Viscoelastic Response Solutions of Multilayered

Viscoelastic Response Solutions of Multilayered Asphalt Pavements

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

Yanqing Zhao1; Yuanbao Ni2; Lei Wang3; and Weiqiao Zeng4 Abstract: This study proposes a procedure for determining the viscoelastic responses of multilayered asphalt pavements. A computer code is developed to determine the elastic responses of pavements subjected to uniform circular pressures according to the layered elastic theory. The results from the code were successfully verified. The viscoelastic behavior of asphalt concrete is characterized using a complex modulus model. Relaxation of asphalt concrete is obtained by replacing the corresponding variables in the complex modulus model and substituting the elastic solution to obtain the step response function in the transformed domain. The time-domain response is solved through numerical transform inversion and is then expressed in the form of an exponential series with the relationship between the exponential series coefficients and the step responses at zero and infinite time being satisfied. The response as a result of an arbitrary loading history is solved using an incremental formulation of the convolution integral, taking advantage of the exponential series representation of the step response function. A discrete formulation of the linear superposition principle is used to solve the response under a moving load. The computer code developed for elastic analysis is extended to implement the viscoelastic analysis procedure and the results are in good agreement with those obtained from finite-element analyses. The procedure developed in this study provides an effective tool for pavement analysis and design. DOI: 10.1061/(ASCE)EM.19437889.0000797. © 2014 American Society of Civil Engineers. Author keywords: Asphalt pavement; Viscoelasticity; Multilayered system; Integral transform.

Introduction The accurate determination of structure responses plays a critical role in the design and performance evaluation of asphalt pavements. Currently, the layered elastic theory is widely used to determine the responses in pavement design. Burmister (1943) first derived the analytical solutions for a two-layered elastic system and subsequently extended them to a three-layered system (Burmister 1945a, b, c). Over the years, the theory has been extended to an arbitrary number of layers (Huang 2003). Asphalt concrete (AC) is a typical viscoelastic material exhibiting time- and rate-dependent behavior. The assumption made in the layered elastic theory that AC is linear elastic simplifies the analysis but may lead to inaccurate response results. The newly developed National Cooperative Highway Research Program (NCHRP) 1-37A mechanistic-empirical pavement design guide (MEPDG) uses a frequency-dependent dynamic modulus to characterize the mechanical properties of AC; however, the response is still computed based on elastic theory [Applied Research Associates (ARA) 2004]. The FEM has been successfully used to perform viscoelastic analysis (Elseifi et al. 2006; Al-Qadi and Wang 2009; Dubois et al. 2012). However, it 1

Associate Professor, School of Transportation Engineering, Dalian Univ. of Technology, Dalian 116024, China (corresponding author). E-mail: [email protected] 2 Research Assistant, School of Transportation Engineering, Dalian Univ. of Technology, Dalian 116024, China. E-mail: [email protected] 3 Lecturer, School of Mathematical Science, Dalian Univ. of Technology, Dalian 116024, China. E-mail: [email protected] 4 Research Assistant, School of Transportation Engineering, Dalian Univ. of Technology, Dalian 116024, China. E-mail: [email protected] Note. This manuscript was submitted on October 11, 2013; approved on March 18, 2014; published online on April 9, 2014. Discussion period open until September 9, 2014; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399/04014080(8)/$25.00. © ASCE

is time consuming and a full implementation of such an approach for routine design is inefficient. Hopman (1996) developed a computer code, VEROAD, to analyze the viscoelastic responses of multilayered asphalt pavements in which AC is characterized using the Burgers model. In VEROAD, all time-dependent parameters and equations are transformed to the Fourier domain and the solutions are then inversely transformed to the time domain. However, the Burgers model adopted in VEROAD may not be able to accurately characterize the properties of viscoelastic materials. The viscoelastic responses of pavements can be approximately determined using the collocation method (Huang 2003). However, the collocation scheme is problem dependent and it may not apply to general complicated cases (Chen et al. 2009). Chen et al. (2009) proposed a dual-parameter method for analyzing viscoelastic responses, which is called dual parameter because asphalt concrete is characterized using the Prony series expressions of both the relaxation modulus and creep compliance. By virtue of the Prony series of the dual parameters, Chen et al. (2009) derived a semianalytical viscoelastic solution in their study. One concern with using the Prony series of the relaxation modulus or creep compliance in viscoelastic analyses is that many Prony terms (usually more than 10) are necessary for accurate characterization of AC. Thus, inverse analyses, such as backcalculation of viscoelastic properties from falling-weight deflectometer (FWD) tests, are difficult. In this study, a procedure for determining the viscoelastic responses of multilayered asphalt pavements is developed. The viscoelastic step response functions are obtained from the elastic solutions by using the elastic-viscoelastic correspondence principle and numerical Laplace inversion. As a result of arbitrary loading histories and moving loads, the responses are then determined from the step response functions according to the Boltzmann superposition principle. The HavriliakNegami (HN) model is used in the procedure, which can accurately characterize the viscoelastic properties of AC and has fewer model coefficients than the Prony series. A computer code is developed to facilitate the viscoelastic analysis.

