Reliability Analysis of Turbine Blade–Disk Dovetail Joints Considering Failure Correlation

Abstract

The service environment of the turbine blade–disk dovetail joint structure in aero-engines is complex. Uncertainties in material properties and geometry, as well as the failure correlations among multiple locations or components, make reliability assessment challenging. First, a probabilistic life modeling method based on linear heteroscedastic regression is proposed, and the Manson–Coffin probabilistic life models of DD6 and FGH96 alloys at 650 °C are established. Then, the Copula function is introduced to characterize the failure dependence structure, and the effectiveness of the method is verified through numerical examples. Fatigue-critical locations of the dovetail are identified, and a Kriging surrogate model is established to obtain the probabilistic stress distribution at the critical locations. Subsequently, the Copula method is employed to conduct reliability analysis of dovetail structures. The results show that the reliability of multiple dovetails considering correlation lies between that of a single dovetail and that under the assumption of complete independence. Moreover, the life of the entire disk dovetail structure is significantly influenced by the number of dovetails and the required reliability level. Finally, the study is extended to the blade–disk dovetail multi-component system. The results indicate that when correlation is considered, the reliability of both components decreases, and the overall structural life is dominated by the dovetail component with the lower life. The analytical method proposed in this paper provides theoretical support and engineering reference for the reliability design and life assessment of aero-engine rotor structures.

1. Introduction

Turbine blades and turbine disks in aero-engines operate for long periods under conditions of high temperature, high rotational speed, and complex cyclic loads, and which are key components that determine the structural safety and service life of the engine. Due to significant uncertainties associated with material manufacturing processes, geometric dimensions, and other factors, the fatigue life of turbine components usually exhibits considerable statistical dispersion. Consequently, the use of probabilistic methods for life prediction and reliability assessment has become an important approach in the structural design and life management of aero-engines. Early studies on structural reliability were based on random variable theory and probabilistic safety assessment, establishing the fundamental framework for fatigue reliability analysis [1,2,3,4]. With the continuous improvement of aero-engine performance requirements, the reliability issues of critical components such as turbine blades and turbine disks under complex coupled loads and multiple failure modes have become increasingly prominent, and traditional deterministic life evaluation methods can no longer satisfy the requirements for high-reliability and long-life design [5,6,7,8]. Therefore, establishing probabilistic life models that can reflect the randomness of materials and conducting failure correlation analysis at the system level are of great significance for achieving accurate reliability assessment of turbine structures [9,10,11,12].

In terms of probabilistic life modeling, traditional fatigue life analysis methods generally employ S–N curves or strain–life models for deterministic regression. However, numerous experimental results have demonstrated that fatigue life exhibits significant randomness and dispersion [13,14,15,16,17,18]. Wirsching and Torng [19] proposed a fatigue reliability analysis method based on statistical theory, in which fatigue life is treated as a random variable for probabilistic modeling. Pascual and Meeker [20] used statistical regression methods to estimate probabilistic life curves from fatigue test data. Castillo and Fernández-Canteli [21] proposed a unified statistical model to describe the distribution characteristics of metal fatigue life. Subsequently, Mahadevan and co-workers conducted a series of studies on probabilistic fatigue modeling. For example, Zhang and Mahadevan [22] established a probabilistic fatigue life model based on Bayesian methods, and Liu and Mahadevan [23] further proposed a multiscale fatigue reliability analysis framework that incorporates the uncertainty of material microstructures into life prediction models. In addition, Sankararaman and Mahadevan [11] systematically investigated model uncertainty in fatigue life prediction and pointed out that the structure of statistical models has a significant influence on reliability results. However, in practical fatigue test data, the degree of life dispersion often varies with load levels, exhibiting evident heteroscedastic characteristics. Davidian and Carroll [24] systematically investigated parameter estimation problems of heteroscedastic regression models in statistical regression theory, while Carroll and Ruppert [25] proposed the use of weighted regression and transformation methods to address non-constant variance problems. Therefore, introducing a linear heteroscedastic regression model into probabilistic fatigue life modeling can more accurately describe the statistical variation in life distributions with load levels, providing an important theoretical basis for establishing probabilistic life models of turbine blade and turbine disk materials.

On the other hand, in complex structural systems, failures among different components or multiple critical locations are often not mutually independent but exhibit certain statistical correlations [26,27]. Song and Der Kiureghian [28] pointed out in structural reliability analysis that statistical correlations among multiple random variables can significantly influence the calculation of system failure probabilities. Song and Kang [29] proposed system reliability evaluation methods that consider correlations among random variables. For fatigue structural systems, Zhu and Huang [30] conducted studies on fatigue reliability modeling and indicated that statistical dependence may exist among multiple failure modes. To characterize the dependence structure among random variables, Sklar [31] proposed the Copula function theory, which provides a unified framework for modeling the joint distributions of multidimensional random variables. Subsequently, Nelsen [32] and Joe [33] further developed the mathematical theory and application methods of Copula functions. In recent years, Copula functions have been widely applied in the field of engineering reliability. For example, Genest and Favre [34] summarized the application methods of Copula functions in multivariate statistical modeling; Navarro [35] applied Copula functions to system reliability analysis and established a reliability evaluation model considering failure correlations; Serkan [36] further investigated the dependence among multiple failure modes in structural systems using Copula-based approaches. In aero-engine structures, similar failure correlations also widely exist. For instance, multiple teeth at the blade root of turbine blades operate under similar loading and structural conditions, and their fatigue lives may exhibit significant correlations. In blade–disk dovetail joints, failure coupling may also occur between the blade root and the disk slot due to the common load transfer mechanism. Therefore, introducing Copula functions in system reliability analysis to describe the failure correlations among multiple components or locations is of great importance for obtaining more reasonable reliability evaluation results.

To address the above issues, this study conducts probabilistic life modeling and failure-correlated reliability analysis of turbine blade–disk structures. First, a probabilistic life modeling method based on linear heteroscedastic regression is proposed, and probabilistic life models are established for turbine blade materials and turbine disk materials, respectively. Second, Copula functions are introduced to construct the dependence structure among random variables, and a parameter estimation method for Copula models is presented and verified through numerical examples. On this basis, failure-correlated reliability analysis is further carried out for the multi-tooth blade root structure of turbine blades, and the method is extended to the multi-component failure-correlated reliability assessment of turbine blade–disk dovetail structures. The results of this study provide new theoretical methods and engineering references for the reliability design and life prediction of critical rotor structures in aero-engines.

2. Probabilistic Life Modeling Based on Linear Heteroscedastic Regression

2.1. Linear Heteroscedastic Regression

The Manson–Coffin equation, shown in Equation (1), is one of the most widely used models in low-cycle fatigue life analysis for engines. Its expression can be divided into two components: the elastic strain line and the plastic strain line. In this equation, Δε_t denotes the applied strain range, E is the elastic modulus of the material, and N_f represents the fatigue life. The parameters σ’_f, ε’_f, b, c are material constants.

$${\displaystyle \frac{\Delta {\epsilon }_t}{2}}={\displaystyle \frac{\Delta {\epsilon }_e}{2}}+{\displaystyle \frac{\Delta {\epsilon }_p}{2}}={\displaystyle \frac{{{\sigma }^{\prime }}_f}{E}}{(2N_f)}^b+{{\epsilon }^{\prime }}_f{(2N_f)}^c,$$

(1)

The core of the Manson–Coffin equation is that, in a double-logarithmic coordinate system, the fatigue life exhibits a linear relationship with the elastic strain amplitude and the plastic strain amplitude, respectively. In the logarithmic coordinate system, the elastic and plastic lines can be expressed as follows:

$$\left\{\begin{array}{c}\lg (2N_f)={\displaystyle \frac{1}{b}}\lg ({\displaystyle \frac{\Delta {\epsilon }_e}{2}})-{\displaystyle \frac{1}{b}}\lg ({\displaystyle \frac{{{\sigma }^{\prime }}_f}{E}})\\ \lg (2N_f)={\displaystyle \frac{1}{c}}\lg ({\displaystyle \frac{\Delta {\epsilon }_p}{2}})-{\displaystyle \frac{1}{c}}\lg ({{\epsilon }^{\prime }}_f)\end{array}\right.,$$

(2)

In engineering practice, situations are often encountered where the standard deviation varies linearly or can be approximated as linear within a certain range of the independent variable [37]. In such cases, traditional homoscedastic regression analysis is no longer applicable. Therefore, a linear heteroscedastic regression method can be employed to perform a probabilistic treatment of the life model.

