Effect of Cross-Anisotropy of Hot-Mix Asphalt ... - SAGE Journals

Effect of Cross-Anisotropy of Hot-Mix Asphalt Modulus on Falling Weight Deflections and Embedded Sensor Stress–Strain M. U. Ahmed, R. A. Tarefder, and M. R. Islam In this study, the effects of cross-anisotropy on asphalt pavement responses are examined. A dynamic finite element model (FEM) was developed in ABAQUS to simulate pavement responses under falling weight deflectometer (FWD) and truck loads on a pavement section on I-40 at Mile Post 141 in New Mexico. This section was recently instrumented with strain gauges, moisture probes, and pressure cells. Pavement response (i.e., stress, strain, deflection) from the instrumented section is compared with the FEM-predicted values. Two combinations of cross-anisotropy are considered. The first combination considers cross-anisotropy of modulus in every layer of the pavement, and the second combination considers it only in the hot-mix asphalt layer. Time–deflection histories, stress, and strain are predicted from the FEM under FWD and truck loads. Results show that predicted deflections, stress, and strain are highly sensitive to cross-anisotropy. Predicted deflections, stress, and strain increase with a decrease in n-value, defined by the ratio of horizontal to vertical modulus of elasticity. Analyses are performed for two shapes of loading area: semicircle–rectangle and rectangle. Predicted stress and strain are shown to be larger for the rectangular shape of the loading area than for the semicircle–rectangle area. This study recommends including modulus anisotropy in FWD backcalculation and in pavement design.

and if not, it is anisotropic (7). A special type of material anisotropy is cross-anisotropy in which properties along any two directions are the same but property in the third direction varies. Generally, a pavement contains cross-anisotropic material in different layers. Cross-anisotropy results from vertical compaction during construction that is followed by the repetition of a vertical wheel load (8). As a result, the modulus of elasticity can be considered to be the same along two directions on the horizontal plane (horizontal and transverse) and different along the vertical direction. In the past decade, several studies to predict pavement responses have considered pavement materials as isotropic under dynamic load (5, 9, 10). A study by Masad et al. covered both isotropic and anisotropic materials to determine the pavement response (11). In that study, an axisymmetric finite element model (FEM) was developed and subjected to a static-loading condition to simulate a Benkelman beam load. Cross-anisotropy was also investigated by Oh et al. (12). They developed four cross-anisotropy material models that were based on laboratory tests. These models were then implemented in a static FEM to determine the pavement response under falling weight deflectometer (FWD) and multidepth deflectometer tests. Choi et al. developed a 3-D static FEM by using anisotropic behavior of hotmix asphalt (HMA) (13). Their model was used to determine the pavement response under truck loading. Base and subgrade materials in their model were considered to be isotropic. Recently, Al-Qadi et al. considered cross-anisotropy for all the layers except the HMA (14). They performed 3-D dynamic finite element analysis (FEA) by using an implicit algorithm to simulate FWD test results. Previous studies have mostly used FWD or other types of deflection data to validate FEM. Very few researchers have used field-collected stress and strain to validate their numerical models. Appea used field instrumentation data to validate the model (15). Recently, an axisymmetric FEM was developed to determine the response of an instrumented pavement section under dynamic load by Howard and Warren (16). The literature cited here makes clear that several studies considered material cross-anisotropy in a static FEA (11–13). However, material cross-anisotropy was rarely considered in dynamic FEA (5, 9, 10). Though material cross-anisotropy was considered in a dynamic FEA by Al-Qadi et al., their model was not validated by pavement response data from the field (14). They used FWD deflection data to validate their model. Shape of the loading area in their study was different from the FWD test load shape (10, 14). In the FEM, FWD load was idealized as a rectangle by Al-Qadi et al. (14) and as a quarter circle by Ahmed and Tarefder (10).

