Strain flexibility identification of bridges from long-gauge strain measurements

Mechanical Systems and Signal Processing 62-63 (2015) 272–283

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Strain flexibility identification of bridges from long-gauge strain measurements Jian Zhang a,b,n, Qi Xia b, YuYao Cheng b, ZhiShen Wu a,b a b

Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Nanjing 210096, China International Institute for Urban Systems Engineering, Southeast University, Nanjing 210096, China

a r t i c l e in f o

abstract

Article history: Received 9 August 2014 Received in revised form 13 February 2015 Accepted 18 February 2015 Available online 24 March 2015

Strain flexibility, defined as the strain response of a structure's element to a unit input force, is import for structural safety evaluation, but its identification is seldom investigated. A novel long-gauge fiber optic sensor has been developed to measure the averaged strain within a long gauge length. Its advantage of measuring both local and global information of the structure offers an excellent opportunity of developing the strain flexibility identification theory. In this article, the method to identify structural strain flexibility from long-gauge dynamic strain measurements is proposed. It includes the following main steps: (a) macro strain frequency response function (FRF) estimation from macro strain measurements and its feature characterization; (b) general strain modal parameter identification; (c) scaling factor calculation, and (d) strain flexibility identification. Numerical and experimental examples successfully verify the effectiveness of the proposed method. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Long-gauge fiber optic sensor Distributed sensing Impact testing Strain flexibility Structural identification

1. Introduction Structural health monitoring (SHM) aims at providing in-time information concerning structural safety condition by installing sensors and processing measured data [1,2]. Operational modal analysis (OMA), a technology to process the accelerations from ambient vibration tests, has been developed over 50 years [3–7]. Modal identification results (frequencies, damping ratios and mode shapes) from the OMA technology have direct relations with structural intrinsic parameters (mass, stiffness, damping), but they are too global to detect structural local damages. More detailed structural parameters, for instance, structural flexibility/stiffness, are much more useful for structural safety evaluation. Even there were investigations to identify structural flexibility from ambient vibration data [8,9], the more reliable way is to perform multiple-reference impact tests for flexibility identification [10,11]. Both accelerations and impacting forces are simultaneously measured during the impact test, thus the magnitudes of estimated frequency response functions (FRFs) are comparable to the analytical ones calculated from structural intrinsic parameters. In contrast, the FRF magnitudes estimated from ambient vibration test data are different with the analytical values [12]. This unique feature of the impact test method guarantees the flexibility characteristic be accurately identified. Brown and Witter [13] reviewed the multiple-reference impact test methods and the related flexibility identification theory. Aktan et al. [14] identified the flexibility matrices of a few short/middle span bridges by performing impact tests. Zhang et al. [11] successfully developed a flexibility identification

n

Correspondence to: Key Laboratory of C&PC Structures of the Ministry of Education, Southeast University, Sipailou 2, Nanjing 210096, China E-mail address: [email protected] (J. Zhang).

http://dx.doi.org/10.1016/j.ymssp.2015.02.021 0888-3270/& 2015 Elsevier Ltd. All rights reserved.

