Numerical study of earth pressure distribution - UM...

Numerical study of earth pressure distribution

behind cantilever retaining walls

by

WONG KA HOU

Final Year Project Report submitted in partial fulfillment

of the requirement of the Degree of

Bachelor of Science in Civil Engineering

2013/2014

Faculty of Science and Technology

University of Macau

i

DECLARATION

I declare that the project report here submitted is original except for the source

materials explicitly acknowledged and that this report as a whole, or any part of this

report has not been previously and concurrently submitted for any other degree or

award at the University of Macau or other institutions.

I also acknowledge that I am aware of the Rules on Handling Student Academic

Dishonesty and the Regulations of the Student Discipline of the University of Macau.

Signature : ____________________________

Name : ____________________________

Student ID :____________________________

Date : ____________________________

ii

APPROVAL FOR SUBMISSION

This project report entitled “Numerical study of earth pressure distribution behind

cantilever retaining walls” was prepared by WONG KA HOU in partial fulfillment

of the requirements for the degree of Bachelor of Science in Civil Engineering at the

University of Macau.

Endorsed by,

Signature : ____________________________

Supervisor : Prof. Lok Man Hoi

iii

ACKNOWLEDGEMENTS

I would like to thank Professor Lok for his guidance and help in this project. He is so

kind to correct my mistakes and his ideas always inspire me.

I would also thank my parents for their support all the time.

iv

ABSTRACT

In some of the design guides, the method of designing a cantilever retaining wall

gives different approaches in terms of the geometry of the wall. By considering the

length of the heel, either Rankine’s or Coulomb’s earth pressure theories are applied

in different cases. However, this may lead to drastic change of earth pressure due to

small change in heel length.

In this study, numerical analysis of finite element method is applied to investigate the

influence to the behavior of the soil behind the cantilever wall. Different heel lengths

together with wall frictions are assigned to the model.

From the analysis results, the lateral pressure behind the wall and the failure mode of

the soil are different between smooth wall and rough wall cases. However, the

influence due to the heel length is not obvious.

On the other hand, the construction sequence of the backfill in the simulation and the

location to examine the earth pressure also affect the results of analysis.

v

CONTENT

DECLARATION ........................................................................................................... i

APPROVAL FOR SUBMISSION .............................................................................. ii

ACKNOWLEDGEMENTS ...................................................................................... iii

ABSTRACT ............................................................................................................. iv

LIST OF FIGURES ................................................................................................... vii

LIST OF TABLES ....................................................................................................... x

CHAPTER 1 INTRODUCTION.............................................................................. 1

CHAPTER 2 LITERATURE REVIEW .................................................................. 2

2.1 Design method of Cantilever walls ................................................................ 2

2.2 Fundamental earth pressure theories ............................................................ 4

2.3 Previous studies on earth pressure behind gravity and cantilever walls ... 8

2.3.1 Analytical Studies .................................................................................... 8

2.3.2 Experimental Studies ............................................................................. 11

2.3.3 Numerical Studies.................................................................................. 15

CHAPTER 3 METHODOLOGY .......................................................................... 19

3.1 Finite Element Method & CRISP ................................................................ 19

3.2 Model Setup ................................................................................................... 19

3.2.1 Model configurations ............................................................................. 20

3.2.2 Meshing and Boundary Conditions ....................................................... 20

3.2.3 Construction Stages ............................................................................... 21

3.3 Material Models............................................................................................. 23

3.3.1 Mohr Coulomb Elastic-perfectly plastic model ..................................... 23

3.3.2 Duncan and Chang Hyperbolic model................................................... 24

3.3.3 Interface elements .................................................................................. 26

3.4 Interpretation of Results ............................................................................... 27

vi

CHAPTER 4 PRELIMINARY STUDIES ............................................................. 28

4.1 Verification of active pressure from numerical simulation ....................... 28

4.2 Variation of Factor of Safety with heel length ............................................ 32

CHAPTER 5 ANALYSIS AND DISCUSSION ..................................................... 36

5.1 Parametric study ........................................................................................... 36

5.2 Results ............................................................................................................ 37

5.2.1 Lateral earth pressure at different heel length ....................................... 38

5.2.3 Contours of deviatoric strain at different heel length ............................ 46

5.2.4 Status plots at different heel length ....................................................... 54

5.3 Discussion ....................................................................................................... 62

5.3.1 Effect on construction sequence ............................................................ 62

5.3.2 Effect on wall friction ............................................................................ 65

5.3.3 Effect on heel length .............................................................................. 65

CHAPTER 6 CONCLUSION ................................................................................ 72

REFERENCE ............................................................................................................ 74

vii

LIST OF FIGURES

Figure 2-1, Influence of heel length on analysis method (From GeoGuide 1) ......................................... 2

Figure 2-2, Stability criteria for retaining walls (From GeoGuide 1) ....................................................... 3

Figure 2-3, Mohr-Coulomb failure envelope (From Das) ........................................................................ 4

Figure 2-4, Rankine lateral earth pressure distribution (From Das) ......................................................... 5

Figure 2-5, Geometry of the soil wedge in active case (From Das) ......................................................... 6

Figure 2-6, Geometry of the soil wedge in Passive case (From Das) ....................................................... 7

Figure 2-7, Calculated and experimental results (From Wang 2000) ....................................................... 9

Figure 2-8, Variation of lateral earth pressure (From Bang 1985) ......................................................... 10

Figure 2-9, Comparison of lateral earth pressure (From Bang 1985) ..................................................... 10

Figure 2-10, Cross section at complete backfill (From Matsuo 1978) ................................................... 11

Figure 2-11, Change of resultant force of earth pressure (From Matsuo 1978) ..................................... 11

Figure 2-12, Distribution of earth pressure in the vertical direction (From Matsuo 1978) ..................... 12

Figure 2-13, Sensitive analysis of earth pressure at rest (From Matsuo 1978) ....................................... 13

Figure 2-14, Summary of test conditions (From Fang 1986) ................................................................. 13

Figure 2-15, Coefficients of horizontal active thrust related with soil density and internal friction angle

(From Fang 1986) ........................................................................................................................... 14

Figure 2-16, Points of application of active thrust under different wall movement (From Fang 1986) . 14

Figure 2-17, Modeling of retaining system (From Clough 1971)........................................................... 15

Figure 2-18, Earth pressure distribution (From Clough 1971) ............................................................... 15

Figure 2-19, Variation of total lateral earth pressure with wall displacement (From Clough 1971) ...... 16

Figure 2-20, Geometry of a cantilever retaining wall (From Goh 1993) ................................................ 17

Figure 2-21, Lateral earth pressure distribution at wall stem and virtual wall back (From Goh 1993) .. 17

Figure 2-22, Lateral earth pressure smaller than the Rankine’s active pressure (From Goh 1993) ........ 18

Figure 2-23, Proposed earth pressure profiles for the wall stem and virtual wall back (From Goh 1993)

........................................................................................................................................................ 18

Figure 3-1, Configuration of calculation ................................................................................................ 20

Figure 3-2, Meshing and boundary conditions of the model .................................................................. 21

Figure 3-3, Construction squence of the model (cont.) .......................................................................... 22

Figure 3-4, Stress-strain relationship of elastic-perfectly plastic model (From Crisp Technical manual)

........................................................................................................................................................ 23

Figure 3-5, Mohr-Coulomb Failure Surfaces in three dimensional space (From Crisp Technical manual)

