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Publication Number:  FHWA-HRT-12-031    Date:  August 2012
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 6. UNBOUND MATERIALS MODELS

Resilient Modulus of Unbound Materials

The following model recommended for predicting the resilient modulus of unbound materials is based on the constitutive equation for modeling resilient modulus behavior when subjected to various stress states:

M subscript r equals k subscript 1 times P subscript a times open parenthesis theta divided by P subscript a closed parenthesis raised to the power of k subscript 2 times open parenthesis tau subscript oct divided by P subscript a closed parenthesis raised to the power of k subscript 3.

 

Figure 100. Equation. Mr.

Where:

[any value] =
Bulk stress = [any value].
[any value] = Principal stress.
[any value] = Confining pressure.
Pa = Atmospheric pressure.
[any value] = Octahedral normal stress =1/3 ([any value]).
k1, k2, k3 = regression constants that are a function of soil properties, as defined in figure 101 through figure 103 of this report.

This model can be used for various soil types, and the model attributes (k1, k2, and k3) for a given soil type remain the same regardless of stress state. Furthermore, models used to predict constitutive model attributes for a given set of soil properties are recommended to characterize resilient modulus behavior rather than developing models individually for each possible combination of expected stress states.

Constitutive Model Parameter k1

k subscript 1 equals 1446.2 minus 4.56764 times PCTHALF plus 4.92 times LL minus 27.73 times OPTMOIST.

Figure 101. Equation. Prediction model 18 for k1.

Model statistics for k1 are as follows:

Constitutive Model Parameter k2

k subscript 2 equals 0.45679 minus 0.00073376 times PCTNO80 minus 0.00269 times LL plus 0.00060555 times PCTGRVL plus 12.97 times D subscript 10.

Figure 102. Equation. Prediction model 19 for k2.

Model statistics for k2 are as follows:

Constitutive Model Parameter k3

k subscript 3 equals -0.188 for fine-grained soils and k subscript 3 equals -0.153 for coarse-grained soils.

Figure 103. Equation. Prediction model 20 for k3.

Where:

PCTHALF = Percent passing 1/2-inch sieve. LL = Liquid limit, percent. OPTMOIST = Optimum moisture content, percent. PCTNO80 = Percent passing No. 80 sieve. PCTGRVL = Percent gravel fraction (0.078- to 2.36-inch size). D10 = Maximum particle size of the smallest 10 percent of soil sample.

In the development of these models, a wide range of k1, k2, and k3 parameters were used, which varied by soil class. Histograms showing the distribution of k1, k2, and k3 values by soil class are shown in figure 104 through figure 106, respectively.

This graph is a bar chart showing the resilient modulus parameter k subscript 1for unbound material types included in the data used to develop the unbound resilient modulus model. The data are categorized by the American Association of State Highway and Transportation Officials soil classification, which, starting from the left, are A-1-a, A-1-b, A-2-4, A-2-6, A-2-7, A-3, A-4, A-5, A-6, A-7-5, and A-7-6. The resilient modulus parameter k subscript 1 is plotted on the y-axis from 0 to 1,400. The values plotted are labeled on the solid bars as follows: 862, 858, 834, 994, 1132, 792, 841, 877, 897, 677, and 845, respectively.

Figure 104. Graph. Resilient modulus parameter k1 for unbound material types included in the model development database.

This graph is a bar chart showing the resilient modulus parameter k subscript 2 for unbound material types included in the data used to develop the unbound resilient modulus model. The data are categorized by the American Association of State Highway and Transportation Officials soil classification, which, starting from the left, are A-1-a, A-1-b, A-2-4, A-2-6, A-2-7, A-3, A-4, A-5, A-6, A-7-5, and A-7-6. The resilient modulus parameter k subscript 2 is plotted on the y-axis from 0 to 0.8. The values plotted are labeled on the solid bars as follows: 0.7, 0.6, 0.5, 0.3, 0.4, 0.6, 0.3, 0.2, 0.2, 0.1, and 0.2, respectively.

Figure 105. Graph. Resilient modulus parameter k2 for unbound material types included in the model development database.

This graph is a bar chart showing the resilient modulus parameter k subscript 3 for unbound material types included in the data used to develop the unbound resilient modulus model. The data are categorized by the American Association of State Highway and Transportation Officials soil classification, which, starting from the left, are A-1-a, A-1-b, A-2-4, A-2-6, A-2-7, A-3, A-4, A-5, A-6, A-7-5, and A-7-6. The resilient modulus parameter k subscript 3 is plotted on the y-axis from -0.9 to 0. The values plotted are labeled on the solid bars as follows: -0.1, -0.3, -0.6, -0.5, -0.3, -0.4, -0.8, -0.5, -0.8, -0.6, and -0.8, respectively.

Figure 106. Graph. Resilient modulus parameter k3 for unbound material types included in the model development database.

Model prediction accuracy and reasonableness were evaluated by reviewing the plot of predicted and measured resilient modulus for all individual resilient modulus test values used in model development as presented in figure 107. Figure 108 presents a plot of measured and predicted resilient modulus versus bulk stress for all fine- and coarse-grained materials included in the model development database.

