Modification of the Asphalt Institute Bituminous Mix Modulus Predictive Equation

Transcription

1 Transportation Research Record Viscoelastic Materials. Trans., Society of Rheology, Vol. 11, No. 2, 1966, pp J.S. Lai and o. Anderson. Irrecoverable and Recoverable Nonlinear Viscoelastic Properties of Asphalt Concrete. HRB, Highway Research Record 468, 1973, pp W.N. Findly, J.s. Lai, and K. Onaran. Nonlinear Creep and Relaxation of Viscoelastic Materials with an Introdction to Linear Viscoelasticity. North Holland, Amsterdam, v.w. Cho and W.N. Findly. Creep and Creep Recovery of 304 Stainless Steel Under Combined Stress and Representation by a Viscos Viscoelastic Model. ASME, Jornal of Mechanics, No. 80, Pt1b/ication of this paper sponsored by Committee on Characteristics of Bi t mi nos Paving Mixtres to Meet Strctral Reqirements. Modification of the Asphalt Institte Bitminos Mix Modls Predictive Eqation JOHNS. MILLER, JCOB UZAN, AND MATTHEW W. WITCZAK The dynamic modls test reslts for five bitminos mix types (crshed stone, gravel, slag, sand-low P 200, sand-high P2ool were compared with modls vales predicted by sing the Asphalt Institte regression model. Reslts of this comparison showed an excellent correlation for crshed stone, which was the primary mix type in the model development. For other materials, slag and sand, the reslts were poor. A correction methodology was developed based on statistical methods to minimize the mean sqare error between the measred modli and those predicted by sing the model. The model param eter sed as the means for correction was the percentage of asphalt content, chosen becase it had been shown that the asphalt content range sed in model development was narrower than the range encontered in the laboratory stdy and that, mathematically, it had the greatest effect on the resltant modls of all the model variables. A niqe, material-dependent constant was calc lated for each of the five mix types in the stdy that wold be sbtracted from the actal asphalt content of the mix for calclating the modls. A more de sirable, generalized method for determining the correction constant was devel oped based on the difference between the Marshall optimm asphalt content and the actal asphalt content of the mix instead of relying on mix nomencla 1Ure (e.g., slag asphalt, sand asphalt). The correction scheme prodced a cor relation coefficient of for all 1179 data points sed in this stdy. This is an excellent reslt for practical engineering applications. With the trend toward application of elastic theory to problems in flexible pavement evalation and design, the development of accrate stress-strain relations for asphalt concrete mixtres is important. Frthermore, asphal tic concrete modls vales have been correlated with sbstittion ratios or layer coefficients in empirical design methods (.!_,ll so that an accrate modls characterization serves both methods. However, direct laboratory modls characterization nder conditions similar to those encontered in the field (dynamic repeated loading, temperatre, and load rate) normally reqires sophisticated and expensive dynamic laboratory eqipment (1_). To avoid costly laboratory procedres, alternative methods of determining modls by sing physical and mechanical properties of the mixtre were developed. Among these methods were se of the Marshall stability-flow qotient and the development of the Shell nomograph and the Asphalt Institte predictive model. The Marshall stability-flow qotient was sggested by Nijboer (_!) and recommended for se in high temperatre ranges by Hekelom and Klomp (_?.) Nijboer's formla was as follows: S6o'c,4 sec; 1.6(stability/flow) (I) where S is given in kilograms per sqare centimeter, stability in kilograms, and flow in millimeters. McLeod (_.) sggested a variation on the Nijboer formla: Modls; 40 (stability/flow) (2) where modls is given in ponds per sqare inch, staoility in ponds, and flow in inches. These formlas se rotine laboratory-determined properties of a mix. The Shell nomograph originally developed by Van der Poel (1.l permitted determination of the stiffness of asphalt cement at a particlar load rate and temperatre as a viscoelastic characteristic as distingished from elastic modls. Hekelom and Klomp <.