variability of mineral composition, particle gradation, and compaction degree, as well as measurement errors and scale effects in laboratory testing, the shear strength parameters of dam construction materials are inherently uncertain and should be treated as random variables. Therefore, it is essential to assess the impact of uncertainties in dam material strength parameters on the reliability of slope stability.

Several scholars have studied the reliability analysis of earth rock dam slope stability, but their research has primarily focused on the reliability calculation of a single slip mode (i.e., a single slip surface) [1,2,3,4,5,6]. In reality, the slip mode of the slope is also uncertain, and multiple potential slip modes may exist. This necessitates a system reliability analysis of slope stability, but related studies remain scarce. Earth rock dam slopes fall within the broader category of soil slopes, on which significant research on system reliability analysis has been conducted [7,8,9,10,11,12,13]. These studies have addressed the system reliability analysis of slopes with multiple slip modes (slip surfaces), but the calculations were primarily based on the rigid body limit equilibrium method, which mainly considers circular slip surfaces.

The rigid body limit equilibrium method is widely used in practical engineering due to its simplicity and practicality. However, it cannot account for the influence of the nonlinear stress-strain relationship of soil on the internal stress state and stability of the slope, and the assumed slip surface location and shape are predefined rather than derived from the analysis. To address these limitations, Zienkiewicz proposed the finite element strength reduction method (FEM-SRM), which progressively reduces the strength parameters (internal friction angle and cohesion) of geotechnical materials in an elastoplastic finite element framework until the soil mass reaches a limiting equilibrium state. This approach allows for the consideration of the nonlinear stress-strain behavior of soil, eliminates the need to predefine the slip surface location and shape, and provides a means to describe the entire process of slope failure development [14]. Some researchers have applied FEM-SRM to slope and earth rock dam slope reliability analyses [15,16,17], but none have extended it to system reliability analysis.

Due to the use of natural construction materials, the shear strength parameters of earth rock dam bodies exhibit significant randomness, while slope failure mechanisms and slip modes are highly complex. This necessitates a system reliability analysis of slope stability. This study proposes a novel system reliability analysis method for earth rock dam slope stability based on FEM-SRM. The core concept involves first identifying slope slip pathways using FEM-SRM, where each pathway consists of multiple elements undergoing shear failure. Each slip pathway is treated as a parallel system composed of failure elements, and the response surface method is used to compute the reliability of individual failure elements and their correlation coefficients. The stepwise equivalent linearization method is then employed to evaluate the reliability of slip pathways and their interdependencies. Since each slip pathway represents a failure mode of the slope, the overall system reliability of slope stability is determined using a series system composed of slip pathways. In this study, the Ditlevsen narrow bound method is used to compute the system reliability of slope stability.

2. Identification of Slip Pathways in Dam Slope

The failure modes of embankments mainly include seepage failure, landslide instability, structural instability, and overflow failure. Seepage failure is one of the core threats to dam safety, which is manifested in the forms of pipe surge, soil flow, contact erosion, and osmotic collapse. For example, pipe gushing is caused by the continuous erosion of fine particles under the action of seepage to form a penetrating channel, while the flow soil is the overall floating or loss of local soil due to the action of permeability, and this kind of damage often occurs in the heterogeneous soil layer or the weak part of the anti-seepage structure. Landslide instability is mostly caused by steep slopes, insufficient soil strength or external loads (such as sudden drop in water level and earthquakes), which is manifested as a large-scale slide of the dam along the potential sliding surface. Structural instability is associated with material aging, uneven settlement, or overloading, which can lead to crack propagation or even overall collapse. In addition, flooding will directly wash away the downstream slope of the dam, resulting in a sudden increase in the risk of dam failure.

