Dynamic Composite Cracked Element Model

8/11/2019 Dynamic Composite Cracked Element Model

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Mechanical Systemsand

Signal ProcessingMechanical Systems and Signal Processing 23 (2009) 415431

Dynamic condition assessment of a cracked beam with thecomposite element model

Z.R. Lu a , S.S. Law b,

a School of Engineering, Sun Yat-sen University, Guangzhou, Guangdong Province, Peoples Republic of Chinab Civil and Structural Engineering Department, Hong Kong Polytechnic University, Yuk Choi Road, Hunghom, Kowloon, Hong Kong

Received 5 September 2007; received in revised form 16 February 2008; accepted 24 February 2008Available online 29 February 2008

Abstract

Existing models of local damage in a beam element are usually formulated as a damage in a single element,and the coupling effect between adjacent damages is simply ignored. This coupling effect is larger in the case of a nemesh of nite elements or when there is a high density of damage in the structure. This paper studies such effect frommultiple cracks in a nite element in the dynamic analysis and local damage identication. The nite beam element isformulated using the composite element method [P. Zeng, Composite element method for vibration analysis of structure,Journal of Sound and Vibration 218 (1998) 619696] with a one-memberone-element conguration with cracks where theinteraction effect between cracks in the same element is automatically included. The accuracy and convergence speed of the proposed model in computation are compared with existing models and experimental results. The parameter of theChristides and Barr [One dimensional theory of cracked BernoulliEuler beams, International Journal of MechanicalScience 26 (1984) 639648] crack model is found needing adjustment with the use of the proposed model. The responsesensitivity-based approach of damage identication is then applied in the identication of single and multiple crackdamages with both simulated and experimental data. Results obtained are found very accurate even under noisyenvironment.r 2008 Elsevier Ltd. All rights reserved.

Keywords: Composite element; Crack; Response sensitivity; Coupling; Damage; Identication

1. Introduction

The effect of cracks on the dynamic behavior of structures has been a subject of active research inmechanical, aeronautic and civil engineering for decades. Numerous crack models for a cracked beamcan be found in the literature. The simplest one is a reduced stiffness (or increased exibility) in a niteelement to simulate a small crack in the element [35]. Another simple approach is to divide the crackedbeam into two-beam segments joined by a rotational spring that represents the cracked section [68].This spring hinge model, combined with fracture mechanics, has been most popular amongst researchers.An improved version of this model [9] leads to a closed-form solution giving the natural frequencies and

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0888-3270/$- see front matter r 2008 Elsevier Ltd. All rights reserved.doi: 10.1016/j.ymssp.2008.02.009

Corresponding author. Tel.: +852 2766 6062; fax: +8522334 6389.E-mail address: [email protected] (S.S. Law).
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mode shapes of the cracked beam directly. Other researchers [1012] solved the differential equations withcompatible boundary conditions satisfying the crack conditions. Also Krawczuk and Ostachowicz [13] andLee and Chung [14] have developed the exibility matrix for a beam element with a crack using the energymethod.

The above models do not give a clear description on the crack and its effect on its adjacent material.

They are either localized at a cross-section or averaged over an entire nite element. This problem wasstudied by Christides and Barr [2] who developed the one-dimensional vibration theory for the lateralvibration of a cracked EulerBernoulli beam with one or more pairs of symmetric cracks. The perturbation inthe stress induced by the crack is incorporated through a local function, which has assumed an exponentialdecay with distance from the crack, and the parameter involved in the function was determined fromexperiments. This cracked beam vibration theory was used for the prediction of the fundamental naturalfrequency of a simply supported beam with transverse open cracks in the middle of the beam. The naturalfrequency of the cracked beam is obtained by adopting a two-term trial function in the RayleighRitz method.Shen and Pierre [15] reconsidered the model of Christides and Barr through a Galerkin procedure [16] in whichthe deection of the cracked beam was expanded in a series of functions. However, the convergence is veryslow for this type of problem, and a technique is proposed to increase the convergence speed by adding asupplementary function to a classical set of suitable innitely differentiable co-ordinate functions. Later,Messina [17] revisited the Galerkin procedure using the global piece-wise smooth functions. It is reported thatthe convergence speed of Galerkin procedure can be more than doubled as compared with that based on theSP technique [15].

However, the coupling effect of adjacent cracks/damages has not been included in the abovestudies, and this problem would be very acute with many closely spaced cracks or when the member ismodeled with a ne mesh of nite elements. Since existing nite element conguration cannot accommodatethe interaction zones of individual crack with its discretized element, an open crack model [2] isincluded in a composite element [1] in the one-memberone-element nite element model of the structure.The element shape function is retained to dene displacement at any location of the element. The interactioneffect between cracks in the element is automatically included. Free vibration analysis is conductedon the cracked beam model and results are compared with those from other existing methods and from

experiments. The dynamic response sensitivities with respect to the crack parameters are used in thesensitivity-based model updating analysis for the condition assessment. Results from two numericalsimulations and experimental study show that the proposed method is accurate in identifying the cracks evenunder noisy condition.

