HighwayEngineering]
SCE Journal of Civil Engineerir Vol. 5, No. 4 / December 2001 pp. 387-395
Analysis of Stress-Dependent Behavior in Conventional Asphalt Pavements By Seong-Wan Park*
Abstract Most low-volume roads are primarily thin flexible pavements with unbound base and subgrades whose materials are nonlinear and stress-dependent even under low traffic stresses. A need therefore exists for predicting more realistic pavement response based on proper constitutive models and the computational method. In this study, the non-linear stress-dependent finite dement program (FERUT) for pavement analysis was developed. Stress-dependent models for the resilient modulus and Possion's Ratio of the unbound pavement materials were incorporated into the finite element model to predict the resilient behavior within the pavement layers under specified wheel loads. The results from this study show that the developed finite element model with the stress-dependency concept is suitable for taking into accoum for the effects of the non-linearity and stress-dependency of the unbound pavement layers in a pavement.
Keywords: asphalt pavements, stress-dependent modulus, stress-dependent Poisson's ratio, resilient behavior, unbound base, finite element method
1, Introduction When a wheel load is applied to a conventional flexible pavement system, unbound base layer should be distributing wheel loads over subgrades. The behavior of unbound pavement materials is highly non-linear even under low traffic stresses, as well as being dependent on its stress history and current stress state. To consider these behaviors realistically in the mechanistic-based pavement analysis, a non-linear finite element program was developed to predict the non-linear stressdependent pavement responses. The stress-dependent behavior of pavement materials was accounted for non-linear resilient response using the Universal Soil Model (Uzan 1992). in addition, stress-dependent Poisson's ratio was predicted on the basis of the energy concept (Lade and Nelson, 1988, Lytton et al., 1993). To verify the prediction models used in finite element approach, the model developed was compared to other pavement stmcatral models widely adopted and the field data from the Multiple-Depth-Deflectometers (MDD).
2. Models for the Stress-Dependent Behavior The strain resulting from traffic load on a typical pave-
merit should be nearly completely recoverable and proportional to the applied load. Although there is some permanent deformation related to each load application, this strain is normally small and causes only long term deterioration of the pavement structure. The deformation can therefore be considered as elastic and this elastic or resilient modulus, M , is defined as the ratio of the repeated deviatoric stress to the recoverable part of the axial strain resulting from repeated load tests as shown in Equation 1 (Haang, 1993). Mr = era eT
where: (rd =the repeated deviatoric stress, and e, =the recoverable strains obtained from repeated load triaxial test. It is well known that the behavior of unbound pavement materials in conventional flexible pavements is non-linear and stress-dependent. The most commonly used non-linear elastic modulus model for characterizing stiffening behaviors of unbound granular materials is the k-0 model as shown in Equation 2 (Hicks and Monismith. 1971), Mr =- kj 0 ~2
* Member, Senior Researcher, Highway Facilities Research Group, Korea Institute of Construction Technology, Korea (E-ma~l:
[email protected]) The manuscript for this paper was submitted for review on July 27, 2001. Vol. 5, No. 4/December 2001
- 387 -
(1)
(2)
Seong- Wan Park
where: 0 =bulk stress or first stress invariant, and k,, k2=regression model constants obtained from repeated load lriaxial test. For characterizing softening behaviors of fine-grained soils, however, the k-o-~ model as shown in Equation 3 has been widely used.
