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All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability
Transportation Research Part C 114 (2020) 84–98
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Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability
T
Osman Erman Gungor , Imad L. Al-Qadi ⁎
Dept. of Civil and Environmental Engineering, Univ. of Illinois at Urbana-Champaign, 205 N. Mathews Ave., Urbana, IL 61801, United States
ARTICLE INFO
ABSTRACT
Keywords: Autonomous truck platooning Optimization Pavement Life-cycle cost Infrastructure Autonomous and connected trucks
With the introduction of connected and autonomous trucks (CATs), truck platooning is expected to be more feasible and prevalent. The reported benefits of the truck platooning include regularizing traffic, reducing congestion, increasing highway safety, and decreasing fuel consumption and emission. Truck platooning may, however, decrease pavement longevity because it would cause channelized load application and hinder the healing properties of asphalt concrete. This study proposes a centralized control strategy that converts the pavement-related challenges of truck platooning into opportunities. This strategy leverages the auto-pilot technologies in CATs by optimizing the lateral position of each platoon or group of platoons. The efficiency of the proposed control strategy was demonstrated in a case study. Results showed that pavement lifecycle costs could be reduced up to 50% by controlling the lateral position of the platoons for each day.
1. Introduction Passenger cars and shuttles have primarily driven connected and automated vehicle (CAV) technology research, but recently there has been significant interest in developing connected and autonomous trucks (CATs). In fact, many major CAV technology manufacturers, such as Google-owned Waymo, have launched subsidiaries or departments for CAT development. CATs are expected to bring many advantages into freight industry such as improving operational efficiency of freight shipments and overcoming shortage in truck drivers. The introduction of CATs may result in drastic changes in freight shipment operation. One important change is the formation of truck platoons, a convoy of trucks traveling in close distance. With the advancements in intelligent technologies used in CATs that enable the connection among vehicles and between vehicles and infrastructure, truck platooning is set to become a feasible, efficient and prevalent practice in the future. Truck platooning has benefits and potential challenges. Reducing congestion and braking/ accelerating, and improving safety, traffic flow, and fuel efficiency are some of the expected benefits of platooning (Alam et al., 2015; Bonnet and Fritz, 2000; Nowakowski et al., 2015; Tsugawa et al., 2016; Browand et al., 2004; Lu and Shladover, 2011; Tsugawa et al., 2011; Tsugawa, 2014; Eilers et al., 2015; Lammert et al., 2014; Humphreys et al., 2016; Ramezani et al., 2018). However, platooning, if implemented without caution, may accelerate pavement damage accumulation. Platooning is expected to cause channelized truck loading application because the lateral positions of trucks in a platoon are expected to be similar as opposed to scattered lateral position of human-driven trucks. This channelized loading is expected to increase the damage accumulation rate and ultimately reduce the pavement’s service life. Noorvand et al. (2017), Chen et al. (2019)
⁎
Corresponding author. E-mail address: [email protected] (O.E. Gungor).
https://doi.org/10.1016/j.trc.2020.02.002 Received 17 August 2019; Received in revised form 29 January 2020; Accepted 1 February 2020 0968-090X/ © 2020 Elsevier Ltd. All rights reserved.
Transportation Research Part C 114 (2020) 84–98
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Fig. 1. Optimization of lateral position of platoons using V2I communication (Gungor et al., in preparation).
acknowledged this challenge and quantified of the impact of trucks’ positionings within a lane. Additionally, the time between two consecutive truck loads will be shorter because of reduced inter-vehicle distance in platoons, which may hinder the self-healing characterization of asphalt concrete (AC) and consequently reduce pavement service life. This study proposes a centralized control strategy that converts the pavement-related challenges of truck platooning into opportunities. This strategy leverages the auto-pilot technologies in CATs by optimizing the lateral position of each platoon or group of platoons. Fig. 1a demonstrates a default scenario where the platoons are aligned with each other, which may lead to accelerated damage accumulation within pavements. Fig. 1b shows the proposed control strategy, where the platoons communicate with a center (either with a cloud or an infrastructure). In the proposed control strategy, the position of each platoon (or group of platoons) is adjusted to minimize the pavement damage and consequently increase the pavement service life. The de-centrailized control strategy, where the lateral position of each truck in a platoon was optimized considering the trade-off between the truck aero-dynamics and pavement damage, was introduced by the authors elsewhere (Gungor et al., in preparation). In the de-centralized optimization, while the lateral shift of trucks in a platoon increased the pavement service life, the fuel efficiency of trucks was comprised (Fig. 2). The centralized optimization, on the other hand, maximizes the fuel efficiency because there is no truck’s lateral shifting in a platoon. Application of a centralized strategy, however, may require significant investment because it assumes existence of reliable, centralized vehicle-to-infrastructure (V2I) communication as opposed to the de-centralized strategy’s vehicle-to-vehicle (V2V) communication requirement, which is already part of the platooning technology.
Fig. 2. Optimization of lateral position of platoons using V2I communication (Gungor et al., in preparation). 85
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It is important to note that the ideal implementation of platooning requires to study more variables in addition to infrastructure damage. For example, Sun and Yin (2019) investigated a behavioral stability of platooning which addresses the willingness of vehicles to join a platoon. Behavioral stability may especially be important for platoons formed by CATs owned by different companies. Also, this study developed a fair allocation mechanism to redistribute to platooning benefits to incentive vehicles to join a platoon. Calvert et al. (2019) studied the effects of truck platooning on traffic flow. The results did not show any positive impact of platooning on traffic flow. The study also concluded that the platoons of two or three trucks have a negligible impact on traffic efficiency. Furthermore, the scheduling of truck platoons to maximize the efficiency (Milanés and Shladover, 2014; Wang, 2018; Luo et al., 2018) and string stability for improving safety (Larson et al., 2016; Boysen et al., 2018; Chen et al., 2017) have been studied and should be considered for optimal control of platoons. 2. Pavement design and analysis To solve the optimization problem given in Fig. 1b, one needs to simulate the effects of any arbitrary positioning of platoons on pavement service life. In other words, the lateral position of each platoon should be an explicit input while simulating the damage accumulation within the pavement. Additionally, the impact of reduced resting time should be considered. To the best of the authors’ knowledge, no pavement design guidelines account for these two variables as explicit inputs. The authors have introduced a framework called Wander 2D that modifies mechanistic-empirical design approach (MEPDG, 2004) to consider the lateral position of trucks and between-vehicle distances. In this framework, pavement structural responses are calculated using ABAQUS. (i.e., mechanistic part). Later, the computed responses are linked to pavement damage through empirical transfer functions (i.e., empirical part) and other steps included in Wander 2D. The summary of Wander 2D is presented below. The details can be found elsewhere (Gungor and Al-Qadi, in preparation). 2.1. Pavement responses to loading In this step, the critical pavement responses are computed under a tire load at different transverse locations. This study uses 3D advanced pavement finite element (FE) models that Al-Qadi and his coworkers developed and have been continously improving over the past two decades (Elseifi et al., 2006; Yoo and Al-Qadi, 2007; Wang and Al-Qadi, 2010; Al-Qadi and Wang, 2012; Gungor et al., 2016a,b; Al-Qadi et al., 2018; Castillo and Al-Qadi, 2018). The developed models utilized a commercial software, ABAQUS. In the models, the asphalt concrete (AC) was considered as linear viscoelastic and simulated using Prony coefficients. For thick pavement sections (AC thickness is greater than five inches), the base layer was considered linear elastic. For thin pavement sections where the stresses on base layer are high, anisotropic-stress dependency was incorporated into the models. The tire-loading was simulated by measured 3D contact stresses which allowed capturing the effects of longitudinal and transverse contact forces on the pavement responses. Infinite elements were used at the boundaries of the model to capture the exponentially decaying behavior of pavement responses in a computationally efficient manner. Mesh sensitivity analysis was used to determine the mesh characteristics. While a refined mesh was used in the wheel path, a relatively coarser mesh was preferred between infinite elements and wheel path (i.e., transition zone) to reduce the computational time. Fig. 3 illustrates the model developed in ABAQUS. Fig. 3 illustrates the model developed in ABAQUS.
Fig. 3. 3D pavement finite-element model (Gungor et al., 2017). 86
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2.2. Differential damage computation The damage-prediction framework developed by MEPDG consists of two steps. The first step predicts damage from the strains, which transformed into damage, that occurs after N number of load repetitions. The outcome of this step is rutting at the mid-depth of each layer and fatigue damage index for fatigue cracking prediction. As loading input and climate conditions change (e.g., change in modulus because of temperate variation or axle loads), they alter the value of the computed strain which requires this step to be repeated. The second step involves combination of differential damages to obtain the accumulated damage. The following sections explain the computation of the parameters in the differential damage computation. 2.2.1. Damage index Damage index (DI) is used for predicting fatigue cracking and computed using Eq. (1).
DI =
n (1)
N
where n = number of load repetition derived from traffic data; N = vector of computed allowable number of repetitions using transfer functions at each transverse location; and DI = vector of computed damage index at each transverse location. The formula used for calculating the number repetition to failure is given in Eq. (2).
