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Structural damage identification using system dynamic properties
Computers and Structures 83 (2005) 2185–2196 www.elsevier.com/locate/compstruc
Structural damage identification using system dynamic properties Ma Ge, Eric M. Lui
*
Department of Civil and Environmental Engineering, Syracuse University, Syracuse, NY 13244-1190, United States Received 13 July 2004; accepted 16 May 2005
Abstract A damage detection method is presented for the identification and quantification of damage that leads to a change in the structureÕs mass and/or stiffness properties. The proposed method requires the use of finite element to model the structure in its undamaged state as well as information on the dynamic properties such as frequencies and mode shapes of the structure in its damaged state. The technique is applicable to any structure that can be accurately modeled using the finite element method and whose frequencies and mode shapes can be reliably measured. A structure pseudo force vector derived from the residual force method is described to locate the damaged regions in the structure. A matrix condensation approach in conjunction with a proportional damage model is then employed to quantify the damage by calculating the change in stiffness and mass properties of the damaged elements in the structure. The validity of the method is demonstrated by applying it to three structures: a beam, a frame and a plate. It is shown that if the amount of damage is not excessively large, the proposed method can be used to detect damage in these structures even when the measured system dynamic properties are slightly erroneous. 2005 Elsevier Ltd. All rights reserved. Keywords: Damage detection; Structural dynamics; Identification model; Severity model; Finite element
1. Introduction Non-destructive evaluation and condition assessment of our aging infrastructure have received much attention over the past two decades as a result of the realization that early detection and timely repair of structural damage can enhance the overall safety and prolong the service life of a structure. Because proper condition assessment and regular maintenance are vital to the
* Corresponding author. Tel.: +1 315 443 2311; fax: +1 315 443 1243. E-mail address: [email protected] (E.M. Lui).
long-term health of a structure, there is always the need to develop and implement simple but relatively accurate methods of damage detection that not only are capable of evaluating the integrity of the structure, but can be performed in a relatively speedy and inexpensive manner. While various non-destructive evaluation techniques have been proposed or are in use, this paper addresses the use of system dynamic properties in locating and quantifying damage. Damage often causes changes in the natural frequencies, vibration modes and strain energy of the structure [19]. An overview of the various damage detection methods using systemÕs modal parameters was given by Salawu [20]. Generally speaking,
0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.05.002
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these methods can be categorized into two approaches: the first approach relates the variation in strain energy of the structure to changes in structural frequencies, and the second approach relates the change of stiffness and mass properties of the structure to changes in structural frequencies and mode shapes. Gudmundson [8] introduced a first order perturbation method to predict cracks, notches, or other geometrical changes, and mathematically showed that the change in eigenfrequency was related to the change in strain energy of the system. Lee and Chung [16] applied GudmundsonÕs theory to identify the location and severity of an edge crack in a cantilever beam. Stubbs and Osegueda [21], and Kim and Stubbs [14] introduced a damage-localized method using variations in element and modal stiffness, and when combined with GudmundsonÕs theory, successfully identified damage in a cracked beam. Cornwell et al. [7] extended StubbsÕ damage-localized method to solve a 2-D problem. Virtually all of the energy based damage detection models described above are applicable only for cases in which the structures are relatively simple, the damage is relatively small (such as cracks), and the system mass is unaltered by the damage. If the structure is more complex or when the damage is more severe, the correlation between the location and severity of damage with changes in the strain energy and the natural frequencies of the structure becomes very complicated. A more versatile method is to use measured frequencies and mode shapes of the damaged and undamaged structures to determine the changes in structural stiffness and mass, and then to correlate these changes with damage. A procedure of matrix optimization using measured modal data was proposed by Baruch [1], and Berman and Nagy [2], to solve for the least change in their values when damage occurred. The method was improved by Kabe [12] through the introduction of a set of constraints in the optimization procedure. Chen and Garba [5] proposed a matrix inversed approach to calculate changes in the system stiffness due to structural damage. Liu [18] introduced a proportional damage identification model based on minimizing an error norm of the eigenequation. Lim and Kashangaki [17] located damage by computing the Euclidean distances between the measured mode shapes and the best achievable eigenvectors. Kaouk and Zimmerman [13] used modal force error criteria to locate damage and used the minimum rank perturbation theory to identify the magnitude of damage for both the undamped and damped cases. It should be noted that all these damage identification methods required the use of a set of mode shapes and frequencies associated with the damaged structures. Consequently, the accuracy of these approaches largely depends on the precision of the measured data. Cawley and Adams [4] proposed a sensitivity based method to detect and estimate damage using only the
measured frequencies of the damaged structure. To locate the damage, theoretical frequency change ratios of two different modes were calculated by introducing damage at selected positions on a finite element model. Numerical results were compared with the measured ratios. Damage was localized when the two ratios were equal. One drawback of this method is that considerable computation time is needed for its implementation because damage is required to be introduced to every possible element. Another drawback of the approach is that stiffness change can only be calculated as an equivalent removed area, so damage is only represented in a qualitative sense. Hassiotis and Jeong [9] presented a damage detection method based on quadratic optimization to calculate a damage factor for each element. This damage factor was then used to locate and estimate the magnitude of damage present in the structure. This method only requires knowledge of an incomplete set of frequencies for the damaged structure. The accuracy was shown to be high when damage was small and the number of damaged elements was not excessive. By using a new optimization procedure based on Markov parameters, Hassiotis [11] extended the method to include detection in a larger number of damaged elements. In both cases, it was assumed that damage caused only stiffness but not mass change in the structure. Damage detection methods that use measured frequencies and mode shapes are easier to formulate and more robust in their application when compared to energy based methods. A disadvantage is that accurate mode shapes are often difficult to obtain, although accuracy in mode shape measurements does not have as much an effect on identifying the damage location as on determining the damage severity. Because it is usually easier to control errors induced in frequency measurements than in mode shape measurements, damage detection methods that only make use of system frequencies are more desirable. An undesirable aspect of using just the measured system frequencies in damage detection is that more extensive computations are needed. To reduce the amount of computations needed, a method based on the matrix condensation approach is presented in this paper to locate damage and identify its severity. The method takes into consideration both mass and stiffness changes when damage occurs. The proposed method uses finite element modeling and locates damage by the use of a pseudo structure residual force. Matrix condensation is then applied to extract the degrees-of-freedom associated with the damaged elements. Damage severity is evaluated using a proportional damage model that makes use of the measured frequencies of the damaged structure. The validity of the method is demonstrated by applying the procedure to detect damage in a beam, a frame and a plate structure.
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2. Damage detection If a structure is properly modeled using finite element, structural damage manifests itself mathematically in the stiffness and mass matrices, and physically in its dynamic properties such as natural frequencies and mode shapes. Consider the equation of motion for an undamped n degrees-of-freedom (dof) structure given by M€ u þ Ku ¼ fðtÞ
ð1Þ
where M and K are the n · n unconstrained structure mass and stiffness matrices, and u and €u are the n · 1 displacement and acceleration vectors, respectively, f(t) is the n · 1 excitation force vector. Under free vibration (i.e., f(t) = 0) a characteristic equation can be derived from Eq. (1) and written as [6] ðK ki MÞui ¼ 0
ð2Þ
where ki is the ith mode eigenvalue (which is equal to the square of the natural frequency xi) of the structure, and ui is the natural mode shape (or eigenvector) that corresponds to xi. Eq. (2) can be used to determine the natural frequencies and mode shapes of the structure and it forms the basis for many damage detection methods that make use of modal properties in damage detection and identification. In what follows, a damage detection method based on finite element modeling of the structure in its undamaged state and measured modal properties of the structure in its undamaged and damaged states will be presented. A schematic of this method is shown in Fig. 1. The original (undamaged) structure is first modeled using finite element. This finite element model is then refined so that the calculated frequencies and/or mode shapes will match those of the original structure. When damage occurs in the structure, changes in the measured frequencies and mode shapes will result. Based on measured frequencies and mode shapes of the damaged structure, the location of damage is identified using
Original
Damaged
(Undamaged)
Structure
Structure
Measured
Measured
Measured
Eigenvalues (Frequencies)
Eigenvectors
Eigenvalues
and/or
(Mode Shapes)
(Frequencies)
Eigenvectors (Mode Shapes)
Finite
Refined
Damage
Damage
Element
Finite
Location
Severity
Model
Element
Model
Model
Model
Fig. 1. Damage detection model.
