Dynamic Method of Neutral Axis Position Determination and ... - MDPI

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Dynamic Method of Neutral Axis Position Determination and Damage Identification with Distributed Long-Gauge FBG Sensors Yongsheng Tang 1, * and Zhongdao Ren 2 1 2

*

School of Urban Rail Transportation, Soochow University, Suzhou 215137, China Department of Civil Engineering, Yanshan University, Qinhuangdao 066004, China; [email protected] Correspondence: [email protected]; Tel.: +86-512-6760-1052

Academic Editors: Christophe Caucheteur and Tuan Guo Received: 25 November 2016; Accepted: 10 February 2017; Published: 20 February 2017

Abstract: The neutral axis position (NAP) is a key parameter of a flexural member for structure design and safety evaluation. The accuracy of NAP measurement based on traditional methods does not satisfy the demands of structural performance assessment especially under live traffic loads. In this paper, a new method to determine NAP is developed by using modal macro-strain (MMS). In the proposed method, macro-strain is first measured with long-gauge Fiber Bragg Grating (FBG) sensors; then the MMS is generated from the measured macro-strain with Fourier transform; and finally the neutral axis position coefficient (NAPC) is determined from the MMS and the neutral axis depth is calculated with NAPC. To verify the effectiveness of the proposed method, some experiments on FE models, steel beam and reinforced concrete (RC) beam were conducted. From the results, the plane section was first verified with MMS of the first bending mode. Then the results confirmed the high accuracy and stability for assessing NAP. The results also proved that the NAPC was a good indicator of local damage. In summary, with the proposed method, accurate assessment of flexural structures can be facilitated. Keywords: neutral axis position; dynamic macro-strain; modal macro-strain; long-gauge FBG sensors; structural performance assessment

1. Introduction Flexural structures such as beams are the most common structures in civil engineering. However, they are often damaged due to severe environment, over loading, original structural flaws, earthquake, typhoon or other factors. Therefore, it is important to ensure the safety of these structures. For addressing this issue, the concept of structural health monitoring (SHM) has been proposed and widely studied for structural maintenance and management in the last 30 years [1–4]. In the SHM systems, some key structural parameters should be selected to be monitored and assessed. The neutral axis is the axis about which bending occurs in a beam or a composite section. As a key parameter, the neutral axis position (NAP) is so important that it is needed in most theories of structural design. Moreover, the neutral axis position serves as a potential indicator of the structure’s safety condition. For instance, Griffin et al. [5] measured the neutral axis locations of two strengthened bridges, from which the retrofitting effects were verified as the neutral axis moved as expected after reinforcement. Many other researchers have already investigated how to determine the neutral axis position from static or live traffic loading tests [6–13]. The static loading test generally costs much labor and time, and it is not suitable for long-term structural monitoring. Therefore, attention has been focused

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on live traffic loading tests for neutral axis position determination. Gutiérrez et al. [6] identified the neutral axis location of a bridge beam with reinforced concrete decks and carbon Fiber reinforced polymer (CFRP) box girders directly using the dynamic strain data. The results showed that the Sensors 2017, 17, 411 2 of 20 identified neutral axis position fluctuated very much especially when the excitation is not strong neutral axis location bridge beam with concrete decksby and carbon enough. Elhelbawey et of al.a [7] evaluated the reinforced neutral axis locations using theFiber staticreinforced and dynamic polymer (CFRP) box girders directly using the dynamic strain data. The results showed that the strains measured at the girder’s bottom, middle and top with four methods. There were obvious identified neutral axis position fluctuated very much especially when the excitation is not strong axis variations in results from different methods. Chakraborty and Dewolf [8] estimated the neutral enough. Elhelbawey et al. [7] evaluated the neutral axis locations by using the static and dynamic locations through the traffic loading test, the results of which also showed large variations when using strains measured at the girder’s bottom, middle and top with four methods. There were obvious different time windows of the test data. Gangone et al. [9] obtained the neutral axis location with variations in results from different methods. Chakraborty and Dewolf [8] estimated the neutral axis strains measured by wireless strain sensors under live truck loading, which passed the bridge at about locations through the traffic loading test, the results of which also showed large variations when 16.1~24.1 axis was Gangone found toethave a strong relationship withlocation the loading using km/h. differentThe timeneutral windows oflocation the test data. al. [9] obtained the neutral axis position. Similar work has also been performed with a conclusion that the dynamic effect was a likely with strains measured by wireless strain sensors under live truck loading, which passed the bridge reason to induce the result variation [10,11]. at about 16.1~24.1 km/h. The neutral axis location was found to have a strong relationship with the loading position. work has also been with a conclusion that the dynamic effect axis It is seen from Similar the literature review thatperformed the biggest problem in determining the neural was afrom likelythe reason to induce the is result [10,11]. position dynamic strain thatvariation the results have obvious variations. Different explanations It brought is seen from the literature that the problem in determining the especially neural axiswhen have been forward: (1) the review uncertainty forbiggest strain measurements is obvious position from the dynamic strain is that the results have obvious variations. Different explanations the absolute value of the measured strain is small [9]; (2) force types influence the strain distribution have been brought forward: (1) the uncertainty for strain measurements is obvious especially when along the cross section, such as dynamic effects, torsion effects and axial forces [8]; (3) the loading the absolute value of the measured strain is small [9]; (2) force types influence the strain distribution magnitude and loading location change the composite action between the girders and the deck, which along the cross section, such as dynamic effects, torsion effects and axial forces [8]; (3) the loading affects the neutral position [7]; and (4) nonlinear structure magnitude andaxis loading location change the composite actionperformance between the influences girders andthe theneutral deck, axis location as affects well when large strain occurs [12,13]. also performance proposed strategies to the improve which the neutral axis position [7]; and Researchers (4) nonlinear have structure influences the accuracy of the neural estimation: (1) the[12,13]. maximum strain was evaluate the neutral axis location asaxis wellposition when large strain occurs Researchers haveused also to proposed strategies to improve accuracy of the to neural axis position the maximum strain was [9]; neutral axis location as itthe was considered be more reliableestimation: [8]; (2) the(1) average value was selected used to evaluate the neutral axis location as it was considered to be more reliable [8]; (2) the average (3) the root-mean square (RMS) value was proposed with a relatively good stability [7]. value selected [9]; (3) the indeed root-mean square (RMS)with valuethese was proposed a relatively good and Somewas improvements have been achieved methods.with However, reliability stability [7]. accuracy of the neural axis position assessment using dynamic strain still needs further exploration. Some improvements have indeed been achieved with these methods. However, reliability and Therefore, a new method is developed to determine the neutral axis position in this paper. In the accuracy of the neural axis position assessment using dynamic strain still needs further exploration. proposed method, dynamic strain istofirst acquired with the long-gauge sensors, Therefore, a newthe method is developed determine the neutral axis position in macro-strain this paper. In the considered as method, a useful the waydynamic to detect localisdamage. Thenwith the modal macro-strain (MMS)sensors, is generated proposed strain first acquired the long-gauge macro-strain fromconsidered strain time andtoisdetect used local to calculate neutral axis macro-strain location. The proposed method has ashistories a useful way damage.the Then the modal (MMS) is generated the merit of high stability asand it uses thetoMMS which robustaxis to location. loading The situations and noises.has Finally, from strain time histories is used calculate theis neutral proposed method the merit of high stability as it uses the MMS which is robust to loading situations and noises. Finally, some experiments are conducted to verify the effectiveness of the proposed method. some experiments are conducted to verify the effectiveness of the proposed method.

