Research Article Dynamic Responses of …downloads.hindawi.com/journals/mpe/2015/951502.pdfResearch Article Dynamic Responses of Continuous Girder Bridges with Uniform Cross-Section

Research ArticleDynamic Responses of Continuous Girder Bridges with UniformCross-Section under Moving Vehicular Loads

Qingfei Gao1 Zonglin Wang1 Hongyu Jia2 Chenguang Liu1 Jun Li13

Binqiang Guo4 and Junfei Zhong1

1School of Transportation Science and Engineering Harbin Institute of Technology Harbin 150090 China2Department of Civil Engineering Southwest Jiaotong University Emei Campus Emeishan 614202 China3School of Architecture Engineering and Technology Heilongjiang College of Construction Harbin 150050 China4Zhejiang Provincial Institute of Communications Planning Design and Research Hangzhou 310000 China

Correspondence should be addressed to Qingfei Gao gaoqingfei 1986163com

Received 13 November 2014 Accepted 26 November 2014

Academic Editor Yun-Bo Zhao

Copyright copy 2015 Qingfei Gao et al This 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

To address the drawback of traditional method of investigating dynamic responses of the continuous girder bridge with uniformcross-section under moving vehicular loads the orthogonal experimental design method is proposed in this paper Firstly someempirical formulas of natural frequencies are obtained by theoretical derivation and numerical simulation The effects of differentparameters on dynamic responses of the vehicle-bridge coupled vibration system are discussed using our own program Finally theorthogonal experimental design method is proposed for the dynamic responses analysisThe results show that the effects of factorson dynamic responses are dependent on both the selected position and the type of the responses In addition the interaction effectsbetween different factors cannot be ignored To efficiently reduce experimental runs the conventional orthogonal design is dividedinto two phases It has been proved that the proposed method of the orthogonal experimental design greatly reduces calculationcost and it is efficient and rational enough to study multifactor problems Furthermore it provides a good way to obtain morerational empirical formulas of the DLA and other dynamic responses which may be adopted in the codes of design and evaluation

1 Introduction

The dynamic response of bridges due to moving vehicularloads has been a subject of interest to engineers more than100 years Owning to the uncertain transient and nonlin-ear characteristics of the vehicle-bridge coupled vibrationsystem the phenomenon of dynamic response is one thatrequires fairly complexmathematical treatment to adequatelycharacterize the motion and forces within the structureHowever for simplicity it will instead be conceptual treat-ment using a common format one that describes dynamicresponse as a fraction or multiple of the response thatwould be obtained if the same forces or loads were appliedstatically This is in fact the same approach that mostdesign specifications use to calculate the effects of dynamiclive loading When using such an approach the amount of

response above the static value is typically called the dynamicincrement and is found by multiplying the static value byan ldquoimpact fractionrdquo Alternatively the total response can beexpressed as a multiple of the static value using an impactfactor [1] In recognition of the complex behavior associatedwith the dynamic response from vehicular loading severalauthors have observed that the term ldquoimpactrdquo is too limitedand therefore not descriptive of the actual behavior [2ndash5]Instead the current trend is to replace the term ldquoimpactfractionrdquo with ldquodynamic load allowancerdquo which representsthe response from all types of vehicular dynamic effectsnot solely impact In this study the term ldquodynamic loadallowancerdquo is adopted

It has been proved that many different definitions ofdynamic load allowance are adopted by various researchersand engineers [6 7] Based on the design theory of the bridge

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 951502 29 pageshttpdxdoiorg1011552015951502

2 Mathematical Problems in Engineering

in Chinese current code the dynamic load allowance in thispaper is defined as follows

120583 (119909119887) =119910119889max (119909119887 119909V1)

119910119904max (119909119887 119909V2)minus 1 (1)

where 119909119887 is the location of the selected section 120583(119909119887) denotesthe dynamic load allowance of the selected section of thebridge and 119910119889max(119909119887 119909V1) and 119910119904max(119909119887 119909V2) respectivelydenote the maximum dynamic response and the maximumstatic response of the selected section Furthermore 119909V1 and119909V2 mean the location of the vehicle when the dynamicresponse and the static response reach the maximum valueIt should be noted that 119909V1 and 119909V2 are not the same in mostof the time

The dynamic response of a bridge to a crossing vehicleis a complex problem affected by the dynamic characteristicsof both the bridge and the vehicle and by the bridge surfaceconditions [1] Extensive research and development has beencarried out to understand the vibration of bridges as aresult of natural sources of vibration and to determinethe dynamic allowance as a function of the fundamentalfrequency due to its uniqueness [8ndash11] So it is importantto find a reliable method to estimate the natural frequencyof bridges Komatsu et al evaluated the natural vibrationresponses of straight and curved I- or box-girder bridgesusing Vlasovrsquos beam theory [12 13] Cantieni conductedan experimental study to establish a reliable relationshipbetween the fundamental frequency and maximum spanlength of bridges [14] Meanwhile Billing presented tablesof natural frequencies for straight continuous symmetricmultispan uniform beams of two through six spans forvarious span ratios [15] Additionally in order to reducethe significant difference between the estimation of thenatural frequency obtained from the codes and theoreticalmethods or numerical simulation Gao et al proposed anumerical improved method for straight bridges [16 17]Also the empirical expressions are established by Mohseniet al using regression analysis to determine the fundamentalfrequencies of skew continuous multicell box-girder bridges[18] However some limitations still exist for these methodsto actually be used in practice

Over the last 60 years a significant amount of researchhas been conducted in the area of bridge dynamics andthis research has been both analytical and experimental inscope Following the collapse of several railway bridges inGreat Britain the first study of vehicle-bridge interactionwas conducted in 1849 by Willis [19] A major experimentalinvestigation of bridge impact loads was initiated in 1958 byAmerican Association of State Highway Officials (AASHO)Eighteen simply supported bridges traversed by the two-axletrucks and the three-axle tractor-trailer combinations wereinvestigated each with a span of 15m [20] Wright and Greenreported on a series of experimental investigations of vehicle-bridge interaction made on 52 highway bridges in Ontariofrom 1956 to 1957 [21] Csagoly et al conducted a secondseries of tests on bridges (11 bridges) in Ontario in 1969 to1971 [22] Page [23] and Leonard [24] reported on a seriesof dynamic tests conducted on highway bridges (30 bridges)

by the Transport and Road Research Laboratory (TRRL) inEngland Shepard and Sidwell [25] Shepard and Aves [26]andWood and Shepard [27] reported on fourteen bridge testsconducted in New Zealand In 1980 a third series of dynamictesting of highway bridges in Ontario was performed Thetests were conducted on 27 bridges of various configurationsof steel concrete and timber construction [5 28] Cantienipresented results from the dynamic load tests of 226 highwaybridges in Switzerland from the mid-1950s to the early 1980s[14 29] As for the numerical simulation in 1970 Veletsosdeveloped a numerical method for computing the dynamicresponse of highway bridges to moving vehicular loads [30]Guant et al investigated analytically the effects on bridgeaccelerations of several parameters including the propertiesof the bridge and the vehicle as well as the initial conditionsof the roadway [31] Inbanathan and Wieland performed ananalytical investigation of the dynamic response of a simplysupported box girder bridge due to vehicle moving across thespan [32] Coussy et al presented an analytical study of theinfluence of random surface irregularities on the dynamicresponse of bridges [33] Recently Bakht and Pinjarkarpresented a literature review of bridge dynamics with aparticular focus on the dynamic testing of highway bridges[7] And Paultre et al presented a comprehensive reviewof analytical and experimental findings on bridge dynamicsand the evaluation of the dynamic load allowance [19]All these studies have proved that many of the parametersinteract with one another further complicating the issue andconsequentlymany research studies have reported seeminglyconflicting conclusions

Therefore themethod of orthogonal experimental designin batches [34ndash38] has been proposed in this paper It can beused for studying both the main effects and the interactioneffects Meanwhile due to the processing in batches it greatlyreduces the calculation cost

In this study the natural frequencies of the simplysupported girder bridge and the continuous girder bridgewith uniform cross-section are firstly investigated based onthe theoretical derivation and the numerical simulation Andthen the effects of different parameters on the dynamicresponses of the vehicle-bridge coupled vibration systemare discussed by our own program VBCVA Finally theorthogonal experimental design method is proposed forthe dynamic responses analysis including the case withoutinteraction and the case considering interaction

2 Natural Frequencies of MultispanContinuous Girder Bridges

In civil and transportation engineering dynamic charac-teristics of structures always consist of natural frequenciescorresponding vibration modes and damping ratios Thenatural frequency is one of the most important dynamiccharacteristics due to its key role in determining dynamicresponses of the structure under dynamic loads such aswindearthquake and moving vehicles Also the impact factor ordynamic load allowance which is used for considering thedynamic increments generated by moving vehicular loads is

Mathematical Problems in Engineering 3

Table 1 Three different types of boundary of single-span bridge

Boundary Hinged and hinged Fixed and hinged Fixed and fixed

Schematic plot

Boundary description

120593119899(0) = 0 12059310158401015840

119899(0) = 0

120593119899(119871) = 0 12059310158401015840

119899(119871) = 0

120593119899(0) = 0 1205931015840

119899(0) = 0

120593119899(119871) = 0 12059310158401015840

119899(119871) = 0

120593119899(0) = 0 1205931015840

119899(0) = 0

120593119899(119871) = 0 1205931015840

119899(119871) = 0

Note 120593 = 0means the deflection is zero 1205931015840 = 0means the rotation is zero and 12059310158401015840 = 0means the moment is zero

Table 2 The results of single-span bridges with different boundary conditions

Boundary Vibration modes Frequency equation First three frequencies

Hinged and hinged 120593119899(119909) = 1198601 sin120572119899119909 sin120572119899119871 = 0

1205721 =120587

119871

1205722 =2120587

119871

1205723=3120587

119871

Fixed and hinged

120593119899(119909) = 119860

1[sin120572

119899119909 minus sinh120572

119899119909

+sinh120572119899119871 minus sin120572119899119871cos120572119899119871 minus cosh120572119899119871

(cos120572119899119909 minus cosh120572119899119909)]

tan120572119899119871 minus tanh120572119899119871 = 0

1205721 =3927

119871

1205722 =7069

119871

1205723 =10210

119871

Fixed and fixed

120593119899(119909) = 1198601[sin120572119899119909 minus sinh120572119899119909

+cos120572119899119871 minus cosh120572119899119871sin120572119899119871 + sinh120572119899119871

(cos120572119899119909 minus cosh120572119899119909)]

3 minus 2 cos120572119899119871 cosh120572119899119871 = 0

1205721 =4739

119871

1205722 =7853

119871

1205723 =10996

119871

given as the function of natural frequencies inmost codes andspecifications for bridge design In addition the serviceabilityof bridges especially for pedestrian bridges is closely relatedto the natural frequency Therefore the natural frequency isfirstly studied in this paper and it is the basis of dynamicresponses analysis

21 Simply Supported Girder Bridges The boundary of thesimply supported girder bridge is assumed as pinned jointin both ends However according to inspections of existingbridges we found that the ends could not completely rotatewith the degeneration of rubber bearings For convenienceof engineers and researchers three types of boundary areinvestigated which are shown in Table 1

As for the Euler beam with uniform cross-section if thedamping is ignored the vibration governing equation can besimplified as

119898 119910 + 1198641198681205974119910

1205971199094= 119901 (119909 119905) (2)

where the 119901 denotes the force and119898 119864 119868 119910 119909 and 119905 denoterespectively the mass per unit length elasticity modulusmoment of inertia displacement location and time

Based on the method of separation of variables thesolution can be listed as follows

120596119899 = 1205722

119899radic119864119868

119898(3)

120593119899 (119909) = 1198601 sin120572119899119909 + 1198602 cos120572119899119909

+ 1198603 sinh120572119899119909 + 1198604 cosh120572119899119909(4)

where 120596119899 is the 119899th natural frequency (rads) 120593119899 is the 119899thvibrationmode and 120572119899 is the constant parameter determinedby both the cross-section and the boundary conditions Inaddition119860111986021198603 and1198604 are constant parameters whichare determined by the boundary condition

Submitting (4) into different types of boundaries inTable 1 then the results can be seen in Table 2

