Incorporating spatial variability of pavement foundation layers stiffness in reliability-based mechanistic-empirical pavement performance prediction

Transportation Geotechnics 17 (2018) 1–13

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Transportation Geotechnics journal homepage: www.elsevier.com/locate/trgeo

Incorporating spatial variability of pavement foundation layers stiffness in reliability-based mechanistic-empirical pavement performance prediction

T

⁎

Ahmad Alhasana,b, , Ayman Alic,1, Daniel Offenbackerd,1, Omar Smadia,b,2, Colin Lewis-Becke,3 a

Civil, Construction and Environmental Engineering, Iowa State University, USA Center for Transportation Research and Education, Iowa State University, USA c Center for Research and Education in Advanced Transportation Engineering Systems, Rowan University, USA d Civil and Environmental Engineering, Rowan University, USA e Department of Statistics, Iowa State University, USA b

A R T I C LE I N FO

A B S T R A C T

Keywords: Mechanistic-empirical pavement performance Risk/reliability surface Spatial statistics Intelligent compaction Stochastic finite element

Adequate pavement quality and performance are critical for road users’ safety, ride comfort, vehicle operation and travel delay costs, and vehicle durability. An accurate and robust pavement design is essential for realistic life cycle cost analysis, as well as overall management of the infrastructure. Compared to deterministic design methods, probabilistic methods are more realistic and can capture the inherent uncertainty in pavement and foundation materials; and loading conditions. In this study, spatial variability and systematic measurement errors in foundation layers’ (including the base and subgrade layers) stiffness are incorporated in reliabilitybased mechanistic-empirical (ME) pavement performance models. Geospatial models are used to characterize both, the spatial variability and systematic measurement errors. To predict the long term pavement performance, the geospatial models were used to construct stochastic finite element (FE) models, which were then used to predict the performance based on the mechanistic-empirical pavement design guide equations (MEPDG). It is found that the typical covariance functions, also known as semivariograms or variograms, should be handled carefully when used in probabilistic performance modeling. Separating the inherent spatial variability from other uncertainties is necessary for performing risk and reliability analysis. Moreover, incorporating the inherent spatial variability in the stochastic FE models can alter the location of the critical response as described in the MEPDG.

Introduction Adequate pavement performance is critically related to road users’ safety, ride comfort, vehicle operation cost, travel delay cost, and vehicle durability [5,7,16]. A robust pavement design should allow for accurate and representative pavement performance predictions, which are essential to perform realistic life cycle cost analysis, and manage the infrastructure [14,36]. Pavement design can be framed either in a deterministic approach, where fixed loading and material conditions are assumed; or a probabilistic approach, where distributions of loading and material conditions are considered [21]. Compared to deterministic

design approaches, probabilistic approaches are more realistic and representative of the varying and uncertain nature of pavements, foundation materials, and loading conditions [14,35]. In 1993, the American association of state highway and transportation officials (AASHTO) published a Guide for Design of Pavement Structures [1]. The empirical design equations presented in the guide, were derived based on the AASHO Road Test conducted in 1958-60 [15]. Despite the simplicity and practicality of empirical design methods, they are limited to the range of conditions used to derive the design relations [6]. Shortly after the release of the AASHTO 1993 design guide, researchers realized the need to utilize mechanistic-

⁎ Corresponding author at: Center for Transportation Research and Education, Iowa State University of Science and Technology, 2711 South Loop Drive, Suite 4700, Ames, IA 50010-8664, USA. E-mail addresses: [email protected] (A. Alhasan), [email protected] (A. Ali), off[email protected] (D. Offenbacker), [email protected] (O. Smadi), [email protected] (C. Lewis-Beck). 1 The Center for Research and Education in Advanced Transportation Engineering Systems, Rowan University, 107 Gilbreth Parkway, Mullica Hill, NJ 08062, USA. 2 Center for Transportation Research and Education, Iowa State University of Science and Technology, 2711 South Loop Drive, Suite 4700, Ames, IA 50010-8664, USA. 3 Department of Statistics, Iowa State University of Science and Technology, 2438 Osborn Dr, Ames, IA 50011-1090, USA.

https://doi.org/10.1016/j.trgeo.2018.08.001 Received 17 May 2018; Received in revised form 30 June 2018; Accepted 5 August 2018 Available online 06 August 2018 2214-3912/ © 2018 Elsevier Ltd. All rights reserved.

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that the sources of uncertainty in geotechnical properties are due to inherent spatial variability, measurement error, and transformation uncertainty (i.e., model bias). Lua and Sues [23] presented one of the earliest efforts to assess the reliability of airfield pavement response and life prediction using a stochastic finite element (FE) with the inclusion of spatial variability. In their study it was mentioned that probabilistic FE models, with spatial variability, are more accurate representation of the true physical condition, and that research is needed to explore the effect of three-dimensional random spatial variability on pavement response and life. Dilip and Babu [10] generated spatially correlated random fields, following a Latin Hypercube sampling technique, to represent a three layers pavement system; namely AC, base, and subgrade layers. The random fields were then used in finite difference simulations to quantify the pavement response and design reliability at varying conditions. From the study it was concluded that ignoring spatial variability can lead to inaccurate assessment of the pavement performance. With the recent developments in intelligent compaction (IC), assessing the stiffness of foundation layers with high coverage became a possibility [38,43,47]. Savan et al. [38] presented a benefit-cost analysis on the application of intelligent compaction for transportation construction. The benefit-cost analysis demonstrated that the use of IC reduces compaction costs by as much as 54% and results in a US $15,385 annual savings per 1.6 km throughout the roadway’s life. IC technology provides a spatial map of response measures such as machine drive power (MDP), compaction meter value (CMV), and vibratory modulus (EVIB). These measures correlate to the stiffness of the compacted materials. The correlations between the material stiffness and IC measures are variable and project site dependent [46]. One of the questions remains unsolved: how to incorporate the dense data acquired using IC technologies into pavement design or performance prediction [48]. Significant efforts have been carried to address the impact of foundation conditions uncertainty on the pavement performance and design. However, there is a very limited number of studies that could successfully incorporate the impact of spatial variability into reliabilitybased M-E pavement performance prediction, which controls pavement design. Moreover, there are no clear definitions or procedures describing the process to include different sources of uncertainty in reliability-based pavement performance prediction and design. In this study, a detailed discussion on the uncertainty in foundation conditions will be presented. Furthermore, a mathematically robust procedure to incorporate these uncertainties into reliability-based M-E flexible pavement performance prediction models will be outlined. MC simulations will be implemented to generate stochastic FE models based on actual data acquired from a previous IC study [46]. Due to the limited information provided on the correlation between IC measures and stiffness measures, the implementation will focus on the impact of inherent spatial variability and measurement errors.

