Figure 1 Converge of Simple Grid Mesh
It is important to study the results of any numerical procedure in order to provide validation that solutions are accurate. EverStressFE is designed to analyze complex pavement structures for which analytical solutions do not necessarily exist. Thus, one way to validate the model is to perform relatively simple analyses in order to facilitate comparisons with available analytical solutions. Another way to validate the model is to analyze a complex structure for which experimental measurements are available. This section details these validation efforts.
A series of FE models were solved and a limited set of results was compared to theoretical values in order to investigate the accuracy and convergence of the FE solution. Theoretical values were obtained for vertical strain, radial strain, and vertical displacement from equations presented by Huang (1993) for a load applied over a single circular area through a flexible plate to a homogeneous half-space. The parameters required to solve the theoretical solution at a specific depth include: elastic modulus, Poisson’s ratio, uniform tire contact stress, and radius of tire contact area. A single Poisson’s ratio and elastic modulus were used in the FE model and a single circular wheel load was applied. Infinite elements were applied normal to the plane of maximum finite depth as well as the vertical planes away from the applied wheel load. Fixed displacement symmetry boundary conditions were applied normal to the planes of symmetry. The finite length, width, and depth of the FE model were always fixed at the same value, but the value was varied to examine its effect. For all results presented, the theoretical solution was taken directly beneath the center of the applied wheel load at a depth equal to 2.5 times the radius of the wheel load.
First, the finite dimensions were arbitrarily fixed at a value equal to ten times the radius of the applied wheel load and the length of the finite elements was examined. All meshes were generated by implementing a simple element grid throughout the finite domain, such that every finite element was a cube. The results are shown in Figure 1 below. It is clear that the solutions are converging for all three of the parameters. However, the solution for vertical displacement is converging to a value that is about 3.3% less than the theoretical solution. The FE mesh with the smallest element size contained about 140,000 nodes and required about 1.3 GB of memory to solve. It is apparent that further refinements in the mesh will have little effect on the solution.
Figure 1 Converge of Simple Grid Mesh
Thus, the second set of analyses was conducted with a locally refined mesh. For all meshes, the plan-view dimensions of the locally refined area were taken as three times the radius of the applied wheel load, which followed recommendations by Clapp (2007). The finite dimensions were again fixed at a value equal to ten times the radius of the applied wheel load. The number of element divisions in the coarse regions was fixed at 6, such that only the element size within the locally refined region was varied. The results are shown in Figure 2. It was found that all three of the parameters converged to relatively constant values when the element size was about 20% (0.05/mm in Figure 2) of the radius of the applied wheel load. The solutions for vertical strain, radial strain, and vertical displacement were within 1.2%, 0.8%, and 3.3% of the theoretical solutions, respectively. Additional reductions in element size did not significantly change these values.
Figure 2 Convergence of Locally Refined Mesh
Finally, the finite extents of the FE model were examined. Both infinite elements and fixed boundary conditions were examined here to highlight the effect of this. The length/width of the elements in the locally refined region were fixed at about 17% of the load radius, which is more refined recommended in the previous paragraph. The ratio of the finite domain size to the radius of the applied load was varied between 4 and 40 and the results are shown in Figure 3. Convergence of the strains is much more rapid when infinite elements are used as opposed to fixed boundary conditions. The strains have converged to within about 1% of theoretical values with a finite domain size equal to ten times the radius of the wheel load with infinite elements. The vertical displacements are under-predicted by 3.3% and 22% for the meshes with infinite and fixed boundaries, respectively, with a finite domain size equal to ten times the radius of the wheel load. This large discrepancy clearly demonstrates the benefits of using infinite elements.
Figure 3 Convergence as a Function of Finite Mesh Size
The recommendations for FE meshing based on this study are as follows:
The strains predicted by the mesh based on these recommendations (the mesh with finite size 10 times the radius of the applied load as shown in Figure 3) are shown in Figure 4 below as well as the theoretical solutions through a depth of 10 times the radius of the applied load. It appears that the vertical and radial strains are accurately predicted at all depths.
Figure 4 Comparison of final FE Model to Analytic Solution
Starting with the recommendations presented in the previous section, a series of models were solved using realistic flexible pavement properties in order to more closely examine the effect of meshing parameters on the accuracy and convergence of the solution. Specifically, the goal of this study was to determine values for meshing parameters that would yield minimal solution times while retaining accurate solutions. All meshing parameters were examined, although parameters corresponding to X and Y (plan view) dimensions were always held equal in each direction.
