Detection of Multiple Cracks Using Frequency Measurements 2003 Engineering Fracture Mechanics

Detection of multiple cracks using frequency measurements

D.P. Patil, S.K. Maiti *

Department of Mechanical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, India

Received 3 April 2002; received in revised form 9 July 2002; accepted 14 July 2002

Abstract

A method for detection of multiple open cracks in a slender Euler–Bernoulli beams is presented based on frequency

measurements. The method is based on the approach given by Hu and Liang [J. Franklin Inst. 330 (5) (1993) 841],

transverse vibration modelling through transfer matrix method and representation of a crack by rotational spring. The

beam is virtually divided into a number of segments, which can be decided by the analyst, and each of them is con-

sidered to be associated with a damage parameter. The procedure gives a linear relationship explicitly between the

changes in natural frequencies of the beam and the damage parameters. These parameters are determined from the

knowledge of changes in the natural frequencies. After obtaining them, each is treated in turn to exactly pinpoint

the crack location in the segment and determine its size. The forward, or natural frequency determination, problems are

examined in the passing. The method is approximate, but it can handle segmented beams, any boundary conditions,

intermediate spring or rigid supports, etc. It eliminates the need for any symbolic computation which is envisaged by Hu

and Liang [J. Franklin Inst. 330 (5) (1993) 841] to obtain mode shapes of the corresponding uncracked beams. The

proposed method gives a clear insight into the whole analysis. Case studies (numerical) are presented to demonstrate

the method effectiveness for two simultaneous cracks of size 10% and more of section depth. The differences between the

actual and predicted crack locations and sizes are less than 10% and 15% respectively. The numbers of segments into

which the beam is virtually divided limits the maximum number of cracks that can be handled. The difference in the

forward problem is less than 5%.

� 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Transfer matrix method; Vibration based detection of multiple cracks; Vibration modelling of cracked beam; NDT of

cracks

1. Introduction

To help in a continuous safety assessment of a machine or structure it is very necessary to constantly

assess the health of its critical components. This calls for a continuous assessment of changes in their static

and/or dynamic behaviour. The changes have very often their origin in local reduction of structural stiffness

caused by cracks or crack-like defects. The development of a crack does not necessarily make a component

instantly useless, but it is a signal that its behaviour has to be monitored more carefully. Such monitoring

Engineering Fracture Mechanics 70 (2003) 1553–1572

www.elsevier.com/locate/engfracmech

*Corresponding author. Tel.: +91-22-578-2545/572-7526; fax: +91-22-578-3480/572-6875.

E-mail address: [email protected] (S.K. Maiti).

0013-7944/03/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0013-7944(02)00121-2

mail to: [email protected]

can play a significant role in assuring an uninterrupted operation in service by the component. This has

made the vibration based monitoring of components consisting of cracks or crack-like defects in service

very important and generated a lot of interest in the study of vibration of components with one or more

cracks. Very good reviews have been presented by Dimarogonas and Paipetis [1], Entwistle and Stone [2],Wauer [3], Gasch [4], Dimarogonas [5] and Salawu [6].

The method of crack detection can based on longitudinal vibration [7–9], or transverse vibration [10–24],

or a coupling of transverse, longitudinal and torsional vibrations [1,25–29]. Majority of the studies consider

open cracks, thereby nonlinearity due to crack closing or material nonlinearity, if any, is neglected. There

are also studies, e.g. Chati et al. [30], Schlums and Dual [31], etc., reported considering closing cracks. It is

shown that the consideration of a fatigue crack as an open crack rather than a closing crack gives rise to an

under prediction of crack size [31].

The method of detection based on transverse vibration of components offers promise for inspection ofhigh volume and low cost components feasible [32]. The choice of using the natural frequency as a basis in

the development of NDE is most attractive due to the fact that the natural frequencies of a beam can be

measured from one single location in the beam, thus offering scope for development of a fast and global

NDE [5,7,32,33] technique than the conventional ultrasonic, X-ray, etc.

The modelling of vibration and representation of crack plays a very significant role in the success of

crack detection [34]. The modelling of transverse vibration has been done within the framework of either

the Euler–Bernoulli or Timoshenko beam theory. The equations have been solved directly, or using the

Frobenius method [24,35], or the Bessel function approach [36,37], Sometimes the transfer matrix method(TMM) [38–41] has been used to simplify the process of solution for complex geometries and continuous

beams.

In the event of a coupling of longitudinal, lateral, transverse and torsional vibrations the crack is rep-

resented in terms of a 6� 6 matrix. In the case of a coupling of the longitudinal and lateral vibrations this

matrix reduces to a 2� 2 matrix. Further, when the axial motion is negligible it can be represented by a

single rotational spring. This favoured the representation of an open crack by a massless rotational spring

and it is found to be very effective for a crack size more than 10% of section depth [17].

The potential of finite element method has also been exploited in this context for handling individualcomponents or systems. Approaches for prediction of crack are given by Cawley and Adams [10,42],

Thambiratnam and Zhuge [43], Chichalkar [44], Yang et al. [45], etc. Thambiratnam and Zhuge has

proposed an element for analysis of beams on elastic foundation. Various beam geometric features, sup-

porting conditions, external or internal cracks, normal or inclined cracks, etc., have received attention

[23,24,35,43,46,47]. Beams on elastic foundation [43,46,47], discontinuous supports [38,46], etc., have also

been studied.

The crack detection has been mostly facilitated by frequency or mode shape data. To facilitate crack

detection sometimes the problem has been formulated as an optimization problem, e.g. minimizationof difference between the theoretical and experimental data on a parameter related to the dynamic be-

haviour [48]. The search engine is based on the system-identification technique [49] or genetic algorithm

[48,50,51]. In the genetic algorithm based search, the theoretical modelling helps to generate a guess data

set.

Most studies concerning crack detection deal with single crack. The case of multiple cracks has not

received the same degree of attention. This type of problems have been addressed by Sekhar [29], Ruotolo

and Surace [48], Ostachowicz and Krawczuk [52], Liang et al. [53], Hu and Liang [54], Tsai and Wang [55],

Kisa and Brandon [56], and Zheng and Fan [57,58]. Theoretically, the approaches can be extended to anynumber of cracks and boundary conditions, some of these lead to an increase in size of characteristic

determinant. For example, for a cantilever beam with two cracks, the characteristic determinant is of size

12� 12 [52]. For every single additional crack the size of the determinant increases by 4. An improved

analytical method for calculating natural frequencies of a uniform beam with a large number of cracks is

1554 D.P. Patil, S.K. Maiti / Engineering Fracture Mechanics 70 (2003) 1553–1572

proposed by Shifrin and Ruotolo [59]. They represent the cracks by the massless rotational springs. Their

method leads to a characteristic equation of size (nþ 2) when there are n cracks.

