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Structural damage identification based on best achievable flexibility change
Applied Mathematical Modelling 35 (2011) 5217–5224
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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Short communication
Structural damage identification based on best achievable flexibility change Q.W. Yang ⇑, B.X. Sun Department of Civil Engineering, Shaoxing University, Shaoxing 312000, PR China
a r t i c l e
i n f o
Article history: Received 20 April 2010 Received in revised form 1 April 2011 Accepted 5 April 2011 Available online 16 April 2011 Keywords: Damage identification Flexibility change Best achievable
a b s t r a c t A new method based on best achievable flexibility change is presented in this paper to localize and quantify damage in structures. Central to the damage localization approach is the computation of the Euclidean distances between the measured flexibility change and the best achievable flexibility changes. The location of damage can be identified by searching for a value that is considerably smaller than others in these distances. With location determined, a simple extent algorithm is then developed. Three examples are used to demonstrate the efficiency of the method. Results indicate that the proposed procedure may be useful for structural damage identification. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The basic idea of vibration-based damage identification is that changes in geometry and physical properties of structures will cause changes in structural modal properties, mainly changes in the natural frequencies (or the square root of eigenvalues) and the mode shapes (i.e., the eigenvectors). An excellent review on all methods can be found in the book of Mottershead and Friswell [1] and in two reports of the Los Alamos National Laboratory [2,3]. Other comprehensive reviews on the topic can be found in Refs. [4–11]. An important class of damage identification methods is based on the change in the flexibility matrices of the undamaged and the damaged structures. As in practice it would be always difficult to excite the structural high modal frequencies, the flexibility-based methods offer the practical attraction of only requiring measurements of the changes in a few of the lowerfrequency modes. Pandey and Biswas developed a method for locating damage in beam type structures using changes in the flexibility matrix of the structure [12,13]. Zhao and DeWolf demonstrated that the algorithms using modal flexibility, derived from frequencies and mode shapes, are very sensitive to local damage [14]. Bernal computed a set of load vectors from the flexibility matrix change, designated as damage location vectors to localize damage [15–17]. Jaishi and Ren presented a damage detection method by finite element model updating using modal flexibility residual [18,19]. Stutz et al. presents a flexibility-based continuum damage identification approach [20]. Yan and Golinval studied a damage localization method by combining flexibility and stiffness methods [21]. Yang and Liu defined a damage localization criterion to locate structural damage, which makes use of the change of flexibility matrix and the stiffness matrix of the intact structure [22]. Perera and Ruiz used the modal flexibility as one objective function in formulation of the multiple objective damage identification problems [23,24]. Yang derived the sensitivity of flexibility matrix using Neumann series expansion and presented a mixed sensitivity method to identify structural damage by combining the eigenvalue sensitivity with the flexibility sensitivity [25]. In this paper, using the concept of best achievable flexibility change, a new method is developed to determine both the location and magnitude of structural damage. As will be shown in Section 2, the key point of the damage location algorithm lies in the formulation of the best achievable flexibility change. The damage is located by calculating the Euclidean distances ⇑ Corresponding author. Tel.: +86 575 88326229; fax: +86 575 88341503. E-mail address: [email protected] (Q.W. Yang). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.04.010
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Q.W. Yang, B.X. Sun / Applied Mathematical Modelling 35 (2011) 5217–5224
between the flexibility change obtained by the measured modes and the best achievable flexibility changes. With location determined, a simple algorithm using the best achievable flexibility change is also developed to determine the extent of damage. The use of the best achievable flexibility change can filter the measured flexibility change from the contamination by truncation and noise. The developed theory is validated in Section 3 with three numerical examples. The results obtained show that the location and magnitude of local damage can be identified by the proposed method using the noisy and incomplete modes. In the following theoretical development, it is assumed that structural damages only reduce the system stiffness matrix and structural refined FEM has been developed before damage occurrence. 2. Theoretical development 2.1. Damage localization using the best achievable flexibility change For the intact and the damaged structures, the global stiffness and flexibility matrices will satisfy the following relationship:
F u K u ¼ F d K d ¼ I;
ð1Þ
