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Incorporating the Effect of Moisture Variation on Resilient Modulus for Unsaturated
Fine-Grained Subgrade Soils
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Submitted to:
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94th Transportation Research Board Annual Meeting
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January 2015
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Washington, D.C.
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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
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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.
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KEYWORDS: resilient modulus, matric suction, unsaturated soil, moisture variation, subgrade
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soil.
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INTRODUCTION
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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.
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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
𝜃 𝑘2
𝑀𝑟 = 𝑘1 𝑃𝑎 (𝑃 )
93
𝑎
𝑘3
𝜏
( 𝑃𝑜𝑐𝑡 + 1)
𝑎
(1)
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where: Pa = atmospheric pressure; θ = bulk stress = σ1 + σ2 + σ3; τoct = octahedral shear stress
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=
√2
3
(𝜎1 − 𝜎3 ) when σ2 = σ3; and k1, k2, k3 = model regression constants.
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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.
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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.
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LABORATORY TESTING PROGRAM AND RESULTS
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Resilient Modulus (Mr)
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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
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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
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155
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%
Plasticity
Passing
Index
No. 200
(PI)
Sieve
7
68.9
17
43.8
26
95.4
53
95.7
%
Silt
%
Clay
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13
13
18
35
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MDD* OMC*
AASHTO USCS
(N/m3) (%)
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17.3
15.8
12.3
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16
22
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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
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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.
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(a)
(b)
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(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%
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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
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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.
𝜓𝑇 = −
200
𝑅𝑇
𝑉0 ∗𝜔𝑣
𝑈
ln(𝑈 𝑣 )
(2)
𝑣0
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where: 𝜓𝑇 = 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
𝑈
equation (2), 𝑣 represents Relative Humidity (RH).
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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.
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𝑈𝑣𝑜
𝜃𝑤 = [1 −
𝜓
)
𝜓𝑟
1,000,000
ln(1+
)
𝜓𝑟
ln(1+
]
𝜃𝑠
𝜓 𝑛
𝑎
𝑚
(3)
(𝑙𝑛(𝑒+( ) ))
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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.
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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’
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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.
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FIGURE 3 Predicted SWCC Curves for Each Soil Type with Measured Values
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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.
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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
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ANALYSIS OF RESULTS
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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.
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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:
𝑏 −𝑎
log 𝐹𝑢 = 𝑎 +
1+𝑒
−𝑏
ln(
)+𝑘𝑚 (𝑆 −𝑆𝑜𝑝𝑡 )
𝑎
(4)
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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.
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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.
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𝑀𝑟
𝑀𝑟𝑜𝑝𝑡
= −0.0009𝑥 2 − 0.0511𝑥 + 1
(5)
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where: x = (S - Sopt) (%), The equation is valid for the range of −30 ≤ 𝑥 ≤ 10.
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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.
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(a)
(b)
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FIGURE 6 Measured versus Predicted Mr Values Utilizing; a) Equation 4, EICM
model; b) Equation 5
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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
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where a meniscus forms (due to difference between air pressure and water pressure), and the
surface tension at the air-water-soil interface (20).
𝜎 ′ = (𝜎 − 𝑢𝑎 ) + 𝜒(𝑢𝑎 − 𝑢𝑤 )
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(6)
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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.
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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.
𝜃 𝑘2
𝑀𝑟 = (𝑘1 𝑃𝑎 (𝑃 )
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𝑎
𝑘3
𝜏
( 𝑃𝑜𝑐𝑡 + 1) ) + 𝛼(𝑢𝑎 − 𝑢𝑤 )𝛽
𝑎
(7)
where: α and β = fittings constants.
𝜃+𝜒𝜓𝑚 𝑘2
𝑀𝑟 = 𝑘1 𝑃𝑎 (
321
(𝑈𝑎 −𝑈𝑤 )𝑏 .55
𝜏
( 𝑃𝑜𝑐𝑡 + 1)
𝑎
𝑘3
(8)
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Where: 𝜒𝑤 = (
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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
)
𝑃𝑎
)
; (ua – uw)b = air-entry pressure; and ua – uw = matric suction.
𝑀𝑟 = 𝛼(𝑢𝑎 − 𝑢𝑤 )𝛽
(9)
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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.
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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
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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
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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
𝜏𝑓 = 𝑐 ′ + (𝜎𝑛 − 𝑢𝑎 ) tan 𝜙′ + (𝑢𝑎 − 𝑢𝑤 ) tan 𝜙 𝑏
(10)
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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.
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381
382
383
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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
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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.
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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.
𝜏 = [𝑐 ′ + (𝜎𝑛 − 𝑢𝑎 ) tan 𝛷′] + (𝑢𝑎 − 𝑢𝑤 )[(𝛩𝑘 )(𝑡𝑎𝑛𝛷′)]
403
(11)
𝜃−𝜃
404
where: 𝛩 = 𝜃 −𝜃𝑟 ; 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.
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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
𝑠
𝑟
𝑀𝑟 = 𝑘1 𝑃𝑎 (
(𝜃+𝛩𝑘 𝛹)
𝑃𝑎
𝑘2
)
𝑘3
𝜏
( 𝑃𝑜𝑐𝑡)
𝑎
(12)
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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.
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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
𝜃𝑤 =
𝜃𝑠
𝜓 𝑛
𝑎
𝑚
(13)
(𝑙𝑛(𝑒+( ) ))
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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
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431
used to determine the desaturation zone of a SWCC.
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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.
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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
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452
453
454
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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.
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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)
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466
467
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472
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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.
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477
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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
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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
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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). The authors would like to express their thanks to Mark Morvant, Zhongjie
Zhang, and Gavin Gautreau at LTRC for providing valuable help and support in this study.
The authors appreciate LTRC for allowing utilization of their Geotechnical Research
laboratory and equipment.
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