It can be seen from Table 2 that the natural frequencyis higher when there are more constraints and the corre-sponding stiffness is higher The 119899th natural frequency of thesingle-span bridge with different boundary conditions can besummarized as

119891119899 =120587

2(119899 + 025119887

119871)

2

radic119864119868

119898 (5)


where 119887 is used for denoting different boundary conditionsand 119871 is the span length When the boundary is hinged andhinged the parameter 119887 = 0 As for the other two conditions119887 = 1 and 119887 = 2 in turn

For common bridges themoment of inertia 119868 is positivelycorrelated with the factor ℎ3 (ℎ is the height of the beam) andthe area119860 is positively correlated with the hight ℎ Thereforethe 119899th natural frequency 119891119899 is negatively related to thespan length 119871 And it successfully explains the phenomenonthat most empirical formulas regressed by fielding test forestimating natural frequencies are negatively correlated withthe span length 119871

22 Continuous Girder Bridges with Uniform Spans As for2-span 3-span and 4-span continuous girder bridges withuniform cross-section and uniform spans the natural fre-quency and corresponding vibration modes are given by Gaoet al [16] Based on a similarmethodmore general frequencyequation can be listed as follows

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus2119866119899 119867119899 0 sdot sdot sdot 0 0

119867119899 minus2119866119899 119867119899 sdot sdot sdot 0 0

0 119867119899 minus2119866119899 119867119899 0 0

d d d

0 0 0 119867119899 minus2119866119899 119867119899

0 0 0 sdot sdot sdot 119867119899 minus2119866119899

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0 (6)

where

119866119899 =cosh120572119899119871sinh120572119899119871

minuscos120572119899119871sin120572119899119871

119867119899 =1

sinh120572119899119871minus

1

sin120572119899119871

(7)

For common bridges the number of spans is not morethan eight Equation (6) is solved by the general softwareMATLAB and the frequency equations and the naturalfrequencies are listed in Tables 3 and 4

It can be seen fromTable 4 that the first natural frequencyof the multispan continuous girder bridge with uniformspans is completely the same as that of the correspondingsimply supported girder bridge Also for bridges with dif-ferent spans the fundamental vibration modes are retainedHowever the number of other vibration modes amongfundamental vibration modes is more for the bridge withmore spans

In the Chinese code D60-2004 for bridge design theimpact factor is defined as the function of the naturalfrequency For continuous girder bridges the impact factorof the positive moment in mid-span is different from that ofthe negativemoment in pier-top section which is determinedby the curvature of differentmodes According to the analysisof vibration modes distribution for positive moment in mid-span section the firstmode is significant for all bridges How-ever for negative moment in pier-top section the curvatureof the second vertical vibration mode is larger for even-spanbridges and the curvature of the third vertical vibrationmodeis larger for odd-span bridges (Figure 1)

1st mode

4-span bridge 5-span bridge

2nd mode

3rd mode

Figure 1 Vertical vibration modes distribution for bridges withdifferent spans

Table 3 Frequency equations of multispan continuous girderbridges with uniform spans

Number of spans Frequency equation2 119866119899 = 0

3 41198662

119899minus 1198672

119899= 0

4 119866119899(41198662

119899minus 21198672

119899) = 0

5 161198664

119899minus 12119866

2

1198991198672

119899+ 1198674

119899= 0

6 119866119899(161198664

119899minus 16119866

2

1198991198672

119899+ 31198674

119899) = 0

7 641198666

119899minus 80119866

4

1198991198672

119899+ 24119866

2

1198991198674

119899minus 1198676

119899= 0

8 119866119899(641198666

119899minus 96119866

4

1198991198672

119899+ 40119866

2

1198991198674

119899minus 41198676

119899) = 0

So the second natural frequency in the even-span bridgeand the third natural frequency in the odd-span bridge areselected for calculating the impact factor of negative momentin pier-top section Tomake it clear both of them are denotedas 1198912 Based on Table 4 the empirical formula of 1198912 can begiven by

1198912 =1

2120587(120582120587

119871)

2

radic119864119868

119898= 1205822 120587

21198712radic119864119868

119898 (8)

where 120582 is the modified factor and it can be gained by

120582 =

125 119904 = 2

1 + 008 times 22minus119898

119904 = 2119898 2 le 119898 le 4

1 + 036 times 22minus119898

119904 = 2119898 minus 1 2 le 119898 le 4

(9)

where 119904 is the number of spans

23 Continuous Girder Bridges with Nonuniform Spans Themultispan continuous girder bridge is usually seen in urbanareas and other areas with special terrain For urban bridgeswhen the road extends under the bridge the span above theroad will be larger than others For bridges in special terrainareas the piers cannot be constructed at the idealized locationdue to the bad terrain occasionally so the spans should benonuniformly arranged

To our experience of bridge design the length of side spanis always smaller in the multispan continuous girder bridgewith nonuniform spans The span ratio 119903 is introduced hereAnd it has been assumed that only the length of side span isdifferent from others (Figure 2)

119903 =119871 119904

119871119898

(10)


Table 4 Natural frequencies of multispan continuous girder bridges with uniform spans

Number of spans Natural frequencies (Hz)

2 119891119899 =1

21205871198712(1205872 3927

2 41205872 )radic

119864119868

119898

3 119891119899 =1

21205871198712(1205872 3556

2 4298

2 41205872 )radic

119864119868

119898

4 119891119899 =1

21205871198712(1205872 3393

2 3927

2 4463

2 41205872 )radic

119864119868

119898

5 119891119899 =1

21205871198712(1205872 3309

2 3700

2 4153

2 4550

2 41205872 )radic

119864119868

119898

6 119891119899 =1

21205871198712(1205872 3261

2 3556

2 3927

2 4298

2 4601

2 41205872 )radic

119864119868

119898

7 119891119899 =1

21205871198712(1205872 3230

2 3460

2 3764

2 4089

2 4395

2 4634

2 41205872 )radic

119864119868

119898

8 119891119899 =1

21205871198712(1205872 3210

2 3393

2 3645

2 3927

2 4208

2 4463

2 4655

2 41205872 )radic

119864119868

119898

Ls Lm

Ls LsLm

Ls Lm Ls

Ls

Lm

Lm LmLs Lm

Figure 2 Schematic plots of multispan continuous girder bridgeswith nonuniform spans

where 119871 119904 is the length of side span and 119871119898 is the length ofmiddle span

The frequency equation of this type of bridge can be givenby10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

minus119866119899119904minus 119866119899119898119867119899119898

0 sdot sdot sdot 0 0

119867119899119898

minus2119866119899119898119867119899119898sdot sdot sdot 0 0

0 119867119899119898minus21198661198991198981198671198991198980 0

d d d

0 0 0 119867119899119898minus2119866119899119898

119867119899119898

0 0 0 sdot sdot sdot 119867119899119898minus119866119899119904minus 119866119899119898

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0

(11)

When the number of spans is from 2 to 5 the frequencyequations are listed in Table 5

The empirical formulas for estimating natural frequenciesof the bridge with nonuniform spans are listed as

1198911 =1

2120587(1205781120587

119871)

2

radic119864119868

119898 (12a)

1198912 =1

2120587(1205782120587

119871)

2

radic119864119868

119898 (12b)

where 1205781 and 1205782 are modified factors which can be seen inFigure 3

It can be seen from Figure 3(a) that all of modified factorsare not less than 10 In otherwords the first natural frequencyof the multispan continuous girder bridge is larger thanthat of the corresponding simply supported girder bridge

Table 5 Frequency equations of multispan bridges with nonuni-form spans

Number ofspans Frequency equation

2 119866119899119904 + 119866119899119898 = 0

3 (119866119899119904 + 119866119899119898)2minus 1198672

119899119898= 0

4 (119866119899119904+ 119866119899119898)(1198662

119899119898+ 119866119899119904119866119899119898minus 1198672

119899119898) = 0

5 1198662

119899119904(41198662

119899119898minus 1198672

119899119898) + 119866119899119904119866119899119898(8119866

2

119899119898minus 61198672

119899119898) +

(41198664

119899119898minus 51198662

1198991198981198672

119899119898+ 1198674

119899119898) = 0

Because the side span length is smaller than the middle spanlength the stiffness has been increased by the side span Inaddition the modified factor 1205781 of the bridge with two spansis completely the same as that of the bridge with four spansThis is due to the free rotation of the middle support of thebridge with four spans and the first vibration mode of four-span bridge is equal of that of two two-span bridges

It can be seen from the Figure 3(b) that the modifiedfactor 1205782 of 4-span bridge is almost the same as that of 5-spanbridge They are largely different from others and should benoted

Finally it has been emphasized that the empirical formu-las and modified factors proposed in this paper can be usedfor estimating natural frequencies of bridges with uniformcross-section Also according to these formulas the factorsinfluencing natural frequencies are more clearly seen whichmay give a better guideline for the design and evaluation ofdynamic performance of bridges

3 Dynamic Responses Analysis byTraditional Method

Research over the last 40 years has shown that the dynamicresponses of bridges under moving vehicles are influencedby many parameters such as the dynamic characteristicsof the vehicle the dynamic characteristics of the bridgeand variations in the surface conditions of the bridge and


1205781

16

15

14

13

12

11

1000 02 04 06 08 10

Span ratio r

3-span4-span (2-span)

5-span

(a) Modified factor 1205781

1205782

00 02 04 06 08 10

Span ratio r

38

34

30

26

22

18

14

10

2-span3-span

4-span5-span

(b) Modified factor 1205782

Figure 3 Modified factors of natural frequencies for the bridge with nonuniform spans

approach roadways [1] However most of the studies focusedon the dynamic load allowance in midspan of the bridgeIn this part the dynamic load allowances (DLAs) in theside span and the middle span are discussed respectivelyto obtain the differences of DLAs in variance sections Alsofor comfort analysis the accelerations of the bridge and thevehicle are investigated

Based on the existing research results thirteen parame-ters are considered here According to the traditionalmethodevery parameter is studied respectively In other words if thisparameter is changing all of the other parameters are fixedTherefore the fundamental model of bridge-vehicle coupledsystem is assumed at first and then every parameter is variedfrom the basis of this modelThey are listed in Tables 6 7 and8

According to the numerical simulation the effect ofdifferent parameters on the dynamic response of the vehicle-bridge system is investigated by our own program VBCVA(Vehicle-Bridge Coupled Vibration Analysis) It has to benoted that the pavement roughness model proposed byHwang and Nowak has been adopted in this program [39]

31 Effect of Roughness For actual bridges the pavementroughness is an inevitable factor The effect of roughness ondynamic responses of the vehicle-bridge system is shown inFigure 4

It can be seen from Figure 4 that with the deteriorationof bridge pavement conditions the DLA of the bridge theacceleration of the bridge and the acceleration of the bridgegrow in a linear fashion Moreover the dynamic responseof the middle span is smaller than that of the side span Inaddition the growth rate in the middle span is a little smallerthan that of the side span

A great deal of research shows that the bridge pavementis one of the most vulnerable components due to its smaller

stiffness [40 41] The results obtained from this study showthat the roughness may significantly promote the vibration ofthe bridge traversed by moving vehicular loads This will setup a vicious circle of increasing vibration poorer roughnessand increased stress Additionally the larger vibration accel-eration of the vehicle induced by the poorer roughness willmake the passengers uncomfortable Therefore we shouldpaymore attention to the bridge pavement condition in actualoperation Prompt repair of the damaged bridge pavementwill reduce the unnecessary cost

32 Effect of Span Length As for the continuous girderbridges with uniform cross-section the common span lengthranges from 20m to 40m It has to be emphasized that thecross-sections of bridges with different span length are thesame The effect of span length on dynamic response of thevehicle-bridge system is shown in Figure 5

It can be seen from Figure 5 that as the span lengthincreases the DLA of the middle span firstly decreases andthen keeps steady and the DLA of the side span is on the risewhich is largely different from the former one But the changeof the vibration acceleration in the middle span is the sameas that of the vibration acceleration in the side span Both ofthem slightly decrease and then increase Additionally thevibration acceleration of the vehicle rises at first and goesdown later with the increasing of the span length

For most dynamic responses in Figure 5 there is anextreme point with the increasing of the span length It maybe due to the resonance phenomenon between the bridge andthe vehicleThe vibration natural frequency of the bridgewith25m long span may be more close to that of the vehicle

