Evaluation of the Response Analysis Approach Used in the Mechanistic-Empirical Pavement Design Guide

Abstract

Asphalt concrete (AC) is a typical viscoelastic material exhibiting rate-dependent behavior. The rate-dependency of AC should be properly taken into consideration in pavement response analysis to accurately evaluate pavement performance and life. In the Mechanistic-Empirical Pavement Design Guide (MEPDG), the dynamic modulus master curve is used to account for the rate-dependency of the dynamic modulus of AC. However, the rate-dependent phase angle is ignored and a constant phase angle of 0 is assumed. The partial characterization of rate-dependent properties of AC in the MEPDG may lead to inaccurate results. This study compares the pavement responses computed using the MEPDG approach and the layered viscoelastic theory (LVET) which utilizes the complex modulus master curve to fully characterize the rate-dependent properties of AC. Typical three-layer pavement structures were analyzed at three temperatures (−10°C, 20°C and 50°C) and four speeds (10, 40, 80 and 120 km/h). The results show that the horizontal tensile stresses at the bottom of cement-treated base layer obtained from the two approaches are almost the same, and for other responses analyzed, the results obtained from the MEPDG approach are larger than those from the LVET approach, especially for the responses in the AC layer. The normalized difference of the vertical compressive strain at the mid-depth of the AC layer between the two approaches can be up to 100% and that for the horizontal tensile strain at the bottom of the AC layer can be more than 50%.

The Mechanistic-Empirical Pavement Design Guide (MEPDG) represents a major advancement in pavement engineering. In the MEPDG, the pavement design procedure has been moved from traditional empirical methodology to mechanistic-empirical methodology ( 1 , 2 ). In a mechanistic-empirical pavement design, the critical pavement responses are computed using mechanistic procedures and then they are used to evaluate the pavement performance and life through empirical models. Therefore, accurate determination of pavement responses, such as stress and strain, plays a vital role in the mechanistic-empirical pavement design.

Currently, many programs have been developed to calculate the multilayer pavement responses and most of them are based on the layered elastic theory (LET); these include KENLAYER ( 3 ), BISAR ( 4 ), JULEA ( 5 ), WESLEA ( 6 ) and ERAPAVE ( 7 ). One of the basic assumptions of the LET is that the materials of all layers are linear elastic, and the elastic moduli or Young’s moduli are used to characterize the mechanistic properties of materials. Since asphalt concrete (AC) is a typical viscoelastic material exhibiting rate-dependent behavior, the viscoelastic properties of AC cannot be well characterized by the elastic modulus ( 8 – 13 ). However, because the LET is easy to implement and many computer programs have been developed based on the theory, it is still the most widely used method for the response analysis of asphalt pavements.

The MEPDG uses the LET, or more specifically the JULEA software, to analyze the pavement responses. Theoretically, the LET cannot account for the rate-dependent behavior of AC. An approximate approach is developed in the MEPDG to take into consideration the rate-dependent behavior of AC and LET is still adopted as the response analysis engine in this approach. The core of the approach is that the dynamic modulus master curve is employed to characterize the properties of AC. The approach first determines the load duration at a given depth in the pavement for a given vehicle speed. The load duration is then converted to frequency and the dynamic modulus is determined for the corresponding frequency according to the dynamic modulus master curve. Then the dynamic modulus is used as the elastic modulus in the LET to calculate the pavement responses. As explained later, the MEPDG approach can only partially incorporate the rate-dependent behavior of AC. This study is to investigate the accuracy of the MEPDG approach by comparing its analysis results to those obtained using the layered viscoelastic theory (LVET) which can fully account for the rate-dependent behavior of AC.

MEPDG Approach of Response Analysis and Discussion

The two most important factors that affect the engineering properties of AC are the temperature and the rate of loading. In the MEPDG, the temperature within the asphalt layer is determined using the enhanced integrated climatic model (EICM), and the loading frequency is determined from the stress pulse caused by the moving axle load passing a given point in the pavement. The loading frequency is computed using the following equation:

f = \frac{1}{t}

(1)

where f is the loading frequency; and t is the duration of stress pulse. The duration of stress pulse is related to the vehicle speed and the effective length of the stress pulse, as shown in

t = \frac{L_{eff}}{17.6 v_{s}}

(2)

where v_s is the vehicle speed (mph); and L_eff is the effective length of the stress pulse (inches).

