STRAINS IN ASPHALT PAVEMENTS UNDER CIRCULAR AND RECTANGULAR FOOTPRINTS

The analysis of flexible pavements is usually carried out by the Linear Elastic Theory in axial-symmetric configuration, and by assuming that stresses are uniformly distributed over a circular footprint. The application of these methods requires simple computations, but beside this, the approximation of the real geometry with circular footprints causes erroneous results in the stress-strain response and therefore in the damage analysis. Rectangular footprint is closer to the real shape of the tire/pavement contact area. This paper aims to identify some relations to evaluate the effective strains induced in pavements by rectangular footprint loads by considering an equivalent single or group of circular footprints. In order to achieve this goal, pavement response was evaluated for different variable combinations both by ViscoRoute 2.0 software for the rectangular footprint, and by an elastic model for the circular footprint in axial-symmetric configuration; results obtained by the two analyses were compared in order to develop two different models. The first model proposes a correction coefficient λ of the radius of the circular area to be used for calculating strains beneath the centre of the rectangular footprint. The correction coefficient λ calculated for different variable configurations were related to those variables regarded as significant by means of a multivariate regression. In the second model, in order to calculate strains outside the rectangular footprint, the superposition of circular footprints that better approximates the rectangular ones was determined.


Introduction
In order to correctly evaluate performances of asphalt pavements, such as rutting and fatigue resistance, an accurate computation of vertical and transversal critical strains due to representative loads and environmental conditions is required. For existing pavements, critical strains are obtained by using results of deflection measurements carried out by the Falling Weight Deflectometer; some relationships were developed to calculate critical strains induced in pavements by the circular load plate as a function of deflection parameters (Losa et al. 2008). As far as the design of new pavements is concerning, strains are calculated by mechanistic analysis of pavements. According to the most common methods used in pavement design, including the Mechanistic-Empirical Pavement Design Guide (MEPDG) method, the stress-strain state is computed under linear elastic axial-symmetric conditions, by considering circularshaped footprints and the tire/pavement contact stress uniformly distributed over pavement. Even if these methods are relatively simple and fast, the assumptions adopted lead to not negligible errors in strain calculation, and so, in damage prediction (Blab 1999;De Beer, Fisher 1997;Hajj et al. 2012;Novak et al. 2003;Park et al. 2005aPark et al. , 2005bWang, Machemehl 2006;Weissman 1999;Yue, Svec 1995).
These errors are linked to two approximations: a) the real footprint shape; b) the non-uniform contact stress. This paper is focused on the first issue. In order to make the load model closer to reality, Yoder and Witczak as well as Huang propose a specific size of the equivalent footprint for both the oval and the rectangular shapes. Alkasawneh et al. (2008) demonstrates that different footprint configurations, including the ones already mentioned, bring to significant differences in pavement response.
The simulations performed in this study support this issue. Fig. 1 shows the conventional reference frame adopted in this work. The results obtained point out that under equal load P, kN, and footprint area A, m 2 (therefore equal stress) the use of a circular footprint, rather than a rectangular one, determines an error ∆, %, in computed strains which ranges between 50−100% for longitudinal strain ε x and transversal strain ε y values, and which is about 10% for the vertical strain ε z values. If the longer side of the rectangle is oriented along the x direction, the utilization of a circular footprint leads to underestimate the ε y values and to overestimate both the ε x and ε z values.
The error is maximum for strains computed under the loading area and close to it, whereas it tends to be negligible for distances between 0.4−0.5 m away from the footprint. In fact, around this distance only the load force becomes significant being the effect of load geometry negligible.

