Article

Structural damage identification based on cuckoo search algorithm

Advances in Structural Engineering 2016, Vol. 19(5) 849–859 Ó The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1369433216630128 ase.sagepub.com

HJ Xu, JK Liu and ZR Lu

Abstract A method for structural damage identification based on cuckoo search algorithm is presented. The nonlinear objective function for the damage identification problem is established by utilizing modal assurance criteria and structural natural frequencies. Then, the cuckoo search algorithm is adopted to detect local damages by solving the objective function. Two numerical examples are studied to investigate the efficiency and correctness of the proposed method. Meanwhile, a laboratory work is conducted for further verification. The simulation and experiment results show that the present method can produce more accurate damage identification results even under measurement noise, comparing with genetic algorithm.

Keywords cuckoo search algorithm, damage detection, frequency domain, global optimization, modal assurance criteria

Introduction Negligence of the structural local damages may cause serious engineering problems. Efficient methods to detect and quantify structural damage have drawn wide attention from various engineering fields. The vibration-based damage identification methods have been researched by Fan and Qiao (2011). Doebling et al. (1998) provided a comprehensive summary on the damage detection methods by examining changes in the dynamic characteristics of the structure. Zou et al. (2000) offer an extensive review of the progress on structural condition monitoring and damage identification for composite structures. Previous studies have shown that the associated changes in the structures will result in changes in dynamic characteristics of the system. Therefore, by detecting some of these properties of the damaged structure, the local damages of the structure could be identified. The identification approaches are mainly based on the change in the natural frequencies (Gelman, 2010; Narkis, 1994), mode shapes (Ghafory and Ghasemi, 2013; Wang et al., 2013b), measured modal flexibility (Pandey et al., 1991; Yang, 2011), frequency response function (Pandey and Biswas, 1994; Ren and De Roeck, 2002), or the combination of these methods (Kim et al., 2003; Owolabi et al., 2003; Tomaszewska, 2010). Among all of the structural damage detection techniques, approaches based on dynamic responses have

been a hot research topic during the past 20 years. There are a lot of non-destructive methods in the literatures for structural damage detection in time domain. By utilizing the features of dynamic response sensitivity, Lu and Law (2007) proposed a response sensitivity-based method to identify structural damage effectively. Besides, Fu et al. (2013) and Lu et al. (2013) have done their job in time domain. In recent years, with the development of the computer technique and mathematics, swarm intelligence technique gained its popularity. From a mathematical view, by transferring the local damage identification problem to the objective function optimization, the damage can be obtained by minimizing the objective function. And the objective function was formulated of the difference between the calculated and the measured data from the certain structural system. Dackermann et al. (2014) applied the neural network (NN) for damage detection. Besides, genetic algorithm (GA) as a global optimization method has been used by Friswell et al. (1998) for structural damage detection. Then, some new optimization algorithms such as artificial Department of Applied Mechanics, Sun Yat-sen University, Guangzhou, P.R. China Corresponding author: ZR Lu, Department of Applied Mechanics, Sun Yat-sen University, Guangzhou 510006, Guangdong, P.R. China. Email: [email protected]

850 bee colony algorithm utilized by Hao et al. (2013), ant colony optimization (ACO) used by Yu and Xu (2011), and particle swarm optimization (PSO) applied by Begambre and Laier (2009) have been introduced in damage detection. But these swarm intelligence techniques both have common failing on slow convergence. And some of them may fall into local optimization easily. Apart from the techniques mentioned above, cuckoo search (CS) algorithm is another swarm intelligence technique, which was developed by Yang and Deb (2010) to solve numerical optimization problems. This algorithm is inspired by the reproduction strategy of cuckoos, which is utilized increasingly for its feasibility and efficiency. The CS is simple in concept and has less parameter to adjust and is easy to implement. It has been applied in many areas such as constrained optimization problems and multi-objective optimization problems. Gandomi et al. (2013) introduced CS algorithm to solve structural optimization problems. Yang and Deb (2013) utilized the CS to solve multi-objective design optimization. Walton et al. (2011) modified the CS to reach high convergence to the global minimum problem even at high numbers of dimensions. So far, very few studies have been done on structural damage detection based on CS algorithm. The article mainly deals with the damage detection problem using CS algorithm based on an objective function established in frequency domain. The performance of the certain algorithm is illustrated in two kinds of structures, that is, a dual span simply supported beam and a truss. Besides numerical simulations, a laboratory work is carried out to further validate the effectiveness of the present method. In the identification, only the first few natural frequencies and/or mode shapes are needed. Results show that local structural damages can be identified successfully even under measurement noise, which shows that CS algorithm is very promising for damage detection, compared with the GA.

A brief introduction to CS algorithm The CS method is becoming very popular due to its simplicity, efficiency, and fast convergence properties. First, introduce the CS algorithm. CS algorithm is inspired by the CS for host nests to breed. At the most basic level, cuckoos lay their eggs in the nests of other host birds, which may be of different species. The host bird may discover that the eggs are not its own and either destroy the egg or abandon the nest all together. This has resulted in the evolution of cuckoo eggs which mimic the eggs of local host birds. For simplicity, we use the following three idealized rules:

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2. 3.

Each cuckoo lays one egg at a time, which represents a set of solution coordinates and dumps it in a randomly chosen nest; The best nests with high quality of eggs (solutions) will carry over to the next generations; The number of available host nests is fixed, and a host can discover an alien egg with a probability pa 2 ½0, 1. In this case, the host bird can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.

