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applied sciences Article

Application of the Random Decrement Technique in Damage Detection under Moving Load Hadi Kordestani

ID

, Yi-Qiang Xiang *

ID

, Xiao-Wei Ye and Ya-Kun Jia

College of Architectural and Civil Engineering, Zhejiang University, Hangzhou 310058, China; [email protected] (H.K.); [email protected] (X.-W.Y.); [email protected] (Y.-K.J.) * Correspondence: [email protected]; Tel.: +86-571-8820-8700 Received: 11 April 2018; Accepted: 6 May 2018; Published: 9 May 2018

Abstract: This paper employs the random decrement technique as an output-only method to detect damage from the acceleration signals under a moving load. The random decrement technique is an especial averaging method that produces Random Decrement Signatures (RDS). For this purpose, Arias Intensity (AI) was employed to calculate the energy content of each RDS and substitute each acceleration signal by a scalar invariant value. Normalizing AIs, all RDSs were then updated so as to show a unique energy along the undamaged structure. Once the normalizing factor was computed for the intact structure, the damage was determined by the absolute difference of normalized AIs obtained from each individual RDS along the structure simultaneously. To verify the proposed method, two experimental models of a simply supported beam and a scaled arch bridge were developed under a moving load (vehicle simulation), and acceleration data were recorded. The results of laboratory models proved that the RDSs can accurately detect the damage location using the normalized AI without applying any further frequency filtering. This method needs neither the damage location nor modal parameters in advance, and could properly work in a noisy environment as well. Keywords: random decrement technique; acceleration data; moving load; damage detection; Arias intensity

1. Introduction Generally, modal parameters are identified by measuring input excitations and output responses. These modal parameters are then used as damage indicators in the damage detection process. For simplicity, it is assumed that if the input excitation is considered as zero, the output response behaves as a free response [1]. There are situations in which inputs come from ambient or random excitations and cannot be ignored. In such cases, it is necessary to use output-only modal identification methods. However, these types of methods also have some limitations. One of the common drawbacks of output-only methods is considering the input as zero-mean and random white noise. Utilizing acceleration data to detect the damage location could effectively decrease the cost of Structural Health Monitoring (SHM). Accelerometers can be implemented in any desired location of structure to record the acceleration response with a wide range of frequencies. Generally, SHM methods using acceleration data can be categorized into four groups: 1-transformer-based methods, such as Wavelet or Hilbert-Huang transformers [2–7]; 2-averaging-based methods, such as Random Decrement (RD) or Moving average based approaches [8–11]; 3-filtering-based methodologies, such as Kalman filter [12,13]; and, finally, 4-source separation methods such as Blind Source Separation approach [14–17]. This paper presents a RD-based approach that belongs to the second group. The RD technique is one of the promising output-only time-domain methods introduced by Cole in 1968 [18]. Since then, the RD technique has been received a large amount of attention in Structural Health Monitoring (SHM) when a complete knowledge of random input to the under-operation Appl. Sci. 2018, 8, 753; doi:10.3390/app8050753

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is one of the promising output-only time-domain methods introduced 2 ofby 17 Cole in 1968 [18]. Since then, the RD technique has been received a large amount of attention in Structural Health Monitoring (SHM) when a complete knowledge of random input to the structure is not available. a special averaging method that has no mathematical under-operation structure isRD nottechnique available.isRD technique is a special averaging method that has no basic developed for general cases [19,20]. A general review of the application of the RD technique in mathematical basic developed for general cases [19,20]. A general review of the application of the Operational Modal Analysis (OMA) is presented in this paper. RD technique in Operational Modal Analysis (OMA) is presented in this paper. For the first time, the RD technique was used to calculate a so-called auto-RD function For the first time, the RD technique was used to calculate a so-called auto-RD function fromfrom one one output signal [18]. However, it was later extended to multi-output signals. To find RDSs output signal [18]. However, it was later extended to multi-output signals. To find RDSs for for multi-output signals, it is necessary to select the main signal first; then, the RDSs for the other signals multi-output signals, it is necessary to select the main signal first; then, the RDSs for the other signals can The RDSs RDSs for for the the main can be be consequently consequently found found according according to to the the main main signal. signal. The main signal signal and and other other signals auto-RDS and and cross-RDS, respectively [1]. It is[1]. proved auto-RDS significantly signals are arecalled called auto-RDS cross-RDS, respectively It isthat proved thatcan auto-RDS can decrease noise content; however, cross-RDS incorporates some sort of noises due to its special triggering significantly decrease noise content; however, cross-RDS incorporates some sort of noises due to its condition. To overcome this problem, vectorthis triggering RD technique is employed, which is can calculate special triggering condition. To overcome problem, vector triggering RD technique employed, the auto-RDS for all the signals simultaneously [21–23]. which can calculate auto-RDS for all signals simultaneously [21–23]. The application of OMA using RDS was properly addressed in bothintime andtime frequency domains The application of OMA using RDS was properly addressed both and frequency by Jorge and Brincker [24]. Later, Ku et al. [10] employed RD technique in time and frequency domains domains by Jorge and Brincker [24]. Later, Ku et al. [10] employed RD technique in time and for acceleration data and compared the OMA results. frequency domains for acceleration data and compared the OMA results. The such as most of of which The infrastructure, infrastructure, such as aa bridge, bridge, receives receives various various ambient ambient excitations, excitations, most which are are non-stationary in nature. Although the the RD RD technique technique was was proved proved to non-stationary in nature. Although to work work well well for for stationary stationary input input excitations, Lee et et al. excitations, some some researchers researchers used used it it for for non-stationary non-stationary excitations excitations [19,25,26]. [19,25,26]. Lee al. [25] [25] used used the the RD technique to find the free response of the bridge under traffic loadings. He et al. [27] combined the RD technique to find the free response of the bridge under traffic loadings. He et al. [27] combined RD withwith empirical modemode decomposition and found the modal parameter for a bridge under the technique RD technique empirical decomposition and found the modal parameter for a bridge non-stationary operational excitation. Wu et al. [28] employed the mode separation technique along under non-stationary operational excitation. Wu et al. [28] employed the mode separation technique with RDthe technique and determined the modal parameter of cables in a cable alongthe with RD technique and determined the modal parameter of cables in abridge. cable bridge. A complete systematic procedure for designing and implementing SHM using RD technique A complete systematic procedure for designing and implementing SHMtheusing the RD on an arbitrary system was well-illustrated by Buff et al. [11]. Using more sensors in RD technique technique on an arbitrary system was well-illustrated by Buff et al. [11]. Using more sensors in RD could lead could to more accurate results, butresults, it would beitmore time-consuming as well. Some technique lead to more accurate but would be more time-consuming as researchers well. Some tried to establish RD technique totechnique overcometo this problemthis [19,29]. researchers tried atodecentralized establish a decentralized RD overcome problem [19,29]. Damage may be recognized without finding the modal parameters. Gonzalez and and Hester Damage may be recognized without finding the modal parameters. Gonzalez Hester [8] [8] identified damage in a bridge-type structure directly from the acceleration data without determining identified damage in a bridge-type structure directly from the acceleration data without determining the A scheme showing several damage detection methodsmethods using RDS is presented the modal modalparameters. parameters. A scheme showing several damage detection using RDS is in Figure 1.inAs shown RD-based localization techniques are generallyare divided into presented Figure 1. in AsFigure shown1,in Figure 1,damage RD-based damage localization techniques generally 2divided groups.into In the first group, modal parameters should be estimated first; then, damage locations can be 2 groups. In the first group, modal parameters should be estimated first; then, damage found using these modal parameters as damage indexes. However, in the second group, non-modal locations can be found using these modal parameters as damage indexes. However, in the second parameters are employed as damage indicators locate indicators the possible group, non-modal parameters are employed as to damage to damage. locate the possible damage.