Aims and methodology of the study
This study aims to determine a set of relationships to predict strains induced by any rectangular-shaped footprint by using strains induced by a single or an overlap of circular-shaped footprints.
In order to determine these relationships, different pavement configurations, obtained by changing layer thickness and modulus, and rectangular footprint sizes were considered. For each of these, the linear elastic strains were computed at different depths of the asphalt concrete multi-layers. The strains induced by rectangular footprints in the elastic multilayer were computed with the Visco-Route 2.0 software (Chabot et al. 2010), while the strains induced by circular footprints were calculated with a linear elastic model in axial-symmetric configuration. Preliminarily, it was checked that under the same circular load σ, kPa, and at any depth z, m not significant differences exist between the pavement response of the ViscoRoute 2.0 and that of the linear elastic model in axial-symmetric configuration.
The ViscoRoute 2.0 software uses a semi-infinite multi-layer model of the pavement structure that is composed of horizontal layers in the z-direction. Each layer of the pavement structure is homogeneous. The structure is loaded by one or several loads moving in the x-direction with a certain constant speed. The load is applied in any of the three directions, at the free surface (z = 0) of the system. The load is either punctual or uniformly distributed on a rectangular or an elliptical surface area. The mechanical behaviour of the materials is assumed to be either linear elastic or linear thermo-viscoelastic. In the first case, the mechanical properties are defined by the Young modulus E and the Poisson ratio ν. In the second case, the behaviour is represented by the five viscoelastic coefficients of the Huet-Sayegh model that can be determined by the procedure reported by Losa, Di Natale (2014). A complete model description is reported by Duhamel et al. (2005).
For the purposes of this work, only the linear elastic feature of the software was used, with the representative loading frequency evaluated by the procedure proposed by Losa, Di Natale (2012).

Strains beneath the centre of the rectangular footprint
For each configuration considered, the strains induced by the rectangular footprint, at different depths of the asphalt concrete layers were calculated along the central vertical axis (z axis). Subsequently, the radius of the circular-shaped footprint that produces, for the same uniform vertical contact stress, the same strain values along its central axis was computed. Since this, it becomes possible to modify the radius of the circular footprint, whose area is the same of the rectangular-shaped one, through the following formula (1): , ( where r − equivalent circular footprint radius, m; λ − correction coefficient; a, b − principal axes of the rectangular footprint, m. Simulations were carried out on a pavement model composed of a single asphalt concrete layer and a sub base course over the subgrade. The method proposed by Cohen was applied to determine the minimum required sample size for a multiple regression study, given the desired probability level p, the number of predictors in the model n, the anticipated effect size f 2 , and the desired statistical power level π. The anticipated effect size f 2 was assumed equal to 0.35 (by convention, large effect), whereas the statistical power level π = 80%, the p-value p = 5% and the number of predictors n = 4 ( , s AC , E AC , z). The application of this method shows that the sample must consist of at least 39 combinations. In order to calibrate the model coefficients, 42 configurations were obtained by combining the following parameters: − five pavement structures with different layer thickness and asphalt concrete modulus; − three investigation depths (at , and s, where s − the thickness of the asphalt concrete layer, m); − six different-sized rectangular footprints. Table 1 shows the parameters used to obtain the 42 ombinations.
The single tire load was assumed constant P = 30 kN, while stresses were changed with the parameter a. Through the simulations, it was established that: 1) the sub base modulus in the range considered has negligible influence on the correction coefficient λ; 2) the asphalt concrete stiffness modulus E AC , MPa in the range considered has very scanty influence on the Fig. 1. Dimensions of the rectangular footprint and reference axes correction coefficient λ. For example, by using the correction coefficient λ calculated for E AC = 13 000 MPa in a pavement with E AC = 2000 MPa, the strain error ∆, %, is lower than 2%; 3) two pavements with different thickness s, at the same depth z, and with all the other conditions being equal, provide the same correction coefficient λ. Thus the correction coefficient λ depends on z and not on the ratio ; 4) different-sized rectangular footprints characterized by the same ratio , produce the same correction coefficient λ, which therefore depends on and not individually on a and b; 5) three strains ε x , ε y , ε z require different correction coefficients λ.
The plot of the correction coefficients λ versus one variable at a time, by holding all the other variables constant, shows the function is convex.
Therefore, a multivariate linear regression cannot be applied directly to these variables, but a variable transformation obtained by combining linear and quadratic forms of and z must be introduced.
By proceeding iteratively, the least significant variables (calculated with the t-Student test) were disregarded at each time and new regression coefficients were determined, until the 95% level of significance was obtained for all the remaining variables. The equivalent circular footprint radius to be used for strain calculations along the three main directions x, y, z is determined by the following formulas (2, 3, 4): (3) The above formulas are valid for a ≥ b. In the case a < b, the same formulas are used by changing r x with r y , a with b and viceversa. Fig. 2 shows a comparison between correction coefficients λ x , λ y , λ z , calculated by using formula (1) and those predicted by formulas (2), (3) and (4). As Fig. 2 shows, the proposed model provides an optimal approximation of the correction coefficient λ. The standard error of the estimate Se and the coefficient of determination R 2 are listed in Table 2. It was found that R 2 is very close to the unity and Se is very low.