> developed a relation to translate the bitmen stiffness to a mixtre stiffness based on volme of aggregate and volme of asphalt cement in the mix. McLeod <..> modified the nomograph by changing the entry temperatre criterion. Finally, Claessen and others () prodced a pair of nomographs to be sed together to accomplish the evalation of bitmen and mixtre stiffness sed in the crrent Shell design manal (9). This paper focses on the third method for modls determination, the Asphalt Institte model <.!.Q) By examining dynamic modls vales measred in the laboratory for varios materials and comparing them with vales predicted by the model based on physical properties of the mixtre (asphalt content, percentage passing the No. 200 sieve, volme of voids, and asphalt viscosity), the accracy of the predictor can be established. RESEARCH PROBLEM STATEMENT At present, the two predictive models most commonly sed are the Shell method and the Asphalt Institte eqation. The Shell method is based on stiffness of the asphalt cement determined by entering a nomog raph with factors derived from asphalt softening point and penetration and temperatre and freqency of loading. This bitmen stiffness is then modified by sing a second nomograph to consider volme percentages of aggregate and asphalt cement to give a stiffness or modls vale for the mix. The Asphalt Institte method was initiated by

2 28 Transportation Research Record 911 Figre 1. Reslts of original Asphalt Institte predictive eqation. =- " c n :\ "\ '. -.>.... i':.;.:::;. \...,,.. ',.., "/.,..,,,,,,.... r,... :.:;.,_.,;;. ":'if ( ';..:.. i.:'\,.,,. :>. '..... '. l'.. A. 'Ii. -"-,.,q. \ U. l Measr ed Dynami c Modls [xl 0 5 ps ij. 0 Table 1. Smmary of parameters for asphalt concrete mixes. No. of Data Asphalt P200 Penetration Mix Type Points Sorce Content(%) Volme of Voids (%)!)( 6, 70) at 77 F Crshed stone, inclding 369 Asphalt Institte " two sand mixes Crshed stone 162 University of Maryland Bank rn gravel 162 University of Maryland Slag 162 University of Maryland Hot mix sand asphalt Low 162 University of Maryland High 162 University of Maryland Note: Viscosities in the University o f Maryland stdy we n meas red on fresh, nsed asphalt cement. 3 Single vales of 3.0, 8. 2, and 8.6 are eliminated d e to negligible foflence. Shook and Kallas (11) in 1969 sing a limited nmber of laboratory (dyoimic modls) test reslts. It waa based on the relation between direct measrements of dynamic modls and certain properties of the mixtre. The eqation was sbseqently refined from an expanded data base and then frther modified by Witczak () for se in calclating design crves crrently sed in the Asphalt Institte MS-1 design gide (_! ). The predictive eqation is as follows: log, o IE* I = (P200/f<l ) Vv where )( 6, 70) T exp(l log f)pac T exp(l log f)(pac 0 5 /fl. 1 ) + ( / ) (3) IE*I dynamic modls ( 5 psi), Pzoo = percentage passing the No. 200 sieve, f loading freqency (Hz), Vv 0 volme of voids (%), n(l0 6,70) = viscosity of asphalt cement at 70 F (megapoises), T temperatre of pavement ( F), and Pac percentage of asphalt cement by weight of mix. It is important to note that the data sed to develop the eqation were almost exclsively derived from mixtres of crshed stone and gravel. This predictive eqation was based on 41 different asphaltic mixtres prepared and tested at the Asphalt Institte over years of laboratory research work. Each mix was tested for dynamic modls at a fll factorial of three test temperatres (40, 70, and 0 F) and three load freqencies (1, 4, and 16 Hz). Ths, the eqation was based on 369 data points. For the mixes stdied, Eqation 3 had an excellent correlation coefficient (R 2 = 0.939) and mean sqare error (MSE = ) and therefore was sed with confidence for the material type and ranges of physical properties sed in its development. This is clearly demonstrated in Figre 1, which compares the predicted and measred modli for the original 369 data points tested by the Asphalt Institte. The parameters sed in the eqation are given in Table 1. In 1978, a laboratory stdy was initiated at the University of Maryland for the Maryland State High-

