MODELING RETAINING STRUCTURES IN 2D FINITE ELEMENT …

MODELING RETAINING STRUCTURES IN 2D FINITE ELEMENT ANALYSIS

USING BEAM-SOLID CONTACT ELEMENT

By

CONGYI ZHANG

A thesis submitted in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE IN CIVIL ENGINEERING

WASHINGTON STATE UNIVERSITY

Department of Civil and Environmental Engineering

MAY 2017

© Copyright by CONGYI ZHANG, 2017

All Rights Reserved

© Copyright by CONGYI ZHANG, 2017 All Rights Reserved

ii

To the Faculty of Washington State University:

The members of the Committee appointed to examine the thesis of CONGYI

ZHANG find it satisfactory and recommend that it be accepted.

__________________________________

Christopher McGann, Ph.D., Chair

__________________________________

Balasingam Muhunthan, Ph.D.

__________________________________

William Cofer, Ph.D.

iii

ACKNOWLEDGMENT

First, I would thank my advisor Dr. Christopher McGann for giving me the opportunity to

continuing my study in WSU. He is very considerate, and always gives me timely feedback for

any questions I have. He is also very knowledgeable, and can easily give the solution to the

problem that troubles me for a long time. This work would not be done without his help.

I would also express my many thanks to the other committee members, Dr. Balasingam

Muhunthan and Dr. Bill Cofer. Dr. Muhunthan always encourage me to conquer the barriers no

matter how difficult the situation is. Dr. Cofer always give me helpful suggestions that I need

most for all kinds of problems. Their classes are the most enjoyable ones in my life, especially

“Finite Element” offered by Dr. Cofer and “Numerical Modeling of Geomaterials” offered by Dr.

Muhunthan.

I cannot thank enough to the dear friends I met in WSU: Shenghua Wu, Feng Hao, and

Weiwei Huang, who always give me wonderful support and even call me from time to time after

their graduations. Sara Tohidi and Partha Bhattacharjee are the best mates to work with, I

appreciate their help along the way. There are a lot more friends to thank that I cannot name

respectively, but their help is always deeply remembered.

Last but certainly not least, I would like to express my deepest gratitude for the constant

support, understanding and love that I received from my family during this study.

iv

MODELING RETAINING STRUCTURES IN 2D FINITE ELEMENT ANALYSIS

USING BEAM-SOLID CONTACT ELEMENT

Abstract

By Congyi Zhang

Washington State University May 2017

Chair: Christopher McGann

Retaining structures are widely-used engineering structures, and it is crucial to assess their

abilities to sustain extreme weather conditions and natural hazards. To tackle this issue, we model

cantilever retaining walls and adjacent soil, using the finite element (FE) software OpenSees, under

static loading and seismic loading. In this study, a 2D Beam-Solid contact element has been

implemented into OpenSees to capture the frictional stick-slip and the gapping behavior between

soil and structure. This element has previously been used in limited analyses of sheet pile walls ,

however, it has not yet been applied to cantilever retaining walls. Previous work with this element

has primarily involved static loading cases; it has rarely been used in dynamic loading cases. In

this study, we would further explore the functions of this element, meanwhile parametric study on

wall stiffness and input ground motions also be conducted to verify the robustness of this element.

This FE model of retaining wall should be helpful to the geotechnical engineering design code,

and later help engineers to design better fundamental structures, more significantly it would help

further the understanding of these structures in severe earthquakes.

v

TABLE OF CONTENTS

Page

ACKNOWLEDGMENT .............................................................................................................. iii

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

TABLE OF CONTENTS .............................................................................................................. v

LIST OF TABLES ...................................................................................................................... vii

LIST OF FIGURES ....................................................................................................................viii

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

2 CONTACT ELEMENT APPROACH ...................................................................................10

2.1 Projection of Slave Node to Beam Element .........................................................................10

2.2 Interface Material Properties ................................................................................................14

2.3 Contact Element ...................................................................................................................16

3 MODEL DEVELOPMENT ....................................................................................................20

3.1 Overview of OpenSees and GiD ..........................................................................................20

3.2 Overview of Centrifuge Test Used for Model Validation ...................................................22

3.3 Building the Finite Element Model ......................................................................................25

3.3.1 Simulated Geometry ......................................................................................................25

3.3.2 Elements Considered in FE Models ..............................................................................26

3.3.3 Treatment of Boundary Conditions ...............................................................................37

3.3.4 Analysis Types ..............................................................................................................37

3.4 One Tiny Modeling Trick.....................................................................................................38

4 STATIC ANALYSIS RESULTS AND DISCUSSION .........................................................39

4.1 The Displacement Distributions ..........................................................................................41

4.2 The Stress Distributions ......................................................................................................43

vi

4.2.1 The Horizontal Stress Distributions .............................................................................43

4.2.2 The Vertical Stress Distributions ..................................................................................45

4.2.3 The Shear Stress Distributions .....................................................................................47

4.3 Lateral Earth Pressure Distributions ....................................................................................49

4.3.1 Contact Forces at Slave Nodes .....................................................................................49

4.3.2 Lateral Earth Pressures on Retaining Walls .................................................................51

4.4 Summary ..............................................................................................................................54

5 DYNAMIC ANALYSIS RESULTS AND DISCUSSION ....................................................56

5.1 Analysis Procedures and Modeling Techniques ..................................................................57

5.1.1 Damping Coefficients ...................................................................................................58

5.1.2 Input Earthquake Motions ............................................................................................59

5.1.3 Model System Parameters ............................................................................................61

5.2 Analysis Results ..................................................................................................................62

5.2.1 Accelerations ................................................................................................................62

5.2.2 Bending Moments ........................................................................................................67

5.2.3 Lateral Earth Pressures .................................................................................................75

5.2.4 Soil Shear Stress Response ...........................................................................................83

5.3 Comparison of Numerical Results with M-O Method Results ...........................................87

5.4 Conclusions and Further Model Improvements ..................................................................90

6 SUMMARY AND CONCLUSIONS .......................................................................................92

BIBLIOGRAPHY ........................................................................................................................94

vii

LIST OF TABLES

Table 3.1 FE model properties for stiff retaining walls ................................................................ 27

Table 3.2 FE model properties for flexible retaining walls .......................................................... 27

Table 3.3 Reference parameters for PDMY material ................................................................... 30

Table 3.4 Material properties used in PDMY material ................................................................ 34

Table 5.1 Comparison of computed and theoretical earth pressure on stiff retaining wall .......... 89

Table 5.2 Comparison of computed and theoretical earth pressure on flexible retaining wall .... 89

Table 5 3 Comparison of computed and theoretical wall moment on stiff retaining wall ............ 89

Table 5.4 Comparison of computed and theoretical wall moment on flexible retaining wall ...... 90

viii

LIST OF FIGURES

Figure 1.1 Free body diagram of active seismic loadings on retaining wall (Kramer, 1996) ....... 3

Figure 1.2 Free body diagram of passive seismic loadings on retaining wall (Kramer, 1996) ...... 5

Figure 1.3 Backfill slope angle effect on the seismic active earth pressure coefficient using the

M-O method (Anderson, Martin, Lam, & Wang, 2009)................................................................. 7

Figure 2.1 Contact body representing the contact element configuration .................................... 10

Figure 2.2 Mohr-Coulomb frictional behavior ............................................................................. 14

Figure 3.1 The workflow of OpenSees analysis used in this study. ............................................. 22

Figure 3.2 Centrifuge test configuration, profile view (Al Atik and Sitar, 2008). ....................... 23

Figure 3.3 Centrifuge test configuration, plan view (Al Atik and Sitar, 2008). ........................... 24

Figure 3.4 The mesh of the centrifuge model. Selective refinement was employed such that the

elements near the walls and ground surface are smaller than those near the base........................ 26

Figure 3.5 Yield Surfaces of PDMY material (Elgamal, Yang, Parra, & Ragheb, 2003) ............ 29

Figure 3.6 Constitutive relationship of PDMY material (Yang, Lu, & and Elgamal, 2008) ........ 32

Figure 3.7 Modulus reduction curve measured in experiments (Al Atik & Sitar, 2008) ............. 35

Figure 3.8 Damping ratio curve measured in the experiments (Al Atik & Sitar, 2008) ............... 35

Figure 3.9 The mesh of beam elements and surrounding soil elements ....................................... 36

Figure 3.10 The configuration of the “cross pieces” at the corners of the retaining walls. .......... 38

ix

Figure 4.1 The coarse mesh of the FE centrifuge test model (the upper mesh) and the refined

mesh of the FE centrifuge test model (the bottom mesh) ............................................................. 40

Figure 4.2 The y-displacement (m) configuration of coarse meshed case without “cross pieces”

....................................................................................................................................................... 41

Figure 4.3 The y-displacement configuration (m) of coarse meshed case with “cross pieces”.... 42

Figure 4.4 The y-displacement (m) configuration of refined meshed case .................................. 42

Figure 4.5 The horizontal stress (kPa) of coarse meshed case without “cross pieces” ................ 43

Figure 4.6 The horizontal stress (kPa) of coarse meshed case with “cross pieces”...................... 44

Figure 4.7 The horizontal stress (kPa) of the refined meshed case .............................................. 44

Figure 4.8 The vertical stress (kPa) of coarse mesh case without “cross pieces” ........................ 45

Figure 4.9 The vertical stress (kPa) of coarse mesh case with “cross pieces”.............................. 46

Figure 4.10 The vertical stress (kPa) of the refined meshed case ................................................. 46

Figure 4.11 The shear stress (kPa) of coarse meshed case without “cross pieces” ...................... 47

Figure 4.12 The shear stress (kPa) of coarse meshed case with “cross pieces” ........................... 48

Figure 4.13 The shear stress (kPa) of the refined meshed case .................................................... 48

Figure 4.14 The contact forces (kN) of coarse meshed case without “cross pieces” ................... 50

Figure 4.15 The contact forces (kN) of coarse meshed case with “cross pieces”......................... 50

Figure 4.16 The contact forces (kN) of the refine meshed case ................................................... 51

Figure 4.17 The earth pressure on retaining walls of coarse meshed case without “cross pieces”

....................................................................................................................................................... 52

Figure 4.18 The earth pressure on retaining walls of coarse meshed case with “cross pieces” ... 53

Figure 4.19 The earth pressure on retaining walls of refine meshed case .................................... 54

x

Figure 5.1 The input ground motions of Loma Prieta-SC-1: the upper is from PEER Ground

Motion database and used in numerical modeling; the lower is generated by centrifuge testing. 60

Figure 5.2 The input ground motions of Kobe-PI-2: the upper is from PEER Ground Motion

database and used in numerical modeling; the lower is generated by centrifuge testing ............. 60


Motion database and used in numerical modeling; the lower is generated by centrifuge testing 61

Figure 5.4 Computed accelerations at the tops of retaining walls and free field during Loma

Prieta-SC-1.................................................................................................................................... 64

Figure 5.5 Computed accelerations at the tops of retaining walls and free field during Kobe-PI-2

....................................................................................................................................................... 65


Prieta-SC-2.................................................................................................................................... 66

Figure 5.7 Computed total wall moment time series at the middle and bottom of both south stiff

and north flexible walls during Loma Prieta-SC-1 ....................................................................... 69


and north flexible walls during Kobe-PI-2 ................................................................................... 70


and north flexible walls during Loma Priesta-SC-2 ..................................................................... 71

Figure 5.10 Comparison of computed and recorded moment profiles in the report (Al Atik &

Sitar, 2008) for Loma Prieta-SC-1 ................................................................................................ 72


Sitar, 2008) for Kobe-PI-2 ............................................................................................................ 73

xi


Sitar, 2008) for Loma Prieta-SC-2 ................................................................................................ 74

Figure 5.13 Computed total earth pressure time series at the middle and bottom of both south

stiff and north flexible walls during Loma Prieta-SC-1................................................................ 77


stiff and north flexible walls during Kobe-PI-2 ............................................................................ 78


stiff and north flexible walls during Loma Prieta-SC-2................................................................ 79

Figure 5.16 Comparison of computed and recorded earth pressure profiles of Al Atik and Sitar

(Al Atik & Sitar, 2008) for Loma Prieta-SC-1 ............................................................................. 80


(Al Atik & Sitar, 2008) for Kobe-PI-2.......................................................................................... 81

Figure 5.18 Comparison of computed and recorded earth presure profiles in the report (Al Atik &

Sitar, 2008) for Loma Priesta-SC-2 .............................................................................................. 82

Figure 5.19 Comparison of computed and recorded shear stress time series in the middle of soil

backfill for Loma Prieta-SC-1....................................................................................................... 84


backfill for Kobe-PI-2 ................................................................................................................... 85


backfill for Loma Prieta-SC-2....................................................................................................... 86

1

CHAPTER ONE

INTRODUCTION

A retaining wall is a structure that balances the lateral earth pressure of a slope from its

sliding or collapsing, and it is widely used on transportation routes, harbors and life lines. Failure

of the wall will certainly impair the critical use of these systems, and it will cause risks to the safety

of the public. What’s more, other facilities that must be functional, especially in an unexpected

natural disaster, may slip into a state of paralysis following the failure of the retaining wall. For

example, local economies of port cities rely heavily on importing and exporting shipped goods.