empirical (M-E) design methods [40]. M-E design methods can incorporate a wider range of varying design inputs. In 2004, the national cooperative highway research program (NCHRP) published the mechanistic-empirical pavement design guide (MEPDG). The guide provided consistent procedures that can be used as a nationwide design tool [28]. MEPDG, combines the virtue of mechanistic models, which are based on scientific models describing the pavement response to loading, and the empirical calibration that corrects for idealized mechanistic models assumptions [6]. Since the development of the AASHTO 1993 design guide, a reliability-based design approach was recommended to account for design uncertainties. Following the same concepts, and with the ability to consider a wider range of variables, reliability was introduced more extensively in the MEPDG. Despite the effort to incorporate a reliability approach in the MEPDG, design reliability was identified as one of the future needs to improve in the design guide [28]. In the past three decades, many studies have focused on implementing reliability-based M-E design [4,18,20,21,37]. Some of the earliest efforts to apply reliability concepts to pavement structural design were introduced in the 1970s [8,9]. Several studies have utilized reliability-based design methods to optimize flexible pavements design in terms of cost and performance [17,33,37]. Sanchez-Silva et al. [37], presented a reliability-based model to optimize the design of flexible pavements. In their study, it was concluded that reliability-based design optimization can incorporate other aspects besides the mechanical performance, such as construction and rehabilitation costs as well as financial factors including the discount rates, which are relevant to the decision making process. Various methods have been developed to perform reliability analysis. Typically, reliability-based pavement design is performed using one of the following methods: Monte Carlo (MC) simulation method, point estimate method, first-order second-moment (FOSM) method, Hasofer-Lind first-order reliability method (FORM), and second-order reliability method (SORM) [3,13,20]. In the mentioned reliability methods, excluding the MC simulation method, the reliability index (β) is estimated first and then the probability of failure can be calculated using: Pf = 1−Φ(β ) , where Φ is the standard normal cumulative distribution function [25]. Alternatively, the MC simulation method derives the probability of failure by generating a large number of models representing the varying material and loading conditions, which makes it computationally expensive. However, MC simulation method is the most robust method since it does not impose assumptions on the distribution of the reliability index. Timm et al. [41] incorporated reliability analysis into the M-E design procedure, developed for Minnesota, by generating 5000 design scenarios for flexible pavements using MC simulations. Dilip et al. [10] performed system reliability analysis for a flexible pavement section designed using the M-E design method. Fatigue cracking and rutting were the failure mechanisms considered due to their significant contribution to flexible pavements performance. Reliability analysis was conducted and validated using FORM, SORM, and MC simulation method. In the study, it was shown that the two failure modes were highly correlated, with a correlation coefficient of approximately 0.80; therefore, the consideration of the joint probability of failure is crucial in the reliability analysis of the pavement system. Several studies have reported the validity of FORM and FOSM for reliability-based M-E flexible pavement design in comparison to the MC simulation method [25,26]. Amongst the uncertainties and variabilities in the pavement design inputs is the foundation conditions variability, such as stiffness [21]. There is a prominent evidence that non-uniform or varying foundation conditions have significant impact on the pavement performance [22,23,24,34,39,42,45]. Several researchers have attempted to quantify and model the spatial variability of the foundation conditions. Phoon and Kulhawy [31,32] presented their extensive investigations on geotechnical uncertainties in two papers. In the papers, it was indicated

Data sources To incorporate the impact of spatial variability in foundation layers’ stiffness on flexible pavement performance, spatial statistics will be utilized to characterize that variability. In this paper, the term foundation layer includes the granular aggregate base/subbase and the subgrade layers. The foundation conditions were simulated based on the results provided in White et al. [46]. In 2009, a research team from Iowa State University performed field testing on the US219 project located near Springville, New York to evaluate Caterpillar and Bomag single drum IC rollers. In their report, test bed 1 (TB1), consisted of compacted embankment granular subgrade material with plane dimensions of approximately 18 m × 200 m. The embankment material was underlain by shredded rubber tires at depths < 1 m below grade. The area was divided into eight roller lanes and compacted with three roller passes using the Caterpillar IC roller. MDP40, a rescaled version of 2

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• Constitutive and performance modeling errors: this uncertainty is