Figure 1. 1. General detection using using RD RD technique. technique. Figure General scheme scheme of of damage damage detection

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The aim of this paper is to find the damage using RDS without calculating the modal parameters. To achieve this goal, acceleration signals were first recorded along the structure, and then the RD technique was used to produce both auto- and cross-RDSs. Each RDS shows a special energy content. Thus, a normalizing factor was defined to drag the RDSs to a unique level of energy. Any possible damage to the bridge disturbs the energy content. This causes the nearest RDS to the damage to show a higher normalized energy than the unique level. Consequently, the damage can easily be detected. To verify the proposed method, two experimental models of a simply supported beam and a scaled arch bridge under moving loads (vehicle simulation) were defined and tested in the laboratory. Then, the acceleration data were recorded along the structure. The results of laboratory models proved that the RDSs can detect the damage location using normalized AI without applying any frequency filtering. This method needs neither knowing the damaged place in advance nor identifying the modal parameters. Additionally, it can properly work in noisy environments. 2. Basic Theories and Method The basic concept of RD technique is very simple, especially if the structural response under random load at each time instant, ti , is composed of these two parts: 1. 2.

Deterministic response due to initial displacement and velocity and, Random response due to random excitation.

In the mathematical solution provided for RD technique, the input excitation should be stationary, random white noise, Gaussian, and zero-mean [20]. Considering a linear system with multi-degree of freedom, the corresponding equation of motion can be expressed as below: ..

.

M X (t) + C X (t) + KX (t) = f (t),

(1)

in which M, C, and K are mass, damping, and stiffness matrices, respectively. Parameter f(t) is a random white noise, zero-mean, stationary, and Gaussian input excitation. X(t) is the displacement recorded in the vertical direction. Assuming that Equation (1) is satisfied at instant ti , a simple shift in time (τ) leads to: .. . M X (ti + τ ) + C X (ti + τ ) + KX (ti + τ ) = f (ti + τ ), (2) It is supposed that the input excitation is zero-mean, which can be easily averaged out using the mean of N ( N → ∞) different starting instants. Computing the mean of N different starting instants can be expressed as below: M N

N

..

C

N

.

K

N

∑ i =1 X ( t i + τ ) + N ∑ i =1 X ( t i + τ ) + N ∑ i =1 X ( t i + τ ) =

1 N

N

∑i=1 f (ti + τ ) = 0,

(3)

Defining Y (τ ) as below, Equation (3) turns into a free vibration form as shown in Equation (5). Y (τ ) = ..

1 N

N

∑ i =1 X ( t i + τ ), .

MY (τ ) + CY (τ ) + KY (τ ) = 0,

(4) (5)

In the literature, Y (τ ) is known as Random Decrement Signature (RDS) [1,11,23,24,29]. RD technique is defined as a special averaging method in which N different starting instants, ti , are chosen using a so-called triggering level, as shown in Figure 2. Some default values for triggering levels recommended by different researchers are listed in Table 1.

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Figure2.2.Basic Basicconcept concept of of the the special special averaging Figure averaging in in RD RD technique. technique. Table 1. Triggering levels recommended by different researchers [1,22,24]. Table 1. Triggering levels recommended by different researchers [1,22,24].

Case No. Case1No. 12 2

Description Description Level crossing Positive points Level crossing Positive points

3

Zero crossing with a positive or negative slope

3

4

Zero crossing with a positive or negative slope

4

Local extremum

Local extremum

Relation ( ) =Relation { ( )= } ( T)X= = a}} (ti{) =≤{ X((ti)) < TX (ti ) = { a ≤ X (ti )>0 < } T ti ) = { X (t(i ) )= 0, . ( X)(= ≤ < , X= < 00 .

TX (ti ) = { a ≤ X (ti ) < b, X = 0}

The level-crossing triggering condition was utilized in different studies due to its simplicity. To use the level-crossing triggering condition, the optimum value is assumed as = √2 × standard The level-crossing triggering condition was utilized in different studies due to√its simplicity. deviation [24]. Then, RDS is simply defined as follows: To use the level-crossing triggering condition, the optimum value is assumed as a = 2× standard ∑ follows: ( + ) | ( ) = , ( ) = as (6) deviation [24]. Then, RDS is simply defined In the Equation (6), two parameters 1 areNquestionable, namely N and τ. These two parameters (6) Y (and τ ) =cost,∑especially X (ti +inτ )| X (ti ) = a, can control the consuming time long-term SHM. The minimum acceptable N i =1 value for N is 100, while 1000 seems to give a stable result [30]. For parameter τ, the RDS should have In thelength Equation (6), two aparameters are questionable, and τ. These two parameters can enough to represent full frequency content of thenamely originalNresponse. controlThe the time consuming time and cost, especially in long-term SHM. The minimum acceptable value for and cost in RD technique are mainly affected by three parameters, namely, triggering Nlevel is 100, 1000 to give (τ), a stable [30]. parameter τ, the should have enough (a),while length of seems RD signature and result number of For signals. According toRDS Equation (6), triggering length to represent a full frequency content of the original response. level has the greatest effect on the calculation’s time and increases the number of segments, The time andthe cost in RD technique are mainlyinaffected by three parameters, namely, triggering especially when acceleration data are utilized the analyses. A small change in the parameter a level (a),increase length of signature (τ), and number of signals. According to Equation (6), triggering level could N RD by hundreds. has theAll greatest effectillustrations on the calculation’s and increases the number segments, especially when the above are only time for single time-history data. Inoforder to control the safety the acceleration data are utilizedsuch in the small change distributed in the parameter could increase and health of infrastructures, as analyses. bridges, aAset of sensors along athe structure areN bynecessary. hundreds.Calculating the RDS in multi-channel signals was well-illustrated by Ibrahim [1]. He supposed if illustrations the RDS for are a signal is called thedata. cross-RDS could then be All the that above only for single auto-RDS, time-history In order to control thecomputed safety and usingoftriggering information the auto-RDS. auto and cross-RDSs, thethe following can health infrastructures, such on as bridges, a set ofFor sensors distributed along structureequations are necessary. be used [1]:the RDS in multi-channel signals was well-illustrated by Ibrahim [1]. He supposed that Calculating if the RDS for a signal is called auto-RDS, the cross-RDS could then be computed using triggering ( )= ∑ ( + ) | ( ) = , (7) information on the auto-RDS. For auto and cross-RDSs, the following equations can be used [1]: ( ) 1= ∑N ( + ) | ( ) = , (8) Ymm (τ ) = Xm (ti + τ )| Xm (ti ) = a, (7) ∑ i = 1 N ( ) is the auto-RDS calculated ( ) is the from the main signal ( ( )) and in which cross-RDS calculated from the signal 1 ( ) using the triggering information of the main signal. N Xc (ti + τ )| Xm (ti ) = a, (8) Ymc (τ ) = ∑ i = 1 Figure 3 shows the concept of auto andNcross-RD signature. As shown in Figure 3, the segments of based on the instant times which triggering main signal. incross-RDS which Ymmwill is chosen the auto-RDS calculated from thein main signal (Xm (t)level ) andcrosses Ymc (τ )the is the cross-RDS (τ )be Thus, it is seen that the initial condition of cross-RDS is not equal to the triggering level a. calculated from the signal X (t) using the triggering information of the main signal. Figure 3 shows c

the concept of auto and cross-RD signature. As shown in Figure 3, the segments of cross-RDS will be

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chosen based on the instant times in which triggering level crosses the main signal. Thus, it is seen that the initial of cross-RDS is not equal to the triggering level a. Appl. Sci. 2018, 8, xcondition FOR PEER REVIEW 5 of 17 Appl. Sci. 2018, 8, x FOR PEER REVIEW

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Figure 3. The concept of calculating the cross-RD function. Figure 3. The concept of calculating the cross-RD function.

