Effect of Transient Dynamic Loading on Flexible Pavements

129

A study by Monismith et al. showed that performing a completedynamic analysis for flexible pavement was not necessary (2). Theeffect of mass inertia force can be ignored, and the local dynamicresponse can be determined by a static method using material prop-erties compatible with the rate of loading. Recently, however, theeffect of vehicle dynamics on pavement design has been of moreimmediate interest. As size and payload of a truck increase, investi-gating pavement response to dynamic wheel loads from various tireconfigurations may be needed.

Cebon suggested that the dynamic component of wheel loads mayreduce the service life of flexible pavements (3). Fatigue damage ofthe hot-mix asphalt (HMA) may increase by as much as a factor offour under dynamic wheel loads. Rutting damage from dynamic wheelloads may reduce the service life of pavement by at least 40%. Evenfor a smooth pavement, dynamic loads may result in significantincrease (10% to 15%) in pavement responses. In addition, variationsof dynamic responses under moving loads were presented by Wuand Shen (4), who observed that dynamic displacement is about18% higher than static displacement.

To evaluate dynamic pavement strains under moving traffic loads,Zafir et al. considered a continuum-based finite layer approach (5).The authors concluded that computed strains showed that dynamiceffects of moving loads are important and should not be ignored.

Dynamic load generally can be considered from a viewpoint ofvehicle dynamics or pavements. To a vehicle dynamic specialist,wheel loads are usually described as static, quasi-static, or dynamic.A static load may be explained by the geometry and mass of the loaditself. The quasi-static load would include the stiffness and masswithout damping force in the equation of equilibrium. The dynamicload would involve inertia, damping, stiffness, and mass terms in theequation of motion (6, 7 ).

To a pavement specialist, wheel loads are described as static,moving, or dynamic. In pavement response analysis, only one rep-resentative wheel is usually considered. Hence, most of the studiesare focused on measuring the dynamic tire-to-pavement contactstresses rather than the stochastic distribution of dynamic wheel loadscorresponding to the specific suspension system of the truck. A staticload would mean stationary load, a moving load would mean mov-ing but constant loads, and a dynamic load would mean a load thatis moving as well as changing (6, 7 ).

To define the continuous distribution of dynamic wheel forces withrespect to the pavement profile, intensive field measurements usingall the axles may be needed. However, it is difficult and expensiveto establish useful relationships between dynamic forces and pave-ment damage criteria because of their stochastic nature, although datafrom extensive laboratory and fieldwork may exist. Therefore, toinvestigate the effect of transient local dynamic loads on pavementresponses without extensive field data, a more simplified methodology(that is, a simplified dynamic load function method, or an uncoupled

Effect of Transient Dynamic Loading on Flexible Pavements

Pyeong Jun Yoo and Imad L. Al-Qadi

The effect of transient dynamic loading on flexible pavements was esti-mated. Transient dynamic loads within a tire-to-pavement contact areaare characterized by continuously increasing or decreasing local dynamiccontact stresses, depending on vehicle speed. A transient dynamic loadmodel was successfully incorporated into a three-dimensional finiteelement model. Dynamic flexible pavement responses to one pass of aheavy vehicular load through a dual-tire assembly were calculated. Resultsof this study indicate that the flexible pavement response at differentpavement temperatures varies depending on whether the analysis wasquasi-static or dynamic, where the mass inertia and damping forces by thetransient local dynamic loads are considered in the equation of motion.Results also show that the time-dependent history of the calculatedpavement responses in the dynamic analysis is more comparable to mea-surements in the field. The transverse and longitudinal tensile strains atthe bottom of the hot-mix asphalt and the compressive stress at the topof the subgrade are underestimated when the mass inertia and dampingforces exerted by the transient local dynamic load are ignored.

Many studies on pavement response to loading have been conductedby using the multilayered elastic approach. A stationary, circular wheelload is assumed in this approach, whereas the pavement is actuallysubjected to dynamic wheel loading. The dynamic loading may causegreater pavement damage than the static loading would indicate.However, developing an analytical model that can accurately simu-late a moving heavy vehicle load on pavement surface is difficult.In fact, it is difficult to consider all variables in the interaction betweenheavy wheel loading and pavements. In addition, a single model maynot be able to consider all the interactive parameters between wheelloading and pavement response. Hence, many previous analyticalstudies have tended to consider only a small number of variables,resulting in analysis of the pavement response that is arbitrary at best.