04014080-1

J. Eng. Mech.

J. Eng. Mech.

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

Solutions for Layered Elastic Systems Fig. 1 shows a typical multilayered pavement structure in cylindrical coordinates, where r and z are the cylindrical coordinates in the radial and vertical directions, respectively. Each layer is homogenous, isotropic, and horizontally infinite. The bottom layer is an infinite half-space. The elastic modulus and Poisson’s ratio of the ith layer are Ei and mi , respectively. The pavement is subjected to a uniform circular load with radius a and pressure magnitude q. The elastic solution to the problem in Fig. 1 can be analytically derived from the layered elastic theory. Details of the theory and solution can be found in the literature (Huang 2003). Some key equations and analysis steps are presented subsequently. The stress function in Eq. (1) satisfies the fourth-order differential equation governing the behavior of the axisymmetric problem in Fig. 1 fi ¼

H 3 J0 ðmrÞ h 2mðli 2lÞ Ai e 2 Bi e2mðl2li21 Þ m2 þ Ci mle2mðli 2lÞ 2 Di mle2mðl2li21 Þ

i (1)

where fi 5 stress function for the ith layer; H 5 distance from the surface to the upper boundary of the lowest layer; r 5 r=H; l 5 z=H; J0 5 Bessel function of the first kind and order 0; and m 5 parameter. By substituting the stress function into the governing equation, analytical solutions for the stresses and displacements are obtained. The solution for vertical stress is obtained as follows: n  p sz i ¼ 2mJ0 ðmrÞ ½Ai 2 Ci ð1 2 2mi 2 mlÞe2mðli 2lÞ o þ ½Bi þ Di ð1 2 2mi þ mlÞe2mðl2li21 Þ (2) where J1 5 Bessel function of the first kind and order 1. The superscript asterisk indicates that the stress or displacement is not the actual response as a result of the uniform circular load, q, but is the response as a result of vertical load 2mJ0 ðmrÞ, which was assumed by Burmister (1943) in deriving the solutions for layered elastic systems. The stresses and displacements as a result of load q are obtained as follows by using the Hankel transform: ð‘

p R ¼ qa R J1 ðmaÞdm m

(3)

0

The stress function for each layer has four constants of integration, Ai , Bi , Ci , and Di . Because the stresses and displacements must vanish as l approaches infinity, the constants An and Cn should be equal to zero. For an n-layer system, the total number of unknown constants is 4n 2 2, which must be evaluated by the boundary and continuity conditions. At the upper surface, the two boundary conditions, spz 5 2mJ0 ðmrÞ and tprz 5 0, result in two equations. The continuity condition assumes that the layers are fully bonded with the same vertical and shear stresses and vertical and radial displacements along the interface. These conditions result in four equations for each n 2 1 interface. Thus, the 4n 2 2 unknown constants can be solved from the 4n 2 2 equations. The constants obtained are then used in Eqs. (2) and (3) to compute the stresses and displacements. Once the stress components are available, the strain components can be determined using the elastic constitutive equations (Huang 2003). A computer code for asphalt pavement response analysis (APRA) was developed in this study to calculate the elastic responses according to the equations and procedures previously presented. The APRA code computes the elastic responses through the following steps: (1) set up the boundary and continuity equations for all interfaces (including the surface and the interface at infinity; (2) assemble all interface equations in one linear matrix equation; (3) solve the matrix equation to determine the constants of integration; and (4) evaluate the integral in Eq. (3) using the Gaussian formula with the integration subintervals being defined by the zeros of J0 ðrmÞ and J1 ðamÞ. The accuracy of APRA was evaluated by comparing it with BISAR (De Jong et al. 1979) and KENLAYER (Huang 2003). Thirty three-layer pavement structures were analyzed in this study. The thickness and elastic modulus selected for these structures covered their typical ranges as listed in Table 1. The Poisson’s ratios were 0.3, 0.4, 0.2, and 0.45 for AC, unbound aggregate base, chemically stabilized mixture base, and subgrade, respectively. The stresses, strains, and displacements in different directions and at various analysis points were calculated using the three computer codes. It is well known that the poor convergence of layered elastic solutions in the vicinity of the pavement surface may result in large errors in response results. For instance, JULEA, which is used in the MEPDG as the analysis engine, specifies a minimum analysis depth of onefifth of load radius, above which the results may be inaccurate. Therefore, in this study responses below one-fifth of the load radius from the pavement surface were analyzed for comparison. A total of 2,200 responses were computed by each program. The difference between the results obtained from various programs was normalized using the following equation:

p

where R 5 stress or displacement as a result of load 2mJ0 ðmrÞ; and R 5 stress or displacement as a result of load q.