Load is one of the most important factors for prediction of pavement response. Pavement responses such as stress, strain, and deflection inside a pavement structure can be predicted by using a static or dynamic load analysis (1, 2). As static load is not a true representation of roadway moving traffic, several researchers have performed analyses by using dynamic loads (3–5). The basic difference between static and dynamic loads is time dependency. The magnitude of a load applied at a point on the pavement surface is constant over time in the case of a static load. However, the load varies with time in the case of a dynamic load (2, 6). Another factor for prediction of pavement responses is material property, which varies along three directions: vertical, horizontal, and transverse. A material is said to be isotropic in a three-dimensional (3-D) reference system if its property is the same in all three directions, Department of Civil Engineering, University of New Mexico, MSC 01 1070, 1 University of New Mexico, Albuquerque, NM 87131. Corresponding author: M. U. Ahmed, [email protected]. Transportation Research Record: Journal of the Transportation Research Board, No. 2369, Transportation Research Board of the National Academies, Washington, D.C., 2013, pp. 20–29. DOI: 10.3141/2369-03 20

Ahmed, Tarefder, and Islam

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To this end, this study performs 3-D dynamic FEAs of a pavement structure while considering material cross-anisotropy and using the quarter-circle shape of the FWD loading area as well as semicircle– rectangular and rectangular shapes of truck loading areas. In essence, cross-anisotropic behavior of an instrumented flexible pavement section is studied under FWD and actual truck loads.

The plan view of the instrumentation section is shown in Figure 2. Fourteen horizontal asphalt strain gauges were installed at the bottom of the HMA layer. Seven such gauges were placed along the longitudinal direction and seven others in the transverse direction of traffic. Six vertical asphalt strain gauges were installed at the bottom of the HMA layer. Four earth pressure cells were installed at different depths to measure vertical stress.

OBJECTIVES The main goal of this study is to examine the effects of cross-anisotropy on pavement response by dynamic FEA. Specific objectives are the following: • To determine the effects of cross-anisotropy of all layers and only the HMA layer on FWD deflections as well as on stress and strain at the embedded sensor locations in an instrumented pavement section on I-40, Mile Post 141 (MP 141), in New Mexico and • To measure the impacts of semicircular–rectangular and rectangular shapes of tire imprints on the cross-anisotropic response of the pavement. DESCRIPTION OF INSTRUMENTED PAVEMENT SECTION The cross section of the instrumented section (I-40, MP 141) that is used for FEM geometry is shown in Figure 1. The instrumented pavement consists of four layers, such as HMA at the surface; an aggregate layer at the base; a processed, placed, and compacted (PPC) layer; and a subgrade soil layer. The PPC layer is prepared by mixing reclaimed asphalt pavement from the surface as well as aggregate from the base layer and then compacting it in place. Figure 1 shows the elevations of strain gauges and pressure cells from the surface. Total thickness of the HMA layer is 26.7 cm (10.5 in.). This HMA layer consists of three lifts, each with a thickness of 8.9 cm (3.5 in.). The thickness of the base is 15.3 cm (6 in.), and the PPC layer is 20.3 cm (8 in).

DESCRIPTION OF FEM Model Geometry A pavement section can be idealized by two-dimensional (2-D) or 3-D geometry in FEM (14, 17). FEM with 2-D geometry is not compatible with different types of loading areas. For example, the 2-D axisymmetric model is convenient for the circular loading area, whereas the 2-D plane strain model is convenient for the rectangular loading area. In contrast, FEM with 3-D geometry is convenient for any arbitrary shape of loading area. In this study, FWD and truck loading areas are idealized by quarter-circle, rectangular, and semicircle–rectangular areas. In relation to memory space and analysis run time, a quartercube model has the advantage over the others because of its two axes of symmetry. The depth of this model is taken to be 50 times the loading radius so that the effect of the boundary can be ignored (18). The loading radius of the FWD test is 15.3 cm (6 in.). Overall dimensions (i.e., length, width, and depth) of this model are 5.08 × 5.08 × 7.62 m (200 × 200 × 300 in.). The number of layers as well as thicknesses of every layer is assigned in accordance with the earlier description of the instrumented pavement section. Generation of Mesh Figure 3 shows the mesh, load, and boundary conditions of FEM. Mesh is generated automatically so that the analysis time is reduced without compromising the accuracy of the FEM solution. An eightnoded brick element is used for mesh generation. Such an element

