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71 of determination (R²) was 0.95, and the model is applicable to cohesive soils only. (39) MRDS 13 Gupta et al. (2007) proposed the following expression for (40) resilient moduli prediction based on soil suction (ψ) mea- surements. This equation is valid for cohesive soil types tested in their research. The R2 value of this correlation is 0.76, and the resilient modulus used in this expression is where MR is the resilient modulus (MPa), qc is the cone valid for a bulk stress of 83 kPa and an octahedral shear resistance (MPa), fs is the sleeve friction (MPa), σc or σ3 stress of 19.3 kPa. is the confining stress (kPa), σv is the vertical stress (kPa), w is the water content in decimal number format, γ is the d (38) dry unit weight (kN/m3), and γ is the unit weight of water w (kN/m3). The coefficients of determination values for both equations are 0.99 and 0.99, respectively. Direct and In Situ Test–Based Models MRDI 3 (FWD, GeoGauge, and SPA) MRDI 1 (DCP) The 1993 AASHTO design guide recommends a factor of In the direct correlations, several DCP-based relationships 0.33 to be multiplied with the FWD backcalculated moduli are summarized in Table 12. More details on the use of these to determine the design resilient moduli of the subgrades. relationships are given in earlier sections. The terms includ- As described in the earlier sections, this ratio is not unique ing EDCP, E, and MFWD values are representative of stiff- and varies considerably. This variation was attributed to ness measurements from nondestructive studies, whereas different soil types, test conditions, and backcalculation the resilient modulus (MR) is derived from the RLT method. programs that provide different moduli predictions. As a These relationships are mostly empirical in nature, and they result, no quotient factor is recommended here. One should are best applicable for the soils or soil types close to the ones use local experience to determine the resilient moduli. In from which these relationships are derived. Engineering the case of GeoGauge and SPAs, more research is needed to judgment and local experience are prudent when using these develop appropriate factors to determine the design resilient models. Please note that these models are nondimensional moduli. and are unit sensitive. MRDI 2 (CPT) Indirect Models In the case of CPT-type correlations, Mohammad et al. Several other models including those recommended by (2000) formulated the following two correlations (Equations AASHTO test procedures utilize two-, three-, or four- 39 and 40) for predicting the resilient properties of cohesive parameter correlations that account for confining and subgrades. The first expression is valid for overburden stress shearing stresses. Some of these formulations use nondi- conditions and the second expression is valid for both over- mensional forms of stresses by normalizing confining and burden stress and traffic conditions. deviatoric stresses with atmospheric pressures and others Table 12 Summary of DCP Correlations for MRDI 1 Reference Expression Units De Beer and van der Marwe (1991) Log (EDCP) = 3.05 – 1.06 × log (DCP) DCP in mm per blow Chai and Roslie (1998) E in MPa = 2224 × DCP-0.99 DCP in blows per 300 mm Hassan (1996) MR in psi = 7013 – 2040.8 ln (DCPI) DCPI in inches per blow Chen et al. (1999) MR (ksi) = 338 × DPI-0.39 DPI in mm per blow MR in MPa for Sandy Soils = 235.3 × DCPI-0.48 George and Uddin (2006) DCPI in mm per blow MR in MPa for Clays = 532.1 × DCPI-0.49 Abu-Farsakh et al. (2004) ln MFWD = 2.35 + [5.21 /ln PR] PR in mm per blow Chen et al. (2007) MR (ksi) = 78.05 × DPI-0.67 DPI in mm per blow; valid for bases
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72 use direct stress attributes where the model constants are MRI2-3 no longer treated as nondimensional entities. Model con- stants of the correlations consider the nonlinearities in the The following power model uses the deviatoric stress (σd) as subgrade moduli properties (Witzack et al. 1995). the lone stress attribute in the formulation: The following notation system will be used to identify each of the formulations. MRI2 represents an indirect two- (43) parameter resilient modulus, and MRI3 and MRI4 denote indirect three-parameter and four-parameter resilient mod- where k1 and k 2 are model constants, pa is the atmospheric ulus formulations. The following sections describe each of pressure, and σd is the deviatoric stress applied during the these formulations. Because the current practice is to use triaxial test. This formulation is normalized and hence the parameters such as k 1, k 2, k 3, and k 4, the same constant model constants are dimensionless. This model formulation parameters are used for each formulation. In cases in which does not consider confining stress effects on the test results, stresses in the formulations were not normalized, readers