interpreting superpave pg test results with confidence intervals

8
Interpreting SuperPAVE PG test results with confidence intervals Sassan Aflaki , Milad Memarzadeh 1 School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran article info Article history: Received 20 February 2010 Received in revised form 17 December 2010 Accepted 24 December 2010 Available online 20 January 2011 Keywords: Performance grading Iranian bitumen Nonlinear regression Confidence intervals Uncertainty Bitumen selection abstract The innovative method to interpret SuperPAVE PG test results using regression analysis and confidence intervals is evaluated in this research. Till now, most of researches and conclusions in the field of bitumen rheological aspects are based on the mean values of test results. Using the mean values does not show the reliability of the test results and in many cases two bitumen with the same PG show different perfor- mances, due to different data variances. In this study the confidence intervals method is implemented to interpret the results of SuperPAVE PG tests and notify that the mean values of the variables are occa- sionally misleading and always some uncertainty exists when using bitumen. This study presents a new formulation that quantifies the uncertainties in bitumen behavior to obtain the true temperature grade of the bitumen with different confidence levels (75% and 95%). Two samples of bitumen with same PG were evaluated with this formulation and the interesting results were found. The behavior of these samples with same PG was significantly different regarding the higher confidence intervals. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction In order to construct new durable pavements, it is necessary to have the ability to predict the performance of hot mixed asphalt concrete (HMAC) when selecting appropriate asphalt binder. SuperPave performance grading (PG) specifications was developed to relate bitumen properties to field performance and allow the selection of appropriate bitumen for a specific climatic condition [1]. For the past two decades significant researches has been con- ducted on modified asphalt mixtures. Polymers and some other additives can successfully improve the performance of asphalt pavements at low, intermediate and high temperature by increas- ing mixture resistance to fatigue cracking, thermal cracking and permanent deformation [2]. For these reasons, it is necessary to predict the field perfor- mance of asphalt mixes when selecting the bitumen (pure or mod- ified). Binders of equal PG grades were found to vary significantly in their rheological properties, i.e. glass transition behavior, low temperature ductility (i.e., strain at failure), and their strength (i.e., stress at failure) [3]. These results indicate that the qualitative evaluation of test results is not adequate regarding selecting appropriate bitumen. The statistical evaluation of bitumen rheo- logical properties as well as qualitative investigation are studied in recent researches. Aflaki and Memarzadeh utilized the statistical methods in evaluating crumb rubber modification agitation effects on rheological properties of bitumen [4]. Liu et al. [5] evaluated the performance of different modified binders with different CRM con- tents, particle size, and CRM type by using the analysis of variance, ANOVA. As illustrated in Fig. 1, using the mean values of test results and disregarding the data variances always imposed the risk of 50% as the nature of test results; mostly half of the results are less than the mean values. To increase the reliability of evaluating the test results, the significant level should be increased, as shown in Fig. 2 for upper confidence interval. This occurs because of different variations in test results, and dissimilar variances of data. Using the mean values does not show the reliability of the test results and in any cases two bitumen samples with same PG show different per- formances. In this study the confidence intervals method is imple- mented to interpret the results of SuperPAVE PG tests and notify that the mean values of the variables are occasionally misleading and always some uncertainty exists in bitumen selection. Defining the confidence intervals and probabilistic approaches to interpret test results, reduces such errors. For example it will be shown later that some modified bitumen that seems to have same PG, are really different and have contrasting risk in performance if reliability other than 50% is considered; this study introduce a significant method in selecting appropriate bitumen to perform well in pro- jects regarding climate needs. Regression analysis has been used widely in all branches of engineering to develop the empirical relations between two (or more) variables or to determine the theoretical relationship 0950-0618/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2010.12.059 Corresponding author. Tel.: +98 21 61112273; fax: +98 21 66403808. E-mail addresses: afl[email protected] (S. Aflaki), [email protected] (M. Memarzadeh). 1 Tel.: +98 917 1831597. Construction and Building Materials 25 (2011) 2777–2784 Contents lists available at ScienceDirect Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

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Construction and Building Materials 25 (2011) 2777–2784

Contents lists available at ScienceDirect

Construction and Building Materials

journal homepage: www.elsevier .com/locate /conbui ldmat

Interpreting SuperPAVE PG test results with confidence intervals

Sassan Aflaki ⇑, Milad Memarzadeh 1

School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 20 February 2010Received in revised form 17 December 2010Accepted 24 December 2010Available online 20 January 2011

Keywords:Performance gradingIranian bitumenNonlinear regressionConfidence intervalsUncertaintyBitumen selection

0950-0618/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.conbuildmat.2010.12.059

⇑ Corresponding author. Tel.: +98 21 61112273; faxE-mail addresses: [email protected] (S. Aflaki), mema

Memarzadeh).1 Tel.: +98 917 1831597.

