Research on fatigue life of steel wire ropes under impact loads based on double Pareto lognormal distribution

Abstract

Fatigue damage is the most important failure mode of wire ropes, and models of cumulative fatigue damage of wire ropes under multiple impact loads have been proposed based on the principle of microdamage linear accumulation. Fatigue damage of wire ropes under a single impact load obeys lognormal distribution, but when under different impact loads, it presents bilateral trailing phenomenon. In this work, a fitting method of wire rope fatigue life analysis under many different impact loads based on double Pareto lognormal distribution is proposed, which can accurately describe and reasonably explain the above phenomenon. The fitting result of test data calculated by double Pareto lognormal distribution is better than the result calculated by lognormal distribution and power-law distribution. This reveals that the model of double Pareto lognormal distribution can well describe the reliability degradation process of wire ropes, thereby providing guidance on the fatigue life estimation of wire ropes.

Introduction

As key bearing components in engineering project, wire ropes are widely used in the fields of mine, bridge, shipping, aerospace, and aviation because of their flexible spatial spiral structures. A great deal of research activities on the reliability and service life of wire ropes have been carried out due to their wide range of applications. However, there have been no mature methods to assess the reliability and service life of them up to now due to their complicated structures and a variety of stress types that are involved. In engineering fields, the fatigue life of wire ropes is routinely conservatively estimated, leading to a huge amount of waste. In 1988, 8000 wire ropes replaced and collected from laboratories and engineering sites were analyzed in America. It revealed that there was only slight strength loss or even no strength loss among more than 70% of the replaced wire ropes. Meanwhile, based on the statistical results in Japan, the strength of more than 50% of replaced wire ropes could reach 90% of the strength level of new ones. Chinese statistical results also showed that 20%–30% of the consumption on wire ropes could be saved if the state detection was reinforced or improved and the replacement was undertaken timely. ¹ Therefore, research on the accurate methods for assessing the service life of wire ropes is of great significance, which can improve the work efficiency and raise economic benefits of wire ropes.

Fatigue damage is the key failure mode of wire ropes, and how to model the fatigue process and assess the reliability and service life of wire ropes becomes more and more significant. A lot of research on the fatigue of wire ropes has been carried out. Torkar and Arzenšek ² analyzed the failure of hoisting wire ropes, discovered fatigue fracture characteristics of steel wires, and observed the fatigue crack source on the surface caused by decarbonization. Chaplin ³ discussed the effect of wear in detail on the standards for inspection and replacement of wire ropes. Atianxa^4–6 calculated the fatigue strength of steel wires based on their geometrical characteristics, alternating tensile stress, bending stress, and contact stress according to the actual working conditions of wire ropes. Beretta and Boniardi ⁷ calculated the fatigue strength of eutectoid steel wires under two different types of surface treatments based on the proposed calculation method for the fatigue strength of steel wires. Paton et al. ⁸ deeply studied the relationship between the rigidity and fatigue strength of wire ropes during the tensile fatigue process. The failure modes of wire ropes studied by a bending fatigue tester were summarized in the US navy manual of wire rope, ⁹ in which the inner stress of wire ropes was analyzed, and the effect of wire rope diameter and pulley diameter on the failure of wire ropes was presented. Gerdemeli et al. ¹⁰ analyzed fatigue life of axial loaded wire rope strands with finite element method and obtained fatigue life variance of axial loaded 1 + 6 simple strand based on Goodman approach. Wang et al. ¹¹ analyzed 6 × 19 + IWS rope and three-layered strand using finite element method and presented implications to fatigue life estimations of fretted wires. Research on the service life of wire ropes mainly focuses on stress failures under axial static and fatigue loading currently. However, theoretical and experimental research on the service life of wire ropes under multiple different impact loads in practical use has not been carried out.

Aiming at solving the above issue, models and processes of cumulative fatigue damage under different impact loading conditions are proposed and analyzed for wire ropes in this work, including cumulative fatigue damage under a single impact load, repeated action of the same impact load, and a combination of several different impact loads. The fitting of cumulative fatigue damage under combined impact loads is carried out by double Pareto lognormal distribution to assess the fatigue life of wire ropes.