In view of the above problems, a variety of numerical simulation methods are often used in engineering for stability analysis and seepage simulation. As a traditional method, the limit equilibrium method (LEM) evaluates the overall stability of the slope by assuming the sliding surface (such as the arc) and calculating the safety factor, which has the advantage of efficient calculation and few parameter requirements and is widely used in preliminary design, but it cannot consider the details of soil deformation and the coupling effect of seepage. The finite element method (FEM) can simulate complex working conditions more finely, such as analyzing the seepage field distribution through Darcy’s law coupled with the unsaturated seepage model, combining the elastoplastic constitutive model to study the stress-strain response of the dam, and using the Biot theory to realize the fluid–structure interaction analysis of seepage and deformation, so as to reveal the progressive failure mechanism under the sudden drop of water level or seismic load. However, FEM is highly dependent on material constitutive models and computational resources. The discrete element method (DEM) and particle flow method (PFC) focus on the meso-scale and visually demonstrate the microscopic mechanisms, such as soil particle migration and skeleton reconstruction, in the pipe surge process by simulating the mechanical behavior of inter-particle contact, which is suitable for studying the local seepage failure or collapse process. However, its calculation cost is high, and the microscopic parameters need to be calibrated by experiments. In practical engineering, LEM and FEM are often combined to improve efficiency and accuracy (e.g., FEM is used to analyze the seepage field, and then LEM is used to calculate stability), while DEM is mostly used for mechanism studies or the detailed simulation of key parts. In this study, the finite element method was used to calculate the slip channel of the dam slope in combination with a elastoplastic constitutive model of the soil.

The elastoplastic finite element method is based on the mechanics of soil elastoplasticity, overcoming the limitations of the rigid body limit equilibrium method, which cannot account for the nonlinear stress-strain behavior of soil. Additionally, it considers the effects of initial stress, stress pathways, and material anisotropy while accurately modeling complex geometric boundaries, dam structures, and dam construction and reservoir impoundment processes. Consequently, this method provides a more realistic representation of the stress distribution and failure states of soil under loading conditions. The elastoplastic finite element strength reduction method (FEM-SRM) progressively reduces the shear strength parameters of dam materials, causing the dam slope to gradually yield until reaching the ultimate equilibrium state. This process reveals the evolution of slope instability and the final slip mode. In this study, the FEM-SRM is employed to identify slip pathways in earth rock dam slopes. The main steps are as follows:

a.

Establishing a Finite Element Model of the Earth-Rock Dam and Foundation: A global finite element model is constructed for both the dam body and its foundation. The constitutive model for dam materials and the overburden layer of the foundation adopts the ideal elastoplastic model, which is commonly used in engineering applications, with the Mohr–Coulomb criterion employed for yield and failure conditions. The bedrock is modeled using a linear elastic approach. The Newton–Raphson method is used for iterative elastoplastic finite element calculations.

b.

Simulating Construction and Reservoir Operation Conditions: Based on different computational scenarios, the dam filling process, reservoir impoundment, and water level drawdown are simulated to obtain the corresponding stress distributions. These stress states serve as the initial conditions for subsequent strength reduction analyses.

c.

Progressively Reducing Shear Strength Parameters: The shear strength parameters (cohesion and internal friction angle) of dam materials and the foundation’s overburden layer are gradually reduced. During this process, elements with lower strength or higher stress levels fail first. The stress in failed elements is then redistributed to adjacent elements, increasing their stress levels and triggering progressive failure in the slope. When the material strength is reduced to a critical level and the elastoplastic finite element iterations fail to converge, the slope is considered to have reached overall instability. By identifying the elements experiencing shear failure, slip pathways can be determined. These pathways consist of continuous failure zones that penetrate the dam body or extend through both the dam body and foundation. Finally, the element numbers in each slip pathway are recorded, and the reliability of each slip pathway is computed using a parallel system approach.

3. Reliability Calculation of a Single Slip Pathway

3.1. Reliability Calculation of Elements

If a slip pathway of the dam slope consists of m elements in a two-dimensional case, the limit state function for the i-th element can be formulated based on the Mohr–Coulomb strength criterion as follows:

$$Z_i={\displaystyle \frac{{\sigma }_1-{\sigma }_3}{2}}-{\displaystyle \frac{{\sigma }_1+{\sigma }_3}{2}}\sin \varphi -c\cos \varphi \left\{\begin{array}{l}<0, Safe state\hfill \\ =0, Limit state\hfill \\ >0, Failure state\hfill \end{array}\right.\left(i=1,2,\cdots ,m\right)$$

(1)

where σ₁ and σ₃ represent the major and minor principal stresses of the element, respectively, which are obtained as the average values of the stress components at the Gauss integration points of the element. c and ϕ denote the cohesion and internal friction angle of the soil, respectively, which are treated as random variables.