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the damage location model; and based on the change in frequencies of the structure from its undamaged to damaged state, the severity of damage is quantified using the damage severity model. Details of the damage location and damage severity models are described below. 2.1. Damage location model A residual force method [15] is used in the present formulation to locate damage. This method is based on identifying the difference in modal properties between the original (undamaged) and damaged structures. If subscript d is used to denote the damaged structure, Eq. (2) becomes ðKd kdi Md Þudi ¼ 0
ð3Þ
Assuming damage in the structure will cause a change in the stiffness and mass, DK and DM, from the original undamaged structure, respectively, we have Kd ¼ Ku þ DK Md ¼ Mu þ DM
ð4aÞ ð4bÞ
where Ku and Mu are the stiffness and mass matrices of the undamaged structure, respectively. Substituting Eqs. (4a) and (4b) into Eq. (3) and rearranging, we obtain DKudi kdi DMudi ¼ ðKu þ kdi Mu Þudi
ð5Þ
The left-hand side of the above equation has unit of force, and if we define a residual force vector for mode i as Ri ¼ DKudi kdi DMudi
ð6Þ
Eq. (5) can be written as Ri ¼ ðKu þ kdi Mu Þudi
ð7Þ
If a set of measured natural frequencies and mode shapes are available, the modal residual force vector Ri can readily be calculated from Eq. (7). A close inspection of Ri reveals that if the measured frequencies and mode shapes are uncorrupted by noise, the jth entry of Ri representing the jth degree-of-freedom will be zero if none of the elements associated with this degree-offreedom is damaged, but it will assume a non-zero value if any element associated with this degree-of-freedom is damaged. This observation can be readily deduced from Eq. (6). If all elements associated with the jth degree-offreedom are undamaged, the submatrices in DK and DM associated with this degree-of-freedom are zero, and so the jth entry of Ri will be zero. Therefore, by identifying non-zero entries in this modal residual force vector, one can pinpoint the location where damage has occurred in the structure. This will be demonstrated numerically in the example problems in a later section. In theory, the aforementioned damage detection model works if modal information for as few as just
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one mode is available. However, in practice modal information for more than one mode is usually needed. This is because it is unrealistic to assume that unpolluted frequencies and mode shapes can always be obtained experimentally. Normally, modal displacement values are measured only at selected points (i.e., degrees-offreedom) on a structure and expanded to all degreesof-freedom through an eigenvector expansion algorithm. Experimental errors introduced in the measurements and numerical errors incurred in the mode expansion algorithm both contribute noise to the mode shapes. As a result, some entries in the modal residual force vector Ri will be corrupted, which may lead to inaccuracies in the damage location model. To overcome the effect of noise, a pseudo structure residual force vector is proposed here. If measured values for p (where p > 1) damaged frequencies and mode shapes are available, the pseudo structure residual force vector R is defined as R ¼ fr1 ; r2 ; r3 ; . . . ; rn gT
ture. If damage alters the structural stiffness DK and mass DM matrices, a corresponding change in eigenvalues and eigenvectors of the original structure will result. Denoting these changes as Dki and Dui respectively, the ith mode eigenequation for the damaged structure can be written as ½ðKu þ DKÞ ðki þ Dki ÞðMu þ DMÞ½ui þ Dui ¼ 0 ð10Þ which upon expansion gives ðKu ki Mu Þui þ ðKu ki Mu ÞDui þ ðDK ki DMÞui Dki Mu ui þ ðDK ki DMÞDui Dki ðMDui þ DMui þ DMDui Þ ¼ 0
ð11Þ
The first term is the eigenequation of the corresponding undamaged structure and thus equal to zero. Neglecting higher order terms and pre-multiplying all remaining terms by uTi , we have
ð8Þ
uTi ðKu ki Mu ÞDui þ uTi ðDK ki DMÞui uTi Dki Mu ui ¼ 0
where rj (j = 1, 2, 3, . . ., n) is calculated from the equation " #1=p p Y ðjrj ji Þ ð9Þ rj ¼
ð12Þ
i¼1
in which jrjji is the absolute value of the jth entry of Ri calculated from Eq. (7). Compared to other commonly used approaches such as the Lp norm (i.e., the pthroot-of-the-sum-of-power-p) and when p = 2, the L2 norm (i.e., the square-root-of-the-sum-of-squares), Eq. (9) has the virtue of being able to magnify true signals and suppress false signals in calculating R, provided that the number of true signals outweighs the number of false signals. This can be explained as follows. If a real damage is present it is likely to affect all modes (especially the lower modes), and so non-zero values of jrjji are expected for all values of i. On the other hand, the existence of noise does not usually affect all modes, and so zero values of jrjji are expected for certain values of i. The use of Eq. (9) under this condition will essentially eliminate the noise. Because modal information from more than one mode is used to calculate the pseudo structure residual force vector, the effect of measurement or calculation errors from one mode becomes less severe. By scanning the R vector for large values of rj, one can pinpoint the degrees-of-freedom associated with damage more precisely. 2.2. Damage severity model The severity of damage is determined by establishing a relationship between frequency (or eigenvalue) shift between the damaged and undamaged structures, and stiffness and mass properties of the undamaged struc-
Since Ku and Mu are symmetric matrices, the transpose of Eq. (2) and hence the first term of Eq. (12) is zero. If ui is normalized so that uTi Mui ¼ 1, Eq. (12) can be simplified to uTi DKui ki uTi DMui ¼ Dki
ð13Þ
It should be noted that DK and DM are sparse matrices. The entries are non-zero only at locations where the degrees-of-freedom are associated with damage as identified by the damage location model described in the preceding section. Eliminating the rows and columns that contain zeros, and retaining only the non-zero rows and columns, DK and DM can be condensed to DK 0 and DM 0 , respectively, where DK 0 and DM 0 represent an assembly of the stiffness and mass matrices associated with the damaged elements in a finite element model. By truncating the corresponding normalized mode shape vector from ui to u0i that includes only those degreesof-freedom that are related to the damage elements, Eq. (13) can be written in a condensed form as 0 0 0 0 0T u0T i DK ui ki ui DM ui ¼ Dki
ð14Þ
The above equation can be used to solve for the changes in stiffness and mass of the damaged elements. An advantage of this approach is that the size of the matrices and vectors in Eq. (14) is a function of the size of the damage and not the size of the structure. Even if the structure is very large and complex, the size of the matrices and vectors in Eq. (14) can be small because they are related only to the damaged elements of the structure. To reduce the number of unknowns that need to be solved from Eq. (14), a proportional damage model is used. In a proportional damage model, DK 0 and DM 0
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are expressed as a function of the undamaged element stiffness and mass matrices by the equations DK0 ¼
nd X
Kej dk j
ð15aÞ
j¼1
DM0 ¼
nd X
Mej dmj
ð15bÞ
j¼1
where Kej and Mej are stiffness and mass matrices of the jth element that contribute to the condensed (damaged) stiffness and mass matrices DK 0 and DM 0 , respectively, nd is the number of damaged elements, and dkj and dmj are the proportional stiffness and mass damage modification factors for element j. The summation sign in Eqs. (15a) and (15b) signifies matrix assembly. By expressing the damage in terms of the damage modification factors, the number of unknowns in DK 0 and DM 0 is reduced to two (dkj and dmj) per damaged element. The problem is therefore rendered much less prohibitive. Upon substitution of Eqs. (15a) and (15b) into Eq. (14), we have nd X
e 0 u0T i Kj ui dk j ki
j¼1
nd X
e 0 u0T i Mj ui dmj ¼ Dki
ð16Þ
j¼1
Note that u0i , Kej , Mej and ki can be obtained from a finite element modeling of the structure in its undamaged state. To solve for dkj and dmj, one needs to determine experimentally the ith mode eigenvalue of the damaged structure kdi, from which Dki can be calculated by subtracting ki from kdi. Suppose p damaged eigenvalues are available, application of the Eq. (16) to each measured eigenvalue leads to a matrix equation of the form Adk þ Bdm ¼ Dk
e 0 Aij ¼ u0T i Kj ui
ð18Þ
e 0 Bij ¼ ki u0T i Mj ui
ð19Þ
dk ¼ ½dk 1 ; dk 2 ; . . . ; dk nd T dm ¼ ½dm1 ; dm2 ; . . . ; dmnd
ð20Þ T
T
ð21Þ ð22Þ
Eq. (17) can be written as L D ¼ Dk
ð23Þ
where L ¼ ½A; B
ð24Þ
and D ¼ ½dk; dmT
For a unique solution, the number of measured eigenvalues must equal the total number of unknown stiffness and mass damage modification factors. For instance, if both stiffness and mass damage is considered in every damaged element, the condition for uniqueness of solution is p = 2nd. However, because of accidental human or instrument errors that often arise in any field measurements, it is advisable to have p > 2nd. Eq. (23) is thus a system of overdetermined equations, and an optimization procedure is needed to solve for the unknowns in vector D. In this study, the least square method is used to obtain the best solution. The least square method uses an error norm that minimizes the square of the errors in solving Eq. (23). If we denote this error as E where E ¼ L D Dk
ð26Þ
the condition that needs to be satisfied is oET E ¼ 0; od k
k ¼ 1; 2; . . . ; 2nd
ð27Þ
where dk is the kth entry of vector D. Substituting Eq. (26) into Eq. (27), and doing the necessary matrix manipulation, we obtain D ¼ ðLT LÞ1 LT Dk
ð28Þ
The above equation can be used to solve for all the unknown stiffness and mass modification factors (dk and dm) that make up the vector D. Because the condition dk = dm = 0 signifies no damage, the deviation of dk and dm from zero can be used as an indicator of how severe the damage is. Unlike the damage location model, the implementation of the damage severity model does not require the use of measured eigenvectors of the damaged structure. Thus, any errors in mode shape measurements will not affect the accuracy of the method.
ð17Þ
where A and B are p · nd matrices, dk and dm are nd · 1 vectors; and Dk is a p · 1 vector, given by
Dk ¼ ½Dk1 ; Dk2 ; . . . ; Dkp
2189
ð25Þ
3. Numerical studies Three examples are presented in this section to demonstrate the validity and applicability of the damage detection and damage severity models described in the foregoing. The first example is the detection of a loaded mass on a cantilever beam. The second example is a numerical study of a 10-story, two-bay steel frame with different proportional damage introduced to its members. The third example is the quantification of damage severity of a hole present in a free hanging aluminum plate. All computations were done in MATLAB using finite element stiffness and mass matrices given by Yang [22]. In order to obtain more accurate result, consistent mass matrix was used. While lumped mass matrix works satisfactorily for the damage detection model, it tends to introduce noticeable errors in the damage severity model. For purpose of consistency, consistent mass matrix was used throughout.
