ch07-mechanical design

8/7/2019 ch07-mechanical design

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Introduction Mechanical components and parts often failed

under variable loading that need to be considered

during design

The stresses vary or fluctuate between levels when

the load varies or fluctuates

These fluctuating loads in machine members

produces variable, repeated, alternating, or fluctuatingstresses

Often machine elements fail due to variable loading

may have actual maximum stress below the ultimate

or yield strength of material


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Fatigue in MetalsWhy does a part fail at a stress level below ultimate and

yield strength of the material???

This type of failure is due to repeated stresses for a verylarge number of times and called FATIGUE failure

When a part fails under static loading, they usually

develop a large deflection and provides visible warning

Fatigue failure is sudden, total, and hencedangerous as it gives no warning

Fatigue is much more complicated phenomenon,

not well understood and the engineer must acquire

knowledge about fatigue behavior of material and load


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Fatigue in Metals Fatigue failure appears like a brittle fracture with flat

fracture surfaces, perpendicular to the stress axis, no

necking in the failed area

Fracture feature of a fatigue failure, are different from a

static brittle fracture arising from three stages of

development

Stage I: Initiation of one or more micro-cracks due to cyclicplastic deformation followed by crystallographicpropagation

Stage II: Microcracks becomes macrocracks formingparallel plateau like smooth surfaces separated by

longitudinal ridges


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Sch

ematics of

fatigue fracture

surfaces of

various parts

under different

load conditions


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Causes ofFatigue

Failure Fatigue failure is due to crack formation and propagation

A fatigue crack will typically initiates at a discontinuity in

the materials

Conditions that can accelerate crack initiation are

Residual tensile stresses

Elevated temperature

Temperature cycling

Corrosive environment

Rate and direction of fatigue propagation is primarily

controlled by localized stresses and by the structure of the

material at the crack


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Examples ofFatigue

FailureFatigue fracture

initiated at the end

of keyway

Fatigue fracturesurface of a

pinhole

Fatigue fracturesurface of a

forged connecting

rod

Fatigue fracture

surface of apiston rod


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Fatigue Failure in

Analysisand Design The following structured approaches can be used by

design engineers to consider fatigue in design of

mechanical systems and elements.

Fatigue-Life method to predict failure

Fatigue strength and endurance limit of material

Endurance limit modifying factors

Stress concentration and notch sensitivity

Fluctuating stresses

Combination of loading modes

Varying fluctuating stresses: cumulative fatigue damage


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Fatigue Life Method Three major fatigue life methods used in design are

Stress-life method (most traditional, least accurate forlow cycle fatigue, easiest to implement)

Strain-life method (good for low cycle fatigue)

Linear-elastic fracture mechanics method (assumes a

crack is already present and detected, most practical) These methods predict life in number of cycles of failure, Nfor a specific level of loading

1 N 103 cycles: Low-cycle fatigue

N > 103 cycles: high cycle fatigue


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Stress Life Method

(S-N Diagram)

Endurance limit


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Stress Life Method

(S-N

Diagram) Figure shows the scatterband of S-N curves for mostcommon aluminum alloys withtensile strength < 38 Kpsi

Aluminum does not have anendurance limit, se . Fatiguestrength is reported for N= 5 x108cycles (Appendix A-24)

Stress life method is leastaccurate for low cycle fatigue,easiest to implement for a widerange of application and highcycle fatigue


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Strain-Life Method

One of the best approaches to

explain the nature of fatigue

failure

A fatigue failure almost alwaysbegins at a local discontinuity

(notch, crack, etc.)

When the stress in the

discontinuity exceeds the

elastic limit, plastic strainoccurs, resulting a fatigue

fracture

Figure shows that strength

decreases with stressrepetitions


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Strain-Life Method

If the reversal occurs in the

compressive region, slightly

different results may be obtained

due to fatigue strengthening effect

of compression) The monotonic stress-strain

relations in both tension and

compression are compared with

cyclic stress-strain curve for H-11

steel (660 BHN) and SAE 4142 (400BHN)

It is difficult to predict the fatigue

strength of a material from known

values of monotonic yield and

ultimate strength in low cyclefatigue


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Strain-Life Method

Fatigue ductility coefficientfis the true straincorresponding to fracture in

one reversal

Fatigue strength coefficientfis the true stresscorresponding to fracture in

one reversal

Fatigue ductility exponent Cis the slope of the plastic

strain line

Fatigue strength exponent bis the slope of the elastic

strain line

The Manson-Coffin relationship

between fatigue life and totalstrain amplitude is expressed as

Table 7-1 shows properties of some

high strength steels

c

F

bF NNE

)2()2(2

''

IWI

!(


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The first phase of fatigue cracking is designated as