For the rubble materials, the cohesion c = 0, and the internal friction angle φ is a function of the minor principal stress [18,19,20], that is,

$$\varphi ={\varphi }_0-\Delta \varphi \log \left({\displaystyle \frac{{\sigma }_3}{p_a}}\right)$$

(2)

where φ₀ and Δφ are the nonlinear shear strength parameters of the rubble materials, which are random variables; p_a is the standard atmospheric pressure.

The element limit state function in Equation (1) contains stress terms, which need to be obtained through finite element calculations. Additionally, the material strength parameters of the element affect the element’s stress. Therefore, the element limit state function is a random variable, which is an implicit function of the strength parameters c, φ, φ₀, and Δφ. The reliability calculation problem of a limit state function, being an implicit function of random variables, can be solved using the response surface method [21]. This method constructs an explicit function to replace the original implicit function using surface fitting, and the replacement function is referred to as the response surface function. In this study, a quadratic polynomial without interaction terms is used to construct the response surface function of the element’s limit state function, i.e.,

$$Z_i=a_i+{\displaystyle \sum _{j=1}^n{b}_{ij}{X}_j+}{\displaystyle \sum _{j=1}^n{c}_{ij}{X}_j^2}\hspace{1em}\left(i=1,2,\cdots ,m\right)$$

(3)

where X_j (j = 1, 2, …, n) are the random variables; a_i, b_ij, and c_ij are the undetermined coefficients.

The undetermined coefficients in Equation (3) can be determined using the least squares method based on the designed numerical test points and their corresponding function values. The response surface, excluding cross-terms, has been widely applied to practical engineering problems. Excluding cross-terms may reduce fitting accuracy, but the corresponding computational demands for finite element analyses decrease.

After establishing the response surface function of the element’s limit state function, the reliability index of the element and the corresponding design verification point x_i* can be calculated using the improved first-order second-moment method [22].

3.2. The Stepwise Equivalent Linearization Method for Reliability Calculation of a Slip Pathway Composed of Multiple Elements

Slope failure is generally progressive. However for conservative reasons, the dam slope is assumed to fail suddenly, so the elements within each slip pathway are assumed to fail simultaneously. When the elements constituting a slip pathway fail simultaneously, the overall instability of the dam slope increases. Therefore, the reliability calculation of a slip pathway composed of multiple elements essentially involves the reliability calculation of a parallel system. The reliability analysis of parallel systems has long been a challenging issue in structural reliability analysis. In this study, the stepwise equivalent linearization method is employed to calculate the reliability of the parallel system. This method is an approximation but offers high computational efficiency, and its accuracy is generally sufficient for engineering applications. Below is a brief introduction to the basic principles and related formulas of this method.

3.2.1. Equivalent Linear Limit State Function of Elements

The element limit state function Z = g(X₁, X₂, …, X_n) is expanded to the first order using the Taylor series at the design verification point x_i*. Since g(x₁*, x₂*, …, x_n*) = 0, we have

$$Z\approx {\displaystyle \sum _{i=1}^n\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}}\left(X_i-x_i^{\ast }\right)$$

(4)

By replacing the random variables X_i in the above equation with standard normal distributed random variables u_i, we have

$$X_i=u_i{\sigma }_{X_i}+{\mu }_{X_i}$$

(5)

where σ_Xi and μ_Xi are the standard deviation and mean of the random variable after equivalent normalization at the design verification point, respectively.

