ress-multiaxial fatigue reliability

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Reliability Engineering and System Safety 93 (2008) 456–467

Multiaxial fatigue reliability analysis of railroad wheels

Yongming Liu, Liming Liu, Brant Stratman, Sankaran MahadevanÃ

Department of Civil and Environmental Engineering, Vanderbilt University, Box 6077-B, 306 Jacobs Hall, Nashville TN 37235, USA

Accepted 11 December 2006

Available online 24 January 2007

Abstract

A general methodology for fatigue reliability degradation of railroad wheels is proposed in this paper. Both fatigue crack initiation and

crack propagation life are included in the proposed methodology using previously developed multiaxial fatigue models by the sameauthors. A response surface method in conjunction with design of experiments is used to develop a closed form approximation of the

fatigue damage accumulation with respect to the input random variables. The total fatigue life of railroad wheels under stochastic

loading is simulated, accounting for the spatial and temporal randomness of the fatigue damage. The field observations of railroad wheel

fatigue failures are compared with the numerical predictions using the proposed methodology.

r 2007 Published by Elsevier Ltd.

Keywords: Stochastic; Rolling contact fatigue; Multiaxial fatigue; Reliability

1. Introduction

Damage accumulation due to fatigue, plastic deforma-

tion and wear significantly reduces the service life of

railroad wheels. In recent years, higher train speeds and

increased axle loads have led to larger wheel/rail contact

forces. Also, efforts have been made to optimize wheel and

rail design. This evolution tends to change the major

wheel rim damage from wear to fatigue [1]. Unlike the

slow deterioration process of wear, fatigue causes abrupt

fractures in wheels or tread surface material loss. These

failures may cause damage to rails, damage to train

suspensions and, in some cases, serious derailment of

the train.

Railroad wheels may fail in different ways correspondingto different failure mechanisms [2–4]. Ekberg and Marais

[5] divide the wheel fatigue failure modes into three

different failure types corresponding to different initiation

locations: surface initiated, subsurface initiated and deep

surface initiated fatigue failures. Surface-initiated failures

usually break off a piece of the wheel tread, while

subsurface-initiated failure can destroy the wheel’s integrity

and thus is more dangerous. Subsurface-initiated failure,

also known as shattered rim, is the type of failure studied in

this paper. Studies on surface-initiated fatigue have been

developed elsewhere [6–8].

Shattered rim failures are the results of large fatigue

cracks that propagate roughly parallel to the wheel tread

surface [9,10]. They can grow up to a length of several

hundred millimeters. Ekberg et al. [11] reported that a

shattered rim can initiate from both inclusions and non-

inclusion locations, which indicate that both crack initia-

tion life and propagation life need to be included to predict

the failure life of railroad wheels.

Historically, the methods for the fatigue life prediction

of mechanical/structural components can be divided intoseveral groups. Among others, the fatigue crack initiation

prediction models based on the S–N or e–N curve

approach, and the fatigue crack propagation prediction

models based on fracture mechanics are predominantly

used. The fatigue crack initiation models are appropriate

for the analysis of components with non-crack-like

geometries or without large initial defects. The fatigue

crack propagation models are appropriate for the analysis

of components with crack-like geometries or with large

initial defects. If neither stage (initiation or propagation)

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ÃCorresponding author. Tel.: +1 615322 3040; fax: +1 615322 3365.

E-mail address: [email protected]

(S. Mahadevan).

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dominates during the entire life of the mechanical

components, a total life methodology is required to

accurately predict the component fatigue reliability.

There are two major difficulties in deterministic railroad

wheel fatigue modeling. One is that the wheels are usually

under rolling contact condition, which leads to a non-

proportional multiaxial stress state within the wheels.Proper multiaxial fatigue models are required to handle

this type of fatigue life prediction, which should be

applicable to non-proportional loading conditions. The

other difficulty is how to accurately describe the stress state

in contact analysis. Analytical solutions and simplified 2D

finite element models are not appropriate for the rolling

contact analysis of mechanical components with complex

geometries, such as railroad wheels [12].

