Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 4, pp. 791-798, Warsaw 2013

INFLUENCE OF THE IRREGULARITY COEFFICIENT OF LOADING
ON CALCULATED FATIGUE LIFE

Roland Pawliczek, Krzysztof Kluger

Opole University of Technology, Department of Mechanic and Machine Design, Opole, Poland

e-mail: r.pawliczek@po.opole.pl; k.kluger@po.opole.pl

The paper presents comparison of two methods for determination of the fatigue damage
degree in a material subjected to random loadings with participation of the mean loading
and including the irregularity coefficient of the loading. The authors estimated effectiveness
of the method where the stress history was transformed considering the global mean value
of the load history and the method for which the stress mean value of the distinguished
loading cycles was taken into account.When the irregularity coefficient of loading increases,
we can observe significant divergences of the calculated fatigue life of the material related
to the model of amplitude transformation because of the loadingmean value.

Key words: fatigue, irregularity coefficient, mean stress

1. Introduction

Variable-amplitude and randomloadings, so-called service loadings, are about90%ofall loadings
occurring in practice. The occurrence of an additional static loading of machine elements and
structures is an important element of the service loadings. The additional static value is often
a result of the influence of a dead weight of a working element or a structure, it can be also an
effect of pre-tension of elements transferring loadings (for examplewedge belts in transmissions).
Estimation of the fatigue life under such loadings is difficult – it requires selection of a suitable
estimation model; there are also many difficulties connected with time-consuming calculations,
etc.

In the case of service loadings, the widely applied algorithm for fatigue life determination
includes the following steps (Macha et al., 2005, 2009; Kluger and Łagoda, 2007; Łagoda et al.,
2008):

1. Determination of loading history in form of histories of strain, stress, forces or moments;

2. Determination of amplitudes andmean values of cycles and half-cycles with the rain flow
method;

3. Determination of equivalent values (stress, strain) according to chosen criteria ofmultiaxial
fatigue;

4. Determination of the damage degree according to the chosen hypothesis of damage accu-
mulation;

5. Calculation of fatigue life of the element.

The stress models are the most often applied models of fatigue life estimation, including
influence of static loading. Themodels proposed byGoodman, Gerber andMorow (Kluger and
Łagoda, 2007; Łagoda et al., 2008; Macha et al., 2009) are the most popular because they are
very simple, and it is easy to apply them under amplitude-constant loadings. In the case of
their application for service loadings, calculations require much calculation power and they are



792 R. Pawliczek, K. Kluger

time-consuming as simple transformations of amplitudes must be repeated many times because
of the mean values.
This paper presents test results including the influence of factors resulting from the parame-

ters of time histories not included in the general algorithm of fatigue life determination.

2. Calculation algorithm

Łagoda et al. (2001) compared effectiveness of the algorithm for fatigue life calculation according
to some procedures including the loading mean value in a different way. This paper concerns
loadings with mean values, so the fatigue life is determined according to two methods: method
1 includes the global mean stress history approach, andmethod 2 is the local mean stress of the
selected cycle approach (see Fig. 1).

Fig. 1. Graphical interpretation of the procedure of stress transformation amplitude;
(a) method 1 – global approach, (b) method 2 – local approach

In the global approach, the transformation includes the global mean value of the stress σm.
The transformation is performed by history scaling with the coefficient K dependent on the
assumed transformation model

K =
1

1−
(

σm
A

)c (2.1)

where: A= Rm and c= 1 for the Goodman relationship, A=Rm and c= 2 for the Gerber
relationship, and A= σ′f and c=1 for the Morrow relationship, Rm – tensile strength of the
material, σ′f – fatigue strength coefficient.
In a consequence, we obtain the transformed history with the zero mean value of the stress

and higher instantaneous values. At the next step, when cycles of the transformed history in the
distinguished cycle with the amplitude σaiT are counted, themean value σmiT occurs (Fig. 1a)
– however, it is not taken into account because it causes double transformation.
In the case of the local approach, at first cycles with the given amplitude, σai and the local

mean value σmi are distinguished. Next, each distinguished cycle is transformed (Fig. 1b) and
in a consequence we obtain a cycle with the zero mean value σmiT =0 and the amplitude σaiT
enlarged with the coefficient Ki

Ki =
1

1−
(

σmi
A

)c (2.2)

where A and c are defined like in Eq. (2.1). Let us note that the coefficient value is not constant
and it changes for each distinguished cycle.