Substituting Equation (5) into Equation (4) gives us

$$Z\approx {\displaystyle \sum _{i=1}^n\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}}\left(u_i{\sigma }_{X_i}+{\mu }_{X_i}-x_i^{\ast }\right)$$

(6)

According to the definition of the reliability index, we have

$$\beta ={\displaystyle \frac{{\mu }_Z}{{\sigma }_Z}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}}\left({\mu }_{X_i}-x_i^{\ast }\right)}{\sqrt{{\displaystyle \sum _{i=1}^n{\left({\sigma }_{X_i}\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}\right)}^2}}}}$$

(7)

From the above equation, we have

$${\displaystyle \sum _{i=1}^n\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}}\left({\mu }_{X_i}-x_i^{\ast }\right)=\beta \sqrt{{\displaystyle \sum _{i=1}^n{\left({\sigma }_{X_i}\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}\right)}^2}}$$

(8)

Substituting Equation (8) into Equation (6), we obtain

$$Z\approx \sqrt{{\displaystyle \sum _{i=1}^n{\left({\sigma }_{X_i}\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}\right)}^2}}\left({\displaystyle \sum _{i=1}^n{\alpha }_i{u}_i}+\beta \right)$$

(9)

where

$${\alpha }_i={\displaystyle \frac{{\sigma }_{X_i}\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}}{\sqrt{{\displaystyle \sum _{i=1}^n{\left({\sigma }_{X_i}\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}\right)}^2}}}}$$

(10)

Because of the following:

$$P\left(Z=\sqrt{{\displaystyle \sum _{i=1}^n{\left({\sigma }_{X_i}\frac{\partial g}{\partial X_i}{|}_{X_i=x_i^{\ast }}\right)}^2}}\left({\displaystyle \sum _{i=1}^n{\alpha }_i{u}_i}+\beta \right)\le 0\right)=P\left(f={\displaystyle \sum _{i=1}^n{\alpha }_i{u}_i}+\beta \le 0\right)$$

(11)

Thus, we obtain

$$f={\displaystyle \sum _{i=1}^n{\alpha }_i{u}_i}+\beta$$

(12)

where f is the equivalent linear limit state function of the element.

3.2.2. Stepwise Equivalent Linearization of the Joint Failure Boundary of Multiple Elements

The joint failure boundary of elements i and j can be represented by the following system of simultaneous equations:

$$\left\{\begin{array}{c}{f}_i={\displaystyle \sum _{k=1}^n{\alpha }_{ik}{u}_k+{\beta }_i=0}\\ f_j={\displaystyle \sum _{k=1}^n{\alpha }_{jk}{u}_k+{\beta }_j=0}\end{array}\right.$$

(13)

Based on the condition of equal reliability indices, the failure boundary equivalent to the joint failure boundary can be constructed as follows:

$$f_E={\displaystyle \sum _{k=1}^n{\alpha }_{Ek}{u}_k}+{\beta }_E=0$$

(14)

The coefficient α_Ek and the reliability index β_E in the equation must satisfy the following relationship:

$$\left\{\begin{array}{l}{\displaystyle \sum _{k=1}^n{\alpha }_{Ek}^2=1}\hfill \\ {\beta }_E=-{\mathsf{\Phi }}^{-1}\left(\mathsf{\Phi }\left(-{\beta }_i,-{\beta }_j,{\rho }_{ij}\right)\right)\hfill \end{array}\right.$$

(15)

where ρ_ij is the correlation coefficient between elements i and j, which is calculated as follows:

$${\rho }_{ij}={\displaystyle \sum _{k=1}^n{\alpha }_{ik}{\alpha }_{jk}}$$

(16)

The coefficient α_Ek can be determined by the sensitivity of the reliability index β_E to the mean of the standard normal random variable u_k. If a small increment ε is applied to the mean of the k-th random variable u_k (k = 1, 2, …, n) in Equations (13) and (14), the increment Δβ_i of β_i is α_ikε, the increment Δβ_j of βj is α_jkε, and the increment Δβ_E of β_E is α_Ekε. According to Equation (15), Δβ_E can also be determined by the following formula:

$$\Delta {\beta }_E={\mathsf{\Phi }}^{-1}\left(\mathsf{\Phi }\left(-{\beta }_i,-{\beta }_j,{\rho }_{ij}\right)\right)-{\mathsf{\Phi }}^{-1}\left(\mathsf{\Phi }\left(-{\beta }_i-{\alpha }_{ik}\epsilon ,-{\beta }_j-{\alpha }_{jk}\epsilon ,{\rho }_{ij}\right)\right)$$