For a life model of the following form:

$$\begin{gathered} u~N\left[0,\sigma (x)\right], \\ y=a+dx+u, \end{gathered}$$

(3)

$$\sigma (x)={\sigma }_0\left[1+\theta (x-x_0)\right],$$

(4)

In the above equation, a, d, σ₀, θ, x₀ are parameters to be determined.

Assume that n independent experiments yield the sample data (x₁, y₁), (x₂, y₂), … (x_n, y_n). The estimators of a, b, σ₀ and θ can then be calculated using the following formulas:

$$\widehat{a}=\overline{y}-\widehat{d}\overline{x},$$

(5)

$$\widehat{d}={\displaystyle \frac{l_{xy}}{l_{xx}}},$$

(6)

$${\widehat{\sigma }}_0=\sqrt{{\displaystyle \frac{1}{v}}{\displaystyle \sum _{i=1}^n\frac{{(y_i-\widehat{a}-\widehat{d}{x}_i)}^2}{I^2(x_i,\theta )}}},$$

(7)

$$\widehat{\theta }=l_{yy\theta }+{\displaystyle \frac{l_{xy}^2}{l_{xx}^2}}{l}_{xx\theta }-2{\displaystyle \frac{l_{xy}}{l_{xx}}}{l}_{xy\theta }=0,$$

(8)

In the above equation, v denotes the degrees of freedom of the variance. When θ = 0, v = n − 2, which degenerates into the homoscedastic case; when θ ≠ 0, v = n − 3. The other process parameters are defined as follows:

$$x_0={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_i},$$

(9)

$$I(x_i,\theta )=1+\theta (x_i-x_0),$$

(10)

$$\overline{x}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\frac{x_i}{I^2(x_i,\theta )}}}{{\displaystyle \sum _{i=1}^n\frac{1}{I^2(x_i,\theta )}}}},$$

(11)

$$\overline{y}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n\frac{y_i}{I^2(x_i,\theta )}}}{{\displaystyle \sum _{i=1}^n\frac{1}{I^2(x_i,\theta )}}}},$$

(12)

$$l_{xx}={\displaystyle \sum _{i=1}^n\frac{{(x_i-\overline{x})}^2}{I^2(x_i,\theta )}},$$

(13)

$$l_{yy}={\displaystyle \sum _{i=1}^n\frac{{(y_i-\overline{y})}^2}{I^2(x_i,\theta )}},$$

(14)

$$l_{xy}={\displaystyle \sum _{i=1}^n\frac{(x_i-\overline{x})(y_i-\overline{y})}{I^2(x_i,\theta )}},$$

(15)

$$l_{xx\theta }={\displaystyle \sum _{i=1}^n\frac{{(x_i-\overline{x})}^2(x_i-x_0)}{I^3(x_i,\theta )}},$$

(16)

$$l_{yy\theta }={\displaystyle \sum _{i=1}^n\frac{{(y_i-\overline{y})}^2(x_i-x_0)}{I^3(x_i,\theta )}},$$

(17)

$$l_{xy\theta }={\displaystyle \sum _{i=1}^n\frac{(x_i-\overline{x})(y_i-\overline{y})(x_i-x_0)}{I^3(x_i,\theta )}},$$

(18)

First, let θ₀ be the preliminary estimate of θ. When θ₀ < θ, ${\widehat{\theta }}_0$ > 0; when θ₀ > θ, ${\widehat{\theta }}_0$ < 0. Moreover, θ also satisfies the following condition:

$$-1/(x_{\max }-x_0)<\theta \le 0,$$

(19)

In Equation (19), x_max denotes the maximum value among x_i. Therefore, the bisection method can be conveniently used to determine θ, after which the other unknown parameters can be solved using the obtained value of θ.

2.2. Probabilistic Life Modeling of Turbine Blade Materials

The material used in this study is the DD6 standard-specimen material for low-cycle fatigue testing. Based on standard specimen test data and the linear heteroscedastic regression method, the material parameters of the Manson–Coffin model for [001]-oriented DD6 at 650 °C were obtained, as listed in

Table 1. Material Parameters of the Manson–Coffin Model for [001]-Oriented DD6 at 650 °C.

Parameter	$\raisebox{1ex}{${{\mathit{\sigma }}^{\prime }}_{\mathit{f}}$}\!\left/ \!\raisebox{-1ex}{$\mathit{E}$}\right.$	${{\mathit{\epsilon }}^{\prime }}_{\mathit{f}}$	$\mathit{b}$	c
Value	1.8277	6.6749	−0.0971	−0.6306

.

The comparison between the model predictions and the experimental results is shown in Figure 1.

From the prediction results, it can be seen that the fatigue life predicted by the established Manson–Coffin model falls within a factor-of-two scatter band of the experimental results, indicating high prediction accuracy.

Given that the Manson–Coffin model exhibits high prediction accuracy, it was used to predict the low-cycle fatigue life of DD6 with three different orientations at 650 °C. The corresponding material parameters of the Manson–Coffin model are listed in

Table 2. Material Parameters of the Manson–Coffin Model for DD6 at 650 °C.

Parameter	$\raisebox{1ex}{${{\mathit{\sigma }}^{\prime }}_{\mathit{f}}$}\!\left/ \!\raisebox{-1ex}{$\mathit{E}$}\right.$	${{\mathit{\epsilon }}^{\prime }}_{\mathit{f}}$	$\mathit{b}$	c
[001]	1.8277	6.6749	−0.0971	−0.6306
[011]	0.8448	1.4814	−0.0818	−0.4438
[111]	0.6413	13.901	−0.0881	−0.889

.

After determining that the life model follows the Manson–Coffin equation, the linear heteroscedastic regression method was applied to perform a probabilistic treatment of the low-cycle fatigue life model of DD6. The resulting parameters of the probabilistic Manson–Coffin model are listed in

Table 3. Material Parameters of the Probabilistic Manson–Coffin Model for DD6.

Parameter	$\raisebox{1ex}{${{\mathit{\sigma }}^{\prime }}_{\mathit{f}}$}\!\left/ \!\raisebox{-1ex}{$\mathit{E}$}\right.$	${{\mathit{\epsilon }}^{\prime }}_{\mathit{f}}$	$\mathit{b}$	c
[001]	${10}^{{\displaystyle \frac{2.6979+0.0901u}{10.301+0.2072u}}}$	${10}^{{\displaystyle \frac{1.3074+1.6697u}{1.5858-0.6366u}}}$	${\displaystyle \frac{-1}{10.301+0.2072u}}$	${\displaystyle \frac{-1}{1.5858-0.6366u}}$
[011]	${10}^{{\displaystyle \frac{-0.8955+0.8423u}{12.2251-1.033u}}}$	${10}^{{\displaystyle \frac{0.3846+0.6377u}{2.2533+0.3514u}}}$	${\displaystyle \frac{-1}{12.2251-1.033u}}$	${\displaystyle \frac{-1}{2.2533+0.3514u}}$
[111]	${10}^{{\displaystyle \frac{-2.2125+0.2436u}{11.4673-0.1883u}}}$	${10}^{{\displaystyle \frac{1.2857+0.0131u}{1.1248+0.4421u}}}$	${\displaystyle \frac{-1}{11.4673-0.1883u}}$	${\displaystyle \frac{-1}{1.1248+0.4421u}}$

, where μ follows a standard normal distribution.