The innovative method to interpret SuperPAVE PG test results using regression analysis and confidenceintervals is evaluated in this research. Till now, most of researches and conclusions in the field of bitumenrheological aspects are based on the mean values of test results. Using the mean values does not show thereliability of the test results and in many cases two bitumen with the same PG show different perfor-mances, due to different data variances. In this study the confidence intervals method is implementedto interpret the results of SuperPAVE PG tests and notify that the mean values of the variables are occa-sionally misleading and always some uncertainty exists when using bitumen. This study presents a newformulation that quantifies the uncertainties in bitumen behavior to obtain the true temperature grade ofthe bitumen with different confidence levels (75% and 95%). Two samples of bitumen with same PG wereevaluated with this formulation and the interesting results were found. The behavior of these sampleswith same PG was significantly different regarding the higher confidence intervals.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In order to construct new durable pavements, it is necessary tohave the ability to predict the performance of hot mixed asphaltconcrete (HMAC) when selecting appropriate asphalt binder.SuperPave performance grading (PG) specifications was developedto relate bitumen properties to field performance and allow theselection of appropriate bitumen for a specific climatic condition[1].

For the past two decades significant researches has been con-ducted on modified asphalt mixtures. Polymers and some otheradditives can successfully improve the performance of asphaltpavements at low, intermediate and high temperature by increas-ing mixture resistance to fatigue cracking, thermal cracking andpermanent deformation [2].

For these reasons, it is necessary to predict the field perfor-mance of asphalt mixes when selecting the bitumen (pure or mod-ified). Binders of equal PG grades were found to vary significantlyin their rheological properties, i.e. glass transition behavior, lowtemperature ductility (i.e., strain at failure), and their strength(i.e., stress at failure) [3]. These results indicate that the qualitativeevaluation of test results is not adequate regarding selectingappropriate bitumen. The statistical evaluation of bitumen rheo-logical properties as well as qualitative investigation are studied

ll rights reserved.

: +98 21 [email protected] (M.

in recent researches. Aflaki and Memarzadeh utilized the statisticalmethods in evaluating crumb rubber modification agitation effectson rheological properties of bitumen [4]. Liu et al. [5] evaluated theperformance of different modified binders with different CRM con-tents, particle size, and CRM type by using the analysis of variance,ANOVA.

As illustrated in Fig. 1, using the mean values of test results anddisregarding the data variances always imposed the risk of 50% asthe nature of test results; mostly half of the results are less thanthe mean values. To increase the reliability of evaluating the testresults, the significant level should be increased, as shown inFig. 2 for upper confidence interval. This occurs because of differentvariations in test results, and dissimilar variances of data. Using themean values does not show the reliability of the test results and inany cases two bitumen samples with same PG show different per-formances. In this study the confidence intervals method is imple-mented to interpret the results of SuperPAVE PG tests and notifythat the mean values of the variables are occasionally misleadingand always some uncertainty exists in bitumen selection. Definingthe confidence intervals and probabilistic approaches to interprettest results, reduces such errors. For example it will be shown laterthat some modified bitumen that seems to have same PG, are reallydifferent and have contrasting risk in performance if reliabilityother than 50% is considered; this study introduce a significantmethod in selecting appropriate bitumen to perform well in pro-jects regarding climate needs.

Regression analysis has been used widely in all branches ofengineering to develop the empirical relations between two(or more) variables or to determine the theoretical relationship

Reliability Risk

Mean value

Fig. 1. Risk and reliability regarding to using mean values of test results.

Reliability

Risk

Mean value

Fig. 2. Example of increasing in reliability by increasing the significant level.

