§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

7.3 The method of slices -Concept Fellenius,1927 Bishop -Method

-Critical surface and solution to K min 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 K min

Pressure of Retaining Walls Prof. M Zhang, A A Javadi University of Bradford, UK, Shanghai University 10 th 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 10 th 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.

10 th IACMAG Fig. 1 Slice method for irregular sliding surface and sliding body obtained by the tracking analysis

10 th IACMAG Fig. 2 Forces acted on slice i and the static equilibrium of slice n

10 th IACMAG

Fig. 3. Static equilibrium of slice n about limit passive earth pressure

10 th IACMAG

Table 1. The parameters of materials

10 th IACMAG (a) Displacement vectors

10 th IACMAG (b) Principal stress vectors

10 th IACMAG (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. 10 th IACMAG