numerical modelling of seismic behaviour of earth-retaining walls.pdf

7/22/2019 NUMERICAL MODELLING OF SEISMIC BEHAVIOUR OF EARTH-RETAINING WALLS.pdf

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Istituto Universitario

di Studi Superiori

Universit degli

Studi di Pavia

EUROPEAN SCHOOL FOR ADVANCED STUDIES IN

REDUCTION OF SEISMIC RISK

ROSE SCHOOL

NUMERICAL MODELLING OF SEISMIC BEHAVIOUR OF

EARTH-RETAINING WALLS

A Dissertation Submitted in Partial

Fulfilment of the Requirements for the Master Degree in

EARTHQUAKE ENGINEERING

by

RAJEEV PATHMANATHAN

Supervisors: Prof.P.E.PINTODr.C.G.LAI

Dr.P.FRANCHIN

June, 2006


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The dissertation entitled Numerical Modelling of Seismic behaviour of Earth-Retaining

Walls, by Rajeev, has been approved in partial fulfilment of the requirements for the Master

Degree in Earthquake Engineering.

Prof.Paolo.E.Pinto

Dr.Carlo.G.Lai

Dr.Paolo Franchin


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Abstract

i

ABSTRACT

In current engineering practice the design methods for earth retaining walls under seismic conditions

are mostly empirical. Dynamic earth pressures are calculated assuming prescribed seismic coefficient

acting in the horizontal and vertical directions using the concept of limit equilibrium Mononobe-

Okabe method. A research investigation has been undertaken to determine the dynamically-induced

lateral earth pressures on a flexible diaphragm wall, a flexible cantilever wall, and a gravity retaining

wall with cohesionless and cohesive backfills. Additionally, this report gives information about the

point of application of total and incremental dynamic forces, the deformation or displacement of the

wall and also the bending moment and shear force envelops for structural design of the wall. A series

of non-linear dynamic finite element numerical analyses have been performed using DIANA

(DIsplacement ANAlyzer). The analyses consisted of the incremental construction of the wall and

placement or excavation of the backfill, followed by dynamic response analyses, wherein the soil was

modelled as elasto-plastic. Particular attention has been devoted to ground excitation and

determination of the wall and soil model parameters. The results obtained with DIANA have been

compared through a series of benchmark tests with those determined using simplified techniques for

computing dynamic earth pressure, co-seismic and post-seismic wall displacements.

Keywords: Retaining walls, Mononobe-Okabe, Dynamic earth pressure, Finite-element analysis


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Acknowledgements

ii

ACKNOWLEDGEMENTS

I would like to thank Prof.P.E.Pinto, Dr.C.G.Lai and Dr.P.Franchin for their guidance and their

patience. I would also like to thank everyone at TNO DIANA B.V. for the opportunity to work at the

companys offices in Delft.


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Index

iii

TABLE OF CONTENTS

ABSTRACT .............................................................................................................................................I

ACKNOWLEDGEMENTS.................................................................................................................... II

TABLE OF CONTENTS.......................................................................................................................III

LIST OF FIGURES ............................................................................................................................. VII

LIST OF TABLES............................................................................................................................... XII

1. INTRODUCTION ...............................................................................................................................1

1.1 Purpose of the study ....................................................................................................................1

1.2 Motivation of the study................................................................................................................1

2. LITERATURE REVIEW ....................................................................................................................5

2.1 Dynamic earth pressure computation ..........................................................................................5

2.2 Limit-state analysis......................................................................................................................8

2.2.1 Pseudo-static approaches....................................................................................................8

2.2.1.1 Mononobe-Okabe (1926,1929) .................................................................................8

2.2.1.2 Arango (1969) .........................................................................................................10

2.2.1.3 Choudhury (2002) ...................................................................................................12

2.2.1.4 Ortigosa (2005) .......................................................................................................18

2.2.2 Pseudo-dynamic approach................................................................................................192.2.2.1 Steedman-Zeng (1990)............................................................................................19

2.2.2.2 Choudhury-Nimbalkar (2005).................................................................................20

2.2.3 Displacement-based analysis............................................................................................22

2.2.3.1 Richards-Elms model ..............................................................................................22

2.2.4 Comparison of seismic earth pressure values computed using different approaches.......23

2.3 Closed form solutions using elastic or viscous elastic behaviour..............................................23

2.3.1 Wood (1973).....................................................................................................................23

2.3.2 Veletsos and Younan (1994) ............................................................................................242.4 Numerical analyses....................................................................................................................26


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Index

iv

2.4.1 Al-Homoud and Whitman (1999).....................................................................................26

2.4.2 Green and Ebeling (2003) ................................................................................................27

2.4.3 Psarropoulos, Klonaris, and Gazetas (2005) ....................................................................28

3. TYPE OF RETAINING WALLS ANALYZED ...............................................................................31

3.1 Diaphragm wall-soil system ......................................................................................................31

3.2 Cantilever wall-soil system........................................................................................................32

3.3 Gravity wall-soil system............................................................................................................32

3.4 Properties of soil........................................................................................................................33

3.5 Properties of concrete ................................................................................................................34

4. SELECTION AND PROCESSING OF GROUND MOTION..........................................................35

4.1 Selection criteria ........................................................................................................................35

4.2 List of ground motion ................................................................................................................35

4.3 Characteristics of ground motion selected.................................................................................36

4.4 Processing of the selected ground motions................................................................................40

5. MODELLING ISSUES AND CHOICES..........................................................................................42

5.1 Finite element modeling of soil-structure system......................................................................42

5.1.1 Soil constitutive model .....................................................................................................42

5.1.1.1 Mohr-Coulomb........................................................................................................43

5.1.1.2 Pastor and Zienkiewicz(l986) [P-Z mark III model]...............................................44

5.1.1.3 HiSS soil model [Hierarchical single surface soil model].......................................45

5.1.1.4 Hyperbolic type Osaki model..................................................................................46

5.1.2 Boundaries........................................................................................................................47

5.1.2.1 Elementary boundaries............................................................................................48

5.1.2.2 Local or transmitting boundaries.............................................................................48

5.1.2.3 Consistent boundaries .............................................................................................50

5.1.3 Soil-structure interface .....................................................................................................50

5.1.4 Size of finite element mesh ..............................................................................................51

5.2 Overview of DIANA .................................................................................................................51

5.3 Numerical model .......................................................................................................................52

5.3.2 Details of diaphragm wall numerical model.....................................................................53

5.3.3 Details of cantilever wall numerical model......................................................................55

5.3.4 Details of gravity wall numerical model ..........................................................................58

5.3.5 Model parameters for soil.................................................................................................59

5.3.6 Model parameters for wall................................................................................................61

5.3.7 Interface element ..............................................................................................................61

5.3.8 Dimensions of finite element mesh ..................................................................................61


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Index

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5.3.9 Damping ...........................................................................................................................61

6. DIANA RESULTS AND DISCUSSION ..........................................................................................63

6.1 Data simplification ....................................................................................................................63

6.1.1 Determination of forces assuming constant stress distribution ........................................63

6.1.2 Incremental dynamic forces .............................................................................................65

6.1.3 Reaction height of forces..................................................................................................65

6.1.4 Dynamic earth pressure coefficient ..................................................................................65

6.2 Presentation and discussion of simplified data..........................................................................66

6.2.1 Phased analysis stress distribution....................................................................................66

6.2.2 Dynamic analysis stress distribution ................................................................................70

6.2.2.1 Pressure distribution along the diaphragm wall ......................................................70

6.2.2.2 Pressure distribution along the stem of the cantilever wall .....................................74

6.2.2.3 Pressure distribution along the height of the gravity wall.......................................76

6.2.3 Design lateral earth pressure coefficient and DIANA computed lateral earth pressure

coefficient ..............................................................................................................................................77