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3.1. Example 1: Identification of an added mass in a cantilever beam In this example the damage detection and damage severity models described in the preceding section are used to identify the location and determine the magnitude of a loaded mass on a cantilever beam. The beam is shown in Fig. 2. It was 0.86 m long and had a unit mass of 1.246 kg/m. The cross-section moment of inertia was 1.458 · 109 m4, and the elastic modulus was 69 GPa. ÔDamageÕ was introduced to the beam by inserting a 0.04286 kg point mass (or 4% of the beam mass) at a distance 0.645 m from the fixed support. Using ten elements (with 4 dofs per element—two translational and two rotational) to model the beam, the frequencies associated with free vibration in the transverse direction of the beam before and after the mass was added were calculated and summarized in Table 1. Also shown in the table were the theoretical frequencies computed using closed-form solution given for the free vibration of a prismatic cantilever beam [6]. As can be seen, the numerical and theoretical values compare very well with each other, thus validating the finite element model used for the study. The addition of the mass changes the dynamic properties of the beam. The frequencies of the ÔdamagedÕ beam are given in Table 1. The presence of a loaded mass lowers the natural frequencies of the beam and changes its mode shapes. Using four natural frequencies and mode shapes, the location of the loaded mass was successful detected using the damage detection model (i.e., Eq. (8)) as shown in Fig. 3a, where the structure pseudo force vector R is plotted as a function of the structure degrees-of-freedom (dof). The non-zero values at dof number 15–18 correspond to the element where the mass was added. Loaded Mass
1
2
3
4
5
6
7
8
9
10
0.86 m Fig. 2. A cantilever beam with a loaded mass.
Fig. 3. Detection of a loaded mass on a cantilever beam: (a) uncontaminated mode data; (b) contaminated mode data and (c) comparison of actual and predicted damage.
As alluded to earlier, exact mode shape measurements are very difficult to obtain experimentally. Due to unavoidable errors in instrumentation and measurements, mode shape data are often contaminated with noise. To investigate the ability of the damage detection model to identify damage when the measured mode shapes are contaminated, a ±5% change in several entries of each eigenvector defining a specific mode shape was randomly introduced. This 5% noise was chosen because it represented a reasonable amount of instrument and computation errors. It was also an upper bound error used by Kaouk and Zimmerman [13] in applying
Table 1 Frequencies of the original and modified cantilever beam Frequencies (Hz) Theoretical Finite element Undamaged Damaged Frequency shift (s2) 2 2 Dk ¼ 4p2 ðfdamaged fundamaged Þ
Mode 1
Mode 2
Mode 3
Mode 4
6.799
42.597
119.303
233.772
6.799 6.574
42.607 42.509
119.328 116.498
233.999 228.36
118.8
329.30
26,347.4
102,929.8
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their minimum rank perturbation theory to assess structural damage. The result of the present analysis is shown in Fig. 3b. Despite the existence of noise, spikes in the R values at dof adjacent to the element where the mass was added were clearly visible, indicating the applicability of the model even for cases with imperfect mode shape data. The ability of the damage severity model to quantify the amount of damage is demonstrated in Fig. 3c in which the calculated mass change is compared to its actual value. Substituting the four frequency shift values Dk computed in Table 1 into Eq. (28), dk and dm for element 8 were calculated to be 0 and 0.374, respectively. This translates to an element stiffness change of zero as expected and an element mass change of 0.0401 kg, or 6.4% error in predicting the magnitude of the added mass.
49
50
91,92,93
94,95,96
28
29 47 85,86,87
25
26 45 76,77,78
22
23 43 67,68,69
19
20 41 58,59,60
16
17 39 49,50,51
3.2. Example 2: Detection and quantification of member damage in a steel frame
13
14
37,38,39
40,41,42
A 10-story two-bay two-dimensional steel frame was used in a damage detection study by Hassiotis and Jeong [10]. This steel frame is shown in Fig. 4. It has 50 members and 33 nodes. Each node has 3 degrees-of-freedom (dof)—two translational and one rotational. The geometry and section type for each member as well as the dof for each node are given in the figure. Elastic modulus used for steel was 2 · 105 MPa. This frame was employed in the present study to determine if Eqs. (8) and (28) could be used to detect and quantify different types and levels of damage imposed on one or more of its members. Three simulated damage scenarios were investigated here:
10
11
The frame was first modeled using finite element. Its frequencies and mode shapes before and after damage were determined using an eigenvalue analysis. The results obtained for the three cases are discussed in the following.
27 79,80,81
24 70,71,72
21 61,62,63
W14x22
18 40
46,47,48
Case 1. Three different levels of damage: a 10%, 40% and 90% stiffness reduction were imposed on member 44. Case 2. Three different types and levels of damage: a 10% stiffness reduction in conjunction with a 5% mass reduction, a 90% stiffness reduction in conjunction with a 90% mass reduction, and a 40% stiffness reduction in conjunction with a 20% mass increase were imposed on member 44. Case 3. Multiple member damage: a 10% stiffness reduction in conjunction with a 5% mass reduction was imposed on member 44; simultaneously, a 20% stiffness reduction in conjunction with a 30% mass reduction was imposed on member 26.
88,89,90
42
55,56,57
37
52,53,54
15 38
35
43,44,45
12 36
31,32,33
8
19,20,21
22,23,24
4
5 31
10,11,12
30
44
64,65,66
33
W12x16
46
73,74,75
7
97,98,99
48
82,83,84
28,29,30
2191
34,35,36
34
9 25,26,27
W16x26
6 32
13,14,15
1
2
1,2,3
4,5,6
16,17,18
3 7,8,9
Fig. 4. A ten-story two-bay steel frame (with member and joint dof labeling).
3.2.1. Case 1 The computed eigenvalues for the first four vibration modes before and after damage was introduced are given in Table 2. The structure pseudo force vector R computed from Eq. (8) using four modes for each level of damage with and without contamination was plotted in Figs. 5 and 6, respectively. Noise was introduced by randomly changing different eigenvectors at 7–20 dof by ±5%. As can be seen in Fig. 5, regardless of the level of damage definitive spikes are present in dof number 67–72 that correspond to the damaged member (i.e., member 44) when perfect eigenvectors were used in the computations. However, false signals were present when contaminated eigenvectors were used. This is particularly the case when the level of damage is low. At low level of damage,
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Table 2 Eigenvalues of the undamaged and damaged steel frames (Case 1) Mode
Undamaged eigenvalues
Damaged eigenvalues 10% Stiffness reduction
40% Stiffness reduction
90% Stiffness reduction
1 2 3 4
6.112E01 4.267E+00 1.444E+01 3.398E+01
6.096E01 4.244E+00 1.443E+01 3.392E+01
6.044E01 4.174E+00 1.442E+01 3.373E+01
5.942E01 4.043E+00 1.440E+01 3.337E+01
(a)
(b)
(c) Fig. 5. Damage detection of a multi-story frame using uncontaminated mode data (Case 1): (a) 10% stiffness damage; (b) 40% stiffness damage and (c) 90% stiffness damage.
the effect this damage has on the systemÕs eigenvectors is small. From a strictly numerical standpoint, it becomes difficult to differentiate between actual damage from noise. Although these false signals seem significant at first glance, they can be filtered out using physical reasoning. Because the proposed damage detection model uses dof as the primary parameter to identify damage, one should expect spikes to be present in the R plot for dof associated with both ends of a damage member. If spikes are present at dof for only one end of the member, the signal is most likely a false signal. For instance, an inspection of the R values shown in Fig. 6 indicates that the spikes at dof number 25, 27, 28, and 30 are false
(a)
(b)
(c) Fig. 6. Damage detection of a multi-story frame using contaminated mode data (Case 1): (a) 10% stiffness damage; (b) 40% stiffness damage and (c) 90% stiffness damage.
signals because the R values for the dof associated with the other end of the members are equal to or very close to zero. For cases in which the measured eigenvectors are highly contaminated, spikes may occur at dof on both ends of an undamaged member, leading one to mistakenly conclude that the member is damaged. However, if the frequency measurements are accurate, the use of Eq. (28) (i.e., the application of the damage severity model) will still give zero (or near zero) values for the stiffness and mass modification factors, signifying that no damage has occurred in the member. Once damage was identified, Eq. (28) was then used to calculate the stiffness and mass damage modification
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Table 3 Comparison of stiffness and mass damage modification factors (Case 1) Damage modification factors
dk dm
Damage in member 44 10% Stiffness reduction
40% Stiffness reduction
90% Stiffness reduction
Actual
Calculated
Actual
Calculated
Actual
Calculated
0.10 0
0.101 0
0.40 0
0.423 0.001
0.90 0
1.03 0.014
factors associated with the damage member. The actual and calculated damage modification factors for member 44 are compared in Table 3 for the three levels of damage. Good correlation is observed. 3.2.2. Case 2 This example is used to demonstrate the ability of the model to detect different levels of damage. By applying Eq. (8) in conjunction with the eigenvalues computed for four modes of the frame in its undamaged and damaged states as shown in Table 4, the R plots obtained using uncontaminated and contaminated mode shape or eigenvectors are given in Figs. 7 and 8, respectively. Spikes at dof number 67–72 corresponding to the damaged member (i.e., member 44) are clearly identifiable. A comparison of the actual and calculated damage modification factors is given in Table 5. Again, good correlation is obtained. 3.2.3. Case 3 This example is used to demonstrate the ability of the model to detect damage in multiple members. By applying Eq. (8) in conjunction with the eigenvalues computed for the frame in its undamaged and damaged states as shown in Table 6, the R plots computed using data from six modes with uncontaminated and contaminated mode shape or eigenvectors are given in Figs. 9 and 10, respectively. Spikes at dof number 67–72 corresponding to member 44 and at dof number 76–78 and 85–87 corresponding to member 26 are clearly seen. A comparison of the actual and computed damage modification factor is given in Table 7.