which is a limitation. This model is primarily used for cohe- should note that these constants and their magnitudes will sive soils. not be construed as nondimensional. The constants will have units of the stresses that have been used in the model- MRI2-4 ing analysis. The following bilinear model for resilient modulus was dis- Two-Parameter Models cussed by Thompson and Elliott (1985). MR value increases with the deviatoric stress up to a break point beyond which the MRI2-1 modulus decreases with an increase in deviatoric stress. The following formulation (Equation 44) is recommended, which Dunlap (1963) formulated the following model in which the uses deviatoric stress (σd) as the lone stress attribute: confining stress (σ3) is used as a stress attribute: (44) (41) where k1 and k 2 are model constants. This formulation is where k1, k 2, k 3, and k4 are model constants, and σd is the normalized and hence the model constants are dimension- deviatoric stress applied during the triaxial test. This for- less. This model formulation does not address the devia- mulation is not normalized and hence the model constants toric stress effects on the test results, which are considered are dimensional. This model is primarily used for cohesive important for better modeling or representation of resilient soils. behavior. MRI2-5 MRI2-2 Wolfe and Butalia (2004) developed the following Seed et al. (1967) formulated the following Equation 42 in correlation: which the bulk stress (θ) is used as a stress attribute: (45) (42) where k 1 and k 2 are model constants, pa is the atmospheric where τoct is octahedral shear stress = [(σ1 − σ2) 2 + (σ2 − pressure, σ3 is the minor principal stress, θ is the bulk σ3) 2 + (σ3 − σ1) 2]½/3; and σoct is octahedral normal stress stress = σ1 + σ2 + σ3, and σ1 and σ2 are the major and = (σ1 + σ2 + σ3)/3. This equation is valid for cohesive soils intermediate principal stresses, respectively. This formu- only. lation is normalized and hence the model constants are dimensionless. This model formulation considers devia- Three-Parameter Models toric stress effects on the test results by including their effects in the bulk stress. However, the true influence of Several other models were reported in the literature, which deviatoric stress effects is not represented in this formula- use both stresses (either confining and deviatoric stresses tion. This model is primarily used for granular soils. or bulk or octahedral stresses) that are functions of confin-
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73 ing and deviatoric stresses. Several of these models are pre- sented in the following equations. Again, the same constants (50) are used in each formulation. If a researcher or practitioner uses two of these models in their analysis, it is important to use different constant terms for each model. Otherwise, it where all of the above formulations used the following would result in confusion to the users. stresses as their attributes: The most general form of a three-parameter model is as τoct = octahedral shear stress = [(σ1 − σ2) 2 + (σ2 − σ3) 2 follows (Ooi et al. 2006): + (σ3 − σ1) 2]½/3; (46) θ = bulk stress = σ1 + σ2 + σ3; σd = deviator stress = σ1 − σ3; where f(c) is a function of confinement; g(s) is a func- tion of shear; and k1, k 2, and k 3 are constants. The effects σ1, σ2, σ3 = major, intermediate, and minor principal of confinement in these models can be expressed in terms stresses, respectively; and of the minor principal stress (σ3), bulk stress (θ), or octa- hedral stress (σoct = θ/3), while the parameter options for pa = atmospheric pressure. modeling the effects of shear include the deviatoric stress or octahedral shear stress (τoct). The three-parameter models The three models (Equations 48, 49, and 50) predict zero represented by the Equation 46 are more versatile and apply resilient modulus when a confining pressure of zero is used to all soils (Ooi et al. 2006). in those formulations. Hence, the attributes are revised in Ni et al. (2002), such that they will work for a wide range MRI3-1 of stresses. Uzan (1985) recommended the following formulation: MRI3-5 Ooi et al. (2004) recommended the following two models (47) with slight modifications to the model developed by Ni et al. (2002): MRI3-2 (51) Witczak and Uzan (1988) revised Equation 47 by replacing the deviatoric stress with octahedral shear stress: (52) (48) MRI3-6 This formulation is recommended in the 1993 AASHTO The NCHRP project 1-28 A and MEPDG recommended the design guide. following expression for resilient modulus: MRI3-3 (53) Pezo (1993) recommended the following formulation: This expression is a simplification of the five-parameter (49) model. MRI3-7 MRI3-4 Gupta et al. (2007) recommended the following expres- Ni et al. (2002) recommended the following three-parameter sion for resilient modulus for compacted and unsaturated formulation (Equation 50): subsoils: