Prediction of Fatigue Life of Rubberized Asphalt Concrete Mixtures Containing Reclaimed Asphalt Pavement Using Artificial Neural Networks

(MT/2007/023481) ASCE Journal of Materials in Civil Engineering

Feipeng Xiao1, Serji Amirkhanian2, M., ASCE, and C. Hsein Juang3, M., ASCE

Abstract: Accurate prediction of the fatigue life of asphalt mixtures is a difficult task due to the

complex nature of materials behavior under various loading and environmental conditions. This

study explores the utilization of artificial neural network (ANN) in predicting the fatigue life of

rubberized asphalt concrete (RAC) mixtures containing reclaimed asphalt pavement (RAP). Over

190 fatigue beams were made with two different rubber types (ambient and cryogenic), two

different RAP sources, four rubber contents (0%, 5%, 10%, and 15%), and tested at two different

testing temperatures of 5ºC and 20ºC. The data were organized into 9 or 10 independent

variables covering the material engineering properties of the fatigue beams and one dependent

variable, the ultimate fatigue life of the modified mixtures. The traditional statistical method was

also used to predict the fatigue life of these mixtures. The results of this study showed that the

ANN techniques are more effective in predicting the fatigue life of the modified mixtures tested

in this study than the traditional regression-based prediction models.

CE Database subject headings: Rubberized Asphalt Concrete, Reclaimed Asphalt Pavements,

Artificial Neural Network, Crumb Rubber, Fatigue Life.

____________________ 1Research Associate, Department of Civil Engineering, Clemson University, Clemson, SC

ail: fpxiao@gmail.com 29634-0911. E-m 2Professor, Department of Civil Engineering, Clemson University, Clemson, SC 29634-0911. E-mail: kcdoc@clemson.edu 3Professor, Department of Civil Engineering, Clemson University, Clemson, SC 29634-0911. E-mail: hsein@clemson.edu

Xiao et al. (2007)

INTRODUCTION

Fatigue, associated with repetitive traffic loading, is considered to be one of the most

significant distress modes in flexible pavements. The fatigue life of an asphalt pavement is

related to the various aspects of hot mix asphalt (HMA). Previous studies have been conducted to

understand how fatigue can occur and fatigue life be extended under repetitive traffic loading

(SHRP 1994; Daniel and Kim 2001; Benedetto et al. 1996; Anderson et al. 2001). When an

asphalt mixture is subjected to a cyclic load or stress, the material response in tension and

compression consists of three major strain components: elastic, viscoelastic, and plastic. The

tensile plastic (permanent) strain or deformation, in general, is responsible for the fatigue

damage and consequently results in fatigue failure of the pavement. A perfectly elastic material

will never fail in fatigue regardless of the number of load applications (Khattak and Baladi 2001).

An asphalt mixture is a composite material of graded aggregates bound with a mastic

mortar. The physical properties and performance of HMA is governed by the properties of the

aggregate (e.g., shape, surface texture, gradation, skeletal structure, modulus, etc.), properties of

the asphalt binder (e.g., grade, complex modulus, relaxation characteristics, cohesion, etc.), and

asphalt aggregate interaction (e.g., adhesion, absorption, physiochemical interactions, etc.). As a

result, the properties of asphalt mixtures are very complicated and sometimes difficult to predict

(You and Buttlar 2004, Xiao et al. 2007). However, the properties of its constituents are

relatively less complicated and easier to characterize. For example, aggregate can be considered

as linearly elastic; the asphalt binder can be considered as viscoelastic/viscoplastic. Therefore, if

the microstructure of asphalt mix can be obtained, its properties can be evaluated from the

properties of its constituents and microstructure (Wang et al. 2004, Xiao et al. 2006).

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Xiao et al. (2007)

The recycling of existing asphalt pavement materials produces new pavements with

considerable savings in material, money, and energy. Aggregate and binder from old asphalt

pavements are still valuable even though these pavements have reached the end of their service

lives. Furthermore, mixtures containing reclaimed asphalt pavement (RAP) have been found, for

the most part, to perform as well as the virgin mixtures with respect to rutting resistance. The

National Cooperative Highway Research Program (NCHRP 2001) report provides the basic

concepts and recommendations concerning the components of mixtures, including new aggregate

and RAP materials.

In recent years, more and more states have begun to ban whole tires from landfills, and most

states have laws specially dealing with scrap tires. As a result, it is necessary to find safer and

economical ways for disposing these tires. The civil engineering market involves a wide range of

uses for scrap tires, exemplified by the fact that currently 39 states have approved the use of tire

shreds in civil engineering applications (RMA 2003). Most laboratory and field experiments

indicate that the rubberized asphalt concretes (RAC), in general, show an improvement in durability,

crack reflection, fatigue resistance, skid resistance, and resistance to rutting not only in an overlay,

but also in stress absorbing membrane (SMA) layers (Hicks et al. 1995; Shen et al. 2006).

The particle size and the surface texture of the ground rubber vary in accordance with the

type of grinding, which can be either ambient or cryogenic. Each method has the ability to

produce crumb rubber of similar particle size, but the major difference between them is the

particle morphology. The ambient process often uses a conventional high powered rubber

cracker mill set with a close nip where vulcanized rubber is sheared and ground into a small

particle. The process produces a material with an irregular jagged particle shape. However, the

cryogenic grinding usually starts with chips or a fine crumb. This is cooled using a chiller and

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Xiao et al. (2007)

the rubber, while frozen, is put through a mill. The cryogenic process produces fairly smooth

fracture surfaces. Previous research indicated that the engineering properties of two type rubbers

are significantly different. The interaction effect (IE) and particle effect (PE) are affected by the

method used to produce the crumb rubber (Putman 2005). Putman (2005) pointed out that the

crumb rubber modifier(CRM) binders, containing ambient rubber, resulted in higher IE and PE

values than the CRM binders made with cryogenic rubber. This is due to the increased surface

area and irregular shape of the ambient CRM.

