LOCAL CALIBRATION OF PAVEMENT-ME PERFORMANCE MODELS USING MAXIMUM LIKELIHOOD ESTIMATION By Rahul Raj Singh A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Civil Engineering – Doctor of Philosophy 2024 ABSTRACT The mechanistic empirical pavement design guide (MEPDG) is a state-of-the-art design approach that incorporates material properties, traffic, and climate to estimate the incremental damage using mechanical responses of the pavement. The cumulative damage is used to predict the field distress using empirical transfer functions. The Pavement-ME transfer functions have been nationally calibrated using long-term pavement performance (LTPP) pavement sections and other experimental test section data such as MnRoad. These nationally calibrated models may not represent the construction practices, materials, and climatic conditions of a particular state/region. Studies have calibrated the Pavement-ME transfer functions using the least squares method. Least squares is a widely used simplistic method based on the normal independent and identically distributed (NIID) assumption. Literature shows that these assumptions may not apply to non-normal distributions. This study introduces a new methodology for calibrating the bottom-up cracking, total rutting, and international roughness index (IRI) models in new flexible pavements and the transverse cracking and IRI models in new rigid pavements using Maximum Likelihood Estimation (MLE). The approach in this study includes MLE using synthetic and observed data, and the results are compared with those of the least squares approach. The MLE and least squares methods were also combined with resampling techniques to improve the robustness of calibration coefficients. The data are analyzed from the Michigan Department of Transportation's (MDOT) Pavement Management System (PMS) database to obtain the pavement sections and observed performance data for calibration. Despite several calibration efforts, limited research is available on the impact of calibration on pavement design. The calibrated models using the least squares method were then used for pavement design to estimate the calibration effects and compare them with AASHTO93 designs. Based on the newly calibrated coefficients, 44 new flexible and 44 rigid sections were designed. This study also identifies the controlling distresses for pavement design. It is often not viable to calibrate all coefficients at the same time. Therefore, it is crucial to identify the most sensitive transfer function coefficients. Moreover, the sensitivity also indicates the impact of each coefficient on the performance prediction. Typically, the sensitivity is obtained using a normalized sensitivity index (NSI). This study estimated the sensitivity of the Pavement-ME transfer function coefficients using scaled sensitivity coefficients (SSCs). The results show that MLE outperformed the least squares method for non-normally distributed data, such as transverse cracking and bottom-up cracking models for synthetic and observed data. Using the calibrated models for pavement design showed that, on average, the surface thicknesses using locally calibrated coefficients were thinner by 0.22 and 0.44 inches for flexible and rigid pavements, respectively. Critical design distresses for flexible pavements include bottom-up and thermal cracking. On the other hand, transverse cracking and IRI control the designs for rigid sections. The sensitivity of Pavement-ME model coefficients showed that SSCs provide a more reliable sensitivity on a range of independent variables rather than a point estimate, unlike NSI. Overall, this study helps improve the calibration process for local conditions. This dissertation is dedicated to my parents, Mr. Dinesh Singh and Mrs. Ranjana Singh, and my sister, Shweta Rani. iv ACKNOWLEDGMENTS First, I would like to extend my deepest gratitude to my advisor, Dr. Syed Waqar Haider, for his continuous support and guidance. His experience, thorough knowledge, patience, and involvement in every step of this project helped me complete this thesis successfully and on time. I want to express my gratitude to Dr. M. Emin Kutay for his guidance throughout my Ph.D. journey. I was fortunate to have taken a course with Dr. Kirk D. Dolan, for which I am thankful to him. It helped me build a foundation in parameter estimation, publish papers, and implement it into my thesis. I would also like to thank Dr. Karim Chatti for his continuous feedback and for serving on my Ph.D. committee. I want to thank my parents, Mr. Dinesh Singh and Mrs. Ranjana Singh, and my sister, Shweta Rani, for their immense love and support. I would also like to thank my friends Komal Narendrakumar Thakkar and Divyang Narendrakumar Thakkar for making my journey to the United States easier. I have been blessed to have worked with some brilliant minds in the Department of Civil and Environmental Engineering at Michigan State University. I want to thank Hamad Muslim, Mumtahin Hasnat, Faizan Lali, Mahdi Ghazavi, Poornachandra Vaddy, Farhad Abdollahi, and many others for being part of my academic journey. Finally, I would like to thank God for giving me the strength and this opportunity to be a part of such a prestigious institute and work with such kind and generous people. v TABLE OF CONTENTS CHAPTER 1 - INTRODUCTION .................................................................................................. 1 CHAPTER 2 - LITERATURE REVIEW ....................................................................................... 6 CHAPTER 3 - DATA FOR CALIBRATION .............................................................................. 36 CHAPTER 4 - METHODOLOGY ............................................................................................... 78 CHAPTER 5 - RESULTS AND DISCUSSION ........................................................................ 107 CHAPTER 6 - CONCLUSIONS, RECOMMENDATIONS AND FUTURE SCOPE .............. 137 REFERENCES ........................................................................................................................... 146 APPENDIX ................................................................................................................................. 151 vi CHAPTER 1 - INTRODUCTION 1.1 BACKGROUND The AASHTOWare Pavement Mechanistic-Empirical Design (Pavement-ME) is the latest American Association of State Highway and Transportation Officials (AASHTO) pavement design software edition. It is based on the AASHTO's Mechanistic-Empirical Pavement Design Guide (MEPDG). Pavement-ME is a significant shift from the empirical design process developed and supported by the AASHTO Interim Guide for Design of Pavement Structures (AASHTO 1972) through the AASHTO Guide for Design of Pavement Structures and its 1998 Supplement (AASHTO 1998) (1). While these earlier AASHTO design guides are based on empirically derived performance equations developed using data from the AASHO road test conducted in the 1950s, these have been widely popular for pavement design. About 48 agencies reported using the AASHTO empirical design guides after their refinements provided by AASHTO in 1986 and 1993 (1). Despite the refinements in the material input parameters and the design reliability, the previous design guides' empirical nature limits their performance for the following reasons (2). • The application of the AASHO Road test is limited by its specific geographic location, which does not account for the climatic effects of a different location on pavement performance. • Truck traffic volume has increased significantly since the 1960s, and truck configurations have also changed. • All test sections were built using a single hot mix asphalt (HMA) mixture for flexible pavements and one Portland cement concrete (PCC) mixture for rigid pavements over one subgrade soil type. Recognizing these limitations, the Joint Task Force on Pavements (JTFP) initiated an effort in 1996 to develop the MEPDG using mechanistic pavement design principles. The new mechanistic-empirical (M-E) design procedure offers multiple benefits, taking advantage of improvements in material characterization, axle load spectra, and climate models to predict the pavement's performance. 1 Version 0.7, the research version of the MEPDG software, was first released in July 2004. It was revised several times under different projects funded by the National Cooperative Highway Research Program (NCHRP). The software's revisions included the release of version 0.8 in November 2005, version 0.9 in July 2006, version 1.0 in April 2007, and version 1.1 in September 2009. An MEPDG Manual of Practice was published in 2008, aiming to assist highway agencies in implementing the M-E design method with version 1.0. It was adopted as an interim AASHTO pavement design procedure a year earlier in 2007 (3). Another version of the M-E design was released in April 2011 called Design, Analysis, and Rehabilitation for Windows (DARWin). DARWin-ME software was later named AASHTOWare Pavement METM once AASHTO underwent rebranding in 2013. Currently, the latest version of Pavement-ME software is version 2.6.2, and the online version is version 3.0. In addition, a Backcalculation Tool (BcT) and Calibration Assistance Tool (CAT) have been developed for use with the Pavement-ME software. 1.2 PROBLEM STATEMENT The MEPDG was developed under the NCHRP project 1-37A (4) to overcome the limitations of the AASHTO 1993 method (5). It is an advanced pavement design tool for new and rehabilitated pavements. MEPDG incorporates material properties, traffic, and climate to estimate the incremental damage using mechanical responses of the pavement. The cumulative damage is empirically used to predict the field distress using transfer functions. The transfer functions used in the Pavement-ME have been globally calibrated using the Long-term Pavement Performance (LTPP) pavement sections (6). Although the globally calibrated models provide fair performance predictions for the entire US road network, these may not represent the construction practices, materials, and climatic conditions of a particular state/region. Therefore, nationally calibrated models may underpredict or overpredict the pavement performance in specific states or regions. Recalibration of these models has been recommended for local conditions in the local calibration guide (7). The design distresses in the Pavement-ME include transverse cracking (percentage of slabs cracked), transverse joint faulting (inches), and international roughness index (IRI in inches/mile) for rigid pavements. For flexible pavements, the design distress includes bottom-up cracking (percentage), top-down cracking (percentage), rutting (inches), thermal (transverse) cracking (feet/mile), reflective cracking (feet/mile), and IRI (inches/mile). 2 Several studies have been performed in Michigan in the recent past to characterize climate, traffic, and material properties, as well as to calibrate the performance models to address the local conditions, materials, and construction practices in the Pavement-ME procedure (8-10). While all the material properties and calibration of performance models were addressed to improve the Pavement-ME local applicability and accuracy, there were still some data gaps, specifically for material characterization and pavement construction. Examples of past data gaps include clustered traffic data, HMA mix, and binder properties. Gaps in data need to be estimated (corresponds to Level 3 for Pavement-ME input levels), which may not be accurate for the location; therefore, having the actual values for new projects will likely improve Pavement-ME calibration accuracy. Also, a limited number of rigid pavement sections were available for previous Michigan calibration efforts; therefore, adding more data from new sections would improve the performance model prediction. Most calibration studies have used the least squares approach to calibrate the Pavement- ME transfer functions. Least squares is a widely used simplistic method based on the normal independent and identically distributed (NIID) assumption. The NIID assumption states that observations in a sample are independent, i.e., the occurrence of one does not influence another. Additionally, these observations should have identical probability distributions, i.e., drawn from the same underlying population distribution. Furthermore, the assumption implies that the observed data and error term follow a normal distribution. Literature shows that the least squares method assumptions may not apply to the non-normal distributions. This limits the robustness of the least squares method for transverse cracking in rigid pavements and bottom-up cracking in flexible pavements, which are usually non-normally distributed. The ultimate goal of Pavement-ME calibration is improving pavement designs for local conditions. Despite several calibration efforts, limited research is available on the effect of calibration on pavement design. Estimating the change in design thicknesses and identifying critical distresses using the calibrated models is vital. By understanding which distress types are most relevant to a region, agencies can develop mitigation and maintenance strategies leading to longer pavement service lives. State Highway Agencies (SHAs) often struggle to identify the most critical data collection needs since the Pavement-ME requires several design inputs. Several studies have conducted sensitivity analyses to determine the most sensitive inputs to the distress prediction 3 models for new and rehabilitated pavements to address this issue. However, limited research is available to assess the impact of each calibration coefficient on the predicted pavement distress and performance. These studies quantified the sensitivity of coefficients using a sensitivity index and a typical range of design inputs. The sensitivity metric adopted to accomplish the sensitivity analyses is called the normalized sensitivity index (NSI), defined as the percentage change of predicted distress relative to its global prediction caused by a given percentage change in the coefficient. While NSI can rank the coefficients based on their level of sensitivity, it does not provide information about any potential correlation between them or how accurately these can be estimated. Moreover, since the calculation of NSI requires distress data, its magnitude can change if the data source is changed; hence, the sensitivity ranking of the coefficients may vary, as reported by Dong et al. (11). 1.3 RESEARCH OBJECTIVES The recalibration of the Pavement-ME models is crucial for any SHA implementing M-E design. This includes identifying the suitable Pavement-ME inputs, potential projects, and performance data. It is also essential to verify the feasibility of the calibrated models for pavement design. The main objectives of this study are to (a) calibrate the Pavement-ME models using improved inputs (traffic, HMA and climate) and additional data (potential projects and performance data) for new flexible and rigid pavements, (b) assess the impact of calibrated models on design thicknesses and to identify critical design distresses, (c) apply maximum likelihood estimation (MLE) to calibrate and validate the Pavement-ME models and compare the results with the least squares method, (d) determine the sensitivity of Pavement-ME calibration coefficients over a continuous scale of independent variables using scaled sensitivity coefficients (SSCs) and compare it with the traditional NSI approach. These objectives were accomplished using the pavements and the corresponding performance data from the MDOT Pavement Management System (PMS) database. 1.4 DISSERTATION OUTLINE This dissertation contains six chapters. Chapter 1 outlines the background of the Pavement-ME, the problem statement, and the research objectives. Chapter 2 documents the literature review from previous calibration studies, Pavement-ME transfer functions, and calibration approaches. 4 Chapter 3 discusses the input and performance data used for calibration efforts. This includes data collection efforts, a summary of the performance, and input data for the selected pavement sections for model calibrations. Chapter 4 details the local calibration methods and procedures used in this study. This chapter also includes the methodology used for calculating the SSCs. Chapter 5 presents the local calibration results for the various performance prediction models, including calibration results from the least squares and MLE methods. This chapter also consists of the results from assessing the impact of calibration on pavement design and the SSC plots. Chapter 6 summarizes this study's conclusions, recommendations, and future scope. Each chapter has a summary at the end, which outlines the overall highlights of the chapter. 5 CHAPTER 2 - LITERATURE REVIEW The Pavement-ME provides highway agencies with a practical tool for designing new and rehabilitated pavements. The analyses in M-E principles use primary pavement responses (stresses, strains, and deflections) and incremental damage over time to predict surface distress through transfer functions. The reliability of performance prediction models depends on the accuracy of the transfer functions, which is achieved through calibration and subsequent validation with observed pavement condition data. A satisfactory correlation between measured and predicted performance indicators increases the viability, acceptance, and usage of the MEPDG procedures for pavement analysis and design procedures. Calibration is a mathematical procedure to reduce the difference between predicted and measured distress values. Validation refers to a process that evaluates the performance of mathematical models on an independent dataset (i.e., data not used for model development). This chapter outlines the literature review of calibration approaches, the methodology used in different studies, and the concept of reliability for Pavement-ME predictions. 2.1 IMPLEMENTATION OF PAVEMENT-ME The AASHTO93 empirical pavement design method has been popular and used by highway agencies for several decades (5). Highway agencies are still using it as their current pavement design procedure. The shift from an empirical to a more M-E design method occurred in 2008 after the publication of the MEPDG practice manual and the release of Pavement-ME software (3). The adoption of the Pavement-ME design was further enhanced by publishing the local calibration guide to implement nationally calibrated models for local conditions (7). In recent years, other supplemental tools like the Calibration Assistance Tool (CAT) and Backcalculation Tool (BcT) have helped agencies implement the Pavement-ME design. The adoption of Pavement-ME design started soon after its release, with fifteen state highway agencies (SHA) implementing it within the first few years (1). The implementation significantly increased between 2010 and 2020 and became stagnant due to several challenges. These challenges include the unavailability of input data, pavement sections for calibration, and sufficient good-quality performance data. Some agencies have returned to using their original design practice (usually AASHTO-93) or M-E design in parallel with their original method. As of 2021, nine state 6 agencies are using Pavement-ME as their primary design method for flexible pavements, and thirteen are using it for rigid pavements. Further, nine state agencies use Pavement-ME with other design methods for flexible pavements, whereas eight use it for rigid pavements (12). Figure 2-1 shows the implementation status of the Pavement-ME design for flexible and rigid pavements. (a) Flexible pavements (b) Rigid pavements Figure 2-1 Pavement-ME implementation status (12) 7 2.2 LOCAL CALIBRATION EFFORTS Calibration of Pavement-ME models is an optimization problem. Several researchers have calibrated these models using different optimization methods. This section summarizes the calibration methods and efforts in different states. 2.2.1 Least Squares Method The least squares method is a mathematical technique used to minimize the sum of squared differences between observed and predicted values. Calibration of the Pavement-ME transfer functions is established by minimizing the bias and standard error between the measured and predicted distress. Researchers have used several simplistic and robust approaches leveraging the least squares method for calibration. Hall et al. (2011) used the Microsoft Excel solver function to calibrate the alligator cracking model for flexible pavements in Arkansas (13). Tarefder and Rodriguez-Ruiz (2013) calibrated the rutting, alligator cracking, and longitudinal cracking models for flexible pavements in New Mexico. The process involved changing the calibration coefficients and rerunning Pavement-ME in an iterative process to obtain minimum mean residual error (MRE) and the sum of squared errors (SSE) (14). These calibration efforts have become more robust with the development of computational and statistical techniques. Dong et al. (2020) calibrated the joint faulting model for rigid pavements in Ontario. This study used three different optimization techniques: (1) one at a time using trial and error; (2) Microsoft Excel solver function; (3) Levenberg-Marquardt Algorithm (LMA). Results showed that calibration using approaches (2) and (3) significantly improved the bias and standard error of estimate (SEE) (11). Haider et al. (2020) calibrated the transverse cracking and IRI models for rigid pavements in Michigan. This study used resampling methods like bootstrapping and repeated split sampling for calibration and validation. The results showed that resampling methods provide a more robust calibration than traditional methods, along with confidence intervals of the SEE, bias, and transfer function coefficients (9). Tabesh and Sakhaeifar (2021) calibrated rutting, IRI, top-down, and bottom-up cracking models in Oklahoma using a narrow- down iterative approach in Microsoft Excel solver (15). This study showed significant improvement in the Pavement-ME predictions and flexible pavement designs. All these studies have used the least squares to calibrate these transfer functions using the NIID assumption. Although least squares is a popular and simplistic approach, the assumptions may not be valid, 8 especially for non-normally distributed data. Tables 2-1 and 2-2 summarize the calibration efforts from different states. Table 2-1 Summary of calibration efforts for flexible pavements States Arkansas (16) Colorado (17) Minnesota (18) Montana (19) New Mexico (14) Ohio (20) Oregon (21) South Carolina (22) Utah (23) Washington (24) Arizona (25) Iowa (26) Kansas (27) Michigan (28) North Carolina (29) Texas (30) Wyoming (31) Missouri (32) Georgia (33) Louisiana (34) Virginia (35) Tennessee (36) Oklahoma (15) Number of sections New 38 46 Rehabilitation - 49 39 102 19 13 - 14 21 8 58 35 28 163 46 18 86 6 27 71 53 - 65 5 - 38 - 9 - 42 - - 121 - - - 11 20 33 59 76 - Pavement-ME models Version BU, TD, RUT, TC, BU, RUT, TC, IRI BU, RUT, TC RUT, TC BU, TD, RUT, IRI RUT, IRI RUT, BU, TD, TC RUT, BU, TD RUT BU, TD, RUT BU, TD, RUT, IRI, REF BU, TD, RUT, IRI TD, RUT, IRI BU, TD, RUT, TC, IRI, REF BU, RUT RUT BU, RUT BU, TD, RUT, TC, IRI, REF BU, RUT, TC BU, RUT, REF BU, RUT, IRI BU, TD, RUT, IRI BU, TD, RUT, TC, IRI V1.1 V1.0 V1.0 V0.9 V1.0 V1.0 V2.0.19 V 2.2 V 1.0 V 1.0 DARWin- ME V1.1 - V2.6 DARWin- ME - V2.2 V2.5.5 - V2.0 V2.2.6 V2.1 V2.3 Year 2014 2013 2009 2007 2013 2009 2019 2016 2009 2009 2014 2014 2015 2023 2011 2009 2015 2020 2014 2016 2022 2016 2021 Note: BU = Bottom-up cracking; TD = Top-down cracking; TC = Thermal cracking; RUT = Total rutting; REF = Reflective cracking; IRI = International roughness index Table 2-2 Summary of calibration efforts for rigid pavements States Pavement-ME models Version Number of sections New 25 Rehabilitation 7 65 Colorado (17) Minnesota (18) Ohio (20) South Carolina (22) Arizona (25) Kansas (27) Michigan (28) Wyoming (31) Missouri (32) Georgia (33) Louisiana (34) Idaho (37) Virginia (35) TC, JF, IRI TC IRI TC TC, JF, IRI JF, IRI TC, JF, IRI JF TC, JF, IRI TC, JF TC, JF TC, JF, IRI JF, IRI Note: TC = Transverse cracking; JF = Joint faulting; IRI = International roughness index - - - - 11 - 9 2 - - - 14 6 48 32 46 26 33 9 43 40 17 V1.0 V1.0 V1.0 V 2.2 - V1.3 V2.6 V2.2 V2.5.5 - V2.0 V2.5.3 V1.3 Year 2013 2009 2009 2016 2014 2015 2023 2015 2020 2014 2016 2019 2022 9 2.2.2 Maximum Likelihood Estimation (MLE) Method MLE has been used by several researchers in different fields; limited research is available on the use of MLE to calibrate the Pavement-ME transfer functions. Chen et al. (2021) presented a local calibration model for predicting punchout distress in continuously reinforced concrete pavement (CRCP). This study utilized a Weibull distribution to estimate the number of equivalent single axle loads (ESALs) leading to punchout, employing MLE and a Newton method. The model was validated using data from the LTPP database, demonstrating its efficacy in describing punchout behavior and facilitating predictions for CRCP reliability and rehabilitation planning (38). Haider et al. (2023) showed the robustness of MLE for non-normally distributed data using the MDOT PMS database. The bias for the transverse cracking model in rigid pavements and the bottom-up cracking model in flexible pavements was significantly improved (28). MLE stands out as an advantageous and robust method for parameter estimation as it is based on a well-defined likelihood function rooted in the underlying probability distribution of the data. MLE is computationally efficient, leveraging standard probability distributions, making it usable for multi-dimensional and complex models. MLE excels in estimating parameters for probabilistic models, and it is especially useful in machine learning (39). Unlike the least squares method, MLE shows resilience to outliers as the probability of outliers is very low and offers a potential advantage in the bias-variance tradeoff. The bias-variance tradeoff is used in statistical modeling and machine learning to balance between capturing the underlying pattern in the data (low bias) and resisting sensitivity to fluctuations and noise (high variance). Models with high bias oversimplify data, leading to underfitting, while those with high variance overfit and fail to generalize the model for new data (40). The bias-variance tradeoff highlights the importance of finding the optimal model complexity and employing regularization or ensemble methods to strike the right balance. Understanding this tradeoff is crucial for effective model selection and evaluation, emphasizing the need for ample high-quality training data to minimize bias and variance in overall error. Jose (2023) showed the application of MLE in modeling commodity prices and pricing financial derivatives. This study highlighted estimating model parameters using various methods, with a preference for maximum likelihood when the parametric specification is highly trusted. The comparison in the study evaluates different techniques for obtaining maximum likelihood estimates in the context of Ornstein-Uhlenbeck mean-reverting models based on observations 10 collected at arbitrary points in time (41). Pan and Fang (2002) discussed MLEs for parameters in growth curve models, emphasizing their differences from generalized least squares estimates (GLSE). The special case of Rao's simple covariance structure (SCS), where MLEs coincide with GLSEs, facilitating analytical and tractable statistical inferences in growth curve models, is explored. It also delves into the restricted maximum likelihood (REML) estimate under the assumption of the SCS, offering insights into statistical techniques for analyzing growth curve models (42). Myung (2003) illustrated using MLE, stressing its fundamental role in statistical inference. Moreover, this study emphasized using MLE and its superiority in nonlinear modeling with non-normal data (39). Bauke (2007) showed the limitations of using the least squares method for estimating power-law distribution exponents due to incompatible assumptions with empirical data. It shows the advantages of maximum likelihood estimators, deemed reliable for power-law distributions, with asymptotic efficiency (43). Zhang and Callan (2001) addressed the information filtering systems based on statistical retrieval models, focusing on optimizing dissemination thresholds for document delivery. This study introduced a novel algorithm grounded in the maximum likelihood principle to adjust thresholds by explicitly compensating for bias in relevant information obtained during filtering. Experiments using Text Retrieval Conference (TREC)-8 and TREC-9 filtering track data illustrate the algorithm's effectiveness in jointly estimating parameters and improving system performance. The TREC is an annual series of workshops evaluating information retrieval systems (44). Rayner and MacGillivray (2002) showed the use of numerical maximum likelihood estimation for distributions defined only by quantile functions, focusing on the g-and-k and generalized g-and-h distributions. Despite increased computing power, this aspect of MLE has received limited attention. This study presents and investigates numerical MLE procedures, conducts simulation studies, and emphasizes the need for resampling to obtain reliable estimates for quantile-defined distributions through maximum likelihood (45). Lio and Liu (2020) performed the regression analysis by defining a likelihood function using uncertain measures to represent parameter likelihoods. This study employs MLE for uncertain regression models, simultaneously calculating the uncertainty distribution of the disturbance term. Numerical examples demonstrate the proposed method, emphasizing its applicability to cases with imprecise observations. Future research directions include applying uncertain maximum likelihood to parameter estimation in uncertain differential equations, time series analysis, and hypothesis testing (46). 11 2.3 PAVEMENT-ME PERFORMANCE MODELS The following section presents the formulation of transfer functions for flexible pavement models and the local calibration coefficients for different states. 2.3.1 Performance Models for Flexible Pavements 2.3.1.1. Fatigue cracking (bottom-up) Bottom-up cracking is a load-related distress caused by the repeated axle load. These cracks initiate at the bottom of the asphalt concrete (AC) layer and propagate to the surface. The total cumulative damage DI can be estimated by summing the cumulative damage that is computed using Miner's law (47), as shown in Equation (2-1). 𝐷𝐼 = ∑(Δ𝐷𝐼)𝑗,𝑚,𝑙,𝑝,𝑇 = ∑ ( 𝑛 𝑁𝑓−𝐻𝑀𝐴 ) 𝑗,𝑚,𝑙,𝑝,𝑇 (2-1) where, n = Number of actual axle load applications within a specific time period j = Axle load-interval m = Axle type (single, tandem, tridem, quad) l = Truck type classified in the MEPDG p = Month T = Median temperature for five temperature quintiles used in MEPDG Nf-HMA = Allowable number of axle load applications, which can be computed using Equation (2- 2). 𝑁𝑓−𝐻𝑀𝐴 = 𝐶 × 𝑘1 × 𝐶𝐻 × 𝛽𝑓1(𝜀𝑡)−𝑘2𝛽𝑓2(𝐸𝐻𝑀𝐴)−𝑘3𝛽𝑓3 (2-2) where, εt = Tensile strain at critical AC locations EHMA = Dynamic modulus (E*) of the Hot mix asphalt (HMA), psi k1, k2, k3 = Laboratory regression coefficients, and βf1, βf2, βf3 = local or field calibration constants C = Adjustment factor (laboratory to the field) as shown in Equation (2-3) and Equation (2-4). 𝐶 = 10𝑀 12 (2-3) 𝑀 = 4.84 ( 𝑉𝑏𝑒 𝑉𝑎 + 𝑉𝑏𝑒 − 0.69) (2-4) where, Vbe = Effective binder content by volume, percent Va = In-situ air voids in the HMA mixture (%) CH = Thickness correction factor for bottom-up cracking as shown in Equation (2-5). 𝐶𝐻 = 1 0.000398 + 0.003602 1 + 𝑒(11.02−3.49𝐻𝐻𝑀𝐴) (2-5) where, HHMA = AC layer thickness Once the cumulative damage is calculated, the bottom-up fatigue cracking (%) can be estimated using the transfer function given in Equation (2-6). 𝐹𝐶Bottom = ( 1 60 ) ( 𝐶4 ∗log⁡(𝐷𝐼Bottom ⋅100)) ∗+𝐶2𝐶2 1 + 𝑒𝐶1𝐶1 (2-6) where, FCBottom = Bottom-up fatigue cracking (in the percentage of area) DIBottom = Cumulative damage at the bottom of the AC layer C1, C2, C4 = Transfer function coefficients where C2 is a function of thickness for HMA thickness between 5 and 12 inches C1* and C2* can be determined using Equation (2-7) and Equation (2-8). ∗ ∗ = −2𝐶2 𝐶1 ∗ = −2.40874 − 39.748(1 + 𝐻𝐻𝑀𝐴)−2.856 𝐶2 (2-7) (2-8) Table 2-3 summarizes the local calibration coefficients for bottom-up cracking model among several states. 