Analysis of the Mechanical Behaviour of Asphalt Concretes Using Artificial Neural Networks

)e current paper deals with the numerical prediction of the mechanical response of asphalt concretes for road pavements, using artificial neural networks (ANNs). )e asphalt concrete mixes considered in this study have been prepared with a diabase aggregate skeleton and two different types of bitumen, namely, a conventional bituminous binder and a polymer-modified one. )e asphalt concretes were produced both in a road materials laboratory and in an asphalt concrete production plant.)emechanical behaviour of themixes was investigated in terms ofMarshall stability, flow, quotient, andmoreover by the stiffnessmodulus.)e artificial neural networks used for the numerical analysis of the experimental data, of the feedforward type, were characterized by one hidden layer and 10 artificial neurons. )e results have been extremely satisfactory, with coefficients of correlation in the testing phase within the range 0.98798–0.91024, depending on the consideredmodel, thus demonstrating the feasibility to apply ANNmodelization to predict the mechanical and performance response of the asphalt concretes investigated. Furthermore, a closed-form equation has been provided for each of the four ANN models developed, assuming as input parameters the production process, the bitumen type and content, the filler/bitumen ratio, and the volumetric properties of the mixes. Such equations allow any other researcher to predict the mechanical parameter of interest, within the framework of the present study.


Introduction
In order to design a road superstructure, the so-called pavement, two main tasks have to be accomplished, namely, the mix design of the asphalt concrete to be used for each of the layers of the pavement and the thickness design of the pavement itself.Focusing the attention on the mix design task, currently, all over the world, experimental procedures carried out in a road laboratory are adopted [1][2][3][4][5][6].Actually, a preliminary material characterization has to be performed for both the asphalt concrete components, namely, aggregates and bitumen, as well as a proper experimental mix design procedure is required in order to identify the optimum bitumen content.
e laboratory tests used to evaluate the physical properties and the mechanical resistance of components and mixtures are quite time consuming; moreover, skilled laboratory technicians have to be involved.At the end of the experimental mix design procedure, the best suited formulation of the asphalt mix is identified, in order to meet the pavement service requirements.However, on the basis of a pure experimental approach, if a component type or its amount has to be modified, for instance, for comparison purposes between different materials, to identify the best technological solution for the pavement construction, further laboratory tests cannot be avoided in order to evaluate the different mechanical response of the asphalt mix.e possibility to estimate the mechanical behaviour of the mix, on the basis of a mathematical model of the material's response, would allow us to save time and cost of further experiments.
A material's response model can be elaborated by means of constitutive equations [7][8][9] implemented on a computational platform with the finite element method [10][11][12], rather than with the discrete element method [13][14][15].Such approaches are elaborated on a physical basis because they try to achieve a rational interpretation of the mechanical response of the asphalt concrete under different loading conditions.However, the complexity of such methods is quite high, being related to the proper formulation of the constitutive equations as well as to extensive laboratory trials (which often involve particular test protocols), aimed at properly calibrating and validating such complex mathematical models.A different approach is based on the statistical regression of large experimental data sets, to obtain prediction equations of the material properties considered [16][17][18][19].
More recently, the so-called learning machines, for instance, the artificial neural networks (ANNs), have been used for the modelization of some significant mechanical parameters of road materials [20][21][22][23].e key point of such approaches is given by the possibility to obtain reliable analytical equations for the quantitative estimation of materials properties, in a relatively automatized and easy way, because the complex physical significance of the material's response is not considered; such an advantage, the so-called "black box esffect," represents on the contrary their main issue.e abovementioned black box effect could somehow be associated also with the relatively low attention paid in the civil engineering literature to the mathematical equations behind the artificial neural networks.Actually, several literature papers are simply devoted to the use of such computational tools in a broad variety of engineering applications, but without a proper discussion of the mathematical framework [24][25][26][27][28][29].Moreover, in such papers, the discussion is often limited to the evaluation of the quality of the training and testing phases of the ANN; just few researchers [20,30,31] have at least presented the predictive analytical equations elaborated by means of the ANN.
e main goal of this paper is to provide the analytical expressions for the prediction of the mechanical parameters involved in the mix design of asphalt concretes, on the basis of the ANN modelization of experimental data related to volumetric and composition parameters of the mixes.In order to allow a full understanding of the ANN modelization, a theoretical discussion of the mathematical equations, which actually constitute the backbone of the ANN, is also provided.