3.3. Example 3: Damage detection and quantification of an aluminum plate with a hole The structure used in this investigation was a free hanging aluminum plate with a rectangular hole. The plate was tested by Cawley and Adams [4] using an experimental procedure they devised earlier to test a carbon fiber-reinforced polymer plate [3]. The dimension of the plate was 450 mm · 350 mm · 6 mm. The experimentally obtained frequencies for the plate with and without the hole are given in Table 8, from which the frequency shift can be calculated. Since no experimentally obtained mode shape data were reported, mode shapes generated using finite element was used. The finite element mesh used to model the plate is shown in Fig. 11. The plate was modeled using 4-node 16-dof plate elements [22]. The initial elastic modulus and mass density used for the analysis were 7.1 · 104 MPa and 2.701 kg/m3, respectively. Because the original finite element model did not produce identical frequencies with the test data, the model was refined by assuming the discrepancy between the numerical and test frequencies was the result of inaccuracies in modeling system stiffness. An iterative procedure proposed by Hassiotis and Jeong [10] was used to systematically change the elastic modulus in the finite element model until the numerical and experimental frequencies matched. By applying the proposed damage location model, the R plots computed using data from four modes are shown in Fig. 12. Spikes at dof number 161–168, and 181–188 corresponding to the damage region are clearly
Table 4 Eigenvalues of the undamaged and damaged steel frames (Case 2) Mode
1 2 3 4
Undamaged eigenvalues
6.112E01 4.267E+00 1.444E+01 3.398E+01
Damaged eigenvalues 10% Stiffness reduction 5% Mass reduction
90% Stiffness reduction 90% Mass reduction
40% Stiffness reduction 20% Mass reduction
6.105E01 4.245E+00 1.446E+01 3.395E+01
6.108E01 4.047E+00 1.485E+01 3.406E+01
6.008E01 4.173E+00 1.433E+01 3.359E+01
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(a)
(a)
(b) (b)
(c) (c) Fig. 7. Damage detection of a multi-story frame using uncontaminated mode data (Case 2): (a) 10% stiffness and 5% mass damage; (b) 90% stiffness and 90% mass damage and (c) 40% stiffness and 20% mass damage.
Fig. 8. Damage detection of a multi-story frame using contaminated mode data (Case 2): (a) 10% stiffness and 5% mass damage; (b) 90% stiffness and 90% mass damage and (c) 40% stiffness and 20% mass damage.
Table 5 Comparison of stiffness and mass damage modification factors (Case 2) Damage modification factors
dk dm
Damage in member 44 10% Stiffness reduction 5% Mass reduction
90% Stiffness reduction 90% Mass reduction
40% Stiffness reduction 20% Mass increase
Actual
Calculated
Actual
Calculated
Actual
Calculated
0.10 0.05
0.101 0.050
0.90 0.90
0.995 0.925
0.40 0.20
0.422 0.198
Table 6 Eigenvalues of the undamaged and damaged steel frames (Case 3) Mode
Undamaged eigenvalues
Damaged eigenvalues
1 2 3 4 5 6
6.112E01 4.267E+00 1.444E+01 3.398E+01 6.850E+01 1.302E+02
6.177E01 4.254E+00 1.441E+01 3.393E+01 6.865E+01 1.308E+02
Fig. 9. Damage detection of a multi-story frame using uncontaminated mode data (Case 3).
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Fig. 10. Damage detection of a multi-story frame using contaminated mode data (Case 3).
Table 7 Comparison of stiffness and mass damage modification factors (Case 3) Damage modification factors dk dm
Damage in member 44
Damage in member 26
Actual
Calculated
Actual
Calculated
0.10 0.05
0.112 0.045
0.20 0.30
0.224 0.327
Fig. 11. Damage analysis of an aluminum plate.
seen. Also, by applying the damage severity model using the stiffness and mass matrices from this refined finite element model as well as the frequency shift computed in Table 8, dk and dm were calculated from Eq. (28) to be 0.204 and 0.109, respectively. While dm compares favorably with the actual value of 0.111, dk cannot be directly compared because the dimensions of the hole were not reported in the paper. However, by using the calculated stiffness and mass damage modification factors, the refined finite element model was used to compute the damaged frequencies of the plate. These calculated frequencies were compared with the measured frequencies in Table 9. The good correlation signifies that the calculated damage modifications factors are indeed correct.
Fig. 12. Damage detection of a rectangular plate with a hole.
4. Summary and conclusions A damage detection method capable of identifying the location and computing the severity of the damage was presented. The method made use of the undamaged structureÕs stiffness and mass properties and the damaged structureÕs eigenvalues and eigenvectors to locate and quantify the damage. Using measured eigenvalues and eigenvectors of the damaged structure, regions where damage had occurred were identified using the damage location model. The damage location model
Table 8 Measured frequencies for the undamaged and damaged plates Frequency (Hz)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Undamaged Damaged 2 2 Frequency shift (s2), Dk ¼ 4p2 ðfdamaged fundamaged Þ
125.05 124.64 4041.5
158.94 158.17 9639.6
278.29 277.11 25,873
301.52 301.15 8803.2
361.44 360.72 20,527
466.98 466.17 29,840
Table 9 Comparison of experimental and numerical frequencies Frequencies of damaged plate (Hz) Experiment Finite element
124.64 124.64
158.17 158.17
277.11 277.11
301.15 301.13
360.72 360.69
466.17 466.16
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was derived using the residual force concept in conjunction with a matrix condensation approach. The advantage of this method is that once the damage locations are isolated, the size of the matrices that need to be manipulated is related only to the size of the damage. In theory, only one measured eigenvalue and its associated eigenvector is needed to implement the model. However, to mitigate the effect of measurement errors or noise, a structure pseudo force vector was defined. By using this pseudo force vector, it was shown that the model was capable of identifying structural damage even when the mode shape data were contaminated. Once the damage locations were isolated, the damage severity model was used to determine the severity of the damage. Damage severity was quantified by the use of two modification factors apply to the element stiffness and mass matrices, respectively. Zero values for these factors denote no damage has occurred. Because the damage severity model uses only the structural properties of the undamaged structure and measured eigenvalues of the damaged structure, errors in the mode shape measurements will not affect the accuracy of the method. The validity of the proposed approach was demonstrated by applying these damage models to three structures: a beam, a frame and a plate. As long as suitable finite element models were used to model the dynamic properties of the structures in their undamaged state, and reliable frequency and mode shape data were available for the structures in their damaged state, good results were obtained. References [1] Baruch M. Optimal correction of mass and stiffness matrices using measured modes. Am Inst Aeronaut Astronaut J 1982;20(11):1623–6. [2] Berman A, Nagy EJ. Improvement of a large analytical model using test data. Am Inst Aeronaut Astronaut J 1983;21(8):1168–73. [3] Cawley P, Adams RD. The predicted and experimental natural modes of free-free CFRP plates. J Compos Mater 1978;12:336–47. [4] Cawley P, Adams RD. The location of defects in structures from measurements of natural frequencies. J Strain Anal 1979;14(2):49–57.
[5] Chen JC, Garba JA. On orbit damage assessment for large space structures. Am Inst Aeronaut Astronaut J 1988;26(9):1119–26. [6] Chopra AK. Dynamics of structures—theory and applications to earthquake engineering. 2nd ed. New Jersey: Prentice Hall; 2001. [7] Cornwell P, Doebling SW, Farrar CR. Application of the strain energy damage detection method to plate-like structures. J Sound Vib 1999;224(2):359–74. [8] Gudmundson P. Eigenfrequency changes of structures due to cracks, notches or other geometrical changes. J Mech Phys Solids 1982;30(5):339–53. [9] Hassiotis S, Jeong GD. Assessment of structural damage from natural frequencies measurements. Comput Struct 1993;49(4):679–91. [10] Hassiotis S, Jeong GD. Identification of stiffness reduction using natural frequencies. J Eng Mech 1995;121:1106–13. [11] Hassiotis S. Identification of damage using natural frequencies and Markov parameters. Comput Struct 2000;74:365–73. [12] Kabe AM. Stiffness matrix adjustment using mode data. Am Inst Aeronaut Astronaut J 1985;23(9):1431–6. [13] Kaouk M, Zimmerman DC. Structural damage assessment using a generalized minimum rank perturbation theory. Am Inst Aeronaut Astronaut J 1994;32(4):836–42. [14] Kim JT, Stubbs N. Crack detection in beam-type structures using frequency data. J Sound Vib 2003;259(1): 145–60. [15] Kosmatka JB, Ricles JM. Damage detection in structures by modal vibration characterization. J Struct Eng 1999;125(12):1384–92. [16] Lee YS, Chung MJ. A study on crack detection using eigenfrequency test data. Comput Struct 2000;77:327–42. [17] Lim TW, Kashangaki TAL. Structural damage detection of space truss structures using best achievable eigenvectors. Am Inst Aeronaut Astronaut J 1994;32(5):1049–57. [18] Liu PL. Identification and damage detection of trusses using modal data. J Struct Eng 1995;121(4):599–608. [19] OÕBrian TK. Mechanics of non-destructive testing. New York: Plenum; 1980. [20] Salawu OS. Detection of structural damage through changes in frequency: a review. Eng Struct 1997;19(9):718–23. [21] Stubbs N, Osegueda R. Global nondestructive damage evaluation in solids. Int J Anal Exp Modal Anal 1990;5(2):67–79. [22] Yang TY. Finite element structural analysis. New Jersey: Prentice-Hall; 1986.
