§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

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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

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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.

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