where Fu and Ku are the n n flexibility and stiffness matrices of the undamaged structure, Fd and Kd are the n n matrices of the damaged structure, I is the n n identity matrix. It is noted that both translational and rotational degrees of freedom are considered in the flexibility and stiffness matrices here. As is well known, damage reduces the stiffness and increases the flexibility of structures. Let DF and DK be the exact perturbation matrices that reflect the nature of the structural damage. Then the undamaged model matrices and the damaged model matrices are related as follows:
F d ¼ F u þ DF;
ð2Þ
K d ¼ K u DK:
ð3Þ
In practice, the exact DF cannot be obtained due to the limitation of the modal survey. But DF can be approximated by the first few low-frequency modes as [12,13]
DF ¼
m m X X 1 1 /dj /Tdj /uj /Tuj ; k k uj dj j¼1 j¼1
ð4Þ
where m is the number of measured modes in modal survey, kuj (/uj) and kdj (/dj) are the eigenparameters of the undamaged and damaged structures, respectively. The modes of the damaged structure can be obtained by a modal survey on it, and the modal data of the undamaged structure can be obtained by solving a generalized eigenvalue problem of the undamaged FEM or through a modal test on the intact structure. A detailed discussion of modal analysis techniques is beyond the scope of this paper, and interested readers are referred to Refs. [26–28]. Substituting Eqs. (2) and (3) into (1), and neglecting the smaller product term yields
DF K d ¼ F u DK:
ð5Þ
Rewriting Eq. (5) yields
DF ¼ F u DKF d :
ð6Þ
When damage has occurred in the structure, the stiffness matrix perturbation DK can be expressed as a sum of each elemental stiffness matrix multiplied by a damage coefficient, that is
DK ¼
N X
ai K i ð0 6 ai 6 1Þ;
ð7Þ
i¼1
where Ki is the ith elemental stiffness matrix, ai is its damage parameter, N is the total number of elements. The value of ai is 0 if the ith element is undamaged and ai is 1 or less than 1 if the corresponding element is completely or partially damaged. In Eq. (7), it is assumed that in case of damage, all the elements in the stiffness matrix Ki will be affected by same ratio ai. This assumption is used for mathematical simplicity since the practical damage is very complex. Substituting Eq. (7) into (6), one has
DF ¼
N X
ai F u K i F d :
ð8Þ
i¼1
According to Eq. (8), the changes in the flexibility could be caused by damage at a single member or at multiple members. Assume, for the time being, that the damage is caused by a single member. Without loss of generality, assume that only the ith element is damaged (ai – 0), then Eq. (8) reduces to
DF ¼ ai F u K i F d :
ð9Þ
Let the jth column of DF and Fd be represented by Dfj and fdj, respectively. That is, DF = [Df1 Dfj Dfn] and Fd = [fd1 fdj fdn]. From Eq. (9), we have
Q.W. Yang, B.X. Sun / Applied Mathematical Modelling 35 (2011) 5217–5224
Dfj ¼ ai F u K i fdj
ðj ¼ 1—nÞ:
5219
ð10Þ
Let
Ei ¼ F u K i ;
ð11Þ
cij ¼ ai fdj ;
ð12Þ
Then Eq. (10) simplifies to
Dfj ¼ Ei cij
ðj ¼ 1—nÞ:
ð13Þ
The implication of Eq. (13) is very important. According to the theory of linear algebra [29,30], Eq. (13) is valid only if the vector Dfj is a linear combination of the columns of Ei. In other words, Dfj must lie in the subspace spanned by the columns of Ei. That is to say, if the ith element is damaged, then the vector Dfj will lie exactly in the subspace spanned by the columns of Ei. If not, Dfj would not lie in the subspace spanned by the columns of Ei. According to the matrix theory, we can use the concept of the best achievable vector to evaluate whether or not Dfj lies in the subspace spanned by the columns of Ei. The best achievable vector of Dfj can be computed by
Dfija ¼ Ei ðEi Þþ Dfj
ðj ¼ 1—nÞ;
ð14Þ
where Ei is the matrix Ei where the zero columns have been removed to enhance computational efficiency, and the superscript + denotes the generalized inverse. For j = 1–n, Eq. (14) can be assembled as
DF ai ¼ Ei ðEi Þþ DF; DF ai
½Dfi1a
Dfija
ð15Þ DF ai
Dfina .
where ¼ The matrix is defined as the best achievable flexibility change. If the damage is caused by the ith element, then the matrices DF ai and DF will be identical. If not, the two matrices will be different. We can use the Euclidean distance between the two matrices to evaluate whether or not DF ai equals DF. The distance between the two matrices can be computed using the Frobenious norm
di ¼ kDF DF ai kF ;
ð16Þ
where k kF represents the Frobenious norm. If the perfect data are presented, the damaged element will has zero distance (di = 0) and all others will have nonzero values. For a structure that has N structural members, a damage localization vector, of length N, can be defined as
d¼
d1 di dN ;...; ;... dmax dmax dmax
T ;
ð17Þ
where dmax is the largest value in all distances, i.e., dmax = max (d1 di dN). For the measured data with truncation and di noise, dmax will be equal or close to zero if damage is located in element i and all other coefficients will be populated with nonzero entries. As a result, the location of damage can be determined by searching for a value that is considerably smaller than others in the vector d. 2.2. Damage quantification using the best achievable flexibility change With the most probable damaged element identified by the above method, the extent of the damage can be obtained by computing its damage parameter using the following procedure. Without loss of generality, assume that the number of damaged element is i and the corresponding damage factor is ai. Then Eq. (7) reduces to
DK ¼ ai K i
ð0 6 ai 6 1Þ:
ð18Þ
Substituting Eq. (2) into (6), one has
DF ¼ F u DKF u þ F u DK DF:
ð19Þ
Neglecting the high-order term in Eq. (19), one obtains
DF ¼ F u DKF u :
ð20Þ
Substituting Eq. (18) into (20) yields
DF ¼ ai ðF u K i F u Þ:
ð21Þ
From Eq. (21), the damage parameter ai can be calculated by
ai ¼
sumðDFÞ ; sumðF u K i F u Þ
where sum(A) represents the sum of all entries in the matrix A.