........................................................................................................................................................ 24

Figure 3-6, Hyperbolic stress-strain curve and the transformed linear plot (From Duncan 1970) ......... 25

Figure 3-7, Comparison of hyperbolic and actual stress-displacement curves (From Clough 1971) ..... 26

Figure 3-8, Location of lateral pressure investigated ............................................................................. 27

viii

Figure 4-1, Earth pressure distribution in Clough (1971) ....................................................................... 29

Figure 4-2, Earth pressure distribution obtained from CRISP ................................................................ 30

Figure 4-3, Earth pressure distribution in elastic-perfectly plastic model .............................................. 31

Figure 4-4, Long heel and Short heel ..................................................................................................... 32

Figure 4-5, FS for Long heel and short heal cases (Modified from GeoGuide 1) .................................. 33

Figure 4-6, Location of α and β .............................................................................................................. 34

Figure 4-7, Variation of Factor of Safety with heel length ..................................................................... 35

Figure 5-1, Material in the model ........................................................................................................... 36

Figure 5-2, Lateral earth pressure of rough wall at L = 3 m ................................................................... 39

Figure 5-3, Lateral earth pressure of smooth wall at L = 3 m ................................................................ 39







Figure 5-10, Lateral earth pressure of rough wall at L = 8 m ................................................................. 43

Figure 5-11, Lateral earth pressure of smooth wall at L = 8 m .............................................................. 43

Figure 5-12, Lateral earth pressure of rough wall at L = 10 m ............................................................... 44

Figure 5-13, Lateral earth pressure of smooth wall at L = 10 m ............................................................ 44

Figure 5-14, Lateral earth pressure of rough wall at L = 12 m ............................................................... 45

Figure 5-15, Lateral earth pressure of smooth wall at L = 12 m ............................................................ 45

Figure 5-16, Contours of deviatoric strain of rough wall at L = 3 m ...................................................... 47

Figure 5-17, Contours of deviatoric strain of smooth wall at L = 3 m ................................................... 47









Figure 5-26, Contours of deviatoric strain of rough wall at L = 10 m .................................................... 52

Figure 5-27, Contours of deviatoric strain of smooth wall at L = 10 m ................................................. 52

Figure 5-28, Contours of deviatoric strain of rough wall at L = 12 m .................................................... 53

Figure 5-29, Contours of deviatoric strain of smooth wall at L = 12 m ................................................. 53

Figure 5-30, Status plot of rough wall at L = 3 m .................................................................................. 55

Figure 5-31, Status plot of smooth wall at L = 3 m ................................................................................ 55

ix









Figure 5-40, Status plot of rough wall at L = 10 m ................................................................................ 60

Figure 5-41, Status plot of smooth wall at L = 10 m .............................................................................. 60

Figure 5-42, Status plot of rough wall at L = 12 m ................................................................................ 61

Figure 5-43, Status plot of smooth wall at L = 12 m .............................................................................. 61

Figure 5-44, Lateral earth pressure distribution at virtual back of smooth wall at L = 6 m ................... 62

Figure 5-45, Horizontal movement at the top of the retaining wall at L = 6 m ...................................... 63

Figure 5-46, Lateral earth pressure distribution at virtual back at L = 6 m, with only one point in a layer

........................................................................................................................................................ 64

Figure 5-47, Lateral earth pressure distribution at wall back of rough wall at different heel length ...... 66

Figure 5-48, Lateral earth pressure distribution at wall back of rough wall at different heel length ...... 66

Figure 5-49, Lateral earth pressure distribution at virtual back of rough and smooth wall at different

heel length ...................................................................................................................................... 67

Figure 5-50, Resultant lateral force at wall back and virtual back of rough wall at different heel length

........................................................................................................................................................ 68

Figure 5-51, Resultant lateral force at wall back and virtual back of smooth wall at different heel length

........................................................................................................................................................ 68

Figure 5-52, Resultant lateral force at wall back of rough and smooth wall at different heel length ..... 69

Figure 5-53, Resultant lateral force at virtual back of rough and smooth wall at different heel length .. 70

x

LIST OF TABLES

Table 4-1, Material properties used in Clough (1971) ............................................................................ 28

Table 4-2, Material properties used in CRISP ........................................................................................ 29

Table 4-3, Material properties of Elastic-Plastic model ......................................................................... 31

Table 5-1, Material properties ................................................................................................................ 36

Table 5-2, Properties of Interface elements ............................................................................................ 37

Table 5-4, Dimension of the retaining wall ............................................................................................ 37

1

CHAPTER 1 INTRODUCTION

The design of rigid retaining walls relies heavily on the estimate of earth pressure

distribution. In common practice, the Rankine’s or Coulomb’s active earth pressure is

assumed depending on the geometry of the retaining wall and the interface friction.

However, the method of determining the earth pressure suggested by the design guides

is rather empirical, which may lead to very different design.

In this study, numerical study using finite element method will be carried out for

various retaining wall configurations. The results of earth pressure distribution will be

compared with the Rankine’s and Coulomb’s theory. The effects of different factors on

the earth pressure will be investigated.

In Chapter 2, the common design method of a cantilever retaining wall will be

introduced and the fundamental earth pressure theories and some previous studies

related to the topic will be reviewed.

The finite element method and the procedures of analyses with the computer program

CRISP will be introduced in Chapter 3. The setup of the finite element model and the

material model used will also be discussed.

Some preliminary results are shown in Chapter 4. The method of analysis is verified

by comparing the results with a previous study using the same model. The factor of

safety of a cantilever retaining wall with various heel lengths is calculated according

to the method suggested in the design guides.

Chapter 5 shows the results of the analyses and discussion will be made according to

the results.

2

CHAPTER 2 LITERATURE REVIEW

2.1 Design method of Cantilever walls

In the GeoGuide 1 - Guide to Retaining Wall Design (1st Edition, 1982) by the Hong

Kong Government, an approach of designing a cantilever retaining wall is proposed.

As shown in the figure below, the cantilever wall is separated to two conditions, the

Rankine’s (long heel) and the Coulomb’s (short heel), by considering the heel length

of the wall. The guide also provides the procedures for checking the stability criteria

by estimating the Factor of Safety of a retaining wall, which is also shown below.

Figure 2-1, Influence of heel length on analysis method (From GeoGuide 1)

3

Figure 2-2, Stability criteria for retaining walls (From GeoGuide 1)

This design method is adopted and introduced in Earth Pressure and Earth-Retaining

Structures by Clayton et al in 1993 and the similar method also appeared in the section

of retaining wall in the Bridge Design Specifications (2004) by Caltrans – California

Department of Transportation.

As no earlier reference can be found at this moment, it may be concluded that this

method has been adopted for a period of time and become a common practice for the

design of cantilever retaining wall.

4

2.2 Fundamental earth pressure theories

In the past studies, finite element analysis was applied to investigate the earth pressure

behind the retaining systems and the results were compared with various experiments

and with the common used Rankine’s and Coulomb’s theory.

According to the Mohr-Coulomb failure criterion, the soil fails when the shear stress

exceed the shear envelope. Base on the geometry of the failure envelop, the active and

passive earth pressure coefficients can be estimated.

