[doi 10.1109%2ficrms.2011.5979247] liu, qin; qian, yunpeng; wang, dan; sun, zhili -- [ieee 2011 9th...

7/25/2019 [Doi 10.1109%2Ficrms.2011.5979247] Liu, Qin; Qian, Yunpeng; Wang, Dan; Sun, Zhili -- [IEEE 2011 9th Internation

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An Efficient Method for Strain Fatigue Reliability

Analysis

Qin Liu

Department of Mechanical Engineering and Automation

Northeastern University

Shenyang, 110004, China

[email protected]

Yunpeng Qian, Dan Wang

North China System Engineering Institute

CNGC

Beijing, 100089, China

Zhili Sun

Department of Mechanical Engineering and Automation

Northeastern University

Shenyang, 110004, China

AbstractBased on Manson-Coffin Equation, a strain fatigue

reliability model was built. Because of high nonlinear degree,

the first-order reliability method has convergence difficulty

for the model. An efficient iterative algorithm for strain

fatigue reliability is proposed by using automatic step

adjustment method etc., and the numerical results show that

the proposed method has a good convergence compared with

FORM.

Keywordsstrain fatigue; reliability; iterative algorithm;

automatic step adjustment

I. INTRODUCTION

Low cycle fatigue is one of the most common

mechanisms for mechanic product fatigue fracture. It is

usual that the structure failed by low cycle fatigue is the

key component of mechanic product, so that the products

reliability and durability are mainly affected. Generally, to

a structure being subjected to cycle load, most is in the

range of elastic response, only local dangerous area is in

the range of plastic because of large stress status, and the

strain intensity in the local area determine structure life[1]

.

Therefore, engineers consider that local strain is one of

most main parameters for low cycle fatigue analysis, and

low cycle fatigue is also called strain fatigue. As

experience shows, structure life can be several times in

case of a 10% deviation to strain [2]

. However, the local

strain is largely speculative because there are random loads,

random material performance, and random size in structure.

So, the structure reliability analysis considering random

factors is a hot research area.

Usually, the life model of low fatigue structure is built

by Coffin-Manson equation, which is highly non-linear.

The convergence for solving reliability degree of strain

fatigue structure by using convention reliability way such

as first-order reliability method (FORM) is slow or difficult.

Considering the features of strain fatigue life, an efficient

and steady iterative algorithm for strain fatigue reliability is

proposed by using some optimization methods.

II. STRAINFATIGUE ANALYSIS

The structure life data can be expressed by Morrow

correction equation based on Coffin-Manson equation.

( ) ( )cfbmf

NNE

222

+

=

(1)

Where is total strain range, which can be evaluated by

test, engineering approximation and finite element

methods. mis mean stress of stress cycle,Eis Youngs

modulus, f , f , b, care the fatigue strengthcoefficient, fatigue ductility coefficient, fatigue strength

exponent and fatigue ductility for material, respectively.

Finally,N is the number of cycles to failure.

Neuber method is an engineering approximation

method for evaluating local strain in common use. This

method mainly uses two relational expressions:

stress-strain hysteresis loop expression and Neuber

equation. The stress-strain hysteresis loop expression is:

978-1-61284-666-8/11$26.00 2011 IEEE


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n

KE

+

=

/1

222

(2)

Where is local strain range, n and K are cycleharden index and cycle intensity index for material. And a

modified Neuber equation is:

( )E

SKf2

= (3)

Where Sis nominal stress range,Kfis notch coefficient.

Eliminating in the simultaneous equations: Eq. (2),

Eq. (3), a new equation named Ffunction is appeared.

,,,,,F nKSKEf

2

22

2/11

/122

22

E

SK

KE

SK fn

n

f +

=0 (4)

For m=/2, eliminating , m in the simultaneous

equations: Eq. (1), Eq. (2), a new equation named G

function is appeared.

NbcE mff ,,,,,,,G

( ) ( ) ( ) ( )2

222

22

2

+

cfbfbf NN

ESKN

E=0 (5)

In the same way, eliminating in the simultaneous

equations: Eq. (4), Eq. (5), the concealed relation between

the number of cycles to failure and every parameter will be

expressed by Hfunction, viz.

N= bcnKSKE fff ,,,,,,,,H .

III. STRAIN FATIGUE RELIABILITY MODEL

The stress-strain hysteresis loop expression, Neuberequation and Coffin-Manson equation are uncertain,

becauseE, Kf, S, K , n , f , f , c, b are random

variables. For the sake of convenience, these random

variables are uniformly expressed byx1,x2,x3,x4,x5,x6,x7,

x8,x9, soN= ( )987654321 ,,,,,,,,H xxxxxxxxx .

In order to solve the reliability degree of strain fatigue

structure, the limit state function is developed based on the

theory of structural reliability design.

Z ( )987654321 ,,,,,,,,H xxxxxxxxx Ng (6)

In that way, reliability degreeRexpresses as:

R { }0ZP >

( ) 0,,,,,,,,HP 987654321 > gNxxxxxxxxx (7)

IV. RELIABILITY DEGREE SOLVING

Generally, structural reliability theory provides many

methods such as FORM, second-order reliability method

(SORM)[3]

etc. to get reliability degree. In FORM, means

and standard deviations, which are the first and secondmoments of random variables, are considered and the

complicated limit state surface is replaced by a

hyper-plane in the standard normal space. Since the

probability density in the standard normal space decays

exponentially with distance from the origin, the optimal

point on the limit state surface for fitting the

approximating surface is the point of minimum distance to

the origin. This point is called the most probable point

(MPP) or design point and its distance, i.e. the minimum

distance, from the origin is the reliability index [4]

. When

the limit state surface is high non-linear, see Fig. 1, it is

difficult to choose a suitable step for finding MPP.

