**1. Introduction**

The flow number (*N*) is an empirical way of characterizing a hot-mix asphalt (HMA) mixture's rutting potential. To determine the flow number, a cyclic load in haversine form is applied on a cylindrical specimen axially as shown in Figure 1. The duration of the load pulse is 0.1 s, followed by a rest period of 0.9 s. The permanent axial deformation measured at the end of the rest period is monitored during repeated loading, and the strain is calculated by dividing by the initial gauge length. The test may be conducted with or without confining pressure. However, if confining pressure is used, it is kept constant while the flow number is tested. If confining pressure is used, it remains constant during the test.

**Figure 1.** Flow number test setup.

In the flow number test, the permanent strain at each cycle is measured while constant deviator stress is applied at each load cycle on the test sample. Permanent deformation of asphalt pavements has three stages [1]: (i) primary or initial consolidation; (ii) secondary; and (iii) tertiary or shear deformation.

Figure 2 shows the three stages of permanent deformation. The first stage of deformation is due to the initial filling of voids, particle rearrangement, etc. The second region is the actual deformation of the aggregates, asphalt film, etc. The tertiary region is the zone where drastic shear failure of the mix occurs. The *N*-value is the number of load cycles at which tertiary flow begins, i.e., where permanent deformation occurs non-linearly. Tertiary flow can be differentiated from secondary flow by a marked departure from the linear relationship between cumulative strain and number of cycles in the secondary zone, as shown in Figure 2. It is assumed that in tertiary flow, the specimen's volume remains constant. The *N*-value can be correlated with rutting potential. The higher the flow number, the better the mix is in terms of its rutting resistance.

**Figure 2.** Relationship between permanent strain and number of load cycles [1].

Mathematically, the *N*-value can be determined using the Francken model.

The Francken model is currently built into the Asphalt Mixture Performance Tester (AMPT) software (Federal Highway Administration, Washington DC, USA). At the beginning, the entire permanent strain curve is fitted using nonlinear least squares optimization, as shown in Equation (1). The flow number is then determined from the second derivative of the best fitted curve. The flow number is the number of cycles where the second derivative, Equation (2), changes from negative to positive.

$$
\varepsilon\_P = A \left( n^B \right) + \mathcal{C} \left( \varepsilon^{Dn} - 1 \right),
\tag{1}
$$

where ε*<sup>p</sup>* is the permanent strain (%); *n* is the number of cycles; and *A*, *B*, *C*, and *D* are fitting parameters.

$$\frac{d^2\varepsilon\_p}{dn^2} = AB(B-1)n^{B-2} + CD^2\varepsilon^{Dn},\tag{2}$$

where *<sup>d</sup>*2ε*<sup>p</sup> dn*<sup>2</sup> is the second derivative of permanent strain with respect to the number of loading cycles.

The final evaluation is an evaluation of the rutting resistance of the mixture using the flow number test defined by the American Association of State Highway and Transportation Officials (AASHTO) TP 79 [2] using the AMPT. The test is conducted at the "high" pavement temperature calculated by the long-term pavement performance (LTPP) Bind 3.1 software program (Federal Highway Administration, Washington DC, USA) for a specific project location. An unconfined flow number test with a repeated deviatoric stress of 600 kPa (87 psi) and a contact deviatoric stress of 30 kPa (4.4 psi) was used in this study. The test was conducted on specimens that were short-term conditioned for two hours at the compaction temperature to simulate the binder absorption and stiffening that occurs during construction. The flow number criteria for HMA as a function of the traffic level are summarized in Table 1.

**Table 1.** Flow number criteria for hot-mix asphalt (HMA) [2].


The effects of different mix factors on the flow number have also been studied by different researchers [3–6]. Kaloush [3] determined that the flow number increases with the viscosity of binder and decreases with the test temperature, effective binder content, and air voids. Kvasnak et al. [4] determined that the flow number increases with gyrations and the viscosity of the binder and decreases with voids in mineral aggregates (VMA) when using Wisconsin dense graded mixtures. Both researchers determined that aggregate gradation also affects the flow number. Christensen [5] applied various statistical techniques to relate the flow number with the applied stress level and observed that the flow number decreases with increasing applied deviator stress. Rodezno et al. [6] determined that the flow number increases with the viscosity of the binder; however, it decreases with the test temperature and air voids and is affected by aggregate gradation. The current study was not intended to investigate the viscosity of the binder or the test temperature. Other factors such as the VMA, voids filled with asphalt (VFA), effective binder content, contractors, testing time, mix gradation, and binder types were investigated. There are some other researches on flow number. For example, Irfan et al. [7] evaluated the flow numbers for static and dynamic creep tests and then correlated it with the rutting. Ogundipe [8] studied the flow numbers of lime-modified asphalt concrete. Irfan et al. [9] investigated the flow numbers of fiber-added stone mastic asphalt concrete mixtures. Leiva-Villacorta et al. [10] evaluated the flow numbers for High-modulus asphalt concrete (HMAC) mixtures for use as base course. Tripathi [11] studied the economic benefits of fiber-reinforced asphalt mixtures by flow number including some other tests. Ziari and Divandari [12] developed a flow number prediction model using artificial neural network. Siswanto et al. [13] investigated the flow numbers of Asphalt Concrete Using Crumb Rubber Modified of Motorcycle Tire Waste. All these studies investigated the flow number

under different conditions. However, none of the studies investigated the sensitivity of different mix factors such as such as VMA, void-filled with asphalt (VFA), effective binder content, contractors, testing time, mix gradation, and binder types on the flow number of asphalt concrete.