A comparison between the established probabilistic model and the experimental data is shown in Figure 2, indicating that the model exhibits good predictive accuracy.

2.3. Probabilistic Life Model for Turbine Disk Materials

The experimental material employed in the present work consists of FGH96 alloy forgings extracted from a turbine disk.

For cases where the mean stress is non-zero, the Manson–Coffin model with Morrow’s correction can be applied:

$${\displaystyle \frac{\Delta \epsilon }{2}}={\displaystyle \frac{{{\sigma }^{\prime }}_f-{\sigma }_m}{E}}{\left(2N_f\right)}^b+{{\epsilon }^{\prime }}_f{\left(2N_f\right)}^c,$$

(20)

where σ_m denotes the mean stress.

The parameters for this model can be obtained from uniaxial low-cycle fatigue (LCF) test results using the following procedure:

(1)

Determination of Young’s modulus E: The Young’s modulus of the material at the specific test temperature can be retrieved from material property handbooks.

(2)

Reformulation of the Manson–Coffin life model: The Manson–Coffin model can be reformulated as follows:

$$\ln \left({\displaystyle \frac{\Delta {\epsilon }^e}{2}}\right)=\ln \left({\displaystyle \frac{{{\sigma }^{\prime }}_f}{E}}\right)+b\ln \left(2N_f\right),$$

(21)

$$\ln \left({\displaystyle \frac{\Delta {\epsilon }^p}{2}}\right)=\ln \left({{\epsilon }^{\prime }}_f\right)+c\ln \left(2N_f\right),$$

(22)

In Equations (21) and (22), Δε^e and Δε^p denote the elastic strain range and plastic strain range, respectively. By utilizing the LCF data, the relationship between strain and fatigue life can be established, from which the parameters σ’_f, ε’_f, b and c are determined.

According to the aforementioned procedure, the identified material parameters are summarized in

Table 4. Material parameters of the Manson–Coffin model for FGH96 at 650 °C.

Parameter	$\raisebox{1ex}{${{\mathit{\sigma }}^{\prime }}_{\mathit{f}}$}\!\left/ \!\raisebox{-1ex}{$\mathit{E}$}\right.$	${{\mathit{\epsilon }}^{\prime }}_{\mathit{f}}$	$\mathit{b}$	c
Value	2.2852	1.9294	−0.1615	−0.4035

.

A comparison between the simulated results and experimental data is presented in Figure 3.

From the prediction results, it can be observed that the fatigue life predicted by the established Manson–Coffin model lies within the factor-of-two scatter band of the experimental results, indicating high prediction accuracy.

After determining that the life model follows the Manson–Coffin formulation, the model is probabilistically characterized using linear heteroscedastic regression. The resulting parameters of the probabilistic Manson–Coffin model are listed in

Table 5. Material Parameters of the Manson–Coffin Model for FGH96.

Parameter	$\raisebox{1ex}{${{\mathit{\sigma }}^{\prime }}_{\mathit{f}}$}\!\left/ \!\raisebox{-1ex}{$\mathit{E}$}\right.$	${{\mathit{\epsilon }}^{\prime }}_{\mathit{f}}$	$\mathit{b}$	c
650 °C	${10}^{{\displaystyle \frac{0.5546+0.7456\mu }{10.7187-0.2976\mu }}}$	${10}^{{\displaystyle \frac{2.3834-0.1592\mu }{1.3281+0.4583\mu }}}$	${\displaystyle \frac{-1}{10.7187-0.2976\mu }}$	${\displaystyle \frac{-1}{1.3281+0.4583\mu }}$

, where μ follows a standard normal distribution. The comparison between the established probabilistic model and the experimental data is shown in Figure 4, demonstrating that the model also exhibits good predictive accuracy.

3. Failure Correlation Analysis Based on Copula Functions

3.1. Copula Functions and Measures of Dependence

The joint distribution function of multivariate variables incorporates both the marginal distribution characteristics of individual variables and the dependence information among variables, and serves as a mathematical model for describing the probabilistic properties of variables [38]. The n-dimensional copula function C(u₁, u₂, …, u_n) has the following properties:

(1)

The function CCC satisfies the following mapping relationship: ${[0,1]}^n\to [0,1]$

(2)

The marginal distributions of C(u₁, u₂, …, u_n) have uniform distribution boundaries on [0,1], that is, u_i = F_i(t_i), i = 1, 2, …, n, where F_i(t_i) is a continuous distribution function;

(3)

For any $u_i\in [0,1],i=1,2,\dots ,n$, the copula function C satisfies:

$$\begin{array}{l}C(u_1,u_2\dots u_{i-1},0,u_{i+1}\dots ,u_n)=0\\ C(1,1\dots 1,u,1\dots ,1)=0\end{array},$$

(23)

(4)

The copula function C is an n-dimensional increasing function with well-defined upper and lower bounds, namely the Fréchet bounds, satisfying

$$\begin{array}{c}W(u_1,u_2,\dots ,u_n)\le C(u_1,u_2,\dots ,u_n)\le M(u_1,u_2,\dots ,u_n)\hfill \\ W(u_1,u_2,\dots ,u_n)=\max (u_1+u_2+\dots +u_n-1,0)\hfill \\ M(u_1,u_2,\dots ,u_n)=\min (u_1,u_2,\dots ,u_n)\hfill \end{array},$$

(24)

(5)

If the variables in the copula function C are mutually independent, a product copula exists, satisfying

$$C(u_1,u_2,\dots ,u_n)=u_1\cdot u_2\cdot \dots \cdot u_n,$$

(25)

According to Sklar’s theorem [39], suppose that F_c(t₁, t₂, …, t_n) is the n-dimensional joint cumulative distribution function (CDF) of the variables t₁, t₂, …, t_n. The marginal distributions of t₁, t₂, …, t_n are continuous functions F₁(t₁), F₂(t₂), …, F_n(t_n) (hereafter abbreviated as F₁, F₂, …, F_n). Then there exists a unique copula function C such that

$$F_c(t_1,\dots ,t_n)=C\left(F_1(t_1),\dots ,F_n(t_n)\right),$$

(26)

The joint probability density function (PDF) can be calculated as

$$\begin{array}{c}{f}_c(t_1,t_2,\dots ,t_n)={\displaystyle \frac{{\partial }^n{F}_c\left(t_1,t_2,\dots ,t_n\right)}{\partial t_1\partial t_2\dots \partial t_n}}={\displaystyle \frac{{\partial }^{n}C\left(F_1,F_2,\dots ,F_n\right)}{\partial t_1\partial t_2\dots \partial t_n}}\\ =c\left(F_1,F_2,\dots ,F_n\right)\times {\displaystyle \prod _{i=1}^n{f}_i\left(t_i\right)}\end{array},$$

(27)

where c (F₁, F₂, …, F_n) is the probability density function of the copula function C(F₁, F₂, …, F_n), and f_i (x_i) denotes the PDF of x_i.