MDP measurement to range between 1 and 150, and CMV were obtained from the roller. To establish locally calibrated correlations between IC measures and other common stiffness measures, 8 point tests were performed after pass 3 using the light weight deflectometer using a Zorn setup with 200 mm plate (LWD-Z2), to determine the elastic modulus. In the same report, test bed 6 (TB 6) consisted of a production area with aggregate subbase material placed over the TB1 embankment material. The area, with plane dimensions of approximately 17 m × 200 m was compacted with two roller passes using the Caterpillar IC roller, followed by four roller passes with the Bomag IC roller. EVIB was obtained from the Bomag IC roller. Zorn LWD with 300 mm plate and Briaud’s compaction device (BCD) setup with 150 mm diameter plate, were used to perform point tests at 9 locations after pass 6. More details regarding the IC procedure and theory can be found in the report and other previous studies. It should be mentioned that the subgrade layer showed higher stiffness compared to the base layer in the study. No details regarding the pavement design or traffic loads were provided in the study. To evaluate the impact of foundation stiffness variability solely and to reduce the number of simulations required in the study, the AC pavement layer was assumed to have a uniform and constant thickness hAC = 101.60 mm (4 in); stiffness E = 2068.43 MPa (300 ksi); and a Poisson ratio ν = 0.35. Only average temperature is used to capture the environmental effect on the pavement rutting. Based on historical data in Springville, New York, the average temperature used in the models was 48.50 °F. The design traffic load in equivalent single axel load (ESAL) was estimated for 20 years with 76,000 ESALs in the first year and a 4% growth rate. Intelligent compaction systems refers to the group of technologies that provide continuous assessment of mechanistic soil properties (e.g. stiffness) through roller vibration monitoring, on-the-fly modification of vibration amplitude and frequency, and integrated GPS to provide a complete geographic information system-based record of the site [38]. Some sources in the literature consider systems without the vibration control as IC systems. Amongst the monitoring sensors in an IC system are the accelerometers, which are mounted in or about the drum to monitor the applied compaction effort, frequency, and response from the material being compacted. From the material response to the applied load, specialized software and processing tools provide various types of IC measurement values [38,43,47]. Typically, site specific calibration is required to correlate the IC measurement values to the standard material characteristics. Amongst the point tests used for IC calibration are the LWD and BCD. The light weight deflectometer (LWD) is a hand portable falling weight device. The device consists of a falling weight, typically varies between 10 and 20 kg, which falls freely from a fixed elevation to strike a damping system that transfers the impact load to a loading plate and then into the soil. The measurement displacement is then used to back calculate the soil stiffness at the test location. A detailed description of the test operation can be found elsewhere in the literature [12]. The Briaud’s compaction device (BCD) works by applying a small repeatable load to thin plate in contact with the compacted soil. Radial and axial strain gauges mounted on the plate, measure the resulting deflections due to the repeated load. These measurements are then used to estimate a low strain modulus based on correlations determined from field and laboratory tests [44].

• •

•

•

due to the lack of knowledge in the true mechanical nature of the materials and their long term behavior. Due to the complexity in pavement systems, many of the models used in pavement design and analysis are simplified mechanistic models or empirically derived models. Accurate constitutive models, describing the various types of mechanical response and behavior, will capture the deteriorating behavior of the pavement system due to different loading conditions, and thus will allow for more accurate prediction models. Forecasting uncertainties: all performance prediction models are based on forecasted loading conditions, which includes traffic loads, environmental loads, and any other sources of loading. This forecasting will induce uncertainties in the predictive performance models. Systematic measurement errors: the term error, incorporates two components: bias and variance. Bias is the consistent and uniform shift in the measured values. Bias error can include the lack of reproducibility for human-operated devices. In the other hand, variance describes the fluctuation in the measurements around a mean value, which describes the testing repeatability. These errors can be due to the lack of knowledge in the testing device behavior, lack of repeatability, and the limited number of tests conducted in the sampling domain. Nested errors due to indirect testing techniques: many of the field tests, such as the LWD test or IC measures, do not measure the fundamental mechanical nature of the material. In such cases, the measurements from one test are correlated to the fundamental material characteristics evaluated in the laboratory or highly controlled field tests. Accounting for this uncertainty requires the use and knowledge of random regression models. Inherent spatial variability of material properties: this describes that change in material properties spatially due to the variation in soil formations, historical loading conditions, paving material properties, and compaction effort and practices. Arguably, this variability can be fully described and quantified through extensive testing, and thus can be more properly defined as a source of variability and not uncertainty. However, the testing mechanisms such as IC will carry the systematic measurement uncertainty and the nested errors which includes other nonsystematic errors.

In this study; the inherent spatial variability, with the uncertainty due to systematic measurement errors solely, will be investigated and included in a reliability-based M-E performance model, Spatial analysis To characterize the variability in the spatial domain we take a geostatistical approach to describe and model the data. Soil stiffness maps provided in the IC report [46], suggest the measurement field exhibits stationarity. In a stationary process, the distribution of observations depend only on the absolute distance between points. Thus, if multiple realizations are simulated, the overall mean and variance in the spatial domain will be constant. However, for a given realization the mean and variance estimated over a short window will vary spatially. The covariance function, also referred to as semivariance or semivariogram, characterizes the level of variability between spatial observations (Z) separated by a distance (h) in a given random field. Eq. (1) represents the theoretical covariance as a function of h. Note the covariance is defined as a function of lag distance h rather than the spatial coordinates of the observed variables [27].