6.2.3.1 Comparison of lateral earth pressure coefficients for diaphragm wall....................78

6.2.3.2 Comparison of lateral earth pressure coefficients for cantilever wall .....................81

6.2.3.3 Comparison of lateral earth pressure coefficients for gravity wall .........................83

6.2.4 Point of application of total dynamic forces and incremental dynamic forces.................85

6.2.4.1 Point of application of total dynamic forces for diaphragm wall ............................86

6.2.4.2 Point of application of total dynamic forces and incremental dynamic forces for

cantilever wall........................................................................................................................................88

6.2.4.3 Point of application of total dynamic forces and incremental dynamic forces for

gravity wall ............................................................................................................................................89

6.2.5 Deformation and displacement of the wall.......................................................................91

6.2.5.1 Deformation of diaphragm wall ..............................................................................92

6.2.5.2 Deformation of cantilever wall................................................................................94

6.2.5.3 Deformation of gravity wall....................................................................................95

6.2.6 Bending moment and shear force.....................................................................................96

6.2.6.1 Bending moment and shear force envelop for diaphragm wall...............................96

7. SUMMARY AND CONCLUSIONS ..............................................................................................104

7.1 Diaphragm wall .......................................................................................................................104

7.2 Cantilever wall.........................................................................................................................104

7.3 Gravity wall .............................................................................................................................105

7.4 Problems encountered in DIANA modelling...........................................................................105

7.4.1 Type of element and material constitutive model...........................................................105


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Index

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7.4.2 Interface element and transmitting boundary .................................................................106

REFERENCES ....................................................................................................................................107

8. APPENDIX A STATIC DESIGN OF DIAPHRAGM WALL .....................................................A1

9. APPENDIX B STATIC DESIGN OF CANTILEVER RETAINING WALL .............................. B1

10. APPENDIX C DESIGN OF GRAVITY WALL......................................................................... C1

11. APPENNDIX D ADDITIONAL RESULTS FROM DIANA ANALYSES...............................D1

12. APPENDIX E DIANA COMMAND FILES ...............................................................................E1


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Index

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LIST OF FIGURES

Figure 1.1. Damage to a retaining wall due to excessive displacement 2004 Niigata-Ken Chuetsu

earthquake .......................................................................................................................................2

Figure 1.2. Overturning failure of retaining wall the Shin-Kang Dam due to Chi-Chi earthquake.........3

Figure 1.3. Top layers moved away from backfill along the construction joints due to Chi-chi

earthquake Taiwan ..........................................................................................................................3

Figure 1.4. The upper section of the retaining wall was uplifted by the thrust fault and the retaining

wall was sheared due to Chi-Chi earthquake Taiwan. ....................................................................4

Figure 2.1. Classification of seismic earth pressure computation methods.............................................7

Figure 2.2. Failure surface and the forces considered by Mononobe-Okabe...........................................9

Figure 2.3. Analytical model ....................................................................................................12

Figure 2.4. Free body diagram ..................................................................................................13

Figure 2.5. Composite failure surface and forces considered...................................................14

Figure 2.6. Comparison of passive pressure computation using the method proposed by

Choudhury (Choudhury et al 2002) ..................................................................................18

Figure 2.7. System considered by Steedman-Zeng...................................................................19

Figure 2.8. System considered by Choudhury-Nimbalkar........................................................21

Figure 2.9. Base-excited soil-wall system investigated ............................................................24

Figure 2.10. Mononobe-Okabe active and passive expressions (yielding backfill), Wood

expression (nonyielding backfill), and FLAC (Continued) (after Green & Ebeling 2003)

...........................................................................................................................................28

Figure 3.1. Dimension of diaphragm wall ................................................................................31

Figure 3.2. Dimension of cantilever wall..................................................................................32

Figure 3.3. Dimension of gravity wall ......................................................................................33

Figure 4.1. Imperial Valley (1940) acceleration time-history ..................................................36


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Index

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Figure 4.2. Imperial Valley (1940) Pseudo-Acceleration spectrum corresponding 5% damping

...........................................................................................................................................36

Figure 4.3. Imperial valley (1940) Arias intensity....................................................................37

Figure 4.4. Chi-Chi (1999) acceleration time-history...............................................................37

Figure 4.5. Chi-Chi (1999) Pseudo-Acceleration spectrum corresponding 5% damping ........38

Figure 4.6. Chi-Chi (1999) Arias intensity ...............................................................................38

Figure 4.7. Kobe (1995) acceleration time-history...................................................................39

Figure 4.8. Kobe (1995) Pseudo-Acceleration spectrum corresponding 5% damping.............39

Figure 4.9. Kobe (1995) Arias intensity ...................................................................................40

Figure 5.1. Shape of yield surfaces in J1-J2D space.................................................................46

Figure 5.2. Non-linear constitutive law for soil ........................................................................46

Figure 5.3. The dashpot model proposed by Lysmer & Kuhlemeyer.......................................49

Figure 5.4. Compound parabolic callectors ..............................................................................49

Figure 5.5. Lumped-parameter consistent boundary ................................................................50

Figure 5.6. CQ16E 8-node 2-D plane strain element................................................................52

Figure 5.7. SP2TR 2-node translation spring/dashpot ..............................................................53

Figure 5.8. Finite element mesh for diaphragm wall ................................................................53

Figure 5.9. Deformed mesh at the end of the each phased construction (sand), magnification

factor 50 ............................................................................................................................55

Figure 5.10. Finite element mesh for cantilever wall ...............................................................55

Figure 5.11. Deformed mesh after placing of backfill (sand), magnification factor 150 .........58

Figure 5.12. Finite element mesh for gravity wall ....................................................................59

Figure 5.13. Deformed mesh after constructing and placing backfill (sand)............................59

Figure 6.1. Constant stress distribution approximation across the element (from Green &

Ebeling 2003)....................................................................................................................64Figure 6.2. Horizontal acceleration ah, corresponding dimensionless horizontal inertial

coefficient kh, of a point in the backfill portion of sliding wedge....................................66

Figure 6.3. Pressure distribution along the diaphragm wall in sand at the end of the phased

analysis..............................................................................................................................67

Figure 6.4. Pressure distribution along the diaphragm wall in clay at the end of the phased

analysis..............................................................................................................................67

Figure 6.5. Pressure distribution along the cantilever wall in sand at the end of the phased

analysis..............................................................................................................................68


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Index

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Figure 6.6. Pressure distribution along the cantilever wall in clay at the end of the phased

analysis..............................................................................................................................69

Figure 6.7. Pressure distribution along the gravity wall in sand at the end of the phased

analysis..............................................................................................................................69

Figure 6.8. Pressure distribution along the gravity wall in clay at the end of the phased

analysis..............................................................................................................................70

Figure 6.9. Pressure distribution along the diaphragm wall in sand at the end of the dynamic

analysis (EL Centro) .........................................................................................................71


analysis (Chi-Chi) .............................................................................................................71


analysis (Kobe) .................................................................................................................72

Figure 6.12. Pressure distribution along the diaphragm wall in clay at the end of the dynamic

analysis (EL Centro) .........................................................................................................72


analysis (Chi-Chi) .............................................................................................................73


analysis (Kobe) .................................................................................................................73

Figure 6.15. Comparison of pressures along the stem of the cantilever wall at the end of the

phased and dynamic analysis (sand, EL Centro) ..............................................................74


phased and dynamic analysis (sand, Chi-Chi) ..................................................................75


phased and dynamic analysis (sand, Kobe) ......................................................................75

Figure 6.18. Pressure along the gravity wall at the end of the dynamic analysis (Sand) .........76Figure 6.19. Pressure along the gravity wall at the end of the dynamic analysis (Clay)..........77