While most of the studies dealing with multiple cracks address the forward problem, i.e. determination of

natural frequencies, Sekhar [29], Ruotolo and Surace [48] and Liang and coworker [53,54], address the in-verse analysis, i.e. detection of crack details from the knowledge of dynamic response of the component. The

problem of detection of location and size of a number of cracks in a component simultaneously is much more

involved and complex than the case of a single crack. While solving such a case some of the investigators

consider that crack reduces the section modulus (EI) of a small beam segment around the crack [47]. Some

have obtained the solution employing the TMM [47], modified Fourier series approach [57,58], etc.

Experimental studies have helped to validate some of the models. Mostly cracks have been made by slit-

cutting in specimens. It is reported that, difference between experimental and theoretical results decreases as

the slit width reduces for smaller crack sizes (<10% depth). For larger crack sizes the effect is insignificant.Experimental results have been presented by Cawley and Adams [10] and Ruotolo and Surace [48], on

multiple cracks, and Rizos et al. [17], Cawley and Ray [32], Boltezar et al. [33], Doyle et al. [51] and Lele

and Maiti [60], on single crack.

The TMM is applicable to beams [38,39,61] with a large number of geometric segments, intermediate

supports, etc. This method offers definite advantages over the approaches based on the modal analysis or

FEM [44]. Particularly, size of the characteristic determinant reduces considerably. The size can be reduced

to (nþ 2), where n is the number of intermediate rigid supports, irrespective of the number of beam seg-

ments. How this method can be exploited for handling beams on elastic foundation, or discontinuoussupports, with multiple cracks is open for an investigation. The present study derives motivation from such

an issue.

Liang et al. [53] have approached the problem of multiple cracks by breaking the beam into a number of

segments, each of which can have a crack/damage. The size of crack in a segment decides the extent of the

local effects. They have assumed the effect to be spread uniformly over the whole segment. They have

showed through energy consideration that the changes in natural frequency are linearly related to the

damage parameters. The exact relationship has however been obtained through symbolic computation,

which is basically employed to obtain the mode shapes of the corresponding uncracked beam. Knowledgeof a change in a number of natural frequencies permits determination of the damage parameters and de-

tection of crack details.

The aim of this work is to present a method for modelling transverse vibration of beams with multiple

open cracks by combining the TMM and the rotational spring based representation of a crack, and the

approximate approach of Hu and Liang [54]. The formulation helps to do away with the symbolic com-

putation and obtain the mode shape of the corresponding uncracked beam. Simply supported beam,

cantilever beam, as well as beams on elastic foundation and multiple pin supports are examined. The beams

are considered slender so that the shear deformation and rotational inertia can be neglected. The damping,if any, present in the beam is neglected. Numerical studies are presented to demonstrate the effectiveness

of the method for crack detection. The natural frequencies required as input are computed using a finite

element program. The forward problems are solved in the passing.

2. Formulation

An uniform beam with n cracks located at n ¼ x=L ¼ b1; b2; b3; . . . and bn (0 < b1 < b2 < b3 < � � � <bn < 1), is shown in Fig. 1. Each of the crack can be represented by a rotational spring with stiffness [52]

given by

Ki ¼Ebh2

72pf ðriÞð1Þ

D.P. Patil, S.K. Maiti / Engineering Fracture Mechanics 70 (2003) 1553–1572 1555

Abhishek Kumar Sahu

Highlight

Abhishek Kumar Sahu

Highlight

where Ki is the equivalent spring stiffness for crack i, h is height and b is width of cross-section, rið¼ ai=hÞ isnondimensional crack size and ai is size of the crack. f ðriÞ, called flexibility function, is given by

f ðriÞ ¼ 0:6384ðriÞ2 � 1:035ðriÞ3 þ 3:7201ðriÞ4 � 5:1773ðriÞ5 þ 7:553ðriÞ6 � 7:3324ðriÞ7 þ 2:4909ðriÞ8

ð2ÞThe rotational spring stiffness corresponding to double edge cracks is presented in [52].

For an Euler–Bernoulli beam the governing equation of motion is

o2

ox2EI

o2yðx; tÞox2

� �þ qA

o2yðx; tÞot2

¼ 0 ð3Þ

Through separation of variables yðx; tÞ ¼ ZðxÞ cosðxtÞ, the mode shape equation is obtained:

EId4Zdx4

� qAx2i Z ¼ 0 or

d4Zdx4

� p4Z ¼ 0 ð4Þ

where q is the mass density (kg/m3), A, cross-sectional area (m2), xi, mode i natural frequency (rad/s), E,

modulus of elasticity (N/m2), I, area moment of inertia (m4), and

p4 ¼ qAx2

EIð5Þ

The general solution of Eq. (4) can be written as follows:

ZðxÞ ¼ C1½cosðpxÞ þ coshðpxÞ þ C2½cosðpxÞ � coshðpxÞ þ C3½sinðpxÞ þ sinhðpxÞþ C4½sinðpxÞ � sinhðpxÞ

ð6Þ

Using this relation it is possible to relate displacement Z, slope h ¼ dZ=dx, bending moment M ¼EIðd2Z=dx2Þ and shear force V ¼ EIðd2Z=dx2Þ at the two ends i and i� 1 of an arbitrary segment:

ZhMV

2664

3775

i

¼

Ai BiCi

EIDi

EI

p4Di AiBi

EICi

EIEIp4Ci EIp4Di Ai Bi

EIp4Bi EIp4Ci p4Di Ai

26666664

37777775

ZhMV

2664

3775

i�1

or ZR ¼ RiZL ð7Þ

Fig. 1. Beam with multiple cracks.


where Ri is called the transfer matrix for the segment i, and

Ai ¼cosðpliÞ þ coshðpliÞ

2; Bi ¼

sinðpliÞ þ sinhðpliÞ2p

Ci ¼�½cosðpliÞ � coshðpliÞ

2p2; Di ¼

�½sinðpliÞ � sinhðpliÞ2p3

9>>=>>;

ð8Þ

li ¼ length of the segment

C1 ¼Zi�1

2; C2 ¼ � Mi�1

2EIp2; C3 ¼

hi�1

2p; C4 ¼ � Vi�1

2EIp3ð9Þ

Shear force V acting on a section with its outer normal oriented in the positive x-direction is considered

positive downward.