ND ¼

jRA 2 RB j  100% RA

(4)

where ND 5 normalized difference; and RA and RB 5 responses obtained from programs A and B, respectively. Fig. 2 shows the results of the normalized differences, where the y-axis represents the percentage of analysis cases whose normalized differences are in the range shown on the x-axis. Fig. 2 shows that the normalized Table 1. Structure Parameter Ranges Parameter

Fig. 1. Multilayered pavement structure

© ASCE

Hot-mix asphalt thickness (cm) Base thickness (cm) Hot-mix asphalt modulus (MPa) Base modulus, unbound aggregate (MPa) Base modulus, chemically stabilized mixture (MPa) Subgrade modulus (MPa)

04014080-2

J. Eng. Mech.

Range 10–30 20–40 3,000–15,000 150–350 1,500–6,000 80–150

J. Eng. Mech.

function is a mathematical abstraction. Besides, at very short loading time inertial effects cause considerable complications. However, the impulse response function has theoretical importance. The theoretical relationship given by the following equation can be used to determine the relaxation of LVE materials (Tschoegl 1989):

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

QðsÞ ¼ E p ðivÞjiv5s

Fig. 2. Comparison of normalized differences

differences between APRA and BISAR, as well as those between APRA and KENLAYER, are similar to those between BISAR and KENLAYER. For most of the cases analyzed, the normalized difference was less than 1%. The maximum normalized difference was about 2.5%. The results indicate that APRA is accurate in determining the responses of layered elastic systems.

Method for Viscoelastic Solutions The derivation of the viscoelastic solution for multilayered systems is based on the elastic-viscoelastic correspondence principle. The set of Laplace-transformed governing equations of a viscoelastic boundary-value problem (BVP) constitutes an associated elastic problem (Haddad 1995). According to the correspondence principle, if an elastic solution to a BVP is known, substitution of the appropriate Laplace transforms of the quantities used in the elastic analysis furnishes the viscoelastic solution in the transform plane. The time-domain viscoelastic solution is then obtained by inverting the transform (Tschoegl 1989; Findley et al. 1976). The constitutive relationship of linear viscoelastic (LVE) materials expressed in the Laplace domain is given by the following equation (Tschoegl 1989): (5)

where sðsÞ and ɛðsÞ 5 Laplace transforms of stress and stain histories, respectively; s 5 Laplace variable; and QðsÞ 5 relaxation of the material. A bar over a symbol means that the quantity has been Laplace transformed. The relaxation is the Laplace transform of the impulse response function, which is the response of the material to the unit impulse strain input represented by a delta function. In the theory of a linear time-invariant system, QðsÞ also represents the transfer functions. When compared with the elastic constitutive relationship, Eq. (5) indicates that, according to the correspondence principle, the Young’s modulus used in the elastic analysis should be replaced by the relaxation of the material to obtain the viscoelastic solution in the Laplace domain. In practice, impulse excitations are rarely used to measure viscoelastic properties of a material. A delta © ASCE

where Effip ðivÞ 5 complex modulus; v 5 angular frequency; and pffiffiffiffiffiffi i 5 21. Eq. (6) indicates that the relaxation of LVE materials can be easily obtained by replacing iv in E p ðivÞ with the Laplace variable, s. Another material parameter used in the analysis of the elastic BVP is the Poisson’s ratio, which for AC is usually assumed to be time independent (Al-Qadi and Wang 2009; Elseifi et al. 2006). A constant Poisson’s ratio of 0.3 was used for asphalt concrete in this study. The time dependency of the Poisson’s ratio could easily be taken into account by replacing the Poisson’s ratio in the elastic solution with the Laplace or Carson transform of the time-dependent function, if available. To analyze the viscoelastic response of a pavement structure subjected to an arbitrary loading history or moving load, it is necessary to determine the step response function of the pavement. The step response function, RH ðtÞ, is the response of a pavement to a unit step loading history, obtained as follows:  1 MPa, pðtÞ ¼ 0 MPa,

t$0 t,0

 (7)

The Laplace transform of pðtÞ in Eq. (7) is 1=s. By substituting the relaxation and Laplace transform of the applied load into the elastic solution, the viscoelastic step response function in the Laplace domain is obtained. The solution in the time domain is then obtained through the inverse Laplace transform.