The effective length, $L_{eff}$ , is the length that defines the extent of the stress pulse at the given depth within the pavement, which is schematically shown in Figure 1. Figure 1 presents a typical three-layer asphalt pavement subjected to a load. The sloped lines show the distribution area of the stress produced by the applied load. It is noted that the steepness of these stress distribution lines in different layers is different because the steepness of the stress distribution line is a function of material stiffness of the structural layer. A stiffer layer tends to distribute the stress over a wider area compared with a less stiff layer. However, at present the relationship between the layer stiffness and the slope of the stress distribution is still unclear. A simplified approach, namely, Odemark’s method, is adopted in MEPDG to overcome this problem. In Odemark’s method, the pavement layers above the subgrade are transformed into one layer that has the modulus of the subgrade and has an equivalent thickness. For simplicity, the steepness of the stress distribution lines in the transformed pavement is assumed to be 45°, and then the effective length can be obtained at any depth within the transformed pavement structure ( 2 ).

Figure 1.

Distribution and the effective length (L_eff) of the stress pulse.

In the MEPDG, an AC layer is divided into several sublayers and the loading frequency of each sublayer is determined according to the procedure presented above. Then the corresponding dynamic modulus of each sublayer can be determined based on the dynamic modulus master curve and time–temperature shift factor of that sublayer. These dynamic moduli of the sublayers are used in the LET to compute pavement responses.

The rate-dependent behavior of AC is considered in the MEPDG by incorporating the frequency-dependent dynamic modulus as outlined earlier. However, the MEPDG approach only partially simulates the rate-dependent or viscoelastic behavior of AC. It is well known that two quantities, namely, the dynamic modulus and phase angle, are obtained from the complex modulus test. It is noted that the phase angle is an equally important viscoelastic quantity and is also frequency-dependent. The complex modulus, as defined in Equation 3, is a fundamental property of viscoelastic materials and can fully characterize the rate-dependent behavior of the materials:

E^{*} = E' + iE ″

(3a)

E' = | E^{*} | \cos φ and E ″ = | E^{*} | \sin φ

(3b)

where

$i = \sqrt{- 1}$ is the unit imaginary number;

$E^{*}$ is the complex modulus;

$E'$ is the storage modulus and $E ″$ is the loss modulus;

$| E^{*} |$ is the dynamic modulus; and

$φ$ is the phase angle.

In Equation 3, $E^{*}$ , $| E^{*} |$ and $φ$ are all functions of frequency. It is noted that the dynamic modulus and phase angle functions are not independent of each other, while they need to satisfy the Kronig–Kramers (KK) relation as shown ( 13 ) in

φ (ω) = \frac{2 ω}{π} ℘ \int_{0}^{\infty} \frac{\ln | E^{*} (β) | - \ln | E^{*} (ω) |}{β^{2} - ω^{2}} d β

(4a)

\ln | E^{*} (ω) | = \ln Eg - \frac{2}{π} ℘ \int_{0}^{\infty} \frac{β φ (β) - ω φ (ω)}{β^{2} - ω^{2}} d β

(4b)

where $β$ is the integral variable; and $℘ \int$ is the Cauchy principal value of integral; Eg is the glassy modulus; $ω$ is the angular frequency. The KK relation is a basic requirement for linear viscoelastic material and it ensures that a physical model is causal, indicating that the system cannot respond to an excitation before the excitation is applied ( 13 , 14 ). Therefore, to accurately characterize the viscoelastic behavior of AC, the rate-dependency of both dynamic modulus and phase angle should be considered and they need to satisfy the KK relation at the same time.

Since the LET is employed in the MEPDG, the phase angle is assumed to be rate-independent, or more specifically, the phase angle is always equal to 0. Thus, AC is characterized in the MEPDG as a material with frequency-dependent dynamic modulus and a constant phase angle of 0, and such a material may not exist in the real world because the dynamic modulus and phase angle of the material do not satisfy the KK relation. Being approximate in nature, the MEPDG approach cannot fully characterize the rate-dependent behavior of AC. It is necessary to investigate and evaluate the differences of the critical pavement responses obtained from the MEPDG approach and the approach based on the LVET which is capable of fully taking the rate-dependent behavior of AC into consideration.