At a crack location, there is continuity in Z, M and V and a jump in h. Utilizing these it is possible to

relate the variables on the two sides of the crack:

ZhMV

2664

3775

i

¼

1 0 0 0

0 1 1=Ki 0

0 0 1 0

0 0 0 1

2664

3775

ZhMV

2664

3775

i�1

or ZR ¼ SiZL ð10Þ

where Si is the transfer matrix and Ki is the rotational spring stiffness given by Eq. (1).For a step in a beam, the transfer matrix is just an identity matrix. This follows from the fact that there is

continuity of deflection, slope, moment and shear force at the junction irrespective of the change in cross-

sections. For an intermediate support the beam has a discontinuity in shear force equal to the support

reaction. This fact can be built into the corresponding transfer matrix.

3. Forward problem

For a given beam with geometric features such as a step, uniform segment, crack, etc., the overall

transfer matrix ½H , i.e. ½ZN ¼ ½H ½Z1 (Fig. 1), can be obtained from intermediate transfer matrices through

simple multiplication

½H ¼ ½Rn4�4½Sn�14�4 � � � ½R24�4½S14�4½R14�4 ð11ÞThese matrices are obtainable from Eqs. (7), (10), etc. By inserting the specified boundary conditions, it ispossible to obtain a set of simultaneous equations ½H ½Z ¼ 0. Mostly two parameters out of the four are

zero at the ends. This helps to obtain the characteristic matrix ½H as a matrix of size 2� 2. This makes the

proposed method very favourable. The characteristic equation of vibration is then given by

det½Hðx; b1;b2; b3; . . . ;K1;K2;K3; . . .Þ2�2 ¼ 0 ð12Þwhere x is natural frequency. For a cantilever beam with a single crack, the characteristic equation is as

follows:

H 111 þ

H 211

KH 1

12 þH 2

12

K

H 121 þ

H 221

KH 1

22 þH 2

22

K

¼ 0 ð13Þ

where

H 111 ¼ p4C1C2 þ p4B1D2 þ A1A2 þ p4D1B2; H 2

11 ¼ EIp4A1D2


H 112 ¼ p4D1C2 þ p4C1D2 þ B1A2 þ A1B2; H 2

12 ¼ EIp4B1D2

H 121 ¼ p4C1B2 þ p4B1C2 þ p4A1D2 þ p4D1A2; H 2

21 ¼ EIp4A1C2

H 122 ¼ p4D1B2 þ p4C1C2 þ p4B1D2 þ A1A2; H 2

22 ¼ EIp4B1C2

Ai, Bi, Ci and Di (i ¼ 1; 2) are given by Eq. (8) except li is replaced by Li, where L1 ¼ bL, L2 ¼ ð1� bÞL andb is the nondimensional crack location. Explicitly Eq. (13) for cantilever and simply supported beams

respectively are as follows:

4ð1þ cosh k cos kÞ þ kKfsinh kðcos k þ cos keÞ � sin kðcosh k þ cosh keÞ þ 2 coshðkbÞ sinðkbÞ

� 2 cosðkbÞ sinhðkbÞ � 2 sin½kð1� bÞ cosh½kð1� bÞ þ 2 cos½kð1� bÞ sinh½kð1� bÞg ¼ 0 ð14Þ

and

4 sin k sinh k þ kKfsinh kðcos k � cos keÞ � sin kðcosh k � cosh keÞg ¼ 0 ð15Þ

where k ¼ pL, b ¼ L1=L, e ¼ 2b � 1 and K ¼ ðKLÞ=ðEIÞ. Putting ð1=KÞ ¼ 0 (i.e. compliance ¼ 0) in Eqs.

(14) and (15) the characteristic equations for the corresponding uncracked beams are obtained. Equa-

tion similar to Eq. (15) for a simply supported beam is derived by Narkis [62] through a different approach.

It is a bit difficult to derive Eq. (14) using this approach. Eqs. (14) and (15) can be re-written in a shortform

D1 þkK

D2 ¼ 0 ð16Þ

where

D1 ¼ 4ð1þ cosh k cos kÞ for a cantilever beam

¼ 4 sin k sinh k for a simply supported beam

D2 corresponds to the portion within { } in Eqs. (14) and (15) respectively.

Solving Eq. (12) or (16) it is possible to solve the forward problem and obtain the natural frequencies. In

case there are n intermediate rigid supports, the size of the characteristic determinant (det½H ) increases byn. That is, the matrix size is then (nþ 2). The mode shape corresponding to a natural frequency can be

obtained through Eq. (6).If any of the beam segment is nonuniform, e.g. tapered, it can be handled conveniently by incorporating

the transfer matrix given in Patil and Maiti [61].

4. Procedure for crack detection

For a single span beam with one crack Eq. (12) or (16) can serve as a basis for the detection of its lo-

cation and size. But for the detection of multiple, say n, cracks the number of unknowns are 2n. It is thendifficult to apply Eq. (16) directly. Here the approximate method of Hu and Liang [54] in conjunction with

the TMM can be employed. This is explained in the following:

The Rayleigh quotient is given by

x2 ¼ l ¼12

Z L

0

EIðxÞ d2Zdx2

� �2

dx

12

R L0

qAZ2 dx¼ U

Vð17Þ


where

U ¼ strain energy ¼ 1

2

Z L

0

EIo2Zox2

� �2

dx ¼ 1

2

Z L

0

Wdx; W ¼ EIo2Zox2

� �2

and

V ¼ kinetic energy ¼ 1

2

Z L

0

qAZ2 dx

For small changes in frequency

Dll

¼ DUU

� DVV

ð18Þ

For a beam with a crack located normally to the beam axis and undergoing transverse vibration DV ¼ 0.

Considering a crack to affect the mode shape locally and representing this effect by a damage para-meter S,

DU ¼ 1

2

Z L

0

SWdx ð19Þ

S is assumed to be nonzero only over a span/segment around as crack. Its magnitude depends on crack size.

