Research ArticleAssessing Static Performance of the DashengguanYangtze Bridge by Monitoring the Correlation betweenTemperature Field and Its Static Strains

Gao-Xin Wang12 You-Liang Ding12 Peng Sun3 Lai-Li Wu4 and Qing Yue4

1The Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education Southeast UniversitySipailou Road Xuanwu District Nanjing Jiangsu 210096 China2Department of Civil Engineering Southeast University Sipailou Road Xuanwu District Nanjing Jiangsu 210096 China3Department of Civil and Environmental Engineering Rice University Houston TX 77005 USA4China Railway Major Bridge (Nanjing) Bridge and Tunnel Inspect amp Retrofit Co Ltd Nanjing 210032 China

Correspondence should be addressed to You-Liang Ding civilchinahotmailcom

Received 30 August 2014 Accepted 9 October 2014

Academic Editor Ting-Hua Yi

Copyright copy 2015 Gao-Xin Wang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Taking advantage of the structural healthmonitoring system installed on the steel truss arch girder of Dashengguan Yangtze Bridgethe temperature field data and static strain data are collected and analyzed for the static performance assessment of the bridgeThrough analysis it is found that the static strain changes are mainly caused by temperature field (temperature and temperaturedifference) and train After the train-induced static strains are removed the correlation between the remaining static strains andthe temperature field shows apparent linear characteristics which can be mathematically modeled for the description of staticperformance Therefore multivariate linear regression function combined with principal component analysis is introduced tomathematically model the correlation Furthermore the residual static strains of mathematical model are adopted as assessmentindicator and three kinds of degradation regulations of static performance are obtained after simulation of the residual static strainsFinally it is concluded that the static performance of Dashengguan Yangtze Bridge was in a good condition during that period

1 Introduction

In recent years with development of the structural healthmonitoring technology it is feasible to install sensors formonitoring the performance of bridge structures [1ndash4]Static strain effect caused by combined load actions suchas temperature field and traffic loading can present staticperformance of bridge structures [5ndash7] Relevant researchshowed that daily or seasonal temperature field caused criticalstrain levels even higher than traffic-induced strains [8]so abnormal variation of correlation between temperaturefield and its static strain effect can indicate static per-formance degradation of bridge structures For exampleDuan et al developed linear regression models for structuralperformance evaluation using temperature-strain correlation[8] Li et al obtained probability models for performance

assessment application on cable stayed bridges through sta-tistical analysis of temperature-strain linear relationships [9]However current research efforts in this field are still insuf-ficient in many specific details (1) current research interestsmerely focus on correlation between temperature and staticstrain and have not considered the influence of temperaturedifference on static strain (2) current research methodsmerely concentrate on single variable analysis whereas staticstrain from one monitoring point is actually influenced byboth temperature and temperature difference from manydifferent monitoring points (3) current research progress hasnot definitely pointed out degradation regulations of staticperformancewhen correlation between temperature field andits static strain effect abnormally changes

Therefore using health monitoring system installed onsteel truss arch girder of the Dashengguan Yangtze Bridge

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 946907 12 pageshttpdxdoiorg1011552015946907

2 Mathematical Problems in Engineering

Beijing Shanghai

1

108 108192 192336

1272

1

1

336

2 3 4 5 6 7

Figure 1 Longitudinal structural type of the Dashengguan Yangtze Bridge (unit m)

temperature field data and static strain data from top chordmember arch rib chordmember bottom chordmember anddiagonal web member are continuously collected and thenvariation regularities of their time history curves are analyzedby comparison Wavelet packet decomposition method isintroduced to extract static strains caused by temperatureand temperature difference data and then principal com-ponent analysis is carried out to simplify the multivariatelinear regression model Finally three kinds of degradationregulations of residual static strains are put forward to assessstatic performance of the Dashengguan Yangtze Bridge

2 Bridge Health Monitoring andData Collection

21 Bridge Health Monitoring The bridge monitoring objectfor this research is the famous Dashengguan Yangtze Bridgewhich is designed as the river-crossing channel for theBeijing-Shanghai high speed railway and Nanjing bidirec-tional subway The structural type of its main girder isthe continuous steel truss arch girder with the main spanreaching 336m as shown in Figure 1 According to its 1-1profile as shown in Figure 2(a) the continuous steel truss archgirder is composed of steel truss arch and steel bridge deckMoreover the steel truss arch comprises chordmembers withbox-shaped cross-sections (top chord arch rib chord andbottom chord resp) diagonal web members with I-shapedcross-sections vertical web members and horizontal andvertical bracings as shown in Figures 2(a) and 2(b) the steelbridge deck comprises top plate and transverse stiffeninggirder as shown in Figure 2(a)

In order to get the variation regularity of temperaturefield in steel truss arch girder eight FBG temperature sensorsare installed on the cross-sections of top chord memberdiagonal web member arch rib chord member and bottomchord member respectively as shown in Figures 2(c)sim2(f)(where119882

119894denotes the 119894th temperature sensor 119894 = 1 2 8)

to continuously measure temperature field data Besideseight FBG strain sensors are installed on the same positionswith temperature sensors as shown in Figures 2(c)sim2(f)respectively (where 119884

119894denotes the 119894th strain sensor 119894 =

1 2 8) to continuously obtain axial static strain dataSampling frequency of data collection is set to 1Hz which is

appropriate since the changes of temperature and strain arevery slow and little

22 Data of Temperature Field and Static Strains Data oftemperature field and static strains in certain months arespecially chosen (1) data from March 2013 to October 2013are chosen as training data for modeling the correlation (2)data in November 2013 are chosen as test data for testingthe modeling effectiveness (3) data from March 1st 2014 toMarch 19th 2014 are chosen as assessment data for evaluatingthe static performance Temperature data from temperaturesensor 119882

119894is denoted by 119879

119894 temperature difference data

acquired by 119879119894minus 119879

119895is denoted by 119879

119894119895 and static strain

data from strain sensor 119884119894is denoted by 119878

119894(119894 119895 = 1 2 8)

The history curves of global change and daily change intemperature data and temperature difference data taking 119879

1

and11987912from training data for example are shown in Figures

3 and 4 respectively The history curves of global changeand daily change in static strain data taking 119878

1and 119878

2from

training data for example are shown in Figures 5 and 6respectively (negative values denote compressive static strainand positive values denote tension static strain)

From Figures 3 to 6 some findings can be obtained (1)the history curves of global change and daily change in staticstrain data 119878

1are similar to that of temperature difference

data 11987912

shown in Figures 4 and 5 that is the global curvesof 1198781and 119879

12are both stationary and the daily curves of 119878

1

and 11987912

both change little at night and change obviously atdaytime in wave shape simultaneously indicating that staticstrain data 119878

1mainly vary with temperature difference (2)

the history curves of global change and daily change in staticstrain data 119878

2are similar to that of temperature data119879

1shown

in Figures 3 and 6 that is the global curves of 1198782and 119879

1

are both seasonable and the daily curves of 1198782and 119879

1both

change in the shape of one-cycle sine curve indicating thatstatic strain data 119878

2mainly vary with temperature (3) daily

history curves of 1198781and 1198782contain many sharp peaks caused

by trains as shown in Figures 5(b) and 6(b) with each peakcorresponding to the timewhen one train is across the bridgewhereas the proportion of its influence on static strain isobviously lower than that of temperature and temperaturedifference

Mathematical Problems in Engineering 3

Downstream Upstream15000

1200

068

007

150002Horizontal bracing of top chord member

Vertical bracing

Top plate

Vertical web memberof side truss

Vertical web memberof side truss

Vertical web memberof middle truss

Horizontal bracingof arch rib chord

member

Transverse stiffening girder

2

W7

Y7

W8

Y8

W5Y5

W4

Y4

W6

Y6

W3

W1

Y3

Y1

W2

Y2

(a) 1-1 profile of steel truss arch girder

Diagonal webmember

Shanghai BeijingTop chord member

Bottom chordmember

Vertical webmember

6

6

W7

W8

Y7

5

5

3

3

4

4

W6

W5

W4

W3

W1

W2

Y6

Y5

Y3

Y1Y2

Y4

Y8

Arch ribchord member

(b) 2-2 profile of steel truss arch

Downstream Upstream

1000

1400

850

1000

850

150

150

W1

Y1W2

Y2

(c) 3-3 profile of top chord member

Downstream Upstream

610

610

150

150

1398

W3

Y3W4

Y4

(d) 4-4 profile of diagonal web member

Downstream Upstream

800

650

650

150

150

1400W5

Y5 W6

Y6

(e) 5-5 profile of arch rib chord member

Downstream Upstream

1266

1266

1416

150

150

1400W7

Y7W8

Y8

(f) 6-6 profile of bottom chord member

Figure 2 Layout of temperature and strain sensors on the steel truss arch girder (unit mm)

4 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

5

15

25

35

45

Time (month)

minus5

Tem

pera

ture

(∘C)

(a) Global history curve

0 4 8 12 16 20 245

10

15

20

Time (hour)

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 3 History curves of global change and daily change in temperature data 1198791

3 4 5 6 7 8 9 10 11

5

15

Time (month)

minus25

minus15

minus5

Tem

pera

ture

diff

eren

ce (∘

C)

(a) Global history curve

0 4 8 12 16 20 24

0

5

Time (hour)

minus10

minus5

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 4 History curves of global change and daily change in temperature difference data 11987912

3 4 5 6 7 8 9 10 11

0

100

200

Time (month)

minus300

minus200

minus100

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus120

minus65

minus10

45

100

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 5 The history curves of global change and daily change in static strain data 1198781

Mathematical Problems in Engineering 5

3 4 5 6 7 8 9 10 11Time (month)

50

150

minus250

minus150

minus50

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus100

minus70

minus40

minus10

20

50

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 6 History curves of global change and daily change in static strain data 1198782

C00j

C01j

C02j

C12j C22

j C32j

C11j

C03j C13

j C23j

C33j

C43j C53

j C63j C73

j

Figure 7 The hierarchical tree of decomposition coefficients in three scales

3 Correlation Analysis

31 Extracting Static Strain Caused by Temperature FieldAccording to the analysis above static strain data mainlycontains three parts static strain I caused by temperaturestatic strain II caused by temperature difference and staticstrain III caused by train In order to research the correlationbetween temperature field and its static strain effect staticstrain I and static strain II (denoted by 119878III) must be extractedfrom the static strain data Considering that primary periodof 119878III is one day which is far longer than that of static strainIII wavelet packet decomposition method is introduced toextract 119878III

In detail based on a pair of low-pass and high-passconjugate quadrature filters ℎ(119895) and 119892(119895) of wavelet packetstatic strain data can be decomposed scale by scale into dif-ferent frequency bands and each decomposition coefficientcorresponding to its frequency band and its scale is calculatedas follows [10 11]

119862

1198962119897

119895= sum

119898isin119885

119862

119896+1119897

119898ℎ (119898 minus 2119895)

