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Publication Number: FHWA-HRT-12-031
Date: August 2012

 

User’s Guide: Estimation of Key PCC, Base, Subbase, and Pavement Engineering Properties From Routine Tests and Physical Characteristics

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CHAPTER 3. PCC MODELS

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:

PCC Compressive Strength 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:

Compressive Strength Model 1: 28-day Cylinder Strength Model

The 28-day compressive strength model developed for cylinder strength is as follows:

f subscript c,28d equals 4,028.41841 minus 3,486.3501 times w/c plus 4.02511 times CMC.

Figure 3. Equation. Prediction model 1 for fc,28d.

Where:

f’c,28d = 28-day compressive strength, psi.
w/c = Water to cementitious materials ratio.
CMC = Cementitious materials content, lb/yd3.

The model statistics are shown in table 1. The model was developed using 42 data points, and the prediction has an R2 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 28-day PCC cylinder strength.

Variable

Degrees of Freedom (DF)

Estimate

Standard Error

t-Value

Pr > |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 28-day 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/ft3 increases the 28-day 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.

This graph shows an x-y scatter plot showing the predicted versus the measured values used in the 28-day cylinder compressive strength model. The x-axis shows the measured compressive strength from 0 to 8,000 psi, and the y-axis shows the predicted compressive strength from 0 to 8,000 psi. The plot contains 42 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 good prediction. The measured values range from 3,034 to 7,611 psi. The graph also shows the model statistics as follows: N equals 42, R-squared equals 0.544 percent, and root mean square error equals 871 psi.

Figure 4. Graph. Predicted versus measured for 28-day cylinder compressive strength model.

This graph shows an x-y scatter plot showing the residual errors in the predictions of the 28-day cylinder compressive strength model. The x-axis shows the predicted compressive strength from 0 to 8,000 psi, and the y-axis shows the residual compressive strength from 0 to 2,500 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 appear in the graph: 
y equals -6E minus 0.7x plus 0.0014 and R-squared equals 5E minus 13.

Figure 5. Graph. Residual error plot for 28-day cylinder compressive strength model.

This graph shows the sensitivity of the 28-day compressive strength model to the water/cement (w/c) ratio. The x-axis shows the w/c ratio from 0 to 0.8, and the y-axis shows the predicted compressive strength values from 3,000 to 8,000 psi. The sensitivity shown for w/c ratio ranges from 0.25 to 0.7, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing w/c ratio, the predicted compressive strength decreases.

Figure 6. Graph. 28-day compressive strength model sensitivity to w/c ratio.

This graph shows the sensitivity of the 28-day compressive strength model to the cementitious materials content (CMC). The x-axis shows CMC from 400 to 1,200 lb/yd3, and the y-axis shows the predicted compressive strength values from 3,000 to 8,000 psi. The sensitivity is shown for CMC ranges from 450 to 1,000 lb/yd3, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing CMC, the predicted compressive strength increases.

Figure 7. Graph. 28-day compressive strength model sensitivity to CMC.

Compressive Strength Model 2: Short-Term Cylinder Strength Model

The short-term cylinder compressive strength is expressed as follows:

f subscript c,t equals 6,358.60655 plus 3.53012 times CMC minus 34.24312 times w/c times uw plus 633.3489 times natural log open parenthesis t closed parenthesis.

Figure 8. Equation. Prediction model 2 for fc,t.

Where:

fc,t = Compressive strength at age t years, psi.
CMC = Cementitious materials content, lb/yd3.
w/c = Water to cementitious materials ratio.
uw = Unit weight, lb/ft3.
t = Short-term 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 R2 value of 66.6 percent and an RMSE of 789 psi. The reason for an improved R2 compared to the 28-day strength model is not clear from these analyses.

Table 3. Regression statistics for short-term cylinder strength model.

Variable

DF

Estimate

Standard Error

t-Value

Pr > |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 short-term 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.

 

This graph shows an x-y scatter plot showing the predicted versus the measured values used in the short-term cylinder compressive strength model. The x-axis shows the measured compressive strength from 0 to 12,000 psi, and the y-axis shows the predicted compressive strength values from 0 to 12,000 psi. The plot contains 79 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 good prediction. The measured values range from 2,480 to 10,032 psi. The graph also shows the model statistics as follows: N equals 79, R-squared equals 0.666 percent, and root mean square error equals 789 psi.