A large amount of scatter has been observed in the

fatigue life distribution of railroad wheels, ranging from

several months to several decades. A probabilistic fatigue

analysis is more appropriate in order to consider the large

observed randomness, including various uncertainties in

material properties, structural geometries and applied

loadings. Due to the complex mechanism involved in the

rolling contact fatigue analysis and large number of

random variables affecting the final reliability, a direct

analytical reliability calculation is impractical.

This paper proposes a general methodology for rolling

contact fatigue life prediction under a stochastic loading

process. The fatigue damage within railroad wheels is

treated as a spatial-temporal random field in this study.

A response surface method (RSM) in conjunction with

design of experiments is used to develop a closed form

approximation of the fatigue damage accumulation withrespect to the input random variables. Then Monte Carlo

simulation with the response surface is used for probabil-

istic fatigue life prediction. The numerical predictions are

compared with field observations of wheel failure data. The

proposed methodology is also very valuable for fatigue

damage tolerance design, and maintenance scheduling of

mechanical and structural components.

2. Rolling contact fatigue modeling of railroad wheels

The total fatigue life of railroad wheels in the proposed

methodology is separated into two parts. One is crack

initiation life and the other is crack propagation life.

Mathematically, it can be expressed as

N total ¼ N initiation þ N propogation, (1)

where N total, N initiation and N propogation are the total fatigue

life, the fatigue crack initiation life and the fatigue crack

propagation life, respectively. The details about the

calculation of each part of the fatigue life and the transition

between the fatigue initiation life and the fatigue propaga-

tion life are described below.

2.1. Fatigue crack initiation life model

Liu et al. [12] have developed a fatigue crack initiation

life prediction model for railroad wheels. It combines a

critical plane-based multiaxial fatigue theory [13] with a 3D

finite element model. A detailed derivation and explanation

of the model can be found in the referred paper. Only a

brief illustration is shown here.

First, use the available profiles to build the geometry

model of one wheel and a piece of rail. This model is called

the full model. At the wheel center, a pilot node is

connected to the wheel using rigid link elements. All the

external loading and boundary conditions of the wheel are

applied on the pilot point. On the possible contact areas of

the railhead and the wheel tread, area contact elements are

used corresponding to the geometry mesh of the wheel. The

full model analysis is first performed and the geometrymodel of the contact region is cut out to be a sub-model

with a very fine mesh near the contact surface. The results

of the full model are interpolated on the cut boundaries of

the sub-model and applied as boundary conditions to the

sub-model. The finite element model is shown in Fig. 1.

The stress response from the finely meshed sub-model is

used for fatigue life prediction. As an illustration, the stress

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Cutting Edge

Rail

WheelPilot Node

Contact Element

Z

XY

a

b

Fig. 1. Finite element modeling of wheel/rail contact. (a) Full model; (b) sub model.

Y. Liu et al. / Reliability Engineering and System Safety 93 (2008) 456–467 457



histories of two points (one is 3 mm below the tread

surface, the other is 10 mm below the tread surface) during

half a revolution of the wheel rotation are plotted in Fig. 2

[12]. Fig. 2 shows that the stress history in the wheel under

rolling contact condition is not proportional, which

indicates that the maximum normal stress and maximum

shear stress do not occur simultaneously.

After obtaining the stress history of the wheel, a

previously developed multiaxial fatigue model [13] is used

to calculate the fatigue initiation life and initial crack plane

orientation. An equivalent stress amplitude (S eq) for

fatigue crack initiation life prediction is calculated as [13]

S eq ¼ 1

B

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs1Þ2 þ t2

s

2

þ t3

s

2

þ AsH

s

2s

, (2)

where S eq is the equivalent stress under multiaxial loading.

s1, t2, and t3 are the normal and shear stress amplitudeapplied on the critical plane. The subscripts 1, 2, 3 indicate

the directions of the stress amplitude. The superscript H in

sH indicates the hydrostatic stress amplitude. s is the ratio

of shear and normal stress amplitudes under a specific

fatigue crack initiation life (N initiation). A and B are material

parameters and can be found in our previously developed

multiaxial fatigue model [12]. The equivalent stress ampli-

tude and experimental S–N curve data is used to calculate

the life of wheels under a specific applied vertical loading.