3 Transportation Research Record way Administration (MSHA) to develop dynamic modls characterizations for typical base and sbbase materials sed in pavement systems (]:ll This comprehensive stdy reslted in the dynamic modls evalation of 45 additional mixes containing crshed stone, gravel, slag, and sand aggregates. By sing the same temperatre and freqency factorials as sed by the Asphalt Institte, along with variations in asphalt content and aggregate gradation, 8 additional dynamic modls test reslts were obtained. This nmber, combined with the 369 data points from the previos Asphalt Institte work, allowed for a combined data pool of 1179 sets of data to be sed in the stdy. The major objective of this paper is to se the new data base generated by the University of Maryland stdy to assess the accracy (verify or modify) of the Asphalt Institte predictive eqation for the broader range of material types (asphaltic mixtres) investigated. SOURCES OF DATA General Table l smmarizes the range of aggregate types and associated mix parameters of the laboratory specimens for which the dynamic modls was measred and then predicted by sing Eqation 3. Asphalt I_nstitte Laboratory stdies condcted for se in the work of Witczak () and Kallas and Shook (11) sed a total of 41 different bitminos mixtres. These specimens were tested at fll factorial levels of 1, 4, and 16 Hz and 40, 70, and l00 F, which yielded 369 vales of dynamic modls. The mixtres sed in the Asphalt Institte stdies consisted of crshed stone and gravel aggregates (except for two sand mixes) with asphalt content ranging from 3.0 (only one specimen) to 8.6 (again, only one specimen) and averaging approximately 5.2 percent. Viscosity of asphalt cements ranged from 1.31 to 4.43 megapoises. In general, almost all of the. mixes tested were of a dense-graded crshed aggregate variety. University of Maryland S tdy The specimens sed in the laboratory stdy were made from combinations of five aggregates. The crshed stone, bank rn gravel, and slag were each graded to conform to MSHA specifications pertinent to BF, BI, and BC mixes. In addition, each gradation was represented by six specimens, two for each of three levels of compaction (0, 98, and 95 percent of maximm density determined by the Marshall design method) The two sand mixes were made with three levels of asphalt content (Marshall optimm, +l perc ent, and -1 percent) instead of tne gradations. There were six specimens for each asphalt content consisting of two specimens for each of three compaction levels (98, 96, and 92 percent of maximm). This yielded 90 specimens of 45 different aggregate-gradation (asphalt content) combinations. When the specimens were tested at fll factorial levels of 1, 4, and 16 Hz and 40, 70, and 0 F, the laboratory stdy yielded 8 vales of dynamic modls. A mch broader range of aggregate types and asphalt contents was investigated in this stdy than in the Asphalt Institte work. For the crshed stone and gravel in the Maryland stdy, the asphalt content ranged from 3.9 to 5.6 percent, which was reasonably close to that sed in the original (Asphalt Institte) model development. However, the other three aggregates investigated had average asphalt contents very different from that of the original stdy (slag, 7.8 percent; sand, high P percent; and sand, low P percent]. Eqation l was sed to compte predicted modls vales for all 8 combinations with the aid of the University of Maryland UNIVAC 10/82. All data were plotted to compare the measred dynamic modli with the predicted vales for each data set. RESULTS Co mparative Stdy Figre 2 shows a comparison between the measred dynamic modls reslts for the 8 test points developed in the University of Maryland (UM) stdy and