Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

4, 39, 2001

GENETIC ALGORITHMS IN FATIGUE CRACK DETECTION

Marek Krawczuk
Arkadiusz Żak
Wiesław Ostachowicz

Institute of Fluid Flow Machinery, Polish Academy of Sciences, Gdańsk

e-mail: mk@imppan.pg.gda.pl

This paper presents results on the identification of fatigue cracks in be-
amsvia genetic search technique and changes innatural frequencies.The
location and size of the crack are determined byminimisation of an error
function involving the difference between the calculated and ”measured”
natural frequencies. The simulation studies indicate that the changes in
the natural frequencies and genetic algorithmallows one to estimate the
fatigue crack parameters (location and size) very accurately and fast.

Key words: fatigue crack detection, vibration methods, genetic algo-
rithms

1. Introduction

Damage detection via changing of modal parameters has been a topic of
extensive research over the past few decades, see Fu-KuoChang (1997, 1999).
Damage will cause local changes in the stiffness of a structure, whichwill lead
to changes in its dynamic response. The changes in natural frequencies, mode
shapes or amplitudes of forced vibrations aremost frequently considered as the
damage indicator. Two distinct methodologies have been applied to identify
the damage parameters (location and size) in a structure using vibration data.
The first method is based on finite element model updating and error locali-
sation (Friswell and Mottershead, 1995; Kaouk and Zimmerman, 1993). The
second one assumes a candidate set of possible damage scenarios (i.e. type of
damage, location and size). The calculated changes in dynamic characteristics
for all damage scenarios are comparedwith themeasured ones and the closest
damage case is chosen (Cawley et al., 1978;Krawczuk andOstachowicz, 1997).



816 M.Krawczuk et al.

Thispaperpresents a studyon the identification of a fatigue crack inbeams
usinggenetic search techniqueandchanges innatural frequencies.The location
and size of the fatigue crack are estimated by the minimisation of an output
error with criteria based on changes in natural frequencies. The method is
demonstratedusingamodel of a cracked cantilever beam.The results obtained
from simulation studies indicate the applicability of the presented approach
to damage detection in structures.

2. Genetic algorithm

Thegenetic algorithm is a search technique based on ideas fromthe science
of genetics and the process of natural selection. A simple genetic algorithm
consists of three basic operations: reproduction, crossover andmutation. The
algorithm starts with a randomly generated initial population. Members of
this population are usually binary strings (called chromosomes). Particular
elements of the chromosomes are called the genes. In these strings values of
a variable or variables are coded, which can be a solution to the examinined
problem in the search space. These variables are then used to evaluate the cor-
responding fitness value, which is the objective function. In the next step the
chromosomes are selected for reproduction. Selected processes can be realised
inmanyways (Goldberg, 1989), nevertheless the number of selectedmembers
is a function of their fitness. Thus, the individuals with a higher fitness will
receivemore copies. In order tominimise premature convergence of the initial
populations special scaling methods are applied (Goldberg, 1989). One of the
most widely applied method is the linear scaling proposed by Bagley (1967).
After reproduction the process of crossover is realised. There aremanyways of
implementing this idea (Goldberg, 1989). Generally, crossover with one orma-
ny crossover points can be used. The crossover points are selected randomly,
usually using roulette wheel. This way, by exchanging some portions between
the selected chromosomes (called parents), two new strings (called children)
are created. The final process is mutation. Here, a particular gene in a par-
ticular chromosome is randomly changed. It means that O is changed to 1,
and vice versa. The process of mutation in nature is very rare and, for this
reason, in a genetic algorithm the probability of mutation in a chromosome is
kept on a very low level.



Genetic algorithms in fatigue crack detection 817

3. Objective function

Objective function used in the present paper is based on the changes in
natural frequencies found from ”measurements”. The changes in natural fre-
quencies can be called the classical damage indicators. They are without any
doubt themostwidely damage indicators in the past andnowadays. Themain
reason for their great popularity is that natural frequencies are rather easy to
determinewith a relatively high level of accuracy. In fact, one sensor placed on
a structure and connected to a frequency analyser gives estimates for several
natural frequencies. Further, natural frequencies are sensitive to all kinds of
damage – local and global damage.
The form of objective function is based on the proposed byMessina et al.