Because the critical location is not available, all of the

possible nodes on the radial section of the wheel are

explored. The fatigue crack orientation and location can be

calculated and have been validated with field observations.It has been found [12] that fatigue cracks usually initiate at

some depth below the wheel tread surface and have a

shallow angle with the wheel tread surface. The informa-

tion about the crack location and orientation will be used

for fatigue crack propagation analysis in Section 2.3.

2.2. Transition between fatigue crack initiation life and

propagation life

The concept of fatigue limit is traditionally used in the

fatigue resistance design, which defines a loading criterion

under which no macroscopic crack will form. The concept

of fatigue crack threshold is often used within the damage-

tolerant design approach, which defines a loading criterion

under which the cracks will not grow significantly [14].

A link between the fatigue limit and the fatigue crack

threshold was proposed by Kitagawa and Takahashi [15].

According to the well-known El Haddad model [16], the

fatigue limit can be expressed using the fatigue threshold

and a fictional crack length a. The crack length a represents

the intersection of the smooth specimen fatigue limit and

the linear elastic fracture mechanics (LEFM) fatigue

threshold, i.e.

f À1 ¼K th ffiffiffiffiffiffipa

p , (3)

where f À1 is the fatigue limit of the material and K th is the

fatigue crack threshold for Mode I loading. Eq. (3) is

originally proposed by El Haddad et al. [16] for Mode I

loading. Liu and Mahadevan [17] showed that this idea canbe also used under mixed-mode loadings using their

developed mixed-mode fatigue crack propagation model.

The fictional crack length a can be used to bridge the

gap between the fatigue crack initiation and propagation.

Eq. (3) can be rewritten as

a ¼ 1

p

K th

f À1

2

. (4)

In the proposed method, if the fatigue damage accumula-

tion according to the fatigue crack initiation analysis

reaches a critical value (usually unity), an initial crack is

assumed to be formed with the size determined by Eq. (4).

Then the fatigue crack propagation analysis based on

fracture mechanics can be performed as shown in the next

section.

2.3. Fatigue crack propagation life model

Liu [18] has developed a method for subsurface fatigue

crack propagation of railroad wheels. Only a brief

illustration is shown here and the detailed explanation

can be found in the referred publication. A finite element

model similar to the one for the fatigue crack initiation

analysis (Section 2.1) is developed. The difference is that an

embedded elliptical crack is built into the model. The crack

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-800

-600

-400

-200

0

200

Time step

S t r e s s

( M P a )

Y stress

YZ shear stress

-800

-600

-400

-200

0

200

Time step

S t r e s s

( M P a )

Y stress

YZ shear stress

Fig. 2. Stress history at two locations in the wheel. (a) Point 3 mm below tread surface; (b) point 10 mm below tread surface.




3. Stochastic fatigue damage modeling

The above discussions are for deterministic fatigue

damage modeling. Due to the large uncertainties involved

in fatigue analysis and field observations, a stochastic

fatigue life prediction approach is proposed here, which

includes various uncertainties in material properties and

applied loadings.First consider the stress response at a specific location

and time instant. The equivalent stress amplitude S eq is

calculated using the method described in Section 2.1. At

any fixed location, S eq is a random variable with

probability density function (PDF) of f S eqðS eqÞ. The fatigue

damage caused by the stress amplitude is usually expressed

as a fraction of the total number of cycles to failure:

D ¼ 1

N , (6)

where N is the fatigue life estimation from the S – N curve

under constant stress amplitude S eq. N represents the

material resistance to fatigue loading. It is also a random

variable at a specific stress amplitude. The conditional

PDF of N can be found from experimental data and is

expressed as f N S eqj ðN Þ. The single cycle damage which

considers both the randomness in material resistance and

applied stress amplitude is a random variable whose joint

PDF can be expressed as

f DðDÞ ¼ f D S eqj ðDÞ f S eqðS eqÞ ¼ 1

D2f N S eqj

1

D

f S eq

ðS eqÞ. (7)

For the fatigue damage accumulation process, a damage

accumulation rule is required. In the current study, a linear

damage accumulation rule, known as the Miner’s rule, is

used for its simplicity. Eq. (8) is the general expression for

the Miner’s rule.