Response Analysis Approach Based on LVET

It has been concluded that the viscoelastic properties of AC can be accurately characterized using the modified Havriliak–Negami (MHN) model ( 15 – 17 ). In this study, the MHN model is adopted to represent the engineering property of AC in the LVET analysis. The MHN model has the following form ( 18 ):

E^{*} (ω_{r}) = E_{0} + \frac{E_{\infty} - E_{0}}{{[1 + {(\frac{ω_{0}}{i ω})}^{α}]}^{β}}

(5)

where

$E_{\infty}$ and $E_{0}$ are complex moduli as $ω$ approaches $\infty$ and 0, respectively;

$α$ and $β$ are coefficients associated with relaxation mechanisms;

$ω_{0}$ controls the horizontal positions of master curves; and

$ω_{r}$ is the reduced angular frequency which is defined by

ω_{r} = ω \times α_{T}

(6)

where $α_{T}$ is the time–temperature shift factor and $ω$ is the physical angular frequency. The Williams–Landel–Ferry (WLF) equation as shown in Equation 7 is used in this study to represent $α_{T}$ ( 18 ):

\log α_{T} = \frac{- C_{1} (T - T_{0})}{C_{2} + (T - T_{0})}

(7)

where $C_{1}$ and $C_{2}$ are model coefficients; and $T_{0}$ is the reference temperature. The MHN model in Equation 5 represents the complex modulus of AC. According to the theory of viscoelasticity, the complex modulus shown in Equation 5 can be expressed in relation to the storage modulus, $E' (ω)$ , and the loss modulus, $E ″ (ω)$ , as follows ( 15 ):

E^{'} (ω) = E_{0} + (E_{\infty} - E_{0}) \frac{{(\frac{ω_{r}}{ω_{0}})}^{α β / 2} \cos β θ (ω_{r})}{{[{(\frac{ω_{r}}{ω_{0}})}^{α} + 2 \cos (α π / 2) + {(\frac{ω_{0}}{ω_{r}})}^{α}]}^{β / 2}}

(8a)

E^{″} (ω) = (E_{\infty} - E_{0}) \frac{{(\frac{ω_{r}}{ω_{0}})}^{α β / 2} \sin β θ (ω_{r})}{{[{(\frac{ω_{r}}{ω_{0}})}^{α} + 2 \cos (α π / 2) + {(\frac{ω_{0}}{ω_{r}})}^{α}]}^{β / 2}}

(8b)

\tan θ (ω) = \frac{\sin (α π / 2)}{{(\frac{ω_{r}}{ω_{0}})}^{α} + \cos (α π / 2)}

(8c)

The formulations of the dynamic modulus, $| E^{*} |$ , and the phase angle, $φ$ , can then be computed using the following equations:

| E^{*} | = \sqrt{{E^{'}}^{2} + {E^{″}}^{2}} and \tan φ = E^{″} / E^{'}

(9)

Since the dynamic modulus and phase angle are theoretically obtained from the same complex modulus model in Equation 5, the KK relation between them is satisfied ( 13 , 15 ). Therefore, the MHN model can accurately characterize the rate-dependency of both the dynamic modulus and the phase angle, and this model is used in the LVET approach for AC characterization. It is noted that only the dynamic modulus master curve obtained from Equations (8) and (9) is used in the MEPDG approach.

The LVET analysis procedure suggested by Zhao et al. ( 19 ) was used in this study to determine the critical viscoelastic responses of asphalt pavements to moving vehicular load. The LVET analysis procedure is developed based on the elastic–viscoelastic correspondence principle. According to the correspondence principle, the viscoelastic solution in the Laplace domain is obtained by replacing the elastic modulus or Young’s modulus in the elastic solution with the relaxation of the material. It is noted that the relaxation of AC can be obtained from the complex modulus shown in Equation 5 by replacing $i ω$ with the Laplace variable, $s$ ( 13 ). Then the inverse Laplace transform is performed using the Gaver–Wynn–Rho (GWR) algorithm to determine the viscoelastic step function in the time domain ( 13 , 20 ). The path of moving load is then divided into many small shifts and the response from each shift is obtained from the step response function. Finally, according to the Boltzmann superposition principle, the viscoelastic response to the moving load can be determined by accumulating the response from each shift using the convolution integral. Details of the development and verification of the analysis procedure are documented elsewhere ( 19 ).