For simplicity, this last assumption can be approximated by a fraction pa of the nests being replaced by new nests (with new random solutions at new locations). For a maximization problem, the quality or fitness of a solution can simply be proportional to the objective function. Based on these three rules mentioned above, the basic steps of the CS can be summarized as flee shown in Figure 1. When generating new solutions x(t + 1) a Levy flight of cuckoo i, is performed xi(t + 1) = x(t) i + aLevy(l)

ð1Þ

where a.0 is the step size which should be related to the scales of the problem. In most cases, we can use a = 0(1). The product means entry-wise multiplications. Levy flight essentially provides a random walk while their random steps are drawn from a Levy distribution for large steps Levy ; m = tl , (1\l\3)

ð2Þ

which has an infinite variance with an infinite mean. Here, the consecutive jumps/steps of a cuckoo essentially form a random walk process which obeys a power-law step-length distribution with a heavy tail. In addition, a fraction pa of the worst nests can be abandoned so that new nests can be built at new locations by random walks and mixing. The mixing of the eggs/solutions can be performed by random permutation according to the similarity/difference to the host eggs. Obviously, the generation of step size s samples is not trivial using Levy flights. A simple scheme discussed in detail by Yang and Deb (2010) can be summarized as (t) s = a0 x(t) x Levy(b) ; j i

jnj

1=b

m (t) x(t) j xi

ð3Þ

where m and n are drawn from normal distributions. That is

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Figure 1. Flowchart of the CS algorithm.

m ; N 0, s2u ,

n ; N 0, s2n

ð4Þ

And we choose frequency residual error and modal assurance criteria (MAC) to build up the structural objective function as below.

ð5Þ

Methodology for vibration-based damage identification

with sm =

G(1 + b) sin (pb=2) G½(1 + b)=2b2(b1)=2

1=b sn = 1

Here, G is the standard gamma function. When generating a new egg in Figure 1, a Levy flight is performed starting at the position of a randomly selected egg if the objective function value at these new coordinates is better than another randomly selected egg, then that egg is moved to this new position. The scale of this random search is controlled by multiplying the generated Levy flight by a step size. Yang and Deb (2009) do not discuss boundary handling in their formulation, but an approach similar to PSO boundary handling is adopted here. When a Levy flight results in an egg location outside the bounds of the objective function, the fitness and position of the original egg are not changed. Above all, using Levy flight and fraction pa , CS algorithm can be proposed to various optimization problems effectively and show a high convergence rate to the trace global minimum. In this article, CS algorithm is selected to solve the damage index objective function of the engineering structure. And then we can conclude the location and extent of the damaged structures to carry out the structural damage detection.

The premise for techniques based on vibration responses is that damage causes a change in structural physical properties, mainly in stiffness and damping at the damaged locations.

Modeling structural local damage The eigenvalue equation for a finite element model of the system is

K v2j M uj = 0

ð6Þ

where M and K are the structural mass matrix and stiffness matrix, respectively. And vj is the jth natural frequency and uj is the corresponding mode shape. In the finite element model of the damaged system, the local damage is modeled as reduction in the elemental stiffness of the system, but the other properties remain unchanged. When a structure with nel elements is damaged, the reduction of the stiffness can be evaluated by a set of damage parameters ai (i = 1, 2, . . . , nel). The value range of parameter ai is

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between 0 and 1. When ai = 1, it means that the ith element is intact while ai = 0, and it represents that system is completely damaged. And then the stiffness matrix of the damaged system can be written as Kd =

nel X

f (x) =

s X

ð9Þ 1 MAC uhi , udi + ER vhi , vdi

i=1

kie

ð7Þ

i=1

where Kd is the stiffness matrix of the damaged system, and kie presents the ith elemental stiffness matrix in the global form. The structural damages can be identified by identifying each value in the damage parameter vector fag, based on the damage model mentioned above.

Objective function based on vibration data It is well known that the changes of stiffness will cause the changes in the structural properties, such as natural frequencies and mode shapes. In this study, the objective function for damage detection is based on vibration data, such as natural frequencies and mode shapes. From a mathematical view, such identification problem can be regarded as a nonlinear optimization problem, in which an objective function, for instance the error between the actual measured structural response and the analytic response of a model, is defined and the parameters of such models determined so to optimize the objective function (usually to minimum) (Franco et al., 2004; Wang et al., 2013a). By minimizing the differences between the measured response data and the calculated one, we can locate and quantify the local damage of the certain system. Refer to Wang et al.’s (2013a) research, in this article, first of all, we select the frequency and modal shape data to build up the objective function of the certain problem. Then, we take the natural frequency and the MAC as indexes of damage detection in view of the crack damage, and then the CS algorithm is applied to identifying the structural damage. The detail of how the objective function is built is described in the following. Considering the mode shape data, MAC proposed by Brehm et al. (2010) is utilized. The MAC is expressed as

MACj =

objective function used for damage detection is expressed as

(fuj gT fvM j g) 2 jfuj gj2 juM j j

2

ð8Þ

where uj and uM j are the jth calculated and measured mode shapes, respectively. Based on MAC and taking into account the changes both in natural frequencies and mode shapes, the

where s is the modal order, vhi , uhi , vdi , udi is the ith order frequency and mode shape of the intact and damaged system. Besides, ER(vhi , vdi ) = (vhi vdi )= d hT h vhi j 3 100%, MAC(uhi , udi ) = juhT i ui j=j(ui ui )jj dT d (ui ui )j, i = 1, 2, . . . , s. As described above, when a structure with nel elements is damaged, the reduction of the stiffness can be evaluated by a set of damage parameters ai (i = 1, 2, . . . , nel). The value range of parameter ai is between 0 and 1. When ai = 0, it means that the ith element is intact while ai = 1, it represents that system is completely damaged, and when ai = 0:1, it means the system is 10% damaged. The change in damage parameters indicates the changes in the frequency and modal shape. Then, the calculated nature frequency vi and mode shape ui are related to damage parameters ½a1 , a2 , . . . , anel . In the inverse problem, if the value of damage parameter vector ai matches the given particular damage condition, then the objective function will achieve its minimum value 0. In other words, when the minimum value of the objective function is identified, the corresponding parameter vector fai g will indicate damage status of the structure. Then, based on the value of the identify parameters ai , we can find out the damages of the system. How the structural local damage being detected by the method mentioned above is shown in Figure 2.

Numerical simulations To evaluate the performance of the certain method, we carried out some simulations and a practical experimentation.

