Hybrid reliability analysis of structural fatigue life: Based on Taylor expansion method

Abstract

A new method for computing the failure probability of the fatigue life is proposed, dealing with uncertain problems with both random and interval variables. Using a Taylor expansion and the concept of statistical moment, the first four central moments of the structural fatigue life performance function are obtained. Then, using a second Taylor expansion, the first four central moments are expanded at the midpoint of the interval variable, and the intervals of the statistical moments of the performance function are calculated. The obtained moment information is applied into an Edgeworth series expansion expression, giving the cumulative distribution function of the structural fatigue life performance function and getting its interval of the failure probability. Two numerical examples of growing complexity are employed to demonstrate the feasibility of the proposed approach.

Introduction

A large amount of experimental results and data samples have shown that the dispersion in the fatigue life of the structural components or mechanical parts should not be omitted. The structural fatigue life is affected by many factors, such as material properties, geometry parameters, loading history, and the environmental conditions. A lot of research has been carried out on the reliability analysis of the fatigue life.

The Hasofer–Lind (HL) algorithm ¹ and the Rackwitz–Fiessler (RF) algorithm ² propose a transformation of the non-normal variables into normal variables, which results in an approximate estimation of the failure probability of the fatigue life by solving the so-called reliability index. Wu and Wirsching ³ used the modified first-order second moment method to solve the reliability of the implicit fatigue life performance functions. Wirsching et al. ⁴ applied the same method to analyze the reliability of the crack initiation life and the crack propagation life. Meng and Sha ⁵ proposed a neural network to perform the reliability optimization design of the structural fatigue life. Li and Meng ⁶ developed a robust optimization design method for the structural fatigue life. Gao et al. ⁷ presented the fatigue reliability analysis of a turbine based on the radial basis function neural network (RBFAN). Li and Yao ⁸ proposed the fatigue reliability analysis of structural components based on the stochastic finite element method (SFEM). Meng et al. ⁹ proposed the fatigue reliability analysis of a loader beam based on the RBFAN response surface method. Wirsching ¹⁰ proposed a fatigue reliability of offshore structures. Zhao et al. ¹¹ presented the fatigue reliability of steel bridge components. CG Soares and Ivanov ¹² deal with the reliability of hull girders concentrated on the hull girder collapse mode. Kwon and Frangopol ¹³ proposed a bridge fatigue reliability assessment using probability density functions of the equivalent stress range based on field monitoring data. Soares and Garbatov ¹⁴ presented for the assessment of the reliability of a ship hull with respect to fatigue failure of the ship hull girder. Gao and Xiong ¹⁵ introduced several important aspects in the fatigue/fracture reliability and their application on the practice. In the above-mentioned investigations, the methods proposed for the reliability analysis of structural fatigue life are usually based on probability theory. When there are insufficient sample data to build a probabilistic model, the uncertain information can also be described by interval variables. Qiu and Wang ¹⁶ proposed the interval estimation method to perform the fatigue life reliability analysis and compared it to the probabilistic reliability. However, in practical engineering, different kinds of uncertain variables may exist in the same structures, resulting in a structural fatigue life reliability problem with both random and interval variables. An and An ¹⁷ combined the mean value–modified first-order second moment (MVFOSM) and the interval number operations and calculated the interval of the failure probability of the structural fatigue life. However, the MVFOSM could not consider the distribution of random variables, and the interval operations had an interval expansion problem.

In this article, a simple method for fatigue life reliability analysis was proposed, estimating the fatigue reliability problems with random and interval variables. Using a Taylor expansion and the concept of statistical moment, the first four central moments of the structural fatigue life performance function are obtained. Then, using the same Taylor expansion again, the first four central moments are expanded at the midpoint of the interval variable, and the intervals of the first four central moments of the performance function were achieved. The obtained moment information was applied in the Edgeworth series expansion expression, giving the cumulative distribution function of the structural fatigue life and getting the interval of the failure probability of the structural fatigue life.

Structural fatigue life

According to the stress life method, the structural fatigue life N, under the constant stress ratio R, can be expressed as follows

$${\mathrm{NS}}_{max}^m=C$$

(1)

where $S_{max}$ is the max stress, N is the cycle number under the $S_{max}$ , and m and C are the fatigue parameters of the material. Usually, the material parameter m may be treated as constant value due to its small variations.