As mentioned previously, MEPDG treats AC as a material with frequency-dependent dynamic modulus and a constant phase angle of 0, and thus MEPDG only partially incorporates the rate-dependent behavior of AC. In addition, a simplified procedure based on the Odemark and 45° influence zone method is adopted in MEPDG to calculate the loading frequency. These simplistic assumptions in MEPDG can lead to inaccuracy of AC characterization. The MHN model can fully characterize the rate-dependency of AC and it is used in LVET to calculate the responses of pavements. The difference between the characterization methods for AC in MEPDG and LVET could lead to deviations in the pavement responses, and it is believed that the pavement responses obtained from LVET are more accurate because the rate-dependent behavior of AC is fully considered in LVET. This paper aims to investigate and evaluate the differences between the critical pavement responses obtained from the MEPDG and LVET approaches.

Comparisons of Responses Obtained from MEPDG and LVET Approaches

Two typical three-layer asphalt pavement structures, one with a granular base (GB) and the other with a cement-treated base (CTB), were analyzed in this study. The pavement parameters are listed in Table 1. A dense graded asphalt mixture with a nominal maximum aggregate size (NMAS) of 19.5 mm is used for the AC surface layer. According to the guidelines in AASHTO TP 62-07, complex modulus tests were conducted to determine the complex moduli of the mixture at the temperatures of −10°C, 0°C, 15°C, 30°C, 40°C and 50°C, and at the frequencies of 20, 10, 5, 1, 0.5 and 0.1 Hz ( 21 ). The MHN model coefficients at a reference temperature of 15°C were determined by fitting the analytical form to complex modulus test results and the coefficients are presented in Table 2. Then the dynamic modulus and phase angle master curves can be constructed, as presented in Figure 2. The non-asphaltic layers are treated as elastic layers.

Table 1.

Pavement Structure Parameters

Layer	Thickness (cm)	Modulus (MPa)	Poisson’s ratio
AC surface	20	Viscoelastic	0.3
Base	35	300 (GB)	0.35 (GB)
Base	35	7,000 (CTB)	0.25 (CTB)
Subgrade	na	50	0.4

Note: AC = asphalt concrete; CTB = cement-treated base; GB = granular base; na = not applicable; subgrade is infinite in thickness.

Table 2.

Modified Havriliak–Negami Coefficients of Asphalt Concrete

$E_{\infty}$ (MPa)	$E_{0}$ (MPa)	$α$	$β$	$ω_{0}$	$C_{1}$	$C_{2}$
53,446.4098	66.1563	0.2474	2.1598	21.2525	20.5	148.1

Figure 2.

Dynamic modulus and phase angle master curves (asphalt concrete surface).

Pavement responses were computed in this study at three temperatures (−10°C, 20°C and 50°C) and four speeds (10, 40, 80 and 120 km/h) using MEPDG and LVET approaches. The moving vehicle load is represented by a moving circular load with a constant pressure of 0.7 MPa in this study. For the MEPDG approach, the AC layer was divided into seven sublayers and the dynamic moduli at various temperatures and speeds were determined for each sublayer following the procedure mentioned earlier; the elastic responses were then computed using the LET software BISAR ( 4 ). It is noted that it is not necessary to divide the AC layer into sublayers for the LVET approach, because the effects of stress distribution with depth have been accounted for by the governing equations of the theory.

The critical pavement responses related to the major pavement distress types were analyzed in this study, including the vertical strains at the mid-depth of the AC and GB layers and on top of the subgrade for the permanent deformation of the layers, and the horizontal strain at the bottom of the AC layer and the horizontal stress at the bottom of the CTB layer for the fatigue cracking of the layers ( 2 , 22 ).

Results and Discussion for the Pavement with GB

Figure 3 presents the vertical strain time–history curves at the mid-depth of the AC layer obtained at a temperature of 20°C and a speed of 80 km/h. In this study, a positive value represents tension response and a negative value represents compression response. The time when the load moves to the top of the analysis point is shown in the figure as the vertical line. The strain curve obtained from the MEPDG approach is symmetric and the strain reaches the maximum value when the load is on top of the analysis point. However, the strain curve obtained from the LVET approach is not symmetric and the maximum strain happens after the load passes the analysis point, which are the typical viscoelastic behaviors. In addition, the peak strain value from the MEPDG approach is significantly larger than that from the LVET approach.

Figure 3.