2778 S. Aflaki, M. Memarzadeh / Construction and Building Materials 25 (2011) 2777–2784

between observed data. Oftentimes the theoretical relationship be-tween variables in engineering cannot be derived on the basis oftheoretical considerations; in these cases, the required relationshipmay be established empirically on the basis of experimental orfield observations [6].

In order to evaluate the reliability of bitumen performances, thenonlinear regression analysis and confidence intervals are used.This paper presents the new formulation based on regression andconfidence intervals methods to obtain the true temperature gradeof binders with high confidence. This formulation is useful inselecting the appropriate bitumen due to importance of the projectand environmental conditions regarding reliability concepts.

2. Methodology

2.1. Confidence intervals in regression

As the regression equation gives us the predicted mean value of Y for knownvalues of control variable X, the confidence intervals of the regression equationcan be defined, which should provide some measurements for the range of the trueequation.

Since the regression coefficients a and b are estimated from finite samples ofsize n, they are individually t-distributed with (n � k � 1) degrees-of-freedom[5];therefore, the mean value �yi ¼ EðYjX ¼ xiÞ for only one variable estimated fromthe linear regression equation at X = xi will also have a t-distribution with (n � 2)degree of freedom, accordingly, the statistic will have the t-distribution with(n � 2) degree of freedom [7].

Yi � lY jxi

sY jx

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nþ

ðxi��xÞ2Pðxi��xÞ2

r ð2:1Þ

On this basis, the (1 � a) confidence interval can be obtained, as the regressionparameters a and b are defined as [8]:

hai1�p ¼ a�SffiffiffiffiffiffiffiffiffiffiffiP

x2i

qffiffiffiffiffiffiffiffiffinSxxp t1�p

2;n�2; aþSffiffiffiffiffiffiffiffiffiffiffiP

x2i

qffiffiffiffiffiffiffiffiffinSxxp t1�p

2;n�2

24

35

hbi1�p ¼ b�St1�p

2;n�2ffiffiffiffiffiffiSxxp ; bþ

St1�p2;n�2ffiffiffiffiffiffiSxxp

" #ð2:2Þ

where Sxx ¼Pn

i¼1ðxi � �xÞ2; Syy ¼Pn

i¼1ðyi � �yÞ2; Sxy ¼Pn

i¼1ðxi � �xÞðyi � �yÞ; and S2 ¼Syy��bSxy

n�2 .In this research the one side confidence interval for the regression parameter is

used because of upper limit criteria (maximum value) for G⁄ � Sin d (Eq. (2.3)) andlower limit criteria (minimum value) for G⁄/Sin d (Eq. (2.4)) which are defined as:

ðai1�p ¼ aþSffiffiffiffiffiffiffiffiffiffiffiP

x2i

qffiffiffiffiffiffiffiffiffinSxxp t1�p;n�2

ðbi1�p ¼ bþ St1�p;n�2ffiffiffiffiffiffiSxxp ð2:3Þ

haÞ1�p ¼ a�SffiffiffiffiffiffiffiffiffiffiffiP

x2i

qffiffiffiffiffiffiffiffiffinSxxp t1�p;n�2

hbÞ1�p ¼ b� St1�p;n�2ffiffiffiffiffiffiSxxp ð2:4Þ

2.2. Correlation analysis

It can be seen that if the conditional standard deviation sY|x is small enough(close to zero), the linear regression equation provides a good prediction of Y forknown values of X. However, a better statistical measurement of the linear relation-ship between two random variables X and Y is the correlation coefficient, which wasdefine as

qX;Y ¼CovðX;YÞ

rXrYð2:5Þ

where Cov(X, Y) is the covariance between X and Y.The accuracy of the predicted mean value of Y for the given values of X depends

on the correlation coefficient [6].