1
N = kf 1 CCH Bf 1
kf 2 f 2
t
1 E
kf 3 f 3
(2)
where t = vector of extracted tensile strains at the bottom of AC for each transverse location; E = dynamic modulus of the HMA layer, psi; kf 1, kf 2, kf 3 = global field calibration factors: 0.007566, 3.9492 and 1.281, respectively; f 1, f 2 , f 3 = local calibration factors that are set to 1.0 by default; C = 10 M ;
M = 4.84
(
Vbe Va + Vbe
)
0.69 ;
Vbe = effective asphalt content by volume; % CH = thickness correction term, depending on type of cracking; and Va = percent air voids in the HMA mixture. 2.2.2. Asphalt Concrete (AC) rutting The formula used for calculating the rutting within AC is given in Eq. (3).
= hsub
r1 kz
v 10
(3)
k31 nkr 2 r 2 T kr 3 r 3
where
hsub = thickness of a layer
= vector of predicted rutting at the mid-depth of each layer of AC at each transverse location; = vector of vertical compressive strain at the mid-depth of each layer at each critical transverse location; n = repetition number of the load; T = temperature at the mid-depth of each layer, F o ; k z = thickness correction factor; kr1, kr 2, kr 3 = global field calibration factors: −3.3512, 0.4791 and 1.5606, respectively; and r1, r 2, r 3 ; = local calibration factors that are set to 1.0 by default. v
2.3. Rutting in unbound layers The formula used for calculating the rutting within unbound materials is given in Eq. (4).
= hsub
r 1 k r1
0 r
e
(n)
v
(4)
where 87
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kr1 = global field calibration factors: 2.03 for granular materials and 1.35 for fine-grained materials; r1 = local calibration factor, set to 1 by default; = material properties; 0, , r = resilient strain; = 100.6119 0.017638Wc ;
( )= 0
) a + e 109 1
e(
2
r
= 109
(
C0
(109)
1
Wc = 51.712 C0 = ln
a9
;
1
);
0.3586 GWT 0.1192
1 E 0.64 2555
( )
;
( ); a1 a9
E = resilient modulus of a layer at mid-depth (psi); GWT = ground water table depth (ft); a1 = 0.15; and a9 = 20.
2.4. Curve fitting At this step, responses and damages that are computed at discrete points are converted to a continuous function by curve fitting using the least squares regression with a nonlinear basis function. The target variable for curve fitting differs depending on the type of damage parameter. For fatigue cracking, the curve is fitted to the final computed damage parameter (i.e., the vector of DI). For the rutting computation, the curve is fitted directly to the response vector. The formula for the curve fitting is as follows: p
f (x ) =
ie
( )
i wx h
(5)
i=0
where
f (x ) = fitted curve or function; p = degree of fitted function; x = continuous axis that represents the cross section of a wheel path; wh = wheel path width; = smoothing parameter; and i = regression coefficient that is computed using least square regression. 2.5. Deterministic and probabilistic shifting Two types of lateral shiftings are considered - deterministic and probabilistic. Deterministic shifting can be interpreted as CATs that they don’t exhibit any random lateral movement as they travel. Probabilistic shifting, on the other hand, can incorporate randomness on the lateral position of the vehicles (i.e., wheel wander) at any level. The mathematical formulation for the shifting can be defined as mapping x to x t s , where s is the random variable for wheel wandering and t is the deterministic shifting (Eq. (6)). n
f x, t, s =
ie
(
i xw t s h
)
2
(6)
i=0
In Eq. (6), f (x , t , s ) has become a random function because of s that follows a truncated normal distribution. To compute the resultant damage profile, the expectation of this function (i.e., E [f (x , t , s )]) should be calculated. For deterministic shifting, where s equals zero and t equals a real number, the calculation is straightforward; the t value is substituted in Eq. (6). 2.6. Damage accumulation 2.6.1. Rutting accumulation MEPDG uses a non-linear strain hardening approach to simulate the rutting accumulation after each discrete step. This approach starts with the computation of equivalent repetition, which is defined as the number of repetitions that would cause previously accumulated rutting using current computed vertical strain. Eq. (7) formulates this statement where the only unknown variable is neq . acc (x )
= (Ti ,
vi (x ),
(7)
neq (x ), C )
where 88
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= accumulated rutting profile until ith step considering probabilistic/deterministic shifting; ( x ) = fitted function to extracted compressive strains after applying shifting; vi C = all required constants for rutting computation; neq (x ) = profile of equivalent number of repetitions which amounts to the solution of Eq. (7); and acc (x )
(Ti ,
vi (x ),
neq (x ), C ) = predicted rutting profile using Eq. (3) or 4.
After solving Eq. (7) for neq (x ) , it is added to the number of repetitions of current strain. Next, the total number of repetitions along with the current strain are plugged into the rutting empirical functions to compute the accumulated rutting (Eq. (8)). i (x )
= (Ti ,
vi (x ),
(8)
neq (x ) + ni, C )
where = accumulated rutting profile including the ith step; and ni = number of repetitions at the ith step. i (x )
Finally, total rutting is computed by summing the accumulated rutting at each layer (Eq. (9)). M
RDi (x ) =
ji (x )
(9)
j=1
where M = number of layers; RDi (x ) = total accumulated rutting profile at the ith step. 2.6.2. Fatigue cracking Damage accumulation for fatigue cracking is simulated using Miner’s Law, which is described in Eq. (10). i
Di (x ) =
DIk (x )
(10)
k=1
where Di (x ) = accumulated fatigue damage profile at the ith step. Afterwards, using the transfer functions given in Eqs. (11) and (12), resultant bottom up and top-down fatigue cracking are computed.
FCibottom (x ) =
1 60
C4 1 + eC1 C2 + C2 C2 log (D (x )
(11)
100)
where
FCibottom (x ) = percent of alligator cracking profile that initiates at the bottom of the HMA layers at the ith step; C1, C2, C4 = calibration factors that equal to 1, 1, 6, respectively; C1 = −2C2 ; and C2 = 2.40874 39.748(1 + hac ) 2.856 .
FCitop (x ) = 10.56
1 + eC1
C4 C2log (D (x ) 100)
(12)
where
FCitop (x ) = length of longitudinal cracks profile that initiate at the top of the HMA layer (ft/mi) at the ith step; and C1, C2, C4 = calibration factors that equal to 7, 3.5, 1, respectively. 2.7. Chebyshev approximation to accumulated damage Accumulated damage equations become too complex to store and compute after a couple of accumulation steps because of the use of continuous functions. This creates the need to simplify equations using function-approximation techniques. The developed framework uses Chebyshev approximation, which globally approximates any bounded functions with a desired level accuracy (Eq. (13)). p
g (y )
hk Rk (y )
(13)
k =1
89
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where
g (y ) = function to be approximated, i.e., it is accumulated damage at ith step; Rk (y ) = chebyshev polynomials (Eq. (14)); and hk = chebyshev coefficients. (14)
Rk (y ) = cos (kcos 1 (y )) 2.8. Incorporation of resting period
Reduced between-vehicle distance effects were incorporated using the proposed adjustment factors. The formula of the factors is as follows:
AF =
D (RRT ) D (FRT )
(15)
where AF = adjustment factor; D (FRT ) = damage computed under field resting time (FRT). FRT can be computed by dividing the minimum safe distance between two consecutive trucks by the speed of the trucks; D (RRT ) = damage computed under reduced resting time (RRT). RRT can be computed by dividing the between-vehicle spacing in a platoon by the speed of the trucks. These factors may be developed using performance functions that explicitly take AC resting period into account while simulating damage accumulation within the pavement. The performance functions used to develop the adjustment factors are as follows: For rutting prediction (Motevalizadeh et al., 2018):
= aN b a = 0.6768T 1.4968S 0.2834L0.1788R0.5692 b = 2.2616 + 0.3256log (S ) + 0.033log (L)
0.1274log (R) + 1.1036log (T )
(16)
For fatigue cracking prediction (Beranek and Carpenter, 2009): (17)
Nf = 9500.6(R + 1) + 6146.7 where
Nf = number of allowable repetitions to fatigue failure; R = resting time in sec; L = loading time in sec; S = stress level; and T = temperature. The calculated damage from MEPDG’s equations were multiplied with the corresponding factor for each damage type (i.e., if it is rutting or fatigue cracking). It is important to note that the adjustment factors were only applied to modify the damage induced by the steering axle because platooning reduces the distance between rear axle of a leading truck and steering axle of a trailing truck. The distances between other axles within a truck will stay the same because the shape of the trucks do not change. 2.9. International Roughness Index (IRI) After computing accumulated damage, International Roughness Index (IRI) progression is simulated using the following equation: (18)
IRI (x ) = IRI0 + C1 RDA (x ) + C2 FCAtotal (x ) + C3 TC + C4 SF where
IRI0 = initial IRI after construction, in/mi; RDA (x ) = accumulated rut depth profile adjusted for reduced spacing; FCAtotal (x ) = accumulated fatigue cracking profile, summation of bottom-up and top-down cracking, adjusted for reduced spacing; TC = thermal cracking; SF = site factor; and C1, C2, C3, C4 = 40, 0.4, 0.008, 0.015, respectively.