Parameters setting for CS algorithm The parameter of discovery rate of alien eggs/solutions pa is taken as 0.25. The searching step s is taken as 1. The initial fitness of the objective function is 100. The tolerance of the iteration is set to be 1e27. All the parameters setting is refer to the Yang and Deb (2009)’s research. To include the uncertainty in the measured data and to study the sensitivity of CS to noise, uniformly distributed random noise is added to measured data in the simulation. To simulate measurement noise, the natural frequencies are contaminated with 1% noise, and 10% noise is added to the modal

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Figure 2. Flowchart of the CS algorithm for solving the problem of structural damage detection (the CS algorithm is inside the dashed box).

displacement proposed by Kang et al. (2012) for all the cases. The detail of the measured noise is as below. In this case, we investigate the effect of artificial measurement noise on the identified results. A normally distributed random error with zero mean and a unit standard deviation is added to the calculated acceleration to simulate the ‘‘noisy’’ measurement as shown below _ d€ = d€cal + Ep Noise var(d€cal ) _

ð10Þ

where d€ is the vector of polluted acceleration response, Ep is the noise level, Noise is a standard normal distribution vector with zero mean and unit standard deviation and var(:) is the variance of the time history. Meanwhile, a classic rank-based GA is also implemented here using the MATLABÒ GA Toolboxä. In the GA, colony size is also 50, the rate of crossover was selected equal to 0.8 while a Gaussian function with zero mean value was adopted for mutation strategy with a scale factor of 0.5 and shrink factor of 0.75 (Hinterding, 1995).

Figure 3. A dual span supported beam and cross-section.

A dual span simply supported beam A dual span simply supported beam is studied as an example. Figure 3 illustrates the geometric characteristics of the beam. A total number of elements and nodes are 20 and 21, respectively. Young’s modulus of the system is 210 GPa, mass density is 7:8 3 103 kg=m3 , and the beam length is 10 m. The damage case is studied and the first three natural frequencies and mode shapes are utilized. In this case, it is assumed that element 4 has 20% reduction in stiffness, while element 5 has 15% and element 16 has 10%. And element 4 and element 5 are two adjacent elements.

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1 Genetic algorithm Cuckoo searh algorithm

Fitness value

0 -1 -2 -3 -4 -5 -6

0

500

1000

1500 Iterations

2000

2500

3000

Figure 4. Iteration process of objective function based on cuckoo search algorithm for the beam compared with the genetic algorithm. Table 1. Identified results of the damage parameters of the beams. Element no.

Assumed damage (damage index)

GA (damage index)

CS algorithm (damage index)

2 3 4* 5* 6 7 8 13 14 15 16* 17 18 19

0 0 0.2 0.15 0 0 0 0 0 0 0.1 0 0 0

0.015 (0.015) 0.035 (0.035) 0.231 (0.031) 0.182 (0.032) 0.036 (0.036) 0.025 (0.025) 0.009 (0.009) 0.008 (0.008) 0.034 (0.034) 0.036 (0.036) 0.132 (0.032) 0.031 (0.031) 0.028 (0.028) 0.015 (0.015)

0.007 (0.007) 0.018 (0.018) 0.212 (0.012) 0.161 (0.011) 0.013 (0.013) 0.012 (0.012) 0 (0) 0.01 (0.01) 0.018 (0.018) 0.022 (0.022) 0.096 (0.006) 0.026 (0.026) 0.005 (0.005) 0.002 (0.002)

GA: genetic algorithm; CS: cuckoo search. (b) is form of data (relative error). *means the element of assumed damage.

In noise-free condition, the identified results from CS algorithm converge to the true value with the final fitness value less than 1028. In noise-contaminated condition, their final converged fitness values are 9:2745e8 . It indicates that the identified results are close to the true damage status of the structure. The curves of fitness values for CS are demonstrated in Figure 4; one can find that CS leads to convergence rapidly.

Figure 5. Evolutionary processes of the elements of the beams.

The evolutionary processes of damaged element in noise-contaminated condition are shown in Figure 5, proving all the damaged elements can be found correctly. What needed to be emphasized is that all these data were collected 20 times for average. We consider both the conditions with noise. Damage detection results are shown in Table 1 and Figure 6. From Figure 6, we can find the damage location and damage degrees. As the detection result shown in Table 1, the largest error of GA is 3.5% in sixth element, and the largest error of CS algorithm is 1.8% in third element. Three damaged elements are identified accurately. Then, we can find out the CS algorithm is more effective than GA; besides, CS algorithm is insensitive for the measurement noise and can introduce to engineering application.

A truss structure A 31-bar truss structure shown in Figure 7 is studied as another example. The length of each exterior bar is l = 1 m, and length of interior bar is l = 1:41 m. The section area of each bar is A = 0:004 m2 . Young’s modulus is E = 200 GPa, and mass density is r = 7800 kg=m3 . Total numbers of elements and nodes are 31 and 14, respectively. The first six frequencies and mode shapes are utilized in this case. Three local damages are assumed to locate at the 5th, 12th, and 25th elements, with a reduction of stiffness of 10%, 15%, and 20%, respectively. Damages have been identified accurately in both aspects of location and extent when noise is free. In noise condition, damage locations have been correctly detected with only little false alarms. Figure 8 shows the evolution processes of all damaged indices in noise condition. The evolutionary processes of all elements in noise-contaminated condition are shown in Figure 9. What needed to be stressed is that all these data were collected 20 times for average. The identified results are shown in Figure 10 for both noise-free and

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855 0.25 Assumed damage Genetic algorithm Cuckoo search algorithm

Damage index

0.2

0.15

0.1

0.05

0 0

1

2

3

4

5

6

7

8

9 10 11 Element number

12

13

14

15

16

17

18

19

20

Figure 6. Damage identification result of the dual span supported beam.

Figure 7. A 31-bar truss structure.

Table 2. Identified results of the damage parameters of the 31-bar truss.

1 Genetic Algorithm Cuckoo search Algorithm

0

Fitness

-1 -2 -3 -4

Element no.

