Volumetric Analysis of Asphalt Mixture

Chapter

ABSTRACT

Chapter 12 focuses mainly on the calculation of the volumetric properties as well as compaction properties related to both Marshall mixture design procedure and Superpave mixture design method. This process is called the volumetric analysis of asphalt mixtures. In this analysis, the relationships between the volumetric properties are to be understood. The effect of one property on other properties will be assessed. The intercorrelation between all volumetric properties will be evaluated. In the volumetric analysis, the outputs of the asphalt mixture (all the volumetric properties) should be computed from raw data including weights, specific gravities, and percentages of asphalt binder and aggregate in addition to bulk and theoretical maximum specific gravities of asphalt mixture. Additionally, given some of the volumetric properties of the mixture, the rest of the volumetric properties should be easy to determine. The correlation between all the phases (by volume and by weight) of the composite material (the asphalt mixture) in the phasing diagram should be linked to the volumetric analysis so that the designer would precisely understand these phases and their relationship to mixture design. Therefore, the questions and problems in this part will deal with the volumetric analysis of asphalt mixtures in Marshall and Superpave mixture design methods, the interrelationships between the volumetric properties, and the determination of all the volumetric properties of asphalt mixtures in the design process. 12.1

In a Marshall test, the percentage of asphalt binder by total weight of aggregate is 5.0%. The bulk specific gravity of aggregate (G_sb) = 2.624, the specific gravity of asphalt binder (G_b) = 1.000, and the density of water (γ_w) = 1.000 g/cm³. If the absorbed asphalt (P_ba) is 2.00% by total weight of aggregate, and the voids in total mixture (VTM) is 4.0%, determine the following properties:

The effective asphalt content (P_be)

The bulk specific gravity of the compacted mixture (G_mb)

The voids in mineral aggregate (VMA)

The theoretical maximum specific gravity of the loose mixture (G_mm)

The voids filled with asphalt (VFA)

The effective specific gravity of aggregate solids (G_se) (see Figure 12.1).320

Solution:

The effective asphalt content (P_be) is determined using the following formula:

12.1 P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0001.tif"/>

Where:

P_be = effective asphalt binder content, percent by total weight of asphalt mixture

P_b = percentage of asphalt binder content, by total weight of asphalt mixture

P_ba = absorbed asphalt content, percent by total weight of aggregate

P_s = percentage of aggregate, by total weight of asphalt mixture

But before substituting in the formula, it has to be made sure that all inputs are ready to be used in the formula. The asphalt binder content that is given in the problem is the binder content by weight of aggregate and not by weight of mixture. Therefore, the value should be converted as shown in the procedure below:

12.2 P s = 100 ( 1 + P b -aggregate 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0002.tif"/>

Or:

12.3 P b = P b -aggregate ( 1 + P b -aggregate 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0003.tif"/>

Where:

P_s = percentage of aggregate, by total weight of asphalt mixture

P_b-aggregate = percentage of asphalt binder content, by total weight of aggregate321

⇒

P s = 100 ( 1 + 5.0 100 ) = 95.24 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0004.tif"/>

But,

12.4 P s = 100 − P b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0005.tif"/>

Where:

P_s = percentage of aggregate, by total weight of asphalt mixture

P_b = percentage of asphalt binder content, by total weight of asphalt mixture

⇒

P b = 100 − P s = 100 − 95.24 = 4.76 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0006.tif"/>

Now, all the inputs can be substituted in the formula for effective asphalt binder content shown below:

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0007.tif"/>

⇒

P be = 4.76 − ( 2.0 100 ) 95.24 = 2.86 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0008.tif"/>

The bulk specific gravity of the compacted mixture (G_mb) is determined by doing some manipulation of the formulas and the definition of the VMA as explained in the procedure below:

Notice that the air voids content (VTM) is given, but the VMA and the VFA are both unknowns. Also, the G_mb for the asphalt mixture is unknown. Therefore, none of the available formulas will be ready directly to use to determine any of these unknowns. For this reason, the original definition of the VMA will be used. The definition of voids in mineral aggregate (VMA) is the percentage of total voids volume in the asphalt mixture by the total volume of the mixture. In other words, by looking at the phasing diagram of the asphalt mixture above, the VMA can be written as:

12.5 VMA = 100 ( V a + V be V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0009.tif"/>

Where:

VMA = voids in mineral aggregate, percent by total volume of asphalt mixture

V_a = V_v = volume of air voids in asphalt mixture

V_be = volume of effective asphalt binder in asphalt mixture

V_mb = total volume of asphalt mixture

Again, the voids in total mixture (VTM) is defined as in the following expression:

12.6 VTM = 100 ( V a V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0010.tif"/> 322

Where:

VTM = voids in total mixture, percent by total volume of asphalt mixture

V_a = V_v = volume of air voids in asphalt mixture

V_mb = total volume of asphalt mixture

And the effective asphalt binder content is defined as in the following expression:

12.7 P be = 100 ( W be W mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0011.tif"/>

Where:

P_be = percentage of asphalt binder content, by total weight of asphalt mixture

W_be = weight of effective asphalt binder

W_mb = bulk or total weight of asphalt mixture

Therefore, the VMA can be re-written as:

VMA = 100 ( V a + V be V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0012.tif"/>

⇒

12.8 VMA = 100 ( V a V mb ) + 100 ( V be V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0013.tif"/>

Since the specific gravity of the asphalt binder and the bulk specific gravity of the asphalt mixture are defined as:

12.9 G b = W b V b γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0014.tif"/>

Or:

12.10 G be = W be V be γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0015.tif"/>

Where:

G_b = G_be = specific gravity of asphalt binder

W_b = weight of asphalt binder

V_b = volume of asphalt binder

W_be = weight of effective asphalt binder

V_be = volume of effective asphalt binder

γ_w = density of water (1.000 g/cm³)

12.11 G mb = W mb V mb γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0016.tif"/>

Where:

G_mb = bulk specific gravity of asphalt mixture

W_bm = bulk or total weight of asphalt mixture

V_mb = bulk or total volume of asphalt mixture

γ_w = density of water (1.000 g/cm³)323

Therefore,

12.12 VMA = 100 ( V a V mb ) + 100 ( ( W be γ w G b ) ( W mb γ w G mb ) ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0017.tif"/>

Equation 12.8 can be rewritten as:

12.13 VMA = 100 ( V a V mb ) + 100 ( W be W mb ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0018.tif"/>

But:

VTM = 100 ( V a V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0019.tif"/>

And:

P be = 100 ( W be W mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0020.tif"/>

Therefore,

12.14 VMA = VTM + P be ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0021.tif"/>

This formula along with the original formula for VMA shown below can be used together to determine G_mb:

12.15 VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0022.tif"/>

Where:

VMA = voids in mineral aggregate, percent by total volume of asphalt mixture

G_mb = bulk specific gravity of asphalt mixture

P_s = percentage of aggregate in asphalt mixture, by total weight of asphalt mixture

G_sb = bulk specific gravity of aggregate

From Equation 12.14 and Equation 12.15, the following formula can be derived:

12.16 VTM + P be ( G mb G b ) = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0023.tif"/>

This formula can be used now to determine the G_mb of the asphalt mixture (the only unknown):324

4.0 + 2.86 ( G mb 1.000 ) = 100 ( 1 − G mb × 95.24 100 2.624 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0024.tif"/>

Solving for G_mb provides:

G mb = 2.452 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0025.tif"/>

The voids in mineral aggregate (VMA) can be computed now easily from one of the two formulas above:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0026.tif"/>

⇒

VMA = 100 ( 1 − 2.452 × 95.24 100 2.624 ) = 11.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0027.tif"/>

The theoretical maximum specific gravity of the loose mixture (G_mm) is determined using the formula for the air voids:

12.17 VTM = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0028.tif"/>

Where:

VTM = voids in total mixture (or air voids), percent by total volume of asphalt mixture

G_mb = bulk specific gravity of asphalt mixture (compacted mixture)

G_mm – theoretical maximum specific gravity of asphalt mixture (loose mixture)

⇒

4.0 = 100 ( 1 − 2.452 G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0029.tif"/>

⇒

G mm = 100 × 2.452 100 − 4.0 = 2.554 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0030.tif"/>

The voids filled with asphalt (VFA) is determined using the formula for VFA:

12.18 VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0031.tif"/>

Where:

VFA = voids filled with asphalt, percent by volume of total voids in asphalt mixture

VMA = voids in mineral aggregate, percent by total volume of asphalt mixture

VTM = %V_a = air voids content, percent by total volume of asphalt mixture325

⇒

VFA = 100 ( 11.0 − 4.0 11.0 ) = 63.6 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0032.tif"/>

The effective specific gravity of aggregate solids (G_se) is calculated using the formula for absorbed asphalt content (P_ba):

12.19 P ba = 100 ( G se − G sb G se G sb ) G b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0033.tif"/>

Where:

P_ba = absorbed asphalt content, percent by weight of aggregate

G_se = effective specific gravity of aggregate

G_sb = bulk specific gravity of aggregate

G_b = specific gravity of asphalt binder

⇒

2.0 = 100 ( G se − 2.624 2.624 G se ) 1.000 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0034.tif"/>

Solving for G_se provides the following value:

⇒

G se = 2.769 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0035.tif"/>

The MS Excel worksheet used to solve this problem for a rapid and efficient solution is shown in Figure 12.2. 326

12.2

In a Marshall test, the percentage of asphalt binder by total weight of aggregate is 5.3%. The bulk specific gravity of aggregate (G_sb) = 2.582, the specific gravity of asphalt binder (G_b) = 1.000, and the density of water (γ_w) = 1.000 g/cm³. If the voids filled with asphalt (VFA) is 70.0% and the effective asphalt binder (P_be) is 4.0% by total weight of mixture, determine the following properties:

The absorbed asphalt binder (P_ba)

The bulk specific gravity of the compacted mixture (G_mb)

The voids in mineral aggregate (VMA)

The voids in total mixture (VTM)

The theoretical maximum specific gravity of the loose mixture (G_mm)

The effective specific gravity of the aggregate (G_se)

Solution:

The absorbed asphalt binder (P_ba) is calculated as shown below:

P s = 100 ( 1 + P b -aggregate 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0036.tif"/>

⇒

P s = 100 ( 1 + 5.3 100 ) = 94.97 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0037.tif"/>

But,

P s = 100 − P b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0038.tif"/>

⇒

P b = 100 − P s = 100 − 94.97 = 5.03 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0039.tif"/>

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0040.tif"/>

⇒

4.0 = 5.03 − ( P ba 100 ) 94.97 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0041.tif"/>

⇒

P ba = ( 5.03 − 4.0 94.97 ) 100 = 1.09 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0042.tif"/>

The bulk specific gravity of the compacted mixture (G_mb) is determined as shown below:

P be = 100 ( W be W mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0043.tif"/> 327

But:

G b = G be = W be V be γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0044.tif"/>

And:

G mb = W mb V mb γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0045.tif"/>

⇒

P be = 100 ( G be V be γ w G mb V mb γ w ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0046.tif"/>

Or:

P be = 100 ( G be V be G mb V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0047.tif"/>

Multiplying the above equation by V_ma/V_ma provides:

P be = 100 ( G be V be V ma G mb V mb V ma ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0048.tif"/>

But:

VMA = 100 ( V ma V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0049.tif"/>

And:

VFA = 100 ( V fa V ma ) = 100 ( V be V ma ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0050.tif"/>

V_fa = V_be as a volume phase in the phasing diagram.

Therefore Equation 12.20 becomes:

P be = ( VFA ) ( VMA ) 100 G b G mb https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0051.tif"/>

⇒

12.22 VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0052.tif"/>

And the VMA is also given by:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0053.tif"/> 328

Therefore, a new formula can be written as in the following expression:

100 ( P be VFA ) ( G mb G b ) = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0054.tif"/>

Using this formula, the G_mb can be calculated:

100 ( 4.0 70.0 ) ( G mb 1.000 ) = 100 ( 1 − G mb × 94.97 100 2.582 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0055.tif"/>

⇒

G mb = 100 ( 4.0 70.0 ) ( 100 1.000 ) + 94.97 2.582 = 2.353 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0056.tif"/>

The voids in mineral aggregate (VMA) can be determined easily using any of the formulas above:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0057.tif"/>

⇒

VMA = 100 ( 1 − 2.353 × 94.97 100 2.582 ) = 13.4 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0058.tif"/>

The voids in total mixture (VTM) is also calculated using the formula below:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0059.tif"/>

⇒

VTM = VMA ( 1 − VFA 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0060.tif"/>

⇒

VTM = 13.4 ( 1 − 70.0 100 ) = 4.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0061.tif"/>

The theoretical maximum specific gravity of the loose mixture (G_mm) is computed using the following formula:

VTM = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0062.tif"/> 329

⇒

4.0 = 100 ( 1 − 2.353 G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0063.tif"/>

⇒

G mm = 2.452 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0064.tif"/>

The effective specific gravity of the aggregate (G_se) is determined using the following formula:

P ba = 100 ( G se − G sb G se G sb ) G b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0065.tif"/>

⇒

1.09 = 100 ( G se − 2.582 2.582 G se ) 1.000 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0066.tif"/>

⇒

G se = 1 ( 1 2.582 − 1.05 100 × 1.000 ) = 2.657 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0067.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.3. The worksheet provides a rapid and efficient solution.330

12.3

In a Marshall test, the percentage of asphalt binder by total weight of asphalt mixture is 4.8%. The bulk specific gravity of aggregate (G_sb) = 2.655, the specific gravity of asphalt binder (G_b) = 1.020, and the density of water (γ_w) = 1.000 g/cm³. If the effective asphalt binder content (P_be) is 2.50% by total weight of mixture, and the voids in total mixture (VTM) is 4.0% by total volume of mixture, determine the following properties:

The bulk specific gravity of the compacted mixture (G_mb)

The absorbed asphalt (P_ba)

The voids in mineral aggregate (VMA)

The voids filled with asphalt (VFA)

Solution:

The bulk specific gravity of the compacted mixture (G_mb) is determined as shown below:

Equation 12.16 derived in Problem 12.1 is used to calculate G_mb.

VTM + P be ( G mb G b ) = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0068.tif"/>

⇒

4.0 + 2.50 ( G mb 1.020 ) = 100 ( 1 − G mb × 95.2 100 2.655 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0069.tif"/>

⇒

G mb = 100 − 4.0 2.50 1.020 + 95.2 2.655 = 2.506 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0070.tif"/>

The absorbed asphalt (P_ba) is calculated using the formula below:

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0071.tif"/>

⇒

2.50 = 4.8 − ( P ba 100 ) 95.2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0072.tif"/>

⇒

P ba = 2.42 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0073.tif"/> 331

The voids in mineral aggregate (VMA) is determined using the following VMA formula:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0074.tif"/>

⇒

VMA = 100 ( 1 − 2.506 × 95.2 100 2.655 ) = 10.14 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0075.tif"/>

The voids filled with asphalt (VFA) is calculated using the formula below:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0076.tif"/>

⇒

VFA = 100 ( 10.14 − 4.0 10.14 ) = 60.6 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0077.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.4.332

12.4

In a Marshall design procedure for an asphalt mixture, the weight of aggregate (W_s) is 1200.0 g, and the weight of asphalt binder (W_b) is 76.0 g. The voids filled with asphalt (VFA) value is determined to be 73.0%, and the effective asphalt content by total weight of mixture (P_be) is found to be 3.80%. If the bulk specific gravity of aggregate (G_sb) is 2.568, the specific gravity of asphalt binder (G_b) = 1.030, and the density of water (γ_w) = 1.000 g/cm³, determine the following properties of the asphalt mixture:

The absorbed asphalt binder content (P_ba), percent by weight of aggregate

The effective specific gravity of the aggregate (G_se)

The bulk specific gravity of the compacted asphalt mixture (G_mb)

The voids in mineral aggregate (VMA), percent by total volume of mixture

The voids in total mixture (VTM), percent by total volume of mixture

The theoretical maximum specific gravity of the loose mixture (G_mm)

Solution:

The absorbed asphalt binder content (P_ba) is determined using the formula:

The asphalt binder content as a percentage of the asphalt mixture is defined as:

P b = 100 W b W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0078.tif"/>

Where:

P_b= percentage of asphalt binder by total weight of asphalt mixture

W_b = weight of asphalt binder

W_T = total weight of asphalt mixture

⇒

P b = 100 ( 76.0 1200.0 + 76.0 ) = 5.96 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0079.tif"/>

⇒

P s = 100 − P b = 100 − 5.96 = 94.04 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0080.tif"/>

3.80 = 5.96 − ( P ba 100 ) 94.04 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0081.tif"/>

⇒

P ba = 2.30 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0082.tif"/>

The effective specific gravity of the aggregate (G_se) is determined using the formula below:

⇒

2.30 = 100 ( G se − 2.568 2.568 G se ) 1.030 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0084.tif"/>

⇒

G se = 2.724 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0085.tif"/> 333

The bulk specific gravity of the compacted asphalt mixture (G_mb) is determined using Equation 12.23 derived in Problem 12.2:

⇒

100 ( 3.80 73.0 ) ( G mb 1.030 ) = 100 ( 1 − G mb × 94.04 100 2.568 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0087.tif"/>

⇒

G mb = 100 ( 3.80 73.0 ) ( 100 1.030 ) + 94.04 2.568 = 2.400 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0088.tif"/>

The voids in mineral aggregate (VMA) is calculated using the typical VMA formula shown below:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0089.tif"/>

⇒

VMA = 100 ( 1 − 2.400 × 94.04 100 2.568 ) = 12.1 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0090.tif"/>

The voids in total mixture (VTM) is determined as below:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0091.tif"/>

⇒

VTM = VMA ( 1 − VFA 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0092.tif"/>

⇒

VTM = 12.1 ( 1 − 73.0 100 ) = 3.27 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0093.tif"/>

The theoretical maximum specific gravity of the loose mixture (G_mm) is computed as follows:334

VTM = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0094.tif"/>

⇒

G mm = G mb ( 1 − VTM 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0095.tif"/>

⇒

G mm = 2.400 ( 1 − 3.27 100 ) = 2.481 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0096.tif"/>

The MS Excel worksheet shown in Figure 12.5 is used to solve the problem for quick and efficient solution.

12.5

In a Superpave mixture design method for an asphalt mixture, the percentage of asphalt binder by total weight of asphalt mixture is 5.0%. The bulk specific gravity of aggregate (G_sb) = 2.462, the specific gravity of asphalt binder (G_b) = 1.030, and the density of water (γ_w) = 1.000 g/cm³. If the absorbed asphalt binder is 1.20% by total weight of asphalt mixture, the volume of the air voids in the mixture is 76.0 cm³, and the bulk volume of the aggregate is 1805.0 cm³, determine the following properties:

The effective asphalt binder content (P_be)335

The voids in mineral aggregate (VMA)

The voids filled with asphalt (VFA)

The air voids in the asphalt mixture (V_A or VTM)

The bulk specific gravity of the asphalt mixture (G_mb)

The theoretical maximum specific gravity of the asphalt mixture (G_mm)

Solution:

The phasing diagram of the asphalt mixture shown below will help in solving this problem.

Since volume values are given in the problem, to deal with volumes and weights will be easier in the volumetric analysis of this mixture.

The weight of aggregate is determined from the volume of aggregate given in the problem as shown below (see Figure 12.6):

G sb = W s γ w V sb https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0097.tif"/>

Where:

G_sb = bulk specific gravity of aggregate

W_s = weight of aggregate in asphalt mixture (g)

V_sb = bulk or total volume of aggregate in asphalt mixture (cm³)

γ_w = density of water (1.000 g/cm³)

⇒

2.462 = W s 1.000 ( 1805.0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0098.tif"/>

⇒

W s = 4443.9 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0099.tif"/> 336

The weight of asphalt binder in the mixture is now determined as shown in the procedure below:

P b = 100 W b W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0100.tif"/>

⇒

5.0 = 100 W b ( 4443.9 ) + W b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0101.tif"/>

Since the total weight of asphalt mixture, W_T = W_b + W_s

⇒

W b = 4443.9 ( 100 5.0 − 1 ) = 233.9 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0102.tif"/>

The weight of absorbed asphalt binder is also calculated as follows:

Since the absorbed asphalt binder is given by the total weight of asphalt mixture in this problem, the following formula (which is not typical) will be used:

P ba-mixture = 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0103.tif"/>

Where:

P_ba-mixture = absorbed asphalt content, percent by total weight of asphalt mixture

W_ba = weight of absorbed asphalt binder (g)

W_T = total weight of asphalt mixture (g)

⇒

1.20 = 100 W ba ( 4443.9 + 233.9 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0104.tif"/>

⇒

W ba = 56.1 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0105.tif"/>

Now the following volumes are calculated:

G b = W b V b γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0106.tif"/>

⇒

V b = W b G b γ w = 233.9 1.030 ( 1.000 ) = 227.1 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0107.tif"/>

Also, the definition of the specific gravity of the asphalt binder applies to the absorbed asphalt binder:

12.27 G b = G ba = W ba V ba γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0108.tif"/> 337

⇒

V ba = W ba G b γ w = 56.1 1.030 ( 1.000 ) = 54.5 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0109.tif"/>

Based on the phasing diagram of the three different phases of the asphalt mixture, the total (bulk) volume of asphalt mixture is determined as below:

V mb = V T = V sb − V ba + V b + V a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0110.tif"/>

Where:

V_mb = V_T = bulk or total volume of asphalt mixture

V_sb = bulk or total volume of aggregate

V_ba = volume of absorbed asphalt binder

V_b = volume of asphalt binder

V_a = V_v = volume of air voids in asphalt mixture

Therefore,

V mb = V T = 1805.0 − 54.5 + 227.1 + 76.0 = 2053.6 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0111.tif"/>

According to the phasing diagram, also the volume of effective asphalt binder is equal to:

V be = V b − V ba https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0112.tif"/>

Where:

V_be = volume of effective asphalt binder

V_b = volume of asphalt binder

V_ba = volume of absorbed asphalt binder

⇒

V be = 227.1 − 54.5 = 172.6 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0113.tif"/>

G b = G be = W be V be γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0114.tif"/>

⇒

W be = G b V be γ w = 1.030 × 172.6 × 1.000 = 177.8 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0115.tif"/>

According to the phasing diagram, the volume of voids in mineral aggregate is equal to:

V ma = V be + V a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0116.tif"/>

Where:

V_ma = volume of voids in mineral aggregate in asphalt mixture

V_be = V_fa = volume of effective asphalt binder = volume of voids filled with asphalt

V_a = V_v = volume of air voids in asphalt mixture338

⇒

V ma = 172.6 + 76.0 = 248.6 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0117.tif"/>

Now as all the needed volumes and weights in the phasing system of the asphalt mixture are determined, the required volumetric properties can be calculated easily based on the definitions of these properties as below:

The effective asphalt binder content (P_be) is determined based on its definition or from the asphalt binder content and the absorbed asphalt binder content since both are provided in the problem:

P be = 100 ( W be W mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0118.tif"/>