Figure 6.20. Comparison of active and passive lateral earth pressure coefficient (KDIANA)

back-calculated from DIANA results with values computed using the Mononobe-Okabe

expressions (Sand) ............................................................................................................79

Figure 6.21. Comparison of active and passive lateral earth pressure coefficient (KDIANA)

back-calculated from DIANA results with values computed using the Mononobe-Okabe

expressions (Clay).............................................................................................................80


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Index

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Figure 6.22. Comparison of active lateral earth pressure coefficient (KDIANA) back-

calculated from DIANA results with values computed using the Mononobe-Okabe











Figure 6.26. Point of application of total active dynamic force for diaphragm wall................86

Figure 6.27. Point of application of total passive dynamic force for diaphragm wall in sand .87

Figure 6.28. Point of application of total passive dynamic force for diaphragm wall in clay..87

Figure 6.29. Point of application of total dynamic force at stem and heel section of cantilever

wall....................................................................................................................................88

Figure 6.30. Point of application of incremental dynamic force at stem and heel section of

cantilever wall ...................................................................................................................89

Figure 6.31. Point of application of total dynamic force of gravity wall..................................90

Figure 6.32. Point of application of incremental dynamic force for gravity wall.....................91

Figure 6.33. Deformed shape of the diaphragm wall in sand at different time step (EL Centro)

...........................................................................................................................................92

Figure 6.34. Deformed shape of the diaphragm wall in sand at different time step (Chi-Chi) 92

Figure 6.35. Deformed shape of the diaphragm wall in clay at different time step (EL Centro)...........................................................................................................................................93

Figure 6.36. Deformed shape of the diaphragm wall in clay at different time step (Chi-Chi) .93

Figure 6.37. Annotated deform mesh from EL Centro analysis ...............................................94

Figure 6.38. Relative permanent displacement time-history of base of the cantilever wall in

sand (El Centro) ................................................................................................................95

Figure 6.39. Bending moment envelop for diaphragm wall in sand (EL Centro) ....................96

Figure 6.40. Shear force envelop for diaphragm wall in sand (EL Centro)..............................97

Figure 6.41. Bending moment envelop for diaphragm wall in sand (Chi-Chi) ........................97


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Index

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Figure 6.42. Shear force envelop for diaphragm wall in sand (Chi-Chi)..................................98

Figure 6.43. Bending moment envelop for diaphragm wall in clay (EL Centro).....................98

Figure 6.44. Shear force envelop for diaphragm wall in clay (EL Centro) ..............................99

Figure 6.45. Bending moment envelop for diaphragm wall in clay (Chi-Chi).........................99

Figure 6.46. Shear force envelop for diaphragm wall in clay (Chi-Chi) ................................100

Figure 6.47. Bending moment envelop for cantilever wall in sand ........................................100

Figure 6.48. Bending moment envelop for cantilever wall in clay.........................................101

Figure 6.49. Shear force envelop for cantilever wall in sand .................................................101

Figure 6.50. Shear force envelop for cantilever wall in clay ..................................................102

Figure 6.51. Maximum pressure envelop along the height of the gravity wall in sand..........102

Figure 6.52. Maximum pressure envelop along the height of the gravity wall in clay ..........103


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Index

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LIST OF TABLES

Table 2.1 Comparison of Kpd values obtained by Choudhurys method and available theories in

seismic case for=0, i=0, =30 ................................................................................................17

Table 2.2 Comparison of seismic earth pressure computation using different approach ......................23

Table 3.1 Properties of sand ..................................................................................................................33

Table 3.2 Properties of clay ...................................................................................................................34

Table 3.3 Properties of reinforced concrete...........................................................................................34

Table 4.1 Ground motion.......................................................................................................................35

Table 5.1 Typical parameters for P-Z model.........................................................................................45

Table 5.2 DIANA input properties of sand............................................................................................60

Table 5.3 DIANA input properties of clay ............................................................................................60

Table 5.4 DIANA input properties of concrete .....................................................................................61

Table 6.1 Pressure values along the height of the cantilever wall .........................................................68

Table 6.2 Relative permanent displacement of base of the cantilever wall ...........................................95

Table 6.3 Relative permanent displacement of base and permanent tilt of gravity wall .......................95


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Chapter 1 Introduction

1

1.INTRODUCTION

1.1Purpose of the study

Understanding the behaviour of earth retaining structures in seismic events is one of the oldest

problems in geotechnical engineering. The devastating effects of earthquakes make the

problem more important. Despite the multitude of studies that have been carried out over the

years, the dynamic response of earth retaining structures is far from being well understood. As

a result, current engineering practice lacks conclusive information that may be used in design.

The most commonly used methods to design retaining structures under seismic conditions are

force equilibrium based pseudo-static analysis (e.g. Mononobe-Okabe 1926, 1929), pseudo-

dynamic analysis (Steedman and Zeng 1990), and displacement based sliding block method

(e.g. Richards and Elms 1979).

Even under static conditions, prediction of actual retaining wall forces and deformations is a

complicated soil-structure interaction problem. The dynamic response of even the simplest

type of retaining wall is quite complex. The dynamic response depends on the mass and

stiffness of the wall, the backfill and the underlying ground, the interaction among them and

the nature of the input motions.

The purpose of this study is to develop finite element numerical models to understand the

dynamic behavior of retaining structures, and, in particular, to find the magnitude and

distribution of dynamic lateral earth pressure on the wall, as well as the displacement and

forces induced by horizontal ground shaking. Retaining structures considered include aflexible diaphragm wall, a cantilever wall and a gravity wall. In all the analyses, the soil is

assumed to act as a homogeneous, elasto-plastic medium with Mohr-Coulomb failure

criterion and the walls are assumed to act as linear elastic. The numerical models for the three

walls have been developed using DIANA, a commercially available finite element program.

The numerical analyses encompass the incremental construction of the wall and the placement

of the backfill or excavation of soil, followed by the seismic response analysis. Particular

attention is given to how the ground motions are selected, processed and prescribed as an

external loading to the numerical model.

The results obtained with DIANA model are compared with results from simplified analysis

techniques for computing dynamic earth pressure.

1.2Motivation of the study

Several types of structures, such as various gravity walls and cantilever sheet pile walls, are

used to retain soils in two different levels such as slope and abutments of highway bridges. In

order to evaluate the stability of earth-retaining structures during earthquakes the seismic

earth pressures and their point of application must be estimated.

Many authors have reported [4, 20] numerous cases of damage or failure of bridges inducedby excessive abutment displacement or failure during recent earthquakes. Observed failures of


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2

retaining walls were due to sliding, overturning and loss of the bearing capacity of soil

underlying the wall.

In 1989, the Loma Prieta Earthquake, a severe 7.1 Richter magnitude event, shook the San

Francisco area, causing serious damage to bridges and buildings. In 1994, the Northridgeearthquake, a severe 6.7 Richter magnitude event, shook the densely populated San Fernando

Valley, 20 miles northwest of Los Angeles. Severe damage occurred to buildings, bridges and

freeways. These bridge related damages were mostly due to the failure of retaining walls.

During the October 23, 2004 Chuetsu earthquake, several residential developments

constructed on reclaimed land in Nagaoka city, Niigata Prefecture, have experienced damages

to houses and roads due to seismically-induced failure of artificial fill slopes. Post-earthquake

field reconnaissance surveys revealed that many fill slope failures were caused by the

excessive seismic displacements of the gravity retaining walls supporting the fill material.

Figure (1.1) shows a retaining wall failure due to the excessive displacement, during the

Niigata-Ken Chuetsu earthquake 2004.

Figure 1.1.Damage to a retaining wall due to excessive displacement 2004 Niigata-Ken Chuetsuearthquake

Figure (1.2) shows retaining walls failed due to the overturning, during the Chi-Chi

earthquake (1999).