13 Table 2-3 Local calibration coefficients for bottom-up cracking States Michigan C1 0.67 C2 0.56 Missouri 0.31 C2<5”=1.367, C2>12”=2.067, C2(5”12”)=0.867+0.1* hac Georgia Louisiana Virginia Tennessee Oklahoma (East Region) Oklahoma (West Region) Oklahoma (East region) Oklahoma (West region) 2.2 0.892 0.319 1.023 3.26 4.12 3.26 4.12 2.2 0.892 0.319 0.045 - - - - C4 6000 6000 6000 6000 - 6000 6000 6000 6000 6000 Standard deviation 0.01 + 32.913 1 + 𝑒1.3972−0.9576×log⁡(𝐷) - 1 + 10 1 + 𝑒7.5−6.5×log(𝐷+0.0001) - - - - - - - Alabama 1 4.5 6000 1.1 + 22.9 1 + 𝑒−0.1214−2.0565×log(𝐷+0.0001) North Carolina Wyoming Arkansas Colorado New Mexico Oregon South Carolina Washington 0.2437 0.4951 0.688 0.07 0.625 0.560 0.47 1.071 Pavement-ME v2.6 1.31 0.24377 1.469 0.294 2.35 0.25 0.225 0.47 1 C2<5”=2.1585, C2>12”=3.9666, C2(5”12”)=(0.867+0.25 83*hac)*1 2.3.1.2. Fatigue cracking (top-down) 6000 6000 6000 6000 6000 6000 6000 6000 - - - 0.01 + 15 1 + 𝑒−1.6673−2.4656×log⁡(𝐷) - - - - 6000 1.13 + 13 1 + 𝑒7.57−15.5×log⁡(𝐷+0.0001) Top-down or longitudinal cracking is a load-related distress due to repeated axle load. It appears in the form of cracks parallel to the wheel path and starts at the surface of the AC layer. Old model: The damage calculation for top-down cracking is the same as bottom-up cracking for the old model except for the thickness correction factor and the transfer function, as shown in Equation (2-9) and Equation (2-10). 𝐶𝐻 = 1 0.01 + 12.00 1 + 𝑒(15.676−2.8186𝐻HMA ) (2-9) 14 𝐹𝐶Top = 10.56 ( 𝐶3 1 + 𝑒𝐶1−𝐶2𝐿𝑜𝑔(𝐷𝐼Top ) ) (2-10) where, FCTop = Top-down fatigue cracking (in ft/mile) DITop = Cumulative damage at the top of the AC layer C1, C2, C3 = Transfer function coefficients New model: The new top-down cracking model is based on fracture mechanics concepts (48). It is expressed in percentage rather than ft./mile. The model involves crack initiation and propagation [based on Paris' law (49)]. Crack initiation is defined as a crack length of 7.5 mm (0.3 inches). Equation (2-11) shows the time to crack initiation formulated using regression over longitudinal and alligator cracking data from the LTPP database. 𝑡0 = K𝐿1 1 + 𝑒K𝐿2×100×(a0/2A0)+K𝐿3×HT+K𝐿4×𝐿𝑇+𝐾𝐿5×log10⁡ AADTT (2-11) where, t0 = Time to crack initiation, days HT = Annual number of days above 32oC LT = Annual number of days below 0oC AADTT = Annual average daily truck traffic (initial year) a0/2A0 = Energy parameter KL1, KL2, KL3, KL4, KL5= Calibration coefficients for time to crack initiation The top-down cracking is expressed in percentage using the transfer function, as shown in Equation (2-12). where, 𝐿(𝑡) = 𝐿𝑀𝐴𝑋𝑒 𝐶2𝛽 −( 𝐶1𝜌 𝑡−𝐶3𝑡0 ) (2-12) L(t) = Top-down cracking expressed as total lane area (%) LMAX = Maximum area of top-down cracking (%) – a value of 58% is assumed t = Analysis month in days ρ = Scale parameter for the top-down cracking curve as shown in Equation (2-13). β = Shape parameter for the top-down cracking curve as shown in Equation (2-14). 𝜌 = 𝛼1 + 𝛼2 × Month (2-13) 15 𝛽 = 0.7319 × (log10 Month )−1.2801 (2-14) where, α1⁡and α2 are functions of the climatic zone (wet freeze, wet non-freeze, dry freeze, dry non- freeze) Table 2-4 summarizes the local calibration coefficients of the top-down cracking model. These coefficients have been obtained for the old top-down cracking model. Table 2-4 Local calibration coefficients for top-down cracking States Michigan Tennessee Oklahoma (East Region) Oklahoma (West Region) Iowa Kansas Arkansas New Mexico Oregon South Carolina Washington Pavement-ME v2.3 C1 2.97 6.44 6.6 6.1 0.82 4.5 3.016 3 1.453 0.2 6.42 7 C2 1.2 0.27 4.6 4.23 1.18 - 0.216 0.3 0.097 0.1 3.596 3.5 2.3.1.3. Transverse (thermal) cracking model C3 Standard deviation 1000 300 + 204.54 723 723 1000 36000 1000 1000 1000 3.97 1000 1000 3000 1 + 𝑒7.5−6.5×log(𝐷𝑏𝑜𝑡𝑡𝑜𝑚+0.0001) - - - - - - - - - - Thermal cracking is associated with the contraction of the HMA material due to surface temperature fluctuations. The temperature variations affect the volume changes of the material. Consequently, stress develops due to the continual contraction of the materials and the restrained conditions, which causes thermal cracks. Typically, thermal cracking in flexible pavements occurs due to the temperature drop experienced by the pavement in cold conditions. A thermal crack will initiate when the tensile stresses in the HMA layers become equal to or greater than the material's tensile strength. The initial cracks propagate through the HMA layer with more thermal cycles. The amount of crack propagation induced by a given thermal cooling cycle is predicted using the Paris law of crack propagation. Experimental results indicate that reasonable estimates of A and n can be obtained from the indirect tensile creep-compliance and tensile strength of the HMA per Equations (2-15 and 2-16). 16 𝛥𝐶 = 𝐴(𝛥𝐾)𝑛 (2-15) where, C K = Change in the crack depth due to a cooling cycle = Change in the stress intensity factor due to a cooling cycle A, n = Fracture parameters for the HMA mixture 𝐴 = 𝑘𝑡𝛽𝑡10[4.389−2.52𝐿𝑜𝑔(𝐸𝐻𝑀𝐴𝜎𝑚𝜂)] (2-16) where,  kt = 0.8 [1 + 1 𝑚 ] = Regression coefficient determined through field calibration EHMA = HMA indirect tensile modulus, psi m m = Mixture tensile strength, psi = The m-value derived from the indirect tensile creep compliance curve measured in the laboratory βt = Local or mixture calibration factor The stress intensity factor, K, has been incorporated in the Pavement-ME through a simplified equation developed from theoretical finite element studies using the model shown in Equation (2-17). where, 𝐾 = 𝜎𝑡𝑖𝑝(0.45 + 1.99(𝐶𝑜)0.56) (2-17) tip = Far-field stress from pavement response model at a depth of crack tip, psi Co = Current crack length, feet Equation (2-18) shows the transfer function for transverse cracking in the Pavement-ME. 𝑇𝐶 = 𝛽𝑡1𝑁(𝑧) [ 1 𝜎𝑑 𝐿𝑜𝑔 ( 𝐶𝑑 𝐻𝐻𝑀𝐴 )] (2-18) where, TC βt1 = Observed amount of thermal cracking, ft/500ft = Regression coefficient determined through global calibration (400) N[z] = Standard normal distribution evaluated at [z] σd = Standard deviation of the log of the depth of cracks in the pavement (0.769), in. 17 Cd = Crack depth, in. HHMA = Thickness of HMA layers, in. Table 2-5 summarizes the modified local calibration coefficients for the various states. Table 2-5 Local calibration coefficients for the thermal cracking model States Level 1 Level 2 Level 3 Michigan 0.75 Missouri Oklahoma (East Region) Oklahoma (West Region) 0.61 3 x 10-7 × MAAT4.0319 – 54 3 x 10-7 × MAAT4.0319 – 23 1.5 7.5 - - Arizona Colorado Minnesota Montana Pavement- ME v2.6 - - - - 0.5 - - - 4 - - - 1.5 - 1.85 0.25 3 x 10-7 × MAAT4.0319 3 x 10-7 × MAAT4.0319 3 x 10-7 × MAAT4.0319 Standard deviation Level 1 K: 0.4258*THERMAL +210.08 Level 3 K: 0.7737*THERMAL +622.92 - - - Level 1 K: 0.1468*THERMAL +65.027 Level 2 K: 0.2841*THERMAL +55.462 Level 3 K: 0.3972*THERMAL +20.422 Level 1 K: 0.1468*THERMAL +65.027 - - Level 1 K: 0.14*THERMAL +168 Level 2 K: 0.14*THERMAL +168 Level 3 K: 0.14*THERMAL +168 2.3.1.4. Rutting model Due to axle loads, rutting is the total accumulated plastic strain in different pavement layers (AC, base/sub-base, and subgrade). It is calculated by summing up the plastic strains at the mid-depth of individual layers accumulated for each time increment. Equation (2-19) shows the permanent plastic strain for the AC layer. Δ𝑝(𝐻𝑀𝐴) = 𝜀𝑝(𝐻𝑀𝐴)ℎ𝐻𝑀𝐴 = 𝛽1𝑟𝑘𝑧𝜀𝑟(𝐻𝑀𝐴)10𝑘1𝑟𝑇𝑘2𝑟𝛽2𝑟𝑁𝑘3𝑟𝛽3𝑟 (2-19) where, Δp(HMA) = Permanent plastic deformation in the AC layer εp(HMA) = Accumulated permanent or plastic axial strain in the AC layer/sublayer εr(HMA) = Resilient or elastic strain calculated by the structural response model at the mid-depth of each AC sublayer h(HMA) = Thickness of the AC layer/sublayer N = Number of axle load repetitions T =Pavement temperature 18 kz = Depth confinement factor k1r, k2r, k3r = Global field calibration parameters β1r, β2r, β3r, = Local or mixture field calibration constants The permanent plastic strain can be expressed for the unbound layers, as shown in Equation (2- 20). where, Δ𝑝(𝑠𝑜𝑖𝑙) = 𝛽𝑠1𝑘𝑠1𝜀𝑣ℎ𝑠𝑜𝑖𝑙 ( 𝜀𝑜 𝜀𝑟 𝛽 𝜌 𝑛 ) ) 𝑒−( (2-20) Δp(Soil) = Permanent plastic deformation for the unbound layer/sublayer εo = Intercept determined from laboratory repeated load permanent deformation tests n = Number of axle load applications εr = Resilient strain imposed in laboratory tests to obtain material properties εo, β, and ρ εv = Average vertical resilient or elastic strain in the layer/sublayer and calculated by the structural response model hsoil = Unbound layer thickness ks1 = Global calibration coefficients (different for granular and fine-grained material) βs1 = Local calibration constant for rutting in the unbound layers (base or subgrade) The total rutting is calculated based on Equation (2-21) below: Rut DepthTotal =𝛥𝐻𝑀𝐴 + 𝛥𝐵𝑎𝑠𝑒/𝑠𝑢𝑏𝑏𝑎𝑠𝑒 + 𝛥𝑆𝑢𝑏𝑔𝑟𝑎𝑑𝑒 (2-21) Table 2-6 presents the local calibration coefficients for different states. 2.3.1.5. IRI model (flexible pavements) IRI is a measure of ride quality provided by a pavement surface and affects vehicle operation cost, safety, and driver comfort. The IRI model is based on findings from multiple studies showing that IRI at any age is a function of the initial construction ride quality and the development of different distresses over time that impact ride quality. IRI can be formulated using the initial IRI and distresses (fatigue cracking, transverse cracking, and rutting), as shown in Equation (2-22). 19 Table 2-6 Local calibration coefficients for the rutting model States β1r β2r β3r βgb βsg Michigan 0.945 1.3 0.7 0.0985 0.0367 Missouri Georgia Louisiana Virginia Tennessee (Plain area) Tennessee (Mountain area) Oklahoma (East Region) Oklahoma (West Region) 0.899 - 0.80 0.664 0.111 0.177 - - - - - - - - 0.85 - - - 1.0798 0.5 - 0.151 0.9779 0.3 0.40 0.151 0.196 0.722 1.034 0.159 0.79 0.53 1.48 0.15 1.29 0.21 0.74 1.03 0.23 1.03 Arizona 0.69 1 1 0.14 0.37 Iowa Kansas North Carolina Texas Wyoming Arkansas Colorado Montana New Mexico Ohio Oregon South Carolina Utah Washington - 0.9 0.947 2.39 - 1.20 1.34 1.07 1.1 0.51 1.48 0.240 0.560 1.05 1.15 - 0.862 - - 1 1 - 1.1 - 1.0 1 1 1.109 - - 1.354 0.856 - 0.8 1 - 0.8 - 0.9 1 1 1.1 0.001 - 0.53767 - 0.4 1 0.4 0.01 0.8 0.32 0 2.979 0.604 - 0.001 0.3251 1.5 0.5 0.4 0.5 0.84 0.437 1.1 0.33 0 0.393 0.400 0 Pavement-ME v2.6 0.4 0.52 1.36 1 1 Standard deviation HMA: 0.1126*RUT0.2352 BASE: 0.1145*RUT0.3907 SG: 3.6118*RUT1.0951 - HMA: 0.20*RUT0.55+0.001 - - - - - - HMA: 0.0999*RUT0.174 + 0.001 BASE:0.05*RUT0.115 + 0.001 SG: 0.05*RU0.085 + 0.001 - - - - - - - - - - - - - HMA: 0.24*RUT0.8026+0.001 BASE: 0.1477*RUT0.6711+0.001 SG: 0.1235*RUT0.5012+0.001 𝐼𝑅𝐼 = 𝐼𝑅𝐼𝑜 + 𝐶1(𝑅𝐷) + 𝐶2(𝐹𝐶Total ) + 𝐶3(𝑇𝐶) + 𝐶4(𝑆𝐹) (2-22) where, IRIo = Initial IRI at construction FCTotal = Percent area of fatigue cracking (bottom-up), fatigue cracking (top-down), and reflection cracking in the wheel path TC = Length of transverse cracking (including the reflection of transverse cracks in existing AC pavements) RD = Average rut depth; C1, C2, C3, C4 = Calibration coefficients 20 SF = site factor, which can be expressed as shown in Equation (2-23) to Equation (2-25). 𝑆𝐹 = ( Frost + Swell ) × 𝐴𝑔𝑒1.5 Frost = Ln⁡[( Rain + 1) × (𝐹𝐼 + 1) × 𝑃4] Swell = Ln⁡[( Rain + 1) × (𝑃𝐼 + 1) × 𝑃200] (2-23) (2-24) (2-25) where, SF = Site factor Age = Pavement age (years) FI = Freezing index PI = Subgrade soil plasticity index Rain = Mean annual rainfall P4 = Percent subgrade material passing No. 4 sieve P200 = Percent subgrade material passing No. 200 sieve. Table 2-7 presents the calibrated IRI coefficients in different states. Table 2-8 summarizes the distress thresholds for flexible pavements used in various states. Table 2-7 Local calibration coefficients for the IRI model States Michigan Missouri Virginia Oklahoma (East Region) Oklahoma (West Region) Arizona Kansas Colorado New Mexico Ohio Pavement-ME v2.6 C1 50.3720 58.9 - 5.23 6.46 1.2281 95 35 - 17.6 40 C2 0.4102 0.3 - 0.127 0.187 0.1175 0.04 0.3 - 1.37 0.4 C3 0.0066 0.0072 - 0.013 0.0098 0.008 0.001 0.02 - 0.01 0.008 C4 0.0068 0.0129 0.0392 0.0128 0.023 0.0280 - 0.019 0.015 0.066 0.015 21 Table 2-8 Summary of design thresholds for flexible pavements States Michigan Missouri Louisiana Virginia Tennessee Oklahoma Arizona Kansas Colorado Bottom-up cracking (%) 20 10 15 10 10 20 20 20 10 Top-down cracking (ft/mile) - - - - 2000 - - - 2000 Total rutting 0.5 0.50 0.4 0.4 0.4 0.4 0.4 0.4 0.4 Thermal cracking 1000 1000 500 500 500 630 630 630 1500 IRI 172 172 160 160 160 169 169 169 160 2.3.2 Performance Models for Rigid Pavements 2.3.2.1. Transverse cracking model Transverse slab cracking in the Pavement-ME is calculated as the percentage of slabs cracked, including all severity levels. The mechanism involves independently predicting the bottom-up and top-down cracking and utilizing a probabilistic relationship to combine both, eliminating the possibility of both co-occurring. The fatigue damage for both bottom-up and top-down is defined using Miner's law as given in Equation (2-26): 𝐷𝐼𝐹 = ∑ 𝑛𝑖,𝑗,𝑘,𝑙,𝑚,𝑛,𝑜 𝑁𝑖,𝑗,𝑘,𝑙,𝑚,𝑛,𝑜 (2-26) where, DIF = Total fatigue damage (bottom-up or top-down) ni,j,k,l,m,n,o = Actual load applications applied at age i, month j, axle type k, load level l, the equivalent temperature difference between top and bottom PCC surfaces m, traffic offset path n, and hourly truck traffic fraction o Ni,j,k,l,m,n,o = Allowable number of load applications applied at age i, month j, axle type k, load level l, the equivalent temperature difference between top and bottom PCC surfaces m, traffic offset path n, and hourly truck traffic fraction o The allowable number of load applications is a function of PCC strength and applied stress and is calculated based on Equation (2-27): log⁡(𝑁𝑖,𝑗,𝑘,𝑙,𝑚,𝑛,𝑜) = 𝐶1 ⋅ ( 𝐶2 𝑀𝑅𝑖 𝜎𝑖,𝑗,𝑘,𝑙,𝑚,𝑛,𝑜 ) (2-27) where, 22 MRi = Modulus of rupture of the PCC slab at the age i σi,j,k,l,m,n = Applied stress at the age i, month j, axle type k, load level l, the equivalent temperature difference between top and bottom PCC surface m, traffic offset path n, and hourly truck traffic fraction o C1, C2 = Fatigue life calibration coefficients The fraction of slabs cracked is predicted using Equation (2-28) for both bottom-up and top- down cracking: where, 𝐶𝑅𝐾 = 1 1 + 𝐶4(𝐷𝐼𝐹)𝐶5 (2-28) CRK = Predicted fraction of bottom-up or top-down cracking Once the bottom-up and top-down cracking is estimated, the percentage of slabs cracked is calculated using Equation (2-29). 𝑇𝐶𝑅𝐴𝐶𝐾 = (𝐶𝑅𝐾Bottom-up + 𝐶𝑅𝐾Top-down − 𝐶𝑅𝐾Bottom-up ⋅ 𝐶𝑅𝐾Top-down ) ⋅ 100 (2-29) where, TCRACK = Total transverse cracking (percentage of slabs cracked with all severities) CRKBottom-up = Predicted fraction of bottom-up transverse cracking CRKTop-down = Predicted fraction of top-down transverse cracking Table 2-9 summarizes the transverse cracking model local calibration coefficients in different states. Table 2-9 Local calibration coefficients for the rigid transverse cracking model States Michigan Louisiana Idaho Arizona Minnesota South Carolina Pavement-ME v2.6 C1 - 2.75 2.366 - - 1.25 2 C2 - - 1.22 - - 1.22 1.22 C4 0.23 1.16 0.52 0.19 0.9 - 0.52 C5 -1.80 -1.73 -2.17 -2.067 -2.64 - -2.17 Standard deviation 1.34*CRK0.6593 - - - - - 3.5522*CRK0.3415+0.75 2.3.2.2. Joint faulting model The transverse joint faulting is calculated monthly in the Pavement-ME using the material properties, climatic conditions, present faulting level, pavement design properties, and axle loads 23 applied. Total faulting is the sum of faulting increments from previous months and is predicted using Equations (2-30) to (2-33) below. 𝑚 Fault𝑚 = ∑   𝑖=1 ΔFault𝑖 𝛥 Fault𝑖 = 𝐶34 × ( FAULTMAX𝑖−1 − Fault𝑖−1)2 × DE𝑖 𝑚 𝐹𝐴𝑈𝐿𝑇𝑀𝐴𝑋𝑖 = 𝐹𝐴𝑈𝐿𝑇𝑀𝐴𝑋0 + 𝐶7 × ∑   𝑗=1 𝐷𝐸𝑗 × log⁡(1 + 𝐶5 × 5.0𝐸𝑅𝑂𝐷)𝐶6 FAULTMAX X0 = C12 × δcurling × [log⁡(1 + C5 × 5.0EROD) × log⁡ ( P200 × WetDays Ps )] C6 where, Faultm = Mean joint faulting at the end of month m (2-30) (2-31) (2-32) (2-33) ΔFaulti = Incremental change (monthly) in mean transverse joint faulting during the month i FAULTMAXi = Maximum mean transverse joint faulting for the month i FAULTMAX0 = Initial maximum mean transverse joint faulting EROD = Erodibility factor for base/subbase DEi = Differential deformation energy of subgrade deformation accumulated during the month i δcurling = Maximum mean monthly slab corner upward PCC deflection due to temperature curling and moisture warping., PS = Overburden pressure on the subgrade, P200 = Percent subgrade soil material passing No. 200 sieve WetDays = Average annual number of wet days (greater than 0.1 in rainfall) C1,2,3,4,5,6,7,12,34 = Calibration coefficients C12 and C34 are defined by Equation (2-34) and Equation (2-35): C12 = C1 + C2 × 𝐹𝑅0.25 C34 = C3 + C4 × 𝐹𝑅0.25 (2-34) (2-35) FR = Base freezing index defined as the percentage of time (in hours) the top base temperature is below freezing (32 °F) temperature to the total number of hours in design life Damage in a doweled joint for the current month is estimated using Equation (2-36). 𝑁 Δ𝐷𝑂𝑊𝐷𝐴𝑀𝑡𝑜𝑡 = ∑   𝑗=1 𝐶8 × 𝐹𝑗 𝑛𝑗 ∗ 106𝑑𝑓𝑐 (2-36) 24 where, ΔDOWDAMtot = Cumulative dowel damage for the current month nj = Number of axle load applications for the current increment and load group j for the current month N = Number of load categories fc* = Estimated PCC compressive stress d = Dowel diameter C8 = Calibration constant Fj = Effective dowel shear force induced by axle loading of load category j The faulting model local calibration results for several states are summarized in Table 2-10. Table 2-10 Local calibration coefficients for the faulting model States Wyoming Georgia Louisiana Idaho Arizona Kansas Michigan Wyoming Pavement- ME v2.6 C1 0.5104 0.595 1.5276 0.516 0.0355 - 0.4 0.5104 C2 0.00838 1.636 - - 0.1147 - - 0.00838 C3 0.00147 0.00217 0.00262 - 0.00436 0.00164 - 0.00147 C4 0.08345 0.00444 - - 1.1E-07 - - 0.08345 C5 5999 - - - 20000 - - 5999 C6 0.504 0.47 0.55 2.0389 0.15 - 0.504 C7 5.9293 7.3 - - 0.1890 0.01 - 5.9293 C8 - - - - 400 - - - Standard deviation 0.0831*FAULT0.3426+0.00521 0.07162*FAULT0.368+0.00806 - - 0.037*FAULT0.6532+0.001 - 0.0442*FAULT0.2698 0.0831*FAULT0.3426+0.00521 0.595 1.636 0.00217 0.00444 250 0.47 7.3 400 0.07162*FAULT0.368+0.00806 2.3.2.3. IRI model (rigid pavements) IRI in the Pavement-ME is a linear relationship between the IRI at construction and change in other distresses (transverse cracking, joint faulting, and joint spalling) over time. As a linear relationship of these factors, IRI can be expressed by Equation (2-37). 𝐼𝑅𝐼 = 𝐼𝑅𝐼𝐼 + 𝐶1 × 𝐶𝑅𝐾 + 𝐶2 × 𝑆𝑃𝐴𝐿𝐿 + 𝐶3 × 𝑇𝐹𝐴𝑈𝐿𝑇 + 𝐶4 × 𝑆𝐹 (2-37) where, IRI = Predicted IRI IRII = Initial IRI at the time of construction CRK = Percent slabs with transverse cracking (all severities). SPALL = Percentage of joints with spalling (medium and high severities). TFAULT = Total joint faulting cumulated per mi C1, C2, C3, C4 = Calibration coefficients SF = Site factor, which can be calculated as shown in Equation (2-38) 25 𝑆𝐹 = 𝐴𝐺𝐸(1 + 0.5556 × 𝐹𝐼)(1 + 𝑃200) × 10−6 (2-38) where, AGE = Pavement age FI = Freezing index, °F-days. P200 = Percent subgrade material passing No. 200 sieve. The joint faulting and transverse cracking for IRI calculation are obtained using previously described models. The joint spalling is calculated as shown in Equation (2-39) 𝑆𝑃𝐴𝐿𝐿 = [ 𝐴𝐺𝐸 𝐴𝐺𝐸 + 0.01 ] [ 100 1 + 1.005(−12 × 𝐴𝐺𝐸 + 𝑆𝐶𝐹) ] (2-39) where, SPALL = percentage joints spalled (medium- and high-severities) AGE = pavement age since construction SCF = scaling factor based on site-, design-, and climate-related variables, which is estimated as given in Equation (2-40) 𝑆𝐶𝐹 = −1400 + 350 × 𝐴𝐶𝑃𝐶𝐶 × (0.5 + 𝑃𝑅𝐸𝐹𝑂𝑅𝑀) + 3.4𝑓𝑐 ′0.4 − 0.2( FTcycles ×𝐴𝐺𝐸) +43ℎ𝑃𝐶𝐶 − 536𝑊𝐶𝑃𝐶𝐶 (2-40) where, ACPCC = PCC air content AGE = Time since construction PREFORM = 1 if preformed sealant is present; 0 if not f'c = PCC compressive strength FTcycles = Average annual number of freeze-thaw cycles hPCC = PCC slab thickness; WCPCC = PCC water/cement ratio The flexible pavement IRI local calibration coefficients for various states are summarized in Table 2-11. Table 2-12 shows threshold values used for different distresses in various states. 26 Table 2-11 Local calibration coefficients for rigid IRI model States Michigan Georgia Idaho Virginia Arizona Iowa Kansas Ohio Pavement-ME v2.6 C1 1.198 1.05 0.845 9.55 0.60 0.04 - 0.820 0.8203 C2 3.570 0.5417 0.4417 172.55 3.48 0.04 - 3.7 0.4417 C3 1.4929 1.85 1.4929 - 1.22 0.07 9.38 1.711 1.4929 C4 25.24 33.8 28.24 - 45.20 1.17 70 5.703 25.24 Table 2-12 Summary of design thresholds for rigid pavements States Michigan Missouri Louisiana Idaho Virginia Arizona Kansas Colorado Minnesota Transverse cracking (%) 15 - 10 10 10 10 10 10 15 2.4 LOCAL CALIBRATION PROCESS Joint faulting (in) 0.125 - 0.15 0.15 0.15 0.15 0.15 0.15 0.12 IRI (in/mile) 172 172 160 169 160 169 169 160 - As mentioned, the Pavement-ME uses performance prediction models that are nationally calibrated based on pavement material properties, structure, climate, truck loading conditions, and data from the LTPP program (50). However, these models may not accurately predict pavement performance if the input properties and data used for calibration do not reflect the state's unique conditions. Therefore, it is recommended that each SHA evaluates how well the nationally calibrated models predict field performance. If the predictions are unsatisfactory, local calibration of the Pavement-ME models is recommended to improve the pavement performance predictions that reflect the state's specific field conditions and design practices. The local calibration process confirms that the prediction models can accurately predict pavement distress and smoothness and determines the standard error associated with the prediction equations. This section summarizes the local calibration process per the local calibration guide, 2010 (7) and MEPDG, 2015 (51). 27 Step 1: Selection of input levels The hierarchical input level must be selected before local calibration. This depends on the availability of inputs in the local database and the agency's laboratory and field-testing capabilities. The selection of input levels is a critical step as it impacts the standard error of prediction. Step 2: Develop an experimental plan and sampling strategy The agency needs to develop a statistically sound and practical experimental plan and sampling template for this step. The sampling strategy should consider the local construction, design, and rehabilitation practices. The design matrix should include a wide range of traffic, materials, and climatic inputs. Step 3: Assess the adequate sample size for each distress A reasonable number of sections should be selected for calibration. The minimum sample size for any distress can be estimated using Equation (2-41). 𝑛 = ( 2 𝑍𝛼/2 × 𝜎 𝑒𝑡 ) (2-41) where, Zα/2 = z-value from a standard normal distribution n = Minimum number of pavement sections σ = Performance threshold et = Tolerable bias = Zα/2 × SEE SEE = Standard error of the estimate Step 4: Selection of pavement sections This step involves selecting the pavement sections to populate the experimental matrix developed in Step 2. Selection should include local construction practices, sections with and without overlay, pavements with non-conventional materials, and replicates. To incorporate any time-dependent effects, a minimum of three measured distress data should be available over ten years. In case of section inadequacy, LTPP sections can be added to enhance the database. Step 5: Get Pavement-ME inputs and measured distress data The Pavement-ME inputs and the measured distress data must be extracted from the local agency database based on the hierarchical input level determined in Step 1. The performance data must be converted to the Pavement-ME compatible units if the agency measurements are different. 28 The average maximum distress from the selected sections should exceed 50% of the threshold design criteria to incorporate considerable distress in the calibration process. Any outliers in the performance data should be reviewed, considering the maintenance activities or changes in agency policies. Further field investigation can be conducted to resolve any discrepancies. Step 6: Conduct field and forensic investigation This step aims to collect any missing data and investigate any discrepancies in the input data available in the local database. The testing protocol to be followed should be in accordance with the agency's practices. At the end of this step, the agency should ensure that a reasonable number of samples remain in the experimental matrix. Step 7: Validation of global model coefficients to local conditions For this step, the global coefficients are used to predict each performance measure for all sections included in the experimental matrix. A reliability of 50% should be used for this step. The predicted values are compared with the measured ones to calculate the bias and SEE. A plot of predicted versus measured values is created for each distress to visualize the accuracy of predictions to a line of equality (LOE). For a good fit, the points should lie along the LOE. The measured distress yMeasured and predicted distress xPredicted can be modeled as a linear model as shown in Equation (2-42) where m is the slope, and bo is the intercept. 𝑦Measured = 𝑏𝑜 + 𝑚 × 𝑥Predicted (2-42) Three hypothesis tests are conducted to evaluate the reasonableness of the global model. If any of these hypotheses fail, the models are recalibrated for local conditions: • There is no systematic bias between the measured and predicted distress [Equation (2- 43)]. This can be tested using a paired t-test. 𝐻0: ∑(𝑦Measured − 𝑥Predicted ) = 0 (2-43) • The slope parameter m is 1, and the intercept parameter bo is zero [Equations (2-44) and (2-45)]. 𝐻0: 𝑚 = 1.0 𝐻𝑜: 𝑏𝑜 = 0 (2-44) (2-45) Step 8: Eliminate the local bias for Pavement-ME models This step should eliminate the local bias by systematically changing the model coefficients. The approach should be based on the overall bias, SEE between the predicted and measured values, 29 and the causes associated with them. The calibration coefficients should be incorporated into the calibration process if they depend on material property, site factor, or design features. Table 2-13 summarizes the calibration coefficients affecting the bias and standard error. Table 2-13 Calibration coefficients eliminating standard error and bias (1) Pavement Type Flexible Rigid Distress Total rut depth Fatigue bottom-up cracking Fatigue top-down cracking Thermal cracking IRI Faulting Transverse cracking IRI - JPCP Eliminate Bias 𝑘1𝑟, 𝛽1𝑟, 𝛽𝑠1 𝑘1, 𝐶2 𝑘1, 𝐶2 𝛽𝑓3, 𝑘𝑓3 𝐶4 𝐶1 𝐶1, 𝐶4 𝐽4 Reduce Standard Error 𝑘2𝑟, 𝑘3𝑟, 𝛽2𝑟, 𝛽3𝑟 𝑘2, 𝑘3, 𝐶1 𝑘2, 𝑘3, 𝐶1 𝛽𝑓3, 𝑘𝑓3 𝐶2, 𝐶3, 𝐶4 𝐶1 𝐶2, 𝐶5 𝐽1 Step 9: Estimate the standard error of the estimate After the bias has been eliminated, the SEE is computed between the measured and predicted distress. This SEE must be compared with the global SEE. Table 2-14 shows the recommended value for SEE and bias for different models. Table 2-14 Recommended values for tolerable bias and SEE (28) Pavement Type Flexible Rigid Distress/performance parameter Fatigue cracking (% total lane area) Rutting (inches) Thermal cracking (ft/mile) Thermal Reflection cracking IRI (inch/mile) Transverse cracking (% slabs cracked) Faulting (inch) IRI (inch/mile) Bias 1.5 0.075 200 20 4 0.02 20 SEE 5 0.2 650 65 15 0.07 65 If the SEE is lower than recommended, the calibration coefficients can be accepted and used for design. The hypothesis tests given in step 7 must be validated before accepting the coefficients. If the SEE exceeds the global value, the agency can still accept the coefficients or move to step 10 to eliminate the standard error. Step 10: Eliminate standard error of estimate (SEE) If the standard error of the estimate calculated in step 9 is higher than the recommended global value, it should be eliminated in the local calibration process. The standard error should be estimated for each category of the experimental matrix to identify the effects of any input 30 parameter on the overall standard error. The coefficients resulting in the minimum standard error can be used for design purposes. Step 11: Assessment of the calibration process After the above ten steps have been performed to establish the local calibration coefficients, they should be examined for reasonableness within each category of the experimental matrix and at different reliability levels. 2.5 CONCEPT OF RELIABILITY The Pavement-ME estimates the performance of a pavement using mechanistic models and transfer functions. Although these estimates are rational for pavement design purposes, the actual field measurements may show variability. This variability may come from the uncertainties in estimating the future traffic, material, and construction variability, measurement error, uncertainties due to the use of level 2 and 3 inputs, and errors associated with the model predictions. To incorporate all these variabilities, Pavement-ME uses a reliability-based design. Reliability for any prediction can be defined as the probability of getting a prediction lower than the threshold prediction over the design life, as shown in Equation (2-46). Reliability = P[distress⁡at⁡the⁡end⁡of⁡design⁡life < Critical⁡distress] (2-46) If 100 sections have been designed at 90% reliability, on average, ten of them may fail before the end of design life. Design reliability levels may vary by distress type and IRI or may remain constant for each. It is recommended that the same reliability be used for all performance indicators (51). Except for IRI, reliability for all other models is estimated using a relationship between the standard deviation of measured distress as the dependent variable and mean predicted distress as the independent variable. The basic assumption implies that the error in predicting the distress is normally distributed on the upper side of the prediction (not on the lower side or near zero values). Figure 2-2 shows an example of IRI prediction at 50% reliability (mean prediction), prediction at any desired reliability R, and are associated with the probability of failure. For 90 percent design reliability, the dashed curve at reliability R should not cross the IRI at the threshold criteria throughout the design analysis period. Failing to do so may lead to a failure at the required reliability and indicates that a design modification (such as a pavement thickness increase) should be applied. 31 Figure 2-2 Design Reliability Concept for Smoothness (IRI) 2.6 IMPACT OF CALIBRATION ON PAVEMENT DESIGN Several studies have been conducted to calibrate Pavement-ME transfer functions. Despite several calibration efforts, limited research is available on the effect of calibration on pavement design. Wu et al. (2014) calibrated the Pavement-ME models in Louisiana using Pavement-ME V1.3 (52). A total of 19 JPCP projects selected for this study had two base types: PCC over HMA and PCC over the unbound base. These 19 JPCP projects were designed using the Pavement-ME to estimate the effect on design thicknesses. The results showed that the Pavement-ME designs generated thinner PCC thicknesses (about 2 cm or 7%) compared to the AASHTO93 method (5). Tran et al. (2017) showed the effect of calibration on pavement design using the Missouri Department of Transportation (MoDOT) and Colorado Department of Transportation (CDOT) calibration results. One section, each for flexible and rigid pavement, was selected from existing MoDOT and CDOT projects. On average, the design thickness from local calibration was lower than that from the global model for both flexible and rigid sections (53). Mu et al. (2018) reviewed the effect of calibration on new JPCP design for seven states: Arizona, Colorado, Iowa, Louisiana, Missouri, Ohio, and Washington. The design thicknesses using global and local model coefficients were similar, such that five out of seven states had a difference of 13 mm or less. The Pavement-ME designs were thinner than AASHTO93 designs for high-traffic volume roads (by 50-70mm), whereas the thicknesses were similar for low-traffic volume roads. In rigid pavements, transverse cracking was the controlling distress for most cases 32 probability of failure ()reliabilityR = (1-)IRIavgIRIfailureIRI0mean predictionR = 50 percentprediction at reliability Rprobability of failure ()reliabilityR = (1-)IRIavgIRIfailureIRI0mean predictionR = 50 percentprediction at reliability R except for low-volume roads in Montana, where IRI was the critical distress (54). Singh et al. (2024) used the calibrated models in Michigan for pavement design to estimate the impact of calibration and for comparison with AASHTO93 designs. A total of 44 new flexible and rigid sections were designed. A comparison between AASHTO93 and Pavement-ME designs showed a reduction in HMA and PCC slab thicknesses for the latter approach. On average, the surface thicknesses using locally calibrated coefficients were thinner by 0.22 and 0.44 inches for flexible and rigid pavements, respectively. Critical design distresses for flexible pavements were bottom- up and thermal cracking. On the other hand, transverse cracking and IRI controlled the designs for rigid sections (55). 