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method corresponding to mobile impact testing. Their results illustrated that the predicted static deflections from the identified flexibility agreed well with the measured deflections from static tests. The identified flexibility is important for bridge owners/engineers to understand stiffness distribution of the structure. It can also be used to predict structural deflection under any static load. Flexibility-based indexes have also been proved to be effective for structural damage detection [15,16]. The value of identifying flexibility from acceleration measurements has been realized by researchers and engineers, however, identifying the strain flexibility from strain measurements was rarely investigated. Strain gauge plays an important role in structural monitoring by providing direct-viewing strain responses [17], and strain measurements have been realized to be sensitive to local damages [18–20]. However, traditional strain gauges are point-type sensors which are only able to reveal structural local information. This limitation hinders the development of strain modal theory for global modal parameter identification. The long-gauge fiber optic strain sensor developed recently [3,4,21,22] provides an excellent opportunity of developing the macro strain modal identification theory. It measures the averaged strain within a designed long-gauge length (e.g. 1–2 m), thus it has the feature of measuring both local and global information of the structure. The long-gauge sensor can also be connected in series for distributed sensing. Taking the advantage of the long-gauge FBG sensor as an opportunity, the theory of strain flexibility identification will be investigated by processing dynamic macro strain measurements. The proposed strain flexibility identification method will be an original contribution to the strain modal identification theory. It will also have clear engineering application potential for instance damage detection and safety evaluation of civil infrastructures. In this article, the concept of the long-gauge fiber optic sensor is first presented. Then, the procedure of identifying strain flexibility from dynamic macro strain measurements is theoretically derived. This section includes strain FRF estimation and its feature characterization, strain modal parameter identification, scaling factor calculation, and strain flexibility identification. Numerical and experimental examples are investigated to verify the effectiveness of the proposed method for strain flexibility identification even in the mass unknown condition. Finally, conclusions are drawn. 2. Concept of the long-gauge FBG sensor Traditional point-type strain gauges are limited to local measurements, thus they are not suitable for strain modal analysis for which aims at identifying global modal parameters. A kind of long-gauge FBG sensor as shown in Fig. 1(a) has been developed [22,23] to overcome that limitation. By designing the FBG sensor with a long gauge (e.g., 1–2 m) and fixing its two ends (Fig. 1(a)), the in-tube fiber has the same mechanical behavior of the structure, and hence the strain transferred from the shift of Bragg center wavelength represents the averaged strain over the long gauge length. An improved packaging design has also been developed to enhance the measuring sensitivity by utilizing composite materials to package the optic fiber and to impose deformation within the gage length largely on the essential sensing part of the FBG (Fig. 1(b)). The designed package also has the function to protect the sensor from high temperature, corrosion and humidity in a harsh environment. Due to its long gauge length, the developed sensor has the merit to measure the averaged strain in a large area of structural critical elements. It is much more suitable for strain modal analysis than the traditional point-type strain measurement from traditional strain gauges. The longer the sensor gauge length, the impact measurement within the gauge length will be more averaged. Moreover, the long-gauge sensors can be connected in series to make an FBG sensor array (Fig. 1(c)) for area sensing. The above features offer the developed sensor the advantage of measuring both local and global information of the structure. Therefore, it provides an excellent opportunity for developing the strain modal identification theory, for instance, identifying strain flexibility investigated in this article.

Fig. 1. Packaged long-gauge FBG sensor: (a) schematic picture, (b) actual sensor, and (c) sensor arrays.

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3. Strain flexibility identification method 3.1. Macro strain FRFs Multiple-reference impact test data including impacting forces and long-gauge strain measurements are first studied to estimate macro strain FRFs, which will be the basis for subsequent strain flexibility identification. The basic idea of deriving the macro strain FRFs is using the mapping relationship between macro strains and displacements. For a typical Euler beam element m with two nodes o and p, each node has two degree of freedoms (vertical displacement and rotation). When a vertical force excites the node q of another element, the long-gauge strain within the element m is

ε^ m ðωÞ ¼ μm ðθo ðωÞ  θp ðωÞÞ

ð1Þ

where μm ¼ hm =Lm , hm is the distance between the sensor mounted at the element bottom and the beam neural axis, Lm is the element length, θo and θp are the rotations of nodes o and p, respectively. Frequency response is a quantitative measure of magnitude and phase of the output as a function of frequency, in comparison to the input. Based on the definition, the macro strain FRF is derived by ^ mq ðωÞ ¼ ε^ m ðωÞ ¼ μ ðθo ðωÞ  θp ðωÞÞ ¼ μ ðH oq ðωÞ H pq ðωÞÞ H m m f q ðωÞ f q ð ωÞ

ð2Þ

^ mq is the macro strain FRF of the element m under the vertical excitation at the node q and H oq is the rotational where H displacement FRF of the node o under the vertical excitation at the node q. The symbol b will be used to denote the macro strain related variables throughout the article. It is known that the displacement FRF can be written in the following format [11]: H pq ðωÞ ¼

ϕpr ϕqr

N X

ð3Þ

M r ðω2r  ω2 þ 2jξr ωr ωÞ r¼1

where M r is structural mass in the mode r, ωr and ξr are structural frequency and damping ratio in the mode r, respectively, ϕ is the displacement mode shape, and N is the number of structural modes used. Substituting the above equation to Eq. (2), it is derived that ^ mq ðωÞ ¼ H

XN r¼1

XN μm ðϕor  ϕpr Þ ϕqr ϕ^ mr ϕqr ¼ 2 2 2 r ¼ 1 Mr ðωr  ω þ 2jξr ωr ωÞ M r ðωr  ω2 þ 2jξr ωr ωÞ