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3

Figure 1.2.Overturning failure of retaining wall the Shin-Kang Dam due to Chi-Chi

earthquake

Figure (1.3) shows the top layers moved way from the backfill along the construction joints

due to the inadequate frictional resistance at the top portion of the retaining wall.

Figure 1.3.Top layers moved away from backfill along the construction joints due to Chi-chiearthquake Taiwan

Figure (1.4) shows the wall uplifted by thrust fault, vertical and horizontal displacement of

2.0m and 1.3m respectively.


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4

Figure 1.4.The upper section of the retaining wall was uplifted by the thrust fault and the

retaining wall was sheared due to Chi-Chi earthquake Taiwan.

The 1995 Kobe earthquake provides many opportunities for documenting the behavior of

retaining structures in waterfront areas, along transportation facilities, and throughout various

public and private developments. Many walls failed catastrophically, but some survived

virtually intact. This should provide an opportunity to understand better some important issuesconcerning the design of these structures.

Compiling information on what happened is essential so that researchers can try to understand

why the structures behaved as they did. There are at least few factors that may have

contributed to the movements or failure of earth retaining structures such as: (1) inertial forces

on the wall themselves, (2) dynamic lateral pressures from the backfill in the absence of

liquefaction, (3) static and dynamic pressures associated with the liquefaction of the backfill,

and (4) reduced resistance to sliding because of liquefaction of soils surrounding the base of

the wall in waterfront areas. There are no data that identify directly the relative contribution of

these factors, thus requiring studies using numerical and possibly physical models to clarifytheir interactions. The problem is complex, and developing an understanding of it is so

complex that it will require refined finite-element or finite-different model analyses.


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Chapter 2 Literature review

5

2.LITERATURE REVIEW

2.1Dynamic earth pressure computation

The methods that are used to compute the dynamic earth pressure on the retaining wallsnowadays can be classified into three main groups:

(1)Limit state analyses, in which a considerable relative movement occurs between the

wall and soil to mobilize the shear strength of the soil

(2)Elastic analyses, in which the relative movement in between the soil and wall is

limited, therefore the soil behaves within its linear elastic range. The soil can be

considered as a linear elastic material.

(3)Numerical analyses, in which the soil is modelled with actual non-linear hysteretic

behaviour.

The limit-state analyses were developed by Mononobe and Okabe (Mononobe and Matuo

1929; Okabe 1924). The Mononobe-Okabe approach has several variants (Kapila 1962,

Arango 1969, Seed and Whitman 1970; Richards and Elms 1979; Nadim and Whitman 1983,

Richards et al 1999, Choudhury 2002). A wedge of soil bounded by the wall is assumed to

move as a rigid block, with prescribe a horizontal and a vertical acceleration. This method

was basically developed to calculate the active and passive earth pressure for dry cohesionless

materials by Mononobe-Okabe. The use of a graphical construction, such as Coulomb or

Melbye construction procedure, has been described by Kabila (1962). Arango (1969) has

developed a simple procedure for obtaining the value of the dynamic lateral earth pressurecoefficient for active conditions from standard charts for static lateral earth pressure

coefficient for active condition using Coulomb method.

The contributions for the elastic analyses came from the works done by Matuo and Ohara

(1960), Wood (1973), Scott (1973), Veletsos and Younan (1994a, 1994b, 1997, 2000), Li

(1999), and Ortigosa and Musante (1991). In particular, Wood (1973) analyzed the dynamic

response of homogeneous linear elastic soil trapped in between two rigid walls connected to a

rigid base, providing an analytical exact solution. An approximate model proposed by Scott

(1973) represents the retaining action of the soil by a set of massless, linear horizontal springs.

The stiffness of the springs is defined as subgrade modulus. Veletsos and Younan (1994a,1994b, 1997, and 2000) improved the Scotts model, by using a semi-infinite, elastically

supported, horizontal bars with distributed mass, to include the radiational damping of the soil

and using horizontal springs with constant stiffness, to model the shearing action of the

stratum. Li (1999) included the foundation flexibility and damping into the Veletsos and

Younan analyses. In this study, the rigid wall with viscoelastic backfill is considered to rest on

viscoelastic half-space foundation. Ortigosa and Musante (1991) proposed a simplified

kinematic method, in which the wall is supported in several locations. The possible wall

movement is the flexural deformation. The free-field shear modulus is used to calculate the

subgrade modulus.


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6

Detailed accounts of pervious analytical and experimental studies on the limit state and elastic

analysis matter have been presented by Nazarian and Hadjian (1979), Prakash (1981),

Whitman (1991), and Veletsos and Younan (1995).

In elastic analyses in which the wall is considered to be fixed against both deflection androtation at the base, usually the wall pressure and associated forces computed are 2.5 to 3

times larger than those determined by the Mononobe-Okabe approach, hence elastic solutions

are generally believed to be excessively conservative and inappropriate for use in design

applications. Conclusions from some recent exploratory studies (Finn et al.1989; Siller et al.

1991; Sun and Lin 1995) suggest that the existing elastic solutions are limited to

nondeflecting rigid walls and do not provide for the important effect of wall flexibility. A

recent study by Veletsos and Younan (1994b) concluded that for walls that are rigid but

elastically constrained against rotation at their base, both the magnitude and distribution of the

dynamic wall pressures and forces are quite sensitive to the flexibility of the base constraint.

For realistic base flexibilities these effects may be significantly lower than those computed fornondeflecting rigid walls. Moreover, Li (1999) showed that, if foundation compliance is taken

into account, the computed based shear may be of the same order of that estimated with

Mononobe-Okabe, even for a rigid gravity wall. Therefore, after these studies, the initial

limitations to the elastic approach seem to be overcome and this method might be considered

as a valuable tool for the seismic design of non-yielding walls.

The third group involves nonlinear numerical analysis to find earthquake-induced

deformations of retaining walls. Numerical analyses should be capable of accounting for

nonlinear, inelastic behaviour of the soil and of the interfaces between the soil and wall.

Among the relatively few examples of numerical analyses that are finite element and/or finitedifference methods are those reported by Alampalli and Elgamel (1990), Finn et al (1992), Iai

and Kameoke (1993), Al-Homoud and Whitman (1999), Green and Ebeling (2003)

Psarropoulos, Klonaris and Gazetas (2005) for different type and configurations of retaining

walls.

The flowchart below (Figure 2.1) summarizes the theories and method those are used to

design the retaining walls in dynamic conditions.


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7

MethodsUsedtoAnalyzetheRetainingWalls

inSeismicConditions

LimiteStateAnalysis

Clos

edFormSolution

Numerica

lAnalysis

ForcedBasedAnalysis

DisplacementBased

Analysis

Pseudo-Static

Approach

Pseudo-Dy

namic

Approa

ch

Richards-Elms

model(1979)

Veletsos&Younan

(1994[a,b],1996,

2000)

Wood(1973)

Scott(1973)

Alampalli-Elgamll

(19

90)

Finn

(1992)

Iai-K

ameoke

(1

993)

A

ctivepressure

computation

without

cohession M

ononobe-Okabe

(1926,1929)

Arango(1969)

[modificationfor

Mononobe-Okabe]

Seed-Whitman

(1970)[modification

forMononobe-Okabe]

Soubra(2000)

Morisson-

Ebeling(1995)

Mononobe-Okabe

(1926,1929)

Passivepressure

computation

without

cohession

Kumar(2001)

Choudhury(2002)

Ortigasa(2

005)

Earthpressure

computation

withcohession P

rakash

(1981)

Choudhury(2002)

Richards-Shi

(1999)

[interactionm

odel]

Steedman-Zeng(1990)

[activepressurecom

putation-

onlyhorizontal

seismic

acceleratio

n]