However, the influence of two byproducts (crumb rubber and RAP) mixed with virgin

mixtures together is not yet identified clearly. The interaction of modified mixtures is not well

understood from the stand point of binder properties to field performance. For example,

pavement engineers only know the aged binder will reduce the fatigue life, but the addition of

crumb rubber makes this issue more complicated. Because of the complicated relationship of

these two materials in the modified mixtures, more information will be beneficial in helping

obtain an optimum balance in the use of these materials. The properties of the binder should be

tested in the modified mixtures, containing RAP and crumb rubber, in order to study fatigue

behavior of modified mixtures.

There are two main approaches in the fatigue characterization of asphalt concrete:

phenomenological and mechanistic. One of the most commonly used phenomenological fatigue

models relates the initial response of an asphalt mixture to the fatigue life because only the

mixture response at the initial stage of fatigue testing needs to be measured. In general, fracture

mechanics or damage mechanics with or without viscoelasticity is adopted in the mechanistic

approach to describe the fatigue damage growth in asphalt concrete mixtures (Lee et al. 2000).

Very few fatigue studies of modified asphalt mixtures, including crumb rubber and reclaimed

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Xiao et al. (2007)

asphalt pavements, have been performed in recent years (Raad et al. 2001; Reese Ron 1997). In

addition, the modified asphalt mixtures containing two materials together are not yet studied in

great detail. Many rubberized asphalt pavements are in need of recycling after 15-20 years of

service. Therefore, it is important to obtain the fatigue behavior of these modified mixtures in the

laboratory, so that the performance can be predicted in the field. In addition, the utilization of

these materials will enable the engineers to find an environmentally friendly method to deal with

these materials, save money, energy, and furthermore, protect the environment.

This study explores the feasibility of using a multilayer feed-forward artificial neural

network (ANN) to predict the fatigue life of the modified asphalt mixture. ANN is adaptive

model-free estimator. A neural network is an interconnected network of processing elements that

has the ability to be trained and tested to map a given input into the desired output. Neural

network modeling techniques have been successfully applied to different areas of civil

engineering (Agrawal et al. 1995; Goh 1994; Goh et al. 1995; Juang and Chen 1999; Jen et al.

2002; Kim et al. 2004; Tarefder et al. 2005; Kim et al. 2005). In this paper, an ANN was

developed for the prediction of fatigue life of the modified asphalt mixtures and the results were

compared with experimental values and those determined with the traditional statistical methods.

BACKGROUND

Traditional Statistical Models of Fatigue Life

The fatigue characteristics of asphalt mixtures are usually expressed as relationships

between the initial stress or strain and the number of load repetitions to failure-determined by

using repeated flexure, direct tension, or diametral tests performed at several stress or strain

levels. The fatigue behavior of a specific mixture can be characterized by the slope and relative

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Xiao et al. (2007)

level of the stress or strain versus the number of load repetitions to failure and may be defined in

the following form (Monismith et al. 1985).

cb

f SaN )/1()/1( 00ε= or (1) cbf SaN )/1()/1( 00σ=

Where

fN = number of load application or crack initiation,

00 ,σε = tensile strain and stress, respectively,

So = initial mix stiffness, and

a, b, c = experimentally determined coefficients.

Several models have been proposed to predict the fatigue lives of pavements (Shell 1978;

Asphalt Institute 1981; Tayebali et al. 1994). To develop these models, laboratory results have

been calibrated by applying shift factors based on field observations to provide reasonable

estimates of the in-service life cycle of a pavement based on limiting the amount of cracking due

to repeated loading.

The fatigue models developed by Shell, the Asphalt Institute, and University of

California at Berkeley (SHRP A-003A contractor), respectively, are shown blow:

Shell Equation (Shell 1978)

( )

5

36.0*08.1856.0

−

⎥⎦

⎤⎢⎣

⎡+

=mixb

tf SV

Nε

(2)

Where

fN = fatigue life,

tε = tensile strain,

bV = volume of asphalt binder, and

mixS = mixture stiffness (flexural).

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Xiao et al. (2007)

Asphalt Institute Equation (Asphalt Institute 1981)

( )[ ] 845.0291.369.084.4 **004325.0*10* −−−= mixtVFA

ff SSN ε (3)

Where

VFA = volume of voids filled with asphalt binder, and

fS = shift factor to convert lab test results to field.

SHRP A-003A Equation (Tayebali et al. 1994)

720.2"

0624.3

0077.0 )(**exp*05738.2* −−= SESN VFA

ff ε (4)

Where "0S = initial loss-stiffness.

The fatigue behavior of a specific mixture can be characterized by the slope and relative

level of the stress or strain versus the number of load repetitions to failure, and can be defined by

a relationship of the following form (Tayebali et al. 1994):

e

od

f SxVcMFbaN )()()][exp()][exp( 0⋅⋅= (5)

Where

fN = cycles to failure,

MF = mode factor,

Vo = initial air-void content in percentage,

x = initial flexural strain ( 0ε ) or initial flexural stress ( 0σ ),

So = initial mix stiffness, and

a b, c, d, e = regression constants.

In recent years, several researchers (Tayebali 1992; Read and Collop 1997; Hossain and

Hoque 1999; Birgisson et al. 2004) have used the energy approach for predicting the fatigue

behavior of the asphalt mixtures. This approach is based on the assumption that the number of

cycles to failure is related mainly to the amount of energy dissipated during the test. A major

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Xiao et al. (2007)

advantage of this approach compared with the classical model is that predicting the fatigue

behavior of a certain mix type over a wide range of conditions, based on a few simple fatigue

tests, is possible. Other criteria based on changes in dissipated energy, including dissipated

energy ratio or damage accumulation ratio, were reported in studies by Rowe (1993) and

Anderson et al. (2001).