The triggering level in the main signal causes the auto-RDS to have an initial value equal to a. The level in signal causes the to have an value equal a. The triggering triggering in the the main main the auto-RDS auto-RDS have an initial initial equal to to is a. However, this initiallevel condition cannotsignal occurcauses for cross-RDS. Here,tothe initial value value of cross-RDS However, this initial condition cannot occur for cross-RDS. Here, the initial value of cross-RDS is However, thisaverage initial condition occur for cross-RDS. Here, initial value of cross-RDS equal to the of the N cannot different starting instants of the the signal ( ). Therefore, dueistoequal this equal to the average of different the N different starting instants of the Xsignal ( ). Therefore, due special to this to the average of the N starting instants of the signal t . Therefore, due to this ( ) c special averaging procedure, the auto-RDS contains less noise than the cross-RDS [24]. special averaging procedure, the auto-RDS less noise the cross-RDS [24]. averaging procedure, the auto-RDS containscontains less noise than thethan cross-RDS [24]. As mentioned above, a mathematical solution for RD technique is only available for the As mentioned above, a mathematical solution for RD technique is only available for the As mentioned above, a mathematical solution for RD technique is only available for themost zero-mean, zero-mean, stationary, Gaussian, and random white noise input excitations. However, of the zero-mean, stationary, Gaussian, and random white noise input excitations. However, most ofand the stationary, Gaussian, randomare white noise inputin excitations. However, of the ambient ambient and random and excitations non-stationary nature, such as trafficmost on the bridges. A large ambient and random are non-stationary such traffic onlarge the bridges. A large random areexcitations non-stationary in nature, such in as nature, traffic on the as bridges. bodyparameters of research body of excitations research available on the literature employed RD technique to findAthe modal body of research available on the literature employed RD technique to find the modal parameters available on the employed RD technique to find the modal parameters and damages for and damages for literature non-stationary excitations [19,25,27,28]. and damages for non-stationary excitations [19,25,27,28]. non-stationary excitations [19,25,27,28]. 3. Laboratory Models of Simply Supported Beam and Arch Cable Bridge 3. Laboratory Models of Simply Supported Beam and Arch Cable Bridge 3.1. Experimental Study of the Simply Supported Beam Beam 3.1. Experimental Study of the Simply Supported The accuracy models, namely, a accuracy of of the the proposed proposedmethod methodwas wasverified verifiedusing usingtwo twolaboratory laboratory models, namely, The accuracy of the proposed method was verified using two laboratory models, namely, a simply supported beam and an arch cable bridge. This part illustrates the simply supported beam a simplysupported supportedbeam beamand andan anarch archcable cablebridge. bridge.This Thispart part illustrates illustrates the the simply simply supported beam simply model model constructed constructed in in the the laboratory laboratory of of civil civil and and structural structural health health monitoring, monitoring, Zhejiang Zhejiang University, University, constructed laboratory structural health monitoring, Zhejiang University, Hangzhou, China. The laboratory model consists of a 6-m long I-shaped beam. In the cross-section, The laboratory laboratory model model consists consists of of aa 6-m 6-m long I-shaped I-shaped beam. beam. In In the the cross-section, cross-section, Hangzhou, China. The the beam’s flanges flanges measure measure0.072 0.072m m× × 0.005 m, while the web web measures measures 0.112 0.112m m× × 0.005 m. the beam’s 0.005 m, while the 0.005 m. The The beam beam flanges measure 0.072 m × web measures 0.112 m × is made of structural steel with Young’s modulus modulus of of 200 200 GPa, GPa, aa Poisson Poisson ratio ratio of of 0.3, 0.3, and and aa density density of of is made of 3structural steel with Young’s 7850 kg/m kg/m . In this bogie moving load. When bogie passes thismodel, model,a bogie was was used used to to simulate simulate the When the 7850 kg/m33. .InInthis model, aabogie was used to simulate the moving load. When the bogie passes through the beam, the acceleration data were recorded at 9 stations with a fixed distance m through the beam, the acceleration acceleration data data were recorded at 9 stations with a fixed distance of of 0.6 0.6 m simultaneously [18]. [18]. Figure 4 shows the positions of accelerometers to the simply 4 shows the positions of accelerometers attachedattached to the simply supported simultaneously [18].Figure Figure 4 shows the positions of accelerometers attached to the simply supported beam in the laboratory. The beams, bogie, and few of the accelerometers are shown beam in the laboratory. The beams, bogie, andbogie, few of thefew accelerometers are shown inshown Figure in 5. supported beam in the laboratory. The beams, and of the accelerometers are in Figure 5. A typical crack, manually created on the beam, and its dimensions are shown in Figure 6. A typical manually on created the beam, dimensions shown inare Figure 6. Ininthis figure, Figure 5. crack, A typical crack, created manually onand theits beam, and its are dimensions shown Figure 6. In figure, the delta ratioofofcrack the height crackoftothe the height of the beam. thethis delta represents the represents ratio of thethe height to the of height In this figure, the delta represents the ratio of the height of crack to thebeam. height of the beam.

Figure 4. Locations of accelerometers to record acceleration data. Figure 4. Locations Locations of accelerometers to record acceleration data.

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Figure 5. A general view of the vehicle and accelerometers. Figure 5. 5. A A general general view view of of the the vehicle vehicle and and accelerometers. accelerometers. Figure

Figure 6. A general view of the crack. The right photo shows the width of the crack (around 0.003 m), Figure 6. A general view of the crack. The right photo shows the width of the crack (around 0.003 m), and the6.left height of The the crack for delta = 30% 0.0216 m ≈ 0.021). Figure A photo generalshows view the of the crack. right photo shows the(0.072 width× of 0.3 the=crack (around 0.003 m), and the left photo shows the height of the crack for delta = 30% (0.072 × 0.3 = 0.0216 m ≈ 0.021). and the left photo shows the height of the crack for delta = 30% (0.072 × 0.3 = 0.0216 m ≈ 0.021).

