Linear Elastic Analysis

5.2 Static Analysis

In this section, the results of the analyses are categorized according to the constitutive model used to simulate the responses of granular layers, resulting in four types of analysis.

The analyses in which all materials are assumed to behave linear elastically form the first type of simulation. In this class, mesh refinement and effects of boundary condition are investigated. The results can be compared with the other classes.

The second class of analysis consists of those with nonlinear elastic constitutive models. The different nonlinear elastic models previously reviewed in Chapter 2 are now are implemented in the numerical simulation and the results are presented.

The third class of analysis considers the effect of using a plastic cap to simulate the failure of granular materials. This class of analysis, however, does not assume nonlinear behaviour while in the elastic domain.

The fourth class of analysis considers the nonlinearity of material in the elastic domain while the plastic cap is also used. Finally, the results from all four classes are compared and discussed.

5.2.1 Linear Elastic Analysis

The first part of the linear elastic analysis studies the effects of mesh refinements and model dimensions and the results are compared using the linear elastic software. The main aim of this part of the study is to make decisions about further

171 simulations and determine the effects of simplification, if any. It should be mentioned that the results of this geometrical investigation were published in a paper by ASCE (Ghadimi et al. 2013), and the same results and explanations are reused here with permission from the publisher.

The sample section of layered pavement with the same material properties and loading characteristics are modelled in CIRCLY, KENLAYER and ABAQUS.

Figure 5.1 illustrates the geometrical dimensions of the modelled pavement. The material properties are listed in Table 5.1. All layers are assumed to behave linear elastically under a 0.75 MPa pressure loading, which is applied over a circular area with a 90 mm radius. This is the standard representation of tyre pressure in Austroads (2004).

Figure 5.2 illustrates the geometry of the first constructed model in KENLAYER and CIRCLY. To investigate the effects of geometrical parameters on the analysis, the material properties are assumed to be consistent throughout the whole analysis.

Table 5.1- Material Properties for KENLAYER and CIRCLY Programs Layer Thickness (mm) Elastic Modulus

(MPa) Poisson’s Ratio

Asphalt (AC) 100 or 200 2800 0.4

Granular

(Base/Subbase) 400 or 500 500 0.35

Subgrade Infinite 62 0.4

172 Figure 5.1- GeometryConstructed Model in KENLAYER and CIRCLY

Figure 5.2 illustrates the geometry of the FEM axisymmetric model. The geometry of the model in ABAQUS cannot be the same as in CIRCLY and KENLAYER, as the horizontal and vertical dimensions must be finite in the model. As discussed in Chapter 2, to overcome this problem Duncan, Monismith, and Wilson (1968) suggested a dimension of 50-times R (loading radius) in the vertical and 12-times R in the horizontal direction. Kim, Tutumluer, and Kwon (2009) found good agreement between the results of the FE analysis and KENLAYER when the model dimension was 140-times R in the vertical and 20-times R in the horizontal direction. In this study, the dimensions of the model have been selected as 55.55-times R in the horizontal direction and 166.70-times R in the vertical direction. The same ratio has been selected for the plane strain and three-dimensional models.

AC 10 cm Base 40 cm

Subgrade Contact Radius 9,2 cm

173

Figure 5.2- Geometry of The Axisymmetric Model

The next step is to investigate the mesh density and element type. Figure 5.3 illustrates the constructed FE meshes for the axisymmetric, plane strain and 3-D analyses.

174 Figure 5.3-Mesh Distributions for The Models

175 The vertical boundaries of the finite element model are modelled using roller boundary conditions which permit displacement in the vertical direction but prohibit it in the horizontal direction. The lower boundary of the model is fixed in every direction (called encastré). For axisymmetric and 3-D models, two different types of element and two mesh densifications were modelled separately.

Three axisymmetric models were investigated: the dense axisymmetric model has 12,656 8-node biquadratic axisymmetric quadrilateral elements; a normal axisymmetric model with 2280 of the same elements; and for the same number of elements (2280) a model was constructed with 4-node bilinear axisymmetric quadrilateral elements.

The three-dimensional model was also investigated by varying the element types and mesh density. The first model had 86,730 20-node quadratic brick elements;

the second model had 35,000 of the same elements and the third model had 35,000 8-node linear brick elements.