2. Theory

Fig. 1 shows a simple beam with multiple transverse open cracks along its length. The cracks are assumed tohave uniform depth across the width of the beam, and they do not change the mass of the beam. The model onthe crack is briey introduced before it is incorporated into the composite element to obtain the governingequation for the vibration analysis of the cracked beam.

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1 x 1 x p

x j

hc1 hcphcj

w

d

x L

Fig. 1. Beam with multiple cracks.

Z.R. Lu, S.S. Law / Mechanical Systems and Signal Processing 23 (2009) 415431416


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2.1. Crack model

According to Christides and Barr [2], the variation of bending stiffness EI along the beam length takes upthe form of

EI x EI 0

1 m 1exp 2ajx xcj=d (1)

where E is Youngs modulus of the beam, I 0 wd 3 /12 is the second moment of area of the intact beam,

m 1/(1 C r)3 , C r d c/d is the crack depth ratio and d c and d are the depth of crack and the beam,

respectively, x c is the location of the crack. a is a constant, which governs the rate of decay and it varies withthe crack geometry and material type. Fig. 2 shows the variations of the bending stiffness along the beamcorresponding to different a values. However, Christides and Barr reported from their experiments that thisconstant can be taken as 0.667 in general for different crack congurations. This parameter will be studiedwith the proposed method for beams with different crack depths and material types.

2.2. Composite element incorporating a crack

Composite element [1] is a relatively new tool for nite element modeling. This method is basically acombination of the conventional nite element method (FEM) and the highly precise classical theory (CT). Inthe composite element method (CEM), the displacement eld is enlarged to the sum of the nite elementdisplacement and the shape functions from the classical theory. The displacement eld in the FEM satises thenodal boundary conditions and the analytical functions are obtained from the CT observing also some specialboundary conditions. The enlarged displacement eld can be expressed as

uCEM x; t uFEM x; t uCT x ; t (2)

where uFEM (x,t) and uCT (x,t) are the two parts of the CEM displacement eld with the subscripts deningthose of the FEM and CT, respectively.

Taking a planar beam element as an example, the rst part of the enlarged displacement eld can be

expressed as the product of the shape function vector N (x) and the nodal displacement vector q,uFEM x; t N xqt (3)

where q(t) [n1(t),y1(t),n2(t),y2(t)]T and n and y represent the transverse and rotational displacements,respectively, and

N x 1 3x=L2 2x=L3; x=L 2x=L2 x=L3; 3x=L2 2x=L3; x=L3 x=L2

N 1x; N 2x; N 3x; N 4x (4)

The second part uCT (x,t) is obtained by the multiplication of analytical mode shapes with a vector of coefcient c ( also called the c degrees-of-freedom or c-coordinates),

uCT x ; t f 1xc1t f 2xc2t f N xcN t (5)

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

50

100

150

Location along the beam

E I

= 0.667 = 1.426

Fig. 2. Variation in EI with different a values.

Z.R. Lu, S.S. Law / Mechanical Systems and Signal Processing 23 (2009) 415431 417


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2.3. Convergence property of the CEM

As we know, the accuracy of the solutions of the CEM depends upon two parameters, the number of termsN in the CT and the bending stiffness decay constant a . First of all, the number of terms N is determined by afrequency convergence test [15]. The frequency convergence criterion is dened as

maxi 1;2;3

jD o N i =oN i j o tol (10)

where o i N is the estimation of the i th frequency with N -terms in the CT. D o i

N o i N o i

N 1 is the difference of the i th frequency obtained with the N -terms and ( N 1)-terms. tol is a small number and it is taken as2.0 10 5 in this study.

The parameters of the beam for this study are: Youngs modulus E 200GPa, mass density r 7850kg/m 3 , width b 0.01 m, depth d 0.01m and length L 0.2 m. The location of crack is 0.1m from theleft support. The beam is assumed to be simply supported. Table 1 shows the convergence of the rst threenatural frequencies with different number of terms in CT when the parameter a equals to 0.667 and 1.426. It isseen from the results that only 10 terms are needed to satisfy the frequency convergence criterion. Calculation

for Table 1 also shows the convergence is little dependent on the value of the parameter a which will bediscussed below. This indicates the robustness of the proposed approach with respect to the parameter a.Another comparison of the convergence property study is made with the Galerkin procedure [16] from

different number of terms in the mode functions and results are presented in Table 2 . Both methods convergewith 10 terms in the shape functions with similar modal frequencies indicating similar accuracy.