So, it is crucial to take care of these facilities, because repair and safe design cost is far less than

the business lost caused by their failure.

In this study, we study the behaviors of cantilever retaining walls under seismic loading.

Cantilever walls, unlike gravity walls which resist the lateral earth pressures by rigid-body

translation and rotation, mainly using their flexural strength to withstand the loading. By doing

this, they save a lot of material compared with traditional gravity walls, but meanwhile, their

behavior is more complex. The failure modes of the cantilever retaining wall include excessive

sliding and overturning, flexural strength failure of the wall and global instability. The geometry

of the wall, the relative stiffness and deformation of both the wall and soil, and the input motion

are believed as the key factors contribute most to the wall movement and dynamic earth pressure

which cause the failure. Due to the complexity of the problem, the current state of art is still far

insufficient in depicting the failure mechanism. So, there is a high significance in evaluating the

effects and helping to improve the current design procedure.

2

Mononobe-Okabe Method

Okabe (Okabe, 1926) and Mononobe and Matsuo (Mononobe & Matsuo, 1929) came up

with a pseudo-static theory of seismic earth pressure on retaining walls after investigating the great

Kanto Earthquake in 1923, and it is basically an extension of the Coulomb Theory to add pseudo-

static acceleration forces caused by the earthquakes. Afterwards, people named the theory the

Mononobe-Okabe (M-O) method. The M-O method has been the state of art (Anderson, et al.,

2009) for decades due to the adaption by Seed and Whitman (1970), however numerous studies

have never ceased on doubting its accuracy in practice. At the same time, the M-O method

continuously serves as a baseline to compare with other theories.

Active Earth Pressure Conditions

The free body diagram of the wall-soil system is shown in Figure 1.1. Like the traditiona l

Coulomb Theory, the M-O method assumes the backfill acting as wedge on the retaining wall, and

the soil is dry and cohesionless. In addition to the forces used in Coulomb Theory, 𝑘ℎ𝑊 and 𝑘𝑣𝑊

are pseudo-static acceleration forces induced by the earthquake, whose values are related to the

mass of the wedge and the earthquake magnitude (𝑘ℎ =𝑎ℎ

𝑔; 𝑘𝑣 =

𝑎𝑣

𝑔). The total active thrust can

be expressed as:

𝑃𝐴𝐸 = 0.5𝐾𝐴𝐸𝛾𝐻2(1− 𝑘𝑣) (1.1)

Where 𝐾𝐴𝐸 is the dynamic active earth pressure coefficient, and formulated:

𝐾𝐴𝐸 = 𝑐𝑜𝑠2(𝜙 − 𝜃 − 𝜓)

𝑐𝑜𝑠𝜓𝑐𝑜𝑠2𝜃 cos(𝛿 + 𝜃 + 𝜓) [1 + √sin (𝛿 +𝜙)sin (𝜙− 𝛽 − 𝜓)cos (𝛿 + 𝜃 + 𝜓)cos (𝛽 − 𝜃)]

2

(1.2)

3

Figure 1.1 Free body diagram of active seismic loadings on retaining wall (Kramer, 1996)

Where β and θ are inclined angle of backfill and retaining wall from horizontal and vertical

direction respectively, δ is the interface friction angle, ψ = 𝑡𝑎𝑛−1(𝑘ℎ/(1− 𝑘𝑣)).

The critical failure angle is also determined by Zarrabi (1979) :

𝛼𝐴𝐸 = 𝜙 − 𝜓 + 𝑡𝑎𝑛−1[− tan(𝜙− 𝜓 −𝛽) + 𝐶1𝐸

𝐶2𝐸

(1.3)

Where 𝐶1𝐸 and 𝐶2𝐸 can be expressed as:

𝐶1𝐸 = √tan(𝜙 −𝜓 −𝛽) [tan(𝜙 −𝜓 −𝛽) + cot(𝜙 −𝜓 −𝜃)][1 + tan (𝛿 +𝜓 +𝜃)cot (𝜙−𝜓 −𝜃)]

𝐶2𝐸 = 1 + tan(𝛿 +𝜓 + 𝜃) [tan(𝜙 −𝜓 − 𝛽) + cot (𝜙 − 𝜓 −𝜃)]

The total active thrust can be divided into the static part and the dynamic part, among which the

static part can be calculated by Coulomb Theory, so the dynamic part is:

𝑃𝐴𝐷 = 𝑃𝐴𝐸 − 𝑃𝐴𝑆 (1.4)

4

To calculate the point of application of total active thrust, the acting points of the static part

and dynamic part can be calculated separately, then the superposition method can be used to add

them up. For the static force, based on the triangle distribution assumed by Coulomb Theory, the

point of application should be at H/3 from the bottom of the retaining wall. For the dynamic force,

Seed and Whitman (1970) suggested it should be at 0.6H. So, the point of application of total

active thrust is:

h =

𝑃𝐴𝑆𝐻3 + 𝑃𝐴𝐷(0.6𝐻)

𝑃𝐴𝐸

(1.5)

Passive Earth Pressure Conditions:

The passive earth pressure is derived in the same manner as active earth pressure in M-O method,

the total passive thrust (Figure 1.2) is:

𝑃𝑃𝐸 = 0.5𝐾𝑃𝐸𝛾𝐻2(1 − 𝑘𝑣) (1.6)

Where

𝐾𝑃𝐸 = 𝑐𝑜𝑠2(𝜙 + 𝜃 − 𝜓)

𝑐𝑜𝑠𝜓𝑐𝑜𝑠2𝜃 cos(𝛿 − 𝜃 + 𝜓) [1 +√sin (𝛿 + 𝜙)sin (𝜙+ 𝛽 −𝜓)cos (𝛿 − 𝜃 +𝜓)cos (𝛽 − 𝜃)]

2

(1.7)

5

Figure 1.2 Free body diagram of passive seismic loadings on retaining wall (Kramer, 1996)

The angle of the critical failure surface from horizontal is given by:

𝛼𝐴𝐸 = 𝜓 − 𝜙 + 𝑡𝑎𝑛−1[− tan(𝜙+ 𝜓 +𝛽) + 𝐶3𝐸

𝐶4𝐸]

(1.8)

Where

𝐶3𝐸 = √tan(𝜙 +𝛽 −𝜓)[tan(𝜙 +𝛽 −𝜓)+ cot(𝜙 + 𝜃 −𝜓)][1 + tan (𝛿 +𝜓 −𝜃)cot (𝜙−𝜓 +𝜃)]

𝐶2𝐸 = 1 + tan(𝛿 +𝜓 − 𝜃) [tan(𝜙 −𝜓 + 𝛽) + cot (𝜙 − 𝜓 +𝜃)]

Like the active pressure, the total passive thrust can also be divided into the static part and the

dynamic part

𝑃𝑃𝐷 = 𝑃𝑃𝐸 −𝑃𝑃𝑆 (1.9)

However, it should be noted, if the dynamic component in the direction that makes the total

active thrust bigger, it will make the total passive thrust smaller because the dynamic force is

always in the direction of the acceleration while active and passive thrusts have opponent

directions.

6

Shortcomings of M-O Method

The M-O method serving as the design code has some distinct drawbacks. These

deficiencies are from the assumptions the method is based on, so they are intrinsic problems that

cannot be easily solved. The main problems are listed below:

Soil properties

The first problem comes from the over-simplicity of the assumption on backfill properties.

The M-O method assumes that the soil is a homogenous body of cohesionless sands within the

active and passive pressure wedges. This assumption is obviously too simple and cannot reflect

the real complex properties of backfill. For nonhomogeneous soil, the computer software called

SLIDE (RocScience, 2005) is widely used to calibrate against the M-O method. For cohesive soil,

Richards and Shi (1994) as well as Chen and Liu (1990) discovered a small amount of cohesion

could significantly reduce the total active thrust during earthquake. In practice, it is not always

possible to use a perfectly cohesionless soil, therefore the MO method is often too conservative.

Backfill slope angle

The second problem relates to backfill slope angle. Figure 1.3 shows there is a sharp

augment of the seismic active pressure on retaining wall when the backfill slope angle increases

slightly after a certain value. This behavior is because the backfill slope and the critical failure

surface are nearly parallel with each other, and mathematically resembles the resonating frequency

of free vibration, which adds infinite mass to the active wedge. However, this phenomenon will

never happen in real soil-wall systems.

7

Figure 1.3 Backfill slope angle effect on the seismic active earth pressure coefficient using the

M-O method (Anderson et al., 2009)

Wall height

There is another issue with the M-O method that it does not take wall height effect into

consideration. Due to the nonlinear soil properties in the backfill, the peak ground acceleration

(PGA) should decrease within increasing the wall height. In the NCHRP report by Anderson et

al. (2009), they brought up a fill height-dependent reduction factor:

k𝑎𝑣 = 𝛼𝑘𝑚𝑎𝑥 (1.10)

Where 𝑘𝑚𝑎𝑥 is the PGA, 𝛼 is the fill height-dependent reduction factor, which can be calculated:

𝛼 = 1 + 0.01𝐻[(0.5𝛽) − 1] (1.11)

H is the height of the backfill, 𝛽 = spectrum shape factor = 𝐹𝑣𝑆1/𝑘𝑚𝑎𝑥(dim.)

8

Other issues

As the M-O method is a pseudo-static method based on equilibrium, structural integr ity

and wall displacement have just been roughly considered by a general reduction factor of 0.5.

However, these parameters will vary with changing wall geometries and ground motions, the

general reduction factor of 0.5 maybe suitable for the cases with limited wall sliding and bending,

but is obviously too conservative in other cases with large deformation.

In conclusion, based on the issues listed above except for the backfill slope angle that brings

up the singular value problem, the M-O method is intrinsically conservative. That’s probably why

cantilever retaining wall performs so well during well-documented earthquakes (Lew, et al., 1995)

(Koseki, et al., 1998) (Gazetas, et al., 2004). Although safety and serviceability is the priority in

design, economic efficiency and productivity improvement based on material saving should never

be neglected.

Objective of study

Have seen the shortcomings of M-O method listed above, it is definite worthy for

researchers to investigate these problems of the design method by taking advantage of the

advanced computational capacity these days. This thesis is focused on the earth pressure acting on

the retaining wall through numerical analysis. Firstly, a finite element (FE) model is built taking

wall-soil interaction into consideration. Later, a parametric study on effects like wall stiffness and

ground motion is conducted. Finally, a comparison between modifications of M-O method or other

design methods and numerical results is studied to sufficiently understand the seismic behavior of

retaining wall, and foster a better design code in the future if possible.