M, = k,4
O)
where: ~ra =deviatoric stress, and k3, k4 =regression model constants obtained from repeated load triaxial test. However, several studies have shown that the mode1 Equations 2 and 3 do not accurately predict the response of unbound granular materials (Brown and Pappin 1981, May and Witczak 1981, Uzan 1985). Because the above models assume a constant Poisson's ratio of pavement materials as an initial input dam, and the effect of shear stresses on the resilient properties is not considered, nor are the dimensional problems of model itself. The shear stress component is especially responsible for dimensional change and consequently, permanent deformation. This limitation is inadequate for pavement analysis because shear stress is relatively more significant in thinner flexible pavements. In addition, those two parameter models only represent a limited range of stress paths under the wheel loads. The Universal Soil Model (Uzan, 1992) introduced the effect of octahedral shear slress into the K-O model and added atmospheric pressure as a normalizing factor into the original model to make the stress terms non-dimensional as shown by Equation 4.
where: Pa =atmospheric pressure, /1 = first stress invariant, %~,=octahedral shear stress, and ki --material constants determined by recession analysis from repeated load triaxial test. The octahedral shear stress term is believed to account for dilation effect that takes place when a pavement element is subjected to a large principal stress ratio. Depending on the level of stresses, the first stress invariant or bulk stress term considers the hardening effect associated with higher modulus, while the octahedral shear stress term considers the softening effect. By setting non-linear material properties of k2 or k3 at zero, the model can be simplified as linear elastic, or non-linear hardening, and softening stress-dependent
- 388 -
behaviors respectively. The level of stress dependence in an unbound material is also very significant. The low confinement and high shear stress state represent the stress condition under the edge of the tire loading, whereas the higher confinement and low shear stress represent the condition under the center of the tire loading. The Universal Soil Model is thus favored to use in pavement analysis for unbound materials. For the analysis of conventional flexible pavements, layered elastic analysis approach is commonly used with a constant or fixed Poisson's ratio based on the assumption of isotropic and homogeneous pavement materials. However, the Poisson's ratio of unbound pavement materials is also known to be the stress-dependent and should therefore be considered with the stress-dependent modulus simultaneously in a single framework (Park 2000). For this study, a relationship between the Poisson's ratio and the resilient modulus based on a thermodynamic constraint is adopted (Lade and Nelson 1987). The derivation of equation is based on the principle of the conservation of energy and the path independence of the energy density function. After the study by Lytton et al. (1993) and Liu (1993), a relationship was established between the resilient modulus as expressed in Equation 4 and the thermodynamic constraints to derive an expression that relates the stress state and the rate of change of the Poisson's ratio with a changing stress state. This relationship between Poisson's ratio and the stress state is as illustrated in Equation 5,
lay where: < = k3'= k/ = /~ = Jz =
r k;
Fi ;
Poisson's ratio, k3f2, material parameters, normalized first stress invariant, and normalized second invariant of the deviatoric stress.
3. Non-Linear Stress-Dependent Finite Element Program The program is a modification of a finite element method program of Owen and Hinton (1980), which provides only a basis of the finite element modeling in pavement layers. The modified finite element program is based on the flow theory of plasticity and can account for the stress-dependency of the resilient modulus and Poisson>s ratio. The nonhnear analysis is made using an incremental loading and an iterative solution technique for each load increment. Figure 1 shows the flow d i a g a m of the program. This
KSCE Journal of Civil Engineering
Analysis of Stress Dependent Behavior in ConventionalAsphalt Pavements mesh were restrained in the horizontal direction, write the bottom of the mesh was restrained in both the vertical and horizontal directions to represent the rigid layer. The MolirCoulomb yield criterion can also be used to determine the yield point and to evaluate the potential for pavement damage as well. The stress-dependent Poisson's ralio was determined using Equation 5. However, the analytical solution can be indeterminate for certain combinations of I, and -/2. It was, therefore, decided to include a numerical solution to the stress-dependent Poisson's ratio determined by simple substitutions of partial terms based on the backward difference method. These partial terms are shown in Equations 6 and 7.