Air void (*Va*) is the total volume of the small pockets of air between the coated aggregate particles expressed as a percentage of the bulk volume of the compacted mixture. The volume of the void space among aggregate particles of a mixture that includes the air voids and the effective asphalt content is known as VMA. The portion of the voids in the mineral aggregate that contain asphalt binder is known as VFA. The total asphalt binder content of the mix less the portion of asphalt binder that is lost by absorption into the aggregate is called the effective asphalt content (*Vbe*). This portion of binder is coated on the aggregate surface and takes part in binding aggregates. The total asphalt binder used in a mix is called the asphalt content (AC).

The flow number (*N*) test procedure recommended in the National Cooperative Highway Research Program (NCHRP) project 9-19 is a simple performance test for rutting evaluation of asphalt mixtures. The test showed good correlation with the rutting performance of mixtures at WesTrack, MnROAD, and the Federal Highway Administration's (FHWA's) accelerated loading facility. Subsequent NCHRP studies allowed the development of a provisional standard. AASHTO TP 79 [2] includes test parameters for stress, temperature, specimen conditioning, and minimum flow number criteria that were established for HMA and for warm-mix asphalt (WMA) based on the traffic level.

The current study used the testing conditions and criteria for *N* testing described in AASHTO TP 79 [2] for unconfined tests. The recommended test temperature, determined by LTPP Bind Version 3.1 software, was the average design high pavement temperature at 50% reliability for cities in Colorado. Tests were conducted at a temperature of 55 ◦C with an average deviator stress of 600 kPa (87 psi) and a minimum (contact) axial stress of 30 kPa (4.4 psi). For conditioning, samples were kept in a conditioning chamber at the testing temperature for 12 h prior to testing.

To confirm again, the objectives of this study were to study the effects of mix factors such as VMA, VFA, effective binder content (*Vbe*), contractors, testing time, mix gradation, and binder types on the flow number of asphalt concrete.

#### **2. Materials**

The eleven types of mixtures studied are listed in Table 2 along with their basic information such as nominal maximum aggregate size (NMAS), performance grade (PG) binder type, and number of gyrations used while designing the mixes. Superpave performance grading is reported using two numbers: the first being the average seven-day maximum pavement temperature (◦C) and the second being the minimum pavement design temperature likely to be experienced (◦C). Thus, a PG 64-22 is intended for use where the average seven-day maximum pavement temperature is 64◦C and the expected minimum pavement temperature is −22◦C. The letter "S" denotes an NMAS of 0.75 in. (19 mm). The letters "SX" denote an NMAS of 0.5 in. (12.5 mm). SMA is the abbreviated form of the stone mix asphalt mixture. The numbers in parentheses are the numbers of gyrations used in the mix design. All the mixes were designed following the Superpave requirements for all parameters. Every group of mixes had identical aggregate gradations.


**Table 2.** A list of the eleven mixtures used in this study

S = NMAS of 0.75 in. (19 mm); SX = NMAS of 0.5 in. (12.5 mm); SMA = stone mix asphalt mixture.

#### **3. E**ff**ects on the Flow Number**

#### *3.1. Same Mix by the Same Contractor*

To investigate the variation in the flow number within the work of a single contractor for the same mix, the following mixes were selected randomly. The information regarding the paving contractor, binder supplier, and aggregate pits is kept confidential. The mixes were manufactured in 2014.

The N-values vary from 120 to 531 with an average value of 261 and standard deviation of 125, as shown in Figure 3. The values shown are for each individual specimen. To determine whether these data are statistically significant or not, a one-sample t-test was conducted. The t-test requires the data to be normally distributed. The t-test showed the 95% Confidence Interval (CI) boundaries to be 150 and 372 with a mean value of 261. This means that all the mixes, except for 19655 P21 14 and 19655 P87 14, are statistically the same. Therefore, a conclusion can be made that the same mix may have statistically different flow numbers for the same contractor. Note that Colorado Department of Transportation (CDOT) uses a 10-digit format to express the mix identity, such as 19655 P21 14. The first five numbers denote the project and the last five digits denote the site and specimen number.

#### *3.2. Same Mix by Di*ff*erent Contractors*

To investigate the differences in flow number for the same mix prepared by different contractors, SX(100) PG 76-28 mix was selected. The average flow numbers from four contractors, 19128, 18842, 19458, and 19677, are presented in Figure 4.

**Figure 4.** Flow numbers of a mix by different contractors.

The pairwise comparison test result shows that the mixes by 19128, 18842, and 19458 are statistically the same (Table 3). Therefore, a conclusion can be made that the same mix may have statistically different flow numbers for different contractors. This is due to variations in the aggregate structures, shapes, orientation, smoothness, etc.


**Table 3.** Pairwise comparisons using *t*-tests to determine statistical difference.

#### *3.3. Groupwise Comparison*

The flow number variation for each group of mix is described in this section. An effort was made to examine whether all specimens' flow numbers were statistically equal or not. A 95% Confidence Interval (CI) was used to indicate reliability. Next a statistical regression analysis was conducted to find the effects of different mix factors.