ð4Þ

^ is the macro strain mode shape. By comparing the rotational displacement FRF in Eq. (3) and the macro strain FRF where ϕ in Eq. (4), it is found that their magnitudes and phase angles have the following relation:     ϕpr   ^ ð5aÞ ðωÞj ¼ r H pq ðωÞj=r H   mq ϕ^ mr

r

ϑpq ¼ r ϑ^ mq

   2ξr ωr ω ¼ arctan ω2r  ω2

ð5bÞ

^ mq ðωÞ is similar. It is seen that their magnitudes have where r H pq ðωÞ denotes the component of H pq ðωÞ in the mode r, and r H a linear relationship but their phase angles are same. 3.2. Macro strain modal identification Displacement FRFs can be written as a numerator polynomial divided by a denominator polynomial as follows [5]:  Pi ¼ n N  X Bpi zik Rr Rnr H p ðωk Þ ¼ ¼ Pii¼¼1n þ ð6Þ n i j ω  γ j ω  γ k k r r r¼1 i ¼ 1 Ai z k

where H p ðωk Þ is displacement FRF at the frequency line ωk , k ¼ 1; …Nf ; p is the output number which is from 1 to No ; Rr is residual of the rth mode, and Rnr is its conjugate; Ai and Bpi are unknown polynomial coefficients; n is the order number. For a discrete time domain model, zk ¼ e  jωk Δt (Z-domain) with Δt the sampling period and j the complex conjugate symbol. It is seen that Ai is not varying with the output node number, p. Substituting Eq. (6) into Eq. (2), the macro strain FRF is rewritten as a numerator polynomial divided by a denominator polynomial Pi ¼ n Pi ¼ n i i i ¼ 1 ðBoi  Bpi Þzk i ¼ 1 Bmi zk ^ m ðωk Þ ¼ μ ðHo ðωk Þ  H p ðωk ÞÞ ¼ μ H ¼ ð7Þ Pi ¼ n Pi ¼ n m m i i i ¼ 1 Ai zk i ¼ 1 Ai zk where Bmi ¼ μm ðBoi  Bpi Þ. It is seen that Ai in Eqs. (6) and (7) are same.

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 2 ^ m ðωk Þ Pi ¼ n Ai zi  Pi ¼ n Bmi zi , unknown numerator and denominator By minimizing the least squares error, H i¼1 i¼1 k k coefficients, Bmi and Ai , in Eq. (7) are obtained by solving Jθ ¼ 0

ð8Þ

where 2

β1

3

6 ⋮ 7 6 7 7 θ¼6 6 β 7; 4 No 5

α

2

Z1

6 6 0 6 J¼6 6 ⋮ 4 0

2

1 z1 6 1 z 6 2 7 6 7 6 βp ¼ 6 4 ⋮ 5; α ¼ 4 ⋮ 5; Z p ¼ 6 ⋮ ⋮ 6 4 Bmn An 1 zNf 2

Bm0

0

⋯

0

Z2

⋯

0

⋮

⋱

⋮

0

⋯

Z No

3

2

A0

3

⋯ ⋯ … ⋯

zn1

3

7 7 7 ; ⋮ 7 7 5 znNf zn2

2 3 ^ m ðω1 Þ ^ m ðω1 Þ z1 H H ^1 ℵ 6 _ 7 ^ ^ 6 H H ð ω Þ z m 2 2 m ðω2 Þ ℵ2 7 6 7 ^ 7 ; ℵp ¼ 6 6 ⋮ ⋮ ⋮ 7 6    5 4 ^  ^N ^ ℵ H z ω H m Nf N f m ωN f o

⋯ ⋯ ⋱ ⋯

^ m ðω1 Þ zn1 H n^ z H m ðω2 Þ 2

⋮  ^ m ωN znNf H f

3 7 7 7 7 7 7 5

The model order n can be determined by using the stabilization diagrams method [5]. After denominator polynomial coefficients, α, are solved from Eq. (8), structural modal parameters are extracted from polynomial roots. The problem to solve polynomial roots can be transformed to the following problem: ! 0 I ð9Þ  α 0 ⋯  α n  1 V ¼ ΛV where V is eigenvector, and Λ is eigenvalue. The diagonal coefficients of Λ are denominator polynomial roots, Λr ¼ zr ¼ e  jγ r Δt , qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 where γ r ; γ nr ¼  ξr ωr 7j 1  ξr ωr , r is the number of structure mode. From the calculated eigenvalues, structural frequencies and damping ratios are identified as, ωr ¼ imagðððlogðΛr ÞÞ= ΔtÞÞ, and ξr ¼ realððlogðΛr ÞÞ= ΔtÞ=imagðððlogðΛr ÞÞ= ΔtÞ, respectively. Modal participation factor, Lr , is identified from eigen vector, V: The length of Lr is the number of input Ni , not the number of output No . It should be noted that the extracted modal participation factor Lr from Eq. (9) are displacement modal participation factor in the vertical direction, because Ai in Eqs. (6) and (7) are same and the excited force is in the vertical direction. Frequencies, damping ratios, and displacement model participation factors in the vertical direction have been estimated by far. The methods to further estimate strain mode shapes and strain flexibility are developed below. The displacement FRF for each output can be written as follows:  1 H p ðωk Þ ¼ Lr ϕ ð10Þ 2 1 3 jωk  γ r pr 1 jω1  γ r jω1  γ nr 7

