[American Society of Civil Engineers GeoShanghai International Conference 2010 - Shanghai, China (June 3-5, 2010)] Paving Materials and Pavement Analysis - Evaluation of Gradation

Evaluation of Gradation of RAP Based on Fractal Theory

Xiaoyang Jia1 and Fen Ye2

1Ph. D. candidate, Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 201804, P.R. China; [email protected] 2Associate Professor, Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University, Shanghai 201804, P.R. China; [email protected], Corresponding Author

ABSTRACT: This paper introduced fractal theory to the description of gradation of asphalt mixture. The feasibility of description of RAP gradation by fractal theory was firstly discussed. The fractal dimension calculated by Least Square Method (LSM) was employed to evaluate gradation of RAP. Compaction tests, surface-dry condition method tests and indirect tensile strength tests were then employed to evaluate physical and mechanical performance of RAP mixtures stabilized by foamed asphalt. Relationships between fractal dimension and test results were studied. It seems that the fractal dimensions are of relevance to the performance of cold-recycling asphalt mixture. And this points out a new direction of RAP research field.

INTRODUCTION Cold recycling becomes a technically promising and cost effective method for pavement recycling. Therefore, more attentions are being paid to recycling asphalt pavement (RAP). However, the evaluation of gradation of RAP is still a long way to go. Gradation of RAP varies largely due to the types of structures of pavements, milling speeds of machines, moisture content of pavement, etc. Therefore, it is hard to evaluate types of gradation in traditional way. And this brings difficulties for QA/QC job for cold-recycling project. The shape of RAP is irregular and it is hard to be quantified. However, fractal theory, which can study things in meso-scope, is a useful research tools to quantify things which are irregular. The application of fractal theory into cold-recycling asphalt mixture could reduce or eliminate the inconvenience of evaluation of gradation of RAP and describe RAP in a more objective way. LITERATURE REVIEW Fractal concepts were introduced into the field of road engineering at 1990’s: Carr et al. (1990) reported the use of fractals for the characterization of aggregate shape. Lee et al. (1990) employed fractal dimension to quantify the roughness profile of rocks.

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Copyright ASCE 2010 GeoShanghai 2010 International Conference Paving Materials and Pavement Analysis

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They also described shape and angularity of rocks by fractal dimension from the measured data. Perdomo et al. (1991) applied fractal analysis into the classification of particle shape and surface texture of both crushed and uncrushed aggregates when they studied the effects of aggregates on the rutting of asphalt mixture. Ribble et al. (1992) studied macro/microtexture of aggregate particles through fractal theory, meanwhile, they concluded that the workability of concrete mixtures is directly dependent on the fractal dimension of aggregates. Their test results showed that the larger percentage of crushed particles content is, the larger fractal dimension of aggregate particles will be. Similar results were also obtained by Yeggonc et al. (1996) when they studied rutting of asphalt mixture. They thought that fractal dimension of coarse aggregates had relationship with both static and dynamic creep test of asphalt mixture. Kokkalis (1996) gave an overall presentation of fractal methods in road engineering in his paper. In China, Li et al. (1995) developed fractal gradation formula and compared it with the current gradation formula and they thought that fractal was the essence of aggregates gradation. Huang (2006) and Yang (2008) applied the fractal theory into performance evaluation of asphalt mixture synthetically. FRACTAL THEORY AND ITS APPLICATION IN ASPHALT MIXTURE

Fractal is generally regarded as "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," (Mandelbrot, B.B., 1982) Therefore, an object which has the characteristic of fractal is called self-similarity. An object is self-similar when it can be broken down into an arbitrary number of small pieces, and each of those pieces is a replica of the entire structure. In mathematics, fractal could be described by a recursive formula. According to the features of fractal, if one point set S could be written as

( ) ⎯⎯ →⎯= →0δδδ dd NM

⎩⎨⎧

<∞>

f

f

dddd

when ,when ,0

(1)

Where, Md represents size of point set S; δ represents line element; N(δ) represents times of measurement; and here “df” is defined as fractal dimension only if Md is finite. Therefore, “df” could be calculated by equation (2). Generally, the value of fractal dimension is a fractional rather an integer.