The dissipated energy per cycle, Wi, for a linearly viscoelastic material is given by the

following equation (Rowe 1993; Anderson et al. 2001):

1 1sin( )

n n

i i ii i

W W iπ σ ε δ= =

= =∑ ∑ (6)

Where

W = cumulative dissipated energy to failure,

iW = dissipated energy at load cycle i,

iσ = stress amplitude at load cycle i,

iε = strain amplitude at load cycle i, and

iδ = phase shift between stress and strain at load cycle i.

Research has shown that the dissipated energy approach makes it possible to predict the

fatigue behavior of mixtures in the laboratory over a wide range of conditions based on the

results of a few simple fatigue tests. Such a relationship can be characterized in the form of the

following equation (Tayebali 1992; Read and Collop 1997; Hossain and Hoque 1999; Birgisson

et al. 2004):

ZfNAW )(= (7)

Where

Nf = fatigue life,

W= cumulative dissipated energy to failure, and

A, Z = experimentally determined coefficients.

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Xiao et al. (2007)

Artificial Neural Networks (ANN) Approach

The neural networks approach may be used to develop the predictive models of the

fatigue life of asphalt mixtures considering the interaction of complicated variables. In this study,

a three-layer feedforward neural network, shown in Figure 1, was trained with the experimental

data. This architecture consists of an input layer, a hidden layer, and an output layer. Each of

the neurons in the hidden and output layers consists of two parts, one dealing with aggregation of

weights and the other providing a transfer function to process the output.

For the three-layer network shown in Figure 1, the output of the network, the fatigue life

Nf, is calculated as follows (Juang and Chen 1999):

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛+•+= ∑ ∑

= =

n

k

m

iiikHKTkTf PWBfWBfN

1 10 (8)

Where,

BBo = bias at the output layer,

Wk = weight of the connection between neuron k of the hidden layer and the single

output layer neuron,

BBHK = bias at neuron k of the hidden layer,

Wik = weight of the connection between input variable i and neuron k of the

hidden layer,

Pi = input ith parameter, and

fT = transfer function, defined as:

tetf −+=

11)( (9)

In this study, the backpropagation algorithm was used to train this neural network. The

objective of the network training using the backpropagation algorithm was to minimize the

network output error through determination and updating of the connection weights and biases.

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Xiao et al. (2007)

Backpropagation is a supervised learning algorithm where the network is trained and adjusted by

reducing the error between the network output and the targeted output. The neural network

training starts with the initiation of all of the weights and biases with random numbers. The input

vector is presented to the network and intermediate results propagate forward to yield the output

vector. The difference between the target output and the network output represents the error. The

error is then propagated backward through the network, and the weights and biases are adjusted

to minimize the error in the next round of prediction. The iteration continues until the error goal

(tolerable error) is reached, as illustrated in Figure 2. It should be noted that a properly trained

backpropagation network would produce reasonable predictions when it is presented with input

not used in the training. This generalization property makes it possible to train a network on a

representative set of input/output pairs, instead of all possible input/output pairs (Chen 1999).

Many implementations of the backpropagation algorithm are possible. In the present

study, the Levenberg-Marquart algorithm (Demuth and Beale 2003) is adopted for its efficiency

in training networks. This implementation is readily available in popular software Matlab and its

neural network toolbox (Demuth and Beale 2003). In the present study, ANN is treated as an

analysis tool, just like statistical regression method.

EXPERIMENTAL DATA AND MODEL DEVELOPMENT

Previous research indicated that the stiffness, fatigue life, and cumulative dissipated

energy are associated with various variables (Tayebali et al. 1994). The statistical analysis for

stiffness shows that asphalt and aggregate types, temperature, and air void content significantly

influence the stiffness for all test types, while the asphalt content and stress/strain do not appear

to be a significant factor on the stiffness for flexural beam tests. In general, the importance

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Xiao et al. (2007)

ranking of these factors observed for the cumulative dissipated energy is similar to that observed

for fatigue life.

The experimental data that were obtained by Xiao (2006) are shown in Tables 1 and 2.

The data included in these tables are the average values of the independent and dependent

variables of modified mixtures tested at 5°C (Table 1) and 20°C (Table 2). The independent

variables include traditional variables such as initial flexural strain (ε0) derived from initial a

linear variable differential transformer (LVDT) value of fatigue beam, volume of voids filled

with asphalt binder (VFA) and initial air-void content in percentage (V0) calculated from the

volumetric properties of a compacted fatigue beam, initial energy dissipated per cycle (w0)

calculated from Equation (6), and initial mix stiffness (S0) from initial stress and strain relation,

and additional specific variables, including the percentage of rubber in the binder (Rb) and the

percentage of RAP in the mixture (Rp). The dependent variable in this study includes only the

fatigue life ( ) recorded by the acquisition system. The entries in Tables 1 and 2 include the

experimental data obtained for two types of crumb rubbers, Ambient rubber and Cryogenic

rubber.

fN

Multiple linear regression analyses of these data were performed using Statistical

Analysis System (SAS) with one of the following two forms (named traditional statistical

models):

)(*)or(*)(*)( 000 SLndVVFAcLnbaNLn f +++= ε (10)

Or,

)or(*)(*)( 00 VVFAgwLnfeNLn f ++= (11)

Where

fN = fatigue life (cycles),

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Xiao et al. (2007)

0S = initial stiffness (Pa),

0ε = initial tensile strain (m/m),

VFA = volume of voids filled with asphalt (m3/m3),

0V = initial air-void content in percentage (m3/m3),

0w = initial energy dissipated per cycle (J/m3), and

a, b, c, d, e, f, g = experimentally determined coefficients.