Traffic direction 3.5’’

HMA (third lift), E1, ν1, ρ1, h1

3.5’’

HMA (second lift), E1, ν1, ρ1,

3.5’’

HMA (first lift), E1, ν1, ρ1, h1

6.0’’

Aggregate base, E2, ν2, ρ2, h2

8.0’’

PPC, E3, ν3, ρ3, h3

Vertical strain gauge Horizontal strain gauge

Subgrade, E4, ν4, ρ4, h4

Earth pressure cell FIGURE 1 Instrumented pavement section (I-40, MP 141).

13.5’’

27.5’’

22

Transportation Research Record 2369

East Bound Lane I-40 (MP 141)

EPC and moisture probe on PPC 2V-ASG on top of first lift 2H-ASG on top of first lift 6V-ASG on top of base 12H-ASG on top of base EPC and moisture probe on base EPC and moisture probe on subgrade

3’

2’ 2’ 2’

2’

2’

Temperature Probe

2’

Solar Power

2’ 2’ 2’

Cabinet

2’ 2’

RCC Foundation

2’ 3’

Weather Station

2’ Axle Sensing Strip

3’

EPC on mid-base 5’

Traffic Direction

Passing lane

Driving lane

Shoulder

FIGURE 2 Plan view of instrumented pavement section (I-40) (1 ft = 30.48 cm; EPC = earth pressure cell; V-ASG = vertical asphalt strain gauge; H-ASG = horizontal asphalt strain gauge; RCC = reinforced cement concrete).

10.5” HMA 6” Aggregate base 8” PPC

Subgrade

FIGURE 3 Three-dimensional model for dynamic FEA.


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TABLE 2 Prony Series for ABAQUS– Computer-Aided Engineering (14)

was suggested by many researchers to model a 3-D FEM of flexible pavement (5, 13, 14). Fine mesh is assigned near the load to capture the high concentration of stresses and deflections in that region (19). For the rectangular loading area, a 3-D eight-noded element (C3D8) is used in the entire geometry. A combination of a six-noded wedge element and an eight-noded brick element is used for cases of quarter-circle and semicircle–rectangle loading areas. In this way, distortion of the mesh element is controlled to achieve an accurate solution. A coarse mesh is used in the region far from the loading area (19). Edge-biased structure-meshing patterns available in ABAQUS are used to obtain a smooth transition from fine mesh to coarse mesh. m= Boundary Conditions The bottom of this model is restrained to move along both horizontal and vertical directions (Figure 3). Vertical planes are restrained to move along the horizontal direction. The pavement surface deflects vertically during the FWD test and truck load. Vertical planes are free to move in the vertical direction. Two adjacent layers are assumed to be bonded at their interface (i.e., no slip is allowed, τ ≠ 0, where τ = shear stress). Material Properties Three lifts of HMA layers on top of the pavement model are assumed to have identical properties. The reason for that assumption is that these lifts have been constructed by using the same mix one after another. HMA is considered a viscoelastic material (20, 21). The elastic modulus of HMA as well as of the rest of the layers is backcalculated from FWD test data. The FWD test was conducted on each layer and lifts sequentially, starting from the subgrade. FWD deflections data were analyzed by using backcalculation software ELMOD. Backcalculated layer moduli as well as density and Poisson’s ratio used in backcalculation are summarized in Table 1. No laboratory tests were conducted in this study. HMA viscous properties were obtained from Al-Qadi et al. and are shown in Table 2 (14). Here, G is shear modulus, K is bulk modulus, and τ is reduced relaxation time. These parameters were obtained from the Prony series fitting by Al-Qadi et al. (14). Cross-anisotropy is assigned to materials by varying the magnitude of anisotropy parameters such as n and m, defined as follows: E n = 11 E22