For a segment without crack S ¼ 0 and for a segment with full depth crack S ¼ 1. S ¼ 1 signifies a complete

failure. Eq. (18) can be now written explicitly:

Dxx

¼ 1

2

R L0SWdxR L

0Wdx

ð20Þ

If a beam is divided into m segments and the segment i is assumed to be associated with a damage

parameter Si,

Dxx

¼ 2Xmi¼1

1

I0

ZLi

wdxSi ð21Þ

where I0 ¼ 4RL wdx, Li represents length of segment i. In keeping with Liang et al.’s [53] representation, for

nth frequency

Dxn

xn¼ 2

Xmi¼1

ZLgnðnÞLdnSi; gnðnÞ ¼ Wn=I0n ð22Þ

where gnðnÞ, Wn and I0n are associated with the corresponding natural mode shape. This gives a set of linearsimultaneous equations involving Si. For a beam divided into m segments and q number of known fre-

quencies, Eq. (22) gives rise to a set of simultaneous equations:

Dxx

� �q�l

¼ 2½H q�mfSgm�l ð23Þ

where a typical element of ½H matrix, hij ¼RLjgiðnÞLdn, i ¼ 1; 2; . . . q, and j ¼ 1; 2; . . .m; i refers to the

mode and j refers to the segment number. For example, for a simply supported beam, the nth mode shape

can be written in terms of sinðnpx=lÞ or sinðnpnÞ. Therefore

ZnðnÞ ¼ sinðnpnÞ; Z 0nðnÞ ¼ np cosðnpnÞ; Z 00

n ðnÞ ¼ �n2p2 sinðnpnÞ


Hence

gnðnÞ ¼½�n2p2 sinðnpnÞ2

4R 1

0½�n2p2 sinðnpnÞ2 dn

¼ 1

2sin2ðnpnÞ

hnj ¼ZLj

1

2sin2ðnpnÞLdn ¼ L

4ðn2

�� n1Þ �

1

2np½sinð2npn2Þ � sinð2npn1Þ

�

For solving the inverse problem Eq. (23) is useful. The numerical values of damage parameters Si ob-tained from Eq. (23) furnish information about the state of damage of the segments.

5. Numerical studies

5.1. Forward problem

The proposed method has been tested for beams with cracks of varying sizes at different locations. In thecase of the forward problem, the characteristic equation (12) is used. The material and geometric properties

used are included in tables mentioned subsequently. All computations are done using double precision

arithmetic and MATLAB package.

Tables 1–3 give results for the first three modes of a single span cantilever beam of length l with two

cracks. Here P ¼ pl, where p is the frequency parameter. The first crack is located at b1 ¼ 0:1 and the

location of the second crack is varied. Different sizes for the second crack are also examined. The fre-

quencies obtained are in good agreement with the results of [51]. There is also good agreement with present

FE results. For FE analysis, the beam is discretised mostly by 8-noded isoparametric elements. Around thecrack tip, quarter point singular elements are used. The computations are done using a package [63].

5.2. Inverse problem/crack detection

The first n natural frequencies required for the solution of a problem of multiple crack detection are

computed here by using the earlier mentioned package. All calculations are again done using the MATLAB

package. Fig. 2 shows the flow chart of the procedure. In the calculations some of the parameters Si are lessthan 0. Since a positive Si represents a reduction in section because of a crack, a negative Si indicates anincrease of area of section. Physically, this is unrealistic. To handle the case Si < 0, the corresponding

segment is treated to be free of any damage and Si is set equal to zero. The calculations are repeated till a set

of all positive damage parameters are obtained.

Taking only one damage parameter Si of a segment i as nonzero at a time and keeping all others as zero,

the change in a natural frequency Dxn due to one such damage in the segment is calculated. A variation of

K with crack location b is obtained using Eq. (16) and Dxn. This is repeated for three or more modes. Since,

the stiffness of the spring is independent of the mode, the location where the three or more such curves

intersect gives the crack location and the spring stiffness. The crack size is then obtained from the relation(2).

To facilitate a comparison for predictability, the geometric and material data of Hu and Liang [54] are

used. The beam (Fig. 3a), simply supported at the ends, is divided into ten equal segments (Fig. 3b). The

total number of segments is the same for all the case studies reported here unless otherwise mentioned. The

first five natural frequencies (Table 4) are employed to obtain the equations

Dxx

� �5�1

¼ 2½H 5�10fSg10�1 ð24Þ


Table 1

Comparison of first natural frequency parameter for cantilever (cl-fr) beam with two cracks (uncracked beam pL ¼ 1:8751{1.87104}a)

a2=h b2

0.11 0.2 0.4 0.6 0.8

b ¼ 0:1, a1=h ¼ 0:1 pL ¼ 1:8605{1.8585}

0.1 1.8468 1.8504{1.8500} 1.8564{1.8550} 1.8595{1.8577} 1.8604

0.3 1.7527 1.7788{1.7892} 1.8252{1.8288} 1.8514{1.8509} 1.8597

0.5 1.5834 1.6391{1.6533} 1.7530{1.7599} 1.8305{1.8312} 1.8581

0.7 1.3947 1.4685 1.6422 1.7914 1.8548

b1 ¼ 0:1, a1=h ¼ 0:3 pL ¼ 1:7603(1.7602){1.7737}

0.1 1.7499(1.7496)a 1.7527(1.7506){1.7668} 1.7573(1.7564){1.7709} 1.7596(1.7593){1.7730} 1.7603(1.7601)

0.3 1.6762(1.6762) 1.6971(1.6964){1.7175} 1.7336(1.7337){1.7501} 1.7536(1.7512){1.7678} 1.7598(1.7598)

0.5 1.5356(1.5354) 1.5831(1.5831){1.6037} 1.6772(1.6772){1.6943} 1.7579(1.7321){1.7523} 1.7586(1.7562)

0.7 1.3685(1.3689) 1.4352(1.4356) 1.5868(1.5871) 1.7081(1.6812) 1.7562(1.7558)

b1 ¼ 0:1, a1=h ¼ 0:5 pL ¼ 1:5836(1.5837){1.6004}

0.1 1.5774(1.5774) 1.5791(1.5792) 1.5818(1.5817) 1.5831(1.5829) 1.5835(1.5835)

0.3 1.5318(1.5318) 1.5452(1.5452){1.5659} 1.5678(1.5684){1.5863} 1.5796(1.5791) 1.5832(1.5826)

0.5 1.4365(1.4426) 1.4702(1.4731){1.4906} 1.5333(1.5342){1.5517} 1.5706(1.5708) 1.5826(1.5794)

0.7 1.3094(1.3062) 1.3622(1.3623) 1.4741(1.4762) 1.5530(1.5491) 1.5812(1.5864)

b1 ¼ 0:1, a1=h ¼ 0:7 pL ¼ 1:3901(1.3906)

0.1 1.3868(1.3869) 1.3878(1.3879) 1.3892(1.3884) 1.3898(1.3901) 1.3901(1.3905)

0.3 1.3622(1.3712) 1.3696(1.3721) 1.3819(1.3812) 1.3881(1.3896) 1.3900(1.3900)

0.5 1.3060(1.3061) 1.3268(1.3243) 1.3634(1.3643) 1.3834(1.3831) 1.3896(1.3892)

0.7 1.2216(1.2212) 1.2582(1.2601) 1.3298(1.3303) 1.3743(1.3742) 1.3889(1.3874)

aData quoted within ( ) are from Ref. [52]; data quoted within { } are by present FE analysis.