The accelerometers were attached to the bottom of the beam. Due to use of the flanges as a lane The accelerometers were attached to the bottom of the beam. Due to use of the flanges as a lane for bogie, the bogie couldwere simply run over thebottom beam. Different velocities 0.4, and 0.5 m/s) The accelerometers attached to the the beam. Due to (0.3, use of the flanges as awere lane for bogie, the bogie could simply run over the beam. of Different velocities (0.3, 0.4, and 0.5 m/s) were considered to examine the effect of velocity variation on damage detection accuracy. The for bogie, thetobogie could the simply run of over the beam. Different 0.4, andaccuracy. 0.5 m/s) were considered examine effect velocity variation onvelocities damage(0.3, detection The acceleration data were recorded with a sampling frequency of 500 Hz in this model. Table 2 shows considered todata examine effect of velocity variation on damage detection accuracy. acceleration acceleration were the recorded with a sampling frequency of 500 Hz in this model.The Table 2 shows damage in thefrequency simply supported data werescenarios recordedconsidered with a sampling of 500 Hzbeam in thismodel. model. Table 2 shows damage scenarios damage scenarios considered in the simply supported beam model. considered in the simply supported beam model. Table 2. Two damage scenarios considered in the laboratory model of the simply supported beam. Table 2. Two damage scenarios considered in the laboratory model of the simply supported beam. Table 2. Two damage scenarios considered in the laboratory model of the simply 2supported beam. Scenario 1 Scenario 1 2 Crack depth to beam flange ratio 40% 30% Scenario 1 2 Crack depth to beam flange ratio 40% 30% Location at node 5 (mid-span) at node 7 at node 5 (mid-span) at30% node 7 Crack depth toLocation beam flange ratio 40% Name N5D40 N7D30 Location at node 5 (mid-span) at N7D30 node 7 Name N5D40 Note: The scenarios are designated as N (damage location) D (delta). For example, N5D40 refers to a Name N5D40 N7D30 Note: The scenarios are designated as N (damage location) D (delta). For example, N5D40 refers to a scenario where delta = 40% at node 5. Note: Thewhere scenarios are designated as N (damage location) D (delta). For example, N5D40 refers to a scenario where scenario delta = 40% at node 5. delta = 40% at node 5.

3.2. Experimental Study of the Arch Cable Bridge 3.2. Experimental Study of the Arch Cable Bridge 3.2. Experimental Studybridge of the Arch Cable Bridge A scaled arch model was designed and assembled in the department of civil A scaled arch bridge model was designed and assembled in the department of civil engineering, University, China [31]. and The assembled model is ainhalf-through steel-tube arch bridge A scaledZhejiang arch bridge model was designed the department of civil engineering, engineering, Zhejiang University, China [31]. The model is a half-through steel-tube arch bridge with a main span of 6 m and 14 cables in total. This arch cable bridge is shown in Figure 7.span The Zhejiang University, China [31]. The model is a half-through steel-tube arch bridge with a main with a main span of 6 m and 14 cables in total. This arch cable bridge is shown in Figure 7. The material properties of the bridge, including Young’s modulus, density, and Poisson’s ratio of 6 m and 14 cables in total. This arch cable bridge is shown in Figure 7. The material properties of are the material properties 6of the bridge, including Young’s modulus, density, and Poisson’s6 ratio are 2, 7.698 × 3, and 0.3, 2, assumed as 2.06 × Young’s 10 N/mm 10−5 N/mm respectively. bridge, including modulus, density, and Poisson’s ratio are assumed as 2.06 × 10 N/mm assumed as−52.06 × 1063 N/mm2, 7.698 × 10−5 N/mm3, and 0.3, respectively. 7.698 × 10 N/mm , and 0.3, respectively.

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Figure 7. The arch cable bridge lab model built at Zhejiang University Laboratory.

Figure 7. The arch cable bridge lab model built at Zhejiang University Laboratory. Figure 7. The arch cable bridge lab model built at Zhejiang University Laboratory.

Figure 8 shows the location of accelerometers attached close to the cables on one side of the bridge. isthe composed ofaccelerometers 7 twisted sub-cables. Figure 9toshows the twisted cable. Figure 8Each shows the location attached close to cables on one side Figure 8 cable shows locationofof accelerometers attached close thethe cables on one side of The theof the damage is applied to the model by cutting the small twisted cables. It enables us to simulate 14% to bridge. is composed of 7 twisted sub-cables. Figure 9 shows the twisted cable. bridge. EachEach cablecable is composed of 7 twisted sub-cables. Figure 9 shows the twisted cable. TheThe damage 100% damage in each cable. Cutting one of the twisted cables corresponds to 14% damage ( ≅100% damage applied to by the cutting model bythe cutting small twisted It enables ussimulate to simulate 14%toto is applied toisthe model smallthe twisted cables.cables. It enables us to 14% 0.14%), whilecable. removing the whole twisted cables 100% damage. 100% in each cable. Cutting onetwisted of the equals twisted cables corresponds to damage 14% damage ≅ damage indamage each Cutting one of the cables corresponds to 14% ( 17 ∼ =( 0.14%), 0.14%), while removing the whole twisted cables equals 100% damage. while removing the whole twisted cables equals 100% damage. Accelerometer Accelerometer

1

2

3

4

5

6

7

7 5 4 6 1 2 3 (a) Schematic view of the bridge with positions of accelerometers marked by red circles. (a) Schematic view of the bridge with positions of accelerometers marked by red circles.

(b) Accelerometer attached close to cable. (b) Accelerometer attached to cable. installed on one side of the Figure 8. Side view of the model showing (a) locations of close accelerometers bridge (b) view accelerometers attached close the cables. Figure and 8. Side of the model showing (a)tolocations of accelerometers installed on one side of the

Figure 8. Side view of the model showing (a) locations of accelerometers installed on one side of the bridge and (b) accelerometers attached close to the cables. bridge and (b) accelerometers attached close to the cables.

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Figure 9. The twisted cable composed of 7 small sub-cables.

Figure 9. The twisted cable composed of 7 small sub-cables.

Five different damage scenarios were considered in the laboratory model of arch cable bridge, Figure 9. The twisted cable composed of 7 small sub-cables.

as listed in Table 3. Five different damage scenarios were considered in the laboratory model of arch cable bridge, as listed in Table 3. Five different damage scenarios were considered in the laboratory model of arch cable bridge, Table 3. Five damage scenarios considered in a laboratory model of the bridge.

as listed in Table 3. Table 3. Scenario Five damage scenarios considered in 1a laboratory model of the bridge. 2 Table 3. Five damage cable scenarios considered in a and laboratory bridge. section area loss in twisted 43%, 70%, 100% model of the 43% and 70% Scenario 1 Node 2 2 Node Scenario 1 2 4 Location Second from left side (middle of bridge) sectionsection area loss in loss twisted cable cable 43%, 70%,node and 100% 43%and and 70% area in twisted 43%, 70%, and 100% 43% 70% Node 2 Node Name N2D3-7, N2D5-7, and N2D7-7 N4D3-7 to N4D5-7 Node 2 Node 4 4 Location Location Second node from left side (middle bridge) Note: To simplify the naming process of different scenarios, eachleft condition named of asof N (place of Second node from side was (middle bridge) Name N2D3-7, N2D5-7, and N2D7-7 N4D3-7 to N4D5-7 damage) D (number of small cables removed fromN2D5-7, the twisted For example, means Name N2D3-7, andcable). N2D7-7 N4D3-7N4D1-7 to N4D5-7

Note: 14% To simplify naming process of different scenarios, each condition was as N (place of damage) damagethe at node 4 (1/7 ≅ 14%). The no-damage condition is shown by named WD (without damage). Note: To simplify the naming process of different scenarios, each condition was named as N (place of D (number of small cables removed from the twisted cable). For example, N4D1-7 means 14% damage at node 4 D (number of condition small cables removed from(without the twisted cable). For example, N4D1-7 means (1/7 ∼ 14%). The no-damage is shown by WD damage). =damage) 4. Results and Discussion 14% damage at node 4 (1/7 ≅ 14%). The no-damage condition is shown by WD (without damage).

