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Publication Number: FHWA-HRT-12-030
Date: August 2012

 

Estimation of Key PCC, Base, Subbase, and Pavement Engineering Properties From Routine Tests and Physical Characteristics

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CHAPTER 3. LITERATURE REVIEW (2)

PREDICTION OF CHEMICALLY STABILIZED MATERIAL PROPERTIES

Several types of chemically stabilized materials are used under pavements as base courses, subbase courses, or treated subgrade. These include lean concrete, cement stabilized or treated aggregate, soil cement, lime-cement flyash, and lime-stabilized materials. Typically, the compressive strength of these materials is used for construction QA and modulus and flexural strength for pavement design. However, compressive strength testing is more common than resilient/elastic modulus testing and flexural strength testing. Industry groups and individual researchers have published several correlations to estimate chemically stabilized base elastic modulus and flexural strength from the compressive strength as shown in figure 29 through figure 32. The more common or feasible of these correlations are presented in figure 32. Resilient modulus (Mr) can be estimated conservatively as 20 percent of the unconfined compressive strength (qu).(95)

The LTPP database does not contain modulus test results for stabilized materials. Limited data were available for modulus tests of LCB layers, which were utilized for model development. Therefore, the discussion on stabilized materials in this report is brief.

 

E equals 57,000 times the square root of f prime subscript c.

Figure 29. Equation. E for lean concrete or cement treated aggregate.(96)

Where:

E = Modulus of elasticity, psi.

f'c = Compressive strength, psi tested in accordance with AASHTO T 22.(17)

 

E equals 1,200 times q subscript u.

Figure 30. Equation. E for soil cement.(96)

 

Where:

E = Modulus of elasticity, psi.

qu = Unconfined compressive strength, psi tested in accordance with ASTM D 1633.(97)

 

E equals 500 plus q subscript u.

Figure 31. Equation. E for lime-cement-flyash.(98)

Where:

E = Modulus of elasticity, psi.

qu = Unconfined compressive strength, psi tested in accordance with ASTM C 593.(99)

 

M subscript r equals 0.124 times q subscript u plus 9.98.

Figure 32. Equation. Mr for lime-stabilized soils.(100)

Where:

Mr = Resilient modulus, ksi.

qu = Unconfined compressive strength, psi tested in accordance with ASTM D 5102.(101)

PREDICTION OF UNBOUND MATERIAL AND SOIL RESILIENT MODULUS

Overview

Resilient modulus of unbound materials and soils is a required input in most pavement design procedures. It has a significant effect on the computed pavement responses and, hence, pavement performance. Resilient modulus can be tested directly from the laboratory, backcalculated using nondestructive test data, or obtained through the use of correlations with other material strength and index properties such as the California bearing ratio (CBR), R-value, dynamic cone penetrometer (DCP) value, soil Atterberg limit and gradation properties, AASHTO soil class, etc.

As a result of extensive research into the characterization of resilient modulus characterization conducted over the past four decades, it is now widely recognized that resilient modulus exhibits stress-state dependency, material dependency, and moisture and temperature dependency. About 54 percent of State transportation departments use resilient modulus in routine pavement design.(102) Ideally, resilient modulus should be obtained from laboratory measurements; however, standard test procedures such as AASHTO T 307 and NCHRP 1-28A require substantial time and resources and are not used in routine engineering practice, especially beyond the design phase of the project.(28,103)

Of the several approaches put forth to estimate resilient modulus in the laboratory for design purposes, the one that has gained considerable traction over time is using a universal constitutive model as proposed in NCHRP 1-28A (see figure 33).(103) The strength of this approach is that two of the resilient modulus dependencies, stress-state and material type, can be handled by this model form, which is an improvement over previously used discrete models for coarse- and fine-grained soils which require knowledge of material behavior prior to applying a function to characterize it.

The nonlinear elastic coefficients and exponents of the constitutive model are determined by using linear or nonlinear regression analyses to fit the model to laboratory generated resilient modulus test data.

 

M subscript r equals k subscript 1 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 33. Equation. Mr according to NCHRP Project 1-28A.(103)

Where:

Mr = Resilient modulus, psi.

[any value] = Bulk stress ([any value]).

[any value] = Major principal stress.

[any value] = Intermediate principal stress = [any value] for Mr test on cylindrical specimen.

[any value] = Minor principal stress/confining pressure.

[any value]oct = Octahedral shear stress.

[any value]oct =[any value].

Pa = Normalizing stress (atmospheric pressure).

k1, k2, and k3 = Regression constants (obtained by fitting Mr test data to this equation).

Resilient Modulus Prediction Models

There have been numerous attempts to estimate resilient modulus as a function of soil index or soil strength properties over time. One of the first studies of the resilient properties of soil with the objective of developing correlation equations for predicting resilient modulus from basic soil test data was conducted by Carmichael and Stuart.(104) They used the Highway Research Information Service (HRIS) database and developed two regression models, one for fine-grained soils and another for coarse-grained soils.(104) Regression models were developed for individual soil types according to the Unified Classification System (UCS). Variables used in the models for coarse-grained soils included moisture content and bulk stress. For fine-grained soils, plasticity index, confining, and deviatoric stresses were used as predictive variables.

Drumm et al. conducted a resilient modulus study of cohesive soils based on AASHTO soil classification.(105) They used deviator stress as the main model parameter. The model coefficients were derived as functions of liquid limit of soil, degree of saturation, and unconfined compressive strength.

Two resilient modulus regression models—one each for fine-grained and coarse-grained Mississippi soils—were developed by George.(106) Variables used to predict fine-grained soil resilient modulus included soil dry density, liquid limit, moisture content, and percentage passing the No. 200 sieve. Variables for the coarse-grained soil resilient modulus model included dry density, moisture content, and percentage passing the No. 200 sieve.

A potential benefit of estimating the resilient modulus from physical properties is that seasonal variations in resilient modulus can be estimated from seasonal changes in a material’s physical properties. A few of the models from phenomenological studies that relate resilient modulus to soil properties are presented in figure 34 to figure 48.

Other soil strength parameters have also been used to estimate resilient modulus. Due to its historical significance in pavement design, CBR is the most commonly used soil strength parameter for correlation. Direct correlations between resilient modulus and CBR are included in the MEPDG.(2) Simple correlation equations were developed to predict resilient modulus from standard CBR by several researchers. (See references 107–110.) All these equations are purely empirical and do not depend on soil properties and stress state. Figure 49 through figure 58 present a list of correlations between resilient modulus and other strength parameters.

One should exercise caution in using these correlations because they may not produce the required input to the MEPDG. The resilient modulus input is required to be at optimum moisture content and density of the soil. Most of these correlations do not predict this resilient modulus at optimum moisture content.

The resilient modulus also has been correlated to indices determined from other test devices. A summary of correlations with in-place test methods is presented in figure 59 to figure 67.

 

Log open parenthesis M subscript r closed parenthesis equals -0.111 times w plus 0.0217 times S plus 1.179.

Figure 34. Equation. Mr according to Jones and Witczak.(111)

Figure 34 was developed for clay type A-7-6. The deviator stress is 6 psi, the confining pressure is 2 psi, and Mr is measured in ksi.

