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03 LCF S-N Curves for Notches

Reference : Fundamentals of Metal Fatigue Analysis Ch. 4 Notches

2017. 9

by Jang, Beom Seon


4.3.1 Notch Root Stresses and Strains

Theoretical stress concentration factor Kt

For increasing nominal stress, S, Kt remains the same, until yielding begins.

4.3 STRAIN-LIFE APPROACH

( notch stress)

(nominal stress)K

S

(notch strain)

(nominal strain)K

e

After yielding occurs,

Kσ decreases with respect to Kt

Kε, increases with respect to Kt

2



After yielding occurs, actual local stress < predicted using Kt,

actual local strain > predicted using Kt,

Neuber’s rule

Nominally Elastic Behavior

Limited Yielding

when yielding occurs in the nominal stresses or strains, Hooke’s law

cannot no longer be used ( ).

Small notches have less effect than is indicated by Kt, Topper et al.

proposed


KKK t

SeKeS

K tt

22 ,

2 1, ( , )t

E EK S Ee

S e S S e S

)()(

)(2

loadappliedE

SKrepsonsenotch t

SeK t

2EeS

E

SK f

2)(

SK

eK

3

( )( )t tK S K e



The local stress-strain response at the notch root may vary from the

nominally applied loading.

Extreme example : Nominally applied stress is from 0 to σmax, while

the local stress-strain response is completely reversed.

Due the residual stresses developed as a result of local yielding at

the notch root.


Nominal stress history Local stress-strain response

4



Kt an S must be defined using the same cross-sectional area (Ktnetand Snet or

Ktgrossand Sgross)

Ktgrossincludes the nominal stress increase effect caused by area reduction

and stress concentration effect in the vicinity of hole.

Ktnetincludes only stress concentration effect in the vicinity of around hole


Gross and net section areas ( a= hole radius, W=plate width)

))(())(( grosstnett SKSKgrossnet

Ktnet

ct

P

5


4.3.2 Example of Notch Analysis Using Neuber’s Rule

For given S1, determine notch stress and strain.

What to be satisfied?

• Neuber’s rule.

• Cyclic stress-strain curve of material

Assumptions

1) Nominally elastic behavior

2) Kf instead of Kt.

3) Net section Ktnet and S based on Anet.


Nominal stress versus time Local (notch) stress-strain

given?

)(

)(

notched

e

unnotched

ef

S

SK

Fatigue notch factor is dependent on material type.

eS : fatigue strength

6



Kf is used to correct only endurance limit(Se) of S-N Curve or entire

S-N curve to be conservative ?


Kʹf is used to adjust S1000

The notch effect at short lives (103) is greatly reduced for low strength of soft material.

For high strength material, the notch effect of fatigue life shortening becomes larger.

)(

)(

notched

e

unnotched

ef

S

SK

Relationship between Kʹf and Kf

(K‘ f

-1)/(K

f-1

) fo

r 10

3 c

ycle

Modification of S-N Curve Juvinall approach

fe KS /

'

1000 / fKS

7



Step 1 : Initial Loading

Nominally elastic version of Neuber’s rule


)()(

)(

2

loadappliedE

SKrepsonsenotch

f

givento be determined

curvehyperbolaconstant :

Intersection of cyclic stress-strain curve and Neuber’s hyperbola

'/1

')( n

KE

E

SK

KE

fn

2

'/1

'

)()(

σ can be solved using numerical methods such as iteration technique

8


Step 2 : Nominal Stress Reversal

Nominal stress change (∆S=S1-S2) results in ∆σ

and ∆ε.

P2 is lies at the intersection of the

hyperbola and hysteresis loop.

The origin of the axis in now at point P1.



Intersection of cyclic stress-strain curve and Neuber’s hyperbola

'/1

')

2(

22

n

KE

E

SK

KE

fn

2

)()

2(

2

2

'/1

'

∆σ can be solved using iterative technique

E

SK f

2)(

2C

Nominal stress reversal

E

SK f

2)2/(

22

E

SK f

2)(

9


Example 4.2

A

PS


Q: A notched steel component of a bar : wide : 1.0 in, 0.25 in thick with two semi-circular edge notches with radii of 0.1 in. Determine the life of the component using a Neuber analysis.

Given 1 :

Given 2 :

c

ff

b

f

fNN

E)2()2(

2

'

'

elastic plastic

'

'

169 0.081

1.14 0.67

2.42

f

f

t

kis b

c

K

123.0,154,1030 ''3 nksiKksiE

Given load P

∆S ∆ ∆ε Nf : fatigue life

Neuber’s rule & cyclic stress-strain

relationship

Strain-life equation

Strain-life equation

Hysteresis loop

'/1

')

2(

22

n

KE

10


Example 4.2


Given load P


Step 3 : calculate ∆Snet

For most engineering applications

Step 2: calculate ∆σ

the elastic-plastic form of Neuber’s rule

))(())(( grosstnett SKSKgrossnet

W= 1.0 in, t=0.25,r=0.1 in, P=10 kips

ksiA

PS

net

net 50)25.0)(2.01(

kips10

2

44

2

eSKt

'/1

')

2(

22

n

KE

'/1

'

'/1

'

2 )2

(22

)2

(22

nn

tKEK

S

E

SSK

'/1

')

2(

22

n

K

S

E

Se

42.2

123.0,154,1030 ''3

tK

nksiKksiE

E

SK f

2)(SeKt

2

11


Example 4.2


Given load P


Step 3 : calculate ∆ε

Fully reversed loading → mean stress =0

Step 4: calculate Nf.

0065.0)2

(22

'/1

'

n

KE

c

ff

b

f

fNN

E)2()2(

2

'

'

reversalsN f 50002

67.014.1

081.0169

'

'

c

bkis

f

f

ksiE 31030

12

2 1/0.123 1/ '

3 3

50 50(2.42) (50) ( ) ( )

30 10 154 2 2(30 10 ) 2 154

n

∆=156 ksi

1.67e-3 1.07e-4 2.6e-3 3.9e-3

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