A further study is made by comparing the frequency ratio computed for the beam in [17] using the proposedCEM, the Galerkin method with piece-wise-smooth functions (GPSF) and from Shen and Pierre [15] (SP), andthe results are shown in Table 3 . The frequency ratio is dened as the ratio of natural frequency of the crackedbeam to that of the intact beam. The ratios from the CEM method are between those from GPSF and SP

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Table 1Convergence in the natural frequencies with different number of shape functions

Mode number Number of terms in the shape function

5 7 8 9 10

1 514.7/547.7 (6%) 511.7/546.3 (6.3%) 511.6/546.3 (6.4%) 509.6/544.9 (6.5%) 509.6/544.9 (6.5%)2 2262.9/2286.4 (1%) 2261.7/2286.2 (1.1%) 2260.7/2285.9 (1.1%) 2260.7/2285.7 (1.1%) 2260.7/2285.9 (1.1%)3 4785.4/4957.2 (3.5%) 4761.3/4945.6 (3.7%) 4761.3/4945.7 (3.7%) 4745.4/4935.4 (3.8%) 4745.4/4935.4 (3.8%)

Notes : / denotes values (Hz) calculated with a 0.667 and 1.426, respectively.( ) denotes the percentage difference relative to results for a 0.667.

Table 2Comparison on the convergence performance of CEM with Galerkin procedure [16]

Modenumber

Number of terms in the shape function ( a 1.426)

5 7 8 9 10

1 547.7/547.0 ( 0.13%) 546.3/545.7 ( 0.11%) 546.3/545.5 ( 0.15%) 544.9/544.1 ( 0.15%) 544.9/544.1 ( 0.15%)2 2286.4/2285.9

( 0.02%)2286.2/2285.8( 0.02%)

2285.9/2285.7( 0.01%)

2285.9/2285.7( 0.01%)

2285.9/2285.6( 0.01%)

3 4957.2/4952.2( 0.10%)

4945.6/4940.1( 0.11%)

4945.7/4940.1( 0.11%)

4935.4/4929.3( 0.12%)

4935.4/4929.3( 0.12%)

Notes : / denotes frequencies (Hz) calculated from CEM and Galerkin procedure, respectively.( ) denotes the percentage change relative to CEM results.



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when the same number of terms in the shape function is used, and they are closer to those from GPSFindicating the correctness of the proposed method.

2.4. Determination of the parameter a

Parameter a has been shown in Fig. 2, governing the distribution of the EI along the beam. The affectedregion is large for a small a and vice versa. The parameter a is valid for a crack at any location of thebeam. It is obtained by tting the fundamental natural frequency calculated by CEM to best match theanalytical one from Saez et al. [9]. The parameters of the beam are the same as for the last study andthe crack is assumed to be in the middle span of the beam. The crack depth ratio changes from 0.1 to 0.9 withincrement of 0.1, and the vector of calculated fundamental frequencies are used in the tting using penaltymethod [19]. The parameter a is found to be 1.426 after 12 iterations. In the calculation, 10 terms in theshape functions are used in CEM for the vibration analysis. The drop in the fundamental natural frequency interms of crack depth is shown in Fig. 3 (a). This gure indicates that the agreement between the two methods is

very good when a equals to 1.426. This value of a is further checked when the crack is at 3 L/4 or L /10 of thebeam. The agreement is excellent as shown in Figs. 3 (b) and (c). The matching of the two curves for the latteris excellent for C r in the range of 0.10.8, but the difference increases when C r is larger than 0.8. The aboveresults show that the proposed model predicts the dynamic behavior very well when the crack depth ratio isless than 0.8.

The variations in the ratio of the fundamental natural frequency obtained from a 0.667 [2] arealso shown in the gures, and Christides and Barr is found to underestimate the solutions with thecrack ratios over the range studied. Therefore, a is taken as 1.426 in the proposed composite element crackmodel.

A major source of error with the proposed model is the use of analytical mode shapes in Eq. (5), while therewould be some changes in the mode shapes with damage. This error is investigated as follows. Fig. 4 shows the

normalized mode shape at the fundamental frequency of the beam from the CEM and the analytical one fromSaez et al. [9] with the crack locates at mid-span of the beam and with a crack depth ratio of 0.3 and 0.5. Themode shapes match each other very well with a better matching at the crack location by the proposed model.Results not shown with the crack at 0.1 L of the beam indicate that the mode shape does not change much witha crack depth ratio of 0.6. These indicate that the use of analytical mode shape with the proposed model couldbe acceptable when the crack depth ratio is not exceeding 0.5.

2.5. Forced vibration analysis

The equation of motion of the forced vibration of a cracked beam with n cracks when expressed in terms of the CET method is

M Q C _Q K xL 1 ;d c1 ; . . . ;xL i ;d ci ; . . . xLn ;d cn Q f t (11)

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Table 3Comparison on the converged frequency ratio

Number of terms in theshape function

GPSFs SP CEM

1stmode

2ndmode

3rdmode

1stmode

2ndmode

3rdmode

1stmode

2ndmode

3rdmode

9 0.9433 0.9997 0.9493 0.9443 0.9997 0.9497 0.9547 0.9996 0.958811 0.9354 0.9997 0.9424 0.9388 0.9997 0.9452 0.9513 0.9996 0.956019 0.9165 0.9997 0.9275 0.9239 0.9997 0.9333 0.9370 0.9996 0.944129 0.9130 0.9996 0.9248 0.9164 0.9997 0.9275 0.9181 0.9996 0.929031 0.9129 0.9996 0.9248 0.9157 0.9997 0.9269 0.9140 0.9996 0.926033 0.9129 0.9996 0.9248 0.9152 0.9997 0.9265 0.9139 0.9996 0.9260