 

M subscript r times open parenthesis ksi closed parenthesis equals 4.46 plus 0.098 times open parenthesis C closed parenthesis plus 0.12 times open parenthesis PI closed parenthesis.

Figure 35. Equation. Mr according to Thompson and LaGrow.(112)

Figure 35 was developed for compacted subgrades. The additional moisture susceptibility correction factor is 0.7 for clay, silty clay, and silty clay loam; 1.5 for clay loam; and 2.1 for loam.

 

M subscript r times open parenthesis ksi closed parenthesis equals 6.37 plus 0.034 times open parenthesis percent CLAY closed parenthesis plus 0.45 times open parenthesis PI closed parenthesis minus 0.0038 times open parenthesis percent SILT closed parenthesis minus 0.244 times open parenthesis CLASS closed parenthesis.

Figure 36. Equation. Mr according to Thompson and Robnett.(113)

Figure 36 was developed for fine-grained cohesive soils. The deviator stress is 6 psi, and there is no confinement.

 

M subscript r times open parenthesis ksi closed parenthesis equals 37.4 minus 0.45 times open parenthesis PI closed parenthesis minus 0.62 times open parenthesis w closed parenthesis minus 0.14 times open parenthesis S times 200 closed parenthesis plus 0.18 times open parenthesis sigma subscript 3 closed parenthesis minus 0.32 times open parenthesis sigma subscript d closed parenthesis plus 36.4 times open parenthesis CH closed parenthesis plus 17.1 times open parenthesis MH closed parenthesis.

Figure 37. Equation. Mr according to Carmichael and Stuart for fine subgrade soils containing clay and silts.(104)

Figure 37 was developed for fine subgrade soils containing clay and silts. It was developed using the HRIS database for individual UCS soil types.

 

M subscript r times open parenthesis ksi closed parenthesis equals 0.523 minus 0.0225 times open parenthesis percent W closed parenthesis plus 0.544 times open parenthesis log theta closed parenthesis plus 0.173 times open parenthesis SM closed parenthesis plus 0.197 times open parenthesis GR closed parenthesis.

Figure 38. Equation. Mr according to Carmichael and Stuart for coarse granular soils and aggregate bases.(104)

Figure 38 was created for coarse granular soils and aggregate bases. It was developed using the HRIS database for individual UCS soil types.

 

M subscript r times open parenthesis ksi closed parenthesis equals 11.21 plus 0.17 times open parenthesis percent CLAY closed parenthesis plus 0.20 times open parenthesis PI closed parenthesis minus 0.73 times open parenthesis w subscript opt closed parenthesis.

Figure 39. Equation. Mr according to Elliot et al. with deviator stress of 4 psi.(114)

Figure 39 was developed using cohesive subgrades.

 

M subscript r times open parenthesis ksi closed parenthesis equals 9.81 plus 0.13 times open parenthesis percent CLAY closed parenthesis plus 0.16 times open parenthesis PI closed parenthesis minus 0.60 times open parenthesis w subscript opt closed parenthesis.

Figure 40. Equation. Mr according to Elliot et al. with deviator stress of 8 psi.(114)

Figure 40 was developed using cohesive subgrades.

 

M subscript r times open parenthesis MPa closed parenthesis equals 17.29 times open parenthesis LL divided by w subscript c plus 1 times gamma subscript dr closed parenthesis raised to the power of 2.18 plus open parenthesis P subscript 200 divided by 100 closed parenthesis raised to the power of -0.609.

Figure 41. Equation. Mr according to Rahim for fine-grained soils.(115)

Figure 41 was developed using undisturbed field samples.

 

M subscript r open parenthesis MPa closed parenthesis equals 324.14 open parenthesis gamma subscript dr divided by w subscript c plus 1 closed parenthesis raised to the power of 0.8998 times open parenthesis P subscript 200 divided by log c subscript u closed parenthesis raised to the power of -0.4652.

Figure 42. Equation. Mr according to Rahim for coarse-grained soils.(115)

Figure 42 was developed using undisturbed field samples.

 

M subscript r times open parenthesis psi closed parenthesis equals 30,280 minus 359 times open parenthesis S closed parenthesis minus 325 times open parenthesis sigma subscript d closed parenthesis plus 237 times open parenthesis sigma subscript c closed parenthesis plus 86 times open parenthesis PI closed parenthesis plus 107 times open parenthesis S subscript 200 closed parenthesis.

Figure 43. Equation. Mr according to Farrar and Turner.(116)

Figure 43 was developed using 13 types of fine-grained soils.

 

Log times open parenthesis M subscript r closed parenthesis equals 46.93 plus 0.018 times open parenthesis sigma subscript d closed parenthesis plus 0.033 times open parenthesis delta times gamma subscript d closed parenthesis minus 0.114 times open parenthesis LI closed parenthesis plus 0.468 times open parenthesis S closed parenthesis plus 0.0085 times open parenthesis CLASS closed parenthesis raised to the power of 2 minus 0.0033 times open parenthesis delta times w closed parenthesis raised to the power of 2 minus 0.0012 times open parenthesis sigma subscript c closed parenthesis raised to the power of 2 plus 0.0001 times open parenthesis PL closed parenthesis raised to the power of 2 minus 0.0278 times open parenthesis LI closed parenthesis raised to the power of 2 minus 0.0017 times open parenthesis S closed parenthesis raised to the power of 2 minus 38.44 times open parenthesis log times S closed parenthesis minus 0.2222 times open parenthesis log times sigma subscript d closed parenthesis.

Figure Figure 44. Equation. Mr according to Hudson et al.(117)

Figure 44 was developed using cohesive subgrade soil types A4 through A7-6.

 

M subscript r equals open parenthesis 0.98 minus 0.28 times open parenthesis delta times w closed parenthesis plus 0.29 times open parenthesis delta times w closed parenthesis raised to the power of 2 closed parenthesis times M subscript Ropt.

Figure 45. Equation. Mr according to Li and Selig.(118)

Figure 45 was developed for cohesive soils.

 

M subscript r times open parenthesis kPa closed parenthesis equals -54,105 plus 57,898 times open parenthesis log psi closed parenthesis.

Figure 46. Equation. Mr according to Gupta et al.(119)

Figure 46 was developed for cohesive soils. The valid for bulk stress is 83 kPa, and the octahedral shear stress is 19.3 kPa.

 

M subscript r times open parenthesis psi closed parenthesis equals 1.518 times 10 raised to the power of 30 times open bracket f times open parenthesis S closed parenthesis closed bracket raised to the power of -13.85 times open bracket f times open parenthesis sigma closed parenthesis closed bracket raised to the power 
of -0.272.

Figure 47. Equation. Mr according to Berg et al.(120)

Figure 47 was developed using fine-grained and coarse-grained soils from the Minnesota Road Research Project test site; applicable only to cohesive soils.

 

M subscript r equals F subscript 0 times F subscript 1 times F subscript 2 times F subscript 3 times F subscript 4 times F subscript 5 times F subscript 6.

Figure 48. Equation. Mr according to Pezo and Hudson.(121)

Figure 48 was developed for Texas subgrade data. It is valid for silty to clayey subgrades, and correction factors are provided.

