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.
____________________ 1
Research Associate, Department of Civil Engineering, Clemson University, Clemson, SC 29634-0911. E-mail:
[email protected] 2
Professor, Department of Civil Engineering, Clemson University, Clemson, SC 29634-0911. Email:
[email protected] 3
Professor, Department of Civil Engineering, Clemson University, Clemson, SC 29634-0911. Email:
[email protected]
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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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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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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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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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).
N f = a(1 / ε 0 ) b (1 / S 0 ) c or N f = a (1 / σ 0 ) b (1 / S 0 ) c
(1)
Where N f = number of load application or crack initiation,
ε 0 , σ 0 = 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)
⎡ ⎤ εt Nf = ⎢ ⎥ ⎣ (0.856Vb + 1.08)S mix * 0.36 ⎦ Where N f = fatigue life,
ε t = tensile strain, Vb = volume of asphalt binder, and S mix = mixture stiffness (flexural).
6
−5
(2)
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Asphalt Institute Equation (Asphalt Institute 1981) −0.845 N f = S f * 10 [4.84(VFA−0.69 )] * 0.004325 * ε t−3.291 * S mix
(3)
Where VFA = volume of voids filled with asphalt binder, and
S f = shift factor to convert lab test results to field.
SHRP A-003A Equation (Tayebali et al. 1994) N f = S f * 2.738 E 05 * exp 0.077VFA * ε 0−3.624 * ( S 0" ) −2.720
(4)
Where S 0" = 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): N f = a [exp( b ⋅ MF )] [exp( c ⋅ V0 )] ( x ) d ( S o ) e
(5)
Where N f = cycles to failure,
MF = mode factor, Vo = initial air-void content in percentage, x = initial flexural strain ( ε 0 ) or initial flexural stress ( σ 0 ), 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 7
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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): n
n
i =1
i =1
W = ∑ Wi = ∑ π σ iε i sin(δ i )
(6)
Where W = cumulative dissipated energy to failure,
Wi = dissipated energy at load cycle i,
σ i = stress amplitude at load cycle i, ε i = strain amplitude at load cycle i, and δ i = 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): W = A (N f )Z
(7)
Where Nf = fatigue life, W= cumulative dissipated energy to failure, and A, Z = experimentally determined coefficients.
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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 m ⎧⎪ ⎡ ⎞⎤ ⎪⎫ ⎛ N f = f T ⎨ B0 + ∑ ⎢Wk • f T ⎜ B HK + ∑ Wik Pi ⎟⎥ ⎬ ⎪⎩ k =1 ⎣ i =1 ⎠⎦ ⎪⎭ ⎝
(8)
Where, Bo = bias at the output layer, B
Wk = weight of the connection between neuron k of the hidden layer and the single output layer neuron, BHK = bias at neuron k of the hidden layer, B
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: f (t ) =
1 1 + e −t
(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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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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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 ( N f ) 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. 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):
Ln ( N f ) = a + b * Ln (ε 0 ) + c * VFA ( or V0 ) + d * Ln ( S0 )
(10)
Or, Ln ( N f ) = e + f * Ln ( w0 ) + g * VFA ( or V0 )
Where N f = fatigue life (cycles),
11
(11)
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S 0 = initial stiffness (Pa),
ε 0 = initial tensile strain (m/m), VFA = volume of voids filled with asphalt (m3/m3),
V0 = initial air-void content in percentage (m3/m3), w0 = 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.
N f = f ( Rb , R p , p1 , p 2 , p3 ,VFA or V0 , p4 , p5 , S0 , ε 0 )
(12)
N f = f ( Rb , R p , p1 , p 2 , p 3 ,VFA or V0 , p 4 , p5 , w0 )
(13)
Or,
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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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 = Rb3, 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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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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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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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: y
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.
