journal fatigue

International Journal of Engineering & Technology IJET-IJENS Vol:14 No:04 50

143804-5656-IJET-IJENS © August 2014 IJENS I J E N S

Abstract— This paper presents experimental and numerical

study for fatigue life estimation of a cylindrical specimens made

of Carbon Steel alloys with different content of carbon. The

fatigue experiments were carried out at room temperature,

applying a fully reversed cyclic load with the frequency of 50 Hz

and mean stress equal to zero (R= -1), on a cantilever rotating-

bending fatigue testing machine. The stress ratio was kept

constant throughout the experiment. Stress Life method based on

S-N curves (Stress–Cycle curves) is employed which deals with

relatively high numbers of cycles and therefore addresses High

Cycle Fatigue (HCF), greater than 106 cycles inclusive of infinite

life .The S-N approach used to predict the fatigue lives of tested

material showed reasonable correlation to the experimental data

from the component tests. FEA method was conducted to show

influence of the mean stress value (R ≠ −1) on fatigue strength of

tested materials based on the mean stress correction theories

(Goodman ,Soderberg ,Gerber and the Mean stress curves) .The

main advantage of the proposed study is that the mean stress

effect correction depends on the number of cycles to failure, what

corresponds to the observed changes in experimental results

presented in the literature at fully reversed rotating bending and

the mechanical properties of tested materials.

Index Term— Carbon Steel alloys ,Stress life approach

,Fatigue Life, Mean Stress Theories, FEA

I. INTRODUCTION

Oscillating stresses are far more dangerous for

structural parts and components than a static force applied

once. In the event of frequent repetition of a static load which

is in itself permissible, a machine part may rupture as a result

of material fatigue. As the number of load cycles increases,

the permissible stress level declines. Even stresses which are

below the yield point of the material in the elastic range may

lead to minor plastic deformations as a result of local peak

stresses inside the part. This effect gradually destroys the

material due to the constant repetition and eventually results in

rupture. The absolute number of load cycles is a more

decisive factor for failure than the frequency [1]. Structural

members subjected to in-service cyclic loads exhibit a fatigue

behavior that generally depends on the mean stress values [2].

Mean stress effects have long been studied, as in the early

work of Gerber and Goodman [3], and one might think that all

has been said on the subject that needs to be said.

Nevertheless, several methods of questionable accuracy are

currently in wide use.

The problem of the mean stress effects in Stress-and Strain-

Life Fatigue has been studied practically by developing

empirical relationships for different metals and alloys, various

criteria have been proposed to deal with the mean stress effect

on fatigue life, such as Soderberg, Goodman and Gerber

diagrams [4].

Reference [5] investigated Fatigue Life of an Aircraft

Engine under different load spectrums. Based on the working

S-N curve and linear cumulative damage theory, the

relationship between the load spectrum and the life of an

aircraft engine are investigated by using a key part of the

engine-1st stage turbine disk as a research object and the

conclusion was under different load spectrums, the working

life of the same engine is different based on criteria of turbine

stages.

Mean stress effects in fatigue are usually presented

as stress amplitude versus mean stress plot according to

Haigh . For a particular given cyclic life it is usually observed

that the load amplitude of the endurance limits decreases with

growing mean stress or static load [6].

The objectives of this study were compare fatigue analysis

of various carbon steel alloys perform rotating bending fatigue

at fully reversed loading (R = -1) and the effect of the mean

stress will be analyzed by FEA to determine the most widely

used methods and to identify their success in correlating

fatigue data for engineering metals. The methods considered

are those of Goodman, Soderberg, Gerber, and Mean stress

curves.

II. STRESS LIFE MODEL

The stress-life curve is a graphical representation of fatigue

data. It represents the relationship between fatigue life, in

cycles, and the applied stress amplitude .

Basquin’s relation the most commonly used model and

provides an analytical expression of the S-N curve, for finite

life (low or high cycle fatigue). By use this technique an

estimation of life prediction, with little information on the

material, can be obtained , see [7].