Theory and analysis approach To develop reliability-based pavement performance models, sources of uncertainty should be identified and quantified properly. Sources of uncertainty can be generally due to the lack of knowledge or the fundamental variability found in natural processes. The sources of uncertainty related to pavement design and performance modeling can be identified as follows:

γ 2 (h) =

1 {z (x )−z (x + h)} 2 2

(1)

For regularly spaced N observations with spacing δ , one can define a set of paired observations (zx̂ , zx̂ + h) ; where h = nδ , n = 1, 2, 3⋯, with a total number of pairs m = N −n . The empirical covariance is shown in 3

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Eq. (2). To incorporate uncertainty due to systematic measurement errors, the observed value is considered unbiased random variable equal to the true measurement plus an independent and identically distributed random error ε with a mean E [ε ] = 0 and a constant variance σε2 as shown in Eq. (2.1). It should be noticed that the term true measurement does not imply the true fundamental material characteristic of interest, it indicates a measurement free of repeatability errors.

γ 2̂ (h) =

1 2m

m

∑

{zx̂ −zx̂ + h } 2

(2)

i=1

zx̂ = zx + εx

(2.1)

Under the assumption of strict stationarity where the mean and the global variance are constant E [zx ] = μz , Var (zx ) = σz2 , and using the definition for the coefficient of correlation between two true observations at distance h as ρ (h) = cov (zx , zx + h)/ σz2 , one can show (Appendix A) that the expected value of the empirical covariance function is:

E [γ 2̂ (h)] = σz2−ρ (h) σz2 + σε2

Fig. 1. A typical view of the 3D FE model as shown in ABAQUS interface.

Finite element analysis A cylindrical three-dimensional finite element model was developed to simulate the layered pavement structure using the finite element package ABAQUS. It is acknowledged that the three-dimensional (3D) analysis requires more computational time than the two-dimensional (2D) analysis; however, the 2D analysis is incapable of simulating the spatial variability in the three directions. Fig. 1 shows the three-dimensional finite element model used in this study. The flexible pavement structure was modeled as a cylinder with a radius of 100 in. The modeled pavement system contains three layers: a 4-in. asphalt pavement layer, an 8-in. base layer, and a 100-in. subgrade layer. The cylindrical shaped model was chosen to minimize the number of distorted elements due to the circular load imprint. The height and radius of the model were selected to ensure minimal interaction with the boundary conditions. The edges of the model were constrained with roller supports, while the bottom of the model was constrained with a fixed support. Several researchers have reported successful use of such boundary conditions including Zaghloul and White [49]. Due to the model size and load concentration, a finer mesh was used near the loading area and a coarser mesh was used at the edges of the cylinder. Additionally, a finer mesh was used near the surface of the pavement system due to its vicinity to the loading area. The pavement system model consisted of 10,192 3D, 8-node block elements (C3D8R) to minimize computer run-time and maximize efficiency. Higher node elements and more integration points could have been implemented, but were unnecessary for the focus of this study. The material properties for each element in the subgrade and base layers were assigned based on the results from the MC simulations, described in Section ‘Monte Carlo simulations’. Due to the large number of FE models in the study, all pavement layers were assumed to follow a linear elastic behavior. Although the nonlinear behavior of the foundation and pavement materials has a significant impact on the pavement performance and prediction model accuracy, the computational power needed for a large number of nonlinear models was unattainable in this study. Moreover, previous studies have indicated that although an accurate representation of the pavement response might not be guaranteed when using a linear elastic model, the impact of spatial variability can be captured with a reasonable accuracy using a linear elastic model [11]. The tire loading was modeled using a circular imprint with a uniform pressure distribution. Stiffening effects and other nonlinearities occurring in the tire-pavement interaction were not considered. A 9000 lb load was applied to the circular imprint, this represent the load level from a single of the 18,000 lb standard axle load. The resulting tire pressure in the circular imprint was 80 psi.

(3)

In Eq.(3), the first two terms combined describe the inherent spatial variability, as a function of lag distance h, in the fundamental nature of the material properties. These two terms represent an unbiased estimator of the global spatial variance in the fundamental material characteristics. The last term in the equation, which is the bias in estimating the true measurement variance, presents the variability in spatial measurements due to the systematic measurement errors, which includes the lack of repeatability in testing or the limited readings at short distances. As a limiting case, if the lag distance between two readings reach zero, then the coefficient of correlation between two true measurements will reach one. Accordingly, Eq. (3) will reduce to the variance of the random error E [γ02] = σε2 , which is typically referred to as the nugget effect. In contrast, as the lag distance reaches infinity, the coefficient of correlation will approach zero. Accordingly, the covariance will reach the maximum total variance E [γ∞2 ] = σz2 + σε2 , which is referred to as the sill. Eq. (4) defines the correlation coefficient (ρ (h) ) between measurements as a function of the covariance function [27].

ρ (h) = 1−

γ 2 (h) γ∞2

(4)

The empirical covariance function is only defined at observed distances. To evaluate the covariance at an arbitrary lag distance, a continuous covariance model is estimated using sample data. In the IC study [46], the nested spherical model, shown in Eq. (5), was used to describe the covariance of both the subgrade and base layers. In the model, the range (r) defines the lag distance at which the covariance function is assumed to reach the sill value. For practicality purposes, the range is always finite, and the correlation between measurements at a lag distance greater than the range is set equal to zero. In the case of nested models, spatial variability is described at two scales, reflected by the two ranges in the model r1 and r2. The model coefficients C1 and C2 define the first and second sill values if the nugget effect is ignored. The nested behavior indicates that at least two nested stochastic processes are responsible for generating the spatial field (i.e., the soil stiffness characteristics).