y
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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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. y
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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Shen, J.N., Amirkhanian, S.N. and Xiao, F.P. (2006) “High-Pressure Gel Permeation Chromatography Characterization of Aging of Recycled Crumb-Rubber-Modified Binders Containing Rejuvenating Agents.” Transportation Research Record, Washington D.C., 21-27. Strategic Highway Research Program (1994) “Fatigue Response of Asphalt-Aggregate Mixes.” SHRP-A-404, National Research Council, Washington, D. C. Tarefder, F.A., White, L., and Zaman, M. (2005) “Neural Network Model for Asphalt Concrete Permeability.” Journal of Materials in Civil Engineering, Vol. 17, 19-27. Tayebali, A.A. (1992) “Re-calibration of Surrogate Fatigue Models Using all Applicable A003A Fatigue data.” Technical memorandum prepared for SHRP Project A-003A. Institute of Transportation Studies, University of California, Berkeley. Tayebali, A.A., Tsai, B., and Monismith, C.L. (1994) “Stiffness of Asphalt Aggregate Mixes.” SHRP Report A-388, National Research Council, Washington D.C. Tsoukalas, L.H., and Uhrig, R.E. (1996) “Fuzzy and Neural Approached in Engineering.” John Wiley & Sons Inc., 234. Wang, L.B., Wang, X., Mohammad L., and Wang, Y.P. (2004) “Application of Mixture Theory in the Evaluation of Mechanical Properties of Asphalt Concrete.” Journal of Materials in Civil Engineering, Vol. 16, 167-174. Xiao, F.P. (2006) “Development of Fatigue Predictive Models of Rubberized Asphalt Concrete (RAC) Containing Reclaimed Asphalt Pavement (RAP) Mixtures.” Ph. D dissertation, Clemson University. Xiao, F.P., Amirkhanian, S.N., and Juang, H.J. (2007) “Rutting Resistance of the Mixture Containing Rubberized Concrete and Reclaimed Asphalt Pavement.” Journal of Materials in Civil Engineering, Vol. 19, 475-483
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Xiao et al. (2007)
Xiao, F.P., Putman B.J., and Amirkhanian S.N. (2006) “Laboratory Investigation of Dimensional Changes of Crumb Rubber Reacting with Asphalt Binder.” Proceedings of the Asphalt Rubber 2006 Conference, Palm Springs, USA, 693-715. You, A., and Buttlar, W.G. (2004) “Discrete Element Modeling to Predict the Modulus of Asphalt Concrete Mixtures.” Journal of Material in Civil Engineering, Vol. 16, 140-146.
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Xiao et al. (2007)
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 regressionbased strain-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 4 Comparison of fatigue lives between predicted and measured results using regressionbased 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 regressionbased energy-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC FIG. 11 Comparison of fatigue lives between predicted and measured results using regressionbased 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
Cryogenic rubber
Ambient rubber
Specific Independent 5ºC R b (%) R P (%)
Ln(ε 0 )
Traditional Independent VFA V0 Ln(w 0 )
Ln(S 0 )
Dependent 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
10.251
0.00
0.25
-7.591
0.744
4.92
1.246
16.755
9.214
0.00
0.30
-7.607
0.733
5.69
0.458
16.899
9.955
0.05 0.05 0.05
0.00 0.15 0.25
-7.647 -7.690 -7.576
0.758 0.757 0.738
4.62 4.17 4.49
0.547 0.676 0.797
16.791 16.879 16.825
9.765 9.960 9.652
0.05
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
10.005
0.10 0.10 0.10 0.15 0.15 0.15 0.15
0.15 0.25 0.30 0.00 0.15 0.25 0.30
-7.631 -7.579 -7.646 -7.609 -7.628 -7.595 -7.628
0.762 0.731 0.743 0.773 0.773 0.760 0.744
3.73 5.04 6.68 4.07 3.21 3.77 6.91
0.974 1.613 0.499 0.388 0.881 1.276 0.382
16.906 16.771 16.730 16.673 16.809 16.725 16.788
10.746 9.642 10.391 9.953 10.847 10.424 10.113
0.00
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
10.251
0.00
0.25
-7.592
0.744
4.92
1.246
16.838
9.268
0.00
0.30
-7.607
0.733
5.69
0.458
16.899
9.955
0.05 0.05 0.05
0.00 0.15 0.25
-7.681 -7.620 -7.699
0.667 0.687 0.699
3.91 5.16 6.31
0.692 0.446 0.849
17.036 16.917 16.965
9.593 9.131 9.036
0.05
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
10.230
0.10 0.10 0.10 0.15 0.15 0.15 0.15
0.15 0.25 0.30 0.00 0.15 0.25 0.30
-7.619 -7.590 -7.577 -7.651 -7.599 -7.678 -7.627
0.684 0.713 0.733 0.662 0.700 0.701 0.711
4.67 6.52 7.58 3.07 5.02 4.84 7.29
0.887 0.661 0.784 0.662 0.752 0.716 0.909
16.907 16.809 16.809 16.886 16.808 17.004 16.841
9.564 9.752 8.676 9.838 8.672 8.786 9.503
Note: Rb = the percentage of rubber in the binder; Rp = the percentage of RAP in the mixture; ε 0 = 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; N f = fatigue life.
