Survival Analysis for Asphalt Pavement Performance and Assessment of Various Factors Affecting Fatigue Cracking Based on LTPP Data

Abstract

Pavement performance is the ability of pavement to remain in an acceptable condition to serve the intended users over a period of time. There are several principal, combined factors that affect flexible pavement performance such as environmental conditions, pavement materials, and traffic loads. Vehicle overloading is considered one of the most significant causes of accelerating flexible pavement deterioration, reducing the pavement’s design life, and affecting the overall sustainability of the pavement system. Therefore, researchers are continuously examining pavement systems with a view to finding the most suitable solutions for sustainable development in road construction systems in order to reduce both costs and pollution. In this study, we present a framework to conduct nonparametric and parametric survival analysis for asphalt pavement test sections, to assess the influence of using reclaimed asphalt pavement (RAP) on fatigue service life, to indicate the most significant subset of risk factors (covariates), and to study the effect of overweight axles on flexible pavement performance. All the data concerned were extracted from the long-term pavement performance (LTPP) program. The Kaplan–Meier (KM) survival probability curves of multiple pavement distresses were developed to compare the failure probability for various distresses and to determine the median survival time for each distress. The fatigue survival curves for the test sections using RAP and virgin materials were developed separately and the equality of the two survival curves was tested and affirmed. Several parametric survival analyses were conducted to select the most significant subset of covariates. For fatigue cracking and, after dropping the insignificant predictors, a model was developed to show the quantitative relationship between fatigue failure time and potentially influential factors. The analysis indicated that the increase in the percentage of overloaded axles from 0% to 20% can reduce the fatigue survival life of flexible pavement by up to 55%. In the absence of overweight axles, a one-inch increase in asphalt layer thickness can extend the fatigue service life by about half a year. However, in the presence of 20% of overweight axles, a one-inch increase in thickness can extend the fatigue service life by only 0.22 years. Therefore, additional virgin materials and resources are needed to maintain traffic conditions in the road network and to compensate for the reduction in fatigue service life. Moreover, the effect of the increase in overweight axles from 0% to 15% on reducing the fatigue survival life is found to be similar to the effect of increasing the AADTT tenfold. Therefore, the sustainability of pavement is directly affected by the fatigue survival life.

1. Introduction

Pavement performance is the ability of pavement to remain in an acceptable condition to serve the intended users over a period of time. High-quality pavement contributes towards the sustainability of pavement by reducing costs and pollution and the impact on the community due to noise during repair or construction. There are several principal, combined factors that affect flexible pavement performance such as environmental conditions, pavement materials, and traffic loads. Pavement deterioration modeling is one of the key elements of any pavement management system (PMS) within which pavement maintenance, rehabilitation, or reconstruction plans can be efficiently predicted. A full account of surrounding pavement systems is necessary to prolong the life of the pavement and to improve the sustainability of pavements by conserving virgin materials and energy, providing quieter and smoother riding surfaces, and reducing noise impacts on the surrounding area. A wide variety of methodological approaches have been used for improving the prediction accuracy in estimating pavement performance measures. These methods can be classified into two main distinct categories: deterministic and stochastic approaches.

Most of the deterministic models are mainly based on regression analysis (including linear and nonlinear models). The pavement performance or condition can be predicted as a single mean value in a certain period when dealing with the aggregated impacts of various contributing independent predictors. The deterministic approach in pavement deterioration modeling includes mechanistic models, mechanistic-empirical models, and regression models [1]. In 1987, Paterson developed a nonlinear regression model for flexible pavement roughness progression using data from Brazil [2]. He found that the initial rate of pavement roughness IRI progression depends on the relative relationship between pavement strength and traffic loading. In 2004, Prozzi and Madanat [3] developed a relationship between flexible pavement roughness performance and several contributing factors using a nonlinear multivariate regression model with the use of both field survey data and accelerated pavement test data to estimate riding quality. The study revealed that the initial IRI decreases with an increase in asphalt layer thickness. In addition, Prozzi and Hong [4] used a seemingly unrelated regression estimation (SURE) model to estimate rutting depth and IRI of the pavement surface while accounting for the correlation between the two variables. In 2013, Luo [5] used the autoregression method in pavement performance modeling to improve the accuracy of predictions. A non-linear mixed effects model using different pavement sections and climatic factors to quantify the contribution of these factors to pavement evolution was developed [6]. Also a multi-input prediction model for flexible asphalt pavements specific to four different climatic conditions and two classes of road was proposed [7]. It was found that the deterministic approaches fail to explain the randomness of environmental conditions and traffic loads, the bias from subjective evaluations of pavement conditions, and the measurement errors related to pavement conditions [8].

The stochastic models, which are alternatives to the deterministic approach, expect different lifetime or pavement states distribution of such events [9,10]. In 1997, Li et al. [11] proposed a stochastic model of pavement performance curve that is represented by the Markov transition process. Other stochastic models use the Bayesian inference which uses a combination of prior knowledge and information from historical data to capture uncertainty in performance modeling [12,13]. The Monte Carlo simulation technique is also extensively used in the pavement-related field. Alsherri and George [14] implemented the Monte Carlo simulation technique to estimate the expected life of different pavement sections based on the present serviceability index (PSI). In 2011, Coleri and Harvey [15] used Monte Carlo simulation to predict rut depths for different reliability levels.