Table 2

Comparison of second natural frequency parameter for cantilever beam with two cracks (uncracked beam pL ¼ 4:69409{4.60859}a)

a2=h b2

0.11 0.2 0.4 0.6 0.8

b1 ¼ 0:1, a1=h ¼ 0:1 pL ¼ 4:6808{4.5976}

0.1 4.6707 4.6807{4.5975} 4.6634{4.5839} 4.6577{4.5795} 4.6763

0.3 4.6093 4.6797{4.5962} 4.5420{4.4860} 4.4916{4.4465} 4.6399

0.5 4.5286 4.6779{4.5918} 4.3258{4.2779} 4.1702{4.1427} 4.5427

0.7 4.4700 4.6763 4.1055 3.7975 4.3557

b1 ¼ 0:1, a1=h ¼ 0:3 pL ¼ 4:6025{4.5340}

0.1 4.5968 4.6025{4.5340} 4.5833{4.5192} 4.5797{4.5161} 4.5983

0.3 4.5599 4.6022{4.5334} 4.4485{4.4128} 4.5172{4.3846} 4.5637

0.5 4.5052 4.6016{4.5306} 4.2054{4.1849} 4.415{4.0820} 4.4715

0.7 4.4600 4.6010 3.9511 4.0941 4.2936

b1 ¼ 0:1, a1=h ¼ 0:5 pL ¼ 4:5045{4.4350}

0.1 4.5025 4.5035 4.4828 4.4819 4.5004

0.3 4.4888 4.4965{4.4294} 4.3302{4.2989} 4.3189 4.4679

0.5 4.4646 4.4823{4.4169} 4.0487{4.0387} 3.9972 4.3813

0.7 4.4398 4.4647 3.7430 3.6103 4.2138

b1 ¼ 0:1, a1=h ¼ 0:7 pL ¼ 4:4363

0.1 4.4358 4.4341 4.4129 4.4140 4.4324

0.3 4.4322 4.4168 4.2469 4.2517 4.4013

0.5 4.4247 4.3784 3.9357 3.9289 4.3182

0.7 4.4152 4.3237 3.5857 3.5352 4.1574

aData quoted within { } are by present FE analysis.


The ten damage parameters (S1; S2; . . . S10) are obtained from five equations employing the pseudo-inverse

technique of the MATLAB.

Solution of Eq. (24) gives the following Si’s or fSg matrix after the first iteration:

fSg ¼ fS1; S2; S3; S4; S5; S6; S7; S8; S9; S10gT

¼ f0:00066; 0; 0; 0; 0; 0:05317;�0:00699; 0:03901;�0:00343; 0gT

S7 and S9 are less than zero. These are set equal to zero and re-solution is carried out. This gives after the

second iteration

fSg ¼ fS1; S2; S3; S4; S5; S6; S8; S10gT

¼ f0:00066;�0:00343; 0;�0:00699; 0; 0:05317; 0:03901; 0gT

The results obtained in the subsequent iterations are as follows:

Iteration 3:

fSg ¼ fS1; S3; S5; S6; S8; S10gT

¼ 103 � f1:23876; 0:78202; 0; 0:00006;�0:78198;�1:23879gT

Iteration 4:

fSg ¼ fS1; S3; S5; S6gT ¼ 103 � f0:00001; 0:00003;�5:37444; 5:37447gT

Table 3

Comparison of third natural frequency parameter for cantilever beam with two cracks (uncracked beam pL ¼ 7:85475{7.53401}a)

a2=h b2

0.11 0.2 0.4 0.6 0.8

b1 ¼ 0:1, a1=h ¼ 0:1 pL ¼ 7:8505{7.5304}

0.1 7.8488 7.8373{7.5200} 7.8322{7.5185} 7.8278{7.5124} 7.8199

0.3 7.8385 7.7465{7.4343} 7.7150{7.4259} 7.6833{7.3865} 7.5853

0.5 7.8254 7.5927{7.1331} 7.5455{7.1845} 7.4775{7.1281} 7.0918

0.7 7.8162 7.4496 7.4125 7.3197 6.5402

b1 ¼ 0:1, a1=h ¼ 0:3 pL ¼ 7:8266{7.5095}

0.1 7.8259 7.8112{7.4994} 7.8083{7.4995} 7.8032{7.4900} 7.7958

0.3 7.8214 7.7025{7.4173} 7.6914{7.4167} 7.7427{7.3515} 7.5596

0.5 7.8148 7.5073{7.1167} 7.5230{7.1832} 7.6542{7.0724} 7.0623

0.7 7.8094 7.3133 7.3914 7.4408 6.5023

b1 ¼ 0:1, a1=h ¼ 0:5 pL ¼ 7:7987{7.4306}

0.1 7.7986 7.7805 7.7804 7.7745 7.7676

0.3 7.7981 7.6480{7.3651} 7.6641{7.3858} 7.6200 7.5298

0.5 7.7971 7.3926{7.0910} 7.4971{7.1746} 7.3978 7.0281

0.7 7.7960 7.1168 7.3672 7.2258 6.4584

b1 ¼ 0:1, a1=h ¼ 0:7 pL ¼ 7:7804

0.1 7.7804 7.7603 7.7622 7.7557 7.7492

0.3 7.7804 7.6107 7.6463 7.5977 7.5104

0.5 7.7803 7.3079 7.4804 7.3696 7.0060

0.7 7.7801 6.9620 7.3517 7.1924 6.4301

aData quoted within { } are by present FE analysis.


