1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Incorporating the Effect of Moisture Variation on Resilient Modulus for Unsaturated Fine-Grained Subgrade Soils 36 Submitted to: 37 94th Transportation Research Board Annual Meeting 38 January 2015 39 Washington, D.C. 40 41 42 43 44 45 Murad Y. Abu-Farsakh, Ph.D., P.E. (Corresponding Author) Research Professor Louisiana Transportation Research Center Louisiana State University 4101 Gourrier Avenue Baton Rouge, LA 70808 E-mail: [email protected] Ayan Mehrotra Former MS Student – Louisiana Transportation Research Center Project Manager Professional Service Industries, Inc. Mandeville, LA 70471 E-mail: [email protected] Louay Mohammad Professor Louisiana Transportation Research Center Louisiana State University 4101 Gourrier Avenue Baton Rouge, LA 70808 And Kevin Gaspard Senior Pavement Engineer Louisiana Department of Transportation and Development Baton Rouge, LA 70808 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 Incorporating the Effect of Moisture Variation on Resilient Modulus for Unsaturated Fine-Grained Subgrade Soils ABSTRACT Moisture content and the state of stress, which for unsaturated soil includes the effects of matric suction, exert a significant effect on the resilient modulus (Mr) values of fine-grained subgrade soils. Due to seasonal variation, the moisture content and consequently, matric suction, vary periodically in subgrades. This study aims at investigating the relationship between Mr and matric suction through conducting repeated load triaxial testing on finegrained soils to obtain Mr values and evaluating the Soil Water Characteristic Curves (SWCC), which provides a relationship between suction and degree of saturation. The SWCC curves were evaluated utilizing a combination of two techniques: axis-translation and chilled mirror hygrometer, which allows for representation of the SWCC across the entire range of saturation. It was found that the PI has a significant impact on the matric suction-water content relationship such that the SWCC shifts to the left as the PI value of the soil decreases. The test results indicate a significant relationship between Mr and matric suction. The results were also used to analyze the relationship between Mr and moisture variation in terms of gravimetric water content and degree of saturation, which showed that the degree of saturation is superior to gravimetric water content in its ability to capture the effect of moisture variation on Mr values. A modified constitutive Mr-matric suction model is proposed with the ability to capture the effect of moisture variation on Mr while taking into account the stress state of unsaturated soils. The proposed model demonstrated better performance than the existing models in terms of reducing the number of regression constants and providing a better fit to the measured Mr data. 71 KEYWORDS: resilient modulus, matric suction, unsaturated soil, moisture variation, subgrade 72 soil. 73 INTRODUCTION 74 75 76 77 78 79 80 81 82 83 84 85 86 87 Premature pavement failure is often associated with loss of support in the subgrade layer, especially in regions with soft subgrade soils. Resilient modulus (Mr) quantifies the support provided by different pavement layers including the subgrade layer. The Mr, which is a measure of the stiffness of a material, is considered a fundamental property for pavement design. It serves as a key input parameter for different pavement layers in the Mechanistic Empirical Pavement Design Guide (1). It is up to the design engineer to select an appropriate Mr value representative of the individual pavement layers. For the subgrade layer, the selection of Mr presents a unique challenge since the subgrade lies within the Active Zone that usually experiences periodic changes in moisture conditions. Previous studies (2) have shown that the water content beneath the pavement subgrades is expected to vary accordingly due to seasonal variations. Studies have also shown that the Mr values are significantly impacted by changes in the moisture conditions of soils (3) (4). It is evident that the effect of moisture fluctuation must be taken into account when selecting an appropriate design value of Mr for subgrade soils. 88 89 90 91 92 Mr is a measure of the elastic behavior of geomaterials under cyclic loading, similar to what is experienced by a pavement layer under vehicular loading. Its value is dependent upon several factors such as stress state, density, soil type, and water content. The MEPDG adopted the so called ‘Universal’ model, displayed in equation 1, proposed by Witczak and Uzan (5), to evaluate Mr for geomaterials by taking into account the bulk and shear stresses. 