(17)

Therefore, the calculation formula for α_Ek can be given as

$${\alpha }_{Ek}={\displaystyle \frac{{\mathsf{\Phi }}^{-1}\left(\mathsf{\Phi }\left(-{\beta }_i,-{\beta }_j,{\rho }_{ij}\right)\right)-{\mathsf{\Phi }}^{-1}\left(\mathsf{\Phi }\left(-{\beta }_i-{\alpha }_{ik}\epsilon ,-{\beta }_j-{\alpha }_{jk}\epsilon ,{\rho }_{ij}\right)\right)}{\epsilon }}$$

(18)

The α_Ek calculated from the above formula generally does not satisfy the requirement of the first equation in Equation (15) and needs to be corrected, that is,

$${{\alpha }^{\prime }}_{Ek}={\alpha }_{Ek}/\sqrt{{\displaystyle \sum _{k=1}^n{\alpha }_{Ek}^2}}$$

(19)

At the same time, β_E needs to be corrected as follows:

$${{\beta }^{\prime }}_E={\beta }_E/\sqrt{{\displaystyle \sum _{k=1}^n{\alpha }_{Ek}^2}}$$

(20)

The m elements of the sliding pathway are arranged in increasing order of reliability index. According to Equations (18) to (20), the equivalent failure boundary of the first and second elements is determined. Then, the equivalent failure boundary of this combined failure boundary is determined with the third element’s failure boundary, and so on. Finally, the equivalent failure boundary of multiple elements and the corresponding parallel reliability can be obtained.

Now, a numerical example is used to verify the computational accuracy of the stepwise equivalent linearization method.

Assuming a parallel system consisting of four elements, the known element limit state functions are as follows:

$$\left\{\begin{array}{l}{Z}_1=X_1+2.5\hfill \\ Z_2=X_2+2.5\hfill \\ Z_3=0.2X_1+0.98X_3+3.25\hfill \\ Z_4=0.4X_1+0.9X_2-0.08X_3+0.15X_4+3.25\hfill \end{array}\right.$$

(21)

where X₁, X₂, X₃, X₄ are standard normal random variables.

The element reliability indices and design verification points obtained using the improved first-order second-moment method are shown in

Table 1. Reliability indices for the four elements and associated maximum probable points.

Element Number	Reliability Index	Design Verification Point
		x1*	x2*	x3*	x4*
1	2.5	−2.5	0	0	0
2	2.5	0	−2.5	0	0
3	3.25	−0.65	0	−3.18	0
4	3.25	−1.3	−2.93	0.26	−0.49

.

The element correlation coefficient matrix is as follows:

$$\left[{\rho }_{ij}\right]=\left[\begin{array}{cccc}1& 0& 0.2& 0.4\\ 0& 1& 0& 0.9\\ 0.2& 0& 1& 0\\ 0.4& 0.9& 0& 1\end{array}\right]$$

(22)

Using the dimension reduction method, the reliability index of the parallel system composed of the above four elements is obtained as 5.23 [22]. The reliability index obtained using the stepwise equivalent linearization method is 5.47. The corresponding equivalent linear limit state equation is Z = 0.64X₁ + 0.57X₂ + 0.51X₃ + 0.01X₄ + 5.47.

4. Calculation of the System Reliability of Dam Slope Stability Considering Multiple Slip Pathways

After obtaining the reliability index of each slip pathway of the dam slope ${\beta }_E^i$(i = 1, 2, …, m), the system failure probability of the dam slope can be calculated based on a series system. In this paper, the Ditlevsen narrow bounds method [23] is used to determine the upper and lower bounds of the system failure probability $P_f^s$, and its expression is as follows:

$$P_f^1+{\displaystyle \sum _{i=2}^m\max \left(P_f^i-{\displaystyle \sum _{j=1}^{i-1}{P}_f^{i,j},0}\right)\le P_f^s\le {\displaystyle \sum _{i=1}^m{P}_f^i-{\displaystyle \sum _{i=2}^m\underset{j<i}{\max }\left(P_f^{i,j}\right)}}}$$

