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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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PCC Flexural Strength Models

Validation of Existing Models

Previous models correlating flexural strength to compressive strength have generally used a power model of the following form:

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

 

Figure 38. Equation. Mr.

Where:

a = 7.5 to 11.7 for b = 0.5.
a = 2 to 2.7 for b = 0.67.

Table 11 shows a summary of the models developed. The regressed constants, a and b, were found to be within the range of values reported by the other studies. This validation not only provides feasible models, but it also confirms that the data used reasonably represent the broad range considered in the various studies. The correlations are presented in figure 39 and figure 40 for the power models with exponents of 0.5 and 0.67, respectively.

Table 11. Power models developed for flexural strength prediction using LTPP data for validation.

Model

a

b

R2

N

[any value]

 

10.3022

 

0.50

 

0.446

 

185

 

2.4277

 

0.67

 

0.449

 

185

 

 


 

This figure is an x-y scatter plot showing the predicted versus the measured values used for validating 0.5 power flexural strength model. The x-axis shows the measured modulus of rupture from 100 to 1,300 psi, and the y-axis shows the predicted modulus of rupture from 100 to 1,300 psi. The plot contains 185 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 467 to 1,075 psi. The graph also shows the model statistics as follows: N equals 185, 
R-squared equals 0.4460 percent, and root mean square error equals 84 psi.

Figure 39. Graph. Predicted versus measured for validating 0.5 power flexural strength model.

This graph is an x-y scatter plot showing the predicted versus the measured values used for validating 0.667 power flexural strength model. The x-axis shows the measured modulus of rupture from 100 to 1,300 psi, and the y-axis shows the predicted modulus of rupture from 100 to 1,300 psi. The plot contains 185 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 467 to 1,075 psi. The graph also shows the model statistics as follows: N equals 185, 
R-squared equals 0.4493 percent, and root mean square error equals 111 psi.

Figure 40. Graph. Predicted versus measured for validating 0.667 power flexural strength model.

Flexural Strength Model 1: Flexural Strength Based on Compressive Strength

This model 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:

MR equals 22.7741 times f prime subscript c raised to the power of 0.4082.

Figure 41. Equation. Prediction model 6 for MR.


 

 

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 12. The model was developed using 185 data points, and the prediction has an R2 value of 45.2 percent and an RMSE of 69 psi. Table 13 provides details of the range of data used to develop the model.

Table 12. Regression statistics for flexural strength model based on compressive strength.

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

 

 

Table 13. Range of data used for flexural strength model based on compressive strength.

Parameter

Minimum

Maximum

Average

Compressive strength

1,770

10,032

5,431

Flexural strength

467

1,075

754

 

Figure 42 and figure 43 show the predicted versus measured plot and the residual plot, respectively.

This graph is an x-y scatter plot that shows the predicted versus the measured values for the flexural strength model based on compressive strength. The x-axis shows the measured modulus of rupture from 100 to 1,300 psi, and the y-axis shows the predicted modulus of rupture from 100 to 1,300 psi. The plot contains 185 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 467 to 1,075 psi. The graph also shows the model statistics as follows: N equals 185, R-squared equals 0.452 percent, and root mean square error equals 69 psi.

Figure 42. Graph. Predicted versus measured values for flexural strength model based on compressive strength.

This graph is an x-y scatter plot showing the residual errors in the predictions of the flexural strength model based on compressive strength. The x-axis shows the predicted modulus of rupture from 100 to 1,300 psi, and the y-axis shows the residual modulus of rupture from -500 to 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 follow equations are provided in the graph: y equals 0.0131x minus 10.037, and R-squared equals 0.0001.

Figure 43. Graph. Residual errors for flexural strength model based on compressive strength.

Figure 44 shows a comparison of the power models used to validate the data and 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 5,500-psi compressive strength range.

The American Concrete Institute (ACI) and Portland Cement Association (PCA) models are plotted for comparison. Also plotted are the raw data that were used in the model. Clearly, the ACI equation is conservative for this data. It has also 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 44 clearly 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 are considered in the other models developed in this study.