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3. Optimization The objective of optimization is maximizing the pavement service life by manipulating the lateral position of truck platoons (Eq. (19)). The service life of a pavement is defined as the year when accumulated damage reaches its serviceability limit. When a pavement reaches its service life, it is rehabilitated to recover its functionality and structural capacity. The serviceability limit, also called a rehabilitation triggering criterion, may be determined in terms of roughness (i.e., IRI), rutting and/or cracking. The rehabilitation activity considered in this study is milling and placing AC overlay. In this study, the milled thickness of the pavement is assumed equivalent to that of the overlay. Son and Al-Qadi (2014) reported that after rehabilitation, 80 to 100% of the pavement condition could be recovered. To simplify the computation, full recovery after rehabilitation was assumed. (19)
maxns (Zi ) subject to
Lw
Vw 2
Lw
zi
Vw
(20)
2
where
z i = lateral position of ith platoon or group of platoons (optimization can be done for each platoon or a group of platoons over a specific period time such as day or week); Zi = {z i , z i 2 , z i 3…z1} ; L w = lane width; Vw = vehicle width; ns = service life of the pavement given in year (Eq. (21)). ns = min
Tp i 365
,
Tp j 365
,
Tp k (21)
365
where
i = {i : maxx IRI (x;Zi ) IRIL and maxx IRI (x;Zi 1) < IRIL} j = {j : maxx RDA (x;Zj ) RDL and maxx RDA (x;Zj 1) < RDL} k = {k: maxx FCAbottom (x ;Zk ) FCLbottom and maxx FCAbottom (x;Zk 1) < FCLbottom} IRI (x;Zi ) = accumulated IRI profile after ith platoon (or group of platoons) pass; RD (x;Zi ) = accumulated rutting profile after ith platoon (or group of platoons) pass; FCbottom (x;Zi) = accumulated bottom-up crack profile after ith platoon (or group of platoons) pass IRIL = serviceability limit for IRI; RDL = serviceability limit for rutting; FCLbottom = serviceability limit for fatigue cracking; TP = time duration (e.g., 1 day, 7 days) over which the platoons are grouped and their lateral positions are optimized. TP equals to Ps 365 for optimizing each platoon; AADTT AADTT = annual average daily truck traffic; and Ps = platoon size (e.g. 2 trucks). As seen from Eq. (19), the number of decision variables amounts to the number of grouped platoons, whose lateral positions are optimized. Over a 10-year pavement service life, assuming 5000 trucks per day, platoon sizes of 2 and 100% CAT penetration, the number of decision variables can increase up to 9 million (5000/2 ∗ 365 ∗ 10). This high number makes it almost impossible to solve the objective function; therefore, the authors introduce a greedy-search solution to this problem. 3.1. Greedy-search solution The first step of the greedy-search solution is to abandon jointly optimizing all lateral positions of the platoons, as given in Eq. (19). Instead, it places the platoons one by one (or group by group) while taking into account the previously accumulated damage profile. IRI was chosen as decision damage type for placing the platoons because it also includes rutting and cracking (Eq. (18)). Eq. (22) presents the updated objective function for the greedy search assuming that the previously accumulated IRI profile is computed and available. The lateral position of approaching a platoon (z i ) is optimized based on this profile. If either cracking or rutting reach their servicibility limits, the greedy-search is also terminated and service life is computed (Eq. (22))
min max (IRI (x;z i ) IRI (x ;Zi 1)) zi
(22)
x
The second step is discretizing the search space of lateral position (i.e., z i ). The discretization is demonstrated in Fig. 4. The lateral space is discretized into fourths allowing platoons four travel options for the sake of demonstration. Two platoons are illustrated with one already passed ([i 1]th platoon) and one approaching (ith platoon). The IRI profile accumulated by the (i 1)th platoon is 91
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Fig. 4. Demonstration of the greedy-search solution.
computed using the pavement design approach. The greedy-search optimization starts with placing the ith platoon on all four locations - one by one. Accumulated IRI profile for each location is computed, and the one minimizing the maximum of IRI profiles is chosen as the optimal ith platoon location. This procedure is repeated for all platoons or group of platoons over a specific period time such as day or week. Initial IRI (i.e., IRI0 ) profile is assumed to be uniform through the cross section of the pavement. 4. Life cycle cost analysis Minimizing the pavement damage by shifting the platoons results in increased service life as compared to channelized traffic with no lateral shifting. In this study life cycle cost analysis (LCCA) was performed to quantify the economic impact of the proposed control strategy on pavements. Pavement LCCA has two main parts: agency and user costs. Agency cost covers pavement rehabilitation and construction expenses for transportation agencies. In this study, the only agency costs considered are those associated with pavement service life. For agency cost calculations, an analysis period ( Ap ) of 45 years and a discount rate of 3% were used in accordance with the Illinois Department of Transportation’s (IDOT) LCCA guideline. In addition, threshold values for triggering rehabilitations are assumed to be 160 in/mi (2.5 m/km) of IRI and 0.5 in (12.5 mm) of rutting or 7.5% of bottom-up cracking. The thicknesses of milling and overlay (M&O) were assumed to be 2 in (50 mm). The cost of 2 in (50 mm) of M&O was computed as 9.81 × 104 $/mi-lane (6.13 × 104 $/km-lane) assuming a 12-ft (3.6 m) lane width. Fuel consumption cost because of pavement roughness was included; but truck operation and construction delay costs were not included. To compute the fuel consumption, it is necessary to have a function that converts pavement roughness (i.e., IRI) into energy consumption, which allows convertion to cost. In this study, the function developed by Ziyadi et al. (2018) was used because it meets this requirement (Eq. (23)).
E=
p + k IRI + d v
+ bv + kc IRI + dc v 2
(23)
where E = estimated energy consumption per mile (kj/ mi ); v = speed (mph); k , kc , dc , d b , and p = model coefficients which are given as 1.4, 1.36 × 10 4 , 2.39, 1.9225 × 10 4 , −2.6435 × 10 2 , 8.2782 × 10 4 , respectively, for large trucks. The energy consumption because of change in roughness is calculated as follows:
E = E (IRIt )
(24)
E (IRI0 )
where
E = Additional energy consumption due to pavement roughness; IRIt = IRI at time t; and IRI0 = Initial IRI which is assumed to be 60 in/mi. After computing the E, the additional fuel cost due to IRI can be computed as shown in Eq. (25).
FuC = E x DGR x OGD where 92
(25)
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Fig. 5. Cash flow of LCCA.
FuC = fuel cost; DGR = diesel gallon required per unit energy; and OGD = the cost of one-gallon diesel. Fig. 5 presents the overall cashflow for pavement LCCA. Here, n denotes the service life when the rehabilitation is triggered, and the blue lines represent the agency’s cost while the red lines show the user’s cost. Eq. (22) converts this cash flow to a net present value (NPV), which inputs the pavement service life and IRI and outputs pavement LCC in the present value. Ap n
NPV
n, IRIt
=
C i=1
1 (1 + r )n
Ap i
+
FuC (IRIt ) t=1
1 (1 + r )t (26)
where C = cost of a rehabilitation activity, 9.81 × 104 $/mi-lane (6.13 × 104 $/km-lane); and r = discount rate, 3%. 5. Case study Two pavement sections were considered - thin and thick sections. The thin section has 5-in (125 mm) AC layers over a 6-in (150 mm) base layer, while the thick section has 12-in (300 mm) AC layers (wearing surface, intermediate and binder layers) and a 12-in (300 mm) base layer. Elastic moduli for the base and subgrade are 60 ksi (414 MPa) and 10 ksi (69 MPa), respectively. The AC is modeled as linear viscoelastic and Prony coefficients were used for the AC layers. All viscoelastic parameters are presented in the appendix for each AC layer (i.e., wearing course, intermediate course and binder course). Platoons were assumed to be formed by two class 9 trucks based on the Federal Highway Administration vehicle classification scheme (Fig. 6). According to the weigh-in-motion data analyzed (Al-Qadi et al., 2017), class 9 trucks constitute 70 to 90% of the total truck traffic. The gross weight of the trucks was assumed to be 80 kips (35.7 ton), which is the US overweight limit. Daily truck traffic was assumed to be 5,000 for the thin pavement section and 13,000 for the thick pavement section. Inter-vehicle distances in a platoon was assumed to be 10 ft (3 m). The time interval over which the lateral position of the platoons are optimized was chosen to be 24 h (i.e., daily optimization of the platoons). The other assumed values are in Table 1. 6. Results and discussions The optimum number of hypothetical grids on the road surface (Fig. 4) is a key variable for the performance of the proposed control strategy. Hence, a sensitivity analysis was performed to determine the grids’ size. Service lives were computed for each grid size, which were used to determine the pavement LCC. The results are presented in Fig. 7. The grid size 1 refers to the channelized
Fig. 6. Simulated class 9 truck. 93
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Table 1 Assumed pavement design parameters. Variable Name
Value
Variable Name
Value
Initial IRI AC temperature Lane width Axle width Avg. ann. precip. Freezing Index
60 in/mi (0.95 m/km) 72° F (21° C) 12 ft (3.6 m) 8 ft 39.2 in (995.7 mm) 50
% plasticity index of soil % passing 0.075 mm sieve Percent air voids % passing 0.02 mm sieve Ground water table depth Effc. Asphalt Content
4 7% 4% 2 10 ft (3 m) 4.6%
Fig. 7. Gridline sensitivity analysis (1 mi = 1.6 km).
Fig. 8. Damage propagation with and without optimization (1 mi = 1.6 km and 1 in = 25.4 mm).