2.7 SENSITIVITY OF PAVEMENT-ME COEFFICIENTS SHAs often struggle to identify the most critical data collection needs since the Pavement-ME requires several design inputs. Several studies have conducted sensitivity analyses to determine the most sensitive inputs to the distress prediction models for new and rehabilitated pavements to address this issue (56-62). However, limited research is available to determine the impact of each calibration coefficient on the predicted pavement distress and performance. Kim et al. (2014) conducted a sensitivity analysis for all the Pavement-ME models, determining the sensitivity by changing coefficients one at a time (26). This study performed the analyses using two in-service pavements representing typical Iowa's HMA and JPCP sections. Each calibration coefficient varied from its global value by 20% to 50%. For JPCP, the study concluded that the fatigue model-related calibration coefficients (C1 and C2) in the transverse cracking model are the most sensitive parameters. For the JPCP IRI model, coefficients C1 (related to transverse cracking) and C4 (related to site factor) are sensitive. Coefficient C6 is the most sensitive for the faulting model. For flexible pavements, β2 and β3 are the most sensitive coefficients in fatigue cracking, whereas C1 and C2 are the most sensitive for IRI predictions. Dong et al. conducted a sensitivity analysis on calibration coefficients for the joint faulting model for JPCP sections in Ontario (12). The study also showed that C6 is the most sensitive coefficient, followed by C1 and C2. Both these studies quantified the sensitivity of coefficients using a sensitivity index (NSI) and a typical range of design inputs. Parameter estimation is needed whenever a model is fitted to data to explain a phenomenon and is usually considered the same as curve-fitting or optimization. However, both 33 are distinctly different. While the optimization only focuses on minimizing the sum-of-squares or any other error criterion considering the parameters unimportant, parameter estimation also considers the parameters' errors (63). According to Beck and Arnold, parameter estimation is "a discipline that provides tools for the efficient use of data in the estimation of constants that appear in mathematical models and for aiding in modeling phenomena" (64). Microsoft Excel's Solver® routine is used to estimate the parameters of a linear or nonlinear model but without computing the parameter errors, thus making it acceptable only for curve-fitting (63). However, according to Geeraerd et al., Solver® can accomplish parameter estimation if the sensitivity matrix is formulated and matrix multiplication is employed to compute the parameter errors (65). As per Dolan, the sensitivity matrix or Jacobian (J) is a matrix of the first derivatives of the model for each parameter and has the dimensions of n-by-p, where n and p are the numbers of data points and parameters, respectively (66). Thus, it is essential to know if any or all the parameters in a model are accurate and estimable, i.e., if they are statistically significant, they do not contain zero in the parameter confidence interval (CI). Hence, reporting the CI of any estimated parameter is equally important as the parameter errors. Parameter identifiability depends on the scaled sensitivity coefficients (SSCs) and the minimization of the objective function (63). The SSCs can help determine whether a parameter is estimable and inform about its accuracy in terms of relative error. Several studies have used SSCs in various applications (other than pavements) to estimate the sensitivity of a parameter on a continuous scale of the independent variable (63, 66, 67). The SSCs for the parameters are desired to be significant (the maximum value of SSC should be at least 10% of the largest value of the dependent variable) compared to the model η and uncorrelated with each other (63). The larger the SSC is for a parameter, the greater it will affect the model and the easier it will be to estimate. Moreover, the parameter with the largest SSC will also be the most accurate. However, suppose any of the SSCs are correlated, i.e., the ratio of SSCs of any two parameters is a constant (one is a linear function of the other); those parameters cannot be estimated together (only one can be calculated at a time) as the model η will respond to either of them identically. SSCs help assess a parameter's sensitivity on a continuous scale of the independent variable, highlight collinearity between coefficients, if any, and inform about the accuracy of the parameters, thus enhancing confidence in the parameter estimates. More importantly, determining SSC is a forward problem and does not require data, unlike NSI, which 34 requires the Pavement-ME design inputs (material, traffic, and climate). Overall, using SSC enhances confidence in the parameter estimates, leading to more reliable and informed decision- making in the analysis without data. 2.8 CHAPTER SUMMARY This chapter summarized the calibration approaches, efforts, and transfer function coefficients from different states. Most states used the least squares method to calibrate the Pavement-ME coefficients. Least squares is a simplistic and popular approach based on the NIID assumption. These assumptions may not hold good for non-normally distributed data. Studies in different engineering fields have highlighted the advantages and applicability of the MLE method. This chapter also outlines the transfer functions for different flexible and rigid pavement models. A step-by-step approach for local calibration is described per the local calibration guide. Pavement-ME uses a reliability-based design. The concept of reliability and its application in Pavement-ME design is explained. Several states have calibrated the Pavement-ME models to implement M-E design for local conditions. Despite several calibration efforts, the impact of calibration on pavement design has not been extensively evaluated. This chapter includes a literature review of studies that assessed the effect of calibration on pavement design. This consists of determining the design thicknesses and critical distress for pavement design. This chapter also includes a review of the sensitivity analysis of transfer function coefficients using the traditional NSI approach and describes the applicability of the SSC approach for sensitivity calculation. SSC has been widely used in different fields for parameter estimation and sensitivity calculations. 35 CHAPTER 3 - DATA FOR CALIBRATION 3.1 INTRODUCTION This chapter discusses the inputs and performance data used for the local calibration process. A crucial step in local calibration involves choosing enough pavement sections that accurately represent the prevailing conditions in the area. The next step is to gather the necessary data for each of the selected pavement sections, including information on the pavement performance, maintenance history, and various Pavement-ME inputs (material, traffic, and climate) that directly influence performance predictions. The predictions are then compared to the actual performance of the constructed pavement sections. A pavement section refers to a specific stretch of road corresponding to a construction project, which may include up to two sections (such as different directions on a divided highway) with similar data inputs but varying measured pavement performance, traffic, and initial IRI. The accuracy of the predicted pavement performance in the Pavement-ME software depends on the information used to describe the in- service pavement. Thus, several inputs are essential for analyzing a particular pavement in the design software, particularly those with significant impacts on the expected performance. This chapter outlines the process for selecting pavement sections for local calibration and the steps in obtaining the required information for each pavement section. First, the measured distresses from the MDOT PMS database were converted to Pavement-ME compatible units. Then, the time-series trends of all distress types were evaluated to identify potential projects for calibration. Also, these trends were explained, considering any significant maintenance activities over time. The information about maintenance activities over time will help to model a section in the Pavement-ME, i.e., whether an existing project should be considered a reconstruct or rehabilitated overlay project. The Pavement-ME inputs for these sections were also reviewed to obtain more updated or higher input levels. It's worth noting that a "project" refers to a specific job number in the construction records, while a "section" refers to multiple directions in a divided highway within a project. Hence, the number of sections is always greater than or equal to the number of projects. The project selection process, Pavement- ME inputs, and performance data have been summarized in this chapter. 36 3.2 MDOT PMS DATA MDOT's Pavement Management System (PMS) and other available construction data sources were reviewed to identify the available input levels, units of measured performance data, and best possible estimates. The PMS and other sources were assessed to extract the following data: a. Performance data were evaluated for their measurement process and units and converted to the Pavement-ME compatible units (wherever required). Necessary assumptions were made for these conversions. b. The construction records, plans, job-mix formula (JMF), and other sources were used to identify the pavement cross-sections and material properties during construction. Any unavailable data was acquired from MDOT, or MDOT provided test results for the best possible estimates. c. Traffic data were collected from the construction records and MDOT Transportation Data Management System (TDMS). Level 2 data were used for traffic data based on road type, number of lanes, and vehicle class 9 traffic percentage. d. For Asphalt concrete (AC) mix and binder properties, DYNAMOD software was used, which is based on laboratory tests for Michigan mixes. The most common construction materials in Michigan were used for base, subbase, and subgrade properties. e. For climatic data, the updated NARR files for Michigan have been used (68). 3.2.1 Pavement Condition Measures Compatibilities MDOT provided the PMS data from 1992 to 2019 (sensor data from 1998 to 2019). Biannually, MDOT obtains performance data on their pavement network by utilizing distress and laser-based measurements (sensors) for a 0.1-mile section. The information gathered on pavement distress in MDOT's PMS is categorized by distinct principle distress (PD) codes, where each PD code corresponds to a specific distress type (69). This pavement performance data was extracted for the selected projects and converted to Pavement-ME compatible units (where needed). In addition, MDOT personnel explained the distress calls made for the 2012 – 2017 data were only at the sampled locations (about 29.41% of any 0.1-mile segment of each control section). Therefore, it was suggested that a 0.2941 division factor be considered for those years of measured PMS data. 37 3.2.1.1. Selected distresses The MDOT PMS and sensor database were carefully analyzed, and relevant data were extracted to obtain the required distress information. The current distress manual of MDOT PMS was used to determine all the principle distress (PD) codes corresponding to the predicted distresses in the Pavement-ME. The earlier versions of the PMS manual were also reviewed to ensure accurate data was extracted for all the years. The necessary steps for PMS data extraction include: 1. Identify the PDs that correspond to the Pavement-ME predicted distresses 2. Extract PDs and sensor data for each project 3. Convert (if necessary) MDOT PDs to the units compatible with the Pavement-ME 4. Summarize time-series data for each project and each distress type Tables 3-1 and 3-2 summarize the identified and extracted pavement distresses and conditions for flexible and rigid pavements. This section also presents a detailed discussion of the conversion process for both flexible and rigid pavements. Table 3-1 Flexible pavement distress measurement by MDOT Flexible pavement distress IRI Top-down cracking Bottom-up cracking Thermal cracking Rutting Reflective cracking MDOT principle distresses (PDs) Directly measured 204, 205, 724, 725, 501 234, 235, 220, 221, 730, 731, 501 101, 103, 104, 114, 701, 703, 704, 110, 501 Directly measured No specific PD MDOT units in/mile miles miles No. of occurrences in None Pavement-ME units in/mile Conversion needed? No % area % area ft/mile in % area Yes Yes Yes No N/A Note: Bold numbers represent older PDs that are not currently in use; PD code 501 = No distress Table 3-2 Rigid pavement distress measurement by MDOT Rigid pavement distresses IRI Faulting Transverse cracking MDOT principle distresses Directly measured Directly measured 112, 113, 501 Note: PD code 501 = No distress MDOT units in/mile in No. of occurrences Pavement-ME units in/mile in Conversion needed? No Yes % slabs cracked Yes 38 3.2.1.2. Pavement distress unit conversion for HMA designs It should be noted that the Pavement-ME predicted distresses for the local calibration were only considered. The corresponding MDOT PDs were determined and compared with distress types predicted by the Pavement-ME to verify if any conversions were necessary. MDOT measures pavement distresses related to HMA pavements are listed in Table 3-1. PD code 501 corresponds to no distress condition and has been used in all distresses except rutting and IRI. The conversion process (if necessary) for all distress types is as follows: IRI: The IRI measurements in the MDOT sensor database are compatible with those in the Pavement-ME. Therefore, no conversion or adjustments were needed, and data could be used directly. Top-down cracking: Top-down cracking is load-related longitudinal cracking in the wheel path. The PDs 204, 205, 724, and 725 were assumed to correspond to the top-down cracking in the MDOT PMS database because those may not have developed an interconnected pattern that indicates alligator cracking. Those cracks may show an early stage of fatigue cracking, which could also be bottom-up. Since estimating such cracking based on the PMS data is difficult, these cracks were converted to % area crack and then categorized into bottom-up or top-down cracking based on the thicknesses. The PDs are recorded in miles and need conversion to % area. Data from the wheel paths were summed into one value and divided by the total project length, as shown in Equation (3-1). The lane width was assumed to be 12 ft. The typical wheel path width of 3 feet was assumed as recommended by the LTPP distress identification manual (70). %⁡𝐴𝐶𝑡𝑜𝑝−𝑑𝑜𝑤𝑛 = ⁡ Length⁡of⁡cracking⁡(miles) × width⁡of⁡wheelpaths⁡(feet) Length⁡of⁡section⁡(miles) × Lane⁡width⁡(feet) ⁡ × 100 (3-1) Literature shows that the AC thickness determines whether the crack initiates from the bottom or the top. Therefore, top-down cracking can be a primary distress based on AC layer thickness. The calculated top-down cracking using Equation (3-1) is assigned as either bottom-up or top- down based on the total AC layer thickness. If the thickness exceeds a certain threshold, the cracking is considered top-down cracking; otherwise, it is categorized as bottom-up cracking. These thicknesses were obtained by a mechanistic approach using Mechanistic Empirical Asphalt Pavement Analysis (MEAPA) software. MEAPA was run for different surface types using typical MDOT design inputs, and damage was calculated for the first 12 months for a 39 single axle load of 9000 lb. Threshold thicknesses were determined where the tensile strain at the top of the AC layer is higher than at the bottom. Table 3-3 presents the minimum threshold thicknesses for top-down cracking for each fix type. Table 3-3 Minimum thicknesses for top-down cracking Fix type HMA overlay on rubblized concrete HMA overlay on crushed and shaped HMA New or reconstruct Threshold thickness (in) 6 4 5 Bottom-up cracking: Bottom-up cracking is alligator cracking in the wheel path. The PDs 234, 235, 220, 221, 730, and 731 match this requirement in the MDOT PMS database. The PDs have units of miles; however, to make those compatible with the Pavement-ME alligator cracking units, conversion to the percent of the total area is needed. This can be achieved by using the following Equation (3-2): %𝐴𝐶𝑏𝑜𝑡𝑡𝑜𝑚−𝑢𝑝 = Length of cracking (miles) × width of wheelpaths (feet) Length of section (miles) × Lane width (feet) × 100 (3-2) The widths of each wheel path and lane were assumed to be 3 feet and 12 feet, respectively. The LTPP distress identification manual recommends a typical wheel path width of 3 feet (70). Thermal cracking: Thermal cracking corresponds to transverse cracking in flexible pavements. The transverse cracking is recorded as the number of occurrences, but the Pavement-ME predicts thermal cracking in feet/mile. To convert transverse cracking into feet/mile, the number of occurrences was multiplied by 3 feet for PDs 114 and 701 because these PDs are defined as "tears" (short cracks) that are less than half the lane width. For all other PDs, the number of occurrences was multiplied by the lane width (12 ft). All transverse crack lengths were summed and divided by the project length to get feet/mile, as shown in Equation (3-3). 𝑇𝐶 = ∑ No.⁡⁡of⁡Occurrences⁡×Lane⁡Width⁡(ft⁡) Section⁡Length⁡⁡(miles⁡) (3-3) Thermal cracking predictions in the Pavement-ME are restricted to a maximum value of 2112 ft/mile due to a minimum crack spacing limit of 30 feet. This means Pavement-ME predictions at 50% reliability cannot exceed 2112 ft/mile. Due to this limitation and ARA recommendations, a 2112 ft/mile cutoff was decided where any measured data for a section above 2112 ft/mile was not used for calibration. 40 Rutting: This is the total amount of surface rutting all the pavement layers and unbound sub- layers contribute. The average rutting (left & right wheel paths) was determined for the entire project length. No conversion was necessary. It is assumed that the measured rutting corresponds to the total surface rutting predicted by the Pavement-ME. 3.2.1.3. Pavement distress unit conversion for rigid designs For rigid sections, transverse cracking requires unit conversion. For all other distresses, MDOT records them in the Pavement-ME compatible units. Table 3-2 summarizes the distresses related to rigid sections, and the conversion process is discussed below: IRI: The IRI in the MDOT sensor database does not need any conversion; the values were used directly. Faulting: In the Pavement-ME, faulting is predicted as average per joint. MDOT's sensor data records the number of faults (FaultNum), average faulting (avgFault), and the maximum faulting (FaultMax) for every 0.1-mile segment. The faulting values had some inconsistencies. For the years between 2000 and 2011, faulting values are maximum fault callouts only (not average values). For 2012 and after, both average and maximum fault values are available. A correlation was developed between the maximum and average faulting values using data from 2013 to 2017 to resolve this issue. These correlations were used to estimate the average faulting from 2000 to 2011. Table 3-4 shows the regression equations between average and maximum faulting using the data from 2013 to 2017. These equations are based on the number of faults. It is important to note that ideally, the number of faults cannot be greater than the number of joints, but the number of faults in the database has records where they are more than the number of joints. These pseudo-fault values might come from cracking, spalling, bridge segments, etc. Therefore, the maximum number of fault counts was restricted to 36, and the average faulting to 0.4 inches to address this issue. Accordingly, any 0.1-mile section above these restricted faulting values was omitted from the calibration data. 41 Table 3-4 Correlation equations based on the number of faults FaultNum From 0 2 5 41 To 1 4 40 ALL Equation (y is avgFault, x is FaultMax) y=x y = 0.3438x + 0.03 y = 0.2132x + 0.0377 y = 0.0936x + 0.0777 R-squared (2013-2017data) 1 0.7189 0.6074 0.2476 The average joint faulting is calculated based on the number of faulting in a 0.1-mile section. It is assumed that if the number of faults is less or equal to the number of joints, faulting occurs at the joints only. In that case, the faulting unit conversion equation is as shown in Equation (3-4). If, for any 0.1-mile section, the number of faults is greater than the number of joints, that section is removed (cut) from the calibration data, as previously mentioned. 𝐹𝑎𝑢𝑙𝑡 = FAULnum⁡×FAULi Njoints (3-4) where, FAULnum = Number of faults in a 0.1 mile FAULi =(FAULT_(Avg_Right⁡) ⁡⁡ + ⁡FAULT_(Avg_Left⁡))/2 = Average faulting in a 0.1 mile (inches) Njoints is the number of joints in 0.1-mile (528 ft) segments, i.e., Njoints=528/Joint Spacing. Transverse cracking: The transverse cracking distress is predicted as the percentage of slabs cracked in the Pavement-ME. However, MDOT measures transverse cracking as the number of transverse cracks. PDs 112 and 113 correspond to transverse cracking. The estimated transverse cracking must be converted to the percent slabs cracked using Equation (3-5). % Slabs Cracked = ⁡ ∑ 𝑃𝐷112,113 Section Length (miles)×5280𝑓𝑡 Joint Spacing (ft) ( ) ⁡ × 100 (3-5) 3.2.2 Condition Database for Local Calibration Customized databases were created to efficiently analyze the condition of selected Pavement Distresses (PDs), which included distress and sensor data for multiple years. These databases were compiled using Microsoft Access and allowed for easy extraction of relevant data for projects of any length. The PMS condition data from 1992 to 2019 and sensor data from 1998 to 2019 were included in these databases. MATLAB codes were used to extract performance data 42 for a section of the given length. For divided highways, which can have an increasing and decreasing direction to indicate north/south or east/west bounds, both directions were included in the time-series data and considered separate sections. In contrast, distress data was collected in one direction for undivided highways. 3.3 PROJECT SELECTION CRITERIA For local calibration, selecting in-service pavement sections that represent local pavement design, currently used materials, construction practices, and performance is essential. A set of project selection criteria was established to identify and choose these representative pavement sections. This approach ensured that the selected pavement sections met the required standards and could accurately represent Michigan's pavement network. The process for identifying and selecting pavement sections consists of the following steps: 1. Determine the minimum number of pavement sections required for calibration based on the statistical requirements. 2. Identify all available in-service pavement projects. 3. Extract all pavement distresses (pavement condition data) from the customized database for all identified projects in Step 2. 4. Evaluate the measured performance for all the identified projects. 5. Identify projects with adequate data, age, trend, and the Pavement-ME inputs available to develop a refined list. 3.3.1 Identify the Minimum Number of Required Pavement Sections The MEPDG local calibration guide provides a method to evaluate the minimum number of required sections for each distress type. The minimum number of sections was calculated using Equation (3-6), and the results are summarized in Table 3-5 for each condition measure. The total number of projects available in Table 3-5 are combined projects from the previous calibration study (10) and newly selected projects from the current calibration effort. where; 𝑛 = ( 𝑍𝛼/2 × 𝜎 𝑒𝑡 2 ) 43 (3-6) n  et = The z-value from a standard normal distribution = Minimum number of pavement sections = Performance threshold = Tolerable bias = Standard error of the estimate Table 3-5 Minimum number of sections for local calibration Performance Model Nationally calibrated SEE Z90 Threshold N (required number of sections) Number of sections used Total number of projects available Fatigue, bottom-up (%) Fatigue, top-down (ft/mile or %) Thermal cracking (ft/mile)1 Rutting (in) IRI (in/mile) Transverse cracking (%) Joint faulting (in) IRI (in/mile) 5.01 583 - 0.107 18.9 4.52 0.033 22 Flexible Pavements 20% 2000 or 20% 1.64 1000 0.5 172 Rigid Pavements 15 0.125 172 1.64 16 12 - 22 83 11 14 61 78 133 133 200 178 48 79 48 163 46 Note: Fatigue top-down has been updated in the recent Pavement-ME V2.6. It is expressed in ft/mile for the old model and in % for the updated model. N= minimum number of samples required for a 90% confidence level 1. No SEE, threshold, or N was reported for thermal cracking in the literature 3.3.1 Initial Projects Selection The common pavement types in Michigan include: 1. HMA reconstruct 2. HMA over crush & shaped existing HMA 3. HMA over rubblized PCC 4. JPCP reconstruct It is important to note that HMA over crushed and shaped existing HMA and HMA over rubblized existing PCC projects were analyzed as new reconstructed pavement. Sections were selected for the local calibration based on performance trends and to accommodate a wide range of different inputs, including layer thicknesses, traffic, region, etc. 44 /2Z2ZSEESEE MDOT provided a comprehensive database consisting of all the projects constructed in Michigan. Initially, all existing projects used in previous calibration efforts were reviewed, and additional performance data were extracted where possible. Additional projects were identified that can be potential candidates for the current local calibration effort. The PMS data extraction was completed for all required distress types in a compatible format with the Pavement-ME software. The time series for each pavement section's performance measures was observed to finalize the preliminary list of new potential candidate projects. To ensure a robust and appropriate set of data, the criteria used to identify additional performance data and the selection of new potential pavement projects include: • The pavement section must have at least three measured data points over time. There are some exceptions to this criterion. Bottom-up cracking has relatively fewer data points; some sections with even two points have been included, considering further data points will be collected in the future. The same process was followed for transverse cracking in rigid sections. As previously explained, joint faulting and thermal cracking have been cut at specific values, so these data points are omitted from the calibration database. • At least one of the distresses should have an increasing trend. Any section with decreasing and no or flat trends over time was excluded from the list. • The previous maintenance history was observed for all pavement sections to explain any decrease or flat trend in the time series plot. If there were any major rehabilitation or reconstruction activities, the measured data from the year traffic opened initially to the very last year until the major repair took place are considered. • The last recorded point should have a Distress Index (DI) of at least 5 for a section. DI is calculated by taking a weighted average of different distress types. DI was observed and limited to ensure sufficient distress for calibration and to capture adequate pavement performance trends. Figures 3-1 and 3-2 illustrate examples of distress progressions for a selected and omitted flexible pavement section. The top-down cracking for the initial project selection was evaluated in feet/mile and later converted to a percentage. Similarly, Figures 3-3 and 3-4 present examples of the selected and omitted rigid pavement sections. The vertical dashed red line is the last reported construction, whereas the dotted blue line in the DI plot indicates reported maintenance activities. For example, Figure 3-1 shows the vertical dotted blue line in the DI plot that shows a 45 cold mill and resurface (CM&R) treatment was applied in 2012. In the same figure, the effect of this rehabilitation event can be noticed with a drop in measured distress in individual distress plots. Therefore, in this case, pavement section performance can be considered from 2001 to 2011. It should be noted that generally, minor maintenance [e.g., crack treatment (CT) or joint sealing (JS)] does not affect the time series trend since these minor maintenances represent non- structural fixes. Note that time series plots for rutting show a consistent drop in the 2012-2013 collection years, regardless of whether any maintenance is reported. This is likely due to changes in the data collection process or vendor differences. Based on the criteria mentioned above, a total of 256 flexible sections and 88 rigid sections were initially selected. The performance of the chosen pavement sections was compared with all sections available in the MDOT database (2081 flexible sections and 442 rigid sections) to verify if the chosen sections represent the overall pavements in Michigan. Sections with at least three available data points are considered. Each section was categorized as good, fair, or poor performing based on the performance trend lines modified to reflect Michigan conditions (10). These trend lines are available only for bottom-up cracking, total rutting, and IRI for flexible sections, as well as transverse cracking and IRI for rigid sections. The performance categories depend on the measured performance trend relative to the reference lines. If the measured performance is below the good performance line, it is categorized as a good performing section, between the good and poor line, as fair, and above the poor performance line, as the poor performing section. The performance category was decided based on a previous calibration study (10). When the performance trend passes through more than one category zone, the zone with the maximum points is considered the performance category for that section. Also, the low-performance category is selected in case of an equal number of points for two different categories. Figures 3-5 and 3-6 show example sections for good, fair, and poor categories for IRI performance for flexible and rigid sections, respectively. 46 Figure 3-1 Example of selected flexible section Figure 3-2 Example of an omitted flexible section 47 Figure 3-3 Example of a selected rigid section Figure 3-4 Example of an omitted rigid section 48 A similar method was followed for categorizing sections based on all other distresses. Figures 3- 7 and 3-8 show the distribution of good, fair, and poor sections for rigid and flexible sections based on different distress criteria. Figures 3-7 and 3-8 show that the selected sections satisfactorily represent MDOT all sections for both flexible and rigid pavements. (a) Good section (b) Fair section Figure 3-5 Categorization of flexible sections based on performance trends (c) Poor section The initially selected projects were further refined based on performance, availability of inputs, and initial IRI. The performance data for these initially selected sections is the average for the entire section length. This data is calculated by averaging the performance for every 0.1-mile segment in the project length. Data for every 0.1 mile has been reviewed to estimate performance data extent and reasonableness. Figures 3-9 to 3-13 show performance data for every 0.1-mile segment with years for all flexible sections. 49 (a) Good section (b) Fair section Figure 3-6 Categorization of rigid sections based on performance trends (c) Poor section 50 (a) Transverse cracking (selected sections) (b) Transverse cracking (MDOT all sections) (c) IRI (selected sections) (d) IRI (MDOT all sections) Figure 3-7 Comparison of selected rigid sections with all MDOT sections 51 (a) Bottom-up cracking (selected sections) (b) Bottom-up cracking (All MDOT sections) (c) Total rutting (selected sections) (d) Total rutting (All MDOT sections) (e) IRI (selected sections) (f) IRI (All MDOT sections) Figure 3-8 Comparison of selected flexible sections with all MDOT sections 52 Figure 3-9 Bottom-up cracking at every 0.1-mile segment for flexible sections Figure 3-10 Top-down cracking at every 0.1-mile segment for flexible sections 53 Figure 3-11 Thermal cracking at every 0.1-mile segment for flexible sections Figure 3-12 Rutting at every 0.1-mile segment for flexible sections 54 Figure 3-13 IRI at every 0.1-mile segment for flexible sections Figures 3-14 to 3-16 show the raw performance data for all rigid sections. As previously noted, 2112 ft/mile and 0.4 inches cutoff values were adopted for thermal cracking and joint faulting, respectively. These values were selected based on the raw (0.1-mile segment) data, limitations of the Pavement-ME models, and consensus with MDOT. Moreover, sections with Superpave mixes are only used to calibrate the thermal cracking model to have consistent Level 1 input in the Pavement-ME. Figure 3-14 Transverse cracking at every 0.1-mile segment for rigid sections 55 Figure 3-15 Joint faulting at every 0.1-mile segment for rigid sections Figure 3-16 IRI at every 0.1-mile segment for rigid sections 3.4 SELECTED SECTION PERFORMANCE DATA SUMMARY The measured performance data was extracted for each project, and the necessary conversions were made to ensure compatibility with the Pavement-ME predicted performance, as discussed in Section 3.2. The level of distress was assessed in all pavement sections identified for local 56 calibration. The calibration process entails comparing each chosen project's predicted and measured performance. To have a robust local calibration, the levels of distress must fall within a reasonable range (i.e., above and below threshold limits for each type of distress). Therefore, the distress levels for all projects were compiled and analyzed to determine their respective ranges. This section summarizes the observed performance for the selected flexible and rigid pavement sections. Efforts were undertaken to gather sufficient information to achieve a precise and dependable local calibration of the performance models. Due to changes in construction practices and/or data availability, most sections are less than 20 years old, so it is expected that most sections do not have poor performance or exceed performance thresholds. Furthermore, these represent the average values of the Pavement-ME prediction using 50% reliability. When designing, a higher reliability factor is applied to account for project variability (including climate, traffic, material, and construction), increasing the resulting distress values. Therefore, while designs will correlate with the calibration sections, it should not be anticipated that pavement designs will exactly match the sections used in calibration because of the increased reliability factor. 