the Asphalt Institte predictive eqation. It is obvios that the general agreement is not nearly as good as that shown in Figre l with the original 369 Asphalt Institte data reslts. These reslts necessitated an in-depth stdy of the original Asphalt Institte predictive eqation by material type. Figre 3 shows the excellent agreement achieved for the UM-MSHA crshed-stone mixes. Althogh this material type exhibited an accrate predictive response correlation, Figres 4-7 show the poor correlations for the aggregate types not inclded in the model development. The R 2 vales ranged from 0.09 to 0.70 for the for aggregates o ther than the crshed stone whereas the University of Maryland crshed-stone vales prodced an R 2 of This compares favorably with the R 2 of achieved with the original 41 mixes (, 11) Althogh the correlation vales for th;- slag, the gravel, and both sands were not satisfactory, the trend toward linearity shown in Figres 4-7 strongly indicates that a material-dependent (mix-dependent) adjstment factor can be sed to improve the origi nal predictive eqation. The lack of fit for mixtres other than crshed stone is probably related to the fact that the ranges of mix properties differ significantly from those sed in the original model development. Table l indicates that the percentage of asphalt in the slag and sand mixtres is mch greater than the pper limit of the range sed in the predictive eqation. It is interesting to note that the mixtre properties of the gravel material are all in the range of vales of the original prediction and that measred modli are only slightly higher than predicted (by a factor of 1 to 3). The above reslts appear to indicate that the high percentage of asphalt is a primary case of the lack of fit shown in Figres 4-7. This led to the frther investigation of the possible cases of the lack of fit for non-crshedstone mixtres. A sensitivity stdy was condcted to determine the magnitde of change in the predicted modls as a conseqence of varying a particlar parameter vale. An examination of the change in loglo IE* I verss P200 Vv, and Pac showed that, for the range of vales in the University of Maryland stdy, the greatest change in dynamic modls occrred as the percentage of asphalt increased. The effect of the percentage passing the No. 200 sieve and the volme of voids is relatively small, and they both fall within the range of vales sed in developing the original eqation. As a reslt, a procedre was investigated to correct the predicted modls eqation by sing percentage of asphalt as the primary variable to be corrected. De velopme n t of Correction Factors As stated previosly, examination of the effect of each predictor variable on the reslting dynamic

4 30 Transportation Research Record 911 modls predicted by Eqation 1 indicated that the asphalt content of the mix cased the greatest change in modls. This, along with the large differences in optimm asphalt content determined by Marshall method procedres for each aggregate in relation to the original range of this parameter sed to develop the eqation, tended to confirm the sspicion that asphalt content was the primary case of the lack of fit. Figres 4-7 show that the mag nitde of the lack of fit (and, hence, the magnitde of any correction factor) is apparently material dependent. The method sed to define the correction assmed that it wold be constant for a given aggregate type. This was highly dp.sirahlp. hp.ae the original mathematical form of the eqation cold be sed Figre 2. Overall comparison of predicted verss measred l ,.--r-t""t""t""lrrrr---.--t-r-r--,-rtt., r-r--,,. modli : University of Maryland stdy, ncorrected. n = 8 MSE v. C- c v. - :.: c " "" t. ".. #.j!i I ta..! '.. -;,.,,,_1;, r..: l :L,1\;t....,.. "-'.,... 'lt., -,.c:t: ' -.:;. 6,. L..:_.,.,-....,..-i, (..:I.. ""',,,,...1. : Measred Dynamic Modls (x 5 psi) 0 Figre 3. Comparison of predicted verss measred modli: University of Maryland crshed stone, ncorrected. n 162 MSE 0.09 " v. c Ul ", 0 " :.:,.,,., Q. R '. '..;1r....,,..,.. / ' J',)'..! Measred D)amic I I I I II I Modls (xjo 5 psi) I I! j i 0

