《高等路面设计理论》课程授课教案（讲义）Pavement Roughness on Expansive Clays

9CHAPTERIIPAVEMENTROUGHNESSONEXPANSIVECLAYSINTRODUCTIONPavement roughness traditionallyforms part of the criteria by which the performance and condition of a pavement isevaluated.Thisinformationisthenusedinthedecisionmakingprocessofpavementmanagementsystemswhichschedulethefrequencyandmagnitudeofpavementmaintenance.Methodsofcharacterizingtheroughnessofsurfacestraversedbyvehiclesevolved out of a need to adequately identify and measure parameters which affect the safe and efficient passage of vehiclesoverthetraveled surface.Sayers (19a5)quotesthedefinitionofroughnessas itpertainstotraveledsurfacesas"thedeviationofa surfacefroma trueplanarsurfacewithcharacteristicdimensionthat affectvehicledynamics,ridequality,dynamicloadsanddrainage"(ASTMDefinition of Terms Relating toTraveled SurfaceCharacteristics (E867-82a).Despite thebroadmeaningof this definition,roughness is generally interpreted asa propertyofthelongitudinal vertical profileofaroadwaypavementbasedonmeasurementstakeninthewheelpathsofthetraveledsurfaceRoughnessparametersmaybeconsideredinthreebroadcategories(Sayers1985))(1)Profilebasedmeasurements andstatistics(2) Responsetyperoadroughness measuring systems (RTRRMS),and(3)Subjectivepanel ratings.Althoughthese parametersmay beconsidered in separatecategories,theyareoften used in combination to producea singleindexofperformanceProfile based measurements and statistics:In this category ofroughness evaluation,relative elevations aremeasuredatdiscreteintervalsalongthepavementsurfaceandareprocessedtoproducenumericsthatdescribesomecharacteristicofthetraversedpath.Suchprofilometricmeasurementsaregenerallycarried outthroughrodand level surveys,rollingwheeldevicesorthroughnon-contactcomputerizedprofilingequipmentsuchastheGeneral Motors(GM)690DSurfaceDynamicsProfilometer (Walker and Beck 1988) used by the Texas State Department of Highways and Public Transportation (SDHPT).Profile statistics whichmaybe directly obtainedform'profile measurements include spectral estimates using Fouriertransformtechniques and numericalevaluation offirst or second orderderivativesusing specified base lengths.TheFourieramplitude spectrumhas been used by Velacsoand Lytton-(1981)and McKeen (1981) in thecharacterization of expansiveclayroughnessandbyStoneandDugundji(1965)intheevaluationofthetrafficabilityofnaturalterrain.ExamplesofroughnessmeasuresbasedonthelattertwomethodsaretheslopevarianceusedinthePresent ServiceabilityIndexoftheAASHO

9 CHAPTERII PAVEMENT ROUGHNESS ON EXPANSIVE CLAYS INTRODUCTION Pavement roughness traditionally forms part of the criteria by which the performance and condition of a pavement is evaluated. This information is then used in the decision making process of pavement management systems which schedule the frequency and magnitude of pavement maintenance. Methods of characterizing the roughness of surfaces traversed by vehicles evolved out of a need to adequately identify and measure parameters which affect the safe and efficient passage of vehicles over the traveled surface. Sayers (1 9a5) quotes the definition of roughness as it pertains to traveled surfaces as "the deviation of a surface from a true planar surface with characteristic dimension that affect vehicle dynamics, ride quality, dynamic loads and drainage" (ASTM Definition of Terms Relating to Traveled Surface Characteristics (E867-82a)). Despite the broad meaning of this definition, roughness is generally interpreted as a property of the longitudinal vertical profile of a roadway pavement based on measurements taken in the wheel paths of the traveled surface. Roughness parameters may be considered in three broad categories (Sayers 1985): (1) Profile based measurements and statistics, (2) Response type road roughness measuring systems (RTRRMS), and (3) Subjective panel ratings. Although these parameters may be considered in separate categories, they are often used in combination to produce a single index of performance. Profile based measurements and statistics: In this category of roughness evaluation, relative elevations are measured at discrete intervals along the pavement surface and are processed to produce numerics that describe some characteristic of the traversed path. Such profilometric measurements are generally carried out through rod and level surveys , rolling wheel devices or through non-contact computerized profiling equipment such as the General Motors (GM) 690D Surface Dynamics Profilometer (Walker and Beck 1988) used by the Texas State Department of Highways and Public Transportation (SDHPT). Profile statistics which may be directly obtained form' profile measurements include spectral estimates using Fourier transform techniques and numerical evaluation of first or second order derivatives using specified base lengths. The Fourier amplitude spectrum has been used by Velacso and Lytton- (1981) and Mc Keen (1981) in the characterization of expansive clay roughness and by Stone and Dugundji (1965) in the evaluation of the traffic ability of natural terrain. Examples of roughness measures based on the latter two methods are the slope variance used in the Present Serviceability Index of the AASHO

