# Development and Analysis of High-Modulus Asphalt Concrete Predictive Model

^{*}

## Abstract

**:**

## 1. Introduction

_{4PB}calculation, which were developed as a result of a research project and study at Poznan University of Technology and the Laboratory of the General Directorate for National Roads and Motorways in Poznań, Poland.

## 2. Laboratory Research Program

- Determination of stiffness modulus and phase angle of an asphalt mixture at various temperatures and various loading frequencies for each test temperature by the four-point bending beam test conducted according to [6];
- Determination of shear modulus and phase angle of a binder at various temperatures and various loading frequencies for each test temperature by a dynamic shear rheometer test, carried out according to [15];
- Determination of the resistance to hardening of binders under the influence of heat and air by an RTFOT test performed according to [16];
- Determination of soluble binder content in an asphalt mixture conducted according to [17];
- Determination of particle size distribution carried out according to [18];
- Determination of the maximum density of an asphalt mixture according to [19];
- Determination of bulk density of an asphalt mixture specimen according to [20];
- Determination of air-voids content of an asphalt-mixture specimen according to [21];
- Determination of dimensions of an asphalt-mixture specimen according to [22].

## 3. Tested Asphalt Mixtures and Binders

_{4PB}= 11,000 MPa, the minimum stiffness modulus value specified for HMAC for the base course, and, therefore, HMA 8 and HMA 10 mixtures, were recognized as asphalt concrete (AC). HMA 9, which is an SMA Jena mix (stone mastic asphalt used as a single-layer pavement), was decided to be included in the research program as a comparative mix. The six HMA mixtures (HMA 1–HMA 6) contain different 20/30 penetration-grade bitumen from three different manufacturers. HMA 7 and HMA 9 mixtures contain polymer-modified bitumen and HMA 8 and HMA 10 mixtures used PMB 25/55-80 highly modified bitumen (so-called HIMA), both from the same manufacturer. In all the tested HMA mixtures, limestone filler was used. An adhesion agent was added to all the tested HMA in an amount ranging from 0.3% to 0.5% by weight of the bitumen. Apart from the adhesive agent, no other modifiers were added to the HMA [27].

## 4. Shear Modulus and Phase Angle of Bituminous Binders

- first group: B1 20/30 penetration grade bitumen with significantly higher values of shear modulus than other 20/30 bitumens;
- second group: binders B2 to B5, these are 20/30 penetration grade bitumens;
- third group: polymer-modified bitumen with the lowest values of shear modulus in this temperature range.

## 5. Stiffness Modulus and Phase Angle of Asphalt Mixtures

_{4PB}and the phase angle Φ

_{4PB}of the asphalt mixture.

_{4PB}= 21,080 MPa (mean) is achieved by HMA 7 and HMA 8 with a modified binder with 5.2% bitumen content by weight, while HMA 6 with an unmodified binder with the same bitumen content by weight has a significantly lower (by 3880 MPa) value of the stiffness modulus E

_{4PB}= 17,200 ± 548 MPa, which is, according to the authors’ opinion, an unusual behaviour. At medium temperature (10 °C), the highest value of stiffness modulus E

_{4PB}= 18,530 ± 1137 MPa is achieved by HMA 1, which corresponds well with the value of shear modulus of B1 bitumen. According to statistical tests performed at 10 °C, the stiffness modulus E

_{4PB}of the HMA 1 mix has a higher value than that of HMA 3 (17,606 ± 907 MPa), HMA 4 (16,459 ± 1219 MPa), and HMA 5 (17,698 ± 613 MPa) mixes. In contrast, the E

_{4PB}stiffness modulus values of the HMA 3 and HMA 5 mixtures are equal to each other. However, the relative difference (5%) between the E

_{4PB}stiffness modulus values of HMA 1 and HMA 3 and HMA 5 is much smaller than the relative difference (75%) between the shear modulus values of the bitumen used as a binder, as B1 bitumen has a significantly higher shear modulus value |G*|. This can be explained by the higher volumetric content of bitumen in HMA 1 and the differences in voids content V

_{a}. At 40 °C, HMA 6 with 20/30, bitumen stands out with a significantly higher value of stiffness modulus E

_{4PB}= 2925 ± 429 MPa than mixtures with polymer-modified bitumens (average E

_{4PB}= 1239 MPa).

## 6. Effective Bitumen Content, Air Voids, and Granulation of HMA

_{beff}. In order to calculate the value of this variable, it is necessary to determine the density of bitumen G

_{b}used in the HMA. The authors adopted the values of bitumen density G

_{b}from data made available by bitumen manufacturers. The results of the effective bitumen content V

_{beff}obtained from tests carried out on the basis of [17] are shown in Figure 14.

