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Neural network-based model for prediction of permanent deformation of unbound granular materials
Journal Pre-proof Neural network-based model for prediction of permanent deformation of unbound granular materials Ali Alnedawi, Riyadh Al-Ameri, Kali Prasad Nepal PII:
S1674-7755(19)30719-X
DOI:
https://doi.org/10.1016/j.jrmge.2019.03.005
Reference:
JRMGE 603
To appear in:
Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 3 August 2018 Revised Date:
29 December 2018
Accepted Date: 7 March 2019
Please cite this article as: Alnedawi A, Al-Ameri R, Nepal KP, Neural network-based model for prediction of permanent deformation of unbound granular materials, Journal of Rock Mechanics and Geotechnical Engineering, https://doi.org/10.1016/j.jrmge.2019.03.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. All rights reserved.
Neural network-based prediction model for permanent deformation of unbound granular materials Ali Alnedawi a*, Riyadh Al-Ameri a, and Kali Prasad Nepal a
School of Engineering, Deakin University, Geelong, Victoria 3220, Australia
* Corresponding author E-mail [email protected] [email protected] Phone no. +61 3 52278825 Deakin University Locked Bag 20000 Geelong VIC 3220 Australia
Ali Alnedawi orcid.org/0000-0003-0018-6696 Riyadh Al-Ameri orcid.org/0000-0003-1881-1787 Kali Prasad Nepal orcid.org/0000-0001-7497-1983
Neural network-based model for prediction of permanent deformation of unbound granular materials Ali Alnedawi*, Riyadh Al-Ameri, Kali Prasad Nepal School of Engineering, Deakin University, Geelong, Victoria, 3220, Australia
ARTICLEINFO Article history: Received 3 August 2018 Received in revised form 29 December 2018 Accepted 7 March 2019 Available online Keywords: Flexible pavement design Unbound granular materials Permanent deformation (PD) Repeated load triaxial test (RLTT) Prediction models Artificial neural network (ANN)
ABSTRACT Several available mechanistic-empirical pavement design methods fail to include predictive model for permanent deformation (PD) of unbound granular materials (UGMs), which make these methods more conservative. In addition, there are limited regression models capable of predicting the PD under multi-stress levels, and these models have regression limitations and generally fail to cover the complexity of UGM behaviour. Recent researches are focused on using new methods of computational intelligence systems to address the problems, such as artificial neural network (ANN). In this context, we aim to develop an artificial neural model to predict the PD of UGMs exposed to repeated loads. Extensive repeated load triaxial tests (RLTTs) were conducted on base and subbase materials locally available in Victoria, Australia to investigate the PD properties of the tested materials and to prepare the database of the neural networks. Specimens were prepared over different moisture contents and gradations to cover a wide testing matrix. The ANN model consists of one input layer with five neurons, one hidden layer with twelve neurons, and one output layer with one neuron. The five inputs were the number of load cycles, deviatoric stress, moisture content, coefficient of uniformity, and coefficient of curvature. The sensitivity analysis showed that the most important indicator that impacts PD is the number of load cycles with influence factor of 41%. It shows that the ANN method is rapid and efficient to predict the PD, which could be implemented in the Austroads pavement design method. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. T h is is a n o p e n a c c e s s a r t ic l e u n d e r t he C C BY - N C - N D l ic e n s e ( h t t p : / / c re a t i v e c o m mo n s . o r g/ licenses/by-nc-nd/4.0/).
involve several prediction variables, for instance, deviatoric and confining 1. Introduction It is acknowledged that permanent deformation (PD) of unbound granular materials (UGMs) is the irrecoverable plastic strain, which can cause rutting in flexible pavements (Dai et al., 2007; Bodin and Kraft, 2015; Gu et al., 2016). A flexible pavement is susceptible to PD when it is exposed to long-term repeated traffic loads (Berthelot et al., 2009; Xiao et al., 2012) and especially when the applied traffic loads exceed the designed loads (Werkmeister, 2003; Austroads, 2018). In practice, distresses such as rutting could be observed in UGM layers (Haider et al., 2014; Alnedawi et al., 2017; Leiva-Villacorta et al., 2017). Rutting in base layer occurs due to a shortage of the requisite strength of UGMs (Tutumluer and Pan, 2008; Xiao et al., 2011). Stability and performance of the flexible pavement structure require an evaluation of PD (Rahman, 2015; Alnedawi et al., 2018). The mechanistic-empirical flexible pavement design methods aim to select the optimum combination of layer depth and material type with the minimum development of PD. However, the existing flexible pavement design guide in Australia, e.g. Austroads (2018), only theorises as a performance design criterion, considering the critical behaviours of the maximum vertical compressive strain at the top of the subgrade and the maximum tensile strain at the bottom of the asphalt layer. Surprisingly, the designed base and subbase layers are not included in the performance criterion. The design procedure requires that the UGMs should follow certain specifications by acquiring particular properties, which results in a conservative structural design. The same guide clearly raises the absence of PD constitutive model for UGMs in flexible pavements. Efforts are demanded when establishing or adopting a particular prediction model for specific requirements due to the complex stress–strain responses of UGMs (Werkmeister et al., 2005). Recent studies showed the necessity to develop reliable constitutive models to predict the PD behaviours under repeated loads (Pratibha et al., 2015; Rahman, 2015; Alnedawi et al., 2019a). A suitable PD model could promise a firm backbone for maintenance, design, and cost estimation (Qiao et al., 2015). These types of models could
stresses, load cycles, resilient strain, and shear strength. The common available PD models were derived from regression analysis (Gabr and Cameron, 2013; Azam et al., 2015; Rahman and Erlingsson, 2015a; Gu et al., 2017). Defining the parameters to fit a regression model requires a pre-specified relationship (Tu, 1996). There is no software available that is able to develop nonlinear models with more than two independent variables. Therefore, a wide array of linear and nonlinear equations needs to be evaluated (Romanoschi, 2017). In addition, simplicity of regression models often fails to cover the complexity of UGM behaviour. Thus, recent researches have shifted the focus towards adopting new techniques of computational intelligence systems, which are able to model any arbitrary complex relationships between the dependent and independent variables (Tu, 1996). Artificial neural network (ANN) is one of the most popular computational intelligence systems (Nazzal and Tatari, 2013), which has been acknowledged for the quick and accurate prediction of pavement layer distresses (Rahman et al., 2001). Due to the robustness and ease, ANN has been used to model many complex relationships in pavement engineering (Lee et al., 2003). Ceylan and Gopalakrishnan (2007) employed ANN to predict the fatigue life of hot-mix asphalt and vertical compressive strain on the top of the subgrade based on falling weight reflectometer (FWD) data. Nazzal and Tatari (2013) showed that the ANN-based subgrade models had better prediction coefficients for resilient modulus (Mr) than those obtained by the regression models. Sharma and Das (2008) used ANN to develop a model to predict the remaining life span of the pavement. Park et al. (2009) established an ANN model to determine Mr of UGMs using in situ stresses and material properties, and they found that the ANN model serves as a reliable and simple analytical tool. Kim et al. (2014) used ANN to predict Mr of base layers using stress state and physical properties of the materials. Similarly, Saha et al. (2018) developed ANN models to estimate Mr of UGMs from soil physical properties, with the correlation coefficient R2 greater than 0.9. Alnedawi et al. (2019b) observed the influence of loading frequency using ANN. However, ANN-based models for PD prediction are rarely reported. Several researches have been conducted to compare the regression and ANN models, some of which indicated prominent outcomes of ANN, while the
*Corresponding author. E-mail address: [email protected]
others showed insignificant differences (Hutton, 1992; Buchman et al., 1994;
Alrashydah and Abo-Qudais, 2018; Saha et al., 2018; Taha et al., 2018). It is likely that the preferences of these two types of models are highly influenced by the type of the database. Therefore, empirical comparison is required in the early stages of modelling (Tu, 1996). In this context, following the Introduction, Section 2 describes the experimental study. Then, three existing regression models and laboratory repeated load trixial test (RLTT) data are used for the regression analyses to predict the PD of UGMs. Subsequently, ANN models are developed to predict the PD using the same regression input variables. Next, the predicted PD values from the regression method are compared with the ANN values to assess the robustness of the two methods. Based on several input variables, an ANN model is proposed for the Austroads pavement design method. 2. Experimental studies 2.1. Repeated load triaxial test Developing neural network requires extensive experimental tests. Generally,
Fig. 1. Repeated load triaxial test equipment.