i
[
Preset the Variables
r-+
_P
]
I
Input Data
I
1
Initializations
I
L
Compute Stresses
I
Solve for Stress Dependent Modulus & Poisson's Ratio
Ov al~
__
I
I
Calculzte the Residual Forces_. I
No
Yes
Zq nib i m Cheek I I
i
I
I
I_
o,+,
I
End
1
J
vj =
i
v~z l
(7)
Lk )
x vj_, + tT;-,/x v~_~-~-;- +-I \t11/
2
Ficl. 1. Flow Diagram for 9-0 Non-linear Elastic Finite Element Program Using $tross-Dependenw
1
_k;
J J2 l~J
(8)
ka]
Equation 8 is solved by choosing a step size for increasing 11 and ,/2 and then increasing 1~ and J~ from a fixed boundary condition for which the Poisson's ratio is known (,rooste and Femando 1995), Varying Poisson's ratio values are calculated iteratively unfit the specified convergence during each load increment is accomplished. The program has two convergence criteria. The equilibrium criterion is based on the residual force values. Convergence occurs if the norm of the residual forces becomes less than the specified tolerance multiplied by the norm of the total applied forces. Thus, equilibrium is acquired when the applied loads and the nodal forces equivalem to the internal stress are balanced. For the stress dependency criterion, the process is repeated until the difference in modulus calculated from following iteration and modulus calculated from the previous iteration is satisfied within a given tolerance. Convergence depends on the percentage difference between the new and
algorithm handles both equilibrium and stress-dependency criteria in an incremental scheme. Equilibrium convergence is fast checked based on the residual force values, At each load increment, equilibrium is satisfactory when the applied loads and the nodal forces equivalent to the internal stress are balanced. After equilibrium check is satisfactory, the stress dependency of the moduli and Poisson's ratio checks are performed until the difference in modulus calculated from following iteration and modulus calculated from the previous iteration is satisfied within a given tolerance. Stresses and strains under loads for each season are also assessed, The program uses 8-node quadratic serendipity element and 9 Gaussian points for calculations. A mesh with a maximum 400 elements and 15 number of layers can be analyzed using an axisymmetric geometry and formulation. For the boundary conditions, the sides of the finite element
Vol. 5, No, 4 / December 2001
i
(6)
where l, k = step sizes for increasing I~ and J> and i, j = counter for L and J2. By setting Equations 6 and 7 into Equation 4 and performing some algebraic manipulation, the iterafive formula for determining the Poisson's ratio for a given stress condition can be expressed as Equation 8.
ullI ad+ X ed I
I Next Season and Year Increment
vj- V i-' k
Ov v = 011 --
Modulus & Poisson's Ratio Check No
=
389 -
Seong-WanPark previous values. In general, a 15% difference in the calculated moduli from the current and previous iteration is acceptable. During the calculation of the new modulus and Poisson's ratio, a damping factor is applied to facilitate convergence. At low stress levels, the equation for resilient modulus may result in unreasonable values. To prevent this, cutoff values for both the fu-st stress invariant and octahedral shear stress are specified in the computer program. The factor used is 20.7 kPa. All values of the first stress invariant and octahedml shear stress less than 20.7 kPa, the factors are set to 20.7 kPa. In addition, the model for stress dependent Poisson's ratio was only set to values below 0.48 due to equilibrium convergence and elastic theory. Since it is common to find observed values of Poisson ratio above 0.5 for particulate materials in the laboratory, more confinement still exists in the field than under laboratory confinement. Therefore, it was decided to use with a Poisson's ratio below 0.48. This also gives a more conservative prediction. More detailed information can be found elsewhere (Park. 2000).
