The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
У роботі представлено числовий алгоритм, який дозволяє вибирати різні схеми навантаження конструкції, наприклад, гідроциліндра, відповідно до заданих кривих Велера, що характеризують опір матеріалу на втомне руйнування. Проаналізовано гідроциліндри, які розглянуті в проекті E. C. \"PROHIPP\&quo...
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Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України
2005
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| Zitieren: | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis / T. Bednarek, I. Marczewska, A. Marczewski, W. Sosnowski, H. Jakubczak, J. Rojek // Фіз.-мат. моделювання та інформ. технології. — 2005. — Вип. 2. — С. 70-86. — Бібліогр.: 5 назв. — англ. |
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| author | Bednarek, T. Marczewska, I. Marczewski, A. Sosnowski, W. Jakubczak, H. Rojek, J. |
| author_facet | Bednarek, T. Marczewska, I. Marczewski, A. Sosnowski, W. Jakubczak, H. Rojek, J. |
| citation_txt | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis / T. Bednarek, I. Marczewska, A. Marczewski, W. Sosnowski, H. Jakubczak, J. Rojek // Фіз.-мат. моделювання та інформ. технології. — 2005. — Вип. 2. — С. 70-86. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| description | У роботі представлено числовий алгоритм, який дозволяє вибирати різні схеми навантаження конструкції, наприклад, гідроциліндра, відповідно до заданих кривих Велера, що характеризують опір матеріалу на втомне руйнування. Проаналізовано гідроциліндри, які розглянуті в проекті E. C. \"PROHIPP\", і запропоновано деякі розв'язки задачі про тріщину в області штуцера. Показано, що просочування олії в зоні з’єднання штуцера може бути усунуто після деяких модифікацій конструкції.
In this paper a numerical algorithm is presented to make possible adopting different loading schemes of specific structure at hand for instance hydraulic cylinders, to specific Wцhler curves characterizing fatigue resistance of given material. Hydraulic cylinders investigated under E. C. project \"PROHIPP\" are analyzed and some solutions of the crack problem in oil port area are proposed. Oil penetration in an oil port connection zone can be eliminated after some design modifications.
В работе представлен численный алгоритм, позволяющий выбирать различные схемы нагружения конструкции, например, гидроциллиндра, в соответствии с заданными кривыми Вёлера, характеризующими сопротивление материала усталостному разрушению. Проанализированы гидроциллиндры, рассматриваемые в проекте E. C. \"PROHIPP\", и предложены некоторые решения задачи о трещине в области штуцера. Показано, что просачивание масла в зоне соединения штуцера может быть устранено после некоторых модификаций конструкции.
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| first_indexed | 2025-11-30T16:44:46Z |
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The equivalent amplitude stress as a solution
of mean stress effect problem in fatigue analysis
Tomasz Bednarek1, Izabela Marczewska2, Artur Marczewski3,
Włodzimierz Sosnowski4, Hieronim Jakubczak5, Jerzy Rojek6
1 Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Swietokrzyska, 21 00-049, War-
saw, Kazimierz Wielki University, Institute of Environmental Mechanics and Applied Computer Science, ul. Chod-
kiewicza, 30 85-064, Bydgoszcz, e-mail: bednarek@ippt.gov.pl
2 Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Swietokrzyska, 21 00-049, War-
saw, e-mail: imar@ippt.gov.pl
3 Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Swietokrzyska, 21 00-049, War-
saw, e-mail: asmar@ippt.gov.pl
4 Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Swietokrzyska, 21 00-049, War-
saw, Kazimierz Wielki University, Institute of Environmental Mechanics and Applied Computer Science, ul. Chod-
kiewicza, 30 85-064, Bydgoszcz, e-mail: wsosn@ippt.gov.pl
5 Warsaw University of Technology, Institute of Construction Machinery Engineering, ul. Narbutta, 84 02-524, Warsaw
6 Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. Swietokrzyska, 21 00-049, War-
saw, e-mail: jrojek@ippt.gov.pl
In this paper a numerical algorithm is presented to make possible adopting different loading sche-
mes of specific structure at hand for instance hydraulic cylinders, to specific Wöhler curves cha-
racterizing fatigue resistance of given material. Hydraulic cylinders investigated under E. C. project
«PROHIPP» are analyzed and some solutions of the crack problem in oil port area are proposed.
