Structural Damage Localization by the Principal ... - Semantic Scholar
algorithms Article
Structural Damage Localization by the Principal Eigenvector of Modal Flexibility Change Cui-Hong Li, Qiu-Wei Yang * and Bing-Xiang Sun Department of Civil Engineering, Shaoxing University, Shaoxing 312000, China; [email protected] (C.H.L.); [email protected] (B.X.S.) * Correspondence: [email protected] or [email protected]; Tel.: +86-575-88326229; Fax: +86-575-88341503 Academic Editor: Stefano Mariani Received: 2 February 2016; Accepted: 31 March 2016; Published: 13 April 2016
Abstract: Using the principal eigenvector (PE) of modal flexibility change, a new vibration-based algorithm for structural defect localization was presented in this paper. From theoretical investigations, it was proven that the PE of modal flexibility variation has a turning point with a sharp peak in its curvature at the damage location. A three-span continuous beam was used as an example to illustrate the feasibility and superiority of the proposed PE algorithm for damage localization. Furthermore, defect localization was also performed using the well-known uniform load surface approach for comparison. Numerical results demonstrated that the PE algorithm can locate structural defects with good accuracy, whereas the ULS approach occasionally missed one or two defect locations. It was found that the PE algorithm may be promising for structural defect assessment. Keywords: defect localization; principal eigenvector; modal flexibility variation; deflection; vibration
1. Introduction During the last decades, a large number of research papers have reported on structural defect assessment based on the changes in measured static/dynamic response. The theoretical basis of these algorithms lies in the fact that the static and dynamic responses are functions of structural material properties (such as stiffness, mass, and damping). As a result, if the material properties are changed due to structural defects, then structural responses must also be changed. The relevant literature reviews can be found in references [1–3]. Generally, defect identification algorithms can be divided into model based and non-model based procedures according to using finite element model (FEM) or not. Algorithms [4–16] based on FEM assess structural defects by modifying structural FEM according to structural responses. The modifications of structural FEM will indicate the location and severity of the defect. It is well known that model-based methods are computationally intensive and highly influenced by the quality of structural FEM. However, the precise FEM is often difficult to achieve in engineering practice for the simplified assumptions in the construction of the FEM, which means that the correct results might be missed. The advantages of non-model algorithms lie in their easiness and straightforwardness because structural EEMs are not needed in these approaches. Damage indexes can be established directly by the variations of structural responses before and after damage for assessing structural defects. Zhang and Aktan [17] used the uniform load surface (ULS) of structural flexibility to locate defects. Wu and Law [18,19] discovered that the curvature of ULS is sensitive to local damage in the plate structure. Wang and Qiao [20] made use of the simplified gapped-smoothing technique to improve the accuracy of the ULS algorithm. Choi et al. [21] proposed an elastic damage load theorem to locate defects in simply supported beams. Sung et al. [22] presented a normalized ULS algorithm Algorithms 2016, 9, 24; doi:10.3390/a9020024
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for damage identification in beam-like structures. Other algorithms for defect localization can be found in references [23–27]. In engineering practice, structural response parameters are sometimes difficult to obtain and/or depend on environmental factors (e.g., temperature, humidity, loading at the time of testing, etc.). If the environmental impact is not considered properly, these localization algorithms will probably give incorrect results. In view of this, Limongelli et al. [9,28,29] proposed the interpolation damage detection method (IDDM) to considering the environmental impact. Their investigations verified the effectivity of the IDDM. Manoach et al. [30,31] proposed a new damage index and a method based on the Poincare map for structural damage localization. Numerical and experimental studies confirm the applicability and sensitivity of their method in application. This study proposes a defect localization algorithm based on the principal eigenvector (PE) obtained from structural flexibility variation. The presented algorithm uses PE as a new damage indicator to locate structural defects without FEM. A three-span continuous beam is used as an example to demonstrate the efficiency of the presented PE algorithm in structural defect localization. For several damage scenarios in the structure, defect localization results obtained by the PE algorithm and the ULS procedure are both given for comparison. The numerical result demonstrated that the PE algorithm can locate structural defects with good accuracy, whereas the ULS approach occasionally missed one or two defect locations. It has been shown that the proposed PE algorithm may be more effective at identifying structural damage locations than the ULS approach. 2. Theory 2.1. The Principle of Deflection-Based Damage Localization In this section, the explicit relation between the defect and defect-induced deformation variation has been derived firstly from the theoretical investigations. Then, the basic principle of deflection-based algorithm for structural damage localization has been illustrated. For a n-DOFs structure, let Ku and Kd are the stiffness matrices of the intact structure, Fu and Fd are the flexibility matrices of the damaged structures. Then they will satisfy the following relationship: Fu ¨ Ku “ Fd ¨ Kd “ Inˆn
(1)
where Inˆn is the identity matrix. Generally, structural defects reduce the stiffness and increase the flexibility. If ∆K and ∆F are the changes of stiffness and flexibility matrices, then one has: ∆K “ Ku ´ Kd
(2)
∆F “ Fd ´ Fu
(3)
Fu Ku “ pFu ` ∆FqpKu ´ ∆Kq
(4)
From Equations (1)–(3), one has
Equation (4) can be expanded as Fu Ku “ Fu Ku ´ Fu ∆K ` ∆FpKu ´ ∆Kq
(5)
Fu ∆K “ ∆FpKu ´ ∆Kq
(6)
From Equation (5), one has Substituting Equation (3) into (6) yields Fu ∆K “ ∆FKd
(7)
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According to Equation (1), Equation (7) can be rewritten as ∆F “ Fu ∆KFd
(8)
Multiplying Equation (8) by a load vector l, the defect-induced deformation variation vector (DV) can be defined as dv “ ∆Fl “ Fu ∆KFd l (9) According to the FEM theory, the global stiffness change ∆K can be expressed as the summation of all the elemental stiffness changes, i.e., ∆K “
Ne ÿ
αie Kie , p0 ď αie ď 1q
(10)
i “1
where αie and Kie are the damage coefficient and stiffness matrix of the ith element, Ne is the number of elements in the structural FEM. αie “ 0 denotes the corresponding element is intact, otherwise the element is damaged. According to Equations (9) and (10), we have dv “
Ne ÿ
pFu Kie qpαie Fd lq
(11)
i “1
According to Equation (11), the DV may be due to a single defect or multiple defects. For brevity, the single defect case is studied firstly in the following discussion. If only αie ‰ 0, Equation (11) can be reduced to dv “ pFu Kie qpαie Fd lq (12) Let Ei “ Fu Kie
(13)
ηi “ αie Fd l
(14)
dv “ Ei ηi
(15)
Then Equation (12) simplifies to Using the Linear algebra theory [32,33], Equation (15) has important implications that dv is the linear combination of the column vectors in Ei . From Equation (13), Ei is a sparse matrix because the j j elemental stiffness matrix Kie is very sparse. Assuming ei and k i are the jth nonzero column vector of Ei and Kie , respectively, one has j
j
ei “ Fu k i
(16) j
According to Equation (16), the physical meaning of the vector ei is the deflection of the intact j
j
j
structure by considering k i as a load vector. Thus, ei is defined as the characteristic deflection and k i is defined as the characteristic force of the ith element. It has been shown from Equation (15) that the deflection variation in a structure due to defect is a linear combination of the characteristic deflections for the damaged element. For the multiple-damage case, a similar conclusion can be obtained using the derivation as before. That is to say, structural deflection variation before and after damage is always the linear combination of the characteristic deflections for those damaged elements. Now we begin to discuss the properties of the characteristic force and deflection for structural elements. For convenience, the beam element is employed to illustrate these substantive features. Figure 1 presents a Bernoulli-Euler plane beam element with two nodes giving four DOFs, whose stiffness matrix and nodal displacement vector in local co-ordinates are as follows: ue “ rv1 , θ1 , v2 , θ2 sT
(17)
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Figure 1 presents a Bernoulli-Euler plane beam element with two nodes giving four DOFs, whose Figure 1 presents a Bernoulli-Euler planevector beamin element with two nodes four DOFs, whose stiffness matrix and nodal displacement local co-ordinates are asgiving follows: stiffness matrix and nodal displacement vector in local co-ordinates are as follows: (17) uee = [v1 ,θ1 , v2 ,θ 2 ]TT Algorithms 2016, 9, 24 4 of 18 (17) u = [v1 ,θ1 , v2 ,θ 2 ]
12
»12 6L EI 12 e Ke =EIEI—3 66L L — KKe “= 3 L3—− 12 L L –− ´12 12 6L 6L 6L
6L 6L 2 4L 6L 4 L22 4L − 6L ´6L − 6L 2 L222 2L 2L
− 12 6 L − 12 6 L 2fi − 6 L 6L2 L2 ´12 2 ffi − 6 L 2L 2L ´6L 12 − 6 Lffi ffi 12 − 6 L2fl 12 ´6L − 6 L 42L2 ´6L − 6 L 4L 4L
(18) (18) (18)
is the elastic modulus, I ismoment the moment of inertia, is the element whereE E where is the elastic modulus, I is the of inertia, and Land is theLelement length. length. where E is the elastic modulus, I is the moment of inertia, and L is the element length.
θ1 v θ1 1 1 v1 1
EI EI
v2 v2
θ2 θ2
2 2
L L Figure1.1.AABernoulli-Euler Bernoulli-Eulerplane planebeam beamelement element Figure Figure 1. A Bernoulli-Euler plane beam element e
Thefour fourcolumn columnvectors vectorsofof in Equation arecharacteristic the characteristic the BernoulliThe Ke K ine Equation (18) (18) are the forcesforces for thefor Bernoulli-Euler Theplane four column vectors ofinKlocal in Equation (18) are the characteristic forces for the BernoulliEuler beam element co-ordinates. Figure 2 shows the corresponding plane beam element in local co-ordinates. Figure 2 shows the corresponding load configurationsload of Euler plane beam element in local co-ordinates. Figure 2 shows the corresponding load configurations of these characteristic forces. these characteristic forces. configurations of these characteristic forces. 12 12 1 1 6L 6L
L L
6L 6L 1 1 4L2 4L2
L L
2 2 6L 6L L L
6L 6L 1 1 2L2 2L2
2L2 2L2 2 2 6L 6L 12 12
6L 6L 1 1 12 12
6L 6L 2 2 12 12
L L
4L2 4L2 2 2 6L 6L
Figure 2. Load configurations of the four characteristic forces. Figure 2. Load configurations of the four characteristic forces.