Conversely, given continuous marginal distribution functions F₁, F₂, …, F_n and a copula function C, F(t₁, t₂, …, t_n) represents the joint CDF of the marginal distributions, and for any $u=(u_1,\dots ,u_n)\in {[0,1]}^n$, we have

$$C(u_1,u_2,\dots ,u_n)=F_c(F_1^{-1}(u_1),F_2^{-1}(u_2),\dots ,F_n^{-1}(u_n)),$$

(28)

Here, $F_i^{-1}$ denotes the inverse function of F_i.

For variables with continuous CDFs, the marginal PDFs and the joint PDF can be studied separately, with the dependence structure represented by constructing a suitable copula function. Copula functions allow the marginal distributions and the dependence among variables to be analyzed independently, which reduces the computational complexity of multivariate probabilistic models.

Under operating conditions, centrifugal forces, aerodynamic loads, and thermal loads act simultaneously on the blade and disk, creating a correlated structure at the blade dovetail-disk connection where loads are transferred. Meanwhile, material processing techniques result in symmetrically distributed dimensional deviations. The Gaussian Copula describes a symmetric, asymptotically independent correlation structure.

In the field of mechanical engineering, typical copula functions include elliptical copulas, Archimedean copulas, and mixed copulas.

(1)

Elliptical Copula Functions

Among elliptical copulas, the most commonly used is the Gaussian copula, whose cumulative distribution function (CDF) is given by

$$C(u_1,u_2;{\theta }_c)={\displaystyle {\int }_{-\infty }^{{\Phi }^{-1}(u_1)}{\displaystyle {\int }_{-\infty }^{{\Phi }^{-1}(u_2)}\frac{1}{2\pi \sqrt{1-{\theta }_c^2}}\exp \left[-\frac{r^2-2{\theta }_{c}rs+s^2}{2(1-{\theta }_c^2)}\right]dr}ds},$$

(29)

The probability density function (PDF) is

$$\begin{array}{l}c(u_1,u_2;{\theta }_c)=\frac{1}{\sqrt{1-{\theta }_c^2}}\exp [-\frac{{\xi }_1^2-2{\theta }_c{\xi }_1{\xi }_2+{\xi }_2^2{\theta }_c^2}{2(1-{\theta }_c^2)}];\\ {\xi }_1={\mathrm{\Phi }}^{-1}\left(u_1\right);{\xi }_2={\mathrm{\Phi }}^{-1}\left(u_2\right)\end{array},$$

(30)

where θ_c ∈ [−1, 1] is the correlation parameter. When θ_c = 0, u₁ and u₂ are mutually independent; when θ_c = 1 or −1, the variables are perfectly positively correlated or perfectly negatively correlated, respectively. By setting θ_c = 0.4, the PDF distribution and scatter plot of the function are obtained, as shown in Figure 5. It can be observed that the Gaussian copula function exhibits a clear symmetric dependence structure.

(2)

Archimedean Copula Functions [40]

Archimedean copula functions have the advantages of simple form, strong symmetry, and good associativity. At present, four main types of Archimedean copula functions have been developed, namely the product copula, Clayton copula, Gumbel copula, and Frank copula. They share the following unified form:

$$C(u_1,\dots ,u_n;{\theta }_c)={\psi }^{-1}\left[\psi (u_1)+\dots +\psi (u_n)\right],$$

(31)

where ψ is the Archimedean generator function, and ψ⁻¹ is the inverse function of ψ.

Common forms of bivariate Archimedean copula functions are listed in

Table 6. Four Common Types of Archimedean Generator Functions (Bivariate Case).

Copula Type	Range of θc	ψ(t)	ψ − 1(s)	C(u1, u2; θc)
Product	/	$-\ln t$	$e^{-s}$	$u_1\cdot u_2$
Clayton	θc ≥ 0	$t^{-{\theta }_c}-1$	${\left(1+s\right)}^{-1/{\theta }_c}$	${(u_1^{-{\theta }_c}+u_2^{-{\theta }_c}-1)}^{-1/{\theta }_c}$
Gumbel	θc ≥ 1	${\left(-\ln t\right)}^{{\theta }_c}$	$\exp \left(-s^{1/{\theta }_c}\right)$	$\exp \left\{-{\left[{(-\ln u_1)}^{{\theta }_c}+{(-\ln u_2)}^{\theta }\right]}^{1/{\theta }_c}\right\}$
Frank	${\theta }_c\in$ R	$-\ln {\displaystyle \frac{e^{-{\theta }_{c}t}-1}{e^{-{\theta }_c}-1}}$	$-{\theta }_c^{-1}\ln (1+e^{-s}(e^{-{\theta }_c}-1))$	$-{\displaystyle \frac{1}{{\theta }_c}}\ln \left(1+{\displaystyle \frac{(e^{-{\theta }_c{u}_1}-1)(e^{-{\theta }_c{u}_2}-1)}{e^{-{\theta }_c}-1}}\right)$

. Among them, the product copula is used to represent the dependence characteristics of independent variables, and therefore does not contain a dependence parameter θ_c.

• Clayton Copula Function: When θ_c = 0, the variables u₁ and u₂ are mutually independent. As θ_c increases, the dependence between u₁ and u₂ becomes stronger. By setting θ_c = 2.5, the probability density distribution and scatter plot of the function are obtained, as shown in Figure 6. It can be observed that the Clayton copula function exhibits a high lower tail and a low upper tail, indicating significant lower-tail dependence.
• Gumbel Copula Function: When θ_c = 1, the variables u₁ and u₂ are mutually independent. As θ_c increases, the dependence between u₁ and u₂ becomes stronger. By setting θ_c = 2.5, the probability density distribution and scatter plot of the function are obtained, as shown in Figure 7. It can be observed that the Gumbel copula function exhibits a high upper tail and a low lower tail, indicating significant upper-tail dependence.
• Frank Copula Function: When θ_c = 0, the variables u₁ and u₂ are mutually independent. When θ_c > 0, the variables are positively correlated, and the dependence between u₁ and u₂ increases as θ_c increases. When θ_c < 0, the variables are negatively correlated, and the dependence between u₁ and u₂ becomes stronger as θ_c decreases. By setting θ_c = 10, the density distribution and scatter plot of the function are obtained, as shown in Figure 8. It can be observed that the Frank copula function is symmetric in the upper and lower tails, making it suitable for describing symmetric dependence between variables.

(3)

Mixed Copula Functions

Different types of copula functions can describe different characteristics of dependence structures among variables. In engineering practice, an appropriate copula function or mixed copula function can be selected according to the dependence characteristics of variables to quantify their correlation. The expression of the mixed copula function is given by

$$C(u_1,\dots ,u_n;{\theta }_{c,1},\dots ,{\theta }_{c,k})={\displaystyle \sum _{k=1}^d{\lambda }_k{C}_k(u_1,\dots ,u_n;{\theta }_{c,k})},$$

(32)

where C_k(u₁, …, u_n; θ_c,k) denotes an individual copula function, d represents the number of individual copula functions, and λ_k and θ_c,k denote the weight and dependence parameter of each copula function, respectively. A mixed copula function constructed from the Clayton copula function C₁ and the Gumbel copula function C₂ can be expressed as

$$C(u_1,u_2)=0.3\cdot C_1(u_1,u_2;{\theta }_{c,1})+0.7\cdot C_2(u_1,u_2;{\theta }_{c,2}),$$

(33)

When θ_c,1 = 2.5 and θ_c,2 = 4, the corresponding probability density distribution and scatter plot of the function are shown in Figure 9.