The thickness of the flexible asphalt concrete (AC) layer and base layer were taken as 150 mm and 300 mm, respectively. The subgrade layer was considered to be infinitely thick. However, a portion of the subgrade was also modeled with finite elements (as required by EverStressFE) with a varying total thickness in order to examine the effect of this parameter. The elastic moduli for the three material layers were taken as 3000 MPa, 300 MPa, and 50 MPa, respectively. Poisson’s ratio was fixed at 0.35 for all materials. Displacements were fixed normal to the planes of symmetry, which is always the case with EverStressFE. Infinite elements were utilized along all outer planes. The finite length and width of the pavement section in plan view was varied to examine its effect.
For all analyses, the wheel load was modeled with a single tire having a circular contact patch area. The total applied load was taken as 40 kN and the tire pressure was taken as 690 kPa, which meant that the radius of the contact area was about 135.8 mm.
The parameters varied included the following:
The outputs examined were primarily vertical strains; however the maximum vertical (surface) displacement was also recorded. The chosen vertical strains were the same strains that would be used in rutting prediction models: average strain in the AC layer, average strain in the base layer, strain at the top of the subgrade layer, and strain at a depth of 150 mm into the subgrade layer. Additionally, the vertical strain at the middle of the base layer and the horizontal strain at the bottom of the asphalt layer were examined. All strains were taken directly beneath the center of the applied wheel load (at the planes of symmetry) in plan-view. Outputs were compared to two benchmark solutions. The first was a EverStressFE solution obtained with a very highly refined mesh. The second was a solution obtained with EverStress, which is based on analytical solutions for layered elastic systems.
It was found that length/width of the total plan-view finite region should be 4000 mm in order to ensure a convergent solution. The solutions for total length/width of 4000 mm are only slightly better than with a total length/width of 3000 mm, but the solution times are equal. On the other hand, the solution time for total length/width of 6000 mm is increased drastically due to the fact that multiple Newton iterations are required in the solver. Further, the solutions are practically equal to the solutions with total length/width of 4000 mm.
It was found that the subgrade region should be divided into two sub-layers, with respect to meshing, where each has identical material properties. The upper sub-layer had a fixed thickness of 150 mm, which was chosen so that the region of interest could be separated from the region that is only present for solution convergence. This also places a node at the exact location of the desired strain output at a depth of 150 mm into the subgrade, which is beneficial since solutions are most accurate at nodal locations. The optimal thickness of the lower subgrade sub-layer was found to be 3000 mm, although the solutions and run times for a thickness of 4000 mm were very similar.
The length/width of the refined region in plan view should be about 2-3 times the dimensions of the applied rectangular patch load. For refined region dimensions of 200 mm, 300 mm, and 400 mm, the solutions and run times were all similar. Generally, increasing the size of the refined region results in a decrease in solution accuracy and decreasing the size results in increased solution time, assuming the number of elements is held constant in each case.
The number of element divisions along the length/width of the refined region should be either 6 or 9 in each direction depending on the desired combination of accuracy and solution time. Increasing the number of element divisions from 9 to 12 along the length/width has a mixed effect on the solutions and increases the solution time by over 50%. On the other hand, decreasing the number of element divisions from 9 to 6 along the length/width decreases the solution time by about 48% and appears to have a minimal effect on solution accuracy (all solutions are still within 1% of both benchmark solutions) Nonetheless, it is recommended that at least 9 element divisions along the length/width should be used for general practice, especially if accurate solutions are needed for the AC and base layers. If solution time is the biggest concern (and some solution inaccuracy can be tolerated), using 6 element divisions along the length/width, as opposed to 9, is an efficient way to reduce solution time without significantly degrading the solution. These recommendations are based on the assumption that only typical rutting strains need to be determined accurately. If accurate strain predictions are needed at particular points in the asphalt or base layer (e.g. the maximum horizontal strain anywhere in the asphalt layer), additional refinement may be necessary.
The number of element divisions along the length/width of coarse region should be taken as 6. Increasing this number to 8 causes small mixed results to the solutions, but increases the solution time by over 20%. Decreasing this number to 4 again causes small mixed results to the solutions, but once again surprisingly increases the solution time due to ill-conditioning of the system stiffness matrix that occurs as a result of the poor element aspect ratios. A further reduction of this number to 2 noticeably deteriorates the solutions and again results in increased solution time.