## 7. Predictive Model

_{DM}—stiffness modulus determined by dynamic-modulus test [10

^{5}psi];

_{beff}—effective binder content expressed by volume (v/v) [%];

_{a}—air-voids content [%];

_{200}—percentage of aggregate passing the No. 200 sieve (#0.075 mm) [%];

_{4}—cumulative percentage retained on the No. 4 sieve (#4.76 mm) [%];

_{38}—cumulative percentage retained on the 3/8 in sieve (#9.5 mm) [%];

_{34}—cumulative percentage retained on the 3/4 in a sieve (#19 mm) [%].

_{4PB}—stiffness modulus determined by four-point bending beam test [MPa];

_{beff}—effective binder content expressed by volume (v/v) [%];

_{a}—air-voids content [%];

_{0}—percentage of aggregate passing the 0.063 mm sieve [%];

_{4}—cumulative percentage retained on the 4.0 mm sieve [%];

_{8}—cumulative percentage retained on the 8.0 mm sieve [%];

_{16}—cumulative percentage retained on the 16.0 mm sieve [%];

_{4PB}and the phase angle Φ representing the results of determinations made on the eight HMA mixtures were used to develop model A. The range of variability of the input data for the optimization process, i.e., the variables in the models, is given in Table 5.

_{i}

^{2}), the limiting value of relative errors (P95(re)) and the distribution of absolute errors (Table 6). As one can read in the work [32], it was discussed that in order to be considered a valid model, the distribution of absolute errors, after discarding outliers (nout), should be a normal distribution, and it was considered that the normality of the distribution should be confirmed by both histogram and statistical tests. The coefficients calculated as a result of the optimization procedure were presented in Equation (3) (after inserting them into Equation (2) of model A).

_{4PB}stiffness moduli are greater than the calculated ones, while the opposite is true for the HMA 2 and HMA 6 mixes. The influence of the type of aggregate and, more specifically, the silica content (SiO

_{2}) in the aggregate, which affects the adhesion of the asphalt to the aggregate, can be seen here. For mixtures with more alkaline aggregates (HMA 7 and HMA 8), the determined stiffness moduli of E

_{4PB}are larger than calculated. Therefore, at the stage of further work on the predictive models, the introduction of an additional variable characterizing the silica content of the aggregate used to make the HMA can be considered in the models. In Figure 17, where the test temperatures T are marked, it can be clearly seen that for high temperature, lower values of the stiffness modulus E

_{4PB}were obtained, and for low temperature, higher values of the stiffness modulus E

_{4PB}were obtained, confirming the generally known relationship and confirming the correctness of the developed model A. It can be noted, however, that for the analysed HMA, the values of stiffness moduli E

_{4PB}obtained at 10 °C are only slightly smaller than those obtained at 0 °C. In Figure 18, where the loading frequencies f are marked, no significant relationship is noticed.

_{4PB}stiffness modulus, a normality analysis of the distribution of the model’s absolute errors was performed. The analysis of the normality of the distribution was performed by plotting histograms and performing multiple statistical tests for the normality of the distribution, with a significance level of α = 0.05. The D’Agostino–Pearson test was used as the binding test for the normality of the distribution due to the fact that the power of this test classifies at the medium level. Before analyzing normality, deviating absolute error values were rejected using the Hampel statistical test.

_{4PB}stiffness modulus determinations were carried out. The highest values of absolute errors are shown by determinations of the E

_{4PB}stiffness modulus made at 0 °C, while the lowest values of absolute errors are shown by tests carried out at the highest temperature of 40 °C. Such error distributions are consistent with the achieved E

_{4PB}stiffness modulus values. The largest relative error values are shown by the E

_{4PB}stiffness modulus determined at 40 °C and 30 °C, where the values are so small that even a small difference will result in a significant relative error. Medium relative error values were found in the results of tests carried out at 0 °C. The smallest relative error values were found at 10 °C and 20 °C. The relative error values at 10 °C are within approx. 30% and at 20 °C within approx. 20%, which can be considered satisfactory.

## 8. Concluding Remarks

_{4PB}) and the values obtained from the model (Predicted E

_{4PB}) is very high, the coefficient of determination being R

^{2}= 0.936.

_{4PB}at 0 °C, confirming the results of the work of Bari and Witczak [32]. Similar to the mentioned researchers, the authors look for physical hardening effects here.