9

A 2D contact element will be used in this analysis to simulate the soil-wall interaction

behavior under earthquake scenarios, and it is the first time that this element has been used in

dynamic analysis. Due to this is a pioneering work, the purpose of this study is not to solve all the

problems listed above, but to build a solid numerical model using the 2D contact element, at the

same time, identify the problems with using this element and dig out the potential for the further

research.

10

CHAPTER TWO

CONTACT ELEMENT APPROACH

2D contact element is implemented in OpenSees to capture the soil-structure interaction

behaviors. The purpose of this element is to account for the gapping and stick-slipp ing

performances between soil and its interacting structures, and help others better model the

complexity of geotechnical problems. This discussion of the 2D contact element is divided into

three parts: kinematic relationships, constitutive behaviors, and equilibrium equations.

2.1 Projection of slave node to beam element

Kinematic constraints are the foundation to build the element’s formula, for it can be

visualized directly. The figure below shows the sketch of the contact body, where the beam

element represents the retaining wall and quadrilateral continuum element represents the

surrounding soil.

Figure 2.1 Contact body representing the contact element configuration

11

In Fig. 2.1, The coordinates 𝒙𝒂 and 𝒙𝒃 are the end nodes of the beam element, also viewed

as the master nodes; and 𝒙𝒔 , the slave node, is one node of the quadrilateral element. The point 𝒙𝒄

is the projection point to the beam centerline from the slave node, that needs to be located within

each step. The vectors 𝒂𝟏 ,𝒃𝟏 and 𝒄𝟏 are tangent vectors along the beam centerline, and r is the

half width of the retaining wall.

We use the Hermitian shape functions to define the beam elements, and locate the

coordinate of the projection node as

𝒙𝒄 = 𝝋(𝜉𝑐) = 𝒙𝒂𝐻1(𝜉𝑐)+ 𝒂𝟏𝐿𝐻2(𝜉𝑐) + 𝒙𝒃𝐻3(𝜉𝑐) + 𝒃𝟏𝐿𝐻4(𝜉𝑐) (2.1)

Where

𝐻1 = 1− 3𝜉2 +2𝜉3 (2.2)

𝐻2 = 𝜉 − 2𝜉2 + 𝜉3 (2.3)

𝐻3 = 3𝜉2 − 2𝜉3 (2.4)

𝐻4 = −𝜉2 + 𝜉3 (2.5)

and 𝜉 ∈ [0,1] is the internal coordinate on the beam.

Initially, there is no force acting on the retaining wall, so the beam element remains straight. Thus

the tangent vectors and 𝜉𝑐 can be determined as

𝒂𝟏 = 𝒃𝟏 = (𝒙𝒂 −𝒙𝒃)/𝐿 (2.6)

𝜉𝑐 = (𝒙𝒃 − 𝒙𝒔) ∙ 𝒂𝟏 (2.7)

Where L is the length of the beam, 𝐿 = ‖𝒙𝒂 −𝒙𝒃‖.

Later, after the beam deforms, the tangent vectors will change according to the incrementa l

rotations of the beam endpoints.

12

𝒂𝟏𝒏+𝟏 = 𝒂𝟏

𝒏 +𝜃𝑎𝟏𝒔𝒂𝟏

𝒏 (2.8)

𝒃𝟏𝒏+𝟏 = 𝒃𝟏

𝒏 + 𝜃𝑏𝟏𝒔𝒃𝟏

𝒏 (2.9)

Where n is the nth step during analysis

𝟏𝒔 = [0 −11 0

] (2.10)

And 𝜉𝑐 also changes, to calculate the new 𝜉𝑐 , we use an iterative Newton algorithm between

successive steps.

Firstly, if 𝑥𝑐 is the projection node of the slave node 𝑥𝑠 , geometrically, it must satisfy the

perpendicular condition.

(𝒙𝒔 −𝒙𝒄) ∙ 𝝋,𝜉 (𝜉𝑐) = 0 (2.11)

Where

𝒙𝒄,𝝃 = 𝝋,𝝃(𝜉𝑐) = 𝒙𝒂𝐻1,𝜉 (𝜉𝑐)+ 𝒂𝟏𝐿𝐻2,𝜉 (𝜉𝑐)+ 𝒙𝒃𝐻3,𝜉 (𝜉𝑐)+ 𝒃𝟏𝐿𝐻4,𝜉 (𝜉𝑐) (2.12)

𝐻1,𝜉 = −6𝜉 + 6𝜉2 (2.13)

𝐻2,𝜉 = 1 − 4𝜉 + 3𝜉2 (2.14)

𝐻3,𝜉 = 6𝜉 − 6𝜉2 (2.15)

𝐻4,𝜉 = −2𝜉 + 3𝜉2 (2.16)

Secondly, plug in the old 𝜉𝑐 , and calculate the residual. Here, the old 𝜉𝑐 is the 𝜉𝑐 from the last

step. For example, if we were in Step Two, then 𝜉𝑐 can be calculated by equation (2.7). What’s

more, the residual is equation (2.11) itself.

𝑅𝑘 = (𝒙𝒔𝒌 − 𝒙𝒄

𝒌) ∙ 𝝋,𝜉 (𝜉𝑐𝑘) (2.17)

Thirdly, linearize the residual and loop to the new 𝜉𝑐 . As the residual is a function of variable 𝜉,

its derivative is

13

𝐷𝑅(𝜉𝑐) = [(𝒙𝒔 − 𝝋(𝜉𝑐)) ∙ 𝝋,𝝃𝝃(𝜉𝑐) −𝝋,𝝃(𝜉𝑐) ∙ 𝝋 ,𝝃(𝜉𝑐)]𝑑𝜉 (2.18)

Where 𝝋,𝝃𝝃(𝜉𝑐) can be calculated in the same way as 𝝋,𝝃(𝜉𝑐).

Based on the Newton algorithm, the new 𝜉𝑐 is

𝜉𝑐𝑘+1 = 𝜉𝑐

𝑘 −𝑅𝑘

𝐷𝑅𝑘

(2.19)

Plug 𝜉𝑐𝑘+1 into equation (2.17), if the residual is still not small enough, set 𝜉𝑐

𝑘+1 as 𝜉𝑐𝑘 and do the

same procedure again, until 𝑅(𝜉𝑐𝑘+1) ≤ 𝑇𝑂𝐿, where TOL is a convergence tolerance. Once the

tolerance condition is met, we get the 𝜉𝑐𝑘+1, which is the new 𝜉𝑐 .

Hence, 𝜉𝑐 can be determined during each step. In other words, we have projected the slave node

to the beam centerline. As our targets are the gapping and slipping increment during each step, we

need to project 𝑥𝑐 back to the beam surface and locate the projected node 𝑥Γ.

𝒙𝚪 = 𝒙𝒄 + 𝑟𝒏 (2.20)

Where n is the unit normal vector to the beam centerline

𝒏 =𝒙𝒔 − 𝒙𝒄‖𝒙𝒔 − 𝒙𝒄‖

(2.21)

And the tangent vector at 𝑥Γ can be computed as

𝜾,𝝃 = 𝒙,𝝃 + 𝑟𝒏,𝝃 (2.22)

Where

𝒏,𝝃 = −(𝒄𝟏,𝝃 ∙ 𝒏)𝒄𝟏 (2.23)

𝒄𝟏 =𝒙𝒄,𝝃

‖𝒙𝒄,𝝃‖

(2.24)

14

𝒄𝟏,𝝃 =1

‖𝒙𝒄,𝝃‖[𝒙𝒄,𝝃𝝃 − (𝒄𝟏,𝝃 ∙ 𝒙𝒄,𝝃𝝃)𝒄𝟏]

(2.25)

So far, we can easily calculate the gapping and sliding values between soil node and beam element.

𝑔 = 𝒏 ∙ (𝒙𝒔− 𝒙𝒄) − 𝑟 (2.26)

𝑠 = 𝜾,𝝃 ∙ (𝒙𝒔 − 𝒙𝚪) (2.27)

2.2 Interface material property

The constitutive relationship in the interface is the flesh of the contact element. In

geotechnical problems, it should follow Mohr-Coulomb behavior:

𝑓 = |𝑡𝑠| − 𝜇𝑡𝑛 − 𝑐 ≤ 0 (2.28)

Where 𝑡𝑠 is the tangential (shear) force, and 𝑡𝑛 is the normal (pressure) force. 𝜇 is the frictiona l

coefficient, and c is the cohesion as shown in Fig 2.2.

Figure 2.2 Mohr-Coulomb frictional behavior

In the figure, we can find that the 𝑡𝑠 has two parts: the elastic part and the plastic part. And

its formulation is like the classical plasticity model with linear elastic and idealized flat plastic

15

curves. Its yielding value is determined by the pressure and cohesion: when f<0, the shear of the

interface is smaller than its cohesion, which means the soil and beam stick together, and 𝑡𝑠 remains

in the elastic stage; when f=0, the shear overcomes cohesion, the beam slides around soil, and 𝑡𝑠

goes into the plastic stage.

The contact element (will discuss in section 2.3) and contact material work together. Firstly,

the contact element determines the state of contact and calibrates the incremental slip value based

on equilibrium. Later, if the element is on contact, the material updates the contact forces and slip

using a closest point projection return mapping algorithm. The updated contact forces and slips

are calculated by an imbedded tangential stiffness matrix within the material.

𝝈 = {

𝑡𝑛𝑡𝑠𝑔} = 𝑪 ∙ {

𝑔

∆𝑠𝑡𝑛

} = 𝑪 ∙ 𝜺 (2.29)

Where the raw 휀 can be obtained roughly from contact element and kinematic relationship, 𝝈 is

the updated forces that should be applied to the return mapping algorithm.

𝑪 =𝝏𝝈

𝝏𝜺=

[ 𝜕𝑡𝑛𝜕𝑔

𝜕𝑡𝑛𝜕𝑠

𝜕𝑡𝑛𝜕𝑡𝑛

𝜕𝑡𝑠𝜕𝑔

𝜕𝑡𝑠𝜕𝑠

𝜕𝑡𝑠𝜕𝑡𝑛

𝜕𝑔

𝜕𝑔

𝜕𝑔

𝜕𝑠

𝜕𝑔

𝜕𝑡𝑛 ]

= [

𝐶𝑛𝑔 𝐶𝑛𝑠 𝐶𝑛𝑛𝐶𝑠𝑔 𝐶𝑠𝑠 𝐶𝑠𝑛𝐶𝑔𝑔 𝐶𝑔𝑠 𝐶𝑔𝑛

]

(2.30)

The independence of 𝑔 with 𝑡𝑠 𝑡𝑛, and 𝑡𝑛 having nothing to 𝑡𝑠 make C much simpler:

𝑪 = [0 0 10 𝐶𝑠𝑠 𝐶𝑠𝑛1 0 0

] (2.31)

𝐶𝑠𝑠 and 𝐶𝑠𝑛 are the two unknowns that we should focus on, and each of them relates to one stage

of Mohr-Coulomb model.

16

After obtaining s value from the contact element, we assume the model is in elastic stage, and set

the trial value 𝑠𝑛+1𝑒,𝑡𝑟 = 𝑠𝑛+1

𝑒 = 𝑠, the plastic slip is zero. The shear can be defined easily as

𝑡𝑠 𝑛+1 = 𝐺𝑠𝑛+1𝑒 (2.32)

Plug 𝑡𝑠 𝑛+1 into equation (2.26), If 𝑓𝑛+1𝑡𝑟 <0, the interface behavior is in the elastic stage as what we

expected, and everything is done. 𝐶𝑠𝑠 = 𝐺, 𝑎𝑛𝑑 𝐶𝑠𝑛 = 0.