Most of these deterioration models are used to predict the amount of a given type of distress after a given period of time as a function of a set of covariates. However, the time duration or the number of applied loads until pavement serviceability performance reaches the terminal value (i.e., the failure threshold) is an important parameter. Survival time analysis is one of the most popular methods for estimating the remaining service life of pavements. The survival analysis models the survival time of an event and quantifies the effect of predictors (independent variables) on the survival time [16]. In 2015, Anastasopoulos and Mannering used survival analysis to evaluate the impact of various factors on pavement service life [17]. Estimating the service life can produce sustainable pavement and contribute to the health of the highway sector. Mamlouk and Souliman [18] stated that the design of sustainable asphalt pavement without accumulation of fatigue cracking is an important goal of transportation agencies and should produce good, long-term performance. Loizos and Karlaftis [19] used the log-logistic proportional-hazard model to evaluate the initiation of pavement cracking. The study revealed that the initiation rate is affected by annual average freezing, annual equivalent single axle load, days where the temperature is below 0 °C, and pavement type. In addition, Auiar-Moya et al. [20] used the Weibull proportional-hazard model to evaluate the time to rutting failure of flexible pavements. They found that the failure time is affected by freezing index, air void, binder content, layer thickness, and annual equivalent single axle load (ESAL).

With the continuous growth of freight transportation, vehicle overloading is considered one of the most significant causes of accelerating flexible pavement deterioration, reducing the life span of vehicles and increasing fuel consumption and crash accident rates. This will affect the United Nations in meeting the sustainable development goals. Several researchers have studied the impact of traffic loads and overweight vehicles on pavement performance and the relationship between excess overweight in the transportation network and pavement maintenance. In order to obtain a sustainable pavement, a system dynamics model (SD) was used in order to enable the assessment of the relationship between overweight vehicles and the costs associated with the operational cost of transportation and social costs due to pavement maintenance and traffic accidents [21]. Hasim et al. [22] indicated that the rate of deterioration is dependent on the quantity and variability of traffic loads. Barraj et al. [23] showed that thick pavement structure is recommended if default axle load spectra is used in mechanistic empirical pavement design guide (MEPDG) for areas suffering from scarcity of traffic data. Dey et al. [24] pointed out that fatigue cracking was more sensitive to overweight trucks compared to other pavement distresses such as rutting and roughness. Sebaaly et al. [25] studied the impact of heavy vehicles on low-volume roads. Rys et al. [26] showed that the presence of overweight vehicles will significantly reduce the service life of flexible pavements. Wu et al. [27] assessed the damage to Texas highways due to oversize and overweight loads considering climatic factors. The results indicated that in the early age of a road, higher oversize and overweight loading would bring a faster deterioration rate. The study conducted by Pais et al. [28] found that the pavement life-cycle costs may increase by about 30% due to overweight vehicles, thus affecting the long-term performance and sustainability of pavement. Another study conducted by Sadeghi and Fathali [29] presented some models that describe the behavior of asphalt pavements under overloaded vehicles. Wang et al. [30] concluded that there was a linear relationship between the percentage of overweight trucks and the reduction ratio of pavement life, despite how the pavement structure and traffic loading changed. Zhang et al. [31] used accelerated failure time models to identify the most critical overweighting characteristics. They pointed out that the average pavement rutting life and fatigue life would be extended by about 29% and 43%, respectively, in the absence of overweight vehicles, thereby extending the lifespan and sustainability of pavement.

The use of processed reclaimed asphalt mixes in HMA is another significant factor that could affect the performance of pavement. RAP is considered a useful alternative to virgin materials because it reduces the use of virgin asphalt binder and aggregate and conserves energy required to obtain quality virgin materials. Zhao et al. [32] stated that a key element to generating sustainable pavement designs is the use of RAP material that saves natural resources and reduces energy, greenhouse gas emissions, and costs. Previous evaluations of the fatigue performance of recycled asphalt mixes have shown some controversial findings. In this context, Barros et al. [33], Shu et al. [34], and Noferini et al. [35] reported a decline in the fatigue resistance of recycled asphalt mixes compared to standard hot mix asphalt (HMA), whereas recent studies by Al Qadi et al. [36], Poulikokas et al. [37] and Baraj et al. [38] experimentally found that the fatigue behavior of recycled asphalt mixes is similar to that of HMA.

The long-term pavement performance program, known as LTPP, is the largest pavement performance research program ever undertaken. It was established in 1987 as part of the first Strategic Highway Research Program (SHRP) to collect and store performance data for more than 2400 monitored pavement test sections in different climatic regions of the United States and Canada. The data collection started in 1989. After 25 years, these data are available to the public via the Web through the data portal system: LTPPInfoPave^TM [39]. The LTPP data is usually collected and uploaded periodically on a six-month cycle by four regional contractors. The information management system, where the LTPP database is stored, consists of 16 general data modulus with 430 tables in a simple row-column format in which the columns are referred to as fields and the rows contains records. The main objective of the LTPP program of collecting and storing performance data is to support analysis and develop usable engineering products relevant to pavement management, construction, maintenance and design. Zhang and Wang [40] developed decision tree models using LTPP data to provide enhanced decision-making information in pavement maintenance and design. Wang et al. [41] developed an AdaBoost regression model to improve the prediction ability of international roughness index (IRI) for roads using records from LTPP program. Another research study was conducted by Rezapour et al. [42] to investigate factors contributing to pavement skid resistance using LTPP data. El Ashwah et al. [43] used the LTPP data to calibrate transfer functions used in developing and implementing a simplified Mechanistic Empirical (M-E) pavement design method.