The simple Basquin’s curve is represented by :

Where :

is the fatigue stress amplitude (MPa),

is the number of cycles to failure (MPa),

Mean Stress Correction Effects On the Fatigue

Life Behavior of Steel Alloys by Using Stress

Life Approach Theories

Qasim Bader

1, Emad Kadum

2

1Lecturer at Engineering College, Department of Mechanical, Babylon University, Hilla-Iraq

2MSc student ,Department of Mechanical Engineering, Babylon University, Hilla-Iraq

[email protected] ,

2 [email protected]



The parameters a and b are both constant, depending on the

material and on the geometry, respectively. The coefficient

is approximately equal to the tensile strength. The coefficient

b is the fatigue strength exponent. These coefficients can be

evaluated by use least square method (linearizing the power

law in logarithmic form) ,it is important to mention that the S-

N curve is represented in the log-log scale .

The value of Fatigue limit is not clearly obvious on the S-N

curve; therefore, the Fatigue limit can be calculated by using

the fatigue life estimation equation at 106 cycles.

III. EFFECT OF MEAN STRESS

Many service load histories will have a non-zero mean

stress . Mean stress correction methods have been developed

to eliminate the burden of having to carry out fatigue tests at

different mean stresses, The influence of local mean stress can

be characterized as the influence of stress ratio, R, the ratio of

a local minimum stress to a local maximum stress in a fatigue

load cycle, with the aid of Fig. 1, many relations concerning to

fully reversed loading can be represented as in below [8]:

is the mean stress and the stress amplitude , stress

range Δσ and R a stress ratio.

Constant stress range: (2)

Mean stress:

(3)

Stress amplitude:

(4)

Stress ratio:

(5)

Max. Stress (6)

Min. Stress (7)

For the special case of stress amplitude where the

mean stress is zero, = 0. Such a situation of zero mean

stress is also called completely reversed cycling, and

corresponds to R = −1.In the treatment that follows, we will

first briefly discuss stress-life curves. Following this, we will

present various methods for estimating mean stress effects,

and then we will look at the ability of these methods to

correlate stress life data for various mean stresses. Finally,

concluding remarks are given that are intended to interpret .

A series of fatigue tests can be conducted at various

mean stresses, and the results can be plotted at a series of S-N

curves. This is accomplished by plotting the allowable stress

amplitude for a specific number of cycles as a function of the

associated mean stress. At zero mean stress, the allowable

stress amplitude is the effective fatigue limit for a specific

number of cycles. As the mean stress increases, the

permissible amplitudes steadily decrease. At a mean stress

equal to the ultimate tensile strength of the material, the

permissible amplitude is zero.

Generally there are many theories used to study influence of

the mean stress [9]. In this paper we have been employed three

theories that are commonly used to predict fatigue life under

fluctuating loading;

1. The Goodman theory,

2. The Soderberg theory, and

3. The Gerber theory.

All three can be expressed in mathematical expression as

following :

(

)

Where:

: mean stress

: is the endurance fatigue limit,

σu : is the ultimate tensile strength,

: Tensile yield stress.

However, a graphical representation is considered very

useful, as shown in Fig. 2. These three theories are shown as

lines in the diagram, where the horizontal axis is the mean

stress ( ), and the vertical axis is the alternating stress ( ).

Note that the endurance limit ( ) plotted on the vertical axis.

Also, the yield strength ( ) has been plotted on both the

horizontal and vertical axes, and a yield line drawn to make

sure this design limitation is not omitted. The line connecting

the endurance limit ( ) with the ultimate tensile strength (σu)

represents the Goodman theory. Also, the line connecting the

endurance limit ( ) with the tensile yield strength ( )

represents the Soderberg theory. But for Gerber theory, the

curve connecting the endurance limit ) with the ultimate

tensile strength ( ) represents the Goodman theory which

behaves in a curved manner due to the square exponent in the

second part of eq. (10).

Also from same figure several important points can be

made. Firstly, the Soderberg theory is the most conservative of

the three shown, and is the only one that is completely below

the yield line. Secondly, the Gerber line fits the available test

data which are the best of the three theories, however, it is the

most difficult to draw accurately.