2. Dynamic Method of Neutral Axis Position Determination 2. Dynamic Method of Neutral Axis Position Determination

2.1. Strain Measurement with Long-Gauge Fiber Optic Sensors 2.1. Strain Measurement with Long-Gauge Fiber Optic Sensors

There are many methods to measure strain, of which the strain gauge is the most common type There are many methods to measure strain, of which the strain gauge is the most common type of sensors. Usually the gauge length is only several centimeters, so the measurement is often named of sensors. Usually the gauge length is only several centimeters, so the measurement is often named as point measurement. It means that the strain reflects only the deformation at some point on the as point measurement. It means that the strain reflects only the deformation at some point on the monitored structure. It may not somelocal localdamage damage directly with the strain gauge, monitored structure. It may noteasy easyto tolocate locate some directly with the strain gauge, such such as concrete crack as shown ininFigure measurement strain gauge hardly locates as concrete crack as shown Figure1.1.In In actual actual measurement thethe strain gauge hardly locates at theat the samesame position duedue to to thethe random of the thedamage, damage, it usually needs additional position randomdistribution distribution of soso it usually needs somesome additional complicated techniques to to locate complicated techniques locatethe thedamage. damage.

(a)

(b)

Figure 1. Strain measurement with strain gauge: (a) strain gauge locates at the crack; and (b) strain

Figure 1. Strain measurement with strain gauge: (a) strain gauge locates at the crack; and (b) strain gauge locates at some distance away from the crack. gauge locates at some distance away from the crack.

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Compared with with the the point point sensing, sensing, area area sensing sensing is is proposed proposed along along with with application application of of long-gauge long-gauge Compared Compared with the point sensing, area sensing is proposed along with application of long-gauge sensors [14]. The parameter measured by the long-gauge sensor is the average strain of the gauge gauge sensors [14]. The parameter by the long-gauge sensor is the average strainofoflong-gauge the Compared with the pointmeasured sensing, area sensing is proposed along with application sensors [14]. The parameter measured by the long-gauge sensor is the average strain of the gauge length, named named as macro-strain. macro-strain. The gauge gauge length can be be up up to to 11 m m ~~ 22 m, m, much much larger larger than that of of the the length, as The can than that sensors [14]. The parameter measured by length the long-gauge strain of the length, named as macro-strain. The gauge length can be upsensor to 1 m ~is2the m, average much larger than thatgauge of the common strain gauge. Then the damage information can be obtained by the measured macro-strain common strain gauge. Then the damage information can be obtained by the measured macro-strain length, named macro-strain. gaugeinformation length can be upbetoobtained 1 m ~2 m, larger than that of the common strainasgauge. Then theThe damage can bymuch the measured macro-strain if the the damage damage locates within thedamage sensor.information The probability probability toobtained capture the the damage is greatly greatly increased if locates within the sensor. The to capture damage is increased common strain gauge. Then the can be by the measured macro-strain if if the damage locates within the sensor. The probability to capture the damage is greatly increased when several long-gauge sensors areThe applied at the the same same time (Figure (Figure 2) because because the increased monitored area when several long-gauge sensors are applied at time 2) the monitored area the damage locates within the sensor. probability to capture the damage is greatly when when several long-gauge sensors are applied at the same time (Figure 2) because the monitored area can be be enlarged enlarged to cover cover the entire entire area. can to the area. several long-gauge sensors applied can be enlarged to cover theare entire area.at the same time (Figure 2) because the monitored area can be enlarged to cover the entire area.

Figure2. 2. Area Area sensing sensing with with long-gauge long-gaugesensors. sensors. Figure sensors. Figure 2. Area sensing with long-gauge sensors.

Due to to its its excellent accuracy, high data acquisition speed and the multiplexing multiplexing performance, the Due speed and the performance, the Due to its excellent excellentaccuracy, accuracy,high highdata dataacquisition acquisition speed and multiplexing performance, Due to its excellent accuracy, high data acquisition speed and thethe multiplexing performance, the Fiber Bragg Grating (FBG) sensor is used to develop the long-gauge macro-strain sensor (Figure 3). Fiber Bragg Grating (FBG) sensor is used to develop the long-gauge macro-strain sensor (Figure the Fiber Bragg Grating (FBG) sensor is used developthe thelong-gauge long-gaugemacro-strain macro-strainsensor sensor (Figure (Figure 3). 3). Fiber Bragg Grating (FBG) sensor is used toto develop 3). Every point point in in the the gauge gauge length length presents presents the the same same mechanical mechanical behavior behavior and and hence hence the the strain strain obtained obtained Every Every Every point point in in the the gauge gauge length length presents presents the the same same mechanical mechanical behavior behavior and and hence hence the the strain strain obtained obtained from FBG can represent the average strain over the gauge length, termed as macro-strain. To actualize from FBG can represent the average strain over the gauge length, termed as macro-strain. To actualize from macro-strain. To To actualize from FBG FBG can can represent represent the the average average strain strain over over the the gauge gauge length, length, termed termed as as macro-strain. actualize the long-gauge long-gauge concept concept for for real real sensors, sensors, the the FBG FBG sensor sensor is is fixed fixed at at the the two two ends ends of of aaa plastic plastic tube tube the long-gauge concept for real sensors, the FBG sensor is fixed at the two ends of plastic the the long-gauge concept for real sensors, the FBG sensor is fixed at the two ends of a plastic tube tube surrounded by by aa fiber fiber sheath sheath impregnated impregnated with with epoxy epoxy resin. resin. The The diameter diameter of of the the packaged packaged sensor sensor is is surrounded surrounded surrounded by by aa fiber fiber sheath sheath impregnated impregnated with with epoxy epoxy resin. resin. The The diameter diameter of of the the packaged packaged sensor sensor is is about 1 mm. Meanwhile, the gauge length can be set from 0.1 m to 2 m. about 1 mm. Meanwhile, the gauge length can be set from 0.1 m to 2 m. about about 11 mm. mm. Meanwhile, Meanwhile, the the gauge gauge length length can can be be set set from from 0.1 0.1 m m to to 22 m. m.

Figure 3. 3. Packaged Packaged long-gauge long-gauge Fiber Fiber Bragg Bragg Grating Grating (FBG) (FBG) sensor. sensor. Figure Figure Figure 3. 3. Packaged Packaged long-gauge long-gauge Fiber Fiber Bragg Bragg Grating Grating (FBG) (FBG) sensor. sensor.

2.2. Macro-Strain Macro-Strain Modal Modal Analysis Analysis 2.2. 2.2. 2.2. Macro-Strain Modal Modal Analysis Analysis Like the the point point strain, strain, the the macro-strain macro-strain measurement measurement can can also also be be used used for for strain strain modal modal analysis analysis Like Like Like the the point point strain, strain, the the macro-strain macro-strain measurement measurement can can also also be be used used for for strain strain modal modal analysis analysis and extraction extraction of of the the macro-strain macro-strain mode mode [15]. [15]. and and and extraction extraction of of the the macro-strain macro-strain mode mode [15]. [15].

 mm11  m 1

 mm m jjthth iithth ith LLmm jth Lm

ff f

 mm11  m 1

ppthth pth

Figure Figure 4. 4. Macro-strain Macro-strain measurements measurementsfor forbeam beamstructure. structure. Figure 4. Macro-strain measurements for beam structure. Figure 4. Macro-strain measurements for beam structure.

InFigure Figure4, 4,aaatypical typicalbeam beammodel modelis istaken takenas asan anexample. example.The Thestrain strainεεεmm is is measured measured by by the the strain strain In Figure 4, typical beam model is taken as an example. The strain In In Figure 4, a typical beam model is taken as an example. The strain εmm is measured by the strain sensor covering covering the the zone zonefrom fromthe theiiithth node node to tojjjthth node, node, namely, namely,the thegauge gaugelength lengthLLLmm,, when when force forcefff is is covering the zone from the node to namely, the gauge length when force is sensor sensor covering the zone from the ithth node to jthth node, namely, the gauge length Lmm, when force f is appliedat atthe thepppth th node. node. The can be be expressed expressed as as applied at the th applied The εεεmmm can applied at the pth node. The εm can be expressed as hh mmhm hmmm  vvjj vvi
i  mm  vvjj vvii 
   ηm m v jvj v− ε m = m LLmmvj v−j vvi i  = i  vi LmLm Then, the the equation equation in in the the frequency frequency domain domain is is Then, Then, the the equation equation in in the the frequency frequency domain domain is is Then, ))   vv (( ))  vv (( )) mm ((  m ( )   mmmv jjj ( )  viii ( )   ε m ( ω ) = ηm v j ( ω ) − v i ( ω )