Additional regression analyses were also performed with two additional variables, Rb and

Rp, added to the list of independent variables that were included in Equations 10 and 11.

Moreover, as summarized in Equation 12 or 13, the other additional variables, p1, p2, p3, p4, and

p5, were included through a trial-and-error process. They were added, one by one, to the list of

input variables so as to improve the accuracy of “mapping” between the input and the output. By

adding these variables, which are not truly independent variables but rather, derived variables,

the performance of the statistical regression analyses was greatly improved. The regression

models developed from Equation 12 or 13 which used the data in Table 1 (or 2) were referred to

herein as specific statistical models. However, as shown later, satisfactory results (predictive

models with high coefficient of determination) were difficult to obtain.

The specific statistical models are shown in the following functions.

),,,,,,,,,( 00540321 εSppVorVFApppRRfN pbf = (12)

Or,

),,,,,,,,( 0540321 wppVorVFApppRRfN pbf = (13)

Where

Nf = fatigue life (strain dependent or dissipated energy method),

VFA = the voids filled with the asphalt binder,

V0 = the percentage of air void,

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Xiao et al. (2007)

ε = tensile strain,

S = flexural stiffness,

Rb = the percentage of rubber in the binder,

Rp = the percentage of RAP in the mixture, and

p1 = Rb * Rp, p2 = Rb2, p3 = Rb

3, p4 = Rb * VFA (or V0), and p5 = Rp * VFA (or V0).

As an alternative to regression analyses, a three-layer, feed-forward artificial neural

network (ANN) was trained and tested with the data shown in Table 1 (or Table 2) to perform

fatigue life prediction using the same variables and the prediction models were developed from

Equation 8 and Equation 12 or 13. Comparing to prediction results of statistical traditional and

specific regression models, ANN models achieved the satisfactory results which were discussed

in the following paragraphs.

ANALYSIS OF RESULTS

Strain dependent models

In this study, the traditional strain-dependent model of fatigue life (Equation 10),

obtained through regression analysis, was divided into the voids filled with the asphalt binder

(VFA) and air void (A.V.) models. These models used initial flexural strain, initial mix stiffness,

and VFA or air voids as input variables. Sixteen records, each including four repeated testing

data, were used to develop regression models (Tables 1 and 2). These strain dependent models

are shown in Table 3. The results indicate a poor fit for the fatigue life with low coefficient of

determination (R2< 0.55) and high coefficient of variation (COV) regardless of the type of crumb

rubbers and test conditions. Thus the fatigue life prediction of using these traditional models may

not be reliable. However, use of the specific strain-dependent model (Equation 12), with which

greater R2 values were obtained (Tables 4 and 5), improves the accuracy of fatigue life prediction.

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Xiao et al. (2007)

Similar to the traditional models, the specific fatigue models show greater R2 values at the test

temperature of 20oC, and all R2 values are greater than 0.65. The measured and predicted results

of the fatigue life, derived from these specific models, are shown in Figures 3 and 4, for ambient

rubberized mixtures and cryogenic rubberized mixtures, respectively. It can be seen that for

ambient rubberized fatigue beam, the predicted and measured values are closer to 1:1 line.

Compared to cryogenic rubber, ambient rubberized fatigue beam shows a relatively greater R2

value.

The same data were used to develop the ANN models. The completed ANN model,

expressed in terms of the connection weights and biases in the three-layer topology, can then be

used to predict fatigue life for any given set of data (Rb, Rp, ε0, VFA/ V0, w0, and S0) using

Equation 8. Note that Equation 8 can easily be implemented in a spreadsheet for routine

applications. The sample spreadsheet, shown in Figure 5, uses a macro, a set of spreadsheet

commands, to compute the fatigue life for ambient rubberized mixture at 5ºC base on Equation 8.

Although it takes time to develop the ANN model, use of the ANN-based spreadsheet model for

calculating fatigue life is simple and the execution is very fast. Figures 6 and 7 show the results

obtained with the ANN models (in the form of Equations 8 and 12) for ambient rubberized

mixtures and cryogenic rubberized mixtures, respectively. Although different materials and

testing conditions were used in the project, the predicting performance of the trained neural

network as shown in Figures 6 and 7 is considered satisfactory and significance improvement (in

terms of R2) over those obtained by regression analyses as presented in Figures 3 and 4 is

achieved.

The effect of individual input variables, especially rubber and RAP percentages, on the

fatigue life as reflected in the developed ANN model is examined by a series of sensitivity

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Xiao et al. (2007)

analyses. Basically, the spreadsheet that implements the ANN model, as shown in Figure 5, is

used for this analysis, in which one variable is allowed to vary while all other variables are kept

constant. This process is repeated for analysis of the effect of each variable of concern. Figure 8

shows the results on the effects of rubber and RAP percentages based on the developed ANN

model. Generally speaking, the fatigue life increases as the rubber percentage increases, as

shown in Figure 8(a), for the entire range of the RAP percentage except when it goes beyond

25%. Furthermore, at a given rubber percentage, the fatigue life increases as the percentage of

RAP increases until it reaches to about 10%. As the RAP percentage is greater than 10% in the

mixture, the fatigue life decreases as the RAP content continues to increase. On the other hand,

the fatigue life decreases as the percentage of RAP increases for the rubber percentage of less

than about 13%, as shown in Figure 8 (b). At a given RAP percentage, the fatigue life increases

as the rubber percentage increases.

