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REPORT 
This report is an archived publication and may contain dated technical, contact, and link information 
Publication Number: FHWAHRT12030 Date: August 2012 
Publication Number:
FHWAHRT12030
Date: August 2012 
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As with flexural strength, the development of PCC elastic modulus models required a detailed data assembly appropriate for statistical analyses. These analyses also entailed the validation of existing models as well as the development of new models. Model forms of existing models were utilized to fit the data assembled in this study for the validation. The development of new models was not as straightforward as for the other PCC models, primarily because the physical characteristics of a PCC mixture that affect the elastic modulus are not fully captured within the data used for building a mathematical relationship.
The following are key points to note about flexural strength data:
Existing models correlate elastic modulus to compressive strength and unit weight. Although more recent models have attempted to introduce lithological type of the coarse aggregate and admixture parameters, within the context of this study, there were not adequate data to validate them. Figure 186 to figure 189 summarize the regressed models using LTPP data.
Regressed coefficients for figure 186 are as follows:
Regression statistics for figure 186 are as follows:
Regressed coefficients for figure 187 are as follows:
Regression statistics for figure 187 are as follows:
Regressed coefficients for figure 188 are as follows:
Regression statistics for figure 188 are as follows:
Regressed coefficients for figure 189 are as follows:
Regression statistics for figure 189 are as follows:
The quality of prediction in the validated models is poor, as indicated by the R^{2} values in figure 186 through figure 189. The predicted versus measured plots for these models have not been included in this report, but they show higher predictions for the lower modulus values and lower predictions for the higher modulus values. 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, and, as discussed in chapter 2, it is more indicative of the elastic deformational characteristics of the material. Additionally, it often captures other ITZ characteristics and can be a good indicator of concrete durability. Finally, the data contain modulus measured at a wide range of ages. The new models developed therefore utilized other mix parameters that impact modulus including age.
Several mix design parameters were evaluated for the elastic modulus model 1 in addition to the compressive strength and unit weight parameters. The model development efforts particularly focused on the aggregate type, given the strong influence of the aggregate hardness on the measured elastic modulus values.
This model utilizes a subset of the data used in the model validation process and has only 71 data points compared to 514 observations in the validation models. This was primarily due to the inclusion of aggregate type in the relationship. Coarse aggregate type information is present in both the materials tables and the CTE tables of the LTPP database. Several data inconsistencies were found in comparing the aggregate types listed in these two tables; therefore, for the development of CTE models (discussed later in this chapter), only those cases with the same aggregate type in both tables were used. In other words, the two tables were used to validate the data against each other. This vastly reduced the dataset used. The dataset included both SPS and GPS sections; however, a majority of the data used in this model belonged to GPS sections.
A nonlinear analysis was performed to establish the following equation:
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 which 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 value of approximately 500,000 psi. The nonlinear analyses results are presented in table 39.
Parameter 
Comment 
Estimate 
Standard Error 
Approximate 95 Percent Confidence Limits 

a 
No comment 
4.499 
18.6844 
32.8506 
41.8485 
b 
No comment 
2.3481 
0.8998 
0.5495 
4.1468 
c 
No comment 
0.2429 
0.1224 
0.00173 
0.4875 
d 
D_{agg} for andesite 
1 
N/A 
N/A 
N/A 
e 
D_{agg} for basalt 
0.9286 
0.0956 
0.7374 
1.1197 
f 
D_{agg} for chert 
1.0079 
0.0863 
0.8354 
1.1803 
g 
D_{agg} for diabase 
0.9215 
0.1858 
0.5501 
1.2928 
h 
D_{agg} for dolomite 
1.0254 
0.0624 
0.9006 
1.1501 
i 
D_{agg} for granite 
0.8333 
0.0624 
0.7085 
0.9581 
j 
D_{agg} for limestone 
1 
N/A 
N/A 
N/A 
k 
D_{agg} for quartzite 
0.9511 
0.1082 
0.7349 
1.1674 
l 
D_{agg} for sandstone 
1 
N/A 
N/A 
N/A 
N/A = Not applicable.
The model statistics for table 39 are as follows:
The model form described for the statistics presented in this table is as follows:
Where:
d through l were iteratively determined through the nonlinear process, and andesite, basalt, chert, diabase, dolomite, granite, limestone, quartzite, and sandstone are categorical variables with values 0, 1.
Table 39 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. Also evident from table 39 is that to a very small extent, the statistical optimization has been compromised (note significance of parameters a and c are not within limits) in the interest of developing a model with variables relevant to elastic modulus predictions. Table 40 provides details of the range of data used to develop the model. The R^{2} value is reasonable and therefore presented as a feasible model. Figure 192 and figure 193 show the predicted versus measured plot and the residual plot, respectively.
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 