STAGE I fatigue that involves crystal slip through

several contiguous grain

The second phase or STAGE II of crack extension

advances the crack that can be observed on

micrographs from an electron microscope

Final fracture occurs during stage III fatigue.When

crack is sufficiently long that KIC = KI for the stress

amplitude involved, KIC= critical stress intensity for undamagedmetal In Stage III fatigue, crack accelerates rapidly to failure

Linear-Elastic Fracture

Mechanics Method


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Mechanics (Crack Growth)

Fatigue crack nucleate andgrow when stresses vary, thereis some stress in each cycle

For fluctuating stress range = max- min, the stressintensity range can be defined

Assuming initial crack lengthof ai, crack growth as afunction of number of stresscycles N will depend on

For KIbelow some threshold

value, crack will not grow

aaKt TWFTWWF (!!( )( minmax

Three stress levels:

()3 > ()2> ()1

(KI)3 > (KI)2> (KI)1

Higher stress range produces longer

cracks at a particular cycle count


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Mechanics (Crack Growth)

m

IKC

dN

da)((!

When rate of crack growth percycle da/dN is plotted, the threestages of crack observed

A simplified procedure forestimating the remaining lifefor a cyclically stresses partafter discovery of crack

Assuming a crack isdiscovered in stage II, the crackgrowth can be approximated by

c and m : empirical constant

(!

(!!

f

i

f

i

f

a

am

a

a

m

I

f

N

a

da

C

K

da

CNdN

TWF

1

1

0m

tKCdN

da)((!

ai is the initial

crack length, afis the final crack

length and Nfis

the estimated

no. of cycles forfailure


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The Endurance Limit

Endurance limit is determinedby fatigue testing

Stress testing is preferred tostrain testing

Results of rotating beamtests and simple tension testsof bar or ingot specimen areavailable in literature

Time for variousmicrostructure are also used todetermine endurance limits thatvary from 23-63% of tensilestrength

Endurance limits versus tensile strength

is plotted. Se/ Sut of 0.6, 0.5, 0.4 are

shown.

The horizontal dotted line for Se=107

kpsi is for tenslie strength >207 kpsi


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The Endurance Limit

Mischke analyzed fatigue test data and presented relationshipbetween tensile strength and endurance limit for steel

Steel treated to give different microstructures have differentSe /Sutratios

Ductile microstructures have a higher ratio, Martensite has alower ratio due to very brittle nature and susceptible to crack

Endurance limits of various materials are in Table A-24

Aluminum alloy do not have an endurance limit, Fatiguestrength of Aluminum at 5x108cycles are given in Table A-24

""

e

!

)1460(740)212(107

)1460212(504.0'

MpaSMpaKpsiSKpsi

MpaorKpsiSMpaorKpsiS

S

ut

ut

utut

e

Sutis theminimumtensilestrength


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Fatigue Strength

From N=1 to 103 cycles, in low cycle fatigue regions, thefatigue strength Sf is slightly less than tensile strength Sut

High cycle fatigue extends from 103 cycles to 106or 107cyclesto endurance limit life N

e Methods for approximation of S-N diagram are presented tocalculate the fatigue life of different materials

From true stress-true strain diagram (Eq.3-11), F=

0m

ut

b

ut

F

ut

b

Fcyclesf

SoffractionfS

f

SfS

!!

!!

)10.2(

)10.2()(

3'

3'

103

W

W


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Fatigue Strength

SAE approximation may be used for steel with HB 500

F= Sut+ 50 Kpsi or F = Sut+345 Mpa

The exponent b can be found from a =Se = F(2Ne)b

Methods for approximation of S-N diagram are presented tocalculate the fatigue life of different materials

For Sut=105 kpsi, Se=52.5 kpsi at 106cycles to failure

F= 105+50=155 Kpsi, b= -0.0746, f = 0.837

Sf= 155(2N)-0.0746,Empiriically Sf= a Nb

)2log()/log( '

e

eF

NSb W!

!!

e

ut

e

ut

S

fSb

S

Sfa log

3

1,

)(2


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Endurance Limit

Modifying Factor Martin developed factors to quantify the effects of surfaceconditions, size. loading., temperature and miscellaneous itemsas

Se = Ka Kb KcKe KfSeKa = surface condition modification factor

Kb = size modification factor

Kc= load modification factor

Kd= temperature modification factorKe = reliability factor

Kf= miscellaneous-effect modification factor

Se = rotary beam test specification factor

Se= endurance limit at the critical location


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Endurance Limit

Modifying Factors The Loading factor, Kcfor bending, axial and torsion limitsare Kc=1 (bending), 0.85 (axial) and 0.59 (torsion)

Brittle fracture is a strong possibility at operating temperaturebelow room temperature and yielding may occur at high

temperature as yield strength decreases at higher temperature The temperature factor Kd is the ratio of tensile strength atoperating temperature (ST) to the tensile strength at roomtemperature (S