⇒

P be = 100 ( 177.8 ( 4443.9 + 233.9 ) = 3.80 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0119.tif"/>

Or:

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0120.tif"/>

And notice that:

12.32 P ba-mixture = ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0121.tif"/>

Where:

P_ba = absorbed asphalt binder content, percent by total weight of aggregate

P_s = percentage of aggregate by total weight of asphalt mixture

P_ba-mixture = absorbed asphalt binder content, percent by total weight of asphalt mixture

In other words, the effective asphalt binder content formula can be re-written as:

P be = P b − P ba-mixture https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0122.tif"/>

Therefore, the effective asphalt binder content can be determined directly from the formula since the absorbed asphalt binder content is given by total weight of asphalt mixture and not by total weight of aggregate as below:

P be = 5.0 − 1.20 = 3.80 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0123.tif"/>

The voids in mineral aggregate (VMA) is determined based on the definition of these voids in asphalt mixture as follows:

VMA = 100 ( V ma V mb ) = 100 ( V ma V T ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0124.tif"/>

Where:339

VMA = voids in mineral aggregate, percent by total volume of asphalt mixture

V_ma = volume of voids in mineral aggregate

V_mb = V_T = bulk or total volume of asphalt mixture

⇒

VMA = 100 ( 248.6 2053.6 ) = 12.1 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0125.tif"/>

The voids filled with asphalt (VFA) is also determined in a similar manner using the definition of these voids in the phasing system of the asphalt mixture:

12.35 VFA = 100 ( V fa V ma ) = 100 ( V be V ma ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0126.tif"/>

Where:

VFA = voids filled with asphalt, percent by total volume of voids in asphalt mixture

V_fa= V_be = volume of in voids filled with asphalt = volume of effective asphalt binder

V_ma = volume of voids in mineral aggregate

⇒

VFA = 100 ( 172.6 248.6 ) = 69.4 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0127.tif"/>

The air voids in the asphalt mixture (V_A or VTM) is calculated using the definition as well:

V A = VTM = 100 ( V a V mb ) = 100 ( V a V T ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0128.tif"/>

Where:

V_A = VTM = air voids in asphalt mixture, percent by total volume of asphalt mixture

V_a= V_v = volume of air voids in asphalt mixture

V_mb = V_T = bulk or total volume of asphalt mixture

⇒

V a = 100 ( 76.0 2053.6 ) = 3.7 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0129.tif"/>

The bulk specific gravity of the asphalt mixture (G_mb) is determined from the formula of the VMA as follows:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0130.tif"/>

⇒340

G mb = ( 1 − VMA 100 ) ( P s 100 G sb ) = ( 1 − 12.1 100 ) ( 95.0 100 × 2.462 ) = 2.278 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0131.tif"/>

Or it can be also determined using the definition of the bulk specific gravity (Equation 12.11) as follows:

G mb = W mb V mb γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0132.tif"/>

⇒

G mb = ( 4443.9 + 233.9 ) 2053.6 ( 1.000 ) = 2.278 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0133.tif"/>

The theoretical maximum specific gravity of the asphalt mixture (G_mm) is finally determined using the following formula:

VTM = V a = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0134.tif"/>

⇒

3.7 = 100 ( 1 − 2.278 G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0135.tif"/>

⇒

G mm = 2.365 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0136.tif"/>

Or, it can be also determined using the definition of the theoretical maximum specific gravity as follows:

G mm = W mm V mm γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0137.tif"/>

Where:

G_mm = theoretical maximum specific gravity of asphalt mixture (loose mixture)

W_mm = W_mb = total or bulk weight of asphalt mixture

V_mm = void-less volume of asphalt mixture = total volume of asphalt mixture less air voids volume

⇒

G mm = ( 4443.9 + 233.9 ) 2053.6 − 76.0 = 2.365 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0138.tif"/>

The MS Excel worksheet used to solve this problem in an efficient and easy way is shown in Figure 12.7.341

12.6

In a Superpave design for an asphalt mixture, the percentage of asphalt binder by total weight of asphalt mixture is 5.2%. The bulk specific gravity of aggregate (G_sb) = 2.536, the specific gravity of asphalt binder (G_b) = 1.020, and the density of water (γ_w) = 1.000 g/cm³. If the absorbed asphalt binder (P_ba) is 1.70% by total weight of aggregate, and the voids filled with asphalt (VFA) is 68.3%, determine the following properties of the asphalt mixture:

The effective asphalt binder content (P_be)

The bulk specific gravity of the asphalt mixture (G_mb)

The voids in mineral aggregate (VMA)

The air voids in the asphalt mixture (V_a or VTM)

The theoretical maximum specific gravity of the asphalt mixture (G_mm)

The effective specific gravity of the aggregate (G_se)

The dust proportion (DP) if the percent passing No. 200 (75 μm) sieve is 4.5%.

Solution:

The effective asphalt binder content (P_be) is determined using the formula shown below:

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0139.tif"/>

P s = 100 − P b = 100 − 5.2 = 94.8 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0140.tif"/>

⇒

P be = 5.2 − ( 1.70 100 ) 94.8 = 3.59 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0141.tif"/> 342

The bulk specific gravity of the asphalt mixture (G_mb) is determined as shown in the following procedure:

The expression of Equation 12.23 derived in Problem 12.2 for VMA will be used:

⇒

100 ( 3.59 68.3 ) ( G mb 1.020 ) = 100 ( 1 − G mb × 94.8 100 2.536 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0143.tif"/>

Solving this equation provides the following solution for Gmb:

G mb = 100 ( 100 × 3.59 68.3 × 1.020 + 94.8 2.536 ) = 2.351 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0144.tif"/>

The voids in mineral aggregate (VMA) is determined using the typical VMA formula:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0145.tif"/>

⇒

VMA = 100 ( 1 − 2.351 × 94.8 100 2.536 ) = 12.1 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0146.tif"/>

Or using the formula derived earlier in this chapter:

VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0147.tif"/>

⇒

VMA = 100 ( 3.59 68.3 ) ( 2.351 1.020 ) = 12.1 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0148.tif"/>

The air voids in the asphalt mixture (V_A or VTM) is determined as shown below:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0149.tif"/> 343

⇒

68.3 = 100 ( 12.1 − VTM 12.1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0150.tif"/>

Solving for VTM provides:

⇒

VTM or V A = 12.1 ( 1 − 68.3 100 ) = 3.84 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0151.tif"/>

The theoretical maximum specific gravity of the asphalt mixture (G_mm) is determined using the VTM (V_A) formula:

VTM = V A = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0152.tif"/>

⇒

3.84 = 100 ( 1 − 2.351 G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0153.tif"/>

⇒

G mm = 2.351 ( 1 − 3.84 100 ) = 2.445 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0154.tif"/>

The effective specific gravity of the aggregate (G_se) is calculated using the P_ba formula shown below:

⇒

1.70 = 100 ( G se − 2.536 2.536 G se ) 1.020 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0156.tif"/>

Solving for G_se provides:

G se = 2.648 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0157.tif"/>

The dust proportion (DP) is determined using the typical DP formula:

DP = P 0.075 P be https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0158.tif"/>

Where:

P_0.075 = percent aggregate portion passing sieve No. 200 (0.075 mm = 75 micron), %

P_be = effective asphalt binder content, percent by total weight of asphalt mixture

⇒

DP = 4.5 3.59 = 1.25 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0159.tif"/>

The MS Excel worksheet used to solve this problem in an efficient and easy way is shown in Figure 12.8.344

12.7

If the apparent volume of the aggregate in a Marshall specimen is 405.0 cm³, the volume of the impermeable voids inside the aggregate particles is 50.0 cm³, and the asphalt binder content by total weight of asphalt mixture is 5.1%, determine the effective specific gravity (G_se) of the aggregate, knowing that the weight of the asphalt binder is 65.0 g.