Since the fatigue parameters of the material are mostly obtained under fully reversed loading, equation (1) can be rewritten as follows

$${\mathrm{NS}}_{-1}^m=C$$

(2)

where $S_{-1}$ is the fatigue limit under fully reversed loading.

When the mean stress is taken into consideration, according to the Goodman ¹⁸ formula, its expression is as follows

$$S_{-1}=\frac{S_a}{1-S_m/S_b}$$

(3)

where $S_b$ is the ultimate strength, $S_a$ is the stress amplitude, and $S_m$ is the mean stress, resulting into

$$S_a=\frac{S_{max}-S_{min}}{2}$$

(4)

$$S_m=\frac{S_{max}+S_{min}}{2}$$

(5)

where $S_{\max }$ and $S_{\min }$ are the maximum and the minimum stresses, respectively.

From equations (2) and (3), the following equation may be obtained

$$N=C{\left(\frac{S_b}{S_b-S_m}{S}_a\right)}^{-m}$$

(6)

When the structure is subjected to random loads, the Palmgren–Miner ¹⁹ linear cumulative damage rule is typically used, which states that the cumulative fatigue damage is a linear accumulation of the fatigue damage of each cycle. Therefore, the cumulative fatigue damage D is as follows

$$D=\sum _{i=1}^k\frac{n_i}{N_i}=\frac{1}{C}\sum _{i=1}^k{n_i\left(\frac{S_b}{S_b-S_{m_i}}{S}_{a_i}\right)}^m=\frac{1}{C}\sum _{i=1}^k{n}_i{S}_{e_i}^m$$

(7)

where k is the rank number of the stress amplitude level. To the $i\mathrm{th}$ stress level, the stress amplitude is $S_{a_i}$ , and the mean stress is $S_{m_i}$ , $n_i$ and $N_i$ refer to the actual cycle number and the number of cycles to failure under the $i\mathrm{th}$ stress level, respectively. $S_{e_i}$ corresponds to the equivalent stress of the $i\mathrm{th}$ stress level, and

$$S_{e_i}=\frac{S_b}{S_b-S_{m_i}}{S}_{a_i}$$

(8)

Wirsching ²⁰ suggested that when $D=\Delta$ , the structure would fail. The total fatigue life $T_f$ of the structure is thus written as follows

$$T_f=\frac{\Delta }{D}=\frac{\Delta C}{\sum _{i=1}^k{n}_i{S}_{e_i}^m}$$

(9)

And, according to the reliability theory, the structural fatigue life performance function may be defined as follows

$$Z=\ln T_f-\ln T_D$$

(10)

where $T_D$ is the design life.

Statistical moments of the fatigue life performance function with hybrid variables

For the problem with both random and interval variables, the structural fatigue life performance function $Z=g(\mathit{x},{\mathit{y}}^I)$ can be expressed as follows

$$Z=g(\mathit{x},{\mathit{y}}^I)=g(x_1,\dots ,x_n,y_1^I,\dots ,y_m^I)$$

(11)

where $\mathit{x}=(x_1,x_2,\dots ,x_n)$ is the vector of the random variables, ${\mathit{y}}^I=(y_1^I,y_2^I,\dots ,y_m^I)$ is the vector of the interval variables

$$y_j^c=\frac{(y_j^R+y_j^L)}{2},\Delta y_j=\frac{(y_j^R-y_j^L)}{2}$$

(12)

where $y_j^c$ and $\Delta y_j$ are the midpoint and the radius of the $j\mathrm{th}$ interval variable, respectively. $y_j^L$ and $y_j^R$ are the lower limit and the upper limit of the $j\mathrm{th}$ interval variable, respectively.