( ) ( )⎟⎠⎞⎜

⎝⎛

=−=

δ

δδδ

1ln

lnln

ln NNd f (2)

In asphalt mixture, shapes of fine aggregates are similar to those of coarse ones under the microscope. Therefore, aggregates in asphalt mixture have fractal characteristic. It seems that RAP has the characteristic of self-similarity just like virgin aggregate. Therefore, it is feasible to the application of fractal theory in both virgin aggregates and RAPs such as “black rocks”. Following steps show the application of fractal theory into the evaluation of aggregates in asphalt mixture. Given N(r) represents numbers of particles, diameter of which is over r while M(r) less than r. N0 is the number of total particles. And it has (3). Distribution function of particles could be written as (4).

( ) ( ) 0NrMrN =+ (3)

GEOTECHNICAL SPECIAL PUBLICATION NO. 20352


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( ) ( )0NrMr =ϕ (4)

According to the fractal theory, ( ) DCrrN −= . Then N0 and M(r) will be; ( ) ( ) ( )DD rrCrNrNN −− −=−= maxminmaxmin0 (5)

( ) ( ) ( ) ( )DD rrCrNrNrM −− −=−= minmin (6) Then, distribution function of particle size could be as follows,

( ) ( ) D

DD

DD

rr

rrrr

NrMr

−

−−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−−

==maxmaxmin

min

0

ϕ (7)

When

minmax rr >> , ( )D

rrr

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

max

ϕ (8)

Given that G(r) represents the weight of aggregate particles whose diameter are less than r. G0 is the weight of whole particles. The percentage of aggregates P(r) could be defined as,

( ) ( )0GrGrP = (9)

According to volume equation, ( ) ( ) 3rKrMrG v= , 3

max00 rKNG v= (10)

Where, Kv is the volume parameter for aggregate particle. Therefore, equation (9) could be rewritten as,

( ) ( ) D

v

v

rr

rKNrKrMrP

−

⎟⎟⎠

⎞⎜⎜⎝

⎛==

3

max3

max0

3

(11)

Here, fractal dimension is D which is employed to evaluate the gradation of aggregate particles for the asphalt mixture, r represents the sieve size. D could be calculated by max/ rr and P(r) in double logarithmic coordinates as Equation (12).

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

max

log3logr

rDrP (12)

In double logarithm coordinate, linear fitting could be done by least square method based on P(r) and r/rmax. Slope of linear is 3-D. Then, D and correlation coefficient (R2) could be determined. Ordinarily, D is from 0 to 3.

FRACTAL CHARACTERISTIC OF GRADATION OF RAP

D for RAP at four different milling speeds were calculated by Least Square Method (LSM) and results were listed in Table1. It could be concluded that correlation coefficient (R2) for RAP is from 0.9741 to 0.9935, which means there was a good linear relationship between P(r) and r/rmax in double logarithm coordinate. In other words, the gradation of RAP has the good fractal characteristic.

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Table 1 shows that the D reaches its peak at 8m/min, meanwhile, it reached the lowest at 10m/min. It is different from the common sense that milling speed determines size distribution of RAP particles. Ds for coarse and fine aggregate were calculated in Table 2, respectively. It seems that D for coarse RAP (>4.75mm) has good relationship with milling speed. D for coarse RAP decreased with the increment of milling speed which indicates that milling speed could only determine the coarse particle size distribution of RAP (>4.75mm). D for required gradation by Asphalt Recycling and Reclaiming Association (ARRA) for cold-recycling RAP by foamed-asphalt or emulsified-asphalt agent were also calculated. D is from 2.34 to 2.69 for NMAS-25.0mm. Meanwhile, D for RAP at different speed is from 2.00 to 2.16. It means that the actually gradation of RAP is coarser than what it is required for cold-recycling. Therefore, more fine virgin aggregates should be added in order to meet the gradation requirements for RAP. Generally, range of D for RAP gradation is from 1.90 to 2.10 and the suggested range of D given by some projects or other researchers is from 2.29 to 2.69.(ARRA) So this means that more fine aggregate should be added for RAP.