Figure 2-3, Mohr-Coulomb failure envelope (From Das)

τ = 𝑐′ + 𝜎′ tan 𝜑′ (2-1)

1

2(𝜎1

′ − 𝜎3′) =

1

2(𝜎1

′ − 𝜎3′) sin 𝜑′ + 𝑐′ cos 𝜑′ (2-2)

𝜎3′ = 𝜎1

′𝐾𝑎 − 2𝑐′√𝐾𝑎 (2-3)

𝜎1′ = 𝜎3

′𝐾𝑝 + 2𝑐′√𝐾𝑝 (2-4)

5

Figure 2-4, Rankine lateral earth pressure distribution (From Das)

σ𝑎,𝑝′ = 𝛾𝑧𝐾𝑎,𝑝 (2-5)

𝑃𝑎,𝑝 =1

2𝛾𝐻2𝐾𝑎,𝑝 (2-6)

𝐾𝑎 = cos 𝛽cos 𝛽 − √cos2 𝛽 − cos2 𝜑′

cos 𝛽 + √cos2 𝛽 − cos2 𝜑′ (2-7)

𝐾𝑝 = cos 𝛽cos 𝛽 + √cos2 𝛽 − cos2 𝜑′

cos 𝛽 − √cos2 𝛽 − cos2 𝜑′ (2-8)

The earth pressure distribution is assumed to be linear and the magnitude and the

location of the resultant force can then be estimated. In particular, when the backfill is

horizontal, the coefficient can be reduced to the followings.

𝐾𝑎 =1 − sin 𝜑′

1 + sin 𝜑′= tan2 (45° −

𝜑′

2) (2-9)

𝐾𝑝 =1 + sin 𝜑′

1 − sin 𝜑′= tan2 (45° +

𝜑′

2) (2-10)

In Coulomb’s analysis, the resultant force is obtained directly from the geometry of

the soil wedge and the coefficient of lateral pressure is then calculated.

In active case, from the geometry,

6

Figure 2-5, Geometry of the soil wedge in active case (From Das)

𝑊

sin(90° + 𝜃 + 𝛿 − 𝛽 + 𝜑′)=

𝑃𝑎

sin(𝛽 − 𝜑′) (2-11)

𝑃𝑎 =sin(𝛽 − 𝜑′) 𝑊

sin(90° + 𝜃 + 𝛿 − 𝛽 + 𝜑′)

=1

2𝛾𝐻2

cos(𝜃 − 𝛽) cos(𝜃 − 𝛼) sin(𝛽 − 𝜑′)

cos2 𝜃 sin(𝛽 − 𝛼) sin(90° + 𝜃 + 𝛿 − 𝛽 + 𝜑′)

(2-12)

To obtain the maximum Pa,

𝑑𝑃𝑎

𝑑𝛽= 0 (2-13)

𝑃𝑎 =1

2𝐾𝑎𝛾𝐻2 (2-14)

Where

𝐾𝑎 =cos2(𝜑′ − 𝜃)

cos2 𝜃 cos(𝛿 + 𝜃) [1 + √sin(𝛿 + 𝜑′) sin(𝜑′ − 𝛼)cos(𝛿 + 𝜃) cos(𝜃 − 𝛼)

]

2 (2-15)

7

In passive case, from the geometry,

Figure 2-6, Geometry of the soil wedge in Passive case (From Das)

𝑊

sin(90° + 𝜃 − 𝛿 − 𝛽 − 𝜑′)=

𝑃𝑝

sin(𝛽 + 𝜑′) (2-16)

𝑃𝑝 =sin(𝛽 + 𝜑′) 𝑊

sin(90° + 𝜃 − 𝛿 − 𝛽 − 𝜑′)

=1

2𝛾𝐻2

cos(𝜃 − 𝛽) cos(𝜃 − 𝛼) sin(𝛽 + 𝜑′)

cos2 𝜃 sin(𝛽 − 𝛼) sin(90° + 𝜃 − 𝛿 − 𝛽 − 𝜑′)

(2-17)

To obtain the maximum Pp,

𝑑𝑃𝑝

𝑑𝛽= 0 (2-18)

𝑃𝑝 =1

2𝐾𝑝𝛾𝐻2 (2-19)

Where

𝐾𝑝 =cos2(𝜑′ + 𝜃)

cos2 𝜃 cos(𝛿 − 𝜃) [1 − √sin(𝛿 + 𝜑′) sin(𝜑′ + 𝛼)cos(𝛿 − 𝜃) cos(𝛼 − 𝜃)

]

2 (2-20)

8

2.3 Previous studies on earth pressure behind gravity and cantilever walls

2.3.1 Analytical Studies

Base on the Coulomb’s concept, Wang (2000) derived another set of equations to

estimate the earth pressure coefficients and the height of application of the resultant

pressure. From his analysis,

The pressure distribution

𝑝 =𝐾

𝑐𝑜𝑠 𝛿[(𝑞 −

𝛾𝐻

𝑎𝐾 − 2) (

𝐻 − 𝑦

𝐻)

𝑎𝐾−1

+𝛾𝐻

𝑎𝐾 − 2

𝐻 − 𝑦

𝐻] (2-21)

and the resultant pressure

𝑃 = (𝑞𝐻 +1

2𝛾𝐻2)

𝑠𝑖𝑛(𝜃 − 𝜑) 𝑐𝑜𝑡 𝜃

𝑐𝑜𝑠(𝜃 − 𝜑 − 𝛿) (2-22)

which give the same result when q = 0 with the Coulomb’s theory regardless of the

pressure coefficient K.

The height of application of the resultant pressure

𝐻𝑝 = [1

3+

𝑎𝐾 − 1

3(𝑎𝐾 + 1)] 𝐻

3𝑞 + 𝛾𝐻

2𝑞 + 𝛾𝐻 (2-23)

which equals to 1/3 as in the Coulomb’s theory when q = 0 and K = 1/a.

From the equation, the earth pressure distribution is nonlinear which gave a better

approximation to the real situation than the linear distribution in Coulomb’s theory. A

comparison was also made with the experiment results and a suitable value of K was

applied to fit the data as shown in the following figure.

9

Figure 2-7, Calculated and experimental results (From Wang 2000)

Another study by Bang (1985) described a method of estimating the magnitude and

distribution of the earth pressure behind the retaining wall. It was assumed that the

active state of the backfill soil can be separate from “initial active” to “full active” and

the “intermediate active state” varies linearly. That is, based on the Coulomb’s and

Rankine’s method, the angle of the active wedge Ψ varies linearly in the intermediate

active state at different depth:

Ψ = Φ − Φ(1 − β)z

H (2-24)

And the active thrust and pressure distribution

𝑃𝑎 =1

2𝛾𝑧2

𝐴2

𝐵 (2-25)

𝑝𝑎 = 𝛾𝑧𝐴2

𝐵+

1

2𝛾𝑧2

𝑑𝛹

𝑑𝑧

𝐴(1 + 𝐴2)

𝐵2(𝐵 −

𝑚

2𝐴2) (2-26)

where

𝐴 = 𝑡𝑎𝑛 (45° −𝛹

2) ; 𝐵 = 1 +

𝑚

2(1 − 𝐴2); 𝑚 =

𝑡𝑎𝑛 𝛿

𝑡𝑎𝑛 𝛹 (2-27)

10

Figure 2-8, Variation of lateral earth pressure (From Bang 1985)

From the figure above, the earth pressure against depth is decreasing from the at-rest

condition, initial active state (β = 0.0) and full active state (β = 1.0).