3.2. Parameter Estimation of Copula Functions

Typical distributions such as the normal distribution, lognormal distribution, and Weibull distribution are used to describe the marginal probability density functions (PDFs) of variables. The parameters of these distributions can be estimated using the maximum likelihood estimation (MLE) method [41]. Taking the three-parameter Weibull distribution as an example, assume that the PDF of the i-th variable follows a Weibull distribution, which can be expressed as

$$f_i\left(t\right)={\displaystyle \frac{{\tau }_i}{{\eta }_i^{{\tau }_i}}}\cdot {\left(t-{\gamma }_i\right)}^{\left({\tau }_i-1\right)}\cdot \exp \left(-{\left({\displaystyle \frac{t-{\gamma }_i}{{\eta }_i}}\right)}^{{\tau }_i}\right),$$

(34)

The cumulative distribution function (CDF) and reliability function of the variable can be expressed as

$$F_i\left(t\right)=1-\exp \left(-{\left({\displaystyle \frac{t-{\gamma }_i}{{\eta }_i}}\right)}^{{\tau }_i}\right),$$

(35)

$$R_i\left(t\right)=\exp \left(-{\left({\displaystyle \frac{t-{\gamma }_i}{{\eta }_i}}\right)}^{{\tau }_i}\right),$$

(36)

where γ_i, τ_i, and η_i denote the location, shape, and scale parameters of the Weibull distribution, respectively.

Let ξ_i represent the parameter vector of the Weibull distribution, i.e., ξ_i = (γ_i, τ_i, η_i)^T.

Assume that the i-th variable t_i has mmm sample observations t_i,1, t_i,2, …, t_i,m. The likelihood function of ξ_i is

$$L\left({\alpha }_i\right)=L\left(t_{i,1},t_{i,2},\dots ,t_{i,n},{\mathit{\xi }}_i\right)={\displaystyle \prod _{j=1}^m{f}_i\left(t_{i,j},{\mathit{\xi }}_i\right)},$$

(37)

where f_i(t) is calculated using Equation (34). The estimate of ξ_i is obtained by selecting the value of ${\overline{\mathit{\xi }}}_i$ that maximizes the likelihood function L, i.e.,

$$L\left({\overline{\mathit{\xi }}}_i\right)=\max L\left({\xi }_i\right),$$

(38)

Assume that under the same condition, the n-dimensional variables have sample points t_i,1, t_i,2, …, t_i,m, i = 1, …, n. After calculating the distribution parameters ${\overline{\mathit{\xi }}}_i$, i = 1, …, n of the n variables using Equation (38), the likelihood function of the dependence parameter θ_c can be expressed as

$$L\left({\theta }_c\right)=L\left(\underset{j=1,\dots ,m}{\underbracket{t_{1j},t_{2j},\dots ,t_{nj}}},{\theta }_c\right)={\displaystyle \prod _{i=1}^{n}f\left(t_{i1},t_{i2},\dots ,t_{im},{\overline{\mathit{\xi }}}_i,{\theta }_c\right)},$$

(39)

where $f\left(t_{i1},t_{i2},\dots ,t_{im},{\overline{\mathit{\xi }}}_i,{\theta }_c\right)$ denotes the likelihood function of the i-th variable. The estimate of the dependence parameter ${\overline{\theta }}_c$ can be obtained by MLE, satisfying

$$L\left({\overline{\theta }}_c\right)=\max L\left({\theta }_c\right),$$

(40)

The selection of the copula function affects the fitting accuracy. After obtaining the likelihood function value of the dependence parameters, the Akaike Information Criterion (AIC) [42] can be used to determine the most appropriate copula function. The expression for AIC is

$$\mathrm{AIC}=-2\log L({\overline{\theta }}_c)+2n_c,$$

(41)

where $\log L({\overline{\theta }}_c)$ denotes the log-likelihood estimate obtained from Equation (40), and n_c represents the number of independent parameters in the model. A smaller AIC value indicates a better model fit. Specifically, when the fitting performances of two models differ significantly, Equation (41) is mainly determined by the first term. A smaller AIC implies a larger log-likelihood value and therefore a better model fit. When the fitting performances of two models are similar, the log-likelihood values are close, and Equation (41) is mainly determined by the second term. In this case, the model with fewer independent parameters is preferred.

3.3. Numerical Example Analysis

Consider a series system with two failure modes and perform correlation analysis of the bivariate variables. The limit-state functions of the failure modes are

$$g_1\left(x_1,x_2\right)=3\sqrt{2}-x_1-x_2,$$

(42)

$$g_2\left(x_1,x_2\right)=2-x_2-\exp \left(-{\displaystyle \frac{x_1^2}{10}}\right)+{\left({\displaystyle \frac{x_1}{5}}\right)}^4,$$

(43)

where g₁ and g₂ share the same input variables x₁ and x₂, resulting in failure dependence.

Assume that the input variables x₁ and x₂ both follow standard normal distributions. First, 2000 sets of samples are generated using the Monte Carlo (MC) sampling method. Substituting these samples into the limit-state functions g₁ and g₂ yields 2000 sets of results. The scatter plot and frequency histogram of the limit-state function values are shown in Figure 10, from which a significant correlation between the two failure modes can be observed.

Next, the marginal PDFs of g₁ and g₂ are fitted. According to Equations (42) and (43), g₁ follows a normal distribution, and g₂ is also assumed to follow a normal distribution. The estimated parameters of the marginal PDFs for g₁ and g₂ are listed in

Table 7. Parameter estimation results of the marginal PDFs for g₁ and g₂.

Edge Function	Mean	Standard Deviation
g1	4.208	1.427
g2	1.059	1.009

. After standardizing the data, the four types of copula functions mentioned above are used to analyze the dependence structure. The dependence parameter θ_c is estimated using MLE, and the AIC values are obtained, as shown in

Table 8. Comparison of copula function fitting results.

Function Types	Gaussian Copula	Clayton Copula	Gumbel Copula	Frank Copula
Interrelating Parameter θc	0.7022	1.2422	1.9220	5.7759
L	−679.5	−493.3	−676.1	−646.8
AIC	−1357	−985	−1292	−1350

. As indicated in Table 8, the Gaussian copula function has the smallest AIC value (bolded), and therefore the Gaussian copula is selected to describe the dependence structure.

Using the Gaussian copula function and the Monte Carlo simulation (MCS) method, the predicted scatter plots of the system performance function g are obtained, as shown in Figure 11a. It can be observed that the prediction results obtained by the two methods exhibit a high degree of overlap. Furthermore, the comparison of the predicted results of g under the two methods is shown in Figure 11b, where the correlation coefficient reaches 0.9997, demonstrating the high predictive accuracy of the Gaussian copula-based method.

4. Reliability Analysis of Multi-Tooth Failure in Turbine Blade Fir-Tree Dovetail Structures

4.1. Parametric Model of the Turbine Blade Dovetail Structure

The turbine blade root adopts a fir-tree configuration, and its geometric structure is illustrated in Figure 12. The mass information of a single blade is listed in

Table 9. Mass and Centroid Information of the High-Pressure Turbine Blade.

Name	Single Mass	Single Centroid Radial Height
High Pressure Turbine Blade	200 g	265.9 mm

.

The blade material is DD6, while the material of the connected high-pressure turbine disk is FGH96.

Under normal operating conditions, the loads acting on the high-pressure turbine working blade mainly include thermal load, centrifugal load and aerodynamic load.

In this study, the loads corresponding to the engine design operating condition are selected for the analysis of the blade.

