COMPARISON OF DIFFERENT PROCEDURES TO PREDICT
UNSATURATED SOIL SHEAR STRENGTH
S.K. Vanapalli and D.G. Fredlund
1
Abstract: Several procedures have been proposed in the recent years to predict the shear
strength of an unsaturated soil. The soil-water characteristic curve has been used as a tool
either directly or indirectly in the prediction of the shear strength along with the saturated
shear strength parameters in these procedures. This paper provides comparisons between the
measured and predicted values of unsaturated shear strength using these procedures for three
soils both for limited and large suction ranges. The three soils used in the study for
comparisons have different gradation properties, percentages of clay and plasticity index, I
p
values. The advantages and limitations associated with predicting the shear strength of
unsaturated soils using the procedures is discussed in the paper.
INTRODUCTION
A theoretical framework for unsaturated soil mechanics that parallels saturated soil
mechanics is available in terms of stress state variables, namely; net normal stress, (
σ
n
- u
a
),
and matric suction, (u
a
- u
w
) where
σ
n
is the normal stress, u
a
is the pore-air pressure an u
w
is
the pore-water pressure. (Fredlund and Rahardjo, 1993). The framework is based on
experimental studies that are costly and time consuming. Several advancements have been
made in the prediction of the engineering behavior of unsaturated soils in recent years. The
soil-water characteristic curve has been found to be a useful tool in the estimation of
engineering properties for unsaturated soils. Examples are the coefficient of permeability and
the shear strength functions.
Shear strength forms an important engineering property in the design of numerous
geotechnical and geo-environmental structures such as earth dams, retaining walls,
pavements, liners, covers, etc. Several procedures have been proposed in the literature
during the past five years to predict the shear strength of an unsaturated soil.
__________________________________________________________________________
1
Department of Civil Engineering, University of Saskatchewan, SK, Canada, S7N 5A9
These procedures use the soil-water characteristic curve as a tool either directly or
indirectly along with the saturated shear strength parameters, c’ and
φ
’, to predict the shear
strength function for an unsaturated soil (Vanapalli et al. 1996, Fredlund et al. 1996, Oberg
and Sallfors 1997, Khallili and Khabbaz 1998 and Bao et al. 1998).
The philosophy used in each of the prediction procedures proposed by these
investigators is different. Comparisons have been provided between predicted and measured
values of shear strength for a limited suction range for various soils (i.e., between 0 to 500
kPa).
Escario and Juca (1989) measured the soil-water characteristic curves and the shear
strength of three soils prior to the time when any proposals had been made for the shear
strength functions. The three soils have different gradation properties, percentages of clay
and plasticity indices, I
p
. These results are used in this paper to provide comparisons between
the predicted and measured shear strength values both for a limited suction range and a large
suction range. The study presented in the paper highlights the advantages and limitations
associated with the various procedures for predicting the shear strength of unsaturated soils.
The simple procedures discussed in this paper are of value in bringing the shear strength
theories for unsaturated soils into engineering practice.
EQUATIONS FOR INTERPRETING THE SHEAR STRENGTH OF
UNSATURATED SOILS
Bishop (1959) proposed shear strength equation for unsaturated soils by extending
Terzaghi’s principle of effective stress for saturated soils. Bishop’s original equation can be
arranged as shown below.
(
)
(
)
(
)
(
)
[
]
'tan'tan' φχφστ
waan
uuuc −+−+= [1]
where:
τ
= shear strength of unsaturated soil,
c’ = effective cohesion,
φ
’ = angle of frictional resistance,
(
σ
n
- u
a
) = net normal stress,
(u
a
- u
w
) = matric suction, and
χ
= a parameter dependent on the degree of saturation
The value of
χ
was assumed to vary from 1 to 0, which represents the variation from a
fully saturated condition to a total dry condition. Several investigators found limitations with
respect to the quantification of the parameter χ both theoretically and experimentally.
Fredlund et al. (1978) have proposed a relationship to explain the shear strength of
unsaturated soils in terms two independent stress state variables as shown below:
(
)
(
)
b
waan
uuuc φφστ tan'tan' −+−+=
[2]
The shear strength contribution due to matric suction,
φ
b
, was initially assumed to be
linear based on the analysis of limited results published in the literature. Later experimental
studies performed over a large range of suction values have shown that the variation of shear
strength with respect to soil suction is non-linear (Gan et al. 1988 and Escario and Juca
1989).
Equation [1] can be applied for both the linear and non-linear variation of shear
strength with respect to suction.
Figure 1. Typical soil-water characteristic curve showing zones of desaturation.
The Relationship between the Soil-Water Characteristic Curve and the Shear Strength
of Unsaturated soils
The soil-water characteristic curve defines the relationship between the soil suction
and either the degree of saturation, S, or gravimetric water content, w, or the volumetric
water content, θ (Figure 1). The soil-water characteristic curve provides a conceptual and
interpretative tool by which the behavior of unsaturated soils can be understood. As the soil
moves from a saturated state to drier conditions, the distribution of the soil, water, and air
phases change as the stress state changes. A typical soil-water characteristic curve with
various zones of desaturation are shown in Figure 1.