Choudhury-Nimbalkar

(2005)[active&

passive

pressurecompu

tation-

vertical&horizontalseismic

acceleratio

n]

Nadim-Whitman

(1983)[stochastic

nonlinearmodel]

M

atuo-Ohara(1960)

Li

(1999)

Ortigosa-Musante

(1991)

Psarropoulos,

Klonaris,&Gazetas

(2

005)

Green&

Ebeling

(20

03)

Al-Ho

moud&

Whitma

n(1999)

(Newmark'sMethod)

Figure 2.1.Classification of seismic earth pressure computation methods


22/159


8

2.2Limit-state analysis

2.2.1Pseudo-static approaches

The seismic stability of earth retaining structures is usually analyzed by the pseudo-static

approach in which the effects of earthquake action are expressed by constant horizontal and

vertical acceleration attached to the mass. The common form of pseudo-static analysis

considers the effects of earthquake shaking by pseudo-static accelerations that produce inertia

forces, Fh and Fv, which act through the centroid of the failure mass in the horizontal and

vertical directions respectively. The magnitudes of the pseudo-static forces are;

(2.1) Wkg

WaF h

hh =

=

(2.2) Wkg

WaF

v

v

v

=

=

where,

ah and av - horizontal and vertical pseudo-static accelerations

kh and kv - coefficients of horizontal and vertical pseudo-static accelerations

W- weight of the failure wedge.

A pseudo-static analysis is relatively simple and very straightforward. Representation of the

complex, transient, dynamic effects of earthquake shaking by a single constant unidirectionalpseudo-static acceleration is obviously quite crude. Experiences have shown that pseudo-

static analysis can be unreliable for soils that build up large pore pressures or show more than

about 15% degradation of strength due to earthquake shaking. [see Kramer, 1996]

2.2.1.1Mononobe-Okabe (1926,1929)

Okabe (1926) [34], Mononobe and Matsuo (1929) [30] were the early pioneers to obtain the

active and passive earth pressure coefficients under seismic conditions. It was an extension of

Coulombs method in the static case for determining the earth pressures by considering the

equilibrium of a triangular failure wedge. The method is now commonly known as Mononobe

- Okabe method. For active and passive cases, planar rupture surfaces were assumed in theanalysis. Figure (2.2) shows the failure surfaces and the forces considered in the analysis.

The Mononobe-Okabe approach is valuable in providing a good assessment of the magnitude

of the peak dynamic force acting on a retaining wall. However, the method is based on three

fundamental assumptions,

1. The wall has already deformed outwards sufficiently to generate the minimum (active)

earth pressure.


23/159


9

2. A soil wedge, with a planar sliding surface running through the base of the wall, is on

the point of failure with a maximum shear strength mobilized along the length the

surface.

3. The soil behind the wall behaves as a rigid body so that acceleration can be assumedto be uniform throughout the backfill at the instant of failure.

Pa

pP

i

i.

.

.

PE

.

.

AE.

.

Active

Passive

plane failure surface

plane failure surface

khW

kvWah=khg

av=kvg

kvW

khW

Figure 2.2.Failure surface and the forces considered by Mononobe-Okabe

The expression for computing the seismic active and passive earth force,Pae,pe, is given by

(2.3) peaevpeae KkHP ,2

, )1(2

1=

And

(2.4)2

5.0

2

2

,

)cos()cos(

sin()sin(1)cos(coscos

)(cos

+++

=

i

i

K peae

m

m

Where

- unit weight of soil

H - vertical height of the wall

Kae,pe- seismic active and passive earth pressure coefficient

- soil friction angle


24/159


10

- wall friction angle

- wall inclination with respect to vertical

i - ground inclination with respect to horizontal

kh - seismic acceleration coefficient in the horizontal direction

kv- seismic acceleration coefficient in the vertical direction

(2.5)

=

v

h

k

k

1tan 1

Seed and Whitman (1970) [43] gave convenient solutions for practical purposes for the

incremental dynamic force in equation (2.3) for the active pressure condition, and gave an

approximate solution for the case of zero vertical acceleration, a vertical wall, horizontal

backfill, and effective friction angle approximately 35. Their approximation can be expressed

as

(2.6) hae k

H

P

8

32

=

in which Pae active wall force increment due to horizontal earthquake load

The approximation is in close agreement with the more exact solution forkh


25/159


11

general solution for any inclination of wall and backfill slope, any angle of wall friction and

any value of angle of friction of backfill material and earthquake acceleration.

The classical Coulomb solution for static earth pressure can be expressed as

(2.7) aa KHP2

2

1=

Where,

(2.8)( )

( )( ) ( )( ) ( )

2

2

2

coscos

sinsin1coscos

cos

++

+

=

i

iKa

Pa

may thus be expressed in the following form

(2.9) caa AHKHP

2

22

cos

1

2

1

2

1==

WhereAc =Ka cos2

On the other hand Mononobe-Okabe value forKae can be written as follows:

(2.10) ( ) ( ) mvaevae AkHKkHP

2

22

coscos

11

2

11

2

1==

WhereAm =Kae coscos2

A comparison of the expression forAm with that forAc shows thatAm can be determine from

the solution forAc by redefining the slope of the back of the wall and as * and the inclination

of the backfill as i*, where

* = + and i* = i+

Thus

(2.11) ( ) ( )

== 2

cos,, iKiAA acm And

(2.12) ( )FkiPP vaae =

1,

Where

( ) ( )

2

2

2

2

coscos

cos

coscos

cos +==

F

Values forFcorresponding to different values for and have been computed by Arango.

Since charts are available for determiningPa for a wide variety of combination of, , i, and


26/159


12

, the corresponding dynamic pressure determined by the Mononobe-Okabe analysis can be

readily obtained.

2.2.1.3Choudhury (2002)

Terzaghi (1943) [50] showed that active earth pressures determined assuming a planar rupture

surface almost match the exact or experimental values of earth pressures, while for the passive

case, when wall friction angle, , exceeds one-third of soil friction angle, , the assumption of

planar failure surface seriously overestimates the passive earth pressures. To correct the error

in the Mononobe-Okabe method for the passive case, Morisson and Ebeling (1995), Soubra

(2000) and Kumar (2001) considered curved rupture surfaces in their analysis of the passive

case. However, all of these analyses were performed only for sands.

Initially Choudhury, Subba Rao and Ghosh gave a complete solution for the distribution of

seismic passive pressure behind rigid retaining walls using the method of horizontal slices by

considering seismic forces in a pseudo-static manner. Only planar rupture surfaces have been

considered and hence wall friction angle has been restricted up to one-third the soil friction

angle. This approach results is the same seismic passive earth pressure coefficients as those

obtained by Mononobe-Okabe approach, besides giving additional information about the

distribution of earth pressures. It has been found that in the seismic case, passive resistance

acts at a point other than at 1/3rd from the base of the wall. Under seismic conditions, the

extension of the failure zone is more than that under static conditions.

Figure (2.3) shows the system considered by Choudhury, Subba Rao and Ghosh, a rigid

retaining wall of vertical height H, supporting dry, homogeneous cohesionless backfill

material with horizontal ground. The seismic forces are considered as pseudo-static forces

along with other static forces. The equilibrium of each elemental slice is considered. It is

assumed that the occurrence of earthquake does not affect the basic soil parameters such as

soil friction angle and soil unit weight .

displacement

y

H

dy

d c

ba

B

CA

PE

Figure 2.3.Analytical model


27/159


13

Pxtan

Px

Rtan

R

PE

dy

d c

ba

py

py+ dpy

(1-kv)dW

dW kh

Figure 2.4.Free body diagram

In figure (2.4), the freebody diagram of an elemental slice shows the action of different

forces. The thickness of the slice is dy, at a depth ofy from the top ground surface. The

vertical pressure py is acting on the top of the element and (py+ dpy) on the bottom of the

element. The reaction px normal to the wall and the shear force pxtan are acting on the

interface between the retaining wall and the backfill material. The normal force R and the

shear force Rtan act on the sliding surface. The other forces are, the weight dW of the

element, the seismic forces dWkh in the horizontal direction and dWkv in the vertical direction.