If 𝑓𝑛+1𝑡𝑟 >0, that means the interface gets into the plastic stage, then additional permanent slip

develops, the elastic part should be truncated as

𝑠𝑛+1𝑒 = 𝑠𝑛+1

𝑒,𝑡𝑟 − ∆𝑠𝑛+1𝑝

(2.33)

∆𝑠𝑛+1𝑝

= ∆𝛾𝑟𝑛+1 (2.34)

Where 𝑟𝑛+1 is the unit vector in the sliding direction, which can be gotten from the plastic flow

rule, combining (2.30) (2.31) (2.32) and (2.26), to make 𝑓𝑛+1𝑡𝑟 = 0 we can get

∆𝛾 =𝑓𝑛+1𝑡𝑟

𝐺

(2.35)

After some transformations, the stiffness terms are thus identified as

𝐶𝑠𝑠 = 0, 𝐶𝑠𝑛 = 𝜇𝑟𝑛+1 (2.36)

2.3 Contact element

As there is friction (non-conservative force) in this system, the contact element formula

cannot be derived by variation method but virtual work method. Firstly, the total virtual work of

the system is calculated, and then linearizing the virtual work expression to get the stiffness matrix.

The total virtual work is

𝛿𝑊𝑐 = 𝑡𝑛𝛿𝑔 + 𝛿𝑡𝑛𝑔− 𝑡𝑠𝛿𝑠 (2.37)

17

Based on the kinematic relationships, the gap can be expressed as

𝑔𝑛+1 = 𝒏𝒏 ∙ [𝒙𝒔𝒏+𝟏 −𝒙𝒂

𝒏+𝟏𝐻1𝑛+ 𝒂𝟏

𝒏+𝟏𝐿𝐻2𝑛 +𝒙𝒃

𝒏+𝟏𝐻3𝑛 + 𝒃𝟏

𝒏+𝟏𝐿𝐻4𝑛] − 𝑟 (2.38)

The variation of gap can be expressed as

𝛿𝑔 = 𝒏𝒏 ∙ [𝛿𝒙𝒔𝒏+𝟏 − 𝜹𝒙𝒂

𝒏+𝟏𝐻1𝑛 + 𝜹𝒂𝟏

𝒏+𝟏𝐿𝐻2𝑛+ 𝜹𝒙𝒃

𝒏+𝟏𝐻3𝑛+ 𝜹𝒃𝟏

𝒏+𝟏𝐿𝐻4𝑛] (2.39)

Where based on equations (2.8) and (2.9)

𝛿𝒂𝟏𝒏+𝟏 = δ𝜃𝑎𝟏

𝒔𝒂𝟏𝒏 (2.40)

𝜹𝒃𝟏𝒏+𝟏 = 𝛿𝜃𝑏𝟏

𝒔𝒃𝟏𝒏 (2.41)

So, the gap can be expressed in the form:

𝛿𝑔 = 𝛿𝒒𝑻𝑩𝒏 (2.42)

Where q is vector of nodal degrees of freedom:

𝛿𝑞 =

{

𝛿𝒙𝒂𝛿𝜃𝑎𝛿𝒙𝒃𝛿𝜃𝑏𝛿𝒙𝒔}

(2.43)

And

𝐵𝑛 =

{

−𝐻1𝒏−𝐻2𝐿(𝟏

𝒔𝒂𝟏)𝒏−𝐻3𝑛

−𝐻4𝐿(𝟏𝒔𝒃𝟏)𝒏

𝒏 }

(2.44)

Based on equation (2.27) the slip can also be defined by vector of nodal degrees of freedom

likewise.

𝑠𝑛+1 = 𝜾,𝜉𝑛 ∙ [𝒙𝒔

𝒏+𝟏 − (𝒙𝒄𝒏+𝟏 − 𝑟𝒏𝒏+𝟏)] (2.45)

The variation of the slip is

18

𝛿𝑠 = 𝜾,𝜉𝑛 ∙ [𝜹𝒙𝒔 − (𝜹𝒙𝒄 − 𝑟𝜹𝒏)] (2.46)

From equations (2.1) (2.40) and (2.41) we know

𝛿𝒙𝒄 = 𝛿𝒙𝒂𝐻1𝑛 + (𝟏𝒔𝒂𝟏)𝐿𝐻2

𝑛𝛿𝜃𝑎 + 𝛿𝒙𝒃𝐻3𝑛 + (𝟏𝒔𝒃𝟏)𝐿𝐻4

𝑛𝛿𝜃𝑏 (2.47)

And the unit normal vector n is associated with the tangent vector 𝒙𝒄,𝝃 .

𝛿𝒙𝒄,𝝃 = 𝛿𝒙𝒂𝐻1,𝜉𝑛 + (𝟏𝒔𝒂𝟏)𝐿𝐻2,𝜉

𝑛 𝛿𝜃𝑎 + 𝛿𝒙𝒃𝐻3,𝜉𝑛 + (𝟏𝒔𝒃𝟏)𝐿𝐻4,𝜉

𝑛 𝛿𝜃𝑏 (2.48)

Plug equations (2.47) and (2.48) into (2.46), the variation of the slip is

𝛿𝑠 = 𝜾,𝜉𝑛 ∙ [𝜹𝒙𝒔 − 𝛿𝒙𝒂(𝑟𝐻1,𝜉

𝑛 + 𝐻1𝑛)− (𝟏𝒔𝒂𝟏)𝐿(𝑟𝐻2,𝜉

𝑛 + 𝐻2𝑛)𝛿𝜃𝑎 −

𝛿𝒙𝒃(𝑟𝐻3,𝜉𝑛 + 𝐻3

𝑛)+ (𝟏𝒔𝒃𝟏)𝐿(𝑟𝐻4,𝜉𝑛 +𝐻4

𝑛)𝛿𝜃𝑏]

(2.49)

It can also be expressed in the vector form

𝛿𝑠 = 𝛿𝒒𝑻𝑩𝒔 (2.50)

Where

𝑩𝒔 =

{

−𝜾,𝜉𝑛 (𝑟𝐻1,𝜉

𝑛 +𝐻1𝑛)

−𝜾,𝜉𝑛 (𝟏𝒔𝒂𝟏)𝐿(𝑟𝐻2,𝜉

𝑛 + 𝐻2𝑛)

−𝜾,𝜉𝑛 (𝑟𝐻3,𝜉

𝑛 +𝐻3𝑛)

−𝜾,𝜉𝑛 (𝟏𝒔𝒃𝟏)𝐿(𝑟𝐻4,𝜉

𝑛 + 𝐻4𝑛)

𝜾,𝜉𝑛

}

(2.51)

Plug equations (2.42) and (2.50) into the virtual work formula (2.37):

𝛿𝑊𝑐 = 𝛿𝒒𝑻𝑩𝒏𝑡𝑛+ 𝛿𝑡𝑛𝑔 − 𝛿𝒒

𝑻𝑩𝒔𝑡𝑠

= 𝛿𝒒𝑻(𝑩𝒏𝑡𝑛 −𝑩𝒔𝑡𝑠) + 𝛿𝑡𝑛𝑔

= 𝛿𝒒∗𝑻𝑹

(2.52)

Where

19

𝜹𝒒∗ = {𝜹𝒒𝛿𝑡𝑛

} (2.53)

𝑹 = {𝑩𝒏𝑡𝑛 − 𝑩𝒔𝑡𝑠

𝑔 } (2.54)

The tangent stiffness matrix of the contact element can be derived from adding a small differentia l

to the virtual work and linearizing it.

∆𝛿𝑊𝑐 = 𝛿𝒒𝑻𝑩𝒏∆𝑡𝑛+ 𝛿𝑡𝑛∆𝑔 − 𝛿𝒒𝑻𝑩𝒔∆𝑡𝑠

= 𝛿𝒒𝑻𝑩𝒏∆𝑡𝑛 +𝛿𝑡𝑛 ∆𝒒𝑻𝑩𝒏 −𝛿𝒒

𝑻𝑩𝒔(𝐶𝑠𝑠∆𝑠 + 𝐶𝑠𝑛∆𝑡𝑛)

= 𝛿𝒒𝑻𝑩𝒏∆𝑡𝑛+ 𝛿𝑡𝑛 ∆𝒒𝑻𝑩𝒏− 𝛿𝒒

𝑻𝑩𝒔𝐶𝑠𝑠𝑩𝒔𝑻∆𝒒−

𝛿𝒒𝑻𝑩𝒔𝐶𝑠𝑛∆𝑡𝑛

= 𝛿𝒒∗𝑻𝑲∆𝒒∗

(2.55)

𝐾 = [−𝑩𝒔𝐶𝑠𝑠𝑩𝒔

𝑻 𝑩𝒏 −𝑩𝒔𝐶𝑠𝑛𝑩𝒏

𝑻 0]

(2.56)

K is the stiffness matrix of the contact element.

20

CHAPTER THREE

MODEL DEVELOPMENT

Numerical models are built in finite element software OpenSees, and they are built upon

the centrifuge tests conducted by Al Atik and Sitar (2008) to examine the efficiency and accuracy

of their capacities to capture the essential features of the wall-soil systems. As described in Chapter

1, the primary purpose of this study is to investigate the function of 2D contact element in dynamic

analysis. The centrifuge test report conducted by the researchers in UC Berkley has comprehens ive

data recorded during the experiments, and the results and conclusions they made have been widely

reviewed and accepted. The report has already been viewed as the state of art on retaining wall

under seismic loading, although it has just published for several years. Besides, there is also an

OpenSees model in their report without using 2D contact element, which can be used as the

reference for our models. This chapter discusses in detail the software used, the sections of the

finite element models, and some techniques that are adopted to build the models.

3.1 Overview of OpenSees and GiD

The finite element models considered in this study are analyzed using the finite element

software OpenSees (Open System for Earthquake Engineering Simulation), which is an open-

source framework developed by the Pacific Earthquake Engineering Research (PEER) (OpenSees,

2007; McKenna, et al., 2010). The most appealing trait of OpenSees is the fact it is an open-source

software, where the whole community can contribute their efforts into enriching the library of

elements and algorithms. While for most commercial FE software, it is very hard to add user-

defined materials or elements, and the ones added can hardly be generally accepted. For this reason,

OpenSees can serve as a platform of exchanging solutions to theoretical and practical problems

21

for researchers. OpenSees is an interpreter, and the model input files are programs written using a

scripting language called “Tcl”. By doing this, people can extend to a very deep part of the

software, and do things like building their own templates, performing parameter studies, and

running the analysis in parallel. OpenSees is a fast and reliable FE software, especially for

nonlinear structural and geotechnical problems.

The commercial pre- and post-processing software GiD (International Center for

Numerical Methods in Engineering (CIMNE), 2008) was used to generate the input file that

OpenSees runs, and GiD was also used to visualize the results of the simulations. GiD has been

developed to cover the common needs of numerical analysis, in the fields from the pre-processing

to post-processing. It is universal, adaptive and user-friendly, and able to do the model mesh and

result visualization in a high speed and quality. By choosing the proper problem type, which is a

set of files configured by a software developer, users can prepare the input file that would be

analyzed by OpenSees. Later, the results data returned by OpenSees can be reformatted to a post-

processing file which can be used by GiD. Some problem types and Matlab scripts for reformatt ing

the files have been previously developed (McGann & Arduino, 2011), and these were used in this

study. The whole process flow chart is shown below.

22

Figure 3.1 The workflow of OpenSees analysis used in this study.

3.2 Overview of the centrifuge test used for model validation

Al Atik and Sitar (2010) conducted experimental and numerical studies on dynamic earth

pressures acting on retaining walls, and they came to several conclusions: firstly, the dynamic earth

pressure increases within the depth in a triangular distribution like the static earth pressure;

secondly, the wall moment is dominated by the inertial force and has phase difference with the

dynamic earth pressure, which means the dynamic earth pressure reaches its maximum when the

wall moment stays in its minimum, and vice versa.

