Evaluation of Dynamic Modulus of HMA Sigmoidal Prediction Models and Optimization by Approach of U.S. Mesh Sieve by AFNOR and LC Mesh Sieve

Pavement design tools are not universal. Indeed, in the sizing of 
pavements in the USA, the prediction models used in the calculation of the 
dynamic modulus of HMA are not adapted to the characterization of the mineral 
skeleton of the HMA mix designed with the French method. This article aims to assess the 
predictive models of the dynamic modulus used in the mechanistic-empirical 
design for their use in the design of bituminous pavements, and to develop new 
predictive models taking into account the sieve series LC and AFNOR standards. 
A total of six types of mixtures were subjected to the determination of complex 
modulus testing by direct tensile-compression on cylindrical specimens (26-700 
LC) over a temperature range (5) and frequency (5) data. Dynamic modulus 
prediction models |E*| are studied 
Witczak model 1999 and model Witczak 2006. These models do not take into 
account the AFNOR or LC mesh sieve, an approach was made in relation to the US 
mesh sieve to replace ρ200 (0.075 mm), ρ4 (4.76 mm), ρ38 (9.5 mm) and ρ34 (19 mm) respectively by 
the AFNOR mesh P0.08 (0.08 
mm), R5 (5 mm), R10 (10 mm) and R14 (14 mm). The result is 
the production of two models whose are evaluated by correlation with the values 
|E*| of modulus measured in the laboratory is 
satisfactory (R2 = 0.83 
respectively R2 = 0.71 and p-value = 0.00). The optimization of 
these approximate models gave new models with the same frame as the original 
models and a better correlation with the data observed in the laboratory 
(respectively R2 = 0. 95 
and R2 = 0.91 p-value = 0.00).


Introduction
In Mechanistical Empirical Design [1] (Hammons, 2007) the bituminous layer at level 2 and 3 requires the use of prediction model because at level 1 (highest reliability level) the dynamic modulus of Hot Mixture Asphalt (HMA) coatings is determined by laboratory tests [2] [3] [4] [5]. However, when these HMA are mix designed according to the French method with aggregates specified with different sieve mesh, the use of American models is no longer possible. This justifies the need to find a solution. One of the main parameters characterizing the (NCHRP) team [6]. In the dynamic modulus prediction approach for HMA, they exist two main methods. The first is called a discrete finite element method 0.16 mm and 0.08 mm. However, a statistical approach remains possible, because Witczak's models are statistical models of sigmoidal type. These models are determined from only database from laboratory tests [4]. In order to make this approach possible similar tests have been carried out.
In this study, the dynamic modulus of asphalt mixtures with aggregate skele- This article will statistically measure the impact of the approach on the prediction of the dynamic modulus and develop an empirical model whose para-meters related to the particle size of the mixture were specified according to Standard French Normalization Association (AFNOR) and Québec Publication (LC) mesh sieve.

Methodology
The objectives of this paper are to assess the Witczak sigmoidal model prediction of the dynamic modulus for HMA by approaching sieve mesh considered and to develop a new predictive model based on the AFNOR standards sieve mesh and LC by a non-linear optimization in Solver-Microsoft Excel.
A total of six mixtures designed according to the Marshall method and validated according to the level 4 of HMA mix design procedures [12] is studied. Two aggregates types and two nominal maximum aggregate size (NMAS) are used in the different formulation with a single type of bitumen (grade 35/50 ERES or PG70/16). The specimens were cored from asphalt plate compacted to LCPC compactor. They have a height of 125 mm and a diameter of 74 mm with a void percentage interval ranging from 2% to 8%. Direct tension-compression test on cylindrical specimens is used for the measurement of the dynamic modulus mixtures studied. The results of the test on the DSR ERES 35/50 bitumen are used for determining the parameters models related to the asphalt binder (A, VTS, η, δ b and G*).

