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Federal Highway Administration Research and Technology
Coordinating, Developing, and Delivering Highway Transportation Innovations
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Publication Number: FHWAHRT12030 Date: August 2012 
Publication Number:
FHWAHRT12030
Date: August 2012 
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The flexural strength model 1 provides the best correlation between compressive strength and flexural strength with the LTPP data. The model form utilizes the power equation. This model will be most useful for cases when the compressive strength of the PCC has been determined through a routine cylinder break.
This model can be expressed as follows:
Where:
MR = Flexural strength, psi.
f'_{c,}= Compressive strength determined at the same age, psi.
The regression statistics for this model are presented in table 33. The model was developed using 185 data points, and the prediction has an R^{2} value of 45.2 percent and an RMSE value of 69 psi. Table 34 provides details of the range of data used to develop the model. The confidence limits are both within acceptable ranges for both the regressed coefficients (i.e., limits are positive numbers). Figure 173 and figure 174 show the predicted versus measured plot and the residual plot, respectively.
Parameter 
Estimate 
Standard Error 
Approximate 95 Percent Confidence Limits 
a 
22.7741 
6.6362 
9.6807 to 35.8674 
b 
0.4082 
0.0338 
0.3416 to 0.4748 
The model statistics for table 33 are as follows:
Parameter 
Minimum 
Maximum 
Average 
Compressive strength 
1,770 
10,032 
5,431 
Flexural strength 
467 
1,075 
754 
Figure 175 shows a comparison of the power models used to validate the data and also to develop a new correlation. Note that the three power models (the new equation developed for this study as well as the validation models) provide close estimates (within 50 psi) in the 4,500 to 5,500psi compressive strength range.
The ACI and PCA models are plotted for comparison. Also plotted in figure 175 are the raw data that were used in the model. Clearly, the ACI equation is very conservative for this data. It also has been found to give a conservative estimate for several large datasets that have been used in flexural strength model prediction. Conversely, the PCA model fits the LTPP data more closely. The reasons for this lack of fit of the current data with the previous models may be too many to fully explain. The data used in models from prior studies often came from mixes batched under controlled laboratory experiments and were typical of paving and structural concrete. The mixes used in the current model developed from LTPP data relies on only mixes proportioned for typical paving operations. Furthermore, the LTPP data used are from many projects widely dispersed around the United States. This in itself makes the models more robust than any previous data used to make similar correlations.
The spread in the raw data about the prediction model in figure 175 indicates that there are factors other than compressive strength that influence the flexural strength of PCC. Among the various factors influencing flexural strength are the mix design parameters and age of the concrete. These variables were considered in the other models developed in this study.
Flexural strength model 2 provides a correlation between flexural strength and mix design parameters, specifically the unit weight and w/c ratio. Age is also a parameter in this model, which helps reduce some of the variability seen in the prediction relative to the predictions shown in figure 175. This model will be most useful for cases when the compressive strength of the PCC is not determined but mix design information is available. Also, the user has the option of predicting the 28day strength value for design or estimating the strength at traffic opening time.
This model can be expressed as follows:
Where:
MR_{t} = Flexural strength at age t years, psi.
w/c = Water to cement ratio.
uw = Unit weight, lb/ft^{3}.
t = Pavement age, years.
The regression statistics for this model are presented in table 35. The model was developed using 62 data points, and the prediction has an R^{2} value of 61.1 percent and an RMSE value of 69 psi. Table 36 provides details of the range of data used to develop the model. Figure 177 and figure 178 show the predicted versus measured plot and the residual plot, respectively.
Variable 
DF 
Estimate 
Standard Error 
tvalue 
P_{r} > t 
VIF 
Intercept 
1 
676.0159 
277.7887 
2.43 
0.0181 
0 
w/c 
1 
1,120.31 
141.3573 
7.93 
< 0.0001 
1.00591 
Unit weight 
1 
4.1304 
1.88934 
2.19 
0.0329 
1.00311 
Ln(age) 
1 
35.74627 
8.78516 
4.07 
0.0001 
1.00619 
The model statistics for table 35 are as follows:
Parameter 
Minimum 
Maximum 
Average 
w/c ratio 
0.27 
0.58 
0.40 
Unit weight 
124 
151 
142 
Pavement age 
0.0384 
1.0000 
0.3169 
Flexural strength 
467 
978 
742 
The data used in the previous model also provided a good correlation by replacing the w/c ratio parameter with CMC. The model is expressed as follows:
Where:
MR_{t} = Flexural strength at age t years, psi.
CMC = Cementitious materials content, lb/yd^{3}.
uw = Unit weight, lb/ft^{3}.
t = Pavement age, years.
The regression statistics for this model are presented in table 37. The model was developed using 62 data points, and the prediction has an R^{2} value of 70.2 percent and an RMSE value of 80 psi. Table 38 provides details of the range of data used to develop the model. Figure 180 and figure 181 show the predicted versus measured plot and the residual plot, respectively.
Figure 182 to figure 185 present the sensitivity of the mix designbased flexural strength models to CMC, w/c ratio, unit weight, and age. Figure 182 and figure 183 show that prediction models in figure 176 and figure 179 do not show any sensitivity to CMC and w/c ratio. For typical values of these parameters, the flexural strength prediction from these two models could show a difference of about 200 psi for extreme values of w/c ratios. However, within a typical range of 0.35 to 0.45, the flexural strength prediction is within 50 psi. Similar trends are evident for the w/c ratio parameter. Therefore, if all details about a mix design are available, it is highly recommended that both models be used to predict flexural strength so that the user has a fair estimate of the MR range. Figure 184 shows that the predictions are close from both models. Likewise, figure 185, which is more or less a flexural strength gain model for a typical mix design, shows close predictions from both models.
Variable 
DF 
Estimate 
Standard Error 
tvalue 
P_{r} > t 
VIF 
Intercept 
1 
24.15063 
236.7606 
0.1 
0.9191 
0 
CMC 
1 
0.55579 
0.05563 
9.99 
< 0.0001 
1.01522 
Unit weight 
1 
2.96376 
1.66087 
1.78 
0.0796 
1.01253 
Ln(age) 
1 
35.54463 
7.68504 
4.63 
< 0.0001 
1.00573 
The model statistics for table 37 are as follows:
Parameter 
Minimum 
Maximum 
Average 
CMC 
388 
936 
668 
Unit weight 
124 
151 
142 
Pavement age 
0.0384 
1.0000 
0.3169 
Flexural strength 
467 
978 
742 