RLTTs are used to test PD of UGMs for material evaluation purpose (Alnedawi et al., 2018). RLTT was recognised as one of the effective laboratory tests that can imitate field conditions (Witczak and Uzan, 1988). The components of the RLTT equipment are highlighted in Fig. 1. Vertical stresses were applied in the form of trapezoidal waveform with loading
Table 1. Stress stages for PD test. Stage Confining pressure, σ3 (kPa) 1 50 2 50 3 50
Deviatoric stress, σd (kPa) 350 450 550
Number of cycles 10,000 10,000 10,000
frequency of 0.5 Hz (Alnedawi et al., 2019c). The rest period was twice the loading time (Alnedawi et al., 2019d). In the testing of protocol, three sequences of repeated deviatoric stresses (σd) with a constant confining pressure (σ3) were applied, as listed in Table 1. The testing procedure was performed as per Austroads (2007). It is worth mentioning that the stress stages in Table 1 involve typical series of repeated deviatoric stress with fixed confining pressure for the purpose of reducing the effect of confining pressure that could affect the incremental PD (Alnedawi et al., 2019e). 2.2. Materials Four different materials as shown in Fig. 2, defined by VicRoads (2011) as high-quality base/subbase materials, were experimentally investigated using RLTTs. The base UGMs were specified as Class 2 (granite and basalt), while the subbase UGMs were specified as Class 3 (basalt) and Class 4 (basalt). Two different particle size distributions were the sources of all basalt materials (Classes 2–4). Modified Proctor protocol was used to measure the maximum dry density (MDD) and optimum moisture content (OMC) in accordance with the Australian Standard AS 1289.5.2.1–2003 (2003). Fig. 3 shows the sieve analysis of the investigated UGMs, and Table 2 tabulates their physical and
Fig. 2. Local UGMs tested.
engineering properties.
Specimens were prepared at different gradations, dry densities, and moisture contents, as listed in Table 3. Two tests for each specimen were conducted. As shown in Fig. 3, the symbols B and G refer to the rock origin, i.e. basalt and granite, respectively, and the numbers (i.e. 1 and 2) after B and G refer to different gradations. 2.3. Preparation of specimens The specimen for the RLTT was prepared according to Austroads (2007), as described in Fig. 4. Specimens were compacted into 8 layers with respect to the design dry density and moisture content using modified compaction effort. The cylindrical mould consisted of three split pieces to produce specimens with height of 200 mm and diameter of 100 mm. The specimens were then enclosed in a rubber membrane and the two ends of the membrane were encircled by O rings. Table 2. Physical and engineering properties of the investigated UGMs. Material D10 D30 D50 D60 Fines < 0.075 Coefficient of (mm) (mm) (mm) (mm) mm (%) uniformity, Cu CL2B1 0.08 1.9 6 8.5 10 106.25 CL2B2 0.68 1.3 3.9 5.5 11 84.62 CL2G 0.1 1 3.2 5.2 8 52 CL3B1 0.06 1.2 3.8 5.9 12 98.33
Coefficient of curvature, Cc 5.31 4.73 1.92 4.07
USCS classification GP-GM GP-GM GW-GM GP-GC
AASHTO classification A-1-a A-1-a A-1-a A-1-a
Compaction (modified) MDD (kg/m3) OMC (%) 2260 8.6 2300 8 2310 6 2260 8.2
Plasticity index 1 1 0 3
CL3B2 CL4B
0.06 0.06
0.7 1.6
2.5 4
4.2 6
Table 3. Experimental testing design. Material Gradation Moisture content (%) -OMC OMC CL2 B1 7.6 8.6 B2 7 8 G 5 6 CL3 B1 7.2 8.2 B2 7.3 8.3
13 11
70 100
1.94 7.11
MDD (kg/m3) +OMC 9.6 9 7 9.2 9.3
GW-GC GP-GC
A-1-a A-1-b
2270 2230
8.3 9.5
3 4
CL3B2, as shown in Fig. 3. The effects of the deviatoric stress and the number of load cycles on PD were significant. For example, CL2B1 (Fig. 5a) prepared at OMC/MDD has PDs of 1.39%, 1.55% and 1.96% under deviatoric stresses
2260 2300 2310 2260 2270
of 350 kPa, 450 kPa and 550 kPa, respectively. It should be noted that the stress levels were applied according to Austroads (2007). However, simulating field condition was challenging due to the complex wheel stresses applied in
Percentage Passing % (by mass)
110 100 90 80 70 60 50 40 30 20 10 0 0.01
both lateral and vertical directions.
CL2B1 = Class 2 (basalt)_1 CL2B2 = Class 2 (basalt)_2 CL2G = Class 2 (granite) CL3B1 = Class 3 (basalt)_1 CL3B2 = Class 3 (basalt)_2 CL4B = Class 4 (basalt)
Surprisingly, subbase materials represented by CL3B1, CL3B2 and CL4B showed a PD resistance similar to base materials in specimens CL2B1, CL2B2 and CL2G. This finding highlights that the current local specification for the subbase UGMs could be highly conservative. Therefore, this paper recommends characterising the UGMs according to their mechanical responses. 3. Regression modelling
0.10
1.00 Particles Size (mm)
10.00
100.00
Regression models are used to predict the rutting depth during pavement design (Siripun et al., 2010). The regression coefficients of such models are
Fig. 3. Sieve analysis of the tested UGMs.
used for PD prediction based on the independent variables (Gu et al., 2016). In addition, these coefficients can also be used to evaluate the material response
2.4. Experimental results The PDs of the tested materials versus load cycles are depicted in Fig. 5. It
(e.g. Qiao et al., 2015; Gu et al., 2016; Qamhia et al., 2017; Romanoschi,
can be seen that the specimens CL2B1, CL2B2, CL2G, CL3B1, CL3B2 and
2017). Generally, any PD regression model has independent variables such as
CL4B at moisture content less than the optimum one tend to have lower PD in
deviatoric and confining stresses, number of applied load cycles, recoverable
comparison with those at and above OMC. Soil suction could be a possible
strain, and shear strength. The cost and time of the prediction process increase
explanation for that trend. In an unsaturated soil, the suction pulls the
as the number of the prediction variables increases (Rahman and Erlingsson,
microstructure of the particles closer together, which results in greater stability
2015b). The implication of regression analysis in this study was used in
and strength. However, compacted materials with the moisture content below
comparison with ANN models. Regression analysis was conducted based on
the OMC are not recommended due to the workability problems, which need
three models in Eqs. (1)–(3), which were developed by Barksdale (1972),
extra effort to reach the desirable dry density. In addition, excessive
Sweere (1990) and Wolff and Visser (1994), respectively. The three selected
compaction effort could lead to an early breakage of the UGM particles.
models can predict the PDs under different load cycles, which were evaluated
Gradation did not show a significant relationship with PD, which could be
and compared with the experimental results.
explained by the close gradations investigated for CL2B1, CL2B2, CL3B1 and
Fig. 4. Fabrication of specimens.
(a) CL2B1.
(b) CL2B2. 3.5 -OMC 3.0
+OMC
PD (%)
2.5
OMC
2.0 1.5 1.0 0.5 0.0
0
5000
10000 15000
20000 25000 30000
Number of load cycles, N (c) CL2G.
(d) CL3B1.