119862

1198962119897+1

119895= sum

119898isin119885

119862

119896+1119897

119898119892 (119898 minus 2119895)

(1)

119862

1198962119897

119895or 1198621198962119897+1119895

denotes the decomposition coefficient withinthe 119896th frequency band and the 2119897th or (2119897 + 1)th scalerespectively which is visually described by a hierarchical treeshown in Figure 7 (only three scales are listed) Each decom-position coefficient can be reconstructed into time-domainsignals with constraints of its own frequency band thereforedecomposition coefficients within primary frequency bandof 119878III can be specially selected and then reconstructedas 119878III

Taking the daily history curve of static strain data 1198781in

Figure 5(b) as an example it is decomposed within 8 scalesand the decomposition coefficient 11986208

119895is specially selected

and then reconstructed as 119878III shown in Figure 8(a) whicheffectively retains main component caused by temperaturefield and removes sharp peaks caused by trains as well Usingthe method above 119878III of each static strain data is extractedsuch as the extraction results of 119878

1from training data shown

in Figure 8(b)

32 Modeling Method In order to better present the cor-relation between 119878III and temperature field from trainingdata two steps are specifically carried out as follows (1)temperature and temperature difference data from suffi-cient solar radiation days whose daily variation trends

6 Mathematical Problems in Engineering

0 4 8 12 16 20 24

45

100

Time (hour)

minus120

minus65

minus10

Stat

ic st

rain

S III

(120583120576)

(a) Daily extraction result 119878III of 1198781

3 4 5 6 7 8 9 10 11

10

80

150

Time (month)

minus200

minus130

minus60

Stat

ic st

rain

S III

(120583120576)

(b) Global extraction result 119878III of 1198781

Figure 8 Extraction results 119878III of static strain data 1198781

are similar to one smooth single-period sine curve asshown in Figure 3(b) [12] are specially selected and then119878III are selected from the same corresponding days (2) dailymaximum and minimum values are furthermore chosenfrom the selected temperature field and 119878III data The scatterplots between 119878III of 119878

1and temperature difference 119879

12

between 119878III of 1198782 and temperature 1198791 are shown in Figures

9(a) and 9(b) respectively both presenting apparent linearcorrelation Moreover considering that 119878III of each staticstrain data may be influenced by many temperature andtemperature difference data 119878III are expressed as follows

119878III =8

sum

119894=1

120572

119894119879

119894+

16

sum

119895=1

120573

119895119863

119895+ 119888 (2)

where 119879119894is the 119894th temperature data from set 119879

1 119879

2 119879

3

119879

4 119879

5 119879

6 119879

7 119879

8 119863119895

is the 119895th temperature differencedata from set 119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24

119879

26 119879

28 119879

46 119879

48 119879

68 120572119894is the 119894th linear expansion coefficient

of 119879119894 120573119895is the 119895th linear expansion coefficient of 119879

119894119895 and 119888 is

the constant term

Considering that total 24 linear expansion coefficientsand one constant term are very complicated for calculationprincipal component analysis is introduced to simplify (2)Generally speaking principal components can be derivedfrom either an algebraic or a geometric viewpoint Given 8random variables 119879

1 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8 with a known

covariance matrix sum the 119896th (119896 = 1 2 8) principalcomponent119875

119896is a linear expression of the 8 random variables

defined as follows [13 14]

119875

119896=

8

sum

119895=1

119906

119895119896119879

119895 (3)

Starting from the maximization of variance of 119875119896subjecting

to 9978881199061015840

119896

997888

119906

119897= 0 (119897 = 1 2 8 119897 = 119896) the algebraic derivation

turns out that 997888119906119896= (119906

1119896 119906

2119896 119906

8119896)

1015840 is an eigenvector ofsum corresponding to the 119896th largest eigenvalue 120582

119896 referred

to as principal component coefficients and principal compo-nent variances respectively After eigendecomposition of thecovariance matrixsum 997888119906

1015840

119896is obtained as follows

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

997888

119906

1015840

1

997888

119906

1015840

2

997888

119906

1015840

3

997888

119906

1015840

4

997888

119906

1015840

5

997888

119906

1015840

6

997888

119906

1015840

7

997888

119906

1015840

8

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0319 0384 0364 0382 0358 0387 0333 0289

0462 minus0341 minus0144 minus0332 0110 minus0365 0333 0531

minus0486 0177 minus0459 minus0363 0110 0461 0238 0330

minus0211 minus0628 0262 minus0060 0641 0213 minus0167 minus0068

0491 0301 minus0335 minus0367 0420 0103 minus0229 0427

minus0138 minus0094 minus0209 0205 0174 minus0212 0739 minus0514

0267 minus0367 minus0580 0551 minus0125 0336 minus0142 0065

minus0269 0275 minus0274 0358 0463 minus0539 minus0271 0264

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(4)

Mathematical Problems in Engineering 7

45 10

0

100

200

minus200

minus100

minus12 minus65 minus1

Temperature difference T12 (∘C)

Stat

ic st

rain

S III

ofS 1

(120583120576)

(a) 119878III of 1198781 and temperature difference 11987912

0 8 16 24 32 40

40

90

140

minus110

minus60

minus10

Stat

ic st

rain

S III

ofS 2

(120583120576)

Temperature T1 (∘C)

(b) 119878III of 1198782 and temperature 1198791

Figure 9 Correlation scatter plots between 119878III and temperature and temperature difference

according to the Pareto diagram of explained variances of119875

119896shown in Figure 10(a) it can be seen that the explained

variance of 1198751is 967 very higher than the other principal

components so 1198751is specially selected with its calculation

equation as follows

119875

1=

997888

119906

1015840

1119879

(5)

where 119879 = [1198791 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8]

1015840 Moreover the prin-cipal components 119877

119898(119898 = 1 2 16) of 16 random

variables 11987912 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26

119879

28 119879

46 119879

48 119879

68 are obtained through eigendecomposition

of covariance matrix [13 14] with the Pareto diagram of their

explained variances shown in Figure 10(b) It can be seen thatthe sum of explained variances of 119877

1 1198772 and 119877

3is 988

so 1198771 1198772 and 119877

3are specially selected with their calculation

equations as follows

[

[

[

119877

1

119877

2

119877

3

]

]

]

=

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863 (6)

where 997888V 1015840119899= (V1119899 V2119899 V

16119899) (119899 = 1 2 3) is an eigenvector

of covariance matrix and their values are

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

= [

0403 0209 0234 minus0108 minus0248 0245 0289 0177 0308 0208 minus0188 minus0077 0288 0148 0341 0302

0369 0119 0239 0088 0385 0299 0311 0019 0019 0019 minus0019 0167 minus0347 0184 minus0232 minus0461

0386 0272 0310 minus0144 minus0055 minus0140 minus0167 minus0341 minus0275 0159 0076 minus0317 minus0078 minus0395 minus0302 0180

]

(7)

119863 = [119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26 119879

28

119879

46 119879

48 119879

68]

1015840Therefore the calculation of 119878III can be simpli-fied as follows

119878III = 12058211198751 +997888

120574

1times3

997888

119877

3times1+ 119888

(8)

where 1205821is performance parameter of 119875

1and 997888120574

1times3is one

vector containing three performance parameters correspond-ing to 997888119877

3times1(9978881205741times3

= [120574

1 120574

2 120574

3] and 997888119877

3times1= [119877

1 119877

2 119877

3]

1015840

specifically)The values of performance parameters 1205821 9978881205741times3

and 119888 can be estimated by multivariate linear regression

method [15] and then the correlation models between 119878IIIand temperature field can be expressed as follows

119878III = 1205821 (997888

119906

1015840

1119879) +

997888

120574

1times3(

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863)+ 119888 (9)

33 Modeling Results and Effectiveness Verification Based onthe modeling method above the correlation between 119878IIIof each static strain data and temperature field is modeledby (9) using training data with corresponding performance

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Beijing Shanghai

1

108 108192 192336

1272

1

1

336

2 3 4 5 6 7

Figure 1 Longitudinal structural type of the Dashengguan Yangtze Bridge (unit m)

temperature field data and static strain data from top chordmember arch rib chordmember bottom chordmember anddiagonal web member are continuously collected and thenvariation regularities of their time history curves are analyzedby comparison Wavelet packet decomposition method isintroduced to extract static strains caused by temperatureand temperature difference data and then principal com-ponent analysis is carried out to simplify the multivariatelinear regression model Finally three kinds of degradationregulations of residual static strains are put forward to assessstatic performance of the Dashengguan Yangtze Bridge

2 Bridge Health Monitoring andData Collection

21 Bridge Health Monitoring The bridge monitoring objectfor this research is the famous Dashengguan Yangtze Bridgewhich is designed as the river-crossing channel for theBeijing-Shanghai high speed railway and Nanjing bidirec-tional subway The structural type of its main girder isthe continuous steel truss arch girder with the main spanreaching 336m as shown in Figure 1 According to its 1-1profile as shown in Figure 2(a) the continuous steel truss archgirder is composed of steel truss arch and steel bridge deckMoreover the steel truss arch comprises chordmembers withbox-shaped cross-sections (top chord arch rib chord andbottom chord resp) diagonal web members with I-shapedcross-sections vertical web members and horizontal andvertical bracings as shown in Figures 2(a) and 2(b) the steelbridge deck comprises top plate and transverse stiffeninggirder as shown in Figure 2(a)

In order to get the variation regularity of temperaturefield in steel truss arch girder eight FBG temperature sensorsare installed on the cross-sections of top chord memberdiagonal web member arch rib chord member and bottomchord member respectively as shown in Figures 2(c)sim2(f)(where119882

119894denotes the 119894th temperature sensor 119894 = 1 2 8)

to continuously measure temperature field data Besideseight FBG strain sensors are installed on the same positionswith temperature sensors as shown in Figures 2(c)sim2(f)respectively (where 119884

119894denotes the 119894th strain sensor 119894 =

1 2 8) to continuously obtain axial static strain dataSampling frequency of data collection is set to 1Hz which is

appropriate since the changes of temperature and strain arevery slow and little

22 Data of Temperature Field and Static Strains Data oftemperature field and static strains in certain months arespecially chosen (1) data from March 2013 to October 2013are chosen as training data for modeling the correlation (2)data in November 2013 are chosen as test data for testingthe modeling effectiveness (3) data from March 1st 2014 toMarch 19th 2014 are chosen as assessment data for evaluatingthe static performance Temperature data from temperaturesensor 119882

119894is denoted by 119879

119894 temperature difference data

acquired by 119879119894minus 119879

119895is denoted by 119879

119894119895 and static strain

data from strain sensor 119884119894is denoted by 119878

119894(119894 119895 = 1 2 8)

The history curves of global change and daily change intemperature data and temperature difference data taking 119879

1

and11987912from training data for example are shown in Figures

3 and 4 respectively The history curves of global changeand daily change in static strain data taking 119878

1and 119878

2from

training data for example are shown in Figures 5 and 6respectively (negative values denote compressive static strainand positive values denote tension static strain)