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where M and K are the system mass and stiffness matrices, which are the same as those shown in Eq. (9), C isthe damping matrix which represents a Rayleigh damping model in this work as

C a1M a2K xL 1 ;d c1 ; . . . ;xL i ;d ci ; . . . xLn ;d cn (12)

where a1 and a2 are constants to be determined from two modal damping ratios. f (t) is the genera-lized force vector. For an external force F (t) acting at the location xF from the left support, f (t) can beexpressed as

f t N 1xF ; N 2xF ; N 3xF ; N 4xF ; f 1xF ; . . . f nxF T F t (13)

The generalized acceleration Q , velocity Q and displacement Q of the cracked beam can be obtained fromEq. (11) by direct integration method. The physical acceleration u (x,t) is obtained from

ux ; t S xT Q (14)

The physical velocity and displacement can be obtained in a similar way.

2.6. Dynamic response sensitivity with respect to the crack parameters

Taking partial derivative of Eq. (11) with respect to the crack location xL i of the i th crack, we have

M q Qq xL i

C q _Qq xL i

K xL 1 ;d c1 ; . . . ;xL i ;d ci ; . . . xLn ;d cn q Qq xL i

a2q K xL 1;d c1 ; . . . ;xL i ;d ci ; . . . xL n ;d cn

q xL i _Q

q K xL 1 ;d c1 ; . . . ;xL i ;d ci ; . . . xLn ;d cn

q xL i Q (15)

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0.1 0.2 0.3

crack at 0.5L

crack at 0.75L

crack at 0.1L

0.4 0.5 0.6 0.7 0.8 0.90

0.5

1

c r a c k

/ 0

Ref. [8] = 1.426

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.2

0.4

0.6

0.8

1

c r a c k

/ 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.2

0.4

0.6

0.8

1

crack depth ratio

c r a c k

/ 0

= 0.667

Ref. [8] = 1.426 = 0.667

Ref. [8] = 1.426 = 0.667

Fig. 3. Comparison on frequency ratio corresponding to different crack depth ratios.



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whereq K xL 1 ;d c1 ; :::;xL i ;d ci ; :::xL n;d cn

q xL i

Z L

0

d 2S T

dx 2q

q xL i

EI 01 1=1 d ci =d

3 1exp 2ajx xL i j=d " #d 2S

dx 2dx (16)

Since the dynamic response has been obtained from Eq. (11) and the right-hand side of Eq. (16) can beobtained from numerical integration, the right-hand side of Eq. (15) can be obtained. Note that Eq. (15) issimilar to Eq. (11), and the dynamic response sensitivity (i.e., the generalized acceleration response sensitivity,velocity response sensitivity and displacement response sensitivity) with respect to the location of the i th crackcan be obtained by direct integration again.

Similarly, the dynamic response sensitivity with respect to the depth of the i th crack can be obtained fromthe following equation:

M q Qq d ci

C q _Qq d ci

K xL 1;d c1 ; . . . ;xL i ;d ci ; . . . xLn ;d cn qQq d ci

a2q K xL1 ;d c1 ; . . . ;xL i ;d ci ; . . . xL n;d cn

q d ci _Q

q K xL1 ;d c1 ; . . . ;xL i ;d ci ; . . . xL n;d cn q d ci

Q

(17)

whereq K xL 1 ;d c1 ; . . . ;xL i ;d ci ; . . . xLn ;d cn

q d ci

Z L

0

d 2S T

dx2

q

qd ci

EI 0

1 1=1 d ci =d 3

1exp 2ajx xL i j=d " #d 2S

dx2 dx (18)

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

1

2

3

4

N o r m a

l i z e d m o

d e s h a p e

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

1

2

3

4

Location along the beam (m)

N o r m a

l i z e

d m o

d e s h a p e

uncrackedproposedref. [8]

crack depth ratio = 0.3

crack depth ratio = 0.5

uncrackedproposedref. [8]

Fig. 4. Comparison of the fundamental mode shape.



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2.7. Identication of the crack location and depth

The vector of parameter to be updated is W [X d c]T , where X xL 1 ;xL2 ; . . . ;xL n and d c d c1 ;d c2 ; . . . ;d cn are the vectors of locations and the crack depths of the n cracks, respectively. In theidentication, one or more generalized responses are used. The measured response is simulated by adding

different levels of articial normally distributed random noise into the one calculated from Eq. (11).The sensitivity-based approach basing on the measured acceleration responses [20] is adopted for theupdating of the vector of parameters W,

d Q P dW (19)

where dW is the vector of perturbations in the updating parameters, d Q Q Q is the differences in the vector

of polluted-generalized acceleration Qand the vector of calculated acceleration Q . Matrix P consists of the

response sensitivity, which is the rst derivative of the dynamic response with respect to the updatingparameters. These derivatives are calculated from Eqs. (15) and (17). For example, when the i th generalizedacceleration is used, the elements in the sensitivity matrix are shown as

P

q Qi t1

q xL1

q Qi t1

q xL2

q Qi t1

q xL n

q Qi t1

q d c1

q Qi t1

q d c2

q Qi t1

q d cnq Q

i t2

q xL1

q Qi t2

q xL2

q Qi t2

q xL n

q Qi t2

q d c1

q Qi t2

q d c2

q Qi t2

q d cn... ..