3.4.1 Flexible Performance Data The magnitude and age distribution for the HMA reconstruct sections (also includes crush and shape and HMA over rubblized PCC) are shown in Figures 3-17 to 3-21. The following observations were made: • Bottom-up cracking: Bottom-up cracking magnitudes are usually low for most sections, with only a seven crossing the threshold of 20% with a maximum of almost 40%. The maximum age ranges from 4 to 20 years. Most sections fall in the good category, as shown in Figure 3-8. • Longitudinal/top-down cracking: Top-down cracking is observed more frequently than bottom-up cracking. More sections have observed top-down cracking compared to bottom-up cracking. The age at maximum distress ranged from 5 to 20 years. • Thermal cracking: Higher thermal cracking values are observed, ranging up to 4000 ft/mile. The design threshold used by MDOT is 1000 ft/mile. The age at which the maximum thermal cracking is observed ranges from 5 to 19 years. Sections with performance grade (PG) binders have been used for thermal cracking calibration. 57 • Rutting: Selected sections do not exhibit significant rutting. All sections were below the threshold of 0.5 inches. The age distribution ranged from 3 to 19 years. Two-thirds of the sections are in the fair performance category, as shown in Figure 3-8. • IRI: The IRI time series is usually flat, with no sections exceeding the 172 in/mile threshold. The maximum observed IRI is 168.5 in/mile. The age at maximum IRI ranged from 5 to 20 years. It is worth noting that a cutoff value of the initial IRI less than or equal to 77 in/mile is selected to calibrate the IRI model. 74% of sections are in good, followed by 25% of sections in fair category. Only 1% of sections showed poor performance. 3.4.2 Rigid Performance Data The magnitude and age distribution for the JPCP rehabilitation projects are shown in Figures 3- 22 to 3-24. The following observations can be made from the figures: • Transverse cracking: A maximum transverse cracking value of 85% is observed, with five sections crossing the distress threshold of 15% slabs cracked. The age distribution ranges from 4 to 20 years. About 72% of these sections fall under the fair performance category, as shown in Figure 3-7. • Transverse joint faulting: Ten sections exceed the joint faulting threshold of 0.125 inches, with a maximum value of 0.17 inches. The age distribution ranges from 8 to 20 years. These observed values for joint faulting have been cut off at 0.4 inches, where a 0.1-mile segment is above 0.4 inches. • IRI: A maximum IRI of 167 in/mile was observed. The age at maximum IRI ranges from 5 to 20 years. It is worth noting that a cutoff value for the initial IRI less than or equal to 82 in/mile is used to calibrate the IRI model. All sections fall under good and fair categories, with none exhibiting poor performance, as shown in Figure 3-7. 58 (a) Time series (b) Age distribution Figure 3-17 Selected flexible sections — Bottom-up cracking (a) Time series (b) Age distribution Figure 3-18 Selected flexible sections — Top-down cracking (a) Time series (b) Age distribution Figure 3-19 Selected flexible sections— Transverse (thermal) cracking 59 (a) Time series (b) Age distribution Figure 3-20 Selected flexible sections — Total rutting (a) Time series (b) Age distribution Figure 3-21 Selected flexible sections — IRI (a) Time series (b) Age distribution Figure 3-22 Selected rigid sections — Transverse cracking 60 (a) Time series (b) Age distribution Figure 3-23 Selected rigid sections — Joint faulting (a) Time series (b) Age distribution Figure 3-24 Selected rigid sections — IRI 3.5 INPUT DATA EXTENT Accurate pavement cross-sectional, traffic, climate, and material input data are essential for adequately characterizing as-constructed pavements since the information directly affects performance prediction accuracy in the Pavement-ME software. Due to the large number of inputs required to characterize a pavement in the Pavement-ME, input data collection can be time-consuming. Moreover, many critical input parameters have three input levels within the Pavement-ME hierarchical structure. The process of collecting as-constructed input data, including the source of the data, how to address missing data, and the selection of input values, is discussed in this section. The best available input level was used for the selected pavement sections. 61 3.5.1 Pavement Cross-Section The pavement cross-sectional information is necessary to characterize the layer thicknesses of the various layers. The cross-sectional information is obtained from the construction records. Typically, in the case of HMA pavements, the drawings provided the asphalt application rate of the HMA layers (dividing the application rate by 110), which was used to determine the HMA lift thicknesses in inches. For the sections used in the previous calibration effort (10), the Pavement-ME inputs data sheet was used to extract design inputs. MDOT provided the drawings (construction plans) for the newly selected sections. The thickness, mix type, traffic, and unbound layer information were included in these drawings. A summary of the design thicknesses for flexible and rigid selected pavement projects is shown in Tables 3-6 and 3-7. Table 3-6 Average flexible pavement thicknesses HMA top course thickness (in.) 1.6 1.6 1.5 1.6 1.6 HMA leveling course thickness (in.) 1.9 2.1 2.1 2.0 2.0 HMA base course thickness (in.) Base thickness (in.) Subbase thickness (in.) 2.0 4.5 3.2 3.0 3.1 7.5 7.1 6.6 3.8 5.7 20.5 16.8 16.4 11.1 15.0 Pavement types Crush and Shape Freeway Non-freeway Rubblized Statewide Average Table 3-7 Average rigid pavement thicknesses Pavement type JPCP Average PCC thickness (in.) 11.4 Average base thickness (in.) 6.9 Average subbase thickness (in.) 12.1 3.5.2 Traffic Inputs The traffic data is a critical input to the Pavement-ME. Level 2 traffic data was used for all sections. MDOT provided a spreadsheet with traffic distribution tables, which was used to extract Pavement-ME inputs for traffic. These tables include: • Vehicle class distribution • Hourly distribution (only for rigid sections) • Monthly adjustment factor • Number of axles per truck • Single axle load spectra 62 • Tandem axle load spectra • Tridem axle load spectra • Quad axle load spectra The inputs (with input categories) required to obtain these tables are summarized in Table 3-8. Table 3-8 Traffic input categories Inputs Categories Percentage of vehicle class 9 Region type COHS type Number of lanes • Less than 45 • 45 to 70 • Above 70 • Rural • Urban • National • Regional • Statewide • 2 • 3 • 4+ The number of lanes was identified from the plans. Wherever the number of lanes was unavailable, they were visually estimated utilizing Google Maps coordinates. The COHS (Corridors of Highest Significance) type was estimated using each project's PR number and beginning and ending milepost. The percentage of class 9 vehicles was estimated for each section using the MDOT Transportation Data Management System (TDMS) website from the following URL: https://mdot.public.ms2soft.com/tcds/tsearch.asp?loc=mdot. For sections where the traffic data was unavailable at the exact location, nearby locations in the same section were used. The range and average two-way AADTT values for all flexible and rigid sections are summarized in Table 3-9. Table 3-9 Ranges of AADTT for all reconstruct projects Road Type Crush and Shape Rubblized HMA Reconstruct (Freeway) HMA Reconstruct (Non-freeway) JPCP Reconstruct Statewide Average Min AADTT Max AADTT Average AADTT 60 173 313 63 150 134 1986 3707 6745 1600 18297 6502 669 1502 2076 431 7141 2381 63 3.5.3 As-constructed Material Inputs The as-constructed material inputs were obtained from the construction records, JMFs, and other test records. Ideally, these inputs are to be recorded at the time of construction. These inputs range between project-specific and statewide average values. This section details the material properties of each pavement structural layer. 3.5.3.1. HMA layer inputs All inputs were collected at the highest hierarchy level; however, the needed data were unavailable for all pavement sections. In that case, the data was collected using other correlations/sources. Data collection for each HMA layer input is as follows: • Dynamic modulus (E*): E* was obtained from the DYNAMOD software developed in a previous study (71). E* for the Superpave mixes was directly obtained from the database. For older mixes (marshal mixes), the volumetric, binder, and gradation information was used to predict the E* using DYNAMOD's Artificial Neural Networks (ANNs). E* was obtained at Level 1. • Binder (G*): G* was also obtained from the DYNAMOD database using the region and binder information. G* was obtained at Level 1. • Creep compliance (D(t)): D(t) was obtained from the DYNAMOD database. D(t) was obtained at Level 1 for Performance grade (PG) sections and Level 3 for other sections. • Indirect tensile strength (IDT): IDT was obtained from the DYNAMOD database at Level 2 for Performance grade (PG) sections and Level 3 for other sections. • AC layer thickness: These were obtained from construction records. Usually, the application rate in lbs/yards2 is available, which can be utilized to obtain the layer thickness, as previously mentioned. • Air voids and binder content: As constructed air voids and binder content were obtained from construction records. Table 3-10 summarizes the average as-constructed air voids for different pavement types. Historical test records were utilized for unavailable data to obtain an average value based on mix type, as shown in Table 3-12. • Aggregate gradation: Gradation was obtained from JMFs. Tables 3-11 summarize the average gradation for the top, leveling, and base layers, respectively, for different 64 pavement types. Historical test records were utilized for unavailable data to obtain an average value based on mix type, as shown in Table 3-12. It is important to note that Level 1 G* and Level 2 IDT data were used to calibrate the thermal cracking model. Table 3-10 As-constructed percent air voids for HMA layers HMA layer Top course Leveling course Base course Road Type Crush and Shape Rubblized HMA Reconstruct Freeway HMA Reconstruct Non-freeway Crush and Shape Rubblized HMA Reconstruct Freeway HMA Reconstruct Non-freeway Crush and Shape Rubblized HMA Reconstruct Freeway HMA Reconstruct Non-freeway Average as-constructed air voids 6.1 6.8 6.6 6.8 6.2 6.4 6.7 6.7 5.8 5.8 6.4 6.8 Table 3-11 HMA layer average aggregate gradation Percent passing sieve size 3/4 100.0 99.4 100.0 100.0 100.0 100.0 99.8 100.0 99.6 99.3 95.8 98.9 3/8 89.7 89.8 92.4 94.6 81.8 87.0 81.3 82.6 77.9 78.9 72.9 76.6 #4 68.4 67.3 67.4 71.4 61.1 67.8 63.3 73.4 60.3 59.9 51.6 57.5 #200 5.2 5.9 5.2 5.3 5.0 5.2 4.8 4.8 4.6 4.8 4.9 4.9 HMA Layer Road type Top course Leveling course Base course Crush and Shape Rubblized HMA Reconstruct (Freeway) HMA Reconstruct (Non-freeway) Crush and Shape Rubblized HMA Reconstruct (Freeway) HMA Reconstruct (Non-freeway) Crush and Shape Rubblized HMA Reconstruct (Freeway) HMA Reconstruct (Non-freeway) Effective AC binder content 11.5 11.9 11.2 11.1 10.6 11.2 10.1 10.2 10.8 10.6 9.4 9.6 65 Table 3-12 MDOT recommended values volumetrics and gradation Mix type Air voids (%) 3E1 4E1 5E1 2E3 3E3 4E3 5E3 2E10 3E10 4E10 5E10 2E30 3E30 4E30 5E30 5.8 6.1 6 4.8 5.8 6.1 6 4.8 5.8 6.1 6 4.8 5.8 6.1 6 Effective binder content (%) 10.8 11.5 12.6 9.7 10.8 11.5 12.6 9.7 10.8 11.5 12.6 9.7 10.8 11.5 12.6 % Passing 3/4" Sieve % Passing 3/8" Sieve % Passing # 4Sieve % Passing #200 Sieve 99.85 100.00 100.00 92.65 99.63 100.00 100.00 94.55 99.78 100.00 100.00 99.00 99.95 100.00 100.00 80.44 87.24 97.14 68.70 77.88 86.91 97.86 73.50 80.27 87.65 98.30 71.80 79.20 88.63 99.00 62.94 70.43 78.23 53.95 60.33 68.66 79.81 59.70 62.78 70.06 81.27 60.60 59.82 66.90 81.24 4.40 5.11 5.63 4.40 4.56 4.92 5.49 4.50 4.84 5.26 5.67 4.20 4.40 4.33 5.68 3.5.3.2. PCC material inputs The Pavement-ME transverse cracking prediction model is very sensitive to concrete strength (compressive or flexural). The PCC material-related inputs were obtained from material testing results. If these results were unavailable, typical MDOT values were used. PCC strength: MDOT collected the concrete core compressive strength (f'c) test data. These tests represent the concrete compressive strength close to the construction time for the selected pavement sections. These test values were used directly for each corresponding project. If compressive strength is unavailable, an average value of 5239 psi was used. This is an average value obtained from the sections with available values. The transverse cracking model in the Pavement-ME directly uses the modulus of rupture (MOR) to estimate the damage. The MOR values were calculated based on the ACI correlation between MOR and f'c (used in the Pavement-ME), as shown by Equation (3-7). Figure 3-25 shows the f'c and estimated MOR distributions. It should be noted that these cores' specific testing age was unavailable; however, all cores were tested after or at least 28 days. The Pavement-ME internally calculates the relationship between f'c and MOR. 𝑀𝑂𝑅 = 9.5 × √𝑓𝑐 ′ (3-7) 66 (a) Compressive strength (f'c) (b) Modulus of rupture (MOR) Figure 3-25 Distribution of concrete strength properties Coefficient of thermal expansion: The CTE input values were obtained from the MDOT recommended values (72). A value of 4.4 in/in/°F×10-6 was used for Bay, Grand, North, Southwest, and Superior regions, whereas 5.0 in/in/°F×10-6 was used for Metro and University regions. 3.5.3.3. Aggregate base/subbase and subgrade input values The aggregate base/subbase and subgrade input values were obtained from the following sources: • Backcalculation of unbound granular layer moduli (73) • Pavement subgrade MR design values for Michigan's seasonal changes (74) The resilient modulus (MR) values for the base and subbase material were selected based on the results from previous MDOT studies (73, 74). The typical backcalculated values for base and subbase MR is 33,000 psi and 20,000 psi, respectively. It is worth noting that crushed and shaped and rubblized sections have been modeled as new flexible pavements. The existing layer has been modeled as a dense aggregate base with an MR of 125,000 psi for crush and shape and 70,000 for rubblized sections. These values were assumed to be the same for all projects since in- situ MR values were unavailable. For base/subbase layers, the software default to "Modify input values by temperature/moisture" was selected. The subgrade material type and resilient modulus were selected based on the Subgrade MR study (73, 74). The study outlined the location of specific soil types and their MR values across the entire State. Annual representative values for 67 subgrade MR were used in Pavement-ME. The recommended design MR value corresponding to the soil type is shown in Table 3-13. Table 3-13 Average roadbed soil MR values Roadbed Type Average MR USCS AASHTO SM SP1 SP2 SP-SM SC-SM SC CL ML SC/CL/ML A-2-4, A-4 A-1-a, A-3 A-1-b, A-3 A-1-b,A-2-4, A-3 A-2-4, A-4 A-2-6, A-6,A-7-6 A-4, A-6, A-7-6 A-4 A-2-6, A-4, A-6, A-7-6 3.5.4 Climatic Inputs Laboratory- determined (psi) 17,028 28,942 25,685 21,147 23,258 18,756 37,225 24,578 26,853 Back- calculated (psi) 24,764 27,739 25,113 20,400 20,314 21,647 15,176 15,976 17,600 Recommended design MR value (psi) 5,200 7,000 6,500 7,000 5,000 4,400 4,400 4,400 4,400 The Enhanced Integrated Climatic Model (EICM) in Pavement-ME requires hourly climatic data. This data includes air temperature, precipitation, relative humidity, percent sunshine, and wind speed. A statistical comparison between Modern-Era Retrospective Analysis for Research and Applications (MERRA) and North American Regional Reanalysis (NARR) data was performed to identify the most suitable climatic data for calibration. Both MERRA and NARR data files used include climatic information for different periods. For that purpose, a common temporal overlap of 13 years was identified for which continuous hourly data is available for all climatic files from September 2000 to September 2013. The MERRA stations falling in the lake region were removed from the database. Moreover, the four closest MERRA stations were identified for each NARR station, and the weighted average (proportional to the distance) for all four stations based on their distances was used for comparison. A total of 29 NARR stations and the four closest corresponding MERRA stations to each have been compared. Table 3-14 shows the SEE, bias, and correlation coefficient (R) between MERRA and NARR for hourly, daily, and monthly data (75). MDOT has been using default Pavement-ME climate data and ground-based climate automated surface observation systems (ASOS) data. This data was reviewed for 68 errors/anomalies and was improved in MDOT's previous study (68). The following observations were made based on the comparison and previous study (68, 75): • MERRA and NARR climatic data are comparable for air temperature followed by humidity and wind speed. Percent sunshine showed a low correlation, and precipitation data is significantly different (i.e., a very low correlation) among all climatic inputs. • The predicted pavement performance using MERRA-2 and NARR climatic data showed good agreement except for thermal cracking in flexible pavement and transverse cracking in rigid pavements. These differences are expected mainly because of sunshine data. • MERRA has anomalies in humidity data. Several humidity values were erroneously higher than 100. • MERRA appeared to be incorrectly estimating precipitation. Specifically, the number of wet days was extremely high, such that the data review showed wet event days in the data on actual dry days. The ground-based stations are more closely aligned with actual wet event days. Furthermore, it was unclear why the percent sunshine was significantly different. Table 3-14 Descriptive statistics for MERRA and NARR data comparison Climatic input Humidity Precipitation Sunshine Temperature Wind speed Descriptive statistics Mean Std. Dev. COV Mean Std. Dev. COV Mean Std. Dev. COV Mean Std. Dev. COV Mean Std. Dev. COV SEE 12.784 0.726 5.68% 0.049 0.005 10.85% 44.614 3.908 8.76% 3.924 0.548 13.98% 3.318 0.946 28.52% Hourly Bias 4.437 2.230 50.27% 0.002 0.000 15.22% -1.457 6.809 -467.39% -0.771 0.766 -99.43% -0.165 1.700 -1029.25% R 0.764 0.035 4.60% 0.062 0.022 34.59% 0.411 0.071 17.27% 0.982 0.006 0.58% 0.752 0.100 13.25% SEE 9.582 1.014 10.58% 0.009 0.001 7.90% 29.317 2.777 9.47% 2.710 0.436 16.08% 2.031 1.097 54.00% Daily Bias 4.437 2.230 50.27% 0.002 0.000 15.22% -1.457 6.809 -467.39% -0.771 0.766 -99.43% -0.165 1.700 -1029.25% R 0.705 0.055 7.86% 0.610 0.045 7.33% 0.570 0.079 13.84% 0.992 0.003 0.31% 0.863 0.105 12.16% SEE 7.387 1.283 17.37% 0.002 0.000 11.21% 11.847 1.788 15.09% 1.837 0.428 23.32% 1.470 1.145 77.92% Monthly Bias 4.437 2.230 50.27% 0.002 0.000 15.22% -1.457 6.809 -467.39% -0.771 0.766 -99.43% -0.165 1.700 -1029.25% R 0.538 0.145 26.96% 0.678 0.059 8.73% 0.821 0.033 4.04% 0.997 0.002 0.20% 0.848 0.145 17.10% Note: 𝑆𝑆𝐸 = √∑(𝑀𝐸𝑅𝑅𝐴−𝑁𝐴𝑅𝑅)2 𝑛−2 ; 𝐵𝑖𝑎𝑠 = ∑(𝑀𝐸𝑅𝑅𝐴−𝑁𝐴𝑅𝑅) 𝑛 In the previous study, additional weather stations were added to improve the climate coverage using ASOS and the Michigan Road Weather Information System (RWIS) as potential data sources (68). Moreover, additional years of climatic data were added from February 2006 to December 2014 to enhance the data. Since the predicted performance did not show significant differences and the NARR data was improved for Michigan climate, the improved MDOT 69 NARR climatic files were used for climatic inputs for both flexible and rigid pavements. The files were downloaded as *.hcd files, which can be read directly in Pavement-ME. The closest weather station to each selected project was used. Table 3-15 Michigan climate station information HCD filename 4847 94849 94889 14815 94871 14822 94847 14853 14826 4854 94860 14858 4839 94814 94893 14833 94815 14836 14840 14841 94817 14845 14847 14850 AMN BAX CFS ERY ESC FKS IRS ISQ IWD LDM MOP OSC PHN RQB SAW City/Location Adrian Alpena Ann Arbor Battle Creek Benton Harbor Detroit Detroit Detroit Flint Gaylord Grand Rapids Hancock Holland Houghton Lake Iron Mountain/Kingsford Jackson Kalamazoo Lansing Muskegon Pellston Pontiac Saginaw Sault Ste Marie Traverse City Alma Bad Axe Caro Newberry Escanaba Frankfort Sturgis Manistique Ironwood Ludington Mount Pleasant Oscoda Port Huron Big Rapids Gwinn Climate identifier Adrian Lenawee County Arpt Alpena Co Rgnl Airport Ann Arbor Municipal Arpt W K Kellogg Airport Sw Michigan Regional Arpt Detroit City Airport Detroit Metro Wayne Co Apt Willow Run Airport Bishop International Arpt Otsego County Airport Gerald R Ford Intl Airport Houghton County Memo Arpt Tulip City Airport Roscommon County Airport Ford Airport Jakson Co-Rynolds Fld Arpt Klmazo/Btl Creek Intl Arpt Capital City Airport Muskegon County Airport Pton Rgl Ap Of Emmet Co Ap Oakland Co. Intnl Airport Mbs International Airport Su Ste Mre Muni/Sasn Fl Ap Cherry Capital Airport Gratiot Community Airport Huron County Memorial Airport Tuscola Area Airport Luce County Airport Delta County Airport Frankfort Dow Memorial Field Airport Kirsch Municipal Airport Schoolcraft County Airport Gogebic Iron County Airport Mason County Airport Mount Pleasant Municipal Airport Oscoda Wurtsmith Airport Saint Clair County Intnl Airport Roben Hood Airport Sawyer International Airport Latitude Longitude 41.868 45.072 42.224 42.308 42.129 42.409 42.215 42.237 42.967 45.013 42.882 47.169 42.746 44.368 45.818 42.26 42.235 42.78 43.171 45.571 42.665 43.533 46.467 44.741 43.322 43.78 43.459 46.311 45.723 44.625 41.813 45.975 46.527 43.962 43.622 44.452 42.911 43.723 46.354 -84.079 -83.581 -83.74 -85.251 -86.422 -83.01 -83.349 -83.526 -83.749 -84.701 -85.523 -88.506 -86.097 -84.691 -88.114 -84.459 -85.552 -84.579 -86.237 -84.796 -83.418 -84.08 -84.367 -85.583 -84.688 -82.985 -83.445 -85.4572 -87.094 -86.201 -85.439 -86.172 -90.131 -86.408 -84.737 -83.394 -82.529 -85.504 -87.39 These files were directly used for rigid sections (since they are default files in the Pavement- ME), and custom stations were formed using these files for flexible sections. Table 3-15 summarizes the climatic files used for calibration. 70 3.5.5 Estimation of Initial IRI Initial IRI is an essential input for IRI prediction and pavement design. Initial IRI is the IRI value right after the construction. It indicates construction and ride quality right after construction. Initial IRI is also an essential part of QC/QA testing. Moreover, higher initial IRI values may lead to a reduction in pavement service life. The IRI model in the Pavement-ME is linear in form, but the measured IRI data may not always be linear. The change in measured IRI with time can be linearly increasing or non-linearly increasing, which may follow an irregular or flat trend. Also, the initial IRI (if available) can be greater or smaller than the first measured IRI data points because of the measurement date. Figure 3-26 shows some examples of measured IRI trends for flexible and rigid sections. (a) Flexible sections (b) Rigid sections Figure 3-26 Examples of measured IRI trends A single backcasting approach may not be applicable for all sections due to the difference in measured IRI trends for each section. Considering the data limitations and challenges, a systematic approach is used to estimate the initial IRI. Five different methods used include: 1. Selecting the IRI at zeroth year (if available). 2. Linear backcasting IRI based on the measured data for the first ten years. 3. Linear backcasting IRI based on the measured data for all available years. 4. Reducing the first measured IRI (after construction) by 5 inches per mile/year up to the zeroth year. 5. Reducing the first measured IRI (after construction) by 5 inches per mile/year if greater than 100; 4 inches per mile/year if between 70 and 100; 3 inches per mile/year if less than 70 up to a zeroth year. 71 It is important to note that the MDOT specification limit of 70 in/mile and 75 in/mile for flexible and rigid pavements are considered. After the initial IRI was obtained using the five methods mentioned above, the final initial IRI was selected based on the following criteria: 1. Use the initial IRI (if available) if it is less than or equal to the specification limit. 2. If the initial IRI (if available) is greater than the specification limit, use the backcasted IRI from other methods, whichever is closest to and lower than the specification limit. 3. If all five methods provide an initial IRI greater than the specification limit, choose the approach with an initial IRI greater than and closest to the specification limit. 4. Subsequently, review data progression to see if the estimated initial IRI fits all available measured data points. Figure 3-27 shows example sections with backcasted initial IRI using different methods. Section 1 has a non-linearly increasing trend, section 2 has an irregular trend, and sections 3 and 4 have linear trends with varying slopes. Different backcasting methods provide significantly different initial IRI values. For example, section 2 has a maximum difference of more than 20 inches/mile among the initial IRI values calculated using various methods. Similar differences can be seen in other sections. Moreover, method 3 for section 2 provides an unrealistic initial IRI value, higher than the first measured data point, due to the nature of the irregular trend. These plots show a need for different backcasting methods for various IRI trends. Figure 3-28 shows a flowchart for selecting the initial IRI using the mentioned approach. For certain flexible sections, the final selected initial IRI was very low (less than 30 in/mile). In that case, an initial IRI value of 30 in/mile is assigned. Moreover, the final selected initial IRI was very high for several flexible and rigid sections. 72 (a) Section 1 (b) Section 2 (c) Section 3 (d) Section 4 Figure 3-27 Illustration showing backcasting of initial IRI Table 3-16 Recommended thresholds based on initial IRI for flexible sections No of sections 380 371 362 356 349 331 295 274 Mean initial IRI (in/mile) 56.1 55.4 54.8 54.4 53.9 52.9 51.0 49.7 IRI less than or equal to 85 82 80 78 75 70 67 65 Table 3-17 Recommended thresholds based on initial IRI for rigid sections IRI less than or equal to 85 82 80 No. of sections 74 65 52 Mean initial IRI (in/mile) 73.7 71.6 69 73 Table 3-18 Summary of initial IRI thresholds Pavement type Flexible Rigid Fix type New Overlay New Overlay Initial IRI threshold (in/mile) 77 82 82 82 Figure 3-28 Flowchart for selection of initial IRI Therefore, some thresholds were selected to keep reasonable initial IRI values. Any section with an initial IRI value higher than the threshold was eliminated from the IRI calibration. Tables 3- 16 and 3-17 show different threshold values for flexible and rigid sections. It is important to note that the sections in Tables 3-16 and 3-17 consist of both new and overlay sections, but only new 74 MD T Sensor data s initial R a ailable es s it less than spec limits Accept o Apply all back-casting methods. Are all options more than spec limits es Select option more than and closest to spec limit. Select option less than and closest to spec limit. o sections have been used in this study. Based on the number of sections available and the average IRI for each cap, different threshold limits were selected for flexible and rigid pavements, as shown in Table 3-18. Figure 3-29 shows the distribution of initial IRI for the selected flexible and rigid sections. The distribution of the initial IRI is acceptable for an optimum IRI model calibration. (a) Flexible sections (b) Rigid sections Figure 3-29 Distribution of initial IRI It is essential to verify the accuracy of the proposed methodology. Five sections from flexible and one from rigid are taken for this purpose. These sections have the initial IRI data available at zero (construction) year. Only one rigid section has initial IRI data available at zero year. Methods 2 to 5 are implemented using measured IRI data from age 1 to 20 (excluding zero-year data). The comparison between the recommended initial IRI based on the proposed methodology and the recorded initial IRI shows a good correlation with an error of less than 8% for all sections. Table 3-19 shows the summary of the validation results. Table 3-19 Summary of validation results Pavement type Flexible Rigid Initial IRI backcasting (in/mile) Method 2 40.6 57.7 42.6 35.8 55.5 70.7 Method 3 40.6 57.7 42.6 34.2 50.9 66.6 Method 3 32.9 52.7 40.5 28.9 48.4 52.0 Method 4 36.9 56.7 44.5 32.9 52.4 56.0 Recommended Initial IRI (in/mile) 40.6 57.7 44.5 35.8 55.5 70.7 Recorded initial IRI (in/mile) 41.2 55.5 45.7 38.9 58.4 72.4 Error (%) 1.4 4.0 2.6 8.0 5.0 2.3 75 3.6 CHAPTER SUMMARY This chapter outlines the data used for local calibration, emphasizing the importance of selecting representative pavement sections and gathering pertinent data for accurate performance predictions. It details the methodology of converting the MDOT PMS data to Pavement-ME compatible units, evaluating distress trends, and considering maintenance history. Key distresses were identified, and databases were created for efficient data extraction. Project selection criteria prioritize sections with adequate data and performance trends. The selected sections were also verified against all MDOT sections to validate if these sections are representative of overall MDOT performance. Sections were categorized as good, fair, or poor based on measured trends relative to reference lines. The results showed that the selected sections represent MDOT pavement sections well. A total of 256 flexible and 88 rigid sections were selected. The number of projects for each performance type and pavement type has also been summarized. This chapter also details each input, source, and possible estimates in case of unavailable data. These inputs include the HMA and PCC material inputs, traffic, climate, and estimation of initial IRI. Table 3-20 summarizes the inputs and corresponding levels for traffic, climate, and material characterization data used for the local calibration. 