Oil penetration in an oil port connection zone can be eliminated after some design modifications.
Key words: fatigue analysis, Wöhler curves, Goodman diagrams, hydraulic cy-
linders, endurance limit.
Nomenclature
ρ density Sat threshold amplitude stress
E Young modulus Sa stress amplitude
ν Poisson ratio Sae equivalent stress amplitude for R = −1
p inner pressure Sm mean stress
pmin minimal values of inner pressure Smax maximal stress
pmax maximal values of inner pressure Smin minimal stress
R stress ratio n exponent in Goodman relationship
N number of cycles f i factor of stress corresponding to initial stress
Su ultimate stress f t factor of stress corresponding to threshold stress
Sai initial amplitude stress
УДК 539.3
70
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2005, вип. 2, 70-86
71
Introduction. There are two objectives of this paper. The first aim eies in presenting a
numerical algorithm, which makes possible adopting different loading schemes of hyd-
raulic cylinders to specific Wöhler curves characterizing fatigue resistance of given
material. An uniaxial loading case is studied in order to confirm the relation between
Goodman and Wöhler curves parameters. The second presenting calculations of hyd-
raulic cylinders investigated under E. C. project «PROHIPP» and to propose some
solution of the crack problem in oil port area, in order to eliminate oil penetration in an
oil port connection zone.
It is well known that majority of fatigue tests is made for symmetric or nearly
symmetric load when stress ratio is
1max
min
max
−===
S
S
p
pR mix . (1)
Number of codes accept input data only for such kind of symmetric load. In
practice majority of structures for instance hydraulic cylinders, works under much
more complex cyclic load characterized by different R values. In particular, some spe-
cific laboratory tests for hydraulic cylinders are made for 0=R when the stresses
changes from 0 to some specific, maximal value. In this case the external pressure is
applied to the cylinder and then the pressure is removed. So the idea of equivalent
amplitude stress is introduced in section 2 in order to find out number of cycles to
failure for loads 1−≠R in situations when test data are provided only for values
1−=R .
Number of cycles in such tests depends on fatigue resistance of the weakest
point of the cylinder. An example of such points is oil ports, where the welding resi-
dual stresses, local shear forces and forces due to oil penetration in the connection gaps
lead to fatigue cracks causing final destruction.
Two kinds of numerical tests are performed. At the beginning the standard fati-
gue tests on workpiece shown in fig. 2 are simulated (Section 3). Goodman and Wöh-
ler curves and idea of equivalent amplitude stress are used in order to calculate number
of cycles to reach the fatigue limit of specimens with non-symmetric ( 1−≠R ) load.
Next in Section 4 the hydraulic cylinders fatigue problem is considered. Oil port
typical deformations and design modifications possibility are shown in fig. 12. It can
be observed that the gap between the oil port and cylinder in the oil port connection
zone grows up with increasing oil pressure. Experiments (fig. 12) also confirm the
crack sensitivity of the weld in this connection zone. Authors propose the improvement
of this bad situation by using special washer or glue (as it is shown in fig. 15) in order
to eliminate excessive gap between the oil port and cylinder surfaces. Such washer or
glue prevents oil penetration thus eliminating the possibility of premature fatigue crack
in neighboring welds.
The washer or glue material should be high temperature resistant — due to wel-
ding process of oil ports.
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
72
1. The idea of equivalent amplitude stress calculations for different stress ratio
The objective of this section is developing the algorithm of calculation of the amplitu-
de stress 1−=R
aeS , which is equivalent to the given value 1−≠R
aS . The assumption is
made that such amplitude stress 1−=R
aeS gives the same number of cycles to fatigue
failure as it is for given 1−≠R
aS . This assumption is based on Goodman and Wöhler
curves dependency (fig. 1). Further the amplitude stress 1−≠R
aS can be obtainned from
the finite element analysis of specific structure at hand loaded by arbitrary non-
symmetric load, 1−≠R .