Solution:

According to the phasing system of the asphalt mixture, the effective volume of aggregate is equal to the apparent volume of aggregate plus the volume of impermeable voids in the aggregate. Therefore:

V se = V sa + V impermeable voids https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0160.tif"/>

Where:

V_se = effective volume of aggregate in asphalt mixture

V_sa = apparent volume of aggregate in asphalt mixture

V _{impermeable voids} = volume of impermeable voids of aggregate in asphalt mixture

⇒

V se = 405.0 + 50.0 = 455.0 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0161.tif"/>

P b = 100 W b W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0162.tif"/> 345

⇒

5.1 = 100 65.0 ( 65.0 + W s ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0163.tif"/>

⇒

W s = 1209.5 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0164.tif"/>

The definition of G_se is given as:

G se = W s V se γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0165.tif"/>

Where:

G_se = effective specific gravity of aggregate in asphalt mixture

W_s = weight of aggregate

V_se = effective volume of aggregate in asphalt mixture

γ_w = density of water (1.000 g/cm³)

⇒

G se = 1209.5 455 ( 1.000 ) = 2.658 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0166.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.9.346

12.8

If the volume of the voids in mineral aggregate of a Marshall specimen (having a weight of 1270.0 g) is 55.0 cm³, and 20% of these voids are air voids, determine the effective asphalt binder content (P_be) knowing that the specific gravity of the asphalt binder is 1.030.

Solution:

Since 20% of the volume of voids in mineral aggregate is air voids, the remaining 80% of the volume of voids in mineral aggregate is the volume of the effective asphalt binder (voids filled with asphalt); in other words, it is equal to:

V be = V fa = ( 1 − 20 100 ) V ma https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0167.tif"/>

⇒

V be = V fa = 55.0 ( 1 − 20 100 ) = 44.0 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0168.tif"/>

G b = G be = W be V be γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0169.tif"/>

⇒

W be = G b V be γ w = 1.030 ( 44.0 ) ( 1.000 ) = 45.3 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0170.tif"/>

P be = 100 W be W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0171.tif"/>

⇒

P be = 100 45.3 1270.0 = 3.57 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0172.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.10.347

12.9

If the voids filled with asphalt of a Marshall specimen by total volume of the specimen is 9.0%, the bulk specific gravity of the specimen is 2.327, and the specific gravity of the asphalt binder is 1.025, determine the effective asphalt binder content (P_be).

Solution:

A formula for VMA derived earlier in Equation 12.22 in Problem 12.2 will be used:

VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0173.tif"/>

The voids filled with asphalt in this problem is given as a percent by total volume of the specimen. To understand the physical meaning of this term, the following derivations must be performed and remarked:

12.41 Voids filled with asphalt, percent by total volume of mixture = ( VFA ) ( VMA ) 100 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0174.tif"/>

Where:

VFA = voids filled with asphalt, percent by total volume of voids in asphalt mixture

VMA = voids in mineral aggregate, percent by total volume of asphalt mixture

The value (VFA)(VMA)/100 is the same value as the voids filled with asphalt as a percent by total volume of asphalt mixture. Using the definitions of VFA and VMA, this value can be written as:

( VFA ) ( VMA ) 100 = ( 100 V fa V ma ) ( 100 V ma V T ) 100 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0175.tif"/>

⇒

( VFA ) ( VMA ) 100 = 100 V fa V T = Voids filled with asphalt, % by total volume of asphalt mixture https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0176.tif"/>

Now from the formula:

VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0177.tif"/>

⇒

P be = ( ( VMA ) ( VFA ) 100 ) ( G b G mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0178.tif"/>

And:

( ( VMA ) ( VFA ) 100 ) = 9.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0179.tif"/>

⇒

P be = 9.0 ( 1.025 2.327 ) = 3.96 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0180.tif"/>

Again, the MS Excel worksheet used to solve this problem is shown in Figure 12.11.348

12.10

In a Superpave asphalt mixture design test, if the voids in mineral aggregate by total volume of asphalt mixture (VMA) is 13.0%, the bulk specific gravity of the asphalt mixture is 2.442, the specific gravity of the asphalt binder is 1.020, the absorbed asphalt binder by weight of aggregate is 1.30%, and the asphalt binder is mixed with the aggregate at a proportion of 1:20 by weight, determine the air voids content in the asphalt mixture (V_a or VTM), and the voids filled with asphalt (VFA).

Solution:

The percentage of asphalt binder by total weight of asphalt mixture is determined as below:

Since the proportion of asphalt binder to aggregate by weight = 1/20, this means that:

W b W s = 0.05 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0181.tif"/>

⇒

W b = 0.05 W s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0182.tif"/>

From Equation 12.24 (in Problem 12.4), since:

P b = 100 W b W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0183.tif"/>

Or:

P b = 100 W b W b + W s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0184.tif"/> 349

⇒

P b = 100 0.05 W s 0.05 W s + W s = 4.76 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0185.tif"/>

P s = 100 − P b = 100 − 4.76 = 95.24 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0186.tif"/>

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0187.tif"/>

⇒

P be = 4.76 − ( 1.30 100 ) 95.24 = 3.52 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0188.tif"/>

The formula derived in Equation 12.22 (in Problem 12.2) and used above in Problem 12.9 will be used to determine the VFA:

VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0189.tif"/>

⇒

VFA = 100 ( P be VMA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0190.tif"/>

⇒

VFA = 100 ( 3.52 13.0 ) ( 2.442 1.020 ) = 64.8 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0191.tif"/>

And the formula for VFA will be used to determine the VTM or V_A:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0192.tif"/>

⇒

VTM or V A = VMA ( 1 − VFA 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0193.tif"/>

⇒

VTM or V a = 13.0 ( 1 − 64.8 100 ) = 4.6 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0194.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.12.350

12.11

If the voids filled with asphalt (VFA) of a Superpave gyratory specimen is 68.0%, the absorbed asphalt binder by weight of aggregate is 1.80%, the asphalt binder content by weight of aggregate is 5.5%, the effective specific gravity of the aggregate is 2.715, and the specific gravity of the asphalt binder is 1.030, determine the following asphalt mixture properties:

The effective asphalt binder content (P_be)

The bulk specific gravity of the specimen (G_mb)

The voids in mineral aggregate (VMA)

The air voids or voids in total mixture (V_a or VTM)

The theoretical maximum specific gravity of the mixture (G_mm)

Solution:

The bulk specific gravity of the aggregate is calculated using the formula shown below:

⇒

G sb = 1 ( P ba 100 G b + 1 G se ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0196.tif"/>

⇒

G sb = 1 ( 1.80 100 ( 1.030 ) + 1 2.715 ) = 2.592 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0197.tif"/> 351

The asphalt binder is given as a percent by total weight of aggregate; therefore, the asphalt binder must be determined as a percent by total weight of asphalt mixture as shown below:

P s = 100 ( 1 + P b -aggregate 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0198.tif"/>

Or:

P b = P b -aggregate ( 1 + P b -aggregate 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0199.tif"/>

⇒

P b = 5.5 ( 1 + 5.5 100 ) = 5.21 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0200.tif"/>

But,

P s = 100 − P b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0201.tif"/>

⇒

P s = 100 − P b = 100 − 5.21 = 94.79 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0202.tif"/>

The effective asphalt binder content (P_be) is determined using the formula shown below:

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0203.tif"/>

⇒

P be = 5.21 − ( 1.80 100 ) 94.79 = 3.51 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0204.tif"/>

The bulk specific gravity of the specimen (G_mb) is determined as shown in the procedure below:

The expression of Equation 12.23 derived in Problem 12.2 will be used to determine the G_mb:

⇒

G mb = 100 ( P be VFA ) ( 100 G b ) + P s G sb https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0206.tif"/> 352

⇒

G mb = 100 ( 3.51 68.0 ) ( 100 1.030 ) + 94.79 2.592 = 2.405 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0207.tif"/>