The definitions of the terms used in figure 34 through figure 48 are as follows:

%CLAY = C = Clay content in percent.

%SILT = Silt content in percent.

%W = w = wc = Moisture content.

[any value]d = Deviation from the Standard Proctor maximum dry density in pounds per cubic foot.

[any value]w = Deviation from the optimum water content in percent.

[any value]dr = Ratio of dry density to maximum dry density.

[any value] = [any value] = Confining stress.

[any value] = Deviator stress.

[any value] = Bulk stress.

[any value] = Soil suction.

CH = 1 for CH type (clay and high plasticity) soil; 0 otherwise.

CLASS = Soil classification (e.g., soil type A7-6 use 7.6 in expression).

cu = Uniformity coefficient.

f(S) = Normalized saturation by a unit saturation of 1 percent.

f([any value]) = Normalized octahedral shear stress by a unit stress of 1 psi.

F0 = 9.8 ksi.

F1 = Moisture content correction factor.

F2 = Relative compaction correction factor.

F3 = Soil plasticity correction factor.

F4 = Age correction factor.

F5 = Confining pressure correction factor.

F6 = Deviator stress correction factor.

GR = 1 for gravelly soils (silty gravel, well-graded gravel, clay gravel, and poorly graded gravel); 0 otherwise.

LI = Liquidity index in percent.

LL = Liquid limit in percent.

MH = 1 for elastic silt type soil; 0 otherwise.

MRopt = Resilient modulus at optimum water content.

PI = Plasticity index.

PL = Plastic limit.

S = Degree of saturation in percent.

SM = 1 for silty sand type soil; 0 otherwise.

S200 = S200 = P200 = Percent material passing the No. 200 sieve.

wopt = Optimum water content.

 

 

M subscript r times open parenthesis psi closed parenthesis equals 1,500 times open parenthesis CBR closed parenthesis.

Figure 49. Equation. Correlation between Mr and CBR according to Heukelom and Klomp.(107)

Figure 49 shows reasonable estimates for fine-grained soils with CBR less than or equal to 10.

 

M subscript r times open parenthesis psi closed parenthesis equals 2,554 times open parenthesis CBR closed parenthesis raised to the power of 0.64.

Figure 50. Equation. Correlation between Mr and CBR according to Powell et al.(122)

Figure 50 is currently included in the MEPDG.

 

M subscript r equals 3,116 times open parenthesis CBR closed parenthesis raised to the power of 0.478.

Figure 51. Equation. Correlation between Mr and CBR according to George.(106)

Figure 51 was developed for cohesionless soils and is recommended for medium clay sands.

 

M subscript r times open parenthesis MPa closed parenthesis equals 10.3 times open parenthesis CBR closed parenthesis.

Figure 52. Equation. Correlation between Mr and CBR according to the Asphalt Institute.(123)

 

M subscript r times open parenthesis psi closed parenthesis equals A plus B times open parenthesis R closed parenthesis.

Figure 53. Equation. Correlation between Mr and R-value according to AASHTO.(12)

Where:

A = 772 to 1,155; recommended 1,000. B = 369 to 555; recommended 555.

 

M subscript r times open parenthesis ksi closed parenthesis equals 1.6 plus 0.038 times open parenthesis R closed parenthesis.

Figure 54. Equation. Correlation between Mr and R-value according to Buu.(124)

Figure 54 was developed for fine-grained soils. The deviator stress is 6 psi, the confinement stress is 2 psi, and the valid for R-value is greater than 20.

 

M subscript r times open parenthesis psi closed parenthesis equals 3,500 plus 125 times open parenthesis R closed parenthesis.

Figure 55. Equation. Correlation between Mr and R-value according to Yeh and Su.(110)

 

M subscript r times open parenthesis psi closed parenthesis equals 1,155 plus 555 times open parenthesis R closed parenthesis.

Figure 56. Equation. Correlation between Mr and R-value according to MEPDG.(2)

Figure 56 is currently included in the MEPDG. The R-value is between 2.3 and 11.

 

M subscript r times open parenthesis ksi closed parenthesis equals 0.86 plus 0.31 times open parenthesis q subscript u closed parenthesis.

Figure 57. Equation. Correlation between Mr and unconfined compressive strength according to Thompson and Robnett.(113)

Figure 57 was developed assuming a bilinear Mr versus deviator stress behavior.

 

M subscript r times open parenthesis KPa closed parenthesis equals a times open parenthesis S subscript u,1 percent closed parenthesis.

Figure 58. Equation. Correlation between Mr and shear strength according to Lee et al.(125)

Figure 58 was developed for clayey subgrade soils and is valid for cohesive soils only where:

 a = Level of deviator stress. Su,1% = Undrained shear strength at 1 percent axial strain.

 

M subscript r times open parenthesis psi closed parenthesis equals 7,013 minus 2,040.8 times open parenthesis natural log times DCPI closed parenthesis.

Figure 59. Equation. Correlation between Mr and DCP test method according to Hassan.(126)

Where DCPI is the DCP index in inches per blow.

 

M subscript r times open parenthesis ksi closed parenthesis equals 338 times open parenthesis DCPI closed parenthesis raised to the power of -0.39.

Figure 60. Equation. Correlation between Mr and DCP test method according to Chen et al.(127)

Figure 60 was developed for subgrade materials using backcalculated FWD modulus values. DCPI is in millimeters per blow.

 

M subscript r times open parenthesis MPa closed parenthesis equals 235.3 times open parenthesis DCPI closed parenthesis raised to the power of 
-0.48.

Figure 61. Equation. Correlation between Mr and DCP test method for coarse-grained sandy soils according to George and Uddin.(128)

For figure 61, DCPI is in millimeters per blow.

 

M subscript r times open parenthesis MPa closed parenthesis equals 532.1 times open parenthesis DCPI closed parenthesis raised to the power of -0.49.

Figure 62. Equation. Correlation between Mr and DCP test method for fine-grained clays according to George and Uddin.(128)

For figure 62, DCPI is in millimeters per blow.

 

M subscript r times open parenthesis ksi closed parenthesis equals 78.05 times open parenthesis DCPI closed parenthesis raised to the power of -0.67.

Figure 63. Equation. Correlation between Mr and DCP test method according to Chen et al.(129)

Figure 63 was developed for both base and subgrade soils materials. DCPI is in millimeters per blow.

 

M subscript r times open parenthesis MPa closed parenthesis equals 151.8 divided by open parenthesis DCPI closed parenthesis raised to the power of 1.096.

Figure 64. Equation. Correlation between Mr and DCP test method, DCPI only according to Mohammad et al.(130)

Figure 64 was developed for soil types A-4, A-6, A-7-5, and A-7-6. A field DCP test and laboratory test were correlated through statistical analysis. DCPI is in millimeters per blow.

Where:

[any value]d = Dry unit weight in kN/m3. w = Water content in percent.

 

M subscript r times open parenthesis MPa closed parenthesis equals 165.5 times open parenthesis 1 divided by DCPI raised to the power of 1.147 closed parenthesis plus 0.0966 times open parenthesis gamma subscript d divided by w closed parenthesis.