The critical directions of these seismic forces are as shown in figure (2.4). The horizontal

planes are assumed as principal planes. Resolving all the forces in the vertical and horizontal

directions, from the boundary conditionpy = 0 aty = 0, and ignoring higher order differentialterms and upon simplifying, the expression for the seismic passive earth pressure at any depth

y is obtained as:

(2.13)( )

( )( )( )

+

= +

+

yHyH

H

Ka

nKp

Ka

Ka

X 1

2

2

Where

pX- seismic passive pressure at any depth y from top acting normal to the wall

K- lateral earth pressure coefficient used for the static analysis

n - (1-kv-kh.b)

b - ( ) +PEcot

a -( )( )

( )( )( )

( )PEPEPEPE

cottan

tantan

cottan1cottan

tantan1cottan

+

++++

- wall friction angle

PE- angle of inclination of the failure plane with horizontal


28/159


14

Integrating over the height of the wall, the total passive resistancePXis given by,

(2.14)( )

( )( )Ka

HnK

Kaa

HnP KaX +

+

= 22

110002

22

The equivalent seismic passive earth pressure coefficient with respect to normal to the wall is

found out as,

(2.15)2

2

H

PK Xpe

=

The critical value ofPE is obtained by minimizing PX with respect to PE keeping all other

parameters constant and it is found to exactly match with the Coulombic values for the static

case.

Recently Choudhury (2002) [6] has given the complete solution for passive earth pressurecoefficients for rigid retaining walls under seismic conditions for variations in parameters,

such as wall inclination, ground inclination, wall friction angle, soil friction angle, wall

adhesion to soil cohesion ratio, and the horizontal and vertical seismic accelerations.

The limit equilibrium method based on a pseudostatic approach was adopted for determining

individually the seismic passive earth pressure coefficients corresponding to own weight,

surcharge and cohesion components. In the determination of each of these components,

composite (logarithmic spiral and planar) failure surfaces were considered. It was considered

that the occurrence of an earthquake does not affect the basic soil parameters: unit cohesion c,

friction angle , and unit weight . Uniform seismic accelerations are assumed in the domainunder consideration.

N

C+Ntan

Pp R

PpcR+PpqR

i

i

y/3 y/2

Pp d

Ppcd+Ppqd

Ca

?i

A

G

E

D

B

Fq

qAG(1-kv)

qAGkh

qGEkv

qGEkh

W2kv

W2kh

W1kh

(1-kv)W1

H/3H/2

H

Figure 2.5.Composite failure surface and forces considered

FB=r0FD=rfFA=L

DG=y

W1=weight ABDGA

W2=weight DGE

q=uniform surcharge


29/159


15

The seismic passive forcePpdwas divided into three components as:

1. Unit weight componentPpd( 0,c=q=0)

2. Surcharge componentPpqd(q 0,=c=0), and

3. Cohesion componentPpcd(c 0,=q=0)

Where , c, and q are the unit weight of the soil, unit cohesion, and surcharge pressure

respectively.

The principle of superposition was assumed to be valid and the minimum of each component

was added to get the minimum seismic passive force. Hence,

(2.16) pcdpqddppd PPPP ++=

The failure surfaces for each of these components that arePpd,PpqdandPpcdwill be different.

A particular single failure surface can be found for which the total seismic passive force is

minimum. For this failure surface, the corresponding components are not necessarily the

minimum values.

It was shown that the error magnitude between the method of superposition considering

minimum of each component and finding the minimum of total earth force was very small,

less than 3%.

In figure (2.5), the failure surface includes portion BD, which is a logarithmic spiral and a

planar portionDE, which is similar to the Rankine passive planar failure surface including the

pseudostatic seismic forces.Fis the focus of logarithmic spiral and is located at a distance L

from A. The initial radius r0 and the final radius rf of the logarithmic spiral are given by

distancesFB andFD, respectively.

For the force system shown in figure (2.5), after satisfying equilibrium equations and Mohr

Coulomb criterion of failure, the exit angle at pointEon the ground surface becomes

(2.17)

+

+=

sin1

tansin

sin21

21tan

21

24

1

11

i

k

k

ik

k v

h

v

h

Forkh=kv=0, Equation. (2.10) yields the same value as given by Rankine and the same value

as given by Kumar (2001) forkv=0.

As given in the figure (2.5), the point of application ofPpdis assumed at a height ofH/3 from

the base of the wall (Chen and Liu 1990), whereasPpqdandPpcdare assumed to act at a height

ofH/2. A uniform surcharge pressure ofq is assumed alongAEweight of the failure wedge

ABDGA is W1. Cohesive force C is assumed to act on the failure surface BD along withnormal forceNand frictional forceNtan . Adhesive force Cais acting on the retaining wall-


30/159


16

soil interfaceAB. Rankine passive forcesPpcR,PpqR, andPpRare assumed to act on the surface

DG. Pseudostatic forces due to seismic weight components for zone DGEare W2khand W2kv

in the horizontal and vertical directions, respectively. Pseudostatic forces due to seismic

weight component for zone ABDGA are W1kh and W1kv in the horizontal and vertical

directions, respectively. Pseudostatic forces due to q.AG.khand q.AG.kv in the horizontal and

vertical directions, respectively, are assumed to act on AG. Similarly pseudostatic forces

q.GE.kh and q.GE.kv in the horizontal and vertical directions, respectively, are assumed to act

on GE.

Considering the moment equilibrium of all forces about the focusF,

1. The seismic passive earth pressure coefficient for the unit weight component Kpd

corresponds to the minimum value of the seismic passive earth force Ppd. The

minimum value ofPpd is obtained by considering different logspirals by varying the

distanceL. The coefficientKpdin normal direction to the wall is then obtained as:

(2.18)2

cos2

H

PK

dp

dp

=

2. The seismic passive earth pressure coefficient for the surcharge component Kpqdcorresponds to the minimum value ofPpqd. The minimum value ofPpqdis obtained by

varyingL. The coefficientKpqdis then obtained as:

(2.19)qH

PK

pqd

pqd

cos=

3. The seismic passive earth pressure coefficient for the cohesion component Kpcdcorresponds to the minimum value ofPpcd. By assumption, the cohesion component

will not be affected by the seismic accelerations, and hence the static and seismic

cohesion values remain the same. The minimum value ofPpcdis obtained by varying

L. The coefficientKpcdis then obtained as:

(2.20)cH

PK

pcd

pcd2

cos=

The total seismic passive forcePpdon the retaining wall of heightHbecomes

(2.21)

cos

1

2

12 2

++= dppqdpcdpd KHqHKcHKP

The forcePpdacts at an angle with the normal to the retaining wall.

The points of applications for passive force under seismic condition were determined by

Choudhury (2002) using the method of horizontal slices. The results showed that the points of

applications values ranges from 0.28H to 0.4H from base of the wall with vertical height H,

for different seismic conditions and wall inclinations.