25
Xiao et al. (2007)
TABLE 2 Average value of each of the variables of modified mixtures tested at 20ºC
Cryogenic rubber
Ambient rubber
Specific Independent 20ºC R b (%) R P (%)
Ln(ε 0 )
Traditional Independent VFA V0 Ln(w 0 )
Ln(S 0 )
Dependent Ln(N f )
0.00
0.00
-7.508
0.739
5.44
0.899
16.364
10.805
0.00
0.15
-7.521
0.749
5.19
0.863
16.430
10.822
0.00
0.25
-7.509
0.744
6.32
0.913
16.166
9.980
0.00
0.30
-7.532
0.733
6.51
0.906
16.451
10.601
0.05 0.05 0.05
0.00 0.15 0.25
-7.532 -7.533 -7.592
0.758 0.757 0.738
5.09 7.28 6.51
1.127 0.819 0.787
16.402 16.375 16.411
11.178 10.674 9.788
0.05
0.30
-7.546
0.733
5.29
0.944
16.495
10.492
0.10
0.00
-7.517
0.765
5.72
0.752
16.315
11.367
0.10 0.10 0.10 0.15 0.15 0.15 0.15
0.15 0.25 0.30 0.00 0.15 0.25 0.30
0.00
0.00
-7.481 -7.525 -7.496 -7.547 -7.536 -7.575 -7.595 -7.508
0.762 0.731 0.743 0.773 0.773 0.760 0.744 0.739
6.86 6.20 6.98 5.68 5.74 8.01 5.20 5.435
0.872 0.702 0.899 0.651 0.729 0.590 0.762 0.899
16.361 16.381 16.448 16.088 16.324 16.351 16.435 16.364
10.687 9.786 10.424 10.701 10.816 9.329 10.579 10.805
0.00
0.15
-7.521
0.749
5.190
0.863
16.430
10.822
0.00
0.25
-7.509
0.744
6.315
0.913
16.166
9.980
0.00
0.30
-7.532
0.733
6.505
0.906
16.451
10.601
0.05 0.05 0.05
0.00 0.15 0.25
-7.509 -7.519 -7.559
0.667 0.687 0.699
5.195 4.040 5.400
0.994 0.978 0.931
16.399 16.457 16.553
10.560 10.577 9.914
0.05
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
10.902
0.10 0.10 0.10 0.15 0.15 0.15 0.15
0.15 0.25 0.30 0.00 0.15 0.25 0.30
-7.493 -7.510 -7.513 -7.505 -7.537 -7.507 -7.554
0.684 0.713 0.733 0.662 0.700 0.701 0.711
5.623 5.820 6.370 3.365 5.300 5.080 7.660
0.947 0.776 0.952 0.649 1.069 0.250 0.412
16.086 16.355 16.159 16.380 16.355 16.551 16.459
10.862 9.879 10.074 10.309 10.479 10.117 9.482
Note: Rb = the percentage of rubber in the binder; Rp = the percentage of RAP in the mixture; ε 0 = 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; N f = fatigue life.