Influence of the irregularity coefficient... 793

The damage degree S(TO) at the observation time TO of the stress history is calculated by
summation of damages from thedistinguished amplitudes of cycles andhalf-cycles σai according
to the following relationship

S(TO)=















k
∑

i=1

ni

NG

(

Zrc
σaiT

)m for σaiT ­ aZrc

0 for σaiT <aZrc

(2.3)

where: σai – stress amplitude of the distinguished cycle, ni – number of cycles with the ampli-
tude σai, m – coefficient of theWöhler curve slope, Zrc – fatigue limit for tension-compression,
NG – limit number of cycles corresponding to Zrc.
While summing up the damages, the amplitudes below the fatigue limit were also included,

assuming a=0.5 (Lachowicz et al., 1995).
There are significant differences between the considered procedures. From Łagoda et al.

(2001) it appears that including the influence of the loading static value in the global approach
(all the loading history) gives similar results of fatigue life calculations like in the case of the
local approach, i.e.within one cycle of loading change.Theglobal approach allows one to shorten
time of calculations, and this is important when the calculation model is applied for estimation
of the fatigue life of elements at real time, or in the case of fatigue tests performed at the stands
with participation of generated loading histories.
In laboratory practice, a reproduced load history should correspond to the service loading as

better as possible. That leads to use some standardized load histories (e.g. Shutz et al., 1990) or
generation of load histories with the use of theMarkovmatrix (Shutz et al., 1990). Additionally,
generation methods must take into account different statistical parameters of the reproduced
loads (Macha and Pawliczek, 2007, 2010). However, there are no clear reasons to say that other
parameters of service courses, not resulting from the considered calculation procedure, influence
its results.
In the case of fatigue behavior of materials, also the coefficient of loading irregularity is an

important parameter. It is defined as

I =
N+σm
N
+
σmax

(2.4)

where: N+σm is the number of transitions through a level of the expected loading value with a
positive slope at the time unit, N+σmax – number of local peaks at the time unit.
Figure 2a presents fragments of variable-amplitude histories with the samemean values and

the history variance but with different values of the irregularity factor (I = 0.4 and I = 0.99
respectively). Both histories were generated based on the sameMarkovmatrix – basic statistical
parameters are the same for both histories. The histograms received for analyzed courses are
presented in Fig. 2b. It can be seen that distribution of the stress values is almost the same for
I =0.4and I =0.99.However, there is avisible differencebetween the timehistory shapes.Such
an effect influences the parameters of the cycles distinguished, when the rain flow algorithm is
applied. For higher values of factor I, cycles counted in the rain flowmethod characterizes with
lower mean stress values and higher amplitudes.When the irregularity coefficient decreases, the
rain flow method detects cycles with higher values of the mean stress and smaller amplitudes.
Such a situation can strongly influence a degree of the accumulated damages in the considered
histories, especially if the mean load is present.
Different values of coefficient I lead to a different type of the load. For I = 1, the cyclic

process can be observed – only one dominant frequency is detected. Decreasing the value of
the irregularity coefficient, we can observe a narrow band random process and a wide band
random process for lower values of this coefficient (Agreskov and Nielsen, 1999; Łagoda et al.,



794 R. Pawliczek, K. Kluger

Fig. 2. Stress histories with coefficients of irregularity I =0.4 and I =0.99; (a) example of the time
courses of the stresses, (b) histograms of the stress values