From Figure 2, it is apparent that all the load configurations are the self-equilibrating force systems. This conclusion is also valid for other types of finite elements. As can be seen in Figure 3, the internal
the statically determinate structure), only the element associated with the characteristic force is deformed, whereas the remaining part of the structure is not deformed and only has rigid body motion. That is to say, the elemental characteristic deflection will, due to its characteristic force, consist of several parts of the rigid displacement, but not the element itself. As stated before, Algorithms 2016, 9, 24 5 of 18 structural deflection change due to defects is a linear combination of the characteristic deflections corresponding to those damaged elements. Thus, the deflection change of the structure without redundant constraints under an arbitrary load will consist of several parts of the rigid displacement force (IF) in most parts of the structure under a characteristic force will be zero, except the element but not the damaged segments. In other words, the turning points between each segment of the rigid associated with this characteristic force. displacement in the shape of the deflection change for the structure are the locations of the defects. 12 6L 1
2
6L IF=0
12 IF=0
IF=0 6L 2L2 1
2
2
4L IF=0
6L IF=0
IF=0 12 6L 1
2 6L
IF=0
12
IF=0
IF=0 6L 4L2 1
2
2
2L IF=0
6L IF=0
IF=0
Figure Figure 3. 3. The internal force (IF) distributions of the structure under under characteristic characteristic forces. forces.
Forshort, the statically structure, derivation the same as that In one canindeterminate conclude from Figuresthe 2 and 3 that:process (1) theis characteristic forceofisthe a statically determinate structure. The only difference is that the IF in some parts of the statically self-equilibrating force; (2) the characteristic force acts only on its own element and not on the rest of indeterminate structure under a characteristic force will be slightly than zero because of the the structure. When the characteristic force is applied to the structuregreater without redundant constraints redundant constraint limitation. Correspondingly, the deflection variation damage willforce consist (i.e., the statically determinate structure), only the element associated withdue theto characteristic is of several parts of the approximate rigid displacement (or the rigid displacement) but not the deformed, whereas the remaining part of the structure is not deformed and only has rigid body motion. damaged segments. Then, the turning points between segment of the approximate rigid That is to say, the elemental characteristic deflection will,each due to its characteristic force, consist of displacement rigid displacement) thethe deflection change the defect locations. As a several parts of(or thethe rigid displacement, butinnot element itself. Asshape statedare before, structural deflection result, the peaksisina the curvature of structural deflection variation will indicate the locations of change duesharp to defects linear combination of the characteristic deflections corresponding to those structuralelements. defects. Thus, the deflection change of the structure without redundant constraints under damaged an arbitrary load will consist of several parts of the rigid displacement but not the damaged segments. In other words, the turning points between each segment of the rigid displacement in the shape of the deflection change for the structure are the locations of the defects. For the statically indeterminate structure, the derivation process is the same as that of the statically determinate structure. The only difference is that the IF in some parts of the statically indeterminate structure under a characteristic force will be slightly greater than zero because of the redundant constraint limitation. Correspondingly, the deflection variation due to damage will consist of several parts of the approximate rigid displacement (or the rigid displacement) but not the damaged segments. Then, the turning points between each segment of the approximate rigid displacement (or the rigid displacement) in the deflection change shape are the defect locations. As a result, the sharp peaks in the curvature of structural deflection variation will indicate the locations of structural defects.
2.2. Deflection Estimated by Modal Flexibility Change and PE Method for Damage Localization In Equation (9), the flexibility change matrix ΔF can be calculated approximately using the first few low-frequency modes, as shown in [34,35] m
ΔF =
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j =1
m
1
λdj
φdjφdjT − j =1
1
λuj
φujφujT
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(19)
2.2. Deflection Estimated by Modal Flexibility Changeλandand PE Method Damage Localization number of measured modes, λ dj areforthe eigenvalues of the structure where m is the uj In and Equation the flexibility matrix ∆F can be calculated approximately using the before after (9), damage, φuj andchange φdj are the corresponding eigenvectors, respectively. For first the few low-frequency modes, as shown in [34,35] damaged structure, the modal data can be acquired through a modal test in practice. For the intact mthrough solving m a generalized eigenvalue problem of the structure, the modal data are acquiredÿ ÿ 1 1 T ∆F “ φ φ ´ φ φT (19) dj dj undamaged FEM or by the modal experiment. λdj λuj uj uj j “1 j“1 Using matrix theory [32,33], the principal eigenvector of ΔF can be given by the solutions of the following where m is theeigenvalue number of problem: measured modes, λ and λ are the eigenvalues of the structure before and uj
dj
after damage, ϕuj and ϕdj are the correspondingeigenvectors, respectively. For the damaged structure, ξ1 ΔF = ξ the modal data can be acquired through a modal (20) λtest in 1practice. For the intact structure, the modal 1 data are acquired through solving a generalizedeigenvalue problem of the undamaged FEM or by the modal experiment. where λ1 is the principal eigenvalue and ξ 1 is the principal eigenvector of ΔF , respectively. Using matrix theory [32,33], the principal eigenvector of ∆F can be given by the solutions of the According to Equation (20), the physical meaning of the vector ξ 1 is the structural deflection following eigenvalue problem: ˆ ˙ ξ1 1 ∆F “ξ ξ 1 as a load vector. Thus, according to (20) variation before and after damage by considering the 1 λ1
λ1
where λ1 is the principal eigenvalue and ξ 1defects is the principal eigenvector ∆F, respectively. According principle stated in Section 2.1, structural can be located in theoffollowing two ways: (1) the to Equation (20), the physical meaning of the vector ξ 1 is the structural deflection variation before turning points in the shape of ξ 1 are the locations of defects; (2) the sharp peaks in the curvature of and after damage by considering λ1 ξ 1 as a load vector. Thus, according to the principle stated in 1 ξ1 indicate ξ1 can be the locations of structural defects.inItthe is known thattwo the ways: curvature of turning computed Section 2.1, structural defects can be located following (1) the points in the using difference approximation as follows: shape the of ξcentral are the locations of defects; (2) the sharp peaks in the curvature of ξ indicate the locations 1 1 of structural defects. It is known that the curvature i +1 of ξ 1 ican be i −1computed using the central difference ζ − 2ζ 1 + ζ 1 approximation as follows: (21) (ζ 1i ) " = 1 2 i `1 i ` ζ i ´1 h ζ ´ 2ζ 2 1 1 (21) pζ 1i q “ 1 2 i i " h where ζ is the i th coefficient of ξ , h is the length of structural element, and (ζ ) is the i th 1
1
1
2
where ζ 1i is the ith coefficient of ξ 1 , h is the length of structural element, and pζ 1i q is the ith curvature curvature coefficient of ξ 1 . coefficient of ξ 1 .
3. 3. Numerical NumericalExample Example The demonstrate the The continuous continuous beam beam in in Figure Figure 44 is is used used to to demonstrate the defect defect localization localization ability ability of of the the presented PE algorithm. presented PE algorithm. A 1
3
5
7
9
11
B 13
15
17
19
21
23
C 25
27
29
31
33
35
D
Figure 4. A A beam beam structure structure with with three three spans. Figure
Note example is is aa statically statically indeterminate indeterminate structure. structure. For Note that that this this example For the the structure, structure, the the elastic elastic 3 Kg/m3,3 1.0416× modulus, density, moment of inertia, and cross-sectional area are 200 GPa, 7.8×10 modulus, density, moment of inertia, and cross-sectional area are 200 GPa, 7.8 ˆ 10 Kg/m3 , −6m4, 0.0025 2 10 In the FEM, the structure divided intois36divided elements, which gives 33 1.0416 ˆ 10´6 m m4,, respectively. 0.0025 m2 , respectively. In the FEM, isthe structure into 36 elements, translational DOFs and 37 rotational DOFs. The length of each segment in the beam FEM is 0.1m. In which gives 33 translational DOFs and 37 rotational DOFs. The length of each segment in the beam the following discussion, only the first four modal parameters with the translational degrees of FEM is 0.1 m. In the following discussion, only the first four modal parameters with the translational freedom used in simulate the incomplete measurements. Defect Defect in theinbeam was degrees ofare freedom areorder used to in order to simulate the incomplete measurements. the beam simulated by reducing the elastic modulus of some structural elements. Five defect scenarios was simulated by reducing the elastic modulus of some structural elements. Five defect scenarios are are assumed assumed in in the the example. example. Scenario Scenario 1: 1: the thedefect defectoccurred occurredin in element element77 with with 20% 20% stiffness stiffness reduction. reduction. Scenario 2: the defect occurred in element 13 with 20% stiffness reduction. Scenario 3: the defects occurred in elements 5 and 17 both with 20% stiffness reduction. Scenario 4: the defects occurred in elements 5 and 31 both with 20% stiffness reduction. Scenario 5: the defects occurred in elements 7, 19 and 33 all with 20% stiffness reduction. Detection results of each damage scenario obtained
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Scenario 2: the defect occurred in element 13 with 20% stiffness reduction. Scenario 3: the defects7 of 18 occurred in elements 5 and 17 both with 20% stiffness reduction. Scenario 4: the defects occurred in elements 5 and 31 both with 20% stiffness reduction. Scenario 5: the defects occurred in elements 7, by the presented PE algorithm and the well-known ULS method are both given to compare the 19 and 33 all with 20% stiffness reduction. Detection results of each damage scenario obtained by the localization performance. presented PE algorithm and the well-known ULS method are both given to compare the localization Using the exact data without noise, the principal eigenvector (PE) of modal flexibility change performance. and the ULS are plotted in Figure 5a,b, respectively. Accordingly, Usingvariation the exact for datadamage withoutscenario noise, the1principal eigenvector (PE) of modal flexibility change the PE curvature and the ULS variation curvature for the case of damage scenario 1 are plotted and the ULS variation for damage scenario 1 are plotted in Figure 5a,b, respectively. Accordingly, the in Figure respectively. PE6a,b, curvature and the ULS variation curvature for the case of damage scenario 1 are plotted in Figure 6a,b, respectively.
(a)
(b) Figure 5. (a) Damage index (principal eigenvector (PE)) map using the first four exact modes when
Figure 5. (a) Damage index (principal eigenvector (PE)) map using the first four exact modes when the the defect in element 7; (b) Damage index (uniform load surface (ULS) variation) map using8 of 17 Algorithms 2016, 9,occurred 24 defect occurred in element 7; (b) Damage index (uniform load surface (ULS) variation) map using the the first four exact modes when the defect occurred in element 7 (×10−4). first four exact modes when the defect occurred in element 7 (ˆ10´4 ).
(a)
(b) Figure 6. (a) Damage index (PE curvature) map using the first four exact modes when the defect
Figure 6. (a) Damage index (PE curvature) map using the first four exact modes when the defect occurred in element 7; (b) Damage index (ULS variation curvature) map using the first four exact occurred in element 7; (b) Damage index (ULS variation curvature) map using the first four exact modes when the defect occurred in element 7 (×10−3). modes when the defect occurred in element 7 (ˆ10´3 ).