Energy dependent models

Similar to the strain dependent models, the traditional energy dependent models of

fatigue life (Equation 11), obtained from regression analysis, were also divided into VFA and air

void (A.V.) models. These traditional energy dependent models are shown in Table 4. The

results also show a poor fit for the fatigue life with low R2 and high COV regardless of the type

of crumb rubbers and test conditions. In the same way, the specific regression energy dependent

models (Equation 13) also achieved the notably greater R2 values than the traditional ones, as

shown in Tables 3, 4, and 5. The measured and predicted results of fatigue life, derived from

these specific regression models, are shown in Figures 9 and 10 for ambient rubberized mixtures

and cryogenic rubberized mixtures, respectively.

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Xiao et al. (2007)

Similarly, Figures 11 and 12 show the results obtained with the ANN models (in the form

of Equations 8 and 13) for ambient rubberized mixtures and cryogenic rubberized mixtures,

respectively. Again, the predicting performance of the trained neural network as shown in

Figures 11 and 12 is considered satisfactory and significance improvement (in terms of R2) over

those obtained by regression analyses as presented in Figures 9 and 10 is achieved.

Based on the limited test data presented, the ANN models (Equation 8 and Equation 12 or

13) are shown to be able to predict the fatigue life of the rubberized mixtures with satisfactory

accuracy. The ANN models appear to outperform both the traditional (Equation 10 or 11) and

the specific regression-based models (Equation 12 or 13) in the prediction of fatigue life of

modified mixtures regardless of the type of crumb rubbers and test conditions.

CONCLUSIONS

The following conclusions were reached based on the limited experimental data

presented regarding the fatigue life of the modified binder and mixtures:

Experimental data on the fatigue life of rubberized asphalt concrete mixtures containing

reclaimed asphalt pavement obtained in this study were used for model development.

The results showed that the traditional regression-based models were unable to predict

the fatigue life of modified mixtures accurately.

ANN approach, as a new fatigue modeling method used in this study, has been shown to

be effective in creating a feasible predictive model. The established ANN-based models

were able to predict the fatigue life accurately, as evidenced by high R2 values regardless

of the type of crumb rubbers and test conditions. The results indicated that ANN-based

models are more effective than the regression models, either the traditional (Equation 10

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Xiao et al. (2007)

or 11) or the specific models (Equation 12 or 13), in predicting the fatigue life. The

ANN models can easily be implemented in a spreadsheet, thus making it easy to apply.

Both strain-dependent and dissipated energy-dependent methods were effective in

predicting the fatigue life of the modified mixtures when additional input variables are

included in the ANN-based models.

ACKNOWLEDGMENTS

The financial support of South Carolina Department of Health and Environmental

Control (SC DHEC) is greatly appreciated. However, the results and opinions presented in this

paper do not necessarily reflect the view and policy of the SC DHEC.

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Xiao et al. (2007)

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LIST OF TABLES

TABLE 1 Average values of independent and dependent variables of modified mixtures tested at 5ºC

TABLE 2 Average values of independent and dependent variables of modified mixtures tested at

20ºC TABLE 3 Strain dependent fatigue prediction models of the mixtures

TABLE 4 Energy dependent fatigue prediction models of the mixtures

TABLE 5 Coefficient of determination (R2) and coefficient of variation (COV) of the specific regression models

23

Xiao et al. (2007)

LIST OF FIGURES

FIG. 1 Example of a three-layer feedforward neural network architecture

FIG. 2 Flowchart illustrating backpropagation training algorithm

FIG. 3 Comparison of fatigue lives between predicted and measured results using regression-based strain-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC

FIG. 4 Comparison of fatigue lives between predicted and measured results using regression-

based strain-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 5 Sample spreadsheet of ANN model based strain-dependent for ambient rubberized

mixtures at 5ºC FIG. 6 Comparison of fatigue lives between predicted and measured results using ANN-based

strain-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 7 Comparison of fatigue lives between predicted and measured results using ANN-based

strain-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 8 Sensitivity analysis of rubber and RAP percentages in ANN fatigue models (a) RAP

analysis; (b) Rubber analysis FIG. 9 Important indexed of input variables in the developed ANN FIG. 10 Comparison of fatigue lives between predicted and measured results using regression-

based energy-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 11 Comparison of fatigue lives between predicted and measured results using regression-

based energy-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 12 Comparison of fatigue lives between predicted and measured results using ANN-based

energy-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 13 Comparison of fatigue lives between predicted and measured results using ANN-based

energy-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC

24

Xiao et al. (2007)

TABLE 1 Average value of each of the variables of modified mixtures tested at 5ºC

Dependent5ºC R b (%) R P (%) Ln(ε 0 ) VFA V 0 Ln(w 0 ) Ln(S 0 ) Ln(N f )

0.00 0.00 -7.642 0.739 3.73 0.790 16.916 10.140

0.00 0.15 -7.625 0.749 4.57 1.200 16.939 10.2510.00 0.25 -7.591 0.744 4.92 1.246 16.755 9.2140.00 0.30 -7.607 0.733 5.69 0.458 16.899 9.9550.05 0.00 -7.647 0.758 4.62 0.547 16.791 9.7650.05 0.15 -7.690 0.757 4.17 0.676 16.879 9.9600.05 0.25 -7.576 0.738 4.49 0.797 16.825 9.6520.05 0.30 -7.589 0.733 5.91 0.744 16.899 9.955

0.10 0.00 -7.869 0.765 3.82 0.393 16.866 10.0050.10 0.15 -7.631 0.762 3.73 0.974 16.906 10.7460.10 0.25 -7.579 0.731 5.04 1.613 16.771 9.6420.10 0.30 -7.646 0.743 6.68 0.499 16.730 10.3910.15 0.00 -7.609 0.773 4.07 0.388 16.673 9.9530.15 0.15 -7.628 0.773 3.21 0.881 16.809 10.8470.15 0.25 -7.595 0.760 3.77 1.276 16.725 10.4240.15 0.30 -7.628 0.744 6.91 0.382 16.788 10.1130.00 0.00 -7.642 0.739 3.73 0.790 16.916 10.140