Comparison of vertical strains at the mid-depth of the asphalt concrete (AC) surface layer (pavement with GB, 80 km/h, 20°C).

The peak compressive strains at the mid-depth of the AC layer obtained at various temperatures and speeds are listed in Table 3 and plotted in Figure 4. The normalized differences (NDs) between the MEPDG and LVET results were computed using Equation 10, and the ND results are also listed in Table 3:

N D = \frac{R_{M E} - R_{V E}}{R_{V E}} \times 100 %

(10)

where $ND$ is the normalized difference; and $R_{ME}$ and $R_{VE}$ are the peak responses obtained from the MEPDG and LVET approaches, respectively. Analysis results for other response types are also presented in Table 3. It is observed in Figure 4 that peak compressive strains at the mid-depth of the AC layer obtained from both the MEPDG and LVET approaches increase significantly as the temperature increases. For example, at a speed of 80 km/h, the peak strain from the LVET approach increases from 7.1 to 434.6 με as the temperature increases from −10°C to 50°C, and the peak strain increases from 7.9 to 793.4 με for the MEPDG approach. The effect of speed on the peak strain is limited at low temperature (−10°C). However, at medium and high temperatures (20°C and 50°C), the peak strain increases considerably as the speed decreases. For example, at 50°C, the peak strain at 10 km/h is about 3.1 times that at 120 km/h for the LVET approach, and about 2.7 times for the MEPDG approach. This is expected because a speed reduction, or, in other words, a loading frequency reduction is equivalent to a temperature increase according to the time–temperature superposition principle.

Table 3.

Peak Responses and Normalized Differences (Pavement with Granular Base)

Type of responses	Speed (km/h)	LVET			MEPDG			Normalized difference (%)
		Temperature (°C)			Temperature (°C)			Temperature (°C)
		–10	20	50	–10	20	50	–10	20	50
Vertical strains at mid-depth of AC surface layer (με)	10	−7.6	−46.2	−1,122.6	−9	−92.5	−1,831	18.3	100.3	63.1
	40	−7.3	−30.2	−596.2	−8.2	−58.2	−1,058.4	13.2	92.6	77.5
	80	−7.1	−25.1	−434.6	−7.9	−47	−793.4	11.3	87.4	82.6
	120	−7	−22.6	−357.9	−7.7	−41.7	−669	10.3	84.6	86.9
Vertical strains at mid-depth of GB base (με)	10	−26	−82.9	−210.8	−26.9	−93.3	−221.5	3.6	12.6	5.1
	40	−24.7	−66.8	−186.6	−25.3	−75.5	−199.6	2.7	13.1	7
	80	−24.2	−60.3	−174.3	−24.8	−68	−187.5	2.4	12.7	7.6
	120	−23.9	−57	−166	−24.4	−64	−180.1	2.1	12.3	8.5
Vertical strains at top of subgrade (με)	10	−56.3	−153.2	−328.7	−57.2	−157.6	−331.8	1.6	2.9	0.9
	40	−53.7	−128.6	−292.6	−54.4	−133.5	−297.1	1.3	3.8	1.5
	80	−52.7	−118.1	−275.1	−53.5	−122.9	−279.2	1.5	4.1	1.5
	120	−52.2	−112.6	−264.6	−52.7	−117.1	−268.8	1	4	1.6
Horizontal strains at bottom of AC surface layer (με)	10	14.4	54.3	185.4	16.2	86.5	200.3	12.6	59.2	8.1
	40	13.7	41.1	171.4	15	64.9	197.1	9.2	57.9	15
	80	13.4	36.3	154.9	14.5	56.2	191.3	7.8	54.9	23.5
	120	13.3	33.8	143.9	14.3	51.8	186.5	7.1	53.2	29.6

Note: AC = asphalt concrete; GB = granular base; LVET = layered viscoelastic theory; MEPDG = Mechanistic-Empirical Pavement Design Guide.

Figure 4.

Peak compressive strains at the mid-depth of the asphalt concrete layer (pavement with granular base).

It is seen from Table 3 and Figure 4 that the peak compressive strains at the mid-depth of the AC layer obtained from the MEPDG approach are significantly greater than those from the LVET approach, especially at medium and high temperatures. At −10°C, the peak strains from the MEPDG approach are 10% ∼ 18% larger than those from the LVET approach, while at 20°C and 50°C, the MEPDG results are 60% ∼ 100% larger than the LVET results. This observation indicates that although the MEPDG approach tries to incorporate the rate-dependent behavior of AC, its analysis results may differ considerably from those obtained from a true viscoelastic analysis.