This graph shows a comparison of all flexural strength models and their sensitivity to changes in compressive strength of the concrete. The x-axis shows the compressive strength from 0 to 12,000 psi, and the data ranges from about 2,000 to 10,000 psi. The y-axis shows the modulus of rupture from 400 to 1,200 psi. The graph consists of five lines, which each represent a different model. The solid triangles connected with a solid line represent the 0.5 power model, the solid diamonds connected with a solid line represent the 0.67 power model, the solid squares connected with a solid line represent the best fit power model the plus signs connected with 
a solid line represent the American Concrete Institute (ACI) model, and the asterisk signs connected with a solid line represent the ACI model. The raw data are plotted as hollow circles.

Figure 44. Graph. Comparison of flexural strength models based on compressive strength.

Flexural Strength Model 2: Flexural Strength Based on Age, Unit Weight, and w/c Ratio

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 44. This model will be most useful for cases when the compressive strength of PCC is not determined but mix design information is available. Also, the user has the option of predicting the 28-day strength value for design or estimating the strength at traffic opening time.

This model can be expressed as follows:

MR subscript t equals 676.0159 minus 1,120.31 times w/c plus 4.1304 times uw plus 35.74627 times natural log open parenthesis t closed parenthesis.

 

Figure 45. Equation. Prediction model 7 for MRt.

Where:

MRt = Flexural strength at age t years, psi.
w/c = w/c ratio.
uw = Unit weight, lb/ft3.
t = Pavement age, years.

The regression statistics for this model are presented in table 14. The model was developed using 62 data points, and the prediction has an R2 value of 61.11 percent and an RMSE of 91 psi. Table 15 provides details of the range of data used to develop the model.


 

Table 14. Regression statistics for flexural strength model based on age, unit weight, and w/c ratio.

Variable

DF

Estimate

Standard Error

t-Value

Pr > |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

 

 

Table 15. Range of data used for flexural strength model based on age, unit weight, and w/c ratio.

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

 

 

Figure 46 and figure 47 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 for the flexural strength model based on age, unit weight, and water/cement (w/c) ratio. The x-axis shows the measured modulus of rupture from 0 to 1,400 psi, and the 
y-axis shows the predicted modulus of rupture from 0 to 1,400 psi. The plot contains 
62 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 467 to 
978 psi. The graph also shows the model statistics as follows: N equals 62, R-squared equals 0.6111 percent, and root mean square error equals 91 psi.

Figure 46. Graph. Predicted versus measured values for flexural strength model based on age, unit weight, and w/c ratio.

This graph is an x-y scatter plot showing the residual errors in the predictions of the flexural strength model based on age, unit weight, and water/cement (w/c) ratio. The x-axis shows the predicted modulus of rupture from 400 to 1,000 psi, and the y-axis shows the residual modulus of rupture from -300 to 300 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 1E minus 0.5x minus 0.0088 and 
R-squared equals 2E minus 10.

Figure 47. Graph. Residual errors for flexural strength model based on age, unit weight, and w/c ratio.

Flexural Strength Model 3: Flexural Strength Based on Age, Unit Weight, and CMC

The model is expressed as follows:

MR subscript t equals 24.15063 plus 0.55579 times CMC plus 2.96376 times uw plus 35.54463 times natural log open parenthesis t closed parenthesis.

 

Figure 48. Equation. Prediction model 8 for MRt.

Where:

MRt = Flexural strength at age t years, psi.
CMC = Cementitious materials content, lb/yd3.
uw = Unit weight, lb/ft3.
t = Pavement age, years.


The regression statistics for this model are presented in table 16. The model was developed using 62 data points, and the prediction has an R2 value of 70.23 percent and RMSE of 80 psi. Table 17 provides details of the range of data used to develop the model.

Table 16. Regression statistics for flexural strength model based on age, unit weight, and CMC.

Variable

DF

Estimate

Standard Error

t-Value

Pr > |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

 


 

Table 17. Range of data used for flexural strength model based on age, unit weight, and CMC.