Figure 9. Graph. Predicted versus measured for short-term cylinder compressive strength model.

This figure shows an x-y scatter plot showing the residual errors in the predictions of the short-term cylinder compressive strength model. The x-axis shows the predicted compressive strength from 0 to 10,000 psi, and the y-axis shows the residual compressive strength from -3,000 to 3,000 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 found in the graph: y equals 9E minus 0.7x minus 0.0082 and R-squared equals 2E minus 12.

Figure 10. Graph. Residual errors for short-term cylinder compressive strength model.

This graph shows the sensitivity of the short-term cylinder compressive strength model to the cementitious materials content (CMC). The x-axis shows CMC from 300 to 1,100 lb/yd3, and the y-axis shows the predicted compressive strength values from 3,000 to 11,000 psi. The sensitivity is shown for CMC ranges from 350 to 1,000 lb/yd3 for strength predictions at 28 days and 1 year. The 28-day strength is plotted using solid squares connected by a solid line, and the 1-year strength is plotted using solid diamonds connected by a solid line. The graph shows that with increasing CMC, the predicted compressive strength increases. The water/cement ratio is -0.4, and the unit weight 
is 145 lb/ft3.

Figure 11. Graph. Short-term cylinder compressive strength sensitivity to CMC.

This graph shows the sensitivity of the short-term cylinder compressive strength model to the water/cement (w/c) ratio. The x-axis shows the w/c ratio from 0 to 0.8, and the y-axis shows the predicted compressive strength values from 2,000 to 8,000 psi. The sensitivity is shown for w/c ratio ranges from 0.25 to 0.70 for strength predictions at 28 days and 1 year. The 28-day strength is plotted using solid squares connected by a solid line, and the 1-year strength is plotted using solid diamonds connected by a solid line. The graph shows that with increasing w/c ratio, the predicted compressive strength decreases. The cementitious materials content is 600 lb/yd3, and the unit weight is 145 lb/ft3.

Figure 12. Graph. Short-term cylinder compressive strength sensitivity to w/c ratio.

This graph shows the sensitivity of the short-term cylinder compressive strength model to the pavement age. The x-axis shows the age in from 0 to 1 year, and the y-axis shows the predicted compressive strength values from 3,000 to 9,000 psi. The sensitivity is shown for pavement ages from 0 to 
1 year, and the data are plotted using solid squares connected by a solid line. The graph shows that as the pavement ages, the predicted compressive strength increases.

Figure 13. Graph. Short-term cylinder compressive strength sensitivity to age.

Compressive Strength Model 3: Short-Term Core Strength Model

The short-term core compressive strength model is as follows:

f subscript c,t equals 98.92962 plus 5.70412 times CMC plus 28.48527 times uw plus 2,570.13151 times MAS times w/c minus 199.84664 times FM plus 611.30879 times natural log open parenthesis t closed parenthesis.

 

Figure 14. Equation. Prediction model 3 for fc,t.

Where:

fc,t = Compressive strength at age t years, psi.
CMC = Cementitious materials content, lb/yd3.
uw = Unit weight, lb/ft3.
MAS = Maximum aggregate size, inch.
w/c = Water to cementitious materials ratio.
FM = Fineness modulus of fine aggregate.
t = Short-term 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 R2 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 short-term core strength model.

Variable

DF

Estimate

Standard Error

t-Value

Pr > |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 short-term 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.

This graph is an x-y scatter plot showing the predicted versus the measured values used in the short-term core compressive strength model. The x-axis shows the measured compressive strength from 0 to 12,000 psi, and the y-axis shows the predicted compressive strength from 0 to 12,000 psi. The plot contains 294 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 good prediction. The measured values range from 1,990 to 11,350 psi. The graph also shows the model statistics as follows: N equals 294, 
R-squared equals 0.6761 percent, and root mean square error equals 1,122 psi.

Figure 15. Graph. Predicted versus measured for short-term core compressive strength model.