3.0
3.0
-OMC 2.5
-OMC
+OMC
2.5
+OMC
OMC
1.5
1.5 1.0
1.0
0.5
0.5 0.0
OMC
2.0
PD (%)
PD (%)
2.0
0.0
0
5000
10000 15000
20000 25000 30000
0
5000
10000 15000
20000 25000 30000
Number of load cycles, N
Number of load cycles, N (e) CL3B2.
(f) CL4B.
Fig. 5. PDs of the tested materials under different load cycles and deviatoric stresses.
PD = a + b log 10 N
(1)
model in Eq. (3) has greater regression coefficients than the other models
PD = aN b
(2)
which could efficiently predict the PD of UGMs.
PD = (cN + a)(1 − e −bN )
(3)
where PD is the permanent deformation; N is the number of load cycles; and a, b and c are the regression coefficients. The PD values of RLTT were used to generate the regression coefficients and to measure R2 for each individual specimen, as shown in Table 4. The R2 values for Eq. (1) could be categorised as fair to good fit. An improvement in the prediction was also observed in Eq. (2) with R2 values greater than 0.9 for each of CL2G (+OMC and OMC) and CL3B1 (+OMC) specimens. The best-fit was clearly obtained by Eq. (3). Surprisingly, regression models that do not adopt stress as an independent variable showed high R2 values although multiple stresses were applied during the RLTT. For instance, the model in Eq. (3) showed the most excellent goodness of fit with the maximum R2 value of 0.99. This observation can be possible that the Specimen
Eq. (1) (Barksdale, 1972)
4. Neural network modelling ANN is a powerful technique used to model complex relationships that are difficult to be described by conventional equations (Beale et al., 2015). Neural networks are constructed by simple elements working in parallel direction. The links (weights) between these elements mainly govern the network role. A neural network can be trained by regulating the values of the weights between elements to achieve specific function. The ANN procedure comprises collecting data, generating and training the neural network, and evaluating the performance using mean square error (MSE) and regression analysis. The number of hidden neurons should be increased when the network training performance is poor (Saeedi et al., 2016). The
Table 4. Regression parameters for Eqs. (1)–(3). Eq. (2) (Sweere, 1990)
Eq. (3) (Wolff and Visser, 1994)
a −0.51 −1.514 −1.493 −1.109 −1.334 −1.27 −2.752 −1.679 −9.574 −0.633 −0.421 −5.59 −1.248 −2.025 −2.042 −0.951 −0.999 −2.518
CL2B1 (OMC) CL2B1 (−OMC) CL2B1 (+OMC) CL2B2 (OMC) CL2B2 (−OMC) CL2B2 (+OMC) CL2G (OMC) CL2G (−OMC) CL2G (+OMC) CL3B1 (OMC) CL3B1 (−OMC) CL3B1 (+OMC) CL3B2 (OMC) CL3B2 (−OMC) CL3B2 (+OMC) CL4B (OMC) CL4B (−OMC) CL4B (+OMC)
b 0.506 0.583 0.848 0.635 0.503 0.846 0.934 0.611 2.839 0.359 0.206 1.844 0.557 0.712 0.89 0.505 0.397 1.083
R2 0.723 0.713 0.674 0.727 0.725 0.644 0.726 0.699 0.488 0.778 0.844 0.75 0.754 0.674 0.769 0.686 0.729 0.671
a 0.329 0.015 0.214 0.167 0.011 0.307 0.003 0.007 3.6×10−9 0.095 0.035 0.003 0.055 0.004 0.077 0.102 0.015 0.073
R2 0.773 0.844 0.751 0.803 0.858 0.713 0.9 0.85 0.905 0.848 0.914 0.925 0.848 0.844 0.866 0.769 0.849 0.785
b 0.164 0.423 0.234 0.23 0.441 0.207 0.619 0.497 2.046 0.228 0.261 0.677 0.308 0.569 0.319 0.251 0.389 0.342
a 1.18 0.426 1.327 1.009 0.343 1.532 0.36 0.354 −0.342 0.57 0.275 0.572 0.614 0.337 0.939 0.727 0.324 1.072
b 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
R2 0.927 0.952 0.915 0.952 0.957 0.897 0.959 0.942 0.793 0.971 0.99 0.97 0.964 0.932 0.974 0.929 0.957 0.93
c 2.4×10−5 2.82×10−5 4.14×10−5 3.04×10−5 2.42×10−5 4.18×10−5 4.5×10−5 4.97×10−5 0 1.68×10−5 9.35×10−6 8.79×10−5 2.64×10−5 3.51×10−5 4.2×10−5 2.46×10−5 1.9×10−5 5.34×10−5
network architecture is composed of input, feed-forward sigmoid hidden and
weights are equivalent to the regression coefficients, bias is comparable to
linear output layers. As shown in Fig. 6, each neuron from the inputs was
intercept parameter, and errors are corresponding to the residuals in regression.
weighted with a proper W value. The transfer function (f) is calculated as the
Similar to the regression models in Table 4, 18 ANN models were developed
sum of the weighted inputs and the bias. Subsequently, the output was
following the architecture shown in Fig. 7. One hidden layer was used in this
generated through the transfer function. The linear output layer is generally
study, since there was no advantage to use multiple hidden layers (Tu, 1996;
used for function fitting or nonlinear regression problems (Beale et al., 2015).
Ashtiani et al., 2018). The number of hidden neurons was chosen after trying different ANN structures in terms of the number of neurons. Fig. 8 shows that five neurons in the hidden layer were found to be the most appropriate topology (i.e. lowest MSE).
Fig. 7. ANN structure for one input. Fig. 6. Neural network architecture.
2.0 1.8
factors and biases based on MSE (Saha et al., 2018). MSE is one of the most
1.6
widely used loss function, which calculates the square difference between the
1.4
actual and predicted values. For example, the MSE between the output a and the target output t in the network is defined as 1 N 1 N MSE = ∑i =1 e i2 = ∑ i =1 (t i − ai ) 2 N N
(4)
Iteration of this algorithm can then be written as X i +1 = X i − α i g i
MSE value (10−4 )
The process of training a neural network includes adjusting the weight
1.2 1.0 0.8 0.6 0.4
(5)
0.2
where Xi is a vector of current weights and biases, gi is the current gradient,
0.0
and αi is the learning rate. Iteration of Eq. (5) is required until the network converges. The training
0
1
2
3
4
5
6
7
8
9
10
Number of hidden neurons Fig. 8. ANN performance for one input model.
stops when the validation error increases for six iterations. Once the network is trained and validated, it can be used to calculate the network output for each input.
MATLAB was used to define the fitting problem. Inputs and target data were imported to the network. The dataset was subdivided randomly into 70%
Many types of ANN architecture are available, for instance, simple static
for training purposes, 15% for validation by stopping the training process
multilayer perception ANN architecture which consists of one input layer, one
before overfitting, and 15% for model generalisation. Model was trained using
or more hidden layers, and one output layer. In this section, the developed
training dataset in order to generate the weights and it was validated using data
ANN models were compared with the regression models. For this purpose, the
to keep the process independent. Individual validation dataset was arranged to
input of the ANN should be the same as the independent variable of the
assure that there are no conflict and bias with the training dataset and to
regression model (i.e. number of load cycles, N), and the output must be the
examine the capability of generalised models.
same as the dependent variable (i.e. PD) as well. In addition, the connection
The network was trained using a Levenberg-Marquardt back-propagation
ANN models performed better than the regression models as illustrated in
algorithm (i.e. optimisation algorithm), which requires more memory but less
Fig. 10. The highest R2 values were observed for ANN models, followed by the
time. This algorithm avoids most of the serious limitations (Marquardt, 1963).
regression models in Eqs. (3), (2) and (1), respectively. This finding was in
The Hessian matrix was approximated to derive the Levenberg-Marquardt
line with the previous studies which suggested a superior prediction
algorithm which is shown as
performance of the ANN than the regression models (Nazzal and Tatari, 2013;
X i +1 = X i − [ JT J + µI]−1 JTe
(6)
Alrashydah and Abo-Qudais, 2018; Ghasemi et al., 2018; Saha et al., 2018;
where X is the parameter vector, J is the Jacobian matrix, e is a vector of
Taha et al., 2018). A possible explanation for the high prediction accuracy is
network errors, I is the identity matrix, and µ is a scalar.
