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§7 Stability of slopes
• Introduction
• Stability analysis of granular materials• The method of slices• Development for the analytical method
7.1 Introduction
7.2 Stability analysis of granular materials
1cos'
sin
lctgWT
WT
sin
cos'
W
cltgW
T
TK
7.3 The method of slices
-Concept
Fellenius,1927 Bishop
dW
Rls
M
MK
s
r
iii WN cos
iii WT sin
iii cltgNs
RTcltgNM
RTM
iiir
is
'
i
iii
s
r
T
TcltgN
M
MK
'
-Method
-Critical surface and solution to Kmin
•The center of the critical arc φ=0
φ=0 φ≠0
Table of the values of angles α , β
•The radius of the critical arc
•Solution to Kmin
Pressure of Retaining Walls
Prof. M Zhang, A A JavadiUniversity of Bradford, UK, Shanghai University
10th IACMAG
7.4 Development for the analytical methods
—— Numerical Analysis to Earth
Introduction
Formation of earth pressure
Principle for calculation based on numerical analysis
Active sliding earth pressure
Passive earth pressure
Example
Conclusions
10th IACMAG
Outline
Some drawbacks in traditional theories of earth pressure
To ignore the stress/strain state. Do not calculate the displacement of the wall,
though they always claim that it plays a very important role.
Based on the limit equilibrium theory, but they give no evidence that limit state takes place in the fracture surface.
Pay no attention to the difference of earth pressure calculation between the fill retaining wall and the cutting one.
-A new method- a combination of numerical analysis with limit equilibrium is proposed.
Formation of earth pressure
To clarify: real cause and actual process
of formation of earth pressure
Relationship: earth pressure
– stress state
– construction sequence
Division of earth pressure
Deforming earth pressure
( In some areas, there exists shear failure, but no significant sliding surface occurs.)
Sliding/limit earth pressure ( After sliding surface occurs )
Principle for calculation based on numerical analysis
The stress state and the deformation are the most important factors that cause the formation of the earth pressure.
Until limit state occurs, the retaining wall has to be subjected to a deforming earth pressure that can be obtained by finite element analysis.
As the shear failure makes a fracture surface (or plastic band), the wall will be subjected to a sliding earth pressure.
The entire earth pressure activity is usually the mixture of the active sliding one and the passive one.
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Fig. 1 Slice method for irregular sliding surface andsliding body obtained by the tracking analysis
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Fig. 2 Forces acted on slice i and the static equilibrium of slice n
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cos)tan(sin
)tan(cos)tan(sin
nn
nnnnnnnnna
WPPE
)(tancossin 11 iiiiiiiiii PlcWWP
iiiiii tan)sin()cos( 11
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Fig. 3. Static equilibrium of slice n about limit passive earth pressure
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)tan(sincos
)tan(cos)tan(sin
nn
nnnnnnnnnp
WPPE
uk pp
)(tancossin 31 iiiiiiiiii PlcWWP
iiiiii tan)sin()cos( 11
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Property ValueUnit weight of soil 16.5 kN/m3
Unit weight of wall 24.0 kN/m3
Cohesion 15.0 kPa 21.0K 160.0n 0.65Rf 0.85Poisson's ratio 0.31
Table 1. The parameters of materials
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(a) Displacement vectors
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(b) Principal stress vectors
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(c) The distribution maximum principal stress 1.
Fig. 4. Results of a fill retaining wall
Conclusions
The traditional theories of earth pressure for retaining walls, based on the limit equilibrium method, are incorrectly assumed that the earth pressure is independent of the stress/strain state in the soil around the wall. Furthermore, it is not calculated, although the soil deformation is emphasized.
The proposed method to calculate the earth pressure, based on numerical analysis, can overcome the weakness of the traditional theories.
10th IACMAG