The curves are determined experimentally by obtaining a

series of S-N curves for different load ratio values (varying the

Fig. 1. Fully Reversed Loading



load ratio will result in varying the ratio of the mean to

alternating stress components).

Fig. 2. Graphical representation

of Gerber, Goodman and Soderberg theories

IV. EXPERIMENTAL WORK

The experimental work included assessment of

fatigue life specifications by using stress life approach for

three alloys of carbon steel (low ,medium and high) treated

commercially supplied from the local market . Those types of

steel alloy have a wide application in industry. The chemical

composition test of the alloys was done by use the device

Spectrometer type and the results was within the specification

limits and as shown in table 2.

MECHANICAL TESTS

tensile test was conducted using the microcomputer

controlled electronic universal testing machine .Average value

of four readings for each test have been taken to satisfy an

additional accuracy.

There are two hardness tests have been done in this

investigation Brinell's and Vicker's Hardness test. The

average value of four readings was recorded, the results of

basic mechanical properties for carbon steel alloys are given in

table (3). ; For more details about the Tensile and Hardness

tests procedure see [10].

FATIGUE TEST

In the revolving fatigue testing machine, a rotating

sample which is clamped on one side is loaded with a

concentrated force. The load is applied at one end of the

sample and with the help of a motor, rotation about its own

axis is achieved. Due to this rotation , a load reversal condition

is achieved at two opposite sides on the circumference of the

specimen. A triangular bending moment is developed in the

specimen. Following a certain number of load cycles, the

sample will rupture as a result of material fatigue A cantilever

bending fixture was designed to test the steel specimens based

on the critical (i.e. failure) location. Cantilever bending was

used in order to minimize the magnitude of the applied loads

necessary to achieve the desired nominal stresses. Stresses at

which the material fails below the load cycle limit are termed

fatigue limit .

S-N curves are plotted by using software of Fatigue

instrument presented in PC which is connected directly to

fatigue machine, for more details about the fatigue test

Features Theory

Most experimental data falls between

Goodman and Gerber theories

It is usually a good choice for brittle

materials

It is not bounded when using negative

mean stresses

Goodman

Generally the most conservative

It is not bounded when using negative

mean stresses

Soderberg

It is usually a good choice for ductile

materials

It is bounded when using negative mean

stresses.

Gerber

Fe S P Mn Si C Item

Bal. 0.021 0.012 0.61 0.27 0.21 LCS

Bal. 0.023 0.009 0.74 0.31 0.37 MCS

Bal. 0.012 0.006 0.33 0.24 0.77 HCS

Material Property Value

LCS

σu (MPa) 470

σy (MPa 350

Elongation [%] 26

Modula's of Elasticity (Gpa) 202

Brinell Hardness (HB) 135

Vickers Hardness (HV) 142

MCS

σu (MPa) 575

σy (MPa 480

Elongation [%] 18

E (Gpa) 206

Brinell Hardness (HB) 172

Vickers Hardness(HV) 180

HCS

σu (MPa) 675

σy (MPa 510

Elongation [%] 15

E Gpa) 200

Brinell Hardness (HB 200

Vickers Hardness(HV) 220

TABLE III

MECHANICAL PROPERTIES OF CARBON STEEL

ALLOYES

TABLE II CHEMICAL COMPOSITION OF THE SELECTED

MATERIALS (MASS％)

TABLE I

COMPARION OF MEAN STRESS COREECTION THEORIES



specimens geometry and procedure , see [10].

V. CORRELATION OF STRESS-LIFE DATA

For a set of amplitude-mean-life data, it is useful to employ

Least Square Method where in many branches of applied

mathematics and engineering sciences we come across

experiments and problems, which involve two variables. For

example, it is known that the Stress amplitude of a steel

specimens in S-N curve varies with the cycles of failure N

according to the Basquin’s formula Here a and b

are the constants to be determined. For this purpose we take

several sets of readings of stress amplitude and the

corresponding Cycles. The problem is to find the best values

for a and b using the observed values of and N, thus, the

general problem is to find a suitable relation or law that may

exist between the variables x and y from a given set of

observed values,(xi , yi ), i = 1, 2,..........,n . Such a relation

connecting x and y is known as empirical law. For above

example, and y = N.

