Stability Charts for Undrained Clay Slopes in Overload

Stability Charts for Undrained Clay
Slopes in Overload Conditions Based on
Upper Bound Limit Analysis
Tang Gaopeng
School of Civil Engineering, Central South University, Changsha, China
e-mail: [email protected]
Zhao Lianheng*
Associate professor, School of Civil Engineering, Central South
University, Changsha, China
*Corresponding author, e-mail: [email protected]
Gao Liansheng, Luo Wei
School of Civil Engineering, Central South University, Changsha, China
ABSTRACT
The load of slope crests often influences the stability of slopes near undrained cohesive soil
foundation pits. Establishing a rapid evaluation method for assessing the stability of slope in
overload conditions has remarkable practical value. Based on upper bound limit analysis,
rotational failure mechanism of undrained clay slopes has been analyzed. Considering the
influences of stiff stratum and overload conditions on the stability of slopes, stability charts
for undrained clay slopes have been established with integrated parameters of stiff stratum
depth, load value and slope angle. The stability coefficient of undrained clay slopes subjected
to overload and stiff stratum depth can rapidly be obtain using the suggested charts; therefore,
the safety factors of such slopes can be obtained with the known parameters. Moreover, the
proposed charts also identify the types of rotational failure mechanisms associated with the
stability coefficients. The present work can be a beneficial supplement to the existing stability
charts for simple undrained clay slopes.
KEYWORDS:
upper bound limit analysis; stability charts; stiff stratum; undrained
clay slope
INTRODUCTION
Traditional stability charts are still routinely used in engineering practices as convenient and
efficient tools for estimating slope safety. Quite a few studies have been made to construct
stability charts that require no iterations to evaluate the safety factors. Taylor (1937) used the
friction circle method to construct the stability charts of undrained clay slopes. Michalowski
(1995) utilized a log-spiral failure mechanism to arrive at his stability charts based on the
kinematic approach of limit analysis. Baker (2003) consummated the Taylor’s work about
stability charts and presented a convenient alternative representation of Taylor’s stability charts in
which critical slip circles were associated with stability coefficients. Taking into account
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compound circles comprising two circular arcs separated by a straight line at the interface with
the stiff stratum, Steward (2011) proposed five types of failure mechanisms, used SLOPE/W to
obtain stability coefficients, and determined the type of critical failure circles. Finally, a stability
design chart was proposed.
Currently, the present constructed stability charts are generally based on the limit equilibrium
method; and keeping in mind the there are few discussions regarding undrained clay slopes in
present stability charts, overload effects are not taken into account. In this study, the rotational
failure mechanism of undrain clay slopes were analyzed on the basis of upper bound limit
analysis. Stability charts for undrain clay slopes are except to be presented considering the
influences of stiff stratum and overload conditions on the stability of slopes. To some extent, it
can provide a useful reference to supplement and complete previous stability charts.
STABILITY COEFFICIENT
To assess the stability of slopes and to obtain the safety index of slopes with given geometric
slope and physical soil parameters, a dimensionless stability coefficient, N, has been introduced to
present the stability of slope. Here, N is defined as
N
=
γH
=
cd
γ HF
c
(1)
where γ = the unit weight of the undrained cohesive soil, H=slope height, cd = undrained
cohesion mobilized along the failure log-spiral within the clayey soil, c= undrained cohesion of
the clayey soil, and, F = safety factor; The safety factor, F, can be understood as follows: the
slope transits to a critical limit equilibrium state when the shear strength parameters of the
potential undrained sliding clay are reduced F times.
STABILITY CHARTS FOR UNDRAINED CLAY SLOPES
Slope failure modes
Three type of slope failure modes have been identified according to a traditional log-spiral
critical slip surface that is not divided by stiff stratum (specifically, when the internal friction
angle of soil is zero, i.e., undrained clay, the log-spiral failure surface becomes cylindrical.).
These failure modes are below toe circle, deep toe circle, and shallow circle as shown in Figure 1.
Here H = slope height, ndH = the depth from slope crest to stiff stratum, β = slope angle,
andβˊ = the angle formed by lines ACˊ and CCˊ. In Fig. 1 (a) the critical slip surface of the slope
passing through below the toe, and such a circle is known as a below toe circle; the critical slip
surface of the slope passing through the toe and the lowest point of it is below the point of C as
shown in Fig.1 (b) which is known as deep toe circle; In Fig.1 (c) the critical slip surface of the
slope passing through the toe and the lowest point of it is C which is known as a shallow toe
circle.
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A
H
¦Â
¨@ ¦ Â
ndH C¨@ C
Stiff stratum
(a)
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q
A
H
C
q
q
H
nd H ¦ Â
nd H
¦Â
C
Stiff stratum
Stiff stratum
(b)
(c)
Figure 1: Three types of slope failure modes: (a) below toe circle; (b) deep toe circle;
(c) shallow toe circle
Stability chart for undrained clay slopes
A non-linear programming program has been given to calculate the stability coefficient of
undrained clay slope based on upper bound limit analysis regardless of exterior conditions, and
relevant stability charts have been produced. The results are illustrated in Figure 2.