Table 1. Fractal Parameters for RAP at Different Milling Speed (NMAS-25)

Parameter Milling Speed (m/min) 4 6 8 10

D 2.07 2.11 2.16 2.00 R2 0.975 0.9741 0.9838 0.9935

Table 2. D of Coarse and Fine Aggregate of RAP at Different Milling Speed

Type Milling Speed (m/min) 4 6 8 10

Coarse RAP 2.56 2.50 2.40 2.14 Fine RAP 1.88 1.92 2.03 2.01

RAW MATERIALS

In this paper, foamed-asphalt mixture was involved. The raw materials employed were listed as follows. Recycling asphalt pavement RAP is the combination of waste asphalt pavement materials milled by milling-planing machine and virgin fine aggregates (0-4.75mm). Three test gradations of RAP were involved in this paper showed in FIG.1. All D for test gradations of RAP were calculated and listed in Table 3. And D for the RAP, coarse RAP (>4.75mm) and fine RAP (<4.75mm) were calculated respectively.



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Table 3. D of RAP for Different Test Gradations

Items No. of Test Gradation 1 2 3

Type of gradation RAP 2.48 2.50 2.54

Coarse RAP 2.57 2.61 2.67 Fine RAP 2.46 2.47 2.48

Asphalt binder According to the < Technical Specifications for Construction of Highway Asphalt Pavement, JTG F40-2004>, 70# asphalt (A-level) was utilized as foamed asphalt. The optimum foaming conditions in which the foamed asphalt mixture is mixed were determined by foaming experiment: foaming temperature is 150℃; foaming water content is 1.5%; expansion ratio is 11; half life is 12s.

0

20

40

60

80

100

0.01 0.1 1 10 100

Sieve Size(mm)

Pass

ing

Perc

enta

ge(%

)

No.1No.2

No.3

FIG. 1. Gradation curves of RAP

SPECIMENS PREPARATION AND MEASUREMENT

In this paper, compaction tests were first employed to determine optimum moisture content and maximum dry density of RAP. Then mixing water content could be determined according to optimum moisture content. Specimens were compacted according to Marshall Methods, 75 times for each side. All specimens were cured for 72 hours at 40°C. After the specimens were prepared, the bulk relative densities of specimens were measured by surface-dry method tests. Then indirect tensile stress (ITS) was measured at 25°C. RELEVANCE OF RAP PERFORMANCE AND FRACTAL DIMENSION Relevance of physical performance of RAP and fractal dimension In Table 4, it seems that both the optimum water content, maximum dry density and bulk volume density increase with the increment of fractal dimension.



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Reasons for this lie in the structure of mixture of foamed asphalt stabilized RAP. Larger D for RAP leads to the finer gradation. So the specific surface area for RAP will be higher. RAP then would absorb more water in the process of compaction which results in higher optimum water content of RAP. As for the maximum dry density and bulk volume density of RAP, the increment of D makes the gradation of RAP finer. So there will be an increment in the number of particles in unit volume which results in the increase in density. It seems that D for fine RAP mostly affect physical performance of RAP which means the distribution of fine RAP in the asphalt mixture will have more influence on the compaction characteristic of RAP material than coarse RAP. And this will give theoretical reference to the pavement construction. Relevance of mechanical performance of RAP and fractal dimension Indirect tensile strength test (ITST) was employed to evaluate the mechanical performance of RAP. Table 4 indicated that relationships of fractal dimension of RAP and ITS at different foamed asphalt content were different. For 1.6% foamed asphalt content, ITS reaches its peak at the RAP fractal dimension of 2.50. Then ITS decreases with the increment of fractal dimension. For 2.0% foamed asphalt content, there was also a peak when D=2.50. However, D has slight influence on ITS after D=2.50. As for 2.5% foamed asphalt content, ITS increased with the increment of D. In foamed asphalt mixture, asphalt mastic which is composed of asphalt foam mixed with fine aggregates and filler contributes the cohesion for asphalt mixture. With the increase of D, the gradation of RAP becomes finer. At low foamed asphalt content, there is not enough foamed asphalt mixed with fine aggregates and fillers to form asphalt mastic. Therefore, ITS is relatively low with the increase of D (for example, D=2.54). At higher asphalt content, there will be enough asphalt to form asphalt mastic which results in the higher ITS value. This showed that the increment of both D and asphalt content will lead to the increment of ITS. And this would determine the characteristic of compaction of RAP. CONCLUSIONS