4. Results and Discussion 4.1. Preliminary Data of Simply Supported Beam

4. Results and Discussion The beam was loaded using a 27.5 kg bogie at different speeds (0.3, 0.4, and 0.5 m/s) to simulate 4.1. Preliminary Data of Simply Supported Beam 4.1. Preliminary Data of Simply Supported Beam the traffic moving load. All acceleration data were recorded at a sampling frequency of 500 Hz. The The was loaded using a 27.5 kg different speeds (0.3,0.4, 0.4, and 0.5 m/s) tonode simulate nodebeam 1 (according to Figure 4) was considered asat locationspeeds of the (0.3, main signal. Auto-RDS for The beam was loaded using a 27.5 kgbogie bogie atthe different and 0.5 m/s) to simulate 1 and cross-RDSs for Nodes 2–9 were calculated using level-crossing triggering condition. As Hz. the traffic moving load. data were recorded at a sampling frequency ofThe 500 the traffic moving load.All All acceleration acceleration data were recorded at a sampling frequency of 500 Hz. mentioned previously, to use the level-crossing triggering condition, the optimum value was The node (according Figure 4) was considered as the location of signal. the main signal.forAuto-RDS node 11 (according to to Figure 4) was considered as the location of the main Auto-RDS node tocross-RDSs be a = for [24]. In this work, this optimum value was computed √2 × standard 1assumed and cross-RDSs Nodes 2–9 deviation were calculated using level-crossing triggering condition. As for node 1 and for Nodes 2–9 were calculated using level-crossing triggering condition. for all signals, and the minimum oflevel-crossing these values was chosencondition, as triggering condition ofvalue the main mentioned previously, to use the triggering the optimum was As mentioned previously, to use the level-crossing triggering condition, the optimum value was signal. This step√ was made by more than 2000 segments (N > 1000) for each RDS. Figurecomputed 10 shows assumed × standard deviation [24]. thiswork, work, this optimum optimum value assumed to betoa be = a= 2×√2standard deviation [24]. InInthis this value was was computed for three signals of and the intact beam, namely, thevalues acceleration signal as recorded in node 1, the auto-RDS of for all signals, the minimum of these was chosen triggering condition of main all signals, and the minimum of node these5 values was chosen asThe triggering condition ofwhole thethe main signal. node 1, and the cross-RDS of (middle of the beam). length of RDS in the of this signal. This step was made by more than 2000 segments (N > 1000) for each RDS. Figure 10 shows This step was made by more than 2000 segments (N > 1000) for each RDS. Figure 10 shows work signals is 5 s. of the intact beam, namely, the acceleration signal recorded in node 1, the auto-RDS ofthree three

signals of the intact beam, namely, the acceleration signal recorded in node 1, the auto-RDS of node 1, node 1, and the cross-RDS of node 5 (middle of the beam). The length of RDS in the whole of this and the cross-RDS of node 5 (middle of the beam). The length of RDS in the whole of this work is 5 s. work is 5 s.

(a) Response acceleration data at node 1 under moving load, speed = 0.3 m/s2 (a) Response acceleration data at node 1 under moving load, speed = 0.3 m/s2 Figure 10. Cont.


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(b) Auto-RDS data at node 1 under moving load, speed 0.3 m/s2 (b) Auto-RDS data at node 1 under moving load, speed 0.3 m/s2

2 (c) Cross-RDS data at node 5 under moving load, speed 0.3 m/s 2

(c) Cross-RDS data at node 5 under moving load, speed 0.3 m/s

Figure 10. 10. (a) (a) Response node1, 1,(b)(b) auto-RDS of node 1, (c) and (c) Figure Responseacceleration acceleration data data recorded recorded atatnode auto-RDS of node 1, and Figure 10. (a) Response acceleration data recorded at node 1; (b) auto-RDS of node 1; and (c) cross-RDS cross-RDS of node 5. 5. cross-RDS of node of node 5.

4.2. Preliminary Data of Arch CableBridge Bridge 4.2. Preliminary Data of Arch Cable 4.2. Preliminary Data of Arch Cable Bridge bridge loaded truckat at different different weights speeds (90,(90, 110,110, andand 130 kg, The The bridge waswas loaded bybya atruck weightsand and speeds 130and kg,0.3, and 0.3, 0.4, and 0.5 m/s) to simulate the traffic moving load. The acceleration signal in the mid-span of and theof0.3, loaded by a truck different weights and speeds (90, 110, 130 kg, 0.4, The and bridge 0.5 m/s)was to simulate the trafficatmoving load. The acceleration signal inand the mid-span the arch0.5 cable bridge is shownthe in traffic Figure moving 11a. All the seven accelerometers simultaneously record the 0.4, and m/s) to simulate load. The acceleration signal in the mid-span of the arch cable bridge is shown in Figure 11a. All the seven accelerometers simultaneously record the acceleration signal with ain sampling rate ofAll 1000 Hz. Figure 11b,c representssimultaneously the auto-RDS of node 1 the arch cable bridge shown Figurerate 11a.of the seven accelerometers acceleration signaliswith a sampling 1000 Hz. Figure 11b,c represents the auto-RDSrecord of node 1 and cross-RDS of node 4 (middle of the bridge) calculated from the signal shown in Figure 11a, acceleration signal Hz. Figure 11b,cfrom represents the auto-RDS of node 1 and and respectively. cross-RDS ofwith nodea sampling 4 (middlerate of of the1000 bridge) calculated the signal shown in Figure 11a, cross-RDS of node 4 (middle of the bridge) calculated from the signal shown in Figure 11a, respectively. respectively.

(a) Response acceleration data at node 1 under moving load, speed = 0.3 m/s2

(a) Response acceleration data at node 1 under moving load, speed = 0.3 m/s2

(b) Auto-RDS data at node 1 under moving load, speed = 0.3 m/s2

(b) Auto-RDS data at node 1 under moving load, speed = 0.3 m/s2 Figure 11. Cont.

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(c) Cross-RDS data at node 4 under moving load, speed = 0.3 m/s2 2 Figure 11. (a) acceleration data4recorded at nodeload, 1, (b) auto-RDS of node 1, and (c) (c)Response Cross-RDS data at node under moving speed = 0.3 m/s Figure 11. (a) Response acceleration data recorded at node 1, (b) auto-RDS of node 1, and (c) cross-RDS cross-RDS of node 5. of node11. 5. (a) Response acceleration data recorded at node 1, (b) auto-RDS of node 1, and (c) Figure

