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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 (12)

PCC Elastic Modulus Models

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:

Validation of Existing Models

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.

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

Figure 186. Equation. Ec as a function of square root of compressive strength.

Regressed coefficients for figure 186 are as follows:

Regression statistics for figure 186 are as follows:

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

Figure 187. Equation. Model form for E as a function of compressive strength with slope and intercept.

Regressed coefficients for figure 187 are as follows:

 

Regression statistics for figure 187 are as follows:

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

Figure 188. Equation. Ec.

Regressed coefficients for figure 188 are as follows:

Regression statistics for figure 188 are as follows:

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

Figure 189. Equation. E as function of unit weight and compressive strength.

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 R2 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.

Elastic Modulus Model 1: Model Based on Aggregate Type

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:

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 c closed parenthesis raised to the power of 0.2429 closed parenthesis times D subscript agg.

Figure 190. 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 which 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 value of approximately 500,000 psi. The nonlinear analyses results are presented in table 39.

Table 39. Regression statistics for elastic modulus model based on aggregate type.

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

Dagg for andesite

1

N/A

N/A

N/A

e

Dagg for basalt

0.9286

0.0956

0.7374

1.1197

f

Dagg for chert

1.0079

0.0863

0.8354

1.1803

g

Dagg for diabase

0.9215

0.1858

0.5501

1.2928

h

Dagg for dolomite

1.0254

0.0624

0.9006

1.1501

i

Dagg for granite

0.8333

0.0624

0.7085

0.9581

j

Dagg for limestone

1

N/A

N/A

N/A

k

Dagg for quartzite

0.9511

0.1082

0.7349

1.1674

l

Dagg 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:

E equals open parenthesis a times open parenthesis UW closed parenthesis raised to the power of b times open parenthesis f prime c closed parenthesis raised to the power of c closed parenthesis times open parenthesis d times Andesite plus e times Basalt plus f times Chert plus g times Diabse plus h times Dolomite plus i times Granite plus j times Limestone plus k times Quartzite plus l times Sandstone closed parenthesis.

Figure 191. Equation. E.

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, 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. 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 R2 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.

Table 40. 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

 

. 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 zero to 8,000,000 psi, and the y-axis shows the predicted values from zero 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 192. 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 zero 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 193. Graph. Residual errors for elastic modulus model based on aggregate type.

 


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