(1)

i

Gi

Ki

τi

1 2 3 4

0.631 0.251 0.0847 0.0267

0.631 0.251 0.0847 0.0267

0.078 0.816 5.68 139

G12 E22

(2)

where E11 = modulus of elasticity on horizontal plane, E22 = modulus of elasticity on vertical plane, and G12 = shear modulus of elasticity. For cross-anisotropy, E11 = E33, where E33 = modulus of elasticity on the horizontal plane along the transverse direction. In addition, the shear modulus is assumed to be the same over three orthogonal planes. That is, G12 = G23 = G13 = G. Layers under the HMA layer (i.e., base, PPC, and subgrade) are assumed to be linear elastic. Backcalculated moduli summarized in Table 1 are used as elastic moduli. Cross-anisotropy in the modulus of elasticity is assigned to these layers by varying the n-value. Loading Type Two loading conditions were selected for the dynamic analysis. These were FWD and truck loading. Figure 4 shows the load magnitude over time during the load application. For FWD loading (Figure 4a), the highest value of the load was 40 kN (9 kips), and the load duration was 25 µs. When this load was applied on the circular area with a radius of 15.24 cm (6 in.), maximum pressure was found to be 548.82 kPa (79.6 psi). Pressure applied by an 80.07-kN (18-kip) truck was assumed to follow the shape in Figure 4b and was 827.4 kPa (120 psi) (22). Test Matrix To investigate the effect of HMA cross-anisotropy, two groups of analyses were performed in this study. In group one, the n-value was varied for every layer. In group two, n was varied for only the HMA layer. The values of n were 0.3, 0.5, 0.7, and 1.0 (isotropic). The value of m was chosen to be 0.38 according to Kim and was not varied in this study (23).

TABLE 1 Pavement Layer Material Material HMA Granular aggregate PPC Fine soil

Modulus of Elasticity (ksi)

Density (lb/ft3)

Poisson’s Ratio

250 30 25 15

145 135 120 110

0.35 0.4 0.4 0.45

Note: 1 ksi = 6.89 MPa; 1 lb/ft3 = 16.02 kg/m3.

DYNAMIC ANALYSIS In dynamic analysis, a pavement section is idealized by a mass, a damper, and a spring subjected to a force that varies in time (t). Then, the general equation of the pavement system can be written as shown in Hagström and Johansson (24): Mu + Cu + Ku = F ( t )

(3)

24


10 Loading amplitude

1.0

Load (kips)

8 6 4 2 0

0

5

10 15 20 Time (milliseconds) (a)

25

3.11 in. 4.15 in. 3.11 in.

0.8 0.6 0.4 0.2 0.0

0

0.02 0.03 0.04 0.01 Duration of wheel load (seconds) (b) 9.03 in.

6.22 in.

6.22 in.

(c)

(d)

FIGURE 4 Loading pattern and area: (a) FWD load, (b) truck load, (c) tire imprint area for semicircle–rectangle, and (d) tire imprint area for rectangle (1 kip = 4.45 kN).