2005b; Petrucci andZuccarello, 2004). Using the power spectrumdensity of the randomprocess,
it is possible to estimate irregularity factor I and the spectral method of the fatigue damage
accumulation analysis can be defined based on the frequency domain (Łagoda et al., 2005a,b;
Petrucci and Zuccarello). The authors shown that the spectral method can be effectively used
to estimate fatigue life of the component with similar or better accuracy comparing to the
time domain method. In Petrucci and Zuccarello (2004), relationships between the moments
of the probability density function of the fatigue cycles of the equivalent stress, two spectral
parameters, the common irregularity factor of the stress process and a bandwidth parameter
weredetermined. In thepaperbyŁagoda et al. (2005a) theauthorspresent results of fatigue tests
of specimens subjected tomultiaxial random fatigue for which the spectral based algorithm for
fatigue life estimation was used. A loading with Gaussian distribution and narrow- and broad-
band frequency spectrawere chosen for tests and the spectralmethodbasedon theassumptionof
the linear Serensen-Kogayev hypothesis was used. It was shown that under amultiaxial random
loading, the results of fatigue life calculated according to the considered algorithms in frequency
and time domains are well correlated with the results of experiments.

The linear hypothesis used in the spectral methods allow analyzis of the fatigue behavior of
materials only for the high cycle fatigue range – for time domain methods, it is easier to use a
selected elasto-plastic theory to estimate the low cycle fatigue stress history.

It has to be indicated, that the authors of the papersmentioned above used the irregularity
factor as a parameter which characterizes the investigated load histories, however there is no
clear suggestion how it affects the calculations. There is no answer for the accuracy problem in
generation of the load history according to the irregularity coefficient of the load history – how



Influence of the irregularity coefficient... 795

small differences between the irregularity factors for the service load and load simulated in a
fatigue test influences the results of fatigue life estimation.

3. Simulation tests

Simulations were performed in order to determine the influence of the coefficient of loading
irregularity combined with the mean load value on the damage accumulation degree. The tests
were realized for histories with different values of the coefficient of irregularity and different
expected values.
Stress histories with the following expected values were generated: σm = 0,150,200 and

225MPa. The coefficients of loading irregularity I = 0.1,0.2, . . . ,0.99 were assumed for each
level of the expected value. The damage degree was calculated according to procedures of the
global and local approaches to themean stress value.Table 1 contains some strength and fatigue
properties of 10HNAP steel applied in the calculations (Łagoda et al., 2001).

Table 1. Strength and fatigue properties of 10HNAP steel

Re Rm A10 Z E ν σ
′

f Zrc NG
[MPa] [MPa] [%] [%] [GPa] [–] [MPa] [MPa] [cycle]

418 566 30.7 36.5 215 0.29 746 252.3 1.28 ·106

The influence of the stress expected value on the accumulated damage degree was included
with three models described with Eqs. (2.1) and (2.2) for the global and local approach, re-
spectively. Table 2 presents the results of calculations of damage accumulation degree for the
considered stress histories.
The relation between the accumulated damage degree and the coefficient of history irre-

gularity is shown in Fig. 3. When the stress mean value is σm = 0 (Fig. 3a), the calculation
results are identical independently of the chosen transformation relationship. In all cases, it is
observed that for each investigated relationship (Goodman,Gerber andMorrow) bothmethods,
local and global approach, give the same results for damage degree accumulation. Generally, the
damage degree increases as the coefficient of irregularity increases by some orders. An increase
in the mean loading causes a significant increase in the accumulated damage degree for the
Goodmanmodel comparing to theGerber andMorrow, where theGerber andMorrow transfor-
mationmodels give similar results (Fig. 3b-d).Additionally, themean stress sigmam =150MPa
(Fig. 3b) leads to a smaller value of the accumulation damage degree by a few orders comparing
to σm =200MPa (Fig. 3c) and σm =225MPa (Fig. 3d).
It can be observed, that for lower factor I (I < 0.5) the accumulated damages do not change

so strong like for a higher value of the irregularity factor (I > 0.5).