The voids in mineral aggregate (VMA) is calculated using the VMA formula as follows:

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0208.tif"/>

⇒

VMA = 100 ( 1 − 2.405 × 94.79 100 2.592 ) = 12.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0209.tif"/>

The air voids or voids in total mixture (V_A or VTM) is determined using the formula shown below:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0210.tif"/>

⇒

VTM or V a = VMA ( 1 − VFA 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0211.tif"/>

⇒

VTM or V a = 12.0 ( 1 − 68.0 100 ) = 3.9 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0212.tif"/>

The theoretical maximum specific gravity of the mixture (G_mm) is determined using the formula below:

VTM = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0213.tif"/>

⇒

G mm = G mb ( 1 − VTM 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0214.tif"/>

⇒

G mm = 2.405 ( 1 − 3.9 100 ) = 2.502 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0215.tif"/>

The MS Excel worksheet used to perform the computations to determine all mixture properties in this problem is shown in Figure 12.13.353

12.12

If the voids filled with asphalt of a Superpave gyratory specimen by total volume of the specimen is 10.0%, the bulk specific gravity of the specimen is 2.322, the effective specific gravity of the aggregate is 2.654, and the specific gravity of the asphalt binder is 1.025, and the asphalt binder content by total weight of mixture is 5.2%, determine the voids in mineral aggregate (VMA) and percentage of air voids (V_A or VTM) in the specimen.

Solution:

According to the formula in Equation 12.41 derived in Problem 12.9 that is shown below:

Voids filled with asphalt, percent by total volume of mixture = ( VFA ) ( VMA ) 100 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0216.tif"/>

10.0 = ( VFA ) ( VMA ) 100 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0217.tif"/>

And using the formula in Equation 12.22 derived in Problem 12.2 and shown below:

VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0218.tif"/>

⇒

(Equation 12.43 in Problem 12.9).

But:

( VFA ) ( VMA ) 100 = 10.0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0220.tif"/>

⇒

P be = 10.0 ( 1.025 2.322 ) = 4.41 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0221.tif"/>

P s = 100 − P b = 100 − 5.2 = 94.8 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0222.tif"/>

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0223.tif"/>

⇒

P ba = 100 ( P b − P be P s ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0224.tif"/>

⇒

P ba = 100 ( 5.2 − 4.41 94.8 ) = 0.83 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0225.tif"/>

But:

⇒

12.48 G sb = 1 ( P ba 100 G b + 1 G se ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0227.tif"/>

⇒

G sb = 1 ( 0.83 100 ( 1.025 ) + 1 2.654 ) = 2.598 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0228.tif"/>

Now the voids in mineral aggregate (VMA) can be determined using the VMA formula shown below:

⇒

VMA = 100 ( 1 − 2.322 × 94.8 100 2.598 ) = 15.3 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0230.tif"/>

The percentage of air voids (V_A or VTM) is also determined using the VFA formula as follows:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0231.tif"/>

⇒

VMA − VTM = ( ( VFA ) ( VMA ) 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0232.tif"/>

But again:

( VFA ) ( VMA ) 100 = 10.0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0233.tif"/>

⇒

VMA − VTM = 10.0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0234.tif"/>

⇒

VTM or V A = VMA − 10.0 = 15.3 − 10.0 = 5.3 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0235.tif"/>

Or:

One can think of this in another way. The volume of voids in mineral aggregate is simply equal to the volume of air voids and the volume of voids filled with asphalt. If all these percentages are computed as a percent by total volume of asphalt mixture, then it can be written as:

12.50 VMA = VFA % by volume of mixture + V A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0236.tif"/>

Where:

VMA = voids in mineral aggregate, percent by total volume of asphalt mixture

VFA_{%by volume of mixture} = voids filled with asphalt, percent by total volume of asphalt mixture

V_A or VTM = air voids, percent by total volume of asphalt mixture

But a subscript is added to the VFA because typically the VFA is determined as a percentage by volume of total voids in the asphalt mixture and not by total volume of asphalt mixture. So, in this case, it is determined by total volume of mixture. Therefore, it can be added directly to the V_A (or VTM), which is also determined as a percentage by total volume of asphalt mixture. The result is the VMA as a percentage by total volume of asphalt mixture.

VMA = VFA % by volume of mixture + V A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0237.tif"/> 356

⇒

15.3 = 10.0 + V A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0238.tif"/>

⇒

V _A = 5.3% (This is the same answer as before).

The MS Excel worksheet used to perform the computations of this problem and which does the computations in an easy and efficient way, is shown in Figure 12.14.

Another solution for this problem is explained below:

12.51 G mm = P mm P s G se + P b G b https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0239.tif"/>

Where:

G_mm = theoretical maximum specific gravity of asphalt mixture

P_mm = total loose asphalt mixture, percentage by total weight of asphalt mixture = 100%

P_s = percentage of aggregate by total weight of asphalt mixture

G_se = effective specific gravity of aggregate

P_b = percentage of asphalt binder by total weight of asphalt mixture

G_b = specific gravity of asphalt binder

⇒

G mm = 100 94.8 2.654 + 5.2 1.025 = 2.451 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0240.tif"/> 357

VTM = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0241.tif"/>

⇒

VTM or V A = 100 ( 1 − 2.322 2.451 ) = 5.3 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0242.tif"/>

VMA = VFA % by volume of mixture + V A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0243.tif"/>

⇒

VMA = 10.0 + 5.3 = 15.3 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0244.tif"/>

The MS Excel worksheet used to perform the computations of the second solution of this problem is shown in Figure 12.15.

12.13

In a Marshall mixture design test, if the absorbed asphalt binder by total weight of mixture is 1.50%, and the asphalt binder content by total weight of aggregate is 5.0%, determine the effective asphalt binder content by weight of asphalt mixture (P_be).

Solution:

Since the given asphalt binder content is a percentage by total weight of aggregate, the asphalt binder content as a percent by total weight of asphalt mixture (P_b) is determined first as shown below:

P s = 100 ( 1 + P b -aggregate 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0245.tif"/> 358

Or:

⇒

P b = 5.0 ( 1 + 5.0 100 ) = 4.76 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0247.tif"/>

The absorbed asphalt binder is given by total weight of mixture and the typical value is normally given as a percentage by total weight of aggregate (P_ba). Therefore,

12.52 P ba-% by weight of mixture = P ba ( P s 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0248.tif"/>

Where:

P_{ba-% by weight of
mixture} = absorbed asphalt binder content, percent by total weight of asphalt mixture

P_ba = absorbed asphalt binder content, percent by total weight of aggregate

P_s = percentage of aggregate, by total weight of asphalt mixture

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0249.tif"/>

⇒

P be = P b − P ba-% by weight of mixture https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0250.tif"/>

⇒

P be = 4.76 − 1.50 = 3.26 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0251.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.16.359

12.14

If the voids in mineral aggregate (VMA) of a Superpave gyratory specimen is 16.0% and 25% of these voids are air voids, determine the voids filled with asphalt (VFA).

Solution:

Since 25% of the VMA is air voids, and knowing that both the air voids content (V_A) and the VMA are determined as a percent of the total volume of the asphalt mixture, then:

V A = 25 100 ( VMA ) = 0.25 ( 16.0 ) = 4.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0252.tif"/>

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0253.tif"/>

⇒

VFA = 100 ( 16.0 − 4.0 16.0 ) = 75.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0254.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.17.360

12.15

If the total volume of a Marshall specimen is 515.0 cm³, the theoretical maximum specific gravity of the loose mixture (G_mm) is 2.566, and the voids in total mixture (VTM) by total volume of mixture is 4.0%, determine the weight of the specimen (in grams).