26
Xiao et al. (2007)
TABLE 3 Strain dependent fatigue prediction models of the mixtures Ambient
VFA (5oC)
Traditional Predicting Model 1.6 2.2 N f = 1.0 E (−13) * ε 0 * e 20.2*VFA * S 0 −0.2
A.V. (5oC)
N f = 4.8E (−2) * ε 0
VFA (20oC)
N f = 3.3E (5) * ε 0
A.V. (20oC)
N f = 1.9 E (26) * ε 0
6.5
*e
*e 6.8
−0.1*V0
20.9*VFA
* S0 *S
−0.4*V0
COV
0.37
66%
0.09
28%
0.36
82%
0.53
106%
0.13
47%
0.32
78%
0.27
51%
0.36
61%
0.7
1.9 0
* S0
0.2
* e −0.1*VFA * S 0
2.9
−0.2*V0
* S0
2.3
* S0
0.02
*e
R2
Cryogenic
VFA (5oC)
N f = 1.3E (5) * ε 0
A.V. (5oC)
N f = 3.6 E (11) * ε 0
VFA (20oC) A.V. (20oC)
6.7
N f = 1.7 E (34) * ε 0
7.2
9.1
N f = 3.9 E (20) * ε 0
*e
*e
3.4
−0.5*VFA
*e
−0.2*V0
* S0
0.6
Note: N f = fatigue life ε 0 = 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
VFA (5oC)
Traditional Predicting Model 0.01 N f = 0.74 * e13.8*VFA * ε 0 −0.1*V0
A.V. (5oC)
N f = 4.9 E (4) * e
VFA (20oC)
N f = 4.1E (−4) * e
A.V. (20oC) Cryogenic
N f = 1.1E (5) * e
*ε0
21.6*VFA
−0.3*V0
N f = 1.1E (4) * e −0.2*VFA * ε 0
A.V. (5oC)
N f = 2.4 E (4) * e
VFA (20oC) A.V. (20oC)
N f = 4.0 E (4) * e
*ε 0
−0.2*VFA
N f = 8.1E (4) * e
−2.5*V0
0.22
57%
0.09
37%
0.49
135%
0.42
121%
0.04
31%
0.21
78%
0.40
84%
0.27
66%
2.6
1.0
VFA (5oC)
0.2*V0
COV
−0.2
*ε 0
*ε 0
R2
0.5
0.4
*ε 0
*ε 0
0.9
1.0
Note: N f = fatigue life ε 0 = 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
Method Strain
Cryogenic
Type Ambient
Rubber Prediction Prediction
Energy Strain Energy
Type VFA A.V VFA A.V VFA A.V VFA A.V
5ºC 2
R 0.95 0.77 0.79 0.66 0.65 0.65 0.52 0.56
20ºC COV 52% 47% 50% 46% 58% 58% 54% 57%
2
R 0.84 0.91 0.81 0.78 0.73 0.73 0.73 0.73
COV 66% 69% 68% 67% 46% 46% 49% 49%
Note: VFA = volume of voids filled with asphalt; A.V. = air voids
29
Xiao et al. (2007)
VFA ε
Nf
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
Out_err < Goal_err (?)