10(CareyandIrick196O)androotmeansquareverticalaccelerationused intheServiceabilityIndexoftheTexasSHDPT(RobertsandHudson1970),Response type road roughness measuring systems (RTRRMS):Theuse of RTRRMS involves the measurement ofvehicleresponsetoroadroughnessandtheinterpretationofthisresponseasavalueonapre-calibratedscale.AnexampleofsuchadeviceistheMaysRideMeter(MRM)developed inTexasbytheRainhartcompany(WalkerandHudson1973).Thistypeof roughness measure is tiedto vehicleresponse,whichvaries among vehicles and alsovaries with time,vehicleconditi onand weather.AsaresultRTRRMS measuresmay provide inconsistent results and requirearelativelycomplicated calibrationtoconvertthemeasuretoastandardscale.Inordertoovercometheinconsistenciesofrealvehicles,mathematicallymodeledquartercarsimulations(QcS)havebeendevelopedwhichsimulatevehicledynamics basedonroad profileelevationorslopeinput(Gillespieetal.1980).ThistypeofmodelprovidesthebasisfortheInternationalRoughnessIndex(Sayersetal.1986)whichwillbepresentedinfollowingsectionsSubjectivepanel ratings:In thelate1950stheAmericanAssociationof State HighwayOfficials (AASHO)(CareyandIrick1960)initiated theconceptofa userbased serviceability rating forhighway pavements.TheAASHO road testproducedaPresent Serviceability Rating (PSR)which was anumber between Oand 5returned byarating panel, based on the perceivedridingqualityoftestsectionstraversedinavehicleoftheirchoice.ApredictivemodelthePresentServiceabilityIndex(PSl).wasalsodevelopedatthesametimetoreproducethePsRbasedonphysicalcharacteristicsoftheroadwaysurface,wherePsl=PSR+ Error.Similarrating systems havebeen developed overthepast threedecades in Texas (Roberts and Hudson 1970,Walkerand Hudson1973)andhavebeen used bytheSDHPTformonitoringof its Highway systems.OverthepastthreedecadessincetheintroductionofthePsl,researchershavebeenconcernedwithfindingtheparameterthat"best'describes pavement roughness as itappliestopavementevaluation.However,thephilosophyof StoneandDugundji(1965)presentedintheiranalysisofthetrafficabilityofnaturalterrain,suggeststhatroughnessmaybebestconsideredavectorquantityofdifferentcomponents,eachdescribingadifferentcharacteristicoftheroadprofile.Thisconcepis reasonableas it is unlikely thata single index can providecomplete information about thedifferentmodes of roughnessdevelopmentthat,likelytooccurinpavements,especiallywhenappliedtoexpansiveclayTwopossiblemodes of roughnessdevelopmentinexpansiveclays werepresented inChapter1,thegilgaiphenomenonandtheedgemoisturevariation.Thegilgaiphenomenonproduceslongperiodwavelengthroughness(12to20ft)(Lyttonetal.1976)andtheedgedeformationassociatedwithshorterwavelengthroughness(5to10ft).Thewavelengthsgenerally

