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Soil slope stability analysis combining shear
strength reduction and slip line method
Supervisor: Yongchang Cai
Ph.D. candidate: Jie Wu
School of Civil Engineering
Tongji University, China
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Research background
 Predicting the stability of soil slopes is a classical problem for
both of practicing engineers and academics.
Picture from: http://www.ccma.vic.gov.au/soilhealth/photos.htm
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Research background
 Two main challenges need to be confronted when
dealing with soil slope
1.
The Determination of potential slip lines of the soil slope
2.
The calculation of corresponding safety factors
Picture from: http://ceae.colorado.edu/~regueiro/images/slope.jpg
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Research background
When the limit equilibrium method is applied
1
The stress strain behavior of soil
are often neglected
2
The calculation models are often
overly simplified
33
Arbitrary assumptions are usually introduced to
ensure static determinacy
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Research background
In the past several decades, there have been growing interests
in Numerical slip line theory, which is developed from the
classical slip line theory. As a particular method for soil slope
stability analysis, the potential slip line surfaces can be tracked
in an efficient way.
Slip line
Start points of the search
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Content
Slip line method for evaluating slope stability
Application in soil slope stability analysis
Combination of Shear strength reduction and
slip line method
Conclusion
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Slip line method for evaluating slope stability
Traditional Slip line theory
For plane problem, when failure occurs, it has two failure planes(lines)
and the angle μ between the failure plane(line) and the principal stress
orientation is specific. As shown in figure below, 1 and 2 represents
the maximum principal stress orientation(θ) and the minimum
principal stress orientation, respectively. And α and β represent two
failure planes(lines).

：

 ：

dy
 tan(  u )
dx
dy
 tan(  u )
dx
7
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Slip line method for evaluating slope stability
Traditional Slip line theory
M-C material
For M-C material:
For Tresca material:
Tresca material
However, Traditional Slip line
theory is based on rigid-plastic
assumption and it can only solve
some simple problems.
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Slip line method for evaluating slope stability
Numerical Slip line theory
 n   y cos 2    x sin 2   2 xy sin  cos 
  (   )sin  cos    (cos 2   sin 2  )
y
x
xy
 n
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Slip line method for evaluating slope stability
 n
0

2 xy

1
 when  x   y，1 = arctan
2
 x  y


 when    ， = 1 arctan 2 xy +90
y
x
1

2
 x  y


 n   3 cos 2    1 sin 2 
 n  ( 3   1 )sin  cos 
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Slip line method for evaluating slope stability
So, the slope safety can be defined as below:
(1 sin 2    3 cos 2  ) tan   c
K
( 3  1 )sin  cos 
For the minimum of K, solve the equation
K
0

 3  1
  (45  arcsin(
))
2c *cot    3   1
Thus, the two surfaces of the smallest anti-shear safety factor
can be expressed as:
 3  1
(45  arcsin(
))
2c *cot    3   1
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Slip line method for evaluating slope stability
By linking the critical slip surface of each point, the slip surface group of
slopes can be obtained. The safety factors of the slip surfaces can be
calculated by the formula below. The slip surface corresponding to the
minimum safety factor is the critical slip surface of slopes stability.
 
n
F 
j 1
ij
tan ij  cij   lij
 
n
j 1
ij
 lij 
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Application in soil slope stability analysis
The following is the procedure of slip line theory to analyze soil
slopes stability, that is, the stress distribution and its value in soil
slopes is calculated by Ansys and then the slip line theory is applied
to find the potential slip surface based on the stress status of the
slopes.
1
Modeling in
Ansys
2
Stress
distribution
in slopes
3
Searching
the
potential
slip line
surfaces
4
critical slip
surface
and safety
factor
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Example 1
This example was provided by Australian Computer
Aided Design Society (Donald and Giam, 1992).
γ（KN/m3） E（Kpa） μ
25
1e7
0.2
c（KPa）
Ψ(°)
1e3
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Reference calculation result:
Geometric model
Safety factor=1
In Limit Equilibrium State
Finite element mesh
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Example 1 (cont.)
Contours of maximum principal stress (Elastoplastic)
Critical slip surfaces (Elastoplastic)
Contours of maximum principal stress 𝜎1 (Elasticity)
Critical slip surfaces (Elasticity)
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Example 1 (cont.)
Orientation
of slip
lineline
(Elasticity)
Orientation
of slip
(Elastoplastic)
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Example 2
Another benchmark from Australian
Computer Aided Design Society
Layer 1
material
γ/KN m-3
Layer 1
19.5
Layer 2
Layer 3
Ψ/°
E/KPa
v
0
38.0
1.0*104
0.25
19.5
5.3
23.0
1.0*104
0.25
19.5
7.2
20.0
1.0*104
0.25
c/KPa
Layer 2
Layer 3
Reference safety factor is 1.400
The orientation of slip line
Plastic range indicated in red color
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Example 2 (cont.)
Critical slip surface, Fs=1.487
Shear strength reduction method, Fs=1.450
Limit equilibrium method, Fs=1.400
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Application in soil slope stability analysis
Based on the calculation results of the 2 examples, some conclusions can
be drawn as follows:
When the soil slope, especially its top area, is not in the plastic state, there
is a dramatic change of the orientation of the slip line and the slip surface
tracked is singular at corresponding area, and the factor of safety is also
not minimum.
1. The shear strength reduction method can be employed to reduce the material
parameters until materials of the slope approaching the plastic state. Then the
slip surface calculated maybe not singular and would be more like the real
one.
2. the defect that the slip surface cannot be easily obtained by the conventional
shear strength reduction method can also be corrected.
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Combination of shear strength reduction and slip line method
The central concept of the SSR method is very simple. Successively reduce
some factors in the shear strengths of slope materials until the finite element
model does not converge to a solution, which means the failure occurs. The
critical factor at which failure occurs is taken to be the factor of safety.
The following is the combination of the shear strength reduction and slip line
method.
1
Modeling in
Ansys
2
Set a range
of different
strength
reduction
factors
3
4
5
Stress
distribution
in slopes
Searching
the critical
slip
surfaces
Comparing
the shape
of different
slip
surfaces
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Combination of shear strength reduction and slip line method
Plastic range indicated in red color, strength reduction factor is 1.3
The critical slip line, safety factor is 1.073
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Combination of shear strength reduction and slip line method
Plastic range indicated in red color, strength reduction factor is 1.4
The critical slip line, safety factor is 1.008
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Combination of shear strength reduction and slip line method
Plastic range indicated in red color, strength reduction factor is 1.45
The critical slip line, safety factor is 0.997
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Conclusions
 The slip line method is time efficient and no priori assumptions have
to be made when it is employed to the stability analyze of the soil slope.
 The slip surface is singular and not as smooth as the real one if
the material, particularly in the top of the slope, is in elastic state
rather than plastic state.
 The combination of strength reduction and slip line method can
correct the defect that the slip surface can not be directly get by the
strength reduction method, and is an effective way to track the slip
surfaces and to evaluate soil slopes stability.
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Thank you very much for your
attention !
Acknowledgements
The authors gratefully acknowledge the support of national science and
technology support program (2011BAB08B01), and program for new century
excellent talents of ministry of education of China (NCET-12-0415).