{

3

( )} { { ( )}

⎧ γ 2̂ + C1 3h −1/2 h + C2 2r 1 r1 ⎪ 0 ⎪ 3h h 3 γ 2̂ (h) = 2 ⎨ γ0̂ + C1 + C2 2r 2 −1/2 r 2 ⎪ ⎪ γ02̂ + C1 + C2 ⎩

3h h 3 −1/2 r 2r 2 2

( )}

, 0 < h ≤ r1 , r1 < h ≤ r2 , h > r2 (5) 4

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Table 1 Summary statistics for IC maps and point test measurements. MV

Layer

MDP40 EVIB ELWD-Z2 (MPa)

Subgrade Base Subgrade

n

6008 27,198 10

Univariate statistics

Spatial statistics

Average

Standard Deviation

Nugget

126.9 80.2 144.8

7.6 18.6 104.4

30 58 110 300 Not enough measurements

Sill1

Range1

Sill2

Range2

16 12

66 430

65 75

Monte Carlo simulations To simulate a spatially correlated random field, we employ a covariance matrix decomposition approach. The first step generates a vector of n independent standard Gaussian observations z Gn . The covariance matrix Σn × n between the elements is then used to generate a positive definite correlation matrix αn × n , as described in Eq. (4). The lag distance between the observations is the Euclidian distance in 3D space. The correlation matrix is then decomposed using the Cholesky decomposition into a lower triangular matrix and its conjugate, (i.e., P = LL∗). Multiplying the matrix L by the observations vector ZGn , results in a single realization of a correlated spatial random field (Zμ0 ). The following relation will shift the mean of the realization to μz and dilate the observations to have a standard deviation σz , Z = μz + Zμ0 × σz [3]. In this study, the spatial characteristics of pavement foundation stiffness are interpreted using the IC measures acquired from the US219 project [46]. For the subgrade layer, the covariance function and distribution moments for MDP40 measurement values (MV) are provided in Table 1. All the MDP40 random fields are then converted to the modulus of elasticity based on Eq. (6) [46]. The summary statistics for the EVIB MVs acquired at the base layer, are provided in Table 1. The correlation used to convert from EVIB MV to a modulus of elasticity is described in Eq. (7) [46]. The uncertainty due to the conversion between IC measurements and the modulus of elasticity is not considered due to the limited information provided on these correlations.

Fig. 2. A typical EVIB profile generated for 1 200 m profile (a) excluding the model nugget and (b) including the model nugget.

10

MDP40 ⎞ E = ELWD − Z2 = ⎛ ⎝ 79.59 ⎠ E = ELWD − Z 3

E −30.6 = VIB 1.49

(6)

windows. Each window will have a different level of spatial variability due to the different localized variance values. To reflect that variability in the localized truncated covariance functions, all lo2 / γ∞2̂ . This calized truncated covariance functions are scaled by σLi scaling will guarantee that the covariance function beyond the range 2 is an unbiased estimator of the local variance lim γt2̂ (h) = σLi , and

(7)

Special precautions should be taken when simulating a random field characterized by a covariance matrix with a nugget. When removing the measurement error component, the true nature of the material properties at the macroscopic and global scale is deterministic for a given point in space, and varies spatially. If the random field is generated using the covariance function with the nugget effect the observations will fluctuate significantly at shorter distances reflecting the uncertainty in the measurement (Fig. 2). To perform localized analyses on short windows sampled from a random field, the sources of spatial variability and uncertainty should be separated. The following steps describe the procedure to generate representative models for the reliability analysis in this study:

h →∞

maintain the same truncated correlation function αt (h) . Step 3: To account for measurement uncertainty, assume the localized mean follows a normal distribution with a mean μLi and variance γ02̂ , which is the nugget value, μLij ~ N (μLi , γ02̂ ) . Generate a 100 localized mean μLij values with the same localized variance 2

2 σLij = σ . This will produce 8000 scenarios representing the inLi herent spatial variability and the measurement uncertainty. The localized variance values are not sampled from a random distribution because these values represent the true inherent variability and thus, given a spatial location in the realization, the variability should be constant across the 100 local scenarios. Yet, the 80 different variance values from Step 3 reflect the variability in the localized covariance function. Step 4: Using the 8000 scenarios generated in Step 3, and assuming the covariance function is isotropic and independent for each layer, generate the 8000 3D spatial random field at the centroids of the FE model elements using a truncated correlation function αt (h) and transformed using Zij = μij + Z μij 0 × σLi . Then transform the IC measurements to elastic modulus values using Eqs. (6) and (7).

Step 1: Generate a 2.5 m × 200 m 2D global realization, which represent the IC measurements for a strip in the compacted stretch with a width equal to the FE model diameter. The realization is generated using a truncated covariance function, γt2̂ (h) = γ 2̂ −γ02̂ , where the terms in the truncated function will reflect the inherent spatial variability in the material properties. The truncated correlation function will accordingly be αt (h) = 1−(γ 2̂ (h)−γ02̂ )/(γ∞2̂ −γ02̂ ). Step 2: Subdivided the global realization into 2.5 m × 2.5 m windows of localized analysis patches. The localized mean μLi and 2 variance σLi values for each window are estimated using the generated points falling in the window. This will produce 80 localized 5

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Performance prediction models

Reliability analysis

Permanent deformation (i.e., rutting), and bottom-up fatigue cracking are the two performance criteria chosen to represent the pavement condition. Top-down fatigue cracking can only be captured using other axle configurations. All the performance prediction models used in the study can be defined as functions of time (t), by estimating the total number of ESAL repetitions applied up until that time (t).

To define the reliability of pavement performance for the two distresses, rutting and fatigue cracking, probability distributions of the severity levels (i.e., rut depth (RD) and percent fatigue cracking (FC)) are defined for each local window in the spatial domain. The overall system reliability considering the combination of both failure mechanisms will not be discussed in this study. Using the MC simulations, 8000 deterministic M-E performance prediction models were generated to represent an exhaustive sample from the probabilistic performance space. From the 80 local analysis windows, 80 different sets of probability density functions (PDF) were defined for each pavement distress severity (S) in a window i at time t using Eq. (13).