Iteration 5:

fSg ¼ fS1; S3; S6gT ¼ f0:00941; 0:031318; 0:050402gT

Fig. 2. Flow chart for inverse problem.

Fig. 3. Simply supported beam containing two cracks and its segmentation for analysis.


All the three damage parameters are greater than zero. This indicates that the beam contains three cracks.

That is, the segments 1, 3 and 6 have cracks.

To predict the crack in segment 1, the set of damage parameter fS1; S3; S6g ¼ ½0:00941; 0; 0 is considered.This is used to calculate Dx=x for the first three natural frequencies from Eq. (24):

ðiÞ Dx1

x1

¼ 0:00304%;Dx2

x2

¼ 0:011445%; andDx3

x3

¼ 0:02332%

The changes in the natural frequencies are of negligible order. This can be interpreted that the segment 1

does not have any crack.

To predict the crack in segment 3, the damage parameter set fS1; S3; S6g ¼ f0; 0:031318; 0g is considered

and the changes Dx=x of the first three natural frequencies are again calculated from Eq. (24):

ðiiÞ Dx1

x1

¼ 0:156587%;Dx2

x2

¼ 0:303075%; andDx3

x3

¼ 0:156588%

These changes are significant. They indicate the existence of a crack in the segment 3. Similarly the last set

fS1; S3; S6g ¼ f0; 0; 0:050402g is considered to locate crack in segment 6. The changes Dx=x in the firstthree natural frequencies are obtained.

ðiiiÞ Dx1

x1

¼ 0:48776%;Dx2

x2

¼ 0:06128%; andDx3

x3

¼ 0:379159%

These changes too are significant and they indicate that element 6 contains a crack.

In order to predict the exact location and size of the crack in segment 3, the frequency changes given by

set (ii) is used to plot the variation of rotational spring stiffness with crack location b over the whole beam

assuming existence of a single crack. To save labour, the variations can be plotted over segment 3 alone.

The variation is plotted for each frequency. Typical K vs. b plots are shown in Fig. 4. Thereby the crack

location, given by the intersection of the three curves, and its size, are obtained. The same is repeated for the

set (iii). The results so obtained are given in Table 5.

The above analysis has also been done using the symbolic computation through symbolic toolboxavailable in the MATLAB. The processing time is 158 s. The processing time by the proposed scheme, i.e.

without the symbolic toolbox, is just 2.5 s.

The similar results for cantilever beams with two cracks, and separate set of geometric and material data,

are presented in Tables 6 and 7.

5.3. Beams on elastic foundation

It has been reported [64] that, at least for cases where natural frequencies of a beam increase due to afoundation, change in a frequency corresponding to a change from without crack to with crack configu-

Table 4

Natural frequencies of simply supported beam with two cracks

Case no. Crack location and size Natural frequencies (Hz)

b1 a1=h b2 a2=h x1 x2 x3 x4 x5

1 Uncracked beam 59.007 236.029 531.065 944.116 1475.182

2 0.25 0.07971 0.45 0.0986 58.625 235.142 528.096 942.515 1469.103

E ¼ 2:8� 1010 N/m2, q ¼ 2350 kg/m3, L ¼ 10:0 m, h ¼ 0:60 m and B ¼ 0:2 m.


rations demonstrates typicality. For a given set of end conditions, this change can be computed first by

considering the foundation and then by neglecting it. The results obtained by the two approaches are very

close and can be considered to be the same. This fact can be exploited in simplifying the approach for crack

detection in such beams. At least numerical computation for obtaining natural frequencies can be done

without invoking the elastic foundation. This has been done in the case studies presented here. Tables 8 and9 shows result for two cracks in a simply supported beam on elastic foundation (Fig. 5).

Fig. 4. Plot of K vs. b curves for simply supported beam.

Table 5

Comparison of actual and predicted crack location and size for simply supported with two cracks of Table 4

Actual data Natural frequencies (Hz) Predicted data

Location b a=h x1 x2 x3 Location b % error Stiffness K a=h % error

Uncracked beam 59.007 236.029 531.065

Crack in 3rd element

0.25 0.07971 58.915 235.314 530.233 0.250 0.0(0.5)a 320.714 0.069 )1.071(0.231)

Crack in 6th element

0.45 0.0986 58.719 235.884 529.051 0.443 )0.7()0.7) 195.631 0.089 )0.96()0.01)a Errors quoted in Ref. [54].


5.4. Beams on multiple pin supports

For a beam on number of supports (more than 2), the mode shape for each segment has to be given

separately. For example, for a beam on three pin supports (Fig. 6), the mode shape for the segments 1 to 2

and 3 to 4 can be written in terms of Z, h,M and V at their left ends. That is, using Eqs. (6) and (9), noting

that Z1 ¼ 0, M1 ¼ 0 and Z3 ¼ 0,

Table 7

Comparison of actual and predicted crack location and size for cantilever beams of Table 6

Case no. Actual data Natural frequencies (Hz) Predicted data

b a=h x1 x2 x3 b % error a=h % error

2 0.1 0.15 278.649 1779.577 5022.165 0.067 )3.3 0.17017 2.0

0.4 0.3 284.462 1752.797 5000.407 0.417 1.7 0.24004 )6.0

3 0.1 0.35 252.719 1778.804 5042.086 0.138 3.8 0.35893 0.893

0.4 0.15 286.500 1757.798 5057.842 0.453 5.3 0.2109 6.09

4 0.1 0.25 269.239 1736.110 4939.163 0.047 )5.3 0.22921 )2.0790.5 0.25 286.954 1772.268 5052.409 0.438 )6.2 0.18761 )6.239

5 0.15 0.40 252.688 1778.773 5042.048 0.138 )1.2 0.35908 )4.0920.45 0.5 273.482 1533.036 4945.268 0.477 2.7 0.46802 )3.198

6 0.2 0.3 272.283 1809.357 4857.243 0.259 5.9 0.29970 )0.030.5 0.4 277.939 1609.985 4983.808 0.469 )3.1 0.39939 )0.06

7 0.2 0.5 239.561 1780.849 4482.823 0.2558 5.58 0.50369 0.369

0.5 0.4 273.940 1629.290 4832.764 0.428 )7.2 0.40967 0.967

8 0.25 0.25 277.064 1811.410 4919.637 0.259 0.9 0.25543 0.054

0.55 0.3 286.805 1695.515 5017.985 0.561 1.1 0.28921 )1.079

9 0.25 0.2 289.919 1815.800 5067.216 0.184 )6.6 0.14268 )5.7320.55 0.45 284.154 1592.381 4958.972 0.575 2.5 0.40052 )4.948