One can see that the transnational DOFs numbered 7 and 8 are the turning points in Figure 5 and the sharp peaks in Figure 6, respectively. In the beam model, the seventh and the eighth transnational DOFs are exactly corresponding to the seventh element. This means that the seventh segment could be successfully detected to be the defect area by inspecting the turning points in Figure 5, or by inspecting the sharp peaks in Figure 6. Note that the human inspection and intervention is
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(b) 7 and 8 are the turning points in Figure 5 and One can see that the transnational DOFs numbered the sharp6.peaks in Figure 6, respectively. In the beam model, and modes the eighth transnational Figure (a) Damage index (PE curvature) map using the the firstseventh four exact when the defect DOFs are exactly corresponding to the seventh element. This means that the seventh segment could be occurred in element 7; (b) Damage index (ULS variation curvature) map using the first four exact successfully detected to be the defect area by inspecting the turning points in Figure 5, or by inspecting modes when the defect occurred in element 7 (×10−3). the sharp peaks in Figure 6. Note that the human inspection and intervention is required in the above damage localization process. One can see that the transnational DOFs numbered 7 and 8 are the turning points in Figure 5 Damage scenario 2 is used to verify the validity of the two algorithms when the defect occurred and the sharp peaks in Figure 6, respectively. In the beam model, the seventh and the eighth near the supported boundary. Figures 7 and 8 showed the detection results of this scenario. One can transnational DOFs are exactly corresponding to the seventh element. This means that the seventh conclude that the damaged element 13 can be detected by inspecting the turning points in Figure 7, or segment could bethe successfully by inspecting sharp peaksdetected in Figureto8.be the defect area by inspecting the turning points in Figure 5, or by inspecting the3–5 sharp peaks Figurethe 6. PE Note that thewith human inspection andthe intervention is Damage cases are used to in compare algorithm the ULS method for multiple required in the above damage process. damage scenarios. Figures 9localization and 10 showed the results of damage localization for damage case 3. From Figures 9a and 10a, one can see that the damaged 5 and 17 were clearly by Damage scenario 2 is used to verify the validity of elements the two algorithms when theidentified defect occurred PE curvature. From Figure 9b, 7the ULS variationthe method fails toresults detect the damaged element nearPE theand supported boundary. Figures and 8 showed detection of this scenario. One can 17. It can be seen from Figure 10 that the PE curvature provides comparatively better damage location conclude that the damaged element 13 can be detected by inspecting the turning points in Figure 7, predictions than ULSpeaks curvature. or by inspecting thethe sharp in Figure 8.
(a)
(b) Figure 7. (a) Damage index firstfour fourexact exactmodes modes when defect occurred in Figure 7. (a) Damage index(PE) (PE)map mapusing using the first when the the defect occurred in element Damage index(ULS (ULSvariation) variation) map four exact modes when the defect element 13; 13; (b) (b) Damage index mapusing usingthe thefirst first four exact modes when the defect −4).´4 ). occurred in element 13(ˆ10 occurred in element 13(×10
(a)
(b) Figure 7. (a) Damage index (PE) map using the first four exact modes when the defect occurred in Algorithmselement 2016, 9, 13; 24 (b) Damage index (ULS variation) map using the first four exact modes when the defect 9 of 18
occurred in element 13(×10−4).
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(a) Figure 8. Cont.
(b) Figure 8. (a) Damage index (PE curvature) map using the first four exact modes when the defect occurred in element 13; (b) Damage index (ULS variation curvature) map using the first four exact modes when the defect occurred in element 13 (×10−3).
Damage cases 3–5 are used to compare the PE (b) algorithm with the ULS method for the multiple damage scenarios. Figures 9 and 10 showed the results of damage localization for damage case 3. Figure 8. (a) Damage index (PE curvature) map using the first four exact modes when the defect Figure 8. (a) (PE see curvature) using elements the first four exact when identified the defect by From Figures 9aDamage and 10a,index one can that themap damaged 5 and 17 modes were clearly occurred in element 13; (b) Damage index (ULS variation curvature) map using the first four exact occurred in elementFrom 13; (b) Damage (ULS variation curvature) map usingthe thedamaged first four exact PE and PE curvature. Figure 9b, index the ULS variation method fails to detect element modes when the defect occurred in element 13 (×10−3´3 ). modes when the defect occurred in element 13 (ˆ10 ). 17. It can be seen from Figure 10 that the PE curvature provides comparatively better damage location predictions than the3–5 ULSare curvature. Damage cases used to compare the PE algorithm with the ULS method for the multiple damage scenarios. Figures 9 and 10 showed the results of damage localization for damage case 3. From Figures 9a and 10a, one can see that the damaged elements 5 and 17 were clearly identified by PE and PE curvature. From Figure 9b, the ULS variation method fails to detect the damaged element 17. It can be seen from Figure 10 that the PE curvature provides comparatively better damage location predictions than the ULS curvature.
(a)
(a)
(b) Figure 9. (a) Damage index (PE) map using the first four exact modes when the defects occurred in Figure 9. (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 17; (b) Damage index (ULS variation) map using the first four exact modes when the elements 5 and 17; (b) Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements 5 and 17 (×10−4). defects occurred in elements 5 and 17 (ˆ10´4 ).
(b) Figure 9. (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 17; (b) Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements 5 and 17 (×10−4).
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(a) (a)
(b) (b) Figure 10. (a) Damage index (PE curvature) map using the first four exact modes when the defects Figure 10. (a) Damage index (PE curvature) map using the first four exact modes when the defects occurred and 17; (b)curvature) Damage index using thethe firstdefects four Figure 10.in (a)elements Damage5 index (PE map (ULS usingvariation the first curvature) four exact map modes when occurred in elements 5 and 17; (b) Damage index (ULS variation−3 curvature) map using the first four exact modes when the5 defects in elements 5 and 17 (×10 curvature) ). occurred in elements and 17;occurred (b) Damage index (ULS variation map using the first four exact modes when the defects occurred in elements 5 and 17 (ˆ10−3´3 ). exact modes when the defects occurred in elements 5 and 17 (×10 ).
Figures 11–14 presented the damage localization results for damage case 4 using the exact modal
Figures 11–14 presented localization for11damage damage case usingthe the exact modal data and the data with 3%the noise, respectively. Fromresults Figures and 12,case both damage localization Figures 11–14 presented thedamage damage localization results for 4 4using exact modal methods can achieve satisfactory by usingFrom error-free data. data and thethe data with 3% respectively. From Figures 11and and12, 12both bothdamage damagelocalization localization data and data with 3%noise, noise,results respectively. Figures 11 methods can achieve methods can achievesatisfactory satisfactoryresults resultsby byusing using error-free error-free data. data.
(a)
(a)
(b) Figure 11. (a) Damage index (PE) map using the (b) first four exact modes when the defects occurred in elements 5 and 31; (b) Damage index (ULS variation) map using the first four exact modes when the Figure 11. (a) Damage index (PE) map using −4). the first four exact modes when the defects occurred in defects in elements 5 and 31 (×10 Figure 11. occurred (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 31; (b) Damage index (ULS variation) map using the first four exact modes when the elements 5 and 31; (b) Damage index (ULS variation) map using the first four exact modes when the −4). defects occurred in elements 5 and 31 (×10´4
defects occurred in elements 5 and 31 (ˆ10
).
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(a) (a)
(b) (b) Figure Damage index(PE (PEcurvature) curvature) map map using using the when thethe defects Figure 12. 12. (a) (a) Damage index the first firstfour fourexact exactmodes modes when defects Figure 12.in(a) Damage5 index (PE map (ULS using variation the first four exact modes whenthe thefirst defects occurred elements and 31; (b)curvature) Damage index curvature) map using four occurred in elements 5 and 31; (b) Damage index (ULS variation curvature) map using the first four occurred in elements and 31;occurred (b) Damage index (ULS variation map using the first four exact modes when the5 defects in elements 5 and 31 (×10−3curvature) ). ´3 exact modes when thethe defects 31 (×10 (ˆ10 −3). ). exact modes when defectsoccurred occurredininelements elements 55 and and 31
(a) (a)
(b) (b)
Figure 13. (a) Damage index (PE) map using the first four modes with 3% noise when elements 5 and Figure 13. (a) Damage index (PE) map using the first four 3% noise when elements and 31 are (b)index Damage mapmodes using with the first modes with 3% 5noise Figure 13. damaged; (a) Damage (PE)index map(ULS usingvariation) the first four modes with 3%four noise when elements 5 and 31 are damaged; (b) Damage index (ULS variation) map using the first four modes with 3% noise −4 ). when elements 31 areindex damaged 31 are damaged; (b)5 and Damage (ULS(×10 variation) map using the first four modes with 3% noise when −4 when elements 5 and 31 are damaged ´4(×10 ).
elements 5 and 31 are damaged (ˆ10
).
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(a) (a)
(b)
(b) Figure 14. (a) Damage index (PE curvature) map using the first four modes with 3% noise when Figure 14. (a) Damage index (PE curvature) map using the first four modes with 3% noise when Figure5 14. index (PE curvature) map (ULS usingvariation the first four modes with noise elements and(a) 31 Damage are damaged; (b) Damage index curvature) map3% using thewhen first four elements 5 and 31 are damaged; (b) Damage index (ULS variation curvature) map using the first four elements 5 and 31 are damaged; (b) Damage index (ULS variation curvature) map using the first four −3). modes with 3% noise when elements 5 and 31 are damaged (×10 modes with 3% noise when elements 5 and 31 are damaged (×10−3). modes with 3% noise when elements 5 and 31 are damaged (ˆ10´3 ).
From Figure 13,13, one of PE PEand andULS ULSvariation variation identify From Figure onecan cansee seethat that the the indexes indexes of cancan stillstill identify the the From Figure 13, one can see that the indexes of PE and ULS variation can still identify the damaged elements 5 and evenifif3% 3%noise noise is considered. considered. According to to Figure 14, 14, it isitapparent that that damaged elements 5 and 3131 even According Figure is apparent damaged elements 5 and 31 even if 3% noise is considered. According to and Figure 14, it is apparent that the curvature indexes are seriously affected by the measurement error, the damaged segments the curvature indexes are seriously affected the measurement error, and the damaged segments the curvature indexes by arethese seriously affected by with the measurement error, and the damaged segments cannot be determined curvature maps cannot be determined by these curvature maps withnoise. noise. cannot be determined by these curvature maps with noise. Figures 15–18 presented the damage localization results for case 5 using the exact modal Figures 15–18 presented the damage localization resultsfor fordamage damage case 5 using the exact modal damage localization results 5 using the see exact modal data Figures15–18 and the datapresented with 3% the noise, respectively. From Figuresdamage 15a andcase 16a, one can that the datadata andand thethedata 3%noise, noise, respectively. Figures 15a and 16a, one can see that the datawith with 3% respectively. FromFrom Figures 15a and one can see that the damaged damaged elements 7, 19 and 33 were clearly identified by PE and PE16a, curvature. From Figure 15b, the damaged elements 7, 19 and 33 were clearly identified by PE and PE curvature. From Figure 15b, the elements 7, 19 and 33 were identified by PE and PE curvature. Figure 15b, ULS variation method fails clearly to detect the damaged element 19. It can From be concluded thatthe theULS PE ULSvariation variationcan method to detect the damaged element 19. Itthe canULS bethat concluded that the PE failsfails to identify detect the damaged element 19. It can be concluded the PE algorithm algorithmmethod precisely multiple defect locations, whereas method occasionally can precisely identify multiple defect locations, whereas the ULS method occasionally missed multiple algorithm can precisely identify multiple defect locations, whereas the ULS method occasionally missed multiple damage locations. damage locations. missed multiple damage locations.