0.00 0.15 -7.626 0.749 4.57 1.200 16.939 10.2510.00 0.25 -7.592 0.744 4.92 1.246 16.838 9.2680.00 0.30 -7.607 0.733 5.69 0.458 16.899 9.9550.05 0.00 -7.681 0.667 3.91 0.692 17.036 9.5930.05 0.15 -7.620 0.687 5.16 0.446 16.917 9.1310.05 0.25 -7.699 0.699 6.31 0.849 16.965 9.0360.05 0.30 -7.608 0.711 5.40 0.899 17.067 10.051

0.10 0.00 -7.556 0.675 4.13 0.997 16.782 10.2300.10 0.15 -7.619 0.684 4.67 0.887 16.907 9.5640.10 0.25 -7.590 0.713 6.52 0.661 16.809 9.7520.10 0.30 -7.577 0.733 7.58 0.784 16.809 8.6760.15 0.00 -7.651 0.662 3.07 0.662 16.886 9.8380.15 0.15 -7.599 0.700 5.02 0.752 16.808 8.6720.15 0.25 -7.678 0.701 4.84 0.716 17.004 8.7860.15 0.30 -7.627 0.711 7.29 0.909 16.841 9.503

Traditional IndependentSpecific Independent

Am

bien

t rub

ber

Cry

ogen

ic ru

bber

Note: Rb = the percentage of rubber in the binder; Rp = the percentage of RAP in the mixture; 0ε = initial flexural strain; VFA = volume of voids filled with asphalt binder; V0 = initial air-void content in percentage; w0 = initial dissipated energy; S0 = initial mix stiffness; = fatigue life. fN

25

Xiao et al. (2007)

TABLE 2 Average value of each of the variables of modified mixtures tested at 20ºC

Dependent20ºC R b (%) R P (%) Ln(ε 0 ) VFA V 0 Ln(w 0 ) Ln(S 0 ) Ln(N f )

0.00 0.00 -7.508 0.739 5.44 0.899 16.364 10.8050.00 0.15 -7.521 0.749 5.19 0.863 16.430 10.8220.00 0.25 -7.509 0.744 6.32 0.913 16.166 9.9800.00 0.30 -7.532 0.733 6.51 0.906 16.451 10.6010.05 0.00 -7.532 0.758 5.09 1.127 16.402 11.1780.05 0.15 -7.533 0.757 7.28 0.819 16.375 10.6740.05 0.25 -7.592 0.738 6.51 0.787 16.411 9.7880.05 0.30 -7.546 0.733 5.29 0.944 16.495 10.4920.10 0.00 -7.517 0.765 5.72 0.752 16.315 11.3670.10 0.15 -7.481 0.762 6.86 0.872 16.361 10.6870.10 0.25 -7.525 0.731 6.20 0.702 16.381 9.7860.10 0.30 -7.496 0.743 6.98 0.899 16.448 10.4240.15 0.00 -7.547 0.773 5.68 0.651 16.088 10.7010.15 0.15 -7.536 0.773 5.74 0.729 16.324 10.8160.15 0.25 -7.575 0.760 8.01 0.590 16.351 9.3290.15 0.30 -7.595 0.744 5.20 0.762 16.435 10.5790.00 0.00 -7.508 0.739 5.435 0.899 16.364 10.805

0.00 0.15 -7.521 0.749 5.190 0.863 16.430 10.8220.00 0.25 -7.509 0.744 6.315 0.913 16.166 9.9800.00 0.30 -7.532 0.733 6.505 0.906 16.451 10.6010.05 0.00 -7.509 0.667 5.195 0.994 16.399 10.5600.05 0.15 -7.519 0.687 4.040 0.978 16.457 10.5770.05 0.25 -7.559 0.699 5.400 0.931 16.553 9.9140.05 0.30 -7.573 0.711 6.405 0.794 16.535 10.117

0.10 0.00 -7.491 0.675 5.385 0.875 16.100 10.9020.10 0.15 -7.493 0.684 5.623 0.947 16.086 10.8620.10 0.25 -7.510 0.713 5.820 0.776 16.355 9.8790.10 0.30 -7.513 0.733 6.370 0.952 16.159 10.0740.15 0.00 -7.505 0.662 3.365 0.649 16.380 10.3090.15 0.15 -7.537 0.700 5.300 1.069 16.355 10.4790.15 0.25 -7.507 0.701 5.080 0.250 16.551 10.1170.15 0.30 -7.554 0.711 7.660 0.412 16.459 9.482

Traditional Independent

Am

bien

t rub

ber

Cry

ogen

ic ru

bber

Specific Independent

Note: Rb = the percentage of rubber in the binder; Rp = the percentage of RAP in the mixture; 0ε = initial flexural strain; VFA = volume of voids filled with asphalt binder; V0 = initial air-void content in percentage; w0 = initial dissipated energy; S0 = initial mix stiffness; = fatigue life. fN

26

Xiao et al. (2007)

TABLE 3 Strain dependent fatigue prediction models of the mixtures

Ambient Traditional Predicting Model R2 COV

VFA (5oC) 2.2

0*2.206.1

0 ***)13(0.1 SeEN VFAf ε−= 0.37 66%

A.V. (5oC) 7.0

0*1.02.0

0 ***)2(8.4 0 SeEN Vf

−−−= ε 0.09 28%

VFA (20oC) 9.1

0*9.205.6

0 ***)5(3.3 SeEN VFAf ε= 0.36 82%

A.V. (20oC) 2.0

0*4.08.6

0 ***)26(9.1 0 SeEN Vf

−= ε 0.53 106% Cryogenic

VFA (5oC) 9.2

0*1.07.6

0 ***)5(3.1 SeEN VFAf

−= ε 0.13 47%

A.V. (5oC) 3.2

0*2.02.7

0 ***)11(6.3 0 SeEN Vf

−= ε 0.32 78%

VFA (20oC) 02.0

0*5.01.9

0 ***)34(7.1 SeEN VFAf

−= ε 0.27 51%

A.V. (20oC) 6.0

0*2.04.3

0 ***)20(9.3 0 SeEN Vf

−= ε 0.36 61% Note: = fatigue life fN 0ε = initial flexural strain; VFA = volume of voids filled with asphalt binder; V0 = initial air-void content in percentage; S0 = initial mix stiffness; A.V. = air voids

