Fatigue Reliability Analysis for the Manned Cabin of Deep Manned Submersibles Based on the Unified Fatigue Life Prediction Method

,,

(1.China Ship Scientific Research Center,Wuxi 214082,China;2.Hadal Science and Technology Research Center, Shanghai Ocean University,Shanghai 201306,China)

Fatigue Reliability Analysis for the Manned Cabin of Deep Manned Submersibles Based on the Unified Fatigue Life Prediction Method

WANG Ying-ying1,WANG Fang2,CUI Wei-cheng2

(1.China Ship Scientific Research Center,Wuxi 214082,China;2.Hadal Science and Technology Research Center, Shanghai Ocean University,Shanghai 201306,China)

Fatigue is generally the most common failure mechanism of metal structures,and it is widely known that there always exist uncertainties in structures because of variations in the process of design,fabrication,manufacture,and operation.Those uncertainties may be transmitted into the fatigue life of structures under cyclic loads,and fatigue reliability analysis methods are developed to accommodate those uncertainties in the prediction of fatigue life.In this paper,the feasibility to combine the unified fatigue life prediction(UFLP)method and the reliability analysis methods is carried out to offer a safety guidance in the usage of the structures.The limit state function is established based on the UFLP,and the effect of every parameter in fatigue life estimation is evaluated,then the fatigue reliability analysis based on a series of experimental data is carried out as an example by means of both the Monte-Carlo simulation and the JC method,the results of which are in good agreement with each other.

fatigue;crack growth;fracture mechanics

0 Introduction

Fatigue is generally the most common failure mechanism of metal structures.It has been reported that 80%-90%of all observed failures in mechanical and structural systems can be accounted for by fatigue[1].In reality,there always exist uncertainties in structures because of variations in the process of design,fabrication,manufacture,and operation,etc[2],and those uncertainties will have their own effects on the fatigue life of the structures.It may be too dangerous or over-conservative to use deterministic methods to predict the fatigue life.To overcome these drawbacks,the fatigue reliability analyses should be carried out to accommodate those uncertainties in the prediction of fatigue life[3].Reliability is the capacity of a structureto fulfill its specified functions in an assigned period[4].In the literature,the fatigue reliability refers to the probability that the structure will not fail due to metal fatigue.Fatigue reliability analysis tries to ensure that the occasions leading to catastrophe are extremely unlikely for a given targeted probability of failure[5].Many studies have been carried out to develop fatigue analysis methods[2,6-14],and lots of achievements have been made.

A unified fatigue life prediction(UFLP)method was proposed for fatigue analysis which has been proved to be a satisfactory approach based on crack growth rate theory by lots of validation analysis made in a deterministic way[15-16].However,the uncertainties during analysis were neglected which will be made up in the present paper.To achieve this,a brief overview on the fatigue reliability analysis regime is presented and two main kinds of fatigue reliability analysis methods are introduced firstly.Then the unified fatigue life prediction(UFLP)which can explain the crack growth behavior well is introduced.Based on that,the limit state function is established.And an example of the fatigue reliability analysis on a series of experimental data is accordingly carried out to exhibit the process.

1 An overview on the fatigue reliability analysis of the structures

1.1 Fatigue reliability analysis

Fatigue life of metal structures scatters as a result of the inherent uncertainties,and it is better to assess the fatigue life probabilistically rather than deterministically.The most common probabilistic assessment method is the fatigue reliability analysis[3].

Fatigue reliability analyses were applied in composite laminates[17],the highway and railway steel bridges[12,18-22],aerospace[23],offshore platform engineering systems[24-25],as well as many other fields during past several decades.Nowadays,fatigue reliability analyses are more and more commonly used in real practice,and they are usually involved in the optimization design.

In general,fatigue reliability analyses can be classified into two main categories which are respectively based on the conventional damage accumulation theory and fracture mechanics. The conventional method often employs the stress-life curve of the material to evaluate fatigue life[26-27]for high-cycle fatigue,and the strain-life curve according to the strain response of the material for low-cycle fatigue[28-31].As there are always notches,voids and inclusions in the material which can be regarded as the initiation of the cracks or the precracks,it is more accurate to use fracture mechanics to predict the probability whether a crack grows to/over a critical size.

