Service loading test as a source for the fatigue curve parameters’ estimation
Very important question of fatigue curve parameter estimation might be explored from the point of view of service loading. Comparative testing at varied stress levels with subsequent computation might reveal some peculiarities of fatigue curves and substantial changes of their parameters due to serv...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2009
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| Cite this: | Service loading test as a source for the fatigue curve parameters’ estimation / I.V. Gadolina, I.M. Petrova // Проблемы прочности. — 2009. — № 1. — С. 76-82. — Бібліогр.: 4 назв. — англ. |
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| citation_txt | Service loading test as a source for the fatigue curve parameters’ estimation / I.V. Gadolina, I.M. Petrova // Проблемы прочности. — 2009. — № 1. — С. 76-82. — Бібліогр.: 4 назв. — англ. |
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| description | Very important question of fatigue curve parameter estimation might be explored from the point of view of service loading. Comparative testing at varied stress levels with subsequent computation might reveal some peculiarities of fatigue curves and substantial changes of their parameters due to service loading factors.
Проблема определения параметров кривой усталости может быть решена с учетом особенностей эксплуатационного нагружения. Показано, что сравнительные испытания в условиях изменения уровня напряжений и учет полученных результатов в последующих расчетах позволяют определить некоторые особенности кривых усталости и существенные изменения их параметров, связанные с влиянием факторов эксплуатационного нагружения.
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| first_indexed | 2025-12-07T16:53:46Z |
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UDC 539.4
Service Loading Test as a Source for the Fatigue Curve Parameters’
Estimation
I. V. G ado lina an d I. M . P etrova
Blagonravov Institute of Engineering Science, Russian Academy of Sciences, Moscow,
Russia
Very important question offatigue curve parameter estimation might be explored from the point o f
view o f service loading. Comparative testing at varied stress levels with subsequent computation
might reveal some peculiarities offatigue curves and substantial changes o f their parameters due to
service loading factors.
K e y w o r d s : metal fatigue, service loading, damage summation, nonlinear equation
systems, test acceleration
In tro d u c tio n . For the operation life estimation under random or block
loading the fatigue curve (1) w hich describes the dependence o f the ultimate
number o f cycles N on the stress amplitude o by the varied kind o f dependences
(Wohler, Weibull, et al.) is often used
N = f (o , A ), ( 1)
where A is the fatigue curve param eters’ vector which is usually estimated in the
test w ith constant amplitude loading (one step loading).
Usage o f the M ainer fatigue damage accumulation hypothesis
W
S n t
N i (2)i =1 1
for the total life estimation
W
n 2 = 2 n i (3)
i =1
often leads to non-conservative estimation [1]. [In (3): n t is the num ber o f the
cycles w ith amplitude o t and w is the num ber o f the intervals o f the cycle-
counted random loading process or the step num ber o f the block loading.] That
m eans that validity o f (2) is not always approved experimentally. Numerous
investigations show that the fatigue curve is affected by such loading factors as
rare overloads, m anifold loads below the endurance limit, etc.
The aim o f the present work is to estimate the fatigue curve parameters with
the utilization o f the service loading test results.
© I. V. GADOLINA, I. M. PETROVA, 2009
76 ISSN 0556-171X. Проблемы прочности, 2009, № 1
Service Loading Test as a Source
Assume we have the data o f some service loading tests, which were carried
out at s loading levels. For reduction o f errors due to the test data scatter inherent
to fatigue, the assessment should include the statistical aspect. Usually 3 to 5
specimens are tested at each level. The value n 2 in (3) is a m edian value.
Let us define the calculated fatigue curve with the param eter vector A (A
in general differs from A)® as
N * = f i (° , A * ) (4)
satisfying s conditions o f the M ainer linear summation (2):
w
------ = 1
ite -*-4
i=1 N i,1( a > A )
W2
У — П И ____ = 1
1 N > . A > ) ' (5)
•'S
У
=1 N i ,s( ° , A )
= 1.
n i s
Here n j , . . . , n ^ i are the components o f the total life for the 1-st test under
service loading with stress amplitudes, respectively о 11, ..., о ^ 1, n 1s , . . . , n w s
are the same for the test num ber s w ith stresses о 1 s , . . . , о w s. The num ber o f the
items Wk in kth test (k = 1, 2 , . . . , s) corresponds to the num ber o f intervals o f the
cycle-counted random loading processes or to the step num ber o f the block
loading.