Structural damage identification using system dynamic properties Ma Ge, Eric M. Lui
*
Department of Civil and Environmental Engineering, Syracuse University, Syracuse, NY 13244-1190, United States Received 13 July 2004; accepted 16 May 2005
Abstract A damage detection method is presented for the identification and quantification of damage that leads to a change in the structureÕs mass and/or stiffness properties. The proposed method requires the use of finite element to model the structure in its undamaged state as well as information on the dynamic properties such as frequencies and mode shapes of the structure in its damaged state. The technique is applicable to any structure that can be accurately modeled using the finite element method and whose frequencies and mode shapes can be reliably measured. A structure pseudo force vector derived from the residual force method is described to locate the damaged regions in the structure. A matrix condensation approach in conjunction with a proportional damage model is then employed to quantify the damage by calculating the change in stiffness and mass properties of the damaged elements in the structure. The validity of the method is demonstrated by applying it to three structures: a beam, a frame and a plate. It is shown that if the amount of damage is not excessively large, the proposed method can be used to detect damage in these structures even when the measured system dynamic properties are slightly erroneous. 2005 Elsevier Ltd. All rights reserved. Keywords: Damage detection; Structural dynamics; Identification model; Severity model; Finite element
1. Introduction Non-destructive evaluation and condition assessment of our aging infrastructure have received much attention over the past two decades as a result of the realization that early detection and timely repair of structural damage can enhance the overall safety and prolong the service life of a structure. Because proper condition assessment and regular maintenance are vital to the
* Corresponding author. Tel.: +1 315 443 2311; fax: +1 315 443 1243. E-mail address: [email protected] (E.M. Lui).
long-term health of a structure, there is always the need to develop and implement simple but relatively accurate methods of damage detection that not only are capable of evaluating the integrity of the structure, but can be performed in a relatively speedy and inexpensive manner. While various non-destructive evaluation techniques have been proposed or are in use, this paper addresses the use of system dynamic properties in locating and quantifying damage. Damage often causes changes in the natural frequencies, vibration modes and strain energy of the structure [19]. An overview of the various damage detection methods using systemÕs modal parameters was given by Salawu [20]. Generally speaking,
0045-7949/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2005.05.002
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these methods can be categorized into two approaches: the first approach relates the variation in strain energy of the structure to changes in structural frequencies, and the second approach relates the change of stiffness and mass properties of the structure to changes in structural frequencies and mode shapes. Gudmundson [8] introduced a first order perturbation method to predict cracks, notches, or other geometrical changes, and mathematically showed that the change in eigenfrequency was related to the change in strain energy of the system. Lee and Chung [16] applied GudmundsonÕs theory to identify the location and severity of an edge crack in a cantilever beam. Stubbs and Osegueda [21], and Kim and Stubbs [14] introduced a damage-localized method using variations in element and modal stiffness, and when combined with GudmundsonÕs theory, successfully identified damage in a cracked beam. Cornwell et al. [7] extended StubbsÕ damage-localized method to solve a 2-D problem. Virtually all of the energy based damage detection models described above are applicable only for cases in which the structures are relatively simple, the damage is relatively small (such as cracks), and the system mass is unaltered by the damage. If the structure is more complex or when the damage is more severe, the correlation between the location and severity of damage with changes in the strain energy and the natural frequencies of the structure becomes very complicated. A more versatile method is to use measured frequencies and mode shapes of the damaged and undamaged structures to determine the changes in structural stiffness and mass, and then to correlate these changes with damage. A procedure of matrix optimization using measured modal data was proposed by Baruch [1], and Berman and Nagy [2], to solve for the least change in their values when damage occurred. The method was improved by Kabe [12] through the introduction of a set of constraints in the optimization procedure. Chen and Garba [5] proposed a matrix inversed approach to calculate changes in the system stiffness due to structural damage. Liu [18] introduced a proportional damage identification model based on minimizing an error norm of the eigenequation. Lim and Kashangaki [17] located damage by computing the Euclidean distances between the measured mode shapes and the best achievable eigenvectors. Kaouk and Zimmerman [13] used modal force error criteria to locate damage and used the minimum rank perturbation theory to identify the magnitude of damage for both the undamped and damped cases. It should be noted that all these damage identification methods required the use of a set of mode shapes and frequencies associated with the damaged structures. Consequently, the accuracy of these approaches largely depends on the precision of the measured data. Cawley and Adams [4] proposed a sensitivity based method to detect and estimate damage using only the
measured frequencies of the damaged structure. To locate the damage, theoretical frequency change ratios of two different modes were calculated by introducing damage at selected positions on a finite element model. Numerical results were compared with the measured ratios. Damage was localized when the two ratios were equal. One drawback of this method is that considerable computation time is needed for its implementation because damage is required to be introduced to every possible element. Another drawback of the approach is that stiffness change can only be calculated as an equivalent removed area, so damage is only represented in a qualitative sense. Hassiotis and Jeong [9] presented a damage detection method based on quadratic optimization to calculate a damage factor for each element. This damage factor was then used to locate and estimate the magnitude of damage present in the structure. This method only requires knowledge of an incomplete set of frequencies for the damaged structure. The accuracy was shown to be high when damage was small and the number of damaged elements was not excessive. By using a new optimization procedure based on Markov parameters, Hassiotis [11] extended the method to include detection in a larger number of damaged elements. In both cases, it was assumed that damage caused only stiffness but not mass change in the structure. Damage detection methods that use measured frequencies and mode shapes are easier to formulate and more robust in their application when compared to energy based methods. A disadvantage is that accurate mode shapes are often difficult to obtain, although accuracy in mode shape measurements does not have as much an effect on identifying the damage location as on determining the damage severity. Because it is usually easier to control errors induced in frequency measurements than in mode shape measurements, damage detection methods that only make use of system frequencies are more desirable. An undesirable aspect of using just the measured system frequencies in damage detection is that more extensive computations are needed. To reduce the amount of computations needed, a method based on the matrix condensation approach is presented in this paper to locate damage and identify its severity. The method takes into consideration both mass and stiffness changes when damage occurs. The proposed method uses finite element modeling and locates damage by the use of a pseudo structure residual force. Matrix condensation is then applied to extract the degrees-of-freedom associated with the damaged elements. Damage severity is evaluated using a proportional damage model that makes use of the measured frequencies of the damaged structure. The validity of the method is demonstrated by applying the procedure to detect damage in a beam, a frame and a plate structure.
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2. Damage detection If a structure is properly modeled using finite element, structural damage manifests itself mathematically in the stiffness and mass matrices, and physically in its dynamic properties such as natural frequencies and mode shapes. Consider the equation of motion for an undamped n degrees-of-freedom (dof) structure given by M€ u þ Ku ¼ fðtÞ
ð1Þ
where M and K are the n · n unconstrained structure mass and stiffness matrices, and u and €u are the n · 1 displacement and acceleration vectors, respectively, f(t) is the n · 1 excitation force vector. Under free vibration (i.e., f(t) = 0) a characteristic equation can be derived from Eq. (1) and written as [6] ðK ki MÞui ¼ 0
ð2Þ
where ki is the ith mode eigenvalue (which is equal to the square of the natural frequency xi) of the structure, and ui is the natural mode shape (or eigenvector) that corresponds to xi. Eq. (2) can be used to determine the natural frequencies and mode shapes of the structure and it forms the basis for many damage detection methods that make use of modal properties in damage detection and identification. In what follows, a damage detection method based on finite element modeling of the structure in its undamaged state and measured modal properties of the structure in its undamaged and damaged states will be presented. A schematic of this method is shown in Fig. 1. The original (undamaged) structure is first modeled using finite element. This finite element model is then refined so that the calculated frequencies and/or mode shapes will match those of the original structure. When damage occurs in the structure, changes in the measured frequencies and mode shapes will result. Based on measured frequencies and mode shapes of the damaged structure, the location of damage is identified using
Original
Damaged
(Undamaged)
Structure
Structure
Measured
Measured
Measured
Eigenvalues (Frequencies)
Eigenvectors
Eigenvalues
and/or
(Mode Shapes)
(Frequencies)
Eigenvectors (Mode Shapes)
Finite
Refined
Damage
Damage
Element
Finite
Location
Severity
Model
Element
Model
Model
Model
Fig. 1. Damage detection model.