Uddin et al. noticed that the multilayered linear elastic modelunder static loading is questionable for structural response analysis ofa pavement system under moving wheel loads (1). They investigatedthe effects of pavement discontinuities and implicit dynamic analy-sis on surface deflection. This study demonstrated the usefulness ofthree-dimensional finite element (FE) analysis to analyze a pavementsystem under dynamic loads. The analysis was not possible by usingthe traditional multilayered elastic analysis.

Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, 205 North Mathews, MC-250, Urbana, IL 61801. Correspondingauthor: I. L. Al-Qadi, [email protected].

Transportation Research Record: Journal of the Transportation Research Board,No. 1990, Transportation Research Board of the National Academies, Washington,D.C., 2007, pp. 129–140.DOI: 10.3141/1990-15

130 Transportation Research Record 1990

one-solid FE model) needs to be used that would consider the effectof transient local dynamic wheel loads on pavement responses (8, 9).

BACKGROUND

Dynamic Wheel Load

Wheel loads applied to a pavement system when a vehicle is movingare not the same as those applied when the vehicle is at rest. Themoving-vehicle wheel loads may be divided into the sum of staticand continuously changing dynamic loads, depending on various fac-tors, such as pavement surface irregularity, vehicle speed, mass, andsuspension. The vertical oscillation of the vehicle causes the dynamicwheel loads to vary about their mean magnitudes as the vehicle is inmotion (7).

Cebon (10) and Papagiannakis and Taha (11) suggested a simpli-fied dynamic load function as a moving, sinusoidal, point load. Thisapproach was used to investigate the effects of vehicle speed andload frequency on flexible pavement responses. An idealized dynamicload function was proposed as follows:

where

P(t) = idealized dynamic load,f = load frequency (Hz),t = time (s),

A = constant dynamic amplification factor, andD = load duration (s).

Cebon suggested that the 10-Hz loading frequency, which corre-sponds to a vehicle axle-hop frequency, produces greater strains thana static loading case (10). Dynamic amplifications of strain wereunexpected and could not be explained by a static model. Papagian-nakis and Taha suggested another idealized dynamic load functionto calculate pavement responses to dynamic loading (11),

where A(t) is time-dependent dynamic load amplitude and D is theduration of the pulse loading.

The interaction between pavement response and dynamic loadingwas seen only at the high frequency of axle hop and not at the lowfrequency of vehicle body bounce. Therefore, if pavement responsesin the FE analysis can be matched with the dynamic loading fre-quency below 10 Hz, the dynamic amplification factor A in the load-ing function can be assumed as constant (Equation 2). On the otherhand, if the loading frequency above 10 Hz corresponds to thevehicle speed, the loading function may be assumed by the loadingtime and frequency-dependent form as presented in Equation 3(12, pp. 471–485).

Although an exact mathematical function cannot be formulated forthe dynamic axle load exerted by vehicle excitation from body bounceor axle hop, the interaction between loading frequency and pave-ment responses may be considered by using the influence functionmethod with an idealized dynamic loading function.

P x t A tt

D

Dt

Dn+( ) = ( ) +⎛⎝⎜

⎞⎠⎟

− ≤ ≤sin ( )π π2 2 2

3

P t At

D

Dt

D( ) = +⎛⎝⎜

⎞⎠⎟

− ≤ ≤sin ( )2

2 2 22

π π

P t ft( ) = + ( )10 5 2 1sin ( )π

Conventional Influence Function Approach

A conventional influence function approach, which involves theconvolution of the measured, impulsive load function and influencefunction, has been used frequently to calculate pavement responses.For example, the response of a linear, isotropic pavement to fluctu-ating wheel loads moving at a constant velocity can be modeled byusing the convolution integral method as shown in Equation 4 (11):

where

y(t) = pavement response at time t,h(t − τ) = pavement response to an impulse of unit amplitude,

τ = fraction of time t, andf(τ) = input force at time τ.

