Reliability Model of Mechanical Components with Dependent Failure Modes

Abstract

A mechanical component may fail in many modes that are usually not independent. There is generally not a joint probability density function to describe these correlated failure modes. Thus, it is difficult to compute the reliability when considering the correlations between the failure modes. It is supposed that three or more failure modes arise synchronously to be a very small probability event. The relationship between ultimate state functions in different failure modes is established by utilizing linear regression method. A double integration model for reliability of mechanical components with dependent failure modes is built according to stress-strength interference model. In case of square, cube, or exponential relationship between two ultimate state functions, a linear transformation is made. An example of pin that may fail in shear fracture, bruise, or both is discussed. The reliability is compared with that obtained by using Monte Carlo method, which represents that the reliability model with dependent failure modes proposed is correct.

1. Introduction

Mechanical components may fail in many modes such as fatigue, creep, wear, resonance, plastic deformation, and instability which lead to several ultimate state functions. The ultimate state functions are often functions of the same random variables; therefore, they are not independent, and there may be some complex or even unknown relations between them. There is generally not a joint probability density function to describe these correlated failure modes; thus, it is a difficult and hot issue to compute the reliability with dependent failure modes in reliability study [1–3]. In [4–7], each failure mode is assumed to be independent when calculating the reliability with many failure modes; then, the series model of system reliability theory is introduced. This method is thought to be less rigorous in theory because the relevance of ultimate state functions among different failure modes is ignored, and the calculation error caused by it is hard to estimate. According to the relevance between the two failure modes, the bound method of reliability calculation is proposed in [8], which distinguishes the two failure modes into primary failure mode and secondary failure mode; the two failure modes are assumed to be of complete correlation and complete dissociation, and the actual condition of the structure falls in between the two modes, and thus the interval estimation value of relevant reliability is obtained. This method is easy to compute and widely used in the reliability design of civilian industry product [9, 10]; however, it belongs to qualitative analytical method, and only the interval estimation value can be acquired. In [11, 12], the correlation coefficient is applied to characterize the relationship among different failure modes as the nuclear power stations do for the reliability design of critical parts in many failure modes; if they are positively correlated and the correlation is desirable, a satisfying result can be obtained; if they are of negative correlation or the correlation is poor, larger error is obtained. Actually, the correlation coefficient can just illustrate the two failure modes to the extent of linear and cannot represent the real relation between them [13, 14]; for example, 𝑋²+𝑌²=1, where 𝑋 and 𝑌 are strongly correlated, but the correlation coefficient between them is 0. Errors will be introduced when using correlation coefficient to deal with the problems of reliability calculation in many failure modes. In [15, 16], the Copula function is used to describe the correlation of the random variables, and the Copula-Reliability method with dependent failure modes is obtained in which the multiple integral operation is replaced by multiple differential operation to simplify the reliability calculation process; but the approximate correlated relation between each failure modes must be identified firstly to reasonably choose Copula model during the implementation process of this method. Moreover, more sample values and some parameter estimation skills are needed to confirm the relevant parameter. The Monte Carlo method is used to calculate the reliability model of components and systems with dependent failure modes in [17, 18]. It needs no assumption or simplicity of the relation among failure modes, basic random variables, and ultimate state functions, and thus it has a wide application scope; however, when the reliability is high or the number of random variables is large, it needs higher sampling amount. Therefore, the Monte Carlo method is usually used to verify the theory method [14].

It is supposed in this paper that three or more failure modes existing simultaneously is a minimum probability event, the functional relation between ultimate state functions in different failure modes is obtained through linear regression method, double integration model for reliability is established by imitating the stress-strength interference model, and the Monte Carlo method is used to verify the new reliability model.

2. Reliability Model with Multiple Failure Modes

Suppose that a mechanical component is controlled by the vector of random variables 𝐗, 𝐗=𝑋₁,𝑋₂,…,𝑋_𝐻. (1)𝑋_ℎ (ℎ=1,2,…,𝐻) are the basic random variables. Assume that there are 𝑚 probable failure modes of the mechanical components, and the relative ultimate state functions are 𝑔₁(𝐗)=0,𝑔₂(𝐗)=0,…,𝑔_𝑚(𝐗)=0, (2) where 𝑔_𝑖(𝐗) (𝑖=1,2,…,𝑚) are supposed to obey normal distribution for simplicity. Take the event that the 𝑖th failure mode occurs (𝑔_𝑖(𝐗)<0) as 𝐸_𝑖, and the failure probability 𝐹 of the components can be presented as 𝐹=^𝑚_𝑖=1𝑃𝐸_𝑖−_{1≤𝑖<𝑗≤𝑚}𝑃𝐸_𝑖𝐸_𝑗+⋯+(−1)^𝑚−1𝑃𝐸₁𝐸₂⋯𝐸_𝑚. (3)

