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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 ρ_{200} (0.075 mm), ρ_{4} (4.76 mm), ρ_{38} (9.5 mm) and ρ_{34} (19 mm) respectively by the AFNOR mesh P_{0.08} (0.08 mm), R_{5} (5 mm), R_{10} (10 mm) and R_{14} (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 (R^{2} = 0.83 respectively R^{2} = 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 R^{2} = 0. 95 and R^{2} = 0.91 p-value = 0.00).

In Mechanistical Empirical Design [_{b}). Parameters related to the granularity of the bituminous mixture in Witczak’s empirical models are specified according to US mesh screen (US Standard Series) described by the “American Association of highway and Transportation Official” (AASHTO) standards named 37.5 mm, 25.0 mm, 19.0 mm, 12.5 mm, 9.5 mm, 4.75 mm, 2.36 mm, 1.18 mm, 0.60 mm, 0.30 mm, 0.15 mm and 0.075 mm. What makes these models almost impossible to use when the sieve mesh from “Canadian Standard Series” (LC) or from French Standards Association (AFNOR) standards are used to learn about the two 14 mm, 10 mm, 8 mm, 5 mm, 2.5 mm, 1.25 mm, 0.315 mm, 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 [

In this study, the dynamic modulus of asphalt mixtures with aggregate skeletons specified according to AFNOR and Lc sieve are determined by US sieve mesh approach by using the 1999 and 2006 Witczak’s models. The aggregates of Senegal used are basalt from Diack and quartzite from Bakel. Bitumen is 35/50 (AFNOR) grade (or pavement grade PG 70/16) ERES.

This article will statistically measure the impact of the approach on the prediction of the dynamic modulus and develop an empirical model whose parameters related to the particle size of the mixture were specified according to Standard French Normalization Association (AFNOR) and Québec Publication (LC) mesh sieve.

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 [_{b} and G*).

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 [

The complex modulus tests are performed according to standard [

The complex modulus E* is determined by Equation (1).

E * = σ sin ( ω t + φ ) ε sin ( ω t − φ ) = | E * | cos φ + i | E * | sin φ (1)

[

Dynamic modulus |E*| is given by Equation (2) [

| E * | = σ ε = 4 P / π d 2 δ h / h (2)

[

The phase angle is determined by Equation (3).

φ = ω t l ∂ g (3)

(Touhara)

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 1999 Witczak’s model is used in levels 2 and 3 of pavement design. It is given by Equation (4).

log | E * | = − 1.249937 + 0.029232 ρ 200 − 0.001767 ( ρ 200 ) 2 − 0.002841 ρ 4 − 0.058097 V a − 0.0802208 ( V b e f f V b e f f + V a ) + 3.871977 − 0.0021 ρ 4 + 0.003958 ρ 38 − 0.000017 ( ρ 38 ) 2 + 0.005470 ρ 34 1 + e ( − 0.603313 − 0.31335 log ( f ) − 0.395321 log ( η ) ) (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 [^{2} = 0.87 precision and Se/Sy 0.36 in basic arithmetic and assessment of its reliability on the new database is less accurate with R^{2} = 0.73 and Se/Sy = 0.53. These observed differences assigned to differences in granularity and bitumen in the mixtures of the two databases. The results of the recalibration on the reliability of the model Witczak 1999 was satisfactory with an R^{2} of 0.941, and Se/Sy 0.2449 [

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 [

log | E * | = − 0.349 + 0.754 ( | G b ∗ | − 0.0052 ) × ( 6.65 − 0.032 ρ 200 + ( 0.0027 ρ 200 ) 2 + 0.011 ρ 4 − 0.0001 ( ρ 4 ) 2 + 0.006 ρ 38 − 0.00014 ( ρ 38 ) 2 − 0.08 V a − 0.16 ( V b e f f V a + V b e f f ) ) + 2.558 + 0.032 V a + 0.713 ( V b e f f V a + V b e f f ) + 0.0124 ρ 38 − 0.0001 ( ρ 38 ) 2 − 0.0098 ρ 34 1 + e ( − 0.7814 − 0.5785 log | G b * | + 0.8834 log δ b ) (5)

[

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.