YES
NO Update Weights for Output-Layer Neurons
Update Weights for Hidden-Layer Neurons
Backpropagation Training Completed
FIG. 2 Flowchart illustrating backpropagation training algorithm (Juang and Chen 1999)
31
Xiao et al. (2007)
Predicted Fatigue Life (10 Cycles)
4
4
VFA Predicted Air Voids Predicted
4
Predicted Fatigue Life (10 Cycles)
5
2
R =0.95 (VFA) R2=0.77 (A.V)
3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
VFA Predicted Air Voids Predicted R2=0.84 (VFA) 2 R =0.91 (A.V)
0
0
1 2 3 4 Measured Fatigue Life (104Cycles)
5
1
2
3
4
5
6
7
8
9 10
Measured Fatigue Life (104Cycles)
(a)
(b)
FIG. 3 Comparison of fatigue lives between predicted and measured results using regressionbased strain-dependent models for ambient rubberized mixtures (a) 5ºC; (b) 20ºC
32
4
Predicted Fatigue Life (104Cycles)
4
Predicted Fatigue Life (10 Cycles)
Xiao et al. (2007)
VFA Predicted Air Voids Predicted R2=0.65 (VFA) R2=0.65 (A.V)
3
2
1
0 0
1
2
3
4
6 VFA Predicted Air Voids Predicted
5
R2=0.73 (VFA) R2=0.73 (A.V)
4 3 2 1 0 0
4
1
2
3
4
5
6
Measured Fatigue Life (104Cycles)
Measured Fatigue Life (10 Cycles)
(a)
(b)
FIG. 4 Comparison of fatigue lives between predicted and measured results using regressionbased strain-dependent models for cryogenic rubberized mixtures (a) 5ºC; (b) 20ºC
33
Xiao et al. (2007)
1 A B 2 COMMANDS OF EXECUTING EQ.8 3 ARGUMENT("R b ", "R p ", "P 1 ", "P 2 ", "P 3 ","Ln(ε 0 ) ") 4 ARGUMENT("VFA ", "P 4 ", "P 5 ", "Ln(S 0 ) ") 5 R b =(R b +0.019)/0.19; R p =(R p +0.038)/0.38 6 P 1 =(P 1 +0.0056)/0.056; P 2 =(P 2 +0.003)/0.03
C D E F Hidden Layer Weight matrix Hidden 1 Hidden 2 Hidden 3 Bias 8.50785 -9.71422 -2.95993 Input 1 -2.92512 3.77847 0.21774 Input 2 -5.43439 2.23798 1.56790
7 8
P 3 =(P 3 +0.0004)/0.004; Ln(ε 0 ) =(Ln(ε 0 )+7.91)/0.37 VFA =(VFA -0.725)/0.053; P 4 =(P 4 +0.015)/0.15
Input 3 Input 4
9
P 5 =(P 5 +0.028)/0.28; Ln(S 0 ) =(Ln(S 0 ) -16.64)/0.333
Input 5
10 11 12 13
pi1=1/(1+EXP(-(R b *D$5+R p *D$6+P 1 *D$7+P 2 *D$8+ P 3 *D$9+Ln(ε 0 ) *D$10+VFA *D$11+P 4 *D$12+ P 5 *D$13+Ln(S 0 ) *D$14+D$4))) pi2=1/(1+EXP(-(R b *E$5+R p *E$6+P 1 *E$7+P 2 *E$8+
Input 6 Input 7 Input 8 Input 9
14 15 16
P 3 *E$9+Ln(ε 0 ) *E$10+VFA *E$11+P 4 *E$12+ P 5 *E$13+Ln(S 0 ) *E$14+E$4))) pi3=1/(1+EXP(-(R b *F$5+R p *F$6+P 1 *F$7+P 2 *F$8+
Input 10
17 18 19 20
P 3 *F$9+Ln(ε0) *F$10+VFA *F$11+P 4 *F$12+ P 5 *F$13+Ln(S 0 ) *F$14+F$4))) pi4=1/(1+EXP(-(R b *G$5+R p *G$6+P 1 *G$7+P 2 *G$8+ P 3 *G$9+Ln(ε 0 ) *G$10+VFA *G$11+P 4 *G$12+
Bias Hidden 1 Hidden 2 Hidden 3
2.81230 -4.37899
Hidden4
-4.17195
21 22 23 24 25
P 5 *G$13+Ln(S 0 ) *G$14+G$4)))
Hidden 4 -1.94422 -2.58416 -4.47596
-2.05854 2.04824 2.17721
3.93442 3.62199 3.40047 -0.12879 3.49616 -5.61273
-1.07765
3.16621
1.70724 -1.50146
-7.00788 -0.51641 0.78339 -1.49672 0.25014 2.09887
2.25666 -1.66313 2.87061 2.33660 3.63714 -3.44396
0.34972
Output Layer
Z=pi1*D18+pi2*D19+pi3*D20+pi4*D21+D17 Cells B3:B25 are Z=1/(1+EXP(-Z)) macro commands to Ln(F)=2.041*Z+9.007 execute Eq.8 RETURN (F)
G
4.06708 -3.83664
0.22930 2.48973 4.02605
3.67111 -3.83600 -0.36828 Weight matrix: Cells D4: G4 are B HK Cells D5: G14 are W ik