. ..

. ... ..

. ..

.

q Qi t f

q xL1

q Qi t f

q xL2

q Qi t f

q xL n

q Qi t f

q d c1

q Qi t f

q d c2

q Qi t f

q d cn

266666666666664

377777777777775

(20)

where t1 is the beginning of the time history and t f is the end of the time history. Matrix P has the dimensionsof t f 2n. Eq. (19) can be solved with the least-squares method directly as

dW P T

P 1

P T

d Q (21)

Like many other inverse problems, Eq. (19) is an ill-conditioned problem. In order to provide bounds to thesolution, the damped-least-squares method (DLS) [21] is used in the pseudo-inverse calculation. Eq. (19) canbe written in the following form in the DLS method

dW P T P l I 1P T d Q (22)

where l is the non-negative damping coefcient governing the participation of least-squares error in thesolution. When the parameter l approaches zero, the estimated vector dW approaches to the solution obtainedfrom the simple least-squares method from Eq. (21). L-curve method [22] is used in this paper to obtain theoptimal-regularization parameter l .

The proposed method is considered capable to identify the location of the crack whether to the left or to the

right of the mid-point of the beam, as the transmissibility between a crack and the measurement point will bedifferent for different locations. The only exception would be when the crack damage is symmetrical to the mid-point of the beam and the response is obtained at the mid-point when the identied result would not be unique.

3. Applications

3.1. Comparision with existing models and experimental results

Several experimental works in Sinha et al. [18] and Lee and Chung [14] are restudied in the following sectionand results are compared with those obtained from the CEM model. They are

Case 1An aluminum cantilever beam with one crack. Case 2An aluminum freefree supported beam with one crack.

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Case 3A steel freefree supported beam with one crack. Case 4An aluminum freefree supported beam with two cracks.

The geometric and material properties and boundary conditions of the beams are given in Table 4 . Thebeam is discretized into one element and ten shape functions are used in the calculation with the total degrees-of-freedom in the CEM equals 14, while the total degrees-of-freedom in the nite element model is 34 and 56[18] for the beam in Cases 1 and 2, respectively. The beam in Case 3 has been discretized [18] into 21 elementsand the total DOFs is 42. The crack depth in the three beams varies in three stages of 4, 8 and 12 mm. Thecomparison of the predicted natural frequencies of the beam from the proposed model and those in Lee andChung [14] and Sinha et al. [18] and the experimental results are shown in Tables 57 . The proposed model, ingeneral, gives better results than the model in Sinha et al. since the latter crack model is a linear approximationof the theoretical crack model of Christides and Barr. The results from Lee and Chung are seen to becomparable with those from the proposed model.

In Case 4, the beam in Case 2 is tested again with a new crack introduced. The rst crack is at 595 mm fromthe left end with a xed crack depth of 12mm; while the second crack is at 800 mm from the left end with thecrack depth varying from 4 to 12 mm in step of 4 mm. Table 8 gives the rst ve natural frequencies of thebeam by CEM method and compares with those from Sinha et al. and the experimental measurement. The

results from CEM are found closer to the experimental prediction than those in Sinha et al. The abovecomparisons show that the CEM approach of modeling a beam with crack(s) is accurate for the vibrationanalysis. A yet bigger advantage of the model is the much lesser number of DOFs in the resulting niteelement model of the structure.

3.2. Dynamic response and response sensitivity with respect to crack parameters

The parameters of the beam under study are: E 200GPa, b 25 mm, d 20 mm, and L 2000mm. Themass density is r 7850 kg/m 3 . The beam is simply support and with a 3-mm-deep crack at 700 mm from the

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Table 4Properties of beams [18] in the experimental study

Case 1 Case 2 Case 3

Boundary conditions Cantilever Freefree FreefreeMaterial Aluminum Aluminum SteelYoungs modulus 69.79 Gpa 69.79 Gpa 203.91 GpaMass density 2600 kg/m 3 2600kg/m 3 7800 kg/m 3

Poisson ratio 0.33 0.33 0.33Beam length 996 mm 1832 mm 1330 mmBeam width 50 mm 50 mm 25.3 mmBeam depth 25 mm 25 mm 25.3 mmBoundary stiffnesses k t 26.5 MN/m N/A N/A

k y 150KN m/rad

Note: k

t and k y

are the vertical and rotational stiffness of the support of the beam, respectively.