Permanent surface deformation The MEPDG rutting model is used in this study, with the national model calibration factors. In the model, total rut depth (RD) is expressed as the sum of vertical deformations for all layers as shown Eq. (8).

n

nlayers

∑

RD =

Pri, t (S1 ≤ st ≤ S2) =

εpi hi

εp (z ) = εp (Z0 ) e−kz

(9)

In Eq. (9), Z0 is the elevation of the centroid for the lowest central subgrade element in the FE model, z is the distance measured from Z0, and k is a regression constant estimated using the subgrade elements in the FE model. To estimate the total rutting in the subgrade layer, the plastic strain estimated for the central subgrade elements in the FE model are multiplied by the element thickness and added to the deformation δ∞ below the model boundaries as shown in Eq. (10).

δSG = δSG, FE + δ∞ ∞ 0

εp (z ) dz =

Pri, t (s > ST , t ) = 1−CDF(ST , t ) = 1−

(10)

∫Z

∞ 0

εp (Z0 ) e−kzdz =

1 εp (Z0) k

(10.1)

Pr (Ψ = k / N |ST ) =

'

1 + e (c1+ c2logD (t ))

D (t ) =

t

∫0 ∑ i=1

(13)

∫0

(11)

ψT

Prfailure (ST , ΨT , t ) = 1−

(12) 3.9492

1 Nf = 0.00432 × k1 × C ⎛ ⎞ ε ⎝ t⎠ ⎜

⎟

1 1.281 ⎛ ⎞ ⎝E⎠

fS, t (s|t ) ds

l ∈ AC

∑ ∫0

ST

fΨ, S (ψ, s|t ) ds

ψ=0

ni (t ) Nf

ST

(14)

(15)

where Fk is a set of all possible subsets with k integers and AC is the complement for A. For example in the 80 windows, the set possible combinations to have an extent 2/80 is Fk = {(k1, k2), (k1, k3), ⋯, (kN − 1, kN )} . Eq. (15) represents the conditional PMF for having an extent ψ given a severity level S > ST. combining the PMFs at different severity thresholds will form a cumulative risk surface, which describes the risk of failure with an extent ψ , and a given failure threshold. To describe the total probability of failure for the entire section, severity and extent threshold can be used to represent the limit state condition. Accordingly, the joint probability of failure as a function of severity and extent thresholds is defined in Eq. (16), and the reliability is defined correspondingly as shown in Eq. (17).

where C1′ and C2′ are calibration parameters defined in the MEPDG, and D(t) is the accumulated damage at time t. The cumulative damage can be estimated for a load spectrum with varying load categories (LC), at time t using Eq. (12). To simplify the analysis, the load spectrum is presented using a single load category expressed in ESAL. In the equation, the number of repetitions to fatigue cracking (Nf) is based on the asphalt institute model [2]. The number of repetitions to fatigue cracking is a function of the tensile strain (εt ) at the critical location, which is the central element under the tire load at the bottom of the AC layer for bottom-up fatigue cracking; the AC layer modulus (E); and the calibration factors K1 and C described in the MEPDG. LC

ni

∑ ∏ pi,ST ∏ (1−pl,ST ) A ∈ Fk i ∈ A

100 '

∑ j = 1 IS1 → 2

where CDF is the cumulative density function. To estimate the probability of failure at multiple windows, one can define the discrete probability mass function (PMF) for the failure extent by treating the failure events in the local windows as independent Bernoulli events with varying probabilities. The PMF of a sum of independent Bernoulli events that are not necessarily identically distributed follows a PoissonBinomial distribution. Eq. (15) describes the PMF for observing a failure in k windows out of N total windows given a severity threshold ST, and accordingly a spatial extent ψ = k / N .

Bottom-up fatigue cracking To estimate the bottom-up fatigue cracking, the MEPDG model described in Eq. (11) is used. In this study, the severity of fatigue cracking for a given 2.5 m patch is defined as the percentage of cracked area to the patch area, this should not be confused with the extent at the section level.

FC =

1

fS, i, t (s ) dfs ≅

where IS1 → 2 is the indicator function, which equals one if the distress severity falls between S1 and S2 and zero otherwise; and ni = 100 is the number of simulations in window i. To simplify the analysis, for a single global realization, it is assumed that a window reaching a severity level S, does not affect the failure in a neighboring window, even though there is a correlation between the mean stiffness values in the two windows. Thus, one can assume conditional independence in the distress severity between the windows given the localized mean conditions. For a given window, the probability of failure due to a given distress is defined as the probability that the distress severity will exceed a threshold value ST at time t. Using the locally estimated PDFs, Eq. (14) presents the probability of failure at a local window i.

where nlayers is the number of sublayers, εpi is the total plastic strain of sublayer i, and hi is sublayer thickness. The detailed description of the procedure to quantify the plastic strains can be found in the MEPDG. To simplify the analysis and reduce the computational time, all materials were assumed linearly elastic. The actual subgrade layer thickness was unknown, in such case the plastic strain can be approximated using the relation described in Eq. (9) [28].

∫Z

s2

(8)

i=1

δ∞ =

∫s

ψT

R (ST , ΨT , t ) =

∑ ∫0

ψ=0

(12.1) 6

ST

fΨ, S (ψ, s|t ) ds

(16)

(17)

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Fig. 3. Heat map of the generated spatial random fields for (a) EVIB of the base layer and (b) MDP40 of the subgrade layer.