 6 ⋮ ⋮ 7. Similarly, for the macro strain FRF, it is where p is the output node number from 1 to No , 1=ðjωk  γ r Þ ¼ 6 4 5 1 1 derived that jωNs  γ r jωN s  γ nr   1 1 ^ m ðωk Þ ¼ L H μm ðϕo  ϕp Þ ¼ L ϕ^ ð11Þ jωk  γ r jωk  γ r m ^ is the scaled strain mode shape; L is the vertical displacement modal participation factor. The strain mode shape where ϕ m ^ are estimated by writing Eq. (11) for all outputs nodes (p¼1 to N ) in all modes (r ¼1 to N). matrix ϕ o 3.3. Strain flexibility identification Structural flexibility is identified from acceleration-based data in the following way [11]: ! T T XN Lϕ ðLϕ Þn f ¼ H ðωk ¼ 0Þ ¼ þ r¼1  γr  γ nr where Lr ϕr is the residue of the rth mode. In a similar way, structural strain flexibility is derived as 0     1 0 T 1 T n T n T T Nr Nr ^ ^ T Þn L ϕ L ϕ X X L ϕ o p L ϕ Lϕ ðLðϕ p B o C m m A ^f ¼ μ @ þ   þ @ A¼ m m  γr  γ nr  γr  γ nr  γr  γ nr r¼1 r¼1

ð12Þ

T

ð13Þ

It is seen that strain flexibility are calculated by using the identified vertical displacement modal participation factor, L, ^ and other identified modal parameters. It is seen that structural mass is not used in the whole strain mode shape ϕ m derivation procedure, which means that the proposed method works even when structural mass is unknown.

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4. Experimental example of a cantilever beam Multiple-reference impact testing of the cantilever beam as shown in Fig. 2 has been performed to verify the effectiveness of the proposed method for strain flexibility identification. The steel beam has a length of 1.6 m, and a 0.08 m  0.03 m hollow tube cross section with the thickness of 0.0025 m. One end of the beam was oriented on a steel pedestal and another end is free. The beam was divided to 8 elements which were continuously labeled beginning from the fixed end. Eight long-gauge fiber optic sensors were placed on each beam element, and the SM130 Optical Sensing Interrogator was used for strain measurements. The PCB model 086D20 short-sledge impulse hammer was used to hit the right nodes of elements 5 and 7 respectively to excite the structure during the impact test, corresponding structural macro strain responses under each excitation were recorded. The NI PXIe-1082 data acquisition system was used for impacting force measurement. The sampling of the measurement data was set to be 0.001 s. Other than the impact test, a static test was also performed by placing the steel block with a mass of 45.8 kg on the beam tip to measure corresponding static strains of all elements. The typical impacting force and macro strain responses are plotted in Fig. 3(a) and (b) respectively for illustration. Macro strain FRFs were estimated from test data, then they were rewritten into Eq. (8) to solve the unknown coefficient matrix θ consisting of numerator and denominator coefficients, Bmi and Ai . The mode order n was determined by plotting the stabilization diagrams as shown in Fig. 4, in which the cycles denote stable poles while the dots are unstable solutions. Two curves in Fig. 4 are scaled strain FRFs to assist in identifying structural modes from spurious modes arising from observation noise. It is seen that the first two structural modes are clearly identified. Their frequencies were identified by Eq. (9), which are 12.1 Hz and 76.7 Hz. Similarly, damping ratios were identified to be 0.82% and 1.54%. It should be noted that vertical displacement modal participation factors corresponding to impacting locations were identified from Eq. (9) but strain mode shapes were still not identified so far. Eq. (11) was used to identify scaled strain mode shapes, and the identified results of the first two modes are shown in Fig. 5. Once strain mode shapes were identified from Eq. (11), the strain flexibility was calculated by Eq. (13). Fig. 6 plots the identified strain flexibility matrix which has the dimension of 8 by 8 because the cantilever beam has 8 elements as described above. The identified flexibility is useful for structural performance evaluation for instance it can be used to predict structural strain response under any static load. Fig. 7 plots the predicted static strain when a mass of 45.8 kg is placed on the beam tip. Static strains measured from the corresponding static test are also plotted for comparison. It should be noted that the different numbers of modes were used in Eq. (13) to study the effects of mode truncation. The curve