Assumed damage (damage index)

GA (damage index)

CS (damage index)

2 3 4 5* 6 7 10 11 12* 13 14 15 23 24 25* 26 27 28

0 0 0 0.1 0 0 0 0 0.15 0 0 0 0 0 0.2 0 0 0

0.004 (0.004) 0.008 (0.008) 0.036 (0.036) 0.143 (0.043) 0.025 (0.025) 0.015 (0.015) 0.008 (0.008) 0.041 (0.041) 0.187 (0.037) 0.042 (0.042) 0.01 (0.01) 0.005 (0.005) 0.021 (0.021) 0.048 (0.048) 0.245 (0.045) 0.047 (0.047) 0.018 (0.018) 0 (0)

0 (0) 0 (0) 0.011 (0.011) 0.113 (0.013) 0.01 (0.01) 0 (0) 0.004 (0.004) 0.02 (0.02) 0.164 (0.014) 0.011 (0.011) 0 (0) 0 (0) 0.008 (0.008) 0.023 (0.023) 0.211 (0.011) 0.029 (0.029) 0.008 (0.008) 0 (0)

-5 -6 0

500

1000

1500

2000 Iterations

2500

3000

3500

4000

Figure 8. Iteration process of objective function based on cuckoo search algorithm for the truss.

noise-contaminated conditions. The identification errors are listed in Table 2. By comparing the detection result of the 31-bar truss with artificial measurement noise showed in Figure 10 and Table 2, the largest error of GA is 4.8% in 24th element, and the largest error of CS algorithm is 2.9% in 26th element. Results show that the proposed method is efficient on determining the sites and the extents of the structure damages. Then, we can find out the CS algorithm is more effective than GA; besides, CS algorithm is insensitive for the measurement noise and can introduce to engineering application.

GA: genetic algorithm; CS: cuckoo search. (b) is form of data (relative error). *means the element of assumed damage.

Laboratory work The proposed method is further demonstrated with laboratory results from a free-free steel beam as shown

in Figure 11. The beam is hung by two nylon ropes at two ends. The parameters of the beam are as follows:

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Figure 9. Evolutionary processes of damage indices of the truss.

Assumed damage Genetic algorithm Cuckoo search algorithm

Damage index

0.25

0.2

0.15

0.1

0.05

0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

Element number

Figure 10. Damage identification result of the truss.

Figure 11. Sketch of the experimental beam (dimensions are not scaled).

length l = 2:1 m, width b = 0:025 m, and height h0 = 0:019 m; Young’s modulus and mass density of the material are E = 207 GPa and r = 7832 kg=m3 , respectively. The damage is simulated by a crack located at 1.72 m from the left free end, created by a machine saw with 1.3 mm thick cutting blade. The crack depth dc is 3 mm. An impulsive force was applied with an impact hammer model B&K 8202 at 1.2 m from the left end. The sampling frequency is 2000 Hz. Four-second acceleration response data collected by an accelerometer model B&K 4370 at the mid-span of the beam are used to extract the first five natural frequencies of the intact and damaged beam for damage identification. The first five natural frequencies of the intact and the damaged beam with crack are shown in Table 3.

The beam is discretized into 20 Euler beam elements with 2 degrees of freedom at each node. And the local damage is located at the 17th element, with a reduction of stiffness of 10% after finite element modeling. The natural frequencies calculated from the finite element model are quite close to the measured values, which indicate that the initial finite element model is accurate enough for the subsequent damage identification. As only the measured natural frequency data are used in the damage identification, the first term of the objective function (equation (10) is used and the second term is removed by setting uj = 0. Figure 12 shows the evolution processes of all damaged indices in noise condition. The evolutionary processes of all elements in noisecontaminated condition are shown in Figure 13. Then

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Table 3. Modal frequencies of beam from test and analytical results. Modal frequency

First

Second

Third

Fourth

Fifth

Intact

22.50 22.75 (0.25) 22.80 22.75 (0.25)

62.75 62.80 (0.05) 62.50 62.55 (0.05)

123.00 123.50 (0.50) 122.50 122.00 (0.50)

203.25 203.50 (0.25) 202.50 202.25 (0.25)

303.50 304.00 (0.50) 302.50 302.75 (0.25)

Measured FEM Measured CS

Damage

FEM: finite element method. (b) is form of data (relative error).

2 Genetic Algorithm Cuckoo search Algorithm

Fitness

0 -2 -4 -6 -8

0

500

1000

1500

2000 Iterations

2500

3000

3500

4000

Figure 12. Iteration process of objective function based on cuckoo search algorithm for the experimental beam.

Figure 14 and Table 4 show the identified results. One can find that the location of the damage has been identified successfully, and the identified extent of

damaged element 17 is 13.2%. The relative error is 3.2%. By observing Figure 14 and Table 4, it is found that there are two large false identifications. The false alarm in element 3 may due to the reason that it is the symmetric one of element 17. The false alarm in element 16 and element 18 can be explained since element 18 is in immediate adjacent to the damage and the vibration energy in the element would be much more disturbed than those in other elements as discussed by Shi et al. (2000). The natural frequencies of the beam calculated with the identified parameters are shown in Table 3 and they are found matching the experimental values very well indicating the success of the identification.

Figure 13. Evolutionary processes of damage indices for the experimental beam.

0.2 Assumed damage Cuckoo search algorithm

Damage index

0.15

0.1

0.05

0 0

1

2

3

4

5

6

7

8

9 10 11 Element number

Figure 14. Damage identification result of the experimental beam.

12

13

14

15

16

17

18

19

20

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Table 4. Identified results of the damage parameters of the experimental beam. Element no.

Assumed damage (damage index)

CS algorithm (damage index)

1 2 3 4 5 6 10 11 12 13 14 15 16 17* 18 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0.1 0 0

0 (0) 0 (0) 0.047 (0.047) 0.01 (0.01) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0.011 (0.011) 0.037 (0.037) 0.047 (0.047) 0.132 (0.032) 0.034 (0.034) 0 (0)

(b) is form of data (relative error). *means the element of assumed damage.

Conclusion Structural damage identification based on CS algorithm using vibration data is investigated in this work. The proposed method only needs the first few natural frequencies and mode shapes of the structure in the identification. The two numerical simulation work and a laboratory work illustrate that the present approach is correct and efficient for detecting structural local damages even under measurement noise, compared with the GA. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (11172333, 11272361). Such financial aids are gratefully acknowledged.

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859 identification methods. Advances in Structural Engineering 16(9): 1611–1620. Yang QW (2011) A new damage identification method based on structural flexibility disassembly. Journal of Vibration and Control 17(7): 1000–1008. Yang XS and Deb S (2009) Cuckoo search via Levy flights. In: Proceedings of the world congress on nature and biologically inspired computing (NaBIC 2009), Coimbatore, India, 9–11 December, pp. 210–214. New York: IEEE Publication. Yang XS and Deb S (2010) Engineering optimization by cuckoo search. International Journal of Mathematical Modelling and Numerical Optimisation 1(4): 330–343. Yang XS and Deb S (2013) Multiobjective cuckoo search for design optimization. Computers & Operations Research 40: 1616–1624. Yu L and Xu P (2011) Structural health monitoring based on continuous ACO method. Microelectronics Reliability 51(2): 270–278. Zou Y, Tong L and Steven GP (2000) Vibration-based model-dependent damage (delamination) identification and health monitoring for composite structures—a review. Journal of Sound and Vibration 230(2): 357–378.