Their centrifuge test is a perfect example for validation of the beam-solid contact element

for use in dynamic analysis of retaining structures. Their report (Al Atik & Sitar, 2008) contains

extensive experimental and numerical data, which can be used in model validation, and for even

further comparison, they conducted numerical analyses of their centrifuge models. In these

simulations, they used beam elements for the walls and solid elements for the soil, with zero-length

elements to simulate the interaction on the interface between the soil and the wall. The zero-length

element is a node to node element, and the particular formulation used by Al Atik and Sitar (2008)

23

does not take sliding and gapping effects into consideration. Replacing the zero-length element

approach with the 2D contact element described in Chapter 2 provides an interesting opportunity

for model verification as well as validation in this study.

The centrifuge tests conducted by Al Atik and Sitar (2008, 2010) consider two U-shaped

retaining walls sitting on dry medium-dense sand. These walls support the same sand as backfill.

Figure 3.2 and Figure 3.3 show the dimensions and setup of the centrifuge tests:

Figure 3.2 Centrifuge test configuration, profile view (Al Atik and Sitar, 2008).

24

Figure 3.3 Centrifuge test configuration, plan view (Al Atik and Sitar, 2008).

The centrifuge container is 1630 mm long and 786 mm wide, and the two U-shaped

retaining structures are 460.5 mm apart with the more stiff wall on the north and the more flexib le

one on the south. The height of the retaining wall is 180.7 mm. It should be noted all these sizes

are in model scale which is 1/N of the real case. According to the centrifuge scaling law, when

the period, shaking displacements and structural sizes shrink to 1/N, but acceleration magnitudes

and frequencies enlarge to N times, the strength, stiffness, stress and strain will remain the same

as the prototype (Kutter, 1995). The scaling factor is 36 g for these centrifuge tests (Al Atik and

Sitar, 2008).

3.3 Building the finite element model

The finite element model is developed in a text input file that can be read by OpenSees.

The model contained in this input file consists of several components: the nodes and elements

25

representing the geometry of the structure; the materials which can be linear or nonlinear, elastic

or plastic, changing with strain or temperature; the elements, which can consider different kinds

of shape functions; the boundary conditions with varying constraints; the integration and solution

algorithms; and so on. The properties of all these components can be defined in a pre-processing

software, such as GiD with a robust problem type.

3.3.1 Simulated geometry

The geometry of the 2D plane strain centrifuge test model is first drawn in GiD. From this

geometry, a mesh is generated. The base mesh is shown in Figure 3.4. It has the same skeleton

geometry as the real experiment configuration shown in Figure 3.2. The finite element model is

built in prototype size, which is 36 times the model scale. The mesh of the model consists of 1148

soil nodes, 1046 soil elements, 78 beam nodes, 76 beam elements, and 76 2D beam-solid contact

elements. The U-shaped walls were modeled as linear elastic beam elements. A 2D pressure-

dependent, plane strain, elastic-plastic soil material was used to model the dry Nevada sand. The

interaction between the soil and structure is modeled by the 2D contact elements. Details on each

element property are discussed in the following sections.

26

Figure 3.4 The mesh of the centrifuge model. Selective refinement was employed such that the

elements near the walls and ground surface are smaller than those near the base.

3.3.2 Elements considered in FE models

3.3.2.1 Beam elements

The U-shaped retaining structures are modeled using the displacement-based beam-column

elements in OpenSees. The vertical part is divided into 14 elements and the base is divided into 7

elements. The north retaining walls and the south ones are both made of the same material,

however the north walls are thicker than the south, so the north ones are much more stiff. For

simplicity, the north walls are referred to henceforth as the stiff retaining walls, and the south are

called flexible retaining walls. Each beam element has three integration points, and its end nodes

are taken as the master nodes of the contact elements. The details of the properties of the walls

are shown in the tables below (Al Atik and Sitar, 2008).

27

Table 3.1 FE model properties for stiff retaining walls (Al Atik and Sitar, 2008)

North Stiff South Stiff Base

Height (m) 5.67 5.67 -

Width (m) - - 10.86

Thickness (m) 0.30 0.30 0.30

Mass (kg) 3334.34 3452.11 12044.31

Area (𝑚2 ) 0.14 0.14 0.25

E (kPa) 7.0E+07 7.0E+07 7.0E+07

I (𝑚4 ) 2.43E-03 2.43E-03 1.42 E-02

Table 3.2 FE model properties for flexible retaining walls (Al Atik and Sitar, 2008)

North Stiff South Stiff Base

Height (m) 5.67 5.67 -

Width (m) - - 11.32

Thickness (m) 0.30 0.30 0.30

Mass (kg) 2890.39 2937.50 12353.95

Area (𝑚2 ) 0.08 0.08 0.25

E (kPa) 7.0E+07 7.0E+07 7.0E+07

I (𝑚4 ) 4.26E-04 4.26E-04 1.42 E-02

28

3.3.2.2 Soil constitutive model

The soil domain was modeled using SSPquad Elements (McGann et al., 2012). The

SSPquad Element is a four-node quadrilateral element using stabilized single-point reduced

integration. It can repeat all functionalities of Quad Element, and due to its reduced-integra t ion

attribute, is much more efficient in dynamic analysis. Besides, it is also physical hourglass stable,

and free of the shear locking or volumetric locking problems.

The behavior of soil is complicated and a robust constitutive model must be used. In our

numerical model, the PressureDependMultiYield (PDMY) nDMaterial was used as the material

for soil elements (Elgamal et al., 2002). The PDMY material is used to simulate sandy material

which is sensitive to pressure in both static and dynamic loadings. This elastic-plastic material has

taken some response characteristics into consideration, such as dilatancy (volume changes

introduced by shear stress), and non-flow liquefaction (cyclic mobility), which can occur in

saturated sands and silts during seismic events.

In the analysis, the material behaves linear elastically at first. After the static gravity

loading is added, the material is turned to plastic by using the UpdateMaterialStage command in

OpenSees. The plasticity of this material model is developed by the yield function forming a

conical surface just as the Drucker-Prager yield surface. The hardening zone is formed by the

multiple similar yield surfaces (D-P type) with a common apex, among which the exterior is the

failure surface. Figure 3.5 shows the details.

29

Figure 3.5 Yield Surfaces of PDMY material (Elgamal et al., 2003)

The PDMY material is defined with at least 15 parameters in OpenSees. For user

convenience, a table (reproduced in Table 3.3) is provided by Yang et al. (2008) as a quick estimate.

However, these parameters should be chosen carefully depending on the real situations.

30

Table 3.3 Reference parameters for PDMY material (Yang et al., 2008)

Parameters Loose

Sand

(15%-

35%)

Medium

Sand

(35%-65%)

Medium-dense

Sand

(65%-85%)

Dense

Sand

(85%-

100%)

Initial Mass Density (ton/m3) 1.7 1.9 2.0 2.1

Ref. Shear Modulus, Gr (kPa) 5.5x104 7.5x104 1.0x105 1.3x105

Ref. Bulk Modulus, Br (kPa) 1.5x105 2.0x105 3.0x105 3.9x105

Friction Angle (deg.) 29 33 37 40

Peak Shear Strain 0.1 0.1 0.1 0.1

Ref. Confining Pressure (kPa) 80 80 80 80

Pressure Dependent Coefficient 0.5 0.5 0.5 0.5

Phase Transformation Angle

(deg.)

29 27 27 27

Contraction Constant 0.21 0.07 0.05 0.03

Dilation Constant (dilat1) 0 0.4 0.6 0.8

Dilation Constant (dilat2) 0 2 3 5

Liquefaction-Induced Strain

Constants

10 10 5 0

0.02 0.01 0.003 0

1 1 1 0

Void Ratio 0.85 0.7 0.55 0.45

31

In Table 3.3, the meanings of these parameters are:

1. The initial mass density is the density of the medium-dense Nevada sand used in the

experiment.

2. The reference shear modulus Gr is defined as the shear modulus at reference confining

pressure and used to calculate the material shear modulus during elastic stage:

𝐺 = 𝐺𝑟 ∙ (𝑝′

𝑝𝑟′)𝑑

(3.1)

Where 𝑝′ is the effective confinement pressure which is increasing with the depth, 𝑝𝑟′ is

the reference confining pressure, d is the pressure-dependent coefficient which usually

taken as 0.5.

3. The reference bulk modulus Br is defined through the elastic relationship:

𝐵𝑟 =2(1 + 𝜇)

3(1 − 2𝜇)∙ 𝐺𝑟

(3.2)

Where 𝜇 is Poisson’ ratio.

4. The friction angle is determined by peak shear strength based on the Mohr-Coulomb theory.

During the seismic analysis, it is estimated by the modulus reduction curve through the

formula:

𝑠𝑖𝑛∅ =3√3 ∙ 𝜎𝑚/𝑝𝑟

′

6 +√3 ∙ 𝜎𝑚/𝑝𝑟′

(3.3)

Where 𝜎𝑚 is the product of the last modulus and strain pair in the modulus reduction curve.

5. The peak shear strain is an octahedral shear strain when the maximum shear strength is

reached under reference confining pressure 𝑃𝑟′, and it is usually set to 0.1. Figure 3.6 from

32

the OpenSees Manual (Yang et al., 2008) gives a clear view of all the parameters discussed

above.

Figure 3.6 Constitutive relationship of PDMY material (Yang et al., 2008)

6. The phase transformation angle, the contraction constants and the dilation constants are

parameters related to the dilatancy behavior of sand.

7. As there is no water in the sand of the centrifuge tests, all the liquefaction coefficients are

set to zero.

Al Atik and Sitar (2008) performed some tests to calibrate the soil properties to achieve

better modeling results. The final parameters used in their numerical model, and adopted here, are

summarized in Table 3.4. Their work is described as:

1. They set 𝑝𝑟′ as 54 kPa, which is the earth pressure at the middle of the backfill. Gr is

measured by the shear wave velocity:

𝐺𝑟 = 𝜌𝑉𝑠2 (3.4)

33

Where 𝜌 is the mass density of the sand, and 𝑉𝑠 is the shear wave velocity in the backfill

before shaking.

2. The modulus reduction curve is calculated through several shaking experiments. The shear

stress is measured using the 1D shear beam idealization theory (Zeghal, et al., 1995):

𝜕𝜏

𝜕𝑧= 𝜌�̈�

(3.5)

Where �̈� is the horizontal acceleration recorded by the lateral downhole accelerometers.

The shear strain is measured by dividing the horizontal displacement difference of

successive accelerometers by the vertical distance between them, and the horizonta l

displacement is calculated by double integrating the acceleration of each accelerometer.

The modulus reduction curve is shown in Figure. 3.7.

3. Damping ratio curve (Figure 3.8) is also measured by the hysteresis loops, by using the

formula (Kramer, 1996):

𝜉 =𝑊𝑑

4𝜋𝑊𝑠=1

2𝜋

𝐴𝑙𝑜𝑜𝑝𝐺𝑠𝑒𝑐𝛾𝑐

2

(3.6)

Where 𝑊𝑑 is the dissipated energy, 𝑊𝑠 is the maximum strain energy, 𝐴𝑙𝑜𝑜𝑝 is the area of

the hysteresis loop.