4. Sensitivity of Stress-Dependent Modulus The material parameters considered in the sensitivity analysis were the parameters, kl to k3, of the Universal Soil Model. The coefficients, kl, k2, and k3, are determined from resilient modulus tests. Since the calculated stresses are normalized with respect to the atmospheric pressure, these coefficients are dimensionless. From results of the study conducted by Jooste and Femando (1995), the coefficient, kt, was found to have the most influence on the predicted resilient modulus. In general, the higher the kl, the higher the predicted resilient modulus. This is illustrated in Figure 2, which shows predicted resilient moduli for a granular base material at three different values of kl. The data
t~ IlO
-
10o
~
9
i . . . . i. . . . . . . .
i- -
shown were calculated assuming a pavement with a 100 m m thick asphalt concrete surface layer and a 200 m m thick granular base layer. For the base layer, values of 0.6 and --0.3 were assumed for the parameters, k2 and k3, respectively. For a given curve, it was observed that the resilient modulus increases with increasing wheel load, illustrating the hardening effect of increasing confinement on the predicted resilient modulus. This hardening effect is associated with the bulk stress term. Hardening term=(pO) k-~
As the wheel load increases, the confining pressures also increase resulting in higher predicted values for the resilient modulus. The octahedral shear stress also increases with increasing wheel load, which tends to decrease the resilient modulus. However, for the pavement and range of wheel loads considered in Figure 3, the increase in confinement with higher wheel loads more than compensates for the softening effect of the octahedral shear stress. Thus, the resilient modulus is predicted to increase with higher wheel loads in the figure shown. However, the opposite trend may be obtained for another pavements (such as thin pavemeats), where the softening effect of the octahedral shear stress may be more pronounced. The hardening effect o f higher confinement and the softening effect of higher octahedral shear stress can be discemed from Figure 3. The shear stress term in the figure is equal to: Softening term=[ "~ \P,,J
9
1.2
~
9
(10)
As the wheel load increases, the bulk stress term increases because of higher confinement. However, the octahedral shear stress also increases so that the shear stress term diminishes with higher wheel loads. Cortsequendy, while the effect of the higher kl is generally 1.3
~
(9)
'
'"
1.6
"',
[.1
~L..,.
- 1.3 60 ~ B 30
I 4O
~k1=490
~
i t, 50 Whemt Load (k.N) ---~-~kl=700
~ 60
70
t).~ ~
31~
+kl~910
.
I '[~-- ~ g
Fig. 2. Variation in Resilient fVlodulus with Resilient Parameter, K1 (Jooste and Femando 1995)
- 390 -
9
4O
,
9
~ Term
' "'-'.
t.g
-
60 A
So~gTenn
]
Fig, 3. Illustration of the Hardening and the Softening Effect (Jooste and Fernando 1995)
KSCE JomTtal of Civil Engineering
Analysis of Stress-DependentBehavior in ConventionalAsphalt Pavements 6O
-71)
-50-
45
~" -55
-80
40
5O
~9 ~-7o-
'~ -I00
-II0
8
zs 8
-140 3,0
~ -75
35
~ -1:~0 " 40
50 Tire Load (IG~
l~t St~SS I~vadattt
Fig. 4. I[lustration of
60
t h e Effect of
-90 tSq
70
~- Oct. She~ $~:ss
The influence of stress-dependent model was illustrated
Oct.ShearSt~ss. I
Section 12
MDD Module Depth
225-n~a
12S-mm lIMA
~ ~ 9
425-rnm ~ Sandy Gravel .-'4~:~.:i Subgrade ~-.';~.'
I
"~ "'300-ram Curshed Limestone
~ 300-mm
~
4r
~ K - ~ Limestone
~(~;
1.625-m Axmhor
. .~,~
9oo-~~s:';'~ .... ~
S~dy c~v~L Subgrade
~
L
2.02S-m Anchor
Fig. 6. Test Pavement Sections with the MDD Locations COzanand Scullion 1990)
Vol. 5, No. 4 / ~ b e r
2001
400
with the field data measured by the Muld-Depth-Deflectometer (MDD). The data was collected in a study performed by Uzan and Scullion (1990). The Falling-Weight-Deflectometer (FWD) was used to back-calculate the modulus and obtain pavement surface deflections. Each pavement deflection was then predicted with the stress-dependent finite element program developed and compared with the MDD deflections measured in the field. The effect of stressdependent material models to characterize unbound base on deflections was also compared. Two pavements, a thin surfacing over a thick crushed limestone base over a sandy gravel subgrade and a thick surface layer over a thick crushed fimestone base over a sandy gravel subgrade were considered. The pavement layer thicknesses and MDD sensor locations are shown in Figure 6. In Figure 7, the MDD deflections decreased with depth and were less than the FWD surface deflections in the thick pavement. In the thin pavement, the MDD deflections measured within the base layer and subgrade were greater than those measured on the pavement surface as shown in Figure 8. This may indicate that dilation takes place in the granular layers. The dilated material acts like an internal
125-ram ~ "
l~tStw~.zlava.fant .
