Skip to contentUnited States Department of Transportation - Federal Highway Administration FHWA Home
Research Home   |   Pavements Home
REPORT
This report is an archived publication and may contain dated technical, contact, and link information
Publication Number: FHWA-HRT-12-030
Date: August 2012

 

Estimation of Key PCC, Base, Subbase, and Pavement Engineering Properties From Routine Tests and Physical Characteristics

PDF Version (4.44 MB)

PDF files can be viewed with the Acrobat® Reader®

CHAPTER 5. MODEL DEVELOPMENT (15)

CTE Model 1: CTE Based on Aggregate Type (Level 3 Equation for MEPDG)

CTE test data were averaged for each aggregate type, and this constituted level 3 inputs for MEPDG. These averages were determined for the entire set of LTPP data as well as for the subset developed by deleting suspect data. A summary of the data is presented in table 47. Table 47 also lists the average PCC CTE for each aggregate type as found in the literature. The data are all in general agreement, providing a degree of confidence in the level 3 MEPDG input recommendations. The average CTE values determined from the data subset are recommended by this study.

 

Table 47. Prediction model 13 PCC CTE based on aggregate type (x 10-6 inch/inch/°F).

Aggregate Type

Average From Literature

Average From All LTPP Data

Average From Data Used in Model (Recommended)

Basalt

4.85

5.11

4.86

Chert

6.55

6.24

6.90

Diabase

4.85

5.33

5.13

Dolomite

5.75

5.79

5.79

Gabbro

4.85

5.28

5.28*

Granite

4.55

5.62

5.71

Limestone

4.25

5.35

5.25

Quartzite

6.85

6.07

6.18

Andesite

4.85

4.99

5.33

Sandstone

6.05

5.98

6.33

N

228

91

*There were no samples with a Gabbro aggregate type in the data used in the model. Hence, the average from the entire dataset is recommended.

Figure 204 shows a plot of recommended CTE values versus average CTE values obtained from other sources. While they are in fairly good agreement, the values recommended from this study are slightly higher for most cases. This can be explained by the overestimation of CTE during testing.

This graph shows an x-y scatter plot of the established portland cement concrete (PCC) coefficient of thermal expansion (CTE) from the Long-Term Pavement Performance (LTPP) data versus the average PCC CTE values noted from past references. The x-axis shows the average CTE for each aggregate type gathered from a literature review, and the y-axis shows the predicted CTE corresponding to each dataset. The solid diamonds represent the average determined from all LTPP data by aggregate type. The hollow squares represent the average CTE by aggregate type obtained from literature. There is a total of nine points under each category. There is also a line of equality, and the data points are concentrated along the line of equality.

Figure 204. Graph. Comparison of average values from other sources and recommended CTE values based on aggregate type from LTPP data.

CTE Model 2: CTE Based on Mix Volumetrics (Level 2 Equation for MEPDG)

A step-wise linear regression analysis that considered all PCC variables, which performed within a 5 to 10 percent confidence limit, showed that CTE was most sensitive to aggregate types basalt, dolomite, limestone, and quartzite, as well as to coarse aggregate weight, coarse aggregate specific gravity, and cement content. This demonstrates the influence of mix volumetrics and aggregate type on the predicted CTE values. This validated the approach of developing a model that uses the CTE of individual components with a weighted average by their volumetric proportions.

Volume proportions of each component of the PCC mix were computed using the mix proportioning and mix design data, specifically the amount of each component and the specific gravity. The specific gravities of the coarse and fine aggregates were included in the LTPP database. Air content information was available for only 72 of the 91 cases. Verification for the volumetric proportion calculation showed that for most of the sections, the volumetric proportions summed up closely to 1.0. The average was 1.007, and all values were between 0.93 and 1.08. However, two data points with extremely large coarse aggregate contents (greater than 2,700 lb/yd3) resulted in total volume proportions greater than 1.3, which were suspect data. With the deletion of these data points, the average value was 0.998.

The iterative procedures during the statistical analyses revealed that the model was handling only the volumetric proportions of the coarse aggregate adequately. Therefore, the equation was set up to consider the volumetric proportions of the coarse aggregate, VCA, and that of the mortar (1 - VCA). Because the individual aggregate CTE values were not available, aggregate CTE ranges and means from other sources of literature were used to assign a value to this parameter for each aggregate type.