ð22Þ
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In practice, the measured modes are usually truncated and contaminated by noise. So the flexibility change DF obtained by these noisy modes using Eq. (4) is always inaccurate. When the noisy DF is used directly to compute the extent of damage by Eq. (22), unrealized values of the damage parameters can be obtained. To improve the calculation precision, we can use the best achievable flexibility change DF ai instead of DF in Eq. (21) to compute the damage parameter. Then Eq. (21) changes to
DF ai ¼ ai ðF u K i F u Þ:
ð23Þ
From Eq. (23), the damage parameter ai can be calculated by
ai ¼
sumðDF ai Þ : sumðF u K i F u Þ
ð24Þ
2.3. Incomplete measurement The measured modes are usually incomplete in practice because of the limited number of sensors, and the rotational degrees of freedom are difficult to measure. When spatial incompleteness in mode shape measurements occurs, mode shape expansion technique is usually applied to expand the measured mode shapes to match the analytical model. It has been shown that the mode shape expansion process will introduce additional errors in the expanded mode shape. But in the method described above, there is no need to use all the entries of the flexibility change in the damage identification and then the mode shape expansion can be avoided. Supposing the measured mode shapes include r DOFs (r < n), the incomplete flexibility change DF is reduced into a (r r) symmetric matrix. We can take only the corresponding entries in the damage localization and quantification procedures. 2.4. Summary of the whole technique A step-by-step summary of the whole technique is presented as follows: Step 1: Calculate DF using Eq. (4) and Ei for all structural elements using Eq. (12). Step 2: Compute di for all structural elements using Eq. (16) and construct the damage localization vector d by Eq. (17). Step 3: Select the single most probable damaged element by examining d and then calculate its damage parameter using Eq. (24). 3. Numerical examples To illustrate characteristics of the proposed damage detection algorithm, three numerical examples are presented. Example 1 is a spring-mass systems with 3 DOFs, which are used to verify the proposed method for the ideal case with complete and exact modes. Example 2 is a space truss structure and Example 3 is a two-storey frame structure. Example 1. The first example is a spring-mass system with 3 DOFs as shown in Fig. 1. Consider the nominal model of the system to have the parameters ki = 1 (i = 1–3) and mj = 1 (j = 1–3). A single damage case is studied in the example that element 2 is damaged with a stiffness loss of 10%. Using the complete and exact modes, the damage localization vector d can be computed as
d ¼ ½ 0:8167 0 1 :
ð25Þ d2 dmax
Examination of d indicates that a single damage occurred in the element 2 because of ¼ 0. Using Eq. (24), we can obtain the damage extent as a2 = 0.1111. The predicted damage extent has slight derivation from the true value because of neglecting the high-order term in Eq. (19). Example 2. The space truss structure used in this example is shown in Fig. 2. The structure was modeled using 26 truss rod elements and 12 nodes (4 restrained), for a total of 24 DOF. The basic parameters of the structure are as follows: E = 200 GPa, q = 7.8 103 kg/m3, L = 1 m, and A = 0.004 m2. Assume that only the first six modes are used to calculate the flexibility matrix change DF. Mode shapes are contaminated with 5% random noise to simulate measurement error. A single damage is simulated in element 10 with damage parameter of 0.1. Using the proposed method, the damage localization vector d of the single-damage case is shown in Fig. 3(a). Inspection of Fig. 3(a) indicates that a single damage occurred in the element 10
k1
k2 m1
k3 m2
m3
Fig. 1. Spring-mass system for Example 1.
Q.W. Yang, B.X. Sun / Applied Mathematical Modelling 35 (2011) 5217–5224
5221
10
Fig. 2. A space truss structure for Example 2.
10 because that ddmax is far less than the others. Damage extents obtained by Eqs. (24) and (22) are a10 = 0.1072 and a10 = 0.1124, respectively. Compared with the assumed value 0.1, it is obvious that the result obtained by the best achievable flexibility change is more accurate than that obtained by the measured flexibility change. In order to study the impact of the order of the model on the efficiency, Fig. 3(b) gives the damage localization vector d using the first four modes. Compared with Fig. 3(a), it is obvious that the result obtained by the first six modes is more accurate than that obtained by the first four modes. Damage extents obtained by Eqs. (24) and (22) are a10 = 0.1095 and a10 = 0.1189, respectively. According to the above results, it has been shown that the damage detection results become more accurate as the number of modes used in the calculation increases.
Damage location index
Example 3. The third example is a two-storey frame structure as shown in Fig. 4 with a rectangular cross-section of 0.14 m 0.24 m. This frame was modeled with 48 equal elements of 0.2 m in length. Every node has 3 DOFs, an axial displacement, a transverse displacement and a rotation. The properties of this structure are: cross-sectional area A = 0.0336 m2; moment of inertia I = 1.6128 104 m4; Young’s modulus E = 200 GPa; shearing modulus of elasticity G = 1.3461 1010 N/m2; Poisson’s ratio v = 0.3; density q = 2500 kg/m3. Only the first six modes are used in the calculations. Furthermore, only horizontal modal displacements are assumed to be ‘measured’ in the columns while for the beams only the vertical components of mode shapes are ‘measured’. In other words, only the first six modes and 22 out of 66 DOFs are used in the present study to simulate incomplete modal data in the real situation. As before, mode shapes are contaminated with 5% random noise to simulate measurement error. Two damage cases are studied in the example. Case 1 assumed that element 4 is damage with 10% stiffness reduction. Case 2 supposed that element 7 is damaged with 10% stiffness loss. Using the proposed approach, the damage localization vector d is shown in Fig. 5 for case 1. Examination of d indicates that a single damage occurred in the element 4 because its corresponding index in Fig. 5 is far less than the others. Using Eq. (24), its
1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Element number Fig. 3a. The damage localization vector d using the first six modes when element 10 is damaged (Example 2).
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Q.W. Yang, B.X. Sun / Applied Mathematical Modelling 35 (2011) 5217–5224
Damage location index
1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Element number Fig. 3b. The damage localization vector d using the first four modes when element 10 is damaged (Example 2).