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

 

 

Figure 49 and figure 50 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 for the flexural strength model based on age, unit weight, and cementitious materials content (CMC). The x-axis shows the measured modulus of rupture from 0 to 
1,400 psi, and the y-axis shows the predicted modulus of rupture from 0 to 1,400 psi. The 
plot contains 62 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 467 to 
978 psi. The graph also shows the model statistics as follows: N equals 62, R-squared equals 0.7023 percent, and root mean square error equals 80 psi.

Figure 49. Graph. Predicted versus measured values for flexural strength model based on age, unit weight, and CMC.

This graph is an x-y scatter plot showing the residual errors in the predictions of the flexural strength model based on age, unit weight, and cementitious materials content (CMC). The x-axis shows the predicted modulus of rupture from 400 to 1,000 psi, and the y-axis shows the residual modulus of rupture from -300 to 300 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 2E minus 0.6x minus 0.0008 and R-squared equals 6E minus 12.

Figure 50. Graph. Residual errors for flexural strength model based on age, unit weight, and CMC.

Figure 51 through figure 54 present the sensitivity of the mix design-based flexural strength models to CMC, w/c ratio, unit weight, and age. Figure 51 and figure 52 show that prediction models 7 and 8 do not show any sensitivity to CMC and w/c ratio (see figure 45 and figure 48). 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 flexural strength range. Figure 53 shows that the predictions are close from both models. Likewise, figure 54, which is more or less a flexural strength gain model for a typical mix design, shows very close predictions from both models.

This graph shows the sensitivity of the flexural strength predictions to cementitious materials content (CMC). The 
x-axis shows CMC from 0 to 1,200 lb/yd3, and the y-axis shows the predicted modulus of rupture from 0 to 1,400 psi. The graph has two lines: the solid diamonds connected with a solid line represent the flexural strength model as a function of water/cement (w/c) ratio, and the solid triangles connected with a solid line represent the flexural strength model as a function of CMC. The raw data are plotted as hollow squares. The sensitivity is shown for CMC ranges from 
350 to 1,000 lb/yd3. The w/c ratio is 0.4, the unit weight is 145 lb/ft3, and the age is 28 days.

Figure 51. Graph. Sensitivity of flexural strength predictions to CMC.

This graph shows the sensitivity of the flexural strength predictions to 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 modulus of rupture from 0 to 1,400 psi. The graph has two lines: the solid diamonds connected with a solid line represent the flexural strength model as a function of w/c ratio, and the solid triangles connected with a solid line represent the flexural strength model as a function of cementitious materials content (CMC). The raw data are plotted as hollow squares. The sensitivity is shown for w/c ratio in the range of 0.25 to 0.7. CMC is 600 lb/yd3, the unit weight is 145 lb/ft3, and the age is 28 days.

Figure 52. Graph. Sensitivity of flexural strength predictions to w/c ratio.

This graph shows the sensitivity of the flexural strength predictions to unit weight. The x-axis shows the unit weight from 100 to 160 lb/ft3, and the y-axis shows the predicted modulus of rupture from 0 to 1,400 psi. The graph has two lines: the solid diamonds connected with a solid line represent the flexural strength model as a function of water/cement (w/c) ratio, and the solid triangles connected with a solid line represent the flexural strength model as a function of cementitious materials content (CMC). The raw data are plotted as hollow squares. The sensitivity is shown for unit weights in the range of 125 to 155 lb/ft3. The w/c ratio is 0.4, CMC is 600 lb/yd3, and the age is 28 days.

Figure 53. Graph. Sensitivity of flexural strength predictions to unit weight.

This graph shows the sensitivity of the flexural strength predictions to age. The x-axis shows age from 0 to 1 year, and the y-axis shows the predicted modulus of rupture from 0 to 1,400 psi. The graph has two lines: the solid diamonds connected with a solid line represent the flexural strength model as a function of water/cement (w/c) ratio, and the solid triangles connected with a solid line represent the flexural strength model as a function of cementitious materials content (CMC). The sensitivity is shown for ages from 0 to 1 year. The w/c ratio is 0.4, CMC is 600 lb/yd3, and the unit weight is 145 lb/ft3.

Figure 54. Graph. Sensitivity of flexural strength predictions to age.

 


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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.
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