(a) Figure 15. Cont.
(a)
Figure 15.Cont. Cont. Figure 15.
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(b) (b)
Figure 15. (a) index (PE) firstfour fourexact exactmodes modes when defects occurred Figure 15. Damage (a) Damage index (PE)map mapusing using the the first when thethe defects occurred in in Figure 15. (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 7, 19 and 33; (b) Damage index (ULS variation) map using the first four exact modes elements 7, 19 and 33; (b) Damage index (ULS variation) map using the first four exact modes whenwhen ´4 −4 elements 7, 19 and 33;in(b) Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements7,7, 19and and 33 (×10 (ˆ10 the defects occurred elements 19 33 ).). the defects occurred in elements 7, 19 and 33 (×10−4).
(a) (a)
(b) (b)
Figure 16. (a) Damage index (PE curvature) map using the first four modes without noise when Figure 16. (a) index (PE curvature) map using the first fourcurvature) modes without noise when elements 19Damage and 33 are damaged; Damage index (ULS mapnoise using the first Figure7,16. (a) Damage index (PE (b) curvature) map using the variation first four modes without when −3 elements 7, 19 33noise are damaged; (b)(b)Damage index variation(×10 curvature) map using four elements modes without when elements 7, 19 and 33 (ULS are damaged ). 7,and 19 and 33 are damaged; Damage index (ULS variation curvature) map using the the firstfirst −3). ´3 four modes without noise when elements 7, 19 and 33 are damaged (×10 four modes without noise when elements 7, 19 and 33 are damaged (ˆ10 ).
(b) Figure 16. (a) Damage index (PE curvature) map using the first four modes without noise when elements Algorithms 2016, 7, 9, 19 24 and 33 are damaged; (b) Damage index (ULS variation curvature) map using the first14 of 18 four modes without noise when elements 7, 19 and 33 are damaged (×10−3).
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Figure 17. Cont.
(b) (b) Figure 17. (a) Damage index (PE) map using the first four modes with 3% noise when elements 19 Figure 17.17. (a)(a) Damage modes with with3% 3%noise noisewhen whenelements elements7,7, 7, Figure Damageindex index(PE) (PE)map mapusing using the the first first four four modes 1919 and 33 are damaged; (b) Damage index (ULS variation) map using the first four modes with 3% noise and 3333 are damaged; map using using the thefirst firstfour fourmodes modeswith with3% 3%noise noise and are damaged;(b) (b)Damage Damageindex index(ULS (ULS variation) variation) map ´4 when elements 7, 19 and 33 are damaged (×10−4−4 ). ). when elements 7, 19 and 33 are damaged (ˆ10 when elements 7, 19 and 33 are damaged (×10 ).
(a) (a)
(b) (b) Figure (a)Damage Damageindex index(PE (PE curvature) curvature) map map using the first Figure 18.18. (a) using the the first first four four modes modeswith with3% 3%noise noisewhen when Figure 18. (a) Damage index (PE curvature) map using four modes with 3% noise when elements 7, 19 and 33 are damaged; (b) Damage index (ULS variation curvature) map using the elements 7, 7, 19 19 and and 33 33 are are damaged; damaged; (b) (b) Damage Damage index index (ULS (ULS variation variation curvature) curvature) map map using using the thefirst first elements first four modes with 3% noise when elements 7, 19 and 33 are damaged. four modes with 3% noise when elements 7, 19 and 33 are damaged. four modes with 3% noise when elements 7, 19 and 33 are damaged.
From Figure 17, one can see that the indexes of PE and ULS variation can still identify the From Figure 17, one can see that the indexes of PE and ULS variation can still identify the damaged elements 7, 19 and 33 even if 3% noise is considered. From Figure 18, it is apparent that the damaged elements 19seriously and 33 even if 3% is considered. From Figure 18, it is apparent that the curvature indexes7,are affected bynoise the measurement error, and the damaged locations cannot curvature indexesbyare seriously affected the noise. measurement error, and the damaged locations cannot be determined these curvature mapsby with be determined by these curvature maps with noise. In order to study the sensitivity of the proposed method to the number of modes, Figures 19 and
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From Figure 17, one can see that the indexes of PE and ULS variation can still identify the damaged elements 7, 19 and 33 even if 3% noise is considered. From Figure 18, it is apparent that the curvature indexes are seriously affected by the measurement error, and the damaged locations cannot be determined by these curvature maps with noise. In order to study the sensitivity of the proposed method to the number of modes, Figures 19 and 20 give the damage localization results of damage case 4 using the first one, two and three modes without noise, respectively. From Figures 19 and 20 one can see that the ULS method is more sensitive to the number of modes than the PE method, and both damage localization methods can achieve reasonable results, even if only the first mode without noise is used in the calculation.
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(a)
(b) Figure 19. (a) Figure 19. (a) Damage Damage index index (PE) (PE) map map using using the the first first one, one, two two and and three three modes modes without without noise, noise, respectively, (b)(b) Damage index (ULS variation) mapmap using the respectively, when when elements elements55and and3131are aredamaged; damaged; Damage index (ULS variation) using −4). first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (×10 the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (ˆ10´4 ).
(a)
(b) Figure 19. (a) Damage index (PE) map using the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged; (b) Damage index (ULS variation) map using the16 of 18 Algorithms 2016, 9, 24 first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (×10−4).
(a)
(b) Figure 20. (a) Damage index (PE curvature) map using the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged; (b) Damage index (ULS variation curvature) map using the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (ˆ10´3 ).
4. Conclusions A new vibration-based method has been developed for structural damage localization, which is based on the principal eigenvector (PE) of modal flexibility change. In order to verify the feasibility of the PE algorithm, a three-span continuous beam was investigated for several damage scenarios with and without noise. Furthermore, the PE method was compared with the well-known ULS method to show its advantages. According to the numerical results, the presented PE method may be more effective at identifying the defect locations of the beam structure than the ULS method. The proposed PE procedure may have good prospects in structural damage localization. Acknowledgments: This work is supported by the Scientific Research Project of Education of Zhejiang Province (Y201430539). Author Contributions: Cui-Hong Li and Qiu-Wei Yang proposed the idea and constructed the principal eigenvector method. Qiu-Wei Yang and Bing-Xiang Sun performed the numerical simulations. Cui-Hong Li and Qiu-Wei Yang wrote the article. Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2.
Doebling, S.W.; Farrar, C.R.; Prime, M.B. A summary review of vibration-based damage identification methods. Shock Vib. Dig. 1998, 30, 91–105. [CrossRef] Carden, E.P.; Fanning, P. Vibration based condition monitoring: A review. Struct. Health Monit. 2004, 3, 355–377. [CrossRef]
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Prinaris, A.; Alampalli, S.; Ettouney, M. Review of remote sensing for condition assessment and damage identification after extreme loading conditions. In Proceedings of the 2008 Structures Congress, Vancouver, BC, Canada, 24–26 April 2008. Ashokkumar, C.R.; Lyengar, N.G.R. Partial eigenvalue assignment for structural damage mitigation. J. Sound Vib. 2011, 330, 9–16. [CrossRef] Yang, Z.C.; Wang, L. Structural damage detection by changes in natural frequencies. J. Intell. Mater. Syst. Struct. 2010, 21, 309–319. [CrossRef] Zhu, H.P.; Li, L.; He, X.Q. Damage detection method for shear buildings using the changes in the first mode shape slopes. Comput. Struct. 2011, 89, 733–743. [CrossRef] Weber, B.; Paultre, P. Damage identification in a truss tower by regularized model updating. J. Struct. Eng. 2010, 136, 1943–1953. [CrossRef] Adewuyi, A.P.; Wu, Z.S. Modal macro-strain flexibility methods for damage localization in flexural structures using long-gage FBG sensors. Struct. Control Health Monit. 2011, 18, 341–360. [CrossRef] Limongelli, M.P. Frequency response function interpolation for damage detection under changing environment. Mech. Syst. Signal Process. 2010, 24, 2898–2913. [CrossRef] Lima, A.M.G.; Faria, A.W.; Rade, D.A. Sensitivity analysis of frequency response functions of composite sandwich plates containing viscoelastic layers. Compos. Struct. 2010, 92, 364–376. [CrossRef] Lu, Z.R.; Law, S.S. Differentiating different types of damage with response sensitivity in time domain. Mech. Syst. Signal Process. 2010, 24, 2914–2928. [CrossRef] Lu, Z.R.; Huang, M.; Liu, J.K. State-space formulation for simultaneous identification of both damage and input force from response sensitivity. Smart Struct. Syst. 2011, 8, 157–172. [CrossRef] Wong, C.N.; Huang, H.Z.; Xiong, J.Q.; Lan, H.L. Generalized-order perturbation with explicit coefficient for damage detection of modular beam. Arch. Appl. Mech. 2011, 81, 451–472. [CrossRef] Yang, Q.W. A new damage identification method based on structural flexibility disassembly. J. Vib. Control 2011, 17, 1000–1008. [CrossRef] Yang, Q.W.; Sun, B.X. Structural damage localization and quantification using static test data. Struct. Health Monit. 2011, 10, 381–389. [CrossRef] Yang, Q.W.; Liu, J.K.; Li, C.H.; Liang, C.F. A Universal fast algorithm for sensitivity-based structural damage detection. Sci. World J. 2013, 2013. [CrossRef] [PubMed] Zhang, Z.; Aktan, A.E. Application of modal flexibility and its derivatives in structural identification. Res. Nondestruct. Eval. 1998, 10, 43–61. [CrossRef] Wu, D.; Law, S.S. Damage localization in plate structures from uniform load surface curvature. J. Sound Vib. 2004, 276, 227–244. [CrossRef] Wu, D.; Law, S.S. Sensitivity of uniform load surface curvature for damage identification in plate structures. J. Vib. Acoust. 2005, 127, 84–92. [CrossRef] Wang, J.; Qiao, P.Z. Improved damage detection for beam-type structures using a uniform load surface. Struct. Health Monit. 2007, 6, 99–110. [CrossRef] Choi, I.Y.; Lee, J.S.; Choi, E.; Cho, H.N. Development of elastic damage load theorem for damage detection in a statically determinate beam. Comput. Struct. 2004, 82, 2483–2492. [CrossRef] Sung, S.H.; Jung, H.J.; Jung, H.Y. Damage detection of beam-like structures using the normalized curvature of a uniform load surface. J. Sound Vib. 2013, 332, 1501–1519. [CrossRef] Tomaszewska, A. Influence of statistical errors on damage detection based on structural flexibility and mode shape curvature. Comput. Struct. 2010, 88, 154–164. [CrossRef] Koo, K.Y.; Lee, J.J.; Yun, C.B.; Kim, J.T. Damage detection in beam-like structures using deflections obtained by modal flexibility matrices. Smart Struct. Syst. 2008, 4, 605–628. [CrossRef] Qiao, P.; Lu, K.; Lestari, W.; Wang, J. Curvature mode shape-based damage detection in composite laminated plates. Compos. Struct. 2007, 80, 409–428. [CrossRef] Fan, W.; Qiao, P. Vibration-based damage identification methods: A review and comparative study. Struct. Health Monit. 2011, 10, 83–111. [CrossRef] Hamze, A.; Gueguen, P.; Roux, P.; Baillet, L. Comparative study of damage identification algorithms applied to a Plexiglas beam. In Proceedings of the 15th International Conference on Experimental Mechanics, Porto, Portuga, 22–27 July 2012; pp. 22–27.