This graph is an x-y scatter plot showing the residual errors in the predictions of the short-term core compressive strength model. The x-axis shows the predicted compressive strength from 0 to 10,000 psi, and the y-axis shows the residual compressive strength from -4,000 to 4,000 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 also provided in the graph: y equals 2E minus 0.7x plus 0.0005 and R-squared equals 7E minus 14.

Figure 16. Graph. Residual errors for short-term core compressive strength model.

This graph shows the sensitivity of the short-term core compressive strength model to the cementitious materials content (CMC). The x-axis shows CMC from 300 to 1,100 lb/yd3, and the y-axis shows the predicted compressive strength from 3,000 to 11,000 psi. The sensitivity is shown for CMC and ranges from 350 to 1,000 lb/yd3 for strength predictions at 28 days and 1 year. The 28-day strength is plotted using solid squares connected by a solid line, and the 1-year strength is plotted using solid diamonds connected by a solid line. The graph shows that with increasing CMC, the predicted compressive strength increases. The water/cement ratio is 0.4, the unit weight is 
145 lb/ft3, the maximum aggregate size is 0.75 inches, and the fineness modulus is 3.0.

Figure 17. Graph. Short-term core compressive strength sensitivity to CMC.

This  graph shows the sensitivity of the short-term core compressive strength model to the unit weight. The x-axis shows the unit weight from 120 to 170 lb/ft3, and the y-axis shows the predicted compressive strength from 3,000 to 9,000 psi. The sensitivity is shown for unit weight ranges from 125 to 155 lb/ft3 for strength predictions at 28 days and 1 year. The 28-day strength is plotted using solid squares connected by a solid line, and the 1-year strength is plotted using solid diamonds connected by a solid line. The graph shows that with increasing unit weight, 
the predicted compressive strength increases. Cementitious materials content equals 600 lb/yd3, the water/cement ratio equals 0.4, maximum aggregate size equals 0.75 inches, and fineness modulus equals 3.0.

Figure 18. Graph. Short-term core compressive strength sensitivity to unit weight.

This graph shows the sensitivity of the short-term core compressive strength model to the maximum aggregate size (MAS). The x-axis shows the maximum aggregate size from 0.2 to 1.2 inches, 
and the y-axis shows the predicted compressive strength from 3,000 to 9,000 psi. The sensitivity is shown for aggregate size ranges from 0.375 to 1 inch for strength predictions at 28 days and 
1 year. The 28-day strength is plotted using solid squares connected by a solid line, and the 
1-year strength is plotted using solid diamonds connected by a solid line. The graph shows 
that with increasing maximum aggregate size, the predicted compressive strength decreases. Cementitious materials content equals 600 lb/yd3, the water/cement ratio equals 0.4, the unit weight equals 145 lb/ft3, and fineness modulus equals 3.0.

Figure 19. Graph. Short-term core compressive strength sensitivity to MAS.

This graph shows the sensitivity of the short-term core compressive strength model to the water/cement (w/c) ratio. The x-axis shows the w/c ratio from 0.2 to 0.8, and the y-axis shows the predicted compressive strength from 3,000 to 9,000 psi. The sensitivity is shown for w/c ratio ranges from 0.25 to 0.70 for strength predictions at 28 days and 1 year. The 28-day strength is plotted using solid squares connected by a solid line, and the 1-year strength is plotted using solid diamonds connected by a solid line. The graph shows that with increasing w/c ratio, the predicted compressive strength decreases. Cementitious materials content is 600 lb/yd3, the unit 
weight is 145 lb/ft3, maximum aggregate size is 0.75 inches, and fineness modulus is 3.0.

Figure 20. Graph. Short-term core compressive strength sensitivity to w/c ratio.

This graph shows the sensitivity of the short-term core compressive strength model to the fineness modulus (FM). The x-axis shows FM from 1 to 6, and the y-axis shows the predicted compressive strength from 3,000 to 9,000 psi. The sensitivity is shown for a FM range of 2 to 4.5, and the strengths are predicted at ages of 28 days and 1 year. The 28-day strength is plotted using solid squares connected by a solid line, and the 1-year strength is plotted using solid diamonds connected by a solid line. The graph shows that with increasing FM, the predicted compressive strength decreases. Cementitious materials content is 600 lb/yd3, the water/cement ratio is 0.4, the unit weight is 145 lb/ft3, and maximum aggregate size is 0.75 inches.