33 Effect of Span Ratio From the aspect of mechanicalcharacteristics due to the static load the length of side spanis not larger than that of middle span and not smaller than


Table 6 Parameters of the bridge in the fundamental model

L (m) W (m) I (m4) A (m2) E (MPa) 120588 (kgm3) 120585 X (cm)40 12 354 708 34500 2600 005 20L lengthW width I moment of inertia E modules of elasticity 120588 density 120585 damping ratio and X maximum magnitude of the roughness

Table 7 Parameters of the vehicle in the fundamental model

Parameters ValueMass of truck body 31800 kgMass of front wheel 400 kgMass of middlerear wheel 600 kgPitching moment of inertia 40000 kgsdotm2

Rolling moment of inertia 10000 kgsdotm2

Distance (front axle to center) 460mDistance (middle axle to center) 036mDistance (middle to rear axle) 140mWheel base 180mUpper stiffness (front axle) 1200 kNsdotmminus1

Upper stiffness (middlerear axle) 2400 kNsdotmminus1

Upper damping (front axle) 5 kNsdotssdotmminus1

Upper damping (middlerear axle) 10 kNsdotssdotmminus1

Lower stiffness (front axle) 2400 kNsdotmminus1

Lower stiffness (middlerear axle) 4800 kNsdotmminus1

Lower damping (front axle) 6 kNsdotssdotmminus1

Lower damping (middlerear axle) 12 kNsdotssdotmminus1

Speed 80 kmh

Table 8 Parameters description and their values

Index Description Values NoteA Pavement roughness 1 cm 2 cm 3 cm 4 cm 5 cm Maximum magnitudeB Length of the main span 20m 25m 30m 35m 40mC Ratio between side span and middle span 05 06 07 08 09 10 Changing the side spanD Number of spans 3 4 5 6 7E Weight of the bridge (06 08 10 12 14) times 120588 Changing the densityF Stiffness of the bridge (06 08 10 12 14) times E Changing the modulesG Damping ratio of the bridge (06 08 10 12 14) times 120585H Weight of the vehicle (06 08 10 12 14) times 119872V Including body and tyresI Upper stiffness of the vehicle (06 08 10 12 14) times 119896119904J Upper damping of the vehicle (06 08 10 12 14) times 119888119904K Lower stiffness of the vehicle (06 08 10 12 14) times 119896119905L Lower damping of the vehicle (06 08 10 12 14) times 119888119905M Speed of the vehicle 20sim120 kmh (step = 20 kmh)119872V the whole weight of the vehicle 119896119904 upper stiffness 119888119904 upper damping 119896119905 lower stiffness and 119888119905 lower stiffnessThe values of all the parameters are obtainedfrom the fundamental model (Table 7)

half of the length of the middle span So the span ratio of theside span and the middle span ranges from 05 to 10 in thisstudy The effect of span ratio on dynamic responses of thevehicle-bridge system is shown in Figure 6

It can be seen from Figure 6 that the change of the DLA isalmost the same as that of the acceleration of the bridge In theside span they slightly decrease and then increase and thenkeep steady In the middle span they firstly go up and thengo down In addition the vibration acceleration of the vehicle

body keeps steady at first and then it goes downMoreover itcan be found that all of these dynamic responses reach theirextreme value when the span ratio is 09 In China as forthe fabricated girder bridge the span ratio is equal to 10 incommon And as for the cast in place the span ratio usuallyranges from 06 to 08

To determine the span ratio in design of the bridgeboth the static and dynamic characteristics should be paidattention to


400

350

300

250

200

150

1001 2 3 4 5

Roughness amplitude (cm)

Side spanMiddle span

DLA

(a) DLA of the bridge

1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)

(b) Acceleration of the bridge

1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000

(c) Acceleration of the vehicle

Figure 4 Effect of roughness on dynamic responses of the vehicle-bridge system

34 Effect of Spans Number For middle- and small-spancontinuous girder bridges the number of spans is alwaysranging from 3 to 7 owning to the expansion or contractionaccording to the change of temperature The effect of spansnumber on dynamic responses of the vehicle-bridge systemis shown in Figure 7

It can be seen from Figure 7 that the effect of spansnumber on the DLA of the bridge is not significant especiallyfor the DLA in the side span However the vibration acceler-ation of the bridge decreases with the increasing of the spansnumber Moreover the change of the vibration accelerationin the side span is much more similar to that in the middlespan Additionally the vibration acceleration of the vehiclebody is little influenced by the spans number of the bridge asthe fundamental natural frequency stays the same for bridgeswith different number of spans

Based on the conclusion in this study the dynamic per-formance is better for the bridge with more spans However

the expansion or contraction induced by the change of thetemperature may be larger when there are more spans As aresult it has to be synthetically considered in the design

35 Effect of Bridge Weight The bridge weight is changedaccording to amending the density of the material Thismethod can make sure that the area and the stiffness of thecross-section of the bridge are invariable The effect of bridgeweight on dynamic responses of the vehicle-bridge system isshown in Figure 8

It can be seen from Figure 8 that as the bridge weightincreases the DLA in the side span keeps steady while theDLA in the middle span jumpily decreases However thechange of the vibration acceleration in the side span and thatin themiddle span are almost the sameThey slightly increaseand then decrease Additionally the vibration acceleration ofthe vehicle body goes up and down and the range is small


220

200

180

160

140

120

10020 25 30 35 40

DLA

Length of the main span L (m)



20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40



Figure 5 Effect of span length on dynamic responses of the vehicle-bridge system

In natural itmay be related to the dynamic characteristicsof the bridgeThe natural frequency is lower when the bridgeweight is bigger As for the vehicle-bridge coupled vibrationsystem the resonance phenomenon will appear when itsnatural frequencies are so approximate

36 Effect of Bridge Stiffness The bridge stiffness is changedaccording to amending the elasticity modules of the materialThismethod canmake sure that the area and theweight of thebridge are invariableThe effect of bridge stiffness on dynamicresponses of the vehicle-bridge system is shown in Figure 9

It can be seen from Figure 9 that with the increasing ofthe bridge stiffness the DLA in the side span continues torise while the DLA in the middle span slightly decreasesand then increases However the change of the vibrationacceleration in the side span and that in the middle spanare almost the same They fluctuate and the range is small

Additionally the bridge stiffness has little influence on thevibration acceleration of the vehicle body

Similarly the natural frequency is higher when the bridgestiffness is bigger Also as for the vehicle-bridge coupledvibration system the resonance phenomenon will appearwhen its natural frequencies are so approximate

37 Effect of Bridge Damping As for the concrete structurethe damping ratio is assumed to be 005 In this study it floatsup and down at 40 percent The effect of bridge damping ondynamic responses of the vehicle-bridge system is shown inFigure 10

It can be seen from Figure 10 that the damping ratio haslittle influence on the DLA of the bridge But the vibrationacceleration of the bridge significantly decreases with theincreasing of the damping ratio Similarly the vibrationacceleration of the vehicle body continues to go down as thedamping ratio increases


220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1


Figure 6 Effect of span ratio on dynamic responses of the vehicle-bridge system

Damping is one of the most important dynamic char-acteristics of the bridge Up till now it cannot be obtainedby calculation The only way to obtain the damping of thestructure is by measuring in the fielding test Howeveraccording to the results in this paper the damping is not soimportant for calculating the DLA But it is closely related tothe vibration acceleration of the bridge and the vehicle whichmay influence the inertia force of the bridge and the ridingcomfort Therefore the damping should still be noticed

38 Effect of Vehicle Weight To investigate the effect of thevehicle weight on dynamic responses of the bridge the factorof the vehicle weight is introduced and the fundamentalweight is assumed to be 35 tons In this study the factor rangesfrom 06 to 14 at the step of 02 The effect of vehicle weighton dynamic responses of the vehicle-bridge system is shownin Figure 11

It can be seen from Figure 11 that when the factor issmaller than 10 the DLA in the side span slightly decreasesand then increases while the DLA in the middle span firstlydecreases and then keeps steady When the factor is biggerthan 10 the DLA in the side span decreases with the increas-ing of the vehicle weight but the DLA in the middle spanincreases with the increasing of the vehicle weight Howeveras the vehicle weight increases the vibration acceleration inthe side span goes down while that in the middle span goesup In addition the vibration acceleration of the vehicle bodydecreases with the increasing of the vehicle weight

Obviously the effect of vehicle weight on dynamicresponses in different positions is not the same Limitingthe vehicle weight can effectively reduce the static stress ofthe bridge but its influence on the dynamic responses isnot clearly enough In particular the change of the vibrationacceleration in different positions of the bridge is completelythe opposite


240

220

200

180

160

140

120

100

DLA

Number of spans3 4 5 6 7



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000



Figure 7 Effect of spans number on dynamic responses of the vehicle-bridge system

39 Effect of Upper Stiffness To investigate the effect of theupper stiffness on dynamic responses of the vehicle-bridgesystem the factor of the upper stiffness is introduced In thisstudy the factor ranges from 06 to 14 at the step of 02 Theeffect of upper stiffness on dynamic responses of the vehicle-bridge system is shown in Figure 12

It can be seen from Figure 12 that the DLA of the bridgealmost goes down with the increasing of the upper stiffnessof the vehicle However as the upper stiffness of the vehicleincreases the vibration acceleration of the bridge goes upMoreover the change of the acceleration in the side spanis faster than that in the middle span In other words theeffect of the upper stiffness on the dynamic response of theside span is much more sensitive In addition the vibrationacceleration of the vehicle body continues to go up with theincreasing of the upper stiffness of the vehicle

As a result the upper stiffness of the vehicle not onlyinfluences the dynamic responses of the bridge but also

significantly affects the vibration acceleration of the vehiclebody which is closely related to the comfort of passengersand the safety of the goods Therefore to find the friendlystiffness of the vehicle is urgent and efficient in the area ofriding comfort analysis

310 Effect of Upper Damping To investigate the effect of theupper damping on dynamic responses of the vehicle-bridgesystem the factor of the upper damping is introduced In thisstudy the factor ranges from 06 to 14 at the step of 02 Theeffect of upper damping on dynamic responses of the vehicle-bridge system is shown in Figure 13

It can be seen from Figure 13 that all of the dynamicresponses of the vehicle-bridge system decease with theincreasing of the upper damping of the vehicleThe change oftheDLA in the side span is faster than that in themiddle spanHowever the change of the vibration acceleration in the sidespan is almost the same as that in themiddle span It has to be


240

220

200

180

160

140

120

10006 08 1 12 14

Factor of bridge mass

DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


Figure 8 Effect of bridge weight on dynamic responses of the vehicle-bridge system

emphasized that the influence of the upper damping on thevibration acceleration of the vehicle is the most significantIn other words the shock absorber system of the vehicle is soimportant in its design

311 Effect of Lower Stiffness To investigate the effect of thelower stiffness on dynamic responses of the vehicle-bridgesystem the factor of the lower stiffness is introduced In thisstudy the factor ranges from 06 to 14 at the step of 02 Theeffect of lower stiffness on dynamic responses of the vehicle-bridge system is shown in Figure 14

It can be seen from Figure 14 that as the lower stiffnessof the vehicle increases the change of the DLA in the sidespan and that in the middle span are entirely the oppositeThe former one slightly increases and then decreases whilethe latter one slightly decreases and then increases Howeverthe vibration acceleration of the bridge goes up with theincreasing of the lower stiffness Moreover the change of

the vibration acceleration in the side span is faster than thatin themiddle span Additionally the vibration acceleration ofthe vehicle body goes up firstly and then keeps steady

Comparing Figure 12 with Figure 14 it can be found thatthe effect of the upper stiffness is almost the same as theeffect of the lower stiffness Moreover the former one is moresignificant than the latter one

312 Effect of Lower Damping To investigate the effect of thelower damping on dynamic responses of the vehicle-bridgesystem the factor of the lower damping is introduced In thisstudy the factor ranges from 06 to 14 at the step of 02 Theeffect of lower damping on dynamic responses of the vehicle-bridge system is shown in Figure 15

It can be seen from Figure 15 that as the lower dampingincreases all of dynamic responses of the vehicle-bridgesystem are falling But the changing range of them is small