(1) (1) (1) (1)

(2) (2) (2) (2)

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In Equations (1) and (2), vi and vi (w) are the rotational displacement for the ith node in the time and frequency domain, respectively. hm is the distance from the strain sensor to the neutral axis. Frequency response function (FRF) is considered to be related with structure properties only. ε ( ω ), it can be achieved by For the macro-strain FRF Hmp ε Hmp (ω )

  ηm v j ( ω ) − v i ( ω ) ε m (ω ) d d = = = ηm ( Hjp (ω ) − Hip (ω )) f p (ω ) f p (ω )

(3)

d ( ω ) is the traditional rotational displacement FRF, which can be expressed as The Hip

Hijd (ω ) =

N

∑

2 r =1 Mr ( ωr

ϕir ϕ jr − ω 2 + 2iξ r ωr ω )

(4)

where ϕir is the rth rotational displacement mode at the ith node. Mr and ω r are the mass and resonant frequency for the rth order mode, respectively. ξ r = Cr /(2Mr ωr ), while Cr is damping efficient for the rth mode. Finally, the macro-strain FRF can be expressed as N

ε Hmp (ω ) =

∑

r =1

N ηm ( ϕ jr − ϕir ) ϕ pr δmr ϕ pr = ∑ 2 − ω 2 + 2iξ ω ω ) Mr (ωr2 − ω 2 + 2iξ r ωr ω ) M ( ω r r r r r =1

(5)

Compared with Equation (4), the similarity is found in Equation (5) except parameter δmr . The absolute value at the peak for the rth order mode can be written as ϕ pr ε δ r Hmp (ω = ωr ) = mr 2 2Mr ξ r ωr

(6)

Here, for a given mode and a given excited point, ϕ pr /(2Mr ξ r ωr2 ) is a constant. Therefore, δmr is the parameter determining the macro-strain mode shape. As is well known, using the Fourier Transform (FT), the relationship can be obtained at the rth order mode as v ( ωr ) ϕir = i (7) ϕ jr v j ( ωr ) It is written to be as Equation (8) for the macro-strain mode. δmr ε m ( ωr ) = δnr ε n ( ωr )

(8)

where εm (wr ) is the modal macro-strain (MMS) for the mth monitored element at the rth mode, namely the absolute value of the magnitude in the frequency spectrum when the frequency is wr . Therefore, the macro-strain modal analysis can be implemented as the traditional displacement modal analysis using the macro-strain measurements. 2.3. Neutral Axis Position Determination with MMS As shown in Figure 5, the plane section assumption is used in each beam-like structure. It means that there are linear relationships among the strains along the section depth which are determined by the structure property itself. Based on this assumption, the neutral axis position coefficient (NAPC) ψ is applied to express the linear relationship ε(t) = ψε0 (t)

(9)

where ε(t) and ε0 (t) are the strains at different depths in the same section. Besides the section, it is easily understood that the plane section can be also applied to analyze some small part of the flexural

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structure. In other words, the strain distributes linearly with an averaged neutral axis for all parts. 0 Then strains Sensorsthe 2017, 17, 411 ε (t ) and ε (t ) here are the macro-strains. 5 of 20  ( t ) H h

Neutral axis

M

Neutral axis

H

 ( t )

h

M

 (t )

 (t )

(a)

(b)

Figure 5. 5. Strain Strain distribution distribution along along the the cross cross section section with with sensors sensors installed installed at: at: (a) (a) different different sides sides of of the the Figure neutral axis; and (b) the same side of the neutral axis. neutral axis; and (b) the same side of the neutral axis.

The neutral axis depth h (Figure 5) can be easily obtained by some basic geometrical knowledge. The neutral axis depth h (Figure 5) can be easily obtained by some basic geometrical knowledge. There are two cases for the strain measurement: (1) sensors installed at different sides of the neutral There are two cases for the strain measurement: (1) sensors installed at different sides of the neutral axis; and (2) sensors installed at the same side of the neutral axis. For these two cases, the equations axis; and (2) sensors installed at the same side of the neutral axis. For these two cases, the equations to to calculate the neutral axis depth are Equations (10) and (11), in which H means the vertical distance calculate the neutral axis depth are Equations (10) and (11), in which H means the vertical distance between the two sensors. between the two sensors. ψ h h=  HH (10) (10) ψ +  11 ψ h h=  HH ψ −  11

(11) (11)

NAPC In Equations Equations (12) (12) and and (13), (13), NAPC should should be be first first solved solved to to determine determine the the neutral neutral axis axis location. location. In with the Fourier transform (FT), the NAPC can be expressed by the modal macro-strain (MMS) ε(( ww)r ). with the Fourier transform (FT), the NAPC can be expressed by the modal macro-strain (MMS) r .

ε((ω )  ) =

+  w ∞ 

+ w∞

 i t ε((tt ))ee− iωtd t   dt =

−∞



 i t   ψε0 (t( )t )ee−iωt ddtt = ψ

−∞

+ w∞ 



 i t  ( ) ) e iωt  0(ω ε0(t)( et )− dtd t=ψε



(12) (12)

−∞

 (w )   (13) ε(w( wr )) ψ= 0 (13) ε ( wr ) The NAPC ψ contains enough spatial information of the neutral axis. If damage occurs within The NAPCzone, ψ contains enough information the neutral If damage within the the monitored the neutral axisspatial will move and theofNAPC ψ willaxis. change as well.occurs Therefore, it has monitored zone, the neutral axis will move and the NAPC ψ will change as well. Therefore, it has the the potential to be a new index to detect structural damage. potential to paper, be a new index to detect structural In this only the first bending mode damage. is applied to implement the method as it can be easily In this paper, only the first bending mode is bending applied to implement the method it can be easily and accurately identified. Moreover, in the first mode, the structure is in aasperfect bending and identified. Moreover, in the firstmethod bending the the structure is infrom a perfect state,accurately improving the accuracy of the proposed bymode, ignoring influence somebending factors, state, improving the accuracy of the proposed method by ignoring the influence from someanalysis factors, such as torsion. Most of the measurement noise is also naturally filtered during the modal such as torsion. Most of the measurement noise is also naturally filtered during the modal analysis process with the Fourier transform. Lastly the macro-strain mode is of no relationship with the process the Fourier Lastly thethe macro-strain is ofisnonot relationship theloading. loading loadingwith magnitude and transform. position and hence MMS basedmode method influencedwith by the magnitude and position and hence the MMS based method is not influenced by the loading. Therefore, Therefore, compared with the traditional methods based on the strain in time domain, the proposed compared witha the traditional based on the strain in time domain, the proposed method method offers better assessingmethods performance. r

r

offers a better assessing performance.

3. Verification with FE Models 3. Verification with FE Models 3.1. FE Model Description Description 3.1. FE Model Solid45 was wasselected selectedto to establish simply-supported Element (FE) model beam with model with Solid45 establish simply-supported FiniteFinite Element (FE) beam ANSYS. ANSYS. The model detailed model information is shown inFrom Figure From thealong direction alonglength, the beam The detailed information is shown in Figure 6. the6.direction the beam the length, the element length was 100 mm, while it was 50 mm for the other two directions. Thealong five element length was 100 mm, while it was 50 mm for the other two directions. The five elements elements width at the middle span, shown in selected Figure 6c, selected as damaged elements the widthalong at the the middle span, shown in Figure 6c, were as were damaged elements by changing the by changing the elastic modulus. The damage cases are listed in Table 1, where ‘Stiffness loss’ only elastic modulus. The damage cases are listed in Table 1, where ‘Stiffness loss’ only means the relative means the relative stiffness loss of the damaged elements. In the experiments the strain measuring gauge length was 300 mm, 3 times as long as the element length, completely covering the damaged zone. The dynamic loads were implemented at the point on the beam up-surface.

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stiffness loss of the damaged elements. In the experiments the strain measuring gauge length was 300 mm, 3 times as long as the element length, completely covering the damaged zone. The dynamic Sensorswere 2017, 17, 411 6 of 20 loads implemented at the point on the beam up-surface.