34

Table 3.4 Material properties used in PDMY material (Al Atik & Sitar, 2008)

Model Parameters Dry Medium-Dense Nevada sand

Initial Mass Density (𝑘𝑔/𝑚3 ) 1692

Ref. Shear Modulus, Gr (kPa) 5.30E+04

Poisson’s Ratio 0.3

Ref. Bulk Modulus, Br (kPa) 1.15E+05

Ref. Confining Pressure (kPa) 54

Peak Shear Strain 0.1

Pressure Dependent Coeffient 0.5

Shear Strain and G/Gmax pairs Based on Figure 3.7

Friction Angle (degree) 35

Phase Transformation Angle (degree) 27

Contraction Constant 0.05

Dilation Constants 𝑑1 = 0.6, 𝑑2 = 3.0

Liquefaction Induced Strain Constants 0

Number of Yield Surfaces 11

Void Ratio 0.566

35

Figure 3.7 Modulus reduction curve measured in experiments (Al Atik & Sitar, 2008)

Figure 3.8 Damping ratio curve measured in the experiments (Al Atik & Sitar, 2008)

3.3.2.3 Beam-solid contact elements

The contact element formulation has been introduced in Chapter 2. The discussion here

shows how it is implemented into the overall FE model. Figure 3.9 shows the mesh of beam

36

elements and surrounding solid elements, the contact elements are arranged between them. There

are several issues that should be noted when using the contact elements. Firstly, the widths of the

contact elements should be exact equal to the beam widths in the mesh, a small discrepancy can

cause a big problem, especially in dynamic analysis. Secondly, the slave node should ideally be

located at the middle of the two master nodes, and the contact element would fail if the slave node

slipped out past the bounds of the corresponding beam element. Thirdly, the penalty method is

better than the Lagrange method for enforcing the contact constraints in dynamic case, as the

penalty approach is a more relaxed approach, however the penalty parameter should be carefully

chosen. Last but not least, as it is assumed that there is no friction on the surface of the wall, the

friction coefficient has been set to zero in the numerical model.

Figure 3.9 The mesh of beam elements and surrounding soil elements

37

3.3.3 Treatment of boundary conditions

3.3.3.1 Bottom boundary conditions

All the nodes of the bottom line of the centrifuge model are fixed in both horizontal and

vertical directions, which means there is no movements allowed at the bottom of the model. This

is consistent with the assumption of what takes place in the centrifuge box.

3.3.3.2 Side boundary conditions

The flexible shear-beam model container (FSB2) was used in the centrifuge test, whose

shear rings of the laminar box are very flexible. Most researchers (e.g., Wilson et al., 1997; Lai et

al., 2004; Yang et al., 2004) agree that it does not affect the testing results due to its flexibility.

Hence, its stiffness and mass are neglected in the numerical model. The two lateral sides of the

centrifuge model are connected using the equalDOF command in OpenSees to achieve periodic

lateral boundary conditions, which means the nodes at the same height of both sides have the same

displacements in both vertical and horizontal directions.

3.3.4 Analysis types

Two analyses are carried out based on the numerical model: static analysis and dynamic

analysis, among which dynamic analysis is the emphasis. The static case is analyzed as a proof-

to-concept case, and it is also used by the dynamic case to study the dynamic earth pressures acting

on the retaining walls. The dynamic case is what we are interested in, multiple variables like

accelerations, earth pressures, beam elements etc. will be carefully studied.

38

3.4 One tiny modeling trick

There is one tiny modeling trick that should be mentioned here in our numerical model:

the “cross pieces”. As can be seen in Figure 3.10, the corners of the retaining structures are

modeled like crosses, where two fictitious beam elements are added to connect each corner point.

By doing this the corner is constrained from moving around, which is perhaps more realistic

because there are stresses around the corners in the field. The fictitious beam elements have the

same stiffness as the nearby real ones, but are without mass. Without these additional “cross piece”

elements, the soil nodes in the corner of the walls are not in contact with the walls as they are not

within the bounds of any beam elements. The implications of this modeling trick are considered

in subsequent chapters through comparison of models defined with and without these additiona l

elements.

Figure 3.10 The configuration of the “cross pieces” at the corners of the retaining walls.

39

CHAPTER FOUR

STATIC ANALYSIS RESULTS AND DISSCUSSION

The primary purpose of the static analysis phase is to verify the model set-up. There is no

sense in shaking this model if it can’t even handle static gravity loads, also this study needs a

benchmark to verify that the transient gravity phase that kicks off the transient analyses is correct.

These models are also used to explore the distributions of lateral earth pressures and forces on the

retaining walls as returned by the contact elements and simple lateral earth pressure theories. Chin

(2015) concluded that the 2D contact element provide best response under static conditions

compared with other connecting elements in OpenSees to simulate the wall-soil interface

behaviors. For all these reasons, there is a strong need to do the static analysis first.

To better understand the stresses around the corners of the U-shaped retaining structures,

a numerical centrifuge model with the refined mesh is built. Figure 4.1 compares the initial coarse

mesh of the FE centrifuge model and later refined mesh of the same configuration. In the refined

mesh, the retaining walls and surrounding soils are meshed more densely around the corner of the

retaining structures. The lengths of beam elements are reduced to 0.2m near the corner while are

0.5m-1m in other parts of the walls. To reduce the computational problems caused by a sharp

change of the adjacent element lengths around the corner, the element lengths are concentrated in

the GiD mesh generation step using a weighting factor. As can be seen in the refined mesh, the

solid element lengths are increasing smoothly with increasing distance away from the retaining

structure corner. The refined mesh model has 1975 nodes, 1700 solid elements, 118 beam elements

and 118 contact elements, almost twice the nodes and elements in the coarse mesh model.

40

Figure 4.1 The coarse mesh of the FE centrifuge test model (the upper mesh) and the refined

mesh of the FE centrifuge test model (the bottom mesh)

As the “cross pieces” trick introduced in Chapter 3 also deals with the behavior of the

corner of the retaining structure, three cases including the coarse meshed FE centrifuge test model

without cross pieces, the coarse meshed FE centrifuge test model with cross pieces, and refine

meshed FE centrifuge test model are all taken into analysis. The displacements, the stresses and

contact force distributions of all three cases are compared, and the evaluation of the “cross pieces”

is also included in the later sections.

41

4.1 The displacement distributions

The vertical displacement configurations are shown in Figures 4.2-4.4. From the figures,

the displacements in the vertical direction do not change too much, which reflects a good estimate

reached by all three cases. The displacement on the top of the flexible retaining wall is larger than

that of the stiff retaining wall, this is likely due to the lower bending stiffness of the flexib le

retaining wall leading to larger structural deformation and correspondingly larger soil

deformations. The maximum displacements in the y-direction are around two centimeters for all

cases, which means the slave nodes still stay at the midrange of the beam elements (i.e., contact

should not be lost in the beam-solid contact elements due to the slave nodes moving “out of

bounds”). The case with “cross pieces” (Figure 4.3) has larger displacements than other two cases,

this may due to the “cross pieces” making the corner point much stiffer, leading to more

deformation at the top of the retaining walls. The displacements do not differ too much between

the refined mesh case and coarse mesh case, which reflects that they are both acceptable.

Figure 4.2 The y-displacement (m) configuration of coarse meshed case without “cross pieces”

42

Figure 4.3 The y-displacement configuration (m) of coarse meshed case with “cross pieces”

Figure 4.4 The y-displacement (m) configuration of refined meshed case

43

4.2 The stress distributions

4.2.1 The horizontal stress distributions

The horizontal stresses are shown in Figures 4.5-4.7. From the figures, the horizontal stresses

increase within the depth of the soil. The corners of the flexible retaining walls have more

concentrated stresses, while the horizontal stresses around the stiff ones are more evenly

distributed. For the case having “cross pieces” added, both the corners of the stiff and flexib le

retaining walls generate concentrated stresses, this is also due to the large stiffness of the “cross

pieces”. Again, there is not much difference of horizontal stresses between coarse mesh and

refined mesh cases. Dividing the horizontal stresses by the vertical stresses shown in Figure 4.8-

4.10, the coefficient of earth pressure at rest is found to be approximately 0.42, and it is almost the

same with the value calculated from the friction angle, which is 0.426. Note that away from the

free-field the model does not represent true “at rest” conditions.

Figure 4.5 The horizontal stress (kPa) of coarse meshed case without “cross pieces”

44

Figure 4.6 The horizontal stress (kPa) of coarse meshed case with “cross pieces”

Figure 4.7 The horizontal stress (kPa) of the refined meshed case

45

4.2.2 The vertical stress distributions

The vertical stress distributions returned by the three considered model configurations are

shown in Figures 4.8-4.10. As shown, the vertical stresses are almost the same for all three cases,

although the “cross pieces” and the refinement of mesh may cause some local perturbation in the

stress contours, the overall trends are consistent for all three cases.

the vertical and horizontal stress distributions shown in these two sections being consistent

with expectations for static gravity loads in these three models.

Figure 4.8 The vertical stress (kPa) of coarse mesh case without “cross pieces”

46

Figure 4.9 The vertical stress (kPa) of coarse mesh case with “cross pieces”

Figure 4.10 The vertical stress (kPa) of the refined meshed case

47

4.2.3 The shear stress distributions

Figures 4.11-4.13 show the contours of the shear stress in the soil. The refined mesh is able

to more accurately capture the shear stresses at the corners of the retaining structure, while the

coarse mesh cases do not capture the high shear stresses at the corners as well (note the differences

in the magnitude scale between Figure 4.13 and Figures 4.11-4.12), and tend to enlarge the high

shear stress areas.

Figure 4.11 The shear stress (kPa) of coarse meshed case without “cross pieces”

48

Figure 4.12 The shear stress (kPa) of coarse meshed case with “cross pieces”

Figure 4.13 The shear stress (kPa) of the refined meshed case

49

4.3 Lateral earth pressure distributions

4.3.1 Contact forces at slave nodes

Figures 4.14-4.16 show the contact forces acting on the slave nodes of the beam-solid

contact elements that define the wall-soil interface. These forces are pulled directly from the

recorded information returned by the contact elements. The contact forces of the flexible retaining

walls are larger than the stiff retaining walls, and they are more in a parabolic shape while the

contact forces of stiff retaining walls are distributed in a triangular shape. Due to the large stiffness

of the “cross pieces” the contact forces are smaller at the top of the retaining walls than other cases

(see Figure 4.15). The contact forces get smaller with depth in the refined mesh case, as the element

tributary area gets smaller, but the contact pressures do continue to get higher with increasing

depth. At the bottom of the retaining wall, the earth pressures get smaller because the earth pressure

under the bottom bar of U-shaped structure is very small which should balance the earth pressure

from the other side of the retaining wall.

50

Figure 4.14 The contact forces (kN) of coarse meshed case without “cross pieces”

Figure 4.15 The contact forces (kN) of coarse meshed case with “cross pieces”

51

Figure 4.16 The contact forces (kN) of the refine meshed case

4.3.2 Lateral earth pressures on retaining walls

The earth pressure on the stiff and flexible retaining walls are compared with the classic

Rankine earth pressure theory. In these models, the friction coefficient for the beam-solid contact

elements is set to zero, thus a frictional contact interface is defined and the Rankine theory is an

appropriate comparison. At the top of the retaining walls, there is some gap between the soil and

walls, so the earth pressure is almost zero. At the middle of the retaining walls, as the wall can

bend the earth pressure is close to the Rankine active earth pressure. At the bottom of the retaining

walls, the wall cannot move too much so the earth pressure is equal to the static earth pressure.

The flexible retaining walls have larger deformations so their earth pressure is in a curvier shape,

with some passive earth pressure generated at the bottom part.

52

Compare the lateral earth pressure distributions of these three model configurations, we

find the “cross pieces” case has more force concentration at the bottom part of the wall, which is

not that accurate compared with other two mesh cases. The refined mesh case has accurate results,

but the earth pressure distributed wavier than the coarse mesh case. The coarse mesh case is still

the best among them.

Figure 4.17 The earth pressure on retaining walls of coarse meshed case without “cross pieces”

53

Figure 4.18 The earth pressure on retaining walls of coarse meshed case with “cross pieces”

54

Figure 4.19 The earth pressure on retaining walls of refined mesh case

4.4 Summary

Several variables are compared around three cases with changed mesh and added “cross pieces”

in this chapter, some conclusions can be found:

1st, the “cross pieces” significantly add stiffness to the corners of the retaining structures,

which will cause large deformation of the walls and concentrated stresses at the corners.