From Figures 3 to 6 some findings can be obtained (1)the history curves of global change and daily change in staticstrain data 119878

1are similar to that of temperature difference

data 11987912

shown in Figures 4 and 5 that is the global curvesof 1198781and 119879

12are both stationary and the daily curves of 119878

1

and 11987912

both change little at night and change obviously atdaytime in wave shape simultaneously indicating that staticstrain data 119878

1mainly vary with temperature difference (2)

the history curves of global change and daily change in staticstrain data 119878

2are similar to that of temperature data119879

1shown

in Figures 3 and 6 that is the global curves of 1198782and 119879

1

are both seasonable and the daily curves of 1198782and 119879

1both

change in the shape of one-cycle sine curve indicating thatstatic strain data 119878

2mainly vary with temperature (3) daily

history curves of 1198781and 1198782contain many sharp peaks caused

by trains as shown in Figures 5(b) and 6(b) with each peakcorresponding to the timewhen one train is across the bridgewhereas the proportion of its influence on static strain isobviously lower than that of temperature and temperaturedifference

Mathematical Problems in Engineering 3

Downstream Upstream15000

1200

068

007

150002Horizontal bracing of top chord member

Vertical bracing

Top plate

Vertical web memberof side truss

Vertical web memberof side truss

Vertical web memberof middle truss

Horizontal bracingof arch rib chord

member

Transverse stiffening girder

2

W7

Y7

W8

Y8

W5Y5

W4

Y4

W6

Y6

W3

W1

Y3

Y1

W2

Y2

(a) 1-1 profile of steel truss arch girder

Diagonal webmember

Shanghai BeijingTop chord member

Bottom chordmember

Vertical webmember

6

6

W7

W8

Y7

5

5

3

3

4

4

W6

W5

W4

W3

W1

W2

Y6

Y5

Y3

Y1Y2

Y4

Y8

Arch ribchord member

(b) 2-2 profile of steel truss arch

Downstream Upstream

1000

1400

850

1000

850

150

150

W1

Y1W2

Y2

(c) 3-3 profile of top chord member

Downstream Upstream

610

610

150

150

1398

W3

Y3W4

Y4

(d) 4-4 profile of diagonal web member

Downstream Upstream

800

650

650

150

150

1400W5

Y5 W6

Y6

(e) 5-5 profile of arch rib chord member

Downstream Upstream

1266

1266

1416

150

150

1400W7

Y7W8

Y8

(f) 6-6 profile of bottom chord member

Figure 2 Layout of temperature and strain sensors on the steel truss arch girder (unit mm)

4 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

5

15

25

35

45

Time (month)

minus5

Tem

pera

ture

(∘C)

(a) Global history curve

0 4 8 12 16 20 245

10

15

20

Time (hour)

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 3 History curves of global change and daily change in temperature data 1198791

3 4 5 6 7 8 9 10 11

5

15

Time (month)

minus25

minus15

minus5

Tem

pera

ture

diff

eren

ce (∘

C)

(a) Global history curve

0 4 8 12 16 20 24

0

5

Time (hour)

minus10

minus5

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 4 History curves of global change and daily change in temperature difference data 11987912

3 4 5 6 7 8 9 10 11

0

100

200

Time (month)

minus300

minus200

minus100

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus120

minus65

minus10

45

100

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 5 The history curves of global change and daily change in static strain data 1198781

Mathematical Problems in Engineering 5

3 4 5 6 7 8 9 10 11Time (month)

50

150

minus250

minus150

minus50

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus100

minus70

minus40

minus10

20

50

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 6 History curves of global change and daily change in static strain data 1198782

C00j

C01j

C02j

C12j C22

j C32j

C11j

C03j C13

j C23j

C33j

C43j C53

j C63j C73

j

Figure 7 The hierarchical tree of decomposition coefficients in three scales

3 Correlation Analysis

31 Extracting Static Strain Caused by Temperature FieldAccording to the analysis above static strain data mainlycontains three parts static strain I caused by temperaturestatic strain II caused by temperature difference and staticstrain III caused by train In order to research the correlationbetween temperature field and its static strain effect staticstrain I and static strain II (denoted by 119878III) must be extractedfrom the static strain data Considering that primary periodof 119878III is one day which is far longer than that of static strainIII wavelet packet decomposition method is introduced toextract 119878III

In detail based on a pair of low-pass and high-passconjugate quadrature filters ℎ(119895) and 119892(119895) of wavelet packetstatic strain data can be decomposed scale by scale into dif-ferent frequency bands and each decomposition coefficientcorresponding to its frequency band and its scale is calculatedas follows [10 11]

119862

1198962119897

119895= sum

119898isin119885

119862

119896+1119897

119898ℎ (119898 minus 2119895)

119862

1198962119897+1

119895= sum

119898isin119885

119862

119896+1119897

119898119892 (119898 minus 2119895)

(1)

119862

1198962119897

119895or 1198621198962119897+1119895

denotes the decomposition coefficient withinthe 119896th frequency band and the 2119897th or (2119897 + 1)th scalerespectively which is visually described by a hierarchical treeshown in Figure 7 (only three scales are listed) Each decom-position coefficient can be reconstructed into time-domainsignals with constraints of its own frequency band thereforedecomposition coefficients within primary frequency bandof 119878III can be specially selected and then reconstructedas 119878III

Taking the daily history curve of static strain data 1198781in

Figure 5(b) as an example it is decomposed within 8 scalesand the decomposition coefficient 11986208

119895is specially selected

and then reconstructed as 119878III shown in Figure 8(a) whicheffectively retains main component caused by temperaturefield and removes sharp peaks caused by trains as well Usingthe method above 119878III of each static strain data is extractedsuch as the extraction results of 119878

1from training data shown

in Figure 8(b)

32 Modeling Method In order to better present the cor-relation between 119878III and temperature field from trainingdata two steps are specifically carried out as follows (1)temperature and temperature difference data from suffi-cient solar radiation days whose daily variation trends

6 Mathematical Problems in Engineering

0 4 8 12 16 20 24

45

100

Time (hour)

minus120

minus65

minus10

Stat

ic st

rain

S III

(120583120576)

(a) Daily extraction result 119878III of 1198781

3 4 5 6 7 8 9 10 11

10

80

150

Time (month)

minus200

minus130

minus60

Stat

ic st

rain

S III

(120583120576)

(b) Global extraction result 119878III of 1198781

Figure 8 Extraction results 119878III of static strain data 1198781

are similar to one smooth single-period sine curve asshown in Figure 3(b) [12] are specially selected and then119878III are selected from the same corresponding days (2) dailymaximum and minimum values are furthermore chosenfrom the selected temperature field and 119878III data The scatterplots between 119878III of 119878

1and temperature difference 119879

12

between 119878III of 1198782 and temperature 1198791 are shown in Figures

9(a) and 9(b) respectively both presenting apparent linearcorrelation Moreover considering that 119878III of each staticstrain data may be influenced by many temperature andtemperature difference data 119878III are expressed as follows

119878III =8

sum

119894=1

120572

119894119879

119894+

16

sum

119895=1

120573

119895119863

119895+ 119888 (2)

where 119879119894is the 119894th temperature data from set 119879

1 119879

2 119879

3

119879

4 119879

5 119879

6 119879

7 119879

8 119863119895

is the 119895th temperature differencedata from set 119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24

119879

26 119879

28 119879

46 119879

48 119879

68 120572119894is the 119894th linear expansion coefficient

of 119879119894 120573119895is the 119895th linear expansion coefficient of 119879

119894119895 and 119888 is

the constant term

Considering that total 24 linear expansion coefficientsand one constant term are very complicated for calculationprincipal component analysis is introduced to simplify (2)Generally speaking principal components can be derivedfrom either an algebraic or a geometric viewpoint Given 8random variables 119879

1 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8 with a known

covariance matrix sum the 119896th (119896 = 1 2 8) principalcomponent119875

119896is a linear expression of the 8 random variables

defined as follows [13 14]

119875

119896=

8

sum

119895=1

119906

119895119896119879

119895 (3)

Starting from the maximization of variance of 119875119896subjecting

to 9978881199061015840

119896

997888

119906

119897= 0 (119897 = 1 2 8 119897 = 119896) the algebraic derivation

turns out that 997888119906119896= (119906

1119896 119906

2119896 119906

8119896)

1015840 is an eigenvector ofsum corresponding to the 119896th largest eigenvalue 120582

119896 referred

to as principal component coefficients and principal compo-nent variances respectively After eigendecomposition of thecovariance matrixsum 997888119906

1015840

119896is obtained as follows

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

997888

119906

1015840

1

997888

119906

1015840

2

997888

119906

1015840

3

997888

119906

1015840

4

997888

119906

1015840

5

997888

119906

1015840

6

997888

119906

1015840

7

997888

119906

1015840

8

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0319 0384 0364 0382 0358 0387 0333 0289

0462 minus0341 minus0144 minus0332 0110 minus0365 0333 0531

minus0486 0177 minus0459 minus0363 0110 0461 0238 0330

minus0211 minus0628 0262 minus0060 0641 0213 minus0167 minus0068

0491 0301 minus0335 minus0367 0420 0103 minus0229 0427

minus0138 minus0094 minus0209 0205 0174 minus0212 0739 minus0514

0267 minus0367 minus0580 0551 minus0125 0336 minus0142 0065

minus0269 0275 minus0274 0358 0463 minus0539 minus0271 0264

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(4)

Mathematical Problems in Engineering 7

45 10

0

100

200

minus200

minus100

minus12 minus65 minus1

Temperature difference T12 (∘C)

Stat

ic st

rain

S III

ofS 1

(120583120576)

(a) 119878III of 1198781 and temperature difference 11987912

0 8 16 24 32 40

40

90

140

minus110

minus60

minus10

Stat

ic st

rain

S III

ofS 2

(120583120576)

Temperature T1 (∘C)

(b) 119878III of 1198782 and temperature 1198791

Figure 9 Correlation scatter plots between 119878III and temperature and temperature difference

according to the Pareto diagram of explained variances of119875

119896shown in Figure 10(a) it can be seen that the explained

variance of 1198751is 967 very higher than the other principal

components so 1198751is specially selected with its calculation

equation as follows

119875

1=

997888

119906

1015840

1119879

(5)

where 119879 = [1198791 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8]

1015840 Moreover the prin-cipal components 119877

119898(119898 = 1 2 16) of 16 random

variables 11987912 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26

119879

28 119879

46 119879

48 119879

68 are obtained through eigendecomposition

of covariance matrix [13 14] with the Pareto diagram of their

explained variances shown in Figure 10(b) It can be seen thatthe sum of explained variances of 119877

1 1198772 and 119877

3is 988

so 1198771 1198772 and 119877

3are specially selected with their calculation

equations as follows

[

[

[

119877

1

119877

2

119877

3

]

]

]