13
14
12 13 12
15 15
14
16 16 17 17
11 11
18
10
19
18
10
19
9 9
20
8
7
6
20
5 5 4
6
7
8 21
4
21 3
22 22
3 2
23
2
23 1
24
1
24
Damage location index
Fig. 4. A two-storey frame structure for Example 3.
1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Element number Fig. 5. The damage localization vector d of case 1 (Example 3).
Damage location index
Q.W. Yang, B.X. Sun / Applied Mathematical Modelling 35 (2011) 5217–5224
5223
1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Element number Fig. 6. The damage localization vector d of case 2 (Example 3).
damage extent can be computed as a4 = 0.1087. Fig. 6 presents the damage localization vector d for case 2. It is obvious that element 7 is damaged by inspecting Fig. 6 and its damage parameter can be calculated using Eq. (24) as a7 = 0.1091. The above results show that the proposed method can identify both the location and extent of structural damage.
4. Conclusions A new method for structural damage identification was developed in this study, which is based on the best achievable concept. The best achievable flexibility change is a projection of a measured flexibility change onto the subspace that is defined by the undamaged analytical model. Damage location can be determined by the Euclidean distance between the measured flexibility change and the best achievable flexibility change. With location identified, the magnitude of damage can be computed by a simple method using the best achievable flexibility change. Three numerical examples are used to exercise this process and measurement noise is also simulated in damage detection. The results show that the proposed method can determine accurately both the location and magnitude of structural damage with the incomplete and noisy modes. It should be noted that we have assumed only a single damage occurred in the structures for the three numerical examples. Future research on the technique should be carried out to tackle the multiple damage case and to demonstrate the procedure using experimentally measured data. Acknowledgements The authors thank the reviewers for a thorough and careful reading of the original paper. Their comments are greatly appreciated and have helped to improve the quality of the paper. This work is supported by Zhejiang Province Natural Science Foundation (Y1110949), the scientific research project of Shaoxing City (2010A23006), and the National Natural Science Foundation of China (No: 40772194). References [1] J.E. Mottershead, M.I. Friswell, Model updating in structural dynamics: a survey, J. Sound Vib. 167 (2) (1993) 347–375. [2] M.I. Friswell, J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. [3] Y. Zou, L. Tong, G.P. Steven, Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures – a review, J. Sound Vib. 230 (2) (2000) 357–378. [4] F.K. Chang, Structural Health Monitoring, Current Status and Perspectives, Technomic Publishing Co., Inc., Lancaster, Pennsylvania, 1997. [5] O.S. Salawu, Detection of structural damage through changes in frequency: a review, Eng. Struct. 19 (9) (1997) 718–723. [6] S.V. Modak, T.K. Kundra, B.C. Nakra, Comparative study of model updating methods using simulated experimental data, Comput. Struct. 80 (2002) 437– 447. [7] A. Alvandi, C. Cremona, Assessment of vibration-based damage identification techniques, J. Sound Vib. 292 (2006) 179–202. [8] A. Prinaris, S. Alampalli, M. Ettouney. Review of remote sensing for condition assessment and damage identification after extreme loading conditions, in: Proceedings of the 2008 Structures Congress, 2008. [9] S. Kanev, F. Weber, M. Verhaegen, Experimental validation of a finite-element model updating procedure, J. Sound Vib. 300 (2007) 394–413. [10] Q.W. Yang, J.K. Liu, Structural damage identification based on residual force vector, J. Sound Vib. 305 (2007) 298–307. [11] J.D. Collins, G.C. Hart, T.K. Hasselman, B. Kennedy, Statistical identification of structures, AIAA J. 12 (2) (1974) 185–190. [12] A.K. Pandey, M. Biswas, Experimental verification of flexibility difference method for locating damage in structures, J. Sound Vib. 184 (2) (1995) 311– 328. [13] A.K. Pandey, M. Biswas, Damage detection in structures using changes in flexibility, J. Sound Vib. 169 (1994) 3–17. [14] J. Zhao, J.T. DeWolf, Sensitivity study for vibrational parameters used in damage detection, J. Struct. Eng., ASCE 125 (4) (1999) 410–416. [15] D. Bernal, Load vectors for damage localization, J. Eng. Mech. 128 (1) (2002) 7–14. [16] D. Bernal, B. Gunes, Damage localization in output-only systems: a flexibility based approach, in: IMAC-XX, Los Angeles, California, 2002, pp. 1185– 1191. [17] D. Bernal, B. Gunes, Flexibility based approach for damage characterization: benchmark application, J. Eng. Mech. 130 (1) (2004) 61–70. [18] B. Jaishi, W.X. Ren, Damage detection by finite element model updating using modal flexibility residual, J. Sound Vib. 290 (2006) 369–387. [19] B. Jaishi, W.X. Ren, Structural finite element model updating using ambient vibration test results, J. Struct. Eng. 131 (4) (2005) 617–628.