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Fig. 9. Pavement LCC cost decomposition (1 mi = 1.6 km).
traffic; no lateral shifting of platoons. The pavement LCC logarithmically decaying with respect to the number of hypothetical gridlines. Gridline 6 appears a reasonable choice. For both pavement sections, the proposed control strategy reduces the total pavement LCC by approximately 50% as compared to the channelized traffic. Fig. 8 presents the damage accumulation progress for two pavement sections under two conditions - optimized (opt) and channelized (chan) - for the optimum gridline number (i.e., gridline 3). In the figure’s legend, index A refers to the thick pavement section and index B refers to the thin pavement section. The accumulated damage is reduced because of the optimization; this suggests an increase in pavement service life. Lateral shifting affects rutting progression the most, which appears to be the governing failure criterion for both sections. On the other hand, the effect of lateral shifting on fatigue cracking is less significant than that on rutting. This is even less pronounced for thick pavement sections. It is interesting to note that IRI progression becomes more linear after some specific. This can be investigated to accelerate the optimization for the future studies. The cost decomposition of pavement LCC is presented in Fig. 9. The optimization of the platoons significantly reduces the agency cost for both pavements. By contrast, there is an increase on user costs which indicates the trade-off between the agency cost and user cost. Using hard threshold for triggering rehabilitation activities caused this trade-off. Bai et al. (2015) suggested a framework that optimizes the triggering values for rehabilitation. This framework may be combined with the proposed control strategy in the future. Fig. 10 demonstrates the sensitivity analysis of inter-vehicle spacing (i.e., resting time) on the optimization. The results were produced based on a channelized case. As seen from the plots, the longer resting period, the lengthier pavement service is. This increase in pavement service life manifests itself into as reduction in pavement life cycle cost (LCC). The sudden decreases in LCC are caused by reduction in number of rehabilitation cycles. For example, for thick pavement section, number of rehabilitation cycles are reduced to two from three when spacing increases to 20 ft (6 m). However, as inter-vehicle spacing in platoon increases, truck fuel efficiency is compromised. This trade-off between pavement damage and aerodynamic efficiency can be added to the objective function for optimizing the resting time. This will be addressed in a future study. As mentioned previously, the platoons are optimized daily in this paper. In other words, all the platoons traveling each day are assigned to an optimized lateral position which produces a thousand of solutions. Therefore, Fig. 11 demonstrates the optimization solution for the first 10 days of the first year (Fig. 11a) and fourth year (Fig. 11b). For the first ten days, the optimization results are
Fig. 10. Sensitivity analysis on spacing (1 ft = 0.3 m).
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Fig. 11. Optimized lateral positions (1 in = 25 mm).
irregular (i.e., not uniform). The strain-hardening approach used for rutting accumulation is the reason behind this irregularity. This approach captures the fact that rutting is accumulated at higher rates at the beginning because of AC is unaged and the binder is relatively softer. Therefore, trucks travelling at the early life of the pavements cause more rutting. This observation is also illustrated in Fig. 8b. As the time passes, the rutting accumulation slows down because of densification and stiffer binder. Hence, other distresses have become dominant in the IRI computation (Eq. (18)). Therefore, the optimization solutions become uniform (Fig. 11b). In summary, the distribution of the optimization solution depends on the time and pavement design criteria. For example, if AC mixes are design specifically for high rutting resistance, one may expect the solutions to become more uniform. The time complexity of the optimization algorithm is (nm ) , where n is the number of platoons (or group of platoons) whose position is to be optimized and m is the number of hypothetical grid lines on a lane. Each iteration (i.e., computing the pavement damage for a candidate position) takes approximately 0.11 s. Because the responses are computed offline (i.e., before the optimization algorithm starts), this time does not include pavement response computation under traffic and environmental loadings. Therefore, adding more pavement structures with different climatic conditions does not change the time complexity of the algorithm. 7. Summary and conclusions Truck platooning penetration is expected to increase with the advancement in CAT. The potential benefits of the truck platooning include but not limited to, regularizing traffic, reducing congestion, increasing highway safety, decreasing fuel consumption and emission. However, truck platoons may accelerate the pavement damage because of the developed of channelized load application and hindering the healing properties of AC. This study introduces a centralized control strategy that converts pavement challenges related to platooning into opportunities. In this control strategy, the lateral position of the platoons is optimized to decelerate pavement damage accumulation and consequently increase pavement service life. The proposed control strategy was applied on a case study. The results showed pavement LCC may be reduced by approximately 50%. A holistic evaluation of the proposed strategy may be needed. The study may include representative pavement sections and traffic and climatic data from actual pavement projects to use in the optimization framework. Acknowledgments This publication is based on the results conducted in cooperation with the University of Transportation Center (UTC), Illinois Center for Transportation of the University of Illinois at Urbana-Champaign. The authors would like to acknowledge the assistance provided by many individuals including Yanfeng Ouyang for his inspiring idea of discretiziation. The contents of this paper reflect the view of the authors, who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of University of Transportation Center or Illinois Center for Transportation. This paper does not constitute a standard, specification, or regulation. Appendix A. Wearing course in asphalt layer See Table 2. Instantaneous modulus: 4725.88 ksi. Williams-Landel-Ferry model parameters: T0 = 70 F, C1 = 25.33, C2 = 345.8.
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Table 2 Prony coefficients for the wearing course in asphalt layer. Prony Coefficient (ksi)
Relaxation Time (sec)
275.31 270.22 458.48 605.26 741.71 802.22 674.63 437.90 257.18 88.53 73.30
0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000
Appendix B. Intermediate course in asphalt layer See Table 3. Instantaneous modulus: 4769.27 ksi. Williams-Landel-Ferry model parameters: T0 = 70 F, C1 = 25.33, C2 = 345.8.
Table 3 Prony coefficients for the intermediate course in asphalt layer. Prony Coefficient (ksi)
Relaxation Time (sec)
475.31 412.46 652.58 742.81 791.66 697.78 485.96 263.62 140.15 44.32 40.02
0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000
Appendix C. Binder course in asphalt layer See Table 4. Instantaneous modulus: 4292.39 ksi. Williams-Landel-Ferry model parameters: T0 = 70 F, C1 = 18.5, C2 = 213.3.
Table 4 Prony coefficients for the binder course in asphalt layer. Prony Coefficient (ksi)
Relaxation Time (sec)
671.53 166.71 643.53 596.65 703.52 630.14 437.24 236.34 117.40 44.76 23.21
0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000
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References Al-Qadi, I. et al., 2017. Development of overweight vehicle permit fee structure in illinois. Illinois Center for Transportation/Illinois Department of Transportation. Al-Qadi, I., Wang, H., 2012. Impact of wide-base tires on pavements: results from instrumentation measurements and modeling analysis. Transp. Res. Rec. J. Transp. Res. Board 2304, 169–176. Al-Qadi, I.L., et al., 2018. Impact of wide-base tires on pavements: a national study. Transp. Res. Rec. 2672 (40), 186–196. Alam, A., et al., 2015. Heavy-duty vehicle platooning for sustainable freight transportation: a cooperative method to enhance safety and efficiency. IEEE Control Syst. Mag. 35 (6), 34–56. Bai, Y., et al., 2015. Optimal pavement design and rehabilitation planning using a mechanistic-empirical approach. EURO J. Transp. Logist. 4 (1), 57–73. Beranek, S., Carpenter, S.H., 2009. Fatigue Failure Testing in Section f. Center for Transportation, Illinois ICT-09-058. Bonnet, C., Fritz, H., 2000. Fuel consumption reduction in a platoon: Experimental results with two electronically coupled trucks at close spacing. SAE Technical Paper. Boysen, N., Briskorn, D., Schwerdfeger, S., 2018. The identical-path truck platooning problem. Transp. Res. Part B: Methodol. 109, 26–39. Browand, F., McArthur, J., Radovich, C., 2004. Fuel saving achieved in the field test of two tandem trucks. California PATH Program. Institute of Transportation Studies, University of California, Berkeley, CA. Calvert, S., Schakel, W., van Arem, B., 2019. Evaluation and modelling of the traffic flow effects of truck platooning. Transp. Res. Part C: Emerg. Technol. 105, 1–22. Castillo, D., Al-Qadi, I., 2018. Importance of heterogeneity in asphalt pavement modeling. J. Eng. Mech. 144 (8), 04018060. Chen, D., et al., 2017. Truck platooning on uphill grades under cooperative adaptive cruise control (cacc). Transp. Res. Procedia 23, 1059–1078. Chen, F., et al., 2019. Assess the impacts of different autonomous trucks’ lateral control modes on asphalt pavement performance. Transp. Res. Part C: Emerg. Technol. 