RESULTS AND DISCUSSION

where M = mass of system, C = damping coefficient, K = stiffness of system, and . u, u, ü = displacement, velocity, and acceleration, respectively. Two algorithms used to solve the differential equation (Equation 3) are explicit and implicit algorithms (25). The explicit algorithm solves the equation at a later time than that for the state of the system at the current time. In contrast, the implicit algorithm solves the equation by using both the current and later states. Some advantages of the explicit algorithm are the following: large matrices need not be generated because the equations are decoupled; no matrix inversion is required because M and C are diagonal; and no iteration is required. Therefore, analysis time is short, but the algorithm is conditionally stable. The implicit algorithm is unconditionally stable. However, the implicit algorithm demands a long analysis time because stiffness matrices must be assembled and the matrix inverted to solve Equation 3. In this study, the explicit algorithm is used in ABAQUS. Stability of the solution from the explicit algorithm is achieved by controlling the time step during analysis. The time step is determined from the length of the smallest element and the wave velocity as follows (26): ∆tstable ≈

Lmin cd

(4)

where Lmin is the length of the smallest element and cd is the wave velocity defined as Equation 5 (27): cd =

E ρ

(5)

where E is the bulk modulus and ρ is the density. The time step used as ABAQUS input is kept smaller than the value determined from Equation 4 to assure the stability of the solution.

FWD Deflection Figure 5a compares the time–deflection histories directly under the FWD loading point by considering cross-anisotropy of all layers. The isotropic model (n = 1) shows the lowest value of peak deflection, which increases with a decrease in n-values. The peak of the time–deflection history is the closest to the field value for n = 0.3. Figure 5b shows surface deflections from the FEM simulation and the field by considering cross-anisotropy in the HMA layer only. Peak deflection increases with a decrease in n-values. However, field peak deflection matches with deflection for n = 0.5. Therefore, one may conclude that surface deflection at the loading point always increases with a decrease in n-values. The cross-anisotropic effect on deflections at the other FWD sensors is shown in Figure 6. In the FWD test, seven geophones were used to measure the surface deflection at 0, 20.3, 30.5, 45.7, 61.0, 91.4, and 152.4 cm (0, 8, 12, 18, 24, 36, and 60 in.). Figure 6a is for cross-anisotropy in all layers and shows that deflection decreases gradually with an increase in n-values at the first [0 cm (0 in.)] and second [20.3 cm (8 in.)] sensors. However, this trend is not strictly followed by the sensors farther from the load. Figure 6b is for crossanisotropy in the HMA layer and shows a decreasing trend with an increase in n-values. This trend is observed to the sixth [91.4 cm (36 in.)] sensor. At the seventh [152.4 cm (60 in.)] sensor, deflection is almost the same for all n-values. Therefore, the effect of crossanisotropy diminishes as the distance from the FWD load increases. In addition, deflection attributable to HMA cross-anisotropy is greater than that for all-layer cross-anisotropy at the loading point. Embedded Sensor Stress–Strain Semicircle–Rectangle Loading Area The longitudinal strain (horizontal strain) is plotted in Figure 7, a and b, for n-values. In both cases of cross-anisotropy, the figure


25

Time (milliseconds) 10 15 20 25

5

30

35

-1 -3 -5 -7

Field n=0.3 n=0.7 n=1

-9 -11 -13

0 Surface deflection (mil)

Surface deflection (mil)

0

(a)

Time (milliseconds) 10 15 20 25

5

35

30

-1 -3 -5 -7

Field n=0.3 n=0.7 n=1

-9 -11 -13

(b)

12 10

n=0.3

8

n=0.5

6

n=0.7

4

n=1

2 0 0

8

12 18 24 36 Radial offset (in.) (a)



FIGURE 5 Comparison of time–deflection histories between FEM and FWD tests: (a) cross-anisotropy of all layers and (b) cross-anisotropy of HMA layer (1 mil = 25.4 μm).

12 n=0.3

10

n=0.5

8

n=0.7

6

n=1

4 2 0

60

0

8

12 18 24 36 Radial offset (in.) (b)

60

250

n=0.3 n=0.5 n=0.7 n=1

200 150 100 50 0 0

0.01

0.02 0.03 Time (seconds)

0.04

0.05

0.04

0.05

Longitudinal strain (µe)


FIGURE 6 Effect of cross-anisotropy on peak deflections at sensors: (a) cross-anisotropy of all layers and (b) cross-anisotropy of HMA layer.