The influence function approach is a simplified method for solvingthe convolution integral, such as Equation 4. This approach assumesthat dynamic loads remain constant during the influential time tin Equation 3. Papagiannakis and Taha (11) calculated pavementresponse, R(t) at time t, as the convolution integral of a pavementresponse function, ψ(t − ξ), which is multiplied by the time deriva-tive of loading function P(ξ), where ξ is the fraction of time t, as inEquation 5:

The influence function can be calculated by using a multilayerelastic theory or the viscoelastic material properties of HMA. Theinfluence function, ψ(t − ξ), for HMA may be obtained from thecreep compliance function to calculate pavement responses depend-ing on loading time. To derive the time-dependent strain of �(t) byusing the creep compliance function of D(t) as an influence functionof ψ(t − ξ) in Equation 5, �(t) can be expressed as follows:

where β is the fraction of time t. If the three-parameter Kelvin–Voigtmodel is used as a time-dependent creep compliance model, thecreep compliance function can be defined as follows (13):

where

D(t) = creep compliance function of three-parameter Kelvin–Voigt model,

D0 = instantaneous creep compliance,D1 = retarded response of individual Kelvin unit,

τ = retardation time, andη∞ = dashpot constant.

Thus, a pavement analyst may calculate the pavement damage atthe end of a prescribed period without considering mass inertia anddamping forces exerted by the dynamic load (7 ).

D t D D ett( ) = + −( ) +−

∞0 1 1 7τ

η( )

� t D tP

td

o

t

( ) = −( ) ∂ ( )∂∫ β

ββ ( )6

R t tP

td

o

t

( ) = −( ) ∂ ( )∂∫ ψ ξ

ξξ ( )5

y t h t f d( ) = −( ) ( )−∞

∞

∫ τ τ τ ( )4

Yoo and Al-Qadi 131

Transient Dynamic Load Excitation

To estimate the excitation level of transient dynamic loads exertedby a simplified dynamic loading pulse such as a half-cycle of sinu-soidal loading pulse, or continuous ramp loading pulse, a system’sresponse to dynamic loads can be calculated by formulating a single-degree-of-freedom (SDOF) model, as shown in Figure 1.

To investigate the dynamic amplification of a system’s responseto transient local dynamic loads, a function describing the dynamicamplification factor, which is normalized deformation, u(t)/(ust)0,must be derived. The equation of motion of the SDOF model withhalf-cycle sinusoidal load can be written as a forced vibration of adamped system:

where

m = mass,c = damping coefficient,u = displacement, andk = stiffness.

Dividing by mass, m, this gives the following:

where

= natural frequency of the system,ϖ = load frequency, andξ = c/(2mωn) = damping ratio or fraction of critical damping.

In general, the equation of motion can be divided into a complemen-tary solution for transient response, uc(t), and three particular solutionsfor steady-state response, up(t), u•

p(t), and u••p (t), as follows (15):

where A and B are constants for the complementary solution, and G1

and G2 are constants for the particular solution.Substituting the three particular solutions into Equation 9 for the

case of ω =ωn ≠ϖ, this means that the natural frequency of the systemis not equal to the frequency of dynamic load, resulting in the functionfor time- and frequency-dependent dynamic response as follows:

u t G t G tp• • sin cos ( )( ) = − −⎡⎣ ⎤⎦ϖ ϖ ϖ ϖ2

12

2 13

u t G t G tp• cos sin ( )( ) = −[ ]ϖ ϖ ϖ ϖ1 2 12

u t G t G tp ( ) = +[ ]1 2 11sin cos ( )ϖ ϖ

u t A t B t ec D Dtn( ) = +[ ] −cos sin ( )ω ω ξω 10

ωn k m=

u t u t u tp t

mn n•• • sin

( )( ) + ( ) + ( ) =2 92 0ξω ωϖ

mu cu ku P t&& &+ + = 0 8sin ( )ϖ

where

as damped frequency, and

(ust)0 = p0/k is the static responses of the system.

The normalized-time variation of dynamic amplification factor inEquation 14, R(t), is plotted in Figures 2a through 2c. Figure 2a showsthe damping ratio dependency of the SDOF system with respect to R(t).As the damping coefficient, c, increases from a value of 0.01 to 0.1,dynamic excitation of the system, which is exerted by the half-cyclesinusoidal force, decreases to the response that is the closest to the sta-tic response; this does not consider dynamic excitation, because moreenergy could be dissipated in the highly damped system by the damp-ing coefficient of the dashpot element (Figure 1).