The statistical moments should be calculated to approximate the cumulative distribution function of the fatigue life performance function expressed in equation (11). Monte Carlo simulation (MCS) is a comprehensive method to calculate the statistical moments of the structural responses. It can deal with any kind of function type without considering the dimensions. However, it needs large amount of simulations to ensure its precision. The point estimation method was first proposed by Rosenblueth, ²¹ Wang et al., ²² and Seo and Kwak ²³ pointed out that this method may be used to estimate the statistical moments and was exploited vastly for engineering applications. The point estimation method is usually affected by the type of the random variables and other factors, leading to a dissatisfactory precision. ²⁴ Therefore, in this article, a Taylor series expansion is applied to the first two items. Assuming that the first four moments of $x_i$ in $\mathit{x}=[x_1,x_2,\dots ,x_n{]}^{\mathrm{T}}$ are ${\mu }_{x_{i1}},{\mu }_{x_{i2}},{\mu }_{x_{i3}},\mathrm{and}{\mu }_{x_{i4}}$ , the first four central moments of the performance function $Z=g(\mathit{x},{\mathit{y}}^I)$ may be defined as follows

$${\mu }_g=g({\mu }_x,{\mathit{y}}^I)+\frac{1}{2}\sum _{i=1}^n\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^I)}{\partial x_i^2}{\mu }_{x_{i2}}$$

(13)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mu }_g^2=}& \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\right]}^2{\mu }_{x_{i2}}\hfill \\ \multicolumn{1}{c}{}& +\sum _{i=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^I)}{\partial x_i^2}{\mu }_{x_{i3}}+\cdots \hfill \\ \multicolumn{1}{c}{}& \frac{1}{4}\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\right]}^2({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)\hfill \\ \multicolumn{1}{c}{}& +\frac{1}{2}\sum _{i=1}^n{\sum _{l=1}^n\left[\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^I)}{\partial x_i\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}\hfill \end{array}$$

(14)

$$\begin{array}{cc}\multicolumn{1}{c}{{\mu }_g^3=}& \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\right]}^3{\mu }_{x_{i3}}+\cdots \hfill \\ \multicolumn{1}{c}{}& +\frac{3}{2}\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\right]}^2\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^I)}{\partial x_i^2}({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)+\cdots \hfill \\ \multicolumn{1}{c}{}& +3\sum _{i=1}^n\sum _{l=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_l}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^I)}{\partial x_i\partial x_l}{\mu }_{x_{i2}}{\mu }_{x_{l2}}\hfill \end{array}$$

(15)

$$\begin{array}{l}{\mu }_g^4={\displaystyle \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\right]}^4}({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)\\ +3{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{l=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_i}\right]}^2{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^I)}{\partial x_l}\right]}^2}}{\mu }_{x_{i2}}{\mu }_{x_{l2}}\end{array}$$

(16)

where ${\mu }_g,{\mu }_g^2,{\mu }_g^3,\mathrm{and}{\mu }_g^4$ correspond to the first four central moments of the structural fatigue life performance function.

If directly using the interval arithmetic operations to compute the intervals in equations (13)–(16), it will lead to the interval expansion. Therefore, the Taylor expansion method can be applied to transform the uncertain interval problem into a certain interval one. If $f(\mathit{x},{\mathit{y}}^I)$ is expanded at the midpoint of the interval variable, omitting the higher order items, the Taylor expansion of $f(\mathit{x},{\mathit{y}}^I)$ is as follows

$$\begin{array}{l}f(\mathit{x},{\mathit{y}}^I)=f(\mathit{x},{\mathit{y}}^c)+{\displaystyle \sum _{j=1}^m\frac{\partial f(\mathit{x},{\mathit{y}}^c)}{\partial y_j}}(y_j^I-y_j^c)\\ =f(\mathit{x},{\mathit{y}}^c)+{\displaystyle \sum _{j=1}^m\left|\frac{\partial f(\mathit{x},{\mathit{y}}^c)}{\partial y_j}\Delta y_j\right|}{e}_{\Delta }\end{array}$$

(17)

where the lower $[f(\mathit{x},{\mathit{y}}^I){]}^L$ and the upper $[f(\mathit{x},{\mathit{y}}^I){]}^R$ limits are as follows

$${[f(\mathit{x},{\mathit{y}}^I)]}^L=f(\mathit{x},{\mathit{y}}^c)-\sum _{j=1}^m\left|\frac{\partial f(\mathit{x},{\mathit{y}}^c)}{\partial y_j}\Delta y_j\right|$$