Figure 6. 6. Finite Finite Element Element (FE) (FE) beam beam model: model: (a) beam; (b) (b) cross cross section; section; and and (c) (c) damaged damaged zone zone and and Figure (a) beam; monitored zone. monitored zone. Table 1. 1. The The damage damage case case for for FE FE models. models. Table

Damage Case I Stiffness loss (%)

Damage Case

Stiffness loss (%)

0

I 0

D1 D2 D3 D1 D2 30 70 100 30

70

D3 100

3.2. Neutral Axis Position 3.2. Neutral Axis Position Through modal analysis, the MMS can be obtained with Equation (6). In this paper, only the first Through analysis, the MMSanalysis. can be obtained with (6). upside In this element paper, only bending modemodal is used to do structural Then with theEquation MMS of the and the the first bending mode is used to doaxis structural analysis. with the MMS of the upside element and downside element, the neutral position can be Then determined with Equation (13). The results of the downside element, the neutral axis position can be Equationthe (13). The results of neutral axis position coefficient (NAPC) were listed in determined Table 2. For with comparison, results of NAPC neutral axisanalysis positionwere coefficient (NAPC) listed inthese Tableresults, 2. For comparison, results method of NAPC with static also listed in thewere table. From the proposedthe dynamic is with static analysis were also listed the table. Fromdeviation these results, the proposed is verified with a good accuracy as theinlargest relative between the modaldynamic analysismethod and static verified a good accuracy as the largest relative deviation between the modal analysis and static analysiswith is less than 1%. analysis is less than 1%. Table 2. The results of neutral axis position coefficient (NAPC). Table 2. The results of neutral axis position coefficient (NAPC).

Damage Case I D1 Modal analysis 1.007 1.047 Damage Case I D1 Static analysis 1.000 1.042 Modal analysis 1.007 1.047 Relative deviation (%) 0.5 Static analysis 1.000 0.7 1.042

Relative deviation (%)

0.7

0.5

D2 1.144 D2 1.141 1.144 0.3 1.141 0.3

D3 1.334 D3 1.330 1.334 0.3 1.330 0.3

3.3. Damage Identification 3.3. Damage Identification As a new index to assess damage, damage sensitivity of the NAPC should be investigated and compared with sometoacknowledged indices, such as resonant frequency, strain mode and curvature As a new index assess damage, damage sensitivity of the NAPC should be investigated and mode. In Table 3, the relative change of the above four indices and neutral axis depth been compared with some acknowledged indices, such as resonant frequency, strain mode and have curvature obtained simulation results of with damage The results show the mode. In from Table the 3, the relative change thedifferent above four indicescases. and neutral axis depth havelargest been relative changes for strain mode. In other words, the strain mode method presents the highest obtained from the simulation results with different damage cases. The results show the largest relative damage sensitivity. Then itIn is other the proposed close to the value of the strain The NAPC changes for strain mode. words, NAPC, the strain mode method presents themode. highest damage results are calculated with the MMS from the down-surface and up-surface. The damage sensitivity of the NAPC is much better than the curvature mode, which is accepted as an index sensitive to local damage. The neutral axis depth is not as sensitive as the NAPC, but it is still much more sensitive than the resonant frequency as the resonant frequency hardly changes under the simulated local damage. Moreover, compared with the strain mode and curvature mode methods, there is no need

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sensitivity. Then it is the proposed NAPC, close to the value of the strain mode. The NAPC results are calculated with the MMS from the down-surface and up-surface. The damage sensitivity of the NAPC is much better than the curvature mode, which is accepted as an index sensitive to local damage. The neutral axis depth is not as sensitive as the NAPC, but it is still much more sensitive than the resonant frequency as the resonant frequency hardly changes under the simulated local damage. Sensors 2017, 17, 411 7 of 20 Moreover, compared with the strain mode and curvature mode methods, there is no need to select an intact referenced element for the proposed method. If some damage happens within the referenced referenced element, the methods may fail to implement damage identification. Therefore, the element, the methods may fail to implement damage identification. Therefore, the proposed NAPC is proposed NAPC is a promising indicator to identify local damage, such as a concrete crack. a promising indicator to identify local damage, such as a concrete crack.

Damage Damage Case Case

D1 D1 D2 D2 D3 D3

Table 3. Comparison results of damage sensitivity (%). Table 3. Comparison results of damage sensitivity (%). Resonant Frequency (Bending Mode) Neutral Resonant Frequency (Bending Mode) Axis NAPC Neutral NAPC First Second Third Fourth Fifth Axis Depth Depth First Second Third Fourth Fifth −0.1 0.0 −0.1 0.0 0.0 1.9 4.0 − 0.1 0.0 − 0.1 0.0 0.0 1.9 4.0 − 0.3 0.0 − 0.3 0.0 −−0.1 0.1 6.3 13.6 −0.3 0.0 −0.3 0.0 6.3 13.6 − 0.8 0.0 − 0.8 0.0 −−0.4 0.4 13.9 32.5 −0.8 0.0 −0.8 0.0 13.9 32.5

Strain Mode Strain Mode [16] [16] 4.2 4.2 15.3 15.3 39.2 39.2

Curvature

Curvature Mode [17] Mode [17]

2.3 2.3 8.5 8.5 22.2 22.2

The factors factors influencing influencing the the damage damage sensitivity sensitivityare arealso also investigated investigatedwith withthe the FE FE modeling modeling results. results. The In the the proposed proposed method method the the sensor sensor gauge gauge length length and and sensor sensor installation installation position position are are the the two two main main In factors. It can be easily understood that the smaller the sensor gauge is, the higher is the damage factors. It can be easily understood that the smaller the sensor gauge is, the higher is the damage sensitivityififthe the sensor the damage zone. Therefore, the influence from installation the sensor sensitivity sensor can can covercover the damage zone. Therefore, the influence from the sensor installation not sensor length is only studied. not sensor gauge lengthgauge is only studied. As shown in Figure 7a, the distance “X” between between the the two two sensors sensors is is taken taken as as the the variable variable to to As shown in Figure 7a, the distance “X” investigate the the damage damage sensitivity sensitivity of of the the NAPC. NAPC. As As the the sensor sensor can can be be easily easily installed installed on on the the beam beam investigate bottom, Sensor 1 is considered fixed. The results are shown in Figure 7b. The strain is too small near bottom, Sensor 1 is considered fixed. The results are shown in Figure 7b. The strain is too small near the neutral axis, so the “X” of 250 mm was not investigated as the neutral axis location was in the the neutral axis, so the “X” of 250 mm was not investigated as the neutral axis location was in the range from from210 210mm mmtoto 290 mm under damage cases. the results it is found that ahigher much range 290 mm under the the damage cases. FromFrom the results it is found that a much higher sensitivity can be obtained when the two sensors are installed on different sides of the neutral sensitivity can be obtained when the two sensors are installed on different sides of the neutral axis. axis.

(a)

(b)

Figure 7. Damage sensitivity with different spatial sensor installations: (a) sensor installation; and (b) Figure 7. Damage sensitivity with different spatial sensor installations: (a) sensor installation; and relative change of neutral axisaxis position coefficient (NAPC). (b) relative change of neutral position coefficient (NAPC).