_{4PB}of HMA itself. A more precise procedure for determining the stiffness modulus E

_{4PB}would contribute to a reduction in uncertainty. The conditions for thermostating the specimens prior to the low-temperature determination of the stiffness modulus E

_{4PB}of HMA need to be defined more precisely in order to further reduce the possible influence of physical hardening. The algorithm for calculating the stiffness modulus E

_{4PB}of HMA also needs to be clearly defined.

_{4PB}, it is also proposed to calculate the limiting values at the significance level α = 0.05 within which the result of the determination should fall, i.e., the uncertainty interval. These limits are proposed to be calculated using the P95(re) statistic. This would be done by subtracting and adding 50% of this value from the calculated value of the stiffness modulus E

_{4PB}of the HMA; in this case, it would be known in which interval with 95% probability the laboratory-determined value of the stiffness modulus E

_{4PB}should fall.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Photos taken during specimen preparation procedure: (

**a**) Rolling device for compacting samples; (

**b**) Samples prepared for testing.

**Figure 4.**Dynamic Shear Rheometer [1].

**Figure 5.**Dynamic Shear Rheometer bitumen test at T = −20 °C and f = 1.59 Hz: (

**a**) Shear modulus |G*|; (

**b**) Phase angle δ.

**Figure 6.**Dynamic Shear Rheometer bitumen test at T = 0 °C and f = 1.59 Hz: (

**a**) Shear modulus |G*|; (

**b**) Phase angle δ.

**Figure 7.**Dynamic Shear Rheometer bitumen test results at T = 10 °C and f = 1.59 Hz: (

**a**) Shear modulus |G*|; (

**b**) Phase angle δ.

**Figure 8.**Dynamic Shear Rheometer bitumen test results at T = 40 °C and f = 1.59 Hz: (

**a**) Shear modulus |G*|; (

**b**) Phase angle δ.

**Figure 9.**Dynamic Shear Rheometer bitumen test results at T = 60 °C and f = 1.59 Hz: (

**a**) Shear modulus |G*|; (

**b**) Phase angle δ.

**Figure 16.**Comparison between measured and calculated values of the stiffness modulus E

_{4PB}of HMA with marked types of asphalt mixtures.

**Figure 17.**Comparison between measured and calculated values of the stiffness modulus E

_{4PB}of HMA with marked temperatures used in the tests.

**Figure 18.**Comparison between measured and calculated values of the stiffness modulus E

_{4PB}of HMA with marked loading frequencies used in the tests.

**Figure 20.**Error distributions of model A for different temperatures: (

**a**) Distribution of absolute errors; (

**b**) Distribution of relative errors.

HMA ID | Type of HMA | Type of Bitumen | Bitumen ID | Effective Content of Bitumen P _{be} (m/m) [%] | Type of Aggregate Used in HMA |
---|---|---|---|---|---|