Figure 2b illustrates the local dynamic excitation of the system inwhich the loading frequency is fixed at 10 Hz, which is equivalentto 0.1s of loading time, and the natural frequency of the system isassumed as varying from 50 to 200 Hz. The dynamic excitation is increased as the natural frequency of the solid approaches theloading frequency of 10 Hz. The natural frequency of a pavementdepends on its structural integrity; typical natural frequency of flexiblepavement is between 6 and 12 Hz (16).

In addition, to estimate dynamic amplification by ramp force thatis continuously increased, as shown in Figure 2c, the input loadingforce in Equation 8 is substituted with the continuous loading pulse,P(t) = P0t/τ. In addition to the half-cycle sinusoidal loading force, thedynamic excitation of the system’s response is developed duringthe ramp-loading pulse.

The FE model in this study considers an uncoupled tire and pave-ment FE model as a one-solid model, and the surface-of-pavementmodel is assumed to be smooth and flat. In the one-solid model withknown contact stress at the pavement surface, dynamic loading ampli-tude could be simplified as continuously increasing, or decreasing,ramp-loading dynamic forces within the tire-to-pavement contactarea at pavement surface. In this case, as shown in Figure 2c, thedynamic excitation exerted by transient local dynamic loads overtime within the contact area may affect pavement responses. In thisstudy, the tire imprint area on the pavement surface does not varywhile the load is moving because the pavement surface is assumedto be flat. Dynamic responses were calculated by using the implicitdynamic analysis feature in the commercial software ABAQUS. Thetransient local dynamic loads were modeled in this study by using aDLOAD subroutine in ABAQUS.

Transient Local Dynamic Contact Stress Model

To develop a transient local dynamic load subroutine for this study,the loading amplitude is simulated by the continuously changingramp-load amplitude within a tire–pavement contact area. When

D

D

= −( ) − ( )⎡⎣

⎤⎦

=

= −

1 1 2

1

2 2 2

2

β ξβ

β ϖ ω

ω ω ξ

,

,

R tu t

u

D tD

st

DD

( ) =( )

( )

= + − −( )

0

2 22 2 1ξβ ωω

ξ ωβ ϖ βcos (( )⎡

⎣⎢

⎤

⎦⎥

+ −( ) −⎡

−sin

sin cos

ω

β ϖ ξβ ϖ

ξωD

tt e

D t t1 22⎣⎣ ⎤⎦ ( )14

P(t) =loading

c=damping coefficient

k=stiffness

Mass.m

FIGURE 1 SDOF model (11, 14).

wheel loads move along a pavement surface, both tire imprint andcontact stress may vary simultaneously. If, however, one assumesthat contact stress will remain constant, then the contact area willvary as dynamic forces are changing. In this case, the irregularitiesof the pavement surface will dominate pavement responses. Incontrast, if one assumes that the contact area will remain constantand vertical stresses will vary as dynamic loads, one can anticipatethe transient local dynamic loading effect as a dominant factor on apavement system’s response (7 ).

The known contact stress field, as shown in Table 1 (14), wastreated as repeated moving loads. Repeated moving dynamic loadswere incorporated into a three-dimensional FE model by using asubroutine. Because the three-dimensional FE model in this study

considered the pavement only as a one-solid, with the known contactstresses, variations in stress from the sidewall stiffness of a rollingtire were not considered in the model. Thus, for a one-solid modelwith measured contact stresses, it was found that dynamic loads willvary with increasing or decreasing ramp-loading amplitudes overtime within the tire–pavement contact area.

Continuous, dynamic ramp-loading amplitudes, Figures 3a and 3b,were assumed to simulate variations in contact stresses under eachtire rib, which are discretized into nine finite elements from A1 to A9in longitudinal direction. Linear ramp-loading amplitude was usedto define the entrance part (front half) and the exit part (rear half) ofa tire’s imprint, respectively. As a vehicle approaches a given elementin the loading path, that element is loaded with the amplitude thatsimulates the increase in loading with time (Figure 3a). Similarly,as the vehicle moves away from the given element, the loadingamplitude that simulates the decrease in loading with time is utilized(Figure 3b). One may notice in these cases that the loading pulses ofthe entrance and exit parts of a tire’s imprint are not constant, andthese pulses are assumed to vary linearly with time (14).

The FE analysis software, ABAQUS, was chosen for implicit,dynamic analysis with the transient, local dynamic contact stressmodel since it allows the user-oriented load model to make a subroutine. To incorporate dynamic loads into the FE model, aFORTRAN subroutine was developed for this study (17 ).

Three-Dimensional FE Model

A three-dimensional FE model was developed to simulate TestSection B at the Virginia Smart Road (Figure 4). The in-plane dimen-sions of the three-dimensional FE model were 2,100 × 2,100 mm.These dimensions were selected to reduce edge effect errors in cal-culation while keeping the elements’ sizes within acceptable limits.No symmetry was considered in the FE model to simulate dual-tireloading and its combined effects on pavement response. A fine meshwas generated around the loading area along the wheelpath, and arelatively coarse mesh was utilized farther away from it. The elementsize at the loading area is 20 mm in the direction of traffic, 16 mmlaterally, and 10 mm vertically. When dynamic loads travel along theloading path at a speed of 8 km/h, the loading time on each elementwas found to be 0.009 s. To improve the rate of convergence, eight-node linear brick reduced-integration elements (C3D8R) were used.For an implicit, dynamic analysis, infinite elements (CIN3D8, eight-node three-dimensional linear infinite element) were used to simulatethe far-field region in the lateral, longitudinal, and vertical boundariesof the three-dimensional FE model.

The developed model simulated the measured tire imprint byconsidering the dimension of each tire rib separately (10 ribs for adual-tire assembly). The tire inflation pressure was 720 kPa. Table 1represents the dimension of tire ribs and the measured contactstresses. The data show that tire inflation pressure and measuredcontact stresses exerted on the pavement surface were different.Measured contact stresses were lower than the tire inflation pressureon the tire edge and greater than the inflation pressure in the middleof the tire rib; this could be reversed at high axle loads. This empha-sizes the importance of considering nonuniform stress distributionsin analysis to ensure the accuracy of calculated pavement responses.

Material Characterization

To characterize the linear viscoelastic behavior of HMA, either ageneralized Kelvin–Voigt solid model that consists of a spring


(a)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5

t/Tn

Damping Ratio = 0.01 Damping Ratio = 0.05 Damping Ratio = 0.1 Sine Load Pulse

R(t

)

(b)

(c)

Loading Frequency = 10 Hz

Natural Frequency= 50 Hz Natural Frequency= 100 Hz Natural Frequency= 200 Hz Half Sinusoidal Load Pulse

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5

t/Tn

R(t

)

Loading Frequency = 10 Hz

0

0.2

0.4

0.6

0.8

1

R(t

)

0 0.5 1 1.5 2 2.5 3

t/Tn

Dynamic Response

Continuous Ramp Load

FIGURE 2 Dynamic amplification: (a) damping ratiodependency, (b) natural frequency dependency, and (c) dynamic ramp-loading force.

Yoo and Al-Qadi 133

and a number of Voigt elements connected in series or a general-ized Maxwell solid model that consists of a spring and a numberof Maxwell elements connected in parallel can be selected. Thegeneralized Maxwell solid model (Prony series), which is a mechan-ical analogue to viscoelastic material behavior, was selected tosimulate the linear viscoelastic behavior of HMA layers in thisstudy (13).

For the considered section, elastic properties (Table 2) wereobtained by conducting falling weight deflectometer evaluations ofopen-graded drainage and subbase layers as well as subgrade andindirect resilient modulus tests on HMA materials at differenttemperatures. In addition, indirect creep compliance tests were

conducted on HMA specimens. Details of the laboratory testing andthe backcalculation have been presented elsewhere (18).