Figure 65. Equation. Correlation between Mr and DCP test method, DCPI, and soil properties according to Mohammad et al.(130)

Figure 65 was developed for soil types A-4, A-6, A-7-5, and A-7-6. Field DCP test and laboratory test were correlated through statistical analysis. DCPI is in millimeters per blow.

Where:

[any value]d = Dry unit weight in kN/m3. w = Water content in percent.

 

M subscript r divided by sigma subscript c raised to the power of 0.55 equals 1 divided by sigma subscript v times open parenthesis 31.8 times open parenthesis q subscript c closed parenthesis plus 74.8 times open parenthesis f subscript s divided by w closed parenthesis, closed parenthesis plus 4.08 times open parenthesis gamma subscript d divided by gamma subscript w closed parenthesis.

Figure 66. Equation. Correlation between Mr and CPT test method, in situ conditions according to Mohammad et al.(131)

Figure 66 was developed for silty clay and heavy clayey cohesive soils.

Where:

qc = Tip resistance (MPa). fs = Frictional resistance (MPa). ut = Total pore pressure.

 

M subscript r divided by sigma subscript 3 raised to the power of 0.55 equals 1 divided by sigma subscript 1 times open parenthesis 47.0 times open parenthesis q subscript c closed parenthesis plus 170.4 times open parenthesis f subscript s divided by w closed parenthesis, closed parenthesis plus 1.70 times open parenthesis gamma subscript d divided by gamma subscript w closed parenthesis.

Figure 67. Equation. Correlation between Mr and CPT test method traffic loading conditions according to Mohammad et al.(131)

Figure 67 was developed for silty clay and heavy clayey cohesive soils.

Where:

[any value] = [any value] = Confining stress (kPa). [any value]v = Vertical stress (kPa). w = Water content. [any value]d = Dry unit weight (kN/m3). [any value]w = Unit weight of water (kN/m3).

Recently, the resilient modulus has been predicted using a two-step approach. First, models to predict parameters k1, k2, and k3 of the constitutive equation are developed. Next, the constitutive equation is used to estimate the resilient modulus. Von Quintus and Killingsworth, Dai and Zollars, and Santha developed prediction equations by regressing the coefficients of selected constitutive equations and relating them to soil physical properties.(132–134) It was observed that the most influential parameters are moisture content, liquid limit, plasticity index, and percent passing the No. 200 sieve.

One of the most comprehensive reviews of the resilient modulus test data measured on pavement materials and soils was made in the LTPP program.(135) A total of 2,014 resilient modulus laboratory tests that passed all the QC checks of the LTPP database (i.e., level E data status, 2000 data release) were used in this review. The study verified that the response characteristics correlate to a form of the constitutive model presented in figure 33. The study correlated resilient modulus data to the physical properties of the materials and found that the physical properties that influenced the resilient modulus varied between the different materials and soils. No one physical property was highly correlated to the modulus. For example, the liquid limit, plasticity index, and the amount of material passing the smaller sieve sizes were found to be important for the lower strength unbound aggregate base/subbase materials, while the moisture content and density were important as related to higher strength materials. Furthermore, the amount of material passing the larger sieve sizes was important for the unbound aggregate base/subbase materials with larger MASs. It also was found that percent clay and test specimen moisture content or density were important for all soil groups, while percent silt was important for all soil groups except gravel. It was also found that to improve correlations, the sample locations (depth) had to be matched while comparing test results for the same test section or site. Based on the quality of correlations developed in this study, it was recommended that the results need to be verified and confirmed after all resilient modulus tests have been completed. Future model refinement was suggested with a more comprehensive dataset.

Appendix CC in the MEPDG presents a detailed discussion of moisture and temperature regime influences on resilient modulus.(2) The values of the resilient moduli at any location and time within a given pavement structure are calculated in the MEPDG as a function of the soil moisture and freeze-thaw influences. The impact of temporal variations in moisture and temperature on resilient modulus are considered through the composite environmental adjustment factor, Fenv. The Enhanced Integrated Climatic model (EICM) is integral to the MEPDG software (the publication by Larson and Dempsey deals with all environmental factors and provides soil moisture, suction, and temperature as a function of time, at any location in the unbound layers from which Fenv can be determined).(136) This moisture prediction capability of the EICM was improved by Zapata and Houston.(137) The resilient modulus at any time or position within the pavement structure is then determined by multiplying the value at optimum with Fenv ( i.e.,[any value]).

Fenv is an adjustment factor, and Mropt is the resilient modulus at optimum conditions (maximum dry density and optimum moisture content) at any state of stress. Variations of resilient modulus with stress and variations of modulus with environmental factors (i.e., moisture, density, and freeze-thaw conditions) are assumed to be independent.

Using the latest testing protocol to measure resilient modulus, AASHTO T 307, Titi et al. performed a comprehensive investigation to estimate the resilient modulus of various Wisconsin subgrade soils and develop basic models to estimate the resilient modulus from soil properties.(28,138) Also, a laboratory testing program was conducted on common subgrade soils to characterize their physical properties. The resilient modulus constitutive equation was used to determine model factors k1, k2, and k3. Titi et al. compared the predictive capability of these models with those developed by Yau and Von Quintus using LTPP data and concluded that the two sets of models did not agree.(138,135) Differences in the test procedures, test equipment, sample preparation, and other conditions involved with development of the LTPP models and the models of this study were cited as potential factors for this lack of correlation. Key prediction models to estimate resilient modulus from soil properties are summarized below.

Regression Constant Estimates Based on Soil Properties

Prediction Model—Mississippi Materials

Based on the results of repeated triaxial load tests performed on 12 fine-grained and coarse-grained Mississippi subgrades, stress ratios were used in a bulk stress log-log model to describe their behavior. Subsequently, soil properties of the same materials were incorporated in the regression constants of the models. Then, the models were verified using a separate subset of nine different Mississippi subgrades with good correlation.(106,139) The resulting equations and constants are shown in figure 68 through figure 73.

 

M subscript R equals k subscript 1 times P subscript a times open parenthesis 1 plus sigma subscript d divided by 1 pus sigma subscript c closed parenthesis raised to the power of k subscript 2.

Figure 68. Equation. Mr of Mississippi materials for fine-grained soils.

 

k subscript 1 equals 1.12 times open parenthesis gamma subscript dr closed parenthesis raised to the power of 1.996 times open parenthesis LL divided by w subscript c closed parenthesis raised to the power of 0.639.

Figure 69. Equation. k1 of Mississippi materials for fine-grained soils.

 

k subscript 2 equals 
-0.27 times open parenthesis gamma subscript dr closed parenthesis raised to the power of 1.04 times open parenthesis w subscript cr closed parenthesis raised to the power of 1.46 times open parenthesis LL divided by pound 200 closed parenthesis raised to the power of 0.47.

Figure 70. Equation. k2 of Mississippi materials for fine-grained soils.

 

M subscript R equals k subscript 1 times P subscript a times open parenthesis 1 plus theta divided by 1 plus sigma subscript d closed parenthesis raised to the power of k subscript 2.

Figure 71. Equation. Mr of Mississippi materials for coarse-grained soils.

