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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
REPORT 
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Publication Number: FHWAHRT12031 Date: August 2012 
Publication Number: FHWAHRT12031 Date: August 2012 
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Existing models correlate elastic modulus to compressive strength and unit weight. The following represent the regressed models using LTPP data for existing model forms:
Figure 55. Equation. E_{c} 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:
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:
Figure 57. Equation. E_{c}.
Regressed coefficients for figure 57 are as follows:
Regression statistics for figure 57 are as follows:
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 R^{2} 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.
The PCC elastic modulus model can be expressed as follows:
Figure 59. Equation. Prediction model 9 for E_{c}.
Where:
E_{c} = PCC elastic modulus,
psi.
UW = Unit weight, lb/ft^{3}.
f'c = Compressive strength.
D_{agg} = 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 D_{agg} 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 R^{2} 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, D_{agg}, has a value of 1.0 for andesite, limestone, and sandstone. Basalt, diabase, granite, and quartzite have lower D_{agg} 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 R^{2} value is reasonable and therefore presented as a feasible model.
Figure 60. Graph. Predicted versus measured for elastic modulus model based on aggregate type.
Figure 61. Graph. Residual errors for elastic modulus model based on aggregate type.
The model can be expressed as follows:
Figure 62. Equation. Prediction model 10 for E_{c,t}.
Where:
E_{c,t} = Elastic modulus at
age t,years.
f'c_{t} = Compressive strength at age t,
years.
t = Age at which modulus is determined, years.
The model uses 371 data points, has an R^{2} 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.
Figure 63. Graph. Predicted versus measured for elastic modulus model based on age and compressive strength.
Figure 64. Graph. Residual errors for elastic modulus model based on age and compressive strength.
Since the 28day compressive strength is usually available for PCC materials, a predictive model based on age and the 28day compressive strength can be useful in many situations. The relationship developed for these variables can be expressed as follows:
Figure 65. Equation. Prediction model 11 for E_{c,t}.
Where:
E_{c,t} = Elastic modulus at
age t, years.
f'c_{28day} = 28day compressive strength.
t = Age at which modulus is determined, years.
The model used 46 data points, had an R^{2} 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 28day 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 28day compressive strength.
Parameter 
Minimum 
Maximum 
Average 
28day 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 shortterm modulus of a project and for supplementing strength estimates used to determine opening time for traffic.
Figure 66. Graph. Predicted versus measured for elastic modulus model based on age and 28day compressive strength.
Figure 67. Graph. Residual errors for elastic modulus model based on age and 28day compressive strength.
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.