cross-RDS of node 5. 4.3. Normalizing the Arias Intensity along the Structure 4.3. Normalizing the Arias Intensity along the Structure As previously pointed out and shown in Figure 1, most of the damage detection methods are 4.3. Normalizing the Arias Intensity along the Structure As previously pointed out and shown in Figure 1, most of the damage detection methods are based on estimating modal parameters through RDSs. These parameters are then used as damage Asonpreviously and shown in Figure 1, These most of the damage methods are based estimatingpointed modal out parameters through RDSs. parameters aredetection then used as damage indicators to identify the damage locations. However, recent research carried out stipulates how to based on estimating parameters through RDSs. recent These research parameters are then used as damage indicators to identifymodal the damage locations. However, carried out stipulates how to find damage without detecting the modal parameters [8,13]. Taking this approach, Arias Intensity indicators to without identify detecting the damage However, recent research outArias stipulates how(AI) to find damage the locations. modal parameters [8,13]. Taking this carried approach, Intensity (AI) is employed as a new damage indicator in this paper to locate the damage without calculating find damage as without the modal [8,13]. this approach, Arias Intensity is employed a new detecting damage indicator in parameters this paper to locateTaking the damage without calculating any any modal parameter. AI is one of the most famous energy criteria defined by Arturo Arias in 1970 (AI) is employed new indicator in this paper to locate the damage without calculating modal parameter.asAIa is onedamage of the most famous energy criteria defined by Arturo Arias in 1970 [32]. [32]. The Arias intensity calculates the energy of the acceleration signal expressed as below: any parameter. AI is one the most famous energy signal criteriaexpressed defined by Arturo Arias in 1970 The modal Arias intensity calculates theofenergy of the acceleration as below: ( ) , signal expressed as below: [32]. The Arias intensity calculates the energy ( of ) =theZacceleration (9) π T 2 m I ( ( ) )== a ((t))dt,, (9) (9) in which IA, T, and a(t) are AI, durationAof sacceleration, acceleration data, respectively. Using this 2g 0 and definition, AI is calculated as a scalar and invariant value. Additionally, it contains the full range of in which and a(t) are of in which IIAA,,T, T, and a(t)duration areAI, AI,duration duration ofacceleration, acceleration,and andacceleration acceleration data, data, respectively. respectively. Using Using this this frequencies and time of the signal. definition, AI is calculated as a scalar and invariant value. Additionally, it contains the full range of definition, AI iswith calculated as a scalar and invariant Additionally, it contains the full rangeand of To begin the proposed method, Equationvalue. (9) was applied to all the recorded signals, frequencies and time duration of the signal. frequencies and time duration of the signal. their corresponding AI values were determined. For instance, Figure 12 shows the results for each To begin with withthe the proposedmethod, method, Equation was applied to the all recorded the recorded signals, and To begin (9)(9) was applied to all signals, and their sensor from both autoproposed and cross-RDSs ofEquation intact structure individually. their corresponding AI values were determined. For instance, Figure 12 shows the results for each corresponding AI values were determined. For instance, Figure 12 shows the results for each sensor sensor from both auto and cross-RDSs intact structure individually. from both auto and cross-RDSs of intactofstructure individually.

(a)

(b)

Figure 12. The AIs calculated from the RDSs for each sensor of the intact (a) (b)structure individually, considering the speed as v = 0.3 m/s. (a) Al calculated for intact bridge and (b) AI calculated for the Figure 12. Figure 12. The The AIs AIs calculated calculated from from the the RDSs RDSs for for each each sensor sensor of of the the intact intact structure structure individually, individually, intact beam. considering the speed as v = 0.3 m/s. (a) Al calculated for intact bridge and (b) considering the speed as v = 0.3 m/s. (a) Al calculated for intact bridge and (b) AI AI calculated calculated for for the the intact beam. intact beam. It is supposed that the damage location can be estimated from the absolute difference value of

the AIs calculated from a structure in damage and non-damage conditions. However, as shown in It is supposed that the damage location can be estimated from the absolute difference value of Figure the AI differs from nodelocation to node.can Thus, it is impossible to find any difference damage without It is12, supposed that the damage be estimated from the absolute value the AIs calculated from a structure in damage and non-damage conditions. However, as shown in normalization. To normalize the AI spectrums, the following equations are used: of the AIs calculated from a structure in damage and non-damage conditions. However, as shown Figure 12, the AI differs from node to node. Thus, it is impossible to find any damage without normalization. To normalize the AI spectrums, the=following(equations are used: ) , (10) =

( )

,

(10)

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in Figure 12, the AI differs from node to node. Thus, it is impossible to find any damage without normalization. To normalize the AI spectrums, the following equations are used: π Appl. Sci. 2018, 8, x FOR PEER REVIEW I A−normalized = 2g s β= =

Z T 0

( βa(t))2 dt,

11 (10) of 17

1000 , , IA

(11) (11)

Equation (10) (10) causes causes the the AIs AIs to to normalize normalize to to1000. 1000. The The parameter parameter ββ is is normalization normalization factor factor and and Equation should be calculated only for structures without damage. Figure 13 and Table 4 show the should be calculated only for structures without damage. Figure 13 and Table 4 show the normalized normalized AIs and corresponding βs. AIs and corresponding βs.

(a)

(b)

Figure 13. 13. The The AIs AIs after after normalization normalization for for the the structure structure under under moving moving load load considering considering the the speed speed as as Figure v = 0.3 m/s. (a) Normalized Al calculated for intact bridge and (b) normalized AI calculated for the v = 0.3 m/s. (a) Normalized Al calculated for intact bridge and (b) normalized AI calculated for the intact beam. beam. intact Table 4. β calculated for all signals under moving load in the laboratory for no-damage condition Table 4. β calculated for all signals under moving load in the laboratory for no-damage condition considering the the speed speed as as vv == 0.3 0.3 m/s. m/s. considering Parameter

Node 1

Parameter 1 β (for bridge) Node 1049

β (for beam) β (for bridge) β (for beam)

294 1049 294

Node 2

Node 3

Node 4

Node 5

Node 2 1306

Node 3 1256

Node 14274

Node 12525

392 1306 392

365 1256 365

402 1427 402

382 1252 382

Node 6 Node 1354 6 342 1354

Node 7 Node 1381 7 442 1381

342

442

Node 8 Node 9 Node Node 9 - 8 397297397

297

Although Table 4 shows the value of β considered to normalize the AI of RDSs, it is limited to the speed of 0.3Table m/s. Considering and velocities leadsthe to aAI different β. Twenty-nine Although 4 shows the different value of weights β considered to normalize of RDSs, it is limited laboratory tests were assumed for the different non-damage models, which twenty-one tests were to the speed of 0.3 m/s. Considering weights and in velocities leads to a different β. implemented on the bridge with 3 different weights (90, 110, and 130 kg) and velocities (0.3, 0.4, and Twenty-nine laboratory tests were assumed for the non-damage models, in which twenty-one tests 0.5 m/s). The remaining tests were on the considering a 27.5 kg weight were implemented on the 8bridge with 3implemented different weights (90,beam 110, and 130 kg) and velocities (0.3, and 0.4, three different velocities (0.3, 0.4, and 0.5 m/s). In each test, the parameter β was calculated for all the and 0.5 m/s). The remaining 8 tests were implemented on the beam considering a 27.5 kg weight and nodes. Figure 14 shows the β for node 1 of both bridge and beam under moving load at different three different velocities (0.3, 0.4, and 0.5 m/s). In each test, the parameter β was calculated for all weights and velocities. the nodes. Figure 14 shows the β for node 1 of both bridge and beam under moving load at different weights and velocities.

implemented on the bridge with 3 different weights (90, 110, and 130 kg) and velocities (0.3, 0.4, and 0.5 m/s). The remaining 8 tests were implemented on the beam considering a 27.5 kg weight and three different velocities (0.3, 0.4, and 0.5 m/s). In each test, the parameter β was calculated for all the nodes. Figure 14 shows the β for node 1 of both bridge and beam under moving load at different Appl. Sci. 2018, 753 12 of 17 weights and8,velocities.

(a)

(b)

Figure 14. Normalizing factor for the structure under moving load at (a) node 1 of the intact bridge and (b) node 1 of the intact beam.