250

n=0.3 n=0.5 n=0.7 n=1

200 150 100 50 0 0

0.01

0

0

0.01

Time (seconds) 0.02 0.03

-2 -4 -6

n=0.3 n=0.5 n=0.7 n=1

-8 -10 -12

(c)

Vertical stress (psi)


(a)

0.00 0

0.01

0.02 0.03 Time (seconds) (b) Time (seconds) 0.02 0.03

0.04

0.05

0.04

0.05

-2 -4 -6

n=0.3 n=0.5 n=0.7 n=1

-8 -10 -12

(d)

FIGURE 7 Effect of cross-anisotropy on longitudinal strain for (a) all layers and (b) HMA layer and on vertical stress for (c) all layers and (d) HMA layer (all for semicircle–rectangle loading areas).

26


150

n=0.5 Field

125 100 75 50 25 0 0

0.01


0.04

0.05

0.04

0.05

in the field is closer to the simulated value at n-values of 0.7 for both cases of cross-anisotropy. Figure 8c shows that vertical stress for cross-anisotropy in all the layers is 59.23 kPa (8.59 psi), while Figure 8d shows that in the HMA layer it is 57.22 kPa (8.3 psi). The field vertical stress is 54.95 kPa (7.97 psi). Therefore, the HMA cross-anisotropy predicts the vertical stress closest to the field value.

Rectangle Loading Area Figure 9, a and b, shows the longitudinal strain variation with n-values for both cases of cross-anisotropy. Similar to the variation in strain observed for the semicircle–rectangle loading area, the longitudinal strain for the rectangle loading area increases as the n-value decreases. In addition, simulation with cross-anisotropy in all layers predicts greater strain than that by cross-anisotropy in the HMA layer. Figure 9, c and d, shows the vertical stress variation with the n-value for both cases of cross-anisotropy. Vertical stress increases as the n-value decreases. Cross-anisotropy in all layers predicts greater vertical stress than that predicted by the HMA layer crossanisotropy. For both cases of cross-anisotropy, the FEM with the rectangle loading area predicts vertical stress greater than with the semicircle–rectangle loading area. Figure 10, a and b, shows the comparison of the longitudinal strain between the field and FEM. These parts of the figure show that the field strain is closest to strain at the n-value of 1.0 (i.e., isotropic model). Figure 10, c and d, shows the comparison of vertical stress between the field and FEM for both cases of cross-anisotropy. Isotropic simulation predicts the vertical stress that is the closest to that obtained from the field instrumentation data.



shows that horizontal strain increases with a decrease in n-values. This decrease in n-values means a decrease in the horizontal E-value while the vertical E-value is constant. Horizontal strain increases as the horizontal modulus decreases. The horizontal strain is noticeably greater for cross-anisotropy in all the layers than that for crossanisotropy in the HMA layer. In the case of cross-anisotropy in all the layers, the longitudinal strain is affected by the n-value in every layer. In contrast, the longitudinal strain layer under the HMA layer is not affected by the n-value in the case of cross-anisotropy in the HMA layer. Therefore, the simulation with cross-anisotropy in all the layers predicts greater longitudinal strain than that from the simulation with HMA cross-anisotropy that is based on compatibility. Vertical stress variations are shown in Figure 7, c and d. Vertical stress increases as the n-value decreases. For n = 0.3, stress is maximum. Stress is minimum for n = 1 (isotropic model). A similar trend is observed in both cases of cross-anisotropy. As noted earlier, vertical deflection increases as the n-value decreases for the same amount of vertical load. An increase in vertical deflection causes an increase in vertical strain. Therefore, vertical stress increases with vertical strain because the vertical E-value remains constant with a decrease in n-values. Figure 8, a and b, compares the longitudinal strain to that collected from the instrumented section. The figure shows that field strain is the closest to strain at an n-value of 0.5 for both cases of cross-anisotropy. In the case of cross-anisotropy in all the layers, the horizontal strain is 140.83 µe (microstrain) (Figure 8a). It is almost equal to the field strain of 141.02 µe. In the case of the HMA layer cross-anisotropy, the longitudinal strain is 127.34 µe (Figure 8b). It is smaller than the field value, and the difference is 13.68 µe. Figure 8, c and d, shows the comparison between the vertical stress from FEM and the field value. The vertical stress measured