31/159


17

Table 2.1Comparison ofKpdvalues obtained by Choudhurys method and available theories in seismiccase for =0, i=0, =30

/ kh kv

Mononobe-

Okabe

Morrison

&Ebeling

Chen

& Liu Soubra Kumar Choudhury

0.0 0.0 4.807 4.463 4.710 4.530 - 4.458

0.1 0.0 4.406 4.240 4.370 4.2020 - 4.240

0.1 0.1 4.350 4.160 - - - 3.890

0.2 0.0 3.988 3.870 4.000 3.900 - 3.860

0.2 0.2 3.770 3.600 - - - 3.020

0.5

0.3 0.0 3.545 3.460 3.590 3.470 - 3.450

0.0 0.0 8.743 6.150 7.100 5.941 5.785 5.783

0.1 0.0 7.812 5.733 6.550 5.500 5.361 5.400

0.2 0.0 6.860 5.280 5.950 5.020 4.902 5.100

0.3 0.0 5.875 4.940 5.300 4.500 4.400 4.750

0.4 0.0 4.830 4.300 - - 3.900 4.100

1.0

0.5 0.0 3.645 3.400 - - 3.200 3.300

Table 2.1 shows a comparison ofKpdvalues obtained from different analyses. For/=0.5, it

is seen that the Choudhurys method results in the least values of the coefficients. However

for/ =1.0, it is not necessarily the least in cases for higherkhvalues. But the difference is

very marginal.


32/159


18

Figure 2.6.Comparison of passive pressure computation using the method proposed by

Choudhury (Choudhury et al 2002)

2.2.1.4Ortigosa (2005)

The Mononobe and Okabe expression has the limitation of not introducing soil cohesion,

which was later included by Prakash (1981) [38], also employing the Coulombs wedge

method. Based on Prakashs expressions, Ortigosa (2005) [31] proposed resolving the

problem uncoupling the static and seismic thrust in the following manner:

1. Determining the resultant static thrust, Pc, including soil cohesion, c, with tension

cracks.

2. Determining the resultant static plus seismic thrust, Pae, with the Mononobe and

Okabe expression, which implicates considering c = 0.

3. Determining the resultant static thrust,P0, by making c = 0.

4. Determining the seismic thrust component as:


33/159


19

(2.22) 0PPP aee =

In this manner, the resultant of the static plus seismic thrust is obtained as:

(2.23) ecec PPP += It is important to point out that the thrust uncoupling is valid ifPc > 0, which equates to

consuming all soil cohesion in the static thrust component. If the cohesion is such that the

critical height of the soil is equal to that of the wall, the uncoupling gives Pc = 0 and if it is

greater givesPc = 0 and an overvalued seismic component.

More recently, Richards and Shi (1994) [42] utilize an interaction model between the

retaining element and the free field seismic movement of the soil in which they incorporate

cohesion.

2.2.2Pseudo-dynamic approach

In this approach, the advancement over the previous approach is that the dynamic nature of

the earthquake loading is considered in an approximate and simple manner. The phase

difference and the amplification effects within the soil mass are considered along with the

accelerations to the inertia.

2.2.2.1Steedman-Zeng (1990)

Steedman and Zeng (1990) [45] considered the harmonic horizontal acceleration of amplitude

ah at base of the wall. For a typical fixed base cantilever wall (figure 2.7) and assuming i =

= kv = 0 for simplicity.

ah(t)

F

Z

Vs

A

B

C

Z

Qh

w

H

h Pae

Figure 2.7.System considered by Steedman-Zeng

Then at depthzbelow the ground surface the acceleration can be expressed as:


34/159


20

(2.24)

=

s

hhV

zHtatza sin),(

Where,

- angular frequency

t- time elapsed

Vs - shear wave velocity

The planar rupture surface, inclined at an assumed angle to the horizontal, is considered in

the analysis along with the seismic force and weight of failure block. The total seismic active

force on the wall is given by,

(2.25))cos(

)sin()cos()()(

++=

WtQtP hae

And

(2.26) [ ])sin(sincos2tan4

)(2

tHg

atQ hh

+=

Where,

(2.27)

sV2

=

(2.28)sV

H= 1

The point of application of the total seismic active force is hdfrom the base of the wall and

given by,

(2.29))sin(sincos2

)cos(cossin2cos2 222

tH

tHHHhd

++

=

This point of application of the seismic force for very low frequency motions (small H/, sothe backfill moves essentially in phase) is athd=H/3. For higher frequency motions, hdmoves

upwards from base of the wall. This solution accounts for non uniformity of acceleration

within the soil mass but disregards dynamic amplification.

2.2.2.2Choudhury-Nimbalkar (2005)

Steedman & Zeng (1990) did not consider the effect of vertical seismic acceleration on the

active earth pressure, which was corrected by Choudhury & Nimbalkar (2005) [7]. Also,

Choudhury & Nimbalkar (2005) considered using pseudo-dynamic method to determine the

seismic passive resistance behind a rigid retaining wall.


35/159


21

The effect of variation of different parameters such as wall friction angle , period of lateral

shaking T, soil friction angle , horizontal and vertical seismic coefficients kh, kv, shear wave

velocity Vs and primary wave velocity Vp are considered in the present analysis. A planar

rupture surface BC, inclined at an angle to the horizontal, is assumed for the analysis to

avoid further complication of the problem.

ah(t)

FPpe

h

H

w

QhQv

Z

C

B

A

Vs , Vp

Z

Figure 2.8.System considered by Choudhury-Nimbalkar

The base is subjected to harmonic horizontal and vertical accelerations of amplitudesah

and

av, the horizontal accelerations at depthzbelow the top of the wall is give in equation (2.15),

and the vertical acceleration can be given as:

(2.30)

=

p

vvV

zHtaa sin

The total horizontal inertia (Qh(t)) force acting on the wall is given in equation (2.17). The

total vertical inertia force acting on the wall is expressed as:

(2.31) ( ) ( )[ ]tHga

tQv

v

sinsincos2tan4 2 +=

Where,

(2.32) pV

2=

(2.33)pV

Ht=

The total (static and dynamic) passive resistance can be obtained by resolving forces on the

wedge: that is,


36/159


22

(2.34)( ) ( ) ( )

( )

+++++

=cos

sincossin vhpe

QQWP

Typical results show the highly non-linear nature of the seismic passive earth pressure

distribution by this pseudo-dynamic method compared with the existing linear seismic passiveearth pressure distribution using a pseudo-static approach. Comparisons of the present method

with the available pseudo-static methods are shown, leading to the minimum seismic passive

resistance by pseudo-dynamic method.

2.2.3Displacement-based analysis

A retaining structure subjected to earthquake motion will vibrate with the backfill soil and the

wall can easily move from the original position due to an earthquake. The methods available

for displacement based analysis of retaining structures during seismic conditions are based on

the early work of Newmark (1965) [see Kramer, 1996]. The basic procedure was developed

for evaluating the deformation of an embankment dam shaken by earthquake based on theanalogy with a sliding block-on-a-plane.

2.2.3.1Richards-Elms model

The model proposed by Richards and Elms (1979) [41] is based on the basic Newmarks

model, developed originally for evaluation of seismic slope stability, modified for the design

of gravity retaining walls. Richards and Elms recommended that the dynamic active earth

force calculated using Mononobe-Okabe method is given by,

(2.35)

+++

+++

=

22

22

)cos()cos(

)sin()sin(1

1

)cos(coscos

)(cos)1(5.0

i

i

kHP vae

The permanent block displacement is given by,

(2.36)4

3

max

2

max087.0y

perma

aVd =

Where,

Vmax- ground velocity

amax- ground acceleration

ay - yield acceleration for the backfill-wall system

(2.37) gW

PPa aeaeby

++=

)sin()cos(tan


37/159


23

2.2.4Comparison of seismic earth pressure values computed using different approaches

It is batter to compare the earth pressure values computed using different approaches. The

table (2.2) shows the comparison. For the computation following data have been used:

H=6m, c=0, =34, =17, =17.3kN/m, kh=0.3, and kv=0.3. A: Forced-based analysis, B:

Displacement-based analysis

Table 2.2Comparison of seismic earth pressure computation using different approach

MethodMononobe-

OkabeChoudhury

Steedman-

Zeng

Richards &

ElmsEC8

Seismic active earth

pressure,Pad(kN/m)145.16 - 141.24 100.92 100.92

p.o.a ofPadfrom base (m) 2.72 - 2.0 2.33 2.33

Seismic passive earth

pressure,Ppd(kN/m)943.99 905.36 - - -

p.o.a ofPpdfrom base - 1.98 - - -

Displacement (mm) - - - 100 100

Remarks A A A B B

2.3Closed form solutions using elastic or viscous elastic behaviour

2.3.1Wood (1973)

The massive gravity walls founded on rock or basement walls braced at both top and bottom,

do not move sufficiently to mobilize the shear strength of the backfill soil. Wood (1973)

analyzed the response of a rigid nonyielding wall retaining a homogeneous linear elastic soil

and connected to a rigid base. For such conditions, Wood established that the dynamic

amplification was insignificant for relatively low-frequency ground motions (that is, motions

at less than half of the natural frequency of the unconstrained backfill), which would include

many or most earthquake problems.

