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Page Benkelman Beam Rebound AC Overlay Model JOINT CSHRP/MANITOBA BAYESIAN APPLICATION 2.0 INTRODUCTION 2.1 What was Manitoba's Problem Statement? The Manitoba Department of Highways & Transportation (MDHT) attempted to model and predict the immediate and long term maximum Benkelman Beam Rebound BBRmax, or deflection, of a flexible pavement after an asphaltic concrete (AC) overlay.
The intent of this general model was to predict the BBRao immediately after the overlay, and the BBRot, over time, to its terminal service life. What is a Benkelman Beam Rebou nd? The Benkelman Beam Rebound is a measure of pavement strength and simulates the rebound of a pavement under an applied load. MDHT currently measures, and uses, Benkelman Beam Rebound values for pavement rehabilitation and for spring load restriction controls throughout the Province. For rehabilitation design purposes, Benkelman Beam Rebounds are the maximum spring rebound readings measured in the outer wheel path of a pavement lane. The maximum BBR over a length of pavement is averaged from a series of readings. The rebound value is calculated as the mean reading, plus two standard deviations. BBR values are temperature corrected to 20°C. 2.2 WHY THE NEED FOR A MODEL? The need for MDHT to develop a design verification model was clear. The BBROverlay model was developed for three reasons.
2.3 WHO WERE THE TEAM MEMBERS AND EXPERTS? There was one lead analyst and two supporting members of Manitoba's Bayesian analysis team. They are as follows:
Three experts from the Manitoba Department of Highways & Transportation (MDHT) contributed their time and talents to encoding the matrices and providing their expertise to the project. The experts had a combined 87 years of design and field experience to offer to this project. They were:
There was minimal direct involvement by the University of Manitoba in this project. However, the Civil Engineering Department was informed of the project and a Civil Engineering Professor was invited to, and attended, the CSHRP training session held in Winnipeg in November, 1994. 3.0 BAYESIAN METHODOLOGY What was the Basic Methodology used? The methodology used in this project followed the ten basic steps outlined in the CSHRP Project Statement. They are:
3.1 WHAT WAS THE MODEL SELECTION? MDHT's original intent was to model BBRmax immediately after overlay, and BBRmax for each year to its terminal service life, as one single expression. However, it soon became difficult to accurately model both the immediate (initiation), and the long term (propagation), BBRmax as a single model. Based on recommendations from the consultant, VEMAX, and with agreement among the experts, the model format was changed from the original submitted in November of 1994, and the expression was separated into a 2stage model. The first model, Model 1, would predict the initial rebound immediately after overlay, and the second model , Model 2, would predict the rebound over time. Model 1 Model and predict the BBRmax immediately after overlay. Model 2 Model and predict the BBRmax each year, to the life of the overlay. 3.2 WHAT WAS THE 2STAGE MODEL DEFINITION, DEPENDENT & INDEPENDENT VARIABLES? Model 1 Model 1 was the firststage initiation model developed to predict the reduction in the maximum BBR of a flexible pavement immediately after overlay. Based on discussions with the Experts, it was considered that the BBRmax immediately after overlay was a function of the overlay thickness, the rebound before the overlay, and the design traffic, expressed in equivalent single axle loads (ESAL), for the overlay. Model 1 consisted of a dependent variable and three independent variables. The functional form of Model 1 is defined as follows:
Model 2 The secondstage Model 2 was the propagation model developed to predict the rebound of an overlay over time. The Experts indicated that the maximum BBR over time was considered a function of the overlay thickness, the rebound immediately after the overlay (modeled and predicted by Model 1), the age of the overlay, the cumulative traffic, expressed in equivalent single axle loads (ESAL), and an environmental factor expressed in annual precipitation. Precipitation was considered by the experts as a proxy for subgrade moisture content. Model 2 consisted of a dependent variable and five independent variables. The functional form of Model 2 was defined as follows:
3.3 WHAT IS MODEL TYPE & FUNCTIONAL FORM? The model type selected for both Models 1 and 2 is empirical. As empirical models, the functional form selected for both models was curvilinear. This curvilinear form for the coefficients satisfied the current restrictions of the XLBayes software (ie only linear functions can be analyzed), while still capturing the relationship that strong and weak pavements behaved differently, and that overlays with low rebound values are affected to a lesser extent by the independent variables than overlays with high rebound values. Semilog equations were selected for both Models 1 and 2. The equations in their preferred linear form, are presented as follows:
3.4 WHAT WERE THE MODEL INPUTS? Data Sources Only field data from the CSHRP long term pavement performance (CLTPP) and SHRP Specific Pavement Studies (SPS5) sites of asphalt concrete overlay of asphalt concrete pavement in Manitoba were used in this analysis. CSHRP and SHRP data was selected for this project because of its accuracy and due to the lack of a good historical rebound data within MDHT. Since the data from the CSHRP and SPS5 sites was only 23 years old, and only two sites were available in Manitoba ( three test sections at the CSHRP site and eight test sections at the SPS5 site), this data source was considered small as well as immature. However, this immature and small dataset was considered a good test for Bayesian analysis. Data Processing: The data for each model was tabulated and preliminary scatter plots of each model's dependent variable, verses each independent variable, were developed. If the dependent variable was found to be correlated to each independent variable then the independent variable was considered to be good for the analysis. If independent variables were found to be correlated to each other, then one of the independent variable was dropped and the other used in the analysis. As an example, preliminary scatter plots showed no difference in the rebound value for milled vs nomilled overlay, nor for recycled vs virgin overlays. Therefore, differentiating between sections that were milled or recycled became unnecessary. Likewise, the structural number of each pavement section was found to be highly correlated to the overlay thickness, and structural number was