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Publication Number: FHWA-HRT-12-031
Date: August 2012

 

User’s Guide: Estimation of Key PCC, Base, Subbase, and Pavement Engineering Properties From Routine Tests and Physical Characteristics

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PCC Elastic Modulus Models

Validation of Existing Models

Existing models correlate elastic modulus to compressive strength and unit weight. The following represent the regressed models using LTPP data for existing model forms:

E subscript c equals a times the square root of f prime subscript c.

Figure 55. Equation. Ec as a function of square root of compressive strength.

 

Regressed coefficients for figure 55 are as follows:

Regression statistics for figure 55 are as follows:

E equals a times the square root of f prime times c plus b.

Figure 56. Equation. Model form for E as a function of compressive strength with slope and intercept.

Regressed coefficients for figure 56 are as follows:

Regression statistics for figure 56 are as follows:

E subscript c equals a times f prime subscript c raised to the power 
of b.

Figure 57. Equation. Ec.

Regressed coefficients for figure 57 are as follows:

Regression statistics for figure 57 are as follows:

E equals a times open parenthesis UW closed parenthesis raised to the power of b times open parenthesis f prime times c closed parenthesis raised to the power of c.

Figure 58. Equation. E as function of unit weight and compressive strength.

Regressed coefficients for figure 58 are as follows:

Regression statistics for figure 58 are as follows:

The quality of prediction in the validated models is poor, as indicated by the R2 values reported for figure 55 through figure 58. This trend is common with elastic modulus models, especially considering that the data used in this study were not generated from controlled laboratory experiments. Also, while compressive strength is the most commonly used strength parameter and correlations with the compressive strength can be implemented most easily, there is an inherent drawback in correlating modulus to compressive strength. Modulus does not test the material to its limits. Instead, it is more indicative of the elastic deformational characteristics of the material. The data contain modulus measured at a wide range of ages. Therefore, the new models developed utilized other mix parameters that impact modulus, including age.

Elastic Modulus Model 1: Model Based on Aggregate Type

The PCC elastic modulus model can be expressed as follows:

E subscript c equals open parenthesis 4.499 times open parenthesis UW closed parenthesis raised to the power of 2.3481 times open parenthesis f prime times c closed parenthesis raised to the power of 0.2429 closed parenthesis times D subscript agg.

Figure 59. Equation. Prediction model 9 for Ec.

Where:

Ec = PCC elastic modulus, psi.
UW = Unit weight, lb/ft3.
f'c = Compressive strength.
Dagg = Regressed constant depending on aggregate type as follows:

The development of the model required the use of a model form that accommodates aggregate type as categorical variables (assigned values of 1, 0). The values for Dagg were initialized to 1.0 at the start of the analyses and allowed to iteratively determine individual values for each aggregate type. The model had 71 observations, an R2 value of 35.8 percent, and an RMSE of approximately 500,000 psi.

The model indicates that the factor that accounts for the aggregate type, Dagg, has a value of 1.0 for andesite, limestone, and sandstone. Basalt, diabase, granite, and quartzite have lower Dagg values and therefore lower modulus values than mixes using andesite, limestone, and sandstone aggregates. Likewise, chert and dolomite have higher values. Table 18 provides details of the range of data used to develop the model.

Table 18. Range of data used for elastic modulus model based on aggregate type.

Parameter

Minimum

Maximum

Average

Compressive strength

1,990

11,310

7,550

Unit weight

137

156

146

Elastic modulus

1,450,000

6,800,000

4,629,646

 

Figure 60 and figure 61 show the predicted versus measured plot and the residual plot, respectively. The R2 value is reasonable and therefore presented as a feasible model.

This graph is an x-y scatter plot showing the predicted versus the measured values for the elastic modulus model based on aggregate type. The x-axis shows the measured values that formed the predictor variable from 0 to 8,000,000 psi, and the y-axis shows the predicted values from 0 to 8,000,000 psi. The plot contains 71 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 fair prediction. The measured values range from 1,450,000 to 6,800,000 psi. The graph also shows the model statistics as follows: N equals 71, R-squared equals 0.3582 percent, and root mean square error equals 499,856 psi.

Figure 60. Graph. Predicted versus measured for elastic modulus model based on aggregate type.

This graph is an x-y scatter plot showing the residual errors in the predictions of the elastic modulus model based on aggregate type. The x-axis shows the predicted elastic modulus 
from 0 to 8,000,000 psi, and the y-axis shows the residual elastic modulus from -2,000,000 to 2,000,000 psi. The points are plotted as solid diamonds, and they appear to show no significant bias (i.e., the data are well distributed about the zero-error line). 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 0.0178x minus 84,187 and R-squared equals 0.0002.

Figure 61. Graph. Residual errors for elastic modulus model based on aggregate type.

Elastic Modulus Model 2: Model Based on Age and Compressive Strength

The model can be expressed as follows:

. E subscript c,t equals 59.0287 times open parenthesis f prime times c subscript t closed parenthesis raised to the power of 1.3 times open parenthesis natural log open parenthesis t divided by 0.03 closed parenthesis, closed parenthesis raised to the power of -0.2118.

Figure 62. Equation. Prediction model 10 for Ec,t.

Where:

Ec,t = Elastic modulus at age t,years.
f'ct = Compressive strength at age t, years.
t = Age at which modulus is determined, years.

The model uses 371 data points, has an R2 value of 26.14 percent, and an RMSE of about 900,000 psi. Table 19 shows the results of the nonlinear analysis, and table 20 provides details of the range of data used to develop the model.