1 2 = 1 ( ) 93 3 ( + 1) (1) 94 where: Pa = atmospheric pressure; θ = bulk stress = σ1 + σ2 + σ3; τoct = octahedral shear stress 95 = √2 3 (1 − 3 ) when σ2 = σ3; and k1, k2, k3 = model regression constants. 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 The selection of Mr design value must account for the effect of moisture content along with the effect of stress state. The moisture content can be a difficult variable to account for due to its tendency to fluctuate periodically. For fine-grained subgrade soils, it is essential to account for the effect of moisture variation on Mr values since a slight increase in the moisture content can have a significant negative impact on the Mr value (3) (6). The MEPDG utilizes the Enhanced Integrated Climatic Model (EICM) to account for the changes in the Mr value due to moisture fluctuation by incorporating an adjustment factor based on the variation of degree of saturation (S), and applying it to the Mr value determined by using equation 1. However, in many cases, the subgrade soil layers can be under an unsaturated state condition due to their shallow depths. In order to accurately describe the stress state behavior in general, and the resilient modulus in specific for unsaturated soils, the effect of matric suction must be incorporated in the design models. Several Mr - stress state constitutive models have been proposed in the literature that incorporate the effect of matric suction when evaluating Mr value for unsaturated subgrades (7) (8) (9) (10). Additionally, previous studies (11) (12) have also shown the impact of matric suction, and consequently moisture variation, on the shear modulus, which is a dynamic stiffness property similar to the Mr albeit at much smaller strains than the Mr, of fine-grained soils. 113 114 115 116 117 118 119 120 121 122 123 124 125 The objective of this paper is to evaluate the existing Mr constitutive models and develop/modify a model for incorporating the effect of moisture variation in estimating Mr values for unsaturated subgrade soils. A comprehensive laboratory testing program was performed to evaluate Mr values by performing Repeated Load Triaxial (RLT) tests on four selected fine-grained soil types with varying plasticity indices (PI) to represent the range of subgrade soils commonly found in the state of Louisiana. The tested soil specimens were prepared at different moisture contents. The Soil Water Characteristic Curves (SWCC) were also evaluated for the four soil types. Methods utilizing gravimetric water content, degree of saturation (S), and matric suction to evaluate changes in Mr values due to moisture fluctuation were analyzed. Based on the laboratory test results and external data, a modified Mr-matric suction constitutive relationship, which incorporates the variation in matric suction due to variation in moisture content for evaluating Mr of unsaturated subgrades soils, was proposed and validated. 126 LABORATORY TESTING PROGRAM AND RESULTS 127 Resilient Modulus (Mr) 128 129 130 131 132 133 134 135 136 137 The selected physical properties for the four soil types tested in this study are provided in Table 1. To evaluate the impact of moisture variation on Mr, the tested soil specimens were prepared at different compacted moisture contents utilizing the Standard proctor effort. The moisture contents of the tested specimens varied from OMC -6% to OMC+6% for A-7 soils and from OMC -3% to OMC+3% for the other three soil types, where OMC represents the optimum moisture content. Triplicate Mr specimens were prepared and tested at each target moisture content in accordance with the AASHTO T-307-99 procedure. This implies testing the subgrade Mr specimens at three confining pressures with five different deviatoric stresses at each confining pressure, which leads to Mr values being evaluated at 15 stress states per test. RLT tests were performed using the Material Testing System MTS 810 device with a 2 138 139 140 141 142 143 144 145 146 147 closed loop servo hydraulic loading system. Figure 1 displays the effect of compacted moisture content on Mr values in terms of normalized water content (w/wopt) and normalized Mr value (Mr/Mropt), where w is the compacted moisture content, wopt is the OMC, and Mropt is the Mr at OMC. The decrease in Mr values with the increase in moisture content can be attributed to weakening of the soil fabric as the moisture content increases. The Mr values in Figure 1 and generally throughout this paper are presented at a bulk stress of 155 kPa and an octahedral shear stress of 13 kPa, which is the recommended stress state for highway subgrade soils per Strategic Highway Research