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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

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CHAPTER 5. MODEL DEVELOPMENT (13)

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

The data used for validation were further reviewed to identify potential sources of error. Since the data covered a wide range of ages, the age parameter was considered for inclusion in the model. A model was developed to predict elastic modulus as a function of age and compressive strength at the corresponding age. The model was not entirely statistically optimized. The regressed constants were adjusted through a trial and error process to provide the best prediction (i.e., to match the measured with the predicted values as close as possible along the line of equality). The model can be expressed as follows:

E subscript c,t equals 59.0287 times open parenthesis f prime 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 194. 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 used 371 data points and had an R2 value of 26.14 percent. The RMSE value for this model is about 900,000 psi. Table 41 shows the results of the nonlinear analysis, and table 42 provides details of the range of data used to develop the model. The measured versus predicted plot and the residuals plot for this model are shown in figure 195 and figure 196, respectively.

 

Table 41. 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

64.7079

b

-0.2118

0.0284

-0.2677

-0.1559

 

The model statistics for table 41 are as follows:

Table 42. 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

 

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 195. 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 scatter 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 196. 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 was developed. A relatively smaller dataset was utilized for this model with only data from SPS sections, as the 28-day compressive strength data was a necessary input. Again, this model utilized a nonlinear analysis, and beyond statistical optimization, the constants determined for this model were adjusted for closest predictions through a trial and error process. The relationship developed for these variables can be expressed as follows:

E subscript c,t equals 375.6 times open parenthesis f prime 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 197. 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 and had an R2 value of 16.32 percent. The RMSE value for this model is about 1,183,400 psi. Table 43 shows the results of the nonlinear analysis, and table 44 provides details of the range of data used to develop the model. The measured versus predicted plot and the residuals plot for this model are shown in figure 198 and figure 199, 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.

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. It is therefore recommended that users exercise caution in using the predictive values for analyses.

Table 43. 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 to 439.3

b

0.00524

0.0714

-0.1388 to -0.1492

 

The model statistics for table 43 are as follows:

Table 44. Range of data used for elastic modulus model based on age and 28-day compressive strength.

Parameter

Minimum

Maximum

Average

28-day compressive strength

3034

7912

5022

Pavement age

0.0384

4.5288

0.9153

Elastic modulus

1,450,000

6,221,000

4,732,101

 

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

 


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