HMA 1 | HMAC 16 | 20/30 | B1 | 5.3 | basalt |

HMA 2 | HMAC 16 | 20/30 | B2 | 5.2 | melaphyre serpentine granite |

HMA 3 | HMAC 16 | 20/30 | B3 | 5.2 | greywacke granite |

HMA 4 | HMAC 16 | 20/30 | B4 | 5.3 | basalt limestone |

HMA 5 | HMAC 16 | 20/30 | B4 | 5.2 | granodiorite limestone basalt |

HMA 6 | HMAC 16 | 20/30 | B5 | 5.2 | granite |

HMA 7 | HMAC 16 | PMB 25/55-60 (modified) | B6 | 5.2 | basalt |

HMA 8 | AC 16 | PMB 25/55-80 (highly modified) | B7 | 5.2 | basalt |

HMA 9 | SMA Jena 16 | PMB 45/80-65 (modified) | B8 | 5.6 | granodiorite |

HMA 10 | AC 16 | PMB 25/55-80 (highly modified) | B7 | 5.6 | basalt |

Temperature T [°C] | −20, −10, 0, 10, 30, 40, 60, 70, 80 |

Frequency f [Hz] | 0.1, 0.2, 0.5, 1, 1.59, 5, 10, 20, 25, 50 |

**Table 3.**Summary of temperature and frequency values used in HMA tests performed using the 4PBB device.

HMA ID | Temperature [°C] | Frequency [Hz] |
---|---|---|

HMA 1 | 10 | 10 |

HMA 2 | 10 | 0.5; 1; 5; 10; 20; 25 |

HMA 3 | 10 | 0.5; 1; 5; 10; 20; 25 |

HMA 4 | 10 | 0.1; 0.5; 1; 5; 10; 20; 25 |

HMA 5 | 10 | 0.1; 0.5; 1; 5; 10; 20; 25 |

HMA 6 | 0; 10; 20; 30; 40 | 0.1; 0.5; 1; 5; 10; 20; 25 |

HMA 7 | 0; 10; 30; 40 | 0.1; 0.5; 1; 5; 10; 20; 25 |

HMA 8 | 0; 10; 30; 40 | 0.1; 0.5; 1; 5; 10; 20; 25 |

HMA 9 | 0; 10; 30 | 0.1; 0.5; 1; 5; 10; 20; 25 |

HMA 10 | 0; 10; 30; 40 | 0.1; 0.5; 1; 5; 10; 20; 25 |

HMA ID | P_{0} [%] | P_{4} [%] | P_{8} [%] | P_{16} [%] |
---|---|---|---|---|

HMA 1 | 8.7 | 50.9 | 30.7 | 1.6 |

HMA 2 | 5.6 | 55.9 | 32.8 | 1.3 |

HMA 3 | 7.6 | 71.4 | 52.5 | 2.1 |

HMA 4 | 7.6 | 52.2 | 31.1 | 0.8 |

HMA 5 | 7.2 | 58.9 | 35.8 | 1.0 |

HMA 6 | 6.4 | 57.7 | 37.3 | 1.7 |

HMA 7 | 6.7 | 50.5 | 33.4 | 1.5 |

HMA 8 | 7.5 | 51.1 | 34.8 | 1.8 |

HMA 9 | 8.7 | 50.9 | 30.7 | 1.6 |

HMA 10 | 5.6 | 55.9 | 32.8 | 1.3 |

Variable | Minimum | Maximum | Average | |
---|---|---|---|---|

|E*| | [MPa] | 290 | 25,268 | 9489 |

Φ | [°] | 3.85 | 48.92 | 19.08 |

|G*| | [MPa] | 0.09 | 133.65 | 30.78 |

δ | [°] | 22.00 | 59.50 | 38.59 |

V_{beff} | [%] | 11.6 | 13.5 | 12.6 |

V_{a} | [%] | 0.5 | 3.8 | 2.2 |

P_{0} | [%] | 5.6 | 10.2 | 7.4 |

P_{4} | [%] | 50.5 | 71.4 | 56.9 |

P_{8} | [%] | 30.7 | 56.6 | 39.0 |

P_{16} | [%] | 0.8 | 2.1 | 1.5 |

$\sum}{{\mathbf{e}}_{\mathbf{i}}}^{\mathbf{2}$ | P95(re) | Nout |
---|---|---|

2,755,451,769 | 66 | 41 |

1,984,652,225 | 52 | 59 |

3,337,871,386 | 64 | 56 |

4,276,307,701 | 58 | 82 |

1,394,365,412 | 50 | 65 |

Parameter | Value for Model A | |
---|---|---|

No. of data points, n | 471 | |

No. of mixes, n_{m} | 8 | |

Σe | [MPa] | 100,538 |

Σ|e| | [MPa] | 556,788 |

m(|e|) | [MPa] | 1182 |

m(re) | [%] | −3 |

P95(re) | [%] | 50 |

Se | [MPa] | 1722 |

SDy | [MPa] | 6826 |

Se/SDy | [-] | 0.25 |

R^{2} | [-] | 0.936 |

aR^{2} | [-] | 0.934 |

Type of Test | The p-Value Statistic |
---|---|

Kolmogorov–Smirnov test | 0.5664 |

Lilliefors test | 0.1488 |

Shapiro–Wilk test | 0.1901 |

D’Agostino–Pearson test | 0.9475 |

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## Share and Cite

**MDPI and ACS Style**

Bartkowiak, M.; Słowik, M.
Development and Analysis of High-Modulus Asphalt Concrete Predictive Model. *Materials* **2023**, *16*, 4509.
https://doi.org/10.3390/ma16134509

**AMA Style**

Bartkowiak M, Słowik M.
Development and Analysis of High-Modulus Asphalt Concrete Predictive Model. *Materials*. 2023; 16(13):4509.
https://doi.org/10.3390/ma16134509

**Chicago/Turabian Style**

Bartkowiak, Mikołaj, and Mieczysław Słowik.
2023. "Development and Analysis of High-Modulus Asphalt Concrete Predictive Model" *Materials* 16, no. 13: 4509.
https://doi.org/10.3390/ma16134509