The effects of model dimensions is now investigated. The first step of this analysis is to compare the elastic solutions and FEM solutions in order to determine the range of induced errors in the approximation. In this comparison it was observed that while there was relative agreement among all constructed models, the plane strain results were out of range and not a reasonable representation of the situation.

176 Figure 5.4- Surface Deflection of Plane Strain Model

Figure 5.4, Figure 5.5 and Figure 5.6 show the contours of surface deflection for the three types of model. It is clear from these figures that although the shape of the contours in plane strain analysis is acceptable, the values are significantly larger than those calculated from the three-dimensional and axisymmetric models.

Figure 5.7 illustrates the comparison between the models in terms of surface deflection. It can be seen that all of the FE models demonstrate the least surface deflection. This is not unusual for it is generally accepted that FEM is slightly stiffer than the actual analytical solution (Helwany 2006). More importantly, there is no meaningful difference between the models with dense or normal meshes.

This implies mesh independency of the FE models.

177 Figure 5.5- Surface Deflection of Axisymmetric Model

Figure 5.6-Surface Deflection of 3-D Model

178 Figure 5.7-Comparison of the Surface Deflections for Different Models

There are four critical responses which play a vital role in flexible pavement design. These responses are surface deflection at the centre of loading, tensile strain at the bottom of the asphalt layer, vertical strain, and stress at the top of the subgrade layer. The values of these four responses for each model are presented in Table 5.2. The plane strain model yielded values considered extreme and not acceptable. For example, when the general surface deflection resulting from the other model was between 0.22 and 0.25 mm, the value calculated by plane strain was 3.228 mm which almost 13 times higher than 0.25 mm. This larger value is attributed to the effect of the loading condition, which was a strip of distributed pressure instead of a circular area as is the case in the 3-D, axisymmetric and analytical solutions. Therefore it can be said that the plane strain assumption will lead to an overestimation of pavement damage.

179 Table 5.2- Comparison of the Results for Different Models

Model

The remaining models were in general agreement, however, the results for the 3-D model with 8-noded elements showed greater discrepancies. The error is especially large for the stress calculation in the subgrade layer (12200.4 Pa compared with 11110 Pa calculated in CIRCLY or 11120 Pa calculated in KENLAYER). This difference is due to the inner approximation of the element

180 (linear interpolation of calculated responses in nodes). Therefore a 4-noded brick should be used with finer mesh density to reach an acceptable approximation.

The second part of the analysis investigates the effect of variations in layer thickness and the influence they have on different models.

Here, three different pavement structures have been studied. The first structure is the same as Figure 5.1 with a 10 cm asphalt layer and a 40 cm base layer. The second structure has a 20 cm asphalt layer and a 40 cm base layer. The third has a 10 cm asphalt layer and a 60 cm base layer. Finite element modelling is in axisymmetric 8-noded element with normal mesh density.

Table 5.3 displays the results of four critical responses for three different pavement structures. There is acceptable agreement among the results, however, the calculated horizontal strain at the bottom of the asphalt layer was higher from ABAQUS than from KENLAYER and CIRCLY, but the difference is less than 10%.

It can be seen that the thickness of the layer does not produce a significant discrepancy in the FEM model, and there is general agreement among the numerical models with other programs.

181 Table 5.3-Effect of Layer Thickness on The Numerical Approximation

Model

The following points summarize the results:

182 (1) In comparison to the analytical solution, axisymmetric solution and 3-D model, the plane strain model results in an extremely severe response by the pavement system. Therefore, this simplification should be used only with extreme caution.

(2) The 8-node brick elements lead to a stiffer medium and the results of the analysis has a range of 10% approximation. Therefore mesh refinement is necessary for proper approximation.

(3) The 8-node axisymmetric elements or 20-node brick elements provide a close approximation to the currently used linear elastic solutions.

Based on this initial analysis, the next step of the linear elastic analysis deals with the two constructed models of layered flexible pavements with different materials and loading which are investigated in the following sections.

This part of the study investigates two different types of loading, two different layer thicknesses and two different material properties. Figure 5.8 demonstrates the simulated axles of loading. A single axle dual tyre loading is assumed to have a 9 ton allowable load capacity and a tandem axle dual tyre loading has a 17 ton allowable load capacity. The load is divided by the number of tyres which have a pressure of 0.75 MPa. Then, in accordance with Figure 2.18, the contact area between the tyre and asphalt is calculated in each case.