350
Fig. 5. Illustration of the Effect of Base Thickness on Oand t~,~
5. The Effect of Stress-Dependent Material Models
400-mm Crushed~ Limestone ~'~.~ ~
303 2~0 Base Thicl~ss (ram)
L~
to increase the predicted resilient modulus, the effects of k2 and k3 depend on the interactions between these coefficients, the applied loads, and the pavement geometry. The tendency of a material to stiffen with increasing confinement, 2, is related to k2. These effects are illustrated in Figures 4 and 5. Figure 4 shows that the magnitudes of the predicted stress invariants, 2, and octahedral shear stress, ~ , increase as the applied load increases for a given pavement smacture. While the increase in the magnitude of 2 wilt result in an increase in the bulk stress term, the increase in J0~ will diminish the shear stress term thus counteracting the effect of higher coM-mement on the predicted resilient modulus. Figure 5 shows the effects of pavement geometry on the induced stresses where the base thickness is varied while the applied load and other variables are kept constant. As expected, the magnitudes of 2 and vo~ decrease as the base thickness increases.
25-ram HMA
200
I
L o ~ o n 0 and r,~
Section 11
3s
- 391
Seong- Wan Park.
Table 2. Average Percent Error of Deflections on Section 11
25o [ [ 200 ~ [~
~;~ FWDSurfaceDeflleetion ~ MDDat L25-mm ~k- MDDat 425-mm ~ MDD at 725-mm
i
~150]~:= 100
~
i
an~ 12 Pavement
Section
llehOrat 2025-alrn
It 12
Model for Base Layer
% Error of Deflections
FWD
MDD
No Stress-Dependency
48.6
41.9
With Stress-Dependency
6.6
7.O
No Stress-Dependency
39.9
35.7
With Stress-Dependency
4.8
10.1
o 0.5
0,0
L0 1.5 OffsetDistancefromLead (m)
2.0
Fig, 7. Measured Surface and Depth Deflections on Thick Pavements (Uzan and Sculli.on1990) 5O0 1 ,~ [\ 400 1 \
+ FW'DSurfaceD~fle.c~on ~ MDD at 225-mm -A--MDD at 575-mm
30O 2OO 100 04
", 0-5
0.0
,
.
9 0
1.0 1.5 OffsetDistancefTomLoad (m)
2.0
6. Comparison of Pavement Response Models
Fig. 8, Measured Surface and Depth Deflections on Thin Pavements (Uzan and Scullion 1990) Table 1, Backcalculated Resilient Material Property for Sections 11 and 12
Resilient Pavement Section Material Property 11
Base
Subbase
Subgrade
Kt
69,000
4480
5t70
1930
K2
0,0
0.255
0,255
0.0
K3
0.0
0.255
0.255
0.0
K] 12
AC
138,D00 5860
material properties until the average percent error in deflections were less than 10%. The linear and non-linear resilient material properties were then used for the analysis as shown in Table 1. The model for asphalt concrete layer and subgrades were assigned as linear elastic. For the base layer, model was set as both linear elastic and non-linear stressdependent elastic model. The error values fi~m each calculation are shown in Table 2. The vertical deflections from the finite element approach with stress-dependent models for unbound pavement materials is much closer to field deflections in the pavements. Based on these results, the behavior of unbound base layer can be predicted using nonlinear stress-dependent approach to model realistic behaviors m pavements.