The following ranges were used to determine the minimum, maximum, and mean for each aggregate type:

 

The sensitivity of the w/c ratio, a proxy variable to account for the porosity of the paste and hence its ability to expand or contract with change in temperature, was evaluated using the following model form:

CTE subscript PCC equals A times CTE subscript CA times V subscript CA plus open parenthesis B times w/c plus C closed parenthesis times V subscript mortar.

Figure 205. Equation. CTEPCC.

Where:

CTEPCC = CTE of the PCC material, inch/inch/°F.

CTECA = CTE of the coarse aggregate, inch/inch/°F.

VCA = Volumetric proportion of the coarse aggregate.

Vmortar = Volumetric proportion of the mortar (1 - VCA).

The regression statistics of this model showed the following:

In subsequent iterations, the analysis procedure attempted to optimize the coarse aggregate CTE value within the range provided above. The model form was reduced to figure 206 or figure 207.

CTE subscript PCC equals CTE subscript CA times V subscript CA plus C times V subscript mortar.

Figure 206. Equation. CTEPCC as a function of volumetric proportions.

CTE subscript PCC equals CTE subscript CA times V subscript CA plus C times open parenthesis 1 minus V subscript CA closed parenthesis.

Figure 207. Equation. CTEPCC as a function of coarse aggregate volumetric proportion.

The model statistics are presented in table 48, and details of the range of data used to develop the model are presented in table 49.

The model was established as follows:

CTE subscript PCC equals CTE subscript CA times V subscript CA plus 6.4514 times open parenthesis 1 minus V subscript CA closed parenthesis.

Figure 208. Equation. Prediction model 14 for CTEPCC.

Where:

CTECA = Constant determined for each aggregate type as shown in table 48.

 

Table 48. Statistical analysis results for CTE model based on mix volumetrics.

Parameter

Comment

Estimate

Standard Error

95 Percent Confidence Limits

c

 No comment

6.4514

0.1889

6.0758

6.827

d

CTECA for basalt

3

0

3

3

e

CTECA for chert

6.4

0

6.4

6.4

f

CTECA for diabase

3.4835

1.2824

0.9337

6.0333

g

CTECA for dolomite

5.1184

0.408

4.3071

5.9297

h

CTECA for gabbro

3.75

N/A

N/A

N/A

i

CTECA for granite

4.7423

0.4188

3.9096

5.5749

j

CTECA for limestone

3.2886

0.3579

2.5771

4.0001

k

CTECA for quartzite

6.1

0

6.1

6.1

l

CTECA for andesite

3.6243

1.4539

0.7336

6.515

m

CTECA for sandstone

4.5

0

4.5

4.5

 

The model statistics for table 48 are as follows:

Table 49. Range of data used for CTE model based on mix volumetrics.

Parameter

Minimum

Maximum

Average

Coarse aggregate content

582

2730

1,811

Coarse aggregate specific gravity

2.42

2.86

2.65

w/c ratio

0

0.71

0.45

Coarse aggregate volume fraction

0.13

0.62

0.41

Mortar volume

0.38

0.87

0.59

 

The model has an R2 value of 44.1 percent and an RMSE value of 0.35 psi. The predicted versus measured plot and the residual error plots are presented in figure 209 and figure 210, respectively.

This graph is an x-y scatter plot showing the predicted versus the measured values used in the coefficient of thermal expansion (CTE) model based on mix volumetrics. The x-axis shows the measured CTE from zero to 8, and the y-axis shows the predicted CTE from zero to 8. The plot contains 89 points, which correspond to the data points used in the model. The graph also shows a 45-degree line that represents the line of equality. The data are shown as solid diamonds, and they appear to demonstrate a good prediction. The measured values range from 4.11 to 
7.31 inch/inch/°F. The graph also shows the model statistics as follows: y equals 0.4228x plus 3.2012 and R-squared equals 0.4415.

Figure 209. Graph. Predicted versus measured for CTE model based on mix volumetrics.