Figure 2: Stability chart for undrained clay slopes
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In Fig. 2 the yellow area represents the failure mode of the below toe circle, and the purple
area represents the failure mode of the deep toe circle. When the slope angle, β, is greater than
54.2°, the failure mode of the slopes is almost the shallow toe circle. When β is less than 54.2°,
the type of failure mode depends on depth factor, nd , and β, when nd=1, the failure mode of the
slopes is almost shallow toe circle. Additionally, if nd = ∞, without consideration of the stiff
stratum effect, β has a slight influence on N when β is greater than 54.2°.
STABILITY CHARTS FOR UNDRAINED CLAY
SLOPES SUBJECTED TO OVERLOAD
In engineering practice, natural or artificial slopes of foundation pits are always affected by
complicated loading, such as vehicle loads on slope crest or loads stacked at tops of slopes.
Particularly for pit excavation of undrained cohesive soil foundations, the load of construction
machinery at a slope crest usually has an important influence on the slope stability. Ignoring or
underrating the load of the slope crest always increases the safety risks. Therefore, considering
the influence of overload conditions on slopes stability, the uniform load of slope crest has been
introduced into failure mechanism as shown in the Fig. 1 to analyze the stability of slopes.
A non-dimensional coefficient, k = q/γh (q represents the vertical load of slope crest), has
been introduced to signify slope in different overload situations, here, k = 0.1, 0.2, and 0.3. A
non-linear programming program has been used to calculate the stability coefficients of undrained
clay slopes in overload condition and to produce the relevant stability charts. The results are
illustrated in the following figures.
Figure 3: Stability chart for undrained clay slopes subjected to overload with k = 0.1
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Figure 4: Stability chart for undrained clay slopes subjected to overload with k = 0.2
Figure 5: Stability chart for undrained clay slopes subjected to overload with k = 0.3
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These figures reveal that the value of overload significantly influences the stability
coefficient and the failure mode type of slopes. The yellow area decreases significantly with
increased k, whereas the purple area increased significantly with increased k. That is, the mainly
type of failure is toe circle with the increased overload.
NUMERICAL EXAMPLES
Numerical example 1
A simple, undrained clay slope is introduced to illustrate the failure modes shown in Fig.1. A
6 m undrained clay slop has a 25° slope angle and there is a stiff stratum as deep as 9m below the
slope crest. The clay has a unit weight of γ = 20kN/m3 and an undrained shear strength of 30kPa.
The stability coefficient, N, can be read from Fig.1 as 6.22 with nd = 1.5 and β = 25°, the type of
failure mode is below toe circle. From Eq. (1):
γ HF 19 × 6 × F
=
30
= 6.22
N= c
Therefore, F = 1.637.
The safety factor is quite similar to that in the methods of Taylor’s (1937) and Steward
(2011)’s methods, which confirms that this stability charts are correct.
Numerical Example 2
An undrained clay slope at overload condition is introduced to illustrate the use of Figs.3-5.
An 8m undrained clay slope has a 30° slope angle and there is a stiff stratum as deep as 12m
below the slope crest. The physical parameters of the soil are γ = 20kN/m3, and, c = 30kPa, and
the vertical load of slope crest q=32kPa. Stability coefficient N can be read from Fig.4 as 4.95
with nd = 1.5, β = 30° and k = q/Rh = 0.2, the type of failure is below toe circle. From Eq. (1):
N=
γ HF 20 × 8 × F
=
= 4.95
c
30
Therefore, F = 0.928.
SUMMARY AND CONCLUSION
A series of charts was produced to assess the stability of undrained clay slopes subjected to
stiff stratum and overload. The stability coefficient was obtained using calculations based on
upper bound limit analysis. These charts allow engineers to conveniently obtain the safety factor
of slope conveniently with the given geometric parameters of the slopes and physical parameters
of the soil.
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ACKNOWLEDGEMENTS
This paper is based on a study financially supported by Natural Science Foundation of China
(Grant No.51208522), China Postdoctoral Science Foundation Projects (No.2012T50708),
Science and Technology Projects of Hunan Science and Technology Department
(No.2012SK3231), Fundamental Research Funds for the Central Universities of Central South
University (2013zzts237).
REFERENCES
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2. Bell, J. M. (1966). “Dimensionless parameters for homogeneous earth slopes.’’ J. Soil
Mech. Found. Div., Am. Soc. Civ. Eng., 92（5）, 51–65.
3. Chen, W. F., Giger, M. W., and Fang, H. Y. (1969). ‘‘On the limit analysis of stability of
slopes.’’ Soils Found., 9(4), 23–32.
4. Michalowski, R. L. (1995). “Slope stability analysis: a kinematical approach.”
Geotechnique, 45(2), 283–293.
5. Michalowski, R. (2002). “Stability Charts for Uniform Slopes.” J. Geotech. Geoenviron.
Eng., 128(4), 351–355.
6. Baker, R. (2003). “A Second Look at Taylor’s Stability Chart.” J. Geotech. Geoenviron.
Eng., 129(12), 1102–1108.
7. Steward, T., Sivakugan, N., Shukla, S., and Das, B. (2011). “Taylor’s Slope Stability
Charts Revisited.” Int. J. Geomech., 11(4), 348–352.
8. Zhao Lian-heng, Li Liang, Yang Feng, Luo Qiang. (2010). “Upper bound analysis of
slope stability with nonlinear failure criterion based on strength reduction technique”
Journal of Central South University of Technology, 17(4): 836-844.
© 2014 EJGE

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