*
The dimension r o f the param eter A depends on the kind o f fatigue life
described by Eq. (1) and is not lower than 2 (r > 2). It defines the num ber o f the
required levels o f testing s (s > r).
Various approaches for the solution o f the equation system (5) relative to
*
unknown A were developed.
*
The D irect Selection o f A B ased on C onsidera tion o f E xperim en ta l
R esults o f C om p ara tiv e Ir re g u la r Testing. The approach is applied when only
one param eter varies, for example, the fatigue limit о r .
For the structural steels under one-step loading often a threshold value
(fatigue limit) is determined. In this case, Eq. (1) in nonlinear form and with
additional param eter о r has the form (r = 3):
® Hereinafter symbol * depicts all values obtained by the comparative irregular testing computation
method.
ISSN 0556-171X. Проблемы прочности, 2009, N2 1 77
I. V. Gadolina and I. M. Petrova
log N = C = m log o for o > o r ,
log N = « for o < o r , (6)
where C and m are the m aterial parameters.
Experiments have shown, that the fatigue limit o r is often decreased due to
operation factors. The design m ethod for operation life estimation based on this
phenom ena has been developed [2]. This m ethod assumes that param eter o r
decreases, and parameters C and m remain constant.
W ith the aim o f experimental verification o f the fatigue curve extrapolation
in the upper part o f the equation (6) below the fatigue limit o r , the irregular
fatigue testing with two different loading blocks were carried out (two irregular
experiments: s = 2). In Fig. 1, those blocks are shown together w ith fatigue
curves. Lines 1 show the initial 8-step loading block, in which two low steps are
below the fatigue limit o r . Here o r was estimated in constant loading test
illustrated by fatigue curve 3. Lines 2 depict the truncated block, in which two
low steps were discarded. I f only two low steps in block 1 do not contribute to
the fatigue damage, the fatigue life expressed in the block numbers w ould remain
the same. The distribution o f fatigue lives expressed in block number
significantly differs. This testifies to the fact, that low steps do contribute into the
fatigue damage. The experimental results confirm that the fatigue limit decreases
*
during the irregular loading, i.e., o r < o r , and fatigue curve changes from 3 to 4.
Fig. 1. Comparative tests under block loading.
*
Selection of A B ased on A ccelerated C oefficient A nalysis k a c . In [3],
the approach o f the acceleration coefficient k ac to the estimating fatigue curve
parameters in gigacycle region ( N > 1 0 ) was introduced. Value o f k ac shows
the ratio o f the fatigue lives at varied loading levels. This approach is applied to
homothetic stress blocks, i.e., for such tests, in which the non-dimensional
distribution ( S i , t t ) is constant (S i = o ^ o max, t t = n / n 2 , o max is the maximum
amplitude in the loading block). The tests’ num ber on varied levels should be
s > 2.
Let us consider the set o f G hypotheses based on the linear summation rule
(2). Comparing the k ac values - experimental and calculated according to
m odified hypotheses - we can choose the best fit hypothesis (providing the best
78 ISSN 0556-171X. npodxeMbi npounocmu, 2009, N9 1
Service Loading Test as a Source
correlation). W hile doing this, the damage estimation is made using by the
m odified G fatigue curves (the num ber o f hypothesis corresponds to the number
o f fatigue curve). The selection o f the m ost appropriate hypothesis is selected by
the m inim um value o f W (g), where g is the order num ber o f the hypothesis
( g = 1, 2, . . . , G ). By definition k ac > 1. I f we have two levels o f irregular testing
( s = 2), then k ac = R m x / R min , where R max and R min are the m edian fatigue
lives under lower and higher loading levels, respectively. Calculated according
the hypothesis g , acceleration coefficients are k̂ ag^ = R mgx/Rmin, where R ^gX
and R are the corresponding fatigue lives calculated by the gth hypothesis.
