The moduli of the unbound layers were found to be stress dependent following the well known relationship:
Equation 7: Non-linear moduli of unbound materials.
| Where |
E is the modulus,
stress is the bulk stress for granular materials and the deviator stress for cohesive materials,
k1, k2 (positive for bulk stress and negative for deviator stress) and p are constants (p = 0.1 MPa). |
More controversially, it was also found that the modulus of the unbound materials varied with the stiffness of the layers above the material. For granular layers this effect is the opposite of what would be expected based on Equation 7. A decrease in the stiffness of the layers above a granular layer would be expected to cause an increase in the bulk stress in the granular material and, therefore, an increase in the modulus, whereas the opposite effect is observed. The effect is in good agreement with the observation made by Richter (2006) that the moduli of granular layers, backcalculated from FWD tests on LTPP Seasonal Monitoring sections, tend to decrease, instead of increase, with increasing bulk stress.
To allow for this effect, the stiffness of each unbound layer was modeled as a function of the bending stiffness of the layers above it:
Equation 8: Modulus of each unbound layer as a function of the bending stiffness of the layers above it.
| Where |
Eo is the modulus (of layer n) at the reference stiffness,
S is the combined bending stiffness of the layers above layer n,
Sref is the reference stiffness (a value of 35003 N·mm was used here),
hi is the thickness of layer i in mm, and
Ei is the modulus of layer i in MPa. |
The Stiffness factor was determined from regression analyses of moduli backcalculated from FWD tests. Stiffness factor represents “fraction of the decrease in the stiffness of the layers above the one under consideration”.
Permanent deformation of unbound layers
The model for permanent deformation of the unbound layers, dp, is given in Equation 9, where MN is the number of load applications in millions, με is the vertical compressive strain at the top of the layer and E is the modulus. The reference constants are μεref = 1000 μstrain and Eref = 40 MPa. The relationship was derived from tests in the Danish Road Testing Machine during the International Pavement Subgrade Performance Study (2005):
Equation 9: Permanent deformation of unbound layers.
The model agreed well with the measured permanent deformations in the unbound materials, but it should be noted that the permanent deformations were all quite small.
Example of Simulation
The example presented here is from Goal 1 (HVS test numbered 503RF). The parameters used for the models given above are shown in Table 1.
Table 1: Parameters used in simulation of test 503RF
The pavement had two layers of conventional dense graded asphalt (AC, top layer 74 mm, bottom layer 88 mm), an aggregate base (AB of 274 mm), and an aggregate subbase (AS of 305 mm) on a clay subgrade. Most of the material parameter values were derived from laboratory tests, the remaining from FWD tests or from calibration using a similar test section (numbered 501RF). A reference temperature of 20 ºC and a reference loading time of 0.015 sec (corresponding roughly to 10 Hz) were used for the AC modulus.
The load was a dual wheel with radial tires at a pressure of 0.69 MPa and a loading speed of approximately 7.6 km/h. The loads were laterally distributed over a width of 1000 mm. The first load level was 40 kN, it was then increased to 80 kN and finally to 100 kN (for most of the load applications).
Some of the resilient deflections measured with the MDDs under a 40 kN wheel load are shown in Figure 3. The legend M is for measured deflections, shown with a fully drawn line, and C is for calculated deflections, shown with a dotted line. Deflections were measured and calculated at the top of the AC (legend 0, for depth 0 mm), close to the top of the AB (legend 137, for depth 137 mm from AC surface) and close to the top of the subgrade (legend 640, for depth 640 mm). The large increase in resilient deflections during the test may be noticed. The first visible cracking was recorded at approximately 650,000 load applications, when almost all of the increase in deflection had already taken place.
Figure 3: Resilient deflections section 503RF, 40 kN.
The first step in the calibration process is to ensure that the response calculated by the mechanistic model is reasonably correct, for the duration of the test. Once the response model results in a satisfactory prediction of the measured resilient deflections, then the empirical relationships for permanent deformation may be calibrated.
The permanent deformation of the asphalt layers, measured and calculated, are shown in Figure 4, and the total permanent deformation at the pavement surface is shown in Figure 5, as measured by MDD 4 at the surface, as the average of the measured surface profile, measured by laser profilometer, and as calculated by CalME.
Figure 4: Permanent deformation of the asphalt layers, measured and calculated.
Figure 5: Permanent deformation at the surface of the pavement.
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