=

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863 (6)

where 997888V 1015840119899= (V1119899 V2119899 V

16119899) (119899 = 1 2 3) is an eigenvector

of covariance matrix and their values are

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

= [

0403 0209 0234 minus0108 minus0248 0245 0289 0177 0308 0208 minus0188 minus0077 0288 0148 0341 0302

0369 0119 0239 0088 0385 0299 0311 0019 0019 0019 minus0019 0167 minus0347 0184 minus0232 minus0461

0386 0272 0310 minus0144 minus0055 minus0140 minus0167 minus0341 minus0275 0159 0076 minus0317 minus0078 minus0395 minus0302 0180

]

(7)

119863 = [119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26 119879

28

119879

46 119879

48 119879

68]

1015840Therefore the calculation of 119878III can be simpli-fied as follows

119878III = 12058211198751 +997888

120574

1times3

997888

119877

3times1+ 119888

(8)

where 1205821is performance parameter of 119875

1and 997888120574

1times3is one

vector containing three performance parameters correspond-ing to 997888119877

3times1(9978881205741times3

= [120574

1 120574

2 120574

3] and 997888119877

3times1= [119877

1 119877

2 119877

3]

1015840

specifically)The values of performance parameters 1205821 9978881205741times3

and 119888 can be estimated by multivariate linear regression

method [15] and then the correlation models between 119878IIIand temperature field can be expressed as follows

119878III = 1205821 (997888

119906

1015840

1119879) +

997888

120574

1times3(

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863)+ 119888 (9)

33 Modeling Results and Effectiveness Verification Based onthe modeling method above the correlation between 119878IIIof each static strain data and temperature field is modeledby (9) using training data with corresponding performance

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Downstream Upstream15000

1200

068

007

150002Horizontal bracing of top chord member

Vertical bracing

Top plate

Vertical web memberof side truss

Vertical web memberof side truss

Vertical web memberof middle truss

Horizontal bracingof arch rib chord

member

Transverse stiffening girder

2

W7

Y7

W8

Y8

W5Y5

W4

Y4

W6

Y6

W3

W1

Y3

Y1

W2

Y2

(a) 1-1 profile of steel truss arch girder

Diagonal webmember

Shanghai BeijingTop chord member

Bottom chordmember

Vertical webmember

6

6

W7

W8

Y7

5

5

3

3

4

4

W6

W5

W4

W3

W1

W2

Y6

Y5

Y3

Y1Y2

Y4

Y8

Arch ribchord member

(b) 2-2 profile of steel truss arch

Downstream Upstream

1000

1400

850

1000

850

150

150

W1

Y1W2

Y2

(c) 3-3 profile of top chord member

Downstream Upstream

610

610

150

150

1398

W3

Y3W4

Y4

(d) 4-4 profile of diagonal web member

Downstream Upstream

800

650

650

150

150

1400W5

Y5 W6

Y6

(e) 5-5 profile of arch rib chord member

Downstream Upstream

1266

1266

1416

150

150

1400W7

Y7W8

Y8

(f) 6-6 profile of bottom chord member

Figure 2 Layout of temperature and strain sensors on the steel truss arch girder (unit mm)

4 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

5

15

25

35

45

Time (month)

minus5

Tem

pera

ture

(∘C)

(a) Global history curve

0 4 8 12 16 20 245

10

15

20

Time (hour)

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 3 History curves of global change and daily change in temperature data 1198791

3 4 5 6 7 8 9 10 11

5

15

Time (month)

minus25

minus15

minus5

Tem

pera

ture

diff

eren

ce (∘

C)

(a) Global history curve

0 4 8 12 16 20 24

0

5

Time (hour)

minus10

minus5

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 4 History curves of global change and daily change in temperature difference data 11987912

3 4 5 6 7 8 9 10 11

0

100

200

Time (month)

minus300

minus200

minus100

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus120

minus65

minus10

45

100

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 5 The history curves of global change and daily change in static strain data 1198781

Mathematical Problems in Engineering 5

3 4 5 6 7 8 9 10 11Time (month)

50

150

minus250

minus150

minus50

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus100

minus70

minus40

minus10

20

50

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 6 History curves of global change and daily change in static strain data 1198782

C00j

C01j

C02j

C12j C22

j C32j

C11j

C03j C13

j C23j

C33j

C43j C53

j C63j C73

j

Figure 7 The hierarchical tree of decomposition coefficients in three scales

3 Correlation Analysis

31 Extracting Static Strain Caused by Temperature FieldAccording to the analysis above static strain data mainlycontains three parts static strain I caused by temperaturestatic strain II caused by temperature difference and staticstrain III caused by train In order to research the correlationbetween temperature field and its static strain effect staticstrain I and static strain II (denoted by 119878III) must be extractedfrom the static strain data Considering that primary periodof 119878III is one day which is far longer than that of static strainIII wavelet packet decomposition method is introduced toextract 119878III

In detail based on a pair of low-pass and high-passconjugate quadrature filters ℎ(119895) and 119892(119895) of wavelet packetstatic strain data can be decomposed scale by scale into dif-ferent frequency bands and each decomposition coefficientcorresponding to its frequency band and its scale is calculatedas follows [10 11]

119862

1198962119897

119895= sum

119898isin119885

119862

119896+1119897

119898ℎ (119898 minus 2119895)

119862

1198962119897+1

119895= sum

119898isin119885

119862

119896+1119897

119898119892 (119898 minus 2119895)

(1)

119862

1198962119897

119895or 1198621198962119897+1119895

denotes the decomposition coefficient withinthe 119896th frequency band and the 2119897th or (2119897 + 1)th scalerespectively which is visually described by a hierarchical treeshown in Figure 7 (only three scales are listed) Each decom-position coefficient can be reconstructed into time-domainsignals with constraints of its own frequency band thereforedecomposition coefficients within primary frequency bandof 119878III can be specially selected and then reconstructedas 119878III

Taking the daily history curve of static strain data 1198781in

Figure 5(b) as an example it is decomposed within 8 scalesand the decomposition coefficient 11986208

119895is specially selected

and then reconstructed as 119878III shown in Figure 8(a) whicheffectively retains main component caused by temperaturefield and removes sharp peaks caused by trains as well Usingthe method above 119878III of each static strain data is extractedsuch as the extraction results of 119878

1from training data shown

in Figure 8(b)

32 Modeling Method In order to better present the cor-relation between 119878III and temperature field from trainingdata two steps are specifically carried out as follows (1)temperature and temperature difference data from suffi-cient solar radiation days whose daily variation trends

6 Mathematical Problems in Engineering

0 4 8 12 16 20 24

45

100

Time (hour)

minus120

minus65

minus10

Stat

ic st

rain

S III

(120583120576)

(a) Daily extraction result 119878III of 1198781

3 4 5 6 7 8 9 10 11

10

80

150

Time (month)

minus200

minus130

minus60

Stat

ic st

rain

S III

(120583120576)

(b) Global extraction result 119878III of 1198781

Figure 8 Extraction results 119878III of static strain data 1198781

are similar to one smooth single-period sine curve asshown in Figure 3(b) [12] are specially selected and then119878III are selected from the same corresponding days (2) dailymaximum and minimum values are furthermore chosenfrom the selected temperature field and 119878III data The scatterplots between 119878III of 119878

1and temperature difference 119879

12

between 119878III of 1198782 and temperature 1198791 are shown in Figures

9(a) and 9(b) respectively both presenting apparent linearcorrelation Moreover considering that 119878III of each staticstrain data may be influenced by many temperature andtemperature difference data 119878III are expressed as follows

119878III =8

sum

119894=1

120572

119894119879

119894+

16

sum

119895=1

120573

119895119863

119895+ 119888 (2)

where 119879119894is the 119894th temperature data from set 119879

1 119879

2 119879

3

119879

4 119879

5 119879

6 119879

7 119879

8 119863119895

is the 119895th temperature differencedata from set 119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24

119879

26 119879

28 119879

46 119879

48 119879

68 120572119894is the 119894th linear expansion coefficient

of 119879119894 120573119895is the 119895th linear expansion coefficient of 119879

119894119895 and 119888 is

the constant term

Considering that total 24 linear expansion coefficientsand one constant term are very complicated for calculationprincipal component analysis is introduced to simplify (2)Generally speaking principal components can be derivedfrom either an algebraic or a geometric viewpoint Given 8random variables 119879

1 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8 with a known

covariance matrix sum the 119896th (119896 = 1 2 8) principalcomponent119875

119896is a linear expression of the 8 random variables

defined as follows [13 14]

119875

119896=

8

sum

119895=1

119906

119895119896119879

119895 (3)

Starting from the maximization of variance of 119875119896subjecting

to 9978881199061015840

119896

997888

119906

119897= 0 (119897 = 1 2 8 119897 = 119896) the algebraic derivation

turns out that 997888119906119896= (119906

1119896 119906

2119896 119906

8119896)

1015840 is an eigenvector ofsum corresponding to the 119896th largest eigenvalue 120582

119896 referred

to as principal component coefficients and principal compo-nent variances respectively After eigendecomposition of thecovariance matrixsum 997888119906

1015840

119896is obtained as follows

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

997888

119906

1015840

1

997888

119906

1015840

2

997888

119906

1015840

3

997888

119906

1015840

4

997888

119906

1015840

5

997888

119906

1015840

6

997888

119906

1015840

7

997888

119906

1015840

8

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0319 0384 0364 0382 0358 0387 0333 0289

0462 minus0341 minus0144 minus0332 0110 minus0365 0333 0531

minus0486 0177 minus0459 minus0363 0110 0461 0238 0330

minus0211 minus0628 0262 minus0060 0641 0213 minus0167 minus0068

0491 0301 minus0335 minus0367 0420 0103 minus0229 0427

minus0138 minus0094 minus0209 0205 0174 minus0212 0739 minus0514

0267 minus0367 minus0580 0551 minus0125 0336 minus0142 0065

minus0269 0275 minus0274 0358 0463 minus0539 minus0271 0264

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(4)

Mathematical Problems in Engineering 7

45 10

0

100

200

minus200

minus100

minus12 minus65 minus1

Temperature difference T12 (∘C)

Stat

ic st

rain

S III

ofS 1

(120583120576)

(a) 119878III of 1198781 and temperature difference 11987912

0 8 16 24 32 40

40

90

140

minus110

minus60

minus10

Stat

ic st

rain

S III

ofS 2

(120583120576)

Temperature T1 (∘C)

(b) 119878III of 1198782 and temperature 1198791

Figure 9 Correlation scatter plots between 119878III and temperature and temperature difference

according to the Pareto diagram of explained variances of119875

119896shown in Figure 10(a) it can be seen that the explained

variance of 1198751is 967 very higher than the other principal

components so 1198751is specially selected with its calculation

equation as follows

119875

1=

997888

119906

1015840

1119879

(5)

where 119879 = [1198791 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8]