The wetted area of contact between the soil particles decreases with an increase in the
soil suction. There is a relationship between the rate at which shear strength changes in
unsaturated conditions to the wetted area of water contact between the soil particles or
aggregates. In other words, a relations hip exists between the soil-water characteristic curve
and the shear strength of unsaturated soils.
Different Procedures for Predicting the Shear Strength of an Unsaturated Soil
Vanapalli et al. (1996) and Fredlund et al. (1996) have proposed a more general, non-
linear function for predicting the shear strength of an unsaturated soil using the entire soil-
water characteristic curve (i.e., 0 to 1,000,000 kPa) and the saturated shear strength
parameters as shown below:
(
)
[
]
(
)
(
)
(
)
{
}
[
]
'tan'tan' φφστ
κ
Θ−+−+=
waan
uuuc [3]
where:
κ
= fitting parameter used for obtaining a best-fit between the measured and predicted
values, and
Θ
= normalized water content, θ
w
/θ
s
.
The shear strength contribution due to suction constitutes the second part of [Eq. 3], which
is:
(
)
(
)
(
)
{
}
[
]
'tanφτ
κ
Θ−=
waus
uu [4]
Equation [3] can also be written in terms of degree of saturation, S, or gravimetric water
content, w, to predict the shear strength yielding similar results.
The entire soil-water characteristic curve data (i.e., 0 to 1,000,000 kPa) is required
along with the saturated shear strength parameters in the use of Equation [3]. A best-fit soil-
water characteristic curve can be obtained in terms of a, n, and m parameters using the
equation proposed by Fredlund and Xing (1994) which is shown below:
( )
( )
+
+
+
−=
m
n
r
r
sw
a
h
h
ψ
ψ
θψθ
1expln
1
10
1ln
1ln
1
6
[5]
where:
ψ
= soil suction,
θ
w
= volumetric water content,
θ
s
= saturated volumetric water content,
a = suction related to the inflection point on the curve,
n = soil parameter related to slope at the inflection point,
m = soil parameter related to the residual water content, and
h
r
= suction related to the volumetric residual water content,
θ
r
The shear strength contribution due to suction, tan
φ
b
is equal to tan
φ
’ up to the air-
entry value of the soil. In other words, the conventional equation for estimating the shear
strength of saturated soils can be used up to the air-entry value for unsaturated soils. The
shear strength variation with respect to suction is linear in the boundary effect zone. The
shear strength contribution due to suction, tan
φ
b
is less than tan
φ
’ in the transition zone.
Hence, the shear strength variation in this zone is non-linear. The shear strength value
gradually starts dropping at high values of suction and reaches saturated shear strength value
at 1,000,000 kPa using this equation. Equation [3] is useful to predict the shear strength of
unsaturated over the entire range of suction values of 0 to 1,000,000 kPa (i.e., from a fully
saturated condition to a total dry cond ition). Analysis of experimental results presented and
shown in the Figures 3 to 8 of this paper using Equation [3] will be referred as Procedure 1.
Vanapalli et al. (1996) proposed another equation for predicting the shear strength of
unsaturated soils without using the fitting parameter,
κ
. The equation is given below:
( ) ( )
−
−
−+−+= 'tan'tan'
φ
θθ
θθ
φστ
rs
rw
waan
uuuc
[6]
where:
θ
w
= volumetric water content,
θ
s
= saturated volumetric water content, and
θ
r
= residual volumetric water content.
Equation [6] can also be written in terms of degree of saturation, S, or gravimetric
water content, w, to predict the shear strength yielding similar results. To use this equation
the residual volumetric water content,
θ
r
, has to be estimated from the soil-water
characteristic curve.
A graphical procedure can be used to define the residual state of saturation (Figure
1). The procedure involves first drawing a tangent line through the inflection point on the
straight-line portion of the soil-water characteristic curve. The residual saturation can be
defined as the point where the line extending from 1,000,000 kPa along the curve intersects
the previous line. A computational technique can also be used to determine the residual
suction value. More details about the computational technique are available in Vanapalli et
al. (1998). The shear strength of soil may start to decrease beyond the residual state
conditions. Analysis of experimental results presented and shown in the figures of this paper
using Equation [6] are referred as Procedure 2.
Both the Procedures 1 and 2 are consistent with the stress state variable approach
satisfying the continuum mechanics concepts. The form of the equation is similar to the
Fredlund et al. (1978) equation (i.e., Equation 2).
Oberg and Sallfors (1997) proposed an equation for predicting the shear strength of
primarily non-clayey soils such as sands and silts. The proposed equation can be rearranged
as follows:
(
)
(
)
(
)
(
)
[
]
'tan'tan' φφστ Suuuc
waan
−+−+= [7]
