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Federal Highway Administration Research and Technology
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Publication Number: FHWAHRT12031 Date: August 2012 
Publication Number: FHWAHRT12031 Date: August 2012 
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Prediction models were developed for PCC compressive strength, PCC flexural strength, PCC elastic modulus, PCC tensile strength, and CTE. The following limitations apply to all PCC models:
Compressive strength is considered a fundamental strength parameter and is used at different stages of a project—design, QA, opening time, rehabilitation design, etc. The following models are offered for PCC compressive strength, each of which is discussed in subsections to follow:
The 28day compressive strength model developed for cylinder strength is as follows:
Figure 3. Equation. Prediction model 1 for f_{c,}_{28d}.
Where:
f’_{c,}_{28d}
= 28day compressive strength, psi.
w/c = Water to cementitious materials ratio.
CMC = Cementitious materials content, lb/yd^{3}.
The model statistics are shown in table 1. The model was developed using 42 data points, and the prediction has an R^{2} value of 54.44 percent and a root mean square error (RMSE) of 871 psi. Although it was compromised relative to the models discussed above, it provides a more meaningful model with a superior predictive ability. Table 2 provides details of the range of data used to develop the model.
Table 1. Regression statistics for selected prediction model for 28day PCC cylinder strength.
Variable 
Degrees of Freedom (DF) 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1 
4,028.41841

1,681.71576

2.4

0.0215

0

w/c ratio 
1 
3486.3501

2,152.99857

1.62

0.1134

2.40903

CMC 
1 
4.02511

1.32664

3.03

0.0043

2.40903

Table 2. Range of data used for 28day PCC cylinder strength.
Parameter 
Minimum 
Maximum 
Average 
w/c ratio 
0.27 
0.71 
0.42 
Cementitious content 
376 
936 
664 
Compressive strength 
3,034 
7,611 
5,239 
Figure 4 and figure 5 show the predicted versus measured values and the residuals plot for the model, respectively. Figure 6 and figure 7 show the sensitivity of this model to w/c ratio and CMC. The change in compressive strength appears reasonable for both of the parameters for the range of values evaluated. They are also consistent with the data in the database. Within practical ranges, a change in CMC from 500 to 650 lb/ft^{3} increases the 28day strength from approximately 4,700 to 5,300 psi for a w/c ratio of 0.4. Likewise, a decrease in w/c ratio from 0.5 to 0.35 increases the strength from 4,700 to 5,200 psi.
Figure 4. Graph. Predicted versus measured for 28day cylinder compressive strength model.
Figure 5. Graph. Residual error plot for 28day cylinder compressive strength model.
Figure 6. Graph. 28day compressive strength model sensitivity to w/c ratio.
Figure 7. Graph. 28day compressive strength model sensitivity to CMC.
The shortterm cylinder compressive strength is expressed as follows:
Figure 8. Equation. Prediction model 2 for f_{c,t}.
Where:
f_{c,t} = Compressive
strength at age t years, psi.
CMC = Cementitious materials content, lb/yd^{3}.
w/c = Water to cementitious materials ratio.
uw = Unit weight, lb/ft^{3}.
t = Shortterm age up to 1 year.
The regression statistics for this model are presented in table 3, and details of the range of data used to develop the model are presented in table 4. The model was developed using 79 data points, and the prediction has an R^{2} value of 66.6 percent and an RMSE of 789 psi. The reason for an improved R^{2} compared to the 28day strength model is not clear from these analyses.
Table 3. Regression statistics for shortterm cylinder strength model.
Variable 
DF 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1 
6,358.60655