2187
the damage location model; and based on the change in frequencies of the structure from its undamaged to damaged state, the severity of damage is quantified using the damage severity model. Details of the damage location and damage severity models are described below. 2.1. Damage location model A residual force method [15] is used in the present formulation to locate damage. This method is based on identifying the difference in modal properties between the original (undamaged) and damaged structures. If subscript d is used to denote the damaged structure, Eq. (2) becomes ðKd kdi Md Þudi ¼ 0
ð3Þ
Assuming damage in the structure will cause a change in the stiffness and mass, DK and DM, from the original undamaged structure, respectively, we have Kd ¼ Ku þ DK Md ¼ Mu þ DM
ð4aÞ ð4bÞ
where Ku and Mu are the stiffness and mass matrices of the undamaged structure, respectively. Substituting Eqs. (4a) and (4b) into Eq. (3) and rearranging, we obtain DKudi kdi DMudi ¼ ðKu þ kdi Mu Þudi
ð5Þ
The left-hand side of the above equation has unit of force, and if we define a residual force vector for mode i as Ri ¼ DKudi kdi DMudi
ð6Þ
Eq. (5) can be written as Ri ¼ ðKu þ kdi Mu Þudi
ð7Þ
If a set of measured natural frequencies and mode shapes are available, the modal residual force vector Ri can readily be calculated from Eq. (7). A close inspection of Ri reveals that if the measured frequencies and mode shapes are uncorrupted by noise, the jth entry of Ri representing the jth degree-of-freedom will be zero if none of the elements associated with this degree-offreedom is damaged, but it will assume a non-zero value if any element associated with this degree-of-freedom is damaged. This observation can be readily deduced from Eq. (6). If all elements associated with the jth degree-offreedom are undamaged, the submatrices in DK and DM associated with this degree-of-freedom are zero, and so the jth entry of Ri will be zero. Therefore, by identifying non-zero entries in this modal residual force vector, one can pinpoint the location where damage has occurred in the structure. This will be demonstrated numerically in the example problems in a later section. In theory, the aforementioned damage detection model works if modal information for as few as just
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one mode is available. However, in practice modal information for more than one mode is usually needed. This is because it is unrealistic to assume that unpolluted frequencies and mode shapes can always be obtained experimentally. Normally, modal displacement values are measured only at selected points (i.e., degrees-offreedom) on a structure and expanded to all degreesof-freedom through an eigenvector expansion algorithm. Experimental errors introduced in the measurements and numerical errors incurred in the mode expansion algorithm both contribute noise to the mode shapes. As a result, some entries in the modal residual force vector Ri will be corrupted, which may lead to inaccuracies in the damage location model. To overcome the effect of noise, a pseudo structure residual force vector is proposed here. If measured values for p (where p > 1) damaged frequencies and mode shapes are available, the pseudo structure residual force vector R is defined as R ¼ fr1 ; r2 ; r3 ; . . . ; rn gT
ture. If damage alters the structural stiffness DK and mass DM matrices, a corresponding change in eigenvalues and eigenvectors of the original structure will result. Denoting these changes as Dki and Dui respectively, the ith mode eigenequation for the damaged structure can be written as ½ðKu þ DKÞ ðki þ Dki ÞðMu þ DMÞ½ui þ Dui ¼ 0 ð10Þ which upon expansion gives ðKu ki Mu Þui þ ðKu ki Mu ÞDui þ ðDK ki DMÞui Dki Mu ui þ ðDK ki DMÞDui Dki ðMDui þ DMui þ DMDui Þ ¼ 0
ð11Þ
The first term is the eigenequation of the corresponding undamaged structure and thus equal to zero. Neglecting higher order terms and pre-multiplying all remaining terms by uTi , we have
ð8Þ
uTi ðKu ki Mu ÞDui þ uTi ðDK ki DMÞui uTi Dki Mu ui ¼ 0
where rj (j = 1, 2, 3, . . ., n) is calculated from the equation " #1=p p Y ðjrj ji Þ ð9Þ rj ¼
ð12Þ
i¼1
in which jrjji is the absolute value of the jth entry of Ri calculated from Eq. (7). Compared to other commonly used approaches such as the Lp norm (i.e., the pthroot-of-the-sum-of-power-p) and when p = 2, the L2 norm (i.e., the square-root-of-the-sum-of-squares), Eq. (9) has the virtue of being able to magnify true signals and suppress false signals in calculating R, provided that the number of true signals outweighs the number of false signals. This can be explained as follows. If a real damage is present it is likely to affect all modes (especially the lower modes), and so non-zero values of jrjji are expected for all values of i. On the other hand, the existence of noise does not usually affect all modes, and so zero values of jrjji are expected for certain values of i. The use of Eq. (9) under this condition will essentially eliminate the noise. Because modal information from more than one mode is used to calculate the pseudo structure residual force vector, the effect of measurement or calculation errors from one mode becomes less severe. By scanning the R vector for large values of rj, one can pinpoint the degrees-of-freedom associated with damage more precisely. 2.2. Damage severity model The severity of damage is determined by establishing a relationship between frequency (or eigenvalue) shift between the damaged and undamaged structures, and stiffness and mass properties of the undamaged struc-
Since Ku and Mu are symmetric matrices, the transpose of Eq. (2) and hence the first term of Eq. (12) is zero. If ui is normalized so that uTi Mui ¼ 1, Eq. (12) can be simplified to uTi DKui ki uTi DMui ¼ Dki
ð13Þ
It should be noted that DK and DM are sparse matrices. The entries are non-zero only at locations where the degrees-of-freedom are associated with damage as identified by the damage location model described in the preceding section. Eliminating the rows and columns that contain zeros, and retaining only the non-zero rows and columns, DK and DM can be condensed to DK 0 and DM 0 , respectively, where DK 0 and DM 0 represent an assembly of the stiffness and mass matrices associated with the damaged elements in a finite element model. By truncating the corresponding normalized mode shape vector from ui to u0i that includes only those degreesof-freedom that are related to the damage elements, Eq. (13) can be written in a condensed form as 0 0 0 0 0T u0T i DK ui ki ui DM ui ¼ Dki
ð14Þ
The above equation can be used to solve for the changes in stiffness and mass of the damaged elements. An advantage of this approach is that the size of the matrices and vectors in Eq. (14) is a function of the size of the damage and not the size of the structure. Even if the structure is very large and complex, the size of the matrices and vectors in Eq. (14) can be small because they are related only to the damaged elements of the structure. To reduce the number of unknowns that need to be solved from Eq. (14), a proportional damage model is used. In a proportional damage model, DK 0 and DM 0
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are expressed as a function of the undamaged element stiffness and mass matrices by the equations DK0 ¼
nd X
Kej dk j
ð15aÞ
j¼1
DM0 ¼
nd X
Mej dmj
ð15bÞ
j¼1
where Kej and Mej are stiffness and mass matrices of the jth element that contribute to the condensed (damaged) stiffness and mass matrices DK 0 and DM 0 , respectively, nd is the number of damaged elements, and dkj and dmj are the proportional stiffness and mass damage modification factors for element j. The summation sign in Eqs. (15a) and (15b) signifies matrix assembly. By expressing the damage in terms of the damage modification factors, the number of unknowns in DK 0 and DM 0 is reduced to two (dkj and dmj) per damaged element. The problem is therefore rendered much less prohibitive. Upon substitution of Eqs. (15a) and (15b) into Eq. (14), we have nd X
e 0 u0T i Kj ui dk j ki
j¼1
nd X
e 0 u0T i Mj ui dmj ¼ Dki
ð16Þ
j¼1
Note that u0i , Kej , Mej and ki can be obtained from a finite element modeling of the structure in its undamaged state. To solve for dkj and dmj, one needs to determine experimentally the ith mode eigenvalue of the damaged structure kdi, from which Dki can be calculated by subtracting ki from kdi. Suppose p damaged eigenvalues are available, application of the Eq. (16) to each measured eigenvalue leads to a matrix equation of the form Adk þ Bdm ¼ Dk
e 0 Aij ¼ u0T i Kj ui
ð18Þ
e 0 Bij ¼ ki u0T i Mj ui
ð19Þ
dk ¼ ½dk 1 ; dk 2 ; . . . ; dk nd T dm ¼ ½dm1 ; dm2 ; . . . ; dmnd
ð20Þ T
T
ð21Þ ð22Þ
Eq. (17) can be written as L D ¼ Dk
ð23Þ
where L ¼ ½A; B
ð24Þ
and D ¼ ½dk; dmT
For a unique solution, the number of measured eigenvalues must equal the total number of unknown stiffness and mass damage modification factors. For instance, if both stiffness and mass damage is considered in every damaged element, the condition for uniqueness of solution is p = 2nd. However, because of accidental human or instrument errors that often arise in any field measurements, it is advisable to have p > 2nd. Eq. (23) is thus a system of overdetermined equations, and an optimization procedure is needed to solve for the unknowns in vector D. In this study, the least square method is used to obtain the best solution. The least square method uses an error norm that minimizes the square of the errors in solving Eq. (23). If we denote this error as E where E ¼ L D Dk
ð26Þ
the condition that needs to be satisfied is oET E ¼ 0; od k
k ¼ 1; 2; . . . ; 2nd
ð27Þ
where dk is the kth entry of vector D. Substituting Eq. (26) into Eq. (27), and doing the necessary matrix manipulation, we obtain D ¼ ðLT LÞ1 LT Dk
ð28Þ
The above equation can be used to solve for all the unknown stiffness and mass modification factors (dk and dm) that make up the vector D. Because the condition dk = dm = 0 signifies no damage, the deviation of dk and dm from zero can be used as an indicator of how severe the damage is. Unlike the damage location model, the implementation of the damage severity model does not require the use of measured eigenvectors of the damaged structure. Thus, any errors in mode shape measurements will not affect the accuracy of the method.
ð17Þ
where A and B are p · nd matrices, dk and dm are nd · 1 vectors; and Dk is a p · 1 vector, given by
Dk ¼ ½Dk1 ; Dk2 ; . . . ; Dkp
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ð25Þ
3. Numerical studies Three examples are presented in this section to demonstrate the validity and applicability of the damage detection and damage severity models described in the foregoing. The first example is the detection of a loaded mass on a cantilever beam. The second example is a numerical study of a 10-story, two-bay steel frame with different proportional damage introduced to its members. The third example is the quantification of damage severity of a hole present in a free hanging aluminum plate. All computations were done in MATLAB using finite element stiffness and mass matrices given by Yang [22]. In order to obtain more accurate result, consistent mass matrix was used. While lumped mass matrix works satisfactorily for the damage detection model, it tends to introduce noticeable errors in the damage severity model. For purpose of consistency, consistent mass matrix was used throughout.