Program Protocol (SHRP) P-46 (3). TABLE 1 Selected Physical Properties and Classifications for the Soil Types Tested Soil Type P-7 P-17 P-26 P-53 148 149 150 151 152 153 154 155 156 157 158 159 160 161 % Plasticity Passing Index No. 200 (PI) Sieve 7 68.9 17 43.8 26 95.4 53 95.7 % Silt % Clay 52 63 61 13 13 18 35 84 MDD* OMC* AASHTO USCS (N/m3) (%) 17 17.3 15.8 12.3 17 16 22 35 A-4 A-6 A-7-6 A-7-6 ML CL CL CH *Maximum Dry Density (MDD) and Optimum Moisture Content (OMC) Based on Standard Proctor (ASTM D698) FIGURE 1 Normalized Mr Values (Mr/Mropt) versus Normalized Water Content (w/wopt) While there is a decrease in Mr value with increasing moisture content, this relationship can also be considered inversely, i.e. an increase in Mr value as the moisture content decreases. It can be explained that the soil becomes stiffer, since Mr is analogous to stiffness, as the water content decreases. This increase in stiffness can best be appreciated by observing the effect of deviatoric stress on Mr for tests conducted on the dry side of optimum versus tests conducted on the wet side of optimum, especially for the A-7 soils. Figures 2(a) through 2(d) display the results of Mr tests performed at OMC - 6% and OMC +3% for soil P-26 and OMC - 6% and OMC + 6% for soil P-53, respectively. From these figures, it is evident that 3 162 163 164 165 166 167 168 169 170 171 172 173 174 175 Mr decreases sharply with increasing cyclic stress when the soils are on the wet side of optimum, indicating an overall decrease in the stiffness of the soil specimen. However, the decrease in Mr with increasing cyclic stress for specimens on the dry side of optimum is not as pronounced as on the wet side of optimum. As will be discussed later in this paper, this decrease in stiffness on the wet side of optimum can be attributed to a decrease in matric suction. A-7 soils tend to develop significant magnitudes of matric suction on the dry side of optimum, therefore the decrease is matric suction and consequently decrease in stiffness is drastic as the soils enter the wet side of optimum moisture content range. The results presented in Figures 1 and 2 represent Mr specimens with varying compaction moisture contents; therefore, the decrease in stiffness (i.e., Mr) could be attributed to both, a decrease in the matric suction and a change in the soil structure since soil structure is affected by the compaction moisture content, especially as moisture content increases from dry to wet of optimum. 176 (a) (b) 177 178 179 180 181 (c) (d) FIGURE 2 Mr Values at Different Confining Pressures versus Cyclic Stress; a) P-26 at OMC -6%; b) P-26 at OMC +3%; c) P-53 at OMC -6%; d) P-53 at OMC +6% 182 183 The SWCC describes the relationship between suction and water content for a given soil type. The total suction consists of two components, osmotic suction and matric suction. The Matric Suction 4 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 matric suction defines the decrease in the thermodynamic potential of soil pore-water due to capillarity and short-range adsorption (13), while the osmotic suction is attributed to the presence of dissolved salts in the pore water. The SWCC curves, for the four soil types in this paper, were established by utilizing the axis-translation and chilled-mirror hygrometer techniques. The axis-translation technique involves maintaining pore-water pressure (uw) at a constant value (i.e., atmospheric pressure) while applying positive air-pressure (ua), thus inducing a matric suction (ua – uw). The SWC-150, Fredlund device, manufactured by GCTS Testing Systems Inc., was utilized to apply the axis-translation technique in this study. The axis-translation technique is limited to applying a maximum matric suction of 1,500 kPa due to the limitations of the high air-entry value (HAE) ceramic disk. To generate an accurate SWCC, data points are needed near the residual saturation conditions (14), which for finegrained soils occur well above the 1,500 kPa matric suction. Therefore, the chilled-mirror hygrometer technique was utilized to measure the data points above the 1,500 kPa matric suction. The chilled-mirror hygrometer technique indirectly measures the total suction through measuring the relative humidity (RH). The relationship between the total suction and RH, as proposed by Fredlund and Rahardjo (15), is provided by equation 2. = − 200 0 ∗ ln( ) (2) 0 201 202 203 204 where: = total suction, R = universal gas constant (J/mol K), T = absolute temperature (K), V0 = specific volume of water (m3/kg), ωv = molecular mass of water vapor (g/mol), Uv = partial pressure of pore-water vapor (kPa), and Uv0 = saturation pressure of water (kPa). In equation (2), represents Relative Humidity (RH). 