Figures 5 and 6 present the peak compressive strains at the mid-depth of GB layer and on top of the subgrade, respectively. It is seen that the effects of temperature and speed on the peak strains are similar to those presented in Figure 4 for the AC layer. For all the temperatures and speeds analyzed, the peak strains obtained from the MEPDG approach are larger than those from the LVET approach. However, the NDs between the two sets of results are not as significant as those for the AC layer. For the peak strains on top of the subgrade, the results from the MEPDG and LVET are close and the differences are less than 5%. For the peak strains at the mid-depth of the GB layer, the differences are less than 10% at −10°C and 50°C and less than 15% at 20°C. However, for the peak compressive strain at the mid-depth of the AC layer, the MEPDG results can be 100% larger, as discussed previously.

Figure 5.

Peak compressive strains at the mid-depth of the granular base (GB) layer (pavement with GB).

Figure 6.

Peak compressive strains on top of the subgrade (pavement with granular base).

Figure 7 presents the peak horizontal tensile strains at the bottom of the AC layer obtained at various temperatures and speeds. A preliminary analysis shows that the horizontal tensile strains at the bottom of the AC layer in the longitudinal direction are more critical than those in the transverse direction. Thus, only the strains in the longitudinal direction are shown in the figure. Similar to the vertical strain results of various layers, the peak horizontal strains at the bottom of the AC layer obtained from the MEPDG approach for various temperatures and speeds are larger than the corresponding LVET results. At −10°C and 50°C, the NDs of peak strains obtained from the two approaches are less than 30%. However, at 20°C, the MEPDG results are more than 50% larger than the LVET results. It is well recognized that the medium temperature is the critical temperature for fatigue cracking of AC. These results raise concerns about the accuracy of the MEPDG approach in distress predictions.

Figure 7.

Peak horizontal tensile strains at the bottom of the asphalt concrete layer (pavement with granular base).

Results and Discussion for the Pavement with CTB

For the pavement with CTB, the major distresses are the permanent deformation of the AC layer and the fatigue cracking of the CTB layer ( 2 ). Accordingly, the responses analyzed in this study are the vertical strain at the mid-depth of the AC layer and the tensile stress at the bottom of the CTB layer.

The peak compressive strains at the mid-depth of the AC layer and peak tensile stresses at the bottom of the CTB layer obtained from both the MEPDG and LVET approaches are listed in Table 4. Figure 8 presents the peak compressive strains at the mid-depth of the AC layer obtained at various temperatures and speeds. It is seen that the trends in Figure 8 are similar to the results presented in Figure 4 for the pavement with GB. The peak strain values obtained from the MEPDG approach are considerably larger than those from the LVET approach, especially at medium and high temperatures. At 20°C and 50°C, the MEPDG results are 55% ∼ 100% larger than the LVET results.

Table 4.

Peak Responses and Normalized Differences (Pavement with Cement-Treated Base)

Type of responses	Speed (km/h)	LVET			MEPDG			Normalized difference (%)
		Temperature (°C)			Temperature (°C)			Temperature (°C)
		–10	20	50	–10	20	50	–10	20	50
Vertical strains at mid-depth of AC surface layer (με)	10	−7.8	−53.4	−1,263.1	−9.2	−105.5	−1982.6	18.4	97.7	57
	40	−7.4	−33.6	−698.2	−8.3	−65	−1,173.6	13.2	93.2	68.1
	80	−7.2	−27.4	−516.1	−8	−51.9	−890.5	11.3	88.9	72.6
	120	−7.1	−24.5	−428.6	−7.9	−45.7	−756	10.3	86.4	76.4
Horizontal stresses at bottom of CTB layer (MPa)	10	0.075	0.122	0.183	0.075	0.122	0.182	0.5	0	−1
	40	0.073	0.109	0.179	0.073	0.11	0.176	0.4	0.9	−1.7
	80	0.072	0.104	0.175	0.073	0.105	0.172	0.3	1	−2
	120	0.072	0.101	0.173	0.072	0.102	0.169	0.3	1.1	−2.2

Note: AC = asphalt concrete; CTB = cement-treated base; LVET = layered viscoelastic theory; MEPDG = Mechanistic-Empirical Pavement Design Guide.