However, the Hfunction in this paper is a high non-linear

equation.

Figure 1. Illustration of the iteration process for solving reliability

degree

The nonlinearity of strain fatigue limit status equation

should be taken into account, by reading a large number of

references about structure reliability algorithms[5~7]

, an

efficient algorithm for strain fatigue reliability analysis is


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proposed base on analyzing the influence of iterative step.

The algorithm consists of the following steps:

1) Transform the original random variable vector to a

standard normal vector;

2) Set k=0 and choose an initial point x(0)

for random

variables, for example, set mean value of every variable,

and initial step length50 (recommend 1~10 of

mean value ), and adjusting coefficient c for step length,

c=1.2~1.5;

3) Calculate the direction cosine vector of the limit

state surface from Eq. (8);

( )

( )

=

+

=9

1

2)(

)(

)()(

)1(

j i

kk

j

i

kk

i

k

Xi

x

Zx

x

Zx

x

x

( )9,,2,1 =i (8)

Where ixZ / can be defined by Eq. (9), also benumerically solved with difference method.

ix

Z

N

xx ii

+

G

GG

N

xx ii

+

G

FFGG

(9)

4) Calculate reliability index from Eq. (10);

( ) ( )

( )

=

+

=+

=9

1

)(

)1(

9

1

)()(

)(

)1(

i i

k

kXi

i

k

i

i

kk

k

xZ

xx

ZZ

x

xx

(10)

5) Calculate limit point vector xfrom Eq. (11);

)1()1()1( +++ = kkXik

ix ( )9,,2,1 =i (11)

6) If)()1( kk xx + < , is the prescribed

acceptable error limit, stop the iterative and)1( +k is the

strain fatigue index, otherwise, go next step;

7) If k=0, set k=k+1, repeat 3) ~ 6), otherwise, compare

)()1( kk xx + and )1()( kk xx , if the former is

greater, adjust=/c, and set k=k+1, return to 3).

V. EXAMPLE

An axle works under cyclic loading, the maximum

value of nominal stress is Smax=400MPa, and the minimum

value is Smin=-200MPa. Set the notch coefficient Kf= 1.5,

in accordance with shape dimension of the axle. The

material is steel Q235A, and some parameters are as

follows: E=192000MPa, K =1125.9MPa, n =0.193,

f =0.26, f =935.9MPa, c=-0.47, b=-0.095. The

dispersion of material parameter E and stress-strain

hysteresis loop is ignored, and the variation coefficient of

other variables is 0.03. Calculate the reliability degree of

axle after suffering 104cycles load.

Three methods, Monte Carlo Method (MCM), FORMand the method in this paper, are used to solve the

reliability degree of axle. Table I lists some information of

solving process and results. Comparing these data, it is

easy to find that the method in this paper is an efficient and

robust method.

TABLE I. RELIABILITY DEGREE RESULTS FOR 3METHODS

Method Iteration

times

Reliability

degree

Remarks

MCM 10000 0.9912 /

FORM 100 / No convergence

New method 8 0.9904 /

VI. CONCLUSIONS

For strain fatigue reliability model is highly nonlinear,

FORM, which is used to calculate the reliability index in

reliability analysis, may fail to converge, one of the prime

reasons is that selected iterative step is not proper. In this

paper, an easy iterative method, which introduces

automatic step adjustment to speed up the convergence of

iterative process, is proposed. Through an example, it is


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show that the method is efficient and accurate to get the

strain fatigue reliability degree.

Above analytical process and example, input load data

is constant amplitude loading. If load data is block

spectrum, the probabilistic damage of every level load need

to calculate before building strain fatigue reliability model,then all damage may be added according to fatigue damage

accumulation theory. Another way, using statistic theory,

load data can be processed to constant amplitude loading,

which obeys a kind of distribution.

ACKNOWLEDGMENT

The work was supported by National Defense Basic

Scientific Research Project of China (No D0920060310),

which is gratefully acknowledged by the authors.

REFERENCES

[1] H. Xu. Fatigue Intensity Beijing: Higher Education Press,198

[2] W. X. Yao. Fatigue Life Prediction of Structures. Beijing: NationalDefense Industry Press, 2003

[3] L. Q. Li. Mechanic Reliability Design and Analysis. Beijing:National Defense Industry Press, 1998

[4] Hasofer AM, Lind NC. Exact and invariant second moment codeformat. J Eng Mech Division ASCE 100:111121

[5] Q. Qin, D. J. Lin, G. Mei. Theory and Applications - ReliabilityStochastic Finite Element Methods. Beijing :Qinghua UniversityPress, 2006

[6] J. X. Gong, P. Yi. A Robust Iterative Algorithm for StructuralReliability Analysis. Structural and Multidisciplinary Optimization,Published online, 31 October 2010.

[7] Aldebenito M A, Schuller G I. A Survey on Approaches forReliability-based Optimization. Structural and MultidisciplinaryOptimization, 2010, 42(5):645-663.

[doi 10.1109%2ficrms.2011.5979247] liu, qin; qian, yunpeng; wang, dan; sun, zhili -- [ieee 2011 9th...

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