dropped as a potential variable in this analysis. Surprisingly, the two independent variables, overlay thickness and cumulative traffic, were not highly correlated. Prior Data Sources: Three MDHT experts with experience in overlay design and rehabilitation were selected to encode their judgements using the full matrix orthogonal approach. An encoding package was developed using the format suggested by CSHRP and given to the experts. The experts had very little problems providing responses for the matrices. A copy of the encoding package and the experts' responses is attached in Appendix A. For the analysis, the expert data was not combined. Transformations of Field and Expert Data Field and expert data was tabulated and transformed into semilog tables to be analysis using XLBAYES. The original, and transformed, field and expert data is shown in Appendix B. There were no major problems developing data statistics or compatibility problems between the field and expert data. However, as indicated previously the field database was small and the data was immature. 4.0 ANALYSIS The XLBAYES software was used to develop coefficients and statistical parameters for the combination of field and expert data in Model 1 and Model 2. With the developed coefficients, sensitivity analyses were run to determine the magnitude of the effects of each independent variable on the predicted BBR value. The sensitivity analysis was performed and compared between experts. The statistical output and probability graphs for each model, expert, and each iteration, is presented in Appendix C, D, E, and F. The models were evaluated based on the rationality of the sign and magnitude of the individual regression coefficients, the statistical significance of the coefficients, and the standard error of the model. The models were further evaluated to determine which information source (ie. Prior or Field Data) the Posterior reflected. 4.1 MODEL ITERATIONS & RESULTS & GENERAL FINDINGS Two iterations were performed for each model, and each expert, and the results analysed. The general findings of the analysis are discussed below. Model 1, Iteration 1 Expert and field data were used to generate coefficients and statistical parameters for Prior, Data, and Posterior equations for Model 1. The first iteration of Model 1 resulted in the cumulative traffic variable with an incorrect sign in the Data equation, and a difference between experts on the significance of cumulative traffic in Model 1. The analysis also indicated that the Posterior equations predominantly reflected the Prior information in the first iteration of Model 1. An evaluation of the statistical output of the first iteration of Model 1, and a comparison table of each expert's equations are summarized in Tables 1 and 2, respectively. In order to improve the results of the first iteration of Model 1, it was suggested by VEMAX at the Spring workshop in Ottawa, that the cumulative traffic variable be dropped. This was due to the fact that in the first stage model, there is no accumulation of traffic immediately after the overlay. After further consultation with the experts, traffic was removed and a second iteration of the model was performed. Iteration 2 A second iteration of Model 1 was performed without the independent traffic variable, and new coefficients and statistical parameters were calculated. The second iteration of Model 1 had regression coefficients which were rational in both sign and magnitude, and were statistically significant. The residual variance of the first and second iterations were similar. The evaluation of the statistical output of the second iteration of Model 1, and a comparison table of each expert's equations are summarized in Tables 3 and 4, respectively. The second iteration of Model 1 was considered to be superior than the first, and consequently, the second iteration of Model 1 was adopted by MDHT. Final Model 1 Selection Since the second iteration of Model 1 was adopted by MDHT , the three expert Posterior equations for the second iteration of Model 1 are shown below. It is recognized that if these models were to be used as a design tool, ultimately a single expert's equation would have to be selected. However, due to time and resource constraints, this final step was not taken.
Model 2, Iteration 1 Prior, Data, and Posterior model coefficients and statistical parameters were generated using the field and expert data in the first iteration of Model 2. The results of the first iteration for Model 2 produced incorrect signs for the cumulative traffic and annual precipitation variables in the Data model. In addition, two of the three experts' Prior models indicated that the overlay thickness variable was not statistically significant. Once again the Posterior models predominantly reflected the Prior information. An evaluation of the statistical output of the first iteration of Model 2, and a comparison table of each expert's equations are summarized in Tables 5 and 6, respectively. In order to improve the results of Model 2 after the first iteration, the experts were consulted to review the significance and signs of the independent variables. Since overlay thickness was not statistically significant, the experts agreed that the thickness variable should be dropped. A second iteration of Model 2 was run without the thickness variable. Iteration 2 The second iteration of Model 2 was run without the independent overlay thickness variable. The second iteration of Model 2 produced models with all variables in the Posterior and Prior equations having the correct sign, rational magnitudes, and similar residual variances as the first iteration. The Data model however still had an incorrect sign for the traffic and precipitation variables. The evaluation of the statistical output of the second iteration of Model 2, and a comparison table of each expert's equations are summarized in Tables 7 and 8, respectively. It was suggested, by VEMAX, that the incorrect sign in the Data model may be a function of the small and premature data set, and that both variables should be maintained in the model since the experts collectively agreed on their importance. The continued collection of field data at the sites may result in a larger database which could cause a "natural" correction of the signs. No further iteration of Model 2 was performed. Final Model 2 Selection Since the second iteration of Model 2 was considered to be superior to the first, the second iteration of Model 2 was adopted by MDHT. The three expert Posterior equations for the second iteration of Model 2 are shown below. Once again, it is recognized that if these models were to be used as a design tool, ultimately a single expert's equation would have to be selected.