that the ANN with the help of the hidden neurons is capable of modelling any
After the network is trained and validated, the neural network can be used to
arbitrary complex nonlinear relationships between the inputs and outputs,
predict the output for any given input. It is worth mentioning that the tan-
whereas the regression models are only able to predict the dependent variable
sigmoid transfer function (i.e. activation function) was used in this network:
based on pre-specified relationships.
t ansig(n ) = 2 / (1 + e −2 n ) − 1
(7)
The aforementioned ANN and regression models were individually
where tansig is the tan-sigmoid transfer function, and n is a S-by-Q matrix of
developed for each case in terms of the type of the materials and the moisture
net input (column) vectors.
content, which was a time-consuming process. Therefore, the challenge
Fig. 9 shows the ANN models’ performance against the measured PD for
between the ANN and regression models was having single model, which is
the tested UGMs under different moisture contents. The data should
capable of predicting the PD of UGMs with different numbers of load cycles
congregate along the line of equity (45° dashed line) when perfect prediction is
(N), deviatoric stresses (σd), moisture contents (Mc), and gradations (i.e. Cc and
achieved, where the predicted values are equal to the measured ones. The best
Cu). To conduct such prediction using a regression model, it is needed to have
fit between outputs and targets was represented by a solid line and the
a pre-specified formula consisting of these independent variables. Developing
correlation coefficient (R2). When R2 = 1, this shows that there is a perfect
such regression models is time-consuming and not be able to model the
linear regression between outputs and targets. It can be clearly seen that the
nonlinearity of the UGMs. The next section attempts to develop a single ANN
ANN models have excellent fits for all cases with R2 greater than 0.99.
model, which is able to predict the PD of UGMs based on the aforementioned variables.
(a) CL2B1.
(b) CL2B2.
(c) CL2G.
(d) CL3B1.
(e) CL3B2.
(f) CL4B.
R2
Fig. 9. Predicted versus measured PDs (%) at OMC, −OMC, and + OMC from left to right, respectively.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Eq. (1) Eq. (2) Eq. (3) ANN
Fig. 10. Comparison of prediction accuracy between regression and ANN models.
5. ANN prediction model for flexible pavements ANN is a rapid and efficient approach to predict the pavement performance (Tutumluer and Meier, 1996; Ceylan and Gopalakrishnan, 2007; Sharma and Das, 2008; Park et al., 2009; Diab and You, 2013; Ceylan et al., 2014; Kim et al., 2014; Alrashydah and Abo-Qudais, 2018; Gu et al., 2018; Saha et al., 2018). In this section, we attempted to predict the PD of the UGMs for a range of materials using ANN modelling. Similar to Section 4, Eqs. (4), (6) and (7) were used as the loss function, optimisation algorithm, and activation function, respectively. The developed ANN architecture is depicted in Fig. 11, which
includes one input layer with five neurons (i.e. N, Mc, σd, Cc and Cu), one hidden layer with 12 neurons, and one output (i.e. PD). Fig. 12 illustrates the ANN performance at different numbers of hidden neurons.
both test and validation errors had close characteristics. Thirdly, no significant overfitting was noticed at epoch 85, where the smallest MSE was observed (best performance of the validation set). For further verification, Fig. 14b shows the error histogram of neural network performance. The histogram displays the outliers, which lay outside most of the other values in a set of data. The results show that most of the errors were distributed between −0.0304 and 0.021. The agreement between the training and test errors suggests that the proposed model showed a good generalisation capacity. However, it is not recommended to use the proposed model when the values are out of the training range. 7.1
Fig. 11. ANN structure for five inputs.
The database of the RLTT was used to derive the ANN models. Fig. 13 shows the network outputs (i.e. predicted PD) with respect to targets (i.e.
MSE value (10−3 )
6.1 5.1 4.1 3.1 2.1
measured PD) for each of training, validation, and test sets. The neural network showed a superior predictability with correlation coefficients of 0.998, 0.997 and 0.993 for training, validation, and test sets, respectively. It appears
1.1 0.1
5
6
7
that the proposed model was positively trained to compute PD as a function of
8
9
10
11
12
13
14
15
16
17
Number of hidden neurons
N, σd, Mc, Cc and Cu.
Fig. 12. ANN performance of model with five inputs.
Fig. 14a shows the performance of the proposed ANN model considering training, validation, and test errors. During each epoch, MSE values were
6. Sensitivity analysis
computed. The result was suitable for three reasons. Firstly, the final MSE was 0.936×10−3
(small
error).
Secondly,
ANN has been described as a mysterious box due to the invisible relationships of the neural network (Montano and Palmer, 2003). Demonstrating the clear relationships between inputs and outputs is not an easy task (Tu, 1996). Generally, better prediction could be observed when there are many inputs. Nevertheless, network prediction efficiency might be declined when some of the inputs are irrelevant and redundant (Getahun et al., 2018). This could result in high noise and degradation at the training phase. Garson (1991) and Milne (1995) attempted to investigate the ANN neurons using the sensitivity analysis (see Eq. (8)). This equation is also adopted in this paper to examine the effect of each input neurons on the output: L
Q ik =
N
∑ j =1| w ijv jk | /(∑r | w rj |) N L N ∑i =1 ∑ j =1| w ijv jk | /(∑r | w rj |)
(8)
where Qik is the importance factor of the inputs, wij is the weight between the ith input neuron and the jth hidden neuron, vjk is the weight between the jth hidden neuron and the kth output neuron.
Fig. 13. Comparison of measured and ANN predicted PD (%).
(a) Training process.
(b) Error histogram. Fig. 14. ANN performance for the five inputs model.
Table 5 lists the exported weights obtained by the neural network from the
behaviour using the ANN model based on the wide range of N, Mc, σd, Cc and
input neurons (i.e. N, Mc, σd, Cc and Cu) to the hidden neurons, and from the
Cu without the need for costly and time-consuming tests. However, it is not
hidden neurons to the output neuron (i.e. PD). The influence factor was
recommended to use the proposed model for values out of the training range.
calculated according to these weights. As shown in Fig. 15, it is found that N is
Sensitivity analysis was conducted to investigate the effect of each input
the most significant variable that affects the PD of UGMs with influence factor
variable on the output PD. The results showed that N is the most significant
of 41%, followed by Mc, Cc, Cu and σd.
variable that affects the PD of UGMs with influence factor of 41%, followed
Table 5. Weights exported from ANN. N Mc σd -0.224 -0.191 0.724 0.495 -0.029 -4.295 -1.868 -0.519 -4.489 0.15 0.162 -0.133 -4.985 -4.127 6.577 3.674 4.732 -0.879 -0.347 -0.391 -4.801 -0.041 0.041 -4.092 -1.876 -0.675 -4.5 3.463 -2.687 -0.415 -0.202 0.031 -1.61 -0.567 -0.871 -1.916
15% and 12%, respectively. It was concluded that the ANN method is rapid
by the influence of Mc, Cc, Cu and σd, with the influence factors of 17%, 15%, Cc -0.246 7.009 -6.184 -0.572 4.177 -3.035 -1.96 -8.221 1.877 -0.009 1.521 1.334
Cu -1.339 0.966 -4.825 -1.298 -2.258 6.262 -3.813 -1.047 2.245 0.272 3.475 1.273
PD (%) 0.874 -0.157 1.106 1.525 -0.047 -0.04 -0.339 -0.717 -1.068 0.541 1.41 -0.171
and efficient to predict PD, which could be implemented in the Austroads pavement design method. Further RLTT with different stress combinations is needed to increase the prediction range of the proposed model.
Conflict of interest The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
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Ali Alnedawi obtained his BSc and MSc degrees in Highway and Transportation Engineering from AlMustansiriya University, Iraq, in 2005 and 2012, respectively. He worked as a civil engineer from 2005 to 2015, and he is currently a PhD student at Deakin University, Australia. His research interests include pavement engineering and management, and geotechnical engineering.