Dinitiation ¼XK

i ¼1

Di ¼XK

i ¼1

ni

N i , (8)

where K is the number of loading blocks, ni is the i th

applied loading cycle. For stochastic loading during a

certain time period, not only is the stress amplitude S eq a

random variable, but also the number of cycles ni at the

stress amplitude S eq. Nagode and Fajdiga [19] proved that

the conditional PDF ( f ni S eqj ðni Þ) of number of cycles ni at

the stress amplitude level S eq can be modeled by a normal

distribution based on the DeMoivre–Laplace principle,

with the mean and standard deviation expressed as

m ¼ 1 þ ðT À 1Þð1 À F ðS eqÞÞ,s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiðT À 1Þð1 À F ðS eqÞÞF ðS eqÞ

p , ð9Þ

where T is the total number of load cycles during each

block, F

ðS eq

Þis the cumulative density function (CDF) of

the stress amplitude which can be obtained from cycle

counting techniques such as the rain-flow counting method

[18]. The joint PDF of the total damage at a specific

location can be expressed as

f DtinitiationðDtinitiationÞ ¼ 1

D2f N i S eqj

1

D

f ni S eqj ðni Þ f S eq

ðS eqÞ.

(10)

When the fatigue damage equals or exceeds unity, we

assume that the initial fatigue crack is formed. We check

the damage accumulation at different locations. If fatigue

damage exceeds unity at one location, the number of

loading blocks is the fatigue crack initiation life of the

ARTICLE IN PRESS

Fig. 5. Crack shape comparison between numerical prediction and field observations. (a) Field observations of crack shape; (b) numerical prediction of

crack shape.




structure. Eq. (11) is a general expression for the structural

fatigue crack initiation criterion

Dstructure ¼ maxðDtotal;x j Þ ¼ G ðx j ; N initiation; R1 ; . . . ; RP Þ ¼ 1,

(11)

where x j is the coordinate at different locations, N initiation is

the number of loading cycles to fatigue crack initiation, R1

through Rp, are random variables which affect the fatigue

damage in the structure. Solving Eq. (11) for N initiation, we

obtain

N initiation ¼ f i x j ; R1 ; . . . ; RP

À Á. (12)

Eq. (12) shows that the fatigue crack initiation life is a

function of geometric locations and input random vari-

ables. The analytical solution for N initiation using Eq. (12) is

rather complicated and sometimes impractical. We use

Monte Carlo simulations to calculate the probabilistic

fatigue crack initiation life.

Once the fatigue crack is initiated, we use the fatigue

crack propagation model described in Section 2.3 to

calculate the fatigue crack propagation life. The equivalent

stress intensity amplitude K eq is calculated using the

described method. At a specific point at the crack tip, K eq

is a random variable with the PDF of f K eqðK eqÞ. The crack

growth rate at a specific SIF amplitude is also a random

variable with the conditional PDF of f dadN

K eqj ðda=dN Þ.Following a similar procedure as the stochastic fatigue

crack initiation life prediction, the single cycle fatigue crack

length increment Da is a random variable and its joint PDF

can be expressed as

f DðDaÞ ¼ f dadN K eqj

da

dN

f K eq ðK eqÞ. (13)

During the entire loading history, the crack propagation

length is added to the initial crack length. When the crack

reaches the critical length, the mechanical component is

assumed to fail. The failure criterion is expressed as

atotal ¼ ai þXR

m¼1

DamXac, (14)

where R is the number of loading cycles, Dam is the crack

length increment during each loading cycle. In Eq. (14), ai ,

Dam and ac are random variables. ai is calculated using

Eq. (4) and is related to material properties. ac is obtainedusing field observations of failed components (failed

railroad wheels in the current study). am is calculated using

Eq. (13) and is related to the applied stochastic loading and

material properties. am is an implicit function of crack

propagation life N propagation and other random variables and

can be expressed as

am ¼ f ðN propogation; R1 ; . . . ; R pÞ. (15)

Substitute Eq. (15) into Eq. (14) and solve for N propagation,we obtain

N propagation ¼ f pðai ; ac; R1 ; . . . ; R pÞ. (16)

Eq. (16) shows that the fatigue crack propagation life is a

function of several random variables. Again, the analytical

solution for N propagation using Eq. (16) is rather complicated

and sometimes impractical. We use Monte Carlo simula-

tions to calculate the probabilistic fatigue crack propaga-

tion life.

