Post on 17-Jan-2023
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Topics in Ship Structures
03 LCF S-N Curves for Notches
Reference : Fundamentals of Metal Fatigue Analysis Ch. 4 Notches
2017. 9
by Jang, Beom Seon
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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
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4.3.1 Notch Root Stresses and Strains
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
4.3 STRAIN-LIFE APPROACH
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
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4.3.1 Notch Root Stresses and Strains
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.
4.3 STRAIN-LIFE APPROACH
Nominal stress history Local stress-strain response
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4.3.1 Notch Root Stresses and Strains
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
4.3 STRAIN-LIFE APPROACH
Gross and net section areas ( a= hole radius, W=plate width)
))(())(( grosstnett SKSKgrossnet
Ktnet
ct
P
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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.
4.3 STRAIN-LIFE APPROACH
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
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4.3.2 Example of Notch Analysis Using Neuber’s Rule
Kf is used to correct only endurance limit(Se) of S-N Curve or entire
S-N curve to be conservative ?
4.3 STRAIN-LIFE APPROACH
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
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4.3.2 Example of Notch Analysis Using Neuber’s Rule
Step 1 : Initial Loading
Nominally elastic version of Neuber’s rule
4.3 STRAIN-LIFE APPROACH
)()(
)(
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
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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.
4.3.2 Example of Notch Analysis Using Neuber’s Rule
4.3 STRAIN-LIFE APPROACH
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)(
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Example 4.2
A
PS
4.3 STRAIN-LIFE APPROACH
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
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Example 4.2
4.3 STRAIN-LIFE APPROACH
Given load P
∆S ∆ ∆ε Nf : fatigue life
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
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Example 4.2
4.3 STRAIN-LIFE APPROACH
Given load P
∆S ∆ ∆ε Nf : fatigue life
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