5170
2070
K2
0.0
0.255
0.255
0,0
K3
0.0
0.255
0.255
0.0
pressure to uplift the surface and pushes the subbase and subgrade down. Thus, the deflections observed in thin pavements cannot be explained by conventional linear elastic techniques. It is, therefore, expected that the pavement response is more stress-dependent in thin pavements than in thick pavements. Several computer runs were made to determine the parameters of the FWD surface deflections and MDD depth deflections using the finite element program with different
- 392 -
The developed finite element model was verified by comparing the results obtained from the BISAR program. B]SAR (De Jong e t al., 1973) is a proven multi-layered linear elastic program. In order to evaluate pavement responses of a conventional flexible pavement system, single axle loads of 40 kN and tire pressure of 689 kPa were applied in the analysis, which corresponds to the radius of the loaded area of 136 ram. A uniformly disn-ibuted toad acting over a circular area was assumed. The side boundary was 10 times the tire radius and the vertical boundary was placed at a depth of 2.0 m. In this analysis, the responses of the pavements were calculated assuming a pavement with a 100-ram thick asphalt concrete surface layer and a 200-ram thick granular base layer, Average values were assumed for the resilient parameters, kl, k2 and k3, respectively, as shown in Figure 9. The responses of the pavement were determined at the near center and edge of the loaded area. Comparisons were made at the closest element, and the closest Gaussian points within the quadrilateral finite element mesh. All evaluation points in the layered approach were specified to the same coordinates as the finite element locations. The finite element model was used to consider the non-linear stress
KSCE Joumat of Civil Engineering
Analysisof Stress-DependentBehavior in Conventional Asphatt Pavements dependent behavior of unbound materials. Three different response models were considered and compared in these analyses. 80 kN Wheel Load, R = 136 m m
i
J,
AC
k2 =0,i k3 = 0.0
~
1
*
CL
...
Evaluation Positions Near Center Loading, R = 5.1 m m Near Edge Loading, R = 130.8 mm
k l = 50,000
IN) turn
I I I
Er = 4 9 9 5 . 3 MPa < = 0.35
Base 200
mrs ,
k t = 700.0 k2 = 0.6 -0.3
Er = 69.9 MPa
k3 =
< = 0,35
Subgrade
k l = 400.0 k2 = 0.0 k3
=
E~ = 40.(I, MPa
<
-0.3
=
0,40
Radial
Distance
Fig. 9. Load and Pavement Parameters Used in Analysis 136
R=
1) Non-linear 2-dimensional finite element model with stress-dependency (SFE), 2) Non-linear 2-dimensional finite element model without stress-dependency (FE), and 3) Layered linear elastic model (LLE). All moduli and the Poisson's ratio for a linear layered elastic and non-linear elastic models without stress-dependency were supposed to be constant for each layer, as shown m Figure 9. Figures 10 and 11 present that the modulus and Poissoffs ratio values varied in the pavement layer under the wheel loads by the stress-dependent finite element method at the near center. In reality, since the stress state may change from point m point, the modulus and Poisson's ratio of pavement materials varies both vertically and horizontally. The stress dependent finite element model can allow the values of moduli and Poisson's ratio m vary in both horizontal and vertical at each element mesh. The vertical and radial stresses with depth beneath the near center of the loaded area are shown in Figures 12 and 13. It is clear that the use of non-linear stress-dependent
~ m
CL
m m
AC B~
.......
6526.2 6:g 14,7 7239.8 7356.t 7404.7 ................................................................................. 5926.9 ~932,5 6457.7 6528.7 6562.6 ................................................
AC
~OO
4267.1
mm
...............
426.7.1 r . . . . . . . . . . . . . . .
~. . . . . . . . . . . . . . .