1,213.09762

5.24

< 0.0001

0

CMC 
1 
3.53012

0.90968

3.88

0.0002

2.15941

w/c × unit weight 
1 
34.24312

11.00358

3.11

0.0026

2.152

Ln(age) 
1 
633.3489

87.49625

7.24

< 0.0001

1.00604

Table 4. Range of data used for shortterm cylinder strength model.
Parameter 
Minimum 
Maximum 
Average 
w/c ratio 
0.27 
0.69 
0.43 
Cementitious content 
376 
936 
660 
Unit weight 
124 
151 
143 
Pavement age 
0.0384 
1.0000 
0.3081 
Compressive strength 
2,480 
10,032 
5,256 
Figure 9 and figure 10 show the predicted versus measured plot and the residual plot, respectively. Figure 11 through figure 13 show the sensitivity of this model to CMC, w/c ratio, and age, respectively. The trends are all reasonable. Figure 11 and figure 12 show the change in compressive strength at two ages, 28 days and 1 year, which are almost at the lower and upper bounds of ages included in this model. The plot in figure 13 can be considered a strength gain curve for typical unit weight and w/c ratios used in mix designs.
Figure 9. Graph. Predicted versus measured for shortterm cylinder compressive strength model.
Figure 10. Graph. Residual errors for shortterm cylinder compressive strength model.
Figure 11. Graph. Shortterm cylinder compressive strength sensitivity to CMC.
Figure 12. Graph. Shortterm cylinder compressive strength sensitivity to w/c ratio.
Figure 13. Graph. Shortterm cylinder compressive strength sensitivity to age.
The shortterm core compressive strength model is as follows:
Figure 14. Equation. Prediction model 3 for f_{c,t}.
Where:
f_{c,t} = Compressive
strength at age t years, psi.
CMC = Cementitious materials content, lb/yd^{3}.
uw = Unit weight, lb/ft^{3}.
MAS = Maximum aggregate size, inch.
w/c = Water to cementitious materials ratio.
FM = Fineness modulus of fine aggregate.
t = Shortterm age up to 1 year.
The regression statistics for this model are presented in table 5. The model was developed using 294 points, and the prediction has an R^{2} value of 67.61 percent and an RMSE of 1,122 psi. Table 6 provides details of the range of data used to develop the model.
Table 5. Regression statistics for shortterm core strength model.
Variable 
DF 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1

98.92962

1,544.34064

0.06

0.949

0

CMC 
1

5.70412

0.36589

15.59

< 0.0001

1.23548

Unit weight 
1

28.48527

10.59672

2.69

0.0076

1.0182

MAS × w/c ratio 
1

2,570.13151

538.267

4.77

< 0.0001

1.2201

FM 
1

199.84664

120.68288

1.66

0.0988

1.01426

Ln(age) 
1

611.30879

45.08962

13.56

< 0.0001

1.00026

Table 6. Range of data used for shortterm core strength model.
Parameter 
Minimum 
Maximum 
Average 
w/c ratio 
0.27 
0.69 
0.42 
Cementitious content 
376 
999 
670 
Unit weight 
120 
163 
144 
MAS 
0.375 
1.000 
0.683 
FM 
2.50 
4.37 
3.05 
Pavement age 
0.0380 
2.2160 
0.4230 
Compressive strength 
1990 
11,350 
5,596 
Figure 15 and figure 16 show the predicted versus measured plot and the residual plot, respectively. Figure 17 through figure 22 show the sensitivity of this model to CMC, unit weight, MAS, w/c ratio, FM, and age, respectively.
Figure 15. Graph. Predicted versus measured for shortterm core compressive strength model.
Figure 16. Graph. Residual errors for shortterm core compressive strength model.
Figure 17. Graph. Shortterm core compressive strength sensitivity to CMC.
Figure 18. Graph. Shortterm core compressive strength sensitivity to unit weight.
Figure 19. Graph. Shortterm core compressive strength sensitivity to MAS.
Figure 20. Graph. Shortterm core compressive strength sensitivity to w/c ratio.
Figure 21. Graph. Shortterm core compressive strength sensitivity to fine aggregate FM.
Figure 22. Graph. Shortterm core compressive strength sensitivity to age.
The compressive strength for cores at all ages is estimated as follows:
Figure 23. Equation. Prediction model 4 for f_{c,t}.
Where:
f_{c,t} = Compressive
strength at age t years, psi.
w/c = Water to cementitious materials ratio.
CMC = Cementitious materials content,
lb/yd^{3}.
uw = Unit weight, lb/ft^{3}.
t = Shortterm age in years.
The regression statistics for this model are presented in table 7. The model was developed using 580 data points, and the prediction has an R^{2} value of 55.38 percent and an RMSE of 992 psi. Table 8 provides details of the range of data used to develop the model.
Table 7. Regression statistics for all ages core strength model.
Variable 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
6,022.44