The process of finding the equation of the curve of best fit,

which may be most suitable for predicting the unknown

values, is known as curve fitting. Therefore, curve fitting

means an exact relationship between two variables by

algebraic equations. There are following methods for fitting a

curve. The graphical method has the drawback in that the

straight line drawn may not be unique but principle of least

squares provides a unique set of values to the constants and

hence suggests a curve of best fit to the given data. The

method of least square is probably the most systematic

procedure to fit a unique curve through the given data

points[11].

VI. FINITE ELEMENT ANALYSIS

Due to limited experimental data available at different

mean stresses or r-ratio’s exist , several empirical options have

been adopted including Gerber, Goodman and Soderberg

theories which use static material properties (yield stress,

tensile strength) along with S-N data to account fatigue

analysis for any mean stress [12].

By the finite element analysis method and the

assistance of ANSYS Workbench software, It is able to

analyze the different components from varied aspects such as

fatigue life data. Solid hexahedral elements (solid185), with 8

nodes were considered. The mechanical properties and stress

life data obtained by experiments and specimens with different

content of carbon are modeled ,the element meshes were

generated, boundary condition corresponding to maximum

loading condition was given and stress analysis with constant

amplitude fully reversed loading and with mean stress

correction theories was applied to the structure .For the

simulation of fatigue analysis the finite element model has

been developed and the fatigue test with equivalent terms and

cyclical loads was performed to ensure the validity of the

finite element model. After the validation, design variables

were defined and the influence of the mean stress on Fatigue

life was estimated. results are generated for fatigue loading

fatigue at different stress amplitude . Analysis of different

stress, life damage , biaxility and safety factor for selected

materials have been done. Fig. 3 explain model with maximum

equivalent stresses generated in specimen of Low carbon steel.

VII. RESULTS

The cylindrical specimen under certain consideration is

being subjected to bending stress due to vertical component

force, so as it's rotates there is a fluctuation of stress. Cyclic

fatigue properties of a material are often obtained from

completely reversed, constant amplitude tests and actual

components seldom experience this pure type of loading,

hence if the loading is other than fully reversed, a mean stress

exists and may be accounted for by using a mean stress

correction .

Fatigue life analysis through ANSYS was trailed to predict

the fatigue life of carbon steel alloy specimens with different

content of carbon under effect of the mean stress using stress

life data obtained by the experiments as shown in Fig. 4 & 5.

The results from the S-N life predictions are found then the

Tensile strength to Fatigue limit ratio were calculated, the

results are summarized along with fatigue test in tables 4 &5.

More details of the fatigue life predictions are given by

Williams and Fatemi in [13].