240

220

200

180

160

140

120

10006 08 1 12 14

Factor of bridge stiffness

DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


Figure 9 Effect of bridge stiffness on dynamic responses of the vehicle-bridge system

Also the dynamic responses in the side span are significantlylarger than those in the middle span especially for the DLA

Comparing Figure 13 with Figure 15 it can be found thatthe effect of the upper damping is almost the same as theeffect of the lower dampingMoreover the former one ismoresignificant than the latter one

313 Effect of Vehicle Speed As for the loading truck thespeed ranges from20 kmh to 120 kmh at the step of 20 kmhThe effect of vehicle speed on dynamic responses of thevehicle-bridge system is shown in Figure 16

It can be seen from Figure 16 that as the speed increasesboth the DLA in the side span and the DLA in the middlespan increase firstly and then decrease But the critical speedsof them are not the same and the former one is biggerthan the latter one However the vibration acceleration ofthe bridge goes up with the increasing of the vehicle speedSimilarly the vibration acceleration of the vehicle body alsoincreases

Therefore it has to be noted that limiting the speeddirectly is not a rational way to ensure the safety of the bridgewhich is the common way during the maintenance of the oldor damaged bridge

Obviously the effect of every parameter on the dynamicresponses of the vehicle-bridge system can be studied by theprogram VBCVA When one parameter is studied the otherparameters should be fixed But according to this methodthe results may not occasionally be the same as the actualcondition because the interaction effects among differentparameters are not considered

4 Numerical Simulations Based onthe Orthogonal Experimental Design

There are two ways for the analysis of factors influencing thedynamic responses of the vehicle-bridge coupled vibrationsystem full factorial designs and fractional factorial designs


220

200

180

160

140

120

10006 08 1 12 14

Factor of bridge damping ratio

DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


Figure 10 Effect of bridge damping on dynamic responses of the vehicle-bridge system

Full factorial designs reveal whether the effect of each factordepends on the levels of other factors in the experimentThisis the primary reason for multifactor experiments One fac-torial experiment can show ldquointeraction effectsrdquo that a seriesof experiments each involving a single factor cannot [42]However fractional factorial designs permit investigation ofthe effects of many factors in fewer runs than a full factorialdesign

Data from fractional factorial designs are interpretedbased on the following two assumptions The first one is thesparsity of important effects Only a few of the many possibleeffects are prominent Even if many effects are nonzero weexpect a few to stand out as much larger than the restThe other one is the simplicity of important effects Maineffects andor two-factor interactions are more likely to beimportant than high-order interactions This is also knownas the hierarchical ordering principle

Therefore considering the high efficiency the orthogonalexperimental design one common type of the fractionalfactorial design is adopted in this study including thenumerical simulation without interaction and the numericalsimulation with interaction

41 Orthogonal Experimental Design According to the orth-ogonal experimental design two objectives can be attainedThe first objective is to select fewer typical combinationsfor the test And the other objective is to obtain the correctconclusions by the scientific method based on the limitedcombinations The orthogonal experimental design consistsof the design of test plan and the process of test data

As for the design of test plan the main procedures arelisted as follows [43] Firstly the objective and the test indexare clearly proposed Secondly the factors and their levels are


Factor of vehicle mass

220

200

180

160

140

120

10006 08 1 12 14

DLA



Factor of vehicle mass06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


Figure 11 Effect of vehicle weight on dynamic responses of the vehicle-bridge system

determined based on the comprehension in this research fieldother than the knowledge of statistics Thirdly the rationalorthogonal table is selected including the table consideringinteraction and the table without interaction Fourthly thetable header is designed Finally the test plan is formed

As for the process of the test data two methods areadopted including the range analysis and the varianceanalysis [44] The range analysis is much simpler and moreconvenient However the test error cannot be estimated andthe reliability of the results cannot be determined Also itcannot be applied in the fields of regression analysis andregression design But the variance analysis can remedy thedefects above

Themethod of range analysis is also called the visual anal-ysismethod and the119877 (the abbreviation of the range)methodIt consists of two procedures calculation and judgment Thediagram can be seen in Figure 17

In Figure 17 the 119870119895119898 is the sum of the test index with the119898th level in the 119895th column and 119896119895119898 is the average of the119870119895119898In addition the 119877119895 is the difference between the maximumand the minimum value of the 119896119895119898

119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)

The 119877119895 is represented for the changing amplitude of thetest index with the changing of the levels of the 119895th factorTherefore it can be concluded that the influence of the 119895thfactor on the test index is much more significant if the valueof 119877119895 is bigger Sometimes for more clarity the trend plot isgiven

Variance analysis is more rigorousThere are four steps torealize the variance analysis [45 46]


220

200

180

160

140

120

10006 08 1 12 14

DLA

Factor of upper stiffness



06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)



Figure 12 Effect of upper stiffness on dynamic responses of the vehicle-bridge system

Step 1 As for every factor calculate the sum of square ofdeviations (119878119895) the degree of freedom (dof 119891119895) and thevariance estimation (120590119895)

Step 2 Estimate the variance of the error (120590119890)

Step 3 Obtain the test static 119865 and compare 119865with its criticalvalue 119865120572 for given significance level 120572

Step 4 For simplicity the variance analysis table is listedincluding the process and the results Consider

119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)

in which 119910119894 is the index result of the 119894th run and 119910119895119896 is theindex result of the 119895th factor with the 119896th level Also 119878119890 and119891119890denote respectively the sum of square of deviations and thedegree of freedomof the error It has to be noted that the errorhas resulted from all of the vacant columns in the orthogonalarray Also the accuracy increases with the increasing dof ofthe error [47]Therefore if the significance level of one factoris larger than 025 it can be included as the error

In this study the orthogonal table 11987127(313) is used for

the numerical simulation without interaction while theorthogonal table11987116(2

15) is used for the numerical simulationwith interaction


06 08 1 12 14

220

200

180

160

140

120

100

DLA

Factor of upper damping



06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)



Figure 13 Effect of upper damping on dynamic responses of the vehicle-bridge system

42 Numerical Simulation without Interaction There arethirteen factors possibly affecting dynamic responses of thevehicle-bridge coupled vibration system A large numberof studies [8] have proved that the roughness is the mostsignificant factorTherefore the roughness is not arranged inthe orthogonal table The other twelve factors are arrangedin the orthogonal table once in a column And the residualcolumn is thought as the test error

The results obtained from the VBCVA are listed inTable 9The range analysis of the data can be seen in Table 10and Figure 18 The variance analysis of the data is shown inTable 11

It can be concluded that the influence of factors ondifferent indices is listed as follows

(i) For DLA in the side span of the bridge consider

B gt H gt C gt J gt D gt E gt K gt F gt I gt L gt G gt M (15)

(ii) For DLA in the middle span of the bridge consider

D gt C gt E gt H gt I gt G gt B gt K gt L gt M gt J gt F (16)

(iii) For vibration acceleration in the side span of thebridge consider

D gt B gt E gt J gt M gt I gt C gt F gt K gt G gt L gt H (17)

(iv) For vibration acceleration in the middle span of thebridge consider

D gt B gt C gt E gt J gt M gt I gt H gt G gt F gt L gt K (18)

(v) For vibration acceleration of the vehicle body con-sider

H gt I gt K gt B gt C gt J gt D gt L gt F gt E gt G gt M (19)


220

200

180

160

140

120

100

DLA

Factor of lower stiffness06 08 1 12 14



050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)



Figure 14 Effect of lower stiffness on dynamic responses of the vehicle-bridge system

It has to be noted that the roughness is assumed as thesignificant factor Obviously among other factors the mostimportant factors affecting the different dynamic responsesare not the same

43 Numerical Simulation with Interaction As mentionedearlier the interaction almost exists in all of physical phe-nomena When the interaction is so small it can be ignoredin application However as for research we do not knowwhether the interaction can be ignored or not at first

Based on the results from the above section eight mostimportant factors influencing theDLA are selected and someinteractions between them are investigated They are thepavement roughness (A) the length of the main span (B)the ratio between the side span and the middle span (C)the number of spans (D) the weight of the bridge (E) theweight of the vehicle (H) the upper stiffness of the vehicle(I) and the upper damping of the vehicle (J) Meanwhiledue to the calculation cost and the existing orthogonal array

the number of levels is determined as two for each factor Toavoid the mixture the most important factor is arranged atfirst It can be seen in Table 12

The results obtained from the VBCVA are listed inTable 13The range analysis of the data can be seen in Table 14The variance analysis of the data is shown in Table 15



A gt I gt B times C gt D gt J gt H gt D times C gt E gt A times D

gt B gt A times C gt A times B gt D times B gt C(20)


B gt A gt D gt D times B gt A times B gt H gt E gt A times D

gt J gt B times C gt I gt D times C gt C gt A times C(21)


220

200

180

160

140

120

100

DLA

Factor of lower damping06 08 1 12 14



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)



Figure 15 Effect of lower damping on dynamic responses of the vehicle-bridge system


B times C gt A gt D gt C gt I gt J gt B gt E gt A times D

gt A times B gt D times C gt A times C gt D times B gt H(22)


A gt B gt A times B gt B times C gt I gt H gt D gt D times B

gt A times D gt J gt E gt A times C gt D times C gt C(23)


A gt I gt H gt B gt D times C gt D times B gt J gt A times B

gt B times C gt A times D gt D gt A times C gt E gt C(24)

Similarly for different indices themost important factorsare not the same Also it should be noted that someinteractions are not ignored

44 Comparison with the Current Code In the current codeof China (General Code for Design of Highway Bridges andCulverts) [48] the dynamic load allowance (120583) is defined asthe function of the natural frequency It is given by

120583 =

005 119891 lt 15Hz01767 ln119891 minus 00157 15Hz le 119891 le 14Hz045 119891 gt 14Hz

(25)

where 119891 is the fundamental natural frequency of the bridgeIts unit is Hz


Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120


Figure 16 Effect of vehicle speeds on dynamic responses of the vehicle-bridge system

R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm

Figure 17 The diagram of the 119877method

TheDLA calculated by the programVBCVA is comparedwith the DLA obtained based on the current code in ChinaThe results are shown in Figure 19

It can be seen from Figure 19 that the differences betweenthem are large especially for the cases of the frequency from

2Hz to 4Hz It may be related to the natural frequency of thevehicle In other words the dynamic load allowance definedin the current code may be not rational enough To betterconsider the dynamic effects induced by moving vehicles inthe design phase some revisionsmay be needed in the future


Table 9 Results of numerical simulation without interaction

Run 1198911 (Hz) DLA119904 DLA119898 119886119887119904 (ms2) 119886119887119898 (ms2) 119886V (ms2)1 1482 1431 2436 02499 07623 348692 1657 1156 2574 03564 11550 292013 1816 1275 2317 02758 08258 309414 885 1302 1107 03288 02249 414585 990 1523 1775 01865 01588 489166 1084 1287 1283 02236 01858 265507 531 1772 1600 00899 01211 395788 594 1175 1045 01257 01683 302829 650 1429 1515 01237 01288 2374610 421 1218 1751 02646 01737 2719711 470 1489 2152 01145 01109 3759012 515 1719 1661 01441 00963 4530813 460 1326 1380 00784 01045 2184414 514 1685 1699 01005 01617 4491715 563 1323 1329 01537 01436 2786316 401 1434 1520 02780 01964 2560817 449 1491 1815 01407 01176 3674518 491 1412 1264 01521 01462 2636019 242 1293 1190 00866 02490 2588020 271 1413 1649 01057 02816 2702821 296 1522 2226 00794 03196 3036222 261 1979 1949 03605 04664 3619523 292 1319 2116 03079 04740 2940124 320 1886 2257 01505 02254 2871925 253 1818 1965 04168 02730 2165026 282 1904 1214 02497 02277 2096227 309 2981 2232 05926 03882 37986

Table 10 Range analysis of numerical simulation without interaction

Index B C D E F G H I J K L MDLA119904 0418 0322 0256 0247 0187 0070 0328 0164 0281 0244 0079 0049DLA119898 0248 0421 0513 0369 0132 0265 0309 0284 0139 0187 0179 0161119886119887119904 (ms2) 0103 0055 0175 0099 0052 0037 0015 0063 0088 0047 0031 0074119886119887119898 (ms2) 0276 0245 0299 0240 0044 0046 0047 0084 0093 0015 0016 0090119886V (ms2) 0526 0477 0401 0309 0342 0141 1099 0640 0457 0595 0394 0083