Solution:

VTM = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0255.tif"/>

⇒

G mb = G mm ( 1 − VTM 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0256.tif"/>

⇒

G mb = 2.566 ( 1 − 4.0 100 ) = 2.463 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0257.tif"/>

G mb = W mb V mb γ w = W T V T γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0258.tif"/>

(Equation 12.11 used in Problem 12.1).

⇒

W T = G mb V mb γ w = 2.463 ( 515 ) ( 1.000 ) = 1268.6 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0259.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.18.361

12.16

If the voids filled with asphalt of a Superpave gyratory specimen by total volume of the specimen is 7.0%, the specific gravity of the asphalt binder is 1.015, and the effective asphalt binder content (P_be) by total weight of specimen is 3.00%, determine the bulk specific gravity of the specimen (G_mb).

Solution:

The formula of Equation 12.22 derived in Problem 12.2 for VMA will be used to solve for G_mb as shown below:

VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0260.tif"/>

⇒

G mb = ( ( VFA ) ( VMA ) 100 ) ( G b P be ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0261.tif"/>

According to the formula in Equation 12.41 derived in Problem 12.9 and shown below:

⇒

( VFA ) ( VMA ) 100 = 7.0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0263.tif"/>

⇒

G mb = 7.0 ( 1.015 3.00 ) = 2.368 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0264.tif"/>

The MS Excel worksheet used to solve this problem is shown in Figure 12.19.362

12.17

If the voids filled with asphalt of a Marshall specimen by total volume of the specimen is 6.0%, the voids in total mixture (VTM) is 4.0%, the asphalt binder content by total weight of specimen is 4.8%, the theoretical maximum specific gravity of the mixture (G_mm) is 2.488, and the specific gravity of the asphalt binder is 1.020, determine the bulk specific gravity of the aggregate (G_sb), the effective binder content by weight of specimen (P_be), and the voids filled with asphalt as a percent by volume of total voids (VFA).

Solution:

The formula that relates the VMA, air voids, and VFA as a percent of total mixture will be used:

VMA = VFA % by volume of mixture + V A or VTM https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0265.tif"/>

⇒

VMA = 6.0 + 4.0 = 10.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0266.tif"/>

VTM = 100 ( 1 − G mb G mm ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0267.tif"/>

⇒

G mb = G mm ( 1 − VTM 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0268.tif"/>

⇒

G mb = 2.488 ( 1 − 4.0 100 ) = 2.388 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0269.tif"/> 363

P s = 100 − P b = 100 − 4.8 = 95.2 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0270.tif"/>

VMA = 100 ( 1 − G mb × P s 100 G sb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0271.tif"/>

⇒

12.55 G sb = ( G mb P s 100 ) ( 1 − VMA 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0272.tif"/>

⇒

G sb = ( 2.388 × 95.2 100 ) ( 1 − 10.0 100 ) = 2.526 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0273.tif"/>

Again, the formula of Equation 12.22 derived in Problem 12.2 for VMA will be used to solve for P_be as shown below:

VMA = 100 ( P be VFA ) ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0274.tif"/>

⇒

But:

⇒

( VFA ) ( VMA ) 100 = 6.0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0277.tif"/>

⇒

P be = 6.0 ( 1.020 G mb ) = 2.56 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0278.tif"/>

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0279.tif"/> 364

⇒

VFA = 100 ( 10.0 − 4.0 10.0 ) = 60.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0280.tif"/>

The MS Excel worksheet used to perform all the computations of this problem to determine the volumetric properties of the Marshall specimen easily and efficiently is shown in Figure 12.20.

12.18

In a Marshall asphalt mixture design test, if the voids in total mixture (VTM) is 4.2%, the bulk specific gravity of the mixture is 2.444, the specific gravity of the asphalt binder is 1.030, the absorbed asphalt binder by total weight of aggregate is 1.20%, and the asphalt binder content by total weight of mixture is 4.0%, determine the voids in mineral aggregate (VMA), and the voids filled with asphalt (VFA).

Solution:

The percentage of aggregate by the total weight of the specimen is determined simply using the formula below:

P s = 100 − P b = 100 − 4.0 = 96.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0281.tif"/>

The effective asphalt binder content by the total weight of the specimen (P_be) is also determined using the formula for P_be as follows:

P be = P b − ( P ba 100 ) P s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0282.tif"/> 365

⇒

P be = 4.0 − ( 1.20 100 ) 96.0 = 2.85 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0283.tif"/>

The expression of Equation 12.14 derived in Problem 12.1 for VMA will be used herein to determine VMA as shown below:

VMA = VTM + P be ( G mb G b ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0284.tif"/>

⇒

VMA = 4.2 + 2.85 ( 2.444 1.030 ) = 10.96 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0285.tif"/>

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0286.tif"/>

⇒

VFA = 100 ( 11.0 − 4.2 11.0 ) = 61.7 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0287.tif"/>

The MS Excel worksheet used to perform all the computations of this problem and to determine the VMA and VFA of the specimen easily and efficiently is shown in Figure 12.21.366

12.19

If the voids filled with asphalt of a Superpave gyratory specimen by total volume of the specimen is 9.0%, and 25.0% of the voids in mineral aggregate (VMA) is air voids, determine the VMA and the VTM (V_a = the percentage of air voids by total volume of specimen).

Solution:

The formula that relates the VMA, air voids, and VFA as a percent of total mixture will be used (Equation 12.50 derived in Problem 12.12):

VFA % by volume of mixture = 9.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0289.tif"/>

But the air voids content is equal to 25.0% of the VMA, therefore:

V A = VTM = 0.25 VMA https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0290.tif"/>

⇒

VMA = 9.0 + 0.25 VMA https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0291.tif"/>

⇒

VMA = 12.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0292.tif"/>

⇒

V A = VTM = 0.25 VMA = 0.25 ( 12.0 ) = 3.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0293.tif"/>

Or:

V A = VMA − VFA % by volume of mixture = 12.0 − 9.0 = 3.0 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0294.tif"/>

See the MS Excel worksheet shown in Figure 12.22.367

12.20

In a Marshall specimen having a total volume of 510.0 cm³, the volume of the absorbed asphalt is 15.0 cm³, the volume of air voids is 30.0 cm³, and the weight of the asphalt binder is 60.0 g, determine the VMA and G_mb if the asphalt binder content by total weight of specimen is 4.8% and the specific gravity (G_b) of the asphalt binder is 1.030.

Solution:

The phasing system of the asphalt mixture specimen is important in this problem to understand the volumetric analysis and the relationships between the volumes of the different phases in the system and the weights of these phases (see Figure 12.23).

The volume of asphalt binder is calculated from the weight of the asphalt binder as follows:

G b = W b V b γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0295.tif"/>

⇒

V b = W b G b γ w = 60.0 1.030 ( 1.000 ) = 58.3 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0296.tif"/>

The volume of effective asphalt binder (same as the volume of voids filled with asphalt) is simply the volume of asphalt binder minus the volume of absorbed asphalt binder based on the phasing system of the asphalt mixture specimen:

V be = V fa = V b − V ba https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0297.tif"/>

Where:

V_be = V_fa = volume of effective asphalt binder (voids filled with asphalt)

V_b = volume of asphalt binder

V_ba = volume of absorbed asphalt binder368

Therefore,

V be = V fa = 58.3 − 15.0 = 43.3 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0298.tif"/>

The volume of voids in mineral aggregate is also equal to the volume of effective asphalt binder (voids filled with asphalt) plus the volume of air voids according to the phasing system of the mixture:

12.57 V ma = V be + V a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0299.tif"/>

Where:

V_ma = volume of voids in mineral aggregate

V_be = V_fa = volume of effective asphalt binder (voids filled with asphalt)

V_a = volume of air voids

⇒

V ma = 43.3 + 30.0 = 73.3 cm 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0300.tif"/>

VMA = 100 ( V a + V be V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0301.tif"/>

⇒

VMA = 100 ( 73.3 510.0 ) = 14.4 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0302.tif"/>

VTM = 100 ( V a V mb ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0303.tif"/>

⇒

VTM = 100 ( 30.0 510.0 ) = 5.9 % https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0304.tif"/>

Now to determine the G_mb of the specimen, the total weight of the specimen is required. It can be determined from the following formula:

P b = 100 W b W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0305.tif"/>

⇒

W T = 100 W b P b = 100 60.0 4.8 = 1250.0 g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0306.tif"/>

Now, the G_mb is calculated using the formula below:

G mb = W mb V mb γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0307.tif"/> 369

⇒

G mb = 1250.0 510.0 ( 1.000 ) = 2.451 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0308.tif"/>

The MS Excel worksheet used to perform the computations of the volumetric analysis of the Marshall specimen in this problem in a rapid and efficient way is shown in Figure 12.24.