GENERAL FINDINGS The general fit of the models to the field and expert data in both Models 1 and 2 appeared to be reasonable, and all experts agreed on the magnitude and significance of the variables. From the analysis, the Posterior model usually reflected the Prior and not the field Data model. This is due to the fit of the data to the functional form selected and the small sample size of the field database. 4.2 SENSITIVITY ANALYSIS GENERAL FINDINGS In order to determine the effects of the independent variables on the dependent variable in each model, sensitivity analysis of the predictions and the change in degree of freedom were determined and analysed. The prediction sensitivity analyses, shown in Figures 18, clearly indicate which variables contribute significantly to the predictions made by the models. On the second iteration of each model, the experts agreed on the magnitude and sensitivity for each variable. The predictions generated by the Data model were less sensitive to changes in the variable settings than the Prior or the Posterior models. The effect of varying the assumption on the degree of freedom (dof) for Expert l's Priors (eg. Model 1Iteration 1 and Model 2Iteration 2) is reflected in the change in standard errors for the Posterior models, as shown in Figures 910. Increasing the dof on the Prior reduces the standard error of the Posterior predictions. This implies that the fit of the functional form is better on the Prior data than on the field data. Conservative estimate of the base case dof for the Prior results in conservative estimates of the standard error for the Posterior model. 5.0 DISCUSSION AND CONCLUSIONS 5.1 CONCLUSION The twostage model to predict BBRoverlay model using the Bayesian approach produced relatively good BBR prediction equations. The Prior, Data, and Posterior models generated from the analysis showed good statistical parameters and clearly indicated the importance and magnitude of the impacts of each variable on the BBR prediction. The sensitivity analysis was very useful and indicated the magnitude and significance of each variable on the BBR prediction. The Bayesian analysis successfully used expert judgement information in conjunction with limited field data to produce useable first models. 5.2 GENERAL IMPRESSIONS/SATISFACTION WITH BAYESIAN METHODOLOGY Bayesian Methodology: The Bayesian methodology and its process appeared to have merit in the development and analysis of performance models. From the Manitoba application it was possible to see the effects and significance of each independent variable on the dependent BBR variable. It was also possible to supplement small data sets with expert judgement as a first step to developing useable expressions. However from the Manitoba analysis, it was not clear that a small data set had significant influenced or could be manipulated to influence the Posterior model. It appeared that a small, but good data set, would always be dominated by the Prior in the Bayesian process. Despite the above comment, the Bayesian application appeared to be a useful tools for developing experiments and determining significant variables. Software: The Bayesian software XLBAYES was relatively easy to use, if one knew and had worked in EXCEL. The plots of the graphs were useful and it was important that files were dynamic and therefore changes could easily be made. Some parts of the analysis appeared to have the "black box" syndrome. It was difficult to verify if the output of the software was valid or the weights placed on field or Expert data. It was also felt that the statistical output should have included an R^2 value as an indicator of the "goodness of fit" for the Prior and Posterior models. Joint Applications: The joint applications provided Manitoba with a good opportunity to evaluate a real world model with Bayesian methodology and compare the process with other Provincial agencies. There was good feedback from the Consultant during the process and at the Workshop in Ottawa. 5.3 FUTURE MODELLING NEEDS/DIRECTION? As more field data is collected, the Manitoba Department of Highways & Transportation will continue to update the developed 2stage models using the Bayesian methodology. In addition, we plan to compare the results and of the predictions from the model to the Department's design equation. This will assist in verification of the design results. It is also hoped that the Bayesian approach may be evaluated within other Branches of the Manitoba Department of Highways. References: (1) Manitoba Department of Highways "Pavement Design Manual" (2) Report by Alberta Transportation Council of Asphaltic Concrete on granular base courses. (3) "Bayesian Methodology for Verifying Recommendations to Minimize Asphalt Pavement Distress." NCHRP Report 213, June 1979 (4) "Strains due to Load in Frozen and Thawed Flexible Pavements." University of Minnesota (5) Training Manual in Bayesian Methods and Software. CSHRP, 1994 |