Two points can be deduced from Figure 14. First, in our models, β follows a linear pattern and has an inverse relationship with velocity. Secondly, β has no relationship with changes in the weights. For instance, there is no relationship between different weights at speed v = 0.4 m/s. Using the linear relationship between β and velocity may lead to a new β for other velocities. This linear relationship can be defined as follows: β = xϑ + y, (12) in which ϑ is the speed of moving load. The two coefficients x and y can be estimated for each node. Figure 14 shows how to use a trend line to estimate x and y for node 1. The results are reported in Table 5 showing x and y for all the nodes of the beam and bridge. Table 5. The coefficients x and y calculated for all nodes of the beam and bridge for the no-damage condition. Parameter

Node 1

Node 2

Node 3

Node 4

Node 5

Node 6

Node 7

Node 8

Node 9

Beam

x y

−203 356

−498 535

−241 441

−377 514

−348 475

−228 432

−457 567

−444 525

−238 378

Bridge

x y

−1116 1389

−1954 1901

−1687 1769

−2249 2112

−1689 1766

−1874 1925

−1949 1975

-

-

Equation (12) and Table 5 provide a baseline for these laboratory models and normalize the AI at different velocities. 4.4. Locating the Connection Loss in the Cables for the Arch Cable Bridge When damage occurs, the cable experiences a localized loss of stiffness. This stiffness reduction will change the distribution of energy in the structure. Consequently, β cannot normalize the AIs calculated from all the signals to 1000. It is assumed that the stiffness reduction in the cable amplifies the acceleration signal on damage location more than other places along the bridge. Therefore, the higher amplitude leads to an AI greater than 1000 after normalization. Figure 15 shows the normalized AI for N2D7-7, in which cable in node 2 receives full damage by losing the connection. It is clear that, except node 2, which is slightly higher than 1000, all other nodes show a result less than 1000. Since all the AIs were previously normalized to 1000 in the bridge without damage, it is possible to deduct 1000 from all the AIs and find the damage according to positive amounts. Figure 15 is re-plotted as a bar chart (Figure 16a). Figure 16b shows the normalized AI along the bridge after subtracting 1000 for the case N2D7-7 (for simplicity, negative values are considered zero).

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Figure 15. Normalized AI for 100% damage in cable 2 under moving load. Figure 15. 15. Normalized Normalized AI AI for for 100% 100% damage damage in in cable cable 22 under under moving moving load. load. Figure

(a) Normalized AI values (b) Normalized AI values after subtracting 1000 (a) Normalized AI values (b) Normalized AI values after subtracting 1000 Figure 16. Distribution of normalized AI along the bridge (a) for the node N2D7-7 and (b) after Figure 16. Distribution of normalized AI subtracting 1000 from the result. Figure 16. Distribution ofN2D7-7 normalized AI along along the the bridge bridge (a) (a) for for the the node node N2D7-7 N2D7-7 and and (b) (b) after after subtracting the N2D7-7 N2D7-7 result. result. subtracting 1000 1000 from from the

4.5. Detecting Small Damage Ratios in Laboratory Models 4.5. Detecting Small Damage Ratios in Laboratory Models 4.5. Detecting Damage Ratios in Laboratory So far, itSmall is explained how 100% damageModels to a cable, such as losing the connection between a So far, it is explained how 100% damage to using a cable, as losing the connection a cableSo and could be located in the RDSs AIsuch asasdamage indicator. There are,between however, far,the it isbridge, explained how 100% damage to a cable, such losing the connection between a cable cable and the bridge, could be located in the RDSs using AI as damage indicator. There are, however, some conditions in which a cable dueindicator. to some reasons such as corrosion. and the bridge, could be located in partially the RDSsloses usingitsAIstiffness as damage There are, however, some some conditions in which a cable partially loses its stiffness due to some reasons such as corrosion. As shown in inwhich Figurea cable 9, each cable isloses composed of 7due twisted thinreasons cables.such Therefore, it is possible to conditions partially its stiffness to some as corrosion. As shown As shown in Figurefrom 9, each cable is composed of 7 In twisted thin cables. Therefore, it is possible to model the damage 14% to 100% in one cable. this section, the detection of small damage is in Figure 9, each cable is composed of 7 twisted thin cables. Therefore, it is possible to model the model the damage from 14% of to the 100% in one cable. In is this section,through the detection of small damage is addressed and the capability proposed method assessed laboratory results. damage from 14% to 100% in one cable. In this section, the detection of small damage is addressed and addressed the100% capability of the proposed method is assessed through laboratory results. A caseand with damage in the is equivalent to a cut through the capability of the proposed method isbeam assessed through laboratory results.the whole beam, which A case with 100% damage in the beam is equivalent to a cut through the wholeInstead, beam, which doesAnot Hence, 100-percent damage is not caseseem withrational. 100% damage in the the beam is equivalent to ascenario cut through thedescribed. whole beam, which other does does notratios seem are rational. Hence, the 100-percent damage scenario is not2. described. Instead, other damage targeted as shown in Figure 6 and illustrated in Table not seem rational. Hence, the 100-percent damage scenario is not described. Instead, other damage damage ratios targeted in Figure 6 and while illustrated in Table 2. are zero along the beam. 17 are shows the AIasin inshown the damage the other values ratiosFigure are targeted as shown Figure 6 and location, illustrated in Table 2. Figure 17 shows the AI in the damage location, while the other values are zero along the beam. It should theinaim the proposed is other to findvalues the maximum normalized AI Figurebe17noticed shows that the AI the of damage location,method while the are zero along the beam. It should be noticed that the aim of the proposed method is to find the maximum normalized AI along thebebeam. Due normalizing the AIs, subtracting AIs could yield much It should noticed thattothe aim of the proposed method is to1000 find from the maximum normalized AI better along along the beam. DueInto normalizing the AIs, subtracting 1000 from AIs could yield much better results (Figure 16b). the beam, subtraction of 1000 from the AIs leads to positive values only in the the beam. Due to normalizing the AIs, subtracting 1000 from AIs could yield much better results results (Figure 16b). In the beam, subtraction of 1000 from the AIs leads to positive values only in the nodes close to damage location. (Figure 16b). In the beam, subtraction of 1000 from the AIs leads to positive values only in the nodes nodes close to damage location. close to damage location.

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(a)

(b)

Figure 17. Distribution of AIs along the beams for all accelerometers assuming (a) Case N5D40 (b) Figure 17. Distribution of(a) AIs along the beams for all accelerometers assuming (a) Case N5D40 from from beam model: delta = 40 at Node 5 and (b) Case N7D30 from beam model: delta = 30 at beamFigure model: delta = 40 at Node 5 and (b) Case N7D30 from beam model: delta = at Node 7. 17. Distribution of AIs along the beams for all accelerometers assuming (a)30Case N5D40 Node 7.