(1992) Damage Location Assurance Criterion (DLAC)

DLAC(s)=
|(δΩ)⊤δωs|

2

(

(δΩ)⊤δΩ
)(

(δωs)⊤δωs
) (3.1)

where δΩ is the trial ”experimental” frequency change vector and δωs is the
theoretical frequency change for the damage at the location s.
DLAC values lie in the range 0 to 1, with 0 indicating no correlation and

1 indicating the exact match between the patterns of the frequency changes.
The value of s, giving the highest DLAC values, determines the predicted
damage location and size.

4. Model of cracked beam

Themodel of cracked, cantilever beam is presented in Fig.1

For each part of the presented beam the following equation of transverse
vibration can be written. In this equation the shear effect is omitted

EJ
∂4yi(x,t)

∂x4
+ρA

∂2yi(x,t)

∂t2
=0 i=1,2 (4.1)

where

E – Youngmodulus
J – moment of inertia of the beam cross-section
ρ – material density
A – cross-sectional area of beam.



818 M.Krawczuk et al.

Fig. 1. Dimensions of the cracked cantilever beam

The solution to equation (4.1) can be expressed in the following form

yi(x)=Ai sin(kx)+Bicos(kx)+Ci sinh(kx)+Dicosh(kh) i=1,2
(4.2)

where the parameters ki correspond to the natural frequencies of the analysed
beam.

The boundary conditions for the analysed structure can be expressed in
the following form:

— fixed end (x=0)

y1(0)= 0 y
′

1(0)= 0

— crack (x= e=L1/L)

y1(e)= y2(e) y
′
2(e)−y

′
1(e)= cy

′′
2(e)

y′′1(e) = y
′′
2(e) y

′′′
1 (e) = y

′′′
2 (e)

— free end (x=1.0)

y′′2(1)= 0 y
′′

2(1)= 0

where c is the stiffness of the elastic element modelling the crack, see Osta-
chowicz and Krawczuk (1991).

Taking into account the boundary conditions and form of the solution to
equation of motion (4.1) the characteristic equation of the problem can be



Genetic algorithms in fatigue crack detection 819

formulated. This equation allows one to determine the natural frequencies of
the cracked beam

det

















1 0 1 0 0 0 0 0
0 1 0 1 0 0 0 0

ch(ke) sh(ke) cos(ke) sin(ke) −ch(ke) −sh(ke) −cos(ke) −sin(ke)
sh(ke) ch(ke) −sin(ke) cos(ke) A1 A2 A3 A4
ch(ke) sh(ke) −cos(ke) −sin(ke) −ch(ke) −sh(ke) cos(ke) sin(ke)
sh(ke) ch(ke) sin(ke) −cos(ke) −sh(ke) −ch(ke) −sin(ke) cos(ke)
0 0 0 0 chk chk −cosk −sink
0 0 0 0 shk chk sink −cosk

















=0

(4.3)
where: sh= sinh, ch= cosh,A1 = ckcosh(ke)−sinh(ke),A2 = ck sinh(ke)−
cosh(ke), A3 =−ckcos(ke)+sin(ke), A4 =−cksin(ke)+cos(ke).
From the above equation it arises that changes in the natural frequencies

will be functions of the crack depth and location along the beam.

5. Numerical examples

Thenumerical calculations were carried out for a cracked, cantilever beam.
The length of beam was 0.4m, width 0.02m and height 0.02m. The beam
was made of steel with Young’s modulus 2.1 · 1011N/m2 and mass density
7860kg/m3.

Two cases were tested:

Case 1 – the crack had depth up to 5% of the total beam height, and was
located 0.04m from the fixed end (e=L1/L=0.1),

Case 2 – the crack had depth up to 2% of the total beam height, and was
located 0.12m from the fixed end (e=L1/L=0.3).

In all cases the population had 10 members. Onemember had 30 bits (15
for each variable). During numerical calculations it was assumed that the pro-
bability of crossover was 95% and the probability ofmutation was 0.05%. The
crack depth and location were identified using the eigensensitivity approach
described in Section 3. The first four natural frequencies were used in the nu-
merical tests. Only one run of the genetic algorithm was used for each case.
The results of numerical calculations are presented in Fig.2 to Fig.5.
From Figures 2-5 it arises that the genetic algorithm correctly locates the

damage andalso correctly estimates its size. The convergence to proper results
was obtained after no more than 110 populations in the first case and 85
populations in the second one.