2,028.37

2.97

0.0032

0

w/c ratio 
854.46

675.86

1.26

0.2069

2.15941

CMC 
4.8656

0.5737

8.48

< 0.0001

2.152

Unit weight 
68.5337

13.4368

5.1

< 0.0001

1.00604

Ln(age) 
533.15

22.3343

23.87

< 0.0001

1.00026

Table 8. Range of data used for all ages core strength model.
Parameter 
Minimum 
Maximum 
Average 
w/c ratio 
0.00 
0.72 
0.43 
Cementitious content 
354 
999 
615 
Unit weight 
120 
163 
145 
Pavement age 
0.0380 
45.3840 
6.4320 
Compressive strength 
1,990 
11,750 
6,430 
Figure 24 and figure 25 show the predicted versus measured plot and the residual plot, respectively.
Figure 24. Graph. Predicted versus measured for all ages core compressive strength model.
Figure 25. Graph. Residual errors for all ages core compressive strength model.
Figure 26 through figure 29 show the sensitivity of this model to w/c ratio, CMC, unit weight, and age, respectively. Again, the sensitivity plots showing the variation in core compressive strength with changes in w/c ratio, CMC, and unit weight are presented for 28 days, 1 year, and 20 years. The rate of strength gain clearly is much higher in the short term (28 days to 1 year) than during the next 19 years. Figure 29 can be treated as the strength gain relationship representative of a typical mix (w/c of 0.4, CMC of 600 lb/yd^{3}, and unit weight of 145 lb/ft^{3}).
Figure 26. Graph. All ages core compressive strength sensitivity to w/c ratio.
Figure 27. Graph. All ages core compressive strength sensitivity to CMC.
Figure 28. Graph. All ages core compressive strength sensitivity to unit weight.
Figure 29. Graph. All ages core compressive strength sensitivity to age.
The model developed for the longterm strength is expressed as follows:
Figure 30. Equation. Prediction model 5 for f_{c,LT}.
Where:
f_{c,LT} = Longterm
compressive strength, psi.
CMC = Cementitious materials content, lb/yd^{3}.
uw = Unit weight, lb/ft^{3}.
The regression statistics for this model are presented in table 9. The model was developed using 201 data points, and the prediction has an R^{2} value of 18.03 percent and an RMSE of 1,179 psi. Table 10 provides details of the range of data used to develop the model.
Table 9. Regression statistics for longterm core strength model.
Variable 
DF 
Estimate 
Standard Error 
tValue 
P_{r} > t 
VIF 
Intercept 
1 
3,467.3508

1,720.49637

2.02

0.0452

0

Cementitious 
1 
3.63452

1.38354

2.63

0.0093

1.024

(Unit weight)^{2} 
1 
0.42362

0.06634

6.39

< 0.0001

1.024

Table 10. Range of data used for longterm core strength model.
Parameter 
Minimum 
Maximum 
Average 
Cementitious content 
354 
781 
550 
Unit weight 
134 
156 
147 
Compressive strength 
4,315 
11,750 
7,655 
Figure 31 and figure 32 show the predicted versus measured plot and the residual plot, respectively. This model does not have a good predictive ability (see figure 31). While there is no significant bias, the error in prediction is fairly high (see figure 32). This model needs to be used with caution. Additionally, other means to verify the value would be necessary, such as core tests.
Figure 31. Graph. Predicted versus measured for longterm core compressive strength model.
Figure 32. Graph. Residual errors for longterm core compressive strength model.
The compressive strength models, like any other empirical model, reproduce the trends present in the datasets used for each correlation. It is highly recommended that a user estimate the strength based on as many models as possible with the information available at the time of analysis. This might provide a fair assessment of the ranges of compressive strength likely for the project and at different ages.
Figure 33 through figure 37 show the relationship between compressive strength and CMC, w/c ratio, and unit weight, respectively. Figure 36 and figure 37 show the strength gain at short and longterm ages, respectively. Note that relationships have been plotted for typical values for all variables, and the raw data used in the models do not necessarily lie on the plots.
Figure 33. Graph. Model compressive strength prediction for varying CMC.
Figure 34. Graph. Model compressive strength prediction for varying w/c ratio.
Figure 35. Graph. Model compressive strength prediction for varying unit weights.
Figure 36. Graph. Strength gain in the shortterm predicted by three models.
Figure 37. Graph. Longterm strength gain predicted by the models.
The following observations can be made:
These observations illustrate the benefit of comparing predictions made by the various models available to obtain the range of strength that each project or observation could develop. Any other information to substantiate or validate the strength predictions should be utilized whenever possible, such as strength values from other projects that have used similar materials and mix design.