Assumption of model

Wire rope failure is affected by many factors such as wear, corrosion, and fatigue caused by the integrative action of bending, stretching, and vibrations. Wire rope failure under impact loads is induced by the performance degradation caused by cumulative fatigue damage under the integrative action of different impact loads. Damage of wire ropes caused by each impact load is stochastic due to the uncertainty of stress levels and other environmental factors.

Damage of wire rope caused by impact stress load

Wire rope failure is induced by the cumulative fatigue damage under the integrative action of different impact loads. Damage of wire ropes under a single impact load is first discussed as follows.

Damage on a wire rope under the kth (k = 1, 2, &, K) magnitude of different impact loads $S_k$ is denoted by $\Delta D^{(k)}$ . As the material, structure, and processing of a wire rope and single impact load are random variables, the corresponding damage of wire rope $\Delta D^{(k)}$ is also a random variable. The mean and variance of wire rope damage caused by a single impact stress load, noted by $E[\Delta D^{(k)}]$ , $\mathrm{Var}[\Delta D^{(k)}]$ , respectively, are expressed as follows

$$E[\Delta D^{(k)}]={\mu }_k$$

(1)

$$\mathrm{Var}[\Delta D^{(k)}]={\sigma }_k^2$$

(2)

When the wire rope failure is induced by the cumulative fatigue damage under the repeated action of an impact load $S_k$ , cumulative fatigue damage under repeated impact load can be calculated based on the principle of microdamage linear accumulation. Damage under n times of a repeated load is the accumulation of damage under every single load according to the principle of microdamage linear accumulation. ¹² Therefore, cumulative fatigue damage $D^{(k)}(n)$ under n times of repeated impact load $S_k$ can be calculated as follows

$$D^{(k)}(n)=\sum _{i=1}^n\Delta D_i^{(k)}$$

(3)

where $\Delta D_i^{(k)}$ expresses the damage of wire ropes under the ith $(i=1,2,\dots ,n)$ impact load $S_k$ .

The damage of steel wire gradually increases with the increase in load action times. $D^{(k)}(i)(i=1,2,\dots ,n)$ expresses the damage after the ith $(i=1,2,\dots ,n)$ impact load $S_k$ , so $D^{(k)}(1)<D^{(k)}(2)<\cdots <D^{(k)}(n)$ is a bunch of random variable sequences obviously. $D^{(k)}(i)-D^{(k)}(i-1)(i=1,2,\dots ,n)$ expresses the ith damage increment, and $D^{(k)}(0)$ expresses the amount of original damage. The increment of damage complies with the rule of proportional effect model,^13,14 which means that the ith damage increment $D^{(k)}(i)-D^{(k)}(i-1)$ is in proportion to $D^{(k)}(i-1)$ . The failure will occur when the damage reaches $D^{(k)}(n)$

$$\frac{D^{(k)}(i)-D^{(k)}(i-1)}{D^{(k)}(i-1)}=X_i,i=1,2,\dots ,n$$

(4)

where ratio $X_i(i=1,2,\dots ,n)$ is a random variable, and is assumed independent of each other, but it may be not identically distributed, and the distribution can also be unknown. The following formula can be obtained if the summation of $X_i$ is carried out

$$\sum _{i=1}^n{X}_i=\sum _{i=1}^n\frac{D^{(k)}(i)-D^{(k)}(i-1)}{D^{(k)}(i-1)}$$

(5)

$D^{(k)}(i)-D^{(k)}(i-1)$ expresses the microdamage increment at each stage and is denoted by $\Delta D^{(k)}(i)$ . Then, the above formula can be expressed as follows

$$\sum _{i=1}^n{X}_i=\sum _{i=1}^n\frac{\Delta D^{(k)}(i)}{D^{(k)}(i-1)}$$

(6)