1.2 Two kinds of fatigue reliability analysis methods

As introduced above,the uncertainties due to intrinsic flaws and extrinsic conditions will be drawn into the fatigue life of the structures[32].Reliability is the capacity of a structure to fulfill its designed functions in an assigned period[4].The fatigue reliability is the reliability that the structure will not fracture due to metal fatigue.Its analysis will combine probabilistic theory and fatigue life prediction methods.

1.2.1 Reliability index

The failure probability of a structure is Pf,whereand β is the reliability index,is the state limit function[33].is the standard normal cumulative probability function,then

Therefore,it can be deduced thatIt is easy to reach the conclusion that with a larger β,a structure is of a higher reliability.

From geometric perspective,the random variables in the original space areand they can be transformed into the normal random variablesin the standard normal space(u space)by means proposed by Rosenblatt[35].It is defined that the shortest distance from the origin to the limit-state surface in u space is the reliability index[36],written as β.As aforementioned,in the u space,β can be calculated as follows:

The solution to Eq.(2)is the most probable point(MPP for short)[34]uMPP.

1.2.2 Fatigue reliability analysis methods

1.2.2.1 Monte-Carlo method

There are several methods to carry out reliability analysis,among which the Monte-Carlo method is quite straightforward[32].Monte-Carlo method is based on the law of great numbers and it can be used without any statistical information[37]while in the fatigue analysis,the distribution ought to be known as a priori.The calculation flow chart of Monte-Carlo method is shown in Fig.1.Theoretically speaking,the Monte-Carlo method will not confront either nonlinear approximation or equivalent normalization,and so given adequate simulation it can obtain any accuracy, in some sense the Monte-Carlo method can offer a reference to the reliability analysis[18]. The Monte-Carlo method is convenient to use because it is flexible for whatever types of distributions and whatever forms of constraint functions[2].However,the efficiency of Monte-Carlo method is not high,especially when the simulation number N is large.If z1-α/2denotes the 1-α/2 quantile of the standard nor-mal distribution and Pfis the failure probability,ε is the percentage error,there exists[2,38],

Fig.1 The flow chart of Monte-Carlo method calculating reliability index

It can be seen from Eq.(3)that the Monte-Carlo simulation number will be very large to obtain a high accuracy,which would therefore cause expensive computational and time burden.Several more efficient sampling methods were developed to improve the simulation efficiency,such as Descriptive Sampling[39-40]and Importance Sampling[41-42],besides that the principle of JC method is another efficient method widely used in fatigue reliability analyses.

1.2.2.2 JC method

The JC method is based on the idea of MPP,and is originally developed in the regime of reliability analysis[36].Nowadays,the JC method is not only a main method in the regime of reliability analysis but is widely used in the field of optimization design[43].Expand the limit state function into Taylor series at MPP[32,44],and the terms whose orders are more than 1 are neglected

And the mean value and variance can be expressed as:

The reliability index is,

And the coordinates of the MPP are,

2 Unified fatigue life prediction method(UFLP)

It is very pivotal to calculate the crack growth rate in the fatigue performance predictionframe[19].A lot of crack growth models have been proposed to describe the crack growth rate curves till now.Paris law is the earliest and most widely used crack growth model based on the linear elastic fracture mechanics.Paris law has a simple form and is convenient to use,but it can not account for the effects of the load ratio[45],the mean stress and the load sequence,and can be applied for the stable growth stage only[46].

Thereafter hundreds of crack growth models were developed to explain more phenomena in crack growth[47-53].On the basis of McEvily model[51],the authors’group proposed a unified crack growth rate formula[15]:

where

The fatigue life prediction approach based on the unified crack growth rate model introduced above is then called UFLP,which is proved to have the capacity of explaining various fatigue phenomena,such as specimen thickness effect,compressive to compressive loading effect and overload effect,etc[15].

3 The limit state function

3.1 The probabilistic fatigue model

In the case of failure due to fatigue,the state limit functions can be established by the probabilistic fatigue models which describe the fatigue performance[54],such asgis the uncertain variables.The subscript‘N’refers to the total stress cycles;aNand KNare respectively the corresponding crack length and the stress intensity factor at N cycles.Similarly,acand KICare respectively the corresponding values of the crack length and the stress intensity factor at the critical state of fracture.