(18)

$${[f(\mathit{x},{\mathit{y}}^I)]}^R=f(\mathit{x},{\mathit{y}}^c)+\sum _{j=1}^m\left|\frac{\partial f(\mathit{x},{\mathit{y}}^c)}{\partial y_j}\Delta y_j\right|$$

(19)

where $e_{\Delta }=[-1,1]$ . From equations (18) and (19), the lower and upper limits of the first few central moments of the structural fatigue life may be calculated as follows

$${\left[{\mu }_g\right]}^L=g({\mu }_x,{\mathit{y}}^c)-{\displaystyle \sum _{j=1}^m\left|\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial y_j}\Delta y_j\right|}+\frac{1}{2}{\displaystyle \sum _{i=1}^n\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}}{\mu }_{x_{i2}}-\frac{1}{2}{\displaystyle \sum _{j=1}^m\left|\frac{\partial {\displaystyle \sum _{i=1}^n\frac{{\partial }^{2}g({\mu }_x,y^c)}{\partial x_i^2}}{\mu }_{x_{i2}}}{\partial y_j}\Delta y_j\right|}$$

(20)

$${\left[{\mu }_g\right]}^R=g({\mu }_x,{\mathit{y}}^c)+\sum _{j=1}^m\left|\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial y_j}\Delta y_j\right|+\frac{1}{2}\sum _{i=1}^n\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}{\mu }_{x_{i2}}+\frac{1}{2}\sum _{j=1}^m\left|\frac{\partial \sum _{i=1}^n\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}{\mu }_{x_{i2}}}{\partial y_j}\Delta y_j\right|$$

(21)

$$\begin{array}{c}{\left[{\mu }_g^2\right]}^L=\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2{\mu }_{x_{i2}}-\sum _{j=1}^m\left|\frac{\partial \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2{\mu }_{x_{i2}}}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ +\sum _{i=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}{\mu }_{x_{i3}}-\sum _{j=1}^m\left|\frac{\partial \sum _{i=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ +\frac{1}{4}\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)-\frac{1}{4}\sum _{j=1}^m\left|\frac{\partial \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ +\frac{1}{2}\sum _{i=1}^n{\sum _{l=1}^n\left[\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}-\frac{1}{2}\sum _{j=1}^m\left|\frac{\partial {\sum _{i=1}^n\sum _{l=1}^n\left[\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}}{\partial y_j}\Delta y_j\right|\hfill \end{array}$$

(22)

$$\begin{array}{c}{\left[{\mu }_g^2\right]}^R=\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2{\mu }_{x_{i2}}+\sum _{j=1}^m\left|\frac{\partial \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2{\mu }_{x_{i2}}}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ +\sum _{i=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}{\mu }_{x_{i3}}+\sum _{j=1}^m\left|\frac{\partial \sum _{i=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ +\frac{1}{4}\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)+\frac{1}{4}\sum _{j=1}^m\left|\frac{\partial \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ +\frac{1}{2}\sum _{i=1}^n{\sum _{l=1}^n\left[\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}+\frac{1}{2}\sum _{j=1}^m\left|\frac{\partial {\sum _{i=1}^n\sum _{l=1}^n\left[\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}}{\partial y_j}\Delta y_j\right|\hfill \end{array}$$

(23)

$$\begin{array}{cc}\multicolumn{1}{c}{{\left[{\mu }_g^3\right]}^L=}& \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^3{\mu }_{x_{i3}}-\sum _{j=1}^m\left|\frac{\partial \left[\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^3{\mu }_{x_{i3}}\right]}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ \multicolumn{1}{c}{}& +\frac{3}{2}\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)+\cdots -\frac{3}{2}\sum _{j=1}^m\left|\frac{\partial [\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)]}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ \multicolumn{1}{c}{}& +3\sum _{i=1}^n\sum _{l=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}{\mu }_{x_{i2}}{\mu }_{x_{l2}}+\cdots -3\sum _{j=1}^m\left|\frac{\partial \left[\sum _{i=1}^n\sum _{l=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}{\mu }_{x_{i2}}{\mu }_{x_{l2}}\right]}{\partial y_j}\Delta y_j\right|\hfill \end{array}$$

