Quantifying the Influence of Geosynthetics on Pavement Performance (2017)

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N-1 APPENDIX N. DEVELOPMENT OF ARTIFICIAL NEURAL NETWORK MODELS FOR PREDICTING GEOSYNTHETIC-REINFORCED PAVEMENT PERFORMANCE The current Pavement ME Design software predicts pavement performance based on the computed critical pavement responses from a linear isotropic and layered elastic program. In other words, the determination of critical pavement responses is the key to forecasting pavement performance. The finite element models developed in this project are sufficiently accurate to compute the critical responses of geosynthetic-reinforced pavement structures. However, these models were developed using the software ABAQUS, which is not compatible with the Pavement ME Design embedded software DARWin-ME. Furthermore, replacing the current Pavement ME Design software with the developed finite element models to compute the critical responses of the arbitrary user-inputted geosynthetic-reinforced pavement structures is impractical at the moment. Therefore, there is a need to predict the responses of any given geosynthetic-reinforced pavement structure based on computation by the developed finite element models for a wide range of geosynthetic-reinforced pavement structures. To satisfy this need, the artificial neural network (ANN) approach is used to predict the critical responses of geosynthetic-reinforced pavement structures. The ANN models allow establishing the correlations between the input variables, iX , and the output variables, jY , through the inter-connected neurons (i.e., weight factor, jiw ) (1). Note that the input variables, iX , and the output variables, jY , are usually normalized to ix and jy , respectively, which are the values between 0 and 1. Herein, the output variables, jY , represent the computed critical pavement responses, including the tensile strain at the bottom of the asphalt concrete, and the compressive strain within the asphalt concrete, base layer, and subgrade. The selection of the input parameters, iX , is based on the sensitivity analysis of the developed finite element models. The identified input parameters to the ANN model include the layer thickness, the modulus of the paving material, the location of the geosynthetic, and the type of geosynthetic. The correlations developed by the ANN models between the normalized input parameters, ix , and the normalized output variables, jy , are shown in Equation N-1. 1 n j ji i i y f w x =     =  (N-1) where f is a transfer function, which normally uses a sigmoidal, Gaussian, or threshold functional form; and jiw is the unknown weight factors. Developing a neural network model specifically requires the determination of the weight factors, jiw , in Equation 1. The ANN model determines these weight factors, jiw , through two major functions: training and validating. The training dataset is used to determine the trial weight factors, jiw , and the validating dataset is employed to examine the accuracy of the model prediction. A robust ANN model normally

N-2 requires a large database of input and output variables (2). Thus, generating the input and output variables database is the first step in developing the ANN model. Experimental Computational Plan for ANN Models To generate the database of the numerical model inputs and the corresponding computed critical pavement responses, the computation of multiple cases was performed based on the developed geosynthetic-reinforced and unreinforced finite element models. Tables N-1 and N-2 show the selected input parameters as well as their values for the geosynthetic-reinforced pavement structures and the corresponding unreinforced pavement structures, respectively. Based on these experimental computational plans, the number of the computed geosynthetic- reinforced pavement models was 5832, and the number of the computed unreinforced pavement models was 486. As shown in Table N-1, two geosynthetic types (geogrid and geotextile) and two geosynthetic locations (middle and bottom of base course) were taken into account in the computation of the multiple cases. The pavement responses database was divided into five categories, including • The geogrid placed in the middle of the base layer (GG-M). • The geogrid placed at the bottom of the base layer (GG-B). • The geotextile placed in the middle of the base layer (GT-M). • The geotextile placed at the bottom of the base layer (GT-B). • The unreinforced one (NG). Each category of pavement response database corresponded to one set of neural network models. Table N-1. Selected Input Parameters for Geosynthetic-Reinforced Pavement Structures Influential Factors Level Input Values Load Magnitude 1 9 kip HMA Thickness 3 2, 4, and 6 inches HMA Modulus 3 300, 450, and 600 ksi Base Thickness 3 6, 10, and 15 inches Base Vertical Modulus 3 20, 40, and 60 ksi Base Anisotropic Ratio 2 0.35 and 0.45 Geosynthetic Location 2 Middle and Bottom of Base Course Geosynthetic Type 2 Geogrid and Geotextile Geogrid Sheet Modulus 3 1200, 2400, and 3600 lb/in Geotextile Sheet Modulus 3 1800, 3600, and 5400 lb/in Subgrade Modulus 3 5, 15, and 25 ksi Note: The number of total cases was 5832.

N-3 Table N-2. Selected Input Parameters for Unreinforced Pavement Structures Influential Factors Level Input Values Load Magnitude 1 9 kip HMA Thickness 3 2, 4, and 6 inches HMA Modulus 3 300, 450, and 600 ksi Base Thickness 3 6, 10, and 15 inches Base Vertical Modulus 3 20, 40, and 60 ksi Base Anisotropic Ratio 2 0.35 and 0.45 Subgrade Modulus 3 5, 15, and 25 ksi Note: The number of total cases was 486. Selection of ANN Algorithms A three-layered neural network architecture consisting of one input layer, one hidden layer, and one output layer was constructed as shown Figure N-1. The input parameters are listed in Tables 1 and 2, except the geosynthetic location and the geosynthetic type. The output variables were the critical pavement responses, including the tensile strain at the bottom of the asphalt concrete, and the compressive strains within the asphalt concrete, base course, and subgrade. The hidden layer assigned 20 neurons to establish the connection between the output layer and the input layer. In this study, the transfer function used a sigmoidal functional form, which is shown in Equation N-2 (3). ( ) ( ) 1 1 expi i f I Iϕ = + − (N-2) where iI is the input quantity; and ϕ is a positive scaling constant, which controls the steepness between the two asymptotic values 0 and 1. The constructed neural network structure was programmed using the software MATLAB R2013a (4). The training algorithm used the Levenberg-Marquardt back-propagation method to minimize the mean squared error (MSE) (5). The gradient descent weight function was employed as a learning algorithm to adjust the weight factors, jiw (6).