Modelling with Artificial Neural
Networks.Artificial neural networks (ANNs), also known as learning machines, represent a computational approach to develop predictive models for the desired parameter, whatever the complexity of the system under investigation, given a robust experimental data set for the training of the ANN [32].ey try to simulate the functioning of biological ones, in particular those of the brain, processing the input data through "neurons" [33][34][35].
ere are different types of ANN; in this study, the feedforward networks have been considered.For such ANN type, the learning of the network is supervised; therefore, for each input vector provided, the corresponding output vector (target) is known.e learning phase consists in optimizing the connections of the ANN so that for each input considered, the network returns a calculated value as close as possible to the target one [35].
An ANN of this type is structured with different neurons, divided into layers; these neurons are connected so that those of the same layer are not linked to each other and that none of the possible paths could touch twice the same neuron.erefore, the feedforward ANN structure is given by the following: (i) An initial layer with p neurons (input layer), where p is the number of input variables (ii) A final layer of c neurons (output layer), where c is the number of output variables (iii) At least one intermediate layer, or hidden layer, with m number of neurons, independent of how many belong to input or output e input layer stores the incoming signals that are introduced through a vector x i for each data set: e output layer provides the calculated values through a vector y i : ( e hidden layer is devoted to the calculations that formally connect the input x i with the output y i .
Each neuron of the hidden layer works according to a simple mathematical model proposed by McCulloch and Pitts [36].A weighted sum of the values of the input variables is computed through the weights that are associated with each connection: e value w 0 is called bias and corresponds to the activation value of the neuron; assuming x 0 � 1, such an expression can be simplified as follows: e output value from the neuron is calculated by applying an appropriate transfer function to such a value: e transfer function can be of different types depending on the desired model, for instance, linear, Heaviside step function, sigmoidal, or hyperbolic tangent.e last one has been used in the current study; it has the following expression: e above steps describe the functioning of a single neuron, while the network, to determine the optimal values of the weights of each connection, follows an iterative procedure, the so-called training of the neural network [33][34][35].Given a set of first attempt values of the weights, the ANN computes the activation values of the neurons and 2 Advances in Civil Engineering subsequently the final output; this phase is known as the forward pass.en, the value of the calculated output is compared with the expected value (target) so that the ANN can proceed to adjust the weights through an optimization algorithm; this phase is called backward pass.

e Forward Pass.
Considering an ANN with p neurons in input, a single hidden layer with m neurons, and an output layer with c neurons, the network processes the information based on the procedure described in the following.
For each of the i observations, or data sets (x i ), whose p coordinates are introduced each one in a neuron of the input layer, the activation value of each neuron j of the hidden layer is calculated: where the exponent (1) identifies the weights and the activation value of the first step, that is, between the input layer and the hidden one.Introducing again a fictitious value For each (x i ), the activation values a (1)   j are then transferred to the next level through a transfer function f: e procedure is repeated between the hidden layer and the output layer by calculating the activation value for each of the output c neurons as follows: a (2)  k � a (2)   k w (2)  jk z k �  m j�0 w (2)  jk f a (1) where the exponent (2) identifies the weights and the activation value of the second step, having made an assumption similar to the first step, that is, f(a (1) j ) � 1. e output of the network introduces a further transfer through a function g, not necessarily of the same type of f; therefore, the calculated output value at the final kth neuron is w (2)  jk f