Since the structure of the high-pressure turbine working blade is relatively complex, in order to improve the efficiency of simulation analysis, a three-dimensional finite element model is established using the high-pressure turbine disk together with the sector under a single blade, and the tetrahedral SOLID187 element with mid-side nodes is adopted. The calculation coordinate system is a Cartesian coordinate system, in which the x-axis is parallel to the engine axis and the direction along the airflow is positive, the y-axis is positive from the blade back to the blade basin, and the z-axis is parallel to the blade stacking axis.

The axial and circumferential displacements of the front mounting edge of the high-pressure turbine disk are constrained. The axial end surfaces of the high-pressure turbine blade and turbine disk are selected for nodal coupling in the axial direction. The cyclic symmetry surface of the high-pressure turbine disk is coupled in the radial, circumferential, and axial directions, and standard contact is established on the dovetail tooth contact surfaces between the high-pressure turbine blade and the turbine disk.

Temperature load and centrifugal load are applied to the high-pressure turbine disk and blade, and circumferential and axial forces are applied to the blade basin side to simulate aerodynamic load.

Elastic finite element simulation analysis is carried out for the high-pressure turbine working blade, and the first principal stress of its dovetail is shown in Figure 13.

According to Figure 13, the position with the maximum first principal stress of the dovetail of the high-pressure turbine working blade is the third pair of fir-tree teeth (the pair close to the blade extension root is regarded as the first pair of dovetail teeth), located at the tooth root on the blade basin side, which is the dangerous point for the low-cycle fatigue life of the blade.

In addition, by comparing the stress distribution of each dovetail tooth, it can be found that the maximum stress appears on one side, and there are differences in both the stress magnitude and distribution between the two sides. The reasons are as follows:

(1)

The structural shape of the dovetail. The included angle between the front and rear end surfaces of the blade dovetail and the dovetail teeth is not a right angle, and the sizes of the ventilation holes of the dovetail are different and unevenly distributed.

(2)

The force state of the dovetail tooth contact surface. Theoretically, the contact surfaces of the dovetail teeth should bear the same force in design, but due to the aerodynamic force during operation, the force on the blade basin side is relatively larger. Generally, adjustment is made through the blade shroud amount design, but its influence cannot be completely eliminated.

The data for the stress concentration locations of the turbine blade/disk dovetail joint structure are shown in

Table 10. Stress Concentration Parameters of the High-Pressure Turbine Blade Dovetail.

Location	${\mathit{\sigma }}_{\mathit{m}\mathit{a}\mathit{x}}$/MPa	T/°C	E/GPa	K/MPa
Tooth Root of Tenon Teeth	1295	650	105.5	2106

.

When constructing a parametric model, extracting geometric parameters is the primary step. The extracted geometric parameters should possess both completeness and conciseness, which should comprehensively cover the structural geometric information while avoiding redundancy.

Based on this, the geometric parameters of the fir-tree root structure are divided into structural parameters and tooth profile parameters. The former is used to describe the basic form and spatial position of the dovetail structure, while the latter is used to define the specific profile of each pair of dovetail teeth.

The cross-sectional shape of the turbine blade dovetail structure used in a certain gas turbine can be referred to in Figure 14. From top to bottom, the section successively presents the first, second, and third teeth, and the tooth profile parameters used by these three pairs of dovetail teeth are identical. There are three tooth-root filets between each dovetail tooth.

For the high-pressure turbine working blade dovetail, the key dimensions include: the profile of the fir-tree tooth working surface; the tooth-root filet of the dovetail tooth; the wedge angle; the inclination angle of the groove bottom and the filet at the edge of the dovetail ventilation hole.

The dispersion of structural dimensions is usually determined by the upper and lower limits of the design tolerance, and the actual machining process determines its probability distribution characteristics.

At present, there is a lack of dimensional measurement data for the above key locations. Therefore, the allowable dimensional tolerance ±Δx of machined parts in the design drawings is used to estimate the standard deviation of the dimensions (assuming the dimensions follow a normal distribution), as shown in Equation (44).

$${\sigma }_x={\displaystyle \frac{\Delta x}{3}},$$

(44)

Referring to the dimensional tolerances of the fir-tree tooth-root filet and the filet at the edge of the blade ventilation hole in the design drawings of the high-pressure turbine working blade, the dispersion of the key dimensions is estimated through the design tolerance, and the dimensional values and randomization parameters are shown in

Table 11. Key dimensional tolerances and discretization parameters.

Size Position	Dimension Tolerance (mm)	Discretization Mean (mm)	Discretization Standard Deviation (mm)
Tenon tooth root rounding (above)	$R{0.7}_{-0.05}^0$	0.675	0.017
Tenon tooth root rounding (middle)	$R{1.5}_0^{+0.05}$	1.525	0.017
Tenon tooth root rounding (below)	$R{0.7}_{-0.05}^0$	0.675	0.017
Tenon vent hole edge rounded	$R{2}_{-0.5}^{+0.5}$	2	0.333

.

In addition, since the turbine blade is a casting, its weight has certain deviations, which directly affect the centrifugal force of the blade. The parameterization is based on the third pair of dovetail teeth, and the parameterization of blade weight is realized through the conversion of blade density. The key dimensions are shown in

Table 12. Key dimensions of the turbine blade fir-tree root structure.

Serial Number	Name	Code
1	Tenon tooth root rounded	R1
2	Tenon tooth root rounded	R2
3	Tenon tooth root rounded	R3
4	Tenon tooth root rounded	R4
5	Tenon tooth root rounded	R5
6	Tenon tooth root rounded	R6
7	Tooth pitch	D
8	Wedge angle	φ
9	Inclination angle of groove bottom	δ
10	Gap size between tenon teeth	D1
11	Gap size between tenon teeth	D2
12	Gap size between tenon teeth	D3

.

The Kriging model is composed of a parametric model and a non-parametric model. The parametric model is a regression analysis model, and the non-parametric model is a random distribution.

The Kriging model predicts a point mainly by using the information of known variables around the point, and estimates the unknown information of the point through the weighted combination of information within a certain range around the point. The selection of weights is determined by minimizing the error variance of the estimated value.

The form of the Kriging model is as follows:

$$y(x)+f(x)+z(x),$$

(45)

In Equation (45), y(x) is the response function to be fitted, and f(x) is a deterministic part providing a global approximation of the simulation, which is generally expressed by a polynomial of xxx. In this paper, a second-order polynomial is adopted. z(x) provides an approximation of the local deviation of the simulation, which is a random function with mean zero and variance σ², representing the local deviation of the global model and obtained through interpolation of sample points.

The covariance matrix of z(x) represents the degree of its local deviation and has the following form:

$$\mathrm{cov}[Z(x),Z(x_i)]={\sigma }^{2}R(\theta ,x,x_i),$$

(46)

In Equation (46), x_i is the input part of the i-th training sample point; R($\theta$, x, x_i) is the correlation function with parameter θ; and n is the number of sample points. R($\theta$, x, x_i) is a Gaussian function.

$$R(x,x_i)=\exp (-{\theta }_k|x_k-x_k^i{|}^2),$$

(47)

Based on the samples, the unknown parameter θ_k is obtained by solving the equations so that the prediction variance of the Kriging model is minimized.

According to the surrogate model, the sensitivity analysis results of the influence of key dimensions and blade weight on the stress at the critical location of the blade dovetail can be obtained, as shown in Figure 15 and

Table 13. Sensitivity influence of key dimensions of the turbine blade dovetail structure.