76 Table 3-20 Summary of input levels and data source Input Vehicle class distribution Hourly distribution Monthly adjustment factor Number of axles per truck Single, tandem, tridem, and quad axle load distribution AADTT Vehicle class 9 percentage Traffic Cross- section layers (new and existing) HMA thickness PCC thickness Base thickness Subbase thickness Layer materials Mix properties HMA mixture aggregate gradation Binder properties Strength (f'c, MOR) CTE MR MR Soil properties HMA PCC Base/ subbase Subgrade Climate Pavement- ME input level 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 3 3 Mix of all levels 1 Data source level Input source 2 2 2 2 2 1 1 1 1 1 1 Mix of 2 and 3 1 or 3 3 1 or 3 2 3 3 3 1 MDOT specified traffic per cluster data From design drawings MDOT TDMS website Project-specific HMA thicknesses based on design drawings Project-specific PCC thicknesses based on design drawings Project specific base thicknesses based on design drawings Project-specific subbase thicknesses based on design drawings MDOT HMA mixture characterization study (DYNAMOD database) Project-specific mixture gradation data obtained from data collection or average statewide values MDOT HMA mixture characterization study (DYNAMOD database) Project specific testing values or average statewide value MDOT recommended values Recommendations from MDOT unbound material study Soil-specific MR values per MDOT subgrade soil study Location-based soil type per MDOT subgrade soil study Closest available climate station Note: Data source Level 1 is project-specific data Data source Level 2 inputs are based on regional averages in Michigan Data source Level 3 inputs are based on statewide averages in Michigan 77 CHAPTER 4 - METHODOLOGY 4.1 INTRODUCTION Local calibration of the Pavement-ME models aims to optimize the model coefficients by minimizing bias and standard error, which is achieved by matching the predicted and measured distress. Bias in the predictions signifies if there is a systematic over or under-prediction, whereas standard error shows the scatter and variability. Figure 4-1 shows a representation of bias and standard error. This chapter highlights each model's calibration methods and approaches, the reliability calculation, and the sensitivity analysis of Pavement-ME model coefficients. (a) High bias, high standard error (b) Low bias, high standard error (c) High bias, low standard error (d) Low bias, low standard error Figure 4-1 Schematic representation of bias and standard error (10) 78 The details for inputs, performance data, and project selection have already been discussed in Chapter 3. Once the data is extracted, it can be used to run the Pavement-ME files (.dgpx files) and generate outputs (structural responses). The process for local calibration is summarized below: (a) Run the Pavement-ME (using global model coefficients) and extract critical responses and predicted distresses. (b) Compare the predicted distress with measured distress. (c) Based on the results from Step 2, test the accuracy of the global models and the need for local calibration. (d) If predictions using global models are satisfactory, local calibration is not required, and global models can be accepted. Local calibration is needed if the global model has significant bias and standard error. (e) Check your calibration results by validating them on an independent set of sections not used for calibration. (f) Estimate the reliability equations based on the calibrated model predictions and measured distress. Before locally calibrating the Pavement-ME models, it is vital to determine the need for calibration. This includes testing the accuracy of the global model predictions at a reliability of 50%, which is the mean prediction. Once the predictions from the global model are obtained, they are compared with measured values to calculate bias and standard error. A plot of predicted versus measured values is created for each distress to visualize the accuracy of predictions to a line of equality (LOE). Testing the global model also includes hypothesis testing. For a good fit, the points should lie along the LOE. The measured distress yMeasured and predicted distress xPredicted can be modeled as a linear model as shown in Equation (4-1), where m is the slope, and bo is the intercept. 𝑦Measured = 𝑏𝑜 + 𝑚 × 𝑥Predicted (4-1) Three hypothesis tests are conducted to evaluate the reasonableness of the global model. If any of these hypotheses fail, the models are recalibrated for local conditions: • There is no systematic bias between the measured and predicted distress [Equation (4-2)]. This can be tested using a paired t-test. 79 𝐻0: ∑(𝑦Measured − 𝑥Predicted ) = 0 • The slope parameter m is 1 (Equation (4-3)). 𝐻0: 𝑚 = 1.0 • The intercept parameter bo is zero (Equation (4-4)). 𝐻𝑜: 𝑏𝑜 = 0 4.2 CALIBRATION APPROACHES (4-2) (4-3) (4-4) The empirical Pavement-ME transfer functions can be of two types: (a) model that directly calculates the magnitude of surface distress, and (b) model that calculates the cumulative damage index rather than actual distress magnitude. Approach 1: For specific models (e.g., fatigue cracking, rutting, transverse cracking, and IRI), damage is directly obtained from Pavement-ME outputs. The transfer functions predict distress from the damage and have been calibrated using the MATLAB program outside the Pavement- ME. Different resampling techniques and MLE have been used to calibrate these functions. Genetic Algorithm (GA) has been used to optimize transfer function coefficients using MATLAB program for this approach. These MATLAB codes are available from the author upon request. GA is an evolutionary optimization technique that can converge towards a global minimum solution even with local minima. GA involves the following operations: • Initialization: GA generates solutions by randomly selecting a subset inside the allowed search space called the population. • Selection: The generated solutions are selected based on the value of the objective function. • Generation of offspring: New solutions are created using the selected solutions or populations (offspring) based on two main processes: mutation and crossover. • Termination: This process continues till the termination criteria for the given population or the number of generations is reached. Approach 2: The Calibration Assistance Tool (CAT) calibrated the models (e.g., thermal cracking and joint faulting) where the damage is not obtained from the Pavement-ME outputs. These models predict distress by calculating cumulative damage over time. One can't use the resampling techniques or the MLE method for this approach. 80 Based on the model, two different calibration approaches have been followed (as shown in Table 4-1): Table 4-1 Model transfer functions and calibration approaches (28) Pavement type Performance prediction model Fatigue cracking – bottom up Fatigue cracking – top down Approach II I   Model transfer functions 1 60 𝐹𝐶𝐵𝑜𝑡𝑡𝑜𝑚 = ( ) ( 1 + 𝑒𝑪𝟏𝐶1 𝐾𝐿1 1 + 𝑒𝑲𝑳𝟐×100×(𝑎0/2𝐴0)+𝑲𝑳𝟑×𝐻𝑇+𝑲𝑳𝟒×𝐿𝑇+𝑲𝑳𝟓×𝑙𝑜𝑔10⁡ 𝐴𝐴𝐷𝑇𝑇 ∗+𝑪𝟐𝐶2 6000 ∗𝐿𝑜𝑔(𝐷𝐼𝐵𝑜𝑡𝑡𝑜𝑚·100)) 𝑡0 = 𝐿(𝑡) = 𝐿𝑀𝐴𝑋𝑒 −( 𝑪𝟏𝜌 𝑡−𝐶3𝑡0 ) 𝑪𝟐𝛽 Flexible pavement Rigid pavement HMA   𝛥𝑝(𝐻𝑀𝐴) = 𝜀𝑝(𝐻𝑀𝐴)ℎ𝐻𝑀𝐴 = 𝜷𝟏𝒓𝑘𝑧𝜀𝑟(𝐻𝑀𝐴)10𝑘1𝑟𝑛𝑘2𝑟𝜷𝟐𝒓𝑇𝑘3𝑟𝜷𝟑𝒓 Rutting Base/subgrade  Thermal cracking IRI Transverse cracking    𝛥𝑝(𝑠𝑜𝑖𝑙) = 𝜷𝒔𝟏𝑘𝑠1𝜀𝑣ℎ𝑠𝑜𝑖𝑙 ( 𝜀𝑜 𝜀𝑟 𝛽 𝜌 𝑛 ) ) 𝑒−( 𝐴 = 10𝒌𝒕𝛽𝑡(4.389−2.52𝐿𝑜𝑔(𝐸𝐻𝑀𝐴𝜎𝑚𝜂)) 𝐼𝑅𝐼 = 𝐼𝑅𝐼𝑜 + 𝑪𝟏(𝑅𝐷) + 𝑪𝟐(𝐹𝐶𝑇𝑜𝑡𝑎𝑙) + 𝑪𝟑(𝑇𝐶) + 𝑪𝟒(𝑆𝐹) 𝐶𝑅𝐾𝐵𝑈/𝑇𝐷 = 100 1 +  𝑪𝟒(𝐷𝐼𝐹)𝑪𝟓 𝑇𝐶𝑅𝐴𝐶𝐾 = (𝐶𝑅𝐾𝐵𝑜𝑡𝑡𝑜𝑚−𝑢𝑝 + 𝐶𝑅𝐾𝑇𝑜𝑝−𝑑𝑜𝑤𝑛 − 𝐶𝑅𝐾𝐵𝑜𝑡𝑡𝑜𝑚−𝑢𝑝 ⋅ 𝐶𝑅𝐾𝑇𝑜𝑝−𝑑𝑜𝑤𝑛) ⋅ 100% 𝑚 𝐹𝑎𝑢𝑙𝑡𝑚 = ∑ 𝛥𝐹𝑎𝑢𝑙𝑡𝑖 𝑖=1 𝛥𝐹𝑎𝑢𝑙𝑡𝑖 = 𝐶34 × (𝐹𝐴𝑈𝐿𝑇𝑀𝐴𝑋𝑖−1 − 𝐹𝑎𝑢𝑙𝑡𝑖−1)2 × 𝐷𝐸𝑖 𝑚 Transverse joint faulting  𝐹𝐴𝑈𝐿𝑇𝑀𝐴𝑋𝑖 = 𝐹𝐴𝑈𝐿𝑇𝑀𝐴𝑋0 + C7 × ∑ 𝐷𝐸𝑗 × 𝐿𝑜𝑔(1 + C5 × 5. 0𝐸𝑅𝑂𝐷)𝑪𝟔 𝑗=1 𝐹𝐴𝑈𝐿𝑇𝑀𝐴𝑋0 = C12 × 𝛿curling × [𝐿𝑜𝑔(1 + 𝐶5 × 5.0𝐸𝑅𝑂𝐷) × 𝐿𝑜𝑔( C12 = C1 + C2 × 𝐹𝑅0.25 C34 = C3 + C4 × 𝐹𝑅0.25 𝐼𝑅𝐼 = 𝐼𝑅𝐼𝑜 + 𝑪𝟏(𝐶𝑅𝐾) + 𝑪𝟐(𝑆𝑃𝐴𝐿𝐿) + 𝐶3(𝑇𝐹𝐴𝑈𝐿𝑇) + 𝑪𝟒(𝑆𝐹) 𝑃200 × 𝑊𝑒𝑡𝐷𝑎𝑦𝑠 𝑝𝑠 𝑪𝟔 )] *Bold font indicates calibration coefficients IRI  4.3 CALIBRATION METHODS This study used different methods (least squares and MLE) to demonstrate and compare calibration differences for normally and non-normally distributed data. For example, the measured transverse cracking in rigid pavements is typically non-normally distributed, with most data points near zero, whereas IRI is close to a normal distribution. Both methods have their advantages and limitations. It is important to note that thermal cracking, top-down cracking in flexible pavements, and joint faulting in rigid pavements were not calibrated using the MLE method. 81 The measured data is limited to the MDOT PMS database. Apart from the measured data, this study also used synthetic data as it provides the freedom to generate any distribution with random errors. This methodology also validates a more generic use of MLE on a dataset outside the measured data. Before calibration using measured data, synthetic data was created to show the applicability of the MLE approach. For this purpose, DIBottom was generated using an exponential distribution with 𝜆 = 0.3 to generate synthetic bottom-up cracking data in flexible pavements. DIBottom was used to calculate bottom-up cracking for 355 points, the same number of points as the measured data. A value of C1 = 0.254, C2 = 0.730 (for total AC thickness (T) < 5 in.), and C2 = (0.867+0.2583* T)*0.238 (for 5 in. <= T <= 12 in.) were used for calculation of bottom-up cracking. The assumption of an exponential distribution and the value of 𝜆 is based on the measured bottom-up cracking data. The generated synthetic data is close to the measured data but follows a smooth exponential distribution curve. Two different datasets were created, one without variability (no change introduced in the generated data) and one with a uniformly distributed random variability applied on each data point between -50 % and 50%. A similar methodology created synthetic data for transverse cracking in rigid pavements. Initially, an exponentially distributed 𝐷𝐼𝐹 was generated using 𝜆 = 0.1. The generated 𝐷𝐼𝐹 was then used to calculate transverse cracking. About 237 points were generated for the synthetic data, the same as for measured transverse cracking data. The selection of a suitable method and distribution is based on several parameters. Negative log-likelihood (NLL) was calculated for the MLE and least squares methods, the formulation for which is presented in the proceeding sections. Besides the NLL values, four other statistical parameters were used as selection criteria for the most suitable model. These are the Standard Error of Estimate (SEE), bias, Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC). AIC is a statistical measure used for model selection that balances the goodness of fit with the complexity of the model, as shown in Equation (4-5). BIC is a similar criterion that penalizes model complexity more strongly, often leading to more efficient model selection, as shown in Equation (4-6). where, 𝐴𝐼𝐶 = 2𝑆 − 2𝐿𝐿 𝐵𝐼𝐶 = 𝑙𝑛⁡(𝑛)𝑆 − 2𝐿𝐿 82 (4-5) (4-6) n = Number of data points 𝑆 = Number of parameters of distribution (for example 𝑆 = 1 for exponential distribution) 𝐿𝐿 = Log-likelihood value 4.3.1 Calibration Using the Least Squares Method The least squares method is a popular technique used in various statistics, mathematics, and engineering fields to fit mathematical models to data. Its primary aim is to minimize the sum of the squares of the residuals between observed and predicted values. It follows the NIID assumption, which may not apply to non-normally distributed data. This method was employed to estimate the parameters of the Pavement-ME transfer functions. The fundamental idea behind the least squares method is to find the line (or curve) that best fits a set of data points by minimizing the sum of the squared differences between the observed data points and the corresponding values predicted by the model. The bias and SEE values were minimized using the least squares method, as shown in Equations (4-7) and (4-8) 𝑆𝐸𝐸 = √ ∑(𝑦 − 𝑦̂)2 𝑛 − 1 𝐵𝑖𝑎𝑠 = ∑(𝑦 − 𝑦̂) (4-7) (4-8) where, 𝑦 = Measured data 𝑦̂ = Predicted data n = Number of data points 4.3.2 Calibration Using the Maximum Likelihood Estimation (MLE) Method MLE is a powerful statistical technique for parameter estimation in various fields, including biology, physics, economics, and engineering. MLE was used to calibrate the bottom-up cracking, total rutting, and IRI models in flexible pavement and transverse cracking and IRI models in rigid pavements. MLE seeks to estimate the parameters of a probability distribution that best describes the observed data based on the likelihood function. The likelihood function measures the probability of the observed data following a known distribution. MLE finds the set of distribution parameters that maximize the likelihood function. Consider a dataset X = (x1, x2, 83 ..., xn) that is generated by a probability distribution with parameters θ. The likelihood function L(θ|X) is the joint probability density function of the obser ed data, gi en the distribution parameters as shown in Equation (4-9). L(θ|X) = P(X|θ) = P(x1, x2, … . , xn|θ) (4-9) Here, P denotes the probability density function, and the likelihood function measures the probability of observing the data X given the distribution parameters 𝜃. The goal of MLE is to find the set of distribution parameters 𝜃 that maximizes the likelihood function between dataset X and the assumed distribution. In practice, it is often easier to work with the log-likelihood function so that the product of likelihood values becomes a summation; one can do this by taking the natural logarithm of the likelihood function. The log-likelihood function is given by Equation (4-10). where; 𝑙(θ|X) = log L(θ|X) = log P(X|θ) = log ∏ P(𝑥𝑖|θ) = ∑ log P(𝑥𝑖|θ) (4-10) П = Product operator Σ = Summation operator Taking the logarithm of the likelihood function also simplifies the computation of the derivative required for optimization. One can solve the optimization problem by finding the values of 𝜃 that maximize the log-likelihood function. This can be done using numerical optimization algorithms, such as gradient descent, Newton's, or quasi-Newton methods. These algorithms require the derivative of the log-likelihood function for the distribution parameters. Numerical optimization algorithms iteratively update the values of the distribution parameters to find the maximum of the log-likelihood function. The optimization process continues until the algorithm converges to a maximum of the log-likelihood function. The MLE obtained from the optimization process represents the most likely estimates of the distribution parameters that can explain the observed data. The calibration process for MLE involves the following steps: Step 1: Assume the initial values of the transfer function coefficients to calculate the predicted cracking. Step 2: Fit a known distribution (for example, exponential, gamma, etc.) to the predicted cracking and estimate the distribution parameters. 84 Step 3: Calculate the NLL between the known distribution parameters in Step 2 and the measured values. Step 4: Repeat Steps 1 to 3 to minimize the NLL value. Step 5: Coefficients with minimum NLL are the desired coefficients. Four distributions were used for this analysis: gamma, log-normal, exponential, and negative binomial. The Probability Density Function (pdf)/ Probability Mass Function (pmf) of these distributions is shown in Equation (4-11) to (4-14), respectively. • Gamma distribution 𝑓(𝑥) = 𝑥𝛼−1𝑒−𝑥/𝛽 𝛽𝛼Γ(𝛼) • Log-normal distribution 𝑒 𝑓(𝑥) = −((ln⁡((𝑥−𝜃)(𝑚))2/(2𝜎2)) (𝑥 − 𝜃)𝜎√2𝜋 𝑥 > 𝜃; 𝑚, 𝜎 > 0 • Exponential distribution • Negative binomial distribution 𝑓(x) = 𝜆e−𝜆x 𝑃(𝑋 = 𝑥 ∣ 𝑟, 𝑝) = ( 𝑥 − 1 𝑟 − 1 ) 𝑝𝑟(1 − 𝑝)𝑥−𝑟, 𝑥 = 𝑟, 𝑟 + 1, …, (4-11) (4-12) (4-13) (4-14) The formulation of the maximum likelihood function for exponential distribution is shown below. A similar approach was used for other distributions. Equation (4-15) shows the pdf for exponential distribution. Comparing it with Equation (4-13), here 𝜆 = 1 𝛽 , which is the rate parameter, and x is the observed value. The likelihood function for a set of independent and identically distributed observations from the exponential distribution is obtained by taking the product of the individual probability density functions shown in Equations (4-16) and (4-17). 𝑓(𝐱, 𝛽) = −𝐱 ) ( 𝛽 𝑒 1 𝛽 ; 𝐱 > 0 𝑁 𝐿(𝛽, 𝐱) = 𝐿(𝛽, 𝑥1, … , 𝑥𝑁) = ∏   𝑖=1 𝑓(𝑥𝑖, 𝛽) 𝑁 𝐿(𝛽, 𝐱) = ∏   𝑖=1 −𝑥𝑖 ( 𝛽 ) 𝑒 1 𝛽 85 (4-15) (4-16) (4-17) It is common to work with the log-likelihood function instead of the likelihood function to simplify the calculation. The log-likelihood function is obtained by taking the natural logarithm of the likelihood function, as shown in Equation (4-18). 𝑁 ℒ(𝛽, 𝐱) = log⁡ (∏   𝑖=1 −𝑥𝑖 ( 𝛽 ) ) 1 𝛽 𝑒 (4-18) Simplifying Equation (4-18) using properties of the log is shown in Equations (4-19) to (4-21). Equation (4-21) shows the negative log-likelihood of exponential distribution used for calibration. 𝑁 ℒ(𝛽, 𝐱) = log⁡ (∏   𝑖=1 −𝑥𝑖 ( 𝛽 ) 𝑒 1 𝛽 𝑁 (log⁡ ( ) = ∑   𝑖=1 ) + log⁡ (𝑒 −𝑥𝑖 ( 𝛽 ) )) 1 𝛽 ℒ(𝛽, 𝐱) = 𝑁log⁡ ( 1 𝛽 𝑁 ) + ∑   𝑖=1 ( −𝑥𝑖 𝛽 ) ℒ(𝛽, 𝐱) = −𝑁log⁡(𝛽) + 1 𝛽 𝑁 ∑   𝑖=1 − 𝑥𝑖 (4-19) (4-20) (4-21) To estimate the value of 𝛽 at the maxima of log-likelihood, Equation (4-21) can be differentiated. Equations (4-22) to (4-24) show the estimation of 𝛽 at the maxima of log- likelihood. ∂ℒ ∂𝛽 = ∂ ∂𝛽 (−𝑁log⁡(𝛽) + 1 𝛽 𝑁 ∑   𝑖=1 − 𝑥𝑖) = 0 ∂ℒ ∂𝛽 = − 𝑁 𝛽 + 1 𝛽2 ∑   𝑁 𝑖=1 𝑥𝑖 = 0 𝛽 = ∑  𝑁 𝑖=1 𝑥𝑖 𝑁 ¯ = 𝐱 (4-22) (4-23) (4-24) Figure 4-2 shows the flow chart of the methodology used and the final selection of the optimum method and distribution. 4.4 RESAMPLING TECHNIQUES Various sampling techniques were used to calibrate Pavement-ME transfer functions. The least squares and MLE methods were combined with these techniques to improve the robustness of 86 the estimated parameters. All these techniques have been used for models calibrated using Approach 1. For models calibrated using Approach 2, no sampling or traditional split sampling has been used in the CAT tool. 1. No sampling: This technique considers the entire dataset (all available measured data points and corresponding damage) and was used for both Approaches 1 and 2. 2. Traditional split sampling: The dataset is randomly divided into two parts—70% of the data for the calibration set and the rest 30% for the validation set. The optimization is performed only on the calibration set, and the obtained coefficients are applied to an independent validation set. This method was used for both Approaches 1 and 2. 3. Repeated split sampling: This technique is like traditional split sampling but with 1000 resamples, where a different data set was picked up each time for calibration (70%) and validation (30%). This method was used only for Approach 1. 4. Bootstrapping: Bootstrap resampling is used to draw 1000 bootstrap samples from the original dataset with replacement. Each bootstrap resamples the original data with the same sample size but may contain some duplicate observations. This method estimates a sampling distribution and confidence intervals for a population parameter, even when the underlying population distribution is unknown. 87 Figure 4-2 Flowchart of calibration methodology Traditional no-sampling or split sampling technique provides a convenient approach to selecting pavement sections from the calibration database. Though these techniques are easy to implement and can be used for any Pavement-ME model, they might impose some limitations. Resampling 88 techniques have several advantages over traditional approaches. Since these are non-parametric techniques, the model parameters can be estimated without making assumptions about the data distribution. The distribution of the model coefficients and error parameters can be estimated instead of the point estimate. This can give a better estimation of parameters within desired confidence intervals. Since a new sample is created every time, the outliers or sections controlling the calibration process can be identified. Though these resampling techniques have several advantages over traditional approaches, there are also certain limitations. Bootstrapping cannot be used for small datasets or when the independence assumption is unmet. Resampling techniques also require higher computing power and time and can be used only for those performance models where the damage and other inputs are available from Pavement-ME. Table 4-2 summarizes the advantages and limitations of all calibration techniques. Table 4-2 Summary of calibration techniques Technique No sampling Split sampling Repeated split sampling Bootstrapping Advantages • Computationally efficient • Applicable even for small sample size • Computationally efficient • Provides validation Limitations • Provides point estimates • It may not be suitable for non- normally distributed data • Provides point estimates • It may not be suitable for non- normally distributed data • Provides confidence intervals • Provides validation • Identifies outliers • Distribution assumption is not required • Provides confidence intervals • • Distribution assumption is not Identifies outliers required • Computationally time-consuming • It cannot be used for smaller sample size It may not be suitable for non- normally distributed data • • Computationally time-consuming • It cannot be used for smaller sample size It may not be suitable for non- normally distributed data • 4.5 FLEXIBLE PAVEMENT MODEL COEFFICIENTS The design distress in the Pavement-ME includes bottom-up cracking, top-down cracking, rutting, thermal (transverse) cracking, reflective cracking, and IRI. The calibration of each model and the specific coefficients calibrated has been discussed in this section. 89 4.5.1 Fatigue Cracking Model (Bottom-up) The fatigue cracking (bottom-up) model was calibrated by optimizing the C1 and C2 coefficients (see Table 4-1). In Pavement-ME v2.6, coefficient C1 is a single value, whereas coefficient C2 has three different values depending on the total HMA thickness. Table 4-3 shows the global values for C1 and C2. Table 4-3 Global values for bottom-up cracking model coefficients Calibration coefficient C1 C2 Hac : Total HMA thickness in inches Global values 1.31 Hac < 5 in. : 2.1585 5 in. <= Hac <=12 in.: (0.867 + 0.2583 × Hac) ×1 Hac > 12 in.: 3.9666 Notably, no sections were selected for the bottom-up calibration with a total HMA thickness of more than 12 inches. The coefficient C2 was calibrated separately for the thickness ranges less than 5 inches and 5 to 12 inches, respectively. For a thickness range of 5 to 12 inches, only the multiplying factor 1 (marked in bold here: (0.867 + 0.2583 × Hac) ×1) was calibrated, while other values (0.867 and 0.2583) were kept at global values. A single value was used for a thickness range of more than 12 inches. The Hac was kept at 12 inches, and the multiplying factor 1 was kept at the calibrated value obtained for the 5 to 12-inch thickness range. The crack initiation time is affected by C1, whereas the slope of the bottom-up cracking curve is affected by C2. Consequently, the calibration was performed using two approaches: (a) combined measured bottom-up and top- down cracking and (b) bottom-up cracking only. MLE was used for approach (a), whereas least squares was used for both methods. 4.5.2 Fatigue Cracking Model (Top-down) The top-down cracking model has been modified in the Pavement-ME v2.6. The model consists of a crack initiation function that calculates the time to crack initiation and a crack propagation function that calculates the percent lane area cracked. This makes it a total of eight coefficients combined from both functions. Since the actual crack initiation time was not known, it was not possible to calibrate the crack initiation model separately. So, a single function was used by substituting the crack initiation function with the crack propagation function. Initially, an attempt was made to change all eight coefficients simultaneously. This approach had some challenges: 90 • The model has some mathematical limitations. High values for C3 cause mathematical errors when using it in Pavement-ME. • No current literature exists for the top-down cracking model calibration. Therefore, estimating the range for each coefficient to be used in optimization was difficult. • The model has many coefficients with coefficient values ranging from 0.011 to 64271618. This makes the optimization challenging to converge. As mentioned above, four coefficients from the crack initiation function (kL2, kL3, kL4, kL5) and two coefficients from the crack propagation function (C1, C2) have been calibrated based on the model's understanding and limitations. 4.5.3 Rutting Model Due to axle loads, rutting is the total accumulated plastic strain in different pavement layers (HMA, base/sub-base, and subgrade). It is calculated by summing up the plastic strains at the mid-depth of individual layers accumulated for each time increment. In the Pavement-ME, rutting is predicted separately for the layers (HMA, base, and subgrade). The total rutting is the sum of rutting from all layers. The AC rutting model has three coefficients (β1r, β2r, β3r). β 1r is a direct multiplier and was calibrated using optimization outside the Pavement-ME. In this model, β2r and β3r are power to the pavement temperature and the number of axle load repetitions. Calibration of β2r and β3r cannot be done outside of the Pavement-ME and requires running the Pavement-ME multiple times or optimizing these in the CAT tool. Initially, β2r and β3r values were used from the previous calibration effort, and β1r was calibrated (10). This calibration approach provided reasonable results; therefore, β2r and β3r from the previous calibration were accepted, and only β1r was calibrated. The unbound layers (base and subgrade) rutting model have one calibration coefficient each (βs1). Since βs1 is a direct multiplier, it can be calibrated using optimization outside the Pavement-ME without running the software or CAT tool. Since both base and subgrade have the same model and calibration coefficient, the base calibration coefficient is referred to as βs1, and the subgrade coefficient is referred to as βsg1. The rutting model in the Pavement-ME was calibrated using the following two methods: • Method 1: Individual layer rutting calibrations — The measured rutting from individual layers was matched against the Pavement-ME predictions (β1r, βs1, and βsg1 were 91 calibrated separately) for this approach. The total measured rutting was multiplied by the percent contribution from each layer to obtain measured rutting for the individual layer. Figure 4-3 shows the percentage contribution estimated using transverse pavement profile analysis. The width and depth of the measured rut channel were used to determine the seat of rutting and rutting in individual layers. AC layer rutting contributes more than 70% to all pavement types [based on transverse profile analysis (10)]. Pavement-ME has separate standard error equations for rutting in the individual layers. This method evaluated the standard error equations for rutting in each layer. • Method 2: Total surface rutting calibration — The total measured rutting was calibrated against the sum of individual predicted rutting (i.e., β1r, βs1, and βsg1 were calibrated simultaneously). (a) Overall (b) HMA reconstruct freeway sections (c) HMA reconstruct non-freeway sections (d) HMA over rubblized PCC Figure 4-3 Transverse profile analysis for total rutting (10) 4.5.4 Thermal Cracking Model The thermal cracking model in the Pavement-ME has three different levels for the calibration coefficient. These levels are based on the level of HMA input. Level 1 G* and Level 2 IDT have been used to calibrate the thermal cracking model. This corresponds to Level 1 thermal cracking calibration coefficients. Both G* and IDT values were obtained from the DYNAMOD software database. In the DYNAMOD database, G* and IDT values are available only for sections with Performance grade (PG) binder type. Therefore, sections with PG binder type (Superpave mixes) 92 11.7%5.9%82.4%0%20%40%60%80%100%% Base% SG% HMASeatof ruttingPercentof projects12.7%7.2%80.2%0%20%40%60%80%100%% Base% SG% HMASeatof ruttingPercentof projects11.1%5.9%83.0%0%20%40%60%80%100%% Base% SG% HMASeatof ruttingPercentof projects7.5%2.0%90.5%0%20%40%60%80%100%% Base% SG% HMASeatof ruttingPercentof projects have been used to calibrate the thermal cracking model. In the Pavement-ME v2.6, the calibration coefficient kt is originally a function of the mean annual air temperature (MAAT), whereas, in v2.3, it was a single representative value. Two different approaches were used for calibration: (a) Using the CAT tool, an initial attempt was made to calibrate kt (using the original equation as a function of MATT). (b) A second attempt was made to calibrate kt by running the Pavement-ME multiple times with different kt values of 0.25, 0.65, 0.75, 0.85, 0.95, and 1.35. This time, single values for kt were used, which were not a function of MAAT. kt as a function of MAAT resulted in contradictory results when comparing Michigan temperature extremes, where thermal cracking at cold temperatures was either reduced or equal to thermal cracking at warm temperatures. Moreover, ARA recommends using a single kt value if this is more suitable for the agency and its local conditions. Based on these results, the kt value based on the second approach was recommended. It is important to note that for this calibration, the average thermal cracking for a section was cut at 2112 ft/mile. 4.5.5 IRI Model for Flexible Pavements IRI is a linear function of initial IRI, rut depth, total fatigue cracking, transverse cracking, and site factor. The initial IRI was obtained from linear backcasting based on the time series trend for each section, as described in Chapter 3. The fatigue cracking, rutting, and transverse cracking models were calibrated before calibrating the IRI model. Since all inputs to the IRI model could be obtained, it was calibrated outside Pavement-ME. IRI has a closed-form solution and does not require a standard error equation in the Pavement-ME. The standard error for IRI is calculated using the standard error of its components. 4.6 RIGID PAVEMENT MODEL COEFFICIENTS The design distresses in the Pavement-ME include transverse cracking (percentage of slabs cracked), transverse joint faulting (inches), and international roughness index (IRI) for rigid pavements. The calibration methodology for each model is discussed in this section. 93 4.6.1 Transverse Cracking Model The coefficients C4 and C5 (shown in Table 4-1) were optimized to calibrate the transverse cracking model. These coefficients were calibrated outside the Pavement-ME and without the CAT tool. C4 affects the crack initiation time, and C5 affects the slope of the transverse cracking curve. 4.6.2 Transverse Joint Faulting Model The joint faulting model in the Pavement-ME consists of eight coefficients. Joint faulting could not be predicted using the available inputs outside the Pavement-ME; therefore, it was calibrated using the CAT tool. CAT tool has a limitation on the run time and the total combinations of coefficients that can be calibrated simultaneously. Therefore, it was essential to identify the most sensitive coefficients. Several research studies (11, 26) show that out of the eight calibration coefficients for the faulting model, C6 is the most sensitive. C1 is the next sensitive coefficient, followed by C2. Using this sequence of sensitivity of the different coefficients, C1 and C6 were calibrated together. The calibrated coefficients from C1 and C6 were kept fixed, and C2 was calibrated. In this sequence, the three most sensitive coefficients were calibrated. As previously noted and explained in Chapter 3, the joint faulting (for every 0.1-mile segment) was cut at 0.4 inches for calibration. 4.6.3 IRI Model for Rigid Pavements IRI in rigid pavements is a linear function of initial IRI, transverse cracking, joint spalling, faulting, and site factor. The initial IRI was obtained from linear backcalculation based on the time series trend for each section. The transverse cracking and joint faulting models were calibrated before calibrating the IRI model. Since all inputs to the IRI model could be obtained, it was calibrated outside Pavement-ME without rerunning it or using the CAT tool. IRI has a closed-form solution and does not require a standard error equation in Pavement-ME. The standard error for IRI is calculated using the standard error of its components. 4.7 CALCULATION OF DESIGN RELIABILITY Pavement-ME uses a reliability-based design, as explained in Chapter 2. Reliability is added to the mean prediction to incorporate input or performance data variability. It is expressed as a 94 function of the predicted performance and derived using the predicted and measured performance data. A step-by-step approach to estimating the reliability of transverse cracking for rigid pavements is shown below as an example. A similar approach was used for the reliability of all other models except IRI in the Pavement-ME. Step 1: All predicted and measured data points are grouped by creating bins on the predicted cracking. The number of data points in each group should be equivalent to reduce bias in the results. Step 2: The average and standard deviation of measured and predicted cracking are computed for each group. The grouping is performed after finalizing the calibration coefficients (global or local) to obtain the predicted performance. Table 4-4 shows the number of data points, bin ranges, and descriptive statistics. Table 4-4 Reliability analysis for transverse cracking in rigid pavements (example) Cracking range (%) 0-0.5 0.5-2 2-5 5-10 10-50 No. of data points 46 31 44 29 12 Average Measured Cracking 0.84 1.41 3.53 1.45 15.06 Average Predicted Cracking 0.54 1.35 3.13 12.18 26.52 Standard dev. of Measured Cracking Standard dev. of Predicted Cracking 0.86 1.51 3.76 8.93 14.96 0.29 0.25 0.72 1.58 1.22 Step 3: A relationship is determined between the standard deviation of the measured cracking on the y-axis and the average predicted cracking on the x-axis. Figure 4-4 shows the fit model to the grouped data in steps 1 and 2. Equation (4-25) shows the relationship between the standard deviation of the measured cracking and the average predicted cracking (when using the no- sampling technique). 