Fiber optic sensor

NI PXIe-1082 SM130 Interrogator Accelerometer

0.8

150

0.6

100 Micro Strain

Force(KN)

Fig. 2. Experiment layout of the cantilever beam test.

0.4 0.2

0 -50

0 -0.2

50

0

10

20 Time(sec)

30

-100

0

10 20 Time(sec)

Fig. 3. Typical impacting force and macro strain response.

30

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20

Model Order

15

10

5

0

0

150

50 100 Frequency (Hz) Fig. 4. Stabilization diagrams.

1

0.5

0

-0.5 1st Mode 2nd Mode -1

0

2

4 Element Number

6

8

Fig. 5. Identified strain mode shapes.

-6

x 10 2 1 0 -1 10

Ele m

10 5

Nu mb er

5 0

0

N Elem

u mb

er

Fig. 6. Identified strain flexibility.

labeled by “1 mode” means only the first mode was used for strain flexibility identification by Eq. (13), while the curve labeled by “2 modes” means the first 2 modes were used. It is seen that the predicted strains from the stain flexibility using two modes agree well with the measurements, which illustrates the accuracy of the identified strain flexibility.

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-4

x 10

6

Measurement 1 Mode 2 Modes

Strai n

4

2

0

0

2

4 Elem Num ber

6

8

Fig. 7. Static strain prediction results.

2 3

5

5

8 8

7 3

18

14 11

Fig. 8. Multiple span beam bridge model and the impact test scheme.

5. Multiple-span beam bridge example A three-span simple-supported beam bridge constructed of steel and reinforced concrete is investigated to test the proposed method for complex structures. The main steel beam has an I-shaped cross section with a 0.5 m height, 0.3 m flange width, 0.08 m flange thickness, and 0.06 m web thickness. Reinforced concrete bridge deck with a thickness of 0.25 m is supported on the main beams to handle road traffic. The bridge consists of three spans with the lengths of 4.5 m, 6.4 m, and 3.6 m. Long-gauge FBG sensors were installed on all beam elements of three spans. For instance, a total of 18 FBG sensors with a gauge length of 0.5 m were instrumented on each beam element's bottom of two main beams of the first span. Similarly, 24 and 14 long-gauge sensors were installed on main beams of the second and third spans respectively, which induced the total number of sensors is 56. Impacting locations in the multiple-reference impact test is shown in Fig. 8. For instance, the impacting forces were applied to the nodes 3, 7, 11, and 14 on the first span. The three-span simple-supported beam bridge was modeled in the SAP2000 software, and its multiple-reference impact test data were simulated through the finite element dynamic analyses. 10% white noise was added to both the macro strain responses and the impacting forces as the observation noise, in which 10% means the standard deviation of the noise is 10% of that of the simulated data [24]. Multiple-reference impact test data including dynamic macro strain responses and impacting forces were first processed to estimate the macro strain FRFs of three spans as shown in Fig. 9. Then, they were used in Eq. (8) to solve the unknown polynomial coefficients matrix through the least squares method. Basic structural modal parameters including frequency, damping, and vertical displacement modal participation factor were extracted from Eq. (9) by using the identified polynomial coefficients. Stabilization diagrams of three spans as shown in Fig. 10 were used to select stable poles for structural modal identification. After vertical displacement modal participation factors were identified from Eq. (9), they were used together with the estimated strain FRFs to calculated strain modal shapes through Eq. (11). To verify the accuracy of the calculated strain mode shapes, they were used to synthesize the strain FRFs and compared to the estimated strain FRF values as plotted in Fig. 11. Agreements between them illustrate the accuracy of the identification results. It should be noted that the structural identification process was performed in two frequency bands in order to improve the identification accuracy [12], thus the synthesized strain FRF curve consists of two parts as shown in Fig. 11. The following modal parameters have been identified so far: structural frequencies, damping ratios, vertical displacement modal participation factors, and strain mode shapes. From those parameters, the structural strain flexibility matrices of three spans were identified from Eq. (13) as plotted in Fig. 12. Strain flexibility f^ mj denotes structural strain response of the m elements under a unit force at the middle point of the j element. Static strain response of the structure under static loads