3.4. Performance under Noise 3.4. Performance under Noise Performance under noise is another issue to be investigated. As shown in Figure 8, the white Performance under noise is another issue to be investigated. As shown in Figure 8, the white noise was input into the strain results for investigation. The results of the proposed method are shown noise was input into the strain results for investigation. The results of the proposed method are shown in Figure 9. The conclusion is that the added noise shows no influence on the proposed method even in Figure 9. The conclusion is that the added noise shows no influence on the proposed method even when the noise is up to 5%. However, the results will be influenced greatly by the noise when the strain is directly used to calculate the neutral axis position with the traditional method as Equation (9). Strain larger than 30 με was only applied in the calculation to reduce the assessment error caused by the strain measurement accuracy. From the results in Figure 10, the traditional method can hardly be implemented to assess the neutral axis position even with 2% noise, while it works well without noise. Therefore, the proposed method is a perfect choice to be applied in neutral axis position measuring

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when the noise is up to 5%. However, the results will be influenced greatly by the noise when the strain is directly used to calculate the neutral axis position with the traditional method as Equation (9). Strain larger than 30 µε was only applied in the calculation to reduce the assessment error caused by the strain measurement accuracy. From the results in Figure 10, the traditional method can hardly be implemented to assess the neutral axis position even with 2% noise, while it works well without noise. Therefore, the proposed method is a perfect choice to be applied in neutral axis position measuring underSensors live 2017, traffic loads especially. 17, 411 8 of 20 8 of 20

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Sensors 2017, 17, 411 100 100 No noise No noise 80 100 2% noise 80 2%noise noise No 60 5% noise 80 60 5% noise noise 2% 40 60 40 5% noise 20 40 20 2000 -20 0 -20 -40 -20 -40 -60 -40 -60 -80 -60 -80 -100 -80 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -100 1.0 1.2 1.4 1.6 1.8 2.0 -100 0.0 0.2 0.4 0.6 0.8Time (s) 0.0 0.2 0.4 0.6 0.8 Time 1.0 (s) 1.2 1.4 1.6 1.8 2.0

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(a) (a)Time (s) (a) (b)local part focused. Figure 8. Typical strain results at Case I: (a) the whole time history; and (b) Figure 8. Typical strain results CaseI:I:(a) (a)the the whole whole time (b)(b) local partpart focused. Figure 8. Typical strain results atatCase timehistory; history;and and local focused. Figure 8. Typical strain results at Case I: (a) the whole time history; and (b) local part focused. 60000 60000

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Neutral axisaxis depth (mm) Neutral axisdepth depth (mm) Neutral (mm)

(b) (b) (a)of the proposed method under noise: (a) NAPC; and (b) Figure 9. Performance (b) neutral axis depth. Figure 9. Performance of the proposed method under noise: (a) NAPC; and (b) neutral axis depth. Figure 9. Performance of the proposed method under noise: (a) NAPC; and (b) neutral axis depth. Figure 9. Performance of the proposed method under noise: (a) NAPC; and (b) neutral axis depth. 390 390 390 360 360 360 330 330 330 300 300 300 270 270 270 240 240 240 210 210 210

No noise No noise 2% noise 2%noise noise No 5% noise 5% noise noise 2% 5% noise True value True value True value

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80 100 120 80 100 120 Strain () Strain (  ) 80 100 120 (b) Strain () (b)

(a) Figure 10. Performance of the traditional method under noise at Case:(b) (a) I; and (b) D3. Figure 10. Performance of the traditional method under noise at Case: (a) I; and (b) D3. Figure 10. Performance of the traditional method under noise at Case: (a) I; and (b) D3.

140 140 140

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Figure 10. Performance of the traditional method under noise at Case: (a) I; and (b) D3. 4. 4. Verification Verification with with aa Steel Steel Beam Beam 4. Verification with a Steel Beam 4.1. 4.1. Experiment Experiment Description Description 4.1. Experiment Description A steel beam A steel beam (Figure (Figure 11) 11) was was prepared prepared for for some some tests tests to to verify verify the the assessment assessment accuracy accuracy of of the the proposed method. Considering the experience from the above RC beam tests, the damage A steelmethod. beam (Figure 11) wasthe prepared for some to verify assessment accuracy the proposed Considering experience from tests the above RCthe beam tests, the damageofwas was simulated by some from shown in 12, proposed the experience from theflange aboveas beam tests, the damage was simulated method. by cutting cuttingConsidering some parts parts away away from the the downside downside flange asRC shown in Figure Figure 12, guaranteeing guaranteeing the same neutral axis location during the dynamic and static tests. simulated by cutting parts away the from the downside flange as shown in Figure 12, guaranteeing the same neutral axissome location during dynamic and static tests.

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In Figure 11a, the steel beam is 5.6 m long with an “I” type cross section, divided into 28 elements Figure 11a, the of steel 5.6 8melements, long withEan “I” type cross section, divided into 28 elements with anInelement length 200beam mm.isThe 1 to E8, were selected as monitored parts with with an element length of 200 mm. The 8 elements, E 1 to E8, were selected as monitored parts with long-gauge FBG sensors bonded on both up-surface (F13 to F83) and down-surface (F11 to F81).9 One Sensors 2017, 17, 411 of 20 long-gauge FBG sensors bonded on both up-surface (F 13 to F83) and down-surface (F11 to F81). One more sensor, F32, was installed on the beam side surface to verify the plane section hypothesis. The more sensor, F32, was installed onfrom the beam sidetosurface to with verify plane section hypothesis. The reflected wavelength of FBG ranges 1525 nm 1580 nm thethe sensitivity of about 1.2 pm/με. reflected wavelength of FBG ranges from 1525 nm to 1580 nm with the sensitivity of about 1.2 pm/με. 4. Verification withstatic a Steel Beam In Figure 11b, load was applied with some steel block, while the dynamic force was input In Figure 11b, static load applied steel block, the while thepoints dynamic force was input by using the hammer to knock was at the beam’swith top some surface. Besides four in the figure, the 4.1. Experiment Description by using the hammer to knock at the beam’s top surface. Besides the four points in the figure, hammer was also used to knock at some random positions at one measure time to simulate the randomthe hammer was also usedthe to11) knock some positions at measure time to simulate the of random dynamic cases. During dynamic testsrandom thefor strain was by (Micron Optics Inc., A steel beam (Figure wasatprepared some testsmeasured to one verify theSM-130 assessment accuracy the dynamic cases. During the dynamic tests the strain was measured by SM-130 (Micron Optics Inc., Atlanta, GA, USA) with a samplingthe frequency of 1000 Hzthe as shown Table presentswas the proposed method. Considering experience from above in RCFigure beam12a. tests, the 4damage Atlanta, GA, USA) with a sampling frequency of 1000 Hz as shown in Figure 12a. Table 4 presents damage cases, D1 tosome D8, where the damage the relative lossinof the element flexural-the simulated by cutting parts away from thequantity downsideisflange as shown Figure 12, guaranteeing damage cases, axis D1 to D8, where is tests. the relative loss of the element flexuralstiffness. the same neutral location duringthe thedamage dynamicquantity and static stiffness.

(a) (a)

(b) (b)

Figure 11. Experimental description for the steel beam tests (Unit: mm): (a) sensor installation at steel Figure 11. Experimental description forfor thethe steel beam tests (Unit: mm): (a)(a) sensor installation at steel Figure description steel beam tests (Unit: mm): sensor installation at steel beam; and11. (b)Experimental loading position. beam; and (b) loading position. beam; and (b) loading position.

(a) (a)

(b) (b)

Figure 12. The steel beam experiment field: (a) dynamic test and data logger; and (b) damage Figure 12. The steel beam experiment field: (a) dynamic test and data logger; and (b) damage simulation. Figure 12. The steel beam experiment field: (a) dynamic test and data logger; and (b) damage simulation. simulation. Table 4. Damage cases for the steel beam test.

In Figure 11a, the steel beamTable is 5.64.mDamage long with “I” divided into 28 elements casesanfor thetype steelcross beamsection, test. Damage Case D1 D2 D4 selected D5 D6 asD7 D8 with an element length of 200 mm. The 8 Ielements, E1 toD3 E8 , were monitored parts with Damage Case I D1 D2 D3 D4 D5 D6 D7 D8 long-gauge FBG sensors bonded both up-surface to F83 (F11 to F81 ). One Damage quantity of on E3 (%) 0 3 5 (F1315 18) and 18down-surface 18 18 18 Damage quantity of E 3 (%) 0 3 5 15 18 18 18 18 18 more sensor, FDamage , was installed on the beam side surface to verify the plane section hypothesis. The quantity of E 7 (%) 0 0 0 0 0 3 5 15 18 32 Damage quantity of from E7 (%)1525 0 nm0 to 1580 0 nm 0 with0 the sensitivity 3 5 15 18 1.2 pm/µε. reflected wavelength of FBG ranges of about 4.2. Experiment Results In Figure 11b, static load was applied with some steel block, while the dynamic force was input Experiment Results by4.2. using the hammer to knock at the beam’s top surface. Besides the four points in the figure, the Figure 13 presents the typical macro-strain results under the single-point and random excitation, hammer was also used to knock at some random positions at onethe measure time toand simulate the excitation, random Figure 13 presents the typical macro-strain under single-point random while the macro-strain based frequency spectrumresults near the first bending mode is obtained with FFT dynamic cases. During the dynamic tests the strain was measured by SM-130 (Micron Optics Inc., the macro-strain frequency spectrum near14). the first bending mode is obtained with FFT to while get MMS to implementbased the proposed method (Figure Atlanta, GA, USA) with a sampling frequency of 1000 Hz as shown in Figure 12a. Table 4 presents the to get MMS to implement the proposed method (Figure 14). damage cases, D1 to D8, where the damage quantity is the relative loss of the element flexural-stiffness.