The history of standard fatigue test goes back to Wöhler who designed and built
the first rotating that is beam test machine that produced fluctuating stress of a constant
amplitude in test specimens [1, 2]. In tests Wöhler established a material property,
known today as the fatigue limit. When specific fatigue data are missing, for example,
number cycles to failure for the given stress ratio, one can use, in some range, empi-
rical relation between amplitude stress and fatigue life N as a linear approximation of
the S−N curve in log-log coordinates. This range is described by two points: initial
amplitude stress ( aiS ) at about 1000 cycles and threshold stress ( atS ) at approximately
2 millions cycles. Such linear Wöhler S-N curve for steel is shown on the left side of
fig. 1 and can be represented (for R = – 1) by the equation
( )[ ] ( )[ ][ ] ( )[ ]ai
Ri
it
ai
R
at
R SNN
NN
SS
a NS
1
11 log)log()log(
)log()log(
loglog
10)(
−=
−=−= +−
−
−
= , (2)
R=-1
Sm
Su
Sat
(R=-1)
Sa
(R=-0.5)
Sm
(R=-0.5)
working point
Goodman diagram Sai
(R=-1)
Sae
(R=-1)
N
Ni Nt
Sa
N
Wöhler curve Sa(N) for R=-1
R=-0.5
Wöhler curve for R=-0.5
Sat
(R=-0.5)
Sat
(R= -1) =ft Su
Sai
(R=-1)=fi Su
Sai
(R=-0.5)
Sa
Fig. 1. Wöhler and Goodman diagrams dependency
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2005, вип. 2, 70-86
73
where 31eNi = and 62eNt = . Initial amplitude stress ( )
ai
RS 1−= for 31eNi = number of
cycles is smaller than the ultimate stress uS
( )
i
u
ai
R fSS =−= 1 . (3)
Value of if can be found in literature [1, 3, 4]. In this paper if equals 0,9. The
value of threshold amplitude stress ( )
at
RS 1−= for 62eNt = number of cycles is obtained
by equation
( )
t
u
at
R fSS =−= 1 , (4)
where tf is the decreasing factor of stress corresponding to the stress threshold value.
Value of tf changes between 0,05 and 0,5 and depends on material properties and
stress concentration factor [1, 3, 4]. In this paper tf is equal 0,5. In following text
simplified notation aiai
R SS =−= )1( and ( )
atat
R SS =−= 1 will be used.
Substituting the amplitude stress in the considered point of the structure by wor-
king point ( )
a
R
a SNS 1)( −== and modifying the equation (2) we obtain the number of
cycles to failure N
( )[ ] ( )[ ]
)log()log(
)log()log()log()log()log(log)log(log 11
10
atai
i
at
t
ai
t
a
Ri
a
R
SS
NSNSNSNS
N −
−+− −=−=
= . (5)
In this case ( )
max
1 SS a
R =−= due to the assumption of 1−=R load scheme.
Wöhler curves are obtained for the load scheme 1−=R by bending fatigue tests
and rarely the axial-load tests ( 0=R ). However thisese zero or −1 mean stress ratio is
not typical for real industrial components working under cyclic load.
On the basis of a on value of ultimate stress uS of the material and a value of the
amplitude stress for different 1−≠R we can obtain the equivalent value of the ampli-
tude stress for 1−=R from Goodman diagram (see fig. 1 right). Then this value can
be applied in the equation (5) in order to calculate number of cycles to failure N .
Goodman diagram (fig. 1) is represented by the line between amplitude stress
( )
a
RS 1−= ( mS equals 0) and ultimate stress of the material on mS axis and the equation (6)
takes plase
( ) ( )
−= −=−≠
n
u
m
a
R
a
R S
SSS 111 (6)
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
74
When 1=n the equation (6) is Goodman equation, at 2=n we obtain Gerber
equation. Here we use safe Goodman case 1=n or 2,1=n . This relationship permits
to obtain the equivalent amplitude stress aeS for any load scheme ( 1−=R ).
From the typical finite element analysis we can get maximal stress at the most
stressed point of the structure.