The surface mixture consisted of an aggregate with a maximumnominal size of 9.5 mm and 4.6% of PG 70-22 binder. The basemixture consisted of an aggregate with a maximum nominal sizeof 25.0 mm and 4.5% of PG 64-22 binder. To determine the Pronyseries for HMA layers, results from indirect creep compliance testsof field-core specimens, at different temperatures (5°C, 25°C, and40°C), were used. Table 3 represents the Prony series parametersfor the surface HMA after a curve fitting. The time-dependentrelaxation of HMA may be ruled by the dimensionless shear relax-ation modulus (G), dimensionless bulk relaxation modulus (K),

(a)

(b)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4

Time Step

Ver

tical

Con

tact

Str

ess

(MP

a)

A6

A7

A8

A9

Time Step

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1 2 3 4

Ver

tical

Con

tact

Str

ess

(MP

a)

A1

A2

A3

A4

A5

FIGURE 3 Dynamic variation of normal tire pressures with time steps: (a) entrancepart of tire imprint elements and (b) exit part of tire imprint elements.

TABLE 1 Pressure and Tire Dimensions at Inflation Pressure 720 kPa for Dual-Tire Assembly

R1a G1b R2 G2 R3 G3 R4 G4 R5

Vertical pressure (kPa) 593.5 — 1,091.8 — 1,205.6 — 1,082.9 — 588.5

Length (mm) 140 — 180 — 180 — 180 — 140

Width (mm) 33 11.4 30 14.6 32 14.6 30 11.4 33

aR1: tire rib. bG1: tire groove (width between two ribs).


Surface Mix (38 mm)

Base Mix (150 mm)

Asphalt- Treated Drainage Layer(O GDL – 75 mm)

21A Cement Stabilized Base(21B – 150 mm)

21B Aggregate Subbase(21B – 175 mm)

Subgrade - Foundation

588 mm

Infinite Boundary of Bottom

1

2

3

(a)

(b)

450

250

250

1600 2100LoadingArea

1200

2100

450

FIGURE 4 Pavement design and layout: (a) structural design and three-dimensional FE model,and (b) in-plane dimensions, mm.

TABLE 2 Material Characterization: Elastic Material Properties

Temperature = 5°C Temperature = 25°C Temperature = 40°C

Resilient Modulus Poisson’s Resilient Modulus Poisson’s Resilient Modulus Poisson’sMix Type (MPa) Ratio (MPa) Ratio (MPa) Ratio

Surface mix 9,155 0.22 4,230 0.33 1,905 0.36

Base mix 8,930 0.23 4,750 0.30 1,790 0.35

Open graded drainage layer 4,830 — 2,415 0.30 965 —

Cement-treated subbase (21-A) 11,000 0.25 11,000 0.25 11,000 0.25

Granular subbase (21-B) 310 0.35 310 0.35 310 0.35

Subgrade 262 0.35 262 0.35 262 0.35

Yoo and Al-Qadi 135

and relaxation time (τ) for the wearing surface. Similarly, the Pronyseries were defined for the base mixture by using creep compliancetest data for that layer.

In addition, the field-measured temperature of the base layer,32°C, was adopted as a reference temperature for shifting themaster curve, while the wearing surface is at 40°C in the field.Details about laboratory test setup, conversion process for the Pronyseries, and additional material properties have been presentedelsewhere (18).

Pavement Response Validation

There is no standard method for dynamic analysis to simulate thedynamic loading effect on pavement responses, so a verification ofcalculations resulting from the simplified dynamic load model needsto be conducted by comparing pavement responses to measureddynamic responses due to moving wheel loads. Field measurementsunder air-sprung drive axles were obtained from the test program atthe Virginia Smart Road, Section B, which is used as the pavementdesign for this study (see Figure 4). To establish a parallel line ofcomparison, the experimental results were shifted to a referencetemperature (25°C) by using temperature shift factors developed

from the field measurements. The shift model has been presentedelsewhere (18).

The longitudinal strains (Figures 5a and 5b) show the typical lon-gitudinal strain responses; as the load approaches the measuringpoint, strain responses are in compression, and they then change totension as the load is leaving. In addition, a loading cycle resultedin some residual strain at the end (19).

Figures 5a and 5b present the comparison between measuredand calculated longitudinal strains at the bottom of the wearing sur-face and at the bottom of the base layer at a vehicle speed of 8 km/hat 25°C. When compared to the measured response, the dynamicanalysis predicted the strain at an error range of less than 5% inall cases.