 

k subscript 1 equals 0.12 plus 0.90 times open parenthesis gamma subscript dr closed parenthesis minus 0.53 times w subscript cr closed parenthesis minus 0.017 times open parenthesis pound 200 closed parenthesis plus 0.314 times open parenthesis log times c subscript u closed parenthesis.

Figure 72. Equation. k1 of Mississippi materials for coarse-grained soils.

 

k subscript 2 equals 0.226 times open parenthesis gamma subscript dr times w subscript cr closed parenthesis raised to the power of 1.2385 times open parenthesis pound 200 divided by log times c subscript u closed parenthesis raised to the power of 0.124.

Figure 73. Equation. k2 of Mississippi materials for coarse-grained soils.

The definitions of the parameters used in figure 68 through figure 73 are as follows:

Pa = Atmospheric pressure.

[any value] = Deviator stress = [any value] -[any value].

[any value] = Major principal stresses.

[any value] = Minor principal stresses.

[any value] = Confining stress.

[any value] = Ratio of dry density to maximum dry density.

LL = Liquid limit in percent.

wc = Moisture content in percent.

wcr = Ratio of moisture content to optimum moisture content.

#200 = Percent material passing the No. 200 sieve.

[any value] = Bulk stress = [any value] +[any value] +[any value].

[any value] = Intermediate principal stress.

cu = Uniformity coefficient.

Prediction Model—Georgia Materials

The following equation was developed as a combination of the bulk and deviator stress models in an effort to improve the predicted response of resilient modulus test results by including both axial and shear effects.

 

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 sigma subscript d divided by P subscript a closed parenthesis raised to the power of k subscript 3.

Figure 74. Equation. Mr of Georgia materials.

Santha estimated values for the regression constants in terms of soil properties using a multiple correlation analysis.(134) The resulting constants for the analyzed Georgia granular soils are shown in figure 75 to figure 77.

 

Log times k subscript 1 equals 3.479 minus 0.07 times open parenthesis w subscript c closed parenthesis plus 0.24 times open parenthesis w subscript cratio closed parenthesis plus 3.681 times open parenthesis COMP closed parenthesis plus 0.011 times open parenthesis percent SILT closed parenthesis plus 0.006 times open parenthesis percent CLAY closed parenthesis minus 0.025 times open parenthesis SW closed parenthesis minus 0.039 times gamma subscript s plus 0.004 times open parenthesis SW raised to the power of 2 divided by percent CLAY closed parenthesis plus 0.003 times open parenthesis gamma subscript s raised to the power of 2 divided by P subscript 40 closed parenthesis. 

Figure 75. Equation. k1 of Georgia materials.

 

k subscript 2 equals 6.044 minus 0.053 times open parenthesis w subscript opt closed parenthesis minus 2.076 times open parenthesis COMP closed parenthesis plus 0.0053 times open parenthesis SATU closed parenthesis minus 0.0056 times open parenthesis percent CLAY closed parenthesis plus 0.0088 times open parenthesis SW closed parenthesis minus 0.0069 times open parenthesis SH closed parenthesis minus 0.027 times gamma subscript s plus 0.012 times open parenthesis CBR closed parenthesis plus 0.003 times open parenthesis SW raised to the power of 2 divided by percent CLAY closed parenthesis minus 0.31 times open parenthesis SW plus SH divided by percent CLAY closed parenthesis.

Figure 76. Equation. k2 of Georgia materials.

 

k subscript 3 equals 3.752 minus 0.068 times open parenthesis w subscript c closed parenthesis plus 0.309 times open parenthesis w subscript cratio closed parenthesis minus 0.006 times open parenthesis percent SILT closed parenthesis plus 0.0053 times open parenthesis percent CLAY closed parenthesis plus 0.026 times open parenthesis SH closed parenthesis minus 0.033 times gamma subscript s minus 0.0009 times open parenthesis SW raised to the power of 2 divided by percent CLAY closed parenthesis plus 0.00004 times open parenthesis SATU raised to the power of 2 divided by SH closed parenthesis minus 0.0026 times open parenthesis CBR times SH closed parenthesis.

Figure 77. Equation. k3 of Georgia materials.

Where:

wc = Moisture content.

wopt = Optimum moisture content.

wcratio = wc /wopt.

COMP = Degree of compaction.

SATU = Percent saturation.

%SILT = Silt content in percent.

%CLAY = Clay content in percent.

SW = Percent swell.

SH = Percent shrinkage.

CBR = California bearing ratio.

[any value] = Dry density.

P40 = Material passing No. 40 sieve in percent.

 

Prediction Model—Louisiana Materials

An equation similar to Uzan’s model using the octahedral normal stress instead of the bulk stress was used to study eight types of Louisiana soils.(140) Figure 79 shows the resilient modulus of Louisiana materials.

 

M subscript R equals k subscript 1 times P subscript a times open parenthesis sigma subscript oct 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 78. Equation. Mr of Louisiana materials.

Linear regression was then applied to estimate the values of the regression constants in terms of soil properties. The resulting equations are shown in figure 79 through figure 81.

 

Log times k subscript 1 equals -0.679 plus 0.0922 times open parenthesis w subscript c closed parenthesis plus 0.00559 times open parenthesis gamma subscript s closed parenthesis plus 3,054 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis plus 2.57 times open parenthesis w subscript cratio closed parenthesis plus 0.00676 times open parenthesis LL closed parenthesis plus 0.0116 times open parenthesis PL closed parenthesis plus 0.022 times open parenthesis percent SAND closed parenthesis plus 0.0182 times open parenthesis percent SILT closed parenthesis.

Figure 79. Equation. k1 of Louisiana materials.

 

Log k subscript 2 equals -0.887 plus 0.0044 times open parenthesis w subscript c closed parenthesis plus 0.0093 times open parenthesis gamma subscript s closed parenthesis plus 0.264 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis plus 0.305 times open parenthesis w subscript cratio closed parenthesis plus 0.00877 times open parenthesis LL closed parenthesis plus 0.00665 times open parenthesis PL closed parenthesis plus 0.116 times open parenthesis percent SAND closed parenthesis plus 0.00429 times open parenthesis SILT closed parenthesis.

Figure 80. Equation. k2 of Louisiana materials.

 

Log times k subscript 3 equals -0.638 plus 0.00252 times open parenthesis w subscript c closed parenthesis plus 0.00207 times open parenthesis gamma subscript s closed parenthesis plus 0.61 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis plus 0.152 times open parenthesis w subscript cratio closed parenthesis plus 0.00049 times open parenthesis LL closed parenthesis plus 0.00416 times open parenthesis PL closed parenthesis plus 0.00311 times open parenthesis percent SAND closed parenthesis plus 0.00413 times open parenthesis percent SILT closed parenthesis.

Figure 81. Equation. k3 of Louisiana materials.

Where:

[any value]oct = Octahedral normal stress [any value].

[any value]oct = Octahedral shear stress [any value].

wc = Moisture content.

wopt = Optimum moisture content.

wcratio = wc /wopt.

LL = Liquid limit.

PL = Plastic limit.

%SILT = Silt content in percent.

%SAND = Sand content in percent.

[any value] = Dry density in kN/m3.

[any value] = Optimum density in kN/m3.