150

n=0.5 Field

125 100 75 50 25 0 0

0.01

(a)

0 -1 -2 -3 -4 -5 -6 -7 -8 -9

0.01

Time (seconds) 0.02 0.03

n=1 n=0.7 Field (c)

0 Vertical stress (psi)


0

0 -1 -2 -3 -4 -5 -6 -7 -8 -9

0.01

0.02 0.03 Time (seconds) (b) Time (seconds) 0.02 0.03

0.04

0.05

0.04

0.05

n=1 n=0.7 Field (d)

FIGURE 8 Comparison of longitudinal strain for cross-anisotropy for (a) all layers and (b) HMA layer and of vertical stress for cross-anisotropy for (c) all layers and (d) HMA layer (all between field and FEM with semicircle–rectangle loading area).

27

350

n=0.3 n=0.5 n=0.7 n=1

300 250 200 150 100 50




0

350

n=0.3 n=0.5 n=0.7 n=1

300 250 200 150 100 50 0

0

0.01


0.04

0.05

0

0.01

0.04

0.05

0

0.01

(a)

Time (seconds) 0.01

0.02

0.03

0

0

-3

-3

-6 -9 n=0.3 n=0.5 n=0.7 n=1

-12 -15 -18

0.04

0.05

0.04

0.05

Time (seconds)



0

0.02 0.03 Time (seconds) (b)

0.03

-6 -9

n=0.3 n=0.5 n=0.7 n=1

-12 -15 -18

(c)

0.02

(d)

160 140 120 100 80 60 40 20 0

n=1 Field

0

0.01

0

0.01

0.02 0.03 Time (seconds) (a)



FIGURE 9 Effect of cross-anisotropy on longitudinal strain for (a) all layers and (b) HMA layer and on vertical stress for (c) all layers and (d) HMA layer (all for rectangle loading area).

160 140 120 100 80 60 40 20 0

n=1 Field

0.04

0.05

0

0.01

0.04

0.05

0

0.01

-1 -3 -5 -7

n=1

-9

Field

-11

(c)



0.03

0.04

0.05

0.04

0.05

Time (seconds)

Time (seconds) 0.02

0.02 0.03 Time (seconds) (b)

0.02

0.03

-1 -3 -5 -7

n=1

-9 -11

Field (d)

FIGURE 10 Comparison of longitudinal strain for cross-anisotropy for (a) all layers and (b) HMA layer and of vertical stress for cross-anisotropy for (c) all layers and (d) HMA layer (all between field and FEM with rectangle loading area).

28


TABLE 3 Stress–Strain Comparison Between Semicircle–Rectangle Loading Area and Rectangle Loading Area Cross-Anisotropy in All Layers

n-Value Variation 0.3 0.5 0.7 1

Cross-Anisotropy in HMA Layer

Horizontal Strain (µe)

Vertical Stress (psi)

Horizontal Strain (µe)

Vertical Stress (psi)