Dynamic Flexible Pavement Responses

In this study, the transient local dynamic load, which correspondsto changing contact pressure within a tire imprint, is assumed toresult from the movement of nonuniform normal stress on a smoothpavement surface.

Figures 6a and 6b show the calculated dynamic pavement responsesat the bottom of HMA underneath five ribs of one tire of a dual-tire

TABLE 3 Material Characterization: Prony Series Parameters

5°C 25°C 40°C

No.* G* K* τ* G K τ G K τ

1 0.72265 0.72265 1.00E + 00 0.77465 0.77471 1.00E − 02 0.64082 0.64107 1.00E − 04

2 0.21045 0.21045 1.00E + 01 0.16498 0.16493 1.00E − 01 0.27554 0.27566 1.00E − 03

3 0.04609 0.04610 1.00E + 02 0.03979 0.03975 1.00E + 00 0.04144 0.04148 1.00E − 02

4 0.01485 0.01485 1.00E + 03 0.01383 0.01382 1.00E + 01 0.01654 0.01654 1.00E − 01

5 0.00354 0.00354 1.00E + 04 0.00274 0.00273 1.00E + 02 0.01304 0.01307 1.00E + 00

6 0.00078 0.00078 1.00E + 05 0.00171 0.00171 1.00E + 03 0.01128 0.01128 1.00E + 01

7 0.00098 0.00098 1.00E + 06 0.00204 0.00204 1.00E + 04 0.00017 0.00016 1.00E + 02

8 0.00058 0.00058 1.00E + 07 0.00002 0.00002 1.00E + 05

9 0.00002 0.00002 1.00E + 08 0.00001 0.00001 1.00E + 06

No.*: number of Prony series; G*: dimensionless relaxation modulus; K*: dimensionless bulk relaxation modulus; and τ*: relaxation time.

-80

-40

0

40

80

160

120

0 0.05 0.1 0.15 0.2 0.25

Time (sec)

Long

itudi

nal S

trai

n (µ

)

Measured

Calculated

(a)

Measured

Calculated

-80

-40

0

40

80

160

120

0 0.05 0.1 0.15 0.2 0.25 0.30

Time (sec)

Long

itudi

nal S

trai

n (µ

)

(b)

FIGURE 5 Comparison between measured and calculated longitudinal strain (a) at bottom of wearing surface and (b) at bottom of HMA.

assembly. These are similar to the time-dependent behaviors of thefield measurements.

To estimate the differences in pavement response between thedynamic and the quasi-static analysis, only the maximum responsesin the middle of the dual tire’s ribs (Rib 3) were selected. Figures 7through 9 present the dynamic and quasi-static pavement responsesat various temperatures.

Dynamic excitation at 5°C (Figure 7) is the highest, especially inthe transverse strain and the shear strain, because the natural fre-quency of the pavement solid at low temperature may be the closestto the frequency of the dynamic loading, which has a loading time of0.009 s at finite elements in the loading area. In other words, thefrequency at this temperature may approach the resonant state wheredynamic excitation is maximized over time.

It is worth noting that there are significant differences in pavementresponse between the dynamic and the quasi-static analysis; first, asthe temperature increases (Figure 7 at 5°C, Figure 8 at 25°C, andFigure 9 at 32 °C), the pavement response also increases since theHMA is stiffer at lower temperatures than at high temperatures.

Second, all the pavement responses caused by the dynamic ampli-fications exerted by the transient local dynamic loads are greaterthan those of the quasi-static analysis. In addition, the higher exci-tation at low temperature (Figure 7a) is diminished as the temper-ature increases. In a direct integration method, the equation ofmotion (Equation 9) is integrated through time implicitly. Theequation of motion of a single degree of freedom system includesthe inertia and viscous damping terms. These additional termsmay play a role as excitation or damp-out source to the excitationof a system (14). The HMA is generally more pliable to externalforces at a higher temperature. Hence, greater damping than thosefrom low temperatures would be applied to the system to attenuatethe excitation.

