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VIBRATION FATIGUE ANALYSIS OF A CANTILEVER BEAM USING<br />
ABSTRACT<br />
DIFFERENT FATIGUE THEORIES<br />
Yusuf ELDOĞAN †‡ , Ender CIGEROGLU †<br />
† Middle East Technical University, 06800, Ankara, Turkey<br />
‡ ASELSAN MGEO Inc., 06750, Ankara, Turkey<br />
In this study, <strong>vibration</strong> <strong>fatigue</strong> <strong>analysis</strong> <strong>of</strong> a <strong>cantilever</strong> <strong>beam</strong> is performed <strong>using</strong> an in-house numerical code. Finite<br />
element model (FEM) <strong>of</strong> the <strong>cantilever</strong> <strong>beam</strong> verified by tests is used for the <strong>analysis</strong>. Several <strong>vibration</strong> <strong>fatigue</strong><br />
theories are used to obtain <strong>fatigue</strong> life <strong>of</strong> the <strong>cantilever</strong> <strong>beam</strong> for white noise random input and the results<br />
obtained are compared with each other. Fatigue life calculations are repeated for <strong>different</strong> damping ratios and the<br />
effect <strong>of</strong> damping ratio is studied. Moreover, <strong>using</strong> strain data obtained from <strong>cantilever</strong> <strong>beam</strong> experiments, <strong>fatigue</strong><br />
life <strong>of</strong> the <strong>beam</strong> is determined by utilizing time domain (Rainflow counting method) and frequency domain<br />
methods, which are compared with each other. In addition to this, <strong>fatigue</strong> tests are performed on <strong>cantilever</strong> <strong>beam</strong><br />
specimens and <strong>fatigue</strong> life results obtained experimentally are compared with that <strong>of</strong> in-house numerical code. It<br />
is observed that the accuracy <strong>of</strong> the damping ratio is very important for accurate determination <strong>of</strong> <strong>fatigue</strong> life.<br />
Furthermore, for the case considered, it is observed that the <strong>fatigue</strong> life result obtained from Dirlik method is<br />
considerably similar to that <strong>of</strong> Rainflow counting method.<br />
Keywords: Vibration Fatigue Theories, Rainflow Counting, Dirlik Method, Probability Density Function,<br />
Frequency Domain Fatigue Theories<br />
INTRODUCTION<br />
In <strong>vibration</strong> environments, if structures are designed only considering static requirements, due to dynamic<br />
characteristic <strong>of</strong> the environment, generally failure occurs. Hence, in order to avoid this kind <strong>of</strong> failures, dynamic<br />
characteristic <strong>of</strong> the environment and the structure should be considered. If the loading is cyclic, usually such<br />
failures occur even though stresses caused by these cyclic loadings are smaller than yield or ultimate strength <strong>of</strong><br />
the material. These kinds <strong>of</strong> failures caused by cyclic loadings are called "Fatigue". However, especially in<br />
military environments, generally structures are exposed to random loading; hence, the resulting stresses are not
cyclic. In order to avoid <strong>fatigue</strong> due to random loading, <strong>fatigue</strong> life <strong>of</strong> structures should be calculated <strong>using</strong><br />
<strong>vibration</strong> <strong>fatigue</strong> approaches in time and frequency domains.<br />
In the following sections, firstly, brief information about <strong>vibration</strong> <strong>fatigue</strong> approaches used in this study is<br />
described. These <strong>fatigue</strong> life estimation approaches are employed in order to determine the <strong>fatigue</strong> life <strong>of</strong> an<br />