103, 17–29. Eilers, S. et al., 2015. Companion–towards co-operative platoon management of heavy-duty vehicles. In: 2015 IEEE 18th International Conference on Intelligent Transportation Systems. IEEE, 1267–1273. Elseifi, M.A., Al-Qadi, I.L., Yoo, P.J., 2006. Viscoelastic modeling and field validation of flexible pavements. J. Eng. Mech. 132 (2), 172–178. Gungor, O.E., Al-Qadi, I.L., in preparation. Wander 2d: A flexible pavement design framework for autonomous and connected trucks. Gungor, O.E., et al., 2016a. In-situ validation of three-dimensional pavement finite element models. In: The Roles of Accelerated Pavement Testing in Pavement Sustainability. Springer, pp. 145–159. Gungor, O.E., et al., 2016b. Quantitative assessment of the effect of wide-base tires on pavement response by finite element analysis. Transp. Res. Rec. J. Transp. Res. Board 2590, 37–43. Gungor, O.E., et al., in preparation. Oplat: Optimization of lateral position of autonomous trucks in a platoon. Humphreys, H.L. et al., 2016. An evaluation of the fuel economy benefits of a driver assistive truck platooning prototype using simulation. SAE Technical Paper. Lammert, M.P. et al., 2014. Effect of platooning on fuel consumption of class 8 vehicles over a range of speeds, following distances, and mass. SAE Int. J. Commer. Veh. 7 (2014-01-2438), 626–639. Gungor, O.E., Al-Qadi, I.L., Gamez, A., Jaime, H., 2017. Development of adjustment factors for MEPDG pavement responses utilizing finite-element analysis. J. Transport. Eng., Part A: Syst. 143. https://doi.org/10.1061/JTEPBS.0000040. Larson, J., Munson, T., Sokolov, V., 2016. Coordinated platoon routing in a metropolitan network. In: 2016 Proceedings of the Seventh SIAM Workshop on Combinatorial Scientific Computing. SIAM, pp. 73–82. Lu, X.Y., Shladover, S.E., 2011. Automated truck platoon control. California PATH Program, Institute of Transportation Studies. University of California, Berkeley, CA UCB-ITS-PRR-2011-13. Luo, F., Larson, J., Munson, T., 2018. Coordinated platooning with multiple speeds. Transp. Res. Part C: Emerg. Technol. 90, 213–225. MEPDG, 2004. Guide for mechanistic-emprical design of new and rehabilitated pavement structures. National Cooperative Highway Research Program. Milanés, V., Shladover, S.E., 2014. Modeling cooperative and autonomous adaptive cruise control dynamic responses using experimental data. Transp. Res. Part C: Emerg. Technol. 48, 285–300. Motevalizadeh, S.M., et al., 2018. Investigating the impact of different loading patterns on the permanent deformation behaviour in hot mix asphalt. Constr. Build. Mater. 167, 707–715. Noorvand, H., Karnati, G., Underwood, B.S., 2017. Autonomous vehicles: assessment of the implications of truck positioning on flexible pavement performance and design. Transp. Res. Rec. J. Transp. Res. Board 2640, 21–28. Nowakowski, C., et al., 2015. Cooperative adaptive cruise control (cacc) for truck platooning: Operational concept alternatives. Ramezani, H., et al., 2018. Microsimulation Framework to Explore Impact of Truck Platooning on Traffic Operation and Energy Consumption: Development and Case Study. University of California, Berkeley, CA. Son, S., Al-Qadi, I.L., 2014. Engineering cost–benefit analysis of thin, durable asphalt overlays. Transp. Res. Rec. 2456 (1), 135–145. Sun, X., Yin, Y., 2019. Behaviorally stable vehicle platooning for energy savings. Transp. Res. Part C: Emerg. Technol. 99, 37–52. Tsugawa, S., 2014. Results and issues of an automated truck platoon within the energy its project. In: 2014 IEEE Intelligent Vehicles Symposium Proceedings. IEEE, pp. 642–647. Tsugawa, S., Jeschke, S., Shladover, S.E., 2016. A review of truck platooning projects for energy savings. IEEE Trans. Intell. Veh. 1 (1), 68–77. Tsugawa, S., Kato, S., Aoki, K., 2011. An automated truck platoon for energy saving. In: 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE, pp. 4109–4114. Wang, H., Al-Qadi, I.L., 2010. Impact quantification of wide-base tire loading on secondary road flexible pavements. J. Transp. Eng. 137 (9), 630–639. Wang, M., 2018. Infrastructure assisted adaptive driving to stabilise heterogeneous vehicle strings. Transp. Res. Part C: Emerg. Technol. 91, 276–295. Yoo, P.J., Al-Qadi, I.L., 2007. Effect of transient dynamic loading on flexible pavements. Transp. Res. Rec. 1990 (1), 129–140. Ziyadi, M., et al., 2018. Vehicle energy consumption and an environmental impact calculation model for the transportation infrastructure systems. J. Cleaner Prod. 174, 424–436.
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Contents lists available at ScienceDirect
Transportation Research Part C journal homepage: www.elsevier.com/locate/trc
All for one: Centralized optimization of truck platoons to improve roadway infrastructure sustainability
T
Osman Erman Gungor , Imad L. Al-Qadi ⁎
Dept. of Civil and Environmental Engineering, Univ. of Illinois at Urbana-Champaign, 205 N. Mathews Ave., Urbana, IL 61801, United States
ARTICLE INFO
ABSTRACT
Keywords: Autonomous truck platooning Optimization Pavement Life-cycle cost Infrastructure Autonomous and connected trucks
With the introduction of connected and autonomous trucks (CATs), truck platooning is expected to be more feasible and prevalent. The reported benefits of the truck platooning include regularizing traffic, reducing congestion, increasing highway safety, and decreasing fuel consumption and emission. Truck platooning may, however, decrease pavement longevity because it would cause channelized load application and hinder the healing properties of asphalt concrete. This study proposes a centralized control strategy that converts the pavement-related challenges of truck platooning into opportunities. This strategy leverages the auto-pilot technologies in CATs by optimizing the lateral position of each platoon or group of platoons. The efficiency of the proposed control strategy was demonstrated in a case study. Results showed that pavement lifecycle costs could be reduced up to 50% by controlling the lateral position of the platoons for each day.
1. Introduction Passenger cars and shuttles have primarily driven connected and automated vehicle (CAV) technology research, but recently there has been significant interest in developing connected and autonomous trucks (CATs). In fact, many major CAV technology manufacturers, such as Google-owned Waymo, have launched subsidiaries or departments for CAT development. CATs are expected to bring many advantages into freight industry such as improving operational efficiency of freight shipments and overcoming shortage in truck drivers. The introduction of CATs may result in drastic changes in freight shipment operation. One important change is the formation of truck platoons, a convoy of trucks traveling in close distance. With the advancements in intelligent technologies used in CATs that enable the connection among vehicles and between vehicles and infrastructure, truck platooning is set to become a feasible, efficient and prevalent practice in the future. Truck platooning has benefits and potential challenges. Reducing congestion and braking/ accelerating, and improving safety, traffic flow, and fuel efficiency are some of the expected benefits of platooning (Alam et al., 2015; Bonnet and Fritz, 2000; Nowakowski et al., 2015; Tsugawa et al., 2016; Browand et al., 2004; Lu and Shladover, 2011; Tsugawa et al., 2011; Tsugawa, 2014; Eilers et al., 2015; Lammert et al., 2014; Humphreys et al., 2016; Ramezani et al., 2018). However, platooning, if implemented without caution, may accelerate pavement damage accumulation. Platooning is expected to cause channelized truck loading application because the lateral positions of trucks in a platoon are expected to be similar as opposed to scattered lateral position of human-driven trucks. This channelized loading is expected to increase the damage accumulation rate and ultimately reduce the pavement’s service life. Noorvand et al. (2017), Chen et al. (2019)
⁎
Corresponding author. E-mail address: [email protected] (O.E. Gungor).
https://doi.org/10.1016/j.trc.2020.02.002 Received 17 August 2019; Received in revised form 29 January 2020; Accepted 1 February 2020 0968-090X/ © 2020 Elsevier Ltd. All rights reserved.
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Fig. 1. Optimization of lateral position of platoons using V2I communication (Gungor et al., in preparation).
acknowledged this challenge and quantified of the impact of trucks’ positionings within a lane. Additionally, the time between two consecutive truck loads will be shorter because of reduced inter-vehicle distance in platoons, which may hinder the self-healing characterization of asphalt concrete (AC) and consequently reduce pavement service life. This study proposes a centralized control strategy that converts the pavement-related challenges of truck platooning into opportunities. This strategy leverages the auto-pilot technologies in CATs by optimizing the lateral position of each platoon or group of platoons. Fig. 1a demonstrates a default scenario where the platoons are aligned with each other, which may lead to accelerated damage accumulation within pavements. Fig. 1b shows the proposed control strategy, where the platoons communicate with a center (either with a cloud or an infrastructure). In the proposed control strategy, the position of each platoon (or group of platoons) is adjusted to minimize the pavement damage and consequently increase the pavement service life. The de-centrailized control strategy, where the lateral position of each truck in a platoon was optimized considering the trade-off between the truck aero-dynamics and pavement damage, was introduced by the authors elsewhere (Gungor et al., in preparation). In the de-centralized optimization, while the lateral shift of trucks in a platoon increased the pavement service life, the fuel efficiency of trucks was comprised (Fig. 2). The centralized optimization, on the other hand, maximizes the fuel efficiency because there is no truck’s lateral shifting in a platoon. Application of a centralized strategy, however, may require significant investment because it assumes existence of reliable, centralized vehicle-to-infrastructure (V2I) communication as opposed to the de-centralized strategy’s vehicle-to-vehicle (V2V) communication requirement, which is already part of the platooning technology.