This method was used to compare with model teat and it made a good agreement with

the experiment result as shown in the following figure.

Figure 2-9, Comparison of lateral earth pressure (From Bang 1985)

11

2.3.2 Experimental Studies

Large scale experiments were performed by Matsuo et al (1978) to obtain the active

and passive pressure behind the wall. Different kinds of materials were used as

backfill, the silty sand and two kinds of slag.

Figure 2-10, Cross section at complete backfill (From Matsuo 1978)

The experiment lasted for a long period. The wall moved outward to achieve the

active state about 60 days after the backfill was compacted and the earth pressure were

monitored for 20 days. Then the wall moved in ward to the original position to obtain

the passive state, as shown in the following figure.

Figure 2-11, Change of resultant force of earth pressure (From Matsuo 1978)

12

It can be seen that the earth pressure increase gradually to the at rest pressure with

time. Therefore, the retaining system should be better to design with the at rest

pressure of increase the factor of safety of the active pressure.

Distribution of earth pressure was measured as shown below and the location of the

application of resultant force can then be calculated.

Figure 2-12, Distribution of earth pressure in the vertical direction (From Matsuo 1978)

The experiment results were compared with the finite element analysis, which shows

that the influence of the unit weight and the elastic modulus inputted in the analysis

are very small but the Poisson’s ratio affects a lot, as shown in the following figure.

The passive state were failed to reach due to the lack of power of the equipment and

therefore only the active pressure were discussed in this experiment.

13

Figure 2-13, Sensitive analysis of earth pressure at rest (From Matsuo 1978)

Another experiments by Fang and Ishibashi (1986) obtained the active pressure of

different soil density (or internal friction angle) from 3 types of wall movement. The

backfill and the retaining system were placed on a shaking table and different densities

of the backfill were reached by the acceleration and duration of shaking, as shown in

the following figure.

Figure 2-14, Summary of test conditions (From Fang 1986)

14

The relationship of the soil density (or internal friction angle) and the coefficient of

active thrust and the height of application of the resultant force were shown in the

figures below.

Figure 2-15, Coefficients of horizontal active thrust related with soil density and internal friction angle

(From Fang 1986)

Figure 2-16, Points of application of active thrust under different wall movement (From Fang 1986)

15

2.3.3 Numerical Studies

A study by Clough and Duncan (1971) made use of their hyperbolic soil and interface

model to investigate the lateral earth pressure behind a gravity wall. Different wall

movement and wall friction is simulated. The model in the study and the result is

shown below.

Figure 2-17, Modeling of retaining system (From Clough 1971)

Figure 2-18, Earth pressure distribution (From Clough 1971)

16

Figure 2-19, Variation of total lateral earth pressure with wall displacement (From Clough 1971)

Many of the previous researches were investigating and analysing the lateral earth

pressure behind a plain wall, or gravity wall, by rotating and translating it to achieve

the active state. The effect due to the construction sequence is also ignored, i.e. the

back fill were suddenly appeared in previous studies.

Unlike the others, Goh (1993) has done an analysis for the soil behaviour behind the

cantilever retaining walls. Instead of moving the wall, the active earth pressure is

achieved by the self-balance of the model itself, where the earth pressure from the soil

pushes the retaining wall away and the displacement of the wall leads the soil to a

balance state, which is usually the active state.

The following figure shows the geometry of a cantilever retaining wall.

17

Figure 2-20, Geometry of a cantilever retaining wall (From Goh 1993)

In this study, several configurations of the cantilever wall were applied in the analysis

and the earth pressure right behind the stem and at the virtual wall back were obtained

and compared with Rankine active pressure. The lateral earth pressure distribution

behind the stem is usually different from the virtual wall back, as shown in the

following figure.

Figure 2-21, Lateral earth pressure distribution at wall stem and virtual wall back (From Goh 1993)

However, in some situations, the lateral pressure obtained in the analysis would be

smaller than the active pressure by Rankine’s method, as in the left side of the

following figure.

18

Figure 2-22, Lateral earth pressure smaller than the Rankine’s active pressure (From Goh 1993)

At the end of this study, it was proposed a simplified earth pressure profiles for the

wall stem and virtual wall back at smooth wall and rough wall conditions.

Figure 2-23, Proposed earth pressure profiles for the wall stem and virtual wall back (From Goh 1993)

19

CHAPTER 3 METHODOLOGY

3.1 Finite Element Method & CRISP

Finite element method (FEM) is a well-known method to solve boundary value

problems numerically. It can deal with different non-linear stress-strain relations of

materials and geometrical configurations with complex boundaries, construction

sequence etc. It has been used as common tool for design and analysis of different

engineering problems, especially for geotechnical analysis. Finite element modelling

has the advantages that parameters can be varied easily and details of stresses and

deformations throughout the system may be studied.

In this study, a finite element analysis program called SAGE CRISP is applied for

analysis.

CRISP was written and developed since 1975 by Mark Zytynski and his colleagues in

the Cambridge University Engineering Department Soil Mechanics Group. It was

previously called “MZSOL” and “CRISTINA”. The new version with a graphical user

interface under Microsoft Windows was launched in 1995 and the version 5.3 used in

this study was released in 2004.

3.2 Model Setup

In this study, various configurations of the cantilever retaining wall and soil properties

will be applied in the finite element analysis to investigate the earth pressure behind

the retaining wall.

20

3.2.1 Model configurations

A 2D plane strain model is applied in the analysis and the in situ gravity level is set to

1 G (9.81 N/m2), as shown in the figure below. To obtain a more accurate result, the

displacement norm tolerance is set to 0.001 and the force norm tolerance is 0.002.

Maximum iterations allowed is set to a larger number of 1000 to assure the accuracy

can be reached.

Figure 3-1, Configuration of calculation

3.2.2 Meshing and Boundary Conditions

The model is set to be 50 m width and 20 meters height. 10 m of the soil is the

foundation and 10 m is the soil backfill. The typical element size is 1 m × 1 m and the

elements next to the wall are smaller as 0.5 m × 0.5 m in order to get more details

about the behaviour of soil near the wall. Eight-node non-consolidation element is

applied in the model. Elements at the edge are lengthened so as to prevent the edge

effect.

21

Figure 3-2, Meshing and boundary conditions of the model

Original Mohr Coulomb Elastic-perfectly plastic material model is assigned for the

soil elements and isotropic elastic material model is assigned for the element of the

retaining wall.

The boundaries are set to pin support at the bottom edge and roller support at the two

edges, as shown in the above figure together with the meshing of the model.

3.2.3 Construction Stages

There are eleven steps to simulate the construction of the retaining wall and the back

fill and each step is performed in five increments. The construction sequence is shown

in the figures below.

0 – In situ condition

1 – Construction of retaining wall

Figure 3-3, Construction squence of the model

22

2 – 1st step of back filling

3 – 2nd

step of back filling

4 – 3rd


5 – 4th


6 – 5th


7 – 6th


8 – 7th


9 – 8th


10 – 9th


11 – last step of back filling

Figure 3-3, Construction squence of the model (cont.)

23

3.3 Material Models

In the study, two constitutive material models have been used to to simulate the soil

behavior. Mohr Coulomb Elastic-perfectly plastic model is mainly used in this study

and the Duncan hyperbolic model is used for comparison with the previous studies.

Besides, isotropic elastic model is applied for the concrete retaining wall and the slip

element model in the program is used at the interface.