5 Transportation Research Record instead of having to develop a different eqation. The simplest and most desirable correction wold be to adjst directly (add or sbtract) the percentage-of-asphalt parameter sed i n the model. The reasoning for this lies in the comparison between the rang e of asphalt contents sed in the original model development and the UM-MSHA crshed-stone data and the ranges of optimm asphalt contents determined for the for additional UM-MSHA mixes. In the development of the predictive model, a relatively narrow range of asphalt contents was sed and excellent reslts were achieved. In contrast, the University of Maryland stdy reslted in the se of widely different ranges of optimm asphalt con- Figre 4. Comparison of predlc:ted verss measred modli: gravel, ncorrec:ted , , , ".-T,---..,.---, T""T,. n 16: MSE R V1 c. c " ".; Cl Q) Q) "- "" Measred Dynamic Modls (xl0 5 psi) 0 Figre 5. Comparison of predic:ted verss measred modli: 0 slag, ncorrec:ted. ::::; c "" 2 " " -Cl :i.,,., ] 0.. "" Tl 162 MSE R ,...,,,, = '\. ".. Jo L--L-'--L-'-. '----L--L-'-''--' Measred Dynamic Modls (x 5 psi) J j

6 32 Transportation Research Record 911 tents according to the material sed as aggregate and acnieved only fair to poor agreement. In addition, an examination of Eqation 3 shows that the terms containing percentage of asphalt tend to be negative and their magnitde tends to increase as higher vales of temperatre, freqency, and asphalt Figre 6. Comparison of predicted verss measred modli: ,., ,..,.,, sand, low P200. ncorrected. 11 = 162 MSE R v, "- Vl c )( v. -g ::;:., <> "" "- content are sed. Becase temperatre and freqency vales were the same in both sets of data, it is l i Kely that tne original model can only tolerate minor deviations in asphalt content and still predict reasonable modls vales. To determine the specific correction to be ap- 1t' : r:: :1-'-.,.... '.'f. I I I.". ' I,...,.. O,l.. _.._.._._.._._...._L-''-'-_._...._..._.._ Measred Dynamic Modls (xl0 5 psi) 0 Figre 7. Comparison of predicted verss measred modli: 0 sand, high P2oo. ncorrected. o - 11 = 162 MSE O. 058 R I.... :. :.f. / ;- I - : _l '"I - '..,.. " --I.. ;..,,'... _; J I I Measred Dynamic Mod l s (x 5 psi) 0

7 Transportation Research Record plied to asphalt content, Eqation 3 was rewritten to isolate the terms containing percentage of asphalt content: where a is a correction constant and Cl= (P 200 /r' ) Vv Joo6, 70 > + ( /r' ) C2 = T exp(l log f) - [ T exp(l Jog f)/f 1 1] Becase the correction constant, a, shold apply to the entire range of parameters for a given mix type, a statistical optimization procedre was sed to achieve the best fit. Draper and Smith (14) present the method of optimizing a nonlinear fnction (which describes Eqation 4). The reslts can be achieved by an iterative process designed to minimize the mean sqare error (MSE). Essentially, a correction constant is chosen and the sm of the sqared differences between the measred dynamic modls and the predicted modls (calclated ny sing the chosen constant) is compted from n 2 MSE; = (i/n).l (log1 0 IE* lmeas - Cl+ C2(Pac - <>;)J0 5 ] j= 1 J where n is the nmber of vales of modls. By iteration, the vale of the MSE is determined for a range of correction vales. This procedre reslts in a crve with a niqe minimm MSE for a single a. Figre 8 shows the graphical reslts of MSEi (4) (5) verss "i for each of the aggregates in the UM-MSHA stdy. All display clearly defined minimm vales on the convex fnctions. Analysis of Effects of Correction Factors Table 2 gives the corrections developed for each type of aggregate in the University of Maryland stdy. The vale listed for crshed stone approaches 0.0 when the Asphalt Institte data are inclded and therefore is not inclded in the analysis. The corrected vales achieved for slag and both sands redce the asphalt content to vales within the limits of the range sed in developing Eqation 3. The bitminos mix with bank rn gravel achieved the poorest fit