2.2.1. Estimation of the correlation coefficientFor a set of n pairs of observations, the estimate of the correlation coefficient is

qX;Y ¼1

n� 1

Pni¼1ðxi � �xÞðyi � �yÞ

sX sY¼ 1

n� 1

Pni¼1xiyi � n�x�y

sX sYð2:6Þ

where �x; �y are the sample means and sX and sY are the sample standard deviations forX and Y. The value of q ranges from �1 to +1. If the estimated qX;Y is large (close to±1.0), there is a strong linear relationship between X and Y. From Eqs. (2.3) and (2.8),the estimated correlation coefficient is calculated as

q ¼Pðxi � �xÞðyi � �yÞPðxi � �xÞ2

sX

sY

� �¼ b

sX

sYð2:7Þ

By using Eq. (2.7) in Eq. (2.6) the correlation coefficient becomes:

q2 ¼ 1� ðn� 2Þðn� 1Þ �

s2Y jx

s2y

ð2:8Þ

In this paper, the r2 parameter was used, that is better known as the linearityindication of the variables and it is closely related to the correlation coefficientq2. When n becomes great r2 is a good prediction for q2.

This parameter is defined as [6]:

r2 ¼ 1�s2

Y jx

s2Y

ð2:8Þ

2.3. Nonlinear regression

Functional relationships between engineering variables are not always linear.Observed data for the variables which are obtained from experiments, may showthe nonlinear behavior [6].

Table 1Complex shear modulus G⁄ � Sin d measurement test results.

Temperature (�C) G⁄ � Sin d (Pa)

eg7rp eg4rp ep13rp

19 5,507,400 6,157,400 5,888,70019 5,444,100 6,160,500 5,956,10019 5,362,300 6,175,000 5,990,00022 4,251,200 4,408,000 4,472,30022 4,237,400 4,389,400 4,518,00022 4,169,700 4,385,400 4,491,00025 3,458,200 3,118,500 3,362,40025 3,334,100 3,142,400 3,353,60025 3,290,100 3,136,500 3,357,70028 2,664,500 2,185,000 2,491,000

S. Aflaki, M. Memarzadeh / Construction and Building Materials 25 (2011) 2777–2784 2779

Nonlinear regression is usually based on an assumed nonlinear function of themean value for the dependent variable, Y as a function of the independent variableX, with certain undetermined coefficients that must be evaluated on the basis of theobserved data. The simplest type of nonlinear functions for the regression of Y on Xis

EðY jxÞ ¼ aþ bgðxÞ ð2:9Þ

where g(x) is a predetermined nonlinear function of x.By defining a new variable x0 = g(x), Eq. (2.10) becomes

EðY jxÞ ¼ aþ bx0 ð2:10Þ

It is clear that the Eq. (2.11) is of the same mathematical form as the linearregression equation of Eq. (2.1). So the linear regression analysis can be used forinterpreting this kind on nonlinear regression.

In this study, nonlinear regression method is used. Different functions are eval-uated for fitness on our data and finally the exponential function was used to makethe dependent variable, G⁄ � Sin d and G⁄/Sin d, linear. The G⁄ � Sin d, binder’s fatigueparameter, varies with Temperature (T) (As shown in Fig. 3) by this formula:

G� � Sind ¼ aebT ð2:11Þ

where a and b are regression constants.When the logarithm function is operated on the formula above, the following

formula is generated:

lnðG� � SindÞ ¼ ln aþ bT ð2:12Þ

The G⁄/Sin d, binder’s rutting parameter, varies with temperature (T) (As shownin Fig. 3) according to this formula:

G�=Sind ¼ cegT ð2:13Þ

where c and g are regression constants.When the logarithm function is operated on the formula above, the following

formula is generated:

lnðG�=SindÞ ¼ ln cþ gT ð2:14Þ

28 2,630,900 2,178,300 2,497,80028 2,589,300 2,174,100 2,518,40031 1,518,600 1,495,800 1,852,90031 1,517,700 1,490,600 1,859,00031 1,482,300 1,480,800 1,848,80034 1,149,500 999,670 1,360,00034 1,041,600 993,150 1,360,20034 1,078,100 991,570 1,366,40037 753,350 668,450 1,002,50037 710,350 669,960 999,01037 689,510 665,160 992,680

3. Materials and test methods

Eleven unmodified bitumen samples, including PG 64-16, 70-16, 58-22 and 64-22, were obtained from seven bitumen produc-tion plants in Iran. Detailed test results and contrast them toclimatic needs was conducted by Aflaki and Tabatabaee [2]. TheSuperpave testing protocol (ASTM D6373) was used to evaluatethe bitumen samples [9].