5 Conclusions

In this study the natural frequencies of the simply supportedgirder bridge and the continuous girder bridge with uniformcross-section are firstly investigated based on the theoreticalderivation and the numerical simulation And then the effectsof different parameters on the dynamic responses of thevehicle-bridge coupled vibration system are discussed by ourown program VBCVA Finally the orthogonal experimen-tal design method is proposed for the dynamic responses

analysis including the case without interaction and the caseconsidering interaction The conclusions obtained in thisstudy are summarized as follows

(1) The empirical formulas on natural frequency of thesingle-span bridge with different boundary condi-tions are obtained based on the theoretical derivationIt can be used for the simply supported girder bridgeespecially for some old bridges where bearing and


Table 11 Various analysis of numerical simulation without interaction

(a) Dynamic load allowance in the side span (DLA119904)

Factor 119878119895 dof Initial Modification120590119895 119865 120572 120590119895 119865 120572

B 08830 2 04415 1101 010 04415 995 001C 04764 2 02382 594 025 02382 537 005D 03311 2 01656 413 025 01656 373 010E 02776 2 01388 346 025 01388 313 010F 01697 2 00848 212 gt025 mdash mdashG 00237 2 00119 030 gt025 mdash mdashH 05193 2 02596 648 025 02596 585 005I 01290 2 00645 161 gt025 mdash mdashJ 03551 2 01775 443 025 01775 400 005K 02684 2 01342 335 025 01342 302 010L 00327 2 00163 041 gt025 mdash mdashM 00121 2 00060 015 gt025 mdash mdashError 00802 2 00401 00444Sum 35581 26

(b) Dynamic load allowance in the middle span (DLA119898)


B 02762 2 01381 407 025 01381 236 025C 08983 2 04492 1323 010 04492 767 001D 12312 2 06156 1813 010 06156 1052 001E 06344 2 03172 934 010 03172 542 005F 01002 2 00501 148 gt025 mdash mdashG 03232 2 01616 476 025 01616 276 025H 05709 2 02855 841 025 02855 488 005I 03674 2 01837 541 025 01837 314 010J 00934 2 00467 138 gt025 mdash mdashK 01627 2 00814 240 gt025 mdash mdashL 01598 2 00799 235 gt025 mdash mdashM 01183 2 00592 174 gt025 mdash mdashError 00679 2 00340 00585Sum 50039 26

(c) Vibration acceleration in the side span (119886119887119904)


B 00477 2 00239 1699 010 00239 1257 001C 00135 2 00068 482 025 00068 357 010D 01598 2 00799 5692 005 00799 4210 001E 00531 2 00266 1892 010 00266 1399 001F 00121 2 00061 431 025 00061 319 010G 00069 2 00034 244 gt025 mdash mdashH 00010 2 00005 037 gt025 mdash mdashI 00181 2 00091 645 025 00091 477 005J 00465 2 00232 1656 010 00232 1225 001K 00114 2 00057 405 025 00057 299 025


(c) Continued


L 00045 2 00022 160 gt025 mdash mdashM 00249 2 00125 888 025 00125 657 005Error 00028 2 00014 00019Sum 04024 26

(d) Vibration acceleration in the middle span (119886119887119898)


B 03544 2 01772 3063 005 01772 4746 001C 03096 2 01548 2676 005 01548 4146 001D 05061 2 02531 4374 005 02531 6778 001E 03331 2 01666 2879 005 01666 4461 001F 00093 2 00046 080 gt025 mdash mdashG 00115 2 00057 099 gt025 mdash mdashH 00102 2 00051 088 gt025 mdash mdashI 00315 2 00158 272 gt025 00158 422 005J 00403 2 00202 348 025 00202 540 005K 00012 2 00006 010 gt025 mdash mdashL 00011 2 00006 010 gt025 mdash mdashM 00402 2 00201 348 025 00201 539 005Error 00116 2 00058 00037Sum 16601 26

(e) Vibration acceleration of the vehicle body (119886V)


B 13451 2 06726 1243 010 06726 1711 001C 10364 2 05182 958 010 05182 1318 001D 08230 2 04115 760 025 04115 1047 005E 04458 2 02229 412 025 02229 567 005F 06294 2 03147 581 025 03147 801 005G 00936 2 00468 086 gt025 mdash mdashH 54978 2 27489 5079 005 27489 6993 001I 18752 2 09376 1732 010 09376 2385 001J 09799 2 04899 905 010 04899 1246 001K 15926 2 07963 1471 010 07963 2026 001L 08165 2 04082 754 025 04082 1038 005M 00340 2 00170 031 gt025 mdash mdashError 01082 2 00541 00393Sum 152776 26

boundary conditions are no longer the same as thedesign state

(2) The empirical formulas on the first two naturalfrequencies of the continuous girder bridge withodd span are given by the numerical simulationsSimilarly the empirical formulas on the first and thethird natural frequencies of the continuous girder

bridge with even span are obtained They can beused for calculating the dynamic load allowance inthe mid-span cross-section and the pier-top cross-section based on the current code of bridge design inChina respectively Also according to these formulasthe factors influencing natural frequencies are moreclearly seen which may give a better guideline for


Table 12 Design of the table header

Column 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Factor A D A times D B A times B D times B E C A times C D times C H B times C I J Non

Table 13 Results of numerical simulation with interaction

Run 1198911 (Hz) DLA119904 DLA119898 119886119887119904 (ms2) 119886119887119898 (ms2) 119886V (ms2)1 740 1212 1123 02164 00981 163982 634 1170 1163 00726 00651 168303 373 1259 1583 00565 01310 147094 320 1192 1181 00802 01128 127885 515 1184 1149 01469 00606 128146 495 1279 1363 00505 00857 183667 325 1195 1629 00770 01351 166098 312 1245 1783 00932 01486 134019 661 1504 1410 03277 01753 3243810 567 1355 1421 01105 01190 3259011 417 1343 1426 01617 02024 2487012 357 1496 2011 02926 03618 3218913 576 1638 1404 02203 02013 3470714 553 1299 1488 01322 01227 2746615 290 1357 3016 01019 03588 2929716 279 1685 2495 02080 03878 28570

Table 14 Range analysis of numerical simulation with interaction

Index A D A times D B A times B D times B E C A times C D times C H B times C I JDLA119904

0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088

the design and evaluation of dynamic performance ofbridges

(3) The dynamic responses in the side span including theDLA and the vibration acceleration are always largelydifferent from those in the middle span In additionthe effects of different factors on different dynamicresponses of the vehicle-bridge system are variousIn other words the effects of factors on dynamicresponses are dependent on both the selected positionand the type of the responses (DLA or vibrationacceleration)

(4) Based on the traditional method when one factor isstudied the others should be fixed Some significantfactors affecting the dynamic responses of the vehicle-bridge system are roughly identified and selectedThey are the pavement roughness the length of themain span the ratio between the side span and themiddle span the number of spans the weight of thebridge and the vehicle and the upper stiffness anddamping of the vehicle

(5) The orthogonal experimental design is introducedin this study for the dynamic responses analysisof the vehicle-bridge coupled vibration system Toefficiently reduce the experimental runs the conven-tional orthogonal design is divided into two phasesIn the first phase the main effects (single factor) areanalyzed without any interaction effects Based onthe results from the first phase the interaction effectsof some of the most important factors are discussedin the second phase It has been proved that theinteraction effects cannot be ignored

In the end it has to be emphasized that the pro-posed method of the orthogonal experimental design greatlyreduces calculation cost And it is efficient and rationalenough to studymultifactor problemsThe proposedmethodis used for not only the analysis of influence factors but alsothe analysis of regression And it can be applied in all typesof bridges other than just the girder bridge Furthermoreit provides a good way to obtain more rational empiricalformulas of the DLA and other dynamic responses whichmay be adopted in the codes of design and evaluation


Table 15 Various analysis of numerical simulation with interaction


Factor 119878119895 dof Initial Modification

120590119895 119865 120572 120590119895 119865 120572

A 02348 1 02348 2547 025 02348 2458 001D 00077 1 00077 083 gt025 00077 080 gt025A times D 00028 1 00028 030 gt025 00028 029 gt025B 00011 1 00011 012 gt025 00011 011 gt025A times B 00001 1 00001 001 gt025 mdash mdashD times B 00001 1 00001 001 gt025 mdash mdashE 00029 1 00029 032 gt025 00029 031 gt025C 00001 1 00001 001 gt025 mdash mdashA times C 00001 1 00001 001 gt025 mdash mdashD times C 00036 1 00036 039 gt025 00036 037 gt025H 00045 1 00045 049 gt025 00045 047 gt025B times C 00506 1 00506 549 gt025 00506 530 010I 00675 1 00675 733 025 00675 707 005J 00074 1 00074 080 gt025 00074 078 gt025Error 00092 1 00092 00095Sum 03924 15



A 08551 1 08551 563 gt025 08551 444 010D 05659 1 05659 372 gt025 05659 294 015A times D 00991 1 00991 065 gt025 00991 051 gt025B 13252 1 13252 872 025 13252 688 005A times B 02134 1 02134 140 gt025 02134 111 gt025D times B 03714 1 03714 244 gt025 03714 193 015E 01582 1 01582 104 gt025 01582 082 gt025C 00017 1 00017 001 gt025 mdash mdashA times C 00015 1 00015 001 gt025 mdash mdashD times C 00057 1 00057 004 gt025 mdash mdashH 01941 1 01941 128 gt025 01941 101 gt025B times C 00179 1 00179 012 gt025 mdash mdashI 00138 1 00138 009 gt025 mdash mdashJ 00395 1 00395 026 gt025 00395 021 gt025Error 01520 1 01520 01926Sum 40147 15



A 00363 1 00363 5914 gt025 00363 067 010D 00052 1 00052 846 gt025 00052 010 015A times D 00018 1 00018 301 gt025 00018 003 gt025B 00027 1 00027 432 025 00027 005 005A times B 00015 1 00015 239 gt025 00015 003 gt025D times B 00003 1 00003 045 gt025 00003 001 015E 00021 1 00021 344 gt025 00021 004 gt025C 00045 1 00045 736 gt025 mdash mdash


(c) Continued


A times C 00011 1 00011 177 gt025 mdash mdashD times C 00013 1 00013 212 gt025 mdash mdashH 00003 1 00003 042 gt025 00003 000 gt025B times C 00423 1 00423 6895 gt025 mdash mdashI 00043 1 00043 701 gt025 mdash mdashJ 00027 1 00027 443 gt025 00027 005 gt025Error 00006 1 00006 00541Sum 01069 15



A 00745 1 00745 17869 005 00745 2712 001D 00035 1 00035 829 025 00035 126 gt025A times D 00022 1 00022 535 gt025 00022 081 gt025B 00518 1 00518 12428 010 00518 1886 001A times B 00141 1 00141 3375 025 00141 512 010D times B 00027 1 00027 658 025 00027 100 gt025E 00006 1 00006 138 gt025 mdash mdashC 00001 1 00001 025 gt025 mdash mdashA times C 00003 1 00003 066 gt025 mdash mdashD times C 00002 1 00002 059 gt025 mdash mdashH 00037 1 00037 881 025 00037 134 gt025B times C 00067 1 00067 1600 025 00067 243 025I 00064 1 00064 1536 025 00064 233 025J 00011 1 00011 271 gt025 mdash mdashError 00004 1 00004 00027Sum 01683 15



A 90320 1 90320 43616 005 90320 32317 001D 00016 1 00016 008 gt025 mdash mdashA times D 00039 1 00039 019 gt025 mdash mdashB 02298 1 02298 1110 025 02298 822 005A times B 00180 1 00180 087 gt025 00180 065 gt025D times B 00423 1 00423 204 gt025 00423 151 gt025E 00005 1 00005 002 gt025 mdash mdashC 00001 1 00001 000 gt025 mdash mdashA times C 00011 1 00011 006 gt025 mdash mdashD times C 00842 1 00842 406 gt025 00842 301 025H 02321 1 02321 1121 025 02321 831 005B times C 00041 1 00041 020 gt025 00041 015 gt025I 03842 1 03842 1855 025 03842 1375 001J 00312 1 00312 150 gt025 00312 111 gt025Error 00207 1 00207 00279Sum 100859 15