de to both inherent scatter (Figre 4) and the fact that the corrected asphalt content was slightly below the range sed in the model development. Table 3 smmarizes the MSE and R 2 vales determined with and withot the correction factor applied to the original Asphalt Institte eqation. It can be seen that a very significant increase in R 2 and decrease in MSE are obtained with the correction factor for all asphalt mixes investigated. The correlation coefficient obtained for the combined data when no asphalt corrections were sed was R 2 = When each material correction factor was applied, the correlation improved dramatically to R 2 = and MSE = For the combined 1179 data points, this translates into a standard error of 20.6 percent (of expected modls). Figre 9 shows the reslts of the corrected UM-MSHA reslts (8 points). Figre 1 shows the original comparison of the 369 data points sed in the Asphalt Institte analysis. Figre 8. Optimization of corrections to asphalt content by mix type Sand-Low P O.OlS (l J ; v. :.: (1.0 E; 0. (l(\ " i:: :.: ll'l-crshed Stone 0 O.O ls 0. Olt> o. 0 Slag (l. 012 Sand-High P Asphalt Correction, a

8 34 Transportation Research Record 911 Sggested I mp1eme nt ati on of Resl t s The analysis presented in this stdy clearly shows that the original Asphalt Institte modls eqation achieved excellent reslts for both the Asphalt In- Table 2. Smmary of compted correction factors by mix type. Correction Factor Asphalt Range Asphalt Content Mix Type (%) Asphalt Range Crshed stone Asphalt Institte University of Maryland Bank rn gravel Slag Sand Low P High P Becase the University of Maryland crshedstone correction approaches 0.0 when combined with AsphaJt Institte data, H is not considered in frther analyses, Table 3. Smmary of error analyses. Mix Type Crshed.stone Asphalt Institte University of Maryland Gravel Slag Sand Low P 200 High P200 Average No Correction MSE Correction Applied MSE SE(%) No correction applied to University of Maryland crshed stone; vales reflect Asphalt Institte and University of Maryland data taken together. sti tte and UM-MSHA crshed-stone mixes. However, the stdy also clearly identified the need to modify frther the original Asphalt Institte eqation in order to predict accrately dynamic modls for a broad range of bitminos mixtres. In the methodology developed, it was shown that a niqe correction factor cold be calclated for each mix typp. stdied. It was alfjo shown that implementation of the derived correction factors sbstantially improved the correlation between predicted and measred modli for mixes that otherwise wold show poor predicted vales (Table 3). Becase the different correction factors corresponded to different mix types, it is obvios that the correction factor was material (mix and type) dependent. Becase of this, a modification scheme for the original Asphalt Institte eqation was developed that cold provide accrate predicted modli for mixes not inclded in the UM-MSHA stdy. It is a widely accepted fact tha Marshall optimm asphalt content is material dependent and can be taken as a global variable to describe material behavior. Logically, if the two material var i al:>les--the correction factor and Marshall optimm asphalt content--vary in the same way and are strongly correlated, a simple relation can be developed to express the nknown variable (a) in terms of the readily available optimm asphalt content. The relation netween a and optimm as;phalt content is shown in Figre. The reslts indicate a linear relation between the two variables. From a practical viewpoint, the eqation can be approximated by 'Pop (6) Sbstittion of this eqation into the original Asphalt Institte eqation (Eqation 4) yields log IE* I= Cl+ C2(Pac - Po pt+ 4.0) 0 5 (7) Figre 9. Overall comparison of predicted verss measred 0 modli corrected by mix typh. n 8 MSE R c:.. 0 ":J "" 0 " " I j, 1, I Measred Dynamic Modls (x!0 5 psi)