Table 5Comparison of natural frequencies (Hz) of the aluminum cantilever beam with one crack (Case 1)

Mode No crack d c1 4 mm at x 1 275mm d c1 8 mm at x 1 275mm d c1 12mm at x 1 275mm

Exp. Proposed Exp. [18] [14] Proposed Exp. [18] [14] Proposed Exp. [18] [14] Proposed

1 20.000 19.900 20.000 19.640 19.822 19.763 19.750 19.382 19.580 a 19.562 19.000 19.164 a 19.048 a 19.2382 124.500 124.530 124.250 124.106 124.410 124.302 124.063 123.689 124.008 a 123.971 123.000 123.343 a 123.147 a 123.4403 342.188 345.202 340.813 340.758 a 343.920 340.978 336.875 336.094 339.263 336.195 326.563 332.383 a 329.937 a 333.4734 664.375 665.682 662.813 663.020 a 663.539 663.187 662.313 660.584 661.299 662.121 660.313 658.641 656.975 659.060

a Denotes result better than that from the proposed method.



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estimates on the crack locations with cracks at different locations on the beam. Convergence of computation isconsidered achieved, when the norm of relative difference between two sets of successively identiedparameters equals 1.0 10 5 .

3.3.1. Case 1: identication of single crack The identied results shown in Table 9 indicate that the parameters of the crack can be identied with very

good accuracy under different noise levels. This indicates the identied results are not sensitive to the articialnoise.

3.3.2. Case 2: identication of two cracksThe cracked beam in the last study is retained with another 5-mm-deep crack introduced at 400 mm from

the left support of the beam. Although the identied results for the two cracks shown in Table 9 are not asgood as those for the single crack, the identied parameters of the two cracks are still very good underdifferent noise levels with a maximum relative error or around 2.4% in the crack location and 3.7% in thecrack depth.

4. Experimental verication

A freefree steel beam was tested in the laboratory to verify the proposed method. The length, width andheight of the beam are 2.1, 0.025 and 0.019 m, respectively, and the elastic modulus and mass density of thematerial are 207 GPa and 7.83 103 kg/m 3 , respectively. The beam is suspended at its two ends as shown inFig. 6 . It was modeled with one Euler-Bernoulli beam element with 10 c-dofs. The rst ve natural frequenciesof the intact beam are computed from the proposed model and compared with those obtained from modal test

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.2

0.1

0

0.1

0.2

0.3

0.4

Time (s)

Q 5 / x

L

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 130

20

10

0

10

20

30

Time (s)

Q

5 / d

c

..

..

Fig. 5. Acceleration response sensitivity with respect to different crack parameters (a) with respect to crack location and (b) with respect tocrack depth.



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and they are shown in Table 10 . A sinusoidal force at the frequency of half of the rst natural frequency of thebeam was applied at the nodal point of the rst vibration mode of the beam 480 mm from the left free end withan exciter model Ling Dynamic LDS V450. Acceleration in the horizontal direction obtained with a B&K4370 accelerometer at the middle of the beam was used to identify the crack parameters. Sampling frequency is2000 Hz. Time history of the input sinusoidal force was also recorded for calculating the numerical response of the beam. Rayleigh damping is adopted and the experimental modal damping ratios of 0.007 and 0.01 for therst two modes are included in the calculation. The convergence tolerance in the identication is same as thatfor the simulation study.

It is well-known that modeling error in the initial intact structure has, in general, signicant effect on theaccuracy of the identied results. In most cases, the initial model is updated rst to obtain a goodrepresentation of the intact structure. Table 10 shows that these two sets of analytical and experimentalfrequencies match each other very well, indicating that the initial model of the beam is good enough for crackidentication.

Two adjacent cracks are introduced at 1.66 and 1.72 m from the left free end with 3 and 9 mm crack depth,respectively, and they are created using a machine saw with 1.3-mm-thick cutting blade. The two cracks areclose to each other to make sure the interaction effect between them exists. The rst ve measured naturalfrequencies of the damaged beam are found from modal test and they are shown in Table 10 . The crackparameters, namely, crack depth and location are taken as the unknowns to be identied in the inverseanalysis. In the inverse identication, the rst three seconds of the measured acceleration response at mid-spanof the beam is used. Initial guess of the crack depth is taken as zero and the crack location is initially estimatedto be at mid-span. Good prediction on the crack parameters are obtained after 38 iterations. The identied

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19 mmCracks

25mm

AccelerometerShaker

Fig. 6. Conguration of freefree steel beam for forced vibration test.