820 M.Krawczuk et al.

Fig. 2. Crack depth as a function of the number of populations – case 1

Fig. 3. Crack location as a function of the number of populations – case 1



Genetic algorithms in fatigue crack detection 821

Fig. 4. Crack depth as a function of the number of populations – case 2

Fig. 5. Crack location as a function of the number of population – case 2



822 M.Krawczuk et al.

6. Conclusions

The combined genetic algorithm and the eigensensitivity criterion DLAC
as an objective function has been used to identify the location and size of the
crack from ”experimental” vibration data. The genetic algorithm presented
in this paper is very simple. Nevertheless, the obtained results are promising.
Thenumberof calculationswhichareneeded for damagedetection ismuch less
than that in classical search algorithms. For this reason, the time of numerical
calculations is shorter.

Future works should be devoted to implementation of processes which are
observed in nature to this algorithm. For example elitism, where the best
solution is always passed on to the next generation, is a particular feature for
which good results have been reported (Goldberg, 1989). It is also planned to
check other vibration criteria being applied to structural health monitoring
and based on changes in mode shapes and amplitudes of forced vibrations as
objective functions. Such comparative analysis should explain which damage
indicator ismost sensitive to changes in stiffness of a structure due to damage.

References

1. Adams R.D., Cawley P., 1979, The Localisation of Defects in Structures
fromMeasurements of Natural Frequencies, Journal of Strain Analysis, 14, 2,
49-57

2. Bagley J.D., 1967, The Behavior Addaptive Systems which Employ Genetic
and Correlation Algorithms, Ph.D. Thesis, University ofMichigan

3. Cawley P., Adams R.D., Pye C.J., Stone B.J., 1978, A Vibration Tech-
nique for Non- Destructivly Assessing the Integrity of Structures, Journal of
Mechanical Engineering Sciences, 20, 2, 93-100

4. Friswell M.I., Mottershead J.E., 1995, Finite Element Model Updating
in Structural Dynamics, Kluwer Academic Publishers

5. Fu-KuoChang, 1997,Structural HealthMonitoring: Current Status and Per-
spectives, Technomic Pub., Basel

6. Fu-KuoChang, 1999,Structural HealthMonitoring: Current Status and Per-
spectives, Technomic Pub., Basel

7. GoldbergD., 1989,GeneticAlgorithms in Search,Optimization andMachine
Learning, Addison-Wesley Publishing Company, Reading



Genetic algorithms in fatigue crack detection 823

8. KaoukM., ZimmermanD.C., 1993,Evaluation of theMinimumRankUpda-
te inDamageDetection:AnExperimentalStudy.Proc. of the 11th International
Modal Analysis Conference, 1061-1068

9. KrawczukM.,OstachowiczW., 1997,Damage Indicators forDiagnostic of
Fatigue Cracks in Structures by Vibration Measurements – a Survey, Journal
of Theoretical and Applied Mechanics, 34, 2, 307-326

10. Messina A., Jones I.A., Williams E.J., 1992, Damage Detection and Lo-
calisation Using Natural Frequency Changes, Proc. of the 1st Conference on
Identification, 67-76

Algorytmy genetyczne w detekcji pęknięć zmęczeniowych

Streszczenie

Wpracy przedstawionowyniki identyfikacji położenia i wielkości pęknięć zmęcze-
niowych metodą algorytmów genetycznych z wykorzystaniem zmian częstości drgań
własnych. Położenie i wielkość pęknięcia poszukiwanominimalizując funkcję celuwy-
korzystującą różnicemiędzy częstościamimierzonymi i obliczanymi.Wyniki symulacji
wskazują, że zmiany częstości drgań własnych i algorytm genetyczny pozwalają wy-
znaczaćparametrypęknięcia zmęczeniowego(położenie iwielkość) szybko i dokładnie.

Manuscript received April 28, 2000; accepted for print December 17, 2000