10 (Carey and Irick 1960) and root mean square vertical acceleration used in the Serviceability Index of the Texas SHDPT (Roberts and Hudson 1970). Response type road roughness measuring systems (RTRRMS): The use of RTRRMS involves the measurement of vehicle response to road roughness and the interpretation of this response as a value on a pre-calibrated scale. An example of such a device is the Mays Ride Meter (MRM) developed in Texas by the Rainhart company (Walker and Hudson 1973). This type of roughness measure is tied to vehicle response, which varies among vehicles and also varies with time, vehicle condition and weather. As a result RTRRMS measures may provide inconsistent results and require a relatively complicated calibration to convert the measure to a standard scale. In order to overcome the inconsistencies of real vehicles, mathematically modeled quarter car simulations (QCS) have been developed which simulate vehicle dynamics based on road profile elevation or slope input (Gillespie et al. 1980). This type of model provides the basis for the International Roughness Index (Sayers et al. 1986) which will be presented in following sections. Subjective panel ratings: In the late 1950s the American Association of State Highway Officials (AASHO) (Carey and Irick 1960) initiated the concept of a user based serviceability rating for highway pavements. The AASHO road test produced a Present Serviceability Rating (PSR) which was a number between 0 and 5 returned by a rating panel, based on the perceived riding quality of test sections traversed in a vehicle of their choice. A predictive model the Present Serviceability Index (PSI). was also developed at the same time to reproduce the PSR based on physical characteristics of the roadway surface, where PSI = PSR + Error. Similar rating systems have been developed over the past three decades in Texas (Roberts and Hudson 1970, Walker and Hudson 1973) and have been used by the SDHPT for monitoring of its Highway systems. Over the past three decades since the introduction of the PSI, researchers have been concerned with finding the parameter that "best" describes pavement roughness as it applies to pavement evaluation. However, the philosophy of Stone and Dugundji (1965) presented in their analysis of the trafficability of natural terrain, suggests that roughness may be best considered a vector quantity of different components, each describing a different characteristic of the road profile. This concept is reasonable as it is unlikely that a single index can provide complete information about the different modes of roughness development that, likely to occur in pavements, especially when applied to expansive clay. Two possible modes of roughness development in expansive clays were presented in Chapter 1, the gilgai phenomenon and the edge moisture variation. The gilgai phenomenon produces long period wavelength roughness (12 to 20 ft) (Lytton et al. 1976) and the edge deformation associated with shorter wavelength roughness (5 to 10 ft). The wavelengths generally

associatedwith pavem6nt roughness duetostructural deterioration of thepavementaregenerallylessthan4ftSinceexpansivedayroughness is associated with specific wavelengths, it isdesirable to use amethod of roughness evaluation that issensitivetothatrangeofwavelengthsandwhichiscapableof identifyingsuchwavelengthsInthesectionsthatfollow,variousmethodsofquantifyingroughnessarepresented.Thisisfollowedbyareviewofpastresearch whichhave applied some of thesemethods in the evaluationofpavement roughness overexpansive claysMEASURESOFROUGHNESSSpectralanalysisThe term spectral analysis is generally used to describe all techniquesfor summarizing time series functions byseparatingthesefunctions intotheirfrequencycomponents.Spectralanalysismethodshavebeenroutinelyapplied inthestatistical analysisofoscillatorytimedependentphenomena.Inthefieldsofcommunications,mechanicsandacoustics,electrical current,displacementandacousticpressurearecharacterizedasrandomlydistributedvariables inthetimedomainInapplyingspectralanalysistoroadprofiledata,ftisassumedthatthesurfaceelevationX(t)canberepresentedbyarandomlydistributeddeviation X(t)-μfroman expectedvalueμ.Theexpectedvaluein this casebeingthemeanordatumaboutwhichprofile elevation points are distributed.Tnis datum isusually set tozero throughfilteringtechniques duringmeasurement.The timedomain is replacedbythespatial ordistancedomain andconditions of stationarityand ergodicity areassumedtobeupheld.Roadprofileelevations arenottrulyrandomvariables and theydocontaindesignedvertical curves andinthecaseofexpansive clays are susceptibleto the development of periodicdeformations.In addition,as roadways generally traversespatiallyvariablesoil andsiteconditionstheconditionsofstationarityandergodicityarealsounlikelytobeupheldoverlongdistances.Thishowever,doesnot invalidatetheuseofsuchmethodsasthespatialvariability inspectralestimates isexploitedto identify areas of different roughness activity.In orderto approach stationarity and ergodicityover characteristic lengthsfilteringanddetrendingtechniquesareappliedduringmeasurementandanalysis.TypicalrelativeelevationprofilesofroughandsmoothpavementsectionsmeasuredaspartofthisresearchareillustratedinFigure3FourieramplitudespectrumIfx(t)is consideredtobeaperiodicfunctionofperiodT,thenforanyvalueof t