205 206 207 208 209 210 211 212 213 214 The WP4C Dewpoint Potentiameter manufactured by Decagon Devices, Inc. was utilized to apply the chilled mirror hygrometer technique. The WP4C can measure matric suction up to 450 MPa; however, its accuracy decreases when measuring matric suction below 1 MPa. Therefore, data points below 1,500 kPa matric suction were obtained from the Fredlund device. A technique similar to that utilized by Nam (16) was utilized to create and test specimens via WP4C. Since the WP4C measures the total suction, specimens were prepared with de-aired/distilled water to minimize the effect of osmotic suction, since measuring the matric suction was the main objective of this study. A nonlinear least squares optimization technique proposed by Fredlund and Xing (17) (described in equation 3) was utilized to present the complete SWCC curves. 215 = [1 − ) 1,000,000 ln(1+ ) ln(1+ ] (3) ((+( ) )) 216 217 where: θw is the volumetric water content; θs is the saturated water content; Ψ is the matric suction; and Ψr, a, n, and m are fitting parameters. 218 219 220 221 222 223 224 225 226 227 Figure 3 provides the SWCC curves obtained for the four different soil types used in this study. From this figure, pertinent information such as air-entry value (AEV), saturation water content, and residual water content can be obtained. Generally, a shift to the left is seen in the SWCC curves as the PI value of the soil decreases. This is expected since higher PI soils generally have a larger clay fraction, which leads to a larger water holding capacity due to the adsorption and surface charge properties of clay particles (13). McQueen and Miller (18) presented the concept that different ‘regimes’ dominate the water holding for certain suction ranges in a SWCC. Sandy and silty soils tend to derive most of their water holding capacity from the ‘capillary’ regime; while for clay soils, due to their mineralogy, a significant portion of the water holding capacity may come from the ‘adsorbed film’ and ‘tightly adsorbed’ 5 228 229 230 regime, which are in the higher suction range. This helps explain the shift to the left, or narrowing of the desaturation range, displayed by the SWCC curves with respect to the decrease in PI value of the soil. 231 232 233 234 235 236 237 238 239 240 241 242 243 244 FIGURE 3 Predicted SWCC Curves for Each Soil Type with Measured Values 245 246 247 FIGURE 4 Graphical Procedure to Obtain Residual Water Content Another advantage of obtaining SWCC curves spanning the entire range of saturation, for each soil type, was having the ability to evaluate the residual condition. Residual water content is generally defined as the water content where a large increase in suction causes a relatively small decrease in water content. However, obtaining a value to represent the residual water content can be difficult since it has no clear definition. Figure 4 illustrates the graphical procedure utilized in this study to obtain a value to represent the residual water content. Table 2 presents the residual water content values obtained for the four soil types along with the air-entry values and the saturated water contents; the parameters that are important in accurately describing a SWCC curve. It should be noted from Table 2 that, in general, there is a decrease in the residual water content value with the decrease in PI value. The physical meaning of the residual water content and its significance in the Mr-matric suction relationship will be discussed later in this paper. 6 248 249 Table 2: Selected parameters obtained from the SWCC curves. Soil Saturated Water Air-Entry Value Type Content (kPa) Residual Water Content P-53 0.56 43 0.18 P-26 0.41 65 0.11 P-17 0.32 42 0.08 P-7 0.34 10 0.06 Note: Water content given is volumetric 250 251 252 ANALYSIS OF RESULTS 253 254 255 256 257 258 259 260 261 262 263 264 265 The gravimetric water content is a popular property in geotechnical engineering and commonly utilized to create correlations with other soil properties since it is widely accepted and easy to measure. The normalized Mr values obtained for the four soil types, compacted at different moisture contents, tested in this study were first correlated with the variation of gravimetric water content from optimum conditions. Figure 5 (a) displays an exponential trend relationship for the (w – wopt) versus the normalized Mr value. The correlation is considered adequate but not excellent with a coefficient of determination (R2) less than 70%. As can be seen, the gravimetric water content, by itself, is not enough to accurately capture the effect of moisture variation on Mr values. In this study, the degree of saturation parameter was realized as a viable alternative to gravimetric water content to represent the moisture conditions of the soil. The degree of saturation can be evaluated without extensive laboratory testing and provides a better description of the soil state since it takes into account both the effects of density and moisture content. 