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This work aims to look for a simplifying surface that can represent the effect of the dual wheels on the variation of the stress and deformation state prevailing during the passage of traffic loads. This was facilitated by the results of Thiam (2016) [4] obtained on the distribution of the vertical contact stress in the space described by the dual wheels. The analysis of the results of this study, on all the loading circles considered, shows that the radius loading circle equal to 0.181 m makes it possible to most probably represent the effect of the dual wheels. With this new surface, the effect of the dual wheels can be determined in 2D. The choice of this load is confirmed by a study in case of overload. Thus, the single axle with dual wheels is represented by a simplified diagram equipped on each side by a disk of radius 0.181 m. These results are obtained using a numerical simulation under Cast3M with a gravelly lateritic pavement.

The concept of equivalent single-wheel load has long been used in the structural design of the pavement. In order to design the exchange of these loads at the equivalent single wheel load (ESWL), calculation methods have been proposed by some authors like [

The study of the distribution of the vertical contact stress in 3D space by [

Modeling the load of a half-axle of reference shows the non-uniformity of the vertical contact stress on the footprint of dual wheels. This allowed us to quantify the maximum deformation of the roadway. This deformation is maximum in the center of each twinning disk. The study of an equivalent surface consists in looking for a single circle of loading to simulate in the most probable or approximate manner the effects of vertical stress on the footprint of dual wheels.

For this, we consider two (2) cases of loadings which are described as follows:

• The load 1 is a circle of radius equal to 0.312 m whose center is the middle of the center distance. This circle is tangent to each twinning disk, and this circle represents a circumscribed circle for the two circular imprints;

• The load 2 is a circle of radius equal to 0.187 m passing through the center of each twinning disk whose center is midway between the axes.

The evolution of the deformation will be observed at the center of each load. Also, an analysis of the evolution of deformation as a function of depth will be used to better understand the details of the study. For this, we will represent the deformation at each layer of the roadway to know the answer at each level during a solicitation. It is important to emphasize that the study of these loads is done by referring to the evolution of the deformation obtained at the center of a dual wheel. The studies conducted as part of this work are essentially based on a numerical simulation using the Cast3M finite element calculation code.

The modeling of a structure leads, in general, is to be interested in several aspects of which the most usual is the geometry and its discretization. The modeling is based on a 3D structure of flexible pavement, consisting of a surface layer made of 80 mm thick bituminous materials, a lateritic gravelly basecoat treated with 2% 250 mm thick cement, a laterite gravelly foundation layer of 250 mm thickness and a platform of infinite thickness. The simulations were performed considering the hypothesis of the non-linearity of the base layer and that of the foundation, while the bituminous layer and the platform have a linear behavior. The input parameters obtained by [

The construction of the finite element model under Cast3M is done in this

Layers of roadway | Thickness (m) | Linear model | Parameters of Uzan nonlinear model | |||
---|---|---|---|---|---|---|

υ () | E(MPa) | k_{1} (KPa) | k_{2} () | k_{3} () | ||

Bituminous | 0.08 | 0.35 | 1300 | − | − | − |

Base 2% | 0.250 | 0.25 | 84 | 279074 | 0.65 | −0.56 |

Raw foundation | 0.250 | 0.25 | 62 | 837276 | 0.13 | −0.33 |

Platform | 10 | 0.25 | 30 | − | − | − |

study using elements at 8 knots (cub8) on a quarter of the pavement. The carriageway is loaded with a half-axle with the dual wheels spaced 37.5 cm apart and a single wheel with a radius of 12.5 cm. The charges will be applied to finely meshed fingerprints located at the surface layer. The description of this geometry has been adopted by [

The loading of a half-axle is equal to 65 KN. This load will be distributed and applied to the two fingerprints of the dual wheels finely meshed located at the surface layer. For each load considered, the entire load of a reference half-axle is applied. For the boundary conditions, the horizontal displacements are blocked in the transverse directions, and the vertical and horizontal displacements are blocked in the lower part of the platform. The images of the 3D mesh of the roadway for each load considered and that of the dual wheels are given respectively by

The results obtained using these loadings, at each layer, are presented in Figures 4(a)-4(d).