Fig. 2. Optimization of lateral position of platoons using V2I communication (Gungor et al., in preparation). 85
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It is important to note that the ideal implementation of platooning requires to study more variables in addition to infrastructure damage. For example, Sun and Yin (2019) investigated a behavioral stability of platooning which addresses the willingness of vehicles to join a platoon. Behavioral stability may especially be important for platoons formed by CATs owned by different companies. Also, this study developed a fair allocation mechanism to redistribute to platooning benefits to incentive vehicles to join a platoon. Calvert et al. (2019) studied the effects of truck platooning on traffic flow. The results did not show any positive impact of platooning on traffic flow. The study also concluded that the platoons of two or three trucks have a negligible impact on traffic efficiency. Furthermore, the scheduling of truck platoons to maximize the efficiency (Milanés and Shladover, 2014; Wang, 2018; Luo et al., 2018) and string stability for improving safety (Larson et al., 2016; Boysen et al., 2018; Chen et al., 2017) have been studied and should be considered for optimal control of platoons. 2. Pavement design and analysis To solve the optimization problem given in Fig. 1b, one needs to simulate the effects of any arbitrary positioning of platoons on pavement service life. In other words, the lateral position of each platoon should be an explicit input while simulating the damage accumulation within the pavement. Additionally, the impact of reduced resting time should be considered. To the best of the authors’ knowledge, no pavement design guidelines account for these two variables as explicit inputs. The authors have introduced a framework called Wander 2D that modifies mechanistic-empirical design approach (MEPDG, 2004) to consider the lateral position of trucks and between-vehicle distances. In this framework, pavement structural responses are calculated using ABAQUS. (i.e., mechanistic part). Later, the computed responses are linked to pavement damage through empirical transfer functions (i.e., empirical part) and other steps included in Wander 2D. The summary of Wander 2D is presented below. The details can be found elsewhere (Gungor and Al-Qadi, in preparation). 2.1. Pavement responses to loading In this step, the critical pavement responses are computed under a tire load at different transverse locations. This study uses 3D advanced pavement finite element (FE) models that Al-Qadi and his coworkers developed and have been continously improving over the past two decades (Elseifi et al., 2006; Yoo and Al-Qadi, 2007; Wang and Al-Qadi, 2010; Al-Qadi and Wang, 2012; Gungor et al., 2016a,b; Al-Qadi et al., 2018; Castillo and Al-Qadi, 2018). The developed models utilized a commercial software, ABAQUS. In the models, the asphalt concrete (AC) was considered as linear viscoelastic and simulated using Prony coefficients. For thick pavement sections (AC thickness is greater than five inches), the base layer was considered linear elastic. For thin pavement sections where the stresses on base layer are high, anisotropic-stress dependency was incorporated into the models. The tire-loading was simulated by measured 3D contact stresses which allowed capturing the effects of longitudinal and transverse contact forces on the pavement responses. Infinite elements were used at the boundaries of the model to capture the exponentially decaying behavior of pavement responses in a computationally efficient manner. Mesh sensitivity analysis was used to determine the mesh characteristics. While a refined mesh was used in the wheel path, a relatively coarser mesh was preferred between infinite elements and wheel path (i.e., transition zone) to reduce the computational time. Fig. 3 illustrates the model developed in ABAQUS. Fig. 3 illustrates the model developed in ABAQUS.
Fig. 3. 3D pavement finite-element model (Gungor et al., 2017). 86
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2.2. Differential damage computation The damage-prediction framework developed by MEPDG consists of two steps. The first step predicts damage from the strains, which transformed into damage, that occurs after N number of load repetitions. The outcome of this step is rutting at the mid-depth of each layer and fatigue damage index for fatigue cracking prediction. As loading input and climate conditions change (e.g., change in modulus because of temperate variation or axle loads), they alter the value of the computed strain which requires this step to be repeated. The second step involves combination of differential damages to obtain the accumulated damage. The following sections explain the computation of the parameters in the differential damage computation. 2.2.1. Damage index Damage index (DI) is used for predicting fatigue cracking and computed using Eq. (1).
DI =
n (1)
N
where n = number of load repetition derived from traffic data; N = vector of computed allowable number of repetitions using transfer functions at each transverse location; and DI = vector of computed damage index at each transverse location. The formula used for calculating the number repetition to failure is given in Eq. (2).
1
N = kf 1 CCH Bf 1
kf 2 f 2
t
1 E
kf 3 f 3
(2)
where t = vector of extracted tensile strains at the bottom of AC for each transverse location; E = dynamic modulus of the HMA layer, psi; kf 1, kf 2, kf 3 = global field calibration factors: 0.007566, 3.9492 and 1.281, respectively; f 1, f 2 , f 3 = local calibration factors that are set to 1.0 by default; C = 10 M ;
M = 4.84
(
Vbe Va + Vbe
)
0.69 ;
Vbe = effective asphalt content by volume; % CH = thickness correction term, depending on type of cracking; and Va = percent air voids in the HMA mixture. 2.2.2. Asphalt Concrete (AC) rutting The formula used for calculating the rutting within AC is given in Eq. (3).
= hsub
r1 kz
v 10
(3)
k31 nkr 2 r 2 T kr 3 r 3
where
hsub = thickness of a layer
= vector of predicted rutting at the mid-depth of each layer of AC at each transverse location; = vector of vertical compressive strain at the mid-depth of each layer at each critical transverse location; n = repetition number of the load; T = temperature at the mid-depth of each layer, F o ; k z = thickness correction factor; kr1, kr 2, kr 3 = global field calibration factors: −3.3512, 0.4791 and 1.5606, respectively; and r1, r 2, r 3 ; = local calibration factors that are set to 1.0 by default. v
2.3. Rutting in unbound layers The formula used for calculating the rutting within unbound materials is given in Eq. (4).
= hsub
r 1 k r1
0 r
e
(n)
v
(4)
where 87
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kr1 = global field calibration factors: 2.03 for granular materials and 1.35 for fine-grained materials; r1 = local calibration factor, set to 1 by default; = material properties; 0, , r = resilient strain; = 100.6119 0.017638Wc ;
( )= 0
) a + e 109 1
e(
2
r
= 109
(
C0
(109)
1
Wc = 51.712 C0 = ln
a9
;
1
);
0.3586 GWT 0.1192
1 E 0.64 2555
( )
;
( ); a1 a9
E = resilient modulus of a layer at mid-depth (psi); GWT = ground water table depth (ft); a1 = 0.15; and a9 = 20.
2.4. Curve fitting At this step, responses and damages that are computed at discrete points are converted to a continuous function by curve fitting using the least squares regression with a nonlinear basis function. The target variable for curve fitting differs depending on the type of damage parameter. For fatigue cracking, the curve is fitted to the final computed damage parameter (i.e., the vector of DI). For the rutting computation, the curve is fitted directly to the response vector. The formula for the curve fitting is as follows: p
f (x ) =
ie
( )
i wx h
(5)
i=0
where
f (x ) = fitted curve or function; p = degree of fitted function; x = continuous axis that represents the cross section of a wheel path; wh = wheel path width; = smoothing parameter; and i = regression coefficient that is computed using least square regression. 2.5. Deterministic and probabilistic shifting Two types of lateral shiftings are considered - deterministic and probabilistic. Deterministic shifting can be interpreted as CATs that they don’t exhibit any random lateral movement as they travel. Probabilistic shifting, on the other hand, can incorporate randomness on the lateral position of the vehicles (i.e., wheel wander) at any level. The mathematical formulation for the shifting can be defined as mapping x to x t s , where s is the random variable for wheel wandering and t is the deterministic shifting (Eq. (6)). n
f x, t, s =
ie
(
i xw t s h
)
2
(6)
i=0
In Eq. (6), f (x , t , s ) has become a random function because of s that follows a truncated normal distribution. To compute the resultant damage profile, the expectation of this function (i.e., E [f (x , t , s )]) should be calculated. For deterministic shifting, where s equals zero and t equals a real number, the calculation is straightforward; the t value is substituted in Eq. (6). 2.6. Damage accumulation 2.6.1. Rutting accumulation MEPDG uses a non-linear strain hardening approach to simulate the rutting accumulation after each discrete step. This approach starts with the computation of equivalent repetition, which is defined as the number of repetitions that would cause previously accumulated rutting using current computed vertical strain. Eq. (7) formulates this statement where the only unknown variable is neq . acc (x )
= (Ti ,
vi (x ),
(7)
neq (x ), C )
where 88
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= accumulated rutting profile until ith step considering probabilistic/deterministic shifting; ( x ) = fitted function to extracted compressive strains after applying shifting; vi C = all required constants for rutting computation; neq (x ) = profile of equivalent number of repetitions which amounts to the solution of Eq. (7); and acc (x )
(Ti ,
vi (x ),
neq (x ), C ) = predicted rutting profile using Eq. (3) or 4.
After solving Eq. (7) for neq (x ) , it is added to the number of repetitions of current strain. Next, the total number of repetitions along with the current strain are plugged into the rutting empirical functions to compute the accumulated rutting (Eq. (8)). i (x )
= (Ti ,
vi (x ),
(8)
neq (x ) + ni, C )
where = accumulated rutting profile including the ith step; and ni = number of repetitions at the ith step. i (x )
Finally, total rutting is computed by summing the accumulated rutting at each layer (Eq. (9)). M
RDi (x ) =
ji (x )
(9)
j=1
where M = number of layers; RDi (x ) = total accumulated rutting profile at the ith step. 2.6.2. Fatigue cracking Damage accumulation for fatigue cracking is simulated using Miner’s Law, which is described in Eq. (10). i
Di (x ) =
DIk (x )
(10)
k=1
where Di (x ) = accumulated fatigue damage profile at the ith step. Afterwards, using the transfer functions given in Eqs. (11) and (12), resultant bottom up and top-down fatigue cracking are computed.
FCibottom (x ) =
1 60
C4 1 + eC1 C2 + C2 C2 log (D (x )
(11)
100)
where
FCibottom (x ) = percent of alligator cracking profile that initiates at the bottom of the HMA layers at the ith step; C1, C2, C4 = calibration factors that equal to 1, 1, 6, respectively; C1 = −2C2 ; and C2 = 2.40874 39.748(1 + hac ) 2.856 .