9 Transportation Research Record Figre. Sggested relation between optimm asphalt content and correction factor. Optimm Arh3lt Content l R ",: 0. "- " 0.., C! c L Optimm Asphalt Content, % (Marshal I Method) Table 4. Reslts of correction methodology applied to University of Maryland stdy. Asphalt Content (%) Analysis of Correction Mix Type Actal Corrected 8 R2 MSE Crshed stone Gravel Slag S ndb Low P High P Note: Nmber of data points for all mix types = Correctd according to Eqation 7 ; i.e., asphajt content= actal asphalt content b - Ol)ll luuffi Sand U JJhalt ixes were designed by sing 51 samples each at Marshall optimm, Marshall optimm+ LO, and MarshaJI optimm - J.0. Table 5. Comparison of analysis for all data with and withot corrections. Uncorrected Corrected Mix Type R2 MSE R2 MSE Crshed.stoneb Gravel p Slag Sand Low P High P Corto c. t cd according to liqtioo '7. Incldes both Aspholt lo.s hle and University of Maryland data. Asphalt lnslitle data were not corrected de to lack or Marshall optimm asphalt content vale. Eqation 7 implies the following: 1. All material types have similar behavior arond cheir respective optimm asphalt contents. In essence, one may view the tt correction parameter as a shifting factor to converge all modls-pac crves into a niqe master crve. 2. For all mixes prepared at Pac = Popt the modls can be compted whe n Pac = 4. O is sbstitted in the original Asphalt Institte eqation (Eqation 3) This does not mean that the modls is independent of the asphalt content, becase the actal asphalt content mst eqal the optimm one. Becase common practice reqires that Pac P0p a se parate stdy of the correlation of the predicted and measred modli was condcted. The reslts are given in Table 4, in which it can be observed that the R 2 and MSE vales are qite satisfactory for the broad range of materials sed in the stdy. It shold be noted that the 4.0 vale corresponds to a fictitios material (practically the crshed-stone and gravel material) with a 4.0 percent optimm asphalt content. However, it shold be noted that Eqation 7 is not limited to actal asphalt contents eqal to the optimm ones. A stdy of the goodness of prediction was condcted by sing all UM-MSHA samples at all asphalt contents sed in the stdy. Table s gives the reslts of the statistical smmary (R" and MSE) for the predicted vales withot and with the correction factor by sing Eqation 6 incorporated into the model. It can be observed that, althogh the corrected R 2 vales (correction based on optimm asphalt content) are not as high as those given in Table 3 (correction based on specific mix type), they are nonetheless qite satisfactory from a practical engineering viewpoint. As Table 5 indicates, the greatest improvement with the correction is achieved for the slag.and sand mixes. All of these oitminos mixtres had optimm asphalt contents mch larger than the original Pac range sed in the Asphalt Institte stdy on crshed-stone and gravel mixes. The form of this recommended eqation obviosly indicates that certain conditions related to the range of Pac vales that reslt in negative vales within the brackets mst be avoided. In fact, it is recommended that the range of Pac and Popt combinations sed shold be sch that Pac - Pac - SUMMARY Popt > -1.S (minimm) Popt < -2.5 (maximm) The stdy described in this paper was based on com-

10 36 Transportation Resear ch Record 911 pacisons between nearly 1200 dynamic modls vales measred in the laboratory and predicted vales calclated by sing the Asphalt Institte regression eqation. The reslts of the comparisons showed a general lack of fit for aggregate mixes, other than crshed stone, whose optimm asphalt content (Marshall method) was otside the range of percent sed to develop the original eqation. A procedre was developed to minimize the MSE between measred and predicted modls vales wh i le redcing the asphalt content by a constant amont for a given aggregate type. This led to the development of niqe correction factors applied directly to the percentage of asphalt variable in the original eqation foe specific mixes. A more generalized correction factor was also developed that relates the magnitde of this vale