10 0.35 0.35 272.793 1731.028 4693.480 0.344 )0.6 0.3828 3.28

0.6 0.5 283.083 1550.699 4935.122 0.582 )1.8 0.43993 )6.007

Table 6

Natural frequencies obtained by FE analysis for cantilever beam with two cracks


b1 a1=h b2 a2=h x1 x2 x3 x4 x5

1 Uncracked beam Analytical 289.925 1816.931 5087.459 9969.389 16480.12

FEM 288.670 1751.346 4680.559 8641.52 13370.73

2 0.1 0.15 0.4 0.3 271.931 1649.858 4545.492 8513.229 12708.17

3 0.1 0.35 0.4 0.15 248.039 1654.086 4611.596 8581.089 12927.16

4 0.1 0.25 0.5 0.25 265.013 1625.863 4660.053 8296.765 13811.38

5 0.15 0.40 0.45 0.5 229.482 1405.505 4445.952 7336.405 12451.00

6 0.2 0.3 0.5 0.4 259.042 1536.825 4558.170 7527.449 13005.59

7 0.2 0.5 0.5 0.4 222.321 1527.623 4197.771 7122.095 12681.79

8 0.25 0.25 0.55 0.3 272.689 1624.409 4463.050 8064.036 13511.55

9 0.25 0.2 0.55 0.45 271.594 1476.876 4435.421 7866.328 12674.92

10 0.35 0.35 0.6 0.5 264.696 1399.212 3902.912 8407.615 11214.34

E ¼ 210 GPa, q ¼ 7860 kg/m3, h ¼ 0:020 m, B ¼ 0:012 m and L ¼ 0:24 m.


Z1ðxÞ ¼h1

2½sinðpxÞ þ sinhðpxÞ � V1

2EIp3½sinðpxÞ � sinhðpxÞ; 06 x6 L1 ð25Þ

Z2ðxÞ ¼ � M3

2EIp2½cosðpxÞ � coshðpxÞ þ h3

2p½sinðpxÞ þ sinhðpxÞ � V3

2EIp3½sinðpxÞ � sinhðpxÞ; L1 6 x6 L

ð26Þ

Table 8

Natural frequencies obtained by FE analysis for beams with two cracks on elastic foundations and end supports


b1 a1=h b2 a2=h x1 x2 x3 x4 x5


FEM 49.039 187.557 419.573 742.124 1152.584

2 0.2 0.15 0.5 0.15 48.595 186.746 415.921 740.932 1147.389

3 0.2 0.15 0.5 0.25 48.246 186.741 412.682 740.917 1138.838

4 0.2 0.15 0.5 0.35 47.671 186.736 407.572 740.889 1125.764

5 0.25 0.25 0.5 0.40 47.072 185.181 403.079 741.997 1110.963

6 0.30 0.35 0.5 0.35 47.021 183.304 407.960 735.612 1099.087

7 0.32 0.40 0.62 0.28 47.181 180.877 418.065 716.950 1119.032

8 0.4 0.25 0.65 0.25 47.896 185.152 417.625 725.847 1145.719

9 0.4 0.45 0.65 0.45 45.624 178.577 412.901 686.454 1130.309

10 0.45 0.25 0.65 0.5 46.215 180.247 415.006 702.017 1118.332

E ¼ 210 GPa, q ¼ 7860 kg/m3, h ¼ 0:02 m, B ¼ 0:012 m and L ¼ 0:24 m.

Table 9

Comparison of actual and predicted crack location and size for beams of Table 8


Location b a=h x1 x2 x3 Location b % error a=h % error

2 0.2 0.15 46.817 187.049 421.359 0.256 5.6 0.10749 )4.2510.5 0.15 46.506 187.321 419.301 0.443 )5.7 0.19898 4.898

3 0.2 0.15 46.818 187.060 421.372 0.256 5.6 0.10607 )4.3930.5 0.25 46.141 187.137 416.746 0.443 )5.7 0.27926 2.926

4 0.2 0.15 46.867 187.368 421.253 0.117 )8.3 0.08738 )6.2620.5 0.35 45.517 186.824 412.385 0.444 )5.6 0.37736 2.736

5 0.25 0.25 46.766 186.653 420.898 0.255 0.5 0.14862 )10.130.5 0.4 45.059 186.594 409.183 0.443 )5.7 0.42309 2.309

6 0.3 0.35 46.428 184.043 417.864 0.251 )4.9 0.29998 )5.0020.5 0.35 45.305 186.718 410.902 0.443 )5.7 0.39665 4.665

7 0.32 0.40 45.599 181.119 416.615 0.303 )1.7 0.42193 2.193

0.62 0.28 46.295 187.215 417.825 0.557 )6.3 0.24915 )3.085

8 0.4 0.25 46.395 187.265 418.526 0.443 4.3 0.22692 )2.3080.65 0.25 46.214 185.344 421.198 0.651 0.1 0.29262 4.262

9 0.4 0.45 46.088 187.111 416.380 0.443 4.3 0.2886 )16.140.65 0.45 44.249 178.923 419.139 0.657 0.7 0.54059 9.059

10 0.45 0.25 46.183 187.158 417.040 0.443 )0.7 0.27147 2.147

0.65 0.5 44.745 180.545 419.660 0.656 0.6 0.4928 )0.720


Using the transfer matrix of the first segment, and Z1 ¼ 0 and M1 ¼ 0 at left hand support,

ZhMV

2664

3775

2

¼

A1 B1

C1

EID1

EI

p4D1 A1

B1

EIC1

EIEIp4C1 EIp4D1 A1 B1

EIp4B1 EIp4C1 p4D1 A1

266666664

377777775

ZhMV

2664

3775

1

¼

B1

D1

EI

A1

C1

EIEIp4D1 B1

EIp4C1 A1

266666664

377777775

hV

� �1

ð27Þ

ZhMV

2664

3775

3

¼

B1

D1

EI0

A1

C1

EI0

EIp4D1 B1 0

EIp4C1 A1 1

266666664

377777775

hVR

24

35

1

ð28Þ

and Z3 ¼ 0, V1 ¼ �ðB1=ðD1=EIÞÞh1. R is the reaction at the intermediate support. The constants A1;B1; . . .are obtained from Eq. (7) by putting li ¼ L1. Similarly, using the transfer matrix of second segment and Eq.(28)

ZhMV

2664

3775

4

¼

A2 B2

C2

EID2

EI

p4D2 A2

B2

EIC2

EIEIp4C2 EIp4D2 A2 B2

EIp4B2 EIp4C2 p4D2 A2

266666664

377777775

B1

D1

EI0

A1

C1

EI0

EIp4D1 B1 0

EIp4C1 A1 1

266666664

377777775

hVR

24

35

1

¼

D11 D12

D2

EI

D21 D22

C2

EID31 D32 B2

D41 D42 A2

266666664

377777775

hVR

24

35

1

and Z4 ¼ 0, M4 ¼ 0,

R ¼ � D11

�� D12

B1

ðD1=EIÞ

�1

ðD2=EIÞ

�h1

Fig. 6. Beam on three pin supports.

