Advances in Civil Engineering

Volume 2018, Article ID 1650945, 17 pages

https://doi.org/10.1155/2018/1650945

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

^{1}Polytechnic Department of Engineering and Architecture, University of Udine, Via del Cotonificio 114, 33100 Udine, Italy^{2}Department of Civil Engineering, Aristotle University of Thessaloniki, University Campus, 54124 Thessaloniki, Greece^{3}Department of Civil, Environmental and Architectural Engineering, University of Padua, Via Marzolo 9, 35131 Padua, Italy

Correspondence should be addressed to Nicola Baldo; ti.duinu@odlab.alocin

Received 27 February 2018; Revised 12 June 2018; Accepted 19 June 2018; Published 12 July 2018

Academic Editor: Victor Yepes

Copyright © 2018 Nicola Baldo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The current paper deals with the numerical prediction of the mechanical response of asphalt concretes for road pavements, using artificial neural networks (ANNs). The 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. The asphalt concretes were produced both in a road materials laboratory and in an asphalt concrete production plant. The mechanical behaviour of the mixes was investigated in terms of Marshall stability, flow, quotient, and moreover by the stiffness modulus. The 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. The results have been extremely satisfactory, with coefficients of correlation in the testing phase within the range 0.98798–0.91024, depending on the considered model, thus demonstrating the feasibility to apply ANN modelization 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.

#### 1. 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 [1–6]. 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. The 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. The 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 [7–9] implemented on a computational platform with the finite element method [10–12], rather than with the discrete element method [13–15]. 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 [16–19].

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 [20–23]. The 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. The 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 [24–29]. 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.

The 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.

#### 2. Theory and Calculation

##### 2.1. 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]. They try to simulate the functioning of biological ones, in particular those of the brain, processing the input data through “neurons” [33–35].

There 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. The 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. Therefore, the feedforward ANN structure is given by the following:(i)An initial layer with neurons (input layer), where 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

The input layer stores the incoming signals that are introduced through a vector for each data set:

The output layer provides the calculated values through a vector :

The hidden layer is devoted to the calculations that formally connect the input with the output .

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:

The value is called bias and corresponds to the activation value of the neuron; assuming = 1, such an expression can be simplified as follows:

The output value from the neuron is calculated by applying an appropriate transfer function to such a value:

The transfer function can be of different types depending on the desired model, for instance, linear, Heaviside step function, sigmoidal, or hyperbolic tangent. The last one has been used in the current study; it has the following expression:

The 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 [33–35]. Given a set of first attempt values of the weights, the ANN computes the activation values of the neurons and subsequently the final output; this phase is known as the forward pass. Then, 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.

##### 2.2. The Forward Pass

Considering an ANN with 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 , whose 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 *x*_{0i} *=* 1, it follows that

For each , the activation values are then transferred to the next level through a transfer function :

The 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: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, .

The output of the network introduces a further transfer through a function , not necessarily of the same type of ; therefore, the calculated output value at the final *k*th neuron is

##### 2.3. The Backward Pass

The ANN proceeds updating the weights of the connections that are the only modifiable parameters, in fact the values of the components of each are fixed; therefore, the interpolating function depends only on the weights of the individual connections. These 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 values are given, this objective function depends only on the weights of the single connections. The weight vector therefore, given the complexity of the function , is calculated iteratively through an optimization algorithm [33–35]; the simplest one is the gradient descent algorithm. At each iteration, the weight vector is updated according to the following equation:where the subscript indicates the number of the iteration and the quantity , which updates the weights, is a vector that moves along the descending path, characterized by a faster reduction of the function , that is, its gradient in the vector space generated by the weights. Therefore, 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 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:

Thus, the partial derivatives can be calculated for each and subsequently added together; basically each can be considered as the component of a vector. The 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

Therefore, the partial derivative of the generic term of with respect to the generic weight resultswhere and 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 with respect to a weight of the hidden layer and using the chain rule, it can be written as follows:

The first derivative of the hyperbolic tangent function has the following property [35]:

Hence, using such an expression and deriving with respect to a generic *j*th activation value of the hidden layer, it follows that

The second partial derivative instead can be expressed as

Therefore, by rearranging these equations, the final value of the partial derivative of the 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.

##### 2.4. Training Algorithm

The 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 aswhere is the Hessian matrix and is the Jacobian matrix that contains the first derivatives of errors () with respect to weights. The gradient is instead calculated aswhere is the vector of network errors. The Jacobian matrix can be calculated through the equations described above. The values of the weights are updated according to an iterative procedure similar to that of the gradient descent but modified as follows:where is the identity matrix. It can be observed that if the scalar increases, it returns to having the gradient descent algorithm with small; the parameter is changed at each iteration; in particular, it is reduced to speed up the convergence to the solution.

#### 3. Materials and Methods

The type of asphalt mixture considered in the current study was dense asphalt concrete (AC) with diabase aggregates and conventional or modified bitumen. The AC mixtures came from three different projects carried out in Greece, having various bitumen contents and aggregate gradations. The 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. The rest of the ACs were produced into a stationary asphalt plant as final mixture production.

##### 3.1. Aggregates

The 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.