Figure 21. Graph. Short-term core compressive strength sensitivity to fine aggregate FM.

This graph shows the sensitivity of the short-term core compressive strength model to the pavement age. The 
x-axis shows the pavement age from 0 to 1.4 years, and the y-axis shows the predicted compressive strength from 3,000 to 9,000 psi. The sensitivity is shown for pavement ages 
from 0 to 1 year, and the data are plotted using solid squares connected by a solid line. The graph shows that as the pavement ages, the predicted compressive strength increases. Cementitious materials content is 600 lb/yd3, the water/cement ratio is 0.4, the unit weight is 145 lb/ft3, maximum aggregate size is 0.75 inches, and fineness modulus is 3.0.

Figure 22. Graph. Short-term core compressive strength sensitivity to age.


 

Compressive Strength Model 4: All Ages Core Strength Model

The compressive strength for cores at all ages is estimated as follows:

f subscript c,t equals -6,022.44 minus 854.46 times w/c plus 4.8645 times CMC plus 68.5337 times uw plus 533.15 times natural log open parenthesis t closed parenthesis.

Figure 23. Equation. Prediction model 4 for fc,t.

Where:

fc,t = Compressive strength at age t years, psi.
w/c = Water to cementitious materials ratio.
CMC = Cementitious materials content, lb/yd3.
uw = Unit weight, lb/ft3.
t = Short-term 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 R2 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

t-Value

Pr > |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.

This graph is an x-y scatter plot showing the predicted versus the measured values used in the all ages core compressive strength model. The x-axis shows the measured compressive strength from 0 to 12,000 psi, and the y-axis shows the predicted compressive strength from 0 to 
12,000 psi. The plot contains 580 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 good prediction. The measured values range from 1,990 to 11,750 psi. The graph also shows the model statistics as follows: N equals 580, 
R-squared equals 0.5538 percent, and root mean square error equals 992 psi.

Figure 24. Graph. Predicted versus measured for all ages core compressive strength model.

This 
graph is an x-y scatter plot showing the residual errors in the predictions for the all ages core compressive strength model. The x-axis shows the predicted compressive strength from 0 to 12,000 psi, and the y-axis shows the residual compressive strength from -6,000 to 6,000 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 -0.002x plus 79.723 and R-squared equals 5E minus 0.6.

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/yd3, and unit weight of 145 lb/ft3).

This graph shows the sensitivity of the all ages core compressive strength model to the water/cement 
(w/c) ratio. The x-axis shows the w/c ratio from 0.2 to 0.8, and the y-axis shows the predicted compressive strength from 3,000 to 11,000 psi. The sensitivity is shown for w/c ratio ranges from 0.25 to 0.70 for strength predictions at 28 days, 1 year, and 20 years. The 28-day strength 
is plotted using solid squares connected by a solid line, the 1-year strength prediction data are shown using solid triangles connected with a solid line, and the 20-year strength data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing w/c ratio, the predicted compressive strength decreases. Cementitious materials content is 600 lb/yd3 and the unit weight is 145 lb/ft3.

Figure 26. Graph. All ages core compressive strength sensitivity to w/c ratio.

This graph shows the sensitivity of the all ages core compressive strength model to the cementitious materials content (CMC). The x-axis shows CMC from 300 to 1,100 lb/yd3, and the y-axis shows the predicted compressive strength from 3,000 to 11,000 psi. The sensitivity is shown for CMC ranges from 350 to 1,000 lb/yd3 for strength predictions at 28 days, 1 year, and 20 years. The 
28-day strength is plotted using solid squares connected by a solid line, the 1-year strength prediction data are shown using solid triangles connected with a solid line, and the 20-year strength data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing CMC, the predicted compressive strength increases. The water/cement ratio is 0.4, and the unit weight is 145 lb/ft3.

Figure 27. Graph. All ages core compressive strength sensitivity to CMC.