The value of the mean stress mS in the considered structure point can be obtai-
ned from the equation
2
)1(max RSS m +
= (7)
where maxS is a maximal stress value obtained for any R from FEM analysis. Then
( )
a
RS 1−≠ can be calculated from the following relations
( )
ma
R SSS −=−≠
max
1
or
( )
max
max
1 2
)1( SRSS a
R +
+
−=−≠ . (8)
These values can be used in the modified equation (6)
( )
( )
( )nu
m
a
Rae
R
SS
S
S
−
= −≠
−=
1
1
1 (9)
After substituting the values from the equations (7) and (8) into the equation (9)
we obtain the equivalent amplitude stress as a function of a maximal stress value from
FE analysis for any stress ratio R
( )
( ) ( )
( )
n
u
R
RRae
R
S
S
R
SRS
S
+−
++−
=
−≠
−≠−≠
−=
max
1
max
1
max
1
1
)1(
2
11
)1(5,0
. (10)
When we use Goodman relationship (n = 1) the equation (10) takes more simpler
( )
( )
( ) )1(2
)1(
max
1
max
1
1 −−
−
=
−≠
−≠
−= RSS
SRS
S
Ru
uRae
R . (11)
Then the value of the equivalent amplitude stress calculated from the equations (10)
or (11) can be substituted into the equation (5), where ( )
ae
R
a
R SS 1)1( −=−= ≡ , in order to
calculate the number of cycles to failure of the analyzed structure loaded by arbitrary
non-symmetric load with any stress ratio 1−≠R .
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75
2. Numerical examples. Damage analysis of uniaxially loaded specimen
The following example shows the capability of the algorithm in comparison with some
experimental results available [5].
Fig. 2. The considered specimen
Tab. 1. Number of cycles to failure calculated and obtained by experiment [5]
Lp. Sm
[MPa]
Sa
[MPa] R Number of cycles to faliure
(experiment)
Number of cycles to
faliure (computation)
1 75 250 -0,538 439300; 402500; no faliure no failure
2 75 270 -0,565 358200; 854700; 318700 857 294
3 75 290 -0,589 252300; 376300; 379700 340 262
4 75 310 -0,610 54800; 123400; 45000 143 639
5 150 270 -0,286 172100; 121500; 233100 143 963
6 150 290 -0,318 124300; 41900; 60500 57 139
7 225 230 -0,011 413900; 204500; 545200 125 280
Fig. 3. The S-N curve and equivalent stress amplitudes ae
RS )1( −= for Sm = 75 [MPa]
1000
am
pl
itu
de
st
re
ss
S
a [M
Pa
]
(lo
ga
ry
th
m
ic
sc
al
e)
100
102 103 104 105 106 107
N (logarythmic scale)
N = 143639 N = 340262
N = 857293 no fatigue
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
76
Fig. 4. Goodman diagram for load schemes with Sm = 75 [MPa]
Fig. 5. Goodman diagram for load schemes with Sm = 150 [MPa] and Sm = 225 [MPa]
Uneasily loaded specimen (fig. 2) was analyzed experimentally in [5]. The geo-
metrical siges are shown in fig. 2. The length of the specimen is 0,019 [m]. Material of
the specimen is 10HNAP steel, parameters are as follows: density ρ = 7800 [kg/m3],
Young modulus E = 2,10e+11 [Pa], Poisson ratio ν = 0,3, yield stress Re = 414 [MPa]
and ultimate stress Su = 556 [MPa]. Exponent factor n in the equation (6) is assumed as 1,2.
Three different value of mean stress are considered: 75, 150 and 225 [MPa]. In the case
am
pl
itu
de
st
re
ss
S
a [M
Pa
]
4,5·102
8,3401 =−=
ae
RS
8,3181 =−=
ae
RS
8,2961 =−=
ae
RS
8,2741 =−=
ae
RS
2,5·102
2,0·102
1,5·102
1,0·102
5,0·101
0 0
working points
0 100 200 300 400 500 600
mean stress Sm = 75 [MPa]
Su
Sa = 250 Sa = 270 Sa = 290 Sa = 310
4,5·102
0,3661 =−=
ae
RS
4,3441 =−=
ae
RS
7,3401 =−=
ae
RS
2,5·102
2,0·102
1,5·102
1,0·102
5,0·101
0,0
am
pl
itu
de
st
re
ss
S
a [M
Pa
]
working points
0 100 200 300 400 500 600
mean stress Sm = 75 [MPa]
Su
Sa = 270 Sa = 290 Sa = 230
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2005, вип. 2, 70-86
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of mS = 75 [MPa] four different stress amplitudes are assumed: aS = 250, 270, 290 and
310 [MPa], in case of mS = 150 [MPa] stress amplitudes are aS = 270 and 290 [MPa]
and in the case of mS = 225 [MPa] stress amplitude equals 230 [MPa] [5].