B1 B2 B3 C1 C2 C3 D1 D2 D3 E1 E2 E3 F1 F2 F3 G1 G2 G3 H1 H2 H3 I1 I2 I3 J1 J2 J3 K1 K2 K3 L1 L2 L3 M1 M2 M3

Factors and levels

Inde

x

20

16

12

(a) Trend plot between factors and the DLA in side span


Factors and levels

Inde

x

25

21

17

(b) Trend plot between factors and the DLA in middle span


Factors and levels

Inde

x

04030201

(c) Trend plot between factors and the acceleration in side span


Factors and levels

Inde

x 05

03

01

(d) Trend plot between factors and the acceleration in middle span


Factors and levels

Inde

x

40353025

(e) Trend plot between factors and the acceleration of the vehicle body

Figure 18 Trend plot between the factors and the tested index

35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)

Code D60-2004 (China)VBCVA (DLA in side span)VBCVA (DLA in middle span)

(a) The sample without interaction

35

30

25

20

15

10

DLA

Frequency (Hz)2 3 4 5 6 7 8


(b) The sample with interaction

Figure 19 Comparison with the current code


Conflict of Interests

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

Acknowledgments

The research reported herein was sponsored by the ChinaScholarship Council (the 2013 China State-Sponsored Post-graduate Study Abroad Program) the National Natural Sci-ence Foundation of China (nos 51308465 50678051 and51108132 U1234208) and the Fundamental Research Fundsfor the Central Universities (no 2682014CX004EM) Theauthors would like to express their deep gratitude to all thesponsors for the financial aid

References

[1] D I McLean and M L Marsh Dynamic Impact Factorsfor Bridges Transportation Research Board Washington DCUSA 1998

[2] ASCE Committee on Loads and Forces on Bridges of the Com-mittee on Bridges of the Structural Division ldquoBridge loadingresearch neededrdquo ASCE Journal of the Structural Division vol108 no 5 pp 1012ndash1020 1982

[3] ASCE Committee on Loads and Forces on Bridges of the Com-mittee on Bridges of the Structural Division ldquoRecommendationdesign loads for bridgesrdquo Journal of the Structural Division vol107 no 7 pp 1161ndash1213 1981

[4] P F Csagoly andRADorton ldquoThedevelopment of theOntarioHighway Bridge Design Coderdquo Transportation Research Recordno 655 pp 1ndash12 1978

[5] J R Billing and R Green ldquoDesign provisions for dynamicloading of highway bridgesrdquo Transportation Research Recordvol 950 pp 94ndash103 1984

[6] D B Ashebo Evaluation of dynamic loads for highway bridges[PhD thesis] The Hong Kong Polytechnic University HongKong 2006

[7] B Bakht and S G Pinjarkar Dynamic Testing of High-way BridgesmdashA Review Ministry of Transportation OntarioCanada 1989

[8] OHBDC Ontario Highway Bridge Design Code Ontario Min-istry of Transportation andCommunications Ontario Canada1983

[9] M Samaan J B Kennedy and K Sennah ldquoImpact factorsfor curved continuous composite multiple-box girder bridgesrdquoJournal of Bridge Engineering vol 12 no 1 pp 80ndash88 2007

[10] K M Sennah X Zhang and J B Kennedy ldquoImpact factors forhorizontally curved composite box girder bridgesrdquo Journal ofBridge Engineering vol 9 no 6 pp 512ndash520 2004

[11] J Senthilvasan D P Thambiratnam and G H BrameldldquoDynamic response of a curved bridge under moving truckloadrdquoEngineering Structures vol 24 no 10 pp 1283ndash1293 2002

[12] S Komatsu and H Nakai ldquoFundamental study on forcedvibration of curved girder bridgesrdquo Transactions of the JapanSociety of Civil Engineers vol 2 pp 37ndash42 1970

[13] KM Sennah Load distribution factor and dynamic characteris-tics of curved composite concrete deck-steel cellular bridges [PhDthesis] University of Windsor Ontario Canada

[14] R Cantieni Dynamic Load Testing of Highway Bridges Trans-portation Research Board Washington DC USA 1984

[15] J R Billing Estimation of the Natural Frequencies of ContinuousMulti-Span Bridges Ontario Ministry of Transportation andCommunications Ontario Canada 1979

[16] Q F Gao Z L Wang Y Liu and B Q Guo ldquoModified formulaof estimating fundamental frequency of girder bridge withuniform cross-sectionrdquoAdvanced Engineering Forum vol 5 pp177ndash182 2012

[17] Q Gao Z Wang and B Guo ldquoModified formula of estimatingfundamental frequency of girder bridge with variable cross-sectionrdquo Key Engineering Materials vol 540 pp 99ndash106 2013

[18] I Mohseni A K A Rashid and J Kang ldquoA simplified methodto estimate the fundamental frequency of skew continuousmulticell box-girder bridgesrdquo Latin American Journal of Solidsand Structures vol 11 no 4 pp 649ndash658 2014

[19] P Paultre O Chaallal and J Proulx ldquoBridge dynamics anddynamic amplification factorsmdasha review of analytical andexperimental findingsrdquo Canadian Journal of Civil Engineeringvol 19 no 2 pp 260ndash278 1992

[20] AASHO The AASHO Road Test Highway Research BoardWashington DC USA 1962

[21] D T Wright and R Green Highway Bridge Vibrations PartII Ontario Test Program Queenrsquos University Ontario Canada1963

[22] P F Csagoly T I Campbell and A C Agarwall Bridge Vibra-tion Study Ministry of Transportation and CommunicationsOntario Canada 1972

[23] J Page Dynamic Wheel Measurements on Motorway BridgesTransport and Road Research Laboratory London UK 1976

[24] D R Leonard Dynamic Tests on Highway Bridges Transportand Road Research Laboratory London UK 1974

[25] R Shepard and G K Sidwell ldquoInvestigations of the dynamicproperties of five concrete bridgesrdquo in Proceedings of the4th Australian Conference on the Mechanics of Structures andMaterials pp 261ndash268 University of Queensland BrisbaneAustralia 1973

[26] R Shepard and R J Aves ldquoImpact factors for simple concretebridgesrdquo in Proceedings of the Institution of Civil Engineers vol55 pp 191ndash210 1973

[27] J H Wood and R Shepard ldquoVehicle induced vibrationsrdquo RoadResearch Unit Bulletin 44 1979

[28] J R Billing ldquoDynamic loading and testing of bridges inOntariordquo Canadian Journal of Civil Engineering vol 11 no 4pp 833ndash843 1984

[29] R Cantieni Dynamic Load Tests on Highway Bridges inSwitzerlandmdash60 Years of Experience Federal Laboratories forTesting of Materials EMPA Dubendorf Switzerland 1983

[30] A S Veletsos ldquoAnalysis of dynamic response of highwaybridgesrdquo ASCE Journal of the Engineering Division vol 93 no5 pp 593ndash620 1970

[31] J T Guant T Aramraks M J Gutzwiller and R H LeeHighway Bridge Vibration Studies Transportation ResearchBoard Washington DC USA 1977

[32] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987

[33] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989

[34] W Cui X Li S Zhou and J Weng ldquoInvestigation on pro-cess parameters of electrospinning system through orthogonal


experimental designrdquo Journal of Applied Polymer Science vol103 no 5 pp 3105ndash3112 2007

[35] F Kaitai and M Changxing Orthogonal and Uniform Experi-mental Design Tsinghua University Press Beijing China 2001

[36] W F Kuhfeld R D Tobias and M Garratt ldquoEfficient experi-mental design with marketing research applicationsrdquo Journal ofMarketing Research vol 12 pp 545ndash557 1994

[37] S Chen X Hong and C J Harris ldquoSparse kernel regressionmodeling using combined locally regularized orthogonal leastsquares and D-optimality experimental designrdquo IEEE Transac-tions on Automatic Control vol 48 no 6 pp 1029ndash1036 2003

[38] A H Jesus Z Dimitrovova and M A G Silva ldquoA statisticalanalysis of the dynamic response of a railway viaductrdquo Engineer-ing Structures vol 71 pp 244ndash259 2014

[39] E S Hwang and A S Nowak ldquoSimulation of dynamic load forbridgesrdquo Journal of Structural Engineering vol 117 no 5 pp1413ndash1434 1991

[40] W Huang X Zhang and G Hu ldquoNew advance of theoryand design on pavement for long-span steel bridgerdquo Journal ofSoutheast University (Natural Science Edition) vol 32 no 3 pp480ndash487 2002 (Chinese)

[41] P K Chatterjee T K Datta and C S Surana ldquoVibration ofcontinuous bridges under moving vehiclesrdquo Journal of Soundand Vibration vol 169 no 5 pp 619ndash632 1994

[42] R Mee A Comprehensive Guide to Factorial Two-Level Experi-mentation Springer Berlin Germany 2009

[43] L Q RenOptimumDesign andAnalysis of Experiments HigherEducation Press Beijing China 2nd edition 2003 (Chinese)

[44] J Cheng H Zhu S Zhong F Zheng and Y Zeng ldquoFinite-timefiltering for switched linear systems with a mode-dependentaverage dwell timerdquoNonlinear Analysis Hybrid Systems vol 15pp 145ndash156 2015

[45] T P Ryan Modern Engineering Statistics Solutions Manual toAccompany John Wiley amp Sons Hoboken NJ USA 2000

[46] D C Montgomery Design and Analysis of Experiments JohnWiley amp Sons Hoboken NJ USA 2008

[47] D Zhang and L Yu ldquoExponential state estimation for Marko-vian jumping neural networks with time-varying discrete anddistributed delaysrdquo Neural Networks vol 35 pp 103ndash111 2012

[48] Ministry of Communication of the Peoplersquos Republic of ChinaJTG D60-2004 General Code for Design of Highway Bridges andCulverts China Communication Press Beijing China 2004(Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


in Chinese current code the dynamic load allowance in thispaper is defined as follows

120583 (119909119887) =119910119889max (119909119887 119909V1)

119910119904max (119909119887 119909V2)minus 1 (1)

where 119909119887 is the location of the selected section 120583(119909119887) denotesthe dynamic load allowance of the selected section of thebridge and 119910119889max(119909119887 119909V1) and 119910119904max(119909119887 119909V2) respectivelydenote the maximum dynamic response and the maximumstatic response of the selected section Furthermore 119909V1 and119909V2 mean the location of the vehicle when the dynamicresponse and the static response reach the maximum valueIt should be noted that 119909V1 and 119909V2 are not the same in mostof the time

The dynamic response of a bridge to a crossing vehicleis a complex problem affected by the dynamic characteristicsof both the bridge and the vehicle and by the bridge surfaceconditions [1] Extensive research and development has beencarried out to understand the vibration of bridges as aresult of natural sources of vibration and to determinethe dynamic allowance as a function of the fundamentalfrequency due to its uniqueness [8ndash11] So it is importantto find a reliable method to estimate the natural frequencyof bridges Komatsu et al evaluated the natural vibrationresponses of straight and curved I- or box-girder bridgesusing Vlasovrsquos beam theory [12 13] Cantieni conductedan experimental study to establish a reliable relationshipbetween the fundamental frequency and maximum spanlength of bridges [14] Meanwhile Billing presented tablesof natural frequencies for straight continuous symmetricmultispan uniform beams of two through six spans forvarious span ratios [15] Additionally in order to reducethe significant difference between the estimation of thenatural frequency obtained from the codes and theoreticalmethods or numerical simulation Gao et al proposed anumerical improved method for straight bridges [16 17]Also the empirical expressions are established by Mohseniet al using regression analysis to determine the fundamentalfrequencies of skew continuous multicell box-girder bridges[18] However some limitations still exist for these methodsto actually be used in practice