S1674-7755(19)30719-X
DOI:
https://doi.org/10.1016/j.jrmge.2019.03.005
Reference:
JRMGE 603
To appear in:
Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 3 August 2018 Revised Date:
29 December 2018
Accepted Date: 7 March 2019
Please cite this article as: Alnedawi A, Al-Ameri R, Nepal KP, Neural network-based model for prediction of permanent deformation of unbound granular materials, Journal of Rock Mechanics and Geotechnical Engineering, https://doi.org/10.1016/j.jrmge.2019.03.005. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. All rights reserved.
Neural network-based prediction model for permanent deformation of unbound granular materials Ali Alnedawi a*, Riyadh Al-Ameri a, and Kali Prasad Nepal a
School of Engineering, Deakin University, Geelong, Victoria 3220, Australia
* Corresponding author E-mail [email protected] [email protected] Phone no. +61 3 52278825 Deakin University Locked Bag 20000 Geelong VIC 3220 Australia
Ali Alnedawi orcid.org/0000-0003-0018-6696 Riyadh Al-Ameri orcid.org/0000-0003-1881-1787 Kali Prasad Nepal orcid.org/0000-0001-7497-1983
Neural network-based model for prediction of permanent deformation of unbound granular materials Ali Alnedawi*, Riyadh Al-Ameri, Kali Prasad Nepal School of Engineering, Deakin University, Geelong, Victoria, 3220, Australia
ARTICLEINFO Article history: Received 3 August 2018 Received in revised form 29 December 2018 Accepted 7 March 2019 Available online Keywords: Flexible pavement design Unbound granular materials Permanent deformation (PD) Repeated load triaxial test (RLTT) Prediction models Artificial neural network (ANN)
ABSTRACT Several available mechanistic-empirical pavement design methods fail to include predictive model for permanent deformation (PD) of unbound granular materials (UGMs), which make these methods more conservative. In addition, there are limited regression models capable of predicting the PD under multi-stress levels, and these models have regression limitations and generally fail to cover the complexity of UGM behaviour. Recent researches are focused on using new methods of computational intelligence systems to address the problems, such as artificial neural network (ANN). In this context, we aim to develop an artificial neural model to predict the PD of UGMs exposed to repeated loads. Extensive repeated load triaxial tests (RLTTs) were conducted on base and subbase materials locally available in Victoria, Australia to investigate the PD properties of the tested materials and to prepare the database of the neural networks. Specimens were prepared over different moisture contents and gradations to cover a wide testing matrix. The ANN model consists of one input layer with five neurons, one hidden layer with twelve neurons, and one output layer with one neuron. The five inputs were the number of load cycles, deviatoric stress, moisture content, coefficient of uniformity, and coefficient of curvature. The sensitivity analysis showed that the most important indicator that impacts PD is the number of load cycles with influence factor of 41%. It shows that the ANN method is rapid and efficient to predict the PD, which could be implemented in the Austroads pavement design method. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. T h is is a n o p e n a c c e s s a r t ic l e u n d e r t he C C BY - N C - N D l ic e n s e ( h t t p : / / c re a t i v e c o m mo n s . o r g/ licenses/by-nc-nd/4.0/).
involve several prediction variables, for instance, deviatoric and confining 1. Introduction It is acknowledged that permanent deformation (PD) of unbound granular materials (UGMs) is the irrecoverable plastic strain, which can cause rutting in flexible pavements (Dai et al., 2007; Bodin and Kraft, 2015; Gu et al., 2016). A flexible pavement is susceptible to PD when it is exposed to long-term repeated traffic loads (Berthelot et al., 2009; Xiao et al., 2012) and especially when the applied traffic loads exceed the designed loads (Werkmeister, 2003; Austroads, 2018). In practice, distresses such as rutting could be observed in UGM layers (Haider et al., 2014; Alnedawi et al., 2017; Leiva-Villacorta et al., 2017). Rutting in base layer occurs due to a shortage of the requisite strength of UGMs (Tutumluer and Pan, 2008; Xiao et al., 2011). Stability and performance of the flexible pavement structure require an evaluation of PD (Rahman, 2015; Alnedawi et al., 2018). The mechanistic-empirical flexible pavement design methods aim to select the optimum combination of layer depth and material type with the minimum development of PD. However, the existing flexible pavement design guide in Australia, e.g. Austroads (2018), only theorises as a performance design criterion, considering the critical behaviours of the maximum vertical compressive strain at the top of the subgrade and the maximum tensile strain at the bottom of the asphalt layer. Surprisingly, the designed base and subbase layers are not included in the performance criterion. The design procedure requires that the UGMs should follow certain specifications by acquiring particular properties, which results in a conservative structural design. The same guide clearly raises the absence of PD constitutive model for UGMs in flexible pavements. Efforts are demanded when establishing or adopting a particular prediction model for specific requirements due to the complex stress–strain responses of UGMs (Werkmeister et al., 2005). Recent studies showed the necessity to develop reliable constitutive models to predict the PD behaviours under repeated loads (Pratibha et al., 2015; Rahman, 2015; Alnedawi et al., 2019a). A suitable PD model could promise a firm backbone for maintenance, design, and cost estimation (Qiao et al., 2015). These types of models could
stresses, load cycles, resilient strain, and shear strength. The common available PD models were derived from regression analysis (Gabr and Cameron, 2013; Azam et al., 2015; Rahman and Erlingsson, 2015a; Gu et al., 2017). Defining the parameters to fit a regression model requires a pre-specified relationship (Tu, 1996). There is no software available that is able to develop nonlinear models with more than two independent variables. Therefore, a wide array of linear and nonlinear equations needs to be evaluated (Romanoschi, 2017). In addition, simplicity of regression models often fails to cover the complexity of UGM behaviour. Thus, recent researches have shifted the focus towards adopting new techniques of computational intelligence systems, which are able to model any arbitrary complex relationships between the dependent and independent variables (Tu, 1996). Artificial neural network (ANN) is one of the most popular computational intelligence systems (Nazzal and Tatari, 2013), which has been acknowledged for the quick and accurate prediction of pavement layer distresses (Rahman et al., 2001). Due to the robustness and ease, ANN has been used to model many complex relationships in pavement engineering (Lee et al., 2003). Ceylan and Gopalakrishnan (2007) employed ANN to predict the fatigue life of hot-mix asphalt and vertical compressive strain on the top of the subgrade based on falling weight reflectometer (FWD) data. Nazzal and Tatari (2013) showed that the ANN-based subgrade models had better prediction coefficients for resilient modulus (Mr) than those obtained by the regression models. Sharma and Das (2008) used ANN to develop a model to predict the remaining life span of the pavement. Park et al. (2009) established an ANN model to determine Mr of UGMs using in situ stresses and material properties, and they found that the ANN model serves as a reliable and simple analytical tool. Kim et al. (2014) used ANN to predict Mr of base layers using stress state and physical properties of the materials. Similarly, Saha et al. (2018) developed ANN models to estimate Mr of UGMs from soil physical properties, with the correlation coefficient R2 greater than 0.9. Alnedawi et al. (2019b) observed the influence of loading frequency using ANN. However, ANN-based models for PD prediction are rarely reported. Several researches have been conducted to compare the regression and ANN models, some of which indicated prominent outcomes of ANN, while the
*Corresponding author. E-mail address: [email protected]
others showed insignificant differences (Hutton, 1992; Buchman et al., 1994;
Alrashydah and Abo-Qudais, 2018; Saha et al., 2018; Taha et al., 2018). It is likely that the preferences of these two types of models are highly influenced by the type of the database. Therefore, empirical comparison is required in the early stages of modelling (Tu, 1996). In this context, following the Introduction, Section 2 describes the experimental study. Then, three existing regression models and laboratory repeated load trixial test (RLTT) data are used for the regression analyses to predict the PD of UGMs. Subsequently, ANN models are developed to predict the PD using the same regression input variables. Next, the predicted PD values from the regression method are compared with the ANN values to assess the robustness of the two methods. Based on several input variables, an ANN model is proposed for the Austroads pavement design method. 2. Experimental studies 2.1. Repeated load triaxial test Developing neural network requires extensive experimental tests. Generally,
Fig. 1. Repeated load triaxial test equipment.