Serial Number	Name	Code	Influence of Sensitivity
1	Tenon tooth root rounded	R1	3.7
2	Tenon tooth root rounded	R2	1.9
3	Tenon tooth root rounded	R3	1.3
4	Tenon tooth root rounded	R4	3.2
5	Tenon tooth root rounded	R5	1.1
6	Tenon tooth root rounded	R6	1.2
7	Tooth pitch	D	15.2
8	Wedge angle	φ	6.9
9	Inclination angle of groove bottom	δ	5.1
10	Gap size between tenon teeth	D1	10.2
11	Gap size between tenon teeth	D2	9.6
12	Gap size between tenon teeth	D3	7.7

.

Based on the established surrogate model, 10,000 samples are generated by Latin hypercube sampling, and the probability distribution of the maximum stress (S1) at the left and right tooth-root filets of the third pair of dovetail teeth of the turbine blade/disk dovetail joint structure can be obtained.

The maximum stress at the left tooth-root filet of the turbine blade dovetail structure is 1295 MPa, with a standard deviation of 9.85 MPa, while the maximum stress at the right tooth-root filet is 1100 MPa, with a standard deviation of 9.11 MPa.

The specific probability distribution data of the maximum stress of the turbine blade dovetail joint structure are shown in Figure 16 and Figure 17, and

Table 14. Probability distribution parameters of the maximum stress at the dangerous location of the dovetail.

	Distribution Type	Mean (MPa)	Standard Deviation (MPa)
The maximum stress of left tooth root rounding	Normal distribution	1295	9.85
The maximum stress of right tooth root rounding	Normal distribution	1100	9.11

.

4.2. Reliability Analysis of Multi-Tooth Fir-Tree Root

First, from the previous chapter, the stress probability distribution parameters at the dangerous locations of the turbine blade dovetail structure are obtained: the maximum stress at the left tooth-root filet is 1295 MPa, and the maximum stress at the right tooth-root filet is 1100 MPa. Then, the equivalent stress is determined using the point method of the Theory of Critical Distances, which introduces a material characteristic length L and takes the stress at a specific distance from the location of maximum stress gradient as the equivalent stress. Finally, the equivalent stress is substituted into the Morrow-modified Manson–Coffin equation (Equation (20)) to calculate fatigue life, where the stress–strain relationship is established through the elastic modulus E. The total strain range is obtained by dividing the equivalent stress by the elastic modulus and subsequently decomposed into elastic and plastic strain components. For multiaxial stress states, the maximum principal stress criterion is adopted as a simplification. The deterministic life of the left and right tooth roots of a single dovetail is calculated, as shown in

Table 15. Fatigue life at the design point of the dangerous location of the dovetail.

Parts	Life/Cycle
Left tooth root	741
Right tooth root	3021

.

Using the Monte Carlo method, the life relationship between the left and right teeth is obtained through joint sampling, as shown in Figure 18, which shows that there is strong correlation. The probability life distributions of the two teeth are plotted, as shown in Figure 19. The tooth life distribution is significantly lower than that of the right tooth, and in the overlapping part, both dangerous locations may fail.

The comparison between the fitting results of the life cumulative distribution function and the traditional Monte Carlo method shows that the Copula model has significant advantages in describing complex dependence structures.

The traditional Monte Carlo method relies on independent sampling or simple correlation assumptions, which may lead to underestimation of tail risks, especially in multivariate joint distribution modeling. In contrast, the Copula-based fitting constructs conditional dependence relationships hierarchically, which can more accurately capture nonlinear and asymmetric dependence between variables (such as tail dependence).

Empirical analysis shows that for high-dimensional life data, the estimation error of the extreme quantile (such as the 99% confidence level) of the Copula model is about 20–30% lower than that of the Monte Carlo method, and the computational efficiency can be improved through vine structure optimization.

However, the Monte Carlo method still has practicality in implementation simplicity and small-sample scenarios, while the superiority of the Copula becomes more prominent with the increase in data dimension and dependence complexity.

This comparison provides a quantitative basis for model selection in life risk evaluation.

The Copula function is used to characterize the common-cause failure correlation of the two dovetail teeth, and the AIC (Akaike Information Criterion) method is used to select the optimal model, as shown in

Table 16. Reliability analysis model selection.

Type	θc	AIC
Gaussian	0.9883	−24243
Clayton	12.1945	−20766
Gumbel	36.9783	−23940
Frank	8.7851	−22293

. The minimum value corresponds to the model with the highest accuracy; therefore, the Gaussian coupling parameter is selected to characterize the correlation.

Using the established Copula function to characterize the correlation between the two teeth, the reliability distribution considering correlation is calculated. Life under different reliability levels is shown in

Table 17. Fatigue life under different reliability levels.

	50% Reliability Life	90% Reliability Life	99% Reliability Life	99.87% Reliability Life
Single tenon life/cycle	1058	410	200	130
Multi-tenon life/cycle	830	330	155	102

.

Combined with the figures and tables, it can be seen that the reliability distribution considering correlation is lower than that of a single tooth root distribution, and higher than the completely independent case, and is greatly affected by the most dangerous tooth. Therefore, in engineering practice it is necessary to consider the influence of correlation.

On the basis of the reliability analysis of multi-tooth correlation of the dovetail, the reliability considering the correlation of multiple dovetails of the whole disk is further analyzed. Correlation analysis is carried out for the whole-disk dovetails, and the multi-dovetail reliability curve is obtained as shown in Figure 20.

The life of the whole disk depends on the life of the dangerous location with the lowest life, but the correlation among different dangerous locations also needs to be considered. After considering correlation, the whole-disk life decreases slightly, and in engineering practice the influence of multi-location correlation on life needs to be considered.

Further study is carried out on the influence of different numbers of dovetails on turbine disk life. Let N_i be the life with multiple dovetails, and the life of a single dovetail be N₁. The life distributions under different numbers of dovetails are simulated, and the variation in N_i/N₁ with the number of teeth under the same reliability is plotted, as shown in Figure 21.

Under the same reliability, as the number of dovetails increases, N_i/N₁ gradually decreases. When the number of dovetails is less than 10, the decrease is significant, and when it is greater than 10, it tends to become stable. Moreover, under the same number of dovetails, the decrease increases with the increase in reliability.

5. Reliability Analysis of Turbine Blade–Disk Multi-Component Failure

5.1. Parametric Analysis of the Turbine Blade–Disk Fir-Tree Root and Slot Structure

The structural geometry of the blade-root and disk-slot assembly is illustrated in Figure 22.

Regarding the boundary conditions, the axial and circumferential displacements of the front mounting flange of the high-pressure turbine disk are constrained. The axial degrees of freedom (DOFs) of the nodes on the axial end faces of the HPT blade and disk are coupled. On the cyclic symmetry planes of the HPT disk, the radial, circumferential, and axial DOFs are coupled. Standard contact interactions are established at the contact interfaces between the HPT blade and the disk tenon teeth.

Thermal and centrifugal loads are applied to the HPT disk and blades, while circumferential and axial forces are applied to the pressure side of the blades to simulate aerodynamic loading.

Elastic finite element simulation is performed on the HPT working blade–disk assembly. The first principal stress distribution of the mortise-tenon joint is shown in Figure 23. Data regarding the stress concentration regions of the turbine blade joint are summarized in

Table 18. Parameters of stress concentration regions in the HPT working blade joint.

Location	${\mathit{\sigma }}_{\mathit{m}\mathit{a}\mathit{x}}$/MPa	T/°C	E/GPa	K/MPa	${{\mathit{\epsilon }}^{\prime }}_{\mathit{f}}/\%$
Tooth root of tenon teeth	1307 (S1)	650	105.5	2401	0.748

.