55

2nd, there is not much difference caused by refining mesh at the corners of the retaining

structures. Although shear stress at the corners will have more accurate estimate by refining

the mesh, it is not the variable that our modeling is interested in.

3rd, the coarse mesh without “cross pieces” is good and accurate enough comparing with

other cases, so it should be used in the dynamic analysis.

56

CHAPTER FIVE

DYNAMIC ANALYSIS RESULTS AND DISSCUSSION

As the numerical model described in Chapter 4 has proved its capacity in capturing the

soil-structure behaviors of the retaining wall under static loading, this chapter will discuss the

resilience of the numerical model under seismic loading. Although the contact element described

in Chapter 2 has been successfully applied in many static scenarios before, it has never been used

in any dynamic analysis. The functionality of this element remains unpredictable, particular ly

during large acceleration input motions. In this dynamic analysis, the element behavior is assessed

using various results, including the accelerations in the free field soil and on the retaining walls,

the total and dynamic earth pressures acting on the walls, the wall moments during the earthquakes,

among others. By doing this, we want to probe the problems that the 2D contact element has in

dynamic analysis and conclude some useful trends of dynamic soil-structure system behaviors

which are still unclear and disputable these days.

This chapter will be divided into four sections, firstly some more modeling procedures and

techniques beside which are already discussed in Chapter 3 will be introduced. Secondly, the

results will be compared with their counterparts in Al Atik and Sitar (2008), including the

accelerations from the free field soil and the retaining walls, the dynamic bending moments of the

retaining walls, the dynamic earth pressure acting on the retaining walls, and the soil shear stress

during the earthquakes. Thirdly, the earth pressure results will also be compared with the Monobe-

Okabe (M-O) theory implemented in most retaining structure the design codes. Finally, useful

suggestions will be given to both the future 2D contact element development and improvement of

the numerical model.

57

5.1 Analysis procedures and modeling techniques

During dynamic analysis, the analysis type is transient, which is different from the static

analysis in Chapter 4. The whole analysis has been divided into three successive steps:

The elastic gravity step: During this step, the gravity is added via elemental body forces

in the soil and as nodal forces in the beam elements representing the walls. For this init ia l

analysis stage, the pressure-dependent multi-yield (PDMY) soil constitutive model is set

to a linear elastic state. Very large time steps are used in analysis to dissipate the inertia l

gravity transient developed by ‘switching on’ the gravity loads. The inertial gravity

transient residue is assumed as completely damped out when the accelerations in the soil

domain remain uniform and are approximately zero.

The elastic-plastic gravity step: At this step, no new load is applied, and the PDMY soil

material is turned to an elastoplastic state. Based on PDMY material property (Yang, 2000),

the stress point could fall out of the failure envelop after elastic stage, and introduce a force

imbalance in the FE equation system. Large numerical damping coefficients should be used

in small time steps to quickly dissipate the imbalanced force. Gradually larger time steps

are considered until the stress contours match those returned by the static analysis of

Chapter 4.

The dynamic step: At this step, the input earthquake motion is added, while the soil is

kept in an elastoplastic state. Time is reset to zero, as no gravity transient or imbalanced

force are left in the system. The dynamic analysis voyage is finally set sail.

There are several modeling techniques that have not been covered in the previous Chapter

3, including the damping coefficients, the input earthquake motions, and the model system

58

parameters. These techniques play an important role in dynamic analysis and should be carefully

chosen. In this section, we will discuss in details on these modeling techniques.

5.1.1 Damping coefficients

Damping effect is complex in dynamic analysis, on one side it troubles numerous

researchers in finding correct damping coefficients, on the other side it is a lubricant that can make

dynamic analysis convergent and smooth. In the current analyses, there are three damping effects,

including hysteresis damping, viscous damping, and numerical damping that need to be considered.

The hysteresis damping is generated by the soil material, and it can be measured through the shear

stress-strain loops. It is defined by the modulus reduction curve of the soil material in the numerica l

model. The viscous damping is modeled using Rayleigh damping (via the Rayleigh command in

OpenSees). The Rayleigh damping matrix is a combination of the stiffness-proportional matrix

and the mass-proportional matrix as [𝐶] = 𝛼 ∙ [𝑀]+ 𝛽 ∙ [𝐾] , where 𝛼 and 𝛽 are mass

proportional coefficient and stiffness proportional coefficient respectively. The damping ratio is

set as 3.5%, and the initial frequency is set as 0.2 Hz and the second frequency is set as 2.53 Hz.

The α and β values can be back calculated by the formula (Charney, 2008):

{𝜉1𝜉2} =

1

2[1/𝑤1 𝑤11/𝑤2 𝑤2

] {𝛼𝛽}

(5.1)

Where 𝜉 is the damping ratio, 𝑤1 and 𝑤2 are the circular frequencies that can be chosen randomly.

In this case, 𝜉1 = 𝜉2 = 0.035,𝑤1 = 2𝜋×0.2,𝑤2 = 2𝜋×2.53 . It should be noted that this

command overwrites any existing damping coefficients at the nodes and elements. To achieve high

computational consistency the Rayleigh command should be defined once for all the steps.

Numerical damping is very crucial in the elastic-plastic step. A high numerical damping coeffic ient

can damp out the force imbalance when the stress point falls out the failure envelope. The

59

numerical damping is generated by careful selection of the coefficients in the Newmark integrator.

For the Newmark method, as is well known, unconditional stability of computation would be

achieved when 𝛽 = 0.25(𝛾+ 0.5)2, and there would be no numerical damping when 𝛾 = 0.5 and

𝛽 = 0.25. In our analysis, we set 𝛾 = 3.6 during the elastic-plastic step, for very high stress

imbalance occurred behind the tip of the retaining wall. And we set 𝛾 = 0.6 during dynamic step,

where adding a little numerical damping can improve the computational efficiency without

significantly impairing the accuracy.

5.1.2 Input earthquake motions

Three ground motions selected from the PEER Ground Motion database including Loma

Prieta-SC-1, Kobe-PI-2, and Loma Prieta-SC-2 are used as the input earthquake motions in the

model. These three shaking events are also adapted by Al Atik and Sitar (2008), however, the input

earthquake motions in their report were recorded during centrifuge testing at the base of the model

container, and there is a high limit of frequency the centrifuge device can go to. Consequently, the

centrifuge testing was acting like a high frequency pass filter and the high frequency segments of

the database signal are eliminated. As the resulting ‘filtered’ motions applied during the centrifuge

tests were not available, the corresponding unaltered ground motions from the database were used

as the input earthquake motions here. The differences are shown in the figures below. All the input

earthquake motions are applied to the base of the model using the OpenSees UniformExcita t ion

command, and the peak ground accelerations vary from 0.41 to 0.8 g.

60


Motion database and used in numerical modeling; the lower is generated by centrifuge testing.

Figure 5.2 The input ground motions of Kobe-PI-2: the upper is from PEER Ground Motion

database and used in numerical modeling; the lower is generated by centrifuge testing

61


Motion database and used in numerical modeling; the lower is generated by centrifuge testing

5.1.3 Model system parameters

In the last part of the analysis, there are some system parameters that need to be defined

with respect to how the numerical system of equations is solved within OpenSees. The constraint

equations are enforced by using the transformation method. The norm displacement increment test

is used to check the convergence status at end of each loading step, the limiting value (convergence

tolerance) is set as 2.0e-4 m after a maximum of 35 iterations. The Newton-Raphson method is

used to solve nonlinear algebraic equations. The reverse Cuthill-Mckee scheme node numbering

scheme was used to order the matrix equations and number the nodal degrees of freedom. The

system of equations is stored and solved by the ProfileSPD command which implies the system

matrix is symmetric positive definite. As previously discussed, the Newmark method was utilized

62

for numerical integration. After all the above parameters defined, there is nothing left to implement

the analysis, and the calculated results are shown in the next section.

5.2 Analysis Results

The analysis results are obtained after running the generated input files in OpenSees for

several scenarios. In this section, the computed responses for the three primary scenarios analyzed

are discussed. These scenarios consist of different shaking events: Loma Prieta-SC-1, Kobe-PI-2,

and Loma Prieta-SC-2 as shown and discussed in Figures 5.1-5.3. Characteristics of time series

of soil and structure acceleration, bending moments and earth pressures on the walls, as well as

soil shear stresses and strains are compared with the results of Al Atik and Sitar (2008), both with

respect to the measured/observed results from centrifuge testing and as computed using a separate

numerical analysis. These comparisons will be a solid base for future application and development

of the 2D contact element in the dynamic analysis.

5.2.1 Accelerations

The horizontal acceleration responses on the tops of the retaining walls and the free field

are shown in Figures 5.4-5.6 for the considered seismic events (Loma Prieta-SC-1, Kobe-PI-2, and

Loma Prieta-SC-2, respectively). The magnitudes of accelerations on the top of the stiff retaining

walls are smaller than those on the top of the flexible retaining walls, which means that the top of

the flexible retaining wall also has larger velocities and displacements. As all the retaining walls

are subjected to the same seismic loading, these observations are reasonable based on the elastic

beam theory. The magnitudes of accelerations on the free field are larger than those input

accelerations which is also a very reasonable observation, for the sand material in our numerica l

model can be classified as soft and is expected to amplify the acceleration magnitudes. The phases

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of the accelerations on the tops of stiff and flexible retaining walls, and on the top of free field are

similar while a bit different from that of the input earthquake accelerations. This phenomenon is

likely due to wave propagation effects, as the soil and walls have different vibration properties.

Compared with the computed and recorded acceleration results of Al Atik and Sitar (2008), the

current results agree well in terms of magnitude. As the input accelerations are different in phase

(their input accelerations was recorded directly on the base of the centrifuge container, of which

high frequencies were cut off), there is no meaning to compare the phase contents.

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Prieta-SC-1

65

Figure 5.5 Computed accelerations at the tops of retaining walls and free field during Kobe-PI-2

66


Prieta-SC-2

67

5.2.2 Bending moments

Figures 5.7-5.9 show the computed total wall moment time series of selected points on

both the south stiff wall and north flexible wall during the three earthquake events. The positive

direction of the moment for both retaining walls is defined as the wall rotating away from the soil.

The phase of the moment time series is in accordance to the phase of the acceleration time series

shown in Figures 5.4-5.6, especially for the moment on the stiff retaining walls. The static

moments after the earthquake are generally larger than the static moments before shaking due to

the soil densification. It should also be noted that the phase of the moment time series of the north

flexible retaining wall is opposite to that of the south stiff retaining wall, this observation will be

discussed later.

Figures 5.10-5.12 present the moment profiles of both the south stiff retaining wall and

north flexible retaining wall at three interesting time points during the three shaking events: the

starting time point before the shaking when the retaining wall is still static, the fiercest shaking

time points when the maximum moments are reached by the north stiff wall and south flexib le

wall respectively, and cool-down time point when both retaining walls settled to quiet static phase

after the shaking. Compared with the testing results measured by Al Atik and Sitar (2008), the

computed moments are generally larger, especially for the ones at the bottom of retaining walls

after shaking. Despite that, there are some trends that match very well between the computed

moment profiles and the recorded ones from testing, which include the moment of the stiff

retaining wall is slightly larger than that of the flexible retaining wall, and the distribution shape

of the moment is parabolic.

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Overall, the current FE models do capture some traits shared by the testing, such as the

increasing static moments after shaking due to soil densification, and the parabolic distributions of

static and total moments along the depth of the walls. However, it is observed that the current

models generally calculate larger values than what have been measured in the centrifuge. In some

areas, such as the bottom of retaining wall after shaking, the overestimation can be quite severe.