The Test Dynamic Shear Rheometer
DSR is a test for measuring the rheological stiffness and elasticity binders and bituminous mastics through the dynamic shear modulus G* and the δ phase angle [13]. It applies to high temperatures and intermediaries usually an old bitumen from "Rolling Thin Film Oven Test" (RTFOT). To input data requirements for writing prediction models studied, the DSR tests were performed at the same temperature (55˚C, 40˚C, 30˚C, 20˚C and 10˚C) and frequencies (10 Hz, 5 Hz, 1 Hz, 0.3 Hz and 0.1 Hz) than the dynamic modulus is testing.

Direct Tension-Compression Test on Cylindrical Specimen
The complex modulus tests are performed according to standard [13] entitled "Determination of the complex modulus of HMA" by using direct tension-compression equipment (TCD) on cylindrical specimens ( Figure 1). E* is determined at small strains, at different frequencies and temperatures, in order to characterize the linear viscoelastic behavior of the mix. E* is a complex number which consists of two parameters, namely the dynamic modulus (|E*|), which is the standard of E*, and the phase angle (δ), which is the argument of E*. The |E*| is used for pavement design and δ can appreciate the viscoelastic behavior of the asphalt.
where E* is the complex modulus (kPa); |E*| is the dynamic modulus (kPa); φ is the phase angle (rad); σ is the total axial stress (kPa); ε is the total axial deformation (m/m); ω is the period (2•π•f) (rad); t is time (sec); "I" is the imaginary number; t lag is time to shift between s and e (sec); P is the axial load (kN); d is the diameter of the test piece (m);. ∆h is the axial displacement (m); and h: height for measuring. ∆h (100 mm) (m).

The Model 1999 Witczak's Model
The 1999 Witczak's model is used in levels 2 and 3 of pavement design. It is given by Equation (4 [4]. With |E*| is the dynamic modulus (105 psi); η is the viscosity of the binder (106 poise); f is the loading frequency in Hz; ρ 200 is the percentage passing through a sieve 0.075 mm (No. 200); ρ 4 cumulative percentage of the sieve 4.76 mm (No. 4); ρ 38 is the cumulative percentage of the sieve 9.5 mm (3/8 in); ρ 34 is the cumulative percentage of the sieve, 19 mm (3/4 in); V a is the air void percentage; V beff is effective binder content in percentage volume.
In fact in the report of NCHRP 1-37A it was developed by recalibration of the dynamic modulus of the prediction model developed by Witczak and Fonseca

The Model Witczak 2006
However, the 1999 Witczak model poses a problem related to the fact the rigidity of the binder is characterized by a viscosity that does not take into account the effects of the charging frequency. In this model the frequency it is considered as another independent variable entered the predictive equation. However, the viscosity of the binder depends on the charging frequency. Thus changes in the load frequency induce changes of viscosity of the binder. From this point of view the scenario presented by the 1999 model where binder viscosity remains constant when the load frequency changes cannot be conceived in reality. In 2006 Bari and Witczak taking into account remarks cited above set of 7400 modulus measurements from 346 mixtures of HMA presents a new model in which the viscosity of the binder and the loading frequency considered directly in the model 1999 are replaced by the shear modulus |G*| and the phase angle. δ b of the binder [11]. This model is described by Equation (5 [11]. where |E*| is the dynamic modulus (psi); |G*| is the dynamic shear modulus of the binder; δ b is the binder phase angle; ρ 200 = % of sieving 200; ρ 4 = % cumulative screen oversize 4; ρ 38 = % cumulative screen oversize 3/8; ρ 34 = % refusal cumulative ¾ screen; V a = % air void; V beff = effective binder content.