FCitop (x ) = 10.56
1 + eC1
C4 C2log (D (x ) 100)
(12)
where
FCitop (x ) = length of longitudinal cracks profile that initiate at the top of the HMA layer (ft/mi) at the ith step; and C1, C2, C4 = calibration factors that equal to 7, 3.5, 1, respectively. 2.7. Chebyshev approximation to accumulated damage Accumulated damage equations become too complex to store and compute after a couple of accumulation steps because of the use of continuous functions. This creates the need to simplify equations using function-approximation techniques. The developed framework uses Chebyshev approximation, which globally approximates any bounded functions with a desired level accuracy (Eq. (13)). p
g (y )
hk Rk (y )
(13)
k =1
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where
g (y ) = function to be approximated, i.e., it is accumulated damage at ith step; Rk (y ) = chebyshev polynomials (Eq. (14)); and hk = chebyshev coefficients. (14)
Rk (y ) = cos (kcos 1 (y )) 2.8. Incorporation of resting period
Reduced between-vehicle distance effects were incorporated using the proposed adjustment factors. The formula of the factors is as follows:
AF =
D (RRT ) D (FRT )
(15)
where AF = adjustment factor; D (FRT ) = damage computed under field resting time (FRT). FRT can be computed by dividing the minimum safe distance between two consecutive trucks by the speed of the trucks; D (RRT ) = damage computed under reduced resting time (RRT). RRT can be computed by dividing the between-vehicle spacing in a platoon by the speed of the trucks. These factors may be developed using performance functions that explicitly take AC resting period into account while simulating damage accumulation within the pavement. The performance functions used to develop the adjustment factors are as follows: For rutting prediction (Motevalizadeh et al., 2018):
= aN b a = 0.6768T 1.4968S 0.2834L0.1788R0.5692 b = 2.2616 + 0.3256log (S ) + 0.033log (L)
0.1274log (R) + 1.1036log (T )
(16)
For fatigue cracking prediction (Beranek and Carpenter, 2009): (17)
Nf = 9500.6(R + 1) + 6146.7 where
Nf = number of allowable repetitions to fatigue failure; R = resting time in sec; L = loading time in sec; S = stress level; and T = temperature. The calculated damage from MEPDG’s equations were multiplied with the corresponding factor for each damage type (i.e., if it is rutting or fatigue cracking). It is important to note that the adjustment factors were only applied to modify the damage induced by the steering axle because platooning reduces the distance between rear axle of a leading truck and steering axle of a trailing truck. The distances between other axles within a truck will stay the same because the shape of the trucks do not change. 2.9. International Roughness Index (IRI) After computing accumulated damage, International Roughness Index (IRI) progression is simulated using the following equation: (18)
IRI (x ) = IRI0 + C1 RDA (x ) + C2 FCAtotal (x ) + C3 TC + C4 SF where
IRI0 = initial IRI after construction, in/mi; RDA (x ) = accumulated rut depth profile adjusted for reduced spacing; FCAtotal (x ) = accumulated fatigue cracking profile, summation of bottom-up and top-down cracking, adjusted for reduced spacing; TC = thermal cracking; SF = site factor; and C1, C2, C3, C4 = 40, 0.4, 0.008, 0.015, respectively.
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3. Optimization The objective of optimization is maximizing the pavement service life by manipulating the lateral position of truck platoons (Eq. (19)). The service life of a pavement is defined as the year when accumulated damage reaches its serviceability limit. When a pavement reaches its service life, it is rehabilitated to recover its functionality and structural capacity. The serviceability limit, also called a rehabilitation triggering criterion, may be determined in terms of roughness (i.e., IRI), rutting and/or cracking. The rehabilitation activity considered in this study is milling and placing AC overlay. In this study, the milled thickness of the pavement is assumed equivalent to that of the overlay. Son and Al-Qadi (2014) reported that after rehabilitation, 80 to 100% of the pavement condition could be recovered. To simplify the computation, full recovery after rehabilitation was assumed. (19)
maxns (Zi ) subject to
Lw
Vw 2
Lw
zi
Vw
(20)
2
where
z i = lateral position of ith platoon or group of platoons (optimization can be done for each platoon or a group of platoons over a specific period time such as day or week); Zi = {z i , z i 2 , z i 3…z1} ; L w = lane width; Vw = vehicle width; ns = service life of the pavement given in year (Eq. (21)). ns = min
Tp i 365
,
Tp j 365
,
Tp k (21)
365
where
i = {i : maxx IRI (x;Zi ) IRIL and maxx IRI (x;Zi 1) < IRIL} j = {j : maxx RDA (x;Zj ) RDL and maxx RDA (x;Zj 1) < RDL} k = {k: maxx FCAbottom (x ;Zk ) FCLbottom and maxx FCAbottom (x;Zk 1) < FCLbottom} IRI (x;Zi ) = accumulated IRI profile after ith platoon (or group of platoons) pass; RD (x;Zi ) = accumulated rutting profile after ith platoon (or group of platoons) pass; FCbottom (x;Zi) = accumulated bottom-up crack profile after ith platoon (or group of platoons) pass IRIL = serviceability limit for IRI; RDL = serviceability limit for rutting; FCLbottom = serviceability limit for fatigue cracking; TP = time duration (e.g., 1 day, 7 days) over which the platoons are grouped and their lateral positions are optimized. TP equals to Ps 365 for optimizing each platoon; AADTT AADTT = annual average daily truck traffic; and Ps = platoon size (e.g. 2 trucks). As seen from Eq. (19), the number of decision variables amounts to the number of grouped platoons, whose lateral positions are optimized. Over a 10-year pavement service life, assuming 5000 trucks per day, platoon sizes of 2 and 100% CAT penetration, the number of decision variables can increase up to 9 million (5000/2 ∗ 365 ∗ 10). This high number makes it almost impossible to solve the objective function; therefore, the authors introduce a greedy-search solution to this problem. 3.1. Greedy-search solution The first step of the greedy-search solution is to abandon jointly optimizing all lateral positions of the platoons, as given in Eq. (19). Instead, it places the platoons one by one (or group by group) while taking into account the previously accumulated damage profile. IRI was chosen as decision damage type for placing the platoons because it also includes rutting and cracking (Eq. (18)). Eq. (22) presents the updated objective function for the greedy search assuming that the previously accumulated IRI profile is computed and available. The lateral position of approaching a platoon (z i ) is optimized based on this profile. If either cracking or rutting reach their servicibility limits, the greedy-search is also terminated and service life is computed (Eq. (22))
min max (IRI (x;z i ) IRI (x ;Zi 1)) zi
(22)
x
The second step is discretizing the search space of lateral position (i.e., z i ). The discretization is demonstrated in Fig. 4. The lateral space is discretized into fourths allowing platoons four travel options for the sake of demonstration. Two platoons are illustrated with one already passed ([i 1]th platoon) and one approaching (ith platoon). The IRI profile accumulated by the (i 1)th platoon is 91
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Fig. 4. Demonstration of the greedy-search solution.
computed using the pavement design approach. The greedy-search optimization starts with placing the ith platoon on all four locations - one by one. Accumulated IRI profile for each location is computed, and the one minimizing the maximum of IRI profiles is chosen as the optimal ith platoon location. This procedure is repeated for all platoons or group of platoons over a specific period time such as day or week. Initial IRI (i.e., IRI0 ) profile is assumed to be uniform through the cross section of the pavement. 4. Life cycle cost analysis Minimizing the pavement damage by shifting the platoons results in increased service life as compared to channelized traffic with no lateral shifting. In this study life cycle cost analysis (LCCA) was performed to quantify the economic impact of the proposed control strategy on pavements. Pavement LCCA has two main parts: agency and user costs. Agency cost covers pavement rehabilitation and construction expenses for transportation agencies. In this study, the only agency costs considered are those associated with pavement service life. For agency cost calculations, an analysis period ( Ap ) of 45 years and a discount rate of 3% were used in accordance with the Illinois Department of Transportation’s (IDOT) LCCA guideline. In addition, threshold values for triggering rehabilitations are assumed to be 160 in/mi (2.5 m/km) of IRI and 0.5 in (12.5 mm) of rutting or 7.5% of bottom-up cracking. The thicknesses of milling and overlay (M&O) were assumed to be 2 in (50 mm). The cost of 2 in (50 mm) of M&O was computed as 9.81 × 104 $/mi-lane (6.13 × 104 $/km-lane) assuming a 12-ft (3.6 m) lane width. Fuel consumption cost because of pavement roughness was included; but truck operation and construction delay costs were not included. To compute the fuel consumption, it is necessary to have a function that converts pavement roughness (i.e., IRI) into energy consumption, which allows convertion to cost. In this study, the function developed by Ziyadi et al. (2018) was used because it meets this requirement (Eq. (23)).
E=
p + k IRI + d v
+ bv + kc IRI + dc v 2
(23)
where E = estimated energy consumption per mile (kj/ mi ); v = speed (mph); k , kc , dc , d b , and p = model coefficients which are given as 1.4, 1.36 × 10 4 , 2.39, 1.9225 × 10 4 , −2.6435 × 10 2 , 8.2782 × 10 4 , respectively, for large trucks. The energy consumption because of change in roughness is calculated as follows:
E = E (IRIt )
(24)
E (IRI0 )
where
E = Additional energy consumption due to pavement roughness; IRIt = IRI at time t; and IRI0 = Initial IRI which is assumed to be 60 in/mi. After computing the E, the additional fuel cost due to IRI can be computed as shown in Eq. (25).
FuC = E x DGR x OGD where 92
(25)
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Fig. 5. Cash flow of LCCA.