3.3.1 Mohr Coulomb Elastic-perfectly plastic model

To simulate the non-linear behaviour of soil, the bilinear stress-strain relationship is

applied for the elastic-perfectly plastic, as shown in the following figure. Based on the

Mohr Coulomb failure criterion, the soil is assumed to be elastic and then become

perfectly plastic once the element is yield.

Figure 3-4, Stress-strain relationship of elastic-perfectly plastic model (From Crisp Technical manual)

F(σ) =1

2(σ1 − σ3) +

1

2(σ1 + σ3) sin φ − c cos φ (3-1)

The main criterion of this model is as the above function. The soil element acts

elastically when σ < 0 and it becomes plastic when σ ≥ 0.

24

Figure 3-5, Mohr-Coulomb Failure Surfaces in three dimensional space (From Crisp Technical manual)

The elastic-perfectly plastic models provide a means of imposing limiting shear stress

(i.e. soil strength) on what would otherwise be a wholly elastic response. However,

there is no hardening or softening of the yield surface during plastic yielding, which

means the shape and size of the yield surface remain constant.

3.3.2 Duncan and Chang Hyperbolic model

Duncan and Chang (1970) developed a hyperbolae stress-strain curve to simulate the

nonlinear behaviour of soil and estimate the earth pressure in the soil elements by

performing incremental analysis which material properties were assigned for each

step. The hyperbolic stress-strain curve and the transformed linear plot are shown

below.

Similar concept and formulation for joint elements were then developed by Clough

and Duncan (1971) representing the soil-structure interface.

25

Figure 3-6, Hyperbolic stress-strain curve and the transformed linear plot (From Duncan 1970)

The following equations were derived to describe the stress-strain behavior:

Tangent stiffness

𝐾𝑠𝑡 = 𝐾𝐼𝛾𝑤 (𝜎𝑛

𝑝𝑎)

𝑛

(1 −𝑅𝑓𝜏

𝜎𝑛 tan 𝛿)

2

(3-2)

Tangent modules

𝐸𝑡 = [1 −𝑅𝑓(1 − sin 𝛷)(𝜎1 − 𝜎3)

2𝑐 cos 𝛷 + 2𝜎3 sin 𝛷] 𝐾𝑝𝑎 (

𝜎3

𝑝𝑎)

𝑛

(primary loading) (3-3)

𝐸𝑡 = 𝐾𝑢𝑟𝑝𝑎 (𝜎3

𝑝𝑎)

𝑛

(unloading and reloading) (3-4)

where KI and K are dimensionless stiffness numbers

γw is the unit weight of water

σn is the normal stress

pa is the atmospheric pressure

n is the stiffness exponent

Rf is the failure ratio

τ is the shear stress

ν is the Poisson’s ratio

26

The modules values were updated in each step by applying the stress value in the

beginning of the increment. By choosing the initial value of the parameters, the

problem can be simulated in the finite element analysis, as shown in the figure below.

Figure 3-7, Comparison of hyperbolic and actual stress-displacement curves (From Clough 1971)

3.3.3 Interface elements

Goodman’s interface element is currently used in CRISP to allow slip to occur

between different materials or materials having different properties. The shear stress

along the interface element is limited by the specified shear strength. After yielding,

the residual shear strength is used in the calculation of element stiffness. When the

element is in tension, the normal stiffness and the shear stiffness are multiplied by

1/10000 so that the element can simulate separation.

The interface element is characterised by the material property matrix, as shown

below.

{∆𝜎∆𝜏

} = [𝐷] {∆휀∆𝛾

} = [𝑘𝑛 00 𝑘𝑠

] {∆휀∆𝛾

} (3-5)

where kn and ks are the elastic normal and shear stiffness

27

3.4 Interpretation of Results

In this study, lateral pressures at two locations are being investigated – the pressure

right behind the retaining wall and at the virtual back, as shown in the following

figure.

Figure 3-8, Location of lateral pressure investigated

The earth pressure at the final stage of the analysis is obtained for different heel length

and different friction angle between the retaining wall and the backfill soil. Besides,

contours of deviatoric strain and status plots are also obtained after the analysis.

Wall back Virtual back

28

CHAPTER 4 PRELIMINARY STUDIES

4.1 Verification of active pressure from numerical simulation

A simple model has been built to verify the method of analysis and the earth pressure

is compared with the past analysis by Clough and Duncan (1971). The soil model by

Duncan and Chang (1970) and the same parameters of soil are applied as in their

analysis and the properties of the interface elements are chosen as the equivalent slip

elements in the program. The tilting of the wall is divided in to 30 increments in the

analysis, so that each of the result is comparable with their analysis.

In this simulation, the fiction angle of zero and equal to the soil internal friction angle

is assigned and parameters used are show below.

Table 4-1, Material properties used in Clough (1971)

Material Parameters Value

Soil Back fill

Unit weight (pcf) 100

At rest coefficient 0.43

Cohesion (psf) 0

Friction angle 35

Primary loading modulus 720

Unloading-reloading modulus 900

Modulus exponent 0.5

Failure ratio 0.8

Poisson’s ratio 0.3

Wall interface

Smooth Rough

Friction angle 0.1 35

Stiffness number 1 75000

Stiffness exponent 0 1

Failure ratio 1 0.9

29

Table 4-2, Material properties used in CRISP


Soil Back fill


At rest coefficient 0.43

Cohesion (psf) 0

Friction angle 35

Primary loading modulus 720

Unloading-reloading modulus 900

Modulus exponent 0.5

Failure ratio 0.8


Wall interface

Smooth Rough

Friction angle 0 35

Cohesion (psf) 0 0

Stiffness number (psf) 27300 27300

Stiffness exponent (psf) 28846 28846

Residual shear stiffness (psf) 0 0

Figure 4-1, Earth pressure distribution in Clough (1971)

30

Figure 4-2, Earth pressure distribution obtained from CRISP

From the figures, it can be seen that the distribution and the magnitude of the lateral

pressure in the two analyses are close to each other and the difference may due to the

different program in used.

On the other hand, the same analysis is conducted again with the elastic-perfectly

plastic model. To compare with the hyperbolic model, equivalent soil parameters are

assigned and the result is shown below.

0

2

4

6

8

10

0 50 100 150 200 250 300 350 400 450

De

pth

fro

m s

urf

ace

, ft

Horizontal earth pressure, psf

Δ/H = 0 Δ/H = 0.0006 Δ/H = 0.0014 Δ/H = 0.0023 Rankine

0

2

4

6

8

10

0 50 100 150 200 250 300 350 400 450

De

pth

fro

m s

urf

ace

, ft



δ = φ

δ = 0

31

Table 4-3, Material properties of Elastic-Plastic model


Soil Back fill


Cohesion (psf) 0

Friction angle 35

Elastic modulus (psf) 72225


Wall interface

Smooth Rough

Friction angle 0 35

Cohesion (psf) 0 0

Stiffness number (psf) 27300 27300

Stiffness exponent (psf) 28846 28846

Residual shear stiffness (psf) 0 0

Figure 4-3, Earth pressure distribution in elastic-perfectly plastic model

As shown in the figure, under the same conditions, the lateral pressure of elastic-

perfectly plastic model in the intermediate increments is smaller and closer to the

active state. It is reasonable as the stress from elastic plastic model is larger the

hyperbolic model at the same strain, which means the elastic plastic material yields

0

2

4

6

8

10

0 50 100 150 200 250 300 350 400 450

De

pth

fro

m s

urf

ace

, ft



δ = 0

32

earlier than the hyperbolic material, as shown in the stress-strain curve below.