4267.1
4267,1
r ................
~. . . . . . . . . . . . . . . .
................................... ..~.O0-
:
4267.1
4267.1
4267.1
4267.I
96,3
103.9
~09.6
t13.4
115,3
87.0
94,6
97 9
t00.7
101.6
4267+I
Base 200 m m
~. . . . . . . . . . . . . . . .
426.7.1
-200
e'-, ~,90 j
=
: Sttbgz~de
80,6
i
88.4
~i
90.6
91.4
]92.5
64A
i
64,I.
i
64,l
64.1
i
Subgrade
r
-800 ! 0
i00
200
300
400
500
600
700
Vertical Stress (kPa)
64.1
I
..............
~. ...............
2 2 2 . 3 mm
~ .................
]63.3 mm
~
l133mm
................
}. ..............
67.8 mm
22,6 mm
RadiaI D i s t a n c e
Fig. 10. Variations of Predicted Resilient Modulus, M, (Modulus in k P a )
Fig. 12. Vertical Stresses at the Near Center of Loading with Depth
-100-
R= 136 mra
CL
_ ~ 2 . L - - - - - - f- - - ~ 0.3"1 ~ 0,475
AC
..............
I00 mm
!
0,373
[
0357
0.430
0.375
'~ .................................
i
0.480
0,480
0,4~0
0.393
0,382
i i
0.380
0,378
0,378
0.462
0.4~0
~
0.450
O.r
0.4~0
i
0.480
............... i 0.480 i .
Subgrad 9
0,351 ~ ...............
0.4~0
0 480
200 m m
0.356 ~. . . . . . . . . .
0,407 2 ~ 2 . 3 mrn
i.
.
.
.
.
.
.
.
.
.
.
.
o.4so
..........................
! .
.
.
.
0.410
1 6 5 . 3 mm
0.4~0
04]2 1 1 3 . 3 mm
/
-600-
0.41~ 67.8 mm
0.480
700-
7i .............. o4so i
-S0( -15
0,413
-I0
2 2 . 6 mm
-5
5 10 Radial $tre~ (kPa)
15
20
25
[ ~ l ~ SFE ,,i..~ FE ~t--- LLE t
R adi~l D i s l a n c c
Fig. 13, Radial Stresses at the Near Center Loading (Unbound
Fig. 11. Variations of Predicted Poisson's Ratio
Vol. 5, N o . 4 [ D e c e m b e r 2001
/J/
g -2~176
- ..............
0.480
!i................................. o.4s0 u.aso i
y ~ Y
0.35g
0.480
.............................
Base
0.358
Layers)
- 393
Seong- Wan Park
Table 3. Strains at Selected Locations
Analysis Model
Strain Component
At the Bottom of Surface AC
At the Middle of Base
At the Top of Subgrade
Center
Edge
Center
Edge
Center
Edge
SFE
Ve~cal Radial
449 365
307 t90
1112 385
934 260
838 296
718 222
FE
Vertical Radial
402 363
266 18 t
995 357
817 248
i042 381
918 301
LLE
Vertical Radial
410 368
275 185
981 389
806 276
1014 429
891 345
Note: All strains in 10~ mrrgnun, and the radial strains are m tension. model, SFE, gives slightly higher results than the elastic analysis used because of the variation of the modulus. However, the difference in vertical stress is quite reasonable. It can also be seen that the stresses from the finite element model without stress dependency, FE and the linear elastic model, LLE, are comparable. Figure 13 shows that significant differences occur in the predicted radial stresses in the base layer. It shows that the stress-dependent modulus and Poisson's ratio reduce tension at the bottom of base layer (i.e. tension is negative, and compression is positive) and improves the prediction results reasonably since the unbound materials are unable to handle tensile strength. At the upper half of the base layer, all models remain compressive. However, with the layered elastic model and finite element model without stress dependency, radial stresses change and gain tension in the lower half of the base layer. This is because that higher modulus ratio leads to a decrease in the radial stress (compression) in the upper half of the base layer and an increase in the radial stress (tension) in the bottom half of the layer. Therefore, this causes more bending stresses and produces more tension. At the top of the subgrade, the differences in radial stresses are small and comparable although stress dependency gives a slightly h i d e r result. With the only stress dependent modulus, however, the effects are relatively less than when the case of stress dependent modulus and Poisson's ratio is considered. Theretbre, changing the modulus and Poisson's ratio by the stress dependency approach greatly affects the prediction of pavement responses in flexible pavements. At the near edge loading, the same trends were observed for radial stress as were observed at the center loading case. Since the vertical strains with depth under the wheel loading and the radial (or tensile) strains at the bottom of the AC layer are important and will be used in the performance analysis, comparisons of those between different models were made. Table 3 summarizes the strains at the selected positions. Similar results were obtained in the AC layer