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Table 4. Damage cases for the steel beam test. Damage Case

I

D1

D2

D3

D4

D5

D6

D7

D8

Damage quantity of E3 (%) Damage quantity of E7 (%)

0 0

3 0

5 0

15 0

18 0

18 3

18 5

18 15

18 18

4.2. Experiment Results Figure 13 presents the typical macro-strain results under the single-point and random excitation, while the macro-strain based frequency spectrum near the first bending mode is obtained with FFT to get MMS to implement the proposed method (Figure 14).

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Figure 13. Typical macro-strain results for the steel beam test under: (a) single-point excitation; and Figure13.13.Typical Typical macro-strain results the steel beam test under: single-pointexcitation; excitation;and and Figure macro-strain results forfor the steel beam test under: (a)(a) single-point (b) random excitation. (b) random excitation. (b) random excitation. 15000 15000

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Figure 14. Typical results of macro-strain based frequency spectrum for the steel beam test. Figure14. 14.Typical Typicalresults resultsofofmacro-strain macro-strainbased based frequency spectrum the steel beam test. Figure frequency spectrum forfor the steel beam test.

With the obtained MMS, the plain section is verified at first (Figure 15). Then the NAPC is With the obtained MMS, the plain section is verified at first (Figure 15). Then the NAPC is Withfrom the obtained MMS, the 16). plain is verified at firstcorrelation (Figure 15). Then theare NAPC is extracted the results (Figure In section the figures all the linear coefficients larger extracted from the results (Figure 16). In the figures all the linear correlation coefficients are larger extracted from the results (Figure 16). In the figures all the linear correlation coefficients are larger than than 0.99, indicating excellent stability of the NAPC obtained with the proposed dynamic method. than 0.99, indicating excellent stability of the NAPC obtained with the proposed dynamic method. 0.99,NAPC indicating excellent stability of the NAPC obtained with the proposed dynamic method. The The results are included in Figure 17 for all the monitored elements. From the relative change The NAPC results are included in Figure 17 for all the monitored elements. From the relative change NAPC results are included in Figure 17 for all the monitored elements. From the relative change in in Figure 17b especially the damaged elements can be detected easily even when the element stiffness in Figure 17b especially the damaged elements can be detected easily even when the element stiffness Figure 17b especially thethe damaged can index be detected easilylocal evendamage. when the element stiffness loss is only 3%, proving NAPC iselements an effective to identify loss is only 3%, proving the NAPC is an effective index to identify local damage. loss is only 3%, proving the NAPC is an effective index to identify local damage. F33 F33

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With the obtained MMS, the plain section is verified at first (Figure 15). Then the NAPC is extracted from the results (Figure 16). In the figures all the linear correlation coefficients are larger than 0.99, indicating excellent stability of the NAPC obtained with the proposed dynamic method. The NAPC results are included in Figure 17 for all the monitored elements. From the relative change Figure 17b especially the damaged elements can be detected easily even when the element stiffness Sensorsin 2017, 17, 411 11 of 20 loss is only 3%, proving the NAPC is an effective index to identify local damage. F33

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Figure 15. Modal macro-strain (MMS) distribution along the depth of the element E3 of the steel beam

Figure 15. Modal macro-strain (MMS) distribution along the depth of the element E3 of the steel beam at Case: (a) I; and (b) D8. at Case: (a) I; and (b) D8.

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Figure 16. NAPC extracting of the steel beam from dynamic tests for: (a) E1; and (b) E3.

Figure 16. NAPC extracting of the steel beam from dynamic tests for: (a) E1 ; (b) and (b) E3 . Figure 16. NAPC extracting of the steel beam from dynamic tests for: (a) E1; and E3.

(a) (a)

(b)

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Figure 17. NAPC results of the steel beam: (a) absolute value; and (b) relative change.

Figure NAPC resultsofofthe thesteel steel beam: beam: (a) and (b) (b) relative change. Figure 17. 17. NAPC results (a)absolute absolutevalue; value; and relative change.

With the obtained NAPC the neutral axis depth was calculated (Table 5). For the undamaged With theexcept obtained NAPC depth waschange calculated (Tablethan 5). For undamaged elements E4 and E8, the the neutral relative axis measurement is smaller 1%,the verifying the elements except E 4 and E8, the relative measurement change is smaller than 1%, verifying the effectiveness of the proposed method. However, it is a little larger for E4 and E8, about 2%. The process effectiveness theparts proposed method. However, it is asmall littledamage larger for E4 and 8, about 2%. The process of cutting of steel to make damage caused some within theEtwo elements. of cutting steel parts to make damage caused some small damage within the two elements. Then the results from the dynamic method are compared with the results from the static tests, theinresults from thethe dynamic method are compared therelative resultsdifference from theisstatic asThen shown Table 6. From results, the largest absolute valuewith of the 2.6%tests, for

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With the obtained NAPC the neutral axis depth was calculated (Table 5). For the undamaged elements except E4 and E8 , the relative measurement change is smaller than 1%, verifying the effectiveness of the proposed method. However, it is a little larger for E4 and E8 , about 2%. The process of cutting steel parts to make damage caused some small damage within the two elements. Table 5. The results of neutral axis depth with modal analysis (mm). Damage Case

I

D1

D2

D3

D4

D5

D6

D7

D8

E1 E2 E3 E4 E5 E6 E7 E8

50.0 49.5 49.9 50.0 49.7 50.1 49.7 50.6

50.2 49.7 51.5 50.2 49.8 50.3 49.8 50.8

50.2 50.0 52.3 50.3 49.8 50.5 49.7 50.7

50.2 49.8 56.3 50.3 49.7 50.3 49.7 50.8

50.3 49.8 57.4 50.7 49.8 50.4 49.8 50.9

50.3 49.7 57.4 50.8 49.8 50.2 51.5 50.8

50.1 49.6 57.3 50.5 49.8 50.2 52.1 50.9

50.4 49.9 57.4 50.8 49.8 50.5 56.6 51.5

50.3 49.8 57.4 50.7 49.8 50.3 57.4 51.5

Then the results from the dynamic method are compared with the results from the static tests, as shown in Table 6. From the results, the largest absolute value of the relative difference is 2.6% for E5 at D5. It means that the relative error is less than 3% for the proposed dynamic method. Thus the high accuracy has been verified for assessing the neutral axis position with the proposed method. Table 6. The relative difference compared with static tests (%). Damage Case

I

D1

D2

D3

D4

D5

D6

D7

D8

E1 E2 E3 E4 E5 E6 E7 E8

0.6 −0.4 0.5 −0.7 1.0 0.3 0.3 1.0

0.5 0.1 0.5 0.1 1.5 1.0 1.0 1.8

−0.2 0.2 −0.1 0.1 1.0 1.1 0.0 1.2

0.8 0.7 1.8 0.7 2.2 1.7 1.0 2.2

0.5 0.0 1.2 0.1 1.7 1.4 0.5 1.8

1.0 0.4 1.7 0.7 2.6 1.5 1.7 2.4

−1.1 −1.2 −0.2 −1.4 0.6 −0.3 −0.5 0.8

−0.6 −0.7 0.3 −0.2 0.5 0.8 0.4 1.8

−0.3 −0.2 0.8 −0.1 1.2 0.9 0.5 1.7

5. Verification with a Reinforced Concrete Beam 5.1. Experiment Description One reinforced concrete (RC) beam as shown in Figure 18a is prepared to investigate the actual performance of the proposed method. The total beam length is 4100 mm, while the length between the two supports is 3600 mm. The rectangle section’s width is 120 mm, while the depth is 240 mm. The section is reinforced with two steel rebars with a diameter of 14 mm at downside and two of 12 mm at upside. Some other detailed information can be found in Figure 18a. In Figures 18b and 19b, long-gauge FBG sensors (Figure 3) were installed to measure the macro-strain. F11 ~F51 were bonded at the beam bottom, while F13 ~F53 were at the beam side 20 mm away from the top. One more sensor F32 was installed at the beam side of the monitored element E3 , 60 mm away from the bottom to prove the plane section. The gauge length is 300 mm, same for all the 11 sensors. The data logger is SM-130 made by Micron Optics Inc. (Atlanta, GA, USA) with a sampling frequency of 1000 Hz applied in the tests as shown in Figure 19a.

