1. Introduction

2. Materials and Method

Measurements were made under an axle load of 10, 13 and 16 tons over a linear one kilometer (PK21 + 500 to PK22 + 500) cut into three zones. In each zone, seven measurements of fifty meters apart were made in staggered rows (Figure 3). The results have been subdivided into “homogeneous” sections with calculation of mean, standard deviation and characteristic d90 deflection by zone. The pavement structure on which the deflection measurements were made is shown in Figure 4.

3. Results and Discussion3.1. Results of Deflection Measurement by Benkelman Beam Method and Determination of the Characteristic Deflection

The results of the deflection measurements from the Fatick-Kaolack road with modulation of the axle load of 10, 13 and 16 tons are given in Table 1.

The results showed that the values read on the beam in the case of the load of 10 tons to the axle vary between 7 and 16. They are maximum on the sides and minimum on the axis. In the same way it is found that the values read on the beam in the case of the loading of 13 and 16 tons to the axle vary respectively between 5 to 19 and 4 to 24. They are maximum on the sides and minimum on the axis. Therefore, the deflection seems to increase as a function of the axle load.

The characteristic deflection of the homogeneous zone is the arithmetic mean of individual value, to which is added the standard deviation of the distribution of values multiplied by a coefficient (K for example). The representative deflection will have the value d90 at which 10 percent of the results are considered

Results of deflection measurements from PK21 + 050 to PK22 + 050 with load variation
10 tons13 tons16 tons
PK21 + 050right15PK21 + 050right17PK21 + 050right15
PK21 + 100axis8PK21 + 100axis10PK21 + 100axis10
PK21 + 150left15PK21 + 150left15PK21 + 150left24
PK21 + 200right15PK21 + 200right19PK21 + 200right15
PK21 + 250axis3PK21 + 250axis6PK21 + 250axis7
PK21 + 300left10PK21 + 300left13PK21 + 300left15
PK21 + 350right9PK21 + 350right11PK21 + 350right8
PK21 + 400axis6PK21 + 400axis2PK21 + 400axis11
PK21 + 450left7PK21 + 450left12PK21 + 450left14
PK21 + 500right15PK21 + 500right15PK21 + 500right13
PK21 + 550axis7PK21 + 550axis6PK21 + 550axis9
PK21 + 600left7PK21 + 600left14PK21 + 600left20
PK21 + 650right10PK21 + 650right12PK21 + 650right7
PK21 + 700axis6PK21 + 700axis8PK21 + 700axis12
PK21 + 750left9PK21 + 750left5PK21 + 750left16
PK21 + 800right12PK21 + 800right8PK21 + 800right4
PK21 + 850axis7PK21 + 850axis9PK21 + 850axis10
PK21 + 900left16PK21 + 900left18PK21 + 900left20
PK21 + 950right12PK21 + 950right10PK21 + 950right13
PK22 + 000axis8PK22 + 000axis6PK22 + 000axis9
PK22 + 050left8PK22 + 050left9PK22 + 050left16

superior. And for the calculations, the formula used is given by the following relation (Bolivar Lobo Carneiro, 1966  ):

d ( 90 ) = d m + 1.3 * σ

where dm is the average deflection and σ is the standard deviation.

Figure 5 represents the evolution of the deflection of the three zones as a function of the axle load. Also, Table 2 gives the slope and correlation coefficient.

It can be seen that deflection increases with the increase of the axle load, and the trend of these curves shows a linear evolution with slope coefficients of straight lines equal to 1.7. However, the variation of the deflection is not regular, which is why in zone 3 there is a slight increase of the deflection between 10 tons and 13 tons, then a sharp increase for 16 tons, hence the coefficient of determination R2 is equal to 0.8033. The latter determines how well the regression equation is

Summary of slope and correlation coefficients
Number of LocationLocationSlope coefficientCorrelation coefficient
Zone 1Left1.76940.9903
Zone 2Axis1.72360.9631
Zone 3Right1.70130.8033

adapted to describe the distribution of points. For Zone 3, the line does not exactly describe the distribution of the measurements. This could be explained by a lack of implementation of the materials in this zone.

3.2. Simulation with the Finite Element Software Cast3M

Cast3M is a finite element calculation software for the mechanics of structures and fluids. It is a high-level instrument that can be used as a support for the design, sizing and analysis of structures and components. With this in mind, Cast3M integrates not only the resolution processes (solver) but also the construction functions of the system model (preprocessor) and exploitation of the results (post-processing). Cast3M is a “toolbox” software that allows the user to develop functions that meet his own needs.