There are several potential issues in the existing pavement deterioration models. The first issue is that most of the regression models that have been developed do not consider correlations between the independent variables. The second issue is related to the input variables selection that would affect the performance of pavement for which the variables in most existing models may not reflect the effect of overweight axles with other potentially influential factors. The third issue is that the pavement failures that occur during the given monitoring period are only considered in model development and the observations in which the time-to-event is unknown, are dropped from the analysis.

Using data extracted from the long-term pavement performance (LTPP) program, this study aims to conduct a non-parametric and parametric survival analysis for the selected flexible pavement test sections and to indicate the most significant subset of risk factors (covariates) under various pavement distresses. The selected distresses for this study are fatigue cracking, longitudinal wheel path and non-wheel path cracking, and transverse cracking, as shown in Figure 1. The non-parametric Kaplan–Meier method was used to assess the influence of using RAP vs standard HMA in asphalt mixes. For fatigue pavement cracking, the most appropriate parametric model was specified to predict the survival times and to evaluate the relationship between fatigue deterioration cracking and its potential influential factors. The percentages of overweight axles in the test section lane were considered to evaluate the impact of overweight vehicles on fatigue cracking and to investigate the relations with other factors. Thus, researchers can reach a good understanding of the relationship between these factors and the pavement systems to define the most appropriate sustainability practices. The outcomes of this research should contribute towards increasing the service life and sustainability of pavement.

2. Methodology

2.1. Survival Analysis

Generally, survival analysis or time-to-event analysis is a collection of statistical procedures used for data analysis. The outcome variable is referred to as the survival time (in year, months, weeks, or days) from the beginning of follow-up of an individual until the occurrence of the event of interest. In this research, the event of interest refers to the pavement failure when the measured pavement deterioration indicator drops below the acceptable threshold level.

Table 1. MEPDG Recommended Performance Failure Criteria for Pavement Design.

Pavement Condition or Distress Type	Threshold Values for Primary Roads
Alligator cracking	20% of lane area
Longitudinal cracking	132.6 m/km (700 ft/mi)
Transverse cracking for hot mix asphalt (HMA)	132.6 m/km (700 ft/mi)

lists the used mechanistic empirical pavement design guide (MEPDG) pavement failure thresholds for each failure distress mode separately.

Survival analysis is considered a “censored regression”. Censored is defined as the incomplete observed responses during the observation period of an experiment when some information is available about a subject’s event time, but we do not know the exact survival time. The exclusion of these censored data can cause bias in the analyzed results according to SAS Institute Inc. [46].

In summary, there are three different types of censoring used in this analysis:

• Left-censored: data can occur when the pavements section’s true survival time is less than or equal to that pavements section’s observed survival time. In other words, if a pavement is left censored at time “t”, the failure event occurs between time 0 and t before the study began, but the exact time of occurrence is not known.
• Right-censored: most survival data used in this study is right-censored. Data can occur when the event has not occurred during the study or before the termination of data collection. In this case, the true survival time is equal to or greater than the observed survival time.
• Interval-censored: the pavement failed within a certain specified time interval but the exact true failure time is unknown.

In this study, the percentages and the numbers of censored data for the extracted data are indicated in

Table 2. Percentages of censored pavement test sections.

Pavement Distress Indicator	Percentages of the Complete Cracked Section	Percentages of the Censored Pavement Section
Fatigue Cracking	81% (159)	19% (37)
Wheel path Longitudinal Cracks	62% (108)	38% (67)
Non-wheel path Longitudinal Cracks	55% (84)	45% (69)
Transverse Cracks	64% (112)	36% (60)

.

2.2. Survival Function:

The survival function, S(t), defines the probability (P) that the pavement section failure has not occurred at time t, which can be expressed as:

$$\mathrm{S}\left(\mathrm{t}\right)= \mathrm{P}\left(\mathrm{T} \ge \mathrm{t}\right)=1- \mathrm{F}\left(\mathrm{t}\right)=1-{{\displaystyle \int }}_0^{\mathrm{t}}\mathrm{f}\left(\mathrm{u}\right)\mathrm{du}$$

(1)

where T is the pavement service life, F(t) is the cumulative distribution function of the pavement service life, and f(u) is the density function of the pavement failure [47].

The hazard function, h(t), defines the instantaneous potential per unit time for the failure event to occur, given that the pavement section has survived up to time t. The hazard function can be expressed as:

$$\mathrm{h}\left(\mathrm{t}\right)=\underset{\Delta \mathrm{t}\to 0}{\lim }\left(\frac{\mathrm{P}(\mathrm{t}\le \mathrm{T}\le \mathrm{t}+\Delta \mathrm{t}}{\Delta \mathrm{t}}\right)=\frac{\mathrm{f}\left(\mathrm{t}\right)}{\mathrm{S}\left(\mathrm{t}\right)}$$

(2)

The Kaplan–Meier (KM) Method or the product-limit method [48] is the most popular nonparametric modal. It does not assume an underlying failure distribution of the data and it is often used to develop survival curves. In Kaplan–Meier method, it defines survival probability S(t) as follows:

$$\mathrm{S}\left(\mathrm{t}\right)={{\displaystyle \prod }}_{{\mathrm{t}}_{\mathrm{i}}\le \mathrm{t}}\frac{{\mathrm{n}}_{\mathrm{i}}-{\mathrm{d}}_{\mathrm{i}}}{{\mathrm{n}}_{\mathrm{i}}}$$

(3)

where t_i is the time of ith pavement failure, d_i is the number of pavement sections that failed at time t_i, and n_i is the number of pavement sections that survived just before time t_i.