from beam model: delta = 40 at Node 5 and (b) Case N7D30 from beam model: delta = 30 at Node Figure 18 shows small damage found a cableusing usingRDSs RDSsand and normalized normalized AI. Figure 18 7. shows howhow small damage cancan bebe found inin a cable AI.In In this this figure, the damage occurs in node 2 (second cable from left) and node 4 (middle of the bridge) figure, the damage occurs in node 2 (second cable from left) and node 4 (middle of the bridge) with 18 shows howratios. small damage can be found in a cable using RDSs and normalized AI. In withFigure different damage As in observed all these scenarios, the amount maximum of AI different damage ratios. As observed all theseinscenarios, the maximum of amount normalized this figure, the occurstoindamage node 2location. (second cable from left) and node 4 (middle of the bridge) normalized AI damage appears close appears close to damage location. with According different damage observed all these scenarios, the maximum of to Figure ratios. 18, it is As proved that theinproposed method can locate the damageamount accurately. According to Figure 18, it is that the proposed method can locate the damage accurately. normalized AI appears close to proved damage location. The small positive amount of normalized AI in other places (nodes 1, 3, and 7) shows that the The small positive amount normalized other (nodescan 1, 3, andthe 7) shows the damage According to Figure it is as proved that proposed method locate damage accurately. damage can affect thoseof18, nodes well. AI Theinthe fact thatplaces the proposed method can providethat acceptable can affect those nodes as well. of The fact that the proposed method in The small amount normalized AI in other17places 1,provide 3, is and 7)acceptable shows thatresults the results in apositive noisy environment is illustrated in Figures and 18.(nodes Thiscan result relevant for velocities damage can affect those nodes as well. The fact that the proposed method can provide acceptable a noisy environment is illustrated in Figures 17 and 18. This result is relevant for velocities between between 0.3 and 0.5 m/s. Using the proposed model in a noisy environment at a speed more than 0.5 results in a noisy environment is illustrated and 18. This result is relevant velocities 0.3 and m/s. Using thea proposed model in in Figures a noisy17 environment at a speed moreforthan 0.5 m/s is m/s0.5 is accompanied by margin of error. between 0.3 and 0.5 m/s. Using the proposed model in a noisy environment at a speed more than 0.5 It should mentioned that the proposed model could be used as a real-time damage detection accompanied by a be margin of error. m/s is accompanied by a margin of error. method if be the mentioned optimized values of truck velocity, triggering number locationdamage of It should that the proposed model could level, be used as aand real-time It should beand mentioned that the are proposed model could be used as a real-time damage detection accelerometers, length of signal calculated in advance. detection method if the optimized values of truck velocity, triggering level, number and location method if the optimized values of truck velocity, triggering level, number and location of of accelerometers, ofsignal signalare arecalculated calculated in advance. accelerometers,and and length length of in advance.

(a) Case N2D7-7

(b) Case N4D7-7

(a) Case N2D7-7

(b) Case N4D7-7

(c) Case N2D5-7

(d) Case N4D5-7

(c) Case N2D5-7

(d) Case N4D5-7 Figure 18. Cont.

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(e) Case N2D3-7

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(f) Case N4D3-7

Figure 18. Normalized AI for six different damage scenarios obtained from laboratory models. (a,c,e)

Figure 18. Normalized AI for six different damage scenarios obtained from laboratory models. Cases N2D7-7, N2D5-7, and N2D3-7 of laboratory model; (b,d,f) cases N4D7-7, N4D5-7, and N4D3-7 (a,c,e) Cases N2D7-7, N2D5-7, and N2D3-7 of laboratory model; (b,d,f) cases N4D7-7, N4D5-7, and of a laboratory model. N4D3-7 of a laboratory model.

4.6. Noise Effect

4.6. Noise Effect

The uncertainty imposed by the noise has not been discussed in this paper. However, it can be claimed that the results may by notthe be highly affected thediscussed incorporated noise.paper. To further argue this The uncertainty imposed noise has not by been in this However, it can be claim, a closer look atmay the proposed methodaffected is addressed here. Based on thenoise. frequency content,argue two this claimed that the results not be highly by the incorporated To further different cases (1)method noise and have the same frequencycontent, and (2) two claim, a closer lookcan at be theassumed: proposed is structural addressedresponse here. Based on the frequency noise and structural response do not contain similar frequency content. In the first case, since the different cases can be assumed: (1) noise and structural response have the same frequency and (2) noise noise removal can distort the main response, it is not possible to attenuate the noise. However, in the and structural response do not contain similar frequency content. In the first case, since the noise second case, the incorporated noise can be efficiently reduced by using frequency filtering methods. removal distort thecan main response, it is not possible to attenuate noise. However, in the second The can RD technique act as a de-noising technique. It means that thethe proposed method essentially case,attenuates the incorporated noise can be efficiently reduced by using frequency filtering methods. The RD the second category of noise. However, to deal with the first category of noise, the loading technique can act as as a de-noising that the proposed method essentially attenuates parameters such velocity cantechnique. be changed It someans as to get a structural response with frequency content far away from noise frequency contents. it is with not possible, accuracy proposed method would the second category of noise. However, to Ifdeal the firstthe category ofofnoise, the loading parameters be significantly suchthen as velocity can be affected. changed so as to get a structural response with frequency content far away One way to detect the If incorporated noise in is to separately from noise frequency contents. it is not possible, thelaboratory accuracy tests of proposed methodrecord wouldthe then be acceleration data without loading. In this research, all the signals begin with a pre-loading segment significantly affected. that lasts for few seconds. This issue can be seen in Figures 10 and 11 of the paper, in which the main One way to detect the incorporated noise in laboratory tests is to separately record the acceleration loading approximately starts after 5 s. Considering the frequency content of these pre-loading data segments, without loading. In this research, all the signals begin with a pre-loading segment that lasts for it is found that the similarity between the frequency contents of noise and structural few seconds. This issue cantobebeseen in Figures 10that andthe 11majority of the paper, in which thethe main loading response is small enough ignored. It means of noises belong to second approximately after 5bes.removed Considering frequency content of these pre-loading segments, it category andstarts thus could by RD the technique.

is found that the similarity between the frequency contents of noise and structural response is small 5. Conclusions enough to be ignored. It means that the majority of noises belong to the second category and thus could be This removed RD technique. paperby proposed an output-only method in which the random decrement technique was employed to produce auto- and cross-RDSs under moving load. Then, AI was calculated to find the

5. Conclusions energy content of each individual RDS. These AIs were normalized, and a parameter β was defined

as a normalization factor, highlighting thein maximum normalized along thetechnique structure, was This paper proposed anthereby output-only method which the random AI decrement which could show the vicinity of damage location accurately. Since the proposed method could employed to produce auto- and cross-RDSs under moving load. Then, AI was calculated to find the calculate the β for an intact structure using a linear pattern, there was no need to calculate the energy content of each individual RDS. These AIs were normalized, and a parameter β was defined as response of the non-damage structure. a normalization factor, highlighting maximum normalized AI along the structure, which The main resultsthereby of the research can be the summarized as follows:

could show the vicinity of damage location accurately. Since the proposed method could calculate 1. It was proved that the damage location could be detected using acceleration data, recorded the β for an intact structure using a linear pattern, there was no need to calculate the response of the along a structure. non-damage structure. 2. Investigating the effect of load-moving velocity, it was shown that the normalizing factor had The an main results of relationship the researchwith canvelocity. be summarized as follows: inverse linear 1. 2.

It was proved that the damage location could be detected using acceleration data, recorded along a structure. Investigating the effect of load-moving velocity, it was shown that the normalizing factor had an inverse linear relationship with velocity.

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The accuracy of the proposed method was examined through two models, namely a simply supported beam and an arch cable bridge. It was indicated that the proposed method could easily find the location of a single damage on a noisy signal.

At the end of the research, it is noticed that the proposed method could locate the damage with no need to calculate the modal parameters; this could effectively reduce the time and cost of SHM through this method compared to other approaches to SHM. Hence, the proposed method provided an efficient, real-time, modal-free tool for damage detection of complex structures. This topic will be further studied by the authors in their future research under seismic loadings. At this time, the proposed method is only verified for structures subjected to the moving load. Other type of excitations, such as seismic loading, will be considered in our future works. Author Contributions: All the authors contributed to the research. H.K. conceived and designed the experiments. He, together with Y.-K.J., performed the laboratory models and recorded the data. Y.-Q.X. and X.-W.Y. guided the research and provided advice to ensure the accuracy of results. The paper was finally written by H.K. and confirmed by all the authors. Acknowledgments: Special thanks to Zhejiang University, which supported this work, and provided financial sources (Cyrus Tang foundation in China and Natural Science Foundation of China (NSFC, No. 51178416)). Conflicts of Interest: The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

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