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-8

10

-10

10

-12

10

-14

10

0

50

100

150

200

250

300

0

50

100

150

200

250

300

0

50

100

150

200

250

300

-8

10

-10

10

-12

10

-14

10

-8

10

-10

10

-12

10

-14

10

Fig. 9. Estimated macro strain FRFs of three spans.

can be predicted from the identified strain flexibility. For instance, Fig. 13(a)–(c) illustrates predicted static strain responses of three spans when uniform forces of 1.0  106 kN are loaded on all nodes of the structure. For comparison, the exact static response values from finite element static analyses are also plotted in Fig. 13. Agreements between the predicted and exact strain values successfully verified the accuracy of the identified strain flexibility. Effect of modal truncation is also studied in Fig. 13. For instance, the “2 modes” denotes that the first two modes were used to identify strain flexibility in Eq. (13). It is seen that only the first mode is not sufficient to obtain accurate identification results. Fig. 14 illustrates the predicted static strain values of the second span when the uniform forces of 1.0 kN on the nodes of only one side main beam. It is seen that another main beam still have strain responses even though there were no loads on itself. This is because the static loads on one main beam transfer to another main beam through the concrete deck.

6. Conclusion Limitations of traditional point-type strain gauges for local strain measurements hinder the development of the strain modal identification theory. Utilization of a kind of long-gauge FBG sensor for distributed sensing and a strain flexibility identification method processing dynamic macro strain measurements have been proposed. Based on the research so far, the following conclusions are drawn: (1) The long-gauge strain sensor has the advantage of measuring not only local but also global information of the structure, thus its output is much more suitable for strain modal analyses than traditional point-type strain measurements. (2) A method to process the dynamic macro strain measurement from the multiple-reference impact testing has been

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8

6

4

2

0

0

50

100

150

200

250

8

6

4

2

0

0

40

80

120

160 180

8

6

4

2

0

0

60

120

180

240

300

Fig. 10. Stabilization diagrams of three spans.

proposed. It not only identified strain mode shapes, but also identified structural strain flexibility even when structural mass is unknown. (3) Numerical and experimental examples have been investigated. The results of them successfully verified the effectiveness of the proposed method for strain flexibility identification. The effects of mode truncation on strain flexibility identification results have also been studied. (4) An important issue when applying the proposed method to real structures is whether the bridge can be sufficiently excited by human-made impacting forces. Due to this concern, novel impacting devices producing large-amplitude and wide-frequency-band impacting forces have been developed by the authors and others [7,11]. Application of them to real structures especially to short/middle span bridges will be studied in the future work.

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x 10

5

-10

Identified Synthesized

4 3

2 1

0

0

1

x 10

50

100

150

200

250

-9

Identified Synthesized

0.8 0.6

0.4 0.2

0

4

0

x 10

50

100

150

200

-10

Identified Synthesized

3

2

1

0

0

50

100

150

200

250

300

Fig. 11. Comparisons of identified and synthesized strain FRFs of three spans.

281

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-10

x 10

-11

x 10

-11

x 10

2

15 10 5

15

1

10

0

5

0

0

-1 30

-5 20

20

20

10

W

10 0

Len

gth

idt

30

h

15

20

10 0

10

Leng

0

W

th

idt

10

h

Fig. 12. Identified strain flexibility of three spans.

1 0.8

Strain

0

0.6 0.4 Test

0.2

1 Mode 5 Modes

0 0

6

12 Element Number

18

1.6

Strain

dth

1

0.4 Test 1 Mode 4 Modes

-0.2 0

5

10 15 Element Number

20

25

0.7 Test 2 Modes 5 Modes

0.6

Strain

Wi

-5 20

0.4

0.2

0 0

2

4

6

8

10

12

Element Number Fig. 13. Predicted static strains under uniform loads of three spans.

14

10 5 0

0

Len

gth

J. Zhang et al. / Mechanical Systems and Signal Processing 62-63 (2015) 272–283

x 10

-6

Test 1 Modes 4 Modes

1.2

Strain

283

0.6

0

0

5

10 15 Element Number

20

25

Fig. 14. Predicted static strains of the second span under one-side loads.

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