Structural damage identification based on cuckoo search algorithm

Advances in Structural Engineering 2016, Vol. 19(5) 849–859 Ó The Author(s) 2016 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1369433216630128 ase.sagepub.com

HJ Xu, JK Liu and ZR Lu

Abstract A method for structural damage identification based on cuckoo search algorithm is presented. The nonlinear objective function for the damage identification problem is established by utilizing modal assurance criteria and structural natural frequencies. Then, the cuckoo search algorithm is adopted to detect local damages by solving the objective function. Two numerical examples are studied to investigate the efficiency and correctness of the proposed method. Meanwhile, a laboratory work is conducted for further verification. The simulation and experiment results show that the present method can produce more accurate damage identification results even under measurement noise, comparing with genetic algorithm.

Keywords cuckoo search algorithm, damage detection, frequency domain, global optimization, modal assurance criteria

Introduction Negligence of the structural local damages may cause serious engineering problems. Efficient methods to detect and quantify structural damage have drawn wide attention from various engineering fields. The vibration-based damage identification methods have been researched by Fan and Qiao (2011). Doebling et al. (1998) provided a comprehensive summary on the damage detection methods by examining changes in the dynamic characteristics of the structure. Zou et al. (2000) offer an extensive review of the progress on structural condition monitoring and damage identification for composite structures. Previous studies have shown that the associated changes in the structures will result in changes in dynamic characteristics of the system. Therefore, by detecting some of these properties of the damaged structure, the local damages of the structure could be identified. The identification approaches are mainly based on the change in the natural frequencies (Gelman, 2010; Narkis, 1994), mode shapes (Ghafory and Ghasemi, 2013; Wang et al., 2013b), measured modal flexibility (Pandey et al., 1991; Yang, 2011), frequency response function (Pandey and Biswas, 1994; Ren and De Roeck, 2002), or the combination of these methods (Kim et al., 2003; Owolabi et al., 2003; Tomaszewska, 2010). Among all of the structural damage detection techniques, approaches based on dynamic responses have

been a hot research topic during the past 20 years. There are a lot of non-destructive methods in the literatures for structural damage detection in time domain. By utilizing the features of dynamic response sensitivity, Lu and Law (2007) proposed a response sensitivity-based method to identify structural damage effectively. Besides, Fu et al. (2013) and Lu et al. (2013) have done their job in time domain. In recent years, with the development of the computer technique and mathematics, swarm intelligence technique gained its popularity. From a mathematical view, by transferring the local damage identification problem to the objective function optimization, the damage can be obtained by minimizing the objective function. And the objective function was formulated of the difference between the calculated and the measured data from the certain structural system. Dackermann et al. (2014) applied the neural network (NN) for damage detection. Besides, genetic algorithm (GA) as a global optimization method has been used by Friswell et al. (1998) for structural damage detection. Then, some new optimization algorithms such as artificial Department of Applied Mechanics, Sun Yat-sen University, Guangzhou, P.R. China Corresponding author: ZR Lu, Department of Applied Mechanics, Sun Yat-sen University, Guangzhou 510006, Guangdong, P.R. China. Email: [email protected]

850 bee colony algorithm utilized by Hao et al. (2013), ant colony optimization (ACO) used by Yu and Xu (2011), and particle swarm optimization (PSO) applied by Begambre and Laier (2009) have been introduced in damage detection. But these swarm intelligence techniques both have common failing on slow convergence. And some of them may fall into local optimization easily. Apart from the techniques mentioned above, cuckoo search (CS) algorithm is another swarm intelligence technique, which was developed by Yang and Deb (2010) to solve numerical optimization problems. This algorithm is inspired by the reproduction strategy of cuckoos, which is utilized increasingly for its feasibility and efficiency. The CS is simple in concept and has less parameter to adjust and is easy to implement. It has been applied in many areas such as constrained optimization problems and multi-objective optimization problems. Gandomi et al. (2013) introduced CS algorithm to solve structural optimization problems. Yang and Deb (2013) utilized the CS to solve multi-objective design optimization. Walton et al. (2011) modified the CS to reach high convergence to the global minimum problem even at high numbers of dimensions. So far, very few studies have been done on structural damage detection based on CS algorithm. The article mainly deals with the damage detection problem using CS algorithm based on an objective function established in frequency domain. The performance of the certain algorithm is illustrated in two kinds of structures, that is, a dual span simply supported beam and a truss. Besides numerical simulations, a laboratory work is carried out to further validate the effectiveness of the present method. In the identification, only the first few natural frequencies and/or mode shapes are needed. Results show that local structural damages can be identified successfully even under measurement noise, which shows that CS algorithm is very promising for damage detection, compared with the GA.

A brief introduction to CS algorithm The CS method is becoming very popular due to its simplicity, efficiency, and fast convergence properties. First, introduce the CS algorithm. CS algorithm is inspired by the CS for host nests to breed. At the most basic level, cuckoos lay their eggs in the nests of other host birds, which may be of different species. The host bird may discover that the eggs are not its own and either destroy the egg or abandon the nest all together. This has resulted in the evolution of cuckoo eggs which mimic the eggs of local host birds. For simplicity, we use the following three idealized rules:

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2. 3.

Each cuckoo lays one egg at a time, which represents a set of solution coordinates and dumps it in a randomly chosen nest; The best nests with high quality of eggs (solutions) will carry over to the next generations; The number of available host nests is fixed, and a host can discover an alien egg with a probability pa 2 ½0, 1. In this case, the host bird can either throw the egg away or abandon the nest so as to build a completely new nest in a new location.