In this paper, fractal theory was introduced into the analysis of characteristic of gradation of RAP. D was adopted to evaluate gradation of RAP. Foamed asphalt stabilized RAP mixture was the research object in this paper. From the tests, it seems that there is a good relationship between D and performance of RAP mixtures. The main conclusions in this paper are as follows.

• Gradation of RAP has the fractal characteristic. • Milling speed only determine distribution of coarse RAP (>4.75mm). • Optimum water content and density increase with the increment of D. • Increment of D and asphalt content will increase ITS.



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Table 4. Fractal dimension of test gradations

Items No. of Test Gradation 1 2 3

Optimum water content (%) 5.8 6.4 6.6 Maximum dry density (g/cm3) 2.16 2.18 2.23

Bulk volume density @ different foamed asphalt content (%)

(g/cm3)

1.6 2.18 2.21 2.25 2.0 2.21 2.23 2.28 2.5 2.23 2.25 2.3

ITS @ different foamed asphalt content (%)

(MPa)

1.6 0.3 0.53 0.45 2.0 0.36 0.56 0.54 2.5 0.33 0.55 0.65

REFERENCES Mandelbrot, B.B. (1982). The Fractal Geometry of Nature. W.H. Freeman and

Company. ISBN 0-7167-1186-9. Lee, Y.H., Carr, J.R., Barr, D.J., and Haas, C.J.(1990). “Fractal dimension as a

measure of the roughness of rock discontinuity profiles”, International journal of rock mechanics and mining sciences & geomechanics abstracts, Vol 27 (6): 453-464.

Ribble, C., Szecsy, R. and Zollinger ,D.G..(1992) “Aggregate macro shape and micro texture in concrete mix design”. ASCE Spring Meeting, New York.

Carr, J. R., Norris, G. M., and Newcomb, D. E., (1990) “Characterizations of aggregate shape using fractal dimension”. TRR 69th. Annual Meeting, Washington. DC.

Perdomo, D. and Button, J. W., (1991) “Identifying and correcting rut susceptible asphalt mixtures”. TRR 1259, TRB, Washington, DC.

Yeggonc M., Button, J. W. and Zollinger, D. G., (1996) “Fractals of aggregates correlated with creep in asphalt concrete”. ASCE Journal of Transportation Engineering. Vol. 122 (1): 22-28

Kokkalis, A. G., (1996) “Fractal principles in highway and pavement engineering”. 4th National Conference on the Complexity and Chaotic Dynamic of Non-linear Systems, Patra, Greece.

Li, G., and Deng, X. (1995). “On Fractal Gradation of Aggregates”. Journal of Chongqing Jiaotong Institute, Vol. 14 (2): 38-43. In Chinese.

Huang, J. (2006). “Research on asphalt mixtures performances by nonlinearity theories”. PhD thesis. Tongji University. Shanghai, P.R.China. In Chinese.

Yang, R. (2008). “Study on the design theory and method for asphalt mixture based on fractal theory”. PhD thesis. Tongji University. Shanghai, P.R.China. In Chinese.



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Documents

[American Society of Civil Engineers GeoShanghai International Conference 2010 - Shanghai, China (June 3-5, 2010)] Paving Materials and Pavement Analysis - Evaluation of Gradation