When $\Delta D^{(k)}(i)\to 0$ , and n is sufficiently large, the limitation of the above formula is as follows based on the definition of integral

$$\sum _{i=1}^n{X}_i\approx \underset{D^{(k)}(0)}{\overset{D^{(k)}(n)}{\int }}\frac{\mathrm{dx}}{x}={\ln x|}_{D^{(k)}(0)}^{D^{(k)}(n)}=\ln D^{(k)}(n)-\ln D^{(k)}(0)$$

(7)

It can also be written as follows

$$\ln D^{(k)}(n)\approx \sum _{i=1}^n{X}_i+\ln D^{(k)}(0)$$

(8)

As the damage increment under every impact load $X_i(i=1,2,\dots ,n)$ is mutually independent, the distribution of $\sum _{i=1}^n{X}_i$ converges to normal distribution according to the central limit theorem. Therefore, $\ln D^{(k)}(n)$ is gradually close to normal distribution when n is sufficiently large, which means that $\ln D^{(k)}(n)$ is a random variable gradually close to logarithmic normal distribution.

Cumulative damage of wire rope $D^{(k)}(n)$ under the same stress obeys logarithmic normal distribution, and the expectation and variance of $D^{(k)}(n)$ , noted by $E(D^{(k)}(n))$ and $\mathrm{Var}(D^{(k)}(n))$ , respectively, can be expressed as follows according to the character of the sum of independent random variables

$$E(D^{(k)}(n))=\sum _{i=1}^{n}E[\Delta D_i^{(k)}]=\sum _{i=1}^{n}E[\Delta D^{(k)}]=n{\mu }_k$$

(9)

$$\mathrm{Var}(D^{(k)}(n))=\sum _{i=1}^n\mathrm{Var}[\Delta D_i^{(k)}]=\sum _{i=1}^n\mathrm{Var}[\Delta D^{(k)}]=n{\sigma }^2$$

(10)

Damage of wire rope caused by integrative action of different impact loads

If a wire rope is under two different loads, the corresponding cumulative damage is denoted by $D^{(1)}(n),D^{(2)}(n)$ , the both of which obey logarithmic normal distribution. The total cumulative damage is calculated as follows based on the principle of microdamage linear accumulation, $D=D^{(1)}(n)+D^{(2)}(n)$ . If $f_X,f_Y$ is the probability density function of $D^{(1)}(n),D^{(2)}(n)$ , respectively, the probability density function of $D$ can be calculated as follows

$$f_D(d)=\underset{-\infty }{\overset{\infty }{\int }}f(d-y,y)\mathrm{dy}$$

(11)

where $f$ is the joint probability density function of $D^{(1)}(n)$ and $D^{(2)}(n)$ . As the cumulative damage at each stage is mutually independent, the probability density function of $D$ can be calculated by convolution formula as follows

$$\begin{array}{c}\multicolumn{1}{c}{f_D(d)=f_X*f_Y=\underset{-\infty }{\overset{\infty }{\int }}{f}_X(d-y)f_Y(y)\mathrm{dy}}\\ \multicolumn{1}{c}{=\underset{-\infty }{\overset{\infty }{\int }}{f}_X(x)f_Y(d-x)\mathrm{dx}}\end{array}$$

(12)

where * expresses convolution.

The mathematical expectation and variance of $D$ , noted by $E(D)$ and $\mathrm{Var}(D)$ , respectively, can be calculated as follows

$$\begin{array}{c}E(D)=E(D^{(1)}(n))+E(D^{(2)}(n))\hfill \\ \mathrm{Var}(D)=\mathrm{Var}(D^{(2)}(n))+\mathrm{Var}(D^{(2)}(n))\hfill \end{array}$$

(13)

It can be obtained that the expectation and variance of cumulative damage $D^{(k)}(n)$ show a linear increase under the same stress level. The expectation and variance are cumulative under two-step loading, as shown in Figure 1.

Figure 1.

The relationship between the distribution of cumulative damage and the cycle numbers under two-step loading.