The results obtained, in the different loading cases, show that the evolution of the deformations differs from one layer to another. It decreases according to the depth of the layer at the level of the roadway. This may be because the effect of the load is attenuated with the depth and distance of the area requested. However, it is important to note that the opposite trend is observed at the bottom of the foundation layer to a certain depth of the platform. This may be related to several factors (material properties, interface phenomenon, etc.) that are not covered in this study. The results of these simulations show that the load 1 is very low to represent the effect of the dual wheels at the two (2) layers of pavement (bituminous layer, base layer). By contrast to the load 2, the evolution of the deformation is close to that obtained with the dual wheels. This is related to the size of the loading circle. As the applied load remains the same, the stress is accentuated with the reduction of the surface of the loading circle. From the load 2,

we study a variation of the radius towards the loading center (that is to say a decrease of the radius) to see if there is another surface closer than that of the loading 2. Figures 5(a)-5(d) show the results obtained with the different radius considered.

It can be seen from these results that, for the different loading circles considered, the evolution of the deformation is the same at the level of the base layer, the foundation and the platform. However, there is a deformation gap at the level of the bituminous layer with the different circles of loadings. With the radius loading circle equal to 0.181 m, the maximum deformation obtained at the surface of the bituminous layer corresponds to that obtained with the dual wheels for this layer. The evolution of the latter is similar and close to that observed with dual wheels. The difference between the two deformations can be expressed

in relative error according to the formula of equation 1.

Relative error ( % ) Present value − Reference value Reference value × 100 (1)

It should be noted that these relative errors are common with the other load circles studied. The difference between the different loading circles is found on the deformation obtained at the surface of the bituminous layer.

That’s why our choice will be on the 0.181 m radius loading circle to represent the effect under the dual wheels footprint. This loading will be noted the loading 3 for the rest of the work whose image of the geometry is given by

Otherwise, [

Level | Roof of the base layer | Roof of the foundation layer | Roof of the platform |
---|---|---|---|

Relative error (%) | 13 | 22 | 8 |

predict the impact of the overload on the deformations and the deflections at the level of the roadway. So it is also important to study the evolution of the deformation with the load 3 in case of overload. For this, we consider a total load of a single axle with dual wheels equal to 20 tons. Half of this load (100 kN) will be applied at the level of the loading 3. This half of the load will be distributed on the two disks representing the half-axle to be able always to make a comparison between the two loadings. Figures 8(a)-(8d) show the results obtained in case of overload.

The analysis of these results shows that even in case of overload, the evolution of the deformation obtained with the load 3 is very close and similar to that of

the dual wheels. This allows us to confirm our choice on the load 3. The relative error in case of overload is presented in

By these facts, the footprint of the dual wheels of a half-axle, described with the aid of two 0.125-m disks, of 0.375 m between-axis, can be represented by a single circular surface of equal radius at 0.181 m (load 3). Given the non-uniformity of the vertical contact stress on the dual wheels footprint, this new surface will represent the effect under the center of a pairing disk where the vertical contact stress is maximum. Therefore this equivalent surface will allow us to determine the effect of the dual wheels on the behavior of pavements in 2D.

Thus, in the space of 3D, the single axle with the dual wheels can be represented by two circles of radius equal to 0.181 m. The loading of the dual wheels is represented with a single disk. So axle will be equipped on each side by a disk of radius 0.181 m.

According to [

The mesh obtained, with the simplified loading of the single axle with the dual wheels, is shown in

In three-dimensional space, [

Level | Surface of the bituminous layer | Roof of the base layer | Roof of the foundation layer | Roof of the platform |
---|---|---|---|---|

Relative error (%) | 3 | 14 | 23 | 8 |

represent the two circles of loadings of the twinned wheels, several circles of loadings considered. By referring to results of [

the dual wheels with other types of road to be able to pronounce on the validity of these types of loading.

Thiam, B.B., Samb, F. and Dione, A. (2018) Determination of an Equivalent Loading Circle Which May Represent the Loading of the Dual Wheels. Open Journal of Civil Engineering, 8, 234-244. https://doi.org/10.4236/ojce.2018.82018