1015840 Moreover the prin-cipal components 119877

119898(119898 = 1 2 16) of 16 random

variables 11987912 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26

119879

28 119879

46 119879

48 119879

68 are obtained through eigendecomposition

of covariance matrix [13 14] with the Pareto diagram of their

explained variances shown in Figure 10(b) It can be seen thatthe sum of explained variances of 119877

1 1198772 and 119877

3is 988

so 1198771 1198772 and 119877

3are specially selected with their calculation

equations as follows

[

[

[

119877

1

119877

2

119877

3

]

]

]

=

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863 (6)

where 997888V 1015840119899= (V1119899 V2119899 V

16119899) (119899 = 1 2 3) is an eigenvector

of covariance matrix and their values are

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

= [

0403 0209 0234 minus0108 minus0248 0245 0289 0177 0308 0208 minus0188 minus0077 0288 0148 0341 0302

0369 0119 0239 0088 0385 0299 0311 0019 0019 0019 minus0019 0167 minus0347 0184 minus0232 minus0461

0386 0272 0310 minus0144 minus0055 minus0140 minus0167 minus0341 minus0275 0159 0076 minus0317 minus0078 minus0395 minus0302 0180

]

(7)

119863 = [119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26 119879

28

119879

46 119879

48 119879

68]

1015840Therefore the calculation of 119878III can be simpli-fied as follows

119878III = 12058211198751 +997888

120574

1times3

997888

119877

3times1+ 119888

(8)

where 1205821is performance parameter of 119875

1and 997888120574

1times3is one

vector containing three performance parameters correspond-ing to 997888119877

3times1(9978881205741times3

= [120574

1 120574

2 120574

3] and 997888119877

3times1= [119877

1 119877

2 119877

3]

1015840

specifically)The values of performance parameters 1205821 9978881205741times3

and 119888 can be estimated by multivariate linear regression

method [15] and then the correlation models between 119878IIIand temperature field can be expressed as follows

119878III = 1205821 (997888

119906

1015840

1119879) +

997888

120574

1times3(

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863)+ 119888 (9)

33 Modeling Results and Effectiveness Verification Based onthe modeling method above the correlation between 119878IIIof each static strain data and temperature field is modeledby (9) using training data with corresponding performance

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

5

15

25

35

45

Time (month)

minus5

Tem

pera

ture

(∘C)

(a) Global history curve

0 4 8 12 16 20 245

10

15

20

Time (hour)

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 3 History curves of global change and daily change in temperature data 1198791

3 4 5 6 7 8 9 10 11

5

15

Time (month)

minus25

minus15

minus5

Tem

pera

ture

diff

eren

ce (∘

C)

(a) Global history curve

0 4 8 12 16 20 24

0

5

Time (hour)

minus10

minus5

Tem

pera

ture

(∘C)

(b) Daily history curve (March 4th 2013)

Figure 4 History curves of global change and daily change in temperature difference data 11987912

3 4 5 6 7 8 9 10 11

0

100

200

Time (month)

minus300

minus200

minus100

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus120

minus65

minus10

45

100

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 5 The history curves of global change and daily change in static strain data 1198781

Mathematical Problems in Engineering 5

3 4 5 6 7 8 9 10 11Time (month)

50

150

minus250

minus150

minus50

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus100

minus70

minus40

minus10

20

50

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 6 History curves of global change and daily change in static strain data 1198782

C00j

C01j

C02j

C12j C22

j C32j

C11j

C03j C13

j C23j

C33j

C43j C53

j C63j C73

j

Figure 7 The hierarchical tree of decomposition coefficients in three scales

3 Correlation Analysis

31 Extracting Static Strain Caused by Temperature FieldAccording to the analysis above static strain data mainlycontains three parts static strain I caused by temperaturestatic strain II caused by temperature difference and staticstrain III caused by train In order to research the correlationbetween temperature field and its static strain effect staticstrain I and static strain II (denoted by 119878III) must be extractedfrom the static strain data Considering that primary periodof 119878III is one day which is far longer than that of static strainIII wavelet packet decomposition method is introduced toextract 119878III

In detail based on a pair of low-pass and high-passconjugate quadrature filters ℎ(119895) and 119892(119895) of wavelet packetstatic strain data can be decomposed scale by scale into dif-ferent frequency bands and each decomposition coefficientcorresponding to its frequency band and its scale is calculatedas follows [10 11]

119862

1198962119897

119895= sum

119898isin119885

119862

119896+1119897

119898ℎ (119898 minus 2119895)

119862

1198962119897+1

119895= sum

119898isin119885

119862

119896+1119897

119898119892 (119898 minus 2119895)

(1)

119862

1198962119897

119895or 1198621198962119897+1119895

denotes the decomposition coefficient withinthe 119896th frequency band and the 2119897th or (2119897 + 1)th scalerespectively which is visually described by a hierarchical treeshown in Figure 7 (only three scales are listed) Each decom-position coefficient can be reconstructed into time-domainsignals with constraints of its own frequency band thereforedecomposition coefficients within primary frequency bandof 119878III can be specially selected and then reconstructedas 119878III

Taking the daily history curve of static strain data 1198781in

Figure 5(b) as an example it is decomposed within 8 scalesand the decomposition coefficient 11986208

119895is specially selected

and then reconstructed as 119878III shown in Figure 8(a) whicheffectively retains main component caused by temperaturefield and removes sharp peaks caused by trains as well Usingthe method above 119878III of each static strain data is extractedsuch as the extraction results of 119878

1from training data shown

in Figure 8(b)

32 Modeling Method In order to better present the cor-relation between 119878III and temperature field from trainingdata two steps are specifically carried out as follows (1)temperature and temperature difference data from suffi-cient solar radiation days whose daily variation trends

6 Mathematical Problems in Engineering

0 4 8 12 16 20 24

45

100

Time (hour)

minus120

minus65

minus10

Stat

ic st

rain

S III

(120583120576)

(a) Daily extraction result 119878III of 1198781

3 4 5 6 7 8 9 10 11

10

80

150

Time (month)

minus200

minus130

minus60

Stat

ic st

rain

S III

(120583120576)

(b) Global extraction result 119878III of 1198781

Figure 8 Extraction results 119878III of static strain data 1198781

are similar to one smooth single-period sine curve asshown in Figure 3(b) [12] are specially selected and then119878III are selected from the same corresponding days (2) dailymaximum and minimum values are furthermore chosenfrom the selected temperature field and 119878III data The scatterplots between 119878III of 119878

1and temperature difference 119879

12

between 119878III of 1198782 and temperature 1198791 are shown in Figures

9(a) and 9(b) respectively both presenting apparent linearcorrelation Moreover considering that 119878III of each staticstrain data may be influenced by many temperature andtemperature difference data 119878III are expressed as follows

119878III =8

sum

119894=1

120572

119894119879

119894+

16

sum

119895=1

120573

119895119863

119895+ 119888 (2)

where 119879119894is the 119894th temperature data from set 119879

1 119879

2 119879

3

119879

4 119879

5 119879

6 119879

7 119879

8 119863119895

is the 119895th temperature differencedata from set 119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24

119879

26 119879

28 119879

46 119879

48 119879

68 120572119894is the 119894th linear expansion coefficient

of 119879119894 120573119895is the 119895th linear expansion coefficient of 119879

119894119895 and 119888 is

the constant term

Considering that total 24 linear expansion coefficientsand one constant term are very complicated for calculationprincipal component analysis is introduced to simplify (2)Generally speaking principal components can be derivedfrom either an algebraic or a geometric viewpoint Given 8random variables 119879

1 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8 with a known

covariance matrix sum the 119896th (119896 = 1 2 8) principalcomponent119875

119896is a linear expression of the 8 random variables

defined as follows [13 14]

119875

119896=

8

sum

119895=1

119906

119895119896119879

119895 (3)

Starting from the maximization of variance of 119875119896subjecting

to 9978881199061015840

119896

997888

119906

119897= 0 (119897 = 1 2 8 119897 = 119896) the algebraic derivation

turns out that 997888119906119896= (119906

1119896 119906

2119896 119906

8119896)

1015840 is an eigenvector ofsum corresponding to the 119896th largest eigenvalue 120582

119896 referred

to as principal component coefficients and principal compo-nent variances respectively After eigendecomposition of thecovariance matrixsum 997888119906

1015840

119896is obtained as follows

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

997888

119906

1015840

1

997888

119906

1015840

2

997888

119906

1015840

3

997888

119906

1015840

4

997888

119906

1015840

5

997888

119906

1015840

6

997888

119906

1015840

7

997888

119906

1015840

8

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0319 0384 0364 0382 0358 0387 0333 0289

0462 minus0341 minus0144 minus0332 0110 minus0365 0333 0531

minus0486 0177 minus0459 minus0363 0110 0461 0238 0330

minus0211 minus0628 0262 minus0060 0641 0213 minus0167 minus0068

0491 0301 minus0335 minus0367 0420 0103 minus0229 0427

minus0138 minus0094 minus0209 0205 0174 minus0212 0739 minus0514

0267 minus0367 minus0580 0551 minus0125 0336 minus0142 0065

minus0269 0275 minus0274 0358 0463 minus0539 minus0271 0264

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(4)

Mathematical Problems in Engineering 7

45 10

0

100

200

minus200

minus100

minus12 minus65 minus1

Temperature difference T12 (∘C)

Stat

ic st

rain

S III

ofS 1

(120583120576)

(a) 119878III of 1198781 and temperature difference 11987912

0 8 16 24 32 40

40

90

140

minus110

minus60

minus10

Stat

ic st

rain

S III

ofS 2

(120583120576)

Temperature T1 (∘C)

(b) 119878III of 1198782 and temperature 1198791

Figure 9 Correlation scatter plots between 119878III and temperature and temperature difference

according to the Pareto diagram of explained variances of119875

119896shown in Figure 10(a) it can be seen that the explained

variance of 1198751is 967 very higher than the other principal

components so 1198751is specially selected with its calculation

equation as follows

119875

1=

997888

119906

1015840

1119879

(5)

where 119879 = [1198791 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8]

1015840 Moreover the prin-cipal components 119877

119898(119898 = 1 2 16) of 16 random

variables 11987912 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26

119879

28 119879

46 119879

48 119879

68 are obtained through eigendecomposition

of covariance matrix [13 14] with the Pareto diagram of their

explained variances shown in Figure 10(b) It can be seen thatthe sum of explained variances of 119877

1 1198772 and 119877

3is 988

so 1198771 1198772 and 119877

3are specially selected with their calculation

equations as follows

[

[

[

119877

1

119877

2

119877

3

]

]

]

=

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863 (6)

where 997888V 1015840119899= (V1119899 V2119899 V

16119899) (119899 = 1 2 3) is an eigenvector

of covariance matrix and their values are

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

= [

0403 0209 0234 minus0108 minus0248 0245 0289 0177 0308 0208 minus0188 minus0077 0288 0148 0341 0302

0369 0119 0239 0088 0385 0299 0311 0019 0019 0019 minus0019 0167 minus0347 0184 minus0232 minus0461