𝑠𝑒(𝐶𝑅𝐾) = 1.3627(𝐶𝑅𝐾)0.7473 (4-25) 95 Figure 4-4 Fitting curve for the reliability of transverse cracking in rigid pavements (example) Step 4: The reliability is calculated under the assumption that the error in prediction is approximately mormally distributed towards the upper side of the mean distress. The predicted cracking can be adjusted to the desired reliability level using Equation (4-26) 𝐶𝑟 = C50 + 𝑆𝑒 × 𝑍𝑎/2 (4-26) where, Cr= Predicted cracking at reliability r (%) C50 = Predicted cracking at 50% reliability Se = Standard deviation of cracking, which can be estimated using Equation (4-25) 𝑍𝑎/2⁡= Standardized normal deviate (mean = 0; standard deviation = 1) at reliability r Step 5: For the final step, the reasonableness of the model should be verified based on the actual measured data before using the reliability equation for design. The reliability model for IRI is different from that of other models. Since it is a closed- form solution and the variances of different components of IRI are known, the reliability model for IRI is based on the variance analysis of its components. The basic assumption implies that the error in predicting IRI is roughly normally distributed. The total error includes input, repeatability, pure, and model errors. Overall, the IRI prediction error can be estimated by Equations (4-27) and (4-28). IRIpe = IRImeas − IRIpred Var(IRIpe) = Var(IRImeas) + Var(IRIpred) − 2R × √Var(IRImeas) × Var(IRIpred) (4-27) (4-28) 96 where, Var(IRIpe) = Variance in prediction error for IRI (estimated from calibration results) Var(IRImeas) = Variance in measured IRI (estimated from field measurement) Var(IRIpred) = Variance in predicted IRI R = Correlation coefficient between predicted and actual IRI The variance in predicted IRI is the sum of the variance in inputs (cracking, spalling, faulting, and initial IRI) and the variance in model + pure error, as shown in Equation (4-29). Var(IRIpred) = Var(IRIINPUTS) + Var(model + pure⁡error) (4-29) The variance in inputs for the IRI model is shown in Equation (4-30). Var(IRIINPUTS) = VarIRIi + C12 × VarCRK + C22 × VarSpall + C32 × VarFault (4-30) where, Var(IRIINPUTS)= Variance in IRI due to measurement errors for each distresses and initial IRI (estimated from field measurements) VarIRIi = Variance in initial IRI VarCRK = Variance in transverse cracking VarSpall = Variance in joint spalling VarFault = Variance in joint faulting C1, C2, C3 = IRI model coefficients Using Equations (4-28) to (4-30), 𝑉𝑎𝑟(𝑚𝑜𝑑𝑒𝑙 + 𝑝𝑢𝑟𝑒⁡𝑒𝑟𝑟𝑜𝑟) can be determined and used to predict the standard deviation in IRI at any predicted value. The global standard error equations for each model are summarized in Table 4-5. 97 Table 4-5 Global calibration reliability equations for each distress and smoothness model Pavement Type Pavement performance prediction model Fatigue cracking (bottom-up) Standard error equation Fatigue cracking (top-down) 𝑠𝑒(𝑡𝑜𝑝−𝑑𝑜𝑤𝑛) = 0.3657 × 𝐹𝐶𝑡𝑜𝑝 + 3.6563 Flexible pavements Rutting Transverse cracking IRI Transverse cracking Faulting IRI Rigid pavements 𝑠𝑒 = 0.14 × 𝑇𝐶 + 168 Estimated internally by the software 𝑠𝑒(𝐶𝑅𝐾) = 3.5522(𝐶𝑅𝐾)0.3415 + 0.75 𝑠𝑒(𝐹𝑎𝑢𝑙𝑡) = 0.07162(𝐹𝑎𝑢𝑙𝑡)0.368 + 0.00806 Initial IRI Se = 5.4 Estimated internally by the software 4.8 IMPACT OF CALIBRATION ON PAVEMENT DESIGN Calibration aims to improve the Pavement-ME predictions and its usability for local conditions. The calibrated model will impact the local design practices. Additional flexible and rigid pavements (not part of the calibration) were designed to evaluate the impact of the locally calibrated models. The designs were based on calibrated model coefficients and standard error equations obtained using the least squares method. Forty-four (44) new flexible and 44 new rigid sections (JPCP) were designed in the Pavement-ME using the new calibrated models and the coefficients from the previous calibration effort (10). It is important to note that MDOT found the global coefficients more suitable than the local ones for actual designs. Therefore, the global coefficients were used for comparison in the case of rigid sections. Other design properties (base/subbase, subgrade, and climatic properties) were kept the same for flexible and rigid sections except for the traffic levels. These sections were also designed using the AASHTO93 design method. MDOT uses widened lane (lane width = 14 feet) sections for rigid pavements. The widened lane sections were designed as standard width (12 feet) by reducing the thicknesses by up to 1 inch from the final thickness. The lane width was kept at the standard width of 12 feet for flexible sections. Figure 4-5 shows the distribution of layer thicknesses (HMA and PCC), ESALs, and average annual MR for subgrade soil. The ESALs for flexible sections range from 1 to 41 million, whereas for rigid sections range from 1 to 64 million. The average annual MR for 98 ()75715500001131131Bottome(bottomup)..LogFC.s.e−−+=++()080260240001.e(HMA)HMAs..=+()06711014770001.e(Base)Bases..=+()05012012350001.e(SG)SGs..=+ subgrade soil ranges from 3.7 to 6.5 ksi for flexible and rigid sections. Table 4-6 shows the number of sections in different categories. All these flexible and rigid sections were designed in the Pavement-ME V2.6 at 95% design reliability and MDOT recommended thresholds. Table 4-7 shows the MDOT recommended threshold values for all distress types. Since the bottom-up cracking model was calibrated by combining the measured bottom-up and top-down cracking, the top-down cracking prediction was not used for design purposes. Moreover, MDOT does not have a formal design threshold for the new top-down cracking model. The design thicknesses were estimated to evaluate the differences between the newly calibrated model, previous calibrated model, and the AASHTO93 designs. Moreover, the critical design thicknesses were also identified separately for flexible and rigid pavements. 99 (a) ESALs (flexible sections) (b) ESALs (rigid sections) (c) AASHTO93 HMA thickness (flexible (d) AASHTO93 PCC thickness (rigid sections) sections) (e) Subgrade MR (for both flexible and rigid sections) Figure 4-5 Distribution of design inputs for the selected sections 100 Table 4-6 Selected sections for pavement design in different categories Category AASHTO soil classification MDOT region Classification Lane width (applicable to rigid sections only) Description SP CL SC SM SC-SM SP-SM Bay Grand Metro North Superior Southwest University Freeway Non-freeway Widened (14 ft) Standard (12 ft) # of sections 6 18 8 3 4 5 4 8 8 4 3 8 9 28 16 17 27 Table 4-7 MDOT recommended design thresholds for Pavement-ME distress Pavement type Flexible pavements Rigid sections Distress type Bottom-up cracking Top-down cracking Total rutting Thermal cracking IRI Transverse cracking Joint faulting IRI Threshold 20% NA 0.5 inches 1000 ft/mile 172 in/mile 15% 0.125 inches 172 in/mile 4.9 SENSITIVITY ANALYSIS OF PAVEMENT-ME COEFFICIENTS The sensitivity of the Pavement-ME transfer function coefficients is crucial in estimating the impact of each coefficient on the overall performance predictions. It is often not viable to calibrate all coefficients; therefore, only the sensitive ones can be estimated if the sensitivity of each coefficient is known. The sensitivity of the Pavement-ME transfer function coefficients was obtained using SSCs and NSI values for both flexible and rigid pavements. Moreover, they were compared to the NSI values from the literature (26). Four transfer functions were used for flexible pavements: bottom-up cracking, top-down cracking, total rutting, and IRI, whereas two transfer functions were used for rigid pavements: transverse cracking and IRI. 101 4.9.1 Sensitivity Using Normalized Sensitivity Index (NSI) NSI has been typically used for this purpose and is defined as the percentage change of predicted distress relative to its global prediction caused by a given percentage change in the coefficient. The NSI was calculated using Equation (4-31). 𝑁𝑆𝐼 = 𝑆𝑖𝑗𝑘 𝐷𝐿 = ⁡ 𝛥𝑌𝑗𝑖 𝛥𝑋𝑘𝑖 𝑋𝑘𝑖 𝑌𝑗 (4-31) where; 𝑁𝑆𝐼 = Normalized sensitivity index, 𝐷𝐿 = Sensitivity index for input k, distress j, and at point i with respect to a given global 𝑆𝑖𝑗𝑘 prediction 𝛥𝑌𝑗𝑖 = Change in distress j around point i (𝑌𝑗,𝑖+1 − 𝑌𝑗,𝑖−1) 𝑋𝑘𝑖 = Value of input 𝑋𝑘⁡at point i 𝛥𝑋𝑘𝑖 = Change in input 𝑋𝑘⁡around point i (𝑋𝑘,𝑖+1 − 𝑋𝑘,𝑖−1) 𝑌𝑗 = Global prediction for distress j The NSI values were also calculated to compare them with the results from SSCs. These calculations are based on the NCHRP 1-47 study (60) as shown in Equation (4-31). Ten sections, each from flexible and rigid pavements, were selected for NSI calculations. These sections exist in the MDOT PMS database, designed using the AASHTO93 design method. These sections are also part of the selected sections for calibration. It is essential to mention that for NSI calculation, each section was modeled in the Pavement-ME with the necessary design inputs (material, traffic, and climate). These inputs were obtained from construction records, job mix formulas, and other sources. Obtaining the design input is tedious and requires multiple data sources, unlike the calculation of SSCs, which does not require any data. The selected sections have a wide range of thicknesses and traffic. Tables 4-8 and 4-9 show the Pavement-ME inputs for flexible and rigid sections, respectively. Each section was initially run at the global values of transfer function coefficients at 50% reliability. Afterward, each coefficient (one at a time) was varied by -50%, -20%, 20%, and 50%, respectively, from the global values. The change in performance prediction was evaluated for differences in transfer function coefficients to calculate the NSI values. 102 Table 4-8 Design inputs for flexible sections used in NSI calculations Section no. 1 2 3 4 5 6 7 8 9 10 HMA thickness (in.) 8 6.5 10.8 4.3 5.5 5.5 14 10.9 8 6.5 Base thickness (in.) 6 6 6 6 6 6 16 6 4 6 Subbase thickness (in.) 18 18 18 12 15 18 8 8 18 18 AADTT 2-way 2034 685 4315 201 859 959 6745 2065 354 313 Table 4-9 Design inputs for rigid sections used in NSI calculations PCC thickness (in.) 11 9.9 12.2 10.8 9.5 10.8 12.5 11.7 11.3 11 Base thickness (in.) 4 3.9 3.9 4 4 6 16 4 3.9 4 Subbase thickness (in.) 14 10 10 12 12 12 0 10 12 10 Dowel diameter (in.) 1.5 1.25 1.5 1.25 1.25 1.5 1.5 1.5 1.5 1.5 AADTT 2- way 7387 4825 12030 500 2758 10.8 12.5 11.7 11.3 11 Section no. 1 2 3 4 5 6 7 8 9 10 While NSI can rank the coefficients based on their level of sensitivity, it does not provide information about any potential correlation between them or how accurately these can be estimated. Moreover, since NSI calculation requires distress data, its magnitude can change if the data source is changed; hence, the sensitivity ranking of the coefficients may vary (11). 4.9.2 Sensitivity Using Scaled Sensitivity Coefficient (SSC) Unlike NSI calculation, SSCs do not require input data. SSCs were calculated for a continuous range of independent variables, and the results were visualized as SSC plots. The ith sensitivity coefficient of a model, η(x,β), where x is an independent ariable, and β represents the parameter vector, is given by 𝑋𝑖 = ∂ η/∂ βi and indicates the magnitude of change of the response resulting from a small perturbation in the parameter βi (64). An initial parameter value is required if the model is nonlinear in that parameter, i.e., ∂ η/∂ βi = f(βi), and requires an iterative solution using 103 any nonlinear regression algorithm (64). The parameter's SSC is the product of its sensitivity coefficient and the parameter itself, as shown in Equation (4-32). 𝑋𝑖 ′ = 𝛽𝑖 𝜕𝜂 𝜕𝛽𝑖 (4-32) where; ′ = Scaled sensitivity coefficient of the parameter i, 𝑋𝑖 𝛽𝑖 = Estimate of the ith parameter, 𝜕𝜂 𝜕𝛽𝑖 = ith sensitivity coefficient of the model w.r.t 𝛽𝑖. Assume that a model η(x,β) has two parameters, 𝛽1 and 𝛽2. The sensitivity coefficients (𝑋𝑖) and ′) for both parameters are estimated using the following equations [Equations (4-33) to SSC (𝑋𝑖 (4-36)]. Suppose the parameters (β) ha e been estimated using any nonlinear regression algorithm, and the sensitivity coefficient matrix J is obtained. In that case, the SSC for either parameter can be approximated using Equations (4-37) and (4-38). 𝑋1 = 𝜕𝜂 𝜕𝛽1 ≈ 𝜂((1.001 ∗ 𝛽1),⁡𝛽2) − 𝜂(𝛽1,⁡𝛽2) 0.001 ∗ 𝛽1 𝑋1 ′ = 𝛽1 𝜕𝜂 𝜕𝛽1 ≈ 𝜂((1.001 ∗ 𝛽1),⁡𝛽2) − 𝜂(𝛽1,⁡𝛽2) 0.001 𝑋2 = 𝜕𝜂 𝜕𝛽2 ≈ 𝜂(𝛽1, (1.001 ∗ 𝛽2),⁡) − 𝜂(𝛽1,⁡𝛽2) 0.001 ∗ 𝛽2 𝑋2 ′ = 𝛽2 𝜕𝜂 𝜕𝛽2 ≈ 𝜂(𝛽1, (1.001 ∗ 𝛽2),⁡) − 𝜂(𝛽1,⁡𝛽2) 0.001 𝑋1 ′ ≈ 𝛽1 ∗ 𝐽(: ,1) 𝑋2 ′ ≈ 𝛽2 ∗ 𝐽(: ,2) (4-33) (4-34) (4-35) (4-36) (4-37) (4-38) The SSC for a particular coefficient (say βi) is calculated by differentiating the function w.r.t. βi and multiplying it by βi [as shown in Equation (4-32)]. Other coefficients except βi are held constant. A similar approach is used to calculate SSCs for all other coefficients. The mathematical model (transfer function) can often be complicated, especially when differentiating the function. In that case, the SSCs can be approximated numerically to avoid errors in the analytical derivation. An example of the estimation of SSCs using the transverse cracking model [shown in Equation (4-39)] for rigid pavements. 104 𝐶𝑅𝐾 = 1 1 + 𝐶4(𝐷𝐼𝐹)𝐶5 (4-39) where, CRK = Predicted fraction of bottom-up or top-down cracking 𝐷𝐼𝐹 = Total fatigue damage (bottom-up or top-down) C4, C5 = Transfer function coefficients Denoting transverse cracking as a function of 𝐷𝐼𝐹, C4, and C5 [CRK(𝐷𝐼𝐹, C4, C5)], the sensitivity coefficient for C4 (𝑋𝐶4) can be approximated as shown in Equation (4-40). ∂𝐶𝑅𝐾 ∂𝐶4 = 𝑋𝐶4 ≈ 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4 + 𝛿, 𝐶5) − 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4, 𝐶5) 𝛿 × 𝐶4 (4-40) Here 𝛿 is a small quantity (a value of 0.001 was used). The SSC for C4 (𝑋′𝐶4) can be approximated as shown in Equation (4-41). 𝐶4 ∂𝐶𝑅𝐾 ∂𝐶4 = 𝑋′𝐶4 ≈ 𝐶4 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4 + 𝛿, 𝐶5) − 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4, 𝐶5) 𝛿 × 𝐶4 = 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4 + 𝛿, 𝐶5) − 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4, 𝐶5) 𝛿 (4-41) The coefficient C4 was changed by δ to get the first term of the numerator. The second term of the numerator is the transverse cracking at global values. Both these terms were evaluated at a continuous range of 𝐷𝐼𝐹⁡from 0 to 1. This provides a continuous set of 𝑋′𝐶4for each value of 𝐷𝐼𝐹. SSCs for C5 (𝑋𝐶5) was calculated as shown in Equation (4-42). SSCs for each coefficient were plotted with 𝐷𝐼𝐹⁡in the same plot. A similar process was used for all other transfer functions. 𝐶5 ∂𝐶𝑅𝐾 ∂𝐶5 = 𝑋′𝐶5 ≈ 𝐶5 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4, 𝐶5 + 𝛿) − 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4, 𝐶5) 𝛿 × 𝐶5 = 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4, 𝐶5 + 𝛿) − 𝐶𝑅𝐾(𝐷𝐼𝐹, 𝐶4, 𝐶5) 𝛿 (4-42) The SSCs were calculated and plotted using MATLAB codes using one coefficient at a time and considering other coefficients as constant. A wide range of independent variables have been used since calculating SSCs is a forward problem without data. 105 4.10 CHAPTER SUMMARY This chapter detailed the calibration approach used for each Pavement-ME prediction model. Transfer functions have been calibrated based on whether they calculate the distresses directly or based on cumulative damage. It also discusses the different resampling techniques and optimization methods. No sampling, bootstrapping, traditional split sampling, and repeated split sampling techniques have been used for calibration. For calibration validation, traditional and repeated split sampling were used. The calibration methods include the least squares and MLE. The process used for the MLE methodology is also outlined in this chapter. Reliability analysis is detailed, illustrating steps for estimating reliability equations for distress prediction, considering the transverse cracking as an example. Additionally, this chapter discusses the approach to assess the impact of calibration on pavement design based on thicknesses and critical distresses. Sensitivity analysis was conducted using Normalized Sensitivity Index (NSI) and Scaled Sensitivity Coefficients (SSCs), providing insights into the impact of model coefficients on performance predictions. These analyses facilitate the identification of sensitive coefficients crucial for accurate predictions and design decisions. 106 CHAPTER 5 - RESULTS AND DISCUSSION The calibration process adjusts the Pavement-ME model parameters to match observed data better to ensure that the model outputs are reliable and valuable for pavement design. The Pavement-ME models' calibration process can be challenging because of their complexity and the large number of parameters involved. However, technological advancements and data collection methods have made the calibration process more efficient and effective. This chapter documents the results for calibration of each model, pavement design, and sensitivity of the Pavement-ME coefficients. Table 5-1 summarizes the calibration method used for each Pavement-ME model. Table 5-1 Summary of calibration method for each Pavement-ME model Pavement type Pavement-ME model Flexible pavement Rigid pavement Bottom-up cracking: Option a Bottom-up cracking: Option b Top-down cracking Rutting (Method 1) Rutting (Method 2) Thermal cracking IRI Transverse cracking Joint faulting IRI Calibration method MLE ✓ ✓ ✓ ✓ ✓ Least squares ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 5.1 LOCAL CALIBRATION RESULTS FOR FLEXIBLE PAVEMENTS This section presents the results for the local calibration of the bottom-up cracking, total rutting, and IRI models. Bottom-up cracking was calibrated using synthetic and observed data. It is important to note that bottom-up cracking using Option a, rutting using Method 1, top-down and thermal cracking models using observed data were calibrated using the least squares method only, as shown in Table 5-1. The calibration results for these models are shown in Table 5-2. These results correspond to the bootstrap resampling technique. The details of these model calibrations are shown in the Appendix. 107 Table 5-2 Summary of flexible pavement models calibrated using only the least squares method Pavement-ME model Bottom-up cracking (Option a) Top-down cracking Local coefficient 𝐶1 = 0.2320 𝐶2 = 0.6998⁡(hac <5 in) 𝐶2 = (0.867 + 0.2583 ∗ ℎ𝑎𝑐) ∗ 0.2204 (5 in <= hac <=12 in) 𝐶2 = 0.8742⁡(hac >12 in) K𝐿1 = 64271618 K𝐿2 = 0.90 K𝐿3 = 0.09 K𝐿4 = 0.101 K𝐿5 = 3.260 𝐶1 = 0.30 𝐶2 = 1.155 𝐶3 = 1 HMA Base Subgrade 𝛽1𝑟 = 0.148 𝛽2𝑟 = 0.7 𝛽3𝑟 = 1.3 𝛽𝑠1 = 0.301 𝛽𝑠𝑔1 = 0.070 Rutting (Method 1) Thermal cracking 𝐾 = 0.85 Global model Bias SEE Local model Bias SEE 8.30 -4.91 8.73 0.00 6.37 -2.36 5.59 1.60 0.256 0.201 0.080 -0.013 0.042 0.038 0.009 0.118 0.109 0.006 -0.001 -0.000 1225 -812 851 20 5.1.1 Calibration Using Synthetic Data As mentioned in Chapter 4, exponentially distributed synthetic data was generated for bottom-up cracking with and without variability. Figure 5-1 shows the generated data distribution and different fitted probability distributions. The normal distribution legend in Figure 5-1 corresponds to the least squares method, while other distributions are used for the MLE method. The distribution is skewed for Figures 5-1(a) and 5-1(b) so that more data points are less than 5%, showing that the data is not normally distributed. (a) No variability (b) With 50% variability Figure 5-1 Distribution of synthetic data (bottom-up cracking in flexible pavements) 108 Both sets of generated data were calibrated using MLE and least squares methods, as well as the four mentioned sampling techniques. Table 5-3 summarizes both data sets’ no-sampling and bootstrapping calibration results. As previously described, resampling techniques provide confidence intervals for the population parameter. Bootstrapping calibration results in Table 5-3 are the mean values. MLE provides better statistical parameters (SEE, bias, NLL, AIC, and BIC values) for all distributions, except for SEE values in the case of exponential distribution. The SEE values show the variability between predicted and measured data. The higher the SEE value, the more the dispersion along the line of equality. Pavement-ME uses a reliability-based design. Higher values of SEE imply higher reliability imposed over mean Pavement-ME predictions. Gamma distribution provides the best parameter estimates for the MLE method, followed by negative binomial distribution. It is worth mentioning that the parameter estimates (C1 and C2) using MLE are much closer to the assumed initial values than the estimates from the least squares method. Tables 5-4 and 5-5 summarize the results from split sampling and repeated split sampling techniques. Table 5-5 shows the mean values using the repeated split sampling method. Similar results are obtained, where MLE provides better statistical parameters than the least squares method. The gamma and negative binomial distribution for these validation results also offer optimum results with the SEE and bias values significantly lower than the least squares method. It is worth mentioning that the results from resampling techniques provide better parameter estimates, as can be seen in Tables 5-3 and 5-5. Table 5-3 Summary of calibration results for synthetic data in flexible pavements Calibration method No sampling Bootstrapping Distribution Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial With no variability With 50% variability SEE 2.967 0.049 0.000 0.015 Bias 0.000 0.000 0.000 -0.007 NLL AIC 2385 1190 1040 1032 1038 2082 2068 2079 BIC 2393 2086 2076 2087 SEE 7.305 5.690 2.584 2.593 Bias 0.000 0.000 0.000 0.033 NLL AIC 2522 1259 1026 1020 1028 2055 2045 2060 BIC 2529 2059 2052 2068 0.002 -0.001 944 1891 1899 2.561 0.045 941 1886 1894 3.265 3.975 0.000 0.013 0.111 0.000 0.000 -0.007 1235 1015 1008 1032 2473 2032 2020 2068 2481 2036 2028 2076 4.282 4.986 2.553 2.552 0.143 0.000 0.000 -0.148 1269 1010 1006 1020 2542 2022 2016 2044 2550 2026 2024 2051 0.001 0.000 591 1186 1194 2.542 -0.001 683 1369 1377 109 Table 5-4 Summary of validation results using synthetic data in flexible pavements (Split sampling) Data set Distribution Calibration set Validation set Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial SEE 0.231 1.707 0.001 0.020 Bias 0.000 0.000 -0.001 -0.010 0.042 0.018 0.280 2.028 0.001 0.024 -0.041 -0.308 -0.001 -0.013 0.050 0.025 718 712 712 652 352 321 319 319 291 With no variability With 50% variability NLL AIC 1601 798 BIC 1608 1441 1434 1435 SEE 6.962 2.838 2.554 2.569 Bias 0.000 0.000 0.000 0.037 1437 1427 1428 NLL AIC 1743 870 1421 1413 1419 BIC 1750 1425 1420 1426 1309 1316 2.515 0.040 708 645 643 643 586 713 648 648 648 591 8.453 3.258 2.690 2.703 1.057 -0.175 0.145 0.196 2.689 0.129 1309 1316 784 634 633 633 581 790 637 639 639 587 710 705 707 652 390 316 315 315 289 Figure 5-2 compares both data sets’ calibration results using MLE and least squares. Figures 5- 2a and 5-2b show the propagation of bottom-up cracking with damage. The MLE predictions are closer to the synthetic measured data than the least squares predictions. This trend is more evident in Figure 5-2b, with 50% variability. Figures 5-2c, 5-2d, 5-2e, and 5-2f show the distribution of residuals (predicted – measured). Error distribution using MLE is less scattered and closer to zero. Moreover, it is closer to a normal distribution than the least squares method. Table 5-5 Summary of validation results using synthetic data in flexible pavements (Repeated split sampling) Data set Distribution Calibration set Validation set Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial With no variability With 50% variability NLL AIC 1920 958 SEE 4.436 3.825 0.000 0.023 Bias 0.203 0.000 0.000 -0.011 0.001 0.000 4.427 3.856 0.000 0.023 0.196 0.028 0.000 -0.011 0.001 0.000 NLL AIC 1838 917 722 719 719 473 390 308 306 306 201 1445 1443 1442 950 784 617 615 616 406 BIC 1845 1449 1450 1449 957 790 620 620 621 411 SEE 5.146 4.851 2.522 2.515 Bias 0.253 0.000 0.000 -0.059 2.508 0.006 5.147 4.904 2.559 2.544 0.236 0.011 0.009 -0.060 2.547 0.007 720 719 716 501 408 306 305 305 213 BIC 1927 1445 1448 1443 1441 1441 1436 1005 1012 819 614 614 613 431 824 617 619 618 436 5.1.2 Calibration Using Observed Data Based on the above process for synthetic data, the bottom-up cracking, total rutting, and IRI models were calibrated using MLE and least squares methods using observed data from field measurements. This observed data is obtained from MDOT's PMS database. Figure 5-3 shows 110 the distribution of observed data for different distresses and fitted distributions. Bottom-up cracking is the most skewed and non-normally distributed. Total rutting and IRI distributions are slightly skewed but closer to a normal distribution. As previously shown, resampling techniques provide better parameter estimates; therefore, bootstrapping and repeated split sampling results are presented. Bottom-up cracking – Option b: Table 5-6 summarizes bootstrapping and repeated split sampling results for bottom-up model calibration. MLE outperforms the least squares method with lower NLL, AIC, and BIC values for all distributions. The gamma distribution provides the best estimates for the MLE approach. Figure 5-4 shows the calibration results for bottom-up cracking using observed data using the bootstrapping technique for MLE (gamma distribution) and least squares methods. The predicted vs. measured plots show less MLE scatter than the least squares method. 111 (a) Bottom-up cracking vs damage (with no (b) Bottom-up cracking vs damage (with 50% variability) variability) (c) Distribution of residuals (least squares (d) Distribution of residuals (MLE method method with no variability) with no variability) (e) Distribution of residuals (least squares (f) Distribution of residuals (MLE method method with 50% variability) with 50% variability) Figure 5-2 Calibration results for bottom-up cracking using synthetic data 112 (a) Bottom-up cracking (b) Total rutting Figure 5-3 Distribution of observed data for flexible pavements (c) IRI The distribution of residuals for MLE is also closer to zero. In Figures 5-4e and 5-4f, the red dashed line indicates the mean, the blue solid line shows the median, the red dashed line shows the 2.5th and 97.5th percentiles and the solid black line shows the cumulative distribution. Interestingly, the model parameters are normally distributed in the case of MLE, with the bias value consistently closer to zero. Total rutting: Table 5-7 shows the calibration results for the total rutting model. MLE shows better NLL, AIC, and BIC values for all MLE distributions compared to the least squares method. Gamma and negative binomial distributions provide the most feasible results using MLE. It also illustrates a bias-variance tradeoff where the SEE for gamma distribution is slightly higher than the least squares method but has a lower bias value. Figure 5-5 shows the calibration results using 113 observed data for MLE (using gamma distribution) and the least squares methods. The predicted vs. measured plots show slightly less scatter for the MLE method. The residuals for MLE and least squares methods are comparable. Table 5-6 Summary of calibration and validation results for observed data (Bottom-up cracking: Option b) Calibration method Distribution SEE Bias C1 6.678 6.114 6.286 6.650 6.509 -3.769 0.052 0.000 0.000 -0.160 1.310 0.221 0.196 0.094 0.112 C2 (T<5 in.) 2.159 0.716 0.766 1.000 0.974 C2 (T=5 to 12 in.) 1.000 0.234 0.250 0.326 0.318 NLL AIC BIC 3.4E+08 6.8E+08 6.8E+08 2784 1652 1495 1523 1390 825 745 759 2792 1656 1502 1530 5.517 0.424 0.467 0.133 6.183 6.239 6.692 6.503 0.021 0.000 0.000 -0.163 0.210 0.206 0.095 0.113 0.745 0.733 0.997 0.973 5.555 0.443 0.469 0.127 6.224 6.281 6.792 6.597 0.043 0.025 0.064 -0.134 0.210 0.206 0.095 0.113 0.745 0.733 0.997 0.973 5.598 0.439 0.469 0.127 0.043 0.243 0.239 0.326 0.318 0.042 0.243 0.239 0.326 0.318 0.042 870 975 579 525 532 602 420 248 248 228 257 1744 1954 1161 1053 1068 1208 844 497 500 461 519 1752 1961 1164 1060 1075 1215 849 500 505 466 524 Global Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial Bootstrapping Repeated split sampling (Calibration set) Repeated split sampling (Validation set) IRI: Table 5-8 shows the calibration results for the IRI model. The results from the MLE and least squares methods are comparable. The negative binomial distribution provides the best estimates among all distributions for the MLE method. Figure 5-6 shows the calibration results for MLE (negative binomial distribution) and the least squares methods. The predicted vs. measured plot shows slightly less scatter for MLE. The residual distribution between the MLE and least squares methods is comparable. In the case of IRI, the bias is consistently close to zero for the least squares method, showing that it is efficient for a robust calibration. 