Considering that three or more failure modes occur synchronously is an event of very small probability, (3) can be expressed as 𝐹≈^𝑚_𝑖=1𝑃𝐸_𝑖−_{1≤𝑖<𝑗≤𝑚}𝑃𝐸_𝑖𝐸_𝑗. (4)

Failure probability for single failure mode of the components can be obtained as follows: 𝑃𝐸_𝑖=⁰_−∞𝑓_𝑖𝑔_𝑖d𝑔_𝑖, (5) where 𝑓_𝑖(𝑔_𝑖) are the ultimate state functions, and 𝑔_𝑖 are the probability density functions.

There are some inevitable functional relationships between generalized strength and generalized stress for the same components in each failure mode; therefore, the probability that 𝐸_𝑖 and 𝐸_𝑗 occur simultaneously cannot be calculated directly. Suppose that the relationships of ultimate state functions of each failure mode are approximately linear, as is shown in Figure 1. Even if there are other functional relationships between them, it can be translated into linear relationship through linear conversion.

Figure 1 

Relationship of ultimate state functions of different failure modes.

The linear function can be presented as 𝑔_𝑖=𝐴_𝑖−𝑗𝑔_𝑗+𝐵_𝑖−𝑗+𝜀_𝑖−𝑗, (6) where 𝐴_𝑖−𝑗, 𝐵_𝑖−𝑗 are constants; 𝜀_𝑖−𝑗 obey normal distribution 𝑁(0,𝜎_𝑖−𝑗), and the intercept of the line on 𝑔_𝑗 axis is recorded as 𝑎_𝑖−𝑗.

In order to obtain the distribution function of 𝑎_𝑖−𝑗, linear regression analysis is introduced on the 𝑁 sets of observed value (𝑔^𝑛_𝑗,𝑔^𝑛_𝑖)(𝑛=1,2,…,𝑁) of 𝑔_𝑗, 𝑔_𝑖, and the estimation value of the unknown parameter in the straight line equation is acquired. Consider 𝐴_𝑖−𝑗=∑^𝑁_𝑛=1𝑔^𝑛_𝑗𝑔^𝑛_𝑖−(1/𝑁)∑^𝑁_𝑛=1𝑔^𝑛_𝑗∑^𝑁_𝑛=1𝑔^𝑛_𝑖∑^𝑁_𝑛=1𝑔^𝑛_𝑗²−(1/𝑁)∑^𝑁_𝑛=1𝑔^𝑛_𝑗²,𝐵_𝑖−𝑗=∑^𝑁_𝑛=1𝑔^𝑛_𝑗∑^𝑁_𝑛=1𝑔^𝑛_𝑖²−∑^𝑁_𝑛=1𝑔^𝑛_𝑗𝑔^𝑛_𝑖∑^𝑁_𝑛=1𝑔^𝑛_𝑗𝑁∑^𝑁_𝑛=1𝑔^𝑛_𝑗²−(1/𝑁)∑^𝑁_𝑛=1𝑔^𝑛_𝑗²,𝜎²_𝑖−𝑗=1𝑁^𝑁_𝑛=1𝑔^𝑛_𝑖−1𝑁^𝑁_𝑛=1𝑔^𝑛_𝑖²−𝐴²_𝑖−𝑗𝑁^𝑁_𝑛=1𝑔^𝑛_𝑗−1𝑁^𝑁_𝑛=1𝑔^𝑛_𝑗². (7)

Notice that the intercept 𝑎_𝑖−𝑗 of the fitting line on 𝑔_𝑗 axis is 𝑎_𝑖−𝑗𝐵=−^𝑖−𝑗+𝜀_𝑖−𝑗𝐴_𝑖−𝑗. (8) It is thus clear that 𝑎_𝑖−𝑗 obey normal distribution 𝑁(−𝐵_𝑖−𝑗/𝐴_𝑖−𝑗,𝜎_𝑖−𝑗/𝐴_𝑖−𝑗).

It can be concluded from Figure 1 that(1)if 𝐴_𝑖−𝑗>0; 𝑃𝐸_𝑖𝐸_𝑗=𝑃𝑔_𝑗<𝑎_𝑖−𝑗≤0∪𝑔_𝑗<0∩𝑎_𝑖−𝑗𝑔>0=𝑃_𝑗<𝑎_𝑖−𝑗≤0+𝑃𝑔_𝑗<0∩𝑎_𝑖−𝑗>0; (9)(2)if 𝐴_𝑖−𝑗<0; 𝑃𝐸_𝑖𝐸_𝑗=𝑃𝑎_𝑖−𝑗<𝑔_𝑗≤0. (10)

The distribution curves of 𝑔_𝑗, 𝑎_𝑖−𝑗 are presented in Figure 2, in which 𝑓_𝑖−𝑗(𝑎_𝑖−𝑗) and 𝑓_𝑖(𝑔_𝑖) denote the probability density function of 𝑔_𝑗 and 𝑎_𝑖−𝑗, respectively.