For simplicity, this last assumption can be approximated by a fraction pa of the nests being replaced by new nests (with new random solutions at new locations). For a maximization problem, the quality or fitness of a solution can simply be proportional to the objective function. Based on these three rules mentioned above, the basic steps of the CS can be summarized as flee shown in Figure 1. When generating new solutions x(t + 1) a Levy flight of cuckoo i, is performed xi(t + 1) = x(t) i + aLevy(l)

ð1Þ

where a.0 is the step size which should be related to the scales of the problem. In most cases, we can use a = 0(1). The product means entry-wise multiplications. Levy flight essentially provides a random walk while their random steps are drawn from a Levy distribution for large steps Levy ; m = tl , (1\l\3)

ð2Þ

which has an infinite variance with an infinite mean. Here, the consecutive jumps/steps of a cuckoo essentially form a random walk process which obeys a power-law step-length distribution with a heavy tail. In addition, a fraction pa of the worst nests can be abandoned so that new nests can be built at new locations by random walks and mixing. The mixing of the eggs/solutions can be performed by random permutation according to the similarity/difference to the host eggs. Obviously, the generation of step size s samples is not trivial using Levy flights. A simple scheme discussed in detail by Yang and Deb (2010) can be summarized as (t) s = a0 x(t) x Levy(b) ; j i

jnj

1=b

m (t) x(t) j xi

ð3Þ

where m and n are drawn from normal distributions. That is

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Figure 1. Flowchart of the CS algorithm.

m ; N 0, s2u ,

n ; N 0, s2n

ð4Þ

And we choose frequency residual error and modal assurance criteria (MAC) to build up the structural objective function as below.

ð5Þ

Methodology for vibration-based damage identification

with sm =

G(1 + b) sin (pb=2) G½(1 + b)=2b2(b1)=2

1=b sn = 1

Here, G is the standard gamma function. When generating a new egg in Figure 1, a Levy flight is performed starting at the position of a randomly selected egg if the objective function value at these new coordinates is better than another randomly selected egg, then that egg is moved to this new position. The scale of this random search is controlled by multiplying the generated Levy flight by a step size. Yang and Deb (2009) do not discuss boundary handling in their formulation, but an approach similar to PSO boundary handling is adopted here. When a Levy flight results in an egg location outside the bounds of the objective function, the fitness and position of the original egg are not changed. Above all, using Levy flight and fraction pa , CS algorithm can be proposed to various optimization problems effectively and show a high convergence rate to the trace global minimum. In this article, CS algorithm is selected to solve the damage index objective function of the engineering structure. And then we can conclude the location and extent of the damaged structures to carry out the structural damage detection.

The premise for techniques based on vibration responses is that damage causes a change in structural physical properties, mainly in stiffness and damping at the damaged locations.

Modeling structural local damage The eigenvalue equation for a finite element model of the system is

K v2j M uj = 0

ð6Þ

where M and K are the structural mass matrix and stiffness matrix, respectively. And vj is the jth natural frequency and uj is the corresponding mode shape. In the finite element model of the damaged system, the local damage is modeled as reduction in the elemental stiffness of the system, but the other properties remain unchanged. When a structure with nel elements is damaged, the reduction of the stiffness can be evaluated by a set of damage parameters ai (i = 1, 2, . . . , nel). The value range of parameter ai is

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between 0 and 1. When ai = 1, it means that the ith element is intact while ai = 0, and it represents that system is completely damaged. And then the stiffness matrix of the damaged system can be written as Kd =

nel X

f (x) =

s X

ð9Þ 1 MAC uhi , udi + ER vhi , vdi

i=1

kie

ð7Þ

i=1

where Kd is the stiffness matrix of the damaged system, and kie presents the ith elemental stiffness matrix in the global form. The structural damages can be identified by identifying each value in the damage parameter vector fag, based on the damage model mentioned above.

Objective function based on vibration data It is well known that the changes of stiffness will cause the changes in the structural properties, such as natural frequencies and mode shapes. In this study, the objective function for damage detection is based on vibration data, such as natural frequencies and mode shapes. From a mathematical view, such identification problem can be regarded as a nonlinear optimization problem, in which an objective function, for instance the error between the actual measured structural response and the analytic response of a model, is defined and the parameters of such models determined so to optimize the objective function (usually to minimum) (Franco et al., 2004; Wang et al., 2013a). By minimizing the differences between the measured response data and the calculated one, we can locate and quantify the local damage of the certain system. Refer to Wang et al.’s (2013a) research, in this article, first of all, we select the frequency and modal shape data to build up the objective function of the certain problem. Then, we take the natural frequency and the MAC as indexes of damage detection in view of the crack damage, and then the CS algorithm is applied to identifying the structural damage. The detail of how the objective function is built is described in the following. Considering the mode shape data, MAC proposed by Brehm et al. (2010) is utilized. The MAC is expressed as

MACj =

objective function used for damage detection is expressed as

(fuj gT fvM j g) 2 jfuj gj2 juM j j

2

ð8Þ

where uj and uM j are the jth calculated and measured mode shapes, respectively. Based on MAC and taking into account the changes both in natural frequencies and mode shapes, the

where s is the modal order, vhi , uhi , vdi , udi is the ith order frequency and mode shape of the intact and damaged system. Besides, ER(vhi , vdi ) = (vhi vdi )= d hT h vhi j 3 100%, MAC(uhi , udi ) = juhT i ui j=j(ui ui )jj dT d (ui ui )j, i = 1, 2, . . . , s. As described above, when a structure with nel elements is damaged, the reduction of the stiffness can be evaluated by a set of damage parameters ai (i = 1, 2, . . . , nel). The value range of parameter ai is between 0 and 1. When ai = 0, it means that the ith element is intact while ai = 1, it represents that system is completely damaged, and when ai = 0:1, it means the system is 10% damaged. The change in damage parameters indicates the changes in the frequency and modal shape. Then, the calculated nature frequency vi and mode shape ui are related to damage parameters ½a1 , a2 , . . . , anel . In the inverse problem, if the value of damage parameter vector ai matches the given particular damage condition, then the objective function will achieve its minimum value 0. In other words, when the minimum value of the objective function is identified, the corresponding parameter vector fai g will indicate damage status of the structure. Then, based on the value of the identify parameters ai , we can find out the damages of the system. How the structural local damage being detected by the method mentioned above is shown in Figure 2.

Numerical simulations To evaluate the performance of the certain method, we carried out some simulations and a practical experimentation.