Solutions for Layered Viscoelastic Systems

sðsÞ ¼ QðsÞɛðsÞ

(6)

Material Characterization As discussed previously, a complex modulus model is needed to obtain the relaxation of AC for providing the Laplace transformed material properties as required by the correspondence principle. It has been shown that the HN model can accurately characterize the LVE behavior of AC (Zhao et al. 2013) as follows: E p ðivr Þ ¼ E0p þ 

E‘p 2 E0p

1 þ ðv0 =ivr Þa

b

(8)

where a and b 5 coefficients associated with relaxation mechanisms; v0 5 time-temperature shift, which controls the horizontal positions of the LVE master curves; E‘p and E0p 5 complex moduli as vr approaches ‘ and 0, respectively; and vr 5 reduced angular frequency, which accounts for the combined effects of temperature and loading frequency according to the time-temperature superposition principle (Ferry 1980). Here, vr is related to the cyclic frequency, f , as follows: vr ¼ 2pf aT

(9)

where aT 5 time-temperature shift factor, which is modeled as a function of temperature T using the following Williams-Landel-Ferry (WLF) equation:

04014080-3

J. Eng. Mech.

log aT ¼

2C1 ðT 2 T0 Þ C2 þ ðT 2 T0 Þ

(10)

J. Eng. Mech.

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

where C1 and C2 5 model coefficients; and T0 5 reference temperature. The HN model in Eq. (8) is essentially a model for the complex modulus master curve. The analytical forms of the storage and loss modulus master curves of the HN model can be derived from Eq. (8) and have been documented elsewhere (Zhao et al. 2013). Stone mastic asphalt (SMA) with a nominal maximum aggregate size of 12.5 mm was used in this study. For brevity, the mixture is designated as SMA 12.5 mixture. Complex modulus tests were performed on the mixture at temperatures of 4, 15, 25, 40, and 55°C and at frequencies of 25, 20, 10, 5, 1, 0.5, and 0.1 Hz following the guidelines specified in AASHTO TP 62-07 (AASHTO 2007). The complex modulus data set was previously used in evaluating the applicability of the HN model to AC. The details of the material, test results, and data analyses have been documented elsewhere (Zhao et al. 2013). The HN model coefficients, E‘ , E0 , a, b, and v0 , determined for the mixture at a reference temperature of 25°C were 26,317 and 51.1 MPa, and 0.266, 1.693, and 657.1, respectively. The WLF coefficients, C1 and C2 , were 21.7 and 183.5, respectively. The storage and loss modulus master curves of the HN model, as well as the test results before and after the time-temperature shift, are presented in Fig. 3, which indicates that the HN model can accurately characterize the LVE properties of AC contained in the entire complex modulus data set. Once the complex modulus model is known, the relaxation of AC is easily obtained from Eq. (6). The

number of coefficients in the HN model [only five coefficients as given by Eq. (8)] is considerably less than that in the Prony series expression of the relaxation modulus or creep compliance. Numerical Inversion of the Laplace Transform As discussed previously, the viscoelastic step response function of a multilayered system in the Laplace domain is obtained by substituting the appropriate Laplace transforms of the quantities into the elastic solution. However, it seems impossible to determine the time-domain viscoelastic solution through the analytical inversion transform. Numerical Laplace inversion has been extensively studied and various methods have been proposed (Cohen 2007). One of the most powerful and proven methods involves using the following so-called Gaver functionals (Gaver 1966; Valkó and Abate 2004):



k 2k P j k ð21Þ f ½ðk þ jÞt (11) fk ðtÞ ¼ kt j k j50 where t 5 lnð2Þ=t. Valkó and Abate (2004) tested various acceleration schemes for the Gaver functionals and found that the most effective one is the Wynn-Rho algorithm, which is given by the following recursive formulations: ðnÞ

r21 ¼ 0, ðnÞ

ðnÞ

r0 ¼ fn ðtÞ

ðnþ1Þ

rk ¼ rk22 þ

k ðnþ1Þ ðnÞ rk21 2 rk21

(12a) (12b)

The time-domain function is obtained as ð0Þ

f ðtÞ ¼ r2m

(13)

The Gaver-Wynn-Rho (GWR) algorithm given by Eqs. (11)–(13) was used in this study to numerically invert the Laplace transform. Verification of the Step Response Functions The computer code APRA was extended to implement the correspondence principle and GWR algorithm to solve the time-domain viscoelastic response. To verify the results of APRA, a typical threelayer asphalt pavement, with the parameters as given in Table 2, was analyzed using the FEM and APRA program, and the results obtained from the two approaches were compared. The base and subgrade layers were assumed to be linear elastic, while the asphalt layer was treated as LVE material. The SMA 12.5 mixture was used for the asphalt layer. The pavement was subjected to a circular unit step load given by Eq. (7) with a radius of 10 cm. The commercial finiteelement (FE) software ABAQUS 6.10 was used to model the pavement. The axisymmetric model developed had a dimension of 5 m along both the vertical and horizontal directions. The size was selected to minimize the edge effect errors and keep the elements’ sizes within acceptable limits (Thompson 1982). The model was meshed using four-node axisymmetric quadrilateral elements with a fine mesh Table 2. Pavement Structure in the Verification Study Layer