RT), Kd= (ST/ SRT) shown in Table 7-6

As the standard deviation of endurance strength is less than8% , the reliability modification factor Ke = 1-0.08 Za(Transformation variate Z = (X-x)/x), Value of Z shown inTable A-10, Table 7-7: reliability factors for some std. reliability

Miscellaneous-Effect Factor Kfaccounts for reduction inendurance limits due to all other effects


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StressConcentration

and Notch Sensitivity

Stress concentration factor Ktor Kts used with nominal stressto obtain maximum resulting stress due to irregularity or defect

Some materials are not fully sensitive to notches and areduced value of Kt can be used. The maximum stress

max= Kfo or max= Kfs o

Kf is a reduced value of Kt and is called a fatigue stressconcentration factor

Notch sensitivity q defined as

specimenfreenotchinstress

specimennotchedinstressimumKf

!

max

1

1

1

1

!

!

ts

fs

shear

t

f

K

Kqor

K

Kq

The value of q is

between zero and one


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StressConcentration


When q = 0, Kf= 1,the materialhas no notch sensitivity

When q = 1, Kf= Kt, thematerial has full notchsensitivity

Find Ktfrom geometry, find qfrom figure (for bending andaxial loading)

Kf= 1 + q(Kt -1)

Kfs = 1 + qshear(Kts-1)

Notch sensitivity of cast ironis very low (q=0.20)

The above figure is based on NeuberEquation

a is defined as Neuber constant, amaterial constant

ra

KK

f

f

!

1

11


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StressConcentration


ra

KK

f

f

!1

11

r

aq

!

1

1

Notch sensitivity for shearloading is shown in the chart

For a large notch radius q 1

The notch sensitivityequation

The modified Neuberequation for fatigue stressconcentration factor

Table 7-8 gives values of a forsteel for transverse holes,shoulders, and grooves

r

a

K

K

KK

t

t

tf

)1(21

!

r

aq

!

1

1


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Fluctuating Stresses

Fluctuating stresses inmachinery often take the formof a sinusoidal pattern due toforces in some rotating

machinery Periodic load patterns exhibita single maximum (Fmax) andsingle minimum force (Fmin)

The steady (midrange) andalternating components are

22

minmaxminmax FFFand

FFF

am

!

!

min = minimum stressmax= maximum stress

a = amplitude component

m = midrange component

r= range of stresss = static or steady stress


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Fluctuating Stresses

The midrange and alternating stresses are defined as

The stress Ratio

In the absence of notch, a and m are equal to the nominalstresses ao and mo induced by loads Fa and Fm respectively

The steady stress component stress concentration factor Kfm

Kfm = Kf Kf|max,o|< SY

Kfm = (Sy-Kfa,o)/ |m,o| Kf|max, o|>SY

Kfm = 0 Kf |max,o min,o| > 2Sy

m

aAR

W

W

W

W!!

max

min

22

minmaxminmax WWWWW

W

!

! am


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Fatigue Failure Criteria for

Fluctuating Stress

Three methods of plotting the results offatigue resistance tests are

- Modified Goodman Diagram,

- Plot of fatigue failure for midrangestresses

- Master Fatigue diagram

The modified Goodman diagram has themidrange stress plotted along the abcissa

and all other components of stress plottedon the ordinate, with tension in positivedirection


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Fluctuating Stress

Fatigue failure for midrange stresses in both tensile and compressive regions


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Fluctuating Stress

Fatigue diagram showing different failure criteria.

Points above the line indicate failure


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Varying, Fluctuating Stresses:

Cumulative Fatigue Damage Instead of a single fully reversed stresshistory of n cycles, a mechanicalcomponent may be subjected to:

Fully reversed stresses of 1 for n1cycles, 2for n2cycles and so on

Characterization of a cycle takes ona max-min-same max form as shown

Failure loci are expressed in terms ofstress amplitude component a andsteady component m

The Palmgren-Miner cycle rationsummation rules also called the Minersrule is presented as

ni is the number of cycles at stress level

i and Niis the number of cycles to failure

! cN

n

i

i

The parameter c has been

determined by experiment, 0.7


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Varying, Fluctuating Stresses:

Cumulative Fatigue Damage Assume steel with Sut = 80 kpsi, Se

=40

kpsi, f =0.9

The log S-logN diagram is shown in thefigure by a heavy solid line

For a reversed stress of 1

= 60 kpsi, n1= 3000 cycles the endurance limit will be

damaged ( 1 > Se)

Sf= aNb = 129. is the number of cycles atstress level i and Ni is the number ofcycles to failure

.


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.7.34

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.7.35

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.7.36

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.7.37

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.7.38

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.7.39

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.P7.11

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.P7.17

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P7.20

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P7.21

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P7.23

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P7.24

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P7.26

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P7.27

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TA7.5a

.


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TA7.5b

.


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TA7.5c

.


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TA7.5d

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