0386 0272 0310 minus0144 minus0055 minus0140 minus0167 minus0341 minus0275 0159 0076 minus0317 minus0078 minus0395 minus0302 0180

]

(7)

119863 = [119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26 119879

28

119879

46 119879

48 119879

68]

1015840Therefore the calculation of 119878III can be simpli-fied as follows

119878III = 12058211198751 +997888

120574

1times3

997888

119877

3times1+ 119888

(8)

where 1205821is performance parameter of 119875

1and 997888120574

1times3is one

vector containing three performance parameters correspond-ing to 997888119877

3times1(9978881205741times3

= [120574

1 120574

2 120574

3] and 997888119877

3times1= [119877

1 119877

2 119877

3]

1015840

specifically)The values of performance parameters 1205821 9978881205741times3

and 119888 can be estimated by multivariate linear regression

method [15] and then the correlation models between 119878IIIand temperature field can be expressed as follows

119878III = 1205821 (997888

119906

1015840

1119879) +

997888

120574

1times3(

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863)+ 119888 (9)

33 Modeling Results and Effectiveness Verification Based onthe modeling method above the correlation between 119878IIIof each static strain data and temperature field is modeledby (9) using training data with corresponding performance

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

3 4 5 6 7 8 9 10 11Time (month)

50

150

minus250

minus150

minus50

Stat

ic st

rain

(120583120576)

(a) Global history curve

Sharp peak

0 4 8 12 16 20 24Time (hour)

minus100

minus70

minus40

minus10

20

50

Stat

ic st

rain

(120583120576)

(b) Daily history curve (March 4th 2013)

Figure 6 History curves of global change and daily change in static strain data 1198782

C00j

C01j

C02j

C12j C22

j C32j

C11j

C03j C13

j C23j

C33j

C43j C53

j C63j C73

j

Figure 7 The hierarchical tree of decomposition coefficients in three scales

3 Correlation Analysis

31 Extracting Static Strain Caused by Temperature FieldAccording to the analysis above static strain data mainlycontains three parts static strain I caused by temperaturestatic strain II caused by temperature difference and staticstrain III caused by train In order to research the correlationbetween temperature field and its static strain effect staticstrain I and static strain II (denoted by 119878III) must be extractedfrom the static strain data Considering that primary periodof 119878III is one day which is far longer than that of static strainIII wavelet packet decomposition method is introduced toextract 119878III

In detail based on a pair of low-pass and high-passconjugate quadrature filters ℎ(119895) and 119892(119895) of wavelet packetstatic strain data can be decomposed scale by scale into dif-ferent frequency bands and each decomposition coefficientcorresponding to its frequency band and its scale is calculatedas follows [10 11]

119862

1198962119897

119895= sum

119898isin119885

119862

119896+1119897

119898ℎ (119898 minus 2119895)

119862

1198962119897+1

119895= sum

119898isin119885

119862

119896+1119897

119898119892 (119898 minus 2119895)

(1)

119862

1198962119897

119895or 1198621198962119897+1119895

denotes the decomposition coefficient withinthe 119896th frequency band and the 2119897th or (2119897 + 1)th scalerespectively which is visually described by a hierarchical treeshown in Figure 7 (only three scales are listed) Each decom-position coefficient can be reconstructed into time-domainsignals with constraints of its own frequency band thereforedecomposition coefficients within primary frequency bandof 119878III can be specially selected and then reconstructedas 119878III

Taking the daily history curve of static strain data 1198781in

Figure 5(b) as an example it is decomposed within 8 scalesand the decomposition coefficient 11986208

119895is specially selected

and then reconstructed as 119878III shown in Figure 8(a) whicheffectively retains main component caused by temperaturefield and removes sharp peaks caused by trains as well Usingthe method above 119878III of each static strain data is extractedsuch as the extraction results of 119878

1from training data shown

in Figure 8(b)

32 Modeling Method In order to better present the cor-relation between 119878III and temperature field from trainingdata two steps are specifically carried out as follows (1)temperature and temperature difference data from suffi-cient solar radiation days whose daily variation trends

6 Mathematical Problems in Engineering

0 4 8 12 16 20 24

45

100

Time (hour)

minus120

minus65

minus10

Stat

ic st

rain

S III

(120583120576)

(a) Daily extraction result 119878III of 1198781

3 4 5 6 7 8 9 10 11

10

80

150

Time (month)

minus200

minus130

minus60

Stat

ic st

rain

S III

(120583120576)

(b) Global extraction result 119878III of 1198781

Figure 8 Extraction results 119878III of static strain data 1198781

are similar to one smooth single-period sine curve asshown in Figure 3(b) [12] are specially selected and then119878III are selected from the same corresponding days (2) dailymaximum and minimum values are furthermore chosenfrom the selected temperature field and 119878III data The scatterplots between 119878III of 119878

1and temperature difference 119879

12

between 119878III of 1198782 and temperature 1198791 are shown in Figures

9(a) and 9(b) respectively both presenting apparent linearcorrelation Moreover considering that 119878III of each staticstrain data may be influenced by many temperature andtemperature difference data 119878III are expressed as follows

119878III =8

sum

119894=1

120572

119894119879

119894+

16

sum

119895=1

120573

119895119863

119895+ 119888 (2)

where 119879119894is the 119894th temperature data from set 119879

1 119879

2 119879

3

119879

4 119879

5 119879

6 119879

7 119879

8 119863119895

is the 119895th temperature differencedata from set 119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24

119879

26 119879

28 119879

46 119879

48 119879

68 120572119894is the 119894th linear expansion coefficient

of 119879119894 120573119895is the 119895th linear expansion coefficient of 119879

119894119895 and 119888 is

the constant term

Considering that total 24 linear expansion coefficientsand one constant term are very complicated for calculationprincipal component analysis is introduced to simplify (2)Generally speaking principal components can be derivedfrom either an algebraic or a geometric viewpoint Given 8random variables 119879

1 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8 with a known

covariance matrix sum the 119896th (119896 = 1 2 8) principalcomponent119875

119896is a linear expression of the 8 random variables

defined as follows [13 14]

119875

119896=

8

sum

119895=1

119906

119895119896119879

119895 (3)

Starting from the maximization of variance of 119875119896subjecting

to 9978881199061015840

119896

997888

119906

119897= 0 (119897 = 1 2 8 119897 = 119896) the algebraic derivation

turns out that 997888119906119896= (119906

1119896 119906

2119896 119906

8119896)

1015840 is an eigenvector ofsum corresponding to the 119896th largest eigenvalue 120582

119896 referred

to as principal component coefficients and principal compo-nent variances respectively After eigendecomposition of thecovariance matrixsum 997888119906

1015840

119896is obtained as follows

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

997888

119906

1015840

1

997888

119906

1015840

2

997888

119906

1015840

3

997888

119906

1015840

4

997888

119906

1015840

5

997888

119906

1015840

6

997888

119906

1015840

7

997888

119906

1015840

8

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0319 0384 0364 0382 0358 0387 0333 0289

0462 minus0341 minus0144 minus0332 0110 minus0365 0333 0531

minus0486 0177 minus0459 minus0363 0110 0461 0238 0330

minus0211 minus0628 0262 minus0060 0641 0213 minus0167 minus0068

0491 0301 minus0335 minus0367 0420 0103 minus0229 0427

minus0138 minus0094 minus0209 0205 0174 minus0212 0739 minus0514

0267 minus0367 minus0580 0551 minus0125 0336 minus0142 0065

minus0269 0275 minus0274 0358 0463 minus0539 minus0271 0264

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(4)

Mathematical Problems in Engineering 7

45 10

0

100

200

minus200

minus100

minus12 minus65 minus1

Temperature difference T12 (∘C)

Stat

ic st

rain

S III

ofS 1

(120583120576)

(a) 119878III of 1198781 and temperature difference 11987912

0 8 16 24 32 40

40

90

140

minus110

minus60

minus10

Stat

ic st

rain

S III

ofS 2

(120583120576)

Temperature T1 (∘C)

(b) 119878III of 1198782 and temperature 1198791

Figure 9 Correlation scatter plots between 119878III and temperature and temperature difference

according to the Pareto diagram of explained variances of119875

119896shown in Figure 10(a) it can be seen that the explained

variance of 1198751is 967 very higher than the other principal

components so 1198751is specially selected with its calculation

equation as follows

119875

1=

997888

119906

1015840

1119879

(5)

where 119879 = [1198791 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8]

1015840 Moreover the prin-cipal components 119877

119898(119898 = 1 2 16) of 16 random

variables 11987912 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26

119879

28 119879

46 119879

48 119879

68 are obtained through eigendecomposition

of covariance matrix [13 14] with the Pareto diagram of their

explained variances shown in Figure 10(b) It can be seen thatthe sum of explained variances of 119877

1 1198772 and 119877

3is 988

so 1198771 1198772 and 119877

3are specially selected with their calculation

equations as follows

[

[

[

119877

1

119877

2

119877

3

]

]

]

=

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863 (6)

where 997888V 1015840119899= (V1119899 V2119899 V

16119899) (119899 = 1 2 3) is an eigenvector

of covariance matrix and their values are

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

= [

0403 0209 0234 minus0108 minus0248 0245 0289 0177 0308 0208 minus0188 minus0077 0288 0148 0341 0302

0369 0119 0239 0088 0385 0299 0311 0019 0019 0019 minus0019 0167 minus0347 0184 minus0232 minus0461

0386 0272 0310 minus0144 minus0055 minus0140 minus0167 minus0341 minus0275 0159 0076 minus0317 minus0078 minus0395 minus0302 0180

]

(7)

119863 = [119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26 119879

28

119879

46 119879

48 119879

68]

1015840Therefore the calculation of 119878III can be simpli-fied as follows

119878III = 12058211198751 +997888

120574

1times3

997888

119877

3times1+ 119888

(8)

where 1205821is performance parameter of 119875

1and 997888120574

1times3is one

vector containing three performance parameters correspond-ing to 997888119877

3times1(9978881205741times3

= [120574

1 120574

2 120574

3] and 997888119877

3times1= [119877

1 119877

2 119877

3]

1015840

specifically)The values of performance parameters 1205821 9978881205741times3

and 119888 can be estimated by multivariate linear regression

method [15] and then the correlation models between 119878IIIand temperature field can be expressed as follows

119878III = 1205821 (997888

119906

1015840

1119879) +

997888

120574

1times3(

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863)+ 119888 (9)

33 Modeling Results and Effectiveness Verification Based onthe modeling method above the correlation between 119878IIIof each static strain data and temperature field is modeledby (9) using training data with corresponding performance

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0 4 8 12 16 20 24

45

100

Time (hour)

minus120

minus65

minus10

Stat

ic st

rain

S III

(120583120576)

(a) Daily extraction result 119878III of 1198781

3 4 5 6 7 8 9 10 11

10

80

150

Time (month)

minus200

minus130

minus60

Stat

ic st

rain

S III

(120583120576)