 

Prediction Model—LTPP Materials

Figure 82 has gained considerable acceptance over time and is the constitutive model using the same parameters as in figure 33 and proposed in NCHRP Project 1-28A.(103)

 

M subscript r times open parenthesis psi closed parenthesis 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 plus 1 closed parenthesis raised to the power of k 
subscript 3.

Figure 82. Equation. Mr of LTPP materials.

The strength of this equation is its ability to handle two of the resilient modulus dependencies, stress-state and material type. Santha compared a log-log bulk stress model to figure 82 in modeling Georgia granular subgrade soils and concluded that the model presented in figure 82 provided a better representation of laboratory measurements of resilient modulus.(134)

One of the most comprehensive validations of figure 82 using LTPP resilient modulus test data was performed by Yau and Von Quintus.(135) The authors first identified anomalous data, verified if the response characteristics correlated to the constitutive model presented in figure 82, subsequently explored the effect of material type and sampling technique, and performed a nonlinear regression analysis to establish relationships between the k regression constants in figure 82 and various material properties. The resulting regression equations for unbound and subgrade materials are presented in figure 83 through figure 115. The definitions of the variables used in the figures are as follows:

P3/8 = Percent material passing 3/8-inch sieve.

P4 = Percent material passing No. 4 sieve.

P40 = Percent material passing No. 40 sieve.

P200 = Percent material passing No. 200 sieve.

%SILT = Silt content in percent.

%CLAY = Clay content in percent.

LL = Liquid limit in percent.

PI = Plasticity index.

Wopt = Optimum water content in percent.

Ws = Water content of the test specimen in percent

[any value] = Maximum dry unit weight of soil in kg/m3.

[any value] = Dry density of the test specimen in kg/m3.

 

Unbound Base and Subbase Materials

Figure 83 through figure 85 show equations for crushed stone (LTPP material code 303) as follows:

 

k subscript 1 equals 0.736 plus 0.0084 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0088 times open parenthesis LL closed parenthesis minus 0.0371 times open parenthesis W subscript opt closed parenthesis minus 0.00001 times gamma subscript opt.

Figure 83. Equation. k1 of LTPP materials—crushed stone.

 

k subscript 2 equals 2.2159 minus 0.0016 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0008 times open parenthesis LL closed parenthesis minus 0.038 times open parenthesis W subscript opt closed parenthesis minus 0.0006 times gamma subscript opt plus 2.4 times 10 raised to the power of -7 times open bracket gamma squared subscript opt divided by P subscript 40 closed bracket.

Figure 84. Equation. k2 of LTPP materials—crushed stone.

 

k subscript 3 equals -1.172 minus 0.0082 times open parenthesis LL closed parenthesis minus 0.0014 times open parenthesis w subscript opt closed parenthesis plus 0.0005 times gamma subscript opt.

Figure 85. Equation. k3 of LTPP materials—crushed stone.

 

Figure 86 through figure 88 show equations for crushed gravel (LTPP material code 304) as follows:

 

k subscript 1 equals -0.8282 minus 0.0065 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0114 times open parenthesis LL closed parenthesis plus 0.0004 times open parenthesis PI closed parenthesis minus 0.0187 times open parenthesis W subscript opt closed parenthesis plus 0.0036 times open parenthesis W subscript s closed parenthesis plus 0.0013 times gamma subscript s minus 2.6 times 10 raised to the power of -6 times open parenthesis gamma squared subscript opt divided by P subscript 40 closed parenthesis.

Figure 86. Equation. k1 of LTPP materials—crushed gravel.

 

k subscript 2 equals 4.995 minus 0.0057 times open parenthesis LL closed parenthesis minus 0.0075 times open parenthesis PI closed parenthesis minus 0.0470 times open parenthesis W subscript s closed parenthesis minus 0.0022 times gamma subscript opt plus 2.8 times 10 raised to the power of -6 times open bracket gamma squared subscript opt divided by P subscript 40 closed bracket.        

Figure 87. Equation. k2 of LTPP materials—crushed gravel.

 

k subscript 3 equals -3.514 plus 0.0061 times gamma subscript s.

Figure 88. Equation. k3 of LTPP materials—crushed gravel.

Figure 89 through figure 91 show equations for uncrushed gravel (LTPP material code 302) as follows:

 

k subscript 1 equals -1.8961 plus 0.0014 times open parenthesis gamma subscript s closed parenthesis minus 0.1184 times open parenthesis W subscript s divided by W subscript opt closed parenthesis.

Figure 89. Equation. k1 of LTPP materials—uncrushed gravel.

 

k subscript 2 equals 0.4960 minus 0.0074 times open parenthesis P subscript 200 closed parenthesis minus 0.0007 times gamma subscript s plus 1.6972 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis plus 0.1199 times open parenthesis W subscript s divided by W subscript opt closed parenthesis.

Figure 90. Equation. k2 of LTPP materials—uncrushed gravel.

 

k subscript 3 equals -0.5979 plus 0.0349 times open parenthesis W subscript opt closed parenthesis plus 0.0004 times gamma subscript opt minus 0.5166 times open bracket W subscript s divided by W subscript opt closed bracket.

Figure 91. Equation. k3 of LTPP materials—uncrushed gravel.

Figure 92 through figure 94 show equations for sand (LTPP material code 306) as follows:

 

k subscript 1 equals -0.2786 plus 0.0097 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0219 times open parenthesis LL closed parenthesis minus 0.0737 times open parenthesis PI closed parenthesis plus 1.8 times 10 raised to the power of -7 times open parenthesis gamma squared subscript opt divided by P subscript 40 closed parenthesis.

Figure 92. Equation. k1 of LTPP materials—sand.

 

k subscript 2 equals 1.1148 minus 0.0053 times open parenthesis P subscript 3 divided by 8 closed parenthesis minus 0.0095 times open parenthesis LL closed parenthesis plus 0.0325 times open parenthesis PI closed parenthesis plus 7.2 times 10 raised to the power of -7 times open parenthesis gamma squared subscript opt divided by P subscript 40 closed parenthesis.

Figure 93. Equation. k2 of LTPP materials—sand.

 

k subscript 3 equals -0.4508 plus 0.0029 times open parenthesis P subscript 3 dived by 8 closed parenthesis plus 0.0185 times open parenthesis LL closed parenthesis minus 0.0798 times open parenthesis PI closed parenthesis.

Figure 94. Equation. k3 of LTPP materials—sand.

Figure 95 through figure 97 show equations for coarse-grained soil-aggregate mixture (LTPP material code 308) as follows:

 

k subscript 1 equals -0.5856 plus 0.0310 times open parenthesis P subscript 3 divided by 8 closed parenthesis minus 0.0174 times open parenthesis P subscript 4 closed parenthesis plus 0.0027 times open parenthesis P subscript 200 closed parenthesis plus 0.0149 times open parenthesis PI closed parenthesis plus 1.6 times 10 raised to the power of -6 times open parenthesis gamma subscript opt closed parenthesis minus 0.0426 times open parenthesis W subscript s closed parenthesis plus 1.6456 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis plus 0.3932 times open parenthesis W subscript s divided by W subscript opt closed parenthesis minus 8.2 times 10 raised to the power of -7 times open parenthesis gamma subscript opt divided by P subscript 40 closed parenthesis.