Fig. 3. Model with Max. equivalent stresses



Fig. 4. Mean stress correction theories



0

100

200

300

400

500

600

700

0 1000000 2000000 3000000

Stre

ss a

mp

litu

de

MP

a

Cycles to Failure

HCS ,R= - 0.5

Mean stress curvesGerberSoderbergGoodmanFully Reversed

0

200

400

600

800

1000

1200

1400

0 1000000 2000000 3000000

Stre

ss A

mp

litu

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a

Cycles to Failure

HCS ,R = 0.25

Mean Stress CurvesGerberSoderbergGoodmanFully Reversed

0

500

1000

1500

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2500

0 1000000 2000000 3000000

Stre

ss a

mp

litu

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a

Cycles to Failure

HCS ,R= 0.5

Mean stress curves

Gerber

Soderberg

Goodman

Fully Reversed

0

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700

800

900

1000

0 1000000 2000000 3000000

Stre

ss a

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Cycles to Failure

HCS ,R = 0

Mean Stress CurvesGerberSoderbergGoodmanFully Reversed

0

100

200

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600

700

800

0 1000000 2000000 3000000

Stre

ss a

mp

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Cycles to Failure

LCS , R=0

Mean Stress Curves

Gerber

Soderberg

Goodman

Fully Reversed

0

200

400

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0 1000000 2000000 3000000

Stre

ss a

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Cycles to Failure

LCS ,R=0.25


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1400

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Cycles to Failure

LCS , R = 0.5


0

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900

1000

0 1000000 2000000 3000000

Stre

ss a

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litu

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MP

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Cycles to Failure

MCS , R=0

Mean Stress Curves

Gerber

Soderberg

Goodman

Fully reversed



0

200

400

600

800

1000

1200

0 1000000 2000000 3000000

Stre

ss a

mp

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Cycles to Failure

MCS ,R=0.25


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Cycles to Failure

MCS , R = 0.5


0

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400

500

600

0 500000 1000000 1500000 2000000 2500000

Stre

ss a

mp

litu

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MP

a

Cycles to Failure

MCS , R = - 0.5


0

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600

0 1000000 2000000 3000000

Stre

ss a

mp

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MP

a

Cycles to Failure

Goodman Theory , R = 0

High Ccarbon Steel

Medium Carbon Steel

Low Carbon Steel

0

100

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500

0 1000000 2000000 3000000

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ss a

mp

litu

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MP

a

Cycles to Failure

Soderberg Theory , R = 0

High Ccarbon SteelMedium Carbon SteelLow Carbon Steel

0

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0 1000000 2000000 3000000

Stre

ss a

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Cycles to Failure

Mean stress curves theory, R = - 0.5

High Ccarbon Steel

Medium Carbon Steel

Low Carbon Steel

0

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0 1000000 2000000 3000000

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ss a

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Gerber theory, R = - 0.5


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0 1000000 2000000 3000000

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ss a

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Cycles to Failure

Mean stress curves theory, R = 0.5

High Ccarbon Steel

Medium Carbon Steel

Low Carbon Steel



VIII. CONCLUSION

In this work, bending fatigue life of free notched specimens

with different content of carbon was investigated by experiment

0

100

200

300

400

500

600

700

0 1000000 2000000 3000000

Stre

ss a

mp

litu

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MP

a

Cycles to Failure

Gerbr theory, R = 0.5


0

100

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600

0 1000000 2000000 3000000

Stre

ss a

mp

litu

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MP

a

Cycles to Failure

Goodman theory, R = 0.25

High Ccarbon Steel

Medium Carbon Steel

Low Carbon Steel

0

50

100

150

200

250

300

350

400

450

500

0 1000000 2000000 3000000

Stre

ss a

mp

litu

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MP

a

Cycles to Failure

Soderberg theory, R = 0.25

High Ccarbon Steel

Medium Carbon Steel

Low Carbon Steel

(σu /σf) R Theory Mat.

0.497 - 0.5

Goodman

LCS

0.6 0

0.612 0.25

0.69 0.5

0.465 - 0.5

Soderberg 0.537 0

0.564 0.25

0.597 0.5

0.589 - 0.5

Gerber 0.75 0

0.812 0.25

0.865 0.5

0.606 - 0.5

Mean stress

curves

0.900 0

1.209 0.25

1.81 0.5

0.511 - 0.5

Goodman

MCS

0.596 0

0.62 0.25

0.71 0.5

0.496 - 0.5

Soderberg 0.551 0

0.587 0.25

0.643 0.5

0.579 -0.5

Gerber 0.725 0

0.804 0.25

0.853 0.5

0.6 - 0.5

Mean stress

curves

0.91 0

1.199 0.25

1.77 0.5

0.508 - 0.5

Goodman

HCS

0.589 0

0.523 0.25

0.708 0.5

0.488 - 0.5

Soderberg 0.537 0

0.559 0.25

0.593 0.5

0.575 - 0.5

Gerber 0.720 0

0.808 0.25

0.855 0.5

0.6 - 0.5

Mean stress

curves

0.883 0

1.201 0.25

1.773 0.5 (σu/σf)

Fatigue limit

(MPa) S-N Equation Mat.