114 (a) Predicted vs. measured cracking (least (b) Predicted vs. measured cracking (MLE) squares) (c) Distribution of residuals (least squares) (d) Distribution of residuals (MLE) (e) Distribution of parameters (least squares) (f) Distribution of parameters (MLE) Figure 5-4 Calibration results for bottom-up cracking (Option b) using observed data 115 Table 5-7 Summary of calibration results for observed data (Total rutting) Calibration method Bootstrapping Repeated split sampling (Calibration set) Repeated split sampling (Validation set) Distribution Global Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial SEE 0.393 0.084 0.084 0.096 0.093 Bias 0.349 -0.008 0.000 0.000 -0.012 β1r 0.400 0.144 0.173 0.129 0.102 βs1 1.000 0.839 0.859 0.163 0.158 βsg1 1.000 0.523 0.493 0.396 0.490 NLL AIC 7572 3784 4481 2238 4293 2146 4031 2013 4394 2195 BIC 7582 4490 4298 4041 4403 0.079 0.003 0.062 0.879 0.559 2230 4464 4474 0.078 0.085 0.096 0.085 0.000 0.000 0.000 0.000 0.028 0.071 0.124 0.071 1.185 0.835 0.160 0.835 0.634 0.490 0.432 0.490 1948 1503 1412 1503 3901 3007 2827 3007 3909 3012 2836 3012 0.079 0.003 0.063 0.869 0.558 1562 3127 3136 0.080 0.085 0.095 0.085 -0.015 0.000 0.000 0.000 0.028 0.071 0.124 0.071 1.185 0.835 0.160 0.835 0.634 0.490 0.432 0.490 0.080 0.003 0.063 0.869 0.558 840 643 607 643 797 1683 1289 1219 1289 1691 1292 1226 1292 1597 1604 Table 5-8 Summary of calibration results for observed data (IRI - Flexible) Calibration method Bootstrapping Repeated split sampling (Calibration set) Repeated split sampling (Validation set) Distribution SEE Bias C1 C2 C3 C4 NLL AIC BIC Global Normal 22.210 16.246 Exponential 16.406 18.943 18.573 Gamma Log normal Negative binomial Normal 15.866 Exponential 16.419 18.951 18.546 Gamma Log normal Negative binomial Normal 15.935 Exponential 16.433 19.034 18.623 Gamma Log normal Negative binomial 14.306 40.000 0.400 0.008 0.015 41.486 0.433 0.006 -0.630 0.0042 43.033 0.485 0.007 0.008 0.0042 0.0001 40.022 0.312 0.020 1.273 40.026 0.195 0.019 0.00005 0.593 7368 14740 14751 16996 33997 34007 15547 15553 7773 13267 13277 6631 13183 13194 6590 15.606 -0.516 41.727 0.259 0.005 0.00617 7745 15493 15504 0.167 0.000 1.280 0.599 48.841 0.327 0.006 43.485 0.516 0.006 40.038 0.324 0.019 40.017 0.202 0.019 0.00002 0.005 0.0041 0.000 4948 5444 4646 4615 9910 9900 10891 10896 9306 9297 9245 9235 15.671 -0.512 41.714 0.261 0.005 0.00615 5425 10853 10863 0.157 -0.006 1.288 0.586 48.841 0.327 0.006 43.485 0.516 0.006 40.038 0.324 0.019 40.017 0.202 0.019 0.00002 0.0051 0.0041 0.000 2118 2329 1988 1975 4241 4660 3980 3953 4249 4664 3989 3962 15.684 -0.502 41.714 0.261 0.005 0.00615 2009 4023 4031 116 (a) Predicted vs. measured rutting (least (b) Predicted vs. measured rutting (MLE) squares) (c) Distribution of residuals (least squares) (d) Distribution of residuals (MLE) (e) Distribution of parameters (least squares) (f) Distribution of parameters (MLE) Figure 5-5 Calibration results for total rutting using observed data 117 (a) Predicted vs. measured IRI (least (b) Predicted vs. measured IRI (MLE) squares) (c) Distribution of residuals (least squares) (d) Distribution of residuals (MLE) (e) Distribution of parameters (least squares) (f) Distribution of parameters (MLE) Figure 5-6 Calibration results for flexible IRI using observed data 118 5.2 LOCAL CALIBRATION RESULTS FOR RIGID PAVEMENTS This section presents the results for the local calibration of the transverse cracking and IRI models. It is important to note that the joint faulting model was calibrated using the least squares method only, as shown in Table 5-1. Table 5-9 shows the calibration results for the joint faulting model, the details of which are shown in the Appendix. Table 5-9 Summary of rigid pavement models calibrated using only the least squares method Pavement-ME model Local coefficient Global model SEE Bias Local model SEE Bias Joint faulting 𝐶1 = 0.8 𝐶2 = 1.3889 𝐶3 = 0.00217 𝐶4 = 0.00444 𝐶5 = 250 𝐶6 = 0.2 𝐶7 = 7.3 𝐶8 = 400 5.2.1 Calibration Using Synthetic Data 0.06 0.01 0.03 0.00 Transverse cracking data was exponentially generated to study the effectiveness of using MLE with different conditions and distributions. Figure 5-7 shows the generated data with different fitted distributions. The normal distribution legend in Figure 5-7 corresponds to the least squares method. (a) No variability (b) With 50% variability Figure 5-7 Distribution of synthetic data (transverse cracking in rigid pavements) 119 Calibration was performed using the least squares method and MLE for all four mentioned calibration and validation approaches: no sampling, bootstrapping, split sampling, and repeated split sampling. Table 5-10 summarizes calibration results using no sampling and bootstrapping methods. The mean values for the parameters are shown for the bootstrapping results in Table 5- 10. The least squares method is denoted as Normal in Table 5-10. All distributions using MLE perform better than the least squares method in terms of NLL, AIC, and BIC. Except for exponential distribution, other distributions perform better regarding SEE values. Exponential and gamma distributions have lower bias than the least squares method. The gamma distribution is the most suitable distribution for this synthetic data. Compared to the least squares method, it provides better results for all parameters (SEE, bias, NLL, AIC, and BIC values). A similar trend can be observed in the validation results. Tables 5-11 and 5-12 summarize the validation results using split sampling and repeated split sampling, respectively. The mean values for the parameters are shown for repeated split sampling results in Table 5-12. Gamma distribution provides better results than least squares regarding SEE, bias, NLL, AIC, and BIC values. This is more evident in resampling approaches. It is also a helpful illustration of the bias-variance tradeoff. Table 5-10 Summary of calibration results for synthetic data in rigid pavements Calibration method No sampling Bootstrapping Distribution Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial With no variability Bias NLL AIC BIC SEE -0.492 1040 2084 2092 6.996 1007 2016 2019 7.267 0.000 0.000 1965 1972 6.461 980 -0.023 1021 2046 2054 6.467 With 50% variability Bias NLL AIC BIC -0.099 1081 2166 2173 1992 1996 0.000 0.000 1982 1989 -0.369 1024 2053 2060 995 989 SEE 3.586 0.705 0.001 0.040 0.001 0.001 922 1847 1855 6.491 -0.051 920 1844 1851 2.315 4.628 0.001 0.036 -0.234 1016 2036 2043 7.159 1988 1992 8.515 0.000 0.000 1962 1969 6.422 -0.020 1019 2042 2049 6.454 993 979 -0.407 1075 2154 2161 1991 1995 0.000 0.000 1980 1987 -0.377 1023 2049 2056 995 988 0.017 -0.004 563 1131 1138 6.465 -0.107 725 1454 1462 120 Table 5-11 Summary of validation results using synthetic data in rigid pavements (Split sampling) Data set Distribution Calibration set Validation set Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial With no variability Bias 0.000 0.000 0.000 -0.028 NLL AIC 1409 703 1381 690 1367 682 1422 709 SEE 0.000 1.328 0.001 0.051 SEE BIC 1416 6.314 1384 6.435 1374 6.334 1428 6.317 With 50% variability Bias -0.103 0.000 0.000 -0.250 NLL AIC BIC 1461 1455 726 1379 1376 687 1375 1368 682 1423 1417 706 0.000 0.000 638 1280 1286 6.319 0.067 636 1276 1282 0.000 1.429 0.001 0.057 0.000 -0.188 -0.001 -0.034 0.000 0.000 303 304 298 312 284 610 609 600 627 572 614 611 605 632 7.065 7.690 6.855 7.037 -1.324 -1.394 -1.151 -1.460 330 308 307 318 576 7.125 -1.174 284 663 617 617 640 572 668 619 622 644 577 Table 5-12 Summary of validation results using synthetic data in rigid pavements (Repeated split sampling) Data set Distribution Calibration set Validation set Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial With no variability Bias NLL AIC 1457 726 -0.287 1397 697 0.000 1426 711 0.000 1427 711 -0.029 SEE 2.809 4.996 0.000 0.052 0.034 -0.016 465 2.828 5.052 0.000 0.052 -0.288 -0.006 0.000 -0.030 310 298 304 304 0.035 -0.016 199 934 624 598 611 611 402 BIC 1463 1400 1432 1433 940 629 600 616 616 407 With 50% variability Bias 0.005 0.000 0.000 -0.535 NLL AIC BIC 1490 1496 743 1407 1410 702 1391 1398 694 1448 1454 722 SEE 7.139 7.989 6.199 5.891 6.410 0.035 512 1028 1035 7.187 8.071 6.265 6.020 -0.036 -0.017 -0.016 -0.522 6.528 0.016 318 301 296 308 219 640 603 597 620 443 645 606 601 625 448 The gamma distribution is most suitable for MLE and performs better than least squares estimates. Figure 5-8 shows the calibration results using the least squares method and MLE using a gamma distribution. The MLE predictions are closer to the measured data points (synthetic data), whereas the distribution of residuals shows a low scatter. The mean residual value is the model bias, whereas the spread of residuals represents the SEE. The mean SEE and bias values for the gamma distribution are 0.001 and 0.000, whereas, for the least squares method, they are 2.315 and -0.234, respectively, using bootstrap resampling on synthetic data with no variability. The mean bias value for the gamma distribution remains 0.000, whereas for the least squares, it 121 is -0.407, using the bootstrap resampling on synthetic data with 50% variability. This shows the robustness of the MLE method for data with different variabilities. 5.2.2 Calibration Using Observed Data MLE and least squares methods were used to calibrate transverse cracking and IRI transfer functions using observed data obtained from MDOT's PMS data. Sections used for transverse cracking and IRI may differ, as the measured performance trends differ for both. Figure 5-9 shows the observed data distribution with different fitted distributions. Figure 5-9 shows that the transverse cracking data is skewed and non-normally distributed. IRI, on the other hand, is closer to a normal distribution. Transverse cracking: Table 5-13 summarizes the calibration and validation results for transverse cracking. Results for only the resampling approaches (bootstrapping and repeated split sampling) have been shown for brevity. The mean values for the parameters are shown in Table 5-13. MLE using gamma distribution provides the most feasible results with lower parameters (SEE, bias, NLL, AIC, and BIC) than the least squares. A similar trend is observed in the validation results using repeated split sampling (Table 5-13), where the MLE results show better validation parameters than the least squares method. Figure 5-10 shows the calibration results (for bootstrapping) for the least squares method and MLE using a gamma distribution. The predicted vs. measured transverse cracking shows a lower scatter for MLE. The mean bias value for the least squares is -0.410, whereas, for MLE, it is 0.000, using bootstrap results. The SEE values between the least squares and MLE are comparable. Also, the bias distribution for MLE is close to zero, illustrating the robustness of the MLE method. The lower and upper 95th percent confidence limits for the least squares are -0.932 and -0.025, whereas for the MLE, they are -0.001 to 0.001. This shows that MLE consistently has no bias for all 1000 bootstrap samples. Figure 5-10 (e) and (f) show the distribution of each bootstrap sample's SEE, bias, and transfer function coefficients. Bootstrap is used for 1000 resamples with replacement. A different set of parameters are obtained for each sample. These plots provide a distribution of parameters, and the mean value can be used as a more reliable estimate. 122 (a) Transverse cracking vs damage (with no (b) Transverse cracking vs damage (with variability) 50% variability) (c) Distribution of residuals (least squares (d) Distribution of residuals (MLE method method with no variability) with no variability) (e) Distribution of residuals (least squares (f) Distribution of residuals (MLE method method with 50% variability) with 50% variability) Figure 5-8 Calibration results for transverse cracking using synthetic data 123 (a) Transverse cracking (b) IRI Figure 5-9 Distribution of observed data for rigid pavements Table 5-13 Summary of calibration and validation results for observed data (Transverse cracking) Calibration method Bootstrapping Repeated split sampling (Calibration set) Repeated split sampling (Validation set) Distribution Global Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial SEE 5.994 4.022 4.218 3.984 4.363 4.812 4.074 4.225 4.038 4.359 4.883 4.129 4.252 4.124 4.374 4.900 Bias -2.390 -0.410 0.000 0.000 -0.578 -0.166 -0.411 0.000 0.000 -0.577 -0.184 -0.404 0.018 0.023 -0.567 -0.223 C4 0.52 0.476 1.071 0.668 1.406 4.563 0.467 1.091 0.650 1.406 4.704 0.467 1.091 0.650 1.406 4.704 C5 -2.17 -0.962 -0.708 -0.76 -0.654 -0.369 -0.963 -0.682 -0.761 -0.652 -0.363 -0.963 -0.682 -0.761 -0.652 -0.363 BIC AIC NLL 26051 52105 52112 1720 1713 973 970 890 882 790 783 945 938 1207 1200 686 682 628 622 555 548 665 658 548 543 297 294 273 268 245 240 292 287 854 484 439 389 467 598 340 309 272 327 270 146 132 118 142 IRI: Table 5-14 summarizes the calibration results for IRI using the least squares and MLE methods. Table 5-14 shows the mean values for the parameters obtained using bootstrap resampling. MLE using negative binomial shows the most feasible results among all distributions. Interestingly, the least squares method shows satisfactory calibration and validation results, especially with lower SEE and bias values than MLE results. 124 (a) Predicted vs. measured cracking (least (b) Predicted vs. measured cracking (MLE) squares) (c) Distribution of residuals (least squares) (d) Distribution of residuals (MLE) (e) Distribution of parameters (least squares) (f) Distribution of parameters (MLE) Figure 5-10 Calibration results for transverse cracking using observed data 125 Table 5-14 Summary of calibration results for observed data (IRI - Rigid) Calibration method Bootstrapping Repeated split sampling (Calibration set) Repeated split sampling (Validation set) Distribution SEE Bias C1 C2 C3 C4 NLL AIC BIC Global Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial Normal Exponential Gamma Log normal Negative binomial 19.721 11.696 0.820 0.442 1.493 10.208 17.503 17.304 17.772 0.02 1.389 3.953 0.915 1.515 2.518 1.171 1.604 2.546 1.114 2094 4191 4199 2.194 1.612 24.138 2464 4932 4941 2554 5110 5114 1974 3953 3961 1966 3936 3945 0.000 0.000 0.001 0.042 8.208 7.044 7.301 25.24 10.150 -0.312 0.001 2.229 1.471 27.041 1714 3432 3441 10.570 18.108 17.261 17.695 0.000 0.000 0.000 0.038 0.225 2.136 1.510 23.741 1412 2829 2837 1792 3587 3590 7.769 1.478 3.832 0.893 1386 2776 2784 6.789 1.503 2.462 1.201 1381 2766 2773 6.757 1.576 2.391 1.176 10.207 -0.316 0.001 2.227 1.476 26.834 1204 2412 2420 10.654 18.208 17.542 17.825 -0.008 0.225 2.136 1.510 23.741 7.769 1.478 3.832 0.893 0.006 6.789 1.503 2.462 1.201 0.098 6.757 1.576 2.391 1.176 0.080 600 762 590 589 1205 1211 1526 1529 1185 1191 1181 1187 10.394 -0.320 0.001 2.227 1.476 26.834 515 1033 1039 Figure 5-11 shows the calibration results for IRI (using bootstrapping) for the least squares and MLE using a negative binomial distribution. The SEE and bias values for the least squares are 10.208 and 0.000, whereas, for the MLE using negative binomial, they are 10.150 and -0.312, using bootstrap resampling. The predicted vs. measured IRI and distribution of residuals are similar for both methods. Figures 5-11 (e) and (f) show the SEE, bias, and IRI transfer function coefficients distribution for 1000 bootstrap resamples. The least squares method shows lower bias, which can be observed from the distribution of parameters in Figure 5-11. A similar trend is observed in the validation results (Table 5-14), where the least squares method shows better parameter estimates in terms of SEE and bias than the MLE method. 126 (a) Predicted vs. measured IRI (least (b) Predicted vs. measured IRI (MLE) squares) (c) Distribution of residuals (least squares) (d) Distribution of residuals (MLE) (e) Distribution of parameters (least squares) (f) Distribution of parameters (MLE) Figure 5-11 Calibration results for rigid IRI using observed data 127 5.3 IMPACT OF CALIBRATION ON PAVEMENT DESIGN Forty-four pavement sections, each for flexible and rigid sections, were designed to assess the impact of calibration on pavement design. It is important to note that the locally calibrated coefficients and standard error equations used for these designs were obtained using the least squares method. The standard error equations are summarized in Chapter 6. Table 5-15 shows the average design thickness for the 44 flexible and rigid sections. These are the final thicknesses based on the following criteria: • The minimum thickness should be 6.5" for flexible, 9" for JPCP freeway, and 8" for JPCP non-freeway sections. • A maximum difference of ± 1 inch from the AASHTO93 minimum thickness limits. • JPCP widened slab sections were designed as standard width (12 feet), and design thicknesses were reduced by a maximum of 1 inch depending on whether the previous conditions were met. This practice is followed because the slab width is a sensitive parameter in the Pavement-ME, giving impractical (very thin) designs. • The design trials were stopped when a pavement reached a maximum thickness of 16". Few designs fail at even 16", but further increasing the thickness leads to impractical designs. This occurs because a particular design may have inputs (material, traffic, climate) that are not well represented in the global (or local) dataset. Therefore, the Pavement-ME has difficulty providing a practical design outcome. These designs may require changes in the Pavement-ME inputs, and simply changing the thickness cannot achieve a passing design. Furthermore, MDOT is limited by design changes (construction, materials, budget, and design procedures). Therefore, changing the inputs may not be practical. Table 5-15 Summary of final pavement design thicknesses Pavement type Design method Flexible Rigid AASHTO93 Pavement-ME previous model Pavement-ME new calibrated model AASHTO93 Pavement-ME global model Pavement-ME new calibrated model Average 9.17 8.86 8.95 10.07 9.83 9.63 Design thickness (in) Standard deviation 2.20 1.78 2.27 1.67 1.63 1.44 CoV 24% 20% 25% 17% 17% 15% 128 The average design thickness using the newly calibrated models is closer to the AASHTO93 design than the previous model calibration, with an average thickness reduction of 0.22 inches for flexible sections. The average PCC thickness using the new calibrated model is 0.44 inches lower than the AASHTO93 design thickness. Interestingly, for designs using the global model, five sections reached the design thickness of 16 inches, and another five sections reached the design thickness of 6 inches. However, for the design using the locally calibrated model, only one section has a design thickness of 16 inches. Figure 5-12 shows the new calibrated model vs. AASHTO93 design thicknesses. Overall, the average design thickness using the locally calibrated models is slightly lower than the AASHTO93 design thickness for both flexible and rigid sections. (a) Flexible pavement design (b) Rigid pavement design Figure 5-12 New calibrated model vs. AASHTO93 final design thickness The Pavement-ME designs are based on several distresses, but it is crucial to identify the controlling distress. Figure 5-13 shows the contribution of different controlling distresses. The values shown in Figure 5-13 are the percentage of sections (out of 44) having that critical distress. It should be noted that some sections may have more than one controlling distress. Bottom-up and thermal cracking are the controlling distresses for flexible pavements, whereas transverse cracking and IRI are for rigid pavements. Figure 5-14 compares reliability for critical distress in flexible and rigid pavements. The standard deviation for the previously calibrated model is higher than the newly calibrated model for both bottom-up cracking and thermal cracking in flexible pavements. Also, the standard deviation for the newly calibrated model is higher than the global model for transverse cracking in rigid pavements. 129 (a) Flexible sections (b) Rigid sections Figure 5-13 Critical distresses for pavement design (a) Bottom-up cracking (Flexible) (b) Thermal cracking (Flexible) (c) Transverse cracking (Rigid) Figure 5-14 Comparison of reliability for critical distresses A higher standard deviation in predicted performance is expected to produce a thicker design, but the design results (Table 5-15) show that models with higher standard deviation have lower 130 design thicknesses. Therefore, these trends indicate that the difference in design thicknesses can be attributed to the calibration coefficients rather than the reliability of these models. 5.4 SENSITIVITY ANALYSIS OF PAVEMENT-ME MODEL COEFFICIENTS The sensitivity of Pavement-ME model coefficients was estimated using the NSI and SSC methods, as explained in Chapter 4. For NSI calculation, each section was initially run at the global values of transfer function coefficients at 50% reliability. Afterward, each coefficient (one at a time) was varied by -50%, -20%, 20%, and 50%, respectively, from the global values. The change in performance prediction was evaluated for differences in transfer function coefficients to calculate the NSI values. Table 5-16 shows the NSI values for each section in this study and the NSI values from Kim et al. (2014) (26). The NSI values vary significantly among different sections and from Kim et al. (2014). These differences are attributed to the material and climate, ultimately affecting the predicted performance. For example, coefficient C4 in the IRI model for rigid pavements ranges from 0.06 to 0.23. These values correspond to the coefficient categorized as non-sensitive and sensitive, respectively (60). Similarly, C2 in bottom-up cracking ranges from -1.3 to -369.5, corresponding to very sensitive and hypersensitive categories. It is important to note that the magnitude of bottom-up cracking in flexible pavements and transverse cracking in rigid pavements was extremely low (close to zero). This has resulted in very high NSI values for C1 in bottom-up cracking and C5 in transverse cracking. These values are also significantly different from the ones in Kim et al. (2014). This is mainly because of the difference in magnitude of bottom-up and transverse cracking between the two studies. The SSCs were calculated and plotted using MATLAB codes using one coefficient at a time and considering other coefficients as constant. A wide range of independent variables have been used since calculating SSCs is a forward problem without data. Figures 5-15 and 5-16 show the SSCs for flexible and rigid pavements. Transfer functions with multiple independent variables have all independent variables shown in the same plot on the x-axis. 131 Table 5-16 Summary of NSI values for transfer function coefficients Performance model Bottom-up cracking (flexible) Top-down cracking (flexible) Total rutting (flexible) IRI (Flexible) Transverse cracking (rigid) IRI (rigid) C1 C2 C1 C2 C3 𝛽1𝑟 𝛽𝑠1 𝛽𝑠𝑔1 C1 C2 C3 C4 C4 C5 C1 C2 C3 C4 Section no. 1 2 3 4 5 6 7 8 9 10 -61.3 -66.0 -58.5 -88.4 -71 -72.2 -57.1 -2.9 -0.59 -2.42 -0.03 0.23 0.24 0.53 0.09 0.02 0.13 0.26 -2.38 -4E4 0.00 0.01 0.06 0.15 -5.28 -0.63 -2.80 -0.18 0.21 0.27 0.52 0.11 0.03 0.13 0.30 -2.38 -1E5 0.00 0.00 0.37 0.09 -40.8 -0.59 -2.37 -0.01 0.18 0.20 0.62 0.07 0.02 0.13 0.22 -2.38 -1E5 0.00 0.00 0.34 0.11 -14.14 0.00 0.00 0.00 0.13 0.28 0.60 0.08 0.00 0.13 0.21 -2.38 -1E6 0.00 0.01 0.08 0.23 -1.3 -0.6 -2.7 -0.1 0.15 0.23 0.62 0.11 0.02 0.06 0.30 -2.38 -1E6 0.00 0.01 0.38 0.12 -17.9 -0.67 -3.39 -0.64 0.13 0.29 0.58 0.09 0.02 0.00 0.25 -2.38 -8E3 0.00 0.01 0.25 0.13 -369 -0.59 -2.36 0.00 0.19 0.17 0.64 0.01 0.02 0.00 0.32 -2.38 -2E3 0.00 0.00 0.40 0.07 -58.4 -15.1 -0.59 -2.42 -0.02 0.19 0.13 0.68 0.11 0.03 0.00 0.30 -2.38 -3E3 0.00 0.00 0.47 0.10 -61.8 -94.9 -0.72 -4.11 0.00 0.17 0.26 0.57 0.06 0.01 0.00 0.25 -2.37 -1E2 0.00 0.00 0.48 0.06 -867 -35.4 -0.78 -5.01 0.00 0.18 0.29 0.53 0.07 0.01 0.00 0.28 -2.38 -4E4 0.00 0.01 0.18 0.13 Average NSI -146.2 -59.75 -0.58 -2.76 -0.10 0.18 0.24 0.59 0.08 0.02 0.06 0.27 -2.38 -2E5 0.00 0.01 0.30 0.12 Kim et al. (26) -11.3 -2.29 NA NA NA 1 1 1 0.15 0.00 0.00 0.31 -0.08 0.20 0.43 0.02 0.07 0.48 Figures 5-15 and 5-16 show the following observations: • Bottom-up cracking (flexible): C1 is more sensitive than C2, and C1 and C2 are not correlated. Moreover, both C1 and C2 are large enough to be confidently estimated. Coefficients with negative SSCs indicate that an increase in the coefficient will decrease predicted performance. Therefore, an increase in C1 or C2 will reduce bottom-up cracking. • Top-down cracking (flexible): The sensitivity of coefficients changes with the independent variables, which are t (analysis time in days) and t0 (time to crack initiation). Overall, C3 is the most sensitive coefficient, followed by C2. C1 is the least sensitive coefficient. C1 and C2 are correlated, which signifies that only one of them can be estimated with confidence. All coefficients are estimable based on the magnitude of SSCs, and an increase in any of the coefficients will reduce the predicted top-down cracking. • Total rutting (flexible): Total rutting is a linear model between the individual layer rutting. Subgrade rutting coefficient (𝛽𝑠𝑔1) is the most sensitive, followed by the AC rutting coefficient (𝛽1𝑟). The base rutting coefficient ( 𝛽𝑠1) is the least sensitive. SSCs for all coefficients are large enough to be estimable and positive. • IRI (flexible): IRI is a linear relationship between IRI at the time of construction (initial IRI) and other distress (cracking, rutting, etc.). The site factor coefficient is the most sensitive, followed by the total rutting coefficient. The thermal cracking coefficient is the 132 next sensitive coefficient, while the fatigue cracking coefficient is the least sensitive. All coefficients have positive values for SSCs. • Transverse cracking (rigid): C5 is more sensitive than C4, and the change in sensitivity with damage can be clearly observed. C4 and C5 are not correlated, and the SSCs for both coefficients are large enough to be estimated with confidence. • IRI (rigid): The transverse cracking coefficient is the most sensitive, and the joint spalling coefficient is the least sensitive. Moreover, the magnitude of SSCs for joint spalling is very low, indicating that the coefficient cannot be estimated with high confidence. (a) Bottom-up cracking (b) Top-down cracking (c) Total rutting (d) IRI Figure 5-15 SSCs for flexible pavements 133 (a) Transverse cracking (b) IRI Figure 5-16 SSCs for rigid pavements SSCs are highly suitable for showing sensitivity for any range of independent variables. For example, the SSC plot for IRI in Figure 5-16 shows that C1 is the most sensitive coefficient, whereas the NSI values are calculated to show that it is the least sensitive input. This is because of the low values of transverse cracking used to calculate the NSI values. Figure 5-17 shows the SSC plot for IRI in rigid pavements using low values for transverse cracking. It can be observed that at this range of transverse cracking, C1 is the least sensitive coefficient. Figure 5-17 SSCs for IRI on small values of transverse cracking The SSC plot is used to visualize the error in parameter estimation. Moreover, the larger the SSC magnitude, the more confidence in parameter estimation. Calibrating the transverse cracking model in rigid pavements is an example of verification. From Figure 5-16a, C5 should have less 134 estimation error than C4. Error in estimation for any parameter refers to the relative error, i.e., the ratio of standard error and the parameter value. C4 and C5 are not correlated, and the SSCs for both coefficients are large enough to be estimated with confidence. The relative error should be less than 60%; otherwise, the confidence interval of the parameter likely includes zero. In other words, the parameter is not estimable or not statistically different than zero. The selected rigid pavements were used to calibrate the transverse cracking model and validate the applicability of SSCs. The measured performance data is obtained from the PMS records, and the Pavement-ME inputs are obtained from construction records, material testing results, and the Job Mix Formula (JMF). Figure 5-18 shows the predicted vs. measured transverse cracking for global and locally calibrated model coefficients. Table 5-17 summarizes the standard error of estimate (SEE), bias, and relative error. The local calibration significantly improved the model predictions. Moreover, the relative error for C5 is less than that for C4, with both values less than 60%. The relative error values verify the results from the SSC plot, and therefore, both coefficients can be estimated with confidence. (a) Global model (b) Local model Figure 5-18 Predicted vs. measured transverse cracking in rigid pavements Table 5-17 Summary of transverse cracking model calibration Global model SEE Bias Local model SEE Relative standard error Bias Coefficient C4 C5 Value 0.52 -2.17 5.99 -2.39 Value 0.426 -0.953 3.95 -0.40 20.73% 6.14% 135 5.5 CHAPTER SUMMARY This chapter summarizes the calibration results for the flexible and rigid Pavement-ME models. Using synthetic and observed data, local calibration was performed using the least squares and MLE methods. Synthetic data was generated using an exponential distribution for bottom-up cracking in flexible pavements and transverse cracking in rigid pavements. MLE results outperformed the least squares method for both sets of synthetic data. Calibration results using observed data showed that MLE provides better parameter estimates for non-normally distributed data. For normally distributed data, MLE and least squares results were comparable. Forty-four sections each for new flexible and rigid pavements were designed using least squares calibration results to assess the impact of calibration on the pavement design. On average, the surface thicknesses using locally calibrated coefficients were thinner than the AASHTO93 design by 0.22 and 0.44 inches for flexible and rigid pavements, respectively. Critical design distresses for flexible pavements include bottom-up and thermal cracking. On the other hand, transverse cracking and IRI control the designs for rigid sections. NSI and SSC methods were used to evaluate the sensitivity of the Pavement-ME transfer function coefficients. Ten sections each, from flexible and rigid pavements, were used to calculate the NSI values and compared with the literature. Results show that SSCs provide a more reliable sensitivity on a range of independent variables rather than a point estimate, unlike NSI. NSI values showed variability among different sections, depending on the magnitude of predicted performance. 136 CHAPTER 6 - CONCLUSIONS, RECOMMENDATIONS AND FUTURE SCOPE 6.1 KEY FINDINGS This study introduces a novel calibration approach for the Pavement-ME transfer functions using MLE and compares it with the least squares method. The calibration was performed using synthetic and observed field data. The impact of calibration on pavement design was also assessed. Moreover, the sensitivity of the Pavement-ME model coefficients was also evaluated using the traditional NSI and the SSC approach. The following conclusions can be drawn based on the results. • The synthetic and observed data distribution for bottom-up cracking in flexible pavements shows skewness, with most data points below 5%. Fitting different distributions over data shows that bottom-up cracking is non-normally distributed. The distribution of observed data for total rutting and IRI shows slight skewness. Moreover, the distribution is close to normal, especially for IRI. • Calibration results from synthetic data indicate that MLE outperforms the least squares method based on statistical parameters and computational efficiency for flexible pavements. The gamma distribution is the most optimum distribution for MLE, consistently showing SEE and bias values close to zero for the synthetic bottom-up cracking data. The SEE value reduced for MLE results from 3.3 to 0.0 for bootstrapping and 4.4 to 0.0 for repeated split sampling (validation) compared to the least squares results for the dataset with no variability. The dominance of MLE calibration is more evident for datasets with 50% variability, especially in the case of validation. • For the observed data, the gamma distribution is most suitable for bottom-up cracking and total rutting models, whereas the negative binomial is for the IRI model. The predicted vs. measured plots show less scatter for MLE results than the least squares results for all models. The applicability of MLE is more evident for the bottom-up cracking model. The residual distribution is normally distributed and closer to zero. Moreover, the distribution of parameters is close to a normal distribution, and the bias value is consistently zero, showing the robustness of the calibration results. Calibration of the total rutting model using MLE showed a slight improvement compared to the least squares method, whereas IRI calibration 137 results for MLE and least squares methods were comparable. This indicates that MLE is more effective for non-normally distributed data. • The use of MLE on synthetic data for rigid pavements also showed better computational efficiency and applicability to bias-variance tradeoffs compared to the least squares method. The gamma distribution is most suitable for the generated synthetic data for transverse cracking. The mean bias (using bootstrapping) for MLE using gamma distribution is zero for data without and with 50% variability. The SEE values for the least squares method and MLE using gamma distribution are comparable with slightly lower values for MLE. A similar trend is observed in validation results. • In rigid pavements, the gamma distribution is most suitable for transverse cracking using observed data. The mean bias is consistently near zero using the MLE method for transverse cracking. Calibration using MLE significantly reduces the model bias while keeping the SEE comparable (slightly lower) than the least squares method. Calibration results for IRI using the least squares method and MLE are similar, with the least squares method being somewhat better regarding model bias. The negative binomial is the most suitable distribution for the MLE method. • The MLE method is proven most effective for skewed and non-normally distributed data, such as bottom-up cracking in flexible pavements and transverse cracking in rigid pavements. In contrast, the least squares method suits data close to a normal distribution, such as IRI. Prior knowledge of distribution is required for the use of MLE. • Calibration significantly improved performance predictions for both least squares and MLE methods. Resampling methods provide better calibration results with lower SEE and bias and can improve the overall robustness of the MLE approach. • The average design thicknesses using new calibration coefficients were close to AASHTO93 design thicknesses with a reduction of 0.22 and 0.44 inches in flexible and rigid pavements, respectively. The design thickness using new calibration coefficients was less than the AASHTO93 design thickness for 21 sections and equal for 12 of 44 flexible sections. Similarly, the design thickness using new calibration coefficients was less than the AASHTO93 design thickness for 17 and equal for 23 of 44 rigid sections. • Thermal cracking is the most critical distress for flexible sections, with 61.4%, followed by bottom-up cracking, with a 36.4% contribution. The contribution of total rutting was 2.3%. 