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Domaneschi, M.; Limongelli, M.P.; Martinelli, L. Vibration based damage localization using MEMS on a suspension bridge model. Smart Struct. Syst. 2013, 12, 679–694. [CrossRef] Domaneschi, M.; Limongelli, M.P.; Martinelli, L. Damage detection and localization on a benchmark cable-stayed bridge. Earthq. Struct. 2015, 8, 1113–1126. [CrossRef] Manoach, E.; Samborski, S.; Mitura, A.; Warminski, J. Vibration based damage detection in composite beams under temperature variations using Poincare maps. Int. J. Mech. Sci. 2012, 62, 120–132. [CrossRef] Manoach, E.; Trendafilova, I. Large amplitude vibrations and damage detection of rectangular plates. J. Sound Vib. 2008, 315, 591–606. [CrossRef] Herstein, I.N.; Winter, D.J. Matrix Theory and Linear Algebra; Macmillan Publishing Company: Indianapolis, IN, USA, 1988. Datta, B.N. Numerical Linear Algebra and Applications; Brooks/Cole Publishing Company: Pacific Grove, CA, USA, 1995. Pandey, A.K.; Biswas, M. Damage detection in structures using changes in flexibility. J. Sound Vib. 1994, 169, 3–17. [CrossRef] Pandey, K.; Biswas, M. Experimental verification of flexibility difference method for locating damage in structures. J. Sound Vib. 1995, 184, 311–328. [CrossRef] © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).
Structural Damage Localization by the Principal Eigenvector of Modal Flexibility Change Cui-Hong Li, Qiu-Wei Yang * and Bing-Xiang Sun Department of Civil Engineering, Shaoxing University, Shaoxing 312000, China; [email protected] (C.H.L.); [email protected] (B.X.S.) * Correspondence: [email protected] or [email protected]; Tel.: +86-575-88326229; Fax: +86-575-88341503 Academic Editor: Stefano Mariani Received: 2 February 2016; Accepted: 31 March 2016; Published: 13 April 2016
Abstract: Using the principal eigenvector (PE) of modal flexibility change, a new vibration-based algorithm for structural defect localization was presented in this paper. From theoretical investigations, it was proven that the PE of modal flexibility variation has a turning point with a sharp peak in its curvature at the damage location. A three-span continuous beam was used as an example to illustrate the feasibility and superiority of the proposed PE algorithm for damage localization. Furthermore, defect localization was also performed using the well-known uniform load surface approach for comparison. Numerical results demonstrated that the PE algorithm can locate structural defects with good accuracy, whereas the ULS approach occasionally missed one or two defect locations. It was found that the PE algorithm may be promising for structural defect assessment. Keywords: defect localization; principal eigenvector; modal flexibility variation; deflection; vibration
1. Introduction During the last decades, a large number of research papers have reported on structural defect assessment based on the changes in measured static/dynamic response. The theoretical basis of these algorithms lies in the fact that the static and dynamic responses are functions of structural material properties (such as stiffness, mass, and damping). As a result, if the material properties are changed due to structural defects, then structural responses must also be changed. The relevant literature reviews can be found in references [1–3]. Generally, defect identification algorithms can be divided into model based and non-model based procedures according to using finite element model (FEM) or not. Algorithms [4–16] based on FEM assess structural defects by modifying structural FEM according to structural responses. The modifications of structural FEM will indicate the location and severity of the defect. It is well known that model-based methods are computationally intensive and highly influenced by the quality of structural FEM. However, the precise FEM is often difficult to achieve in engineering practice for the simplified assumptions in the construction of the FEM, which means that the correct results might be missed. The advantages of non-model algorithms lie in their easiness and straightforwardness because structural EEMs are not needed in these approaches. Damage indexes can be established directly by the variations of structural responses before and after damage for assessing structural defects. Zhang and Aktan [17] used the uniform load surface (ULS) of structural flexibility to locate defects. Wu and Law [18,19] discovered that the curvature of ULS is sensitive to local damage in the plate structure. Wang and Qiao [20] made use of the simplified gapped-smoothing technique to improve the accuracy of the ULS algorithm. Choi et al. [21] proposed an elastic damage load theorem to locate defects in simply supported beams. Sung et al. [22] presented a normalized ULS algorithm Algorithms 2016, 9, 24; doi:10.3390/a9020024
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for damage identification in beam-like structures. Other algorithms for defect localization can be found in references [23–27]. In engineering practice, structural response parameters are sometimes difficult to obtain and/or depend on environmental factors (e.g., temperature, humidity, loading at the time of testing, etc.). If the environmental impact is not considered properly, these localization algorithms will probably give incorrect results. In view of this, Limongelli et al. [9,28,29] proposed the interpolation damage detection method (IDDM) to considering the environmental impact. Their investigations verified the effectivity of the IDDM. Manoach et al. [30,31] proposed a new damage index and a method based on the Poincare map for structural damage localization. Numerical and experimental studies confirm the applicability and sensitivity of their method in application. This study proposes a defect localization algorithm based on the principal eigenvector (PE) obtained from structural flexibility variation. The presented algorithm uses PE as a new damage indicator to locate structural defects without FEM. A three-span continuous beam is used as an example to demonstrate the efficiency of the presented PE algorithm in structural defect localization. For several damage scenarios in the structure, defect localization results obtained by the PE algorithm and the ULS procedure are both given for comparison. The numerical result demonstrated that the PE algorithm can locate structural defects with good accuracy, whereas the ULS approach occasionally missed one or two defect locations. It has been shown that the proposed PE algorithm may be more effective at identifying structural damage locations than the ULS approach. 2. Theory 2.1. The Principle of Deflection-Based Damage Localization In this section, the explicit relation between the defect and defect-induced deformation variation has been derived firstly from the theoretical investigations. Then, the basic principle of deflection-based algorithm for structural damage localization has been illustrated. For a n-DOFs structure, let Ku and Kd are the stiffness matrices of the intact structure, Fu and Fd are the flexibility matrices of the damaged structures. Then they will satisfy the following relationship: Fu ¨ Ku “ Fd ¨ Kd “ Inˆn
(1)
where Inˆn is the identity matrix. Generally, structural defects reduce the stiffness and increase the flexibility. If ∆K and ∆F are the changes of stiffness and flexibility matrices, then one has: ∆K “ Ku ´ Kd
(2)
∆F “ Fd ´ Fu
(3)
Fu Ku “ pFu ` ∆FqpKu ´ ∆Kq
(4)
From Equations (1)–(3), one has
Equation (4) can be expanded as Fu Ku “ Fu Ku ´ Fu ∆K ` ∆FpKu ´ ∆Kq
(5)
Fu ∆K “ ∆FpKu ´ ∆Kq
(6)
From Equation (5), one has Substituting Equation (3) into (6) yields Fu ∆K “ ∆FKd
(7)
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According to Equation (1), Equation (7) can be rewritten as ∆F “ Fu ∆KFd
(8)
Multiplying Equation (8) by a load vector l, the defect-induced deformation variation vector (DV) can be defined as dv “ ∆Fl “ Fu ∆KFd l (9) According to the FEM theory, the global stiffness change ∆K can be expressed as the summation of all the elemental stiffness changes, i.e., ∆K “
Ne ÿ
αie Kie , p0 ď αie ď 1q
(10)
i “1
where αie and Kie are the damage coefficient and stiffness matrix of the ith element, Ne is the number of elements in the structural FEM. αie “ 0 denotes the corresponding element is intact, otherwise the element is damaged. According to Equations (9) and (10), we have dv “
Ne ÿ
pFu Kie qpαie Fd lq
(11)
i “1
According to Equation (11), the DV may be due to a single defect or multiple defects. For brevity, the single defect case is studied firstly in the following discussion. If only αie ‰ 0, Equation (11) can be reduced to dv “ pFu Kie qpαie Fd lq (12) Let Ei “ Fu Kie
(13)
ηi “ αie Fd l
(14)
dv “ Ei ηi
(15)
Then Equation (12) simplifies to Using the Linear algebra theory [32,33], Equation (15) has important implications that dv is the linear combination of the column vectors in Ei . From Equation (13), Ei is a sparse matrix because the j j elemental stiffness matrix Kie is very sparse. Assuming ei and k i are the jth nonzero column vector of Ei and Kie , respectively, one has j
j
ei “ Fu k i
(16) j
According to Equation (16), the physical meaning of the vector ei is the deflection of the intact j
j
j
structure by considering k i as a load vector. Thus, ei is defined as the characteristic deflection and k i is defined as the characteristic force of the ith element. It has been shown from Equation (15) that the deflection variation in a structure due to defect is a linear combination of the characteristic deflections for the damaged element. For the multiple-damage case, a similar conclusion can be obtained using the derivation as before. That is to say, structural deflection variation before and after damage is always the linear combination of the characteristic deflections for those damaged elements. Now we begin to discuss the properties of the characteristic force and deflection for structural elements. For convenience, the beam element is employed to illustrate these substantive features. Figure 1 presents a Bernoulli-Euler plane beam element with two nodes giving four DOFs, whose stiffness matrix and nodal displacement vector in local co-ordinates are as follows: ue “ rv1 , θ1 , v2 , θ2 sT
(17)
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Figure 1 presents a Bernoulli-Euler plane beam element with two nodes giving four DOFs, whose Figure 1 presents a Bernoulli-Euler planevector beamin element with two nodes four DOFs, whose stiffness matrix and nodal displacement local co-ordinates are asgiving follows: stiffness matrix and nodal displacement vector in local co-ordinates are as follows: (17) uee = [v1 ,θ1 , v2 ,θ 2 ]TT Algorithms 2016, 9, 24 4 of 18 (17) u = [v1 ,θ1 , v2 ,θ 2 ]
12
»12 6L EI 12 e Ke =EIEI—3 66L L — KKe “= 3 L3—− 12 L L –− ´12 12 6L 6L 6L
6L 6L 2 4L 6L 4 L22 4L − 6L ´6L − 6L 2 L222 2L 2L
− 12 6 L − 12 6 L 2fi − 6 L 6L2 L2 ´12 2 ffi − 6 L 2L 2L ´6L 12 − 6 Lffi ffi 12 − 6 L2fl 12 ´6L − 6 L 42L2 ´6L − 6 L 4L 4L
(18) (18) (18)
is the elastic modulus, I ismoment the moment of inertia, is the element whereE E where is the elastic modulus, I is the of inertia, and Land is theLelement length. length. where E is the elastic modulus, I is the moment of inertia, and L is the element length.