Substituting Eqs. (12) and (16) into Eq. (1), we can

obtain the total fatigue life of the mechanical components.

4. Response surface approximation

Due to the expensive computational effort involved in

the rolling contact finite element modeling, the RSM is

used here to approximate the relationship between the

input variables and the output variables using a few sample

points. Based on the parametric studies of the fatigue crack

initiation and the fatigue crack propagation analysis [18],

several important factors are chosen as the input variables.

The output variables are the equivalent stress amplitude for

the fatigue crack initiation analysis and the equivalent SIF

for the fatigue crack propagation analysis. A full factorial

design is used to design the numerical experiments. Thelower, middle and upper design values for five variables are

listed in Table 1. In Table 1, the applied loading is

normalized with the maximum design loading specified by

the Association of American Railroads [20]. A finite

element analysis corresponding to the design of experi-

ments in Table 1 is performed. The equivalent stress

amplitude and the equivalent SIF are used to build the

response surface by regression analysis.

Analysis of the numerical results show that the

maximum equivalent stress not only varies its amplitude

but also its location (depth below the tread surface). A two-

step regression analysis is performed to handle thisproblem. First, for each numerical experiment, a rational

regression function (Eq. (17)) is used to formulate the stress

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Table 1

Design values for the five random variables

Random variables (unit) Lower Middle Upper Fatigue crack stage

Wheel diameter D (in) 28 33 38 Crack initiation

Hardness Ha (BHN) 235 320 405 Crack initiation

Applied loading F 0.4 1.0 1.5 Crack initiation and propagation

Crack depth d (mm) 5 6 8 Crack propagation

Crack length a (mm) 1 5 15 Crack propagation




variation corresponding to the depth.

S eq ¼ 1

P 1

þP 2x

þP 3x2

, (17)

where S eq is the equivalent stress, P 1, P 2 and P 3 are

functions of input random variables (i.e., wheel diameter

D, hardness Ha and applied loading F ), x is the depth

below the tread surface (Fig. 6).

The regression results using Eq. (17) for two specific

numerical examples are plotted in Fig. 6(a). The regression

results for all the numerical examples are plotted in

Fig. 6(b). One point to note is that Eq. (17) is a random

function, which represents the stress variation correspond-

ing to spatial domain. The regression result of Eq. (17) is

used for the fatigue crack initiation life prediction.

The above discussion of stress range approximation

(Eq. (17)) is for fatigue crack initiation life prediction.After a fatigue crack initiates, it continues to propagate to

the final failure. SIF amplitude approximation is required

for fatigue crack propagation life prediction. Liu [18] has

performed a parametric study for the fatigue crack

propagation analysis of railroad wheels and found that

the applied load, the crack length and the crack depth

below the tread surface have significant influences on the

SIF ranges. Based on first principles of fracture mechanics,

a simple formula is proposed [18] to calculate the

equivalent SIF range as

K eq

¼x

ðF

ÀF c

Þ ffiffiffiffiffiffipa

p d

ð2d c

Àd

Þ, (18)

where K eq is the equivalent SIF. F is the applied loading. a

is the half length of the embedded crack along the axis. F c,

d c and x are regression constants. The prediction using

Eq. (18) and the finite element results for the SIF along the

major axis are plotted in Fig. 7. It is seen that the proposed

formula agrees with the finite element results well. Eq. (18)

is used for fatigue crack propagation life prediction.

The regression functions for the S eq and K eq (Eqs. (17)

and (18)) indicate that they are functions of input random

variables and coordinate variables. Their statistical para-

meters depend on these two types of variables. The

analytical statistical expressions (e.g., PDF functions) of

S eq and K eq are then quite complicated. In this paper, we

used Monte Carlo simulation to calculate the probabilisticfatigue life prediction and the analytical expression of S eqand K eq is not required. The final objective is the life

distribution of railroad wheels considering the randomness

of the input random variables (e.g., applied loading,

structural details and material properties, etc.). S eq and

K eq are only ‘‘intermediate’’ variables during the calcula-

tion and their statistical information is not explicitly

quantified.