266 267 268 269 270 271 272 Utilizing Degree of Saturation to Predict Changes in Mr Due to Moisture Variation (a) (b) FIGURE 5 a) Mr/Mropt Versus (w – wopt); Mropt = Mr Value at OMC; wopt = OMC; b) Mr/Mropt versus (S - Sopt) (%) for the Four Soils Types Subjected to Mr Testing. As mentioned earlier, EICM utilizes an adjustment factor, Fu, to adjust Mr at optimum conditions due to changes in moisture content. The Fu factor is defined in equation 4 as: − log = + 1+ − ln( )+ ( − ) (4) 7 273 274 275 276 where: Fu = Mr/Mropt, the ratio of Mr at a given condition to Mr at optimum condition, Mropt;, a = minimum of log FU (-0.5934 for fine-grained); b = maximum of log FU (0.3979 for finegrained); km = regression parameter (6.1324 for fine-grained); (S-Sopt) = variation of degree of saturation (S) from optimum condition (Sopt), expressed as a decimal. 277 278 279 280 The impact of moisture variation on Mr data from this study was also evaluated in terms of changes in the degree of saturation. Figure 5 (b) displays the normalized Mr versus the variation in degree of saturation (S – Sopt). A nonlinear regression analysis yields the best fit line shown in Figure 5 (b) and the corresponding equation 5. 281 = −0.0009 2 − 0.0511 + 1 (5) 282 where: x = (S - Sopt) (%), The equation is valid for the range of −30 ≤ ≤ 10. 283 284 285 286 287 288 289 290 To assess the validity of equation 5, the measured Mr data from this study along with data from Drumm et al. (3) was utilized for comparison. Figures 6 (a) and 6 (b) compare the measured Mr values versus the predicted Mr values obtained utilizing equations 4 and 5, respectively. Both equations provide an acceptable agreement with the measured data, while equation 5 performs slightly better than equation 4 (R2 = 0.8 versus R2 = 0.75). Based on the performance of equations 4 and 5 on the measured Mr data, it can be concluded that degree of saturation can serve as a viable predictor variable for evaluating the changes in the Mr value due to moisture fluctuation. 291 (a) (b) 292 293 294 295 FIGURE 6 Measured versus Predicted Mr Values Utilizing; a) Equation 4, EICM model; b) Equation 5 296 297 298 299 300 301 302 303 304 The key difference between saturated and unsaturated soils is the role played by the porewater pressure (PWP). While PWP is usually positive, with depth, in saturated soils, negative PWP (or matric suction) exists in unsaturated soils due to the presence of air-water-soil interface. The value of PWP is important in determining the effective stresses in soils. For unsaturated soils, the effective stress can be evaluated by applying the model proposed by Bishop (19), given in equation 6, which takes into account the effect of negative PWP (i.e., matric suction). Consequently, the matric suction contributes to ‘suction stress’ in unsaturated soils (13). Suction stress represents the combined macroscopic effects of the air pressure acting on particle surfaces, the water pressure acting on wetted portions of soil particles Comparison of Data with Existing Mr – Matric Suction Constitutive Models 8 305 306 where a meniscus forms (due to difference between air pressure and water pressure), and the surface tension at the air-water-soil interface (20). ′ = ( − ) + ( − ) 307 (6) 308 309 310 where: σ’ = effective stress; σ = total stress; χ = Bishop’s parameter, representing the contribution of matric suction to effective stress; ua = pore-air pressure; and uw = pore-water pressure. 311 312 313 314 315 316 317 318 It is well-known that effective stresses control the strength, stiffness and deformation characteristics of soils. Also, the effective stress for unsaturated soils is highly dependent upon the matric suction, as evidenced in equation 6. Therefore, a sound theoretical framework for Mr, which represents a deformation characteristic of soils, should incorporate the matric suction. The matric suction varies with water content as evidenced by the SWCCs. Therefore, incorporating the matric suction in a Mr model will indirectly capture the effect of moisture variation on Mr. The Mr-matric suction models proposed by Gupta et al. (8) and Liang (9) are given in equations 7 and 8, respectively. 