(23)

where $P_f^i$ is the failure probability of the i-th slip pathway, $P_f^i=\mathsf{\Phi }\left(-{\beta }^i\right)$ is the joint failure probabilities of the i-th and j-th slip pathways, and $P_f^{i,j}=\mathsf{\Phi }\left(-{\beta }_E^i,-{\beta }_E^j,{\rho }_{ij}\right)$, where ρ_ij is the correlation coefficient between the i-th and j-th slip pathways, which can be determined by the following formula:

$${\rho }_{ij}={\displaystyle \sum _{k=1}^n{\alpha }_{Ek}^i{\alpha }_{Ek}^j}$$

(24)

where ${\alpha }_{Ek}^i$ and ${\alpha }_{Ek}^j$ are the coefficients of the equivalent linear limit state equations for the slip pathway i and slip pathway j.

The reliability index of the dam slope stability system corresponding to the system failure probability $P_f^s$ is as follows:

$${\beta }^s=-{\mathsf{\Phi }}^{-1}\left(P_f^s\right)$$

(25)

5. Case Study

5.1. Dam Section and Finite Element Model

The reservoir dam of a hydropower station is an earth core rockfill dam. The top elevation of the dam is 2510.00 m, and the bottom elevation of the core wall at the riverbed is 2198.00 m. A 2 m thick concrete base is set at the bottom, with the maximum dam height being 314 m. The dam crest is 16.00 m wide, the upstream slope is 1:2.0, and at an elevation of 2430.00 m, a 5 m wide service road is provided. The downstream slope is 1:1.9, and an access road is set along the dam slope.

A two-dimensional finite element model of the typical cross-section of the earth core rockfill dam is established as follows: along the river direction, extending approximately one times the dam height upwards and downstream from the dam base; vertically, it extends about 1.5 times the dam height downward from the core wall base. In the structural discretization, the dam body, dam foundation cover layer, and rock mass are all modeled using quadrilateral isoparametric elements (with a few degraded elements), and the entire calculation domain is divided into 1630 elements and 1626 nodes. The dam body is divided into 451 elements and 546 nodes.

Displacement boundary conditions for the finite element model: the upstream and downstream sides have normal constraints, the bottom boundary has a fixed boundary condition, and the top boundary is free. The typical cross-section of the dam and the finite element model mesh are shown in Figure 1.

5.2. Material Parameters and Basic Random Variables

The constitutive model for the dam construction materials and dam foundation cover layer uses the ideal elastoplastic model, with the yield and failure criteria based on the Mohr–Coulomb criterion; the bedrock is modeled using a linear elastic model. Due to the large geometric dimensions of the dam section as well as the small variability of the upstream reservoir water level and the unit weight of the dam construction materials, these are not considered as random variables. Existing works [24,25] show that if spatial variations are not considered, deformation parameters do not significantly impact slope stability and thus slope reliability. In addition, there is a lack of experimental data on the deformation parameters of the dam materials, and they are also not considered as random variables in this study. This paper primarily considers the random characteristics of the shear strength parameters of the dam construction materials. The physical and mechanical parameters of the dam body and dam foundation cover layer materials, as well as the basic random variables for reliability analysis, are shown in

Table 2. Material parameters and basic random variables.

Material Category	Deterministic Material Parameters				Random Material Parameters
	ρ (t/m3)	ρsat (t/m3)	E (MPa)	ν	φ (Normal Distribution)		φ0 (Normal Distribution)		Δφ (Lognormal Distribution)
					Mean Value (°)	Standard Deviation (°)	Mean Value (°)	Standard Deviation (°)	Mean Value (°)	Standard Deviation (°)
Cover Layer③	2.05	2.25	96.10	0.30	31.00	4.65	──	──	──	──
Cover Layer②	2.03	2.23	81.00	0.30	27.00	4.05	──	──	──	──
Cover Layer①	2.06	2.25	105.00	0.30	31.00	4.65	──	──	──	──
Cofferdam and Surcharge	2.07	2.27	80.00	0.30	36.49	1.82	──	──	──	──
Core Wall with Gravelly Soil	2.10	2.33	44.70	0.30	──	──	41.67	6.13	4.70	1.22
Filter Layer I	2.00	2.25	114.10	0.30	──	──	43.38	2.17	3.74	0.77
Filter Layer II	2.02	2.26	139.60	0.30	──	──	45.83	2.59	5.45	2.19
Transition Layer	2.09	2.29	96.00	0.30	──	──	46.11	2.31	5.04	1.13
Upstream Gravel Fill	2.12	2.33	105.00	0.30	──	──	42.75	2.26	3.59	1.24
Downstream Main Gravel Fill	2.09	2.29	123.40	0.30	──	──	48.15	2.95	6.35	1.41
Downstream Secondary Gravel Fill	2.07	2.27	103.40	0.30	──	──	42.75	2.26	3.59	1.24