0

5000

10000

15000

20000

25000

30000

0 10 20 30 40

G*/

Sin(

delt

a) (

Pa)

Temperatu

eg2

Fig. 3. Variation of G⁄/Sin d and G⁄ � Sin d with

In this research some of the modified binders from the mainstudy by Aflaki and Tabatabaee were selected to conduct the con-fidence interval method and compare the results. As stated byAflaki and Tabatabaee a base binder was blended with varyingamounts of modifiers such as crumb rubber, gilsonite, PPA, andSBS as modifiers.

In all cases, mixing temperature and blending speed were se-lected so that a sufficient vortex can be formed in the mixingchamber. Modified samples, eg2, eg4, eg7, were modified by Gil-sonite and ep13 was modified by Polyphosphoric Acid [2].

To produce eg2, eg4, and eg7, fine grained gilsonite was addedto base bitumen in amounts equaling 2%, 4% and 7% of the weightof the bitumen when the base bitumen reached 180 �C and blendedfor 180 min at 4500 rpm using a high shear homogenizer mixer in

0

1000000

2000000

3000000

4000000

5000000

6000000

7000000

50 60 70 80re (ºC)

ep13rp

G*.

Sin(

delt

a) (

Pa)

temperature, the exponential behavior.

Table 2The confidence interval results for G⁄ � Sin d.

Sample G⁄ � Sin d at 22 �C IT (�C)

Average 50% (Pa) 75% (Pa) 95% (Pa) Variation 75% (%) Variation 95% (%) ITMean IT 75% IT 95%

ep13rp 4,493,767 4,650,475 4,694,818 3 4 20.9 21.26 21.35eg4rp 4,394,287 4,837,700 5,443,213 10 24 21 21.73 22.71eg7rp 4,219,433 4,963,889 6,095,248 18 44 20.8 21.93 23.84

Table 3Regression parameters for Eq. (4.3).

Sample ln a b r2

Mean 75% 95% Mean 75% 95%

eg7rp 17.79 17.87 17.98 �0.11 �0.11 �0.10 0.976eg4rp 18.02 18.06 18.13 �0.12 �0.12 �0.12 0.993ep13rp 17.49 17.51 17.49 �0.10 �0.10 �0.097 0.997

2780 S. Aflaki, M. Memarzadeh / Construction and Building Materials 25 (2011) 2777–2784

a closed container to minimize the exposure to air and control theaging. The modified binders were labeled eg2, eg4, and eg7 respec-tively [2].

PPA gel in 1.3% of the weight of the bitumen was added to thebase binder at 160 �C and blended for 60 min using a low shearmixer and the modified binder was labeled ep13.

High service temperature (HT) for a binder is determined as thetemperature at which the complex shear modulus G⁄/Sin d is great-er than 1 kPa for unaged binder and greater than 2.2 kPa for therolling thin film oven (RTFO) aged condition (ASTM D2872) [10].Low service temperature (LT), on the other hand, is 10 �C lowerthan the temperature at which the pressurized aging vessel

13

13.5

14

14.5

15

15.5

16

16.5

15 20 25

ln(G

*sin

(del

ta))

(ln

(Pa)

)

Temper

e

crit

75

95

Da

Lin

IT(Mean) IT(75%) IT(95%)

Fig. 4. ln(G⁄ � Sin d) – temperature regressi

(PAV) (ASTM D6521) [9] aged binder exhibits creep stiffness of lessthan 300 MPa and an m-value of greater than 0.300 at 60 s ofloading (ASTM D6648) [11]. Intermediate service temperature(IT) is equal to HTþLT

2 þ 4; at which the binder’s complex shearmodulus G⁄Sin d in the PAV aged condition should not exceed5000 kPa.

4. Discussions

The formulation of determining the temperature grade withhigher confidence levels is generated and examined trough twoexamples.

Table 1 shows the complex shear modulus G⁄Sin d measure-ment test results for three different modified binders with samePG. As shown in the table the parameter was evaluated at sevendifferent temperatures (19–37 �C) for test samples, eg4rp, eg7rpand ep13rp [2]. All samples have almost the same IT (20.8–21 �C), based on the average of DSR PG test results, as shown in Ta-ble 2.

The regression parameters of these materials for mean, 75% and95% confidence levels were shown in Table 3.