Fig. 3. Linear viscoelastic master curves of the HN model: (a) storage modulus; (b) loss modulus

© ASCE

Asphalt layer Base layer Subgrade

04014080-4

J. Eng. Mech.

Thickness (cm)

Modulus (MPa)

Poisson’s ratio

20 40 —

Viscoelastic 350 100

0.3 0.4 0.45

J. Eng. Mech.

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

around the loading area and a relatively course mesh farther from it. A roller support boundary condition was adopted for the right side to prevent horizontal movements. The bottom of the model was fixed in both horizontal and vertical directions. A comprehensive comparison was made between the viscoelastic responses obtained from APRA and FEM. The stresses, strains, and displacements in different directions and at various analysis points were compared. Figs. 4–6 show the vertical stress and strain at the middle of the AC layer, the horizontal stress and strain at the bottom of the AC layer, and the vertical stress and strain at the top of the subgrade, respectively. These responses are the critical responses for the permanent deformation of the asphalt concrete layer, fatigue cracking of the AC layer, and permanent deformation of the subgrade, respectively (ARA 2004). The results shown in Figs. 4–6, and the results not presented here for brevity, indicate that APRA can accurately analyze the viscoelastic step response function of multilayered asphalt pavements.

superposition principle can be used to analyze the response as a result of an arbitrary loading history, provided that the step response function is known. The Boltzmann superposition principle is in the form of convolution integral as follows: ðt ∂IðtÞ dt OðtÞ ¼ RH ðt 2 tÞ ∂t o

where IðtÞ 5 input loading history; RH ðtÞ 5 step response function; OðtÞ 5 output viscoelastic response; and t 5 integral variable. A direct numerical integration of Eq. (14) is profoundly inefficient because analyses need to be performed for all time steps that precede the time step of interest. The step response function can always be expressed as a Prony series (Huang 2003). The computation complexity of Eq. (14) can be greatly reduced by using the Prony series representation of RH ðtÞ. In this study, the following form of the Prony series is used to represent RH ðtÞ:

Responses to Arbitrary Loading Histories Engineers often need to analyze the responses of pavements under arbitrary load histories, such as the load induced in a FWD test. The load is applied to a fixed area on the pavement surface and the load magnitude varies with time. Because a pavement consisting of linear elastic and viscoelastic layers is a linear system, the Boltzmann

(14)

RH ðtÞ ¼ H0 þ

M P

Hk ½1 2 expð2t=rk Þ

(15)

k51

where H0 5 constant, which is the value of RH ðtÞ as t → 0; and Hk and rk 5 coefficients. Another constant, which is not given in Eq. (15), is H‘ , the value of RH ðtÞ as t → ‘. It is known from Eq. (15) that the constants, H0 and H‘ , and the Prony series coefficients, Hk , are related by H‘ 2 H0 ¼

M P

Hk

(16)

k51

Because a LVE body exhibits elastic behavior as time approaches zero or infinity, constants H0 and H‘ can be determined using the layered elastic theory, with the elastic modulus values of the asphalt layer being the relaxation moduli at zero and infinite time, respectively. The constants of the relaxation modulus, EðtÞ, and complex modulus, E p ðivÞ, are related as follows (Tschoegl 1989): EðtÞjt50 ¼ E p ðivÞjv5‘

Fig. 4. Vertical stresses and strains in the middle of the asphalt layer

EðtÞjt5‘ ¼ E p ðivÞjv50

(17)

Thus, H0 and H‘ were computed using the elastic analysis module in APRA with E‘p and E0p in Eq. (8) and the other parameters listed in Table 2 as input. For comparison, H0 and H‘ were also computed

Fig. 5. Horizontal stresses and strains at the bottom of the asphalt layer

© ASCE

and

04014080-5

J. Eng. Mech.

Fig. 6. Vertical stresses and strains at the top of the subgrade

J. Eng. Mech.

using the BISAR program. The same results were obtained from the two programs. Once H0 and H‘ are known, the Prony series coefficients are obtained by matching the computed RH ðtÞ values with Eq. (15) at discrete values of time, tj , using the collocation method (Tschoegl 1989; Park and Schapery 1999; Schapery 1961). In the collocation method, the coefficients, rk , in Eq. (15) are prespecified and related to the discrete time values, tj , as follows: rj ¼ ltj

(18)

stress at the bottom of the asphalt layer turns into compression at long loading time. Wang et al. (2006) reported similar phenomenon when investigating the viscoelastic responses of asphalt pavements using the displacement discontinuity method. They explained that the horizontal compressive stress was a result of the various rates of increase in the vertical displacement between the top and bottom of the asphalt layer. When the step response function is expressed as a Prony series, the convolution integral in Eq. (14) can be solved using the following incremental formulation (Chazal and Moutou Pitti 2011):