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Q.W. Yang, B.X. Sun / Applied Mathematical Modelling 35 (2011) 5217–5224 L.T. Stutz, D.A. Castello, F.A. Rochinha, A flexibility-based continuum damage identification approach, J. Sound Vib. 279 (2005) 641–667. A. Yan, J.C. Golinval, Structural damage localization by combining flexibility and stiffness methods, Eng. Struct. 27 (2005) 1752–1761. Q.W. Yang, J.K. Liu, A coupled method for structural damage identification, J. Sound Vib. 296 (2006) 401–405. R. Perera, A. Ruiz, C. Manzano, An evolutionary multiobjective framework for structural damage localization and quantification, Eng. Struct. 29 (2007) 2540–2550. R. Perera, A. Ruiz, A multistage FE updating procedure for damage identification in large-scale structures based on multiobjective evolutionary optimization, Mech. Syst. Signal Process. 22 (4) (2008) 970–991. Q.W. Yang, A mixed sensitivity method for structural damage detection, Commun. Numer. Methods Eng. 25 (4) (2009) 381–389. D.J. Ewins, Modal Testing: Theory and Practice, Research Studies Press Ltd., England, 1984. J.N. Juang, Applied System Identification, Prentice-Hall Inc., Englewood Cliffs (NJ), 1994. N.M.M. Maia, J.M.M. Silva (Eds.), Theoretical and Experimental Modal Analysis, Research Studies Press Ltd., England, 1997. I.N. Herstein, D.J. Winter, Matrix Theory and Linear Algebra, Macmillan Publishing Company, Indianapolis, USA, 1988. B.N. Datta, Numerical Linear Algebra and Applications, Brooks/Cole Publishing Company, California, 1995.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Short communication
Structural damage identification based on best achievable flexibility change Q.W. Yang ⇑, B.X. Sun Department of Civil Engineering, Shaoxing University, Shaoxing 312000, PR China
a r t i c l e
i n f o
Article history: Received 20 April 2010 Received in revised form 1 April 2011 Accepted 5 April 2011 Available online 16 April 2011 Keywords: Damage identification Flexibility change Best achievable
a b s t r a c t A new method based on best achievable flexibility change is presented in this paper to localize and quantify damage in structures. Central to the damage localization approach is the computation of the Euclidean distances between the measured flexibility change and the best achievable flexibility changes. The location of damage can be identified by searching for a value that is considerably smaller than others in these distances. With location determined, a simple extent algorithm is then developed. Three examples are used to demonstrate the efficiency of the method. Results indicate that the proposed procedure may be useful for structural damage identification. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction The basic idea of vibration-based damage identification is that changes in geometry and physical properties of structures will cause changes in structural modal properties, mainly changes in the natural frequencies (or the square root of eigenvalues) and the mode shapes (i.e., the eigenvectors). An excellent review on all methods can be found in the book of Mottershead and Friswell [1] and in two reports of the Los Alamos National Laboratory [2,3]. Other comprehensive reviews on the topic can be found in Refs. [4–11]. An important class of damage identification methods is based on the change in the flexibility matrices of the undamaged and the damaged structures. As in practice it would be always difficult to excite the structural high modal frequencies, the flexibility-based methods offer the practical attraction of only requiring measurements of the changes in a few of the lowerfrequency modes. Pandey and Biswas developed a method for locating damage in beam type structures using changes in the flexibility matrix of the structure [12,13]. Zhao and DeWolf demonstrated that the algorithms using modal flexibility, derived from frequencies and mode shapes, are very sensitive to local damage [14]. Bernal computed a set of load vectors from the flexibility matrix change, designated as damage location vectors to localize damage [15–17]. Jaishi and Ren presented a damage detection method by finite element model updating using modal flexibility residual [18,19]. Stutz et al. presents a flexibility-based continuum damage identification approach [20]. Yan and Golinval studied a damage localization method by combining flexibility and stiffness methods [21]. Yang and Liu defined a damage localization criterion to locate structural damage, which makes use of the change of flexibility matrix and the stiffness matrix of the intact structure [22]. Perera and Ruiz used the modal flexibility as one objective function in formulation of the multiple objective damage identification problems [23,24]. Yang derived the sensitivity of flexibility matrix using Neumann series expansion and presented a mixed sensitivity method to identify structural damage by combining the eigenvalue sensitivity with the flexibility sensitivity [25]. In this paper, using the concept of best achievable flexibility change, a new method is developed to determine both the location and magnitude of structural damage. As will be shown in Section 2, the key point of the damage location algorithm lies in the formulation of the best achievable flexibility change. The damage is located by calculating the Euclidean distances ⇑ Corresponding author. Tel.: +86 575 88326229; fax: +86 575 88341503. E-mail address: [email protected] (Q.W. Yang). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.04.010
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Q.W. Yang, B.X. Sun / Applied Mathematical Modelling 35 (2011) 5217–5224
between the flexibility change obtained by the measured modes and the best achievable flexibility changes. With location determined, a simple algorithm using the best achievable flexibility change is also developed to determine the extent of damage. The use of the best achievable flexibility change can filter the measured flexibility change from the contamination by truncation and noise. The developed theory is validated in Section 3 with three numerical examples. The results obtained show that the location and magnitude of local damage can be identified by the proposed method using the noisy and incomplete modes. In the following theoretical development, it is assumed that structural damages only reduce the system stiffness matrix and structural refined FEM has been developed before damage occurrence. 2. Theoretical development 2.1. Damage localization using the best achievable flexibility change For the intact and the damaged structures, the global stiffness and flexibility matrices will satisfy the following relationship:
F u K u ¼ F d K d ¼ I;
ð1Þ
where Fu and Ku are the n n flexibility and stiffness matrices of the undamaged structure, Fd and Kd are the n n matrices of the damaged structure, I is the n n identity matrix. It is noted that both translational and rotational degrees of freedom are considered in the flexibility and stiffness matrices here. As is well known, damage reduces the stiffness and increases the flexibility of structures. Let DF and DK be the exact perturbation matrices that reflect the nature of the structural damage. Then the undamaged model matrices and the damaged model matrices are related as follows:
F d ¼ F u þ DF;
ð2Þ
K d ¼ K u DK:
ð3Þ
In practice, the exact DF cannot be obtained due to the limitation of the modal survey. But DF can be approximated by the first few low-frequency modes as [12,13]
DF ¼
m m X X 1 1 /dj /Tdj /uj /Tuj ; k k uj dj j¼1 j¼1
ð4Þ
where m is the number of measured modes in modal survey, kuj (/uj) and kdj (/dj) are the eigenparameters of the undamaged and damaged structures, respectively. The modes of the damaged structure can be obtained by a modal survey on it, and the modal data of the undamaged structure can be obtained by solving a generalized eigenvalue problem of the undamaged FEM or through a modal test on the intact structure. A detailed discussion of modal analysis techniques is beyond the scope of this paper, and interested readers are referred to Refs. [26–28]. Substituting Eqs. (2) and (3) into (1), and neglecting the smaller product term yields
DF K d ¼ F u DK:
ð5Þ
Rewriting Eq. (5) yields
DF ¼ F u DKF d :
ð6Þ
When damage has occurred in the structure, the stiffness matrix perturbation DK can be expressed as a sum of each elemental stiffness matrix multiplied by a damage coefficient, that is
DK ¼
N X
ai K i ð0 6 ai 6 1Þ;
ð7Þ
i¼1
where Ki is the ith elemental stiffness matrix, ai is its damage parameter, N is the total number of elements. The value of ai is 0 if the ith element is undamaged and ai is 1 or less than 1 if the corresponding element is completely or partially damaged. In Eq. (7), it is assumed that in case of damage, all the elements in the stiffness matrix Ki will be affected by same ratio ai. This assumption is used for mathematical simplicity since the practical damage is very complex. Substituting Eq. (7) into (6), one has
DF ¼
N X
ai F u K i F d :
ð8Þ
i¼1
According to Eq. (8), the changes in the flexibility could be caused by damage at a single member or at multiple members. Assume, for the time being, that the damage is caused by a single member. Without loss of generality, assume that only the ith element is damaged (ai – 0), then Eq. (8) reduces to
DF ¼ ai F u K i F d :
ð9Þ
Let the jth column of DF and Fd be represented by Dfj and fdj, respectively. That is, DF = [Df1 Dfj Dfn] and Fd = [fd1 fdj fdn]. From Eq. (9), we have
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Dfj ¼ ai F u K i fdj
ðj ¼ 1—nÞ:
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ð10Þ
Let
Ei ¼ F u K i ;
ð11Þ
cij ¼ ai fdj ;
ð12Þ
Then Eq. (10) simplifies to
Dfj ¼ Ei cij
ðj ¼ 1—nÞ:
ð13Þ
The implication of Eq. (13) is very important. According to the theory of linear algebra [29,30], Eq. (13) is valid only if the vector Dfj is a linear combination of the columns of Ei. In other words, Dfj must lie in the subspace spanned by the columns of Ei. That is to say, if the ith element is damaged, then the vector Dfj will lie exactly in the subspace spanned by the columns of Ei. If not, Dfj would not lie in the subspace spanned by the columns of Ei. According to the matrix theory, we can use the concept of the best achievable vector to evaluate whether or not Dfj lies in the subspace spanned by the columns of Ei. The best achievable vector of Dfj can be computed by
Dfija ¼ Ei ðEi Þþ Dfj
ðj ¼ 1—nÞ;
ð14Þ
where Ei is the matrix Ei where the zero columns have been removed to enhance computational efficiency, and the superscript + denotes the generalized inverse. For j = 1–n, Eq. (14) can be assembled as
DF ai ¼ Ei ðEi Þþ DF; DF ai
½Dfi1a
Dfija
ð15Þ DF ai
Dfina .
where ¼ The matrix is defined as the best achievable flexibility change. If the damage is caused by the ith element, then the matrices DF ai and DF will be identical. If not, the two matrices will be different. We can use the Euclidean distance between the two matrices to evaluate whether or not DF ai equals DF. The distance between the two matrices can be computed using the Frobenious norm
di ¼ kDF DF ai kF ;
ð16Þ
where k kF represents the Frobenious norm. If the perfect data are presented, the damaged element will has zero distance (di = 0) and all others will have nonzero values. For a structure that has N structural members, a damage localization vector, of length N, can be defined as
d¼
d1 di dN ;...; ;... dmax dmax dmax
T ;
ð17Þ
where dmax is the largest value in all distances, i.e., dmax = max (d1 di dN). For the measured data with truncation and di noise, dmax will be equal or close to zero if damage is located in element i and all other coefficients will be populated with nonzero entries. As a result, the location of damage can be determined by searching for a value that is considerably smaller than others in the vector d. 2.2. Damage quantification using the best achievable flexibility change With the most probable damaged element identified by the above method, the extent of the damage can be obtained by computing its damage parameter using the following procedure. Without loss of generality, assume that the number of damaged element is i and the corresponding damage factor is ai. Then Eq. (7) reduces to
DK ¼ ai K i
ð0 6 ai 6 1Þ:
ð18Þ
Substituting Eq. (2) into (6), one has
DF ¼ F u DKF u þ F u DK DF:
ð19Þ
Neglecting the high-order term in Eq. (19), one obtains
DF ¼ F u DKF u :
ð20Þ
Substituting Eq. (18) into (20) yields
DF ¼ ai ðF u K i F u Þ:
ð21Þ
From Eq. (21), the damage parameter ai can be calculated by
ai ¼
sumðDFÞ ; sumðF u K i F u Þ
where sum(A) represents the sum of all entries in the matrix A.
