UNIVERSITI PUTRA MALAYSIA

DEVELOPMENT OF FINITE ELEMENT CODES FOR SHEAR WALL

ANALYSIS

MASOUD PAKNAHAD

FS 2008 42

DEVELOPMENT OF FINITE ELEMENT CODES FOR SHEAR WALL

ANALYSIS

By

MASOUD PAKNAHAD

Thesis Submitted to the School of Graduate Studies, University Putra Malaysia, in Fulfilment of the Requirements for the Degree of Doctor of Philosophy

April 2008

Dedicated to

To my parents

ii

Abstract of thesis presented to the Senate of University Putra Malaysia in fulfilment of the requirement for the degree of Doctor of Philosophy

DEVELOPMENT OF FINITE ELEMENT CODES FOR SHEAR WALL

ANALYSIS

By

MASOUD PAKNAHAD

April 2008

Chairman: Associate Professor Jamaloddin Noorzaei, PhD

Faculty : Engineering

Shear wall is commonly employed as a principal element to resist lateral loads due to

wind or earthquake forces. An accurate mathematical model for shear wall structure

needs to consider the effect of all components in the system i.e. shear wall,

foundation, and subsoil. In addition to the necessity for accurate modelling of shear

wall-foundation-soil system, the model must be proficient to capture the structural

response of the shear wall building-foundation-soil and it should also be

computationally efficient. Furthermore, since shear wall is a key structural element

which plays a major role to ensure stability of the shear building system under lateral

loads, it is essential to consider the inelastic response of the shear wall buildings.

With the tremendous advancement in computer technology several finite element

software’s are commercially available to researchers and engineers. However, this

would not serve the objectives of the research investigator. In this study, new

elements have been developed, modified constitutive law has been proposed, new

computational algorithm has been utilised.

iii

Furthermore, this investigation focuses on the development of effective and suitable

modelling of reinforced concrete shear walls. The modelling includes physical and

constitutive modelling.

(a) Physical modelling

Most suitable elements had been adopted and some new elements have been

developed for physical modelling of shear wall structural systems. The elements

used for the purpose of mathematical discretisation of the shear wall structural

system are:

• Modified optimal triangle (MOPT) element with drilling degree of freedom to

represent the superstructure and foundation. This element was developed in this

study.

• Interface element to represent the interfacial characteristics between the

foundation represented by MOPT element with three degree of freedom per

node (top continuum) and bottom continuum i.e. soil media idealised as plane

stress /plane strain problem with two degree of freedom per node. This element

was developed to incorporate slip and separation at the interface.

• Super elements are groups of elements which are used to represent different

substructures domain.

• Nine node shell element has been used to represent the superstructure-foundation.

This element is multi-layer in nature to simulate the R/C structures.

• Coupled finite and infinite elements were used to model the near and far field in

the soil media. These elements were employed when the modelling of soil

structure interaction is in mind.

iv

All the above elements have been formulated based on continuum mechanics

principle.

(b) Constitutive modelling

The following constitutive law for different materials are used in the problem of

shear wall building-foundation-soil system.

• Concrete; nonlinearities arising due to crushing, cracking and plastic

yielding of concrete in one or two directions have been considered.

• Steel; the steel reinforcement is assumed to be in a uni-axial stress state

and is modelled as a bilinear material with strain hardening

• Soil; to account for the nonlinear behaviour of the soil media, the

nonlinear elastic model has been used

Based on the above physical and material modelling two finite element codes have

been developed during the course of this study:

(i) Two dimensional finite element analysis of shear wall (2DASW)

The modified optimal triangle element, super element, interface element and

the nonlinear elastic model with different numerical techniques are included

in its elements technology library in this FE code.

(ii) Three dimensional finite element analysis of shear wall (3DASW)

The 3DASW finite element codes deals with constitutive nonlinearities due

to concrete and steel. The multilayer shell and beam elements are available in

the element library of this finite element program.

v

Both above finite element codes are equipped with pre, post processing and

animation graphic facilities. The codes have been written using FORTRAN

language and they are working under FORTRAN-95 Power Station.