2 = (1 ( ) 319 320 3 ( + 1) ) + ( − ) (7) where: α and β = fittings constants. + 2 = 1 ( 321 ( − ) .55 ( + 1) 3 (8) 322 Where: = ( 323 324 325 326 327 328 329 330 The Mr-matric suction relationship displays a non-linear trend. Gutpa et al. (8) proposed that the Mr-matric suction relationship resembles a power function. If Mr values are considered independent of stress state (i.e., both bulk stress and octahedral shear stress are constant) across different moisture contents, then during the regression analysis the terms k1, k2 and k3 in equation 7 become unnecessary. Only the matric suction (ua – uw) varies if the stress state is constant, and equation 7 can be reduced to the form given in equation 9 for the regression analysis. Equation 9 represents a simple power function relationship between Mr and matric suction for the special case of constant stress state. 331 − ) ) ; (ua – uw)b = air-entry pressure; and ua – uw = matric suction. = ( − ) (9) 332 333 334 335 336 337 Figure 7 presents the plots of Mr values obtained at the SHRP P-46 stress state condition (θ = 155kPa, τoct = 13 kPa) but at different moisture contents, for the four soil types, versus the matric suction in a semi-logarithmic scale. The data points for each soil type are fitted with the best fit power regression function. The data in Figure 7 displays an excellent fit to the power function with high R2 values, which demonstrates that the Mr-matric relationship follows a non-linear trend. 338 339 340 341 342 343 344 345 346 The ability of equations 7 and 8 to predict Mr across different stress states was evaluated by performing non-linear regression analysis on the laboratory measured Mr data obtained from this study. Table 3, presented later in the text, provides the regression constants obtained for each soil type and for each model along with the coefficient of determination (R2) values. Based on R2 values, the reader can realize that equation 7 has stronger agreement with the measured data than equation 8, while also possessing the ability to capture the effect of moisture variation. However, equation 7 treats the matric suction as an additional term, which leads to an increased number of fitting parameters, as compared to equation 8 and the ‘Universal’ model. 9 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 FIGURE 7 Measured Mr Values versus Matric Suction for the Four Soil Types Tested The Liang (9) model (equation 8) has a sound approach since it incorporates the effect of matric suction within the bulk stress term. Similar to bulk stress, the matric suction has a stiffening effect on the soil specimen. However, based on R2 values that are presented in Table 3, equation 8 only provided a marginal fit to the measured Mr data, especially for soils P-7 and P-17. The definition of the χ parameter, which represents the contribution of matric suction to effective stress, utilized by Liang (9) was introduced by Khalili and Khabbaz (21). It represents a linear relationship in a log-log scale, which was obtained from the shear strength tests on unsaturated soils. The applicability of this relationship to Mr tests may be questionable since Mr tests are conducted under dynamic loading, while shear strength tests are conducted under static loading conditions. 368 Proposed Modified Mr-Matric Suction Constitutive Relationship 369 370 371 372 373 374 The results of the laboratory testing program were used to develop a model to estimate Mr that takes into account the stress state of unsaturated soils while also capturing the effect of moisture variation. Fredlund et al. (22) proposed a linear function, presented in equation 10, which incorporated the matric suction in predicting the shear strength of unsaturated soils. Equation 10 has two components, the saturated shear strength of the soil and the contribution of matric suction to shear strength. 375 = ′ + ( − ) tan ′ + ( − ) tan (10) 376 377 378 379 where: τf = shear strength of unsaturated soil; c = effective cohesion of saturated soil; ϕ = effective angle of shearing resistance for saturated soil; ϕb = angle of shearing resistance with respect to matric suction; (σn – ua) = net normal stress on the plane of failure; and (ua – uw) = matric suction on plane of failure. 