Figure 95. Equation. k1 of LTPP materials—coarse-grained soil-aggregate mixture.

 

k subscript 2 equals 0.7833 minus 0.0060 times open parenthesis P subscript 200 closed parenthesis minus 0.0081 times open parenthesis PI closed parenthesis plus 0.0001 times open parenthesis gamma subscript opt closed parenthesis minus 0.1483 times open parenthesis W subscript s divided by W subscript opt closed parenthesis minus 2.7 times 10 raised to the power 
of -7  times open parenthesis gamma squared subscript opt divided by P subscript 40 closed parenthesis.

Figure 96. Equation. k2 of LTPP materials—coarse-grained soil-aggregate mixture.

 

k subscript 3 equals -0.1906 minus 0.0026 times open parenthesis P subscript 200 closed parenthesis plus 8.1 times 10 raised to the power of -7 times open parenthesis gamma squared subscript opt divided by P subscript 40 closed parenthesis.

Figure 97. Equation. k3 of LTPP materials—coarse-grained soil-aggregate mixture.

Figure 98 through figure 100 show equations for fine-grained soil-aggregate mixture (LTPP material code 307) as follows:

 

k subscript 1 equals -0.7668 plus 0.0051 times open parenthesis P subscript 4 closed parenthesis plus 0.0128 times open parenthesis P subscript 200 closed parenthesis plus 0.0030 times open parenthesis LL closed parenthesis minus 0.0510 times open parenthesis W subscript opt closed parenthesis plus 1.1729 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis.

Figure 98. Equation. k1 of LTPP materials—fine-grained soil-aggregate mixture.

 

. k subscript 2 equals 0.4951 minus 0.0141 times open parenthesis P subscript 4 closed parenthesis minus 0.0061 times open parenthesis P subscript 200 closed parenthesis plus 1.3941 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis.

Figure 99. Equation. k2 of LTPP materials—fine-grained soil-aggregate mixture.

 

k subscript 3 equals 0.9303 plus 0.0293 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0036 times open parenthesis LL closed parenthesis minus 3.8903 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis.

Figure 100. Equation. k3 of LTPP materials—fine-grained soil-aggregate mixture.

 

Figure 101 through figure 103 show equations for fine-grained soil (LTPP material code 309) as follows:

 

k subscript 1 equals 0.8409 plus 0.0004 times open parenthesis P subscript 40 closed parenthesis plus 0.0161 times open parenthesis PI closed parenthesis.

Figure 101. Equation. k1 of LTPP materials—fine-grained soil.

 

k subscript 2 equals 0.6668 minus 0.0007 times open parenthesis P subscript 40 closed parenthesis minus 0.0139 times open parenthesis PI closed parenthesis.

Figure 102. Equation. k2 of LTPP materials—fine-grained soil.

 

k subscript 3 equals -0.1667 minus 0.0207 times open parenthesis PI closed parenthesis.

Figure 103. Equation. k3 of LTPP materials—fine-grained soil.

Subgrade Soils

Figure 104 through figure 106 show equations for coarse-grained gravel soils as follows:

 

k subscript 1equals 1.3429 minus 0.0051 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0124 times open parenthesis percent CLAY closed parenthesis plus 0.0053 times open parenthesis LL closed parenthesis minus 0.0231 times open parenthesis W subscript s closed parenthesis.

Figure 104. Equation. k1 of LTPP materials—coarse-grained gravel soils.

 

k subscript 2 equals 0.3311 plus 0.0010 times open parenthesis P subscript 3 divided by 8 closed parenthesis minus 0.0019 times open parenthesis percent CLAY closed parenthesis minus 0.0050 times open parenthesis LL closed parenthesis minus 0.0072 times open parenthesis PI closed parenthesis plus 0.0093 times open parenthesis W subscript s closed parenthesis.

Figure 105. Equation. k2 of LTPP materials—coarse-grained gravel soils.

 

k subscript 3 equals 1.5167 minus 0.0302 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0435 times open parenthesis percent CLAY closed parenthesis plus 0.0626 times open parenthesis LL closed parenthesis plus 0.0377 times open parenthesis PI closed parenthesis minus 0.2353 times open parenthesis W subscript s closed parenthesis.

Figure 106. Equation. k3 of LTPP materials—coarse-grained gravel soils.

Figure 107 through figure 109 show equations for coarse-grained sand soils as follows:

k subscript 1 equals  3.2868 minus 0.0412 times open parenthesis P subscript 3 divided by 8 closed parenthesis plus 0.0267 times open parenthesis P subscript 4 closed parenthesis plus 0.0137 times open parenthesis percent CLAY closed parenthesis plus 0.0083 times open parenthesis LL closed parenthesis minus 0.0379 times open parenthesis W subscript opt closed parenthesis minus 0.0004 times open parenthesis gamma subscript s closed parenthesis.

Figure 107. Equation. k1 of LTPP materials—coarse-grained sand soils.

 

k subscript 2 equals 0.5670 plus 0.0045 times open parenthesis P subscript 3 divided by 8 closed parenthesis minus 2.98 times 10 raised to the power of -5 times open parenthesis P subscript 4 closed parenthesis minus 0.0043 times open parenthesis percent SILT closed parenthesis minus 0.0102 times open parenthesis percent CLAY closed parenthesis minus 0.0041 times open parenthesis LL closed parenthesis plus 0.0014 times open parenthesis W subscript opt closed parenthesis minus 3.41 times 10 raised to the power of -5 times open parenthesis gamma subscript s closed parenthesis minus 0.4582 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis plus 0.1779 times open parenthesis W subscript s divided by W subscript opt closed parenthesis.

Figure 108. Equation. k2 of LTPP materials—coarse-grained sand soils.

 

k subscript 3 equals -3.5677 plus 0.1142 times open parenthesis P subscript 3 divided by 8 closed parenthesis minus 0.0839 times open parenthesis P subscript 4 closed parenthesis minus 0.1249 times open parenthesis P subscript 200 closed parenthesis plus 0.1030 times open parenthesis percent SILT closed parenthesis plus 0.1191 times open parenthesis percent CLAY closed parenthesis minus 0.0069 times open parenthesis LL closed parenthesis minus 0.0103 times open parenthesis W subscript opt closed parenthesis minus 0.0017 times open parenthesis gamma subscript s closed parenthesis plus 4.3177 times open parenthesis gamma subscript s divided by gamma subscript opt closed parenthesis minus 1.1095 times open parenthesis W subscript s divided by W subscript opt closed parenthesis.

Figure 109. Equation. k3 of LTPP materials—coarse-grained sand soils.

 

Figure 110 through figure 112 show equations for fine-grained silt soils as follows:

 

k subscript 1 equals 1.0480 plus 0.0177 times open parenthesis percent CLAY closed parenthesis plus 0.0279 times open parenthesis PI closed parenthesis minus 0.370 times W subscript S.

Figure 110. Equation. k1 of LTPP materials—fine-grained silt soils.

 

k subscript 2 equals 0.5097 minus 0.0286 times open parenthesis PI closed parenthesis.

Figure 111. Equation. k2 of LTPP materials—fine-grained silt soils.