Besides, the pressure at the last stage is similar in the two models.

Figure 4-4, Stress-strain curves of the two material model

In this study, the lateral pressure in the final stage is concerned and the variation of

stresses in the intermediate increments is less important. Therefore the elastic-plastic

model which simulates the process faster is better in this analysis.

4.2 Variation of Factor of Safety with heel length

Figure 4-5, Long heel and Short heel

According to the design method mentioned in section 2.1, the Factor of Safety (FS) at

active state with different heel length is calculated.

33

For overturning,

FS =∑ 𝑀𝑅

∑ 𝑀𝑂 (4-1)

For sliding,

FS =∑ 𝐹𝑅

∑ 𝐹𝑑 (4-2)

where

∑ 𝑀𝑅 = ∑ 𝑊𝑖𝑥𝑖 (4-3)

∑ 𝑀𝑂 = 𝑃𝑎ℎ�̅� − 𝑃𝑎𝑣𝑓 (4-4)

∑ 𝐹𝑅 = B𝑐𝑎′ + ∑ 𝑉 tan 𝛿 +

1

2𝑃𝑝 (4-5)

∑ 𝐹𝑑 = 𝑃𝑎ℎ (4-6)

∑ 𝑉 = ∑ 𝑊 + 𝑃𝑎𝑣 (4-7)

𝑃𝑎 =1

2𝛾𝐷2𝐾𝑎 − 2𝑐′𝐷√𝐾𝑎 (4-8)

𝑃𝑝 =1

2𝛾𝐷2𝐾𝑝 + 2𝑐′𝐷√𝐾𝑝 (4-9)

The approach of the value of Pa, Pah and Pav is different and depends on the heel length.

Figure 4-6, FS for Long heel and short heal cases (Modified from GeoGuide 1)

34

Figure 4-7, Location of α and β

A parameter of Ψ is calculated and compared with the angle α for determining

whether it is short heel or long heel, where

Ψ =1

2(90° + φ − β + ε) (4-10)

It is identified as long heel when Ψ ≥ α and short heel when Ψ < α.

For short heel cases, Coulomb’s active coefficient is adopted and the magnitude of the

active force is calculated as follow:

𝑃𝑎ℎ = 𝑃𝑎 cos(𝛿 + 90° − Ψ) (4-11)

𝑃𝑎𝑣 = 𝑃𝑎 sin(𝛿 + 90° − Ψ) (4-12)

where

𝛿 = φ [1 −1

2(

𝛼 − Ψ

90° − Ψ)] (4-13)

ε = sin−1sin 𝛽

sin 𝜑 (4-14)

α and β are shown in Figure 4-6

For long heel cases, Rankine’s active coefficient is adopted and the magnitude of the

active force is calculated as follow:

𝑃𝑎ℎ = 𝑃𝑎 cos(𝛿) (4-15)

𝑃𝑎𝑣 = 𝑃𝑎 sin(𝛿) (4-16)

where

𝛿 = 𝛽 ≤ φ (4-17)

35

After some calculation, the variation of Factor of Safety with heel length is shown in

Figure 4-7.

As shown in the figure, the FS of sliding governs the design at all condition and the

factor of safety for overturning and sliding appears to be discontinuous over the

transition from short heel to long heel walls. This is caused by the different

approaches in the analyses of two types of walls as different geometry of the wall and

earth pressure coefficients were used. However, in reality, the behaviour of the wall

should not change suddenly by slightly increasing the heel length. Therefore, the

finite element analysis in the following chapters will provide further insights of this

problem.

Figure 4-8, Variation of Factor of Safety with heel length

0

1

2

3

4

5

6

7

8

9

10

0 3 6 9 12 15

Fact

or

of

Safe

ty

Heel Length (m)

Overturning Sliding

Short heel Long heel

36

CHAPTER 5 ANALYSIS AND DISCUSSION

5.1 Parametric study

Figure 5-1, Material in the model

In the analysis, Mohr Coulomb Elastic-perfectly plastic model is assigned to the two

layers of soils and isotropic elastic model is for the concrete retaining wall. The

parameters assigned in the analysis are shown in the following tables.

Table 5-1, Material properties

Foundation Soil Backfill Soil Concrete

Elastic modulus, E (GPa) 20 30 270

Poisson’s ratio, ν 0.3 0.3 0.1

Unit weight, γ (kN/m3) 20 20 25

Cohesion, c (kN/m2) 0 0 --

Internal friction angle, Φ 30º 30º --

37

Table 5-2, Properties of Interface elements

Wall interface Base interface

Internal friction angle, Φ 0º (smooth wall) /

30º (rough wall) 30º

Cohesion, c (kN/m2) 0 0

Normal stiffness, kn (GPa) 40.4 40.4

Shear stiffness, ks (GPa) 11.5 11.5

Thickness (m) 0.001 0.001

Table 5-3, Dimension of the retaining wall

Parts Dimension

Height (H) 10 m

Wall thickness (tw) 1 m

Base thickness (tb) 1 m

Heel length (L) varies from 2 m to 15 m

Toe length (T) 2 m

The normal and shear stiffness of the interface elements is calculated according to the

program’s technical manual, where

k𝑛 =𝐸(1 − 𝜈)

(1 + 𝜈)(1 − 2𝜈) (5-1)

k𝑠 =𝐸

2(1 + 𝜈) (5-2)

The heel length of the retaining wall varies from 2 m to 15 m so as to investigate the

effect on the earth pressure due to the heel length.

5.2 Results

After computer analysis, the following pages show part of the figures from the results

of different heel length with rough and smooth walls.

As the displacement of the model is very little, the displacement plots are not shown

in this report.

38

5.2.1 Lateral earth pressure at different heel length

Figure 5-2 to Figure 5-15 shows the lateral earth pressure. The pressure behind the

wall and at the virtual back is obtained from the analysis, as shown Figure 3-8.

From the results, it can be seen that in all case right behind the wall, about 80% of the

height has reached the active state and most of the results of the lateral pressure are

following the traditional theory.

There is a big difference between rough wall and smooth wall that in all length of the

heel, the active pressure of smooth wall is lying on the Rankine’s active pressure (Ka ≈

0.333) and that of rough wall is lying on a line with Ka ≈ 0.275, which is a little

smaller that the Coulomb’s active coefficient (Ka ≈ 0.297). At the bottom of the wall,

the pressure of smooth walls tends to the at-rest pressure and that of rough walls is

reaching a value of about 40.

On the other hand, the lateral earth pressure distribution at virtual back does not form

a smooth line at the overall height, no matter the wall is rough or smooth. However, in

the particular step of the construction, i.e. every meter of soil, the pressure is linear.

From the figures, it can be seen that the variation of heel length have minor effect on

the lateral earth pressure behind the wall. Meanwhile, the lateral pressure at the virtual

back is more or less the same at different heel length

Besides, the pressure distribution at wall back and virtual back acts inversely in the

rough wall cases and smooth wall cases. Where one is going outside to the at rest

pressure and the other one go inside.

39

Figure 5-2, Lateral earth pressure of rough wall at L = 3 m

Figure 5-3, Lateral earth pressure of smooth wall at L = 3 m

10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)

Horizontal stress (kPa)

Behind wall Virtual back At rest Rankine's Active Ka = 0.275

10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)


Behind wall Virtual back At rest Rankine's Active

40



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)


Rough wall (δ = φ) Smooth wall (δ = 0) At rest Active Ka = 0.275

10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



41



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



42



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



43



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



44



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



45



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Elav

atio

n (

m)



46

5.2.3 Contours of deviatoric strain at different heel length

Figure 5-16 to Figure 5-29 shows the contours of deviatoric strain.

As shown in the figures, the contour of deviatoric strain shows differently at different

wall friction. From the figures, it can be seen that the deviatoric strain is concentrated

at a failure surface connecting the top of the wall and the end of the heel but the

behaviour of the failure surface is different.

At the rough wall, the failure surface develops from the end of the heel to about the

top of the wall and it lies near to the wall. When the wall is smooth, after reached to

the wall, the surface reflects to the back fill, which is a big different with the rough

wall. When the heel is long enough, as in Figure 5-28 and Figure 5-29, the surface

does not develop but the deviatoric strain is still concentrated at the back of the rough

wall, while it does not happen in the smooth case.

When the heel length increases, the line of concentration is always having the same

slope. It fails to connect the end of the heel and top of the wall at the heel length of

about 6 m and 7 m, and starts to disappear at a longer heel.

47

Figure 5-16, Contours of deviatoric strain of rough wall at L = 3 m

Figure 5-17, Contours of deviatoric strain of smooth wall at L = 3 m

48



49



50



51



52



53



54

5.2.4 Status plots at different heel length

Figure 5-30 to Figure 5-43 shows the status plots of the soil.

Status plot shows the individual condition of the soil elements. In the status plots, the

colour of green indicates that the soil element has yielded and become active while the

colour of red means that the soil is still in the elastic zone.

The result from the status plots is similar to the contour but shows more details about

the soil.

In the status plots, two failure surfaces can be found where one is developed from the

end of the heel and the other is developed from the bottom of the wall. At some cases,

there is a cross of the two failure surface in the plots.

As in the figures, only the elements on the failure surface have yield in the rough

cases and the whole failure wedge is yield at the smooth cases, together with the

element at the line where the deviatoric strain is concentrated.

It also shows that when the heel length is less than 8 m, there is a failure surface

connecting the heel end to the wall top and this failure surface disappear when the

heel length is larger than 8 m.

Particularly for the smooth walls, after this failure surface disappeared, another failure

surface is developed in the opposite direction. This also happens to the rough walls but

it requires a longer heel.

55

Figure 5-30, Status plot of rough wall at L = 3 m

Figure 5-31, Status plot of smooth wall at L = 3 m

56



57



58



59



60



61



62

5.3 Discussion

5.3.1 Effect on construction sequence

As in the previous figures and the figure shown below, the lateral earth pressure

distribution at virtual back does not form a smooth line at the overall height, no matter

the wall is rough or smooth. However, in the particular step of the construction, i.e.

every meter of soil, the pressure is linear.

The figure shows that the pressure at every meter of soil is linear but not continuous.

The main reason is due to the construction sequence of the back fill.

Figure 5-44, Lateral earth pressure distribution at virtual back of smooth wall at L = 6 m

Figure 5-48 shows the horizontal movement at the top of the retaining wall. The initial

displacement is due to the self-weight of the wall and the unbalance of the base length

of heel and toe.

10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Ele

vati

on

(m

)

Horizontal Stress (kPa)

At rest Rankine's Active

63

As shown in the figure, in the early construction of the backfill, the wall tilts inwards

to the soil backfill, which may increase the pressure of the elements behind. When the

backfill is constructed to about half way, the wall tilts outwards and till the end of the

construction.

Figure 5-45, Horizontal movement at the top of the retaining wall at L = 6 m

The reason of the discontinuous of the pressure distribution may due to this kind of

wall tilting. When the wall tilts inwards, the stress in the element increase and after the

wall tilts outwards, the stress in the element decrease but the outwards displacement of

the element at the virtual back is not enough to achieve the active state, whereas the

elements right behind the wall have enough displacement to reach the active state.

0

1

2

3

4

5

6

7

8

9

10

-0.015 -0.01 -0.005 0 0.005 0.01 0.015

He

igh

of

con

stru

ctio

n (

m)

Horrizontal movement (mm)

64

On the other hand, as the increase of the stress and strain is linear at each increment of

fill, the total strain at the lower layer is obviously larger than the upper layer. But

inside the same layer, due to the wall movement - tilting about the bottom, the strain at

the top of the layer is larger than the bottom.

As a result of these factors, the lateral earth pressure distribution at virtual back is not

linear but shows step by step.

In the previous studies, if only one point at a soil layer, i.e. at the middle, is examined,

the distribution of the lateral pressure would be linear and the effect of the

construction sequence would be ignored, as shown in the following figure.

Figure 5-46, Lateral earth pressure distribution at virtual back at L = 6 m, with only one point in a layer

10

11

12

13

14

15

16

17

18

19

20

0 20 40 60 80 100

Ele

vati

on

(m

)

Horizontal Stress (kPa)

At rest Rankine's Active

65

5.3.2 Effect on wall friction

As shown in the previous figures, there is a big difference that the active pressure of

smooth wall follows Rankine’s active pressure and that of rough wall follows

Rankine’s active pressure. Besides, the contour of deviatoric strain shows differently

at different wall friction. Similar results can be found in the status plots.

The reason of this phenomenon can be explained that when there is friction between

the soil and the wall, the soil is, say, stick to the wall so that the displacement of the

elements near the wall is larger and turn to a more active state, the Coulomb’s active

state. As a result of adhesion, there is stress or strain concentration next to the wall.

Besides, as the soil is stick to the wall, the whole wedge moves away so that only the

elements on the failure surface are yielded.

When there is no friction at the interface, the soil elements activate themselves and

stop at Rankine’s active state. At the same condition, the whole failure wedge turns to

the active state and yielded. As no adhering between the soil and the wall, no

concentration of stress or strain happens at the wall.

5.3.3 Effect on heel length

The following figures show that at any heel length, the earth pressure distribution

would not vary too much. It shows that the soil near the bottom of the retaining wall is

moving away from the active state when the heel length increases, which is reasonable

as the retaining system is more stable having a longer heel.

Meanwhile, the pattern of the lateral pressure at the virtual back is more or less the

same at different heel length. When the heel length increases, the pressure seems to

increase from the active state, which is reasonable with the same reason.

66

Figure 5-47, Lateral earth pressure distribution at wall back of rough wall at different heel length

Figure 5-48, Lateral earth pressure distribution at wall back of rough wall at different heel length

10

11

12

13

14

15

16

17

18

19

20

0 10 20 30 40 50 60 70 80 90 100

Elav

atio

n (

m)


30 50 60 70 80 100 120 At rest Rankine's Active Ka = 0.275

10

11

12

13

14

15

16

17

18

19

20

0 10 20 30 40 50 60 70 80 90 100

Elav

atio

n (

m)


30 50 60 70 80 100 120 At rest Rankine's Active Ka = 0.275

67

Figure 5-49, Lateral earth pressure distribution at virtual back of rough and smooth wall at different heel length

As from the contour of deviatoric strain, when the heel length increases, the l is

always having the same slope and it starts to disappear at about 6 m and 7 m.

On the other hand, the status plots of both rough walls and smooth walls show that

there is a failure surface connecting the heel end to the wall top when the heel length

is less than 8 m and it disappear when the heel length is larger than 8 m. For the

smooth walls, another failure surface is developed in the opposite direction after this

failure surface disappeared.

It is reasonable that as the heel is so long at this stage that the total volume above the

heel is pushed by the soil behind, therefore the failure surface develops out of the

entire soil wedge.

The following figures show the comparison of the resultant lateral force acting behind

10

11

12

13

14

15

16

17

18

19

20

0 10 20 30 40 50 60 70 80 90 100

Elav

atio

n (

m)


30 50 60 70 80 100 120 At rest Rankine's Active

68

the wall and at the virtual wall back.

Figure 5-50, Resultant lateral force at wall back and virtual back of rough wall at different heel length

Figure 5-51, Resultant lateral force at wall back and virtual back of smooth wall at different heel length

Figure 5-50 and Figure 5-51 show the resultant lateral force at different heel length at

280

290

300

310

320

330

340

350

360

370

380

0 2 4 6 8 10 12 14 16

Re

sult

ant

forc

e (

kN/m

)

Heel Length (m)


280

290

300

310

320

330

340

350

360

370

380

0 2 4 6 8 10 12 14 16

Re

sult

ant

forc

e (

kN/m

)

Heel Length (m)


69

the wall back and virtual back of rough wall and smooth wall respectively. From the

figures, when the wall is rough, the resultant force at the wall back is smaller than the

virtual back but it reverses in smooth wall cases, which gives the same results as in the

previous discussion.

Figure 5-52 shows the resultant lateral force at different heel length at the wall back of

rough wall and smooth wall and Figure 5-53 shows the resultant lateral forces at the

virtual back. It shows that the resultant lateral force of smooth wall is larger than the

rough at wall back but it is about the same at the virtual back.

In both of the smooth and rough wall cases, the minimum resultant force at wall back

happens when the heel length is about 6 to 7 meter, while the maximum resultant force

at virtual back happens at the same range.

Figure 5-52, Resultant lateral force at wall back of rough and smooth wall at different heel length

280

290

300

310

320

330

340

350

360

370

380

0 2 4 6 8 10 12 14 16

Re

sult

ant

forc

e (

kN/m

)

Heel Length (m)

Smooth wall Rough wall

70

Figure 5-53, Resultant lateral force at virtual back of rough and smooth wall at different heel length

It is interested that these kinds of behaviour match the criterion mentioned in the

design guides in determining the short heel and long heel cases. The increase at small

heel length may due to the unstable and fluctuation of the pressure and the increase in

the large heel length is because the heel is too long that the pressure is stable even it

does not reach the active state.

However, these findings could not explain the need and the reason of dividing two

conditions for designs. Besides, no other information can show the difference or the

effect between short heel and long heel cases.

280

290

300

310

320

330

340

350

360

370

380

0 2 4 6 8 10 12 14 16

Re

sult

ant

forc

e (

kN/m

)

Heel Length (m)

Rough wall Smooth wall

72

CHAPTER 6 CONCLUSION

In this study, finite element analysis was applied to investigate the lateral earth

pressure behind a cantilever retaining wall of different configuration. Parametric study

on using various wall frictions at the soil structure interface and heel length of the

retaining wall was performed and the result is shown in the previous chapters. The

major findings are as follows:

After the construction of the backfill, the soil behind the wall achieves the active

state in all cases.

The wall tilts inwards in the early stage of construction of the backfill and tilts

outwards afterwards and therefore the construction sequence has influence on the

results.

The interpretation of the analysis data also affects the results, such as taking less

data points provides a smooth and better result but some important details may be

ignored.

The main different between rough wall and smooth wall is that the lateral

pressure behind rough walls follow the Coulomb’s active pressure and the lateral

pressure behind smooth walls follow the Rankine’s active pressure.

The lateral pressure at the virtual back is about the same at both rough wall and

smooth wall cases and it is affected by the construction sequence.

From the results in the previous chapter, the effect of the heel length is not

obvious in terms of earth pressure behind the retaining wall.

73

In the preliminary analysis, the Factor of Safety of long heel and short heel

conditions are not continuous due to different approaches of calculation.

The contours and status plots shows the soil behaviour is continuous in

increasing heel length, which is different from the FS calculation.

The behaviour of the resultant force plots (Figure 5-50 to Figure 5-53) and the

contours and status plots matches the criterion in determining the short heel and

long heel cases.

After the analysis, there is no obvious difference and effect between short heel

and long heel cases as identified in GeoGuide 1 and the result could not explain

the need and reason of the method.

Rankines’s earth pressure theory may be good enough for the design of cantilever

retaining wall.

74

REFERENCE

1. Bang, S. (1985) Active earth pressure behind retaining walls. Journal of

Geotechnical Engineering, American Society of Civil Engineers, Vol.111, No.3,

pp. 407-412.

2. California Department of Transportation (2004) “Bridge Design Specifications”,

Section 5 - Retaining Walls, California Department of Transportation, California.

3. Clayton, C. R. I., Milititsky J. and Woods R. I., (1993) Earth Pressure and Earth-

Retaining Structures, 2nd Edition, Chapman & Hall.

4. Clough, G. W. and Duncan, J. M. (1971) Finite element analyses of retaining

wall behavior. Journal of the Soil Mechanics and Foundations Engineering

Division, American Society of Civil Engineers, Vol. 97, No. SM12, pp. 1657-

1673.

5. CRISP Consortium Ltd, (2001), “Sage Crisp User Manual”.

6. Das, B.M. (2011) Principles of Foundation Engineering, 7th

Ed., Cengage

Learning.

7. Duncan, J. M. and Chang, C. Y. (1970) Nonlinear analysis of stress and strain in

soils. Journal of the Soil Mechanics and Foundations Engineering Division,

American Society of Civil Engineers, Vol. 56, No. SM5, pp. 1629-1653.

8. Fang, Y. S. and Ishibashi, I. (1986) Static earth pressures with various wall

movements. Journal of Geotechnical Engineering, American Society of Civil

Engineers, Vol.112, No.3, pp. 317-333.

9. GCO (1982) Guide to Retaining Wall Design. GeoGuide 1, Geotechnical Control

Office, Engineering Development Department, Hong Kong.

10. Goh, A.T.C. (1993) Behavior of Cantilever Retaining Walls. Journal of

Geotechnical Engineering, American Society of Civil Engineers, Vol. 119,

No.11, pp. 1751-1770.

11. Matsuo, M., Kenmochi S., and Yagi, H. (1978) Experimental study on earth

pressure of retaining wall by field tests. Soils and Foundations, Japanese Society

of Soil Mechanics and Foundation Engineering, Vol. 18, No. 3, pp. 27-41.

12. Nakai, T. (1985) Finite element computations for active and passive earth

pressure problems of retaining wall. Soils and Foundations, Japanese Society of

Soil Mechanics and Foundation Engineering, Vol. 25, No. 3, pp. 98-112.

13. Wang, Y. Z. (2000) Distribution of earth pressure on a retaining wall.

Géotechnique, Vol. 50, No. 1, pp. 83-88.

14. Woods, R. and Rahi, A., (2001), “Sage Crisp Technical Reference Manual”.

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