FuC = fuel cost; DGR = diesel gallon required per unit energy; and OGD = the cost of one-gallon diesel. Fig. 5 presents the overall cashflow for pavement LCCA. Here, n denotes the service life when the rehabilitation is triggered, and the blue lines represent the agency’s cost while the red lines show the user’s cost. Eq. (22) converts this cash flow to a net present value (NPV), which inputs the pavement service life and IRI and outputs pavement LCC in the present value. Ap n
NPV
n, IRIt
=
C i=1
1 (1 + r )n
Ap i
+
FuC (IRIt ) t=1
1 (1 + r )t (26)
where C = cost of a rehabilitation activity, 9.81 × 104 $/mi-lane (6.13 × 104 $/km-lane); and r = discount rate, 3%. 5. Case study Two pavement sections were considered - thin and thick sections. The thin section has 5-in (125 mm) AC layers over a 6-in (150 mm) base layer, while the thick section has 12-in (300 mm) AC layers (wearing surface, intermediate and binder layers) and a 12-in (300 mm) base layer. Elastic moduli for the base and subgrade are 60 ksi (414 MPa) and 10 ksi (69 MPa), respectively. The AC is modeled as linear viscoelastic and Prony coefficients were used for the AC layers. All viscoelastic parameters are presented in the appendix for each AC layer (i.e., wearing course, intermediate course and binder course). Platoons were assumed to be formed by two class 9 trucks based on the Federal Highway Administration vehicle classification scheme (Fig. 6). According to the weigh-in-motion data analyzed (Al-Qadi et al., 2017), class 9 trucks constitute 70 to 90% of the total truck traffic. The gross weight of the trucks was assumed to be 80 kips (35.7 ton), which is the US overweight limit. Daily truck traffic was assumed to be 5,000 for the thin pavement section and 13,000 for the thick pavement section. Inter-vehicle distances in a platoon was assumed to be 10 ft (3 m). The time interval over which the lateral position of the platoons are optimized was chosen to be 24 h (i.e., daily optimization of the platoons). The other assumed values are in Table 1. 6. Results and discussions The optimum number of hypothetical grids on the road surface (Fig. 4) is a key variable for the performance of the proposed control strategy. Hence, a sensitivity analysis was performed to determine the grids’ size. Service lives were computed for each grid size, which were used to determine the pavement LCC. The results are presented in Fig. 7. The grid size 1 refers to the channelized
Fig. 6. Simulated class 9 truck. 93
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Table 1 Assumed pavement design parameters. Variable Name
Value
Variable Name
Value
Initial IRI AC temperature Lane width Axle width Avg. ann. precip. Freezing Index
60 in/mi (0.95 m/km) 72° F (21° C) 12 ft (3.6 m) 8 ft 39.2 in (995.7 mm) 50
% plasticity index of soil % passing 0.075 mm sieve Percent air voids % passing 0.02 mm sieve Ground water table depth Effc. Asphalt Content
4 7% 4% 2 10 ft (3 m) 4.6%
Fig. 7. Gridline sensitivity analysis (1 mi = 1.6 km).
Fig. 8. Damage propagation with and without optimization (1 mi = 1.6 km and 1 in = 25.4 mm).
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Fig. 9. Pavement LCC cost decomposition (1 mi = 1.6 km).
traffic; no lateral shifting of platoons. The pavement LCC logarithmically decaying with respect to the number of hypothetical gridlines. Gridline 6 appears a reasonable choice. For both pavement sections, the proposed control strategy reduces the total pavement LCC by approximately 50% as compared to the channelized traffic. Fig. 8 presents the damage accumulation progress for two pavement sections under two conditions - optimized (opt) and channelized (chan) - for the optimum gridline number (i.e., gridline 3). In the figure’s legend, index A refers to the thick pavement section and index B refers to the thin pavement section. The accumulated damage is reduced because of the optimization; this suggests an increase in pavement service life. Lateral shifting affects rutting progression the most, which appears to be the governing failure criterion for both sections. On the other hand, the effect of lateral shifting on fatigue cracking is less significant than that on rutting. This is even less pronounced for thick pavement sections. It is interesting to note that IRI progression becomes more linear after some specific. This can be investigated to accelerate the optimization for the future studies. The cost decomposition of pavement LCC is presented in Fig. 9. The optimization of the platoons significantly reduces the agency cost for both pavements. By contrast, there is an increase on user costs which indicates the trade-off between the agency cost and user cost. Using hard threshold for triggering rehabilitation activities caused this trade-off. Bai et al. (2015) suggested a framework that optimizes the triggering values for rehabilitation. This framework may be combined with the proposed control strategy in the future. Fig. 10 demonstrates the sensitivity analysis of inter-vehicle spacing (i.e., resting time) on the optimization. The results were produced based on a channelized case. As seen from the plots, the longer resting period, the lengthier pavement service is. This increase in pavement service life manifests itself into as reduction in pavement life cycle cost (LCC). The sudden decreases in LCC are caused by reduction in number of rehabilitation cycles. For example, for thick pavement section, number of rehabilitation cycles are reduced to two from three when spacing increases to 20 ft (6 m). However, as inter-vehicle spacing in platoon increases, truck fuel efficiency is compromised. This trade-off between pavement damage and aerodynamic efficiency can be added to the objective function for optimizing the resting time. This will be addressed in a future study. As mentioned previously, the platoons are optimized daily in this paper. In other words, all the platoons traveling each day are assigned to an optimized lateral position which produces a thousand of solutions. Therefore, Fig. 11 demonstrates the optimization solution for the first 10 days of the first year (Fig. 11a) and fourth year (Fig. 11b). For the first ten days, the optimization results are
Fig. 10. Sensitivity analysis on spacing (1 ft = 0.3 m).
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Fig. 11. Optimized lateral positions (1 in = 25 mm).
irregular (i.e., not uniform). The strain-hardening approach used for rutting accumulation is the reason behind this irregularity. This approach captures the fact that rutting is accumulated at higher rates at the beginning because of AC is unaged and the binder is relatively softer. Therefore, trucks travelling at the early life of the pavements cause more rutting. This observation is also illustrated in Fig. 8b. As the time passes, the rutting accumulation slows down because of densification and stiffer binder. Hence, other distresses have become dominant in the IRI computation (Eq. (18)). Therefore, the optimization solutions become uniform (Fig. 11b). In summary, the distribution of the optimization solution depends on the time and pavement design criteria. For example, if AC mixes are design specifically for high rutting resistance, one may expect the solutions to become more uniform. The time complexity of the optimization algorithm is (nm ) , where n is the number of platoons (or group of platoons) whose position is to be optimized and m is the number of hypothetical grid lines on a lane. Each iteration (i.e., computing the pavement damage for a candidate position) takes approximately 0.11 s. Because the responses are computed offline (i.e., before the optimization algorithm starts), this time does not include pavement response computation under traffic and environmental loadings. Therefore, adding more pavement structures with different climatic conditions does not change the time complexity of the algorithm. 7. Summary and conclusions Truck platooning penetration is expected to increase with the advancement in CAT. The potential benefits of the truck platooning include but not limited to, regularizing traffic, reducing congestion, increasing highway safety, decreasing fuel consumption and emission. However, truck platoons may accelerate the pavement damage because of the developed of channelized load application and hindering the healing properties of AC. This study introduces a centralized control strategy that converts pavement challenges related to platooning into opportunities. In this control strategy, the lateral position of the platoons is optimized to decelerate pavement damage accumulation and consequently increase pavement service life. The proposed control strategy was applied on a case study. The results showed pavement LCC may be reduced by approximately 50%. A holistic evaluation of the proposed strategy may be needed. The study may include representative pavement sections and traffic and climatic data from actual pavement projects to use in the optimization framework. Acknowledgments This publication is based on the results conducted in cooperation with the University of Transportation Center (UTC), Illinois Center for Transportation of the University of Illinois at Urbana-Champaign. The authors would like to acknowledge the assistance provided by many individuals including Yanfeng Ouyang for his inspiring idea of discretiziation. The contents of this paper reflect the view of the authors, who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of University of Transportation Center or Illinois Center for Transportation. This paper does not constitute a standard, specification, or regulation. Appendix A. Wearing course in asphalt layer See Table 2. Instantaneous modulus: 4725.88 ksi. Williams-Landel-Ferry model parameters: T0 = 70 F, C1 = 25.33, C2 = 345.8.
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Table 2 Prony coefficients for the wearing course in asphalt layer. Prony Coefficient (ksi)
Relaxation Time (sec)
275.31 270.22 458.48 605.26 741.71 802.22 674.63 437.90 257.18 88.53 73.30
0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000
Appendix B. Intermediate course in asphalt layer See Table 3. Instantaneous modulus: 4769.27 ksi. Williams-Landel-Ferry model parameters: T0 = 70 F, C1 = 25.33, C2 = 345.8.
Table 3 Prony coefficients for the intermediate course in asphalt layer. Prony Coefficient (ksi)
Relaxation Time (sec)
475.31 412.46 652.58 742.81 791.66 697.78 485.96 263.62 140.15 44.32 40.02
0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000
Appendix C. Binder course in asphalt layer See Table 4. Instantaneous modulus: 4292.39 ksi. Williams-Landel-Ferry model parameters: T0 = 70 F, C1 = 18.5, C2 = 213.3.
Table 4 Prony coefficients for the binder course in asphalt layer. Prony Coefficient (ksi)
Relaxation Time (sec)
671.53 166.71 643.53 596.65 703.52 630.14 437.24 236.34 117.40 44.76 23.21
0.00001 0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000
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