5. Monte Carlo simulation and reliability degradation

A Monte Carlo simulation-based methodology is used to

calculate the probabilistic life distribution and reliability

degradation of railroad wheels in this paper. The response

surface developed in Section 4 is used to calculate the

fatigue crack initiation life and the fatigue crack propaga-

tion life. Various uncertainties from material properties,

wheel geometry, applied loading and crack profiles are

included in the proposed calculation.

5.1. Statistics of input random variables

For the Monte Carlo simulation, the probability

distribution functions of the input random variables are

required. The details are shown below.

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0

100

200

300

400

500

0 5 10 15 20

X (mm)

S

t r e s s ( M P a )

FEA results (33-320-0.4)

FEA results (38-405-1.0)

Regression results

0

100

200

300

400

500

600

700

0 5 10 15 20

X (mm)

S

t r e s s ( M P a )

95% bounds of regression results

90% bounds of regression resultsFEA results

Fig. 6. Comparison between FEA results and regression results for equivalent stress. (a) Individual comparison; (b) overall comparison.

0

5

10

15

20

0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06

FEA results

regression

K e q ( M P a ( m

) 1 / 2 )

(F − F c) πad (2d c −d ) (MPa(m)1/2)

Fig. 7. Comparison between regression results and FEA results for

equivalent SIF.




The median fatigue S – N curve and its 90% confidence

bounds are plotted with the experimental data in Fig. 8.

The fatigue life at a specific stress level is assumed to

follow the lognormal distribution. For wheel diameter,

collected field data shows that it can be described as a

multinomial distribution. The histogram of wheel dia-

meters is plotted in Fig. 9. The current study focuses on the

fatigue degradation behavior of the entire population of

railroad wheels currently in service. Thus we included all

wheels with different diameters. The next step of the

ongoing research is to divide the entire population of

railroad wheels into several groups based on their

similarities (e.g., same diameter, same manufacturing,

similar operation conditions, etc.) using cluster analysis.

Each group will be studied individually to identify the most

critical features affecting the reliability of railroad wheels.

However, that objective is beyond the scope of the current

paper, which mainly presents a general methodology

for the rolling contact fatigue reliability calculation of

railroad wheels.

The applied loading on railroad wheels appears to be a

bimodal distribution (Fig. 10). The reason is that the

service loading can be classified as either empty loaded or

full loaded. In the current study, this distribution is

simplified by a linear combination of lognormal and

Weibull distributions (Eq. (19)).

F bimodalðRLÞ ¼ 0:44F lognormalðRLÞ þ 0:56F weibullðRLÞ, (19)

where RL is the random variable of applied loadings on

railroad wheels. F XXXXX ðRLÞ is the CDF of different

distributions of RL. Eq. (19) uses non-dimensional loading

factor, which is defined as the actual loading divided by

maximum design load.

No experimental data for the material hardness distribu-

tion is available. However, the hardness value for class B

and C railroad wheels is bounded between 277 and 363

[20]. In the current study, we assume the hardness follows a

Beta distribution to represent this bounded distribution.

The PDF of the beta distribution is plotted in Fig. 11.

The fatigue crack propagation curve suggested in

Association of American Railroads [20] is used. The

median and 90% confidence bounds are plotted in

Fig. 12. The crack growth rate at a specific SIF amplitude

is assumed to follow the lognormal distribution. The initial

crack size after the fatigue crack initiation is calculated

using the Kitagawa diagram (Eq. (4)); it depends on the

fatigue limit and the fatigue crack threshold value. In the

current study, the initial crack length is approximated

using a lognormal distribution. The PDF of the initial

crack length is plotted in Fig. 13. The final failure crack

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300

350

400

450

500

4 6

Life (log(N))

S t r e s s ( M P a )

Experimental data

90% bounds

median curve

5 7

Fig. 8. Fatigue S – N curve.

0

20

40

60

80

100120

140

160

28.0 30.0 33.0 36.0 38.0

Wheel diameter (in)

F r e q u e n c y

Fig. 9. Histogram of wheel diameter.

180

160

140

120

100

80

60

40

20

0

0.00 0.64 1.28 1.92 2.56 3.20 3.84 4.48

Loading factor

F

r e q u e n c y

Fig. 10. Histogram of loading factor.