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(a) (a)

(b) (b)

(c) (c) Figure 18. Experiment setup (Unit: mm): (a) reinforced concrete (RC) beam; (b) FBG Figure 18. Experiment Experimentsetup setup(Unit: (Unit: mm): reinforced concrete (b) long-gauge long-gauge FBG Figure 18. mm): (a) (a) reinforced concrete (RC)(RC) beam;beam; (b) long-gauge FBG sensor sensor distribution; and (c) loading positions. sensor distribution; (c) loading positions. distribution; and (c) and loading positions.

(a) (a)

(b) (b)

Figure Figure 19. 19. Experiment Experiment field: field: (a) (a) beam beam specimen, specimen, loader loader and and data data logger; logger; and and (b) (b) long-gauge long-gauge FBG FBG Figure 19. Experiment field: (a) beam specimen, loader and data logger; and (b) long-gauge sensors. sensors. FBG sensors. 18 18 17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 99 88 77 66 55 44 33 22 11 00 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18

In Figure 18c, static load was applied to cause some structural damage, while the dynamic force was input by using the hammer to knock at the beam’s top surface. Besides the four points in the figure, the hammer was also used to knock at some random positions at one measure time to simulate (a) (a) the random dynamic cases. 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 00 11 22 33 44 55 66 77 88 99 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 The experiments were implemented in the following five cases: (1) Case I, dynamic test for the intact beam; (2) Case D1, static loading up to concrete cracking and unloading, then dynamic test; (3) Case D2, static loading to make concrete crack up to 1/4 beam depth and unloading, then dynamic (b) (b)

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test; (4) Case D3, static loading to make concrete crack up to 1/2 beam depth and unloading, then dynamic test; and (5) Case D4, static loading to make the monitored macro-strain up to 1000 µε and unloading, then dynamic test. The actual concrete crack distribution is shown in Figure 20, in which (a) visual inspection and measured by rules from (b) one side of the beam. Due the cracks were identified by to the difficulty to induce the initial damage, the damage is much larger the planFBG for Case D1, Figure 19. Experiment field: (a) beam specimen, loader and data logger; and than (b) long-gauge while the sensors. other three cases are implemented as per the plan. 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

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(c) 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

(d) Figure 20. Concrete crack distribution at Case: (a) D1; (b) D2; (c) D3; and (d) D4.

Figure 20. Concrete crack distribution (d) at Case: (a) D1; (b) D2; (c) D3; and (d) D4.

5.2. Experiment Results Figure 20. Concrete crack distribution at Case: (a) D1; (b) D2; (c) D3; and (d) D4.

5.2. Experiment Results

5.2.1. Macro-Strain and Macro-Strain Based Frequency Spectrum 5.2. Experiment Results

5.2.1. Macro-Strain and Macro-Strain Based Frequency Spectrum

Figure 21 presents the typical dynamic macro-strain results under the single-point excitation and 5.2.1. Macro-Strain Macro-Strain Based Frequency Spectrum random To modal information, theresults measured dynamic macro-strainexcitation was first and Figure 21excitation. presentsand theextract typicalthe dynamic macro-strain under the single-point transformed frequency domain using the classic fastthe Fourier transform (FFT) method. In Figure 22, first Figure 21into presents the typical dynamic macro-strain results under the single-point excitation and random excitation. To extract the modal information, measured dynamic macro-strain was the typical results are obtained near the first bending mode frequency, 33.25 Hz, where the magnitude random excitation. To extract the modal information, the measured dynamic macro-strain was first transformed into frequency domain using the classic fast Fourier transform (FFT) method. In Figure 22, peak value isinto MMS for the measurement. Then the proposed method can bemethod. applied In to Figure extract22, the transformed using thebending classic fast Fourier transform (FFT) the typical resultsthe arefrequency obtaineddomain near the first mode frequency, 33.25 Hz, where the magnitude information of theare neutral axisnear position. the typical results obtained the first bending mode frequency, 33.25 Hz, where the magnitude

peak value is the MMS for the measurement. Then the proposed method can be applied to extract the peak value is the MMS for the measurement. Then the proposed method can be applied to extract the information of the neutral axis position. information of the neutral axis position. 20

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Figure 21. Typical macro-strain results under: (a) single-point excitation; Single-point excitationand (b) random excitation. Magnitude Magnitude

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Figure 22. Typical results of macro-strain based frequency spectrum. Figure 22. Typical results of macro-strain based frequency spectrum.

5.2.2. Verification of Plane Section Hypothesis It is easy to theoretically prove the plane section with the MMS as Equation (12). However, it is most important to implement the verification of the plane section hypothesis with the experimental MMS results. From the results of the element E3 in Figure 23, the MMS distributes linearly at all the cases with one rotation center where the MMS value is zero. Therefore, the plane section has been verified when the MMS is applied. Moreover, the neutral axis lies at the rotation center, as per the definition. From the results in Figure 23, it is obvious that the neutral axis is moving up step by step with loading. 5.2.3. Neutral Axis Position Coefficient As shown in Figure 24 the line was fit between the MMSs from the two sensors at up-surface and down-surface (e.g., long-gauge FBG sensors F31 and F33 for E3 as shown in Figure 18b) to obtain the neutral axis position coefficient (NAPC) ψ. The fitting line slope is just the NAPC, increasing obviously with the damage developing in the figures. The perfect linear correlation has also been found from all the results as all the fitting correlation coefficients are larger than 0.99. In other words, it has been proved that the new index NAPC presents excellent stability, which is of great importance for long-term SHM. In Figure 25a and Table 7, results of NAPC for all the five monitored elements are included, for all the cases. At Case I the NAPC of all the five elements are close to each other as there is no damage happening, while the NAPC increases obviously with the damage developing. Because the damages are different for each element, there are differences among the NAPC results. For damage assessment, the relative change of NAPC is calculated as shown in Figure 25b. In the early stage of development of the concrete crack, the index change is up to 10%, sensitive enough to identify the damage happening, while it is over 50% for the larger damage. However, the detailed investigation of the damage sensitivity will be a future work, especially to detect initial damages that occur at the real structures through their service lives. The resonant frequency is often taken as a useful index to identify structural damage. To explain the sensitivity of the proposed method, the resonant frequency of the first bending mode has also been extracted, namely, 33.25 Hz, 32.38 Hz, 30.42 Hz, 29.33 Hz and 28.87 Hz for the five cases. The relative change is 2.6%, 8.5%, 11.8% and 13.2% for Case D1, D2, D3 and D4. Considering the results in Figure 25b, the damage sensitivity of NAPC is much larger than the resonant frequency, which has also proved that the NAPC is a good local damage index.

most important to implement the verification of the plane section hypothesis with the experimental MMS results. From the results of the element E3 in Figure 23, the MMS distributes linearly at all the cases with one rotation center where the MMS value is zero. Therefore, the plane section has been verified when the MMS is applied. Moreover, the neutral axis lies at the rotation center, as per the Sensors 2017, 17, 411 the results in Figure 23, it is obvious that the neutral axis is moving up step by 16 step of 20 definition. From with loading. F33

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1.026 1.126 1.048 1.103 1.121

1.347 1.318 1.233 1.511 1.216

1.439 1.403 1.321 1.635 1.300

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As shown in Figure 24 the line was fit between the MMSs from the two sensors at up-surface and down-surface (e.g., long-gauge FBG sensors F31 and F33 for E3 as shown in Figure 18b) to obtain the neutral axis position coefficient (NAPC) ψ. The fitting line slope is just the NAPC, increasing obviously with the damage developing in the figures. The perfect linear correlation has also been found from all the results as all the fitting correlation coefficients are larger than 0.99. In other words, Sensorsit2017, 17, 411 17 of 20 has been proved that the new index NAPC presents excellent stability, which is of great importance for long-term SHM.