27

Xiao et al. (2007)

TABLE 4 Energy dependent fatigue prediction models of the mixtures

Ambient Traditional Predicting Model R2 COV

VFA (5oC) 01.0

0*8.13 **74.0 εVFA

f eN = 0.22 57%

A.V. (5oC) 2.0

0*1.0 **)4(9.4 0 −−= εV

f eEN 0.09 37%

VFA (20oC) 6.2

0*6.21 **)4(1.4 εVFA

f eEN −= 0.49 135%

A.V. (20oC) 0.1

0*3.0 **)5(1.1 0 εV

f eEN −= 0.42 121% Cryogenic

VFA (5oC) 5.0

0*2.0 **)4(1.1 εVFA

f eEN −= 0.04 31%

A.V. (5oC) 4.0

0*2.0 **)4(4.2 0 εV

f eEN = 0.21 78%

VFA (20oC) 9.0

0*2.0 **)4(0.4 εVFA

f eEN −= 0.40 84%

A.V. (20oC) 0.1

0*5.2 **)4(1.8 0 εV

f eEN −= 0.27 66% Note: = fatigue life fN 0ε = initial flexural strain; VFA = volume of voids filled with asphalt binder; V0 = initial air-void content in percentage; S0 = initial mix stiffness; A.V. = air voids

28

Xiao et al. (2007)

TABLE 5 Coefficient of determination (R2) and coefficient of variation (COV) of the specific regression models

R2 COV R2 COVVFA 0.95 52% 0.84 66%A.V 0.77 47% 0.91 69%VFA 0.79 50% 0.81 68%A.V 0.66 46% 0.78 67%VFA 0.65 58% 0.73 46%A.V 0.65 58% 0.73 46%VFA 0.52 54% 0.73 49%A.V 0.56 57% 0.73 49%C

ryog

enic

Strain

Energy

RubberType

PredictionMethod

Strain

Energy

PredictionType

5ºC 20ºC

Am

bien

t

Note: VFA = volume of voids filled with asphalt; A.V. = air voids

29

Xiao et al. (2007)

Nf

VFA

ε

S

Rb

Rp

Input layer Hidden Output

Rb = the percentage of rubber in the binder; Rp = the percentage of RAP in the mixture; ε = initial flexural strain; VFA= Volume of Voids filled with asphalt binder; S = initial mix stiffness; Nf= fatigue life.

FIG. 1 Example of a three-layer feedforward neural network architecture

30

Xiao et al. (2007)

Scaling Input/Output Vectors

Assigning Initial Weights

Calculating Output

Out_err = Target- Prediction

Update Weights for Output-Layer Neurons

Update Weights for Hidden-Layer Neurons

Backpropagation Training Completed

Out_err < Goal_err (?) YES

NO

FIG. 2 Flowchart illustrating backpropagation training algorithm (Juang and Chen 1999)

31

Xiao et al. (2007)

0

1

2

3

4

5

0 1 2 3 4 5Measured Fatigue Life (104Cycles)

Pred

icte

d Fa

tigue

Life

(104 C

ycle

s) VFA PredictedAir Voids Predicted

R2=0.95 (VFA)R2=0.77 (A.V)

0123456789

10

0 1 2 3 4 5 6 7 8 9 10

Measured Fatigue Life (104Cycles)

Pred

icte

d Fa

tigue

Life

(104 C

ycle

s) VFA PredictedAir Voids Predicted

R2=0.84 (VFA)R2=0.91 (A.V)

(a) (b)

FIG. 3 Comparison of fatigue lives between predicted and measured results using regression-based strain-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC

32

Xiao et al. (2007)

0

1

2

3

4

0 1 2 3 4

Measured Fatigue Life (104Cycles)

Pred

icte

d Fa

tigue

Life

(104 C

ycle

s) VFA PredictedAir Voids Predicted

R2=0.65 (VFA)R2=0.65 (A.V)

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Measured Fatigue Life (104Cycles)

Pred

icte

d Fa

tigue

Life

(104 C

ycle

s) VFA PredictedAir Voids Predicted

R2=0.73 (VFA)R2=0.73 (A.V)

(a) (b)

FIG. 4 Comparison of fatigue lives between predicted and measured results using regression-based strain-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC

33

Xiao et al. (2007)