0.465 218.672 σ = 3234.4 N - 0.195

LCS

0.449 258 σ = 3816.3 N - 0.195

MCS

0.449 303.5 σ = 4810.3 N - 0.2

HCS

Fig. 5. S- N curve for different stress ratio

TABLE IV

TENSILE STRENGTH TO FATIGUE LIMIT RATIO AT R = -1

TABLE V

TENSILE STRENGTH TO FATIGUE LIMIT RATIO



at R = -1 ,along with FEA method , fatigue analysis based on the

mean stress correction theories, were obtained .

It is observed that the life prediction results by

Experimental and FEA simulation showed close agreement,

within 12% difference for different stress amplitudes and also at

different no. of cycles at R = -1

For the mean stress correction theories ,the results show that

Soderberg and Goodman is the best representation for the

selected materials and with increase of carbon content the

differences of fatigue data obtained will be more.

For positive values of stress ratio, an increase in the R-ratio

increase the number of cycles to initiate a fatigue crack while the

alternating stress is not kept constant. This increasing is due to the

diminution of amplitude loading range when maximum

amplitude is maintained constant. The fatigue life equation is

fitted by regression method [14] .

Also from the results , it is observed that with decreasing stress

ratio (R ≈ -1 ), the S-N curves generated from fatigue test of

round specimen are converge and the overhaul and percentage

error will be less [15].

The accuracy of the predicted life by FEA simulation

depends on the selection of appropriate material model (specially

mesh parameters) and the reliability of the mechanical properties

of materials used .

On the other hand the prediction in this method also depends

on the correctness of the material total S-N curve (simple

regression) generated from experimental results of high cycle

fatigue data .

Fatigue limit increase with increasing of Hardness and Carbon

content and this is applicable with R[16].

According to results found in table 5, it is very important to

recognize that on certain stress ratio ,the Tensile Strength to

Fatigue limit ratio (σu /σf ) remains constant for the same theory

regardless of type of metal (carbon weight ) or the mechanical

properties, with percentage error no more 5 % . and this also

applies in the case of fully reversed bending (R=-1) as shown in

table 4.

REFERENCES

[1] Juli A. Bannantine , Jess J. Comer, James, L. Handbook ,

Fundamentals of Metal Fatigue Analysis , prentice hall, Englewood

Cliffs, New Jersey 07632.

[2] Thomas Svensson, et al., "Mean Value Influence in Fatigue – on the

rational choice of model complexity " , Int. J Fatigue, 2003.

[3] Goodman, J. ,"Mechanics Applied to Engineering" ,Longmans,

Green and Co., London, 1919, pp. 631-636.

[4] Norman E. Dowling, "Mean Stress Effects in Stress-Life and Strain-

Life Fatigue", Society of Automotive Engineers, Inc.2004.

[5] Hong-Zhong Huang, et al., "Fatigue Life Estimation of an Aircraft

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[6] F. Klubberg, et al. , "Fatigue testing of materials and components

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[7] Jaap Schijve , Fatigue of Structures and Materials, Springer ,Second

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[8] ASM, "Handbook of Mechanical Testing", 9th Edition, Vol.8, 1981.

[9] Norman E. Dowling, " Mean Stress Effects in Stress Life and Strain

Life Fatigue ", 2nd SAE Brasil International Conference on Fatigue

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[10] QASIM BADER & EMAD K. NJIM "Effect of V notch shape on

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[11] M.R. Mitchell, "Fundamentals of Modern Fatigue Analysis for

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[12] Bharath .S.O ,Nitesh Kumar , "Fatigue Analysis of Engineering

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[13] Williams, J. and Fatemi, A., "Fatigue Performance of Forged Steel

and Ductile Cast Iron Crankshafts", SAE Technical Paper No. 2007-

01-1001, 2007.

[14] QASIM BADER& EMAD K. NJIM "Effect of Stress Ratio and V

notch shape on Fatigue Life in Steel beam ", International Journal of

scientific and Engineering Research (IJSR) ,Vol. 5, ISSN 2229-5518

, Issue 6, 2014.

[15] Johnson Lim Soon Chong, Adnan Husain and Tee Boon Tuan ,

"Simulation of Airflow in Lecture Rooms" , Proceedings of The

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[16] Bruce Boardman, Fatigue Resistance of Steels, ASM Handbook,

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