Third, in comparing the compressive stresses at the top of the sub-grade, the analysis shows that there are more prominent differencesin both cases; the initial stresses of the quasi-static analysis are higherthan the dynamic. The dynamic analysis resulted in higher residualstresses than the quasi-static analysis at all temperatures within thesame time domain. This may be because the calculation in the quasi-static analysis is based on the spontaneous equilibrium state at allanalysis steps, whereas the dynamic analysis is basically controlledby the equation of motion, which considers mass inertia forces. Inother words, additional time is needed in dynamic analysis for stress


waves to reach observing points. It also takes more time to dampout all the residual stresses in the system than for the quasi-staticanalysis.

Finally, from results of comparing the peak strains and stresses atdifferent layers, responses in the dynamic case are higher than thosein the quasi-static analysis at all considered temperatures. The max-imum variances in both cases are 39% in the tensile strain at thebottom of HMA, 25% in the compressive stress at the top of thesubgrade, and 10% in the longitudinal strain.

CONCLUSIONS

The primary objective for this study was to estimate the effect oftransient local dynamic loads on flexible pavement response due toheavy vehicular loading. Local dynamic loads bounded by the tireimprint are characterized as continuously increasing, or decreasing,transient dynamic loads, depending on vehicle speed. A transientdynamic load model was successfully incorporated into a three-dimensional FE model that was developed to calculate the dynamicflexible pavement response during one pass of a dual-tire assembly(275/80R22.5).

Results from the three-dimensional FE model that was developedin this study were successfully verified against the pavement responsemeasured at the Virginia Smart Road. The study found that quasi-static analysis underestimated the flexible pavement response toheavy vehicular loading within the temperature range and pavementdesign setup of this study. This included the tensile strains at thebottom of the HMA and compressive stress at the top of the subgrade.It was also found that dynamic excitation in the transverse and shearstrains is maximized at the low temperature of 5°C. This is becauseof the low magnitude of damp-out source to the system.

The maximum variances between quasi-static and transient dynamicanalyses are 39% in tensile strain at the bottom of the HMA, 25% incompressive stress at the top of the subgrade, and 10% in longitudi-nal strain. These differences may affect the prediction of pavementperformance. Hence, this study recommends that the mass inertiaand damping forces generated by transient local dynamic loads beconsidered in pavement response analysis. Even small dynamicvariations in tire–pavement contact stresses may increase pavementdamage.

0 0.05 0.10 0.15 0.20 0.25 0.30

RIB 1 RIB 2 RIB 3 RIB 4 RIB 5

-50

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Time (sec)

Long

itudi

nal S

trai

n (µ

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RIB 1 RIB 2 RIB 3 RIB 4 RIB 5

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Tra

nsve

rse

Str

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FIGURE 6 Calculated dynamic pavement responses at bottom of HMA at 32�C: (a) longitudinal strain and (b) transverse strain.

Yoo and Al-Qadi 137

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ompr

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Pa) Quasi-static response

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Lateral Strain Longitudinal Strain

Str

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Quasi-static

Dynamic

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Str

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a)

Compressive Stress

Quasi-static

Dynamic

(f)

FIGURE 7 Calculated dynamic versus quasi-static pavement responses at bottom of HMA at 5�C: (a) transverse strain, (b) longitudinal strain, (c) shear strain, (d ) compressive stress at top of subgrade, (e) peak strains at bottom of HMA, and (f ) peak stresses at top of subgrade.



Dynamic response

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Lateral Strain Longitudinal Strain

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(f)


Yoo and Al-Qadi 139


Dynamic response

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0.00 0.10 0.20 0.30 0.40

Time (sec)

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ral S

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Dynamic response

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Dynamic response

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Dynamic response

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Time (sec)

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30

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a)

(e)

Quasi-static Dynamic

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Tensile Strain Longitudinal Strain

Str

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Quasi-static

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(kP

a)

Compressive Stress

(f)


ACKNOWLEDGMENTS

Financial support was provided by the Michelin Americas Researchand Development Corporation. The authors thank John Melson andSue Nelson for their input. The authors acknowledge the assistancefrom the National Center for Supercomputing Applications at theUniversity of Illinois at Urbana–Champaign.

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The Full-Scale and Accelerated Pavement Testing Committee sponsored publicationof this paper.

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Effect of Transient Dynamic Loading on Flexible Pavements