Table 19. Regression statistics for elastic modulus model based on age and compressive strength.

Parameter Constants

Estimate

Standard Error

Approximate 95 Percent Confidence Limits

a

59.0287

2.8881

53.3495 to 64.7079

b

-0.2118

0.0284

-0.2677 to -0.1559

 

Table 20. Range of data used for elastic modulus model based on age and compressive strength.

Parameter

Minimum

Maximum

Average

Compressive strength

1,990

12,360

7,361

Pavement age

0.0384

45.3836

14.0900

Elastic modulus

1,450,000

6,800,000

4,586,545

 

The measured versus predicted plot and the residuals plot for this model are shown in figure 63 and figure 64, respectively.

This graph is an x-y scatter plot showing the predicted versus the measured values for the elastic modulus model based on age and compressive strength. The 
x-axis shows the measured elastic modulus from 1,000,000 to 9,000,000 psi, and the y-axis shows the predicted elastic modulus from 1,000,000 to 9,000,000 psi. The plot contains 
371 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 fair prediction. The measured values range from 1,450,000 to 6,800,000 psi. The graph also shows the model statistics as follows: N equals 371, R-squared equals 0.2614 percent, and root mean square error equals 949,404.

Figure 63. Graph. Predicted versus measured for elastic modulus model based on age and compressive strength.

This graph is an x-y scatter plot showing the residual errors in the predictions of the elastic modulus model based on age and compressive strength. The x-axis shows the predicted elastic modulus from 1,000,000 to 9,000,000 psi, and the y-axis shows the residual elastic modulus from -4,000,000 to 4,000,000 psi. The points are plotted as solid diamonds, and they show some bias. This plot illustrates a relatively large variation, especially at the lower and upper bounds of the model. 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 0.5885x minus 3E plus 0.6 and R-squared equals 0.4198.

Figure 64. Graph. Residual errors for elastic modulus model based on age and compressive strength.

Elastic Modulus Model 3: Model Based on Age and 28-Day Compressive Strength

Since the 28-day compressive strength is usually available for PCC materials, a predictive model based on age and the 28-day compressive strength can be useful in many situations. The relationship developed for these variables can be expressed as follows:

E subscript c,t equals 375.6 times open parenthesis f prime times c subscript 28-day closed parenthesis raised to the power of 1.1 times open parenthesis natural log times open parenthesis t divided by 0.03 closed parenthesis, closed parenthesis times 0.00524.

Figure 65. Equation. Prediction model 11 for Ec,t.

Where:

Ec,t = Elastic modulus at age t, years.
f'c28-day = 28-day compressive strength.
t = Age at which modulus is determined, years.

The model used 46 data points, had an R2 value of 16.32 percent, and an RMSE of about 1,183,400 psi. Table 21 shows the results of the nonlinear analysis, and table 22 provides details of the range of data used to develop the model.

Table 21. Regression statistics for elastic modulus model based on age and 28-day compressive strength.

Parameter Constants

Estimate

Standard Error

Approximate 95 Percent Confidence Limits

a

375.6

31.4592

312.5

439.3

b

0.00524

0.0714

-0.1388

-0.1492

 

Table 22. Range of data used for elastic modulus model based on age and 28-day compressive strength.

Parameter

Minimum

Maximum

Average

28-day compressive strength

3,034

7,912

5,022

Pavement age

0.0384

4.5288

0.9153

Elastic modulus

1,450,000

6,221,000

4,732,101

 

The measured versus predicted plot and the residuals plot for this model are shown in figure 66 and figure 67, respectively. This model uses data up to an age of 1 year. It is more appropriate for estimating the short-term modulus of a project and for supplementing strength estimates used to determine opening time for traffic.

This graph is an x-y scatter plot showing the predicted versus the measured values for the elastic modulus model based on age and 28-day compressive strength. The x-axis shows the measured elastic modulus from 1,000,000 to 9,000,000 psi, and the y-axis shows the predicted elastic modulus from 1,000,000 to 9,000,000 psi. The plot contains 
46 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 poor prediction. The measured values range from 1,450,000 to 6,221,000 psi. The graph also shows the model statistics as follows: N equals 46, R-squared equals 0.1632 percent, and root mean square error equals 1,183,400 psi.

Figure 66. Graph. Predicted versus measured for elastic modulus model based on age and 28-day compressive strength.

This graph is an x-y scatter plot showing the residual errors in the predictions of the elastic modulus model based on age and 28-day compressive strength. The 
x-axis shows predicted elastic modulus from 1,000,000 to 9,000,000 psi, and the y-axis shows the residual elastic modulus from -3,000,000 to 3,000,000 psi. The points are plotted as solid diamonds, and they appear to show some bias. This plot illustrates a relatively large error, especially at the lower and upper bounds. Thus, the model is recommended for use with careful consideration. 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 0.7443x minus 4E plus 0.6 and R-squared equals 0.623.

Figure 67. Graph. Residual errors for elastic modulus model based on age and 28-day compressive strength.

Limitations of Elastic Modulus Models

An examination of the statistics proposed for determining elastic modulus suggests that they do not possess the predictive ability of the other material parameters presented in this study. The models are considered fair but not excellent. They provide users with an option of moderate estimates when no information about the elastic modulus is available. Therefore, it is recommended that users exercise caution when using the predicted elastic modulus values for analyses.

 


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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.
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