0

0.01

0.02

0.03

270 320 370

Hardness (BHN)

P r o b a b i l i t y d e n s i t

y

Fig. 11. PDF of hardness distribution.




length uses field observation data and is approximated

using a lognormal distribution. The histogram and lognor-

mal fit are plotted in Fig. 14. The statistical properties of the

input random variables are listed in Table 2.

5.2. Probabilistic life distribution and reliability

Using the described statistics of the input random

variables, Monte Carlo simulation can be used to predict

the fatigue life of railroad wheels with the response surface

in Section 4.

Ten thousand Monte Carlo samples are used to simulate

the fatigue failure life of railroad wheels. Field data

regarding the number of cycles to failure of railroad wheels

are collected and compared with the Monte Carlo

simulation results. The empirical CDF and the frequencyhistogram of the numerical fatigue life prediction are

plotted in Fig. 15 together with the field observations. The

fatigue life in Fig. 15 is censored at 2Â 109 cycles because it

is assumed that the wheel would fail due to other failure

mechanisms by the end of this time-period.

In Fig. 15(a), the numerical fatigue life predictions agree

with the field observations reasonably well and capture the

major trend of the life distribution. However, a large

difference is observed at the early life regime, i.e. near the

tail region of the fatigue life distribution. The reason is that

the field data shows two ranges of fatigue life distribution.

This phenomenon can be clearly seen in the frequency

diagram (Fig. 15(b)). It also can be seen that the number of

wheels experiencing premature failure is only a small

fraction of wheels (around 10%) and does not greatly

affect the overall mean fatigue life. However, their effects

are significant at the tail region, which affects the reliability

evaluation.

Berge [21] and Stone and Geoffrey [22] suggest that the

large stress, perhaps due to wheel/rail impact or material

discontinuity, has important effect on the shattered rim

failure. Also, the large on-tread brake loading and the

thermal stresses arising from on-tread friction braking

will reduce the fatigue life of railroad wheels [23]. The

observed premature fatigue failure (earlier failure modal inFig. 15(b)) is possibly due to the above mentioned factors

or their combinations, such as initial defects, brake

loading, and thermal loading.

The brake loading and the thermal loading effects are

beyond the scope of the current study. Only the effect of

initial defects is considered here to predict the premature

fatigue failure. Since no information about the initial defect

geometry and distribution is available at this stage, the

length of the initial defect is assumed to be 3.2 mm (1/8 in)

according to AAR regulations. The location of the initial

defect is assumed to be uniformly distributed between 5

and 8 mm below the tread surface. A Monte Carlo

simulation is used again to calculate the fatigue life of

defective railroad wheels, in which 10% of the failed wheels

are assumed to be controlled by the large initial defects.

The numerical prediction and field observations are plotted

in Fig. 16. It is seen that the numerical prediction are closer

to the field observations if large initial defects are

considered.

The above discussion assumes a deterministic initial

defect length specified in Association of American Rail-

roads [20]. One remaining question is whether the

premature fatigue failure of railroad wheels can be fully

or mostly explained by the existences of the large initial

defects, if the randomness in initial defect size is also

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1.0E-08

1.0E-07

1.0E-061.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

11 0 100

Mean

90% confidence bounds

d a / d N

( m m / c y c l e )

K MPa(m)1/2

Fig. 12. Fatigue crack growth curve.

0

2

4

6

810

12

14

0.05 0.07 0.09 0.11 0.13 0.15

Initial crack length (mm)

P r o b a b i l i t y d e n

s i t y

Fig. 13. PDF of fictional crack length.

Critical crack length (mm)

F r e q u e n c y

350300250200150

9

8

7

6

5

4

3

2

1

0

Fig. 14. Histogram of critical crack length.




considered. For example, if the initial defect is slightly

larger than the specified AAR value (e.g., 4 vs. 3.2 mm),

can the numerical model predict all or most of the

premature failure? Note that this situation is possible in

practical operations if the inspection limitations and errors

are considered. A trial and error method is used to

calibrate the length of the initial defect to match the field

observations. A 10 mm initial defect assumption can match

the MC simulation results with field observations (Fig. 17)

very well. However, a 10 mm defect is too large at the

current manufacturing and inspection technology level and

is not realistic.