(b) Global extraction result 119878III of 1198781

Figure 8 Extraction results 119878III of static strain data 1198781

are similar to one smooth single-period sine curve asshown in Figure 3(b) [12] are specially selected and then119878III are selected from the same corresponding days (2) dailymaximum and minimum values are furthermore chosenfrom the selected temperature field and 119878III data The scatterplots between 119878III of 119878

1and temperature difference 119879

12

between 119878III of 1198782 and temperature 1198791 are shown in Figures

9(a) and 9(b) respectively both presenting apparent linearcorrelation Moreover considering that 119878III of each staticstrain data may be influenced by many temperature andtemperature difference data 119878III are expressed as follows

119878III =8

sum

119894=1

120572

119894119879

119894+

16

sum

119895=1

120573

119895119863

119895+ 119888 (2)

where 119879119894is the 119894th temperature data from set 119879

1 119879

2 119879

3

119879

4 119879

5 119879

6 119879

7 119879

8 119863119895

is the 119895th temperature differencedata from set 119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24

119879

26 119879

28 119879

46 119879

48 119879

68 120572119894is the 119894th linear expansion coefficient

of 119879119894 120573119895is the 119895th linear expansion coefficient of 119879

119894119895 and 119888 is

the constant term

Considering that total 24 linear expansion coefficientsand one constant term are very complicated for calculationprincipal component analysis is introduced to simplify (2)Generally speaking principal components can be derivedfrom either an algebraic or a geometric viewpoint Given 8random variables 119879

1 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8 with a known

covariance matrix sum the 119896th (119896 = 1 2 8) principalcomponent119875

119896is a linear expression of the 8 random variables

defined as follows [13 14]

119875

119896=

8

sum

119895=1

119906

119895119896119879

119895 (3)

Starting from the maximization of variance of 119875119896subjecting

to 9978881199061015840

119896

997888

119906

119897= 0 (119897 = 1 2 8 119897 = 119896) the algebraic derivation

turns out that 997888119906119896= (119906

1119896 119906

2119896 119906

8119896)

1015840 is an eigenvector ofsum corresponding to the 119896th largest eigenvalue 120582

119896 referred

to as principal component coefficients and principal compo-nent variances respectively After eigendecomposition of thecovariance matrixsum 997888119906

1015840

119896is obtained as follows

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

997888

119906

1015840

1

997888

119906

1015840

2

997888

119906

1015840

3

997888

119906

1015840

4

997888

119906

1015840

5

997888

119906

1015840

6

997888

119906

1015840

7

997888

119906

1015840

8

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0319 0384 0364 0382 0358 0387 0333 0289

0462 minus0341 minus0144 minus0332 0110 minus0365 0333 0531

minus0486 0177 minus0459 minus0363 0110 0461 0238 0330

minus0211 minus0628 0262 minus0060 0641 0213 minus0167 minus0068

0491 0301 minus0335 minus0367 0420 0103 minus0229 0427

minus0138 minus0094 minus0209 0205 0174 minus0212 0739 minus0514

0267 minus0367 minus0580 0551 minus0125 0336 minus0142 0065

minus0269 0275 minus0274 0358 0463 minus0539 minus0271 0264

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(4)

Mathematical Problems in Engineering 7

45 10

0

100

200

minus200

minus100

minus12 minus65 minus1

Temperature difference T12 (∘C)

Stat

ic st

rain

S III

ofS 1

(120583120576)

(a) 119878III of 1198781 and temperature difference 11987912

0 8 16 24 32 40

40

90

140

minus110

minus60

minus10

Stat

ic st

rain

S III

ofS 2

(120583120576)

Temperature T1 (∘C)

(b) 119878III of 1198782 and temperature 1198791

Figure 9 Correlation scatter plots between 119878III and temperature and temperature difference

according to the Pareto diagram of explained variances of119875

119896shown in Figure 10(a) it can be seen that the explained

variance of 1198751is 967 very higher than the other principal

components so 1198751is specially selected with its calculation

equation as follows

119875

1=

997888

119906

1015840

1119879

(5)

where 119879 = [1198791 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8]

1015840 Moreover the prin-cipal components 119877

119898(119898 = 1 2 16) of 16 random

variables 11987912 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26

119879

28 119879

46 119879

48 119879

68 are obtained through eigendecomposition

of covariance matrix [13 14] with the Pareto diagram of their

explained variances shown in Figure 10(b) It can be seen thatthe sum of explained variances of 119877

1 1198772 and 119877

3is 988

so 1198771 1198772 and 119877

3are specially selected with their calculation

equations as follows

[

[

[

119877

1

119877

2

119877

3

]

]

]

=

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863 (6)

where 997888V 1015840119899= (V1119899 V2119899 V

16119899) (119899 = 1 2 3) is an eigenvector

of covariance matrix and their values are

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

= [

0403 0209 0234 minus0108 minus0248 0245 0289 0177 0308 0208 minus0188 minus0077 0288 0148 0341 0302

0369 0119 0239 0088 0385 0299 0311 0019 0019 0019 minus0019 0167 minus0347 0184 minus0232 minus0461

0386 0272 0310 minus0144 minus0055 minus0140 minus0167 minus0341 minus0275 0159 0076 minus0317 minus0078 minus0395 minus0302 0180

]

(7)

119863 = [119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26 119879

28

119879

46 119879

48 119879

68]

1015840Therefore the calculation of 119878III can be simpli-fied as follows

119878III = 12058211198751 +997888

120574

1times3

997888

119877

3times1+ 119888

(8)

where 1205821is performance parameter of 119875

1and 997888120574

1times3is one

vector containing three performance parameters correspond-ing to 997888119877

3times1(9978881205741times3

= [120574

1 120574

2 120574

3] and 997888119877

3times1= [119877

1 119877

2 119877

3]

1015840

specifically)The values of performance parameters 1205821 9978881205741times3

and 119888 can be estimated by multivariate linear regression

method [15] and then the correlation models between 119878IIIand temperature field can be expressed as follows

119878III = 1205821 (997888

119906

1015840

1119879) +

997888

120574

1times3(

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863)+ 119888 (9)

33 Modeling Results and Effectiveness Verification Based onthe modeling method above the correlation between 119878IIIof each static strain data and temperature field is modeledby (9) using training data with corresponding performance

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

45 10

0

100

200

minus200

minus100

minus12 minus65 minus1

Temperature difference T12 (∘C)

Stat

ic st

rain

S III

ofS 1

(120583120576)

(a) 119878III of 1198781 and temperature difference 11987912

0 8 16 24 32 40

40

90

140

minus110

minus60

minus10

Stat

ic st

rain

S III

ofS 2

(120583120576)

Temperature T1 (∘C)

(b) 119878III of 1198782 and temperature 1198791

Figure 9 Correlation scatter plots between 119878III and temperature and temperature difference

according to the Pareto diagram of explained variances of119875

119896shown in Figure 10(a) it can be seen that the explained

variance of 1198751is 967 very higher than the other principal

components so 1198751is specially selected with its calculation

equation as follows

119875

1=

997888

119906

1015840

1119879

(5)

where 119879 = [1198791 119879

2 119879

3 119879

4 119879

5 119879

6 119879

7 119879

8]

1015840 Moreover the prin-cipal components 119877

119898(119898 = 1 2 16) of 16 random

variables 11987912 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26

119879

28 119879

46 119879

48 119879

68 are obtained through eigendecomposition

of covariance matrix [13 14] with the Pareto diagram of their

explained variances shown in Figure 10(b) It can be seen thatthe sum of explained variances of 119877

1 1198772 and 119877

3is 988

so 1198771 1198772 and 119877

3are specially selected with their calculation

equations as follows

[

[

[

119877

1

119877

2

119877

3

]

]

]

=

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863 (6)

where 997888V 1015840119899= (V1119899 V2119899 V

16119899) (119899 = 1 2 3) is an eigenvector

of covariance matrix and their values are

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

= [

0403 0209 0234 minus0108 minus0248 0245 0289 0177 0308 0208 minus0188 minus0077 0288 0148 0341 0302

0369 0119 0239 0088 0385 0299 0311 0019 0019 0019 minus0019 0167 minus0347 0184 minus0232 minus0461

0386 0272 0310 minus0144 minus0055 minus0140 minus0167 minus0341 minus0275 0159 0076 minus0317 minus0078 minus0395 minus0302 0180

]

(7)

119863 = [119879

12 119879

34 119879

56 119879

78 119879

13 119879

15 119879

17 119879

35 119879

37 119879

57 119879

24 119879

26 119879

28

119879

46 119879

48 119879

68]

1015840Therefore the calculation of 119878III can be simpli-fied as follows

119878III = 12058211198751 +997888

120574

1times3

997888

119877

3times1+ 119888

(8)

where 1205821is performance parameter of 119875

1and 997888120574

1times3is one

vector containing three performance parameters correspond-ing to 997888119877

3times1(9978881205741times3

= [120574

1 120574

2 120574

3] and 997888119877

3times1= [119877

1 119877

2 119877

3]

1015840

specifically)The values of performance parameters 1205821 9978881205741times3

and 119888 can be estimated by multivariate linear regression

method [15] and then the correlation models between 119878IIIand temperature field can be expressed as follows

119878III = 1205821 (997888

119906

1015840

1119879) +

997888

120574

1times3(

[

[

[

[

997888V1015840

1

997888V1015840

2

997888V1015840

3

]

]

]

]

119863)+ 119888 (9)

33 Modeling Results and Effectiveness Verification Based onthe modeling method above the correlation between 119878IIIof each static strain data and temperature field is modeledby (9) using training data with corresponding performance

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

0

20

40

60

80

100Ex

plai

ned

varia

nces

()

Eight potential comprehensive factors (∘C)P1 P2 P3 P4 P5 P6 P7 P8

(a) Temperature data

0

20

40

60

80

100

Expl

aine

d va

rianc

es (

)

R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16

Sixteen potential comprehensive factors (∘C)

(b) Temperature difference data

Figure 10 Pareto diagram of explained variances of temperature and temperature difference data

Table 1 Performance parameter values 1205821 1205741 1205742 1205743 and 119888 of

correlation models

Static strain data Weighted values of correlation models120582

1120574

1120574

2120574

3119888

119878III of 1198781 0454 minus4711 minus2597 minus4944 minus46806119878III of 1198782 1585 0013 4104 minus1247 minus70238119878III of 1198783 0301 3416 minus2098 2968 minus6884119878III of 1198784 0350 minus0734 0351 0948 minus23597119878III of 1198785 0179 minus4219 1827 3165 minus22049119878III of 1198786 0062 minus3245 8242 minus5398 minus22862119878III of 1198787 minus0367 2411 minus0598 5531 17075119878III of 1198788 0611 2188 minus1130 2018 minus32799

parameter values 1205821 1205741 1205742 1205743 and 119888 shown in Table 1 In

order to test the modeling effectiveness of the correlationmodels temperature and temperature difference data fromtest data are substituted into (9) to calculate simulative 119878IIIafter two-step analysis (detailed in Section 32) and thenthe scattered points between simulative 119878III and monitoring119878III are plotted as shown in Figures 11(a)sim11(f) all of whichpresent good linear correlation Therefore the scatteredpoints are furthermore least-square fitted by linear function