For uniform, constant khapplied throughout the elastic backfill, Wood (1973) developed the

dynamic thrust, PE, acting on smooth rigid nonyielding walls as:

(2.38) 2HkFP hPE =

Where,

FP- Dimensionless thrust factor for various geometry and soil Poissons ratio values


38/159


24

The value ofFP is approximately equal to unity (Whitman, 1991) leading to the following

approximate formulation for a rigid nonyielding wall on a rigid base:

(2.39) 2HkP hE =

As for yielding walls, the point of application of the dynamic thrust is taken typically at a

height of 0.6Habove the base of the wall.

Woods simplified procedures do not account for: (1) vertical accelerations, (2) the typical

increase of modulus with depth in the backfill, (3) the influence of structures or other loads on

the surface of the backfill, (4) the phased response at any given time for the accelerations and

the dynamic earth pressures with elevation along the back of the wall, and (5) the effect of the

reduced soil stiffness with the level of shaking induced in both the soil backfill and soil

foundation.

Depending on the dynamic properties of the backfill as well as the frequency characteristics

of the input ground motion, a range of dynamic earth pressure solutions would be obtained for

which the Mononobe-Okabe solution and the Wood (1973) solution represent a lower and

an upper bound, respectively.

2.3.2Veletsos and Younan (1994)

The system examined by Veletsos and Younan is shown in figure (2.9). It consists of a semi-

infinite, uniform layer of linear viscoelastic material of height h that is free at its upper

surface, is bounded to a rigid base, and is retained along one of its vertical boundaries by a

rigid wall. The wall may be either fixed or elastically constrained against rotation at its base,by a spring of stiffness R. Both the base of the layer and the wall are subjected to a space-

invariant, harmonic, uniform horizontal motion, a(t), at any time t., characteristic by a

frequency and a maximum amplitudeA. Material damping for the medium is considered to

be of the constant hysteretic type, frequency-independent, and the same for both shearing and

axial deformations.

h

y

x

a(t)

R?

Rigid

Wall

Figure 2.9.Base-excited soil-wall system investigated


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The properties of the soil stratum are defined by its mass density (), shear modulus of

elasticity (G), Poissons ratio (), and the material damping factor ().

Veletsos and Younana realized that the Scotts model (1973) fails to provide for the capacity

of the medium between the wall and the far field to transfer forces vertically by horizontalshearing. In addition to the horizontal normal stresses and inertia forces, a horizontal element

of the medium is acted upon along its upper and lower faces by horizontal shearing stress, the

difference of which is given by:

(2.40)

=

= xyxy

hy

1

With being the dimensionless distance given by =y/h. On the assumption that the

horizontal variation of the vertical displacements is negligible, xy can be expressed as:

(2.41)

=u

h

Gxy and 2

2

2

=u

h

G

Where u is the relative horizontal displacement of the medium with respect to the moving

base.

Ifu is expressed by the method of separation of variables as a linear combination of modal

terms, through mathematical manipulation (Veletsos and Younana, 1994b), the nth component

of, denoted as ()n, may be recognized to be:

(2.42) ( ) nnn u2

=

Where un is the nth component of the displacement u and nis the circular frequency of the

stratum, considered to respond as a cantilever shear-beam, given by:

(2.43)h

Vn sn

2

)12(

=

Where n refers to the mode being considered and Vsis the shear wave velocity of the stratum.

The equation (2.33) represents a force per unit of length that is identical to the force induced

by a massless linear spring of stiffness kn given by:

(2.44)( )

2

2

2

2

12

h

Gnk nn

==

This corresponds to modeling the shearing action of the medium, for each modal component,

with a set of horizontal linear spring of constant stiffness kn, connected at their lower ends to

the common base, subjected to the prescribed ground acceleration a(t). The other end of the

spring connected to the medium that may be modeled by a series of semiinfinitely long,

elastically supported horizontal bars with distributed mass (For more detail see [2]).

In 1996, Veletsos and Younan analyzed the response of flexible cantilever retaining walls that

are elastically constrained against rotation at their base for horizontal ground shaking. The


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retaining medium is idealized as a uniform, linear, viscoelastic stratum of constant thickness

and semi-infinite extent in the horizontal direction. The parameters varied including the

flexibilities of the wall and its base, the properties of the retained medium, and the

characteristics of the ground motion.

They discovered that the dynamic pressures depend profoundly on both the wall flexibility

and the foundation rotational compliance, and that for realistic values of these factors the

dynamic pressures are substantially lower than the pressures for a rigid, fixed-based wall. In

fact, they found out that the dynamic pressures may reduce to the level of the Mononobe

Okabe solution if either the wall or the base flexibility is substantial.

More recently (2000), Veletsos and Younan [60] have proposed a solution technique for the

dynamic analysis of flexible both cantilever and top-supported walls.

However, these analytical solutions are based on the assumption of homogeneous retainedsoil, and there are reasons for someone to believe that the potential soil inhomogeneity may

lead to significant changes in the magnitude and distribution of the dynamic earth pressures.

Furthermore, as the presence of the foundation soil layers under the retained system is only

crudely modelled through a rotational spring, these solutions do not account for the potential

horizontal translation at the wall base, which in general may have both an elastic and an

inelastic (sliding) component.

2.4Numerical analyses

Earthquake-induced pressures on retaining walls can also be evaluated using dynamic

response-analyses. A number of computer programs are available for such analyses. Linear orequivalent linear or non-linear analysis can be used to estimate wall pressures. Non-linear

analyses are capable of predicting permanent deformations as well as wall pressure.

2.4.1Al-Homoud and Whitman (1999)

A finite element numerical model has been developed for gravity walls founded on dry sand

by Al-Homoud and Whitman (1999), using Weidlingger Associates two-dimensional (2D)

finite element computer code, FLEX. Dynamic analyses in FLEX are performed using an

explicit time integration technique.

The suggested model for studying the dynamic response of rigid gravity wall can besummarized as follows:

1. The soil (dry sand in this study) is modelled by a 2D finite element grid. This includes

the backfill material and the foundation soil.

2. The gravity retaining wall is modeled as a rigid substructure.

3. The strength and deformation of the soil material are modeled using the viscous cap

constitutive model. This model consists of a failure surface and a hardening cap

together with an associated flow rule. The cap surface is activated only for the soilunder the wall to represent compaction during wall rocking. In addition, viscoelastic


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behavior is provided for the state of stress within the region bounded by these

surfaces, so as to provide for hysteretic-like damping of soil during dynamic loading.

4. Interface (continuum approximation) elements are used between the soil and the wall

(at the back face of the wall and under its base), allowing for sliding and for theopening and closing of the gaps (i.e. debonding and bonding).

5. The f

numerical modelling of seismic behaviour of earth-retaining walls.pdf

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