RLTTs are used to test PD of UGMs for material evaluation purpose (Alnedawi et al., 2018). RLTT was recognised as one of the effective laboratory tests that can imitate field conditions (Witczak and Uzan, 1988). The components of the RLTT equipment are highlighted in Fig. 1. Vertical stresses were applied in the form of trapezoidal waveform with loading
Table 1. Stress stages for PD test. Stage Confining pressure, σ3 (kPa) 1 50 2 50 3 50
Deviatoric stress, σd (kPa) 350 450 550
Number of cycles 10,000 10,000 10,000
frequency of 0.5 Hz (Alnedawi et al., 2019c). The rest period was twice the loading time (Alnedawi et al., 2019d). In the testing of protocol, three sequences of repeated deviatoric stresses (σd) with a constant confining pressure (σ3) were applied, as listed in Table 1. The testing procedure was performed as per Austroads (2007). It is worth mentioning that the stress stages in Table 1 involve typical series of repeated deviatoric stress with fixed confining pressure for the purpose of reducing the effect of confining pressure that could affect the incremental PD (Alnedawi et al., 2019e). 2.2. Materials Four different materials as shown in Fig. 2, defined by VicRoads (2011) as high-quality base/subbase materials, were experimentally investigated using RLTTs. The base UGMs were specified as Class 2 (granite and basalt), while the subbase UGMs were specified as Class 3 (basalt) and Class 4 (basalt). Two different particle size distributions were the sources of all basalt materials (Classes 2–4). Modified Proctor protocol was used to measure the maximum dry density (MDD) and optimum moisture content (OMC) in accordance with the Australian Standard AS 1289.5.2.1–2003 (2003). Fig. 3 shows the sieve analysis of the investigated UGMs, and Table 2 tabulates their physical and
Fig. 2. Local UGMs tested.
engineering properties.
Specimens were prepared at different gradations, dry densities, and moisture contents, as listed in Table 3. Two tests for each specimen were conducted. As shown in Fig. 3, the symbols B and G refer to the rock origin, i.e. basalt and granite, respectively, and the numbers (i.e. 1 and 2) after B and G refer to different gradations. 2.3. Preparation of specimens The specimen for the RLTT was prepared according to Austroads (2007), as described in Fig. 4. Specimens were compacted into 8 layers with respect to the design dry density and moisture content using modified compaction effort. The cylindrical mould consisted of three split pieces to produce specimens with height of 200 mm and diameter of 100 mm. The specimens were then enclosed in a rubber membrane and the two ends of the membrane were encircled by O rings. Table 2. Physical and engineering properties of the investigated UGMs. Material D10 D30 D50 D60 Fines < 0.075 Coefficient of (mm) (mm) (mm) (mm) mm (%) uniformity, Cu CL2B1 0.08 1.9 6 8.5 10 106.25 CL2B2 0.68 1.3 3.9 5.5 11 84.62 CL2G 0.1 1 3.2 5.2 8 52 CL3B1 0.06 1.2 3.8 5.9 12 98.33
Coefficient of curvature, Cc 5.31 4.73 1.92 4.07
USCS classification GP-GM GP-GM GW-GM GP-GC
AASHTO classification A-1-a A-1-a A-1-a A-1-a
Compaction (modified) MDD (kg/m3) OMC (%) 2260 8.6 2300 8 2310 6 2260 8.2
Plasticity index 1 1 0 3
CL3B2 CL4B
0.06 0.06
0.7 1.6
2.5 4
4.2 6
Table 3. Experimental testing design. Material Gradation Moisture content (%) -OMC OMC CL2 B1 7.6 8.6 B2 7 8 G 5 6 CL3 B1 7.2 8.2 B2 7.3 8.3
13 11
70 100
1.94 7.11
MDD (kg/m3) +OMC 9.6 9 7 9.2 9.3
GW-GC GP-GC
A-1-a A-1-b
2270 2230
8.3 9.5
3 4
CL3B2, as shown in Fig. 3. The effects of the deviatoric stress and the number of load cycles on PD were significant. For example, CL2B1 (Fig. 5a) prepared at OMC/MDD has PDs of 1.39%, 1.55% and 1.96% under deviatoric stresses
2260 2300 2310 2260 2270
of 350 kPa, 450 kPa and 550 kPa, respectively. It should be noted that the stress levels were applied according to Austroads (2007). However, simulating field condition was challenging due to the complex wheel stresses applied in
Percentage Passing % (by mass)
110 100 90 80 70 60 50 40 30 20 10 0 0.01
both lateral and vertical directions.
CL2B1 = Class 2 (basalt)_1 CL2B2 = Class 2 (basalt)_2 CL2G = Class 2 (granite) CL3B1 = Class 3 (basalt)_1 CL3B2 = Class 3 (basalt)_2 CL4B = Class 4 (basalt)
Surprisingly, subbase materials represented by CL3B1, CL3B2 and CL4B showed a PD resistance similar to base materials in specimens CL2B1, CL2B2 and CL2G. This finding highlights that the current local specification for the subbase UGMs could be highly conservative. Therefore, this paper recommends characterising the UGMs according to their mechanical responses. 3. Regression modelling
0.10
1.00 Particles Size (mm)
10.00
100.00
Regression models are used to predict the rutting depth during pavement design (Siripun et al., 2010). The regression coefficients of such models are
Fig. 3. Sieve analysis of the tested UGMs.
used for PD prediction based on the independent variables (Gu et al., 2016). In addition, these coefficients can also be used to evaluate the material response
2.4. Experimental results The PDs of the tested materials versus load cycles are depicted in Fig. 5. It
(e.g. Qiao et al., 2015; Gu et al., 2016; Qamhia et al., 2017; Romanoschi,
can be seen that the specimens CL2B1, CL2B2, CL2G, CL3B1, CL3B2 and
2017). Generally, any PD regression model has independent variables such as
CL4B at moisture content less than the optimum one tend to have lower PD in
deviatoric and confining stresses, number of applied load cycles, recoverable
comparison with those at and above OMC. Soil suction could be a possible
strain, and shear strength. The cost and time of the prediction process increase
explanation for that trend. In an unsaturated soil, the suction pulls the
as the number of the prediction variables increases (Rahman and Erlingsson,
microstructure of the particles closer together, which results in greater stability
2015b). The implication of regression analysis in this study was used in
and strength. However, compacted materials with the moisture content below
comparison with ANN models. Regression analysis was conducted based on
the OMC are not recommended due to the workability problems, which need
three models in Eqs. (1)–(3), which were developed by Barksdale (1972),
extra effort to reach the desirable dry density. In addition, excessive
Sweere (1990) and Wolff and Visser (1994), respectively. The three selected
compaction effort could lead to an early breakage of the UGM particles.
models can predict the PDs under different load cycles, which were evaluated
Gradation did not show a significant relationship with PD, which could be
and compared with the experimental results.
explained by the close gradations investigated for CL2B1, CL2B2, CL3B1 and
Fig. 4. Fabrication of specimens.
(a) CL2B1.
(b) CL2B2. 3.5 -OMC 3.0
+OMC
PD (%)
2.5
OMC
2.0 1.5 1.0 0.5 0.0
0
5000
10000 15000
20000 25000 30000
Number of load cycles, N (c) CL2G.
(d) CL3B1.
3.0
3.0
-OMC 2.5
-OMC
+OMC
2.5
+OMC
OMC
1.5
1.5 1.0
1.0
0.5
0.5 0.0
OMC
2.0
PD (%)
PD (%)
2.0
0.0
0
5000
10000 15000
20000 25000 30000
0
5000
10000 15000
20000 25000 30000
Number of load cycles, N
Number of load cycles, N (e) CL3B2.