. The means and standard deviations of friction angle and cohesion for the dam construction materials are estimated based on triaxial test data and engineering experience.

5.3. Identification of Slope Slip Pathways

Since the main body of the earth core rockfill dam in the case study is constructed with fresh rockfill, which has good drainage performance, rapid drawdown of the water level has a minor impact on the dam slope stability. The earthquakes are inactive near the dam site, and thus seismic loading conditions are not critical for the dam slope stability. Because the dam operates most of the service time under normal reservoir water level conditions for power generation, this working condition is adopted for illustrating the proposed method. The dam construction and reservoir impoundment processes are simulated, and seepage calculation is performed to obtain the seepage force and uplift force acting on the units. These forces are then applied as external loads on the corresponding nodes for stress analysis, resulting in the stress distribution of the dam under normal reservoir water level steady-seepage conditions, which serves as the initial stress state for strength reduction. In this study, the FEM-SRM method is utilized to identify the critical slip surfaces in the dam slope. The FEM-SRM method has been widely applied to reveal the evolution of slope failure and the critical slip modes.

By gradually reducing the shear strength parameters of the dam body materials and foundation cover layer, the main failure zones of the dam slope are identified. When the strength reduction factor K = 1, both the dam body and foundation materials remain within the elastic range, with only a few localized failure elements appearing in the weaker overburden layer. At K = 1.2, the failure zone within the foundation overburden expands and extends in both the upstream and downstream directions. Scattered failure elements also begin to emerge at the interface between the upstream core wall and the filter layer. When K = 1.4, additional failure elements appear in the overburden, as well as at the upstream core–filter interface. Meanwhile, failures begin to occur near the contact surface between the main and secondary rockfill zones in the downstream shell. At K = 1.6, failure elements continue to increase in the foundation overburden and upstream core–filter interface. In the downstream shell, the failure zone further propagates along the interface between the main and secondary rockfill zones, and failure begins to appear in the upstream shell. When K = 1.8, significant failure is observed in both upstream and downstream overburden layers. Damage intensifies near the slope surface of the upstream shell, with internal failure elements becoming nearly continuous. In the downstream shell, a potential slip pathway emerges along the contact between the main and secondary rockfill zones and within the foundation overburden layer. When the strength reduction factor K = 2.0, the elastic-plastic finite element iteration does not converge, and the dam and foundation lose their bearing capacity. The final failure state of the dam body and foundation is shown in Figure 2. The shaded part is the position of the element where shear failure occurs, when the elastic-plastic finite element iterative calculation does not converge and there are three main slip channels, as shown in Figure 2.

From the figure, it can be seen that the dam body and foundation are severely damaged. A failure zone is formed in the upstream dam shell, which starts from the dam crest, passes through the embankment zone, and finally emerges at the top of the upstream ballast, as shown in part 1 of Figure 2. Another failure zone also forms within the upstream dam shell, starting from the top of the core wall, passing through the embankment zone, and finally connecting with the upstream cover layer, as shown in part 2 of Figure 2. At the same time, the downstream dam shell along the main and secondary embankment zones, as well as the sliding failure zone at the foundation cover layer, has also formed, as shown in part 3 of Figure 2. The three critical slip surfaces identified with the FEM-SRM method are in agreement with the engineering judgements, and thus they are likely to be critical for the dam.