example <strong>cantilever</strong> <strong>beam</strong>. In addition to this, <strong>fatigue</strong> life <strong>of</strong> the <strong>cantilever</strong> <strong>beam</strong> is determined experimentally, by<br />
utilizing several <strong>cantilever</strong> <strong>beam</strong>s and an average <strong>fatigue</strong> life is determined. The results obtained from <strong>different</strong><br />
<strong>fatigue</strong> approaches are compared with each other and also with the experimental data.<br />
THEORY<br />
In frequency domain, expected <strong>fatigue</strong> damage, ED , caused by random loading is given as follows[1];<br />
n<br />
ED . (1)<br />
N<br />
where, n and N are the number <strong>of</strong> stress cycles applied at a fixed stress amplitude and the number <strong>of</strong> cycles the<br />
material can withstand at applied fixed stress amplitude, respectively.<br />
In order to calculate ED, firstly Probability Density Function (PDF), p S , <strong>of</strong> rainflow stress ranges should be<br />
determined. A typical PDF is shown in Fig. 1 [2].<br />
Fig. 1 Probability Density Function (PDF)<br />
The bin widths, dS and total number <strong>of</strong> cycles in the histogram, S t should be obtained in order to calculate PDF<br />
from a stress range histogram. The opposite work can also be performed <strong>using</strong> PDF. The area shown in Fig. 1,<br />
p S dS,<br />
gives the probability <strong>of</strong> stress ranges between<br />
widths.<br />
p(S)<br />
dS<br />
Stress (S)<br />
dS dS<br />
Si and Si where i indicates the number <strong>of</strong> bin<br />
2<br />
2<br />
Finally, multiplying the probability <strong>of</strong> stress ranges with the total number <strong>of</strong> cycles, S t , in the histogram, the<br />
number <strong>of</strong> total cycles, nS , for a given stress level, S , can be obtained as follows [2]:
n( S) p( S) dS St<br />
. (2)<br />
In order to use Eq. (1), nS is obtained from PDF. For a given stress level, S , total number <strong>of</strong> cycles, NS ,<br />
which cause failure, can be obtained by <strong>using</strong> Wohler curve formulation, which is defined as :<br />
b<br />
N C S <br />
. (3)<br />
In Eq. (3), b is Basquin exponent and C is a material constant.<br />
Finally, by substituting Eq. (2) and Eq. (3) into Eq. (1), expected <strong>fatigue</strong> damage, ED, can be found as:<br />
<br />
<br />
niS St<br />
b<br />
<br />
NS C<br />
. (4)<br />
E D S p S dS<br />
i<br />
i<br />
0<br />
In frequency domain, Power Spectral Density (PSD) obtained from random time histories is the most widely used<br />
expression for the calculation <strong>of</strong> <strong>fatigue</strong> life if <strong>vibration</strong> <strong>fatigue</strong> theory is used [3].<br />
There are many <strong>different</strong> techniques that determine the PDF as functions <strong>of</strong> four spectral moments <strong>of</strong> the PSD<br />
( m0, m1, m2, m 4).These<br />
spectral moments <strong>of</strong> the PSD given in Fig. 2 are expressed as G<br />
k f<br />
k [1]<br />
m<br />
m<br />
n<br />
n<br />
<br />
0<br />
G f df f<br />
k<br />
G k f<br />
k f<br />
, (5)<br />
k 1<br />
where f , G<br />
k f<br />
k <br />
and f are frequency, PSD and frequency increment, respectively.<br />
Fig. 2 Power Spectral Density <strong>of</strong> Stress<br />
In 1954, a very important relationship is developed by S.O Rice [4], for the number <strong>of</strong> upward mean crossings per<br />
second, EP , and peaks per second, E 0 , in terms <strong>of</strong> the spectral moments defined by Eq. (5) in a random<br />
signal. The relationships developed by Rice can be given as follows;<br />
m m<br />
E P E<br />
, 0 <br />
2<br />
Stress<br />
Hz<br />
4 2<br />
. (6)<br />
m2 m0<br />
By utilizing Eq. (6), irregularity factor can be defined as<br />