Wire rope failure is induced by the cumulative fatigue damage under the integrative action of different impact loads. Hence, the above formula can be extended to cumulative damage under K magnitudes of different impact loads. Cumulative damage under each magnitude of impact load is denoted by $D^{(1)}(n),D^{(2)}(n)\cdots D^{(K)}(n)$ . The total cumulative damage under the combined impact loads can be calculated as $D=D^{(1)}(n)+D^{(2)}(n)+\cdots +D^{(K)}(n)$ . Then, the probability density function of $D$ is calculated as follows

$$f_D(d)=f_X*f_Y*\cdots *f_K$$

(14)

The mathematical expectation and variance of $D$ are calculated as follows

$$\begin{array}{c}E(D)=E(D^{(1)}(n))+E(D^{(2)}(n))+\cdots +E(D^{(K)}(n))\hfill \\ \mathrm{Var}(D)=\mathrm{Var}(D^{(1)}(n))+\mathrm{Var}(D^{(2)}(n))+\cdots +\mathrm{Var}(D^{(K)}(n))\hfill \end{array}$$

(15)

The sum distribution of two or more logarithmic normal distributions cannot be expressed usually, and its distribution curve does not fully meet the standard logarithmic normal distribution as shown in Figure 1. The sum distribution presents the bilateral trailing phenomenon in its distribution curve, which becomes especially evident when under the condition of multistage stress accumulation. The major part of the above distribution curve usually obeys logarithmic normal distribution, but the tail of the distribution curve is closer to power-law distribution. Thus, direct use of logarithmic normal distribution to describe cumulative damage under the integrative action of different impact loads will lead to imprecise results, whereas double Pareto lognormal distribution provides a better method to describe the distribution characteristics of bilateral power-law trailing phenomenon.^16,17 Double Pareto lognormal distribution can be formed by the superposition of a variety of logarithmic normal distributions, which is consistent with the characteristics of the cumulative damage distribution under combined impact loads. Thus, the double Pareto lognormal distribution is adopted to analyze cumulative damage under integrative action of different impact loads and finally to acquire more precise fatigue life estimation of wire ropes.

Double Pareto lognormal distribution

Random process $X_t=X_{t-1}{C}_t$ , $X_0=c,c>0$ is assumed, and $C_t$ is the random variable in normal distribution with the parameter of $(\mu ,{\sigma }^2)$ . When $t=T$ , random variable $X_T$ obeys logarithmic normal distribution with the parameter of $(\mu T,{\sigma }^{2}T)$ . If the time t is also regarded as a random variable, and this process is assumed to suspend at a suspension rate $\lambda$ , duration time $t$ of a random process in the whole system will obey exponential distribution with the parameter of $\lambda$ as expressed below

$$f(t)=\lambda e^{-\lambda t},t\ge 0$$

(16)

It is proved that the distribution of random variable $X_t$ in logarithmic normal distribution mixed with exponential distribution shows the bilateral power-law trailing phenomenon and is named double Pareto lognormal distribution. ¹⁷

Probability density function of the double Pareto lognormal distribution is as follows

$$f\left(x\right)=\underset{t=0}{\overset{\infty }{\int }}\lambda e^{-\lambda t}\frac{1}{\sqrt{2\pi t}\sigma x}{e}^{-\frac{{(\ln x-t\mu )}^2}{2t{\sigma }^2}}\mathrm{dt}$$

(17)

where x is a variable, and $\lambda$ , $\mu$ , and $\sigma$ are parameters of the double Pareto lognormal distribution.

Parameter estimation

The service life of a wire rope under the integrative action of different impact loads is related to the coupling of damage under each impact load. It has been demonstrated in the previous study that the service life of a wire rope under a single impact load obeys logarithmic normal distribution, so the service life under combined impact loads is the coupling of multiple logarithmic normal distributions. Double Pareto lognormal distribution is recommended to express such coupling.