This graph shows the sensitivity of the all ages core compressive strength model to the unit weight. The 
x-axis shows the unit weight from 120 to 160 lb/ft3, and the y-axis shows the predicted compressive strength from 3,000 to 11,000 psi. The sensitivity is shown for unit weight ranging from 125 to 155 lb/ft3 for strength predictions at 28 days, 1 year, and 20 years. The 28-day strength is plotted using solid squares connected by a solid line, the 1-year strength prediction data are shown using solid triangles connected with a solid line, and the 20-year strength data 
are plotted using solid diamonds connected by a solid line. The graph shows that with increasing 
unit weight, the predicted compressive strength increases. Cementitious materials content is 
600 lb/yd3, and the water/cement ratio is 0.4

Figure 28. Graph. All ages core compressive strength sensitivity to unit weight.

This graph shows 
the sensitivity of the short-term core compressive strength model to the pavement age. The 
x-axis shows the pavement age from 0 to 20 years, and the y-axis shows the predicted compressive strength from 3,000 to 11,000 psi. The sensitivity is shown for pavement ages from 0 to 1 year, and the data are plotted using solid squares connected by a solid line. The graph shows that as the pavement ages, the predicted compressive strength increases. Cementitious materials content is 600 lb/yd3, the water/cement ratio is 0.4, and the unit weight is 145 lb/ft3.

Figure 29. Graph. All ages core compressive strength sensitivity to age.

Compressive Strength Model 5: Long-Term Core Strength Model

The model developed for the long-term strength is expressed as follows:

f subscript c,LT equals -3,467.3508 plus 3.63452 times CMC plus 0.42362 times uw squared.

 

Figure 30. Equation. Prediction model 5 for fc,LT.

Where:

fc,LT = Long-term compressive strength, psi.
CMC = Cementitious materials content, lb/yd3.
uw = Unit weight, lb/ft3.

The regression statistics for this model are presented in table 9. The model was developed using 201 data points, and the prediction has an R2 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 long-term core strength model.

Variable

DF

Estimate

Standard Error

t-Value

Pr > |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 long-term 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.

This graph is an x-y scatter plot showing the predicted versus the measured values used in the long-term core compressive strength model. The x-axis shows the measured compressive strength from 4,000 to 12,000 psi, and the y-axis shows the predicted compressive strength from 4,000 to 12,000 psi. The plot contains 201 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 good prediction. The measured values range from 4,315 to 11,750 psi. The graph also shows the model statistics as follows: 
N equals 201, R-squared equals 0.1803 percent, and root mean square error equals 1,179 psi.

Figure 31. Graph. Predicted versus measured for long-term core compressive strength model.

This graph is an x-y scatter plot showing the residual errors in the predictions of the long-term core compressive strength model. The x-axis shows the predicted compressive strength from 4,000 to 10,000 psi, and the y-axis shows the residual compressive strength from -4,000 to 4,000 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 0.0384x minus 291.07 and R-squared equals 0.0003.

Figure 32. Graph. Residual errors for long-term core compressive strength model.

Relative Comparison of All Compressive Strength Models

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 long-term 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.

This graph shows the sensitivity of four compressive strength models. The x-axis shows the cementitious materials content (CMC) from 300 to 1,100 lb/yd3, and the y-axis shows the predicted compressive strength from 1,000 to 11,000 psi. The sensitivity is shown for CMC ranging from 350 to 1,000 lb/yd3 for strength predictions at 28 days. The 28-day strength is plotted using different markers for the four models used. The solid diamonds represent the 28-day cylinder model, the solid squares represent the short-term cylinder strength model, the asterisk marks represent the short-term core strength model, and the solid triangles represent the all-ages core strength model. The raw data representing 28-day strengths are plotted as hollow triangles for cylinders and hollow circles for cores. The graph shows that with increasing CMC, the predicted compressive strength increases. The lines are mostly inclined at approximately 30 degrees. The graph also shows that the predictions for all models are within 500 psi of each other for most part. The water/cement ratio is 0.4, the unit weight is 145 lb/ft3, maximum aggregate size is 
0.75 inches, and fineness modulus is 3.0.

Figure 33. Graph. Model compressive strength prediction for varying CMC.