f k<br />
Gk ⋅ fk<br />
Frequency Hz
0 <br />
2<br />
2<br />
E m<br />
<br />
E P m m<br />
0 4<br />
. (7)<br />
Theoretically, this factor can only take a value in the range <strong>of</strong> 0 to 1. If the value is 1, the process is narrow band<br />
otherwise; it tends to be white noise. It should be noted that;<br />
EPT St,<br />
(8)<br />
where, T is <strong>fatigue</strong> life in seconds.<br />
By substituting Eq. (8) into Eq. (4), expected damage formulation is obtained as<br />
<br />
<br />
niS EPT b<br />
<br />
NS C , (9)<br />
E D S p S dS<br />
i<br />
i<br />
0<br />
which can be used in frequency domain applications. By setting ED equal to 1, <strong>fatigue</strong> life, T , can be obtained.<br />
However, it is worth noting that, while calculating <strong>fatigue</strong> damage by <strong>using</strong> Eq. (9), an appropriate cut-<strong>of</strong>f value<br />
should be used for the upper limit <strong>of</strong> integration.<br />
Bendat [5] first proposed a frequency domain solution, which gives conservative results for wide-band<br />
applications. Therefore, Bendat’s solution is also called as a Narrow-Band solution. As a result <strong>of</strong> this, for<br />
Narrow-Band solution, probability density function and expected damage are given as follows:<br />
S<br />
pS e<br />
4 m<br />
<br />
NB<br />
0<br />
S<br />
2<br />
8m0 , (10)<br />
2<br />
<br />
S<br />
niS EPT b S 8m0 NS C . (11)<br />
4<br />
m<br />
E D S e dS<br />
i<br />
i<br />
0<br />
0<br />
In order to solve the problem <strong>of</strong> getting conservative results from narrow-band solutions, many expressions are<br />
investigated. The first expression for expected damage developed by Wirsching [6] is<br />
c<br />
<br />
1 1 <br />
E D E D a a , (12)<br />
NB<br />
where, ED is the expected damage determined by narrow-band solutions and<br />
NB<br />
a 0.926 0.033b,<br />
c 1.587b 2.323,<br />
2<br />
1<br />
. (13)<br />
<br />
Later, Tunna [7] replaced the expression <strong>of</strong> probability density function given in narrow-band solution with the<br />
following one<br />
S<br />
pS e<br />
4 m 0<br />
S<br />
2<br />
8m0 . (14)
Expected <strong>fatigue</strong> damage, ED , given by Eq.(9) can be also described as follows<br />
<br />
E P T<br />
ED Seq<br />
(14)<br />
C<br />
where, S eq is equivalent stress.<br />
In order to find equivalent stress, Hancock [8] first proposed the following expression<br />
b Seq 2 2m 1<br />
Hancock<br />
<br />
2<br />
<br />
<br />
0 <br />
1<br />
b<br />
Chaudhuery and Dover [9] used a <strong>different</strong> expression for equivalent stress<br />
1<br />
b2 b<br />
b b b <br />
2 2 1 2 2<br />
22 2 2 2 2 <br />
S m0 erf eq CandD<br />
However, the simplest form <strong>of</strong> equivalent stress is given by Steinberg [10] as<br />
b b b<br />
1<br />
b<br />
S 0.6832 m0 0.2714 m0 0.0436 m0<br />
<br />
eq Steinberg<br />
(15)<br />
. (16)<br />
<br />
<br />
(17)<br />
The best correlation in order to find probability density function is proposed by Dirlik [11] and it is given as<br />
2<br />
Z Z<br />
Z<br />
2<br />
1 Q 22 R<br />
2<br />
e e D 2<br />
2<br />
3 Ze D D Z<br />
pS ( ) <br />
Q R<br />
2<br />
m<br />
where,<br />
0<br />
2 2<br />
2( xm ) 1D1D1 1D1D2<br />
1 ,<br />
2 2 , 3 <br />
D D D<br />
1<br />
<br />
1R 1R<br />
, (18)<br />
.<br />
Dirlik investigated this method without <strong>using</strong> narrow-band solution. This empirical closed form <strong>of</strong> PDF is<br />
obtained by <strong>using</strong> computer simulations based on Monte Carlo technique [1].<br />
RESULTS<br />
An Aluminum 6061 T6 <strong>beam</strong> shown in Fig. 3 is used for case studies, finite element model (FEM) <strong>of</strong> which is<br />
verified by tests[12].