e Backward Pass.
e ANN proceeds updating the weights of the connections that are the only modifiable parameters, in fact the values of the components of each (x i ) are fixed; therefore, the interpolating function depends only on the weights of the individual connections.ese can be considered as the parameters of an interpolating function for the approximation of the target values t ik with the computed values y ik ; the optimal value of these weights is then calculated minimizing the objective function: Since the t ik values are given, this objective function depends only on the weights of the single connections.e weight vector w therefore, given the complexity of the function E, is calculated iteratively through an optimization algorithm [33][34][35]; the simplest one is the gradient descent algorithm.At each iteration, the weight vector is updated according to the following equation: where the subscript τ indicates the number of the iteration and the quantity Δw, which updates the weights, is a vector that moves along the descending path, characterized by a faster reduction of the function E, that is, its gradient in the vector space generated by the weights.erefore, it can be written as follows: To find this direction, it is necessary to compute the partial derivatives of the objective function and to define the value of η which is a positive real number that should not be too small; otherwise, the calculation time becomes longer, or too big, to avoid the instability of the method.
Specifying the structure of the function E, it can be written as follows: us, the partial derivatives can be calculated for each E i and subsequently added together; basically each E i can be considered as the component of a vector.e partial derivatives must be expressed with respect to the weights, and these are relative to both the hidden and output layers.Considering the partial derivatives of the output layer, if its transfer function is of the linear type, as it has been assumed in the present study, it follows that erefore, the partial derivative of the generic term of E i with respect to the generic weight w (2)  jk results zE i zw (2) where a (1)  j �  p s�0 w (1)  sj x is and f is the transfer function of the hidden layer.In the present study, the hyperbolic tangent function has been assumed for the hidden layer: Instead, deriving E i with respect to a weight of the hidden layer and using the chain rule, it can be written as follows: Advances in Civil Engineering za (1)   j za (1)   j zw (1)   sj . ( e first derivative of the hyperbolic tangent function has the following property [35]: Hence, using such an expression and deriving with respect to a generic jth activation value of the hidden layer, it follows that zy ik za (1)   j � zw (2)  jk f a (1)   j   za (1)   j � w (2)   jk zf a (1)   j e second partial derivative instead can be expressed as za (1)   j zw (1) erefore, by rearranging these equations, the final value of the partial derivative of the E i component with respect to a weight of the hidden layer and of the output layer can be written as In this way, it is possible to evaluate the gradient and to optimize the weight values at each iteration.

Training Algorithm.
e training algorithm adopted in the current study was similar to that of the gradient descent, but slightly modified; it was the backpropagation algorithm of Levenberg-Marquardt [37].Such an algorithm is of the second order but does not require the calculation of the Hessian matrix, which is approximated as where H is the Hessian matrix and J is the Jacobian matrix that contains the first derivatives of errors (E i ) with respect to weights.e gradient g is instead calculated as where e is the vector of network errors.e Jacobian matrix can be calculated through the equations described above.e values of the weights are updated according to an iterative procedure similar to that of the gradient descent but modified as follows: where I is the identity matrix.It can be observed that if the scalar μ increases, it returns to having the gradient descent algorithm with η small; the parameter μ is changed at each iteration; in particular, it is reduced to speed up the convergence to the solution.

Materials and Methods
e type of asphalt mixture considered in the current study was dense asphalt concrete (AC) with diabase aggregates and conventional or modified bitumen.e AC mixtures came from three different projects carried out in Greece, having various bitumen contents and aggregate gradations.e production of some of the AC mixtures was carried out in the laboratory either as part of the mix design procedure or as part of stiffness testing of the design mixture.e rest of the ACs were produced into a stationary asphalt plant as final mixture production.

Aggregates.
e diabase aggregates, depending on the project, came from three different quarries; their characteristic properties, as well as the test protocols used, are given in Table 1.

Bitumen.
Two types of bitumen have been used in the current study, a 50/70 conventional bitumen and a SBS modified bitumen.e characteristic properties of the two bitumen types, along with the test protocols adopted, are reported in Table 2.  Percentage passing (%) Greek AC limits Critical zone of Greek specification for AC AC20-ModP-G1 AC20-ModP-G2 Advances in Civil Engineering e dense asphalt concrete (AC) mixtures had a maximum aggregate size of 20 mm (AC20), in all cases.In detail, the AC20 mixture with 50/70 conventional bitumen (AC20-50/70L) was produced exclusively in the laboratory, while the AC20s with SBS-modified bitumen were produced in the laboratory (AC20-ModL) and in a stationary asphalt plant (AC20-ModP).e specimens of all mixtures were compacted in the laboratory using an impact compactor (EN 12697-30) having a diameter of 100 mm and an average thickness of 63.7 mm.irty specimens for each type of mixture were produced; hence, total ninety specimens were utilized in the current study.e gradations of the AC20-50/70L and AC20-ModL are given in Figure 1. e gradations of the AC20-ModP are given in Figure 2 and correspond to different production dates throughout the project.
Tables 3-5 show specimens' volumetric properties (EN 12697-8), Marshall stability, and Marshall flow values (EN 12697-34), per type of mixture.Furthermore, the Marshall quotient has been computed per each specimen, equal to the ratio between Marshall stability and Marshall flow.Even if it has been previously outlined the partial representativity of the Marshall data with respect to the asphalt concrete behaviour, such a test is still widely adopted given the large experience    Advances in Civil Engineering cumulated over the years [38][39][40][41].Table 5 gives additional details concerning the correspondence between specimens and gradations shown in Figure 2.