This graph shows the sensitivity of four compressive strength models. The x-axis shows the water/cement (w/c) ratio from 0.2 to 0.8, and the y-axis shows the predicted compressive strength from 
1,000 to 11,000 psi. The sensitivity is shown for w/c ratio ranges from 0.25 to 0.70 for strength predictions at 28 days. The 28-day strength is plotted using different markers for the four models used. The solid diamonds represent the 28-day cylinder model, the solid squares represent 
the short-term cylinder strength model, the solid circles marks represent the short-term core strength model, and the solid triangles represent the all ages core strength model. The raw data representing 28-day strengths are plotted as hollow triangles for cylinders and hollow circles for cores. The graph shows that with an increasing w/c ratio, the predicted compressive strength decreases. The four plots have different slopes. The graph also shows that the predictions for 
all models are within 500 psi of each other for w/c ratios of less than 0.50. The raw data are scattered to the top of the models for w/c ratios below 0.38 and between w/c ratios of 0.4 and 0.5. They are spread on both sides of predictions for w/c ratios approaching 0.6. Cementitious materials content is 600 lb/yd3, the unit weight is 145 lb/ft3, and maximum aggregate size is 
0.75 inches.

Figure 34. Graph. Model compressive strength prediction for varying w/c ratio.

This graph shows the model compressive strength prediction for varying unit weights. The x-axis shows the unit weight from 120 to 160 lb/ft3, and the y-axis shows the predicted compressive strength from 1,000 to 11,000 psi. The sensitivity is shown for unit weight ranges from 125 to 155 lb/ft3 for strength predictions at 1 year. The 28-day strength is plotted using different markers for the two models. The solid triangles represent the core all ages model, and the solid squares represent the core short-term model. The raw data are plotted as hollow triangles for cylinders and hollow circles for cores. The graph shows that with increasing unit weight, the predicted compressive strength increases. The two lines have different slopes; the core all ages line is steeper than the core short-term model line. The raw data are spread on both sides of the predictions. Cementitious materials content is 600 lb/yd3, the water/cement ratio is 0.4, maximum aggregate size is 0.75 inches, and fineness modulus is 3.0.

Figure 35. Graph. Model compressive strength prediction for varying unit weights.

This graph shows the sensitivity of three compressive strength models to the pavement age. The x-axis shows the pavement age from 0 to 1 year, and the y-axis shows the predicted compressive strength from 4,000 to 8,000 psi. The models are represented by different markers; the solid triangles represent the cylinder short-term strength, the solid squares represent the core all ages strength, and the solid triangles represent the core short-term strength. The graph shows that with increasing age, the predicted compressive strength increases. Cementitious materials content is 600 lb/yd3, the water/cement ratio is 0.4, the unit weight is 145 lb/ft3, maximum aggregate size is 0.75 inches, and fineness modulus is 3.0.

Figure 36. Graph. Strength gain in the short-term predicted by three models.

This graph shows the sensitivity of long-term strength gain to the pavement age. The x-axis shows the pavement age from 0 to 20 years, and the y-axis shows the predicted compressive strength from 1,000 to 15,000 psi. The models are represented by different markers; the solid squares connected by a solid line represent the core all ages model, and the solid line without any markers represents the long-term strength model. The core all ages model has a steep increase from an 0 to 1 year and has a considerable reduction in slope and is almost a flat line after 10 years. The long-term strength model ranges from 5 to 20 years and is a straight line with zero slope, which indicates that it is not affected by age. The hollow circles represent the raw data. Cementitious materials content is 600 lb/yd3, the water/cement ratio is 0.4, the unit weight is 145 lb/ft3, maximum aggregate size is 0.75 inches, and fineness modulus is 3.0.

Figure 37. Graph. Long-term 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.

 


The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT).
The Federal Highway Administration (FHWA) is a part of the U.S. Department of Transportation and is headquartered in Washington, D.C., with field offices across the United States. is a major agency of the U.S. Department of Transportation (DOT). Provide leadership and technology for the delivery of long life pavements that meet our customers needs and are safe, cost effective, and can be effectively maintained. Federal Highway Administration's (FHWA) R&T Web site portal, which provides access to or information about the Agency’s R&T program, projects, partnerships, publications, and results.
FHWA
United States Department of Transportation - Federal Highway Administration