The number of element divisions through the thickness of the AC layer, base layer, and upper subgrade layer may be taken as 2 if the only solutions of interest are the 4 traditional strains used to predict rutting. Increasing this number has practically no effect on these solutions. However, it is recommended to use 4 elements divisions through the thickness of the lower subgrade layer in order to obtain more accurate (within 1% as opposed to 3%) predictions of the average AC and base layer strains. It is also necessary to use 4 element divisions through the thickness of the base layer if accurate (within 1% as opposed to 12%) predictions are needed for the strain in the middle of the base layer.
Based on the above recommendations, two of the models were solved on a typical high-end desktop PC (as of May, 2005). This computer had an AMD Athlon 64 (3800+) 2.41 GHz processor and 2 GB of RAM. Both FE models had plan-view finite dimensions of 4000 mm, upper subgrade sub-layer thickness of 150 mm, lower subgrade sub-layer thickness of 3000 mm, refined region dimensions of 300 mm, 6 element divisions along the length/width of the coarse region, 2 element divisions through the thickness of the AC layer, 4 element divisions through the thickness of the base layer, 2 element divisions through the thickness of the upper subgrade sub-layer, and 4 element divisions through the thickness of the lower subgrade sub-layer. The first model had 9 element divisions along the length/width of the refined region, whereas the second model had 6. These models required about 122 MB and 91 MB of RAM and took 33 seconds and 17 seconds to solve, respectively.
Two additional analyses were conducted to examine the effect of introducing contact constraints to the FE model. These two FE models were identical to the models described above, except that interface elements with distributed stiffness of 1 N/mm^3 were added at the AC/base interface and the base/subgrade interface. The FE solver used in these analyses uses a nonlinear solution technique to handle the contact constraints, which increases solution time. These models required about 141 MB and 107 MB of RAM and took 60 seconds and 38 seconds to solve, respectively.
The final recommended parameters corresponding to mesh density are those that are set as default values in EverStressFE. The number of elements through the thickness of each layer is increased as compared to recommended values. This is done as a precaution because the recommended values were based on a very simple load case and may not be sufficient for other loads. Note that the default dimensions of the refined region and overall finite region do not correspond to recommended values because recommendations are generally based on relative magnitudes. It may be necessary to study the effect of all meshing parameters to achieve convergence in general cases, especially those that involve unique and/or complex loading.
As further validation, a model was solved with EverStressFE with plan-view finite dimensions of 4000 mm, upper subgrade sub-layer thickness of 150 mm, lower subgrade sub-layer thickness of 3000 mm, refined region dimensions of 300 mm, 9 element divisions along the length/width of the refined region, 6 element divisions along the length/width of the coarse region, 2 element divisions through the thickness of the AC layer, 4 element divisions through the thickness of the base layer, 2 element divisions through the thickness of the upper subgrade sub-layer, and 4 element divisions through the thickness of the lower subgrade sub-layer. The vertical (εzz) and radial (εxx) strains are compared to the EverStress (analytical) solution in Figure 5, where it is clear that solutions are nearly identical.
Figure 5 Comparison of EverStress (Analytical) and EverStressFE Solutions
A recent study was conducted by Henry et al. (in review) to examine the effectiveness of geogrid in relatively thick pavement sections. The underlying code in EverStressFE was extended in this study to account for the geogrid reinforcement. However, the basic code was also directly utilized to predict strains in unreinforced test sections. The test sections in this study consisted of asphalt concrete, base course, and subgrade material.
The modulus of the asphalt concrete layer was measured with independent laboratory testing, but the moduli of the soils were only able to be estimated with Falling Weight Deflectometer (FWD) testing. FWD testing is a widely accepted method of estimating layer moduli of in-situ materials based on measured surface deflections. However, it was not known whether the moduli determined by this method accurately reflected the moduli of the materials in these specific test sections.
In this study, additional instrumentation was embedded within the pavement layers, which provided another means to estimate layer moduli. The strains measured in the test section were compared to those predicted by EverStressFE. Optimal layer moduli were identified as those which provided the best fit between the strains predicted by EverStressFE and measured in the test section. It was found that the optimal soil layer moduli were similar to those obtained from FWD testing. The optimal layer moduli were 81.4 MPa and 50.9 MPa for the base and subgrade layers, respectively, which compare very well to the average FWD values of 87.6 MPa and 51.6 MPa.
The strains predicted by EverStressFE using the optimal layer moduli as well as the measured values are shown in Figure 7 as a function of layer number from the pavement surface. Each layer corresponds to roughly 150 mm of thickness. The strains predicted by EverStressFE are generally in very good agreement with the values measured in the laboratory.
Figure 6 Comparison of Microstrains Predicted by EverStressFE and Measured in a Test Section by Henry et al. (in printing)