Statistical Interpretations
The 1999 and 2006 Witczak's models are nonlinear models (polynomial) as a sigmoidal function. Their development was based on the analysis and optimization of the statistical process.
Statistical analysis was intended to reduce the prediction error by comparing the predicted values with the measured values [5].
The nonlinear optimization is to find the values of the regression coefficients or adjustment parameters used in a model so that the model equation has a minimum error when a set of predicted and measured data are compared (Bari and Witczak, 2006).
The goodness of fit indicates the degree of binding of the adjustment parameters to the prediction model. The nonlinear optimization uses as indicator the determination coefficient R 2 and the ratio of the standard error (Se) and standard deviation (Sy) noted Se/Sy. A good model present a high R 2 (close to unity) and a low Se/Sy. Statistical quality of the correlation is given by R 2 and usually taken p-value of p < 0.005. It is the latter that will be used in the interpretation of our results.
The complex structure of HMA explains the choice of nonlinear optimization to study the predictive models.
The Solver Microsoft Excel is a useful and precise function chosen by most researchers to optimize the nonlinear problems. When the sum of the squared error is minimized, the solution is a biased solution.
The database used to make the calculations are presented as an appendix at the end of the article.
NB: η values were determined from the coefficients ASTM A + VTS. This results in Equations (6) and (7) With |E*| = dynamic modulus (105 psi); η = viscosity of the binder (106 poise); f = frequency in Hz loading; P 200 = percent passing sieve 0.08 mm; R 5 = cumulative percentage of the sieve 5 mm; R 10 = the percentage of cumulative screen oversize 10 mm; R 14 = the percentage of cumulative screen oversize 14 mm; V a = percentage of vacuum; V beff = Binder content effective in percentage volume.
With |E*| = dynamic modulus (105 psi); |G*| is the dynamic shear modulus of the binder; δ b is the phase angle of the binder P 0.08 = percent passing 0.08 mm sieve; R 5 = cumulative percentage of the sieve 5 mm; R 10 = the percentage of cumulative screen oversize 10 mm; R 14 = the percentage of cumulative screen oversize 14 mm; V a = void percentage; V beff = Binder content effective in percentage volume.

Evaluation 1999 Witczak's Approached Model
A correlation is performed on the values of modulus predicted by sieve mesh with the approximate 1999 Witczak's model and the values measured in the laboratory on the mixtures designed with the basalt Diack and the quartzite Bakel by direct tensile/compression tests test on cylindrical specimens. Figure 3 shows a fairly good estimate of the 1999 Witczak's model with a very strong correlation of R 2 = 0.83 and a significant p (p = 0.00). Table 1 shows the optimized coefficients Witczak module 1999 in comparison to the initial coefficients.

Adjustment and Optimization of 1999 Witczak's Approached Model
With |E*| = dynamic modulus (105 psi); η = viscosity of the binder (106 poise); f = frequency in Hz; P 200 = percent passing sieve 0.08 mm; R 5 = cumulative percentage of the sieve 5 mm; R 10 = the percentage of cumulative screen oversize 10 mm; R 14 = the percentage of cumulative screen oversize 14 mm; V a = void percentage; V beff = Binder content effective in percentage volume.
Correlating the predicted modulus values with the new approached model values shows a good correlation with an R 2 = 0.9574 and p = 0.00. Figure 4 illustrate the results of the correlation.  Table 2 shows the optimized coefficients of 2006 Witczak's model in comparison to the initial coefficients. The optimization shows that the coefficient F11 is

Adjustment and Optimization of 2006 Witczak's Approached Model
With |E*| = dynamic modulus (105 psi); |G*| est the dynamic shear modulus of the binder; δ b is the phase angle of the binder; P 0.08 = percent passing 0.08 mm sieve; R 5 = cumulative percentage of the sieve 5 mm; R 10 = the percentage of cumulative screen oversize 10 mm; R 14 = the percentage of cumulative screen oversize 14 mm; V a = void percentage; V beff = Binder content effective in volume percentage.
Correlating the predicted modulus values with the new model shows a good correlation with an R 2 = 0.9175 and p = 0.00. Figure 6 shows the results of the correlation.

Conclusion
The approach of Witczak's sigmoidal model by sieve mesh (1999 and 2006)