associated with pavem6nt roughness due to structural deterioration of the pavement are generally less than 4 ft. Since expansive day roughness is associated with specific wavelengths, it is desirable to use a method of roughness evaluation that is sensitive to that range of wavelengths and which is capable of identifying such wavelengths. In the sections that follow, various methods of quantifying roughness are presented. This is followed by a review of past research which have applied some of these methods in the evaluation of pavement roughness over expansive clays. MEASURES OF ROUGHNESS Spectral analysis The term spectral analysis is generally used to describe all techniques for summarizing time series functions by separating these functions into their frequency components. Spectral analysis methods have been routinely applied in the statistical analysis of oscillatory time dependent phenomena. In the fields of communications, mechanics and acoustics, electrical current, displacement and acoustic pressure are characterized as randomly distributed variables in the time domain. In applying spectral analysis to road profile data, ft is assumed that the surface elevation X(t) can be represented by a randomly distributed deviation X(t)- from an expected value . The expected value in this case being the mean or datum about which profile elevation points are distributed. This datum is usually set to zero through filtering techniques during measurement. The time domain is replaced by the spatial or distance domain and conditions of stationarity and ergodicity are assumed to be upheld. Road profile elevations are not truly random variables and they do contain designed vertical curves and in the case of expansive clays are susceptible to the development of periodic deformations. In addition, as roadways generally traverse spatially variable soil and site conditions, the conditions of stationarity and ergodicity are also unlikely to be upheld over long distances. This however, does not invalidate the use of such methods as the spatial variability in spectral estimates is exploited to identify areas of different roughness activity. In order to approach stationarity and ergodicity over characteristic lengths, filtering and detrending techniques are applied during measurement and analysis. Typical relative elevation profiles of rough and smooth pavement sections measured as part of this research are illustrated in Figure 3. Fourier amplitude spectrum If x(t) is considered to be a periodic function of period T, then for any value of t

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13(1)k=1,2,3...x(t)=x(t ±kT)Thefundamental frequencyf,satisfies7(2)LSuchperiodic datacan be expandedinaFourierseriesaccordingtothefollowing:+E(3)x(t) =(a,cos2mft+b,sin2mf,t)2k-1wherek(4)f=kf,k=1,2,3....Thusx(t)is described in terms of sine and cosinewaves at discrete frequencies spacedf=f,apart Thecoefficients a,and b,are computedby carrying out thefollowing integrations overthe period T from zero to T given by:a. - x0)co2mfdtk=0,1,2...b-x0)sin2mftdtk.= 1,2,3...(5)x0dtp2whereμisthemeanofx(t).Thefunctionx(t)mayalsoberepresentedincomplexexponentialform using Euler's relationship given by:e-e=cose-jsing(6)x(t)Aexp(j2mf,t)where,A-32A-(ax-jb) -x(t)exp(-j2mf,t)dtk=±1,±2,±3,..(7)A[A/ exp(-je,)bxIA/ -2V+bE.e,=tanaTheFourier spectrum is thereforedescribedby Equation7,whereA,is theFourieramplitude

14correspondingtothefrequencyfTransientorrandomstationarydatacannotgenerallybeassumedtobeperiodic withinthe observed interval [o,T].However,thepreviousFourierseriesrepresentations can be extended by considering that what occurs within the discrete intervalTtoapproach infinity.This leads totheFourier integral,(8)x(f) =x(t)e-2mdt-8f<8where X(f) will exist if,(9)Ix(t) / dt < 80Sincex(t) ismeasured overthe finite interval [0,T] then its Fouriertransformexists and isreferredtoas thefiniteFouriertransformgivenby:(10)[x(t)e-mdtX.()=X(fT)=Such finiteFouriertransforms will always existforfinite lengthrecords of stationarydataEquations8and9demonstratethatatall discretefrequencies f,=(k/m),thefiniteFouriertransformyields(11)X(f.T)=TAk±1,±2,±3,..Henceiff isrestrictedtotake ononlythesediscretefrequencies,thenthefiniteFouriertransformwo&ddwill actuallyproduceaFourierseriesofperiodT.Whenx(t)is sampledatpointstapart,therecord lengthbecomesT=NAt,whereNisthesamplesize.This inducesaNyquistcutofffrequencyf=1/2At.Also,thecomputationstreatthedataasifitwereperiodic data withperiod T.The continuous function x(t) is replaced bythe discrete series X,=x(nt)forn=1,2,3...N,andtheFouriertransformX(f)is replacedbythediscreteFouriertransformX,=X(k△f)fork=1,2,3,..N.ThediscreteFouriertransform is symmetric aboutk=N/2,and theFouriertransformpair is given by.Nknj2nk=1,23,.,NX=Atx,expNn-1(12)j2m.knx,=AfxexpK=1,2,3, ...NN1-1