It is assumed that cumulative fatigue $D^{(k)}(n)$ under impact load $S_k$ obeys double Pareto lognormal distribution, so the fatigue life $T_k$ under multiple impact loads also obeys double Pareto lognormal distribution

$$\begin{array}{cc}\multicolumn{1}{c}{f\left(x\right)}& =\underset{t=0}{\overset{\infty }{\int }}\lambda e^{-\lambda t}\frac{1}{\sqrt{2\pi t}\sigma x}{e}^{-\frac{{(\ln x-t\mu )}^2}{2t{\sigma }^2}}\mathrm{dt}\hfill \\ \multicolumn{1}{c}{}& =\underset{t=0}{\overset{\infty }{\int }}\lambda e^{-\lambda t}\frac{1}{\sqrt{t}}\frac{1}{\sqrt{2\pi }\sigma x}{e}^{-\frac{{(\ln x-t\mu )}^2}{2t{\sigma }^2}}\mathrm{dt}\hfill \\ \multicolumn{1}{c}{}& \approx \underset{t=0}{\overset{\infty }{\int }}\lambda e^{-\lambda t}\frac{1}{\sqrt{t}}\mathrm{dt}\frac{1}{\sqrt{2\pi }\sigma x}{e}^{-\frac{{(\ln x-\mu )}^2}{2{\sigma }^2}}\hfill \\ \multicolumn{1}{c}{}& =\sqrt{\lambda \pi }\frac{1}{\sqrt{2\pi }\sigma x}{e}^{-\frac{{(\ln x-\mu )}^2}{2{\sigma }^2}}\hfill \end{array}$$

(18)

then

$$\frac{f\left(x\right)}{\sqrt{\lambda \pi }}=\frac{1}{\sqrt{2\pi }\sigma x}{e}^{-\frac{{(\ln x-\mu )}^2}{2{\sigma }^2}},\frac{f\left(x\right)}{\sqrt{\lambda \pi }}~\mathrm{LN}(\mu ,{\sigma }^2)$$

(19)

$$\log \left(\frac{f\left(x\right)}{\sqrt{\lambda \pi }}\right)~N(\mu ,{\sigma }^2)$$

(20)

The formula indicates that an adjusting parameter $\lambda$ is added on the basis of logarithmic normal distribution, which would make the result more accurate in theory.

The fatigue life $T_k$ under multiple impact loads $S_k$ obeys double Pareto lognormal distribution, the following process is deduced for N sample data $N^{(k)}\stackrel{\wedge }{=}\{N_1^{(k)},N_2^{(k)},N_3^{(k)},\dots ,N_N^{(k)}\}$

$$\begin{array}{c}\log \left(\frac{N^{(k)}}{\sqrt{\lambda \pi }}\right)\stackrel{\wedge }{=}\hfill \\ \{\log \left(\frac{N_1^{(k)}}{\sqrt{\lambda \pi }}\right),\log \left(\frac{N_2^{(k)}}{\sqrt{\lambda \pi }}\right),\log \left(\frac{N_3^{(k)}}{\sqrt{\lambda \pi }}\right),\dots ,\log \left(\frac{N_N^{(k)}}{\sqrt{\lambda \pi }}\right)\}\hfill \end{array}$$

(21)

The above formula obeys normal distribution $N({\mu }_{T_k},{\sigma }_{T_k}^2)$ , and the maximum likelihood estimation of parameter ${\mu }_{T_k},{\sigma }_{T_k}^2$ is as follows

$$\begin{array}{c}{\hat{\mu }}_{T_k}=\frac{1}{N}\sum _{i=1}^N\log \left(\frac{N_i^{(k)}}{\sqrt{\lambda \pi }}\right)\hfill \\ {\hat{\sigma }}_{T_k}^2=\frac{1}{N-1}\sum _{i=1}^N{(\log \left(\frac{N_i^{(k)}}{\sqrt{\lambda \pi }}\right)-{\mu }_{T_k})}^2\hfill \end{array}$$

(22)