Case Study 1<br />
In this case study, <strong>fatigue</strong> analyses and tests are performed <strong>using</strong> the white noise PSD acceleration given in Fig. 4<br />
as base input.<br />
AMPLITUDE (g 2 /HZ)<br />
0.18<br />
0.16<br />
0.14<br />
0.12<br />
0.1<br />
0.08<br />
0.06<br />
0.04<br />
0.02<br />
0<br />
411<br />
350<br />
Fig. 3 Dimensions <strong>of</strong> the Cantilever Beam<br />
PSD INPUT<br />
50 100 150 200 250 300 350 400 450 500<br />
FREQUENCY (HZ)<br />
Fig. 4 PSD Input Acceleration for Fatigue Analysis<br />
Experimental setup is shown in Fig. 5 where the <strong>beam</strong> is fixed to shaker. Since the experimental setup is<br />
configured perpendicular to gravity direction, there is no mean component <strong>of</strong> the stress can be neglected. The test<br />
is stopped when the crack is observed and the time from the beginning to the end <strong>of</strong> the test is accepted to be the<br />
<strong>fatigue</strong> life <strong>of</strong> the <strong>cantilever</strong> <strong>beam</strong>. The test is repeated for seven <strong>cantilever</strong> <strong>beam</strong>s and the average <strong>of</strong> the results is<br />
accepted as the <strong>fatigue</strong> life <strong>of</strong> the <strong>beam</strong>. The <strong>fatigue</strong> life results for the seven test items and average <strong>of</strong> them is<br />
given in Table 1.<br />
3<br />
7<br />
1.5<br />
R 1.5<br />
50
Fig. 5 Experi imental Setup p for Cantilevver<br />
Beam Fatiigue<br />
Test<br />
Ta able 1 Fatigue e Life <strong>of</strong> Testt<br />
Specimens<br />
CONDITION<br />
TE EST ITEM 1<br />
TE EST ITEM 2<br />
TE EST ITEM 3<br />
TE EST ITEM 4<br />
TE EST ITEM 5<br />
TE EST ITEM 6<br />
TE EST ITEM 7<br />
AVE ERAGE LIF FE<br />
FATIGGUE<br />
LIFE (s) )<br />
FFatigue<br />
life <strong>of</strong> o <strong>cantilever</strong> <strong>beam</strong> is calculated<br />
<strong>using</strong> <strong>different</strong> vibbration<br />
fatiguue<br />
theories ass<br />
mentioned iin<br />
Theory<br />
ssection.<br />
The obtained o fatig gue life results s together wit th experimenttally<br />
obtainedd<br />
<strong>fatigue</strong> life aare<br />
given in TTable<br />
2. It<br />
ccan<br />
be conclu uded from the e results obta ained that Dir rlik [11] methhod<br />
generallyy<br />
gives the mmost<br />
accurate results <strong>of</strong><br />
f<strong>fatigue</strong><br />
life. In n addition, co omparison <strong>of</strong> f the location ns <strong>of</strong> crack innitiation<br />
estimmated<br />
from bboth<br />
finite eleement<br />
and<br />
eexperimental<br />
setup is show wn in Fig. 6, which w are in good g agreement<br />
IIn<br />
addition, in n order to obs serve how the e damping rat tio affects thee<br />
<strong>fatigue</strong> life results <strong>of</strong> thee<br />
<strong>cantilever</strong> b<strong>beam</strong>,<br />
four<br />
d<strong>different</strong><br />
fatig gue life analy yses are perf formed <strong>using</strong> 0.5%, 1%, 1.5% and 2% % constant ddamping<br />
ratioos<br />
and the<br />
rresults<br />
includi ing that <strong>of</strong> id dentified damp ping ratio are e given Tablee<br />
3. It is observed<br />
from thee<br />
results that, , damping<br />
rratio<br />
<strong>of</strong> the st tructure affects<br />
the <strong>fatigue</strong> e life significa antly. Hence, , accurate ideentification<br />
<strong>of</strong>f<br />
the dampingg<br />
ratios is<br />
ccrucial<br />
