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CTU in Prague, Faculty of Mechanical Engineering DAF Page 1
Concentration factors
y
x
S
SK t
0
1
x
y
xG
Shape Factor orStress Concentration Factor ofan Elastic Stress
Relative Stress Gradient
CTU in Prague, Faculty of Mechanical Engineering DAF Page 2
http://mechanika.fs.cvut.cz/calculator.php
Concentration factors
15
2
R 4
20
10
111
CTU in Prague, Faculty of Mechanical Engineering DAF Page 3
Concentration factors
https://www.efatigue.com/constantamplitude/stressconcentration/
CTU in Prague, Faculty of Mechanical Engineering DAF Page 4
Fatigue notch factor
0
100
200
300
400
500
600
1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08
Number of cycles [1]
Str
es
s a
mp
litu
de
[M
Pa
]
smooth
notched
FL
FL,N
,
FLf
FL N
K
1t fK K q
Theoretically, if material has high notch sensitivity q
Fatigue notch factor
CTU in Prague, Faculty of Mechanical Engineering DAF Page 5
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.
1 1f tK K q
Notch radius
CTU in Prague, Faculty of Mechanical Engineering DAF Page 6
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 kL
Axial 0.9Bending 1.0Torsion 0.57
CTU in Prague, Faculty of Mechanical Engineering DAF Page 7
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
CTU in Prague, Faculty of Mechanical Engineering DAF Page 8
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.
CTU in Prague, Faculty of Mechanical Engineering DAF Page 9
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
exp
10exp
mDDFL
S d dFL
Vk
V
y
x
S
CTU in Prague, Faculty of Mechanical Engineering DAF Page 10
Fatigue limit modifiing factors
Surface treatment factor, kT
CTU in Prague, Faculty of Mechanical Engineering DAF Page 11
Fatigue limit of a real part
Estimation of the fatigue limit of a real part
,FL L SF S T
FL Nf
k k k k
K
1. CTU in Prague, Faculty of Mechanical Engineering 1. DAF 1. 1. Page 12
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
a
NFLn
,
100
1000
1.E+04 1.E+05 1.E+06 1.E+07
N [1]
a [
MP
a]
alloy steel
CTU in Prague, Faculty of Mechanical Engineering DAF Page 13
Example – Fatigue safe factor calculation
Railway axle
Material: alloy steel 24CrMo4, ASTM 4130
Point A of the potential crack initiation
Experimental strain amplitude measurement (in the point A):
Problem description:
,max 312 microstraina
A
CTU in Prague, Faculty of Mechanical Engineering DAF Page 14
Example – continuation http://fatiguecalculator.com
0,0762 1570 2
b
a f N N
0,07671570 2 10 437.5 MPaFL
CTU in Prague, Faculty of Mechanical Engineering DAF Page 15
Example – continuation http://fatiguecalculator.com
2.09tK
CTU in Prague, Faculty of Mechanical Engineering DAF Page 16
Example – continuation FEM Calculation – CTU Prague
1.95tK
Wheel
Braking disc
Axle
CTU in Prague, Faculty of Mechanical Engineering DAF Page 17
Example – continuation
1.95tK
1 1 1 2.0 1 0.83 1.83f tK K q
15
0
100
200
300
400
500
600
1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08
Number of cycles [1]
Str
es
s a
mp
litu
de
[M
Pa
]
smooth
notched
FL
FL,N
CTU in Prague, Faculty of Mechanical Engineering DAF Page 18
Fatigue Limit
Estimation of the fatigue limit of a real part
,FL L SF S T
FL Nf
k k k k
K
factor k value
loading kL 1.00
surface finish kSF 0.67
size factor kS 0.70
size factor kT 1.00
,
437.5 1.00 0.67 0.70 1.00112.1 MPa
1.83FL N
Example – continuation
CTU in Prague, Faculty of Mechanical Engineering DAF Page 19
Estimation of the nominal stress amplitude
Experimental strain amplitude measurement (in the point A):
,max 512 microstraina
A
206850 0.000312 64.5 MPaa E
Fatigue loadingExample – continuation
CTU in Prague, Faculty of Mechanical Engineering DAF Page 20
Safety factor
Estimation of the safety factor nFL
A 64.5 MPaa
Example – continuation
, 112.1 MPaFL N
0
100
200
300
400
500
600
1,E+03 1,E+04 1,E+05 1,E+06 1,E+07 1,E+08
Number of cycles [1]
Str
es
s a
mp
litu
de
[M
Pa
]
smooth
notched
FL
FL,N
, 112.11.74
64.5FL N
FLa
n
CTU in Prague, Faculty of Mechanical Engineering DAF Page 21
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?
CTU in Prague, Faculty of Mechanical Engineering DAF Page 22
Questions and problems II
.
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 MPasee http://fatiguecalculator.com/cgi-bin/StressShowMatProp.pl
Fig. 1
Other inputs:D= 50 mm outer diameterd= 30 mm inner diameterr= 5 mm notch radius
F= 40 kN maximum loading forceq= 0,8 1 notch sensitivity of the steelR= -1 stress ratio
n L= ? fatigue life safety factorN FL 2,00E+06 number of fatigue limit cyclesRa= 12,5 surface roughness
Example:
Solution
shape factor Kt= 1,71notch factor Kf= 1,57fatigue limit sFL= 7,95E+01safety factor nF= 1,40E+00