Results and discussion

within the asphalt layer despite the uniform AC pavement layer stiffness. These fluctuations within the AC pavement layer conform to the observations from previous studies, where the foundation stiffness spatial variability had a significant impact on the pavement response and performance [11,34,42,45]. Following the analysis procedure described in the MEPDG, the tensile strain at the bottom of the AC layer underneath the tire was considered as the critical tensile strain, which induces bottom-up fatigue cracking. Fig. 7a presents the histogram for the critical tensile strains acquired from the 8000 scenarios. The variation in the critical tensile strains for different foundation conditions and uniform AC pavement layer further confirms the impact of varying foundation conditions. It can be noticed in Fig. 7b that the maximum tensile strain does not necessarily occur at the critical location defined in the MEPDG, however this impact is not investigated further in this study.

Spatial analysis and Monte Carlo simulations Using the procedure described in this paper, and the statistical summaries provided in the IC study report, 2D global realizations were independently generated for the base and subgrade. Fig. 3a and b show heat maps for the global realizations simulated for (a) EVIB of the base layer and (b) MDP40 of the subgrade layer. Notice the random field is smooth reflecting the inherent spatial variability due to the truncated covariance function. Localized weak or stiff pockets are not present in the realization. Including localized weak pockets will alter the analysis procedure and can be achieved by introducing the concepts of clustered spatial fields [30]. Fig. 4a–d present the histograms for the expected value for the lo2 calized means and variances (i.e., μLi and σLi ) estimated by dividing the 2D global realization into 80 localized windows for the two unbound layers. Summary statistics are reported in terms of the sample mean, standard deviation (std), and coefficient of variation (COV). Fig. 5a and b present the histograms of the localized means for the entire 8000 scenarios generated for both layers. The histograms in Fig. 5 match those reported by White et al. [46], therefore the scenarios are expected to represent the true conditions with high accuracy. It is worth mentioning that several IC measures, such as EVIB, are strictly positive and negative values are considered outliers; therefore, the infinite negative bound of the normal distribution is not acceptable; however, there is a very low probability of observing outlier negative values, making the normal approximation a reasonable model.

Pavement performance models Using the FE model responses and the cumulative traffic load at 15 and 20 years, rutting and fatigue cracking severities were estimated for the 8000 scenarios from the MEPDG performance models. Fig. 8a and b show the histograms, with summary statistics, for estimated rutting and fatigue cracking severities, expressed in inches and percentage respectively, for all scenarios after 15 service years. It should be mentioned that although rutting is a downward movement, therefore expressed as a negative deformation, all estimated values are converted to positive values to facilitate the analysis. Fig. 9a and b show the histograms for rutting and fatigue cracking severities after 20 service years. The increase in the mean rutting value from age 15 years to age 20 years was 5.07%, however the percent increase in the fatigue cracking severity was 39.70%. This rapid increase in the fatigue cracking reflects the S-shaped sigmoidal growth curve for fatigue cracking severity. In the other hand, the rutting model used in the study is exponential with a negative power of time reciprocal function, accordingly rutting increases significantly at early service life, during the primary rutting stage, and tampers down in the secondary rutting stage.

Finite element simulations Given the model geometry drafted in ABAQUS, the centroids for all elements in the model defined the domain for the spatial random field generator created in Matlab. The generator utilized the localized truncated covariance functions and the localized means ( μLij ) generated in the MC simulation, to perform a covariance matrix decomposition and generate 8000 FE input files, with the 3D spatial stiffness fields for unbound layers and uniform stiffness for the AC pavement layer. Fig. 6a shows a typical vertical compression strain profile to be used in rutting calculations. The profile represents the compressive strain as a function of depth for the central elements across the model thickness. The fluctuations in compressive strain at varying depths and widths, as shown in Fig. 6b, reflect the impact of spatial variability in the elements’ stiffness values. Moreover, the compressive strain fluctuates

Reliability analysis Subdividing the full global realization into 80 analysis windows, will result in 80 distribution estimates for each distress type at each year. The probability distribution of failure in multiple windows can be estimated using the Poisson-Binomial distribution described in Eq. (15). 7

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Fig. 4. Empirical histograms with summary statistics for (a) the expected localized EVIB mean in the base layer, (b) the expected EVIB variance in the base layer, (c) the expected localized MDP40 mean in the subgrade layer, and (b) the expected MDP40 variance in the subgrade layer.

do not reflect failure, they reflect the probability of not observing any distress. Alternatively the cumulative reliability surface presents the complement probability of observing fatigue cracking with severity greater than ST and at an extent greater than ψT . These cumulative risk and reliability surfaces can be instrumental when considering multiple failure modes and more complex failure criteria, however these surfaces will be presented as hypersurfaces in higher order analyses. To examine the evolution of reliability surfaces over time, heat maps were created for both distresses modes at 15 and 20 service years. Fig. 11a and b present the reliability heat maps for rutting after 15 and 20 service years respectively. It can be noticed at very lenient

In order to estimate the probabilities for the Poisson-Binomial PMF, astronomical number of combinations should be considered. As an alternative, the PMF can be estimated efficiently with high accuracy using discrete Fourier transform [19]. The estimates were performed using the ‘poisbinom’ package in RStudio [29]. Fig. 10a and b show the cumulative risk and reliability surfaces devised for fatigue cracking after 20 service years. The cumulative risk surface presents the probability to observe fatigue crackling at ψ extent level given a severity threshold ST. Any cross section at a given severity threshold presents a proper PMF, which sums up to 1. It should be mentioned that the probabilities at extent ψ = 0 were excluded from the figure since they

Fig. 5. Empirical histograms with summary statistics for the 8000 generated mean values of (a) the EVIB in the base layer and the MDP40 in the subgrade layer. 8

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Fig. 6. (a) A typical compressive strain profile estimated from the central elements in the FE model and (b) a typical FE model cross section with the vertical compressive strain.