Parameters setting for CS algorithm The parameter of discovery rate of alien eggs/solutions pa is taken as 0.25. The searching step s is taken as 1. The initial fitness of the objective function is 100. The tolerance of the iteration is set to be 1e27. All the parameters setting is refer to the Yang and Deb (2009)’s research. To include the uncertainty in the measured data and to study the sensitivity of CS to noise, uniformly distributed random noise is added to measured data in the simulation. To simulate measurement noise, the natural frequencies are contaminated with 1% noise, and 10% noise is added to the modal

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Figure 2. Flowchart of the CS algorithm for solving the problem of structural damage detection (the CS algorithm is inside the dashed box).

displacement proposed by Kang et al. (2012) for all the cases. The detail of the measured noise is as below. In this case, we investigate the effect of artificial measurement noise on the identified results. A normally distributed random error with zero mean and a unit standard deviation is added to the calculated acceleration to simulate the ‘‘noisy’’ measurement as shown below _ d€ = d€cal + Ep Noise var(d€cal ) _

ð10Þ

where d€ is the vector of polluted acceleration response, Ep is the noise level, Noise is a standard normal distribution vector with zero mean and unit standard deviation and var(:) is the variance of the time history. Meanwhile, a classic rank-based GA is also implemented here using the MATLABÒ GA Toolboxä. In the GA, colony size is also 50, the rate of crossover was selected equal to 0.8 while a Gaussian function with zero mean value was adopted for mutation strategy with a scale factor of 0.5 and shrink factor of 0.75 (Hinterding, 1995).

Figure 3. A dual span supported beam and cross-section.

A dual span simply supported beam A dual span simply supported beam is studied as an example. Figure 3 illustrates the geometric characteristics of the beam. A total number of elements and nodes are 20 and 21, respectively. Young’s modulus of the system is 210 GPa, mass density is 7:8 3 103 kg=m3 , and the beam length is 10 m. The damage case is studied and the first three natural frequencies and mode shapes are utilized. In this case, it is assumed that element 4 has 20% reduction in stiffness, while element 5 has 15% and element 16 has 10%. And element 4 and element 5 are two adjacent elements.

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1 Genetic algorithm Cuckoo searh algorithm

Fitness value

0 -1 -2 -3 -4 -5 -6

0

500

1000

1500 Iterations

2000

2500

3000

Figure 4. Iteration process of objective function based on cuckoo search algorithm for the beam compared with the genetic algorithm. Table 1. Identified results of the damage parameters of the beams. Element no.

Assumed damage (damage index)

GA (damage index)

CS algorithm (damage index)

2 3 4* 5* 6 7 8 13 14 15 16* 17 18 19

0 0 0.2 0.15 0 0 0 0 0 0 0.1 0 0 0

0.015 (0.015) 0.035 (0.035) 0.231 (0.031) 0.182 (0.032) 0.036 (0.036) 0.025 (0.025) 0.009 (0.009) 0.008 (0.008) 0.034 (0.034) 0.036 (0.036) 0.132 (0.032) 0.031 (0.031) 0.028 (0.028) 0.015 (0.015)

0.007 (0.007) 0.018 (0.018) 0.212 (0.012) 0.161 (0.011) 0.013 (0.013) 0.012 (0.012) 0 (0) 0.01 (0.01) 0.018 (0.018) 0.022 (0.022) 0.096 (0.006) 0.026 (0.026) 0.005 (0.005) 0.002 (0.002)

GA: genetic algorithm; CS: cuckoo search. (b) is form of data (relative error). *means the element of assumed damage.

In noise-free condition, the identified results from CS algorithm converge to the true value with the final fitness value less than 1028. In noise-contaminated condition, their final converged fitness values are 9:2745e8 . It indicates that the identified results are close to the true damage status of the structure. The curves of fitness values for CS are demonstrated in Figure 4; one can find that CS leads to convergence rapidly.

Figure 5. Evolutionary processes of the elements of the beams.

The evolutionary processes of damaged element in noise-contaminated condition are shown in Figure 5, proving all the damaged elements can be found correctly. What needed to be emphasized is that all these data were collected 20 times for average. We consider both the conditions with noise. Damage detection results are shown in Table 1 and Figure 6. From Figure 6, we can find the damage location and damage degrees. As the detection result shown in Table 1, the largest error of GA is 3.5% in sixth element, and the largest error of CS algorithm is 1.8% in third element. Three damaged elements are identified accurately. Then, we can find out the CS algorithm is more effective than GA; besides, CS algorithm is insensitive for the measurement noise and can introduce to engineering application.

A truss structure A 31-bar truss structure shown in Figure 7 is studied as another example. The length of each exterior bar is l = 1 m, and length of interior bar is l = 1:41 m. The section area of each bar is A = 0:004 m2 . Young’s modulus is E = 200 GPa, and mass density is r = 7800 kg=m3 . Total numbers of elements and nodes are 31 and 14, respectively. The first six frequencies and mode shapes are utilized in this case. Three local damages are assumed to locate at the 5th, 12th, and 25th elements, with a reduction of stiffness of 10%, 15%, and 20%, respectively. Damages have been identified accurately in both aspects of location and extent when noise is free. In noise condition, damage locations have been correctly detected with only little false alarms. Figure 8 shows the evolution processes of all damaged indices in noise condition. The evolutionary processes of all elements in noise-contaminated condition are shown in Figure 9. What needed to be stressed is that all these data were collected 20 times for average. The identified results are shown in Figure 10 for both noise-free and

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Damage index

0.2

0.15

0.1

0.05

0 0

1

2

3

4

5

6

7

8

9 10 11 Element number

12

13

14

15

16

17

18

19

20

Figure 6. Damage identification result of the dual span supported beam.

Figure 7. A 31-bar truss structure.

Table 2. Identified results of the damage parameters of the 31-bar truss.

1 Genetic Algorithm Cuckoo search Algorithm

0

Fitness

-1 -2 -3 -4

Element no.