Based on the above discussions, it is clear that large

initial defects contribute to the premature fatigue failure

but cannot fully explain it. Other effects may also

contribute to the premature fatigue failure of railroad

wheels. The current comparison is only a preliminary study

as the initial defect information is assumed. Future work is

required to study the effect of the initial defect and other

factors in detail.

Once the probabilistic fatigue life of railroad wheels is

obtained, time-dependent reliability can be easily calcu-

lated. The reliability degradation of railroad wheels with

and without considering large initial defects are plotted in

Fig. 18. The current focus is on the high reliability regime,

which is of most interest for fatigue resistance design and

maintenance scheduling. It can be seen that the reliability

degradation behaviors are different for different assump-

tions. The simulation results without considering large

initial defects overestimates the reliability of the railroad

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Table 2

Summary of the statistics of input random variables

Random variable Mean Coefficient of variance Distribution type

Fatigue strength 360.8 MPa 0.03 Lognormal

Wheel diameter 35.5 in 0.01 Multinomial

Loading factor 0.91 0.67 Bimodal: Lognormal+Weibull

Material hardness 306 BHN 0.53 Beta

Fatigue crack threshold intensity factor 3.26MPa(m)1/2 0.42 Lognormal

Fictional crack length 0.1 mm 0.07 Lognormal

Critical crack length 207.6 mm 0.25 Lognormal

0

0.2

0.4

0.6

0.8

1

5 8 10

Fatigue life (log(N))

P r o b a b i l i t y

Field data

Monte Carlo simulation

0

0.05

0.1

0.15

0.2

0.25

0.3

5 7 9 10


F r e q u e n c y

Field data


6 7 9 6 8

ba

Fig. 15. Empirical CDF and frequency histogram of the field data and numerical predictions with no initial defects. (a) CDF; (b) frequency histogram.

0

0.2

0.4

0.6

0.8

1

5 7 9 10


P r o b a b

i l i t y

Field data


0

0.05

0.1

0.15

0.2

0.25

0.3

5 8 9 10


F r e q u e

n c y

Field data


6 8 6 7

ba

Fig. 16. Empirical CDF and frequency histogram of the field data and numerical predictions with 3.2 mm defects. (a) CDF; (b) frequency histogram.




wheels. The difference increases as the reliability level

increases.

6. Conclusions

A general methodology for fatigue reliability degrada-

tion of railroad wheels is developed in this paper. It

combines a characteristic plane-based fatigue crack initia-

tion and propagation model, 3D rolling contact finite

element analysis, RSM and Monte Carlo simulation

technique into a general framework. The non-proportional

loading history within railroad wheels and the large spatial

and temporal randomness are included in the proposed

methodology.

Qualitative comparisons between numerical crack pre-

dictions and field observations show that the proposed

methodology agrees with the field-observed fatigue crack

growth behavior very well. Quantitative comparisons

between numerical life predictions and field observations

show that the proposed methodology can capture the

major trend of fatigue failure of railroad wheels, except for

the large difference at the tail region of the probabilistic

fatigue life distribution. A preliminary investigation shows

that the fatigue life can be approximated as a bi-modal (or

tri-modal) distribution and large initial defects contribute

to the premature fatigue failure of railroad wheels. The

current used inspection regulation by Association of

American Railroads [20] suggests a 3.2 mm initial flaw size

as the inspection limit. The numerical predictions in this

study show that a 3.2 mm initial defect contributes to the

premature fatigue failure and a stricter quality control is

desired for railroad wheels. It is also observed that the

large initial defect cannot fully explain the premature

fatigue failure. The current study assumes that 10%

of wheels contain large initial defects. Future detailed

analysis of initial defect effects need quantitative data on

fraction of wheels containing large defects (e.g., 5% or

10%) and statistical information about defects (e.g., size,

shape and orientation). Other effects, such as brake loading

and thermal loading, need further detailed study.

Acknowledgments

The research reported in this paper was supported by

funds from Union Pacific Railroad and Meridian Railroad(Research Agreement No. 18140, Monitor: Rex Beck). The

support is gratefully acknowledged.

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