Fig. 6. Wöhler curve and equivalent stress amplitudes ae
RS )1( −=
for Sm = 150 [MPa] and Sm = 225 [MPa]
The first case: mS = 75 [MPa].
Four values of the amplitude stress aS are analyzed: aS = 250, 270, 290 and
310 [MPa]. Maximal and minimal stresses ),( minmax SS are: (325,–175); (345,–195);
(365,–215); (385,–235) respectively. Working points and equivalent amplitude stresses
for 1−=R are presented in fig. 4. Number of cycles to failure (obtained from the equa-
tion (5)) are in table 1 (see fig. 3). Taking into consideration very large range of expe-
rimental life of specimen, obtained predicted life of specimens are mostly located in
experiment range.
The second case: mS = 150 [MPa] and mS = 225 [MPa].
Three values of the amplitude stress aS are analyzed: aS = 270 and 290 [MPa]
which corresponds with mS = 150 [MPa] and aS = 230 [MPa] as well as with
mS = 225 [MPa]. Maximal and minimal stresses ),( minmax SS are (420,–120); (440,–
140); (455,–5) respectively. Working points and equivalent amplitude stresses for
1−=R are presented in fig. 5. Number of cycles to failure (obtained from the equation
(5)) are in table 1 (see fig. 6). In this case range of experimental data is very large.
1000
am
pl
itu
de
st
re
ss
S
a [M
Pa
]
(lo
ga
ry
th
m
ic
sc
al
e)
100
N = 57139
N = 125279 N = 143962
102 103 104 105 106 107
N (logarythmic scale)
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
78
The calculated values of number of cycles to failure are close to experimental data, ex-
cept mS = 225 [MPa]. Here a larger value of exponent n in the equation (6) might be used.
Fig. 7. Geometrical shape of the cylinder 1
Fig. 8. Critical zones and critical points in the cylinder 1
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
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79
3. Hydraulic cylinder design, oil ports fatigue and possible design modifications
3.1. Fatigue analysis of the hydraulic cylinder 1. The cylinder shown in fig. 7 was
tested.
The cylinder has two oil ports. The material of the specimen is steel St52. The mate-
rial data are: density ρ = 7800 [kg/m3], Young modulus E = 2,10e+11 [Pa], Poisson ratio
ν = 0,3, initial flow stress Re = 350 [MPa], saturation flow stress Su = 520 [MPa]. Inner pres-
sure is 30 ± 0,08 [MPa]. Stress ratio maxmin ppR= equals 0.
Initial and threshold amplitude stress for steel St52 and 1−=R equals: ( ) =−=
at
RS 1
[MPa] 4689,0 == uS and ( ) [MPa] 23445,01 ==−= u
at
R SS .
Five critical zones determined in the numerical experiment are shown in fig. 8.
Number of cycles to failure (see fig. 9) is obtained as intersection of adopted to fully
symmetrical load scheme )1( −=R stress amplitude with classical Wöhler curve. The
values of maximal, amplitude and mean stress in each of the zones and number of
cycles to failure are shown in table 2.
Tab. 2. Number of cycles to failure in critical zones of the cylinder 1
zone 1 zone 2 zone 3 zone 4 zone 5
Smax
(R=0) [Mpa] 399,13 367 352,10 357,38 376,76
Sa
(R=0) [Mpa] 199,57 183,50 176,05 178,69 188,38
Sm
(R=0) [Mpa] 199,57 183,50 176,05 178,69 188,38
Sae
(R=-1) [Mpa] 323,85 283,57 266,16 272,24 295,39
number of cycles to failure 5,67·104 2,43·105 4,8·105 3,8·105 1,55·105
Fig. 9. Behavior of the Wöhler curve and values of amplitude stress Sae
in critical zones for the cylinder 1
am
pl
itu
de
st
re
ss
S
a [M
Pa
]
(lo
ga
ry
th
m
ic
sc
al
e)
1000
100
102 103 104 105 106 107
N (logarythmic scale)
S-N curve zone 1 zone 2 zone 3 zone 4 zone 5
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
80
The largest value of stress appears in the zone 1, then next in the zone 5 (point 5.1)
and successively in the zone 2 (point 2.1), and in the zones 4 and 3 (points 4.1 and 3.1)
(see fig. 8). Location, where fatigue crack appears, first are points 5.1 and 2.1. That
fact was confirmed in real experiment where the weakest point was in the zone 5.