In the present paper,the state limit function is defined asand the reliability of a component or structure is as follows,

If there is,

3.2 Establishment of the limit-state function

In this paper,the crack growth model is UFLP,integrating Eq.(18)from a0to ac,so that

The limit state function can be expressed in the form of

4 An example of the fatigue reliability analysis

In Eq.(10),A and m can be obtained by fitting,and in general,m is taken as a constant[55-56],and other parameters are obtained either by fitting or by measuring.The effect of each parameter on the fatigue life is different in significance.For a series of experimental data of titanium alloy TC4-ELI with number of 5904 in which the TC specimens are imposed constant amplitude loads,and the parameters are shown in Tab.1.

Tab.1 The values of the parameters involved in the limit state

4.1 The uncertain variables

The effects on fatigue life of each parameter varying with the other parameters unchanged are illustrated in Figs.2-11.In the present analysis,m is treated as a constant according to the suggestions of Fisher[55]and Wirsching[56].Qualitatively,it can be seen from the figures that the fatigue life has a little scatter with variations of σy,σv,σb,re,n,KICand aN.However,it is easily to observe that the effects of A and a0on the fatigue life are significant which can be explained by Eq.(10)that the crack growth rate da/dN is proportional to parameter A,and then the variation in the value of A leads to different fatigue life proportionally.The initial crack length a0decides the crack growth rate in the beginning,i.e.,with the shorter macro-crack a0which is just larger than its threshold value,the crack propagates slower dramatically,therefore the fatigue life will increase significantly.Fig.11 illustrates that the fatigue life will increase(/reduce)by 80%(/almost1/3)when initial length of the crack decreases(/increases)by 16%.Fig.7 depicts the trend that fatigue life changes to a certain degree with variation of Keffth, nevertheless,Keffthis calculated by Eq.(14)where athis obtained through Eq.(15),thus Keffthis calculated through crack growth rate tests[16]in essence,i.e.,Keffthwill not change by a large margin and is not regarded as a variable in this paper.Generally speaking,it is reasonable to choose A and a0to be the uncertain variables in fatigue reliability analysis.It is supposed that both A and a0follow the lognormal distribution[57-59].In marine structures,a0is usually measured by either Non Destructive Evaluation(NDE)or the Equivalent Initial Flaw Size(EIFS)[19]. The test data used in this paper,actually,aim to study the crack fatigue behavior instead of the fatigue reliability,so that we assume the statistical properties of a0,and that is also the reason why the initial length of a0is quite large.In literature,different suggestions on a0havebeen made.For example,Albrecht&Yazdani[60]and Zhao et al[61]suggested the initial length of crack to be a lognormal variable with mean of 0.02 inch and a CoV of 0.5 while Li& Huang[19]suggested the initial length of crack to be a lognormal variable with mean of 1mm and a CoV of 0.3.In the present paper,to demonstrate of the fatigue reliability analysis procedure,it is without loss of generality that we assume the initial length of crack following a lognormal distribution with mean of 19.027 mm(as the crack begins to propagate in test)and a CoV of 0.3.The crack growth test specimens were conducted using an MTS810 servo-hydraulic testing machine.Standard C-T specimens with dimensions of B=12.5 mm;W=100 mm were cut and machined from 90 mm-thick hot rolled thick plate at load ratio of 0.0 to obtain crack growth rate data.

The statistics of A is obtained by MATLAB when fitting the crack growth curve.Then the statistical properties of A and a0are listed in Tab.2.

Tab.2 The statistical properties of the uncertain variables in the limit state

Fig.2 The effect of σyon the a-N curve

Fig.3 The effect of σvon the a-N curve

4.2 The fatigue reliability analysis

Fig.12 illustrates the experimental results and the simulation results of the crack growth. It can be seen that the crack grows at a quite large rate because the value of a0is quite large. Fig.13 illustrates the results of fatigue reliability analyses both by Monte-Carlo simulation and the JC method.Both of the results are in good agreement with each other.However,as the value of a0in the test is relatively large,which means the structure has more serious defects,the reliability index β is relatively small.It can be seen from Fig.13 that the value of β has dropped to 3 after about 130 cycles,i.e.the fatigue probability of the structure is about 0.13%.Fig.13 implies that it is not applicable to expect a high fatigue reliability index with larger crack initial length.The present test is lack of crack growth data in its earlier stage,then Eq.(10)can not be extrapolated to the region in which the crack length shorter than a0.However,it is promising that the fatigue life with a small initial crack size under the same circumstances would be longer and that the fatigue reliability index would maintain a higher level. More work will be done in the future.