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
6
Predicted Fatigue Life (x104cycles)
4
Predicted Fatigue Life (x10 cycles)
Xiao et al. (2007)
VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
5 4 3 2
R2=0.97 (VFA) 2 R =0.95 (A.V)
1 0 0
1
2
3
4
5
6
9 VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
8 7 6 5 4 3 2
R2=0.96 (VFA) R2=0.95 (A.V)
1 0 0
Measured Fatigue Life (x104cycles)
(a)
1 2 3 4 5 6 7 8 4 Measured Fatigue Life (x10 cycles)
9
(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
Predicted Fatigue Life (x10 cycles)
4
3
4
VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
4
Predicted Fatigue Life (x10 cycles)
Xiao et al. (2007)
2
1
2
R =0.91 (VFA) R2=0.84 (A.V)
0 0
1
2
3
6 VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
5 4 3 2
R2=0.92 (VFA) R2=0.97 (A.V)
1 0 0
4
1
2
3
4
5
6
4
4
Measured Fatigue Life (x10 cycles)
Measured Fatigue Life (x10 cycles)
(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
6.0
Fatigue Life (x104cycles)
Fatigue Life (x104cycles)
Xiao et al. (2007)
5.0 4.0 3.0 2.0 8%Rubber 12%Rubber
1.0
10%Rubber
0.0 0
5
10
15
20
25
30
6.0 5.0 4.0 3.0 2.0 10%RAP 20%RAP
1.0 0.0 0
35
15%RAP
5
10
15
Rubber Percentage (%)
RAP Percentage (%)
(a)
(b)
FIG. 8 Sensitivity analysis of rubber and RAP percentages in ANN fatigue models (a) RAP analysis; (b) Rubber analysis
37
20
5
Predicted Fatigue Life (x10 cycles)
VFA Predicted Air Voids Predicted
4
2
R =0.79 (VFA) 2 R =0.66 (A.V)
4
4
Predicted Fatigue Life (x10 cycles)
Xiao et al. (2007)
3 2 1 0 0
10 9 8 7 6 5 4 3 2 1 0
VFA Predicted Air Voids Predicted 2
R =0.81 (VFA) R2=0.78 (A.V)
0
1 2 3 4 5 4 Measured Fatigue Life (x10 cycles)
1
2
3
4
5
6
7
8
9 10
4
Measured Fatigue Life (x10 cycles)
(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
Predicted Fatigue Life (x10 cycles)
3
VFA Predicted Air Voids Predicted
4
2
4
Predicted Fatigue Life (x10 cycles)
Xiao et al. (2007)
R =0.52 (VFA) 2 R =0.56 (A.V)
2
1
0 0
1
2
6 5
VFA Predicted Air Voids Predicted
4
R =0.73 (VFA) R2=0.73 (A.V)
2
3 2 1 0 0
3
1
2
3
4
5
6
4
4
Measured Fatigue Life (x10 cycles)
Measured Fatigue Life (x10 cycles)
(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
Predicted Fatigue Life (x10 cycles)
6 VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
5 4
4
4
Predicted Fatigue Life (x10 cycles)
Xiao et al. (2007)
3 2 R2=0.94 (VFA) R2=0.93 (A.V)
1 0 0
1
2
3
4
5
9 8 7 6 5 4 3 2 1 0
VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
R2=0.92 (VFA) R2=0.90 (A.V)
0
6
1
2
3
4
5
6
7
8
9
4
4
Measured Fatigue Life (x10 cycles)
Measured Fatigue Life (x10 cycles)
(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
Predicted Fatigue Life (x10 cycles)
4 3
4
VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
4
Predicted Fatigue Life (x10 cycles)
Xiao et al. (2007)
2 1
2
R =0.89 (VFA) R2=0.86 (A.V)
0 0
1
2
3
6 VFA (Training) Air Voids (Training) VFA (Testing) Air Voids (Testing)
5 4 3 2
R2=0.92 (VFA) R2=0.93 (A.V)
1 0 0
4
1
2
3
4
5
6
4
4
Measured Fatigue Life (x10 cycles)
Measured Fatigue Life (x10 cycles)
(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