ð22Þ
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In practice, the measured modes are usually truncated and contaminated by noise. So the flexibility change DF obtained by these noisy modes using Eq. (4) is always inaccurate. When the noisy DF is used directly to compute the extent of damage by Eq. (22), unrealized values of the damage parameters can be obtained. To improve the calculation precision, we can use the best achievable flexibility change DF ai instead of DF in Eq. (21) to compute the damage parameter. Then Eq. (21) changes to
DF ai ¼ ai ðF u K i F u Þ:
ð23Þ
From Eq. (23), the damage parameter ai can be calculated by
ai ¼
sumðDF ai Þ : sumðF u K i F u Þ
ð24Þ
2.3. Incomplete measurement The measured modes are usually incomplete in practice because of the limited number of sensors, and the rotational degrees of freedom are difficult to measure. When spatial incompleteness in mode shape measurements occurs, mode shape expansion technique is usually applied to expand the measured mode shapes to match the analytical model. It has been shown that the mode shape expansion process will introduce additional errors in the expanded mode shape. But in the method described above, there is no need to use all the entries of the flexibility change in the damage identification and then the mode shape expansion can be avoided. Supposing the measured mode shapes include r DOFs (r < n), the incomplete flexibility change DF is reduced into a (r r) symmetric matrix. We can take only the corresponding entries in the damage localization and quantification procedures. 2.4. Summary of the whole technique A step-by-step summary of the whole technique is presented as follows: Step 1: Calculate DF using Eq. (4) and Ei for all structural elements using Eq. (12). Step 2: Compute di for all structural elements using Eq. (16) and construct the damage localization vector d by Eq. (17). Step 3: Select the single most probable damaged element by examining d and then calculate its damage parameter using Eq. (24). 3. Numerical examples To illustrate characteristics of the proposed damage detection algorithm, three numerical examples are presented. Example 1 is a spring-mass systems with 3 DOFs, which are used to verify the proposed method for the ideal case with complete and exact modes. Example 2 is a space truss structure and Example 3 is a two-storey frame structure. Example 1. The first example is a spring-mass system with 3 DOFs as shown in Fig. 1. Consider the nominal model of the system to have the parameters ki = 1 (i = 1–3) and mj = 1 (j = 1–3). A single damage case is studied in the example that element 2 is damaged with a stiffness loss of 10%. Using the complete and exact modes, the damage localization vector d can be computed as
d ¼ ½ 0:8167 0 1 :
ð25Þ d2 dmax
Examination of d indicates that a single damage occurred in the element 2 because of ¼ 0. Using Eq. (24), we can obtain the damage extent as a2 = 0.1111. The predicted damage extent has slight derivation from the true value because of neglecting the high-order term in Eq. (19). Example 2. The space truss structure used in this example is shown in Fig. 2. The structure was modeled using 26 truss rod elements and 12 nodes (4 restrained), for a total of 24 DOF. The basic parameters of the structure are as follows: E = 200 GPa, q = 7.8 103 kg/m3, L = 1 m, and A = 0.004 m2. Assume that only the first six modes are used to calculate the flexibility matrix change DF. Mode shapes are contaminated with 5% random noise to simulate measurement error. A single damage is simulated in element 10 with damage parameter of 0.1. Using the proposed method, the damage localization vector d of the single-damage case is shown in Fig. 3(a). Inspection of Fig. 3(a) indicates that a single damage occurred in the element 10
k1
k2 m1
k3 m2
m3
Fig. 1. Spring-mass system for Example 1.
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10
Fig. 2. A space truss structure for Example 2.
10 because that ddmax is far less than the others. Damage extents obtained by Eqs. (24) and (22) are a10 = 0.1072 and a10 = 0.1124, respectively. Compared with the assumed value 0.1, it is obvious that the result obtained by the best achievable flexibility change is more accurate than that obtained by the measured flexibility change. In order to study the impact of the order of the model on the efficiency, Fig. 3(b) gives the damage localization vector d using the first four modes. Compared with Fig. 3(a), it is obvious that the result obtained by the first six modes is more accurate than that obtained by the first four modes. Damage extents obtained by Eqs. (24) and (22) are a10 = 0.1095 and a10 = 0.1189, respectively. According to the above results, it has been shown that the damage detection results become more accurate as the number of modes used in the calculation increases.
Damage location index
Example 3. The third example is a two-storey frame structure as shown in Fig. 4 with a rectangular cross-section of 0.14 m 0.24 m. This frame was modeled with 48 equal elements of 0.2 m in length. Every node has 3 DOFs, an axial displacement, a transverse displacement and a rotation. The properties of this structure are: cross-sectional area A = 0.0336 m2; moment of inertia I = 1.6128 104 m4; Young’s modulus E = 200 GPa; shearing modulus of elasticity G = 1.3461 1010 N/m2; Poisson’s ratio v = 0.3; density q = 2500 kg/m3. Only the first six modes are used in the calculations. Furthermore, only horizontal modal displacements are assumed to be ‘measured’ in the columns while for the beams only the vertical components of mode shapes are ‘measured’. In other words, only the first six modes and 22 out of 66 DOFs are used in the present study to simulate incomplete modal data in the real situation. As before, mode shapes are contaminated with 5% random noise to simulate measurement error. Two damage cases are studied in the example. Case 1 assumed that element 4 is damage with 10% stiffness reduction. Case 2 supposed that element 7 is damaged with 10% stiffness loss. Using the proposed approach, the damage localization vector d is shown in Fig. 5 for case 1. Examination of d indicates that a single damage occurred in the element 4 because its corresponding index in Fig. 5 is far less than the others. Using Eq. (24), its
1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Element number Fig. 3a. The damage localization vector d using the first six modes when element 10 is damaged (Example 2).