Table 9Identied results

True Identied values

Noise free 5% Noise 10% Noise

Single crack Crack location x L (mm) 700 700.8 705.7 707.3Crack depth d c (mm) 3 2.99 2.97 2.94No. of iteration required N/A 28 32 35

Two cracksCrack locations xL 1 /xL 2 (mm) 400/700 400.9/700.4 393.6/709.1 389.6/716.7Crack depths d c1/d c2 (mm) 5/3 4.97/3.02 4.93/2.94 4.89/2.91No. of iteration required N/A 31 45 50Optimal regularization parameter N/A 2.31 10 5 3.42 10 5 8.67 10 5



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locations of the two cracks are 1.641 and 1.702 m, respectively, and the depths are 2.84 and 8.97 mm,respectively, with a maximum error of 1.15% in crack location and 5.33% in crack depth. This shows thehigh accuracy of the proposed crack identication method despite the coupling effect of the two close cracks.The optimal-regularization parameter is 1.75 10 4 . The analytical results of the updated beam with theidentied crack parameters are found very close to the experimental values as shown in Table 10 . The same setof experimental data has been used in the identication of local damage averaged over an element [23], and theanalytical results from the updated beam are also shown in Table 10 for comparison. It is noted thatthe identication with the CEM model on the crack damage gives much higher accuracy than those without

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Table 10The measured and the analytical Natural frequencies (Hz) and relative error of identication (%) of the experimental beam

Crack scenarios Mode no.

1 2 3 4 5

No crack Experimental 22.87 62.76 123.05 203.24 303.45Analytical 22.83/ 0.18 62.74/ 0.03 123.04/ 0.0 203.03/ 0.12 302.86/ 0.2

Two cracksExperimental 22.74 61.77 119.75 198.49 299.50Analytical 22.72/ 0.09 61.84/0.11 120.32/0.46 199.85/0.69 301.11/0.54Analytical [23] 21.71/ 0.05 62.25/0.78 120.73/0.82 200.55/1.04 301.37/0.62

Note : / denotes the modal frequency/relative error.

0 0.5 1 1.5 2 2.5 34

2

0

2

4

Time (second)

F ( N )

0 0.5 1 1.5 2 2.5 30.4

0.2

0

0.2

0.4

Time (second)

A c c e

l e r a

t i o n

( m / s

2 )

Fig. 7. Time history of experimental and calculated acceleration and applied force (a) applied force and (b) acceleration, __ experiment, reconstructed.



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a proper model of the multiple cracks. Fig. 7 compares the time history of the calculated acceleration atmid-span including the identied crack parameters and the corresponding measured acceleration smoothedwith twenty terms orthogonal polynomial function to remove the measurement noise [24]. The time seriesmatch each other very well through out the measured duration.

5. Conclusions

A crack model is incorporated into a composite element for the vibration analysis and for crackidentication. The composite beam element is of one-elementone-member conguration. Modeling with thistype of element would allow the automatic inclusion of interaction effect between adjacent local damages inthe nite element model. The accuracy and convergence of this new composite element has been comparedsatisfactory with existing model and experimental results.

Results obtained from a system identication approach based on the response sensitivity show that anaccurate model is essential for an accurate inverse identication, and the formulation of similar compositeelements incorporating different type of damages could be a useful library of damage models which, is moresuitable for structural damage detection than the existing ones.

Acknowledgment

The work described in this paper was supported by a grant from the Hong Kong Research Grant CouncilProject No. PolyU 5194/05E.

6. Appendix A. The submatrix of the elemental stiffness matrix

k qq R L

0

d 2N 1

dx2 EI x d

2N 1

dx2 dx

R L

0

d 2N 1

dx2 EI x d

2N 2

dx2 dx

R L

0

d 2N 1

dx2 EI x d

2N 3

dx2 dx

R L

0

d 2N 1

dx2 EI x d

2N 4

dx2 dx

R L0

d 2N 2dx 2 EI x

d 2N 1dx 2 dx R

L0

d 2N 2dx 2 EI x

d 2N 2dx 2 dx R

L0

d 2N 2dx 2 EI x

d 2N 3dx 2 dx R

L0

d 2N 2dx 2 EI x

d 2N 4dx 2 dx

R L0

d 2N 3dx 2 EI x

d 2N 1dx 2 dx R

L0

d 2N 3dx 2 EI x

d 2N 2dx 2 dx R

L0

d 2N 3dx 2 EI x

d 2N 3dx 2 dx R

L0

d 2N 3dx 2 EI x

d 2N 4dx 2 dx

R L0

d 2N 4dx 2 EI x

d 2N 1dx 2 dx R

L0

d 2N 4dx 2 EI x

d 2N 2dx 2 dx R

L0

d 2N 4dx 2 EI x

d 2N 3dx 2 dx R

L0

d 2N 4dx 2 EI x

d 2N 4dx 2 dx

2666666437777775

k cq R

L0

d 2N 1dx 2 EI x

d 2 f 1dx 2 dx R

L0

d 2N 1dx 2 EI x

d 2 f 2dx 2 dx . . . R

L0

d 2N 1dx 2 EI x

d 2 f n 1dx 2 dx R

L0

d 2N 1dx 2 EI x

d 2 f ndx 2 dx

R L0

d 2N 2dx 2 EI x

d 2 f 1dx 2 dx R

L0

d 2N 2dx 2 EI x

d 2 f 2dx 2 dx . . . R

L0

d 2N 2dx 2 EI x

d 2 f n 1dx 2 dx R

L0

d 2N 2dx 2 EI x

d 2 f ndx 2 dx

R L

0

d 2N 3

dx2 EI x

d 2 f 1

dx2 dx

R L

0

d 2N 3

dx2 EI x

d 2 f 2

dx2 dx . . .