119878

119904= 119896119878

119898+ 119887 (10)

where 119878119904denotes simulative 119878III 119878119898 denotes monitoring 119878III

and 119896 119887 are fitting parameters with the fitting results shownin Figures 11(a)sim11(f) as well The fitting results show that allthe fitting curves can well describe the linear correlation ofscattered points with 119896 close to 1 and 119887 close to 0 verifyinggood modeling effectiveness of the correlation models

4 Static Performance Assessment

41 Assessment Method Daily maximum and minimum val-ues of temperature and temperature difference from trainingdata are substituted into (9) to calculate simulative 119878III and

then residual static strains are obtained by simulative 119878IIIminus monitoring 119878III Augmented Dickey-Fuller test atnominal significance level of 005 indicates that time historyof residual static strains is one stationary stochastic processaround the central line of 0 120583120576 and within the scope of upperand lower limit [minus50120583120576 50120583120576] such as that of 119878

1shown

in Figure 12(a) which are considered relative initial state ofstatic performance of Dashengguan Yangtze Bridge

In order to investigate change regulations of residualstatic strains under performance degradation performanceparameters 120582

1and 997888120574

1times3in (9) are assumed to increase by

20 eachmonth (20 is an appropriate value to clearly revealthe change regulations of residual static strains under perfor-mance degradation) with 3 kinds of possible combinationsspecifically as follows (1) only 120582

1increases by 20 (2) only

997888

120574

1times3increases by 20 (3) both 120582

1and997888120574

1times3increase by 20

Then residual static strains of 1198781for each combination are

calculated as shown in Figures 12(b)sim12(d) respectively fromwhich three kinds of degradation regulations can be obtained(1) residual static strains of 119878

1from the first combination

gradually deviate from the central line with time and thensurpass the upper limit (2) residual static strains of 119878

1from

the second combination fluctuate more fiercely with timearound the central line and then surpass the upper andlower limit (3) residual static strains of 119878

1from the third

combination can be considered the summation of the firstand second situations Therefore the degradation of staticperformance can be confirmed if one of the degradationregulations exits in time history of residual static strains

42 Assessment Results Using the analytical method abovedaily maximum and minimum values of temperature andtemperature difference from assessment data are substitutedinto (9) to calculate simulative 119878III and then residual staticstrains of each static strain data are acquired by simulative119878III minus monitoring 119878III as shown in Figures 13(a)sim13(f)respectively Through analysis of their change trends allof the residual static strains do not obviously present any

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

35

35

80

80minus100

minus100

minus55

minus55

minus10

minus10

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0946b = minus3632

(a) 1198781

15

15

45

45

75

75minus75minus75

minus45

minus45

minus15

minus15

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0898b = minus2013

(b) 1198782

0

0

30

30

60

60

90

90minus60

minus60

minus30

minus30

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0904b = minus1844

(c) 1198783

5

15

20

30

35

minus40minus30

minus25

minus15

minus10

0

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1126b = 2057

(d) 1198784

minus100minus105 minus75 minus45 minus15

minus65

minus30

5

40

15

75

45 75

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0935b = minus4139

(e) 1198785

minus185minus205 minus145 minus85 minus25

minus130

minus75

minus20

35

35

90

95

Sim

ulat

iveS

III

(120583120576)

k = 0918b = minus4094

Monitoring SIII (120583120576)

(f) 1198786

21 43 65

16

38

60

minus45

minus6

minus28

minus50minus23 minus1

Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 0881

Scattered pointsFitting curve

b = 2148

(g) 1198787

15

30

0

5 15 25minus5minus15minus25minus25

minus15Sim

ulat

iveS

III

(120583120576)

Monitoring SIII (120583120576)

k = 1118

Scattered pointsFitting curve

b = 0615

(h) 1198788

Figure 11 Correlation between simulative 119878III and monitoring 119878III

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

3 4 5 6 7 8 9 10 11

0

40

80

120

Time (month)

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Relative initial state of static performance

3 4 5 6 7 8 9 10 11Time (month)

0

50

100

150

minus100

minus50

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(b) 1205821 increases by 20

3 4 5 6 7 8 9 10 11Time (month)

Static strain residuals

Central lineUpper and lower limit

0

200

400

minus400

minus200

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(c) 9978881205741times3

increases by 20

3 4 5 6 7 8 9 10 11

100

300

500

Time (month)

minus100

minus300

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(d) 1205821 and9978881205741times3

increase by 20

Figure 12 Relative initial state and degradation state of static performance

kind of the degradation regulations indicating that the staticperformance of Dashengguan Yangtze Bridge was in a goodcondition during that period

5 Conclusions

Based on correlation between temperature field and its staticstrain static performance of Dashengguan Yangtze Bridge isassessed by using methods such as wavelet packet decom-position multivariate linear regression modeling principalcomponent analysis and residual strain analysis After theinvestigation some conclusions can be drawn as follows

(1) Static strain data mainly contains three parts staticstrain I caused by temperature static strain II causedby temperature difference and static strain III causedby train The influence proportion of train in staticstrain data is obviously lower than temperature andtemperature difference

(2) Wavelet packet decompositionmethod can effectivelyextract 119878III and remove sharp peaks caused by trainsas well principal component analysis can effectivelysimplify the multivariate linear regression modelsbetween 119878III and temperature field fitting parametersof linear functions verify goodmodeling effectivenessof multivariate linear correlation models

(3) Three kinds of degradation regulations of static per-formance are put forward to assess static performanceof Dashengguan Yangtze Bridge and the results showthat residual static strains of all static strain data donot obviously present any kind of the degradationregulations indicating that the static performance ofDashengguan Yangtze Bridge was in a good conditionduring that period

What should to be mentioned is that the conclusionsabove are based on the monitoring data in certain periodHowever the static performance degradation behavior of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

0 4 8 12 16 20

0

Time (day)

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 1

(120583120576)

(a) Residual static strains of 1198781

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 2

(120583120576)

(b) Residual static strains of 1198782

Stat

ic st

rain

resid

uals

ofS 3

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(c) Residual static strains of 1198783

Stat

ic st

rain

resid

uals

ofS 4

(120583120576)

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

(d) Residual static strains of 1198784

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 5

(120583120576)

(e) Residual static strains of 1198785

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 6

(120583120576)

(f) Residual static strains of 1198786

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 7

(120583120576)

Static strain residuals

Central lineUpper and lower limit

(g) Residual static strains of 1198787

Static strain residuals

Central lineUpper and lower limit

0 4 8 12 16 20Time (day)

0

40

80

120

minus40

minus80

Stat

ic st

rain

resid

uals

ofS 8

(120583120576)

(h) Residual static strains of 1198788

Figure 13 Time histories of residual static strains for each static strain data

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

bridge structures is a very slow process therefore staticperformance assessment of Dashengguan Yangtze Bridgeshould be further analyzed by the accumulation of long-termmeasurement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the National Scienceand Technology Support Program (no 2014BAG07B01) theNational Natural Science Foundation (no 51178100) theFundamental Research Funds for the Central Universitiesand the Innovation Plan Program for Ordinary UniversityGraduates of Jiangsu Province in 2014 (no KYLX 0156) theScientific Research Foundation of Graduate School of South-east University (no YBJJ1441) and Key Program of Ministryof Transport (no 2011318223190 and no 2013319223120)

References

[1] T H Yi H N Li and X D Zhang ldquoSensor placementon Canton Tower for health monitoring using asynchronous-climbing monkey algorithmrdquo Smart Materials and Structuresvol 21 no 12 pp 1ndash12 2012

[2] T-H Yi H-N Li andMGu ldquoRecent research and applicationsof GPS-based monitoring technology for high-rise structuresrdquoStructural Control andHealthMonitoring vol 20 no 5 pp 649ndash670 2013

[3] R Regier and N A Hoult ldquoDistributed strain monitoring forbridges temperature effectsrdquo in Sensors and Smart StructuresTechnologies for Civil Mechanical and Aerospace Systems vol9061 of Proceedings of SPIE San Diego Calif USA March 2014

[4] M Gul H B Gokce and F N Catbas ldquoA characterizationof traffic and temperature induced strains acquired using abridge monitoring systemrdquo in Proceedings of the 2011 StructuresCongress pp 89ndash100 April 2011

[5] M Li W-X Ren Y-D Hu and N-B Wang ldquoSeparating tem-perature effect from dynamic strain measurements of a bridgebased on analytical mode decomposition methodrdquo Journal ofVibration and Shock vol 31 no 21 pp 6ndash29 2012 (Chinese)

[6] A Bueno B D Torres P Calderon and S Sales ldquoMonitoringof a steel incrementally launched bridge construction withstrain and temperature FBGs sensorsrdquo in Optical Sensing andDetection 772620 vol 7726 of Proceedings of SPIE BrusselsBelgium April 2010

[7] B Gu Z-J Chen and X-D Chen ldquoAnalysis of measuredeffective temperature and strains of long-span concrete boxgirder bridgerdquo Journal of Jilin University Engineering andTechnology Edition vol 43 no 4 pp 877ndash884 2013 (Chinese)

[8] Y-F Duan Y Li and Y-Q Xiang ldquoStrain-temperature corre-lation analysis of a tied arch bridge using monitoring datardquo inProceedings of the 2nd International Conference on MultimediaTechnology (ICMT rsquo11) pp 6025ndash6028 July 2011

[9] S Li H Li J Ou and H Li ldquoIntegrity strain response analysisof a long span cable-stayed bridgerdquo Key Engineering Materialsvol 413-414 pp 775ndash783 2009

[10] T Figlus S Liscak A Wilk and B Łazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[11] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature selectionfor optimization of wavelet packet decomposition in reliabilityanalysis of systemsrdquo International Journal on Artificial Intelli-gence Tools vol 22 no 5 Article ID 1360011 2013

[12] Y-L Ding and G-X Wang ldquoEstimating extreme temperaturedifferences in steel box girder using long-term measurementdatardquo Journal of Central SouthUniversity vol 20 no 9 pp 2537ndash2545 2013

[13] H W Wang R Guan and J J Wu ldquoComplete-information-based principal component analysis for interval-valued datardquoNeurocomputing vol 86 pp 158ndash169 2012

[14] A Singh S Datta and S S Mahapatra ldquoPrincipal componentanalysis and fuzzy embedded Taguchi approach for multi-response optimization in machining of GFRP polyester com-posites a case studyrdquo International Journal of Industrial andSystems Engineering vol 14 no 2 pp 175ndash206 2013

[15] A Amiri A Saghaei M Mohseni and Y Zerehsaz ldquoDiagnosisaids in multivariate multiple linear regression profiles monitor-ingrdquo Communications in Statistics Theory and Methods vol 43no 14 pp 3057ndash3079 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of