to the optimm asphalt content of the mix. Althogh the reslts indicated that modification of the original Asphalt Institte eqation was necessary for a wide variety of bitminos mixes, it can also be stated neqivocally that the original eqation is highly satisfactory for dense-graded crshed-stone and gravel mixes. It is only foe other widely differing mixes sch as sand asphalt and slag asphalt mixes that the correcti on (modification) is really necessary. The proposed methods of correction presented in this papar now create a highly accrate predictive eqation that is applicable foe a wide range of mix types and physical properties. The modified eqation sing the mix-dependent correction factor reslted in an R 2 of O. 928, whereas the more generalized correction based on optimm asphalt content yielded an R 2 of These R 2 vales, based on a data base of 1179 laboratory tests condcted at the Asphalt Institte and University of Maryland laboratories over the past decade, point ot the accracy and wide range of applicability of the predictive eqation. REFERENCES 1. H. Takeshita. Considerations on the Strctral Nmber. Proc., 2nd International Conference on the Strctral Design of Asphalt Pavements, Univ. of Michigan, Ann Arbor, Ag C.J. Van Til and others. Evalation of AASHO Interim Gides for Design of Pavement Strctres. NCHRP, Rept. 128, M.W. Witczak and E.J. Yoder. Principles of Pavement Design, 2nd ed. Wiley, New York, L.W. Nij boer. Einige Betrachtngen bec das Macshallverfabren zc Unterschng bitminosec Massen. St4asse nd Atobahn 8, W. Hekelom and A.J.G. Klomp. Road Design and Dynamic Loading. Proc., Assn. of Asphalt Paving Technologists, Vol. 33, N.W. McLeod. The Asphalt Institte's Layer Eqivalency Program. Asphalt Institte, College Park, MD, Res. Series 15, March C. Van dee Poel. A General System Describing the Visco-Elastic Properties of Bitmens and Its Relation to Rotine Test Data. Koninklijke/Shell Laboratorim, Amsterdam, Shell Bitmen Rept. 9, A.I.M. Claessen and others. Asphalt Pavement Design: The Shell Method. Proc., 4th International Conference on the Strctral Design of Asphalt Pavements, Univ. of Michigan, Ann Arbor, Ag Shell Pavement Design Manal: Asphalt Pavements and Overlays for Road Traffic. Shell International Petrolem Co., Ltd., The Hage, Netherlands, M.W. Witczak. Development of Regression Model for Asphalt Concrete Modls foe Use in MS-1 Stdy. Univ. of Maryland, College Park, B.F. Kallas and J.F. Shook. Factors Inflencing Dynamic Modls of Asphalt Concrete. Proc., Assn. of Asphalt Paving 1.'echnologists, Vol. 38, Thickness Design: Asphalt Pavements for Highways and Streets. Asphalt Institte, College Park, MD, Manal Series 1, M.W. Witczak. New Research Proposal: Modls Characterization of MSHA Base-Sbbase Materials foe Use in Pavement Design and Rehabilitation. Univ. of Maryland, College Pack, N.R. Draper and H. Smith. Applied Regression Analysis. Wiley, New Yock, 1966, D iscssion Waheed Uddin In the abstract, and in some places in the text, the athors se the teem correlation coefficient in refer ring to R 2 vales corresponding to the different regression eqations developed in this stdy. In the terminology of statistics, correlation coefficient is referred to as R vale (14, p. 46). In the case of mltiple linear regression analysis, R is the correlation between Y (the response variable) and estimated Y and is also called the mltiple correlation coefficient. R can take any vale between -1 and +l (14, p. 44). On the other hand, R 2 measres the "proportion of total variation abot the mean, Y, explai ned by the regressi on" (14, p. 33). R 2 oan take a vale between zero and 1 d is also called the coefficient of mltiple determination, if the regression eqation contains more than one independent variable. It is sggested that the athors shold be consistent with the proper terminology and se coefficient of mltiple determination in referring to R 2 vales. Pblication of this paper sponsored by Cnmmittee 011 Oraracteristics of Billlminos Paving Mixlllres to Meet Strctral Reqirements.