Therefore, the reliability function of wire rope under impact load $S_k$ is as follows

$$R_k(t)=P(T_k\ge t)=1-\Phi \left(\frac{\log \frac{t}{\sqrt{\lambda \pi }}-{\hat{\mu }}_{T_k}}{{\hat{\sigma }}_{T_k}}\right),k=1,2,\dots$$

(23)

By substituting the estimated values of the parameters into the above equation, the reliability function of a wire rope under different specific impact loads can be acquired. As the service life of a wire rope obeys double Pareto lognormal distribution, the average life $E[T_k]$ under the impact load $S_k$ is as follows

$$E[T_k]=\sqrt{\lambda \pi }\exp ({\hat{\mu }}_{T_k}+\frac{{\hat{\sigma }}_{T_k}^2}{2})$$

(24)

Case study of reliability life assessment on wire rope

The impact loads of a wire rope in actual service vary at a certain stress range. The impact load of a wire rope in use is assumed equivalent to K magnitudes of different impact loads, and the service life in use is assessed by double Pareto lognormal distribution based on the life estimation under the K magnitudes of different impact loads.

A certain type of wire ropes was chosen for this case study. It consists of six threads, which are 10.0 m in length and 32.2 mm in diameter. Five samples were randomly chosen to be tested. The impact experiments of these wire ropes with fixed-ends were carried out under impact loads in actual service ranging from 3.0 × 10⁵ N to 8.0 × 10⁵ N, and impact speed ranges from 30 to 80 m/s. Tensions of both ends of the wire ropes were measured by strain gauge.

First, cluster analysis of actual stress state is conducted for the wire ropes by the method of k-mean clustering. The two-dimensional coordinates of wire rope stress $(X_1,X_2)$ is established by the left maximum tension $Y_1$ and right maximum tension $Y_2$ , where $X_1=max(Y_1,Y_2)$ , $X_2=|Y_1-Y_2|$ . Sample data obtained from 705 effective tests on the five wire ropes are selected to make clustering according to impact loads: S₁ = 3.65 × 10⁵ N, S₂ = 5.01 ×10⁵ N, S₃ = 5.22 × 10⁵ N, S₄ = 6.18 × 10⁵ N, and S₅ = 6.77 × 10⁵ N. Table 1 shows the impact times as well as cumulative times of each wire rope under different impact loads.

Table 1.

Impact times under different impact loads of the five wire ropes.

Rope	Loads					Cumulative times
	S 1	S 2	S 3	S 4	S 5
Wire rope 1	11	7	6	48	21	93
Wire rope 2	6	18	7	20	54	105
Wire rope 3	21	52	7	22	25	127
Wire rope 4	8	32	34	83	29	186
Wire rope 5	6	40	86	51	11	194

The impact stress loading of the wire ropes in actual service can be considered to be equivalent to the integrative action of the five impact loads $S_1,S_2,S_3,S_4,S_5$ .

Then, the service life of wire ropes under the five impact loads is calculated. The service life of the wire ropes under the five impact loads is estimated according to the method reported in the literature, ¹⁵ as shown in Table 2.

Table 2.

Service life estimation under different impact stress loads based on microdamage.

Impact loads	Microdamage, $d^{(k)}$ (times−1)	Service life estimation of wire rope, $N^{(k)}=\frac{1}{d^{(k)}}$ (times)
S 1	0.002	500
S 2	0.004500045	222.22
S 3	0.003500053	285.71
S 4	0.006499838	153.85
S 5	0.00699986	142.86

According to double Pareto lognormal distribution, the adjusting parameter λ needs to be determined. Damage increment of steel wires under the action of impact stress loads $S_1,S_2,S_3,S_4,S_5$ is assumed to obey exponential distribution, and the error function J(λ) is the sum of square error between practical measured data N and observational data f(d,λ). The error function J(λ) converges to the minimum by the nonlinear curve fitting method, as a consequence, the fitting curve of the relation between microdamage and life estimation of wire rope under different impact stress loads is completed. Therefore, $f(\lambda )$ is an exponential distribution with the parameter of $\lambda$

$$J(\lambda )=\sum _i{(f(d_i,\lambda )-N_i)}^2$$

(25)

Figure 2 presents the measured data from Table 2 and the fitted curve based on the formula (16). The value of $\lambda$ in the exponential distribution is 636.8306. Coefficient R² used to evaluate the goodness of fit is calculated to be 0.9996, which is close to 1, indicating that the fitting degree is fairly good. It is shown that the distribution of damage increment under the impact stress loads obeys exponential distribution according to the goodness of fit and the fitting curve.