The modeling is based on a flexible pavement 2D structure consisting of a 50 mm thick asphalt concrete surface layer followed by a 100 mm thick bitumen, a 2.5% grained lateritic base course, 200 mm thick cement, a 200 mm thick lateritic gravel subgrade and a platform of infinite thickness. The Poisson’s ratio is 0.35 for all materials except for the laterite, which is 0.25. The elastic moduli of the bituminous materials are taken from the Alizé software at an equivalent temperature of 30˚C and those of the lateritic gravel are derived from the geotechnical tests. The main parameters are given in Table 3.

The trend of the two curves shows a linear evolution of the deflection. When

Parameters of the different layers
LayerYoung Modulus (MPa)Thickness (mm)Unit Weight (Kg/m3)
Asphalt concrete surface layer1300502718
Bitumen27001002650
Lateritic gravel base course18932002400
Platform layer (Sand)120Infinite1750

the axle load increases, the deflection increases. This increasing is greater in the case of a single wheel than in the case where the effect of twinning is taken into account.

3.3. Results of Calculations by the Rational Method with Alizé

Alize software is developed by LCPC and SETRA to implement the rational method of mechanical dimensioning of road structures. It uses mechanical methods for the analysis of pavement behavior considering a multilayer structure with elastic behavior, linear, homogeneous and isotropic with static loading. The French pavement design method (LCPC-SETRA, 1994  ), based on a multilayer axisymmetric model of Burmister (Burmister, 1943  ), is used in the ALIZE software (LCPC-SETRA, 1998  ). In this static 2D model, each layer has a homogeneous and elastic behavior. The geometrical and elastic characteristics of this pavement structure are the same as for the FEM by Cast3M. The results are shown in Table 4.

3.4. Correlation between in Situ Measurements and Numerical Models

The evolution of the deflection as a function of the axle load is shown in Figure 9 and Figure 10 and the comparison is made on the results measured on the Fatick-Kaolack section and those calculated from Cast3M (Axisymmetric calculation with Req = 0.125 m and Req = 0.181 m) and from Alizé.

The analysis shows that the deflection obtained with Cast3M in the case of a single wheel of radius R = 0.125 m is closer to the deflection measured in situ than that obtained in the case where the effect is taken into account (two twin wheels of equivalent radius R = 0.187 m). For the single wheel, we have a straight line coefficient equal to 1.98 which is close to the slope coefficients of the values measured on the field and the coefficient of determination R2 = 1. So it is to further the research to validate the use of an equivalent radius and to have a better evaluation of this parameter.

Deflection results calculated by Alizé
Deflection (1/100 mm)21.52834.540.9

The analysis of the graphs shows a linear evolution of the deflection calculated with Alizé as a function of the axle load. The results obtained corroborate the measurements in situ because the deflection values vary practically in the same range as the values measured in situ. We can see that the modeling of a structure can allow us to get closer to the reality. The deflection measured on the Fatick-Kaolack section and that calculated by cast3M and Alizé show a linear evolution when the axle load is varied. However, the slight difference observed between the two methods could be related to the accuracy of the calculations built into the two programs. The effect of a single wheel of radius R = 0.125 m is closer to reality.

4. Conclusion and Perspectives

The objective of this study was to understand the evolution of the deflection when the axle load is varied. The analysis of the measurement results shows a linear evolution of the deflection for a variation of the axle load of 10, 13 and 16 tons on the Fatick-Kaolack road. However, in zone 3 it has been found that the linearity of the line does not reflect more than 95% of the distribution of measurements (R2 = 0.83). This can be related to a lack of implemented materials. The maximum deflection measured for a load of 16 tons shows that in the event of an extreme overload rate observed on the weighing stations (18.2 tons, approximately equal to 19 tons), the duration of the life of the pavement structure would be seriously threatened. Then a numerical simulation of the pavement with the Cast3M finite element calculation code and the Alizé pavement design software shows a linear behavior of the surface deformations when the axle load is varied from 10 to 19 tons. The results obtained by Cast3M in the case of a single wheel are closer to reality. This should make it possible to predict the deformation on the pavement surface when an extreme overload rate is recorded. It is also important to take into account the climate and to do measurements to accurately predict deflection in extreme overload conditions related to high temperature and extreme humidity (rainy seasons). Geophysical measurements should also be made to understand the causes of irregularities observed in some areas.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Samb, F., Diop, N., Thiam, B.B. and Ba, M. (2018) Evolution of the Deflection According to Truck Overloading―Comparison between Benkelman Beam Method and FEM of Lateritic Pavements. Open Journal of Civil Engineering, 8, 478-488.

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