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and north flexible walls during Loma Prieta-SC-1

70


and north flexible walls during Kobe-PI-2

71


and north flexible walls during Loma Priesta-SC-2

72

Figure 5.10 Comparison of computed and recorded moment profiles (from Al Atik & Sitar,

2008) for Loma Prieta-SC-1

73

Figure 5.11 Comparison of computed and recorded moment profiles (from Al Atik & Sitar,

2008) for Kobe-PI-2

74

Figure 5. 12 Comparison of computed and recorded moment profiles (from Al Atik & Sitar,

2008) for Loma Prieta-SC-2

75

5.2.3 Lateral earth pressures

Figures 5.13-5.15 show the computed total earth pressure time series of the middle and

bottom points of both south stiff wall and north flexible wall during the three earthquake events.

Note that the quantities shown are the contact forces recorded by the 2D beam-solid contact

elements rather than true earth pressures, but the interpretation is the same. The positive direction

is set as the south direction in the model. From the figures, the phases of the earth pressure time

series are in accordance with each other for the south stiff retaining wall and north flexib le

retaining wall, which means that the overall behavior of the model is consistent. However, it is

not difficult to find some spikes in the earth pressure time series during the shaking events

especially in the high acceleration magnitude portions of the input motions, and when the

accelerations grow, the extreme values increase exponentially. The peak earth pressure value is

far beyond the axis limit when Kobe-PI-2 is adapted as the input motion whose acceleration

magnitude is highest among all the three motions. This phenomenon has not been fully understood

based on the author’s scope of knowledge, one possible reason is the 2D contact element formula

is developed based on the static equilibrium, and during dynamic analysis there is some high-

energy momentum that cannot be fully digested by the 2D contact element. Further element

formula modification or redevelopment is needed to perfect this 2D contact element in dynamic

analysis.

Figures 5.16-5.18 present the earth pressure profiles of both the south stiff retaining wall

and north flexible retaining wall at three interesting time points during the three shaking events

(just like for the moment analysis). The very high-frequency extreme values are filtered out of

these results to provide a clear picture of the pressure distributions. From the figures, some

conclusions can be easily reached. First, the earth pressure distribution before shaking matches

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what was returned by the static analysis, which means the first two steps of analysis (the elastic

gravity step and the elastic-plastic gravity step) perform very well, and the 2D contact element

does work in transient analysis after large time steps used to dissipate the inertial gravity transient.

Second, the maximum dynamic earth pressure distribution is not in a triangle shape as suggested

by the results of Al Atik and Sitar (2008), it is more in an inverse trapezoidal shape. This

observation is very crucial, and it manifests the difference between the 2D contact element and the

zero-length element used by Al Atik and Sitar. For the contact element approach used here, the

wall and soil act as two bodies linked through forces, while in the zero-length element approach,

the wall and soil are largely considered as one body. Third, the static earth pressure at the bottom

of retaining wall gets very large after the shaking. There is no clear explanation for this currently.

This observation may relate to the plastic soil model, of which the fine details are beyond the

author’s knowledge.

Compared with the centrifuge testing results of Al Atik and Sitar (2008), the current

numerical model appears to overestimate the earth pressures. Despite bearing the measure

discrepancy in mind, it is still obvious that the 2D contact element is not perfect in dynamic

analysis. The reason may also due to the momentum stored in the contact element in dynamic

analysis cannot be quickly dissipated in short time steps and/or the consideration for momentum

in the element formulation is lacking.

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stiff and north flexible walls during Loma Prieta-SC-1

78


stiff and north flexible walls during Kobe-PI-2

79


stiff and north flexible walls during Loma Prieta-SC-2

80


(2008) for Loma Prieta-SC-1

81


(2008) for Kobe-PI-2

82


(2008) for Loma Priesta-SC-2

83

5.2.4 Soil shear stress response

Figures 5.19-5.21 present the comparisons of the computed soil stress time series in the

middle of the backfill with those measured by accelerometers during the tests for the three

earthquake events. Taking into account the previously discussed differences in the input motions

for the current analyses and those used by Al Atik and Sitar (2008), the results compare quite well.

As the phase of the input motions are different, the soil stress time series are also different in

phases with those measured, however, the magnitudes of the soil stress time series are generally

in the same level with those interpreted in the tests. This reflects the robustness of the PDMY

material used in the model and accurate depiction of the modulus reduction curve calibrated from

the experiment.

84

Figure 5. 19 Comparison of computed and recorded shear stress time series in the middle of soil

backfill for Loma Prieta-SC-1

85


backfill for Kobe-PI-2

86


backfill for Loma Prieta-SC-2

87

5.3 Comparison of numerical results with M-O method results

The Mononobe-Okabe (M-O) method has been discussed in detail in the first chapter. This

method has been developed for decades, but is still widely accepted as the state of art approach for

seismic retaining wall analysis and design for its conciseness. In order to further assess the results

of the current numerical models, the M-O method is applied to the model retaining walls. Based

on the numerical model the necessary parameters have been determined. The height of the wall is

set as 6.5m. As there is no vertical acceleration the acceleration coefficient in the vertical direction

is set as 0, the acceleration coefficient in the horizontal direction is calculated by dividing the

maximum acceleration of each ground motion by g, which are 0.4168, 0.6711, and 0.4061 for

Loma Prieta-SC-1, Kobe-PI-2, and Loma Prieta-SC-2, respectively. The density and friction angle

are described in Chapter 3, and other parameters are generally set as zero.

Tables 5.1-5.4 compare both the computed earth pressure and wall moment with those

calculated by the M-O method. For the total earth pressure acting on the retaining wall, the

computed results are far bigger than the theoretical results. Many studies have suggested that the

M-O method is overconservative, thus these results suggest that the current model fails to predict

the real total earth pressure acting on the retaining walls (as the current results would then be

excessively over-conservative). The results calculated acting on the flexible retaining walls are

further beyond reasonable limits than those acting on the stiff retaining walls. The possible reason

is again due to the high-energy momentum generated by 2D contact element stay in the system

and produce enlarged pressure values.

88

What is interesting from the results of Tables 5.3-5.4 is that the computed total wall

moments are generally less than the theoretical values. As the pressure is far beyond the normal

value, the moment should follow the same trend too. The moment in the flexible wall is even

smaller, whose acting earth pressure is enlarged to an almost absurd extent. The reason to this

observation is the total wall moment is generated by two sources, one is the acting earth pressure

on the wall and the other is the wall inertial moment during the dynamic loading. As discovered

by other researchers (Al Atik & Sitar, 2010), the moment generated by the wall inertia is always

in the opposite direction to that generated by the earth pressure. In other words, the phase of the

moment time series generated by the wall inertial moment is opposite to that generated by the

acting earth pressure. In section 5.2.2, we find the moment time series acting on the flexib le

retaining wall is in total opposite phase to that acting on the stiff retaining wall, and the moment

time series acting on the stiff retaining wall is in the same phase with the acceleration measured

on the retaining walls. This phenomenon can only be explained as the stiff retaining wall has larger

wall inertial moment due to the larger wall width, and the moment generated by the inertial force

overcomes that created by the acting earth pressure. The flexible retaining wall has smaller wall

inertial moment and the acting earth pressure is much larger, so the total wall moment is dominated

by the acting earth pressure.

Overall, after comparing the theoretical values calculated by M-O method, it is more

convincing that the computed pressure acting the retaining walls is not realistic, and the 2D contact

element cannot accurately depict the soil-wall interface behavior in dynamic analysis. However,

as discussed in section 5.2.3 and further in collusion, this element does have a lot of potential to

adjust to the dynamic environment. As for the wall moment caused by the wall inertial moments

during dynamic analysis which have been neglected for too long, further numerical investiga t ion

89

should be done immediately as its share in the total wall moment during earthquake is clearly

significant.

Table 5.1 Comparison of computed and theoretical earth pressure on south stiff retaining wall

Pressure (kN)

Numerical M-O method

Loma-Prieta-SC-1 788000 211000

Kobe-PI-2 882000 313000


Table 5.2 Comparison of computed and theoretical earth pressure on north flexible retaining wall

Pressure (kN)


Loma-Prieta-SC-1 1086000 211000

Kobe-PI-2 1970000 313000


Table 5.3 Comparison of computed and theoretical wall moment on south stiff retaining wall

Moment (kN*m)



Kobe-PI-2 936 1430


90

Table 5. 4 Comparison of computed and theoretical wall moment on north flexible retaining wall

Moment (kN*m)



Kobe-PI-2 672 1430


5.4 Conclusions and further model improvements

This chapter mainly presents the results of the numerical model in dynamic analysis, results

regarding to the accelerations from the free field soil and the retaining walls, the dynamic bending

moments of the retaining walls, the dynamic earth pressure acting on the retaining walls, and the

soil shear stress during the earthquakes have been discussed in detail; the 2D beam-solid contact

element has clearly been examined thoroughly. Through this effort, several key areas for further

development in modeling this type of problem have been identified. They are listed blow without

emphasis of importance:

1. The corners of the U-shaped retaining structures are modeled as rigid connections. As

suggested by Al Atik and Sitar (2008) the actual joints do have some rotational flexibility.

More accurate simulation can be reached by using some rotational spring connectors.

2. The stress state around the corner of the U-shaped retaining structures has not been

considered, and whether the PDMY material gets into the full plastic stage or not remains

unknown. As very large earth pressure and wall moment is discovered around the bottom

of the retaining walls, these values are worth investigation.

91

3. Different soil properties and different wall heights can be used as parameters to do the

parameter study on the evaluation of the M-O method.

4. The zero-length element can be used in the numerical model as the replacement of the 2D

contact element, to further probe the causes of the element’s dysfunction in the dynamic

analysis.

5. The formulation of the 2D beam-solid contact element can be expanded to include explic it

consideration for dynamic analysis. The findings here suggest that there is a problem of

energy balance (or momentum balance) that needs to be resolved.

92

CHAPTER SIX

SUMMARY AND CONCLUSIONS

After the examination of the numerical model in OpenSees using 2D contact element to

model the retaining wall behaviors under both static and dynamic conditions, this thesis is the trail

blazer in the numerical analysis of earth pressure problems. The summary of this study can be

made:

In Chapter 1, a brief introduction on the current state of art in retaining wall design under

seismic loading --M-O method is conducted, the shortcomings of this method are also listed. The

purpose of this research and expectations of results are also summarized.

In Chapter 2, the formula of 2D contact element is displayed from three aspects includ ing

the kinematic relationships, constitutive behaviors and the equilibrium equations. This chapter is

the foundation of the work in later chapters, and the theoretical essence of this thesis.

Chapter 3 introduces OpenSees and general modeling approaches. In the beginning, the

centrifuge test and the work of Al Atik and Sitar (2008, 2010) is introduced, whose recorded results

are used as the baselines to compare with in later chapters. Then modeling parameters include

boundary conditions, number of nodes/elements, types of elements, material models, etc. are

addressed. The contact element’s application is also demonstrated with figures. At last, some

modelling tricks (cross pieces et al.) are discussed.

In Chapter 4, the results of the static analyses are presented and discussed. Outcomes like

the displacement distributions, the stress distribution and the lateral earth pressures are displayed,

and the lateral earth pressures are also compared with Rankine’s earth pressure theory. All these

93

results are reasonable which proves the robustness of the numerical model. Cases with/without

cross pieces and refined meshes are compared. Finally, the coarse mesh without “cross pieces”

case is selected as the model to be used in dynamic analysis.

Chapter 5 presents the results of the dynamic analyses and it is the core chapter of this

study. First, some modeling approaches that not covered in Chapter 3 are discussed, and the

difference of the input motions used in our numerical and the centrifuge test is addressed. Later,

the results including accelerations on the top of soil, wall bending moments, earth pressure

distributions and soil shear stresses are displayed. After comparing the results of the numerica l

model and the M-O method, the shortcomings and potentials of both have been discussed.

Although satisfying results have not been obtained by this thesis, the significance of this

study is still far reaching and promising, as more advanced computational devices coming in days

and months, more advanced elements simulating complex behaviors will be widely needed. The

first step of application of 2D contact element in dynamic analysis is conducted in this thesis,

further development and implementation should be done to counter the shortcomings that have

been summarized.

94

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