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 [

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.

The analysis of the data consists of 1999 and 2006 Witczak’s model parameters i.e. |G*|, f, η, δ_{b}, ρ_{200}, ρ_{4}, ρ_{38}, ρ_{34}, V_{a} and V_{beff}. They are the explanatory or independent variables to predict the dependent variable log|E*| (explain). However, the values of the sieve meshes are approached P_{0.08} (ρ_{200}), R_{10} (ρ_{38}), R_{14} (ρ_{34}) and R_{5} (ρ_{4}).

NB: η values were determined from the coefficients ASTM A + VTS. This results in Equations (6) and (7) below.

log | E * | = − 1.249937 + 0.029232 P 0.08 − 0.001767 ( P 0.08 ) 2 − 0.002841 R 5 − 0.058097 V a − 0.0802208 ( V b e f f V b e f f + V a ) + 3.871977 − 0.0021 R 5 + 0.003958 R 10 − 0.000017 ( R 10 ) 2 + 0.005470 R 14 1 + e ( − 0.603313 − 0.31335 log ( f ) − 0.395321 log ( η ) ) (6)

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.

log E * = − 0.349 + 0.754 ( | G b * | − 0.0052 ) × ( 6.65 − 0.032 P 0.08 + ( 0.0027 P 0.08 ) 2 + 0.011 R 5 − 0.0001 ( R 5 ) 2 + 0.006 R 10 − 0.00014 ( R 10 ) 2 − 0.08 V a − 0.16 ( V b e f f V a + V b e f f ) ) + 2.558 + 0.032 V a + 0.713 ( V b e f f V a + V b e f f ) + 0.0124 R 10 − 0.0001 ( R 10 ) 2 − 0.0098 R 14 1 + e ( − 0.7814 − 0.5785 log | G b * | + 0.8834 log δ b ) (7)

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.

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.

^{2} = 0.83 and a significant p (p = 0.00).

Régression coefficients | Optimized coefficients | Initial coefficients |
---|---|---|

F1 | 0.152593542 | 1.249937 |

F2 | 0.006186186 | 0.029232 |

F3 | 0.003011471 | 0.001767 |

F4 | 0.002858835 | 0.002841 |

F5 | 0.0000001 | 0.058097 |

F6 | 0.088945463 | 0.0802208 |

F7 | 7.698304773 | 3.871977 |

F8 | 0.00194078 | 0.0021 |

F9 | 0.003636243 | 0.003958 |

F10 | 1.72114E-05 | 0.000017 |

F11 | 0.003804769 | 0.00547 |

F12 | 1.90175288 | 0.603313 |

F13 | 0.205909497 | 0.31335 |

F14 | 0.284943589 | 0.395321 |

Sum of squared deviations | 4.56 |

The new 1999 Witczak’s approached model and optimized is given by Equation (8) below:

log | E * | = − 0.152593542 + 0.006186186 P 0.08 − 0.003011471 ( P 0.08 ) 2 − 0.002858835 R 5 − 0.00000017 V a − 0.088945463 ( V b e f f V b e f f + V a ) + 7.698304773 − 0.00194078 R 5 + 0.003636243 R 10 − 1.72114 E − 05 ( R 10 ) 2 + 0.003804769 R 14 1 + e ( − 1.90175288 − 0.205909497 log ( f ) − 0.2849435891 log ( η ) ) (8)

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.