Figure 8.

Peak compressive strains at the mid-depth of the asphalt concrete layer (pavement with cement-treated base).

Figure 9 presents the peak tensile stresses at the bottom of the CTB layer for various temperatures and speeds. It is seen that the peak stress increases as the temperature increases or the speed decreases. However, the effect of speed on the peak stress at the bottom of the CTB layer is much less than that on the vertical strain at the mid-depth of the AC layer. When the speed decreases from 120 to 10 km/h, the peak stress increases by less than 7% at −10°C and 50°C and by about 20% at 20°C. At the same time, the peak vertical strain at the mid-depth of the AC layer can increase by more than 190% for the same speed reduction. It is also observed that the peak stresses obtained from the MEPDG approach for various temperatures and speeds are almost the same as those obtained from the LVET approach, and the differences between the two sets of results are less than 2.2%.

Figure 9.

Peak horizontal tensile stresses at the bottom of the cement-treated base (CTB) layer (pavement with CTB).

As discussed previously, the LVET approach can realistically simulate the pavement behavior and can provide more accurate pavement responses than MEPDG. Thus, the LVET approach is a more effective tool for the response analysis of asphalt pavements. For example, LVET can be used to analyze the critical stress or strain for investigating the mechanism of pavement distress, and it could provide more accurate results than MEPDG. In addition, in the future the critical response obtained from LVET should be used in pavement design and performance evaluation. However, this is a long process because the pavement design and performance evaluation method should be a coherent system. Adopting the response results from LVET in the pavement design and performance evaluation method requires recalibrating all performance models for field pavement structure sections. This is beyond the scope of this study.

Conclusion

Accurate determination of pavement responses plays a vital role in the performance evaluation and structural design of asphalt pavements. To this end, it is important to take into consideration the rate-dependent or viscoelastic properties of AC in the response analysis. The MEPDG uses the dynamic modulus master curve to account for the rate-dependency of the dynamic modulus of AC, but the rate-dependency of the phase angle is ignored and a constant phase angle of 0 is assumed. Therefore, the MEPDG only partially characterize the rate-dependent properties of AC and may lead to inaccurate results. This study compares the pavement responses computed using the MEPDG approach and the LVET which can take into consideration the full rate-dependent behavior of AC. Two typical three-layer pavement structures were analyzed at three temperatures (–10°C, 20°C and 50°C) and four speeds (10, 40, 80 and 120 km/h). The results show that the peak horizontal tensile stresses at the bottom of the CTB layer obtained from the two approaches are almost the same. For other responses analyzed, including the vertical compressive strains at the mid-depth of the AC and GB layers and on top of the subgrade, and the horizontal tensile strains at the bottom of the AC layer, the peak responses from the MEPDG approach are larger than those from the LVET approach, especially for the responses in the AC layer. The ND of the peak vertical strain at the mid-depth of the AC layer obtained from the two approaches can be as high as 100%, and that of the peak horizontal strain at the bottom of the AC layer can be more than 50%. The differences of peak responses in the non-asphaltic layers are not as significant as those in the AC layer, and they are less than 15%.

Footnotes

Acknowledgements

The authors gratefully acknowledge their financial support.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: Guozhi Fu, Yanqing Zhao; analysis and interpretation of results: Guozhi Fu, Yanqing Zhao, Wanqiu Liu, Changjun Zhou; draft manuscript preparation: Guozhi Fu, Yanqing Zhao, Changjun Zhou. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was sponsored by National Natural Science Foundation of China (51678114), Urumqi Transportation Research Project (JSKJ201806), Inner Mongolia Transportation Research Project (NJ-2014-21, NJ-2015-36), and Shanxi Transportation Research Project (2015-1-22, 2017-1-18).

References

American Association of State Highway and Transportation Officials (AASHTO). Guide for Design of Pavement Structures. AASHTO, Washington, D.C., 1993.

Applied Research Associates (ARA). Guide for Mechanistic-Empirical Design of New and Rehabilitated Pavement Structures. National Cooperative Highway Research Program (NCHRP) Project 1-37A, Final Report. Washington, D.C., 2004.

Huang

Pavement Analysis and Design. Prentice Hall, Englewood Cliffs, NJ, 2003.