Over the last 60 years a significant amount of researchhas been conducted in the area of bridge dynamics andthis research has been both analytical and experimental inscope Following the collapse of several railway bridges inGreat Britain the first study of vehicle-bridge interactionwas conducted in 1849 by Willis [19] A major experimentalinvestigation of bridge impact loads was initiated in 1958 byAmerican Association of State Highway Officials (AASHO)Eighteen simply supported bridges traversed by the two-axletrucks and the three-axle tractor-trailer combinations wereinvestigated each with a span of 15m [20] Wright and Greenreported on a series of experimental investigations of vehicle-bridge interaction made on 52 highway bridges in Ontariofrom 1956 to 1957 [21] Csagoly et al conducted a secondseries of tests on bridges (11 bridges) in Ontario in 1969 to1971 [22] Page [23] and Leonard [24] reported on a seriesof dynamic tests conducted on highway bridges (30 bridges)

by the Transport and Road Research Laboratory (TRRL) inEngland Shepard and Sidwell [25] Shepard and Aves [26]andWood and Shepard [27] reported on fourteen bridge testsconducted in New Zealand In 1980 a third series of dynamictesting of highway bridges in Ontario was performed Thetests were conducted on 27 bridges of various configurationsof steel concrete and timber construction [5 28] Cantienipresented results from the dynamic load tests of 226 highwaybridges in Switzerland from the mid-1950s to the early 1980s[14 29] As for the numerical simulation in 1970 Veletsosdeveloped a numerical method for computing the dynamicresponse of highway bridges to moving vehicular loads [30]Guant et al investigated analytically the effects on bridgeaccelerations of several parameters including the propertiesof the bridge and the vehicle as well as the initial conditionsof the roadway [31] Inbanathan and Wieland performed ananalytical investigation of the dynamic response of a simplysupported box girder bridge due to vehicle moving across thespan [32] Coussy et al presented an analytical study of theinfluence of random surface irregularities on the dynamicresponse of bridges [33] Recently Bakht and Pinjarkarpresented a literature review of bridge dynamics with aparticular focus on the dynamic testing of highway bridges[7] And Paultre et al presented a comprehensive reviewof analytical and experimental findings on bridge dynamicsand the evaluation of the dynamic load allowance [19]All these studies have proved that many of the parametersinteract with one another further complicating the issue andconsequentlymany research studies have reported seeminglyconflicting conclusions

Therefore themethod of orthogonal experimental designin batches [34ndash38] has been proposed in this paper It can beused for studying both the main effects and the interactioneffects Meanwhile due to the processing in batches it greatlyreduces the calculation cost

In this study the natural frequencies of the simplysupported girder bridge and the continuous girder bridgewith uniform cross-section are firstly investigated based onthe theoretical derivation and the numerical simulation Andthen the effects of different parameters on the dynamicresponses of the vehicle-bridge coupled vibration systemare discussed by our own program VBCVA Finally theorthogonal experimental design method is proposed forthe dynamic responses analysis including the case withoutinteraction and the case considering interaction

2 Natural Frequencies of MultispanContinuous Girder Bridges

In civil and transportation engineering dynamic charac-teristics of structures always consist of natural frequenciescorresponding vibration modes and damping ratios Thenatural frequency is one of the most important dynamiccharacteristics due to its key role in determining dynamicresponses of the structure under dynamic loads such aswindearthquake and moving vehicles Also the impact factor ordynamic load allowance which is used for considering thedynamic increments generated by moving vehicular loads is




Schematic plot


120593119899(0) = 0 12059310158401015840

119899(0) = 0

120593119899(119871) = 0 12059310158401015840

119899(119871) = 0

120593119899(0) = 0 1205931015840

119899(0) = 0

120593119899(119871) = 0 12059310158401015840

119899(119871) = 0

120593119899(0) = 0 1205931015840

119899(0) = 0

120593119899(119871) = 0 1205931015840

119899(119871) = 0





1205721 =120587

119871

1205722 =2120587

119871

1205723=3120587

119871

Fixed and hinged

120593119899(119909) = 119860

1[sin120572

119899119909 minus sinh120572

119899119909


(cos120572119899119909 minus cosh120572119899119909)]

tan120572119899119871 minus tanh120572119899119871 = 0

1205721 =3927

119871

1205722 =7069

119871

1205723 =10210

119871

Fixed and fixed

120593119899(119909) = 1198601[sin120572119899119909 minus sinh120572119899119909


(cos120572119899119909 minus cosh120572119899119909)]

3 minus 2 cos120572119899119871 cosh120572119899119871 = 0

1205721 =4739

119871

1205722 =7853

119871

1205723 =10996

119871




119898 119910 + 1198641198681205974119910

1205971199094= 119901 (119909 119905) (2)



120596119899 = 1205722

119899radic119864119868

119898(3)

120593119899 (119909) = 1198601 sin120572119899119909 + 1198602 cos120572119899119909

+ 1198603 sinh120572119899119909 + 1198604 cosh120572119899119909(4)




119891119899 =120587

2(119899 + 025119887

119871)

2

radic119864119868

119898 (5)





10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



0 119867119899 minus2119866119899 119867119899 0 0

d d d

0 0 0 119867119899 minus2119866119899 119867119899


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0 (6)

where

119866119899 =cosh120572119899119871sinh120572119899119871

minuscos120572119899119871sin120572119899119871

119867119899 =1

sinh120572119899119871minus

1

sin120572119899119871

(7)




1st mode


2nd mode

3rd mode




3 41198662

119899minus 1198672

119899= 0

4 119866119899(41198662

119899minus 21198672

119899) = 0

5 161198664

119899minus 12119866

2

1198991198672

119899+ 1198674

119899= 0

6 119866119899(161198664

119899minus 16119866

2

1198991198672

119899+ 31198674

119899) = 0

7 641198666

119899minus 80119866

4

1198991198672

119899+ 24119866

2

1198991198674

119899minus 1198676

119899= 0

8 119866119899(641198666

119899minus 96119866

4

1198991198672

119899+ 40119866

2

1198991198674

119899minus 41198676

119899) = 0


1198912 =1

2120587(120582120587

119871)

2

radic119864119868

119898= 1205822 120587

21198712radic119864119868

119898 (8)


120582 =

125 119904 = 2

1 + 008 times 22minus119898

119904 = 2119898 2 le 119898 le 4

1 + 036 times 22minus119898

119904 = 2119898 minus 1 2 le 119898 le 4

(9)




119903 =119871 119904

119871119898

(10)




2 119891119899 =1

21205871198712(1205872 3927

2 41205872 )radic

119864119868

119898

3 119891119899 =1

21205871198712(1205872 3556

2 4298

2 41205872 )radic

119864119868

119898

4 119891119899 =1

21205871198712(1205872 3393

2 3927

2 4463

2 41205872 )radic

119864119868

119898

5 119891119899 =1

21205871198712(1205872 3309

2 3700

2 4153

2 4550

2 41205872 )radic

119864119868

119898

6 119891119899 =1

21205871198712(1205872 3261

2 3556

2 3927

2 4298

2 4601

2 41205872 )radic

119864119868

119898

7 119891119899 =1

21205871198712(1205872 3230

2 3460

2 3764

2 4089

2 4395

2 4634

2 41205872 )radic

119864119868

119898

8 119891119899 =1

21205871198712(1205872 3210

2 3393

2 3645

2 3927

2 4208

2 4463

2 4655

2 41205872 )radic

119864119868

119898

Ls Lm

Ls LsLm

Ls Lm Ls

Ls

Lm

Lm LmLs Lm




minus119866119899119904minus 119866119899119898119867119899119898


119867119899119898


0 119867119899119898minus21198661198991198981198671198991198980 0

d d d

0 0 0 119867119899119898minus2119866119899119898

119867119899119898


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0

(11)



1198911 =1

2120587(1205781120587

119871)

2

radic119864119868

119898 (12a)

1198912 =1

2120587(1205782120587

119871)

2

radic119864119868

119898 (12b)





2 119866119899119904 + 119866119899119898 = 0

3 (119866119899119904 + 119866119899119898)2minus 1198672

119899119898= 0

4 (119866119899119904+ 119866119899119898)(1198662

119899119898+ 119866119899119904119866119899119898minus 1198672

119899119898) = 0

5 1198662

119899119904(41198662

119899119898minus 1198672

119899119898) + 119866119899119904119866119899119898(8119866

2

119899119898minus 61198672

119899119898) +

(41198664

119899119898minus 51198662

1198991198981198672

119899119898+ 1198674

119899119898) = 0







1205781

16

15

14

13

12

11

1000 02 04 06 08 10

Span ratio r


5-span


1205782

00 02 04 06 08 10

Span ratio r

38

34

30

26

22

18

14

10

2-span3-span

4-span5-span




























Speed 80 kmh








400

350

300

250

200

150

1001 2 3 4 5



DLA


1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)


1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Schematic plot


120593119899(0) = 0 12059310158401015840

119899(0) = 0

120593119899(119871) = 0 12059310158401015840

119899(119871) = 0

120593119899(0) = 0 1205931015840

119899(0) = 0

120593119899(119871) = 0 12059310158401015840

119899(119871) = 0

120593119899(0) = 0 1205931015840

119899(0) = 0

120593119899(119871) = 0 1205931015840

119899(119871) = 0





1205721 =120587

119871

1205722 =2120587

119871

1205723=3120587

119871

Fixed and hinged

120593119899(119909) = 119860

1[sin120572

119899119909 minus sinh120572

119899119909


(cos120572119899119909 minus cosh120572119899119909)]

tan120572119899119871 minus tanh120572119899119871 = 0

1205721 =3927

119871

1205722 =7069

119871

1205723 =10210

119871

Fixed and fixed

120593119899(119909) = 1198601[sin120572119899119909 minus sinh120572119899119909


(cos120572119899119909 minus cosh120572119899119909)]

3 minus 2 cos120572119899119871 cosh120572119899119871 = 0

1205721 =4739

119871

1205722 =7853

119871

1205723 =10996

119871




119898 119910 + 1198641198681205974119910

1205971199094= 119901 (119909 119905) (2)



120596119899 = 1205722

119899radic119864119868

119898(3)

120593119899 (119909) = 1198601 sin120572119899119909 + 1198602 cos120572119899119909

+ 1198603 sinh120572119899119909 + 1198604 cosh120572119899119909(4)




119891119899 =120587

2(119899 + 025119887

119871)

2

radic119864119868

119898 (5)





10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



0 119867119899 minus2119866119899 119867119899 0 0

d d d

0 0 0 119867119899 minus2119866119899 119867119899


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0 (6)

where

119866119899 =cosh120572119899119871sinh120572119899119871

minuscos120572119899119871sin120572119899119871

119867119899 =1

sinh120572119899119871minus

1

sin120572119899119871

(7)




1st mode


2nd mode

3rd mode




3 41198662

119899minus 1198672

119899= 0

4 119866119899(41198662

119899minus 21198672

119899) = 0

5 161198664

119899minus 12119866

2

1198991198672

119899+ 1198674

119899= 0

6 119866119899(161198664

119899minus 16119866

2

1198991198672

119899+ 31198674

119899) = 0

7 641198666

119899minus 80119866

4

1198991198672

119899+ 24119866

2

1198991198674

119899minus 1198676

119899= 0

8 119866119899(641198666

119899minus 96119866

4

1198991198672

119899+ 40119866

2

1198991198674

119899minus 41198676

119899) = 0


1198912 =1

2120587(120582120587

119871)

2

radic119864119868

119898= 1205822 120587

21198712radic119864119868

119898 (8)


120582 =

125 119904 = 2

1 + 008 times 22minus119898

119904 = 2119898 2 le 119898 le 4

1 + 036 times 22minus119898

119904 = 2119898 minus 1 2 le 119898 le 4

(9)




119903 =119871 119904

119871119898

(10)




2 119891119899 =1

21205871198712(1205872 3927

2 41205872 )radic

119864119868

119898

3 119891119899 =1

21205871198712(1205872 3556

2 4298

2 41205872 )radic

119864119868

119898

4 119891119899 =1

21205871198712(1205872 3393

2 3927

2 4463

2 41205872 )radic

119864119868

119898

5 119891119899 =1

21205871198712(1205872 3309

2 3700

2 4153

2 4550

2 41205872 )radic

119864119868

119898

6 119891119899 =1

21205871198712(1205872 3261

2 3556

2 3927

2 4298

2 4601

2 41205872 )radic

119864119868

119898

7 119891119899 =1

21205871198712(1205872 3230

2 3460

2 3764

2 4089

2 4395

2 4634

2 41205872 )radic

119864119868

119898

8 119891119899 =1

21205871198712(1205872 3210

2 3393

2 3645

2 3927

2 4208

2 4463

2 4655

2 41205872 )radic

119864119868

119898

Ls Lm

Ls LsLm

Ls Lm Ls

Ls

Lm

Lm LmLs Lm




minus119866119899119904minus 119866119899119898119867119899119898


119867119899119898


0 119867119899119898minus21198661198991198981198671198991198980 0

d d d

0 0 0 119867119899119898minus2119866119899119898

119867119899119898


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0

(11)