Fig. 10. Goodman diagram and working points for cylinder 1
for different critical zones
Number of cycles to failure in the zone 5 (see fig. 8) equals 1554165 =zoneN . The
difference between results obtained by experiment and numerical test do not exceed 9%.
3.2. Fatigue analysis of the hydraulic cylinder 2. The part of hydraulic cylinder is ana-
lyzed. The geometrical shape is shown in fig. 11. The cylinder has two oil ports.
The material of the cylinder is steel St52. Material data are the same as in pre-
vious example. Inner pressure equals 10 [MPa]. Stress ratio R equals 0. The initial and
threshold stress for steel St52 are taken from previous example.
Deformations and stress redistribution in the connection zone caused by oil penetra-
tion and shear forces inside anoil tube are shown in fig. 12.
Sa [M
Pa
]
ae
RS )1( −= amplitude stresses in each critical
zone when R = – 1
( )m
R
a
R SS )0()0( , == amplitude and mean
stresses for each critical zone
Sm [MPa]
Su
u
at
R SS 45,0)1( =−= amplitude
of threshold stress for St52
steel when R = –1
Goodman diagram for St52 steel
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2005, вип. 2, 70-86
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Fig. 11. Geometrical shape of the cylinder 2
Tab. 3. Number of cycles to failure in critical zones of the cylinder 2 calculated using FEM
zone 1 zone 2 zone 3 zone 4 zone 5
max
)0( =RS [MPa] 350,00 219,00 255,00 130,00 352,00
a
RS )0( = [MPa] 175,19 109,34 127,25 65,25 175,84
m
RS )0( = [MPa] 175,19 109,34 127,25 65,25 175,84
ae
RS )1( −= [MPa] 254,20 138,45 168,48 74,61 265,68
number of cycles to
failure 5,28·105 no fatigue no fatigue no fatigue 4,97·105
Fig. 12. Deformations and stress redistribution in connection zone caused by oil
penetration and shear forces inside oil tube of the cylinder 2
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
82
Fig. 13. Goodman diagram and working points in critical zones for the cylinder 2
The values of the maximal stresses, amplitude and mean stresses for 0=R and
1−=R and number cycles to failure calculated using FEM in each zones are shown in
table 3. The values of the maximal stresses in the zone 1 and zone 5 are almost the same.
Fig. 14. Behavior of the Wöhler curve and values of equivalent amplitude stress Sae
in critical zones for the cylinder 2
Sa [M
pa
]
uSS at
R 45,0)1( =−=
ae
RS )1( −=
R
R
( )m
R
a
R SS )0(,)0( ==
Su
am
pl
itu
de
st
re
ss
S
a [M
Pa
]
(lo
ga
ry
th
m
ic
sc
al
e)
1000
100
102 103 104 105 106 107
N (logarithmic scale)
S-N curve zone 1 zone 2 zone 3 zone 4 zone 5
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2005, вип. 2, 70-86
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These two working points (zone 1 and zone 5) are located above the Goodman
diagram, other points are lying below the Goodman diagram in the safe area (see
fig. 13). Fatigue crack appears in the zones 1 and 5. Cracks do not appear in other zo-
nes. Number of cycles to failure in the zone 1 and zone 5 obtained as intersection of the
equivalent amplitude stress for 1−=R with Wöhler curve is shown in table 3 and in fig. 14.
3.3. Fatigue analysis of the hydraulic cylinder 1 with resistant material in oil ports and
cup. The part of hydraulic cylinder is analyzed. The geometrical shape and material da-
ta are the same as in the example from subsection 4.1. Authors propose modification of
oil ports area by introducing the washer or glue as it is shown in fig. 15. The cylinder
has two oil ports with washers.
Fig. 15. Washer or used in order to prevent oil penetration
Tab. 4. Number of cycles to failure in critical zones
of the cylinder 1 with washer
zone 1 zone 2 zone 3 zone 4 zone 5
max
)0( =RS [MPa] 398,17 365,66 360,42 359,14 357,11
a
RS )0( = [MPa] 199,08 182,83 180,21 179,57 178,55
m
RS )0( = [MPa] 199,08 182,83 180,21 179,57 178,55
ae
RS )1( −= [MPa] 322,59 281,97 275,78 274,29 271,93
number of cycles to
failure 5,92·104 2,59·105 3,30·105 3,50·105 3,85·105
Number of cycles to failure in each zones obtained as intersection of the ampli-
tude stress with Wöhler curve is shown in table 4.