(24)

$$\begin{array}{cc}\multicolumn{1}{c}{{\left[{\mu }_g^3\right]}^R=}& \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^3{\mu }_{x_{i3}}+\sum _{j=1}^m\left|\frac{\partial \left[\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^3{\mu }_{x_{i3}}\right]}{\partial y_j}\Delta y_j\right|+\cdots +\frac{3}{2}\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)+\cdots \hfill \\ \multicolumn{1}{c}{}& +\frac{3}{2}\sum _{j=1}^m\left|\frac{\partial [\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i^2}({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)]}{\partial y_j}\Delta y_j\right|+\cdots +3\sum _{i=1}^n\sum _{l=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}{\mu }_{x_{i2}}{\mu }_{x_{l2}}+\cdots \hfill \\ \multicolumn{1}{c}{}& +3\sum _{j=1}^m\left|\frac{\partial \left[\sum _{i=1}^n\sum _{l=1}^n\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\frac{{\partial }^{2}g({\mu }_x,{\mathit{y}}^c)}{\partial x_i\partial x_l}{\mu }_{x_{i2}}{\mu }_{x_{l2}}\right]}{\partial y_j}\Delta y_j\right|\hfill \end{array}$$

(25)

$$\begin{array}{c}{\left[{\mu }_g^4\right]}^L=\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^4({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)-\sum _{j=1}^m\left|\frac{\partial [\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^4({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)]}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ +3\sum _{i=1}^n\sum _{l=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}+\cdots -3\sum _{j=1}^m\left|\frac{\partial \left[\sum _{i=1}^n\sum _{l=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}\right]}{\partial y_j}\Delta y_j\right|\hfill \end{array}$$

(26)

$$\begin{array}{cc}\multicolumn{1}{c}{{\left[{\mu }_g^4\right]}^R=}& \sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^4({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)+\sum _{j=1}^m\left|\frac{\partial [\sum _{i=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^4({\mu }_{x_{i4}}-3{\mu }_{x_{i_2}}^2)]}{\partial y_j}\Delta y_j\right|+\cdots \hfill \\ \multicolumn{1}{c}{}& +3\sum _{i=1}^n\sum _{l=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}+\cdots +3\sum _{j=1}^m\left|\frac{\partial \left[{\sum _{i=1}^n\sum _{l=1}^n{\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_i}\right]}^2\left[\frac{\partial g({\mu }_x,{\mathit{y}}^c)}{\partial x_l}\right]}^2{\mu }_{x_{i2}}{\mu }_{x_{l2}}\right]}{\partial y_j}\Delta y_j\right|\hfill \end{array}$$

(27)

Edgeworth series

Either the probability density function or the joint probability density function needs to be known in order to calculate the failure probability. Due to insufficient sample data, the exact distribution of the parameter could not be determined, and the distribution type is known to affect the correctness of the results. Even for a given probability distribution type, it is difficult to use the integral method to calculate the failure probability.

After obtaining the statistical moments of the structural fatigue life performance function given in equations (20)–(27), the cumulative distribution function of the performance function can be fitted. To do so, the following methods are available: the maximum entropy method, ²⁵ the polynomial method, ²⁶ the finite mixture density method, ²⁷ the Pearson system, ²⁸ and the saddle point approximation method. ²⁹ These methods cannot avoid singularities, and the obtained numerical solutions may become unstable during the calculations. Therefore, the Edgeworth series is chosen to expand the unknown probability distribution into the expression of a standard normal one, thus obtaining the failure probability of the structure.