380 381 382 383 384 385 However, a subsequent study by Gan et al. (23) showed that the shear strength- matric suction relationship for unsaturated soils follows a non-linear trend. The non-linearity in the Mr-matric suction relationship was noted earlier in this paper. The non-linearity in the matric suction-shear strength relationship can be attributed to the variation in the contribution of matric suction to effective stress. Vanapalli (24) argued that the contribution of matric suction to shear strength is related to the area of water menisci in contact with soil particles, 10 386 387 388 389 390 391 392 393 and as water content decreases the area of contact decreases as well. Initially, the soil is saturated and the area of water in contact with soil particles is continuous. As suction increases to values above the air-entry value, air begins to enter the soil pores, and consequently, the area of water in contact with the soil particles is reduced. This process continues till the residual saturation condition is achieved. At this point, the area of water in contact with the soil particles is discontinuous and insignificant. This concept was furthered by Lu and Likos (20) by presenting a non-linear relationship between suction stress, defined as intergranular stress in unsaturated soils, and matric suction. 394 395 396 397 398 399 400 401 402 Vanapalli et al. (25) proposed a shear strength model for unsaturated soils, presented in equation 11, which accounts for the variation in the contribution of matric suction to shear strength by relating the area of water in contact with the soil particles to the normalized water content. The normalized water content utilized by Vanapalli et al. (25) accounts for the residual water content when the area of water in contact with the soil particles is insignificant. Suction forces act at the particle contacts, and the strength of the force depends on the size of the liquid contact area between the soil particles (26). It can be inferred that the increase in matric suction at the residual stage does not significantly contribute to the increase in the shear strength, and similarly the Mr value. = [ ′ + ( − ) tan ′] + ( − )[( )(′)] 403 (11) − 404 where: = − ; Θ = normalized water content; θ = water content; θr = water content at 405 406 residual condition; θs = water content at saturated condition; and k = fitting parameter to obtain better agreement amongst measured and predicted values. 407 408 409 410 411 412 413 414 415 The model proposed in this paper, displayed in equation 12, represents a modified version of the ‘Universal’ and Liang (9) Mr constitutive models. The proposed model also has distinct similarities with the small strain shear modulus (Go) – matric suction relationship proposed by Sawangsuriya et al. (12). The Sawangsuriya et al. (12) Go-matric suction relationship utilized principles from Vanapalli (24) and Vanapalli et al. (25) that are also included in equation 12 by incorporating the impact of normalized water content on the contribution of matric suction to the Mr value. It should be noted here that evaluating the normalized water content requires data from a SWCC curve, which has been evaluated over the entire range of saturation. 416 = 1 ( (+ ) 2 ) 3 ( ) (12) 417 418 where: k = 1/n; n is obtained from the Fredlund and Xing (17) SWCC fitting model, which represents the rate of change of matric suction due to changes in water content. 419 420 421 422 Equation 12 also establishes an explicit link between the SWCC and the Mr-matric suction relationship by utilizing the exponent k, which is evaluated based on the parameter n. The parameter n is obtained from equation 13, the 3-parameter version of the Fredlund and Xing (17) model. 423 = (13) ((+( ) )) 424 425 426 427 428 The n parameter depends on the ‘slope’ of the SWCC, and it implicitly takes into account the soil type. Leong and Rahardjo (14) demonstrated the effect of the n parameter on the shape of the SWCC curve in the desaturation zone. The n parameter generally captures the rate of change of suction with respect to water content in the desaturation zone of the SWCC. As mentioned earlier, different ‘regimes,’ which are dependent on soil type, can be 11 429 430 431 used to determine the desaturation zone of a SWCC. 432 433 434 435 436 437 438 439 440 441 The proposed model was first evaluated utilizing the laboratory Mr data from this study by performing non-linear regression analysis. Table 3 presents the regression constants obtained from the regression analysis utilizing the proposed model in equation 12, in addition to equation 7 and 8. Figure 8 displays the comparison between the laboratory measured Mr values and the predicted Mr values obtained utilizing equation 12 for the four soil types. The figure clearly demonstrates strong agreement between the measured and predicted Mr values. Based on the R2 values presented in Table 3 and the results displayed in Figure 8, the reader can realize that the proposed model presented in equation 12 performs well in predicting the Mr value of unsaturated soils by incorporating the matric suction, and hence being able to account for the effect of moisture variation on the Mr value through matric suction. 