in the estimation <strong>of</strong> f <strong>fatigue</strong> life.<br />
1320<br />
1260<br />
1250<br />
1380<br />
1200<br />
1400<br />
1200<br />
1287
Table 2 Fa atigue Life <strong>of</strong> Cantilever Be eam Predicted d by Differennt<br />
Vibration FFatigue<br />
Theorries<br />
and Experriments<br />
MET THOD<br />
NARROW-BAND<br />
WIRC CHING<br />
TUN NNA<br />
HANC COCK<br />
KAM and d DOVER<br />
STEIN NBERG<br />
DIR RLIK<br />
EXPER RIMENT<br />
Fi ig. 6 Location ns <strong>of</strong> the Crac ck Initiation for fo Both Finitee<br />
Element andd<br />
Manufacturred<br />
Models<br />
Table 3 Fati igue Life Resu ults <strong>of</strong> Cantilever<br />
Beam foor<br />
Three Diffeerent<br />
Dampinng<br />
Ratios<br />
FATI IGUE THOR RIES<br />
FREQUENCY<br />
Y<br />
DOMAIN<br />
0.5 5%<br />
DAM MPING<br />
RAT TIO<br />
FATIGUE<br />
(s)<br />
9.57E+0 01 1.59E+00<br />
1.59E+0 02 22.65E+00<br />
6.05E+0 02 1.01E+01<br />
3.76E-0 03 66.27E-05<br />
3.76E-0 03 66.27E-05<br />
3.76E-0 03 66.27E-05<br />
9.84E+0 02 1.64E+01<br />
1.29E+0 03 22.15E+01<br />
1%<br />
DAM MPING<br />
RATIO R<br />
NA ARROW-BAN ND 1.47E E+02 2.5 58E+02 44.13E+02<br />
WIRCHING<br />
W G 2.45E E+02 4.2 29E+02 66.87E+02<br />
TUNNA 1.22E E+03 4.2 26E+03 11.08E+04<br />
HANCOCK<br />
H<br />
5.04 4E-03 5.2 25E-03 55.35E-03<br />
KAM M and DOV VER 5.04 4E-03 5.2 25E-03 55.35E-03<br />
STEINBERG<br />
S G 5.04 4E-03 5.2 25E-03 55.35E-03<br />
DIRLIK 1.41E E+03 2.9 90E+03 55.00E+03<br />
% DIFFERRENCE<br />
LIFE FATIIGUE<br />
LIFE<br />
WITHH<br />
(min)<br />
EXPERIMEENTAL<br />
FATIGGUE<br />
LIFE ( (s)<br />
1.5%<br />
DDAMPING<br />
RATIO<br />
93<br />
88<br />
53<br />
100<br />
100<br />
100<br />
24<br />
-<br />
2%<br />
DAMPING<br />
RATIO<br />
6.73E+02<br />
1.12E+03<br />
2.21E+04<br />
6.60E+00<br />
5.44E-03<br />
5.44E-03<br />
8.35E+03<br />
IDENTIFIEED<br />
DAMPINGG<br />
RATIO<br />
9.57E+01<br />
1.59E+022<br />
6.05E+022<br />
3.76E-03<br />
3.76E-03<br />
3.76E-03<br />
9.84E+022
Case Study 2<br />
In this case study, firstly, rainflow counting algorithm is developed according to ASTM E 1048 85 [13] and it is<br />
used in time domain <strong>fatigue</strong> life calculations. In order to compare the <strong>fatigue</strong> life results obtained in time and<br />
frequency domains, stress history <strong>of</strong> the location where strain gage is placed on the <strong>cantilever</strong> <strong>beam</strong> is obtained<br />
both in time and frequency domains. Strain gage cannot be located on the most critical location exactly due to<br />
geometric properties <strong>of</strong> the notch and strain gage; hence, the measured data is used only for comparison <strong>of</strong> time<br />
domain and frequency domain methods. In this case study, strain gage data is obtained by <strong>using</strong> a 0.001 g 2 /Hz<br />
White Noise PSD input. Using developed rainflow algorithm, rainflow counting <strong>of</strong> the stress-time history given in<br />
Fig. 7 is performed and <strong>fatigue</strong> life is obtained. Then, <strong>using</strong> the stress PSD data presented in Fig. 8, <strong>fatigue</strong> life is<br />
calculated utilizing <strong>different</strong> <strong>fatigue</strong> theories and all the results are presented in Table 4<br />
NORMAL STRESS sigmax (MPa)<br />
80<br />
60<br />
40<br />
20<br />
0<br />
-20<br />
-40<br />
-60<br />
STRESS RESPONSE<br />
-80<br />
0 10 20 30<br />
TIME (s)<br />
40 50 60 70<br />
Fig. 7 Stress Data for 0.001 g 2 /Hz White Noise PSD<br />