Semicircle– Rectangle

Rectangle


Rectangle


Rectangle


Rectangle

204.36 140.83 112.99 80.15

317.49 220.40 174.54 125.69

10.24 9.13 8.59 7.10

16.16 13.69 12.45 9.91

162.41 127.34 109.20 80.15

265.79 202.84 169.71 125.69

9.42 8.75 8.30 7.10

14.75 13.03 11.97 9.91

Semicircle–Rectangle Area Versus Rectangle Loading Area Longitudinal (horizontal) strain and vertical stresses determined for the semicircle–rectangle area and the rectangle loading area are compared and summarized in Table 3. The stress–strain simulated by the rectangle loading area is always greater than that simulated by the semicircle–rectangle loading area for both the cross-anisotropic cases. Table 3 shows that the ratio of the horizontal strain of the two different loading areas is about 1.54 to 1.57, whereas the ratio of the vertical stresses varies between 1.30 and 1.56. For isotropic layer material, the vertical stress difference between the two loading areas is small, whereas this difference increases as the effect of crossanisotropy becomes more significant (i.e., as the n-value decreases). A similar trend is observed in the case of the horizontal strain. In the isotropic material, the E-value is the same in all three orthogonal directions. For this reason, the stress variation is small. However, in cross-anisotropic material, the E-value is not the same in the vertical and horizontal directions. This difference leads to an increase in stress and strain variation. APPLICATION Most backcalculation software assumes isotropic material property (modulus of elasticity) for each layer of pavement (2). Surface deflections determined by using isotropic layer moduli may show a significant difference from the field deflections. Consequently, an error minimization algorithm in the backcalculation process may produce a combination of layer moduli that are not representative of field values. This study helps in the understanding of the variation of pavement response attributable to cross-anisotropy. Backcalculation software can be improved by the integration of the findings of this study on cross-anisotropy. The Mechanistic–Empirical Pavement Design Guide (MEPDG) (22) uses backcalculated moduli as inputs. Therefore, it is possible to find how cross-anisotropy affects the design thickness from MEPDG. To consider the effect of cross-anisotropy of different materials, backcalculation and pavement design software should use different n-values for specific layer materials. Typical suggested ranges of n-values for layer materials are 0.7 to 1.0 for HMA, 0.15 to 0.5 for granular aggregate, and 0.15 to 0.5 for PPC (8, 13). The n-value for subgrade soil varies over a wide range because of inherent variability in naturally occurring soils. On the basis of previous studies, the n-value can be assumed to be 0.15 to 1.0 for a subgrade (8, 12, 13).

CONCLUSION Dynamic analysis with cross-anisotropy and subsequent comparison with field instrumentation data show that the cross-anisotropic behavior of pavement material is extremely important. Ignoring the effect of cross-anisotropy may lead to an erroneous analysis resulting from inaccurate FWD backcalculation and MEPDG inputs. The following conclusions can be drawn: • FWD deflections are significantly sensitive to cross-anisotropy. An increase in n-value (ratio of horizontal to vertical modulus of elasticity) causes a decrease in deflection. Clearly, FWD surface deflections change with a change in n-values for a specific combination of layer moduli. Therefore, the n-value of the layer material needs to be varied in the deflection-matching algorithm during the backcalculation process. • Stress–strain data are also sensitive to cross-anisotropy. Both vertical stress and longitudinal strain increase with a decrease in n-values. Predicted stress and strain that consider cross-anisotropy in all layers are higher than those that consider cross-anisotropy in the HMA layer only. Vertical stress and longitudinal strain simulated by cross-anisotropy are close to field-measured values, and therefore, cross-anisotropy in all layers is recommended for FWD back calculation and pavement analysis. The authors recommend that, in the backcalculation process, stress and strain at specific points be included in the error minimization subroutine. Inclusion of stress and strain accompanied by n-value variation will increase the reliability of backcalculated FWD modulus. Similarly, consideration of n-value variation in pavement analysis by MEPDG will ensure that the analysis will be more realistic. • Predicted stress and strain values when the shape of the loading area is considered a rectangle are higher than those when the shape of the loading area is considered a semicircle–rectangle. The semicircle–rectangle shape is recommended for the loading imprint area because the predicted pavement responses from using a loading area of this shape are close to the field-measured response. ACKNOWLEDGMENTS The authors thank the New Mexico Department of Transportation for funding this study. Special thanks go to the members of the field coring and testing team for providing the required field data.


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