De Jong

D. L.

Peutz

M. G. F.

Korswagen

A. R.

Computer Program BISAR. Layered Systems Under Normal and Tangential Surface Load. External Rep. No. AMSR. 0006.73, Koninklijke/Shell Laboratorium, Amsterdam, The Netherlands, 1979.

Uzan

Advanced Backcalculation Techniques. Nondestructive Testing of Pavements and Backcalculation of Moduli. ASTM Special Technical Publications, Philadelphia, PA, 1994, 1198: 3–37.

Van Cauwelaert

F. J.

Alexander

D. R.

White

T. D.

Barker

W. R.

Multilayer Elastic Program for Backcalculating Layer Moduli in Pavement Evaluation. ASTM Special Technical Publications, Philadelphia, PA, 1989, 1026: 171–188.

Erlingsson

Ahmed

A. W.

Fast Layered Elastic Response Program for the Analysis of Flexible Pavement Structures. Road Materials and Pavement Design, Vol. 14, No. 1, 2003, pp. 196–210.

Mohammad

L. N.

Myers

Cooper

Abadie

A Practical Look at Simple Performance Tests: Louisiana’s Experience. Journal of the Association of Asphalt Paving Technologists, Vol. 74, 2005, pp. 557–600.

Pellinen

T. K.

Witczak

M. W.

Stress Dependent Master Curve Construction for Dynamic (Complex) Modulus. Journal of the Association of Asphalt Paving Technologists, Vol. 71, 2002, pp. 281–309.

10.

Bonaquist

Christensen

D. W.

Practical Procedure for Developing Dynamic Modulus Master Curves for Pavement Structural Design. Transportation Research Record: Journal of the Transportation Research Board, 2005, 1929: 208–217.

11.

Rowe

G. M.

Baumgardner

G. L.

Evaluation of the Rheological Properties and Master Curve Development for Bituminous Binders Used in Roofing. Journal of ASTM International, Vol. 4, No. 9, 2007, JAI101016.

12.

Mun

Modeling the Viscoelastic Function of Asphalt Concrete using a Spectrum Method. Mechanics of Time-Dependent Materials, Vol. 14, No. 2, 2010, pp. 191–202.

13.

Tschoegl

N. W.

The Phenomenological Theory of Linear Viscoelastic Behavior: An Introduction. Springer-Verlag, NY, 1989.

14.

John

Causality and the Dispersion Relation: Logical Foundations. Physical Review, Vol. 104, 1956, pp. 1760–1770.

15.

Zhao

Liu

Characterization of Linear Viscoelastic Behavior of Asphalt Concrete using a Complex Modulus Model. Journal of Materials in Civil Engineering, Vol. 25, No. 10, 2013, pp. 1543–1548.

16.

Sun

Huang

Chen

A Unified Procedure for Rapidly Determining Asphalt Concrete Discrete Relaxation and Retardation Spectra. Construction and Building Materials, Vol. 93, 2015, pp. 35–48.

17.

Gudmarsson

Ryden

Benedetto

H. D.

Sauzéat

Tapsoba

Birgisson

Comparing Linear Viscoelastic Properties of Asphalt Concrete Measured by Laboratory Seismic and Tension–Compression Tests. Journal of Nondestructive Evaluation, Vol. 33, No. 4, 2014, pp. 571–582.

18.

Ferry

J. D.

Viscoelastic Properties of Polymers, 3rd ed. John Wiley & Sons, NY, 1980.

19.

Zhao

Wang

Zeng

Viscoelastic Response Solutions of Multilayered Asphalt Pavements. Journal of Engineering Mechanics, Vol. 140, No. 10, 2014, p. 04014080.

20.

Findley

W. N.

Lai

J. S.

Onaran

Creep and Relaxation of Nonlinear Viscoelastic Materials. Dover, Mineola, NY, 1976.

21.

American Association of State Highway and Transportation Officials (AASHTO). Standard Method of Test for Determining Dynamic Modulus of Hot-Mix Asphalt (HMA). AASHTO TP 62-07. Washington, D.C., 2007.

22.

Cao

Zhao

Liu

Ouyang

Comparisons of Asphalt Pavement Response Computed using Layer Properties Backcalculated from Dynamic and Static Approaches. Road Materials and Pavement Design, Vol. 20, No. 5, 2019, pp. 1114–1130.