Assumed damage (damage index)

GA (damage index)

CS (damage index)

2 3 4 5* 6 7 10 11 12* 13 14 15 23 24 25* 26 27 28

0 0 0 0.1 0 0 0 0 0.15 0 0 0 0 0 0.2 0 0 0

0.004 (0.004) 0.008 (0.008) 0.036 (0.036) 0.143 (0.043) 0.025 (0.025) 0.015 (0.015) 0.008 (0.008) 0.041 (0.041) 0.187 (0.037) 0.042 (0.042) 0.01 (0.01) 0.005 (0.005) 0.021 (0.021) 0.048 (0.048) 0.245 (0.045) 0.047 (0.047) 0.018 (0.018) 0 (0)

0 (0) 0 (0) 0.011 (0.011) 0.113 (0.013) 0.01 (0.01) 0 (0) 0.004 (0.004) 0.02 (0.02) 0.164 (0.014) 0.011 (0.011) 0 (0) 0 (0) 0.008 (0.008) 0.023 (0.023) 0.211 (0.011) 0.029 (0.029) 0.008 (0.008) 0 (0)

-5 -6 0

500

1000

1500

2000 Iterations

2500

3000

3500

4000

Figure 8. Iteration process of objective function based on cuckoo search algorithm for the truss.

noise-contaminated conditions. The identification errors are listed in Table 2. By comparing the detection result of the 31-bar truss with artificial measurement noise showed in Figure 10 and Table 2, the largest error of GA is 4.8% in 24th element, and the largest error of CS algorithm is 2.9% in 26th element. Results show that the proposed method is efficient on determining the sites and the extents of the structure damages. Then, we can find out the CS algorithm is more effective than GA; besides, CS algorithm is insensitive for the measurement noise and can introduce to engineering application.

GA: genetic algorithm; CS: cuckoo search. (b) is form of data (relative error). *means the element of assumed damage.

Laboratory work The proposed method is further demonstrated with laboratory results from a free-free steel beam as shown

in Figure 11. The beam is hung by two nylon ropes at two ends. The parameters of the beam are as follows:

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Figure 9. Evolutionary processes of damage indices of the truss.

Assumed damage Genetic algorithm Cuckoo search algorithm

Damage index

0.25

0.2

0.15

0.1

0.05

0 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

Element number

Figure 10. Damage identification result of the truss.

Figure 11. Sketch of the experimental beam (dimensions are not scaled).

length l = 2:1 m, width b = 0:025 m, and height h0 = 0:019 m; Young’s modulus and mass density of the material are E = 207 GPa and r = 7832 kg=m3 , respectively. The damage is simulated by a crack located at 1.72 m from the left free end, created by a machine saw with 1.3 mm thick cutting blade. The crack depth dc is 3 mm. An impulsive force was applied with an impact hammer model B&K 8202 at 1.2 m from the left end. The sampling frequency is 2000 Hz. Four-second acceleration response data collected by an accelerometer model B&K 4370 at the mid-span of the beam are used to extract the first five natural frequencies of the intact and damaged beam for damage identification. The first five natural frequencies of the intact and the damaged beam with crack are shown in Table 3.

The beam is discretized into 20 Euler beam elements with 2 degrees of freedom at each node. And the local damage is located at the 17th element, with a reduction of stiffness of 10% after finite element modeling. The natural frequencies calculated from the finite element model are quite close to the measured values, which indicate that the initial finite element model is accurate enough for the subsequent damage identification. As only the measured natural frequency data are used in the damage identification, the first term of the objective function (equation (10) is used and the second term is removed by setting uj = 0. Figure 12 shows the evolution processes of all damaged indices in noise condition. The evolutionary processes of all elements in noisecontaminated condition are shown in Figure 13. Then

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Table 3. Modal frequencies of beam from test and analytical results. Modal frequency

First

Second

Third

Fourth

Fifth

Intact

22.50 22.75 (0.25) 22.80 22.75 (0.25)

62.75 62.80 (0.05) 62.50 62.55 (0.05)

123.00 123.50 (0.50) 122.50 122.00 (0.50)

203.25 203.50 (0.25) 202.50 202.25 (0.25)

303.50 304.00 (0.50) 302.50 302.75 (0.25)

Measured FEM Measured CS

Damage

FEM: finite element method. (b) is form of data (relative error).

2 Genetic Algorithm Cuckoo search Algorithm

Fitness

0 -2 -4 -6 -8

0

500

1000

1500

2000 Iterations

2500

3000

3500

4000

Figure 12. Iteration process of objective function based on cuckoo search algorithm for the experimental beam.

Figure 14 and Table 4 show the identified results. One can find that the location of the damage has been identified successfully, and the identified extent of

damaged element 17 is 13.2%. The relative error is 3.2%. By observing Figure 14 and Table 4, it is found that there are two large false identifications. The false alarm in element 3 may due to the reason that it is the symmetric one of element 17. The false alarm in element 16 and element 18 can be explained since element 18 is in immediate adjacent to the damage and the vibration energy in the element would be much more disturbed than those in other elements as discussed by Shi et al. (2000). The natural frequencies of the beam calculated with the identified parameters are shown in Table 3 and they are found matching the experimental values very well indicating the success of the identification.

Figure 13. Evolutionary processes of damage indices for the experimental beam.

0.2 Assumed damage Cuckoo search algorithm

Damage index

0.15

0.1

0.05

0 0

1

2

3

4

5

6

7

8

9 10 11 Element number

Figure 14. Damage identification result of the experimental beam.

12

13

14

15

16

17

18

19

20

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Table 4. Identified results of the damage parameters of the experimental beam. Element no.

Assumed damage (damage index)

CS algorithm (damage index)

1 2 3 4 5 6 10 11 12 13 14 15 16 17* 18 19

0 0 0 0 0 0 0 0 0 0 0 0 0 0.1 0 0

0 (0) 0 (0) 0.047 (0.047) 0.01 (0.01) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0 (0) 0.011 (0.011) 0.037 (0.037) 0.047 (0.047) 0.132 (0.032) 0.034 (0.034) 0 (0)

(b) is form of data (relative error). *means the element of assumed damage.

Conclusion Structural damage identification based on CS algorithm using vibration data is investigated in this work. The proposed method only needs the first few natural frequencies and mode shapes of the structure in the identification. The two numerical simulation work and a laboratory work illustrate that the present approach is correct and efficient for detecting structural local damages even under measurement noise, compared with the GA. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (11172333, 11272361). Such financial aids are gratefully acknowledged.

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