The formulation for calculating the IT grade of each binderbased on G⁄ � Sin d with 75% confidence is as follows,

30 35 40

ature (ºC)

g7rp

eria (G*.Sin(delta)<=5000 kPa)

%

%

ta Point

ear (Data Point)

on and confidence intervals for eg7rp.

13

13.5

14

14.5

15

15.5

16

16.5

15 20 25 30 35 40

ln(G

*Sin

(del

ta))

(ln

(Pa)

)

Temperature (ºC)

eg4rp

criteria (G*.Sin(delta)<=5000 kPa)

75%

95%

Data Point

Linear (Data Point)

Fig. 5. ln(G⁄ � Sin d) – temperature regression and confidence intervals for eg4rp.

S. Aflaki, M. Memarzadeh / Construction and Building Materials 25 (2011) 2777–2784 2781

13.5

14

14.5

15

15.5

16

15 20 25 30 35 40

ln(G

*Sin

(del

ta))

(ln

(Pa)

)

Temperature (ºC)

ep13rp

criteria (G*.Sin(delta)<=5000 kPa)

75%

95%

Data Point

Linear (Data Point)

Fig. 6. ln(G⁄ � Sin d) – temperature regression and confidence intervals for ep13rp.

Table 4Complex shear modulus G⁄/Sin d measurement test results.

Temperature (�C) G⁄/Sin d (Pa)

o eg2

46 25,589 24,30046 25,751 23,70346 25,770 23,51052 9075.1 9473.552 9381.5 9218.752 9392.3 9120.558 3684.1 3915.358 3601.8 3832.158 3580.2 3692.864 1488.7 1511.564 1435.8 1507.964 1469.2 1506.970 654.8 665.170 673.1 663.870 628.2 643.2

Table 6Regression parameters of Eq. (4.5).

Sample ln c g r2

Mean 75% 95% Mean 75% 95%

eg2 16.95 16.92 16.88 �0.15 �0.150 �0.151 0.999o 17.14 17.03 16.87 �0.153 �0.155 �0.157 0.995

2782 S. Aflaki, M. Memarzadeh / Construction and Building Materials 25 (2011) 2777–2784

Grade75% ¼lnð5� 106Þ � aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSyy�bSxyÞ:

Px2

inðn�2ÞSxx

r:t0:75;n�2

" #

bþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSyy�bSxyÞðn�2ÞSxx

� �r:t0:75;n�2

� � ð4:3Þ

Table 5The confidence interval results for G⁄/Sin d.

Sample G⁄/Sin d at 64 �C

Average 50% (Pa) 75% (Pa) 95% (Pa) Variatio

o 1464.57 1248.25 888.85 15eg2 1508.77 1470.68 1346.64 3

4

5

6

7

8

9

10

11

40 45 50 55

ln(G

*/si

n(de

lta)

) (

ln(P

a))

Temper

Fig. 7. ln(G⁄ � Sin d) – temperature regres

where a and b are the constants that are found from Eq. (4.4):

b ¼Pn

i¼1xiyi � n�x�yPni¼1x2

i � n�x2¼ Sxy

Sxx

a ¼ �y� b�x ð4:4Þ

For calculating the IT grade of each material based on G⁄ � Sin dwith 95% confidence, t0.95,n�2 must be replaced with t0.75,n�2.

The results are shown in Table 2 and Figs. 4–6. As illustrated inTable 2 and figures the average value of G⁄ � Sin d, the IT of thesebinders are the same and by using average values of test results,the differences between the performances of the binders cannotbe realized. However, by utilizing Eq. (4.3), the minor differencesbetween ITs can be realized.

The risk level depends on the confidence interval, i.e. usingmean values which is 50% confidence interval yields 50% risk to re-sults. Correspondingly using 75% and 95% confidence intervals

HT (�C)

n 75% (%) Variation 95% (%) HTMean HT75% HT95%

39 66.8 65.43 63.2511 66.9 66.56 65.97

60 65 70 75ature (ºC)

o

criteria (G*/Sin(delta)>=1 kPa)

75%

95%

Data Point

Linear (Data Point)

HT(Mean)HT(75%)HT(95%)

sion and confidence intervals for o.