R L

0

d 2N 3

dx2 EI x

d 2 f n 1

dx2 dx

R L

0

d 2N 3

dx2 EI x

d 2 f n

dx2 dx

R L0

d 2N 4dx 2 EI x

d 2 f 1dx 2 dx R

L0

d 2N 4dx 2 EI x

d 2 f 2dx 2 dx . . . R

L0

d 2N 4dx 2 EI x

d 2 f n 1dx 2 dx R

L0

d 2N 4dx 2 EI x

d 2 f ndx 2 dx

26666664

37777775

k cc

R L0

d 2f 1dx 2 EI x

d 2f 1dx 2 dx R

L0

d 2f 1dx 2 EI x

d 2f 2dx 2 dx R

L0

d 2 f 1dx 2 EI x

d 2f n 1dx 2 dx R

L0

d 2f 1dx 2 EI x

d 2f ndx2 dx

R L0

d 2f 2dx 2 EI x

d 2f 1dx 2 dx R

L0

d 2f 2dx 2 EI x

d 2f 2dx 2 dx R

L0

d 2 f 2dx 2 EI x

d 2f n 1dx 2 dx R

L0

d 2f 2dx 2 EI x

d 2f ndx2 dx

..

. ...

... ..

.

R L0

d 2f n 1dx 2 EI x

d 2 f 1dx 2 dx R

L0

d 2f n 1dx 2 EI x

d 2 f 2dx 2 dx R

L0

d 2f n 1dx 2 EI x

d 2f n 1dx2 dx R

L0

d 2 f n 1dx 2 EI x

d 2f ndx2 dx

R L

0

d 2f n

dx2 EI x

d 2f 1

dx2 dx

R L

0

d 2f n

dx2 EI x

d 2f 2

dx2 dx

R L

0

d 2 f n

dx2 EI x

d 2f n 1

dx2 dx

R L

0

d 2f n

dx2 EI x

d 2f n

dx2 dx

266666666664

377777777775

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7. Appendix B. The submatrix of the elemental mass matrix

mqq R

L0 N 1r AN 1dx R

L0 N 1r AN 2dx R

L0 N 1r AN 3dx R

L0 N 1r AN 4dx

R L0 N 2r AN 1dx

R L0 N 2r AN 2dx

R L0 N 2r AN 3dx

R L0 N 2r AN 4dx

R L0 N 3r AN 1dx R

L0 N 3r AN 2dx R L0 N 3r AN 3dx R

L0 N 3r AN 4dx

R L0 N 4r AN 1dx R

L0 N 4r AN 2dx R

L0 N 4r AN 3dx R

L0 N 4r AN 4dx

2666664

3777775

mcq R

L0 N 1r Af 1dx R

L0 N 1r Af 2dx R

L0 N 1r Af n 1dx R

L0 N 1r Af ndx

R L0 N 2r Af 1dx R

L0 N 2r Af 2dx R

L0 N 2r Af n 1dx R

L0 N 2r Af ndx

R L0 N 3r Af 1dx R

L0 N 3r Af 2dx R

L0 N 3r Af n 1dx R

L0 N 3r Af ndx

R L0 N 4r Af 1dx R

L0 N 4r Af 2dx R

L0 N 4r Af n 1dx R

L0 N 4r Af ndx

26666643777775

mcc R

L

0 f 1r Af 1dx

R L

0 f 1r Af 2dx

R L

0 f 1r Af n 1dx

R L

0 f 1r Af ndx

R L0 f 2r Af 1dx R

L0 f 2r Af 2dx R

L0 f 2r Af n 1dx R

L0 f 2r Af ndx

..

. ...

... ..

.

R L0 f n 1r Af 1dx R

L0 f n 1r Af 2dx R

L0 f n 1r Af n 1dx R

L0 f n 1r Af ndx

R L0 f nr Af 1dx R

L0 f nr Af 2dx R

L0 f nr Af n 1dx R

L0 f nr Af ndx

26666666664

37777777775References

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[17] A. Messina, Revisiting Galerkins method through global piece-wise-smooth functions in Christides and Barrs cracked beam theory,Journal of Sound and Vibration 263 (4) (2003) 937944.

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[20] Z.R. Lu, S.S. Law, Identication of prestress force from measured structural responses, Mechanical System and Signal Processing 20(8) (2006) 21862199.[21] A.M. Tikhonov, On the solution of ill-posed problems and the method of regularization, Soviet Mathematics 4 (1963) 10351038.[22] P.C. Hansen, Regularization tools: a Matlab package for analysis and solution of discrete ill-posed problems, Numerical Algorithms

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Dynamic Composite Cracked Element Model