1 A B C D E F G2 COMMANDS OF EXECUTING EQ.8 Hidden Layer3 ARGUMENT("R b ", "R p ", "P 1 ", "P 2 ", "P 3 ","Ln(ε 0 ) ") Weight matrix Hidden 1 Hidden 2 Hidden 3 Hidden 44 ARGUMENT("VFA ", "P 4 ", "P 5 ", "Ln(S 0 ) ") Bias 8.50785 -9.71422 -2.95993 -1.944225 R b =(R b +0.019)/0.19; R p =(R p +0.038)/0.38 Input 1 -2.92512 3.77847 0.21774 -2.584166 P 1 =(P 1 +0.0056)/0.056; P 2 =(P 2 +0.003)/0.03 Input 2 -5.43439 2.23798 1.56790 -4.475967 P 3 =(P 3 +0.0004)/0.004; Ln(ε 0 ) =(Ln(ε0 )+7.91)/0.37 Input 3 -2.05854 3.93442 3.62199 0.229308 VFA=(VFA -0.725)/0.053; P 4 =(P 4 +0.015)/0.15 Input 4 2.04824 3.40047 -0.12879 2.489739 P 5 =(P 5 +0.028)/0.28; Ln(S 0 ) =(Ln(S 0 ) -16.64)/0.333 Input 5 2.17721 3.49616 -5.61273 4.0260510 pi1=1/(1+EXP(-(R b *D$5+R p *D$6+P 1 *D$7+P 2 *D$8+ Input 6 -1.07765 3.16621 1.70724 -1.5014611 P 3 *D$9+Ln(ε 0 ) *D$10+VFA *D$11+P 4 *D$12+ Input 7 -7.00788 -0.51641 2.25666 -1.6631312 P 5 *D$13+Ln(S 0 ) *D$14+D$4))) Input 8 0.78339 -1.49672 2.87061 2.3366013 pi2=1/(1+EXP(-(R b *E$5+R p *E$6+P 1 *E$7+P 2 *E$8+ Input 9 0.25014 2.09887 3.63714 -3.4439614 P 3 *E$9+Ln(ε 0 ) *E$10+VFA*E$11+P 4 *E$12+ Input 10 0.34972 3.67111 -3.83600 -0.3682815 P 5 *E$13+Ln(S 0 ) *E$14+E$4)))16 pi3=1/(1+EXP(-(R b *F$5+R p *F$6+P 1 *F$7+P 2 *F$8+ Output Layer17 P 3 *F$9+Ln(ε0) *F$10+VFA *F$11+P 4 *F$12+ Bias 4.0670818 P 5 *F$13+Ln(S 0 ) *F$14+F$4))) Hidden 1 -3.8366419 pi4=1/(1+EXP(-(R b *G$5+R p *G$6+P 1 *G$7+P 2 *G$8+ Hidden 2 2.8123020 P 3 *G$9+Ln(ε 0 ) *G$10+VFA *G$11+P 4 *G$12+ Hidden 3 -4.3789921 P 5 *G$13+Ln(S 0 ) *G$14+G$4))) Hidden4 -4.1719522 Z=pi1*D18+pi2*D19+pi3*D20+pi4*D21+D1723 Z=1/(1+EXP(-Z))24 Ln(F)=2.041*Z+9.00725 RETURN (F)

Cells B3:B25 are macro commands to execute Eq.8

Weight matrix:Cells D4: G4 are BHK

Cells D5: G14 are Wik

Weight matrix:Cell D17 is Bo

Cells D18: D21 are Wik

FIG. 5 Sample spreadsheet of ANN model based strain-dependent for ambient rubberized mixtures at 5ºC

34

Xiao et al. (2007)

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.97 (VFA)R2=0.95 (A.V)

0123456789

0 1 2 3 4 5 6 7 8 9Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.96 (VFA)R2=0.95 (A.V)

(a) (b)

FIG. 6 Comparison of fatigue lives between predicted and measured results using ANN-based strain-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC

35

Xiao et al. (2007)

0

1

2

3

4

0 1 2 3

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

4

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.91 (VFA)R2=0.84 (A.V)

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.92 (VFA)R2=0.97 (A.V)

(a) (b)

FIG. 7 Comparison of fatigue lives between predicted and measured results using ANN-based strain-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC

36

Xiao et al. (2007)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 5 10 15 20 25 30 35

RAP Percentage (%)

Fatig

ue L

ife (x

104 cy

cles

)

8%Rubber 10%Rubber12%Rubber

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 5 10 15 20

Rubber Percentage (%)

Fatig

ue L

ife (x

104c

ycle

s)

10%RAP 15%RAP20%RAP

(a) (b)

FIG. 8 Sensitivity analysis of rubber and RAP percentages in ANN fatigue models

(a) RAP analysis; (b) Rubber analysis

37

Xiao et al. (2007)

0

1

2

3

4

5

0 1 2 3 4 5Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es) VFA Predicted

Air Voids PredictedR2=0.79 (VFA)R2=0.66 (A.V)

0123456789

10

0 1 2 3 4 5 6 7 8 9 10

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA PredictedAir Voids Predicted

R2=0.81 (VFA)R2=0.78 (A.V)

(a) (b)

FIG. 9 Comparison of fatigue lives between predicted and measured results regression-based energy-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC

38

Xiao et al. (2007)

0

1

2

3

0 1 2 3

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA PredictedAir Voids Predicted

R2=0.52 (VFA)R2=0.56 (A.V)

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA PredictedAir Voids PredictedR2=0.73 (VFA)R2=0.73 (A.V)

(a) (b)

FIG. 10 Comparison of fatigue lives between predicted and measured results regression-based energy-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC

39

Xiao et al. (2007)

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.94 (VFA)R2=0.93 (A.V)

0123456789

0 1 2 3 4 5 6 7 8 9

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.92 (VFA)R2=0.90 (A.V)

(a) (b)

FIG. 11 Comparison of fatigue lives between predicted and measured results using ANN-based energy-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC

40

Xiao et al. (2007)

0

1

2

3

4

0 1 2 3

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

4

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.89 (VFA)R2=0.86 (A.V)

0

1

2

3

4

5

6

0 1 2 3 4 5 6

Measured Fatigue Life (x104cycles)

Pred

icte

d Fa

tigue

Life

(x10

4 cycl

es)

VFA (Training)Air Voids (Training)VFA (Testing)Air Voids (Testing)

R2=0.92 (VFA)R2=0.93 (A.V)

(a) (b)

FIG. 12 Comparison of fatigue lives between predicted and measured results using ANN-based energy-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC

41