4. Conclusions

The presented tests proved that both, global and local approaches to the mean value influence,
give similar values of theaccumulateddamagedegree independentlyof thevalueof the coefficient
of history irregularity. On the other hand, there are some other aspects which have not been
taken into account so far:
• an increase in the coefficient of irregularity causes an increase in the degree of the accu-
mulated fatigue damages by some orders,

• an increase in the loading mean value causes occurrence of significant differences between
the obtained damage degrees for particular transformation relationships. The Goodman
relationship shows the greatest differences.



796 R. Pawliczek, K. Kluger

Table 2.Accumulation degree of fatigue damages

σm
[MPa]

S(TO) ·10
−6

I Local approach Global approach
Goodman Gerber Morrow Goodman Gerber Morrow

0

0.10 0.01 0.01 0.01 0.01 0.01 0.01
0.20 0.54 0.52 0.52 0.51 0.51 0.51
0.30 1.14 1.12 1.12 1.12 1.12 1.12
0.40 1.50 1.48 1.48 1.48 1.48 1.48
0.50 1.90 1.89 1.89 1.89 1.89 1.89
0.60 2.60 2.58 2.59 2.58 2.58 2.58
0.70 3.47 3.42 3.44 3.42 3.42 3.42
0.80 5.06 4.99 5.00 4.97 4.97 4.97
0.90 8.32 8.26 8.26 8.24 8.24 8.24
0.99 13.99 13.90 13.92 13.89 13.89 13.89

150

0.10 0.02 0.00 0.00 0.02 0.00 0.00
0.20 0.74 0.04 0.08 0.70 0.04 0.08
0.30 1.56 0.08 0.18 1.53 0.08 0.18
0.40 2.02 0.10 0.25 2.00 0.10 0.25
0.50 2.55 0.13 0.30 2.54 0.13 0.30
0.60 3.47 0.18 0.41 3.44 0.18 0.41
0.70 4.66 0.24 0.55 4.58 0.23 0.55
0.80 6.82 0.35 0.81 6.70 0.34 0.81
0.90 11.21 0.57 1.33 11.10 0.57 1.33
0.99 18.53 0.94 2.21 18.39 0.94 2.21

200

0.10 13.84 1.36 3.23 13.68 1.36 3.21
0.20 695.50 59.76 140.06 562.86 55.95 132.25
0.30 1400.00 125.73 294.71 1200.00 122.69 290.07
0.40 1700.00 165.02 390.25 1600.00 163.65 391.24
0.50 2100.00 206.97 488.10 2100.00 205.58 486.01
0.60 3000.00 285.98 674.06 2800.00 282.93 668.89
0.70 4000.00 380.86 897.82 3800.00 373.40 882.63
0.80 5900.00 555.69 1300.00 5500.00 544.86 1300.00
0.90 9600.00 919.68 2200.00 9100.00 907.30 2100.00
0.99 15700.00 1500.00 3600.00 15300.00 1500.00 3600.00

225

0.10 15.75 0.58 1.27 15.56 0.58 1.27
0.20 807.33 26.67 55.57 648.44 24.23 52.87
0.30 1600.00 54.82 116.52 1400.00 52.76 115.07
0.40 2050.00 122.24 261.22 2000.00 117.98 257.38
0.50 2500.00 123.50 195.83 2400.00 89.49 195.21
0.60 3400.00 125.02 270.03 3300.00 123.04 268.38
0.70 4800.00 170.62 367.00 4400.00 165.77 361.72
0.80 7000.00 251.76 540.80 6500.00 244.58 533.62
0.90 11500.00 414.95 893.46 10900.00 406.50 886.88
0.99 18300.00 673.77 1500.00 17800.00 665.68 1500.00

Taking into account that the load histories generated in the laboratories should correspond
to the service loads, the irregularity factor must be analyzed, especially for higher values of this
coefficient (I > 0.5) because the presented algorithm for fatigue life estimation is sensitive to
this factor.



Influence of the irregularity coefficient... 797

Fig. 3. Changes of the damage accumulation degree S(TO) versus irregularity factor I for different
values of the mean stress

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Manuscript received June 5, 2012; accepted for print November 12, 2012