Deformations and stress redistribution in the connection zone are illustrated in fig. 16.
Washer or glue used
in order to prevent
oil penetration
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
84
Fig. 16. Deformations and stress redistribution in the connection zone with washer used
in order to prevent oil penetration in the cylinder 1
Fig. 17. Behavior of the Wöhler curve and values of amplitude stress Sae in the critical zones
for the cylinder 1 with resistant material in the port zone
Behavior of the Wöhler curve and values of the equivalent amplitude stress
( )1−=R
aeS in the critical zones are shown in fig. 17. Significant improvement is archi-
ved in the zone 5 (the most endanger to fatigue zone) where fatigue strength increased
and number of cycles was bigger then about 100000 cycles. In the zones 2, 3 and 4
there was a slight improvement in the value of fatigue strength.
am
pl
itu
de
st
re
ss
S
a [M
Pa
]
(lo
ga
ry
th
m
ic
sc
al
e)
1000
100
102 103 104 105 106 107
N (logarithmic scale)
S-N curve zone 1 zone 2 zone 3 zone 4 zone 5
ISSN 1816-1545 Фізико-математичне моделювання та інформаційні технології
2005, вип. 2, 70-86
85
The fatigue in the zone 2 appears later if we use washer between the oil port and
cylinder (see fig. 15). The value of maximal stress is lower then in the case when there
is no modification between the oil port and cylinder surface.
Conclusions. The idea of equivalent amplitude stress is introduced in order to find out
number of cycles to failure for non-symmetric loads in situations when material data
are provided only for stress ratio 1−=R .
In order to solve the crack problem in an oil port area we propose to use the wa-
sher made from temperature resistant material or glue in order to fill up the gap bet-
ween oil port — cylinder surface (see fig. 15). Such washer or glue will prevent oil pe-
netration into the above mentioned gap thus eliminating the possibility of promotive
fatigue crack in neighboring welds.
Life expectancy are values of the moment absolutely theoretical and pending of
the adjustment of the model and experimental validation.
Acknowledgement. «PROHIPP» project is partially funded by the E. C. inside the sixth framework
programme, priority 3 NMP FP62002-NMP-2-SME, Research area 3.4.3.1.5: Support to the develop-
ment of new knowledged based added value products and services in traditionally less RTD intensive
industries.
We thanks the financial contribution of the E. C. and we state that the article reflects
only the personal opinion of the authors.
Authors are indebted to the Roquet SA and to CIMNE for providing experimental data
and numerical code COMET used in these calculations.
References
[1] Zahavi E., Torbilo V. Fatigue design. Life expectancy of machine parts, A Solomon
Press Book, Boca Rato, New York, London, Tokyo, 1996.
[2] Frost N. E., Marsh K. J., Pook L. P. Metal fatigue, Dover publications, INC, Mineola,
New York, 1999.
[3] Brzoska Z. Wytrzymalosc materialow (Strength of materials), PWN, Warsaw, 1983.
[4] Kocanda S. Zmeczeniowe pekanie, Wydawnictwa Naukowo-Techniczne, Warsaw, 1985.
[5] Kluger K., Ladoga T. Fatigue lifetime of 10hnap steel under random tension-compression
with the mean value according to the Dang-Van criterion, Engineering Machines Prob-
lems, Z. 24, 2004, Warsaw, 2004.
Использование эквивалентных амплитудных напряжений
для учета средних напряжений в задачах усталостной прочности
Томаш Беднарек, Изабелла Марчевска, Артур Марчевски, Влодзимеж Сосновски,
Хероним Якубчак, Ежы Роек
В работе представлен численный алгоритм, позволяющий выбирать различные схемы нагру-
жения конструкции, например, гидроциллиндра, в соответствии с заданными кривыми
Вёлера, характеризующими сопротивление материала усталостному разрушению. Проана-
лизированы гидроциллиндры, рассматриваемые в проекте E. C. «PROHIPP», и предложены
Tomasz Bednarek, Izabela Marczewska, Artur Marczewski, Włodzimierz Sosnowski…
The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis
86
некоторые решения задачи о трещине в области штуцера. Показано, что просачивание
масла в зоне соединения штуцера может быть устранено после некоторых модификаций
конструкции.