θ1 v θ1 1 1 v1 1
EI EI
v2 v2
θ2 θ2
2 2
L L Figure1.1.AABernoulli-Euler Bernoulli-Eulerplane planebeam beamelement element Figure Figure 1. A Bernoulli-Euler plane beam element e
Thefour fourcolumn columnvectors vectorsofof in Equation arecharacteristic the characteristic the BernoulliThe Ke K ine Equation (18) (18) are the forcesforces for thefor Bernoulli-Euler Theplane four column vectors ofinKlocal in Equation (18) are the characteristic forces for the BernoulliEuler beam element co-ordinates. Figure 2 shows the corresponding plane beam element in local co-ordinates. Figure 2 shows the corresponding load configurationsload of Euler plane beam element in local co-ordinates. Figure 2 shows the corresponding load configurations of these characteristic forces. these characteristic forces. configurations of these characteristic forces. 12 12 1 1 6L 6L
L L
6L 6L 1 1 4L2 4L2
L L
2 2 6L 6L L L
6L 6L 1 1 2L2 2L2
2L2 2L2 2 2 6L 6L 12 12
6L 6L 1 1 12 12
6L 6L 2 2 12 12
L L
4L2 4L2 2 2 6L 6L
Figure 2. Load configurations of the four characteristic forces. Figure 2. Load configurations of the four characteristic forces.
From Figure 2, it is apparent that all the load configurations are the self-equilibrating force systems. This conclusion is also valid for other types of finite elements. As can be seen in Figure 3, the internal
the statically determinate structure), only the element associated with the characteristic force is deformed, whereas the remaining part of the structure is not deformed and only has rigid body motion. That is to say, the elemental characteristic deflection will, due to its characteristic force, consist of several parts of the rigid displacement, but not the element itself. As stated before, Algorithms 2016, 9, 24 5 of 18 structural deflection change due to defects is a linear combination of the characteristic deflections corresponding to those damaged elements. Thus, the deflection change of the structure without redundant constraints under an arbitrary load will consist of several parts of the rigid displacement force (IF) in most parts of the structure under a characteristic force will be zero, except the element but not the damaged segments. In other words, the turning points between each segment of the rigid associated with this characteristic force. displacement in the shape of the deflection change for the structure are the locations of the defects. 12 6L 1
2
6L IF=0
12 IF=0
IF=0 6L 2L2 1
2
2
4L IF=0
6L IF=0
IF=0 12 6L 1
2 6L
IF=0
12
IF=0
IF=0 6L 4L2 1
2
2
2L IF=0
6L IF=0
IF=0
Figure Figure 3. 3. The internal force (IF) distributions of the structure under under characteristic characteristic forces. forces.
Forshort, the statically structure, derivation the same as that In one canindeterminate conclude from Figuresthe 2 and 3 that:process (1) theis characteristic forceofisthe a statically determinate structure. The only difference is that the IF in some parts of the statically self-equilibrating force; (2) the characteristic force acts only on its own element and not on the rest of indeterminate structure under a characteristic force will be slightly than zero because of the the structure. When the characteristic force is applied to the structuregreater without redundant constraints redundant constraint limitation. Correspondingly, the deflection variation damage willforce consist (i.e., the statically determinate structure), only the element associated withdue theto characteristic is of several parts of the approximate rigid displacement (or the rigid displacement) but not the deformed, whereas the remaining part of the structure is not deformed and only has rigid body motion. damaged segments. Then, the turning points between segment of the approximate rigid That is to say, the elemental characteristic deflection will,each due to its characteristic force, consist of displacement rigid displacement) thethe deflection change the defect locations. As a several parts of(or thethe rigid displacement, butinnot element itself. Asshape statedare before, structural deflection result, the peaksisina the curvature of structural deflection variation will indicate the locations of change duesharp to defects linear combination of the characteristic deflections corresponding to those structuralelements. defects. Thus, the deflection change of the structure without redundant constraints under damaged an arbitrary load will consist of several parts of the rigid displacement but not the damaged segments. In other words, the turning points between each segment of the rigid displacement in the shape of the deflection change for the structure are the locations of the defects. For the statically indeterminate structure, the derivation process is the same as that of the statically determinate structure. The only difference is that the IF in some parts of the statically indeterminate structure under a characteristic force will be slightly greater than zero because of the redundant constraint limitation. Correspondingly, the deflection variation due to damage will consist of several parts of the approximate rigid displacement (or the rigid displacement) but not the damaged segments. Then, the turning points between each segment of the approximate rigid displacement (or the rigid displacement) in the deflection change shape are the defect locations. As a result, the sharp peaks in the curvature of structural deflection variation will indicate the locations of structural defects.
2.2. Deflection Estimated by Modal Flexibility Change and PE Method for Damage Localization In Equation (9), the flexibility change matrix ΔF can be calculated approximately using the first few low-frequency modes, as shown in [34,35] m
ΔF =
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j =1
m
1
λdj
φdjφdjT − j =1
1
λuj
φujφujT
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(19)
2.2. Deflection Estimated by Modal Flexibility Changeλandand PE Method Damage Localization number of measured modes, λ dj areforthe eigenvalues of the structure where m is the uj In and Equation the flexibility matrix ∆F can be calculated approximately using the before after (9), damage, φuj andchange φdj are the corresponding eigenvectors, respectively. For first the few low-frequency modes, as shown in [34,35] damaged structure, the modal data can be acquired through a modal test in practice. For the intact mthrough solving m a generalized eigenvalue problem of the structure, the modal data are acquiredÿ ÿ 1 1 T ∆F “ φ φ ´ φ φT (19) dj dj undamaged FEM or by the modal experiment. λdj λuj uj uj j “1 j“1 Using matrix theory [32,33], the principal eigenvector of ΔF can be given by the solutions of the following where m is theeigenvalue number of problem: measured modes, λ and λ are the eigenvalues of the structure before and uj
dj
after damage, ϕuj and ϕdj are the correspondingeigenvectors, respectively. For the damaged structure, ξ1 ΔF = ξ the modal data can be acquired through a modal (20) λtest in 1practice. For the intact structure, the modal 1 data are acquired through solving a generalizedeigenvalue problem of the undamaged FEM or by the modal experiment. where λ1 is the principal eigenvalue and ξ 1 is the principal eigenvector of ΔF , respectively. Using matrix theory [32,33], the principal eigenvector of ∆F can be given by the solutions of the According to Equation (20), the physical meaning of the vector ξ 1 is the structural deflection following eigenvalue problem: ˆ ˙ ξ1 1 ∆F “ξ ξ 1 as a load vector. Thus, according to (20) variation before and after damage by considering the 1 λ1
λ1
where λ1 is the principal eigenvalue and ξ 1defects is the principal eigenvector ∆F, respectively. According principle stated in Section 2.1, structural can be located in theoffollowing two ways: (1) the to Equation (20), the physical meaning of the vector ξ 1 is the structural deflection variation before turning points in the shape of ξ 1 are the locations of defects; (2) the sharp peaks in the curvature of and after damage by considering λ1 ξ 1 as a load vector. Thus, according to the principle stated in 1 ξ1 indicate ξ1 can be the locations of structural defects.inItthe is known thattwo the ways: curvature of turning computed Section 2.1, structural defects can be located following (1) the points in the using difference approximation as follows: shape the of ξcentral are the locations of defects; (2) the sharp peaks in the curvature of ξ indicate the locations 1 1 of structural defects. It is known that the curvature i +1 of ξ 1 ican be i −1computed using the central difference ζ − 2ζ 1 + ζ 1 approximation as follows: (21) (ζ 1i ) " = 1 2 i `1 i ` ζ i ´1 h ζ ´ 2ζ 2 1 1 (21) pζ 1i q “ 1 2 i i " h where ζ is the i th coefficient of ξ , h is the length of structural element, and (ζ ) is the i th 1
1
1
2
where ζ 1i is the ith coefficient of ξ 1 , h is the length of structural element, and pζ 1i q is the ith curvature curvature coefficient of ξ 1 . coefficient of ξ 1 .