In this case, the following expression is proposed as an optimization
criterion:
W (g) = I k *yc — k (gc) \ ^ min. (7)
For the analysis o f test results w ith s > 2, s — 1 experimental and calculated
acceleration coefficients are determined. For the selection o f the optimum
hypothesis the m inim um o f the additive value o f the optimization criterion is
chosen:
s—1
W (g ) = 2 \ k a c i — kS \ ^ min. (8)
i =1
Here
k ac,i = R i l R min and k<g,i = R(g V R ,
where R t and R (g ) ( i = 1, 2 , . . . , s — 1) are respectively experimental and calculated
according to g th hypothesis fatigue lives under lower stress levels.
*
The example o f estimation o f the m odified fatigue curve slope m is
considered. Here the form o f fatigue curve (1) was taken as
log N = C — m log o , (9)
where C and m are the m aterial parameters ( r = 2).
The example o f laboratory testing o f the specimens under wide-band random
loading at three levels (s = 3) was considered. Based on comparison o f the
calculated acceleration coefficients with experimentally obtained ones, the optimum
*
value o f m = 8 -9 was derived (the initial value, resulted from the constant
amplitude test, reached m = 18. In Fig. 2, the dependence o f the additional
optimization criterion (8) on varied slope param eter is shown. The first minimum
*
o f W is for m = 8-9. A dotted vertical line indicates m value obtained in
regular loading testing. The second m inim um o f the calculated dependence
W ( m = 17) = = 0.95 is located nearby.
N on-L inear E q u atio n System (5) Solution by the L east-S quare M ethod.
The approach is the m ost general. Superimposing unit to the left part, we convert
the system (5) into the form:
ISSN 0556-171X. npoôëeMbi npounocmu, 2009, N9 1 79
I. V. Gadolina and I. M. Petrova
w
2
4 ,1
=1 N i ,1( a , A )
4 ,2w22-
N -2 (a , A " )
1 = 0 ,
1 = 0 ,
( 10)
*1 ,s
=1 N i ,S ( a , A )
- 1 = 0.
%
The solution o f the system (5) will be the values o f parameters A , under
which the deviations from zero (the objective function) o f the Eqs. (10) are
m inim al.
W
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 5 10 15 20 m
• • • • • • • • ̂ •Fig. 2. Optimization additive criterion W on slope parameter m (modeling).
For the fatigue curve in form (6) the system (9) is converted to
w
n i ,1
> — F— l---------- 1= 0 ,^ 10C -m log a i,1
W2
n i ,2
> — F— l---------- 1= 0 ,Z - t 10C-m toga ,,2
1 =1 ( 11)
s
2
i =1 10
C —m lg a , 1 = 0.
n
C
Converting the expressions in denom inator in (11) and denoting 10 = C ',
we get
80 ISSN 0556-171X. Проблемы прочности, 2009, N 1
Service Loading Test as a Source
w
_ L ll log0 i,1
i =1
w
1 = o,
i =1 ( 12)
ÿ — 10 Z j с '
m log о i
i =1
- 1= 0.
The next step is the system (12) solution by the least square method. This
m ethod application is the m ost correct when the values have norm al distribution.
I f this condition is satisfied, the solution will correspond to the maximum
likelihood method. I f the distribution differs from normal, the problem should be
solved by the maximum likelihood method taking into consideration the hypotheses
o f the real random value distribution.
To have fairly representative set by the irregular loading w ith two-parameter
fatigue curve estimation (9) ( r = 2), the num ber o f tested levels s (considering
scatter) should be not less than 3 (s > 3).
For the least square m ethod system solution the standard algorithm in
computation m edia MATLAB 6.5 was used. I f the fatigue curve has only two
parameters, the equation system solution m ight be obtained graphically, using
computation m athematics graphical facilities.
I f the fatigue curve is a step-wise non-linear function (r > 2), for example as
in (6), the random adaptive search m ethod should be used for its solution.
For the considered example o f random loading, the calculated by the least
*
square m ethod result is almost the same as that obtained by A selection based
on the accelerated coefficient analysis k ac.
a, MPa
Fig. 3. Comparison of the experimental and calculated fatigue curves.