Strains outside of the footprint area
The criterion of the equivalent single circular footprint cannot be extended to evaluate strains outside the footprint area. The results of simulations carried out by considering the equivalent single circular footprint for 15 combinations of the rectangular footprint size, depth z and mechanical properties of the elastic multilayer system, showed that a different approach has to be adopted for two reasons: a) the correction coefficient λ to be applied to the circular footprint radius changes significantly as variable changes; b) all other conditions being equal, the plot of strains versus y showed discontinuities and variations in convexity.
A different approach was thus adopted, based on the superposition of the effects produced by a series of circular footprints suitably distributed within the rectangular footprint. In this approach, the problem consists of defining the number of circular footprints, their radius, their position and the load to be assigned to each circular footprint in such a way as to obtain, for y ≠ 0, the same strains as those produced by the rectangular footprint. Table 3 shows the feasible load configurations considered where P is the rectangular footprint load, kN, a and b are the lengths of the principal axles, m.
The results of simulations are reported in Fig. 3. It shows that the configuration No. 4 provides a good fit of ε x plots versus the axis y; in addition, it points out that the configurations No. 5 and No. 6 well fit the plots of ε y and ε z values, respectively.
The load configurations allow to obtain a good fit of the strain plots outside the footprint area, whilst strains beneath the footprint are quite different. As expected, all the geometric configurations provide a good fit of strains far away from the footprint (for y > 0.3 m) demonstrating the load geometry has a negligible effect with increasing distance. For vertical strains, the fit is good even closer to the footprint (about 15 cm). Fig. 4 shows some examples of a comparison between strain values calculated by using the rectangular footprint and values calculated by the circular footprint superposition at different distances y from the footprint centre.
The configurations obtained are valid for a ≥ b. The conversions valid for a < b are obtained by considering a 90° rotation of the footprint.

Conclusions
1. Although circular-shaped footprint is widely used in classical pavement design methods, it brings to considerable errors in both the stress-strain computation and the evaluation of rutting or cracking. The use of rectangular shaped footprints is certainly closer to the real geometry of the tire footprint but, despite this, it is not applicable to usual models of computation. Fig. 3. Plots of strains versus y for different circular footprint super-positions and for a rectangular footprint 2. In order to estimate the difference in strains evaluated by using these two different geometric configurations, some simulations were carried out by an axial-symmetric linear elastic model and a 3D linear elastic model that allows taking into account rectangular footprints.
3. Subsequently, in order to convert a generic rectangular footprint into an equivalent single circular footprint or into a superposition of circular footprints, some equivalence relationships were determined. These two equivalent load configurations (single circular and superposition of circular footprints) produce the same strains in pavements both beneath the centre and outside of the loaded area.
4. The proposed relationships represent a useful tool that allows to consider the rectangular-shaped footprint which is closer to reality if compared to the circular one in evaluating the pavement response; at the same time, these relationships allow to benefit of the simplicity of the axialsymmetric linear elastic methods.