Figure 2.

The exponential distribution fitting curve of damage increment under the stresses.

The values of $\hat{\mu }$ and $\hat{\sigma }$ are estimated according to the actual data and fitting curve, but there still exists some errors. In order to obtain more precise parameter values, the three parameters of $\hat{\mu }$ , $\hat{\sigma }$ , and $\lambda$ are determined by the combination of double Pareto lognormal distribution and particle swarm optimization, combined with formula (22). By comparing the parameter values to the actual data, it can be seen that the damage increment varies with the times of applied stress, so this result fits the actual data curve more accurately, and the resulting service life ranges would be more precise. It is stated that the adjustment of fit curves in this work makes the obtained life results more coincident with the actual life of the steel wires. Therefore, fatigue life of wire ropes under multiple impact stress loads described by the double Pareto lognormal distribution is reasonable.

Formula (22) is a ternary quadratic equation. By substituting the data in Table 1 into Formula (22), $\hat{\mu }$ and $\hat{\sigma }$ are calculated and the results are shown in Table 3.

Table 3.

$\hat{\mu }$ and $\hat{\sigma }$ calculated by formula (22).

${\hat{\mu }}_{T_1}$	${\hat{\mu }}_{T_2}$	${\hat{\mu }}_{T_3}$	${\hat{\mu }}_{T_4}$	${\hat{\mu }}_{T_5}$
6.2215	5.4042	5.6545	5.036	4.9616
${\hat{\sigma }}_{T_1}$	${\hat{\sigma }}_{T_2}$	${\hat{\sigma }}_{T_3}$	${\hat{\sigma }}_{T_4}$	${\hat{\sigma }}_{T_5}$
0.097311	0.019961	0.017242	0.0058673	0.0051351

Then, the service life estimation of the five wire ropes under actual impact stress loads is determined, as shown in Table 4.

Table 4.

Service life estimation of wire rope under different impact loads.

Impact loads	Service life estimation (times)
S 1	506
S 2	224
S 3	291
S 4	149
S 5	144

Interval estimation on the service life of wire ropes (confidence level, 1 − α = 0.95) under different impact stress loads can be determined by the method of Bootstrap, as shown in Table 5.

Table 5.

Interval estimation on service life of wire rope under different impact loads.

Impact loads	Interval estimation on service life (times)
S 1	[497.62, 516.37]
S 2	[284.95, 286.89]
S 3	[221.82, 222.93]
S 4	[153.74, 153.97]
S 5	[142.74, 142.92]

From the service life of wire ropes shown in Table 5, it can be concluded that the interval estimation on the service life of wire ropes determined by the proposed model and double Pareto lognormal distribution can well express the life time evaluation in use and can be directly applied for reliability assessments. The service life prediction based on the double Pareto lognormal distribution is more practical than the estimation under a single impact load for wire ropes. It should be noted that double Pareto lognormal distribution has more parameters than the logarithmic normal distribution, which means that more test data are required to acquire more accurate assessment of those parameters.

Conclusion

As the reliability degradation process of wire ropes under different impact loads is similar to logarithmic normal distribution and presents bilateral trailing phenomenon, double Pareto lognormal distribution is recommended to describe the reliability degradation under multiple impact fatigue loads. Double Pareto lognormal distribution can reasonably explain the above phenomenon. Moreover, it provides a new method for service life assessments from the perspective of random processes. The case demonstrates that double Pareto lognormal distribution can well describe the reliability degradation process for wire ropes and accordingly provide the experience and reference for integrative assessments of wire rope reliability.