Figure 2 

Relationship of 𝑔𝑗 and 𝑎𝑖−𝑗 distribution functions.

It can be concluded from Figure 2 that 𝑃𝑔_𝑗<𝑎_𝑖−𝑗<0=⁰_−∞𝑓_𝑖−𝑗𝑎_𝑖−𝑗^{𝑎𝑖−𝑗}_−∞𝑓_𝑖𝑔_𝑗d𝑔_𝑗d𝑎_𝑖−𝑗,𝑃𝑔_𝑗<0∩𝑎_𝑖−𝑗>0=⁰_−∞𝑓_𝑖𝑔_𝑗d𝑔_𝑗0^+∞𝑓_𝑖−𝑗𝑎_𝑖−𝑗d𝑎_𝑖−𝑗,𝑃𝑎_𝑖−𝑗<𝑔_𝑗≤0=⁰_−∞𝑓_𝑖𝑔_𝑗^𝑔𝑖_−∞𝑓_𝑖−𝑗𝑎_𝑖−𝑗d𝑎_𝑖−𝑗d𝑔_𝑗. (11)(12)(13) Equations (11)–(13) can be solved through numerical integration method.

3. Validation of Reliability Model with Dependent Failure Modes Utilizing Monte Carlo Method

The Monte Carlo method is also called stochastic simulation [19]. It is a numerical computation method that is on the basis of probability statistics theory, in terms of the laws of large numbers, with the utilization of numerical simulation technology and through statistical sampling test or stochastic simulation of relevant random variable to seek for approximate solution of complex engineering and technology problems. The basic idea of the method is first to build the probability model of the problem to be solved, then produce a large number of sample values of random variable according to the statistical property of random variable of probability model, and last through sampling calculation of probability model to get the approximate solution of the problem to be solved. Mathematical difficulties in structure reliability analysis can be avoided in Monte Carlo simulation method, and no matter if the state functions are nonlinear or not, the random variables are normal distribution or not, and the limit state functions are correlated or not, as long as the number of simulation is large enough, accurate failure probability and reliability index can be obtained. Thus, the Monte Carlo simulation method is used to assess the correctness of the new method in this paper.

The hypothesis that three or more failure modes exist simultaneously is a minimum probability event which is unnecessary when using the Monte Carlo method to calculate the reliability with dependent failure modes, and sampling calculation of (14) can be directly used. Consider 𝑅=𝑃^𝑚_𝑖=1𝐸_𝑖. (14)

Concrete steps are listed as follows:(1)initialize loop control variable 𝑖𝑖, sampling amount 𝑟 and safe times 𝑅𝑔, in which 𝑟 is a larger value and the others are zero;(2)generate 𝑁 dimensions vector of random number 𝐗(𝑖𝑖) according to the basic random variables 𝑋_ℎ (ℎ=1,2,…,𝑁);(3)calculate 𝑔₁^(𝑖𝑖),𝑔₂^(𝑖𝑖),…,𝑔_𝑚^(𝑖𝑖);(4)test if ⋂^𝑚_𝑖=1𝑔_𝑖^(𝑖𝑖)>0 (𝑖=1,2,…,𝑚) is satisfied, and if it is the case, 𝑅𝑔^(𝑖𝑖+1)=𝑅𝑔^(𝑖𝑖)+1; (15)(5)test if 𝑖𝑖≥𝑟 is satisfied, and if it is the case, the reliability is 𝑅=𝑅𝑔𝑟. (16) If it is not case, 𝑖𝑖=𝑖𝑖+1, back to the second step;(6)output reliability 𝑅.

The higher is the reliability, the more samples are required in the Monte Carlo method, and the Monte Carlo method is only used to verify the correctness of the model for reliability of components with dependent failure modes mentioned earlier.

4. Examples

Compute the reliability of the pin connection component in Figure 3. Given that 𝐹_𝑎∼𝑁(6000,1500²)𝑁, the diameter of pin 𝑑=10 mm, ℎ₁=ℎ₃=30 mm, and ℎ₂=8 mm. The shear strength [𝜏] and bearing strength of 20 pins are obtained by experiment and listed in Table 1.

Table 1 

The Statistics Data of Shear Strength and Bearing Strength of 20 Pins.

Figure 3 

A pin connection component structure.