(f) CL4B.
Fig. 5. PDs of the tested materials under different load cycles and deviatoric stresses.
PD = a + b log 10 N
(1)
model in Eq. (3) has greater regression coefficients than the other models
PD = aN b
(2)
which could efficiently predict the PD of UGMs.
PD = (cN + a)(1 − e −bN )
(3)
where PD is the permanent deformation; N is the number of load cycles; and a, b and c are the regression coefficients. The PD values of RLTT were used to generate the regression coefficients and to measure R2 for each individual specimen, as shown in Table 4. The R2 values for Eq. (1) could be categorised as fair to good fit. An improvement in the prediction was also observed in Eq. (2) with R2 values greater than 0.9 for each of CL2G (+OMC and OMC) and CL3B1 (+OMC) specimens. The best-fit was clearly obtained by Eq. (3). Surprisingly, regression models that do not adopt stress as an independent variable showed high R2 values although multiple stresses were applied during the RLTT. For instance, the model in Eq. (3) showed the most excellent goodness of fit with the maximum R2 value of 0.99. This observation can be possible that the Specimen
Eq. (1) (Barksdale, 1972)
4. Neural network modelling ANN is a powerful technique used to model complex relationships that are difficult to be described by conventional equations (Beale et al., 2015). Neural networks are constructed by simple elements working in parallel direction. The links (weights) between these elements mainly govern the network role. A neural network can be trained by regulating the values of the weights between elements to achieve specific function. The ANN procedure comprises collecting data, generating and training the neural network, and evaluating the performance using mean square error (MSE) and regression analysis. The number of hidden neurons should be increased when the network training performance is poor (Saeedi et al., 2016). The
Table 4. Regression parameters for Eqs. (1)–(3). Eq. (2) (Sweere, 1990)
Eq. (3) (Wolff and Visser, 1994)
a −0.51 −1.514 −1.493 −1.109 −1.334 −1.27 −2.752 −1.679 −9.574 −0.633 −0.421 −5.59 −1.248 −2.025 −2.042 −0.951 −0.999 −2.518
CL2B1 (OMC) CL2B1 (−OMC) CL2B1 (+OMC) CL2B2 (OMC) CL2B2 (−OMC) CL2B2 (+OMC) CL2G (OMC) CL2G (−OMC) CL2G (+OMC) CL3B1 (OMC) CL3B1 (−OMC) CL3B1 (+OMC) CL3B2 (OMC) CL3B2 (−OMC) CL3B2 (+OMC) CL4B (OMC) CL4B (−OMC) CL4B (+OMC)
b 0.506 0.583 0.848 0.635 0.503 0.846 0.934 0.611 2.839 0.359 0.206 1.844 0.557 0.712 0.89 0.505 0.397 1.083
R2 0.723 0.713 0.674 0.727 0.725 0.644 0.726 0.699 0.488 0.778 0.844 0.75 0.754 0.674 0.769 0.686 0.729 0.671
a 0.329 0.015 0.214 0.167 0.011 0.307 0.003 0.007 3.6×10−9 0.095 0.035 0.003 0.055 0.004 0.077 0.102 0.015 0.073
R2 0.773 0.844 0.751 0.803 0.858 0.713 0.9 0.85 0.905 0.848 0.914 0.925 0.848 0.844 0.866 0.769 0.849 0.785
b 0.164 0.423 0.234 0.23 0.441 0.207 0.619 0.497 2.046 0.228 0.261 0.677 0.308 0.569 0.319 0.251 0.389 0.342
a 1.18 0.426 1.327 1.009 0.343 1.532 0.36 0.354 −0.342 0.57 0.275 0.572 0.614 0.337 0.939 0.727 0.324 1.072
b 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
R2 0.927 0.952 0.915 0.952 0.957 0.897 0.959 0.942 0.793 0.971 0.99 0.97 0.964 0.932 0.974 0.929 0.957 0.93
c 2.4×10−5 2.82×10−5 4.14×10−5 3.04×10−5 2.42×10−5 4.18×10−5 4.5×10−5 4.97×10−5 0 1.68×10−5 9.35×10−6 8.79×10−5 2.64×10−5 3.51×10−5 4.2×10−5 2.46×10−5 1.9×10−5 5.34×10−5
network architecture is composed of input, feed-forward sigmoid hidden and
weights are equivalent to the regression coefficients, bias is comparable to
linear output layers. As shown in Fig. 6, each neuron from the inputs was
intercept parameter, and errors are corresponding to the residuals in regression.
weighted with a proper W value. The transfer function (f) is calculated as the
Similar to the regression models in Table 4, 18 ANN models were developed
sum of the weighted inputs and the bias. Subsequently, the output was
following the architecture shown in Fig. 7. One hidden layer was used in this
generated through the transfer function. The linear output layer is generally
study, since there was no advantage to use multiple hidden layers (Tu, 1996;
used for function fitting or nonlinear regression problems (Beale et al., 2015).
Ashtiani et al., 2018). The number of hidden neurons was chosen after trying different ANN structures in terms of the number of neurons. Fig. 8 shows that five neurons in the hidden layer were found to be the most appropriate topology (i.e. lowest MSE).
Fig. 7. ANN structure for one input. Fig. 6. Neural network architecture.
2.0 1.8
factors and biases based on MSE (Saha et al., 2018). MSE is one of the most
1.6
widely used loss function, which calculates the square difference between the
1.4
actual and predicted values. For example, the MSE between the output a and the target output t in the network is defined as 1 N 1 N MSE = ∑i =1 e i2 = ∑ i =1 (t i − ai ) 2 N N
(4)
Iteration of this algorithm can then be written as X i +1 = X i − α i g i
MSE value (10−4 )
The process of training a neural network includes adjusting the weight
1.2 1.0 0.8 0.6 0.4
(5)
0.2
where Xi is a vector of current weights and biases, gi is the current gradient,
0.0
and αi is the learning rate. Iteration of Eq. (5) is required until the network converges. The training
0
1
2
3
4
5
6
7
8
9
10
Number of hidden neurons Fig. 8. ANN performance for one input model.
stops when the validation error increases for six iterations. Once the network is trained and validated, it can be used to calculate the network output for each input.
MATLAB was used to define the fitting problem. Inputs and target data were imported to the network. The dataset was subdivided randomly into 70%
Many types of ANN architecture are available, for instance, simple static
for training purposes, 15% for validation by stopping the training process
multilayer perception ANN architecture which consists of one input layer, one
before overfitting, and 15% for model generalisation. Model was trained using
or more hidden layers, and one output layer. In this section, the developed
training dataset in order to generate the weights and it was validated using data
ANN models were compared with the regression models. For this purpose, the
to keep the process independent. Individual validation dataset was arranged to
input of the ANN should be the same as the independent variable of the
assure that there are no conflict and bias with the training dataset and to
regression model (i.e. number of load cycles, N), and the output must be the
examine the capability of generalised models.
same as the dependent variable (i.e. PD) as well. In addition, the connection
The network was trained using a Levenberg-Marquardt back-propagation
ANN models performed better than the regression models as illustrated in
algorithm (i.e. optimisation algorithm), which requires more memory but less
Fig. 10. The highest R2 values were observed for ANN models, followed by the
time. This algorithm avoids most of the serious limitations (Marquardt, 1963).
regression models in Eqs. (3), (2) and (1), respectively. This finding was in
The Hessian matrix was approximated to derive the Levenberg-Marquardt
line with the previous studies which suggested a superior prediction
algorithm which is shown as
performance of the ANN than the regression models (Nazzal and Tatari, 2013;
X i +1 = X i − [ JT J + µI]−1 JTe
(6)
Alrashydah and Abo-Qudais, 2018; Ghasemi et al., 2018; Saha et al., 2018;
where X is the parameter vector, J is the Jacobian matrix, e is a vector of
Taha et al., 2018). A possible explanation for the high prediction accuracy is
network errors, I is the identity matrix, and µ is a scalar.