It is known from the Edgeworth series expansion method that the cumulative distribution function $F(g)$ and the probability density function $f(g)$ of the performance function with only random variables could be, respectively, expressed as follows

$$\begin{array}{l}F(g)=\Phi (\overline{g})-\frac{1}{3!}\frac{{\mu }_g^3}{{\left(\sqrt{{\mu }_g^2}\right)}^3}{\Phi }^{(3)}(\overline{g})+\cdots +\\ \frac{1}{4!}\left(\frac{{\mu }_g^4}{{\left[\left(\sqrt{{\mu }_g^2}\right)\right]}^{4^R}}-3\right){\Phi }^{(4)}(\overline{g})+\cdots +\frac{10}{6!}{\left(\frac{{\mu }_g^3}{{\left(\sqrt{{\mu }_g^2}\right)}^3}\right)}^2{\Phi }^{(6)}(\overline{g})-\cdots \end{array}$$

(28)

$$\begin{array}{l}f(g)=\phi (\overline{g})-\frac{1}{3!}\frac{{\mu }_g^3}{{\left(\sqrt{{\mu }_g^2}\right)}^3}{\phi }^{(3)}(\overline{g})+\cdots +\\ \frac{1}{4!}\left(\frac{{\mu }_g^4}{{\left(\sqrt{{\mu }_g^2}\right)}^4}-3\right){\phi }^{(4)}(\overline{g})+\cdots +\frac{10}{6!}{\left(\frac{{\mu }_g^3}{{\left(\sqrt{{\mu }_g^2}\right)}^3}\right)}^2{\phi }^{(6)}(\overline{g})-\cdots \end{array}$$

(29)

where $\overline{g}$ is the standardized performance function, $\overline{g}=(g-{\mu }_g)/(\sqrt{{\mu }_g^2})$ . $\Phi (\overline{g})$ and ${\Phi }^{(i)}(\overline{g})$ denote the standard normal distribution function and its $i\mathrm{th}$ partial derivative, respectively. If there is to be any negative value, it can be amended by the methods in Zhang et al.^30,31 With respect to a complicated structure with both random and interval variables, according to the monotony of the function, ³² the upper limits and the lower limits of the cumulative distribution function $F(g)$ and the probability density function $f(g)$ may be expressed as follows

$$\begin{array}{l}[F(g){]}^R=\Phi ({\overline{g}}^R)-\frac{1}{3!}\frac{{\left({\mu }_g^3\right)}^R}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^3\right]}^R}{\Phi }^{(3)}({\overline{g}}^R)+\cdots +\\ \frac{1}{4!}\left(\frac{{\left({\mu }_g^4\right)}^R}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^4\right]}^R}-3\right){\Phi }^{(4)}({\overline{g}}^R)+\cdots +\\ \frac{10}{6!}{\left(\frac{{\left({\mu }_g^3\right)}^R}{{\left[\left({\sqrt{{\mu }_g^2}}^3\right)\right]}^R}\right)}^2{\Phi }^{(6)}({\overline{g}}^R)-\cdots \end{array}$$

(30)

$$\begin{array}{l}[F(g){]}^L=\Phi ({\overline{g}}^L)-\frac{1}{3!}\frac{{\left({\mu }_g^3\right)}^L}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^3\right]}^L}{\Phi }^{(3)}({\overline{g}}^L)+\cdots +\\ \frac{1}{4!}\left(\frac{{\left({\mu }_g^4\right)}^L}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^4\right]}^L}-3\right){\Phi }^{(4)}({\overline{g}}^L)+\cdots +\\ \frac{10}{6!}{\left(\frac{{\left({\mu }_g^3\right)}^L}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^3\right]}^L}\right)}^2{\Phi }^{(6)}({\overline{g}}^L)-\cdots \end{array}$$

(31)

$$\begin{array}{l}[f(g){]}^R=\phi ({\overline{g}}^R)-\frac{1}{3!}\frac{{\left({\mu }_g^3\right)}^R}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^3\right]}^R}{\phi }^{(3)}({\overline{g}}^R)+\cdots +\\ \frac{1}{4!}\left(\frac{{\left({\mu }_g^4\right)}^R}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^4\right]}^R}-3\right){\phi }^{(4)}({\overline{g}}^R)+\cdots +\\ \frac{10}{6!}{\left(\frac{{\left({\mu }_g^3\right)}^R}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^3\right]}^R}\right)}^2{\phi }^{(6)}({\overline{g}}^R)-\cdots \end{array}$$