The solution of the problem is listed as follows.(1)Force analysis. The force analysis of the pin is illustrated in Figure 4(a). The pin is divided into three parts I, II, and III with assumed plane m-m and plane n-n. Because the force condition of part I is the same as part III, only part I and II are taken as research objects as shown in Figure 4(b). From equilibrium conditions, we can get 𝑄₁=𝑄₂=𝐹_𝑎/2. Obviously, the bearing force of part II is bigger and the bearing area is smaller. Therefore, only part II is possible to fail in the example. (2)Failure mode analysis. The structure has two single failure modes. They are, respectively, 𝐸₁: shear fracture of the pin and 𝐸₂: bruise of the pin. Moreover, the structure is possible to have two failure modes synchronously. Therefore, the failure probability 𝐹 is 𝐹=𝑃𝐸₁+𝑃𝐸₂+𝑃𝐸₁𝐸₂. (17)(3)The limit state functions of the two failure modes are 𝑔₁=[𝜏]−2𝐹_𝑎𝜋𝑑²=[𝜏]−0.0064𝐹_𝑎,𝑔₂=𝜎_𝑗𝑦−𝐹_𝑎𝑑ℎ₂=𝜎_𝑗𝑦−0.0125𝐹_𝑎. (18) The shear strength of the pin [𝜏]∼𝑁(60.5,3²) MPa and the bearing strength of the pin [𝜎_𝑗𝑦]₁∼𝑁(162.9,6.1²) MPa are both obtained by statistics data. The relationship between [𝜏] and [𝜎_𝑗𝑦]₁ is shown in Figure 4, and the relationships between the two limit state functions are 𝑔₁∼𝑁(21.93,9.90);𝑔₂∼𝑁(69.42,20.72). (19)(4)According to the distribution pattern of the basic variables, the accounted functional relation between 𝑔₁ and 𝑔₂ is shown in Figure 5. The unknown parameters are obtained from (7) as 𝐴₂₋₁=1.90;𝐵₂₋₁=26.36;𝜎²₂₋₁=75.45, (20) and intercept 𝑎_𝑖−𝑗 is obtained from (8) as 𝑎₁₋₂∼𝑁(−13.87,4.57). (21)(5)Calculation of failure probability 𝐹 and reliability 𝑅. According to the standard normal distribution list, the failure probabilities 𝑃(𝐸₁) and 𝑃(𝐸₂) are 𝑃𝐸₁=Φ0−21.939.90=1.39×10⁻²,𝑃𝐸₂=Φ0−69.4220.72=5.38×10⁻⁴. (22) The failure probability 𝑃(𝐸₁𝐸₂) can be obtained from (9), (11), and (12) as 𝑃𝐸₁𝐸₂=⁰_−∞14.57√−12𝜋exp2𝑎₂₋₁+13.874.57²_{𝑎2−1−∞}19.90√−12𝜋exp2𝑔₁−21.939.90²d𝑔₁d𝑎₂₋₁+⁰_−∞19.90√−12𝜋exp2𝑔₁−21.939.90²d𝑔_10+∞14.57√−12𝜋exp2𝑎₂₋₁+13.874.57²d𝑎₂₋₁=3.04×10⁻⁴+1.61×10⁻⁵=3.20×10⁻⁴,𝐹=𝑃𝐸₁+𝑃𝐸₂−𝑃𝐸₁𝐸₂=1.41×10⁻²,𝑅=1−𝐹=0.9859. (23)(6)According to (4), conducting 10⁵ samples with the method of Monte Carlo, the reliability 𝑅=0.985613; when the sampling amount is 10⁶, the reliability 𝑅=0.98618, and when the sampling amount is 1.5×10⁷, the reliability 𝑅=0.985943.

(a)

(b)

Figure 4 

Force analysis of the pin.

Figure 5 

Functional relation between 𝑔1 and 𝑔2.

It is clear that the reliability of the pin obtained from the theoretical approach and Monte Carlo simulation shows good agreements; therefore, the double integration model for reliability of mechanical components with dependent failure modes established in this paper is correct.

5. Conclusion

(1)It is supposed that three or more failure modes arise synchronously to be a minimum probability event. The relationship between different ultimate state functions with dependent failure modes is established with the method of linear regression; the double integration model for reliability of mechanical components with dependent failure modes is established by imitating the stress-strength interference model, and it is verified with the method of Monte Carlo.(2)For the simplicity and ease of description of the problem, suppose that the relationship between ultimate state functions is linear. In case of square, cubic, or exponential relationship, linear transformation is used before utilizing this model.

Acknowledgments

This project is supported by the Ministry of Major Science and Technology of Liaoning (Grant no. 2012219020) and Shenyang Key Scientific and Technological Program (Grant no. F12-006-2-00).