3. 3. Numerical NumericalExample Example The demonstrate the The continuous continuous beam beam in in Figure Figure 44 is is used used to to demonstrate the defect defect localization localization ability ability of of the the presented PE algorithm. presented PE algorithm. A 1
3
5
7
9
11
B 13
15
17
19
21
23
C 25
27
29
31
33
35
D
Figure 4. A A beam beam structure structure with with three three spans. Figure
Note example is is aa statically statically indeterminate indeterminate structure. structure. For Note that that this this example For the the structure, structure, the the elastic elastic 3 Kg/m3,3 1.0416× modulus, density, moment of inertia, and cross-sectional area are 200 GPa, 7.8×10 modulus, density, moment of inertia, and cross-sectional area are 200 GPa, 7.8 ˆ 10 Kg/m3 , −6m4, 0.0025 2 10 In the FEM, the structure divided intois36divided elements, which gives 33 1.0416 ˆ 10´6 m m4,, respectively. 0.0025 m2 , respectively. In the FEM, isthe structure into 36 elements, translational DOFs and 37 rotational DOFs. The length of each segment in the beam FEM is 0.1m. In which gives 33 translational DOFs and 37 rotational DOFs. The length of each segment in the beam the following discussion, only the first four modal parameters with the translational degrees of FEM is 0.1 m. In the following discussion, only the first four modal parameters with the translational freedom used in simulate the incomplete measurements. Defect Defect in theinbeam was degrees ofare freedom areorder used to in order to simulate the incomplete measurements. the beam simulated by reducing the elastic modulus of some structural elements. Five defect scenarios was simulated by reducing the elastic modulus of some structural elements. Five defect scenarios are are assumed assumed in in the the example. example. Scenario Scenario 1: 1: the thedefect defectoccurred occurredin in element element77 with with 20% 20% stiffness stiffness reduction. reduction. Scenario 2: the defect occurred in element 13 with 20% stiffness reduction. Scenario 3: the defects occurred in elements 5 and 17 both with 20% stiffness reduction. Scenario 4: the defects occurred in elements 5 and 31 both with 20% stiffness reduction. Scenario 5: the defects occurred in elements 7, 19 and 33 all with 20% stiffness reduction. Detection results of each damage scenario obtained
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Scenario 2: the defect occurred in element 13 with 20% stiffness reduction. Scenario 3: the defects7 of 18 occurred in elements 5 and 17 both with 20% stiffness reduction. Scenario 4: the defects occurred in elements 5 and 31 both with 20% stiffness reduction. Scenario 5: the defects occurred in elements 7, by the presented PE algorithm and the well-known ULS method are both given to compare the 19 and 33 all with 20% stiffness reduction. Detection results of each damage scenario obtained by the localization performance. presented PE algorithm and the well-known ULS method are both given to compare the localization Using the exact data without noise, the principal eigenvector (PE) of modal flexibility change performance. and the ULS are plotted in Figure 5a,b, respectively. Accordingly, Usingvariation the exact for datadamage withoutscenario noise, the1principal eigenvector (PE) of modal flexibility change the PE curvature and the ULS variation curvature for the case of damage scenario 1 are plotted and the ULS variation for damage scenario 1 are plotted in Figure 5a,b, respectively. Accordingly, the in Figure respectively. PE6a,b, curvature and the ULS variation curvature for the case of damage scenario 1 are plotted in Figure 6a,b, respectively.
(a)
(b) Figure 5. (a) Damage index (principal eigenvector (PE)) map using the first four exact modes when
Figure 5. (a) Damage index (principal eigenvector (PE)) map using the first four exact modes when the the defect in element 7; (b) Damage index (uniform load surface (ULS) variation) map using8 of 17 Algorithms 2016, 9,occurred 24 defect occurred in element 7; (b) Damage index (uniform load surface (ULS) variation) map using the the first four exact modes when the defect occurred in element 7 (×10−4). first four exact modes when the defect occurred in element 7 (ˆ10´4 ).
(a)
(b) Figure 6. (a) Damage index (PE curvature) map using the first four exact modes when the defect
Figure 6. (a) Damage index (PE curvature) map using the first four exact modes when the defect occurred in element 7; (b) Damage index (ULS variation curvature) map using the first four exact occurred in element 7; (b) Damage index (ULS variation curvature) map using the first four exact modes when the defect occurred in element 7 (×10−3). modes when the defect occurred in element 7 (ˆ10´3 ).
One can see that the transnational DOFs numbered 7 and 8 are the turning points in Figure 5 and the sharp peaks in Figure 6, respectively. In the beam model, the seventh and the eighth transnational DOFs are exactly corresponding to the seventh element. This means that the seventh segment could be successfully detected to be the defect area by inspecting the turning points in Figure 5, or by inspecting the sharp peaks in Figure 6. Note that the human inspection and intervention is
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(b) 7 and 8 are the turning points in Figure 5 and One can see that the transnational DOFs numbered the sharp6.peaks in Figure 6, respectively. In the beam model, and modes the eighth transnational Figure (a) Damage index (PE curvature) map using the the firstseventh four exact when the defect DOFs are exactly corresponding to the seventh element. This means that the seventh segment could be occurred in element 7; (b) Damage index (ULS variation curvature) map using the first four exact successfully detected to be the defect area by inspecting the turning points in Figure 5, or by inspecting modes when the defect occurred in element 7 (×10−3). the sharp peaks in Figure 6. Note that the human inspection and intervention is required in the above damage localization process. One can see that the transnational DOFs numbered 7 and 8 are the turning points in Figure 5 Damage scenario 2 is used to verify the validity of the two algorithms when the defect occurred and the sharp peaks in Figure 6, respectively. In the beam model, the seventh and the eighth near the supported boundary. Figures 7 and 8 showed the detection results of this scenario. One can transnational DOFs are exactly corresponding to the seventh element. This means that the seventh conclude that the damaged element 13 can be detected by inspecting the turning points in Figure 7, or segment could bethe successfully by inspecting sharp peaksdetected in Figureto8.be the defect area by inspecting the turning points in Figure 5, or by inspecting the3–5 sharp peaks Figurethe 6. PE Note that thewith human inspection andthe intervention is Damage cases are used to in compare algorithm the ULS method for multiple required in the above damage process. damage scenarios. Figures 9localization and 10 showed the results of damage localization for damage case 3. From Figures 9a and 10a, one can see that the damaged 5 and 17 were clearly by Damage scenario 2 is used to verify the validity of elements the two algorithms when theidentified defect occurred PE curvature. From Figure 9b, 7the ULS variationthe method fails toresults detect the damaged element nearPE theand supported boundary. Figures and 8 showed detection of this scenario. One can 17. It can be seen from Figure 10 that the PE curvature provides comparatively better damage location conclude that the damaged element 13 can be detected by inspecting the turning points in Figure 7, predictions than ULSpeaks curvature. or by inspecting thethe sharp in Figure 8.
(a)
(b) Figure 7. (a) Damage index firstfour fourexact exactmodes modes when defect occurred in Figure 7. (a) Damage index(PE) (PE)map mapusing using the first when the the defect occurred in element Damage index(ULS (ULSvariation) variation) map four exact modes when the defect element 13; 13; (b) (b) Damage index mapusing usingthe thefirst first four exact modes when the defect −4).´4 ). occurred in element 13(ˆ10 occurred in element 13(×10
(a)
(b) Figure 7. (a) Damage index (PE) map using the first four exact modes when the defect occurred in Algorithmselement 2016, 9, 13; 24 (b) Damage index (ULS variation) map using the first four exact modes when the defect 9 of 18
occurred in element 13(×10−4).
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(a) Figure 8. Cont.
(b) Figure 8. (a) Damage index (PE curvature) map using the first four exact modes when the defect occurred in element 13; (b) Damage index (ULS variation curvature) map using the first four exact modes when the defect occurred in element 13 (×10−3).
Damage cases 3–5 are used to compare the PE (b) algorithm with the ULS method for the multiple damage scenarios. Figures 9 and 10 showed the results of damage localization for damage case 3. Figure 8. (a) Damage index (PE curvature) map using the first four exact modes when the defect Figure 8. (a) (PE see curvature) using elements the first four exact when identified the defect by From Figures 9aDamage and 10a,index one can that themap damaged 5 and 17 modes were clearly occurred in element 13; (b) Damage index (ULS variation curvature) map using the first four exact occurred in elementFrom 13; (b) Damage (ULS variation curvature) map usingthe thedamaged first four exact PE and PE curvature. Figure 9b, index the ULS variation method fails to detect element modes when the defect occurred in element 13 (×10−3´3 ). modes when the defect occurred in element 13 (ˆ10 ). 17. It can be seen from Figure 10 that the PE curvature provides comparatively better damage location predictions than the3–5 ULSare curvature. Damage cases used to compare the PE algorithm with the ULS method for the multiple damage scenarios. Figures 9 and 10 showed the results of damage localization for damage case 3. From Figures 9a and 10a, one can see that the damaged elements 5 and 17 were clearly identified by PE and PE curvature. From Figure 9b, the ULS variation method fails to detect the damaged element 17. It can be seen from Figure 10 that the PE curvature provides comparatively better damage location predictions than the ULS curvature.
(a)
(a)
(b) Figure 9. (a) Damage index (PE) map using the first four exact modes when the defects occurred in Figure 9. (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 17; (b) Damage index (ULS variation) map using the first four exact modes when the elements 5 and 17; (b) Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements 5 and 17 (×10−4). defects occurred in elements 5 and 17 (ˆ10´4 ).
(b) Figure 9. (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 17; (b) Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements 5 and 17 (×10−4).
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(b) (b) Figure 10. (a) Damage index (PE curvature) map using the first four exact modes when the defects Figure 10. (a) Damage index (PE curvature) map using the first four exact modes when the defects occurred and 17; (b)curvature) Damage index using thethe firstdefects four Figure 10.in (a)elements Damage5 index (PE map (ULS usingvariation the first curvature) four exact map modes when occurred in elements 5 and 17; (b) Damage index (ULS variation−3 curvature) map using the first four exact modes when the5 defects in elements 5 and 17 (×10 curvature) ). occurred in elements and 17;occurred (b) Damage index (ULS variation map using the first four exact modes when the defects occurred in elements 5 and 17 (ˆ10−3´3 ). exact modes when the defects occurred in elements 5 and 17 (×10 ).
Figures 11–14 presented the damage localization results for damage case 4 using the exact modal
Figures 11–14 presented localization for11damage damage case usingthe the exact modal data and the data with 3%the noise, respectively. Fromresults Figures and 12,case both damage localization Figures 11–14 presented thedamage damage localization results for 4 4using exact modal methods can achieve satisfactory by usingFrom error-free data. data and thethe data with 3% respectively. From Figures 11and and12, 12both bothdamage damagelocalization localization data and data with 3%noise, noise,results respectively. Figures 11 methods can achieve methods can achievesatisfactory satisfactoryresults resultsby byusing using error-free error-free data. data.
(a)
(a)
(b) Figure 11. (a) Damage index (PE) map using the (b) first four exact modes when the defects occurred in elements 5 and 31; (b) Damage index (ULS variation) map using the first four exact modes when the Figure 11. (a) Damage index (PE) map using −4). the first four exact modes when the defects occurred in defects in elements 5 and 31 (×10 Figure 11. occurred (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 5 and 31; (b) Damage index (ULS variation) map using the first four exact modes when the elements 5 and 31; (b) Damage index (ULS variation) map using the first four exact modes when the −4). defects occurred in elements 5 and 31 (×10´4
defects occurred in elements 5 and 31 (ˆ10
).