Geometric parameters of the turbine blade–disk joint were extracted following parametric modeling to represent the complete geometry of the structure. The cross-sectional shape of the joint is shown in Figure 24. From top to bottom, the section presents the first, second, and third teeth. The tooth profile parameters for these three pairs of teeth remain consistent, with three root filets between each tooth.

Based on the dimensional tolerances of the tooth root filets specified in the HPT working blade design drawings, the dispersion of critical dimensions was derived. To achieve an automated integration of variable assignment, modeling, and finite element analysis (FEA), the procedure is defined as follows: First, the model is fully parameterized to ensure all geometric dimensions can be expressed and adjusted as parameters. Subsequently, sampling of random variables is conducted using the Latin Hypercube Sampling (LHS) method to ensure the samples sufficiently cover the entire potential design space, thereby enhancing representativeness and reliability. The updated models are then solved using FEA. Following this, a response surface is generated based on the calculation results to characterize the relationship between the model outputs and input parameters. Sensitivity indices for each parameter are generated via the response surface to quantify the impact of critical geometric dimensions on the structural performance of the turbine disk.

For the sensitivity calculation of the shaft geometric parameters, the range for each parameter is set within ±0.1% of its nominal value. LHS is employed for sampling to construct the response surface, while random sampling is used for error analysis, with 12 sampling iterations per error verification group. The maximum equivalent stress at the critical locations of the joint structure is selected as the objective function for sensitivity analysis. The dimensional sensitivity analysis of the key dimensions of the joint structure is presented in

Table 19. Critical dimensions of the turbine blade root structure.

Number	Name	Code
1	Tenon tooth root rounded	R1
2	Tenon tooth root rounded	R2
3	Tenon tooth root rounded	R3
4	Tenon tooth root rounded	R4
5	Tenon tooth root rounded	R5
6	Tenon tooth root rounded	R6
7	Tooth pitch	D
8	Gap size between tenon teeth	D1
9	Gap size between tenon teeth	D2
10	Gap size between tenon teeth	D3

.

Based on the surrogate model, the sensitivity analysis results regarding the influence of critical dimensions and blade weight on the stress at critical locations of the blade tenon are obtained, as shown in Figure 25 and

Table 20. Sensitivity analysis of critical joint dimensions.

Number	Name	Code	Sensitivity Effects
1	Tenon tooth root rounded	R1	2.6
2	Tenon tooth root rounded	R2	2.93
3	Tenon tooth root rounded	R3	2.12
4	Tenon tooth root rounded	R4	2.54
5	Tenon tooth root rounded	R5	1.42
6	Tenon tooth root rounded	R6	2.64
7	Tooth pitch	D	11.2
8	Gap size between tenon teeth	D1	7.8
9	Gap size between tenon teeth	D2	6.5
10	Gap size between tenon teeth	D3	8.7

.

Using the established surrogate model, 10,000 samples were generated via LHS to cover the entire level space. The probability distributions of the maximum stress (S1) at the left and right root filets of the third pair of turbine blade/disk joint teeth were obtained. The results indicate that the maximum stresses at the left and right root filets are 1307 MPa and 1258 MPa, with standard deviations of 6.62 MPa and 6.8 MPa, respectively. Detailed probability distributions of the maximum root stress are shown in Figure 26 and Figure 27 and

Table 21. Probability distribution parameters of the maximum stress at critical joint locations of the mortise.

	Distribution Type	Mean (MPa)	Standard Deviation (MPa)
The maximum stress of left tooth root rounding	Normal distribution	1307	6.62
The maximum stress of right tooth root rounding	Normal distribution	1258	6.8

.

5.2. Multi-Component Failure Reliability Analysis of the Turbine Blade Root and Disk Slot System

The deterministic lives of the tenon and mortise, calculated using the crack initiation life model based on the Theory of Critical Distances (TCD) in Section 2.3, are presented in

Table 22. Probability distribution parameters of the maximum stress at critical joint locations of the tenon.

	Stress (MPa)	Distribution Type	Mean (MPa)	Standard Deviation (MPa)
The maximum stress of tooth root rounding of tenon structure	1295	Normal distribution	1284.4	11.80
Maximum stress of tooth root rounding of mortise structure	1307	Normal distribution	1095.54	9.87

. The fatigue lives at critical points are shown in

Table 23. Fatigue life at design points of critical locations.

Location	Life/Cycle
Tenon tooth root	3021
Tenon groove tooth root	14750

.

A reliability analysis considering the correlation between the tenon and mortise in the joint structure was performed using the Copula method. Given that the stress dispersion for a single turbine blade joint is identical, only material dispersion needs to be considered. The life relationship between the tenon and mortise obtained through joint sampling is shown in Figure 28.

The probability distributions of the fatigue life for the two components are plotted in Figure 29. It is evident that the fatigue life of the tenon is significantly lower than that of the mortise, and failure may occur at either critical location within the overlapping region.

The Copula function is utilized to characterize the common cause failure correlation between the three teeth of the blade root and disk slot. The Akaike Information Criterion (AIC) was employed to select the optimal model. As shown in Table 23, the model with the minimum AIC value represents the highest accuracy; therefore, the Gaussian Copula parameter was selected to characterize the correlation. A comparison of the cumulative distribution function (CDF) fit between the proposed method and the traditional Monte Carlo Method (MCM) is shown in Figure 30.

The established Copula function was used to characterize the correlation between the two teeth of the tenon and mortise, and the reliability distribution considering this correlation was calculated. The reliability distributions for the tenon and mortise lives are plotted in Figure 31 and Figure 32, and the lives under different reliability levels are listed in

Table 24. Fatigue life under different reliability levels.

	50% Reliability Life (Cycle)	90% Reliability Life (Cycle)	99% Reliability Life (Cycle)	99.87% Reliability Life (Cycle)
Tenon (multi-component related)	1601	1121	728	470
Tenon	2599	1960	1437	1095
Tenon groove (multi-component related)	9750	8258	7025	6215
Tenon groove	14761	12451	10570	9330

. Analysis of the figures and tables indicates that the reliability distribution considering correlation is lower than the distribution of a single tooth root but higher than the completely independent case, and it is significantly influenced by the most critical tooth.

6. Conclusions

This study addresses the combined effects of material and geometric variability, as well as the failure correlation among multiple locations/components, in the turbine blade–disk dovetail joint structure of aero-engines. A probabilistic life model based on linear heteroscedastic regression is established, and failure-correlated reliability analysis is conducted using the Copula method. The results indicate that the established Manson–Coffin probabilistic life models for DD6 and FGH96 alloys can effectively characterize the stochastic dispersion characteristics of fatigue life, and the predicted results show good agreement with experimental data.

Furthermore, a parametric finite element model of the fir-tree root structure is developed. Combined with the Kriging surrogate model and Latin hypercube sampling, the stress distribution at critical locations is obtained. After introducing the Copula function to describe the life correlation among multiple critical locations, it is found that the reliability of multi-tooth structures considering correlation is lower than that of a single critical location, but higher than that under the assumption of complete independence. Meanwhile, as the number of dovetails increases and the reliability requirement becomes higher, the overall life of the disk decreases further.

In the analysis of the blade root–disk slot multi-component system, significant failure correlation is also observed between the dovetail and the slot. When the correlation is taken into account, the reliability of both the dovetail and the slot decreases, and the overall system life is mainly governed by the dovetail component with the lower life.

The integrated framework proposed in this study—“probabilistic life modeling–probabilistic stress analysis–correlated failure modeling–system reliability evaluation”—provides a theoretical basis and engineering reference for the reliability design, life prediction, and maintenance decision-making of critical rotor connection structures in aero-engines.