1198911 =1

2120587(1205781120587

119871)

2

radic119864119868

119898 (12a)

1198912 =1

2120587(1205782120587

119871)

2

radic119864119868

119898 (12b)





2 119866119899119904 + 119866119899119898 = 0

3 (119866119899119904 + 119866119899119898)2minus 1198672

119899119898= 0

4 (119866119899119904+ 119866119899119898)(1198662

119899119898+ 119866119899119904119866119899119898minus 1198672

119899119898) = 0

5 1198662

119899119904(41198662

119899119898minus 1198672

119899119898) + 119866119899119904119866119899119898(8119866

2

119899119898minus 61198672

119899119898) +

(41198664

119899119898minus 51198662

1198991198981198672

119899119898+ 1198674

119899119898) = 0







1205781

16

15

14

13

12

11

1000 02 04 06 08 10

Span ratio r


5-span


1205782

00 02 04 06 08 10

Span ratio r

38

34

30

26

22

18

14

10

2-span3-span

4-span5-span




























Speed 80 kmh








400

350

300

250

200

150

1001 2 3 4 5



DLA


1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)


1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816



0 119867119899 minus2119866119899 119867119899 0 0

d d d

0 0 0 119867119899 minus2119866119899 119867119899


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0 (6)

where

119866119899 =cosh120572119899119871sinh120572119899119871

minuscos120572119899119871sin120572119899119871

119867119899 =1

sinh120572119899119871minus

1

sin120572119899119871

(7)




1st mode


2nd mode

3rd mode




3 41198662

119899minus 1198672

119899= 0

4 119866119899(41198662

119899minus 21198672

119899) = 0

5 161198664

119899minus 12119866

2

1198991198672

119899+ 1198674

119899= 0

6 119866119899(161198664

119899minus 16119866

2

1198991198672

119899+ 31198674

119899) = 0

7 641198666

119899minus 80119866

4

1198991198672

119899+ 24119866

2

1198991198674

119899minus 1198676

119899= 0

8 119866119899(641198666

119899minus 96119866

4

1198991198672

119899+ 40119866

2

1198991198674

119899minus 41198676

119899) = 0


1198912 =1

2120587(120582120587

119871)

2

radic119864119868

119898= 1205822 120587

21198712radic119864119868

119898 (8)


120582 =

125 119904 = 2

1 + 008 times 22minus119898

119904 = 2119898 2 le 119898 le 4

1 + 036 times 22minus119898

119904 = 2119898 minus 1 2 le 119898 le 4

(9)




119903 =119871 119904

119871119898

(10)




2 119891119899 =1

21205871198712(1205872 3927

2 41205872 )radic

119864119868

119898

3 119891119899 =1

21205871198712(1205872 3556

2 4298

2 41205872 )radic

119864119868

119898

4 119891119899 =1

21205871198712(1205872 3393

2 3927

2 4463

2 41205872 )radic

119864119868

119898

5 119891119899 =1

21205871198712(1205872 3309

2 3700

2 4153

2 4550

2 41205872 )radic

119864119868

119898

6 119891119899 =1

21205871198712(1205872 3261

2 3556

2 3927

2 4298

2 4601

2 41205872 )radic

119864119868

119898

7 119891119899 =1

21205871198712(1205872 3230

2 3460

2 3764

2 4089

2 4395

2 4634

2 41205872 )radic

119864119868

119898

8 119891119899 =1

21205871198712(1205872 3210

2 3393

2 3645

2 3927

2 4208

2 4463

2 4655

2 41205872 )radic

119864119868

119898

Ls Lm

Ls LsLm

Ls Lm Ls

Ls

Lm

Lm LmLs Lm




minus119866119899119904minus 119866119899119898119867119899119898


119867119899119898


0 119867119899119898minus21198661198991198981198671198991198980 0

d d d

0 0 0 119867119899119898minus2119866119899119898

119867119899119898


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0

(11)



1198911 =1

2120587(1205781120587

119871)

2

radic119864119868

119898 (12a)

1198912 =1

2120587(1205782120587

119871)

2

radic119864119868

119898 (12b)





2 119866119899119904 + 119866119899119898 = 0

3 (119866119899119904 + 119866119899119898)2minus 1198672

119899119898= 0

4 (119866119899119904+ 119866119899119898)(1198662

119899119898+ 119866119899119904119866119899119898minus 1198672

119899119898) = 0

5 1198662

119899119904(41198662

119899119898minus 1198672

119899119898) + 119866119899119904119866119899119898(8119866

2

119899119898minus 61198672

119899119898) +

(41198664

119899119898minus 51198662

1198991198981198672

119899119898+ 1198674

119899119898) = 0







1205781

16

15

14

13

12

11

1000 02 04 06 08 10

Span ratio r


5-span


1205782

00 02 04 06 08 10

Span ratio r

38

34

30

26

22

18

14

10

2-span3-span

4-span5-span




























Speed 80 kmh








400

350

300

250

200

150

1001 2 3 4 5



DLA


1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)


1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2 119891119899 =1

21205871198712(1205872 3927

2 41205872 )radic

119864119868

119898

3 119891119899 =1

21205871198712(1205872 3556

2 4298

2 41205872 )radic

119864119868

119898

4 119891119899 =1

21205871198712(1205872 3393

2 3927

2 4463

2 41205872 )radic

119864119868

119898

5 119891119899 =1

21205871198712(1205872 3309

2 3700

2 4153

2 4550

2 41205872 )radic

119864119868

119898

6 119891119899 =1

21205871198712(1205872 3261

2 3556

2 3927

2 4298

2 4601

2 41205872 )radic

119864119868

119898

7 119891119899 =1

21205871198712(1205872 3230

2 3460

2 3764

2 4089

2 4395

2 4634

2 41205872 )radic

119864119868

119898

8 119891119899 =1

21205871198712(1205872 3210

2 3393

2 3645

2 3927

2 4208

2 4463

2 4655

2 41205872 )radic

119864119868

119898

Ls Lm

Ls LsLm

Ls Lm Ls

Ls

Lm

Lm LmLs Lm




minus119866119899119904minus 119866119899119898119867119899119898


119867119899119898


0 119867119899119898minus21198661198991198981198671198991198980 0

d d d

0 0 0 119867119899119898minus2119866119899119898

119867119899119898


10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

2

3

119899 minus 1

119899

= 0

(11)



1198911 =1

2120587(1205781120587

119871)

2

radic119864119868

119898 (12a)

1198912 =1

2120587(1205782120587

119871)

2

radic119864119868

119898 (12b)





2 119866119899119904 + 119866119899119898 = 0

3 (119866119899119904 + 119866119899119898)2minus 1198672

119899119898= 0

4 (119866119899119904+ 119866119899119898)(1198662

119899119898+ 119866119899119904119866119899119898minus 1198672

119899119898) = 0

5 1198662

119899119904(41198662

119899119898minus 1198672

119899119898) + 119866119899119904119866119899119898(8119866

2

119899119898minus 61198672

119899119898) +

(41198664

119899119898minus 51198662

1198991198981198672

119899119898+ 1198674

119899119898) = 0







1205781

16

15

14

13

12

11

1000 02 04 06 08 10

Span ratio r


5-span


1205782

00 02 04 06 08 10

Span ratio r

38

34

30

26

22

18

14

10

2-span3-span

4-span5-span




























Speed 80 kmh








400

350

300

250

200

150

1001 2 3 4 5



DLA


1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)


1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1205781

16

15

14

13

12

11

1000 02 04 06 08 10

Span ratio r


5-span


1205782

00 02 04 06 08 10

Span ratio r

38

34

30

26

22

18

14

10

2-span3-span

4-span5-span




























Speed 80 kmh








400

350

300

250

200

150

1001 2 3 4 5



DLA


1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)


1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Speed 80 kmh








400

350

300

250

200

150

1001 2 3 4 5



DLA


1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)


1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










400

350

300

250

200

150

1001 2 3 4 5



DLA


1 2 3 4 5



100

040

020

000

060

080

Acce

lera

tiona b

(mmiddotsminus

2)


1 2 3 4 5


Acce

lera

tiona

(mmiddotsminus

2)

800

600

400

200

000










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










220

200

180

160

140

120

10020 25 30 35 40

DLA




20 25 30 35 40



040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)


Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

00020 25 30 35 40












220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










220

200

180

160

140

120

100

DLA

Span ratio


05 06 07 08 09 1


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

Span ratio05 06 07 08 09 1



Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000

Span ratio05 06 07 08 09 1








240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










240

220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




Acce

lera

tiona

(mmiddotsminus

2)

400

300

200

100

000











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)











240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










240

220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)












220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










220

200

180

160

140

120

10006 08 1 12 14


DLA



06 08 1 12 14


040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)



06 08 1 12 14


400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)










220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











220

200

180

160

140

120

10006 08 1 12 14

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




500

06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)







119877119895 = max (1198961198951 1198961198952 119896119895119898) minusmin (1198961198951 1198961198952 119896119895119898) (13)




220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










220

200

180

160

140

120

10006 08 1 12 14

DLA




06 08 1 12 14

040

050

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)








119878 =

119886

sum

119894=1

(119910119894 minus 119910)2=

119886

sum

119894=1

1199102

119894minus1

119886(

119886

sum

119894=1

119910119894)

2

119878119895 =119886

119887

119887

sum

119896=1

(119910119895119896minus 119910)2

=119886

119887

119887

sum

119896=1

1199102

119895119896minus1

119886(

119886

sum

119894=1

119910119894)

2

119865119895 =120590119895

120590119890

=119878119895119891119895

119878119890119891119890

(14)






06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










06 08 1 12 14

220

200

180

160

140

120

100

DLA




06 08 1 12 14

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




06 08 1 12 14

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)


















220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










220

200

180

160

140

120

100

DLA




050

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

















220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










220

200

180

160

140

120

100

DLA




040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)




400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)















120583 =


(25)



Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Vehicle speed (kmh)

220

200

180

160

140

120

100

DLA

20 40 60 80 100 120



Vehicle speed (kmh)

040

030

020

010

000

Acce

lera

tiona b

(mmiddotsminus

2)

20 40 60 80 100 120



Vehicle speed (kmh)

400

300

200

100

000

Acce

lera

tiona

(mmiddotsminus

2)

20 40 60 80 100 120



R method

Calculation

Judgment

Rj

Factor order

Best level

Best combination

Kjm kjm










5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














5 Conclusions
















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(c) Continued



















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















0242 0044 0026 0016 0005 0004 0027 0004 0005 0030 0034 0112 0130 0043DLA119898 0462 0376 0157 0576 0231 0305 0199 0021 0019 0038 0220 0067 0059 0099119886119887119904 (ms2) 0095 0036 0021 0026 0019 0008 0023 0034 0016 0018 0008 0103 0033 0026119886119887119898 (ms2) 0136 0029 0024 0114 0059 0026 0012 0005 0008 0008 0030 0041 0040 0017119886V (ms2) 1503 0020 0031 0240 0067 0103 0011 0004 0017 0145 0241 0032 0310 0088










120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













120590119895 119865 120572 120590119895 119865 120572









(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(c) Continued











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Factors and levels

Inde

x

20

16

12



Factors and levels

Inde

x

25

21

17



Factors and levels

Inde

x

04030201



Factors and levels

Inde

x 05

03

01



Factors and levels

Inde

x

40353025



35

30

25

20

15

100 2 4 6 8 10 12 14 16 18 20

DLA

Frequency (Hz)



35

30

25

20

15

10

DLA








Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Acknowledgments


References


























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Dynamic Responses of …downloads.hindawi.com/journals/mpe/2015/951502.pdfResearch Article Dynamic Responses of Continuous Girder Bridges with Uniform Cross-Section