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(a) (a)
(b) (b) Figure Damage index(PE (PEcurvature) curvature) map map using using the when thethe defects Figure 12. 12. (a) (a) Damage index the first firstfour fourexact exactmodes modes when defects Figure 12.in(a) Damage5 index (PE map (ULS using variation the first four exact modes whenthe thefirst defects occurred elements and 31; (b)curvature) Damage index curvature) map using four occurred in elements 5 and 31; (b) Damage index (ULS variation curvature) map using the first four occurred in elements and 31;occurred (b) Damage index (ULS variation map using the first four exact modes when the5 defects in elements 5 and 31 (×10−3curvature) ). ´3 exact modes when thethe defects 31 (×10 (ˆ10 −3). ). exact modes when defectsoccurred occurredininelements elements 55 and and 31
(a) (a)
(b) (b)
Figure 13. (a) Damage index (PE) map using the first four modes with 3% noise when elements 5 and Figure 13. (a) Damage index (PE) map using the first four 3% noise when elements and 31 are (b)index Damage mapmodes using with the first modes with 3% 5noise Figure 13. damaged; (a) Damage (PE)index map(ULS usingvariation) the first four modes with 3%four noise when elements 5 and 31 are damaged; (b) Damage index (ULS variation) map using the first four modes with 3% noise −4 ). when elements 31 areindex damaged 31 are damaged; (b)5 and Damage (ULS(×10 variation) map using the first four modes with 3% noise when −4 when elements 5 and 31 are damaged ´4(×10 ).
elements 5 and 31 are damaged (ˆ10
).
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(a) (a)
(b)
(b) Figure 14. (a) Damage index (PE curvature) map using the first four modes with 3% noise when Figure 14. (a) Damage index (PE curvature) map using the first four modes with 3% noise when Figure5 14. index (PE curvature) map (ULS usingvariation the first four modes with noise elements and(a) 31 Damage are damaged; (b) Damage index curvature) map3% using thewhen first four elements 5 and 31 are damaged; (b) Damage index (ULS variation curvature) map using the first four elements 5 and 31 are damaged; (b) Damage index (ULS variation curvature) map using the first four −3). modes with 3% noise when elements 5 and 31 are damaged (×10 modes with 3% noise when elements 5 and 31 are damaged (×10−3). modes with 3% noise when elements 5 and 31 are damaged (ˆ10´3 ).
From Figure 13,13, one of PE PEand andULS ULSvariation variation identify From Figure onecan cansee seethat that the the indexes indexes of cancan stillstill identify the the From Figure 13, one can see that the indexes of PE and ULS variation can still identify the damaged elements 5 and evenifif3% 3%noise noise is considered. considered. According to to Figure 14, 14, it isitapparent that that damaged elements 5 and 3131 even According Figure is apparent damaged elements 5 and 31 even if 3% noise is considered. According to and Figure 14, it is apparent that the curvature indexes are seriously affected by the measurement error, the damaged segments the curvature indexes are seriously affected the measurement error, and the damaged segments the curvature indexes by arethese seriously affected by with the measurement error, and the damaged segments cannot be determined curvature maps cannot be determined by these curvature maps withnoise. noise. cannot be determined by these curvature maps with noise. Figures 15–18 presented the damage localization results for case 5 using the exact modal Figures 15–18 presented the damage localization resultsfor fordamage damage case 5 using the exact modal damage localization results 5 using the see exact modal data Figures15–18 and the datapresented with 3% the noise, respectively. From Figuresdamage 15a andcase 16a, one can that the datadata andand thethedata 3%noise, noise, respectively. Figures 15a and 16a, one can see that the datawith with 3% respectively. FromFrom Figures 15a and one can see that the damaged damaged elements 7, 19 and 33 were clearly identified by PE and PE16a, curvature. From Figure 15b, the damaged elements 7, 19 and 33 were clearly identified by PE and PE curvature. From Figure 15b, the elements 7, 19 and 33 were identified by PE and PE curvature. Figure 15b, ULS variation method fails clearly to detect the damaged element 19. It can From be concluded thatthe theULS PE ULSvariation variationcan method to detect the damaged element 19. Itthe canULS bethat concluded that the PE failsfails to identify detect the damaged element 19. It can be concluded the PE algorithm algorithmmethod precisely multiple defect locations, whereas method occasionally can precisely identify multiple defect locations, whereas the ULS method occasionally missed multiple algorithm can precisely identify multiple defect locations, whereas the ULS method occasionally missed multiple damage locations. damage locations. missed multiple damage locations.
(a) Figure 15. Cont.
(a)
Figure 15.Cont. Cont. Figure 15.
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Figure 15. (a) index (PE) firstfour fourexact exactmodes modes when defects occurred Figure 15. Damage (a) Damage index (PE)map mapusing using the the first when thethe defects occurred in in Figure 15. (a) Damage index (PE) map using the first four exact modes when the defects occurred in elements 7, 19 and 33; (b) Damage index (ULS variation) map using the first four exact modes elements 7, 19 and 33; (b) Damage index (ULS variation) map using the first four exact modes whenwhen ´4 −4 elements 7, 19 and 33;in(b) Damage index (ULS variation) map using the first four exact modes when the defects occurred in elements7,7, 19and and 33 (×10 (ˆ10 the defects occurred elements 19 33 ).). the defects occurred in elements 7, 19 and 33 (×10−4).
(a) (a)
(b) (b)
Figure 16. (a) Damage index (PE curvature) map using the first four modes without noise when Figure 16. (a) index (PE curvature) map using the first fourcurvature) modes without noise when elements 19Damage and 33 are damaged; Damage index (ULS mapnoise using the first Figure7,16. (a) Damage index (PE (b) curvature) map using the variation first four modes without when −3 elements 7, 19 33noise are damaged; (b)(b)Damage index variation(×10 curvature) map using four elements modes without when elements 7, 19 and 33 (ULS are damaged ). 7,and 19 and 33 are damaged; Damage index (ULS variation curvature) map using the the firstfirst −3). ´3 four modes without noise when elements 7, 19 and 33 are damaged (×10 four modes without noise when elements 7, 19 and 33 are damaged (ˆ10 ).
(b) Figure 16. (a) Damage index (PE curvature) map using the first four modes without noise when elements Algorithms 2016, 7, 9, 19 24 and 33 are damaged; (b) Damage index (ULS variation curvature) map using the first14 of 18 four modes without noise when elements 7, 19 and 33 are damaged (×10−3).
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(a)
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Figure 17. Cont.
(b) (b) Figure 17. (a) Damage index (PE) map using the first four modes with 3% noise when elements 19 Figure 17.17. (a)(a) Damage modes with with3% 3%noise noisewhen whenelements elements7,7, 7, Figure Damageindex index(PE) (PE)map mapusing using the the first first four four modes 1919 and 33 are damaged; (b) Damage index (ULS variation) map using the first four modes with 3% noise and 3333 are damaged; map using using the thefirst firstfour fourmodes modeswith with3% 3%noise noise and are damaged;(b) (b)Damage Damageindex index(ULS (ULS variation) variation) map ´4 when elements 7, 19 and 33 are damaged (×10−4−4 ). ). when elements 7, 19 and 33 are damaged (ˆ10 when elements 7, 19 and 33 are damaged (×10 ).
(a) (a)
(b) (b) Figure (a)Damage Damageindex index(PE (PE curvature) curvature) map map using the first Figure 18.18. (a) using the the first first four four modes modeswith with3% 3%noise noisewhen when Figure 18. (a) Damage index (PE curvature) map using four modes with 3% noise when elements 7, 19 and 33 are damaged; (b) Damage index (ULS variation curvature) map using the elements 7, 7, 19 19 and and 33 33 are are damaged; damaged; (b) (b) Damage Damage index index (ULS (ULS variation variation curvature) curvature) map map using using the thefirst first elements first four modes with 3% noise when elements 7, 19 and 33 are damaged. four modes with 3% noise when elements 7, 19 and 33 are damaged. four modes with 3% noise when elements 7, 19 and 33 are damaged.
From Figure 17, one can see that the indexes of PE and ULS variation can still identify the From Figure 17, one can see that the indexes of PE and ULS variation can still identify the damaged elements 7, 19 and 33 even if 3% noise is considered. From Figure 18, it is apparent that the damaged elements 19seriously and 33 even if 3% is considered. From Figure 18, it is apparent that the curvature indexes7,are affected bynoise the measurement error, and the damaged locations cannot curvature indexesbyare seriously affected the noise. measurement error, and the damaged locations cannot be determined these curvature mapsby with be determined by these curvature maps with noise. In order to study the sensitivity of the proposed method to the number of modes, Figures 19 and
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From Figure 17, one can see that the indexes of PE and ULS variation can still identify the damaged elements 7, 19 and 33 even if 3% noise is considered. From Figure 18, it is apparent that the curvature indexes are seriously affected by the measurement error, and the damaged locations cannot be determined by these curvature maps with noise. In order to study the sensitivity of the proposed method to the number of modes, Figures 19 and 20 give the damage localization results of damage case 4 using the first one, two and three modes without noise, respectively. From Figures 19 and 20 one can see that the ULS method is more sensitive to the number of modes than the PE method, and both damage localization methods can achieve reasonable results, even if only the first mode without noise is used in the calculation.
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(a)
(b) Figure 19. (a) Figure 19. (a) Damage Damage index index (PE) (PE) map map using using the the first first one, one, two two and and three three modes modes without without noise, noise, respectively, (b)(b) Damage index (ULS variation) mapmap using the respectively, when when elements elements55and and3131are aredamaged; damaged; Damage index (ULS variation) using −4). first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (×10 the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (ˆ10´4 ).
(a)
(b) Figure 19. (a) Damage index (PE) map using the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged; (b) Damage index (ULS variation) map using the16 of 18 Algorithms 2016, 9, 24 first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (×10−4).
(a)
(b) Figure 20. (a) Damage index (PE curvature) map using the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged; (b) Damage index (ULS variation curvature) map using the first one, two and three modes without noise, respectively, when elements 5 and 31 are damaged (ˆ10´3 ).
4. Conclusions A new vibration-based method has been developed for structural damage localization, which is based on the principal eigenvector (PE) of modal flexibility change. In order to verify the feasibility of the PE algorithm, a three-span continuous beam was investigated for several damage scenarios with and without noise. Furthermore, the PE method was compared with the well-known ULS method to show its advantages. According to the numerical results, the presented PE method may be more effective at identifying the defect locations of the beam structure than the ULS method. The proposed PE procedure may have good prospects in structural damage localization. Acknowledgments: This work is supported by the Scientific Research Project of Education of Zhejiang Province (Y201430539). Author Contributions: Cui-Hong Li and Qiu-Wei Yang proposed the idea and constructed the principal eigenvector method. Qiu-Wei Yang and Bing-Xiang Sun performed the numerical simulations. Cui-Hong Li and Qiu-Wei Yang wrote the article. Conflicts of Interest: The authors declare no conflict of interest.
References 1. 2.
Doebling, S.W.; Farrar, C.R.; Prime, M.B. A summary review of vibration-based damage identification methods. Shock Vib. Dig. 1998, 30, 91–105. [CrossRef] Carden, E.P.; Fanning, P. Vibration based condition monitoring: A review. Struct. Health Monit. 2004, 3, 355–377. [CrossRef]
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3.
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