(32)

$$\begin{array}{l}[f(g){]}^L=\phi ({\overline{g}}^L)-\frac{1}{3!}\frac{{\left({\mu }_g^3\right)}^L}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^3\right]}^L}{\phi }^{(3)}({\overline{g}}^L)+\cdots +\\ \frac{1}{4!}\left(\frac{{\left({\mu }_g^4\right)}^L}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^4\right]}^L}-3\right){\phi }^{(4)}({\overline{g}}^L)+\cdots +\\ \frac{10}{6!}{\left(\frac{{\left({\mu }_g^3\right)}^L}{{\left[{\left(\sqrt{{\mu }_g^2}\right)}^3\right]}^L}\right)}^2{\phi }^{(6)}({\overline{g}}^L)-\cdots \end{array}$$

(33)

where ${\overline{g}}^R=(g-{\mu }_g^R)/(\sqrt{{({\mu }_g^2)}^R})$ , and ${\overline{g}}^L=(g-{\mu }_g^L)/(\sqrt{{({\mu }_g^2)}^L})$ . By means of equations (30) and (31), the interval of the failure probability of the structural fatigue life performance function can be obtained.

Numerical examples

A cantilever beam

A cantilever beam is pictured in Figure 1, with its material 16Mn, ³³ length $L=0.762\mathrm{m}$ , width $b=0.0635\mathrm{m}$ , height $h=0.0635\mathrm{m}$ , elastic modulus $E=2.01\times {10}^5\mathrm{MPa}$ , fatigue parameter $m=10.51$ , and the design life $T_D={10}^6$ . Among these parameters, the damage $\Delta$ , the fatigue parameter C, and the ultimate strength $S_b$ are considered as random variables. The load P applied at the free end of the beam is of constant amplitude, with $P_{min}$ and $P_{max}$ being interval variables. The characteristics of these parameters are listed in Table 1.

Figure 1.

A cantilever beam.

Table 1.

Parameters of the cantilever beam.

Uncertain variables	Parameter 1	Parameter 2	Types
$\Delta$	1	0.01	Normal
C	8.48 × 1031	0.01	Normal
$S_b$	585 MPa	0.01	Normal
$P_{min}$	0.01501 kN	0.01516 kN	Interval
$P_{max}$	0.02644 kN	0.02670 kN	Interval

Parameter 1 is the mean value, and Parameter 2 is the variable coefficient for the random variables; and Parameter 1 is the lower limit, and Parameter 2 is the upper limit for the interval variables.

Table 2 provides a comparison of the first four central moments between the MCS method and the one proposed in this article. In Table 3, the failure probability of the fatigue life calculated with the proposed method corresponds to (0.00096, 0.001554), and when calculated with the MCS, it becomes (0.00092, 0.001514), with the relative error being (4.17%, 2.57%). The variation of the failure probability of the fatigue life with respect to the different design life is depicted in Figure 2. It may be seen from this figure that as the design life increases, the failure probability also increases, and when $T_D\ge 1.2\times {10}^6$ , the failure probability increases rapidly. The method proposed in this article does not require the use of multiple integrals to solve the statistical moments of the performance function and does not need to search for the most probable point. This example thus testifies the correctness and efficiency of the proposed approach.

Table 2.

Comparison of the first four central moments.

Moments	Mean value	Variance	Skewness	Kurtosis
MCS	(0.5621, 0.5990)	(0.0330, 0.0338)	(−6.93 × 104, −6.65 × 104)	(0.0033, 0.0035)
This article	(0.5619, 0.5988)	(0.0329, 0.0337)	(−6.89 × 104, −6.62 × 104)	(0.0032, 0.0034)

MCS: Monte Carlo simulation.

Table 3.

Comparison of the failure probability of the fatigue life.

Methods	Interval of the failure probability	Relative error (%)
MCS	(0.00096, 0.001554)	–
This article	(0.00092, 0.001514)	(4.17, 2.57)

MCS: Monte Carlo simulation.

Figure 2.

Relationship between the designing life and the failure probability.