Використання еквівалентних амплітудних напружень для
врахування середніх напружень у задачах на втомну міцність
Томаш Беднарек, Ізабела Марчевська, Артур Марчевські, Влодзімеж Сосновскі,
Гєронім Якубчак, Єжи Роєк
У роботі представлено числовий алгоритм, який дозволяє вибирати різні схеми наванта-
ження конструкції, наприклад, гідроциліндра, відповідно до заданих кривих Велера, що
характеризують опір матеріалу на втомне руйнування. Проаналізовано гідроциліндри, які
розглянуті в проекті E. C. «PROHIPP», і запропоновано деякі розв'язки задачі про тріщину
в області штуцера. Показано, що просочування олії в зоні з’єднання штуцера може бути
усунуто після деяких модифікацій конструкції.
Отримано 04.11.05
|
| id | nasplib_isofts_kiev_ua-123456789-20920 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1816-1545 |
| language | English |
| last_indexed | 2025-11-30T16:44:46Z |
| publishDate | 2005 |
| publisher | Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України |
| record_format | dspace |
| spelling | Bednarek, T. Marczewska, I. Marczewski, A. Sosnowski, W. Jakubczak, H. Rojek, J. 2011-06-10T00:11:26Z 2011-06-10T00:11:26Z 2005 The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis / T. Bednarek, I. Marczewska, A. Marczewski, W. Sosnowski, H. Jakubczak, J. Rojek // Фіз.-мат. моделювання та інформ. технології. — 2005. — Вип. 2. — С. 70-86. — Бібліогр.: 5 назв. — англ. 1816-1545 https://nasplib.isofts.kiev.ua/handle/123456789/20920 539.3 У роботі представлено числовий алгоритм, який дозволяє вибирати різні схеми навантаження конструкції, наприклад, гідроциліндра, відповідно до заданих кривих Велера, що характеризують опір матеріалу на втомне руйнування. Проаналізовано гідроциліндри, які розглянуті в проекті E. C. \"PROHIPP\", і запропоновано деякі розв'язки задачі про тріщину в області штуцера. Показано, що просочування олії в зоні з’єднання штуцера може бути усунуто після деяких модифікацій конструкції. In this paper a numerical algorithm is presented to make possible adopting different loading schemes of specific structure at hand for instance hydraulic cylinders, to specific Wцhler curves characterizing fatigue resistance of given material. Hydraulic cylinders investigated under E. C. project \"PROHIPP\" are analyzed and some solutions of the crack problem in oil port area are proposed. Oil penetration in an oil port connection zone can be eliminated after some design modifications. В работе представлен численный алгоритм, позволяющий выбирать различные схемы нагружения конструкции, например, гидроциллиндра, в соответствии с заданными кривыми Вёлера, характеризующими сопротивление материала усталостному разрушению. Проанализированы гидроциллиндры, рассматриваемые в проекте E. C. \"PROHIPP\", и предложены некоторые решения задачи о трещине в области штуцера. Показано, что просачивание масла в зоне соединения штуцера может быть устранено после некоторых модификаций конструкции. We thanks the financial contribution of the E. C. and we state that the article reflects only the personal opinion of the authors. Authors are indebted to the Roquet SA and to CIMNE for providing experimental data and numerical code COMET used in these calculations. en Центр математичного моделювання Інституту прикладних проблем механіки і математики ім. Я.С. Підстригача НАН України The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis Использование эквивалентных амплитудных напряжений для учета средних напряжений в задачах усталостной прочности Використання еквівалентних амплітудних напружень для врахування середніх напружень у задачах на втомну міцність Article published earlier |
| spellingShingle | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis Bednarek, T. Marczewska, I. Marczewski, A. Sosnowski, W. Jakubczak, H. Rojek, J. |
| title | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis |
| title_alt | Использование эквивалентных амплитудных напряжений для учета средних напряжений в задачах усталостной прочности Використання еквівалентних амплітудних напружень для врахування середніх напружень у задачах на втомну міцність |
| title_full | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis |
| title_fullStr | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis |
| title_full_unstemmed | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis |
| title_short | The equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis |
| title_sort | equivalent amplitude stress as a solution of mean stress effect problem in fatigue analysis |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/20920 |
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