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

where l 5 proportionality constant. Thus, Eq. (15) is reduced to the following system of linear algebraic equations: fAg ¼ fBgfHg

(19)

  and Bjk ¼ 1 2 exp 2tj rk 

(21a)

~ nÞ DOðtn Þ ¼ PðDtn ÞDIðtn Þ þ Oðt

(21b)

PðDtn Þ ¼ H0 þ

where   fAg ¼ RH tj 2 H0

Oðtn Þ ¼ Oðtn21 Þ þ DOðtn Þ

  r Hk 1 2 k ½1 2 expð2Dtn =rk Þ Dtn k51 M P

(21c)

(20)

where fHg 5 vector of unknown coefficients, Hk . Traditionally Eq. (16) is disregarded in the analysis (Huang 2003). This may result in Prony series coefficients that are not in compliance with LVE theories. The constraint imposed by Eq. (16) is essentially another linear equation in which the mathematical form is consistent with Eq. (19). Thus, in this study the constraint was accounted for by adding a number, H‘ 2 H0 , to fAg and adding a row to fBg in which the elements are all 1. The unknown coefficients were then determined by solving the linear equation system. The Prony series coefficients obtained for the horizontal strain and stress at the bottom of the asphalt layer are listed in Table 3. The values of H‘ obtained from the layered elastic analyses for the stain and stress were 3:26 3 1024 and 20:137 MPa, respectively. Table 3 indicates that the Prony series coefficients obtained satisfy the requirement in Eq. (16). Fig. 7 shows the step response functions of strain and stress at the bottom of the asphalt layer obtained from APRA and the Prony series fit, where the solid circles represent the H0 and H‘ values. Fig. 7 indicates that the Prony series can accurately represent the step response functions. Analyses were performed for other responses and the same conclusion was made. It is seen in Fig. 7 that the horizontal tensile

M ~ n Þ ¼ P ½1 2 expð2Dtn =rk ÞFk ðtn Þ Oðt

(21d)

k51

Fk ðtn Þ ¼ Fk ðtn21 Þexpð2Dtn =rk Þ þ

DIðtn21 Þrk Hk ½1 2 expð2Dtn =rk Þ Dtn21

(21e)

where Fk 5 vector of internal state variables, which are updated at the end of each increment. The incremental formulation calculates the output at a particular time step from that at the previous time step. Thus, the computational time and computer storage requirements are greatly reduced. The responses were analyzed for the pavement structure listed in Table 2, which was subjected to a haversine loading history as follows:

 1 1 2pt p 2 (22) pðtÞ ¼ A þ sin 2 2 d 2 where d 5 load duration; and A 5 pressure amplitude. In this study, d and A were assumed to be 0.1 s and 0.7 MPa, respectively. The

Table 3. Prony Series Coefficients of the Step Response Functions Hk Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

© ASCE

rk

Strain 29

5 3 10 5 3 1028 5 3 1027 5 3 1026 5 3 1025 5 3 1024 5 3 1023 5 3 1022 5 3 1021 5 3 10 5 3 101 5 3 102 5 3 103 5 3 104 5 3 105 5 3 106 5 3 107 5 3 108

H0 Stress (MPa)

26

1:71 3 10 1:1 3 1026 2:48 3 1026 4:53 3 1026 8:53 3 1026 1:61 3 1025 3 3 1025 5:28 3 1025 7:28 3 1025 6:13 3 1025 1:4 3 1025 6:03 3 1026 1:25 3 1025 6:97 3 1026 2:65 3 1026 9:89 3 1027 3:18 3 1027 2:41 3 1027

22

21:78 3 10 21:10 3 1022 22:41 3 1022 24:12 3 1022 26:91 3 1022 21:08 3 1021 21:55 3 1021 22:00 3 1021 22:22 3 1021 21:84 3 1021 29:57 3 1022 23:56 3 1022 21:30 3 1022 24:64 3 1023 21:66 3 1023 25:86 3 1024 22:03 3 1024 21:37 3 1024

04014080-6

J. Eng. Mech.

Strain

Stress (MPa)

— — — — — — — — 2:82 3 1025 — — — — — — — — —

— — — — — — — — 1:05 3 10 — — — — — — — — —

J. Eng. Mech.

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

Fig. 7. Step response functions of stress and strain at the bottom of asphalt layer

Fig. 9. Horizontal strains at the bottom of the asphalt layer as a result of a moving load

applied to a circular area at each shift. The following equation for a moving load is obtained: RðtÞ ¼