This graph shows the predicted resilient modulus (Mr) values versus the measured Mr values on an x-y scatter plot. The x-axis shows the measured Mr from 0 to 60,000 psi, and the y-axis shows the predicted Mr from 0 to 60,000 psi. The data are plotted using solid diamond markers. A linear trend line is also plotted. The model statistics are as follows: y equals 0.9388x, R-squared equals 0.5661, and N equals 23,056.

Figure 107. Graph. Plot of measured versus predicted resilient modulus (using k1, k2, and k3 derived from figure 101 through figure 103).

This graph shows the sensitivity of the resilient modulus (Mr) model to the bulk stress for fine grained and coarse grained samples. The x-axis shows the bulk stress from 0 to 100 psi, and the y-axis shows the predicted Mr from 0 to 35,000 psi. The sensitivity is shown for bulk stress ranges from 10 to 95 psi. The graph consists of four plots. The predicted Mr for fine-grained samples are represented by solid diamonds, the measured Mr for fine-grained samples are represented by solid triangles, the predicted Mr are represented by solid squares, and the measured Mr for coarse grained samples is represented by X-marks. All plots connect the markers with a solid line. The graph shows that with increasing bulk stress, the predicted Mr increases.

Figure 108. Graph. Plot showing predicted and measured resilient modulus versus bulk stress for fine- and coarse-grained soils.

 

Sensitivity analysis results are presented in figure 109 through figure 115. The results of the sensitivity analysis are summarized as follows:

This figure is a graph showing the sensitivity of the resilient modulus (Mr) model to the bulk stress for four different American Association of State Highway and Transportation Officials (AASHTO) soil classes. The x-axis shows the bulk stress from 0 to 100 psi, and the y-axis shows the predicted Mr from 0 to 40,000 psi. The sensitivity is shown for bulk stress ranges from 10 to 95 psi. The four soil classes are A-1-b represented by solid diamonds, A-2-4 represented by solid squares, A-6 represented by solid triangles, and A-7-5 represented by X-marks. All plots connect the markers with a solid line. The graph shows that with increasing bulk stress, the predicted Mr increases.

Figure 109. Graph. Effect of material type (AASHTO soil class) on predicted resilient modulus.

This graph shows the sensitivity of the resilient modulus (Mr) model to the percent passing the 0.5-inch sieve. The x-axis shows the percent passing from 0 to 100 percent, and the y-axis shows the predicted Mr value from 20,000 to 35,000 psi. The sensitivity is shown for 15 to 85 percent passing the 0.5-inch sieve, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing percent passing, the predicted Mr decreases.

Figure 110. Graph. Effect of percent passing 1/2-inch sieve on predicted resilient modulus.

This graph shows the sensitivity of the resilient modulus (Mr) model to the liquid limit. The x-axis shows the liquid limit from 0 to 50 percent, and the y-axis shows the predicted Mr values from 20,000 to 35,000 psi. The sensitivity is shown for liquid limit ranges between 6 and 40 percent, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing liquid limit, the predicted Mr increases.

Figure 111. Graph. Effect of liquid limit on predicted resilient modulus.

This graph shows the sensitivity of the resilient modulus (Mr) model to the optimum moisture content. The x-axis shows the optimum moisture content from 0 to 40 percent, and the y-axis plots the predicted Mr values from 5,000 to 35,000 psi. The sensitivity is shown for optimum moisture content ranges between 2 and 30 percent, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing optimum moisture content, the predicted Mr decreases.

Figure 112. Graph. Effect of optimum moisture content on predicted resilient modulus.

This graph shows the sensitivity of the resilient modulus (Mr) model to the percent passing the number 80 sieve. The x-axis shows the percent passing the Number 80 sieve from 0 to 100 percent, and the y-axis shows the predicted Mr values from 20,000 to 35,000 psi. The sensitivity is shown for 15 to 85 percent passing the Number 80 sieve, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing percent passing, the predicted Mr decreases.

Figure 113. Graph. Effect of No. 80 sieve on predicted resilient modulus.

This graph shows the sensitivity of the resilient modulus (Mr) model to the gravel content. The x-axis shows the gravel content from 0 to 80 percent, and the y-axis plots the predicted Mr values from 20,000 to 35,000 psi. The sensitivity is shown for gravel content ranges between 5 and 75 percent, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing gravel content, the predicted Mr increases.

Figure 114. Graph. Effect of gravel content on predicted resilient modulus.

This graph shows the sensitivity of the resilient modulus (Mr) model to the effective size. The x-axis shows the effective size from 0 to 0.015 inches, and the y-axis shows the predicted Mr values from 20,000 to 35,000 psi. The sensitivity is shown for effective sizes between 0 and 0.0125 inches, and the data are plotted using solid diamonds connected by a solid line. The graph shows that with increasing effective size, the predicted Mr increases.

Figure 115. Graph. Effect of effective size on predicted resilient modulus.

 

 

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