Sti ness Modulus Test.
e Sti ness modulus has been evaluated, for all the specimens, in accordance with EN 12697-26, Annex C (IT-CY), assuming the following testing conditions: temperature of 20 °C, target deformation xed at 5 μm, and rise-time equal to 124 ms.
e number of specimens tested for sti ness was ninety (90), that is, thirty (30) for each mixture, that is, AC20-50/70L, AC20-ModL, and AC20-ModP.e sti ness modulus results are presented in Table 6.Advances in Civil Engineering can be seen from Table 6, the AC mixtures with modi ed bitumen resulted in higher sti ness values than the AC mixtures with 50/70 penetration bitumen. is can be attributed to the bene cial e ect of the modi ed bitumen.e smaller sti ness standard deviation was achieved for the plant-produced mixtures.is could be attributed to the machinery of the plant for asphalt production.More homogeneous mixtures are produced in a plant where large quantities are handled, and all production steps are automated, rather than in the laboratory.One of the most sensitive steps in laboratory production and specimen preparation is the lling of the Marshall moulds after asphalt mixing.Since  Advances in Civil Engineering the mould lling is done by hand and the handler may not be the same in every case, it is normal not to have the same homogeneity obtained by automated plant procedures.

Results and Discussion
Considering each specimen sti ness value obtained in the current study, it could be stated that the sti ness values measured are depending mainly on the type of bituminous binder, bitumen content, and voids content.Gradation variations, especially for the plant-produced mixtures, are indirectly taken into account by the voids content.

Arti cial Neural Networks Modelling Results.
In the present study, a three-layer feedforward ANN was used to model each of the four mechanical parameters considered in the experimental investigation, namely, Marshall stability, ow and quotient, as well as Sti ness Modulus.e ANN models have been developed using the MATLAB ® ANN toolbox [42].e architecture of the four ANN models has been optimized with an input layer characterized by seven neurons, one hidden layer with 10 neurons, and an output layer with one neuron (Figure 3).
All the four ANN models have been elaborated on the basis of seven type of input data, namely, bitumen type, bitumen content, ller-bitumen ratio, air voids, voids in the mineral aggregates, voids lled with bitumen, and type of production process.Such input data have been considered of fundamental importance to take into account the di erent production process (laboratory or plant) and to properly represent the composition of the asphalt concretes, which were characterized by the same aggregate type, similar gradation, but prepared with di erent bitumen contents, binder type, and ller-bitumen ratios.en, the single output neuron is associated with the speci c mechanical parameter considered.
e number of neurons used in the hidden layer (10) has been optimized by means of a trial and error procedure; in the literature, it is recommended to use the lower number of neurons that allow us to obtain satisfactory results [32].
e experimental data set used for training and testing of the ANN has included 30 specimens for each of the three type of asphalt concrete (i.e., 90 specimens overall); therefore, all the mixtures, with modi ed or conventional bitumen, prepared in the laboratory or in the plant, have been analyzed together.is was done to obtain, for each of the four mechanical parameters considered, a unique predictive model, whatever the composition of the mixture, within the ambit of the current study.e sampling process is completely random and is performed automatically by MATLAB [38].
Figures 4-7 show the comparison between experimental (target) and predicted (output) data, for the mechanical parameters analyzed.As it can be observed, the predicted values, computed by the ANN models, are very close to the experimental data, for both Marshall parameters and sti ness modulus. is is particularly signi cant form an engineering point of view, considering the di erent characteristics of the mixtures, in terms of composition and mechanical response.
e results of the training, validation, and testing phases of the ANN models can be observed in Figures 8-11.
In order to evaluate the performance of the ANN, two di erent statistical indicators are provided by MATLAB, namely, the coe cient of correlation (R) and the mean square error (MSE) [42]; the closest to 1 the R value and the lower the MSE value, the better the performance of the ANN.Similar performance indicators have been used in previous investigations, focused on the mechanical parameter prediction of asphalt concretes, by ANN analysis [20][21][22][30][31][32]. e lowest MSE value (0.67751) was observed for the prediction of the sti ness modulus, while increasing values have been obtained for Marshall stability, ow, and quotient (0.9962, 1.6377, and 1.6830, resp.).e highest accuracy achieved for the prediction of the sti ness modulus has been also con rmed in terms of correlation coe cient, with an R value in the testing phase equal to 0.98798.However, R values greater than 0.91 were also obtained for the Marshall parameters (stability, ow, and quotient).erefore, it can be concluded that the ANN approach allows us to obtain a satisfactory interpretation of the nonlinear relationships between the input variables considered and the mechanical parameters analysed.
Also previous studies have veri ed the good matching between experimental and ANN model results for sti ness [32] and Marshall stability [21].Nevertheless, other researchers have outlined a worst prediction of Marshall ow and Marshall quotient, with respect to Marshall