Pressure vessel

A pressure vessel is illustrated in Figure 3, with material $\mathrm{Q}235$ , elastic modulus $E=2\times {10}^5\mathrm{MPa}$ , Poisson’s ratio $\mu =0.3$ , fatigue parameter $m=7.4382$ , and design life $T_D={10}^6$ . The main dimensions of the structure are also shown in Figure 3, and the loading history of the vessel is depicted in Figure 4. Since the top area has several holes, it is considered as the most critical area. The fatigue life analysis carried out in the following thus solely focuses on this area. The associated finite element model is pictured in Figure 5. The characteristics of the parameters are listed in Table 4, where $\Delta ,C,\mathrm{and}{S}_b$ are random variables, and $p_1,p_2,\mathrm{and}{p}_3$ are interval variables.

Figure 3.

Structural scheme of the pressure vessel.

Figure 4.

Loading history of the pressure vessel.

Figure 5.

Finite element model of the pressure vessel.

Table 4.

Characteristics of the pressure vessel’s parameters.

Uncertain variables	Parameter 1	Parameter 2	Type
$\Delta$	1	0.05	Normal
C	1022.632	0.01	Normal
$S_b$	407 MPa	0.02	Normal
$p_1$	−0.1919 MPa	−0.1900 MPa	Interval
$p_2$	0.4200 MPa	0.4242 MPa	Interval
$p_3$	0.6520 MPa	0.6585 MPa	Interval

Parameter 1 is the mean value, and Parameter 2 is the variable coefficient for the random variables; Parameter 1 is the lower limit, and Parameter 2 is the upper limit for the interval variables.

The intervals of the maximum stress and the minimum stress under different stages of the loading history were obtained using the finite element codes of the pressure vessel. The upper limits and the lower limits of the first four central moments of the structural performance function were obtained using the moment method and the Taylor expansion, which were given by equations (20)–(27). After substituting the central moments into equations (30) and (31), the cumulative distribution function, the probability density function, and the interval of the failure probability of the fatigue life could be obtained. Table 5 shows a comparison of the first four central moments between MCS and this article. In Table 6, the interval of the failure probability of the fatigue life calculated with the proposed approach is (0.001542, 0.006831), and the one calculated with the MCS is (0.001610, 0.006710), with the relative error being (4.23%, 1.80%), showing a good consistency between the two methods. Furthermore, the relation between the design life and the failure probability is shown in Figure 6. When the design life increases, the failure probability also increases, and as $T_D\ge 1.2\times {10}^6$ , the failure probability increases drastically. The new type of fatigue reliability method proposed in this article can thus solve uncertain problems with both random and interval variables, having a relatively high precision.

Table 5.

Comparison of the first four central moments between MCS and this article.

Moments	Mean value	Variance	Skewness	Kurtosis
MCS	(0.3361, 0.3982)	(0.0165, 0.0168)	(−3.27 × 104, −3.39 × 104)	(8.27 × 104, 8.60 × 104)
This article	(0.3359, 0.3981)	(0.0164, 0.0167)	(−3.15 × 104, −3.30 × 104)	(7.99 × 104, 8.37 × 104)

MCS: Monte Carlo simulation.

Table 6.

Comparison of the failure probability of the structural fatigue life.

Methods	Interval of the failure probability	Relative error (%)
MCS	(0.001610, 0.006710)	–
This article	(0.001542, 0.006831)	(4.23, 1.80)

MCS: Monte Carlo simulation.

Figure 6.

Relationship between the design life and the failure probability.

Conclusion

A structural fatigue life reliability method is proposed in this article, which can accurately solve uncertain problems with both random and interval variables. By combining the Taylor expansion method and the concept of the statistical moments, the expressions of the statistical moments of the performance function were obtained. With respect to the interval variables, by conducting a Taylor expansion at their midpoint, the intervals of the first four central moments were calculated. After substituting the moments into the Edgeworth series expansion formula, the interval of the failure probability was obtained. Two numerical examples of growing complexity clearly showed the relative high precision of the proposed method when dealing with a nonlinear problem or an implicit performance function. The proposed method has the benefit of not requiring the use of multiple integrals to solve the statistical moments of the performance function, does not need to search for the most probable point, and does not need to calculate the inverse of the performance function.

It can transform uncertain interval problems into certain interval problem using a Taylor expansion and avoiding the interval expansion. This hybrid fatigue life reliability analysis approach could utilize the merits of every single reliability model, reflecting the actual safe situation more comprehensively.