138 None of the sections had top-down cracking or IRI as their critical distress. In rigid sections, IRI controlled distress with 77.3%, followed by transverse cracking with a 29.5% contribution. Joint faulting had the most negligible contribution of 6.8%. Comparison between the standard deviation of different models indicates that the differences in design thicknesses come from the calibration coefficients rather than the reliability for both flexible and rigid sections. • The sensitivity analysis showed that NSI values differed for each section in both flexible and rigid pavements. These sections have been designed using different Pavement-ME inputs, resulting in a wide range of performance predictions and, ultimately, a range of NSI values. The bottom-up cracking predictions in flexible and transverse cracking predictions in rigid sections were extremely low (close to zero). This resulted in very high NSI values, which are unreliable. The coefficient C1 for IRI in rigid sections is also zero because of the low magnitude of transverse cracking. NSI values are variable and depend on the magnitude of predicted distresses. Moreover, the Pavement-ME inputs (material, traffic, and climatic) are required for NSI calculations. • SSCs provide a convenient visual representation of the sensitivity of different transfer function coefficients over a continuous range of independent variables, unlike NSI, which is a point estimate. SSCs for transverse cracking and IRI for rigid sections show that the sensitivity changes at different ranges of the independent variable. It also indicates any correlations between different coefficients and confidence in estimation. Calculation of SSCs is a forward problem and does not require any input data. Therefore, a user only needs a mathematical model (the transfer functions) and can calculate SSCs on any range of independent variables. • NSI and SSCs provide a measure of sensitivity, but it is convenient to rank transfer function coefficients for straightforward interpretation. Table 6-1 shows the ranking of transfer function coefficients based on different methods. The order using SSCs is based on the overall sensitivity in the entire range of independent variables. As previously shown, this sensitivity might change for a limited range of independent variables. Coefficients with the same NSI values have been ranked the same. For example, all rutting coefficients in Kim et al. (2014) (26) study have been ranked 1 as they all have the same NSI values. Some models (e.g., bottom-up cracking and transverse cracking) have similar rankings using different 139 methods, whereas others (e.g., IRI for rigid pavements) have significant differences. These differences make it challenging to estimate the most sensitive coefficients truly. Therefore, SSCs can help obtain a continuous range of sensitivity rather than a point estimate. Table 6-1 Rank of transfer function coefficients based on different methods Pavement type Performance model Bottom-up cracking Top-down cracking Flexible Total rutting IRI Transverse cracking IRI Rigid Coefficient NSI SSCs C1 C2 C1 C2 C3 𝛽1𝑟 𝛽𝑠1 𝛽𝑠𝑔1 C1 C2 C3 C4 C4 C5 C1 C2 C3 C4 1 2 2 1 3 3 2 1 3 4 2 1 2 1 4 3 1 2 1 2 3 2 1 2 3 1 2 4 3 1 2 1 1 4 3 2 Kim et al. (2014) (26) 1 2 NA NA NA 1 1 1 2 3 3 1 2 1 2 4 3 1 6.2 RECOMMENDED CALIBRATION COEFFICIENTS Tables 6-2 and 6-3 summarize the recommended calibration coefficients and reliability equations for flexible and rigid pavements. These results were obtained using the least squares method and validated with extensive pavement designs. The detailed results of pavement designs are shown in Chapter 5. 140 Table 6-2 Flexible pavement recommended calibration coefficients and standard error equations Local coefficient Standard error Performance prediction model Bottom-up cracking (Option a) Bottom-up cracking (Option b) Top-down cracking HMA Rutting 𝐶1 = 0.2320 𝐶2 = 0.6998⁡(hac <5 in) 𝐶2 = (0.867 + 0.2583 ∗ ℎ𝑎𝑐) ∗ 0.2204 (5 in <= hac <=12 in) 𝐶2 = 0.8742⁡(hac >12 in) 𝐶1 = 0.2540 𝐶2 = 0.7303⁡(hac <5 in) 𝐶2 = (0.867 + 0.2583 ∗ ℎ𝑎𝑐) ∗ 0.2692 (5 in <= hac <=12 in) 𝐶2 = 1.0678⁡(hac >12 in) K𝐿1 = 64271618 K𝐿2 = 0.90 K𝐿3 = 0.09 K𝐿4 = 0.101 K𝐿5 = 3.260 𝐶1 = 0.30 𝐶2 = 1.155 𝐶3 = 1 𝛽1𝑟 = 0.148 𝛽2𝑟 = 0.7 𝛽3𝑟 = 1.3 Base/subgrade 𝛽𝑠1 = 0.301 𝛽𝑠𝑔1 = 0.070 Thermal cracking IRI 𝐾 = 0.85 𝐶1 = 42.874, 𝐶2 = 0.102 𝐶3 = 0.0081, 𝐶4 = 0.003 𝑠𝑒(𝐵𝑈) = 0.2262 + 14.2349 1 + exp⁡(0.2958 − 0.1441 log(𝐶𝑟𝑎𝑐𝑘)) 𝑠𝑒(𝐵𝑈) = 4.4396 + 25.4391 1 + exp⁡(4.3119 − 2.2778 log(𝐶𝑟𝑎𝑐𝑘)) 𝑠𝑒(𝑇𝐷) = 0.6417 × 𝑇𝑂𝑃 + 0.5014 𝑠𝑒(𝐻𝑀𝐴) = 0.1481(𝑅𝑈𝑇𝐻𝑀𝐴)0.4175 𝑠𝑒(𝑏𝑎𝑠𝑒) = 0.0411(𝑅𝑈𝑇𝑏𝑎𝑠𝑒)0.3656 𝑠𝑒(𝑠𝑢𝑏𝑔𝑟𝑎𝑑𝑒) = 0. 0728(𝑅𝑈𝑇𝑠𝑢𝑏𝑔𝑟𝑎𝑑𝑒) 𝑠𝑒(𝑇𝐶) = 0.1223(𝑇𝐶) + 400.9 Internally determined by the software 0.5456 Table 6-3 Rigid pavement recommended calibration coefficients and standard error equations Performance prediction model Local coefficient Transverse cracking Transverse joint faulting IRI 𝐶4 = 0.415 𝐶5 = −0.965 𝐶1 = 0.6 𝐶2 = 1.611 𝐶3 = 0.00217 𝐶4 = 0.00444 𝐶5 = 250 𝐶6 = 0.2 𝐶7 = 7.3 𝐶8 = 400 𝐶1 = 0.0942 𝐶2 = 1.5471 𝐶3 = 1.7970 𝐶4 = 23.7529 6.3 PRACTICAL IMPLICATIONS Standard error 𝑠𝑒(𝐶𝑅𝐾) = 2.9004(𝐶𝑅𝐾)0.5074 𝑠𝑒(𝐹𝑎𝑢𝑙𝑡) = 0.0919(𝐹𝑎𝑢𝑙𝑡)0.2249 Internally determined by the software This study provides a framework for the local calibration of performance models. Highway agencies can leverage the results for better design and adaptation of the Pavement-ME for local conditions. The critical implications include: 141 • The recorded performance data may have irregularities due to measurement errors and limitations in distress identification. Moreover, the recorded performance data may require conversion to the Pavement-ME units, which involves several assumptions. It may cause anomalies in the measured performance data and may not be practical to use directly for calibration. Therefore, analyzing the raw performance data and filtering it (if required) is recommended for practicality. • It is worth mentioning that the calibration process and pavement design were simultaneously executed. For every set of calibration coefficients, pavements were designed, and the calibration was improved based on the results. Pavement design is one of the most crucial calibration process steps and is often not considered in practice. It is recommended that the calibration results should not be based only on statistical parameters (SEE, bias, etc.) but also on practical engineering judgments. • Identifying critical design distress types is crucial. By understanding which distress types are most relevant to their region, agencies can develop mitigation and maintenance strategies leading to longer pavement service lives. For example, thermal cracking is critical in Michigan for flexible pavements. MDOT can mitigate the occurrence of cracking by using modified and improved binders. • It is recommended that local calibrations be performed every six years when more time series data points (e.g., three data points in Michigan) are available for the already selected and new pavement sections. SSCs can help agencies improve their local calibration process. The advantages and interpretation of the SSC plots are described in Chapters 4 and 5. The following approach is recommended to leverage these SSC plots before starting the local calibration process: • Run Pavement-ME to identify the magnitude of independent variables for each model. For example, one should know the range of damage values for transverse cracking in rigid pavements. • Obtain the sensitivity of each calibration coefficient from the SSC plots for the respective range of independent variables. • Ensure that the SSCs for each coefficient are large enough (the maximum value of SSC should be at least 10% of the largest value of the dependent variable). For example, the maximum SSC values for C4 and C5 are 25% and 38%, respectively, in transverse cracking 142 in rigid pavements. These SSC values exceeded 10% of the maximum predicted transverse cracking. Moreover, the SSCs should not be correlated (SSCs for different coefficients should not show similar trends). • If the SSCs are not large enough, one can not estimate those coefficients with sufficient confidence, i.e., they may be insignificant. On the other hand, if coefficients are correlated, both coefficients cannot be simultaneously estimated. For example, coefficients C1 and C2 in top-down cracking for flexible pavements show a correlation; therefore, only one should be calibrated. Calibration of C2 is recommended since the magnitude of SSC for C2 is higher. • Ensure that the relative error is lower for the more sensitive coefficients and is not more than 60% for any coefficient. • The SSCs can highlight the most significant coefficients for a range of independent variables. That can help in diverting more attention to those coefficients during local calibration. For example, in the rigid IRI model, C1 is the least sensitive for lower transverse cracking (less than 1%), and C1 is the most sensitive for higher transverse cracking. 6.4 REVIEW OF CAT TOOL This study used The CAT tool to calibrate the thermal cracking model in flexible and joint faulting models in rigid pavements. CAT provides a convenient alternative for those models where rerunning Pavement-ME is required. The advantages and limitations of the CAT tool are summarized below: Advantages of CAT • CAT provides good visualization of the input data and experimental matrix of the *.dgpx files. It helps quickly glance at the overall data and identify any outliers or biases. • It has default validation of the optimized coefficients, which helps to verify the model on an independent set of sections. • It provides sufficient descriptive statistics for calibration results and a linear model showing the effect of different Pavement-ME inputs on the overall calibration. • It helps visualize the change in error and bias for each iteration, making it easier to identify local minima in the given range. • It assists in evaluating the impact of the number of bins on the reliability model. 143 Limitations of CAT • Pavement-ME (.dgpx) files, once uploaded, cannot be deleted. Also, trials for optimizing calibration coefficients are run; they cannot be deleted or paused. This makes the nomenclature of the .dgpx files challenging, and trials should be run meticulously. • The limit to the total number of combinations of calibration coefficients is 100. Hence, all calibration coefficients cannot be changed simultaneously for several increments. • Since the number of increments is fixed to 100, the coefficients must be changed systematically by reducing the range provided. Also, not more than three coefficients can be involved in one trial run for a reasonable range and number of increments. It makes the optimization process cumbersome, and some prior experience is required to recalibrate with optimum time and effort. • The computation time is comparatively large. For example, for 100 pavement sections, changing a total of two calibration coefficients with five increments each makes it a total of 100 5 5 = 2500 Pavement-ME runs, which takes a computation time of around 29 hours. Therefore, considerable computational time is required, especially when the number of sections is large. • The same sections cannot be used for different projects using different measured data. Changing the measured data changes it in all existing (already run) projects. 6.5 FUTURE SCOPE OF THIS STUDY The scope of this study is limited to new flexible and rigid pavements. Moreover, bottom-up cracking, total rutting, IRI models for flexible pavements, and transverse cracking and IRI models for rigid pavements were calibrated using the four distributions mentioned: exponential, gamma, log-normal, and negative binomial. Using an exponential distribution, the MLE methodology was validated using synthetic data for bottom-up cracking in flexible and transverse cracking in rigid pavements. The following can be explored as part of future studies: • The MLE approach can be extended to calibrate other Pavement-ME models and models for rehabilitated pavements. Different probability distributions can be explored as part of future research. 144  • This methodology can be validated using synthetic data for different Pavement-ME transfer functions. Moreover, synthetic data can be generated using different distributions and variability. • Further studies can be conducted to estimate the impact of varying calibration approaches on pavement design. • Top-down cracking model calibration improved the SEE and bias but did not provide realistic results, i.e., high SEE. Furthermore, the top-down cracking predictions didn't vary for different sections, producing the same predictions. The Pavement-ME limits the thermal cracking prediction to 2112 ft/mile, but the measured data showed several records of thermal cracking above 2112 ft/mile. Also, the thermal cracking coefficient in the current version is changed and is a function of MAAT. This made the calibration of the thermal cracking model challenging. Due to the model's limitations, although the SEE and bias were improved after local calibration, the thermal cracking model still showed high variability. 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Michigan DOT User Guide for Mechanistic-Empirical Pavement Design. Michigan Department of Transportation Lansing, MI, USA; 2021. 73. Baladi GY, Thottempudi A, Dawson T. Backcalculation of unbound granular layer moduli. 2011. 74. Baladi G, Dawson T, Sessions C. Pa ement subgrade MR design alues for michigan’s seasonal changes, final report. Michigan Department of Transportation, Construction and Technology Division, PO Box. 2009;30049. 75. Singh RR, Haider SW, Kutay ME, Cetin B, Buch N. Impact of Climatic Data Sources on Pavement Performance Prediction in Michigan. Journal of Transportation Engineering, Part B: Pavements. 2022;148(3):04022048. 150 APPENDIX This chapter summarizes the results of the Pavement-ME models calibrated using the least squares method only. These models include bottom-up cracking (Option a), top-down cracking, thermal cracking, and rutting (Method 1) models for flexible pavements and joint faulting models for rigid pavements. BOTTOM-UP CRACKING MODEL (OPTION A) No Sampling In no sampling, the entire dataset was used for calibration. The error between the predicted and measured fatigue cracking was minimized. Figure A-1 shows the predicted versus measured bottom-up for the global and locally calibrated models. The global model underpredicts bottom- up cracking. Table A-1 shows the local calibration results. The SEE is reduced from 8.28 to 8.08, whereas the bias is reduced from -4.90 to 0.17. Figure A-2 shows the fatigue damage curve and the measured and locally predicted bottom-up cracking with time. These measured and predicted cracking values are for the same sections and at the same ages. Figure A-2 shows that local predictions are close to the measured values. (a) Global model (b) Local model Figure A-1 Predicted vs. measured bottom-up cracking (No sampling) 151 (a) Fatigue damage (b) Measured and predicted time series Figure A-2 Local calibration results for bottom-up cracking (No sampling) Table A-1 Local calibration summary for bottom-up cracking (No sampling) Parameter SEE (% total lane area) Bias (% total lane area) C1 C2 (hac < 5 in.) Global model 8.28 -4.90 1.31 2.1585 Local model 8.08 0.17 0.22 0.66 C2 (5 in. <= hac <=12 in.) (0.867+0.2583* hac)*1 (0.867+0.2583* hac)*0.22 Split Sampling Split sampling was used with a random split of 70% sections for the calibration set and the rest 30% for the validation set. Figure A-3 shows the predicted vs. measured bottom-up cracking for the calibration and validation sets. The validation set shows a similar trend as the calibration set. Table A-2 summarizes the local calibration results. Though SEE is higher than the global model, bias is significantly improved from -4.54 to 0.7018 in the validation set. Overall, the validation results are satisfactory. Table A-2 Local calibration summary for bottom-up cracking (split sampling) Parameter SEE (% total lane area) Bias (% total lane area) C1 C2 (hac < 5 in.) Global model 7.76 -4.54 1.31 2.1585 Local model 7.11 -0.47 0.19 0.78 Validation 11.2955 0.7018 0.19 0.78 C2 (5 in. <= hac <=12 in.) (0.867+0.2583*hac)*1 (0.867+0.2583*hac)*0.26 (0.867+0.2583*hac)*0.26 152 (a) Calibration set (b) Validation set Figure A-3 Local calibration results for bottom-up cracking (split sampling) Repeated Split Sampling Like split sampling, repeated split sampling was used with a random split of 70% sections for the calibration set and the remaining 30% for the validation set. This process was repeated 1000 times, where a new random set of calibration and validation sets was picked each time. Repeated split sampling is used to estimate the distribution of different parameters instead of optimizing for a point estimate. Confidence intervals (CI) for each parameter can also be obtained. Tables A-3 to A-5 show the summary for the global model, calibration, and validation sets. It is important to note that coefficient C2 is a function of total HMA thickness (hac). For estimating the confidence intervals and distribution of C2, it was converted to a single value for all HMA thicknesses. Figures A-4 and A-5 present the distribution of model parameters for calibration and validation sets. In Figures A-4 and A-5, the solid blue line shows the median, the dashed red line shows the mean, the solid black line shows the cumulative distribution and the dashed red lines on both sides show the 2.5th and 97.5th percentiles. The mean SEE is reduced from 8.29 to 7.90 for the calibration and 7.93 for the validation set. Similarly, bias was improved from -4.91 to - 0.02 for the calibration and 0.03 for the validation set. Table A-3 Global model summary (Repeated split sampling) Parameter SEE (% total lane area) Bias (% total lane area) C1 C2 (hac < 5 in.) C2 (5 in. <= hac <=12 in.) Global model mean 8.29 -4.91 1.31 2.1585 (0.867+0.2583* hac)*1 Global model median 8.29 -4.91 1.31 2.1585 (0.867+0.2583* hac)*1 153 Global model lower CI 7.63 -5.35 - - - Global model upper CI 8.84 -4.47 - - - Table A-4 Calibration set summary (Repeated split sampling) Parameter Local model mean Local model median SEE (% total lane area) Bias (% total lane area) C1 C2 (hac < 5 in.) C2 (5 in. <= hac <=12 in.) 7.90 -0.02 0.26 0.60 (0.867+0.2583* hac)* 0.19 7.73 0.00 0.25 0.60 (0.867+0.2583* hac)* 0.19 Local model lower CI 6.49 -0.51 0.13 Local model upper CI 9.93 0.42 0.42 0.29 0.89 Table A-5 Validation set summary (Repeated split sampling) Parameter Local model mean Local model median SEE (% total lane area) Bias (% total lane area) C1 C2 (hac < 5 in.) C2 (5 in. <= hac <=12 in.) 7.93 0.03 0.26 0.60 (0.867+0.2583* hac)* 0.19 7.68 0.02 0.25 0.60 (0.867+0.2583* hac)* 0.19 Local model lower CI 6.01 -2.04 0.13 Local model upper CI 10.88 2.27 0.42 0.29 0.89 Figure A-4 Local calibration results for bottom-up cracking – calibration dataset (repeated split sampling) 154 Figure A-5 Local calibration results for bottom-up cracking – validation dataset (repeated split sampling) Bootstrapping Bootstrapping was used as a resampling technique to calibrate the bottom-up cracking model. One thousand bootstrap samples were created, randomly sampling with replacement. Unlike repeated split sampling, in bootstrap, the samples were not split; instead, the entire dataset was used. Bootstrapping also generated CI and distribution of model parameters. Tables A-6 and A-7 summarize the model parameters for global and local models, respectively. SEE is slightly increased, whereas bias is significantly improved after local calibration. Figure A-6 shows the distribution of parameters for the 1000 bootstrap samples. Table A-6 Bootstrapping global model summary Parameter Global model mean Global model median SEE (% total lane area) Bias (% total lane area) C1 C2 (hac < 5 in.) C2 (5 in. <= hac <=12 in.) 8.30 -4.91 1.31 2.1585 (0.867+0.2583*hac)*1 8.30 -4.91 1.31 2.1585 (0.867+0.2583* hac)*1 Global model lower CI 7.38 -5.53 - - - Global model upper CI 9.20 -4.33 - - - 155 Table A-7 Bootstrapping local calibration results summary Parameter Local model mean Local model median SEE (% total lane area) Bias (% total lane area) C1 C2 (hac < 5 in.) C2 (5 in. <= hac <=12 in.) 8.73 0.00 0.23 0.70 (0.867+0.2583* hac)* 0.22 8.30 -0.03 0.20 0.73 (0.867+0.2583* hac)* 0.23 Local model lower CI 6.21 -0.80 0.01 Local model upper CI 12.83 0.68 0.54 0.04 1.29 Figure A-6 Local calibration results for bottom-up cracking (bootstrapping) Summary All calibration approaches have significantly improved the bottom-up cracking model. Table A-8 shows the summary of all sampling techniques. It should be noted that these calibrations were performed with specific limits on the calibration coefficients taken from the literature, as mentioned in Chapter 2. These limits ensure that we get reasonable and practical calibration results. Table A-8 Summary of results for all sampling techniques (Option a) Sampling technique No sampling Split sampling Repeated split sampling Bootstrapping SEE 8.08 7.11 7.90 8.73 Bias 0.17 -0.47 -0.02 0.00 C1 0.22 0.19 0.26 0.23 C2 (hac < 5 in.) 0.66 0.78 0.60 0.70 C2 (5 in. <= hac <=12 in.) (0.867+0.2583* hac)*0.21 (0.867+0.2583* hac)*0.26 (0.867+0.2583* hac)*0.20 (0.867+0.2583* hac)*0.22 156 TOP-DOWN CRACKING MODEL The following section shows the calibration of the top-down cracking model. The model contains crack initiation and crack propagation models. Since the actual crack initiation time is not known, it was not possible to calibrate the crack initiation model separately. So, a single function was used by substituting the crack initiation function with the crack propagation function. Initially, an attempt was made to change all eight coefficients simultaneously. This approach had some challenges: • The model has some mathematical limitations. High values for C3 give mathematical errors in the Pavement-ME output. • There is no current literature available for the top-down cracking model. Therefore, estimating the range for each coefficient to be used in optimization was difficult. • The model has numerous coefficients with coefficient values ranging from 0.011 to 64271618. This makes the optimization challenging to converge. The top-down cracking model was calibrated in Microsoft Excel by combining engineering judgment and the solver function. Four coefficients from the crack initiation function (kL2, kL3, kL4, kL5) and two from the crack propagation function (C1, C2) have been calibrated. No sampling method was used for this calibration. (a) Global model (b) Local model Figure A-7 Predicted vs. measured top-down cracking (No sampling) 157 Figure A-7 shows the predicted vs. measured top-down cracking, and Figure A-8 shows the predicted and measured top-down cracking with time. The predicted and measured top-down cracking does not follow similar trends. Most top-down cracking predictions are limited to a specific time series curve. Table A-9 summarizes model parameters. The SEE and bias are improved. The reliability of the top-down cracking model is estimated by developing a relationship between the standard deviation of the measured cracking, and the mean predicted cracking. Table A-10 outlines the standard error equations for the global and calibrated model. Figure A-8 Measured and predicted top-down-cracking (time series) Table A-9 Calibration results for top-down cracking Parameters SEE Bias KL2 KL3 KL4 KL5 C1 C2 Global model 6.37 -2.36 0.2855 0.011 0.01488 3.266 2.5219 0.8069 Local model 5.59 1.60 0.90 0.09 0.101 3.260 0.30 1.155 Table A-10 Reliability equation for top-down cracking Pavement-ME model Top-down cracking Global model equation 𝑠𝑒(𝑇𝑜𝑝−𝑑𝑜𝑤𝑛) = ⁡0.3657 × 𝑇𝑂𝑃 + 3.6563 Local model equation 𝑠𝑒(𝑇𝑜𝑝−𝑑𝑜𝑤𝑛) = ⁡0.6417 × 𝑇𝑂𝑃 + 0.5014 158 THERMAL CRACKING MODEL The thermal cracking model was calibrated for Level 1 inputs in the Pavement-ME. The model calibration only considered sections with Performance Grade (PG) binder type. The thermal cracking model was calibrated as a single K-value by running Pavement-ME multiple times. Although calibration coefficient K is a function of mean annual air temperature (MAAT), it was calibrated as a single value similar to the previous version of Pavement-ME (version 2.3). For this purpose, the Pavement-ME was run at different K values (0.25,0.65,0.75,0.85, 0.95 and 1.35). SEE and bias were determined for each value of K. Table A-11 summarizes the SEE and bias for the global model and different K values. Based on the SEE and bias, a value of 0.85 is recommended. Recalibration improved the SEE and bias, but thermal cracking predictions still show high variability. Figure A-9 shows the predicted vs. measured thermal cracking for the global and local models at K=0.85. As previously explained in Chapter 3, measured thermal cracking values have been capped at 2112 feet/mile. This means any measured value of more than 2112 feet/mile for sections has been removed from the calibration data. (a) Global model (b) Local model Figure A-9 Predicted vs. measured thermal cracking (at K=0.85) Table A-11 Thermal cracking calibration results Parameter Global model K = 0.25 K = 0.65 K = 0.75 K = 0.85 K = 0.95 K = 1.35 SEE 1225 650 760 813 851 893 1077 159 Bias -812 272 172 106 20 -71 -471 The standard error equations were developed using the standard deviation of the measured cracking and mean predicted cracking, as explained in Chapter 4. Table A-12 summarizes the standard error equations for the global and locally calibrated models. Table A-12 Reliability summary for thermal cracking Pavement-ME model Thermal cracking Global model equation 𝑠𝑒 = 0.14(𝑇𝐶) + 168 Local model equation 𝑠𝑒 = 0.1223(𝑇𝐶) + 400.9 RUTTING MODEL (METHOD 1) No Sampling Pavement-ME predictions for individual layer rutting were matched against measured rutting determined by using the transverse profile analysis results, as discussed in Chapter 4. Figures A- 10 to A-12 show the predicted vs. measured rutting for AC, base, and subgrade layers, respectively. The Pavement-ME under-predicts AC rutting and over-predicts base and subgrade rutting. Table A-13 shows the SEE and bias, whereas Table A-14 shows the calibrated coefficients. Both SEE and bias significantly improved for all layers. (a) Global model (b) Local model Figure A-10 Predicted vs. measured AC rutting (No sampling) 160 (a) Global model (b) Local model Figure A-11 Predicted vs. measured base rutting (No sampling) (a) Global model (b) Local model Figure A-12 Predicted vs. measured subgrade rutting (No sampling) Table A-13 Rutting models SEE and bias Layer HMA rut Base rut Subgrade Global model Local model SEE (in.) 0.2579 0.0426 0.1184 Bias (in.) 0.2015 0.0380 0.1095 SEE (in.) 0.0812 0.0099 0.0062 Bias (in.) -0.0138 -0.0011 -0.0009 Table A-14 Rutting model calibration coefficients Calibration coefficient HMA rutting (br1) Base rutting (bs1) Subgrade rutting (bsg1) Global model 0.4 1.0000 1.0000 Local model 0.1466 0.3003 0.0691 161 Split Sampling Split sampling was performed on 70% of the sections for the calibration set and 30% for the validation set. Figures A-13 to A-15 show the predicted vs. measured for calibration and validation set for different layers. All layers show reasonable validation results. (a) Calibration set (b) Validation set Figure A-13 Predicted vs. measured AC rutting (Split sampling) (a) Calibration set (b) Validation set Figure A-14 Predicted vs. measured Base rutting (Split sampling) 162 (a) Calibration set (b) Validation set Figure A-15 Predicted vs. measured Subgrade rutting (Split sampling) Table A-15 shows the SEE, bias, and model parameters for the global model, and Table A-16 shows the same for the calibration-validation set. Both SEE and bias significantly improved for all layers. Layer HMA rut Base rut Subgrade Table A-15 Rutting global model results SEE 0.2454 0.0872 0.1153 Bias 0.1759 -0.0138 0.1071 Coefficient 0.4 1.0000 1.0000 Table A-16 Rutting local model results Layer HMA rut Base rut Subgrade SEE 0.0962 0.0102 0.0061 Calibration set Bias -0.0165 -0.0012 -0.0008 Coefficient 0.0705 0.2955 0.0705 SEE 0.1008 0.0092 0.0064 Validation set Bias -0.0117 -0.0018 -0.0007 Coefficient 0.0705 0.2955 0.0705 Repeated Split Sampling Repeated split sampling was performed for 1000 split samples with new calibration and validation sets. Figures A-16 to A-18 show the distribution of model parameters for calibration and validation set for different layers. Tables A-17 to A-19 show the SEE, bias, model parameters, CI for the global model, and the calibration and validation sets, respectively. The rutting model significantly improved after local calibration. 163 (a) Calibration set Figure A-16 Distribution of calibration parameters - AC rutting (Repeated split sampling) (b) Validation set 164 (a) Calibration set Figure A-17 Distribution of calibration parameters - Base rutting (Repeated split sampling) (b) Validation set 165 (a) Calibration set Figure A-18 Distribution of calibration parameters - Subgrade rutting (Repeated split sampling) (b) Validation set Table A-17 Global model results (repeated split sampling) Layer Average SEE HMA Base Subgrade 0.2387 0.0426 0.1185 SEE Lower CI 0.2097 0.0409 0.1150 SEE Upper CI Average bias (in.) Bias Lower CI Bias Upper CI 0.2540 0.0440 0.1216 0.1743 0.0380 0.1095 0.1617 0.0367 0.1064 0.1853 0.0394 0.1126 166 Table A-18 Local model calibration results (repeated split sampling) Statistics Average SEE SEE Lower CI SEE Upper CI Average bias (in.) Bias Lower CI Bias Upper CI Average calibration coefficient Calibration coefficient Lower CI Calibration coefficient Upper CI HMA rutting Base rutting Subgrade rutting 0.0966 0.0856 0.1021 -0.0162 -0.0169 -0.0135 0.1757 0.1689 0.1852 0.0099 0.0094 0.0103 -0.0011 -0.0013 -0.0009 0.3003 0.2897 0.3115 0.0062 0.0059 0.0064 -0.0009 -0.0009 -0.0008 0.0693 0.0663 0.0723 Table A-19 Local model validation results (repeated split sampling) Statistics Average SEE SEE Lower CI SEE Upper CI Average bias (in.) Bias Lower CI Bias Upper CI Average calibration coefficient Calibration coefficient Lower CI Calibration coefficient Upper CI Bootstrapping HMA rutting Base rutting Subgrade rutting 0.0971 0.0725 0.1358 -0.0153 -0.0434 0.0174 0.1757 0.1689 0.1852 0.0100 0.0084 0.0119 -0.0011 -0.0041 0.0017 0.3003 0.2897 0.3115 0.0062 0.0053 0.0071 -0.0009 -0.0027 0.0009 0.0693 0.0663 0.0723 Bootstrapping was performed with 1000 bootstrap samples with replacement. Figures A-19 to A- 21 show the distribution of model parameters for AC, base, and subgrade rutting. Tables A-20 and A-21 summarize the calibration results for the global and local models. Model parameter distribution and CI provide a more reliable estimate of model coefficients. Moreover, SEE and bias significantly improved for all layers. 167 Figure A-19 Distribution of calibration parameters - AC rutting (Bootstrapping) Figure A-20 Distribution of calibration parameters - Base rutting (Bootstrapping) 168 Figure A-21 Distribution of calibration parameters-Subgrade rutting (Bootstrapping) Table A-20 Global rutting model summary (Bootstrapping) Layer type Average SEE HMA Base Subgrade 0.2565 0.0425 0.1183 SEE Lower CI 0.2174 0.0396 0.1117 SEE Upper CI 0.3047 0.0456 0.1251 Average bias (in.) 0.2010 0.0380 0.1094 Bias Lower CI 0.1796 0.0355 0.1032 Bias Upper CI 0.2238 0.0408 0.1159 Table A-21 Local rutting model summary (Bootstrapping) Statistics Average SEE SEE Lower CI SEE Upper CI Average bias (in.) Bias Lower CI Bias Upper CI Average calibration coefficient Calibration coefficient Lower CI Calibration coefficient Upper CI Summary HMA rutting 0.0805 0.0677 0.0953 -0.0131 -0.0145 -0.0087 0.1476 0.1363 0.1616 Base rutting 0.0099 0.0091 0.0108 -0.0011 -0.0015 -0.0007 0.3009 0.2803 0.3228 Subgrade rutting 0.0061 0.0057 0.0066 -0.0009 -0.0010 -0.0007 0.0696 0.0639 0.0760 Results for Method 1 are summarized in Table A-22. All calibration approaches have improved the SEE and bias. Bootstrap shows the lowest SEE and bias for all layers. 169 Table A-22 Rutting model calibration results summary Sampling Technique No sampling Split sampling Repeated split sampling Bootstrapping Pavement layer rutting HMA Base Subgrade HMA Base Subgrade HMA Base Subgrade HMA Base Subgrade SEE 0.0812 0.0099 0.0062 0.0962 0.0102 0.0061 0.0971 0.0099 0.0062 0.080 0.010 0.006 Bias -0.0138 -0.0011 -0.0009 -0.0165 -0.0012 -0.0008 -0.0153 -0.0011 -0.0009 -0.013 -0.001 -0.001 Calibration coefficient 0.1466 0.3003 0.0691 0.0705 0.2955 0.0705 0.1757 0.3003 0.0693 0.148 0.301 0.070 JOINT FAULTING MODEL The calibration of the faulting model was performed using the CAT tool. No sampling technique was used for the calibration. In the first step, the most sensitive coefficients, C1 and C6, were simultaneously calibrated. In the next step, C1 and C6 were kept at the calibrated value, and C2 was calibrated. All other coefficients (C3, C4, C5, C7, and C8) were kept at the global values. It should be noted that the measured faulting was cut to 0.4 inches, as mentioned in Chapter 3. Figure A-22 shows the predicted vs. measured joint faulting for the global and local models. Figure A-23 shows the measured and predicted joint faulting with time. In Figure A-23, the predicted faulting is in the same range as measured faulting except for high values for measured faulting. (a) Global model (b) Local model Figure A-22 Calibration results for joint faulting 170 Figure A-23 Measured and predicted joint faulting (time series) Table A-23 summarizes local calibration and the corresponding model parameters. SEE and bias are significantly improved. Table A-23 Summary of faulting model calibration Parameter SEE Bias C1 C2 C3 C4 C5 C6 C7 C8 Global model 0.06 0.01 0.595 1.636 0.00217 0.00444 250 0.47 7.3 400 Local model 0.03 0.00 0.8 1.3889 0.00217 0.00444 250 0.2 7.3 400 The standard error equations were estimated, establishing a relationship between the standard deviation of the measured faulting and mean predicted faulting, as explained in Chapter 4. Table A-24 summarizes standard error equations for the faulting model. Table A-24 Faulting model reliability Pavement-ME model Joint faulting Global model equation 𝑠𝑒(𝐹𝑎𝑢𝑙𝑡) = 0.07162(𝐹𝑎𝑢𝑙𝑡)0.368 + 0.00806 Local model equation 𝑠𝑒(𝐹𝑎𝑢𝑙𝑡) = 0.0902(𝐹𝑎𝑢𝑙𝑡)0.2038 171