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Damage location index
1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Element number Fig. 3b. The damage localization vector d using the first four modes when element 10 is damaged (Example 2).
13
14
12 13 12
15 15
14
16 16 17 17
11 11
18
10
19
18
10
19
9 9
20
8
7
6
20
5 5 4
6
7
8 21
4
21 3
22 22
3 2
23
2
23 1
24
1
24
Damage location index
Fig. 4. A two-storey frame structure for Example 3.
1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Element number Fig. 5. The damage localization vector d of case 1 (Example 3).
Damage location index
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1.2 1 0.8 0.6 0.4 0.2 0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Element number Fig. 6. The damage localization vector d of case 2 (Example 3).
damage extent can be computed as a4 = 0.1087. Fig. 6 presents the damage localization vector d for case 2. It is obvious that element 7 is damaged by inspecting Fig. 6 and its damage parameter can be calculated using Eq. (24) as a7 = 0.1091. The above results show that the proposed method can identify both the location and extent of structural damage.
4. Conclusions A new method for structural damage identification was developed in this study, which is based on the best achievable concept. The best achievable flexibility change is a projection of a measured flexibility change onto the subspace that is defined by the undamaged analytical model. Damage location can be determined by the Euclidean distance between the measured flexibility change and the best achievable flexibility change. With location identified, the magnitude of damage can be computed by a simple method using the best achievable flexibility change. Three numerical examples are used to exercise this process and measurement noise is also simulated in damage detection. The results show that the proposed method can determine accurately both the location and magnitude of structural damage with the incomplete and noisy modes. It should be noted that we have assumed only a single damage occurred in the structures for the three numerical examples. Future research on the technique should be carried out to tackle the multiple damage case and to demonstrate the procedure using experimentally measured data. Acknowledgements The authors thank the reviewers for a thorough and careful reading of the original paper. Their comments are greatly appreciated and have helped to improve the quality of the paper. This work is supported by Zhejiang Province Natural Science Foundation (Y1110949), the scientific research project of Shaoxing City (2010A23006), and the National Natural Science Foundation of China (No: 40772194). References [1] J.E. Mottershead, M.I. Friswell, Model updating in structural dynamics: a survey, J. Sound Vib. 167 (2) (1993) 347–375. [2] M.I. Friswell, J.E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. [3] Y. Zou, L. Tong, G.P. Steven, Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures – a review, J. Sound Vib. 230 (2) (2000) 357–378. [4] F.K. Chang, Structural Health Monitoring, Current Status and Perspectives, Technomic Publishing Co., Inc., Lancaster, Pennsylvania, 1997. [5] O.S. Salawu, Detection of structural damage through changes in frequency: a review, Eng. Struct. 19 (9) (1997) 718–723. [6] S.V. Modak, T.K. Kundra, B.C. Nakra, Comparative study of model updating methods using simulated experimental data, Comput. Struct. 80 (2002) 437– 447. [7] A. Alvandi, C. Cremona, Assessment of vibration-based damage identification techniques, J. Sound Vib. 292 (2006) 179–202. [8] A. Prinaris, S. Alampalli, M. Ettouney. Review of remote sensing for condition assessment and damage identification after extreme loading conditions, in: Proceedings of the 2008 Structures Congress, 2008. [9] S. Kanev, F. Weber, M. Verhaegen, Experimental validation of a finite-element model updating procedure, J. Sound Vib. 300 (2007) 394–413. [10] Q.W. Yang, J.K. Liu, Structural damage identification based on residual force vector, J. Sound Vib. 305 (2007) 298–307. [11] J.D. Collins, G.C. Hart, T.K. Hasselman, B. Kennedy, Statistical identification of structures, AIAA J. 12 (2) (1974) 185–190. [12] A.K. Pandey, M. Biswas, Experimental verification of flexibility difference method for locating damage in structures, J. Sound Vib. 184 (2) (1995) 311– 328. [13] A.K. Pandey, M. Biswas, Damage detection in structures using changes in flexibility, J. Sound Vib. 169 (1994) 3–17. [14] J. Zhao, J.T. DeWolf, Sensitivity study for vibrational parameters used in damage detection, J. Struct. Eng., ASCE 125 (4) (1999) 410–416. [15] D. Bernal, Load vectors for damage localization, J. Eng. Mech. 128 (1) (2002) 7–14. [16] D. Bernal, B. Gunes, Damage localization in output-only systems: a flexibility based approach, in: IMAC-XX, Los Angeles, California, 2002, pp. 1185– 1191. [17] D. Bernal, B. Gunes, Flexibility based approach for damage characterization: benchmark application, J. Eng. Mech. 130 (1) (2004) 61–70. [18] B. Jaishi, W.X. Ren, Damage detection by finite element model updating using modal flexibility residual, J. Sound Vib. 290 (2006) 369–387. [19] B. Jaishi, W.X. Ren, Structural finite element model updating using ambient vibration test results, J. Struct. Eng. 131 (4) (2005) 617–628.
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