6

6.5

7

7.5

8

8.5

9

9.5

10

10.5

45 50 55 60 65 70 75

ln(G

*/Si

n(de

lta)

) (

ln(P

a))

Temperature (ºC)

eg2

Data Point

criteria (G*/Sin(delta)>=1 kPa)

75%

95%

Linear (Data Point)

Fig. 8. ln(G⁄ � Sin d) – temperature regression and confidence intervals for eg2.

S. Aflaki, M. Memarzadeh / Construction and Building Materials 25 (2011) 2777–2784 2783

yield 25% and 5% risk, respectively. As it is clear from Table 2,ep13rp and eg7rp have the same ITs, based on average values.While using the 75% confidence interval results 3% and 18% varia-tion from the average values for ep13rp and eg7rp respectively.These variations are 4% and 44% regarding 95% confidence intervalfor ep13rp and eg7rp respectively. It can be concluded that in spiteof same ITs for ep13rp and eg7rp, the behavior of the ep13rp

regarding 95% and 75% confidence intervals is 90% ¼ ð44%�4%Þ44%

h iand 83% ¼ ð18%�3%Þ

18%

h imore reliable than eg7rp, respectively. This

comparison can be made between other binders in Table 2.Table 4 shows the complex shear modulus G⁄/Sin d measure-

ment test results, as shown in the table the parameter was evalu-ated at five different temperatures for test samples, o and eg2 [2].All samples have almost the same HT (66.8–66.9 �C), based on theaverage of DSR PG test results, as shown in Table 5. The regressionparameter of these binders for mean, 75% and 95% confidence lev-els were shown in Table 6.

The formulation for calculating the HT grade of each binder onG⁄/Sin d with 75% confidence is as follows,

Grade75% ¼lnð103Þ � c�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSyy�gSxyÞ:

Px2

inðn�2ÞSxx

r:t0:75;n�2

" #

g�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðSyy�gSxyÞðn�2ÞSxx

� �r:t0:75;n�2

� � ð4:5Þ

where c and g are the constants that were found from Eq. (4.4).For calculating the HT grade of each material based on G⁄/Sin d

with 95% confidence, t0.95,n�2 must be replaced with t0.75,n�2.The results are shown in Table 5 and Figs. 7 and 8. As shown in

Table 5 and figures the average value of G⁄/Sin d, the HT of thesematerials, are the same and by using average values of test resultsthe differences between the performance of these materials cannotbe realized, but when the confidence intervals method was used

for interpreting the performances of the binders, the minor differ-ences between HTs distinguished. As it is clear from Table 5, eg2and o have the same HTs, based on average values. Using the75% confidence interval results 3% and 15% variation from the aver-age values for eg2 and o respectively. These variations are 11% and39% in case of 95% confidence interval for eg2 and o respectively.

It can be concluded that in spite of same HTs for eg2 and o, thebehavior of the eg2 regarding 95% and 75% confidence intervals is

72% ¼ ð39%�11%Þ39%

h iand 80% ¼ ð15%�3%Þ

15%

h imore reliable than o,

respectively.

5. Conclusion

This research is clearly demonstrated that using the mean valueof the test results for interpreting the characteristics of materialsdoes not efficiently address the performances of binders and doesnot consider data variance. Using confidence intervals could helpresearchers to diagnosis the differences while testing binders anddefining the reliability of characteristic behavior of binders.

The formulation presented in this paper is examined with twodifferent examples and it can be concluded in each case that bind-ers despite of having the same PGs, show different reliability intheir characteristics such as service temperature grade. The pro-posed formulation is more appropriate in obtaining the true tem-perature grade of binders.

Acknowledgements

The authors would like to express their sincere appreciation toProfessor Nader Tabatabaee supervisor of the SuperPAVE labora-tory of Civil Engineering Department, Sharif University of Technol-ogy, and also wish to acknowledge Eng. Zahra Kamali for hervaluable efforts on conducting the laboratory tests. Also, the

2784 S. Aflaki, M. Memarzadeh / Construction and Building Materials 25 (2011) 2777–2784

authors wish to acknowledge and thank Dr. Abbass Babazadeh forhis precious technical comments and Eng. Maral Jalili for providingassistance in reviewing the final manuscript.

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