N P

pi RH ðx0 2 di , y0 , z0 , t 2 ti Þ

(23)

i51

Fig. 8. Horizontal stress and strain at the bottom of asphalt layer under haversine loading

various responses obtained from APRA and FEM were in good agreement. The results of the horizontal stress and strain at the bottom of the asphalt layer are shown in Fig. 8. The computational time needed to evaluate the integral in Eq. (14) using the incremental formulation in Eqs. (21a)–(21e) was less than 1 s, while it took about 80 s to perform a direct integration of Eq. (14). To determine the viscoelastic responses shown in Fig. 8, the computational time for the proposed procedure was about 20 s, while that for the FEM was about 12 min. The procedure proposed in this study provides an efficient tool for evaluating the viscoelastic responses of asphalt pavements. Responses to Moving Loads The solution for the viscoelastic response to a moving load is also based on the Boltzmann superposition principle. By dividing the path of the moving load into small shifts, Khavassefat et al. (2012) derived a discrete formulation of the Boltzmann superposition principle for use in the FEM analysis. The formulation can be simplified for the problem at hand because the load area can be treated as one element. In addition, it is assumed that the load moves along a straight line (in the x-axis direction in this study) and a constant load pressure is © ASCE

where RðtÞ 5 viscoelastic response; N 5 number of shifts, where the first shift is at zero time; x0 , y0 , and z0 5 distances between the analysis point and the load center at zero time in the x-, y-, and z-directions, respectively; di 5 distance along which the load moves in the x-direction at the ith shift; pi 5 pressure magnitude applied at the ith shift; and ti 5 time at which the load moves to the ith shift. When the load moves from shift i to i 1 1, a pressure pi11 is applied at location i 1 1, and simultaneously a pressure pi is unloaded at location i, which is equivalent to applying a negative pi at location i. Therefore, the step response function needs to be evaluated at x0 2 di11 and x0 2 di for loading and unloading, respectively. When multiple wheels are applied, the responses as a result of individual wheel loads are superposed to obtain the total response. The pavement structure listed in Table 2 under dual wheels moving at a speed of 50 km=h was analyzed. The center-to-center distance of the dual wheels was 30 cm and the other wheel parameters were the same as previously used. In this study, x0 was 250 cm, and a constant shift length of 2 cm was used. Fig. 9 shows the horizontal longitudinal (traffic direction) and transverse strains at the bottom of the asphalt layer and under the center of one wheel. The response curves are not symmetric because of the viscoelastic effects. The wheels move to the top of the analysis point at 0.18 s, which is shown in Fig. 9 as the vertical line. It is seen from Fig. 9 that peak strains take place after the wheels leave the analysis point, which is typical viscoelastic behavior. The shapes of the strain curves are similar to those measured from instrumented pavement sections (Elseifi et al. 2006; Al-Qadi and Wang 2009).

Conclusions This study proposed a procedure for analyzing viscoelastic responses of multilayered asphalt pavements. The research started with developing a computer code, APRA, for determining the elastic responses of pavements according to the layered elastic theory. The results from APRA were successfully verified against those from BISAR and

04014080-7

J. Eng. Mech.

J. Eng. Mech.

Downloaded from ascelibrary.org by CURTIN UNIVERSITY on 07/22/14. Copyright ASCE. For personal use only; all rights reserved.

KENLAYER. The elastic-viscoelastic correspondence principle was used to solve the viscoelastic step response function of the pavement. The viscoelastic properties of AC were characterized using the HN model. The relaxation of AC was obtained by simply replacing the corresponding variables in the HN model and substituting the elastic solution to obtain the viscoelastic solution in the Laplace domain. The time-domain step response function was then solved by numerically inverting the Laplace transform using the GWR algorithm. The step response function was expressed in the form of a Prony series and the relationship between the Prony series coefficients and the step responses at zero and infinite time was taken into consideration, such that the results were consistent with linear viscoelastic theory. The response as a result of an arbitrary loading history was determined using an incremental formulation of the convolution integral, taking advantage of the Prony series expression of the step response function. A discrete formulation of the Boltzmann superposition principle was used to solve the responses under moving loads. The viscoelastic analysis procedure was implemented in APRA and the results were in very good agreement with those obtained from FE analyses. The procedure developed in this study provides an effective tool for pavement analysis and design. In addition, because the HN model for characterizing AC has fewer coefficients than the widely used Prony series representation of relaxation modulus or creep compliance, the proposed procedure offers the potential to backcalculate the viscoelastic properties of AC from field nondestructive tests, such as FWD tests.

Acknowledgments This research was sponsored by the Inner Mongolia Transportation Research Project (NJ-2009-11), the Liaoning Transportation Research Project (201309), and the Fundamental Research Funds for the Central Universities (DUT13LK). All support is gratefully acknowledged.