Fig.4 The effect of σbon the a-N curve

Fig.5 The effect of reon the a-N curve

Fig.6 The effect of n on the a-N curve

Fig.7 The effect of Keffthon the a-N curve

Fig.8 The effect of KICon the a-N curve

Fig.9 The effect of aNon the a-N curve

Fig.10 The effect of A on the a-N curve

Fig.11 The effect of a0on the a-N curve

Fig.12 The crack growth rate curve

Fig.13 The reliability index β.vs.N

5 Conclusions

In this paper,the method of fatigue reliability analysis based on the unified fatigue life prediction(UFLP)model has been introduced and the reliability theory to evaluate the fatigue reliability index of a structure is demonstrated.A reliability analysis on a series of experimental data has been carried out for illustration and the main conclusions can be drawn as follows:

(1)The limit state function can be established through several ways,and in this paper it was established bybased on UFLP method which has the capacity of describing the crack growth behavior well.

(2)Several parameters will affect the fatigue life in different levels.It is necessary to separately evaluate the effects of each parameter in limit state function on the fatigue life.In the cases studied presently,only A and a0are regarded as the uncertain variables,and the same procedure can be applied in other circumstances.

(3)Two kinds of methods have been employed in the present paper to assess the fatigue reliability and it turned out that the Monte-Carlo method and the JC method drew similar results,indicating the feasibility of combing UFLP and reliability methods to assess fatigue reliability.

Acknowledgments

This work is supported by the Sci-tech Innovation Funds from CSSRC(Grant No.G3314) and the State Key Program of National Natural Science of China‘Structural Reliability Analysis on the Spherical Hull of Deep-sea Manned Submersibles’(Grant No.51439004).

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基于統(tǒng)一的疲勞壽命預報方法（UFLP）的深潛器載人艙疲勞可靠性分析

王瑩瑩1，王 芳2，崔維成2

（1中國船舶科學研究中心，江蘇 無錫 214082；2上海海洋大學 深淵科學技術研究中心，上海 201306）

疲勞是承受循環(huán)載荷的金屬結構最重要的失效機制，現(xiàn)已有多種金屬疲勞壽命預報方法。實際的設計、制造、加工和操作等工序環(huán)節(jié)中總是存在不確定性，這些不確定性的影響將在結構的疲勞壽命中富集，為此發(fā)展了疲勞可靠性分析。該文討論了將疲勞可靠性分析方法和統(tǒng)一的疲勞壽命預報方法（UFLP）相結合的可行性，為海洋結構物的安全使用提供參考。文中的極限狀態(tài)方程是基于統(tǒng)一的疲勞壽命預報方法（UFLP）得到的，并討論了其中各參數(shù)對疲勞壽命的影響；針對一組TC4-ELI的疲勞裂紋擴展數(shù)據(jù)，分別采用Monte-Carlo法和JC法開展了疲勞可靠性分析，兩種分析方法得到的結果彼此符合得較好。

疲勞；裂紋擴展；斷裂力學

U661.4

:A

王瑩瑩（1983-），女，中國船舶科學研究中心博士研究生；

U661.4

A

10.3969/j.issn.1007-7294.2016.03.010

1007-7294（2016）03-0335-13

王 芳（1979-），女，上海海洋大學副研究員；

崔維成（1963-），男，上海海洋大學教授，博士生導師。

Received date：2015-11-25

Foundation item:Supported by Supported by the Sci-tech Innovation Funds from CSSRC(Grant No.G3314)and the State Key Program of National Natural Science of China‘Structural Reliability Analysis on the Spherical Hull of Deep-sea Manned Submersibles’(Grant No.51439004)

Biography：WANG Ying-ying(1981-),female,Ph.D.student,E-mail:yunbeidou@yeah.net; Wang Fang(1979-),female,associate professor.