5.4. The Reliability and Correlation Coefficients of Slip Pathways

In this study, for constructing the response surface functions of the elements in each slip pathway, 50 finite element analyses were implemented. The coefficients of determination of the constructed response surface functions are all above 0.85, which confirms that ignoring interaction effects does not significantly compromise accuracy. Based on the elements that make up each slip pathway, the reliability indices of the three slip pathways of the dam slope were calculated using the step-by-step equivalent linearization method: the reliability index of the upstream dam slope slip pathway 1 is 6.94, the reliability index of the upstream deep slip pathway 2 is 6.87, and the reliability index of the downstream deep slip pathway 3 is 7.44. The correlation coefficients between the three slip pathways are shown in

Table 3. Correlation coefficients between the sliding pathways.

Sliding Pathway	Upstream Slope	Upstream Deep Layer	Downstream Deep Layer
Upstream Dam Slope	1.00	0.89	0.33
Upstream Deep Layer	0.89	1.00	0.26
Downstream Deep Layer	0.33	0.26	1.00

. From the data listed in the table, it can be seen that among the three slip pathways, the upstream dam slope slip pathway 1 and the upstream deep slip pathway 2 have a high correlation, with a correlation coefficient of 0.89. This is because both slip pathways mainly pass through the upstream embankment area, and their high correlation is reasonable. Since the upstream dam slope slip pathway 1 and the downstream deep slip pathway 3 are located on the upstream and downstream sides of the dam, respectively, their correlation is low, with a correlation coefficient of 0.33. Similarly, the correlation between the upstream deep slip pathway 2 and the downstream deep slip pathway 3 is also low, with a correlation coefficient of 0.26.

5.5. System Reliability Analysis of Dam Slope Stability

Based on the reliability indices and correlation coefficients of each sliding zone, and because the reliability indices of each sliding zone are large (corresponding to a very small failure probability), the difference between the upper and lower bounds of the dam slope system failure probability determined by the Ditlevsen narrow bounds formula is small. Therefore, the average of the upper and lower bounds is used as the dam slope system failure probability, and the corresponding dam slope stability system reliability index is 6.81. Since, in this study, dam slope instability is defined as the simultaneous failure of all units in the sliding zone, according to the “Unified Standard for Structural Reliability Design of Water Conservancy and Hydroelectric Engineering” (GB50199-2013) [26], this type of failure is considered a Category II failure, and the target reliability index for Grade I (important) structures is 4.2. In this example, the reliability index of the dam slope stability system is 6.81, which exceeds the target reliability index required by the standard. When the reliability of the dam slope system does not meet the requirements, relevant measures need to be taken, such as slope weight reinforcement, drainage system optimization, anti-slip structure implantation, slope geometry adjustment, etc. In practical projects, multi-measure collaborative schemes are often adopted, such as the combination of “drainage + ballast + reinforcement”, in conjunction with safety monitoring systems (such as GNSS deformation monitoring and piezometer) to dynamically evaluate the reinforcement effect. This has also been discussed in the literature [27,28].

6. Conclusions

This paper proposes a reliability analysis method for the slope stability system of earth rock dams based on the finite element strength reduction method. The method uses the finite element strength reduction technique to identify the primary sliding zones of the dam slope, treating each sliding zone as a parallel system composed of failure elements. The response surface method is employed to calculate the unit reliability and the correlation coefficients between units. Then, the step-by-step equivalent linearization method is used to calculate the reliability of the sliding zones and the correlation coefficients between the sliding zones. Finally, the Ditlevsen narrow bounds method is used to compute the reliability of the dam slope stability system.

Due to the significant randomness of the dam materials and foundation cover parameters, the mechanism and modes of instability in high earth rock dam slopes are complex and may involve multiple primary sliding modes. Therefore, considering the random uncertainty of material parameters and conducting reliability analysis of the slope stability system based on elastoplastic finite element methods is necessary. This approach can serve as a valuable complement to the deterministic dam slope stability analysis based on rigid limit equilibrium methods. Spatial variations of material properties are common phenomena in natural slopes. For artificially constructed earth rock dams, the densities and gradations of construction materials are controlled, and thus spatial variations of material properties may be not significant. Nevertheless, it is worth exploring random field modeling in future works.