15TheFourieramplitude spectrum obtained fromEquation 12 is the same as thatgiven in Equation7.TwoFourieramplitude spectratypicalofroughan smoothhighwayprofiles are illustrated inFigure 4.These spectra indicate thatthe significant differences in theFourieramplitudesbetweenthetwo surfaces occur inthewavelengthsgreaterthan approximately8feet.Thishassignificant consequences on the interpretation ofparameters using the amplitude spectrum as ameasureofroughness.This is discussedinthe summaryof this chapter.Correlation functionsIfx(t)andy(t)representtworandomstationaryprocesses,the covariancefunctionbetweenx(t)andy(t)foranytimedelayrisgivenby:C()-E[(x()-u)(y(t+t)-u)](13)lim1"[(x) -(y+)-)dt-R,(-,where:R.f0-x0yd(14)Forthegeneral case where x(t)andy(t)represent differentdata,R,(r) in Equation14 is calledthecross-correlation functionbetweenx(t)andy(t).Forthe special casewherex(t)=y(t),(15)Ca(0)-Ra()-where:lim 1x()xt+) t(16)R()"T-TJ。is calledtheauto-correlationfunctionofx(t)and C,(t)theauto-covariancefunction.Thevalueof the auto-correlation function at=O is the mean square value of the data,which is the sumof thevarianceand the square of the mean value of thedata,where,Ra(0)-o+(17)Ro(o)"u

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17Theautocorrelationfunctionalso convergestoaconstant valueequal tothesquareof themeanasrincreasesto infinity.Figure5illustratestheautocorrelationfunctionsofeight128ft.sectionsmakingup1024ft.oftheroughhighwaysection.Traditionallythese functions were used as an indicatorof the degree of accuracy to which the current time history can be used to predicteventsbeyondtheperiodofobservationT.Forexamplethecorrelationfunctionofasinewaveisacosinewavewithanamplitude equal to the mean square value of the original sine wave and of the same singular frequency and wavelength.Thesinewave correlation function remains constant over all timedelays T,suggesting that one can predictfuture values of the datapreciselybased on pastvalues.Fora sinusoidal function this is indeedthecase.Anothertypeof informationwhich couldbe interpreted fromautocorrelationfunctions is thedominantfrequency contentoftheoriginaldata.However,this infomation is more clearly interpreted fromautospectral densityfunctionsorFourieramplitudespectra.Inthe caseofthe sinusoid,thisfrequency isreadilyobtainedas thereciprocalofthedecorrelation time (ordistance)which isgivenbythevalueofthelagT.whenthatfunctionfirstreacheszero,and isequalto/4As thedatafunction resembles less and less a pure sinusoid and becomesmore ofa wide band random signal (as inthecaseofroadprofiles)theinterpretationofthedecorrelationdistancebecomeslessclearandisrelatedonlytothebandwidthB,of dominantfrequencies oftheautospectral densityfunction and thedecorrelation distance T=1(2B)(Bendat and Plersol1980)SpectraldensityfunctionsThe spectral density functionbetween two timehistories x(t) and y(t)oftwo random stationary processes may bedefined as theFouriertransformof thecorrelation functionas follows:S, (f)=R,(t)e-12mf dr(18)Forthe general case where x(t) and y(t) are different SxyY(f)l is called the cross-spectral density function or more simply thecross spectrum.Forthe special caseofx(t)andy(t)beingequal,Sy (f)=R,(t)e =12mf dr(19)

17 The autocorrelation function also converges to a constant value equal to the square of the mean as r increases to infinity. Figure 5 illustrates the autocorrelation functions of eight 128 ft. sections making up 1024 ft. of the rough highway section. Traditionally these functions were used as an indicator of the degree of accuracy to which the current time history can be used to predict events beyond the period of observation T. For example the correlation function of a sine wave is a cosine wave with an amplitude equal to the mean square value of the original sine wave and of the same singular frequency and wavelength. The sine wave correlation function remains constant over all time delays T, suggesting that one can predict future values of the data precisely based on past values. For a sinusoidal function this is indeed the case. Another type of information which could be interpreted from autocorrelation functions is the dominant frequency content of the original data. However, this information is more clearly interpreted from autospectral density functions or Fourier amplitude spectra. In the case of the sinusoid, this frequency is readily obtained as the reciprocal of the decorrelation time (or dist ance), which is given by the value of the lag T0, when that function first reaches zero, and is equal to /4. As the data function resembles less and less a pure sinusoid and becomes more of a wide band random signal (as in the case of road profiles) the interpretation of the decorrelation distance becomes less clear and is related only to the bandwidth B, of dominant frequencies of the autospectral density function and the decorrelation distance T0= 1 (2B) (Bendat and Plersol 1980) Spectral density functions The spectral density function between two time histories x(t) and y(t) of two random stationary processes may be defined as the Fourier transform of the correlation function as follows:    = - -j2mfr Sxy (f) Rxy (t) e dr (18) For the general case where x(t) and y(t) are different SxyY(f)l is called the cross-spectral density function or more simply the cross spectrum. For the special case of x(t) and y(t) being equal,    = - -j2mfr Sxy (f) Rxy (t) e dr (19)

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