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3.1. Example 1: Identification of an added mass in a cantilever beam In this example the damage detection and damage severity models described in the preceding section are used to identify the location and determine the magnitude of a loaded mass on a cantilever beam. The beam is shown in Fig. 2. It was 0.86 m long and had a unit mass of 1.246 kg/m. The cross-section moment of inertia was 1.458 · 109 m4, and the elastic modulus was 69 GPa. ÔDamageÕ was introduced to the beam by inserting a 0.04286 kg point mass (or 4% of the beam mass) at a distance 0.645 m from the fixed support. Using ten elements (with 4 dofs per element—two translational and two rotational) to model the beam, the frequencies associated with free vibration in the transverse direction of the beam before and after the mass was added were calculated and summarized in Table 1. Also shown in the table were the theoretical frequencies computed using closed-form solution given for the free vibration of a prismatic cantilever beam [6]. As can be seen, the numerical and theoretical values compare very well with each other, thus validating the finite element model used for the study. The addition of the mass changes the dynamic properties of the beam. The frequencies of the ÔdamagedÕ beam are given in Table 1. The presence of a loaded mass lowers the natural frequencies of the beam and changes its mode shapes. Using four natural frequencies and mode shapes, the location of the loaded mass was successful detected using the damage detection model (i.e., Eq. (8)) as shown in Fig. 3a, where the structure pseudo force vector R is plotted as a function of the structure degrees-of-freedom (dof). The non-zero values at dof number 15–18 correspond to the element where the mass was added. Loaded Mass
1
2
3
4
5
6
7
8
9
10
0.86 m Fig. 2. A cantilever beam with a loaded mass.
Fig. 3. Detection of a loaded mass on a cantilever beam: (a) uncontaminated mode data; (b) contaminated mode data and (c) comparison of actual and predicted damage.
As alluded to earlier, exact mode shape measurements are very difficult to obtain experimentally. Due to unavoidable errors in instrumentation and measurements, mode shape data are often contaminated with noise. To investigate the ability of the damage detection model to identify damage when the measured mode shapes are contaminated, a ±5% change in several entries of each eigenvector defining a specific mode shape was randomly introduced. This 5% noise was chosen because it represented a reasonable amount of instrument and computation errors. It was also an upper bound error used by Kaouk and Zimmerman [13] in applying
Table 1 Frequencies of the original and modified cantilever beam Frequencies (Hz) Theoretical Finite element Undamaged Damaged Frequency shift (s2) 2 2 Dk ¼ 4p2 ðfdamaged fundamaged Þ
Mode 1
Mode 2
Mode 3
Mode 4
6.799
42.597
119.303
233.772
6.799 6.574
42.607 42.509
119.328 116.498
233.999 228.36
118.8
329.30
26,347.4
102,929.8
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their minimum rank perturbation theory to assess structural damage. The result of the present analysis is shown in Fig. 3b. Despite the existence of noise, spikes in the R values at dof adjacent to the element where the mass was added were clearly visible, indicating the applicability of the model even for cases with imperfect mode shape data. The ability of the damage severity model to quantify the amount of damage is demonstrated in Fig. 3c in which the calculated mass change is compared to its actual value. Substituting the four frequency shift values Dk computed in Table 1 into Eq. (28), dk and dm for element 8 were calculated to be 0 and 0.374, respectively. This translates to an element stiffness change of zero as expected and an element mass change of 0.0401 kg, or 6.4% error in predicting the magnitude of the added mass.
49
50
91,92,93
94,95,96
28
29 47 85,86,87
25
26 45 76,77,78
22
23 43 67,68,69
19
20 41 58,59,60
16
17 39 49,50,51
3.2. Example 2: Detection and quantification of member damage in a steel frame
13
14
37,38,39
40,41,42
A 10-story two-bay two-dimensional steel frame was used in a damage detection study by Hassiotis and Jeong [10]. This steel frame is shown in Fig. 4. It has 50 members and 33 nodes. Each node has 3 degrees-of-freedom (dof)—two translational and one rotational. The geometry and section type for each member as well as the dof for each node are given in the figure. Elastic modulus used for steel was 2 · 105 MPa. This frame was employed in the present study to determine if Eqs. (8) and (28) could be used to detect and quantify different types and levels of damage imposed on one or more of its members. Three simulated damage scenarios were investigated here:
10
11
The frame was first modeled using finite element. Its frequencies and mode shapes before and after damage were determined using an eigenvalue analysis. The results obtained for the three cases are discussed in the following.
27 79,80,81
24 70,71,72
21 61,62,63
W14x22
18 40
46,47,48
Case 1. Three different levels of damage: a 10%, 40% and 90% stiffness reduction were imposed on member 44. Case 2. Three different types and levels of damage: a 10% stiffness reduction in conjunction with a 5% mass reduction, a 90% stiffness reduction in conjunction with a 90% mass reduction, and a 40% stiffness reduction in conjunction with a 20% mass increase were imposed on member 44. Case 3. Multiple member damage: a 10% stiffness reduction in conjunction with a 5% mass reduction was imposed on member 44; simultaneously, a 20% stiffness reduction in conjunction with a 30% mass reduction was imposed on member 26.
88,89,90
42
55,56,57
37
52,53,54
15 38
35
43,44,45
12 36
31,32,33
8
19,20,21
22,23,24
4
5 31
10,11,12
30
44
64,65,66
33
W12x16
46
73,74,75
7
97,98,99
48
82,83,84
28,29,30
2191
34,35,36
34
9 25,26,27
W16x26
6 32
13,14,15
1
2
1,2,3
4,5,6
16,17,18
3 7,8,9
Fig. 4. A ten-story two-bay steel frame (with member and joint dof labeling).
3.2.1. Case 1 The computed eigenvalues for the first four vibration modes before and after damage was introduced are given in Table 2. The structure pseudo force vector R computed from Eq. (8) using four modes for each level of damage with and without contamination was plotted in Figs. 5 and 6, respectively. Noise was introduced by randomly changing different eigenvectors at 7–20 dof by ±5%. As can be seen in Fig. 5, regardless of the level of damage definitive spikes are present in dof number 67–72 that correspond to the damaged member (i.e., member 44) when perfect eigenvectors were used in the computations. However, false signals were present when contaminated eigenvectors were used. This is particularly the case when the level of damage is low. At low level of damage,
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Table 2 Eigenvalues of the undamaged and damaged steel frames (Case 1) Mode
Undamaged eigenvalues
Damaged eigenvalues 10% Stiffness reduction
40% Stiffness reduction
90% Stiffness reduction
1 2 3 4
6.112E01 4.267E+00 1.444E+01 3.398E+01
6.096E01 4.244E+00 1.443E+01 3.392E+01
6.044E01 4.174E+00 1.442E+01 3.373E+01
5.942E01 4.043E+00 1.440E+01 3.337E+01
(a)
(b)
(c) Fig. 5. Damage detection of a multi-story frame using uncontaminated mode data (Case 1): (a) 10% stiffness damage; (b) 40% stiffness damage and (c) 90% stiffness damage.
the effect this damage has on the systemÕs eigenvectors is small. From a strictly numerical standpoint, it becomes difficult to differentiate between actual damage from noise. Although these false signals seem significant at first glance, they can be filtered out using physical reasoning. Because the proposed damage detection model uses dof as the primary parameter to identify damage, one should expect spikes to be present in the R plot for dof associated with both ends of a damage member. If spikes are present at dof for only one end of the member, the signal is most likely a false signal. For instance, an inspection of the R values shown in Fig. 6 indicates that the spikes at dof number 25, 27, 28, and 30 are false
(a)
(b)
(c) Fig. 6. Damage detection of a multi-story frame using contaminated mode data (Case 1): (a) 10% stiffness damage; (b) 40% stiffness damage and (c) 90% stiffness damage.
signals because the R values for the dof associated with the other end of the members are equal to or very close to zero. For cases in which the measured eigenvectors are highly contaminated, spikes may occur at dof on both ends of an undamaged member, leading one to mistakenly conclude that the member is damaged. However, if the frequency measurements are accurate, the use of Eq. (28) (i.e., the application of the damage severity model) will still give zero (or near zero) values for the stiffness and mass modification factors, signifying that no damage has occurred in the member. Once damage was identified, Eq. (28) was then used to calculate the stiffness and mass damage modification
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Table 3 Comparison of stiffness and mass damage modification factors (Case 1) Damage modification factors
dk dm
Damage in member 44 10% Stiffness reduction
40% Stiffness reduction
90% Stiffness reduction
Actual
Calculated
Actual
Calculated
Actual
Calculated
0.10 0
0.101 0
0.40 0
0.423 0.001
0.90 0
1.03 0.014
factors associated with the damage member. The actual and calculated damage modification factors for member 44 are compared in Table 3 for the three levels of damage. Good correlation is observed. 3.2.2. Case 2 This example is used to demonstrate the ability of the model to detect different levels of damage. By applying Eq. (8) in conjunction with the eigenvalues computed for four modes of the frame in its undamaged and damaged states as shown in Table 4, the R plots obtained using uncontaminated and contaminated mode shape or eigenvectors are given in Figs. 7 and 8, respectively. Spikes at dof number 67–72 corresponding to the damaged member (i.e., member 44) are clearly identifiable. A comparison of the actual and calculated damage modification factors is given in Table 5. Again, good correlation is obtained. 3.2.3. Case 3 This example is used to demonstrate the ability of the model to detect damage in multiple members. By applying Eq. (8) in conjunction with the eigenvalues computed for the frame in its undamaged and damaged states as shown in Table 6, the R plots computed using data from six modes with uncontaminated and contaminated mode shape or eigenvectors are given in Figs. 9 and 10, respectively. Spikes at dof number 67–72 corresponding to member 44 and at dof number 76–78 and 85–87 corresponding to member 26 are clearly seen. A comparison of the actual and computed damage modification factor is given in Table 7.