This graph is an x-y scatter plot showing the residual errors in the predictions of the coefficient of thermal expansion (CTE) model based on mix volumetrics. The x-axis shows the predicted CTE from zero to 7, and the y-axis shows the residual error from -2 to 2. The points are plotted as solid diamonds, and there is no significant bias (i.e., the data are well distributed about the zero-error line). This plot illustrates a fair but acceptable error. There appears to be no trend in the data, and the trend line is almost horizontal (i.e., zero slope). The following equations are provided in the graph: y equals -0.0443x plus 0.233 and R-squared equals 0.0014.

Figure 210. Graph. Residual errors for CTE model based on mix volumetrics.

The constant, C, in the model form, determined as 6.4514, is equivalent to CTE of the mortar. (At TFHRC, using the AASHTO TP 60 uncorrected values, a CTE value of 6.2 for mortar containing silica sand was determined. Hence, the value of C is in agreement with the test result.(24)) Since the mortar (all components of the mix design except the coarse aggregate, as per the definition in this equation) occupies a large volume of the matrix, it was necessary for the model to predict higher CTE for increased mortar proportions (or decreasing coarse aggregate proportions). In optimizing the model and selecting the representative CTE for each aggregate type, it was ensured that the CTE of the aggregate is not above C.

Figure 211 and figure 212 show a comparison of the predicted CTE values with average values reported in literature for each aggregate type. The predictions are close, with the model showing a slight bias. The over-prediction observed can be a result of the errors in the CTE test procedure (over-measured CTEs) or could simply reflect the CTE typical of paving mixes. Figure 213 shows the sensitivity of the model to coarse aggregate content. As expected, CTE decreases as the coarse aggregate content increases (or mortar volume decreases). While this is true for most cases, it is also observed that for aggregates with high CTE values, such as chert and quartzite, CTE of the aggregate approaches CTE of mortar, thereby showing little or no sensitivity to coarse aggregate content.

As with all other models, the user is advised to verify model predictions with other sources of information. If possible, both CTE models should be evaluated simultaneously to obtain a range.

This graph is a bar chart showing predicted values using the coefficient of thermal expansion (CTE) model as well as portland cement concrete CTE values typically reported in literature. The values are reported for several aggregate types, and aggregate type is the category on the x-axis. Starting from the left, the aggregates include basalt, chert, diabase, dolomite, gabbro, granite, limestone, quartzite, andesite, sandstone, conglomerate, syenite, diorite, and peridotite. The y-axis shows concrete CTE from zero to 8. The CTE model values are shown as blue bars, and the average values from literature are shown as red bars. For each aggregate type category, the blue bar closely matches the red bar.

Figure 211. Graph. Comparison of CTE model prediction with average values reported in literature for each aggregate rock type.

This graph is an x-y scatter plot showing the predicted coefficient of thermal expansion (CTE) values versus the typical values reported in literature for each aggregate rock type. The x-axis shows the typical aggregate specific CTE value reported in literature from 4 to 8 x10-6 inch/inch/Fahrenheit, and the y-axis shows the portland cement concrete CTE values from Long-Term Pavement Performance data from 4 to 8 x10-6 inch/inch/Fahrenheit. The data are plotted as solid squares and have a linear trend line. The graph also has a 45-degree line of equality plotted as a solid line. The points range from 4 to 7 x10-6 inch/inch/Fahrenheit. The model statistics are also included as follows: y equals 0.4303x plus 3.3629 and R-squared equals 0.6709.

Figure 212. Graph. CTE model prediction versus average values reported in literature for each aggregate rock type.

This graph shows the sensitivity of the coefficient of thermal expansion (CTE) model to coarse aggregate content for different aggregate types. The x-axis shows the coarse aggregate content in the mix from 500 to 3,000 lb/yd3, and the y-axis shows the porland cement concrete CTE from zero to 
7 x10-6 inch/inch/Fahrenheit. The sensitivity is show in the range of 700 to 2,750 lb/yd3. The aggregate types in the order from top to bottom are chert (solid diamonds), quartzite (solid squares), peridotite (x-marks), andesite (solid triangles), and basalt (solid squares). The markers are connected by a solid line for all aggregate types. The two lines representing quartzite and chert remain horizontal, but the lines representing peridotite, andesite, and basalt show a decrease in CTE with increasing coarse aggregate content.

Figure 213. Graph. Sensitivity of the CTE model to coarse aggregate content.

 


The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
FHWA
United States Department of Transportation - Federal Highway Administration