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Advances in Civil Engineering stability [20].According to the results of the present study, all the Marshall parameters, as well as the sti ness modulus, have been predicted by the ANN models with a comparable quality level.

Predictive Equations.
e general equation for the prediction of the mechanical parameter (MP) considered, obtained by the ANN modelization, is reported in the following: where f and b are the ller and the bitumen contents, respectively; V a , VMA, and VFA are the residual air voids, voids in the mineral aggregate, and voids lled with bitumen; T b is the type of bitumen; and PP is the production process type.e other ANN coe cients are reported in Tables 7-10, for the sti ness modulus and Marshall stability, ow, and quotient, respectively.e structure of the equation is characterized by the contribution of the ten arti cial neurons, by means of the ten ANN coe cients.
Such a predictive model can be very useful to obtain an analytical estimation of the mechanical parameters analyzed, without the necessity to perform further experimental tests.
is approach allows us to identify more easily and quickly the best mix design option, Advances in Civil Engineering 13 saving time and resources with respect to a direct laboratory evaluation.

Conclusions
In the present study, the arti cial neural network approach has been used to numerically model the mechanical behaviour of asphalt concretes for road constructions.e mixtures involved in the study were characterized by two di erent types of bituminous binder and production processes, as well as various bitumen contents and consequently di erent volumetric properties.
A feedforward multilayer ANN architecture, characterized by ten neurons in the hidden layer, has been elaborated to predict the sti ness modulus and the Marshall parameters of the mixes considered in the experimental investigation.
e hyperbolic tangent transfer function and a linear one have been assumed for the hidden and the output layers, respectively, whereas the Levenberg-Marquardt optimization algorithm was adopted as training algorithm.
e good results achieved in the testing phase demonstrate that the ANNs have the capability to generalize the complex relationships between input and output data, learned in the training phase, so allowing us to elaborate a satisfactory prediction model for the sti ness modulus, as well as for Marshall parameters, whatever the composition and the production process of the mixtures considered in the study.
A closed-form equation has been elaborated for each of the mechanical parameters studied, to allow other researchers and engineers to obtain an estimation of such parameters, within the type of mixes investigated.
e present study has demonstrated the feasibility to obtain, by means of ANN, predictive models of mechanical parameters (Marshall stability, ow, quotient, and sti ness modulus) very important for the mix design and the performance characterization of asphalt concretes for road pavements.However, even if di erent compositions of the  Advances in Civil Engineering

10 Figure 3 :
Figure3: ANN structure adopted in the current study.

Figure 8 :
Figure 8: Training, validation, testing phases, and all data for the sti ness modulus (MPa) ANN model.

Figure 9 :
Figure 9: Training, validation, testing phases, and all data for the Marshall stability (kN) ANN model.

Figure 10 :
Figure 10: Training, validation, testing phases, and all data for the Marshall ow (mm) ANN model.

Figure 11 :
Figure 11: Training, validation, testing phases, and all data for the Marshall quotient (kN/mm) ANN model.

Table 7 :
ANN coefficients for the stiffness modulus prediction model.

Table 8 :
ANN coefficients for the Marshall stability prediction model.

Table 9 :
ANN coefficients for the Marshall flow prediction model.

Table 10 :
ANN coefficients for the Marshall quotient prediction model.Advances in Civil Engineeringmixes have been studied, it would be very useful to increase the variability of the input parameters, for instance, considering different sources of the aggregates; this requires just a further new training of the ANN with additional experimental data.