3.3. Example 3: Damage detection and quantification of an aluminum plate with a hole The structure used in this investigation was a free hanging aluminum plate with a rectangular hole. The plate was tested by Cawley and Adams [4] using an experimental procedure they devised earlier to test a carbon fiber-reinforced polymer plate [3]. The dimension of the plate was 450 mm · 350 mm · 6 mm. The experimentally obtained frequencies for the plate with and without the hole are given in Table 8, from which the frequency shift can be calculated. Since no experimentally obtained mode shape data were reported, mode shapes generated using finite element was used. The finite element mesh used to model the plate is shown in Fig. 11. The plate was modeled using 4-node 16-dof plate elements [22]. The initial elastic modulus and mass density used for the analysis were 7.1 · 104 MPa and 2.701 kg/m3, respectively. Because the original finite element model did not produce identical frequencies with the test data, the model was refined by assuming the discrepancy between the numerical and test frequencies was the result of inaccuracies in modeling system stiffness. An iterative procedure proposed by Hassiotis and Jeong [10] was used to systematically change the elastic modulus in the finite element model until the numerical and experimental frequencies matched. By applying the proposed damage location model, the R plots computed using data from four modes are shown in Fig. 12. Spikes at dof number 161–168, and 181–188 corresponding to the damage region are clearly
Table 4 Eigenvalues of the undamaged and damaged steel frames (Case 2) Mode
1 2 3 4
Undamaged eigenvalues
6.112E01 4.267E+00 1.444E+01 3.398E+01
Damaged eigenvalues 10% Stiffness reduction 5% Mass reduction
90% Stiffness reduction 90% Mass reduction
40% Stiffness reduction 20% Mass reduction
6.105E01 4.245E+00 1.446E+01 3.395E+01
6.108E01 4.047E+00 1.485E+01 3.406E+01
6.008E01 4.173E+00 1.433E+01 3.359E+01
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(a)
(a)
(b) (b)
(c) (c) Fig. 7. Damage detection of a multi-story frame using uncontaminated mode data (Case 2): (a) 10% stiffness and 5% mass damage; (b) 90% stiffness and 90% mass damage and (c) 40% stiffness and 20% mass damage.
Fig. 8. Damage detection of a multi-story frame using contaminated mode data (Case 2): (a) 10% stiffness and 5% mass damage; (b) 90% stiffness and 90% mass damage and (c) 40% stiffness and 20% mass damage.
Table 5 Comparison of stiffness and mass damage modification factors (Case 2) Damage modification factors
dk dm
Damage in member 44 10% Stiffness reduction 5% Mass reduction
90% Stiffness reduction 90% Mass reduction
40% Stiffness reduction 20% Mass increase
Actual
Calculated
Actual
Calculated
Actual
Calculated
0.10 0.05
0.101 0.050
0.90 0.90
0.995 0.925
0.40 0.20
0.422 0.198
Table 6 Eigenvalues of the undamaged and damaged steel frames (Case 3) Mode
Undamaged eigenvalues
Damaged eigenvalues
1 2 3 4 5 6
6.112E01 4.267E+00 1.444E+01 3.398E+01 6.850E+01 1.302E+02
6.177E01 4.254E+00 1.441E+01 3.393E+01 6.865E+01 1.308E+02
Fig. 9. Damage detection of a multi-story frame using uncontaminated mode data (Case 3).
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Fig. 10. Damage detection of a multi-story frame using contaminated mode data (Case 3).
Table 7 Comparison of stiffness and mass damage modification factors (Case 3) Damage modification factors dk dm
Damage in member 44
Damage in member 26
Actual
Calculated
Actual
Calculated
0.10 0.05
0.112 0.045
0.20 0.30
0.224 0.327
Fig. 11. Damage analysis of an aluminum plate.
seen. Also, by applying the damage severity model using the stiffness and mass matrices from this refined finite element model as well as the frequency shift computed in Table 8, dk and dm were calculated from Eq. (28) to be 0.204 and 0.109, respectively. While dm compares favorably with the actual value of 0.111, dk cannot be directly compared because the dimensions of the hole were not reported in the paper. However, by using the calculated stiffness and mass damage modification factors, the refined finite element model was used to compute the damaged frequencies of the plate. These calculated frequencies were compared with the measured frequencies in Table 9. The good correlation signifies that the calculated damage modifications factors are indeed correct.
Fig. 12. Damage detection of a rectangular plate with a hole.
4. Summary and conclusions A damage detection method capable of identifying the location and computing the severity of the damage was presented. The method made use of the undamaged structureÕs stiffness and mass properties and the damaged structureÕs eigenvalues and eigenvectors to locate and quantify the damage. Using measured eigenvalues and eigenvectors of the damaged structure, regions where damage had occurred were identified using the damage location model. The damage location model
Table 8 Measured frequencies for the undamaged and damaged plates Frequency (Hz)
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Undamaged Damaged 2 2 Frequency shift (s2), Dk ¼ 4p2 ðfdamaged fundamaged Þ
125.05 124.64 4041.5
158.94 158.17 9639.6
278.29 277.11 25,873
301.52 301.15 8803.2
361.44 360.72 20,527
466.98 466.17 29,840
Table 9 Comparison of experimental and numerical frequencies Frequencies of damaged plate (Hz) Experiment Finite element
124.64 124.64
158.17 158.17
277.11 277.11
301.15 301.13
360.72 360.69
466.17 466.16
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was derived using the residual force concept in conjunction with a matrix condensation approach. The advantage of this method is that once the damage locations are isolated, the size of the matrices that need to be manipulated is related only to the size of the damage. In theory, only one measured eigenvalue and its associated eigenvector is needed to implement the model. However, to mitigate the effect of measurement errors or noise, a structure pseudo force vector was defined. By using this pseudo force vector, it was shown that the model was capable of identifying structural damage even when the mode shape data were contaminated. Once the damage locations were isolated, the damage severity model was used to determine the severity of the damage. Damage severity was quantified by the use of two modification factors apply to the element stiffness and mass matrices, respectively. Zero values for these factors denote no damage has occurred. Because the damage severity model uses only the structural properties of the undamaged structure and measured eigenvalues of the damaged structure, errors in the mode shape measurements will not affect the accuracy of the method. The validity of the proposed approach was demonstrated by applying these damage models to three structures: a beam, a frame and a plate. As long as suitable finite element models were used to model the dynamic properties of the structures in their undamaged state, and reliable frequency and mode shape data were available for the structures in their damaged state, good results were obtained. References [1] Baruch M. Optimal correction of mass and stiffness matrices using measured modes. Am Inst Aeronaut Astronaut J 1982;20(11):1623–6. [2] Berman A, Nagy EJ. Improvement of a large analytical model using test data. Am Inst Aeronaut Astronaut J 1983;21(8):1168–73. [3] Cawley P, Adams RD. The predicted and experimental natural modes of free-free CFRP plates. J Compos Mater 1978;12:336–47. [4] Cawley P, Adams RD. The location of defects in structures from measurements of natural frequencies. J Strain Anal 1979;14(2):49–57.
[5] Chen JC, Garba JA. On orbit damage assessment for large space structures. Am Inst Aeronaut Astronaut J 1988;26(9):1119–26. [6] Chopra AK. Dynamics of structures—theory and applications to earthquake engineering. 2nd ed. New Jersey: Prentice Hall; 2001. [7] Cornwell P, Doebling SW, Farrar CR. Application of the strain energy damage detection method to plate-like structures. J Sound Vib 1999;224(2):359–74. [8] Gudmundson P. Eigenfrequency changes of structures due to cracks, notches or other geometrical changes. J Mech Phys Solids 1982;30(5):339–53. [9] Hassiotis S, Jeong GD. Assessment of structural damage from natural frequencies measurements. Comput Struct 1993;49(4):679–91. [10] Hassiotis S, Jeong GD. Identification of stiffness reduction using natural frequencies. J Eng Mech 1995;121:1106–13. [11] Hassiotis S. Identification of damage using natural frequencies and Markov parameters. Comput Struct 2000;74:365–73. [12] Kabe AM. Stiffness matrix adjustment using mode data. Am Inst Aeronaut Astronaut J 1985;23(9):1431–6. [13] Kaouk M, Zimmerman DC. Structural damage assessment using a generalized minimum rank perturbation theory. Am Inst Aeronaut Astronaut J 1994;32(4):836–42. [14] Kim JT, Stubbs N. Crack detection in beam-type structures using frequency data. J Sound Vib 2003;259(1): 145–60. [15] Kosmatka JB, Ricles JM. Damage detection in structures by modal vibration characterization. J Struct Eng 1999;125(12):1384–92. [16] Lee YS, Chung MJ. A study on crack detection using eigenfrequency test data. Comput Struct 2000;77:327–42. [17] Lim TW, Kashangaki TAL. Structural damage detection of space truss structures using best achievable eigenvectors. Am Inst Aeronaut Astronaut J 1994;32(5):1049–57. [18] Liu PL. Identification and damage detection of trusses using modal data. J Struct Eng 1995;121(4):599–608. [19] OÕBrian TK. Mechanics of non-destructive testing. New York: Plenum; 1980. [20] Salawu OS. Detection of structural damage through changes in frequency: a review. Eng Struct 1997;19(9):718–23. [21] Stubbs N, Osegueda R. Global nondestructive damage evaluation in solids. Int J Anal Exp Modal Anal 1990;5(2):67–79. [22] Yang TY. Finite element structural analysis. New Jersey: Prentice-Hall; 1986.