The 95% confidence interval (CI) for the KM and for the median survival time, when the time point at which the probability of survival equals 50%, is calculated using Greenwood’s formula.

2.3. Parametric Survival Analysis

In a parametric survival model, the survival time is assumed to follow several distributions. The most commonly used distributions that are applied in this study are the Gompertz, the generalized gamma, the Weibull, the log-logistic, and the log-normal.

These models are used to investigate the influence of predictors on the survival time or hazard rate. The maximum likelihood estimation method and the likelihood ratio test are used to estimate the survival model and to test the significance of each independent variable, respectively. The accelerated failure-time (AFT) and the proportional hazards (PH) are two models which are often used for adjusting survivor functions for the effects of predictors. Specifically, the underlying assumption for PH models is that the effect of predictors is proportional with respect to hazard, whereas the underlying assumption of AFT models is that the effect of predictors is proportional with respect to the survival time.

2.4. Model Selection Criterion

The corrected Akaike information criterion (AIC) is an approach that can be applied to compare the fit of models with different underlying distributions. The model with the smaller AIC value is considered to be closer to the true distribution. The following formula was used to calculate the AIC value:

AIC = −2 Log(LL) + 2(c + p + 1)

(4)

where LL is the likelihood, c is the number of model covariates and p is the number of model-specific ancillary parameters [49].

In addition to the AIC criteria method, the Cox-Snell residuals are also applied to select the most appropriate parametric model by graphically assessing which model would generate a plot that lies directly on top of the diagonal line [50]. The residuals “r_ci”, that were reported by Cox and Snell [51] and Hosmer and Lemeshow [52], are formed by using the model-based estimate of the empirical cumulative hazard function $\widehat{H}$(t_i) or the survival empirical hazard function $\widehat{S}$(t_i) where:

$${\mathrm{r}}_{\mathrm{ci}}=\widehat{H}\left({\mathrm{t}}_{\mathrm{i}}\right)=-\log \left(\widehat{S}\left({\mathrm{t}}_{\mathrm{i}}\right)\right)$$

(5)

Stata MP/13 software package [53] was chosen to conduct the underlying nonparametric and parametric survival analysis and to provide the analysis needed for this study. A p-value of less than 0.05 was considered statistically significant.

2.5. Preparation of Data

The data extracted for this study were selected from the GPS-1 and GPS-2 experiments in LTPP. The GPS-1 and GPS-2 are commonly constructed pavement types that refer to an asphalt concrete (AC) layer on unbound and bound bases, respectively.

The potential influential predictors and the pavement performance indicators used in this study with its source table in the LTPP database are listed in

Table 3. Models independent variables considered in the analysis.

Category	Variable	Abbreviation	Unit	LTPP Table [39]
Traffic Loads and Overweighting	Annual Average Daily Truck Traffic	AADTT	trucks/day	TRF_MEPDG_AADTT_LTPP_LN
	Annual Average cumulative Single Axle Load (×1000)	KESAL	#	TRF_HIST_EST_ESAL & TRF_MON_EST_ESAL
	Total Axle Weights	W	million lb	TRF_MEPDG_AX_DIST_ANL & TRF_MONITOR_LTPP_LN
	Total Overweight Axles	OW	million lb	TRF_MEPDG_AX_DIST_ANL & TRF_MONITOR_LTPP_LN
	Total Axles Volume	V	#	TRF_MONITOR_LTPP_LN
	Total Overweight Axles Volume	OV	#	TRF_MONITOR_LTPP_LN
	Total Percentage of Overweight Axles	%OA	%	TRF_MEPDG_AX_DIST_ANL & TRF_MEPDG_AX_PER_TRUCK
Environmental	Average Annual Precipitation	Total Precip.	in	CLM_VWS_PRECIP_ANNUAL
	Average Freezing Indices (FI)	FI	°F-days	CLM_VWS_TEMP_ANNUAL
	Average Annual Temperatures	Temp.	°F	CLM_VWS_TEMP_ANNUAL
	Average Total Annual Snowfall	Total Snow	in	CLM_VWS_PRECIP_ANNUAL
	Average number of days above 89 °F	Days > 89 °F	#	CLM_VWS_TEMP_ANNUAL
	Average number of days below 32 °F	Days < 32 °F	#	CLM_VWS_TEMP_ANNUAL
Pavement Materials	Total Thickness of Asphalt Layer	hac	in	SECTION_LAYER_STRUCTURE
	Total Thickness of Base Layer	hb	in	SECTION_LAYER_STRUCTURE
	Total Thickness of Subbase Layer	hsub	in	SECTION_LAYER_STRUCTURE
	Subgrade Material Resilient Modulus	Mr	psi	TST_UG07_SS07_WKSHT_SUM
	Asphalt material (1 if RAP, 0 if standard HMA)	RAP	NA	SECTION_LAYER_STRUCTURE

and

Table 4. Models dependent variables considered in the analysis.