Input at the Critical Location<br />
10 20 30 40 50 60 70 80 90 100<br />
FREQUENCY (HZ)<br />
Fig. 8 Stress PSD Data for 0.001 g 2 /Hz White Noise PSD<br />
Input at the Critical Location<br />
Table 4 Fatigue Life Results Calculated in Time and Frequency Domains<br />
FATIGUE THEORIES FATIGUE LIFE (s) FATIGUE LIFE (h)<br />
FREQUENCY DOMAIN - -<br />
Narrow-band 4.14E+09 1.15E+06<br />
Wirching 6.89E+09 1.91E+06<br />
Tunna 9.95E+09 2.76E+06<br />
Hancock 1.72E+07 4.78E+03<br />
Kam and Dover 1.93E+07 5.37E+03<br />
Steinberg 7.08E+06 1.97E+03<br />
Dirlik 1.40E+10 3.89E+06<br />
TIME DOMAIN - -<br />
Rainflow Counting 1.43E+10 3.97E+06<br />
AMPLITUDE (MPa 2 /HZ)<br />
10 2<br />
10 1<br />
10 0<br />
10 -1<br />
10 -2<br />
10 -3<br />
STRESS RESPONSE PSD
Using the same stress history, it is expected to have similar <strong>fatigue</strong> life results calculated by time and frequency<br />
domain methods. However, when the results in Table 4 are studied, it is observed that Dirlik method gives the<br />
closest result to that <strong>of</strong> Rainflow counting.<br />
CONCLUSION<br />
In this study, firstly descriptions <strong>of</strong> <strong>vibration</strong> <strong>fatigue</strong> approaches are given which are used for <strong>fatigue</strong> life<br />
predictions. An Aluminum 6061 T6 <strong>cantilever</strong> <strong>beam</strong> is used for all case studies. In addition, several <strong>fatigue</strong> tests<br />
are performed in order to compare the results obtained by <strong>fatigue</strong> life prediction approaches with experimentally<br />
obtained <strong>fatigue</strong> lives. It is observed that <strong>fatigue</strong> life obtained <strong>using</strong> Dirlik method is the closest and smaller than<br />
the experimental <strong>fatigue</strong> lives. In addition, by performing <strong>fatigue</strong> life calculations <strong>using</strong> <strong>different</strong> damping ratios,<br />
effect <strong>of</strong> the accurate identification <strong>of</strong> the damping ratio is brought out to be crucial. Finally, <strong>using</strong> <strong>different</strong><br />
loading and strain gage data, <strong>fatigue</strong> life results are obtained in both frequency and timed domains. For time<br />
domain calculations, rainflow counting algorithm is developed and it is observed that the results obtained <strong>using</strong><br />
Dirlik method and rainflow counting.method are very similar to each other.<br />
REFERENCES<br />
[1] N. W. M. Bishop and F. Sherratt, Finite Element Based Fatigue Calculations. NAFEMS, GERMANY,<br />
2005.<br />
[2] N. Bishop, “Fatigue Analysis <strong>of</strong> a Missile Shaker Table Mounting Bracket,” Time.<br />
[3] M. Aykan, “VIBRATION FATIGUE ANALYSIS OF EQUIPMENTS USED IN,” M.Sc. Thesis, Middle<br />
East Technical University, Ankara, 2005.<br />
[4] S. O. Rice, “Mathematical Analysis <strong>of</strong> Random Noise,” Selected Papers on Noise and Stochastic<br />
Processes, Dover, New York, 1954.<br />
[5] J. S. Bendat, “Probability functions for random responses,” NASA report on contract NAS-4590, 1964.<br />
[6] P.H. Wirsching and M. C. Light, “Fatigue under wide band random loading”, Journal <strong>of</strong> Structural<br />
Division,” pp. ASCE, pp. 1593–1607, 1980.<br />
[7] J. M. Tunna, “Fatigue Life Prediction for Gaussian Random Loads at the Design Stage,” Fatigue Fact<br />
Engineering Mat. Struct, vol. vol. 9, no. 3, pp. 169–184, 1986.<br />
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