References AASHTO. (2007). “Standard method of test for determining dynamic modulus of hot-mix asphalt (HMA).” TP 62-07, Washington, DC. ABAQUS 6.1 [Computer software]. Providence, RI, Dassault Systèmes. Al-Qadi, I. L., and Wang, H. (2009). “Full-depth pavement responses under various tire configurations: Accelerated pavement testing and finite element modeling.” J. Assoc. Asphalt Paving Technol., 78, 721–760. Applied Research Associates (ARA). (2004). “Guide for mechanisticempirical design of new and rehabilitated pavement structures.” Final Rep. 1-37A, National Cooperative Highway Research Program, Albuquerque, NM. Burmister, D. M. (1943). “The theory of stress and displacements in layered systems and applications to the design of airport runways.” Proc., 23rd Highway Research Board Annual Meeting, Highway Research Board, Washington, DC, 126–144. Burmister, D. M. (1945a). “The general theory of stress and displacements in layered soil systems. I.” J. Appl. Phys., 16(2), 89–94.

© ASCE

Burmister, D. M. (1945b). “The general theory of stress and displacements in layered soil systems. II.” J. Appl. Phys., 16(3), 126–127. Burmister, D. M. (1945c). “The general theory of stress and displacements in layered soil systems. III.” J. Appl. Phys., 16(5), 296–302. Chazal, C., and Moutou Pitti, R. (2011). “Incremental constitutive formulation for time dependent materials: Creep integral approach.” Mech. Time-Depend. Mater., 15(3), 239–253. Chen, E. Y. G., Pan, E., and Green, R. (2009). “Surface loading of a multilayered viscoelastic pavement: Semianalytical solution.” J. Eng. Mech., 10.1061/(ASCE)0733-9399(2009)135:6(517), 517–528. Cohen, A. M. (2007). Numerical methods for Laplace transform inversion, Springer, New York. De Jong, D. L., Peutz, M. G. F., and Korswagen, A. R. (1979). “Computer program BISAR. Layered systems under normal and tangential surface load.” External Rep. No. AMSR. 0006.73, Koninklijke/Shell Laboratorium, Amsterdam, Netherlands. Dubois, F., Moutou Pitti, R., Picoux, B., and Petit, C. (2012). “Finite element model for crack growth process in concrete bituminous.” Adv. Eng. Software, 44(1), 35–43. Elseifi, M., Al-Qadi, I. L., and Yoo, P. J. (2006). “Viscoelastic modeling and field validation of flexible pavements.” J. Eng. Mech., 10.1061/(ASCE) 0733-9399(2006)132:2(172), 172–178. Ferry, J. D. (1980). Viscoelastic properties of polymers, 3rd Ed., Wiley, New York. Findley, W. N., Lai, J. S., and Onaran, K. (1976). Creep and relaxation of nonlinear viscoelastic materials, Dover, Mineola, NY. Gaver, D. P. (1966). “Observing stochastic processes and approximate transform inversion.” Oper. Res., 14(3), 444–459. Haddad, Y. M. (1995). Viscoelasticity of engineering materials, Chapman & Hall, New York. Hopman, P. C. (1996). “The visco-elastic multilayer program VEROAD.” Heron, 41(1), 71–91. Huang, Y. (2003). Pavement analysis and design, Prentice Hall, Englewood Cliffs, NJ. Khavassefat, P., Jelagin, D., and Birgisson, B. (2012). “A computational framework for viscoelastic analysis of flexible pavements under moving loads.” Mater. Struct., 45(11), 1655–1671. Park, S. W., and Schapery, R. A. (1999). “Methods of interconversion between linear viscoelastic material functions. Part I: A numerical method based on Prony series.” Int. J. Solids Struct., 36(11), 1653–1675. Schapery, R. A. (1961). “A simple collocation method for fitting viscoelastic models to experimental data.” Rep. GALCIT SM 61-32A, California Institute of Technology, Pasadena, CA. Thompson, M. R. (1982). ILLI-PAVE, user’s manual, Transportation Facilities Group, Dept. of Civil Engineering, Univ. of Illinois, Urbana, IL. Tschoegl, N. W. (1989). The phenomenological theory of linear viscoelastic behavior: An introduction, Springer, New York. Valkó, P. P., and Abate, J. (2004). “Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion.” Comput. Math. Appl., 48(3–4), 629–636. Wang, J., Birgisson, B., and Roque, R. (2006). “Effects of viscoelastic stress redistribution on the cracking performance of asphalt pavements.” J. Assoc. Asphalt Paving Technol., 75, 637–675. Zhao, Y., Liu, H., Bai, L., and Tan, Y. (2013). “Characterization of linear viscoelastic behavior of asphalt concrete using complex modulus model.” J. Mater. Civ. Eng., 10.1061/(ASCE)MT.1943-5533.0000688, 1543–1548.

04014080-8

J. Eng. Mech.

J. Eng. Mech.