N-4 Figure N-1. Illustration of Three-Layered Neural Network Architecture Prediction of Pavement Performance The pavement response database was first randomly divided into a training dataset and a validating dataset as the ratio of 80 percent and 20 percent, respectively. The training dataset was used to determine the weight factors, jiw , and the validating dataset was employed to examine the prediction accuracy of the developed neural network. Figures N-2–N-26 show the comprehensive comparisons between the finite element model computed pavement responses and the ANN model predicted pavement responses for the geosynthetic-reinforced and unreinforced pavement structures. The ANN models accurately predicted all of the pavement responses from the validating dataset after the training process. The developed ANN models can be used to interpolate the critical responses of any given geosynthetic-reinforced pavement structure. Input Layer Hidden Layers Output Layer x1 x2 xn yi x3 20 Neurons Sigmoidal Transfer Function Back Propagation Error

N-5 Figure N-2. Comparison of Tensile Strain at the Bottom of the Asphalt Concrete for GG-M Structure

N-6 Figure N-3. Comparison of Average Vertical Strain in the Asphalt Concrete for GG-M Structure

N-7 Figure N-4. Comparison of Average Vertical Strain in the Base Layer for GG-M Structure

N-8 Figure N-5. Comparison of Vertical Strain at the Top of the Subgrade for GG-M Structure

N-9 Figure N-6. Comparison of Vertical Strain at 6 inches below the Top of the Subgrade for GG-M Structure

N-10 Figure N-7. Comparison of Tensile Strain at the Bottom of the Asphalt Concrete for GG-B Structure

N-11 Figure N-8. Comparison of Average Vertical Strain in the Asphalt Concrete for GG-B Structure

N-12 Figure N-9. Comparison of Average Vertical Strain in the Base Layer for GG-B Structure

N-13 Figure N-10. Comparison of Vertical Strain at the Top of the Subgrade for GG-B Structure

N-14 Figure N-11. Comparison of Vertical Strain at 6 inches below the Top of the Subgrade for GG-B Structure

N-15 Figure N-12. Comparison of Tensile Strain at the Bottom of the Asphalt Concrete for GT-M Structure

N-16 Figure N-13. Comparison of Average Vertical Strain in the Asphalt Concrete for GT-M Structure

N-17 Figure N-14. Comparison of Average Vertical Strain in the Base Layer for GT-M Structure

N-18 Figure N-15. Comparison of Vertical Strain at the Top of the Subgrade for GT-M Structure

N-19 Figure N-16. Comparison of Vertical Strain at 6 inches below the Top of the Subgrade for GT-M Structure

N-20 Figure N-17. Comparison of Tensile Strain at the Bottom of the Asphalt Concrete for GT-B Structure

N-21 Figure N-18. Comparison of Average Vertical Strain in the Asphalt Concrete for GT-B Structure

N-22 Figure N-19. Comparison of Average Vertical Strain in the Base Layer for GT-B Structure

N-23 Figure N-20. Comparison of Vertical Strain at the Top of the Subgrade for GT-B Structure

N-24 Figure N-21. Comparison of Vertical Strain at 6 inches below the Top of the Subgrade for GT-B Structure

N-25 Figure N-22. Comparison of Tensile Strain at the Bottom of the Asphalt Concrete for NG Structure

N-26 Figure N-23. Comparison of Average Vertical Strain in the Asphalt Concrete for NG Structure

N-27 Figure N-24. Comparison of Average Vertical Strain in the Base Layer for NG Structure

N-28 Figure N-25. Comparison of Vertical Strain at the Top of the Subgrade for NG Structure

N-29 Figure N-26. Comparison of Vertical Strain at 6 inches below the Top of the Subgrade for NG Structure References 1. Haykin, S.S. (1999). Neural Networks: A Comprehensive Foundation. Prentice Hall, Upper Saddle River, NJ. 2. Wu, Z., Hu, S., and Zhou, F. (2014). Prediction of Stress Intensity Factors in Pavement Cracking with Neural Networks Based on Semi-Analytical FEA. Expert Systems with Applications, Vol. 41, pp. 1021–1030.

N-30 3. Ceylan, H., Bayrak, M.B., and Gopalakrishnan, K. (2014). Neural Networks Applications in Pavement Engineering: A Recent Survey. International Journal of Pavement Research and Technology, Vol. 7, No. 6, pp. 434–444. 4. Demuth, H., and Beale, M. (1998). Neural Network Toolbox for Use with MATLAB. The MathWorks, Natick, MA. 5. More, J. (1978). The Levenberg-Marquardt Algorithm: Implementation and Theory. Numerical Analysis, Vol. 630, pp. 105–116. 6. Amari, S. (1998). Natural Gradient Works Efficiently in Learning. Neural Computation, Vol. 10, No. 2, pp. 251–276.