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the relative change of NAPC is calculated as shown in Figure 25b. In the early stage of development 200 of the concrete crack, the index change is up to 10%, sensitive enough to identify the damage I 150 happening, while it is over 50% for the larger damage. However, the detailed investigation of the D1 D2 damage sensitivity will be a future work, especially to detect initial damages that occur at the real 100 D3 structures through their service lives. D4 50 The resonant frequency is often taken as a useful index to identify structural damage. To explain 0 the sensitivity of the proposed method, the 0 50 resonant 100 150frequency 200 250of the 300 first bending mode has also MMS of F 53 Hz and 28.87 Hz for the five cases. The been extracted, namely, 33.25 Hz, 32.38 Hz, 30.42 Hz, 29.33 relative change is 2.6%, 8.5%, 11.8% and 13.2% for(e) Case D1, D2, D3 and D4. Considering the results in Figure 25b, the damage sensitivity offrom NAPC is much larger the frequency, which has Figure 24. NAPC extracting dynamic tests for: (a) Ethan 1; (b) E 2; (c)resonant E3; (d) E4; and (e) E5. Figure 24. NAPC extracting from dynamic tests for: (a) E1 ; (b) E2 ; (c) E3 ; (d) E4 ; and (e) E5 . also proved that the NAPC is a good local damage index. In Figure 25a and Table 7, results of NAPC for all the five monitored elements are included, for all the cases. At Case I the NAPC of all the five elements are close to each other as there is no damage happening, while the NAPC increases obviously with the damage developing. Because the damages are different for each element, there are differences among the NAPC results. For damage assessment,

(a)

(b)

Figure 25. Results of NAPC: (a) absolute value; and (b) relative change. Figure 25. Results of NAPC: (a) absolute value; and (b) relative change. Table 7. The results of NAPC.

Damage Case E1 E2 E3 E4

I 1.007 1.023 1.020 1.068

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D2 1.347 1.318 1.233 1.511

D3 1.439 1.403 1.321 1.635

D4 1.454 1.410 1.371 1.640

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5.2.4. Neutral Axis Depth The neutral axis depth can be calculated by taking the NAPC results (Table 7) and the vertical distance between the two sensors (Table 8) into Equation (10), as shown in Figure 26a and Table 9. The results show the trend that the neutral axis moves towards the compressive zone with as the concrete cracks develop within the tensile zone. However, the relative change of the neutral axis is about 50% smaller than that of the NAPC (Figure 25b) as shown in Figure 26b. Sensors 2017, 17, 411

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Table 8. The vertical distance between the two sensors (mm).

load Element is unloaded. MeanwhileEthe dynamic Eload is not large the crack Number E3 enough to make E4 E5open. Therefore, 1 2 the actual neutral axis location is different at the dynamic and static loading cases. 223

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Figure 26. Results of neutral depth: (a) absolute value; (b) relative change. Figure 26. Results of neutral axis axis depth: (a) absolute value; and and (b) relative change. Table 9. The results of neutral depth the proposed method (mm). Table 9. The results of neutral axis axis depth withwith the proposed method (mm).

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The neutral axis depth has also been obtained from the static tests (Table 10) for further verification Table 10. The results of neutral axis depth with the static method (mm). of accuracy of the proposed method. Table 11 shows results of the comparison between the neutral axis depth results obtained fromCase the dynamicI and the static Damage D1 tests. From D2 the results, D3 the smallest D4 difference, not over 2.5%, is present at Case I, indicating the excellent accuracy of the proposed dynamic method. E1 111 150 155 156 166 However, the difference increases greatly after the cracks happening. Another trend is that the neutral E 2 110 145 149 149 161from the static axis depth obtained with the proposed dynamic method is about 20% smaller than that 3 152is unloaded. 153 Meanwhile 166 the dynamic tests. The likely reason isEthat the crack 112 closes after 149 the static load E 4 112 149 153 153 161 is different load is not large enough to make the crack open. Therefore, the actual neutral axis location 5 115 149 152 153 160 at the dynamic and staticEloading cases. Table 10. Table The results of comparison neutral axis depth 11. The resultswith withthe thestatic staticmethod method(mm). (%).

Damage CaseI Damage Case E1 E2 E3 E4 E5

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−24.8 −21.0 −24.5 −24.5 −21.1

155 −17.6 149 −17.7 152 153 −20.3 152 −16.0

156 −15.6 149 −15.6 153 −18.3 153 153 −13.3

−20.1

−18.2

166 −20.5 161 −21.3 166 −23.2 161 160 −17.4 −20.8

6. Conclusions and Remarks In this paper, a new method is proposed to determine neutral axis position and identify local damage for flexural structures using modal macro-strain. Based on the research, the following

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Table 11. The comparison results with the static method (%). Damage Case

I

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D4

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1.2 −1.1 −0.7 −2.2 1.0

−24.8 −21.0 −24.5 −24.5 −21.1

−17.6 −17.7 −20.3 −16.0 −20.1

−15.6 −15.6 −18.3 −13.3 −18.2

−20.5 −21.3 −23.2 −17.4 −20.8

6. Conclusions and Remarks In this paper, a new method is proposed to determine neutral axis position and identify local damage for flexural structures using modal macro-strain. Based on the research, the following conclusions can be drawn: (1) (2)

(3)

The linear relationships also exist as the traditional plane section assumption when the modal macro-strain (MMS) of the first bending mode is applied for long-gauge monitored element. The neutral axis position can be determined with high accuracy by using the MMS based method. For example, the neutral depth assessment error is not larger than 2.5% for the Case I of RC beam experiment and all the cases of the steel beam experiment. Meanwhile, the proposed method offers excellent stability because all the correlation coefficients of the fitting lines are larger than 0.99 for all the tests reported in this paper. The neutral axis position coefficient (NAPC) is potentially a good indicator for local damage identification, such as concrete crack, as it presents an excellent damage sensitivity and assessing robustness even under 5% noise. The two sensors used to measure dynamic macro-strain need to be installed at each side of the neutral axis to obtain the best sensitivity.

However, there is still some work to be done in the future to make the proposed method more useful and applicable. Firstly, different types of beam sections and actual bridge models should be investigated to further verify the proposed method as the studied models in this paper are too simple. Secondly, field tests should be implemented to verify the proposed method under live traffic loads. Thirdly, except concrete crack some other damages, such as steel corrosion, should be considered to verify the effectiveness of the proposed method. In a word, there is a lot of research to be implemented to apply the proposed new method. Therefore, the work in this paper is just the beginning. Due to the excellent properties, a positive future can be expected, especially for the application of the proposed method in concrete infrastructures. Acknowledgments: This paper is financially supported by National Natural Science Foundation of China (Grant No. 51508364), Natural Science Foundation of Jiangsu Province (Grant No. BK20150333), National Natural Science Foundation of China (Grant No. 51508363) and University Natural Science Foundation of Jiangsu Province (Grant No. 14KJB580009). Author Contributions: The author, Yongsheng Tang, organized the entire experimental program and drafted the manuscript. The co-author, Zhongdao Ren, checked and re-calculated the part needing structural mechanic. Conflicts of Interest: The authors declare no conflict of interest.

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