Concentration factors


S
x
Shape Factor or
Stress Concentration Factor of
an Elastic Stress
Relative Stress Gradient
G
  Kt 
1   y

  x

S


 x 0
y
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1
Concentration factors
R4
2
1
11
1
5
10
20
http://mechanika.fs.cvut.cz/calculator.php
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DAF
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2
Concentration factors
https://www.efatigue.com/constantamplitude/stressconcentration/
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DAF
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3
Fatigue notch factor
Stress amplitude [MPa]
Fatigue notch factor
600
 FL
  Kf 
 FL, N
smooth
500
notched
400
Theoretically,
if material has high notch sensitivity q
300
Kt  K f
FL
200
FL,N
100
0
1,E+03
 q  1
1,E+04
1,E+05
1,E+06
1,E+07
1,E+08
Number of cycles [1]
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Fatigue notch factor, Kf and notch sensitivity factor, q
Experiments have shown that the effect
of notches is less than that the estimated
effect according a traditional stress
concentration factor, Kt.
The fatigue notch factor, Kf , can be
described as the effective stress
concentration in fatigue.
This effect is dealt with using a notch
sensitivity factor, q.
  K f  1  Kt 1  q
Notch radius
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Fatigue limit modifiing factors
Loading factor, kL
Historically, fatigue limits have been determined from simple
bending tests where there is a stress gradient in the test
specimen. A specimen loaded in tension will have a lower
fatigue limit than one loaded in bending. An empirical
correction factor, called the loading factor, is used to make an
allowance for this effect.
Loading Type
Axial
Bending
Torsion
kL
0.9
1.0
0.57
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Fatigue limit modifiing factors
Surface finish factor, kSF
Fatigue limits are determined from small polished laboratory specimens. A surface
finish correction is made to estimate the fatigue limit of the part with the actual
surface finish
kSF
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Fatigue limit modifiing factors
Size factor, kS
Experimentally, larger parts have lower fatigue limits than smaller parts. Since the
materials data is obtained from small specimens, a correction factor, called the size factor,
is used for larger diameters.
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Fatigue limit modifiing factors
Size factor, kS
For non-circular sections an effective diameter is computed. The effective diameter is
obtained by equating the volume of material subjected to 95% of the maximum stress to a
round bar in bending with the same highly stressed volume


kS 

D
FL
d 10
FL

D
 Vexp

 d 
V 
 exp 
m
S
x
y
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Fatigue limit modifiing factors
Surface treatment factor, kT
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Fatigue limit of a real part
Estimation of the fatigue limit of a real part
 FL, N 
 FL  kL  kSF  kS  kT
Kf
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Safety factor of unlimited fatigue life (permanent strength)
1. Alternating stress (R=-1)
• operational loading stress amplitude a
• fatigue limit of the real part in the critical cross section area FL,N
1000
 a [MPa]
alloy steel
100
1.E+04
1.E+05
1.E+06
1.E+07
N [1]
n 
1.
 FL, N
a
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DAF
1.
1.
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12
Example – Fatigue safe factor calculation
Problem description:
Railway axle
Material: alloy steel 24CrMo4,
ASTM 4130
A

Point A of the potential crack
initiation
Experimental strain amplitude
measurement (in the point A):
 a,max  312 microstrain
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Example – continuation
http://fatiguecalculator.com
 a   f   2N   1570   2N 
b
 FL  1570   2 107 
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0,076
0,076
 437.5
DAF
MPa 
Page
14
Example – continuation
http://fatiguecalculator.com
  Kt  2.09
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Example – continuation
FEM Calculation – CTU Prague
Wheel
Braking disc
Axle
  Kt  1.95
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Example – continuation
  Kt  1.95
Stress amplitude [MPa]
  K f  1  Kt 1  q  1  2.0 1  0.83  1.83
600
smooth
500
notched
400
300
FL
200
FL,N
100
0
1,E+03
1,E+04
1,E+05
1,E+06
1,E+07
1,E+08
Number of cycles [1]
15
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Example – continuation
Fatigue Limit
Estimation of the fatigue limit of a real part
 FL, N 
 FL, N 
 FL  kL  kSF  kS  kT
factor
k
value
loading
kL
1.00
surface finish
kSF
0.67
size factor
kS
0.70
size factor
kT
1.00
Kf
437.5 1.00  0.67  0.70 1.00
 112.1
1.83
 MPa 
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Example – continuation
Fatigue loading
Estimation of the nominal stress amplitude
A
Experimental strain amplitude
measurement (in the point A):
 a,max  512 microstrain
 a  E    206850  0.000312  64.5
CTU in Prague, Faculty of Mechanical Engineering
MPa 
DAF
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Example – continuation
Safety factor
Estimation of the safety factor nFL
 FL, N  112.1
A
Stress amplitude [MPa]
 a  64.5
MPa
MPa
600
smooth
nFL 
500
notched
400
 FL, N 112.1

 1.74
a
64.5
300
FL
200
FL,N
100
0
1,E+03
1,E+04
1,E+05
1,E+06
1,E+07
1,E+08
Number of cycles [1]
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Questions and problems II
1. What is difference between the shape factor and the notch factor?
Write their definition equations.
2. Define the notch sensitivity factor of material and write equation for
it (as a function of shape and notch factor).
3. Depends the stress concentration factor of metals on a material
parameters? And what about of the notch factor?
4. What is the typical value of the stress concentration factor at a large
tensile loaded plate with a round hole in the middle? Is the notch
factor of such plate lower or higher as the shape factor?
5. Is the fatigue limit of a real part the same as the fatigue limit of a
basic material? What other factors could be taken in the account by
an expression of such fatigue limit?
6. What dimension of a shaft has higher the size factor ks ? Shaft with
higher or smaller diameter?
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Questions and problems II
Example:
Estimate
the fatigue limit and the safety factor of the part on the Fig. 1. under harmonic loading.
The material of the part is steel 4130 bar
Su= 778
see http://fatiguecalculator.com/cgi-bin/StressShowMatProp.pl
MPa
Fig. 1
Other inputs:
D= 50
d= 30
r= 5
F= 40
q= 0,8
R= -1
n L= ?
N FL 2,00E+06
Ra= 12,5
mm
mm
mm
kN
1
outer diameter
inner diameter
notch radius
maximum loading force
notch sensitivity of the steel
stress ratio
fatigue life safety factor
number of fatigue limit cycles
surface roughness
CTU in Prague, Faculty of Mechanical Engineering
Solution
shape factor
notch factor
fatigue limit
safety factor
DAF
Kt=
Kf=
sFL=
nF=
1,71
1,57
7,95E+01
1,40E+00
Page
22