12.21

Prove that P be ≠ P b − P ba https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0309.tif"/> despite the fact that from the phasing system of the asphalt mixture V be = V b − V ba https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0310.tif"/> .

And show that:

P be = P b − P ba ( P s 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0311.tif"/>

Solution:

V be = V b − V ba https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0312.tif"/>

The relationship between volume phase, weight, and specific gravity is given as below:

G b = W b V b γ w = W be V be γ w = W ba V ba γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0313.tif"/> 370

Therefore, V be = V b − V ba https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0314.tif"/> becomes:

W be G b γ w = W b G b γ w − W ba G b γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0315.tif"/>

By multiplying both sides of the equation by the term 100 W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0316.tif"/> and cancelling the term 1 G b γ w https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0317.tif"/> from both sides of the equation, the following expression is obtained:

100 W be W T = 100 W b W T − 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0318.tif"/>

Now the definition of P_be, the percentage of effective asphalt binder by total weight of asphalt mixture is given as follows:

P be = 100 W be W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0319.tif"/>

Also, the definition of P_b, the percentage of asphalt binder by total weight of asphalt mixture is given as follows:

P b = 100 W b W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0320.tif"/>

On the other hand, the definition of P_ba, the percentage of absorbed asphalt binder by total weight of aggregate is given as follows:

P ba = 100 W ba W s ≠ 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0321.tif"/>

In other words, the equation 100 W be W T = 100 W b W T − 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0322.tif"/> becomes:

P be = P b − 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0323.tif"/>

And since P ba = 100 W ba W s ≠ 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0324.tif"/> , therefore:

P be ≠ P b − P ba https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0325.tif"/>

But rather:

P be = P b − 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0326.tif"/>

By multiplying the term 100 W ba W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0327.tif"/> in the above equation by the term W s W s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0328.tif"/> , it becomes:

P be = P b − 100 W ba W T W s W s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0329.tif"/> 371

But:

12.61 P ba = 100 W ba W s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0330.tif"/>

And:

P s = 100 W s W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0331.tif"/>

Or:

P s 100 = W s W T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0332.tif"/>

Therefore, the equation P be = P b − 100 W ba W T W s W s https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0333.tif"/> becomes:

P be = P b − P ba ( P s 100 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0334.tif"/>

12.22

Prove that VFA ≠ VMA − VTM https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0335.tif"/> despite the fact that from the phasing system of the asphalt mixture V fa = V be = V ma − V a https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0336.tif"/> .

And show that:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0337.tif"/>

Solution:

V fa = V be = V ma − V a . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0338.tif"/>

By multiplying both sides of the equation by the term 100 V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0339.tif"/> , the following expression is obtained:

100 V fa V T = 100 V ma V T − 100 V a V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0340.tif"/>

Now the definition of VMA, the voids in mineral aggregate as a percent by total volume of asphalt mixture is given as follows:

VMA = 100 V ma V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0341.tif"/>

Also, the definition of VTM, the air voids as a percent by total volume of asphalt mixture is given as follows:

VTM = 100 V a V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0342.tif"/> 372

On the other hand, the definition of VFA, the voids filled with asphalt as a percent by volume of total voids in the asphalt mixture is given as follows:

VFA = 100 V fa V ma https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0343.tif"/>

In other words, the equation 100 V fa V T = 100 V ma V T − 100 V a V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0344.tif"/> becomes:

12.64 100 V fa V T = VMA − VTM https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0345.tif"/>

But since:

VFA = 100 V fa V ma ≠ 100 V fa V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0346.tif"/>

⇒

VFA ≠ VMA − VTM https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0347.tif"/>

100 V fa V T = VMA − VTM https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0348.tif"/>

By dividing both sides of the above equation by the term 100 V ma V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0349.tif"/> , it becomes:

( 100 V fa V T ) ( 100 V ma V T ) = VMA − VTM ( 100 V ma V T ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0350.tif"/>

Or:

( 100 V fa V ma ) ( V T 100 V T ) = VMA − VTM ( 100 V ma V T ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0351.tif"/>

Multiplying both sides of the above equation by 100 provides:

( 100 V fa V ma ) = 100 VMA − VTM ( 100 V ma V T ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0352.tif"/> (12.65)

Knowing that:

VMA = 100 V ma V T https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0353.tif"/> 373 374

And:

VFA = 100 V fa V ma https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0354.tif"/>

The equation ( 100 V fa V ma ) = 100 VMA − VTM ( 100 V ma V T ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0355.tif"/> becomes:

VFA = 100 ( VMA − VTM VMA ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/TNF-CH012_eqn_0356.tif"/>

Figure 12.1 Phasing diagram of a compacted asphalt mixture. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_1_B.tif"/> Figure 12.2 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.1. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_2_B.tif"/> Figure 12.3 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.2. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_3_B.tif"/> Figure 12.4 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.3. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_4_B.tif"/> Figure 12.5 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.4. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_5_B.tif"/> Figure 12.6 Phasing diagram of a compacted asphalt mixture for Problem 12.5. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_6_B.tif"/> Figure 12.7 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.5. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_7_B.tif"/> Figure 12.8 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.6. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_8_B.tif"/> Figure 12.9 Image of the MS Excel worksheet used for the computations of Problem 12.7. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_9_B.tif"/> Figure 12.10 Image of the MS Excel worksheet used for the computations of Problem 12.8. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_10_B.tif"/> Figure 12.11 Image of the MS Excel worksheet used for the computations of Problem 12.9. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_11_B.tif"/> Figure 12.12 Image of the MS Excel worksheet used for the computations of Problem 12.10. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_12_B.tif"/> Figure 12.13 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.11. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_13_B.tif"/> Figure 12.14 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.12. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_14_B.tif"/> Figure 12.15 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.12 (second solution). https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_15_B.tif"/> Figure 12.16 Image of the MS Excel worksheet used for the computation of Pbe of the asphalt mixture in Problem 12.13. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_16_B.tif"/> Figure 12.17 Image of the MS Excel worksheet used for the Computation of VA and VFA of the asphalt mixture in Problem 12.14. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_17_B.tif"/> Figure 12.18 Image of the MS Excel worksheet used for the computations of Problem 12.15. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_18_B.tif"/> Figure 12.19 Image of the MS Excel worksheet used for the computations of Problem 12.16. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_19_B.tif"/> Figure 12.20 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.17. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_20_B.tif"/> Figure 12.21 Image of the MS Excel worksheet used for the computations of Problem 12.18. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_21_B.tif"/> Figure 12.22 Image of the MS Excel worksheet used for the determination of VA and VMA of the asphalt mixture in Problem 12.19. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_22_B.tif"/> Figure 12.23 Phasing diagram of the compacted asphalt mixture in Problem 12.20. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_23_B.tif"/> Figure 12.24 Image of the MS Excel worksheet used for the volumetric analysis of the asphalt mixture in Problem 12.20. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429054297/5885f92e-b0a3-419d-837a-bad4e02024b0/content/fig12_24_B.tif"/>

Volumetric Analysis of Asphalt Mixtures (Marshall and Superpave)

ABSTRACT