Fig. 7. (a) Empirical histogram with summary statistics for the tensile strains estimated at the critical location in the 8000 scenarios and (b) a typical FE model cross section with the horizontal tensile strain.

Fig. 8. Empirical histogram with summary statistics for the estimated (a) rutting severity and (b) fatigue cracking severity after 15 service years.

Summary and conclusions

thresholds (i.e., accepting high severity and extent levels) the system reliability reaches one. On the opposite side the reliability drops to zero for strict thresholds. Fig. 12a and b present the reliability surfaces for fatigue cracking after 15 and 20 service years respectively. It is clear that the surfaces for both distresses are drifting to lower reliability values, however the drift in the fatigue cracking reliability surface is more prominent.

The study presented an extensive geostatistical framework to analyze dense spatial data for their use in reliability-based M-E pavement performance analysis and design. The key findings and remarks from this study are:

• A full procedure to incorporate spatial uncertainty, including the inherent spatial variability and the lack of repeatability, was developed based on rigors mathematical analysis tools.

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Fig. 9. Empirical histogram with summary statistics for the estimated (a) rutting severity and (b) fatigue cracking severity after 20 service years.

Fig. 10. (a) The cumulative risk surface and (b) reliability surface for fatigue cracking after 20 service years.

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Fig. 11. The reliability heat map for pavement rutting after (a) 15 service years and (b) 20 service years.

Fig. 12. The reliability heat map for pavement fatigue cracking after (a) 15 service years and (b) 20 service years.

• Geostatistical analysis provides a valuable tool to characterize and

• Incorporating the inherent spatial variability in the stochastic FE

•

• •

• • •

analyze the inherent spatial variability as well as the uncertainty due to lack of testing repeatability. Covariance functions, also known as semivariograms or variograms, should be handled carefully when used in probabilistic performance modeling. Geostatistical models were sufficient to characterize and summarize the IC outputs. Proper interpretations of the geostatistical models helps in understanding the sources of error and variability in the IC data. Separating the inherent spatial variability from other uncertainties is necessary for deriving risk and reliability surfaces as functions of severity and extent.

models can alter the location of the critical response as described in the MEPDG. The trends in the simulated realizations are clearly reflected in the FE model output. More studies are needed to incorporate the impact of other uncertainty sources in the derivation and estimation of risk and reliability surfaces.

Acknowledgements The authors would like to thank Dr. Petrutza Caragea from the Department of Statistics for her help and support.

Appendix A. Derivation of the expected value of covariance function Following the standard definition of the empirical covariance function at a given lag distance [27], and that an empirical observation with test repeatability error can be defined as zx̂ = zx + εx , Eq. (A.1) defines the empirical covariance function. 11

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γ 2̂ (h) = = =

1 2m 1 2m 1 2m

m

∑i = 1 {(zx + εx )−(zx + h + εx + h )} 2 m

∑i = 1 {(zx + εx )2 + (zx + h + εx + h )2−2(zx + εx )(zx + h + εx + h)} m

∑i = 1 {z x2 + 2zx εx + εx2 + z x2+ h + 2zx + h εx + h + εx2+ h−2zx zx + h−2zx εx + h−2zx + h εx −2εx εx + h}

(A.1)

Taking the expected value of the empirical covariance function, and keeping in mind the linearity and non-multiplicatively properties of the expected value operator, Eq. (A.2) defines all the terms for an expected value.

1 2m

E (γ 2̂ (h)) =

m

∑

{E [z x2] + 2(E [zx ] E [εx ] + Cov (zx , εx )) + E [εx2] + E [z x2+ h] + 2(E [zx + h ] E [εx + h] + Cov (zx + h , εx + h )) + E [εx2+ h]

i=1

−2(E [zx ] E [zx + h] + Cov (zx , zx + h))−2(E [zx ] E [εx + h] + Cov (zx , εx + h))−2(E [zx + h ] E [εx ] + Cov (zx + h , εx ))−2(E [εx ] E [εx + h] + Cov (εx , εx + h ))} (A.2) Since the error term ε , is defined to be independent and identically distributed term with mean zero and constant variance (i.e., E [εx2] = E [εx2+ h]), all terms containing E [εx ], or the covariance between ε and any other terms (e.g., Cov (zx + h , εx ) or Cov (εx , εx + h) ), will cancel out, which reduces Eq. (A.2) to Eq. (A.3).

E (γ 2̂ (h)) =

1 2m

m

∑

{E [z x2]−2E [zx ] E [zx + h] + E [z x2+ h]−2Cov (zx , zx + h) + 2E [εx2]}

(A.3)

i=1

σz2 ,

Under the strict stationarity assumptions, E [zx ] = E [zx + h] and Var (zx ) = Var (zx + h) = Cov (zx , zx + h) = ρ (h) σz2 , E [εx2] = σε2 ; Eq. (A.3) can be further simplified as shown in Eq. (A.4).

E (γ 2̂ (h)) = =

1 2m 1 2m

and based on the definitions

σz2

=

E [z x2]−E 2 [zx ],

m

∑i = 1 {(E [z x2]−E 2 [zx ]) + (E [z x2+ h]−E 2 [zx + h])−2ρ (h) σz2 + 2σε2} m

∑i = 1 {2σz2−2ρ (h) σz2 + 2σε2} = σz2−ρ (h) σz2 + σε2

(A.4)

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