- 394 -
Table 4, Predicted Pavement Performance Using Asphalt Institute Equations
Analysis Model
Fatigue Cracking
Ratting
Center
Edge
Center
Edge
SFE
183,754
1,570,140
81,I59
161,852
FE
163,988
t,614,828
30,674
54,115
LLE
156,291
1,491,049
34,602
61,859
Note: Number of load applications to reach performance criteria. between models and the differences were all within 10%. This means those strains may not have been affected significantly by the stress dependency. By considering the unbound pavement layers as stress dependent, the radial swains obtained below the bottom part o f the base layer are somewhat different but consistent. However, since the concem is tensile slrain at the bottom of AC layer in this study, the results are quite close. According to a comparison of vertical strain at the center loading locations with depth as shown in Table 3, there are significant differences between the with and without stress dependent models, particularly at the base and subgrade layers. At the lower half of the base layer, vertical strains of stress dependent finite element model are about 10% higher than the linear models values. Since the assigned moduli within the unbound base and subgrade layers in the case of linear models are constant, the above results would be expected. For the top of subgrade, vertical strains between models with and without stress dependency are quite different and a/l elastic models gives higher values than stress dependent models. Since, with stress dependency, the base layers have higher moduli values than those without stress dependency, this causes less stress intensity and results in less strains on the top of subgrade. Therefore, it directly affects rutting prediction when the empirical equation is used. The vertical strains on the top of subgrade obtained by using stress dependency are only about 75 to 80% of those obtained using linear models. Table 4 summarizes the estimated per-
KSCE Journal of Civil Engineering
Analysis of Stress-DependentBehavior in ConventionatAsphalt Pavements formance using the Asphalt Institute equadon (1991) with the data obtained at the bottom of AC layer and at the top of subgrade. It is obvious that the estimated number of load applications on rotting is much higher with the stress dependency.
7. Summary and Conclusions The non-hnear finite element program for pavement analysis that combines the stress-dependent moduli and Possion's ratio models was presented. The pavement responses from the stress-dependent finite element model were compared with different pavement response models and field data to validate its appficability. From the results, the developed f'mite element model was compared with the linear elastic model and the accuracy of the finite element model was well established. Analysis illustrated the effect of non-linear response of unbound materials in pavement on conventional response such as tensile strains at asphalt concrete layer and compressive strains at the top of subgrade. As a result, the estimated number of load applications on rutting is much higher and sensitive with the stress dependency. Therefore, this indicates that the stress-dependency approach is more suitable to predict pavement performance particularly in low-volume roads. The developed finite element model with die stressdependency is appropriate for reducing the tension in the bottom half of the unbound base layers, as validated through comparison of the models. Unlike additional or assumed methods to correct horizontal tension in unbound layers, compressive stresses can be obtained only by the use of the selected constitutive material models with the finite element method formulations. Furthermore, this could be more significant when the effects of cross-anisotropy of the unbound layer properties are considered for predicting the compression at the bottom of base.
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