Fig. 5. Beam on continuous elastic foundation.


The constant A2;B2; . . . are obtained from Eq. (7) by putting li ¼ L2. Thus all the parameters (V1; h3;M3; V3are R) are available in terms of h1. Hence the mode shapes involves only one arbitrary constant h1. It can be

assigned any value, e.g. unity for convenience. From the mode shapes gnðxÞ is obtained. In turn the co-

efficients of the linear simultaneous equation (24) are obtained.The crack location and size has been predicted following exactly the steps given for the simply supported

beam earlier. The results for a uniform beam on three pin supports are presented in Tables 10 and 11. The

whole beam has again been divided into 10 segments (i.e. five segments per span). The error in the pre-

diction of crack location is less than 10% and it is less than 15% in the case of prediction of crack size.

Table 10

Natural frequencies obtained by FE analysis for beams with two cracks on three pin supports


b1 a1=h b2 a2=h x1 x2 x3 x4 x5


FEM 516.916 798.657 2022.936 2510.160 4400.137

2 0.25 0.25 0.75 0.15 502.730 784.273 2022.736 2502.486 4288.776

3 0.25 0.25 0.75 0.25 496.523 777.6449 2022.702 2499.195 4245.431

4 0.25 0.25 0.75 0.35 486.858 768.4461 2022.521 2494.133 4176.365

5 0.25 0.25 0.75 0.50 464.7114 751.3703 2021.883 2483.061 4023.732

6 0.30 0.15 0.60 0.25 509.704 795.208 1983.040 2488.788 4314.053

7 0.35 0.25 0.60 0.30 504.645 795.861 1940.238 2461.339 4287.657

8 0.40 0.35 0.70 0.35 490.712 786.783 1934.542 2431.647 4211.034

9 0.45 0.15 0.65 0.15 513.937 794.137 2003.986 2493.895 4377.027

E ¼ 210 GPa, q ¼ 7860 kg/m3, h ¼ 0:02 m, B ¼ 0:012 m and Lspan ¼ 0:3 m.

Table 11

Comparison of actual and predicted crack location and size for beams of Table 10


Location b a=h x1 x2 x3 Location b % error a=h % error

2 0.25 0.25 512.536 806.890 2079.232 0.2705 2.05 0.21935 )3.0650.75 0.15 514.983 806.069 2080.462 0.7812 3.12 0.18520 3.52

3 0.25 0.25 509.595 804.491 2077.754 0.269 1.9 0.25487 0.487

0.75 0.25 511.717 801.839 2078.821 0.7855 3.55 0.23239 )1.761

4 0.25 0.25 508.970 798.280 2077.440 0.211 )3.9 0.26558 1.558

0.75 0.35 502.678 798.851 2074.278 0.735 )1.5 0.32000 )3.0

5 0.25 0.25 511.020 800.935 2078.470 0.2135 )3.65 0.24121 )0.8790.75 0.50 478.481 779.121 2062.117 0.7495 )0.05 0.47113 )2.887

6 0.30 0.15 518.217 811.522 2082.087 0.2735 )2.65 0.12174 )2.8260.60 0.25 516.704 813.347 2051.294 0.652 5.2 0.18844 )6.156

7 0.35 0.25 518.952 812.122 2082.457 0.274 )7.6 0.10279 )14.720.60 0.30 510.680 812.875 2004.652 0.6165 1.65 0.35617 5.617

8 0.40 0.35 507.973 812.664 1983.701 0.3835 )1.65 0.39982 4.982

0.70 0.35 507.528 802.806 2076.715 0.733 3.3 0.27493 )7.507

9 0.45 0.15 520.667 813.164 2081.147 0.426 )2.4 0.06924 )8.0760.65 0.15 520.223 813.622 2078.539 0.617 )3.3 0.08802 )6.198


6. Discussion and conclusions

A technique, combining the TMM and representation of an open edge crack oriented normally to the

axis by a rotational spring, has been proposed to solve the problem of detection of multiple cracks in beamswith varying end conditions (fixed and simple supports) and intermediate support conditions (pin support

and elastic foundation). Smaller size of the characteristic equation that is typical of the TMM is fully

exploited. The steps to obtain the mode shapes for multi-segment uncracked beam which are crucial for

obtaining the function gnðxÞ associated with a span are given. This has helped to extend the scope of the

approximate method given by Hu and Liang [54] for a single segment beam to multi-segment beams.

Additionally, this has helped to eliminate the symbolic computation [53] to determine the uncracked beam

mode shapes. Thereby the problem solving time reduces to an enormous extent. The method gives a clear

insight into the whole analysis. The accuracy of results in the inverse, or crack detection, problem is good.The maximum difference in the prediction of location is less than 10% and the difference in the size is less

than 15%. The maximum difference of results in the forward, or natural frequency determination, problems,

which are solved in the passing, is less than 5% with the published and/or finite element data. The method

however suffers from one important limitation. That is, the maximum number of cracks that can be de-

tected is less than equal to the number of segments into which the beam is virtually divided. This limitation

is inherent in the approximate approach given by Liang et al. [53]. The other shortcomings emanate from

the assumptions; the method cannot be applied to beams where rotational inertia, shear deformation and

damping effects are significant.The present procedure is based on the fact that the effect of a damage parameter extends over a segment

length, which is dictated by the analyst. There is a need to see how this length can be rationally decided.

There is also a scope for examining the limitation on the number of cracks that can be handled in a multi-

crack problem.

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Detection of Multiple Cracks Using Frequency Measurements 2003 Engineering Fracture Mechanics

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