ISSN 0556-171X. Проблемы прочности, 2009, N2 1 81
I. V. Gadolina and I. M. Petrova
In Fig. 3, the fatigue curve obtained for regular loading test is shown (solid
line) together w ith the curve, obtained on the base o f comparative random loading
according to the proposed method.
C onclusions. Based on some examples o f the block and random loading
*
tests, the fatigue curve param eter A estimation by the proposed m ethod has
been performed. This estimation was obtained under conditions which were the
*
m ost congenial to operation ones. Due to this, using A for estimation o f the
service life discounts the error probability. For the fatigue curve in form (6) it was
* * found that the slope fatigue param eter in A is less fitting w ith A ( m < m).
This result agrees w ell with those obtained in [4].
1. A. P. Gusenkov (Ed.), V. P. Kogaev, S tre n g th C a lc u la tio n s u n d e r V a r ia b le
S tr e s s C o n d it io n s [in Russian], M ashinostroenie, M oscow (1993).
2. A. P. Gusenkov (Ed.), I. M. Petrova, I. V. Gadolina, F a tig u e S tre n g th
A n a lis is o f M a c h in e s w ith A c c o u n t o f P r o b a b i l i t y A s p e c ts [in Russian],
M e th o d ic a l G u id e lin e : P r o b le m s o f S tren g th , O p e ra tio n L ife , a n d R e a lia b il ity
o f M a c h in e s , M oscow (1991).
3. I. V. Gadolina, “Tests under irregular loading as a source o f information
about the fatigue curve behavior in the gigacycle region,” in: M e c h a n ic a l
F a tig u e o f M e ta ls (Proc. o f the Colloq.) [in Russian], Ternopil State University,
Ternopil (2006), pp. 141-143.
4. M. E. Garf and E. D. Buglov, “On peculiarities o f fatigue damage accumulation
under a range o f nonstationary loads below fatigue lim it,” V est. M a s h in o s tr .,
No. 6 , 23-25 (1964).
Received 11. 06. 2008
82 ISSN 0556-171X. npodxeMbi npounocmu, 2009, N 1
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| id | nasplib_isofts_kiev_ua-123456789-48473 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T16:53:46Z |
| publishDate | 2009 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Gadolina, I.V. Petrova, I.M. 2013-08-20T04:04:47Z 2013-08-20T04:04:47Z 2009 Service loading test as a source for the fatigue curve parameters’ estimation / I.V. Gadolina, I.M. Petrova // Проблемы прочности. — 2009. — № 1. — С. 76-82. — Бібліогр.: 4 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/48473 539.4 Very important question of fatigue curve parameter estimation might be explored from the point of view of service loading. Comparative testing at varied stress levels with subsequent computation might reveal some peculiarities of fatigue curves and substantial changes of their parameters due to service loading factors. Проблема определения параметров кривой усталости может быть решена с учетом особенностей эксплуатационного нагружения. Показано, что сравнительные испытания в условиях изменения уровня напряжений и учет полученных результатов в последующих расчетах позволяют определить некоторые особенности кривых усталости и существенные изменения их параметров, связанные с влиянием факторов эксплуатационного нагружения. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Service loading test as a source for the fatigue curve parameters’ estimation Применение испытаний с эксплуатационным нагружением для определения параметров кривой усталости Article published earlier |
| spellingShingle | Service loading test as a source for the fatigue curve parameters’ estimation Gadolina, I.V. Petrova, I.M. Научно-технический раздел |
| title | Service loading test as a source for the fatigue curve parameters’ estimation |
| title_alt | Применение испытаний с эксплуатационным нагружением для определения параметров кривой усталости |
| title_full | Service loading test as a source for the fatigue curve parameters’ estimation |
| title_fullStr | Service loading test as a source for the fatigue curve parameters’ estimation |
| title_full_unstemmed | Service loading test as a source for the fatigue curve parameters’ estimation |
| title_short | Service loading test as a source for the fatigue curve parameters’ estimation |
| title_sort | service loading test as a source for the fatigue curve parameters’ estimation |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/48473 |
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