A correlation is performed on the values of modulus predicted by sieve mesh with the approximate 2006 Witczak’s model and the values measured in the laboratory show a fairly good estimate of the model Witczak 1999 with a very strong correlation of R^{2} = 0.71 and a significant p (p = 0.00) (

Régression coefficients | Optimized coefficients | Initial coefficients |
---|---|---|

F1 | 0.354624896 | 0.349 |

F2 | 0.73016561 | 0.754 |

F3 | 0.005481779 | 0.0052 |

F4 | 5.461922604 | 6.65 |

F5 | 0.036114863 | 0.032 |

F6 | 0.002060831 | 0.0027 |

F7 | 0.005137335 | 0.011 |

F8 | 0.000128808 | 0.0001 |

F9 | 0.003963945 | 0.006 |

F10 | 0.000166591 | 0.00014 |

F11 | 0.0000001 | 0.08 |

F12 | 0.18414392 | 0.16 |

F13 | 2.597758994 | 2.558 |

F14 | 0.040197991 | 0.032 |

F15 | 0.596946197 | 0.713 |

F16 | 0.00806974 | 0.012 |

F17 | 0.000106102 | 0.0001 |

F18 | 0.010347472 | 0.0098 |

F19 | 0.847277886 | 0.7814 |

F20 | 0.80288457 | 0.5851 |

F21 | 1.540388406 | 0.8834 |

Sum of squared deviations | 85.49290941 |

canceled by the solver. However, in order to keep the shape intact models, a lower value of this coefficient is chosen so as not to vary the SSD significantly. The new model 2006 Witczak’s approached model optimized is given by Equation (9) below.

log E * = − 0 , 354624896 + 0.73016561 ( | G b ∗ | − 0.005481779 ) × ( 5.461922604 − 0.036114863 P 0.08 + ( 0.002060831 P 0.08 ) 2 + 0.005137335 R 5 − 0.0.000128808 ( R 5 ) 2 + 0.0.003963945 R 10 − 0.000166591 ( R 10 ) 2 − 0.0000001 V a − 0.18414392 ( V b e f f V a + V b e f f ) ) + 2.597758994 + 0.040197991 V a + 0.596946197 ( V b e f f V a + V b e f f ) + 0.00806974 R 10 − 0.000106102 ( R 10 ) 2 − 0.010347472 R 14 1 + e ( − 0.847277886 − 0.80288457 log | G b * | + 1.540388406 log δ b ) (9)

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.

The approach of Witczak’s sigmoidal model by sieve mesh (1999 and 2006) is a simple replacement of US sieve mesh ρ_{200} ( 0.075 mm), ρ_{4} (4.76 mm), ρ_{38} (9.5 mm) and ρ_{34} (19 mm) by AFNOR and Lc sieve mesh respectively P_{0.08} (0.08 mm), R_{5} (5 mm), R_{10} (10 mm) and R_{14} (14 mm).The evaluation Witczak’s model 1999 shows that the sieve mesh approach does not compromise the ability of this model to predict asphalt formulated according to AFNOR and LC methods. The evaluation of the Witczak 2006 model presents a less accurate than the 1999 model. But accuracy remains statistically valid to predict the dynamic modulus of asphalt mixtures. Solver MS-Excel is a great tool adjustment through its nonlinear GRG module. The adjustment of the model approached Witczak 1999 is more accurate than the adjustment of the approximate 2006 model. Dynamic modulus of HMA designed with basalt of Diack and quartzite Diack Bakel is well predicted by Witczak’s of 1999 and 2006 model by approached mesh sieve. For best accuracy, however, the optimized models are recommended. In perspective it will be important for a better reliability of the prediction to calibrate these new models with a larger database including different types of HMA.

The authors would like to acknowledge the research group of the “Ecole de Technologies Supérieure” of Montréal.

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

Aidara, M.L.C., Ba, M. and Carter, A. (2020) Evaluation of Dynamic Modulus of HMA Sigmoidal Prediction Models and Optimization by Approach of U.S. Mesh Sieve by AFNOR and LC Mesh Sieve. Open Journal of Civil Engineering, 10, 195-212. https://doi.org/10.4236/ojce.2020.103017