442 443 444 445 446 FIGURE 8 Comparison between the Laboratory Measured Mr Values and Mr Predicted Values Obtained utilizing Equation 12 for the Four Soil Types Validation of the Proposed Model TABLE 3 Results of Regression Analysis Performed on Laboratory Data Utilizing Equations 7, 8 and 12. P-7 P-17 P-26 P-53 P-7 Equation 7 P-17 P-26 P-53 P-7 Equation 8 P-17 P-26 P-53 Equation 12 k1 52.0 71.7 150.9 225.4 345.5 123.2 165.5 347.1 205.0 58.5 169.6 368.1 k2 1.73 3.17 2.57 1.19 0.76 1.79 1.23 0.8 0.90 1.30 0.90 0.45 k3 -0.88 -7.71 -15.05 -6.06 -1.07 -2.27 -1.49 -1.58 -0.81 -1.2 -1 -1.1 α 968.8 119.4 1398.6 2472.7 β 0.46 0.86 0.36 0.24 n 0.73 1.05 0.76 0.59 k 1.37 0.95 1.32 1.69 RMSE 2 R 553.1 692.4 1090.2 484.1 881.2 1756.6 1449.1 786.6 772.2 579.4 1111.8 694.2 0.92 0.95 0.88 0.94 0.64 0.69 0.81 0.85 0.83 0.97 0.89 0.84 12 447 448 449 450 451 452 453 454 455 456 To further validate the proposed model in equation 12, additional Mr data was collected from Gupta et al. (8) and Liang (9) studies. A non-linear regression analysis was performed on the external collected Mr data, obtained at different moisture contents, to evaluate the performance of equation 12. Figures 9 (a) and 9 (b) display the comparison between the measured Mr values, from Gupta et al. (8) and Liang (9), respectively, versus the predicted Mr values, obtained by performing regression analysis utilizing equation 12. The figures clearly demonstrate excellent agreement between the measured and predicted Mr values with high R2 values. It is very clear that the proposed model in equation 12 performs very well in predicting the Mr values for unsaturated specimens prepared at different moisture contents and tested at different stress states as demonstrated in Figures 8, and 9a, and 9b. 457 458 459 460 461 (a) (b) FIGURE 9 Measured Mr versus Predicted Mr Values Obtained Utilizing Equation 12: a) Measured Mr Data from Gupta et al. (2007); b) Measured Mr Data from Liang (2008) 462 463 464 465 466 467 468 469 470 471 472 473 474 A laboratory testing program was conducted on four soil types to assess the impact of moisture variation on the Mr values of unsaturated fine-grained subgrade soils. RLT tests were conducted on laboratory specimens compacted at various moisture contents to measure the Mr values at various stress states. The SWCC curves for the four soil types were established utilizing the axis-translation and chilled mirror hygrometer techniques, which allowed for the evaluation of the SWCC curves from saturation to residual saturation conditions. The SWCC curves demonstrate that PI has a significant impact on the matric suction-water content relationship such that the SWCC shifts to the left as the PI value of the soil decreases. Analysis of the laboratory test results demonstrated that the effect of moisture variation on Mr values can be captured by applying different relationships; Mrgravimetric water content, Mr-degree of saturation, and Mr-matric suction. It was demonstrated that the degree of saturation is superior to gravimetric water content in its ability to capture the effect of moisture variation on the Mr values. 475 476 477 478 479 480 481 482 The existence of a non-linear relationship between the Mr and matric suction was emphasized in this study. The data obtained from the laboratory testing program was applied to existing Mr-matric suction constitutive models (8) (9), which showed that the existing models provided adequate fit to the laboratory measured Mr data. However, due to certain concerns in the existing models (Gupta et al. (8) model needs five regression constants while the Liang (9) model provided a marginal fit to the laboratory measured data for lower PI soils), a modified Mr-matric suction constitutive model was proposed to evaluate Mr for unsaturated soils. The proposed Mr model accounts for the non-linear contribution of matric CONCLUSION 13 483 484 485 486 487 488 489 suction to the Mr values by utilizing the normalized water content. Also, the proposed model implicitly includes the effect of soil type through incorporating the n parameter. The proposed model was validated utilizing laboratory data obtained from this study, and data collected from external sources available in literature. The proposed model demonstrated its ability to accurately capture the variation in moisture conditions and the effect of stress state on the Mr values for unsaturated soils. 490 ACKNOWLEDGEMENTS 491 492 493 494 495 496 497 498 This research is funded by the Louisiana Transportation Research Center (LTRC Project No. 12-2P) and Louisiana Department of Transportation and Development (State SIO No. 30000425). 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