Category	Variable	Abbreviation	Unit	LTPP Table [39]
Pavement Performance Indicator	Fatigue Cracking	FC	%	MON_DIS_AC_REV
	Non-wheel path Longitudinal Cracks	NWPL	ft/mi	MON_DIS_AC_REV
	Wheel path Longitudinal Cracks	WPL	ft/mi	MON_DIS_AC_REV
	Transverse Cracks	TC	ft/mi	MON_DIS_AC_REV

, respectively. These explanatory variables are briefly summarized as follows:

• Traffic Loads and Overweighting:

There are two types of LTPP traffic data: historical and monitored traffic data [39]. The historical traffic data provide traffic data for the period of time from the original date of pavement construction to the beginning of traffic monitoring in 1990; while the monitoring traffic data provide data for each year after 1990, computed from raw data or estimated by the highway agencies [54]. The traffic data are stored in Traffic Data Module (TRF) of the LTPP database. The potentially influential traffic data extracted for this research to depict the effect of traffic loads and overweighting in the lane of the LTPP test section are:

• The annual average daily truck traffic (AADTT), in trucks/day, extracted from LTPP table (TRF_MEPDG_AADTT_LTPP_LN) according to specific state and section ID.
• The annual average cumulative single axle load (KESAL) extracted from two sources. The first table (TRF_HIST_EST_ESAL) contains estimates of 80 KN (18 kips) ESALs for sections with historical traffic data and the second table (TRF_MON_EST_ESAL) contains annual estimate of ESAL during the period when pavement monitoring measurements were conducted.
• The total axle weights (W) for the 13-bin classified vehicles according to FHWA (federal highway administration)
  The total overweight axles (OW) for the weight of the axles exceed the federal trucks axle weight limit listed in

  Table 5. Federal truck’s weight limits.

  Axle Group	Limits
  Single Axle	20,000 lbs
  Tandem Axle	34,000 lbs
  Gross Weight	80,000 lbs

  . The total axle weights and overweight axles are extracted from table (TRF_MEPDG_AX_DIST_ANL), which contains the annual normalized axle distribution by class and axle group and from table (TRF_MONITOR_LTPP_LN), which contains information about the estimated annual volumes of trucks and axles per LTPP lane (LN).
• The total axles volume (V) and the total overweight axles volume (OV) for all the vehicle classes and axle group. These data are also extracted from table (TRF_MONITOR_LTPP_LN).
• The total percentage of overweight axles (%OA) is calculated using the normalized axle distribution by vehicle class and axle group type. The data are extracted from table (TRF_MEPDG_AX_DIST_ANL) and from table (TRF_MEPDG_AX_PER_TRUCK), which contains the annual average number of axles number by vehicle class and axle type per year.

The cross-correlations and the scatterplot matrix for the extracted traffic variables are shown in Figure 2 with the calculated Pearson correlated coefficients, located above the diagonal of the plot matrix, and were used to verify the multicollinearity of these variables. All the p-values were smaller than the significance level (a = 0.05) so that the correlations were statistically significant. The results revealed that many correlations were quite high. There was no significant evidence of a relationship between the AADTT and the KESAL values, while the AADTT and KESAL were strongly interdependent with the other extracted variables, with the exception of the total percentage of overweight (%OA).

• Environmental Data:

Climatic factors also have a significant effect on pavement deterioration. The climate-related variables chosen for this study were the average annual precipitation, freezing indices (FI), temperatures, and snowfall for the test sections, the average number of days when maximum temperature is above 32 °C (89 °F), and the average number of days when the maximum temperature is below 0 °C (32 °F).

The multicollinearity was also tested for the environmental data and the results are shown in

Table 6. Correlation Matrix of Environmental Variables.

	Total Precip.	Total Snow	Temp.	Days > 89 °F	Days < 32 °F	FI
Total Precip.	1.0000
Total Snow	0.0497	1.0000
Temp.	0.1971	−0.7628	1.0000
Days > 89 °F	−0.2409	−0.5136	0.8118	1.0000
Days < 32 °F	−0.2538	0.7292	−0.9713	−0.6531	1.0000
FI	−0.1861	0.7048	−0.8004	−0.5089	0.7991	1.0000

.

• Pavement Materials:

The pavement materials-related variables used in this study are the subgrade material resilient modulus (Mr) to characterize the subgrade material stiffness, the thicknesses of surface, base, and subbase layers, and the type of materials used in asphalt mixes (RAP or standard HMA). The subgrade resilient modulus was extracted from TST (test modulus), specifically from table (TST_UG07_SS07_WKSHT_SUM), which contains the average resilient modulus data for unbound granular materials.

The other pavement material data available in the LTPP data such as the percentage of asphalt content and air void in the mix were excluded from the selected data due to the high percentage of the missing values for the selected test sections.

2.6. Pavement Performance

The four pavement performance indicators selected for this study were alligator cracks or bottom-up fatigue cracking, non-wheel path longitudinal cracks, wheel path longitudinal cracks, and thermal or transverse cracks. The pavement deterioration data in each test section was examined to identify any incomplete historical data, outliers in data, or any abnormal data due to technical and mechanical errors. The data were extracted from the (MON_DIS_AC_REV) table, which contains distress survey information obtained by manual inspection for asphalt concrete surfaces and this table belongs to the monitoring (MON) module.

2.7. Descriptive Analysis

A univariate analysis was performed to establish the descriptive data of the selected independent variables and the values are shown in

Table 7. Descriptive Analysis of the selected independent variables.

Variables	Mean	Standard Deviation
AADTT	718.336	811.192
KESAL	294.112	396.653
%OA	14.890	6.848
Total Precip.	34.114	17.262
FI	602.257	703.891
Temp.	55.314	11.039
Total Snow	25.926	30.935
hac	7.788	3.465
hb	8.906	6.211
hsub	5.499	7.843
Mr	11,661.193	5341.987
RAP	0.229	0.421

.

3. Results and Analysis

3.1. Non-Parametric Survival Analysis

For the purpose of comparison, Figure 3 presents the survival probabilities or the Kaplan–Meier product limit estimator curves (KMPLE) of the different pavement performance indicators. As illustrated, the calculated survival probabilities decrease with the increase in the pavement age. Moreover, the transverse or the thermal cracking had the lowest survival probability, followed by the non-wheel path longitudinal cracking, the fatigue cracking, and the wheel path longitudinal cracking.

The 95% CIs of KMPLE survival probability for the different pavement distresses at the median survival time and at 5, 10, and 15 years of pavement age are summarized in

Table 8. 95% CIs of KMPLE Survival Probability.

		95% Confidence Intervals of KMPLE Survival Probability
Pavement Distresses	Median Survival Time, in Years	Median Survival Time	5 Years	10 Years	15 Years
Fatigue Cracking	8.43	[0.4009, 0.5889]	[0.7352, 0.8776]	[0.2834, 0.4684]	[0.1050, 0.2664]
Non-wheel path Longitudinal Cracking	5.9	[0.3290, 0.5264]	[0.4229, 0.6094]	[0.151, 0.37545]	[0.0535, 0.2690]
Wheel path Longitudinal Cracking	10.6	[0.3947, 0.6172]	[0.7047, 0.8629]	[0.4321, 0.6483]	[0.2185, 0.4790]
Transverse Cracking	1.96	[0.3639, 0.5659]	[0.0106, 0.2234]	[0.0153, 0.1834]	[0.0112, 0.0425]

. The demonstrated survival functions using the non-parametric survival analysis were not smooth, meaning that the survival probabilities curves are a piecewise constant and they can have unrealistic properties. Over a short period of time, the pavement failure probability may jump by a huge amount. Moreover, it is not easy to incorporate predictors or covariates. It is particularly difficult to describe how the selected pavement test sections with various percentages of overweight axles differ in their survival functions.

3.2. Effect of Using RAP in Asphalt Mixes

The selected pavement sections in this study were divided into two groups according to the type of the used asphalt mixes. Group (1) is related to asphalt mix with standard or virgin materials and group (2) is related to asphalt mix with RAP materials. The survival curves for fatigue lives are estimated separately for each group using the Kaplan–Meier method, and the curves are shown in Figure 4.

Graphically, the survival chance is 25% at time t = 6.13 years for group (1) and at time t = 5.38 years for group (2). The median survival time for group (1) is t = 8.37 years and for group (2) is t = 10.22 years indicating that using RAP materials in asphalt mixes has a better survival rate than using standard HMA materials.

The log rank test was performed to compare the equality of the two survival curves by testing the null hypothesis (H₀) of no difference in survival between using RAP and virgin materials or no difference between the populations in the probability of fatigue failure at any point. The log rank test compares the observed number of events in groups 1 and 2 with what would be expected if the null hypothesis were true.

The log rank test is a form of chi-square and the degree of freedom is 1. It was calculated much the same way as the c² statistic. For this test, the chi-square c² = 0.003 and p-value = 0.956 > 0.05 so the null hypothesis was retained. Therefore, there is no difference in survival curves between using RAP and virgin materials.

3.3. Parametric Survival Analysis

Table 9. The selected subset of covariates with the Log-likelihood and AIC values for each estimated model and for various pavement distress type.

Pavement Distress Indicator	Model	The Most Significant Subset of Covariates (Significant or Marginally Significant at 5%)	Log-Likelihood (LL)	AIC
Fatigue Cracking	Gompertz	%OA, AADTT, KESAL, Total Precip., FI, hac, Mr	−50.902038	119.8041
	Generalized Gamma	%OA, AADTT, KESAL, Total Precip., FI, hac, Mr	−47.825837	115.6517
	Weibull	%OA, AADTT, KESAL, Total Precip., FI, hac, Mr	−47.682033	113.3641
	Log-logistic	%OA, AADTT, KESAL, Total Precip., FI, hac, Mr	−53.071456	124.1429
	Log normal	%OA, AADTT, KESAL, Total Precip., FI, hac, Mr	−56.657391	131.3148
Non-wheelpath Longitudinal Cracks	Gompertz	Temp.	−95.414942	196.8299
	Generalized Gamma	-	-	-
	Weibull	Temp.	−92.772062	191.5441
	Log-logistic	-	-	-
	Log normal	-	-	-
Wheelpath Longitudinal Cracks	Gompertz	AADTT, FI	−141.98669	291.9734
	Generalized Gamma	AADTT, FI	−138.18737	286.3747
	Weibull	AADTT, FI, Mr	−139.47333	288.9467
	Log-logistic	AADTT, FI	−139.39759	286.7952
	Log normal	AADTT, FI, Mr	−138.19604	284.3921
Transverse Cracks	Gompertz	Snowfall, Temp.	−126.30721	260.6144
	Generalized Gamma	Snowfall, Temp.	−115.62259	241.2452
	Weibull	Snowfall, Temp., AADTT	−120.67733	251.3547
	Log-logistic	Snowfall, Temp.	−112.28529	232.5706
	Log normal	Snowfall, Temp.	−116.30418	240.6084

indicates the most significant subset of risk factors (covariates) and the most appropriate parametric model that was identified simultaneously for each pavement performance indicator. The stepwise selection procedure was applied to fit different parametric regression models to the data and included in the AIC procedure was the subset of the significant predictors identified in each fit. Upon the different model assumption, based on the regression analysis conducted using the sample data, the results are as follows:

3.3.1. Transverse and Longitudinal Cracking

Most of the used variables have been excluded from the estimated models since they were found to be statistically insignificant. For the longitudinal non-wheel path cracking, the temperature was the only significant variable when using the Gompertz and Weibull models. For the longitudinal wheel path cracking, the average annual daily truck traffic (AADTT) and the freezing index (FI) were the only significant factors with a p-value less than 0.05 for all used models. The subgrade material resilient modulus (M_r) was also found to be statically significant using Weibull and lognormal distributions. As for the transverse cracking, the snowfall and the temperature were found to be the significant variables for all models, and the AADTT was also considered a significant factor when using the Weibull model.

The calculated AIC and log-likelihood values for all models were found to be similar with slight differences. It is difficult, therefore, to draw a firm conclusion about the most appropriate parametric model for these types of distresses and to indicate the relationship between the significant covariates and the selected distresses.

3.3.2. Fatigue Cracking

Based on the collected data for fatigue cracking distresses and upon the different model assumptions, the independent variables which were significant or marginally significant at 5% significance level included: percentages of overweight axles (%OA), annual average daily truck traffic (AADTT), annual average cumulative single axle (KESAL), total precipitation, freeze index (FI), the thickness of asphalt (h_ac) and subgrade material resilient modulus (M_r). The AIC values indicated the models in the order of fitness were Weibull, generalized gamma, Gompertz, log-logistic, and log-normal.

On the other hand, the plots of Cox-Snell residuals were used as a graphic method to check the assumption of the distribution and to test the overall fitness of the model. The cumulative hazard plots of residuals of the Weibull, generalized gamma, and Gompertz are illustrated in Figure 5, Figure 6 and Figure 7. The differences among the three distributions are small with the Weibull distribution model being slightly better than the others, since the curve is closer to a straight line with unit slope and zero intercepts. Therefore, the results are similar to those obtained using the AIC criterion, thus the Weibull model selected in this study provides the best overall fit. The Weibull model was implemented as both AFT and PH models. The parameterization and ancillary parameters for these distribution models are summarized in

Table 10. Weibull parametric survival model functions.

Weibull Distribution Model
Metric	Survivor Function S(t)	Parametrization	Ancillary Parameters
PH	exp(−ljtpj) (1)	lj = exp(xjb)	p
AFT	exp(−ljtpj) (2)	lj = exp(−pxjb)	p

. Where x_j is a vector of covariates, b is a vector of regression coefficients, and p is the shape parameter (also known as the Weibull slope).

The calculated random parameters for Weibull duration models and the hazard ratios are reported in

Table 11. Weibull parametric regression for pavement failure due to fatigue cracking.

Variable	Coefficient	Std. Error	Z Score	p-Value (Two-Tailed)	Hazard Ratio
%OA	0.1408651	0.0216452	6.51	p < 0.001	1.151269
AADTT	0.0012121	0.0001983	6.11	p < 0.001	1.001213
KESAL	0.0021101	0.0004219	5	p < 0.001	1.002112
Total Precip.	−0.0204645	0.0064971	−3.15	0.002	0.9797434
hac	−0.1272957	0.0373627	−3.41	0.001	0.8804733
Mr	−0.0000395	0.0000215	−1.84	0.066	0.9999605
FI	−0.0003363	0.0001371	−2.45	0.014	0.9996638
_cons	−9.048324	0.9245831	−9.79	p < 0.001	0.0001176
/ln_p	1.273013	0.0818707	15.55	p < 0.001
p	3.571599	0.2924095
1/p	0.2799866	0.0229227

.

All the predictors in the final model were statistically significant at a 95% confidence level except the Mr factor, which was marginally significant but was included in the final model since it improved the fitness of the selected model. The estimated ancillary shape parameter (p) is 3.57 and reported that the Weibull hazard is not constant and it is monotone-increasing. Moreover, the test statistic is 15.5 so we can reject the null hypothesis that the hazard is constant.

Table 11 also reported the hazard ratio which represents the decrease or increase in the hazard rate due to a unit increase of the covariate. A 1% increase in overweight axles can be related to a 15% increase in the hazard rate of fatigue failure cracking. Additionally, the AADTT and KESAL values were also found to be important. A hundredfold increase in the average annual daily truck traffic “AADTT” can be associated with a 13% increase in the hazard ratio and a ten-thousandfold increase in the annual average cumulative equivalent single axle load can be related to a 23.5% increase in the hazard ratio. 25 mm (one inch) increase in asphalt layer thickness corresponds to a 12% decrease in the hazard rate of fatigue failure cracking. A 1000 °F-days increase in freeze index corresponds to a 3.3% decrease in the hazard rate and a one-inch increase in total precipitation can be related to a 2% decrease in the hazard rate.

The final model for estimating survival time can now be expressed as:

S(t\x_j) = exp{−exp (−9.048 + 0.1408 × (%OA) + 0.0012 × (AADTT) + 0.0021 × (KESAL) − 0.0204 × (Total Precip.) − 0.1272 × (h_ac) − 0.00003 × (M_r) − 0.0003 × (FI)) × t^3.571}

(6)

The survival curve of the average pavement section, referring to average values in Table 7 is illustrated in Figure 8. The median survival life (S(t) = 0.5) of the average pavement section is 8.07 years. This value is slightly different from the median survival life obtained using KMPLE survival function (t = 8.43 years).

3.3.3. Influence of Overweight Axles

To investigate the trend of median survival time changes related to a change in overweight axles, the survival time was computed at each desired overweight axle value (0%, 5%, 10%, 15%, 20%, 25%,and 30%) using Equation (6) with other parameters being the average and survival probability equal to 0.5. Figure 9 shows the plot of median survival time for the average pavement with various percentages of overweight axles in order to further investigate the relation between the percentage of overweight axles and the trend of the survival time changes. The increase in percentage of overloaded axles from 0% to 20% can reduce the survival time of the fatigue life up to 55% (from 14.49 years to 6.58 years).This is consistent with findings from a previous study conducted in 2014 by Rys et al. [26], where the analysis showed that the fatigue life of flexible pavement was reduced up to 50% when the percentage of overloaded vehicles increased from 0% to 20%.

Figure 10 presents the relation between the thicknesses of asphalt layer and survival time for different percentages of overweight axles. A one-inch increase in thickness can extend the fatigue service life about half a year in case of 0% overweight axles and about 0.22 years when the percentage of overweigh axles is 20%. Therefore, as the percentage of overweight axles increases, the instantaneous rate of change of service life extension decreases with the increase in asphalt layer thickness. For thicker asphalt layers, the fatigue service life is extended even further. For example, at OA% = 5, the fatigue service life increases about 0.35 years if the thickness increases from 2 inches to 3 inches, while the service life will increase about 0.4 years if the thickness increases from 5 inches to 6 inches which is an increase of 15%.

Figure 11 illustrates the relationship between the pavement survival life and the equivalent single axle load (KESAL) for different scenarios of percentages of overweight axles and for two levels of traffic volume (high and low traffic volume). In this regard, two values for AADTT were used to represent the two different traffic scenarios: AADTT = 200 represents the low traffic volume scenario and AADTT = 2000 represents the high traffic volume scenario. This figure shows that the survival time decreases with the increase in KESAL, AADTT, and the percentages of overweight axles. Moreover, it can be seen from this figure that the survival time curve for the case of %OA = 15 with low traffic volume overlapped over the survival time curve for the case of %OA = 0 with high traffic volume. On the other hand, the two curves for the case of %OA = 5 with high traffic volume and for the case of %OA = 20 with low high traffic volume also overlapped. The effect of the increase in the percentage of overweight axles from 0% to 15% on reducing the median survival time is therefore similar to the effect of increasing the annual average daily truck traffic by 10 times.

4. Conclusions

The study presents a framework for conducting non-parametric and parametric survival analysis for asphalt pavement sections using LTPP data and to assess the influence of overweight axles on flexible pavement performance, particularly on fatigue cracking. This should predict the remaining service life and aid in assessing the sustainability of pavement. The main findings of this study provide researchers with a good understanding of the relationship between the potential influential factors and the pavement system, in order to define the most appropriate sustainability practices. The results are summarized as follows:

• Non-parametric Kaplan–Meier survival probability curves indicated that the thermal transverse cracking had the highest failure probability followed by the non-wheel path longitudinal cracking, the fatigue cracking, and the wheel path longitudinal cracking.
• To assess the influence of using reclaimed asphalt pavement (RAP) on fatigue service life, the survival curves for the test sections using RAP and virgin materials are separately developed to compare the equality of the two survival curves and the results indicated that there is no difference in fatigue survival life. These results encourage the sustainable use of RAP in pavement construction.
• For fatigue cracking, based on the data extracted from the pavement test sections across the United States, only seven independent variables out of seventeen potential influential factors were found to be statistically significant or marginally significant and the Weibull distribution was found to be an effective description to model the data of concern to our study. A final model for estimating the fatigue survival time, including the potential influential factors, is represented in this paper. This model seems to be more suitable for quantifying the effect of the independent variables (covariates) than predicting the survival time.
• By performing the parametric survival analysis, the median survival time for fatigue failure is 8.07 years. Moreover, the fatigue cracking was found to be sensitive to the percentages of overweight axles but it was difficult to draw a firm conclusion for longitudinal and thermal cracking based on the available extracted data and the selected pavement test sections.
• Percentages of overweight axles, AADTT, ESAL, and thickness of the asphalt layer have significant effects on the hazard rate. The increase of the percentage of overloaded axles from 0% to 20% can reduce the survival time of the fatigue life up to 55%.
• A one-inch increase in asphalt layer thickness can extend the fatigue service life by about half a year when there are no overweight axles and about 0.22 years when the percentage of overweight axles is 20%. Therefore, additional virgin materials and resources are needed to maintain traffic conditions in the road network and to compensate for the reduction in fatigue service life. Therefore, using overweight axles will negatively impact the sustainability of pavement and necessitates new design guides.
• The effect of increasing the overweight axles from 0 to 15% on reducing the fatigue survival life is found to be similar to the effect of increasing the annual average daily truck traffic (AADTT) by ten times.