that the ANN with the help of the hidden neurons is capable of modelling any
After the network is trained and validated, the neural network can be used to
arbitrary complex nonlinear relationships between the inputs and outputs,
predict the output for any given input. It is worth mentioning that the tan-
whereas the regression models are only able to predict the dependent variable
sigmoid transfer function (i.e. activation function) was used in this network:
based on pre-specified relationships.
t ansig(n ) = 2 / (1 + e −2 n ) − 1
(7)
The aforementioned ANN and regression models were individually
where tansig is the tan-sigmoid transfer function, and n is a S-by-Q matrix of
developed for each case in terms of the type of the materials and the moisture
net input (column) vectors.
content, which was a time-consuming process. Therefore, the challenge
Fig. 9 shows the ANN models’ performance against the measured PD for
between the ANN and regression models was having single model, which is
the tested UGMs under different moisture contents. The data should
capable of predicting the PD of UGMs with different numbers of load cycles
congregate along the line of equity (45° dashed line) when perfect prediction is
(N), deviatoric stresses (σd), moisture contents (Mc), and gradations (i.e. Cc and
achieved, where the predicted values are equal to the measured ones. The best
Cu). To conduct such prediction using a regression model, it is needed to have
fit between outputs and targets was represented by a solid line and the
a pre-specified formula consisting of these independent variables. Developing
correlation coefficient (R2). When R2 = 1, this shows that there is a perfect
such regression models is time-consuming and not be able to model the
linear regression between outputs and targets. It can be clearly seen that the
nonlinearity of the UGMs. The next section attempts to develop a single ANN
ANN models have excellent fits for all cases with R2 greater than 0.99.
model, which is able to predict the PD of UGMs based on the aforementioned variables.
(a) CL2B1.
(b) CL2B2.
(c) CL2G.
(d) CL3B1.
(e) CL3B2.
(f) CL4B.
R2
Fig. 9. Predicted versus measured PDs (%) at OMC, −OMC, and + OMC from left to right, respectively.
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0
Eq. (1) Eq. (2) Eq. (3) ANN
Fig. 10. Comparison of prediction accuracy between regression and ANN models.
5. ANN prediction model for flexible pavements ANN is a rapid and efficient approach to predict the pavement performance (Tutumluer and Meier, 1996; Ceylan and Gopalakrishnan, 2007; Sharma and Das, 2008; Park et al., 2009; Diab and You, 2013; Ceylan et al., 2014; Kim et al., 2014; Alrashydah and Abo-Qudais, 2018; Gu et al., 2018; Saha et al., 2018). In this section, we attempted to predict the PD of the UGMs for a range of materials using ANN modelling. Similar to Section 4, Eqs. (4), (6) and (7) were used as the loss function, optimisation algorithm, and activation function, respectively. The developed ANN architecture is depicted in Fig. 11, which
includes one input layer with five neurons (i.e. N, Mc, σd, Cc and Cu), one hidden layer with 12 neurons, and one output (i.e. PD). Fig. 12 illustrates the ANN performance at different numbers of hidden neurons.
both test and validation errors had close characteristics. Thirdly, no significant overfitting was noticed at epoch 85, where the smallest MSE was observed (best performance of the validation set). For further verification, Fig. 14b shows the error histogram of neural network performance. The histogram displays the outliers, which lay outside most of the other values in a set of data. The results show that most of the errors were distributed between −0.0304 and 0.021. The agreement between the training and test errors suggests that the proposed model showed a good generalisation capacity. However, it is not recommended to use the proposed model when the values are out of the training range. 7.1
Fig. 11. ANN structure for five inputs.
The database of the RLTT was used to derive the ANN models. Fig. 13 shows the network outputs (i.e. predicted PD) with respect to targets (i.e.
MSE value (10−3 )
6.1 5.1 4.1 3.1 2.1
measured PD) for each of training, validation, and test sets. The neural network showed a superior predictability with correlation coefficients of 0.998, 0.997 and 0.993 for training, validation, and test sets, respectively. It appears
1.1 0.1
5
6
7
that the proposed model was positively trained to compute PD as a function of
8
9
10
11
12
13
14
15
16
17
Number of hidden neurons
N, σd, Mc, Cc and Cu.
Fig. 12. ANN performance of model with five inputs.
Fig. 14a shows the performance of the proposed ANN model considering training, validation, and test errors. During each epoch, MSE values were
6. Sensitivity analysis
computed. The result was suitable for three reasons. Firstly, the final MSE was 0.936×10−3
(small
error).
Secondly,
ANN has been described as a mysterious box due to the invisible relationships of the neural network (Montano and Palmer, 2003). Demonstrating the clear relationships between inputs and outputs is not an easy task (Tu, 1996). Generally, better prediction could be observed when there are many inputs. Nevertheless, network prediction efficiency might be declined when some of the inputs are irrelevant and redundant (Getahun et al., 2018). This could result in high noise and degradation at the training phase. Garson (1991) and Milne (1995) attempted to investigate the ANN neurons using the sensitivity analysis (see Eq. (8)). This equation is also adopted in this paper to examine the effect of each input neurons on the output: L
Q ik =
N
∑ j =1| w ijv jk | /(∑r | w rj |) N L N ∑i =1 ∑ j =1| w ijv jk | /(∑r | w rj |)
(8)
where Qik is the importance factor of the inputs, wij is the weight between the ith input neuron and the jth hidden neuron, vjk is the weight between the jth hidden neuron and the kth output neuron.
Fig. 13. Comparison of measured and ANN predicted PD (%).
(a) Training process.
(b) Error histogram. Fig. 14. ANN performance for the five inputs model.
Table 5 lists the exported weights obtained by the neural network from the
behaviour using the ANN model based on the wide range of N, Mc, σd, Cc and
input neurons (i.e. N, Mc, σd, Cc and Cu) to the hidden neurons, and from the
Cu without the need for costly and time-consuming tests. However, it is not
hidden neurons to the output neuron (i.e. PD). The influence factor was
recommended to use the proposed model for values out of the training range.
calculated according to these weights. As shown in Fig. 15, it is found that N is
Sensitivity analysis was conducted to investigate the effect of each input
the most significant variable that affects the PD of UGMs with influence factor
variable on the output PD. The results showed that N is the most significant
of 41%, followed by Mc, Cc, Cu and σd.
variable that affects the PD of UGMs with influence factor of 41%, followed
Table 5. Weights exported from ANN. N Mc σd -0.224 -0.191 0.724 0.495 -0.029 -4.295 -1.868 -0.519 -4.489 0.15 0.162 -0.133 -4.985 -4.127 6.577 3.674 4.732 -0.879 -0.347 -0.391 -4.801 -0.041 0.041 -4.092 -1.876 -0.675 -4.5 3.463 -2.687 -0.415 -0.202 0.031 -1.61 -0.567 -0.871 -1.916
15% and 12%, respectively. It was concluded that the ANN method is rapid
by the influence of Mc, Cc, Cu and σd, with the influence factors of 17%, 15%, Cc -0.246 7.009 -6.184 -0.572 4.177 -3.035 -1.96 -8.221 1.877 -0.009 1.521 1.334
Cu -1.339 0.966 -4.825 -1.298 -2.258 6.262 -3.813 -1.047 2.245 0.272 3.475 1.273
PD (%) 0.874 -0.157 1.106 1.525 -0.047 -0.04 -0.339 -0.717 -1.068 0.541 1.41 -0.171
and efficient to predict PD, which could be implemented in the Austroads pavement design method. Further RLTT with different stress combinations is needed to increase the prediction range of the proposed model.
Conflict of interest The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
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Ali Alnedawi obtained his BSc and MSc degrees in Highway and Transportation Engineering from AlMustansiriya University, Iraq, in 2005 and 2012, respectively. He worked as a civil engineer from 2005 to 2015, and he is currently a PhD student at Deakin University, Australia. His research interests include pavement engineering and management, and geotechnical engineering.