 

k subscript 3 equals 
-0.2218 plus 0.0047 times open parenthesis percent SILT closed parenthesis plus 0.0849 times open parenthesis PI closed parenthesis minus 0.1399 times open parenthesis W subscript S closed parenthesis.

Figure 112. Equation. k3 of LTPP materials—fine-grained silt soils.

Figure 113 through figure 115 show equations for fine-grained clay soils as follows:

 

k subscript 1 equals 1.3577 plus 0.0106 times open parenthesis percent CLAY closed parenthesis minus 0.0437 times open parenthesis W subscript S closed parenthesis.

Figure 113. Equation. k1 of LTPP materials—fine-grained clay soils.

 

k subscript 2 equals 0.5193 minus 0.0073 times open parenthesis P subscript 4 closed parenthesis plus 0.0095 times open parenthesis P subscript 40 closed parenthesis minus 0.0027 times open parenthesis P subscript 200 closed parenthesis minus 0.0030 times open parenthesis LL closed parenthesis minus 0.0049 times open parenthesis W subscript opt closed parenthesis.

Figure 114. Equation. k2 of LTPP materials—fine-grained clay soils.

 

k subscript 3 equals 1.4258 minus 0.0288 times open parenthesis P subscript 4 closed parenthesis plus 0.0303 times open parenthesis P subscript 40 closed parenthesis plus 0.0251 times open parenthesis percent SILT closed parenthesis plus 0.0535 times open parenthesis LL closed parenthesis minus 0.0672 times open parenthesis W subscript opt closed parenthesis minus 0.0026 times gamma subscript opt plus 0.0025 times open parenthesis gamma subscript s closed parenthesis minus 0.6055 times open parenthesis W subscript s divided by W subscript opt closed parenthesis.

Figure 115. Equation. k3 of LTPP materials—fine-grained clay soils.

Prediction Model—Wisconsin Materials

Titi et al. evaluated the model presented in figure 82 using repeated triaxial test results performed on various Wisconsin subgrade soils and developed correlations between the model’s regression constants and soil properties.(138) Good results were achieved when the statistical analysis was performed for fine-grained and coarse-grained soils separately. Researchers also used the LTPP data to compare their correlations to those developed by Yau and Von Quintus.(135) It was discovered that the two sets of models did not agree. Differences in the test procedures, test equipment, sample preparation, and other conditions involved with development of the LTPP models versus the ones used in the study were cited as potential factors for this lack of agreement. The equations developed by Titi et al. are provided in figure 116 through figure 124.(138) The definitions of the terms used in the figures are as follows:

P4 = Percent material passing No. 4 sieve.

P40 = Percent material passing No. 40 sieve.

P200 = Percent material passing No. 200 sieve.

%SAND = Sand content in percent.

%SILT = Silt content in percent.

%CLAY = Clay content in percent.

LL = Liquid limit.

PL = Plastic limit.

PI = Plasticity index.

LI = Liquidity index.

w = Water content.

wopt = Optimum water content.

[any value] = Dry unit weight.

[any value] = Maximum dry unit weight.

Figure 116 through figure 118 show equations for fine-grained soils as follows:

 

k subscript 1 equals 404.166 plus 42.933 times open parenthesis PI closed parenthesis plus 52.260 times open parenthesis gamma subscript d closed parenthesis minus 987.353 times open parenthesis w divided by w subscript opt closed parenthesis.

Figure 116. Equation. k1 of Wisconsin materials—fine-grained soils.

 

k subscript 2 equals 0.25113 minus 0.0292 times open parenthesis PI closed parenthesis plus 0.5573 times open parenthesis w divided by w subscript opt closed parenthesis times open parenthesis gamma subscript d divided by gamma subscript max closed parenthesis.

Figure 117. Equation. k2 of Wisconsin materials—fine-grained soils.

 

k subscript 3 equals 
-0.20772 plus 0.23088 times open parenthesis PI closed parenthesis plus 0.00367 times open parenthesis gamma subscript d closed parenthesis minus 5.4238 times open parenthesis w divided by w subscript opt closed parenthesis.

Figure 118. Equation. k3 of Wisconsin materials—fine-grained soils.

Figure 119 through figure 121 show equations for non-plastic coarse-grained soils as follows:

 

k subscript 1 equals 809.547 plus 10.568 times open parenthesis P subscript 4 closed parenthesis minus 6.112 times open parenthesis P subscript 40 closed parenthesis minus 578.337 times open parenthesis w divided by w subscript opt closed parenthesis times open parenthesis gamma subscript d divided by gamma subscript dmax closed parenthesis.

Figure 119. Equation. k1 of Wisconsin materials—non-plastic coarse-grained soils.

 

k subscript 2 equals 0.5661 plus 0.006711 times open parenthesis P subscript 40 closed parenthesis minus 0.02423 times open parenthesis P subscript 200 closed parenthesis plus 0.05849 times open parenthesis w minus w subscript opt closed parenthesis plus 0.001242 times open parenthesis w subscript opt closed parenthesis times open parenthesis gamma subscript dmax closed parenthesis.

Figure 120. Equation. k2 of Wisconsin materials—non-plastic coarse-grained soils.

 

k subscript 3 equals -0.5079 minus 0.041411 times open parenthesis P subscript 40 closed parenthesis plus 0.14820 times open parenthesis P subscript 200 closed parenthesis minus 0.1726 times open parenthesis w minus w subscript opt closed parenthesis minus 0.01214 times open parenthesis w subscript opt closed parenthesis times open parenthesis gamma subscript dmax closed parenthesis.

Figure 121. Equation. k3 of Wisconsin materials—non-plastic coarse-grained soils.

Figure 122 through figure 124 show equations for plastic coarse-grained soils as follows:

 

k subscript 1 equals 8,642.873 plus 132.643 times open parenthesis P subscript 200 closed parenthesis minus 428.067 times open parenthesis percent SILT closed parenthesis minus 254.685 times open parenthesis PI closed parenthesis plus 197.230 times open parenthesis gamma subscript d closed parenthesis minus 381.400 times open parenthesis w divided by w subscript opt closed parenthesis.

Figure 122. Equation. k1 of Wisconsin materials—plastic coarse-grained soils.

 

k subscript 2 equals 2.3250 minus 0.00853 times open parenthesis P subscript 200 closed parenthesis plus 0.02579 times open parenthesis LL closed parenthesis minus 0.06224 times open parenthesis PI closed parenthesis minus 1.73380 times open parenthesis gamma subscript d divided by gamma subscript dmax closed parenthesis plus 0.20911 times open parenthesis w divided by w subscript opt closed parenthesis.

Figure 123. Equation. k2 of Wisconsin materials—plastic coarse-grained soils.

 

k subscript 3 equals -32.5449 plus 0.7691 times open parenthesis P subscript 200 closed parenthesis minus 1.1370 times open parenthesis percent SILT closed parenthesis plus 31.5542 times open parenthesis gamma subscript d divided by gamma subscript dmax closed parenthesis minus 0.4128 times open parenthesis w minus w subscript opt closed parenthesis.

Figure 124. Equation. k3 of Wisconsin materials—plastic coarse-grained soils.

 


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