Accuracy and efficiency of the finite element codes has been achieved by analysing

several benchmark examples available in literature. The finite element discretisation

of different numerical examples used for the verification purpose and also

computational efficiency indicate that, the finite element mesh with coarse mesh and

wide range of element aspect ratio using MOPT produced good results. The results

show accuracy and fast rate-convergence of super element and MOPT.

An attempt has been made to highlight the computational efficiency of the developed

2DASW finite element codes by analysing actual shear wall buildings i.e. with and

without opening. Results in terms of displacements and stresses are compared with

those of commercial packages such as SAP-2000 and STAAD-PRO. It was observed

that the contour of stress distribution, calculated by STAAD-PRO, SAP 2000 and the

present study show that FE code using the present MOPT element is comparable in

predicting stress distribution in the shear wall, but the MOPT element display more

precise stress at the connecting beams.

Moreover, the stress distribution evaluated by the developed FE code based on

MOPT element formulation with coarse FE mesh, agrees well with the stress

distributions given by the commercial packages where fine mesh was used. This

comparison further proves the computational efficiency of the proposed formulation

of the MOPT element.

vi

The validation of the 3DASW finite element has been made by analyzing a series of

reinforced concrete (RC) structures studied by earlier researchers experimentally or

analytically. The response of the RC structures in terms load–deflection curve,

cracking pattern and prediction of ultimate load have been compared. It was found

that the results predicted by 3DASW compared reasonably well with those reported

in the literature.

An extensive study has been carried out with respect to two dimensional analysis of

shear wall-foundation-soil system subjected to vertical and lateral loading. It was

observed that the displacement predicted by both linear and nonlinear interactive

analyses showed remarkable differences in the values of displacement and stress

distribution in the shear wall. This can be attributed to stress dependant nature of the

tangent modulus. The computed displacement and stresses in nonlinear interactive

analysis are further increased when compared to the linear interactive analyses. The

effect of the shear wall building –foundation –soil interaction cannot be overlooked.

The application of the 3DASW has been further enhanced by carrying out the

inelastic analysis of the shear wall building. It was observed that;

(i) Cracking patterns were initiated at the tension side at base of the wall and

then spread across width and height of the wall.

(ii) The cracking of concrete has significant influence on redistribution of

normal stress σy. That is narrowing the zone of compression stress and

transfer higher stress to un-cracked zone and reinforcement bars.

(iii) Predefined target displacement is very helpful in proper prediction of load

deflection curves.

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Abstrak tesis yang dikemukakan kepada Senat Universiti Putra Malaysia sebagai memenuhi keperluan untuk ijazah Doktor Falsafah

PEMBANGUNAN KOD UNSUR TERHINGGA UNTUK BANGUNAN

DINDING RICIH

Oleh

MASOUD PAKNAHAD

2008

Pengerusi: Profesor Jamaloddin Noorzaei, PhD

Fakulti : Kejuruteraan

Dinding ricih biasanya digunakan sebagai anggota penting untuk menahan muatan

sisi akibat beban angin atau gempa bumi. Model matematik yang jitu melibatkan

struktur dinding ricih diperlukan bagi mengambil kira kesan ke atas semua

komponen di dalam sistem iaitu dinding ricih, asas dan tanah. Selain daripada

memerlukan model sistem yang jitu, model itu juga mestilah cekap untuk

menggambarkan tindak balas struktur sistem binaan bangunan dinding ricih-asas-

tanah dan ianya mestilah cekap dalam pengiraan. Tambahan lagi, memandangkan

dinding ricih adalah anggota kekunci dalam susunan rangka binaan di mana ia

memainkan peranan penting untuk memastikan kestabilan sistem dinding ricih di

bawah muatan sisi, pertimbangan ke atas tindakbalas tidak kenyal anggota ini dalam

sistem bangunan ricih adalah sangat penting.

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Dengan kemajuan yang besar dalam teknologi perkomputeran, beberapa perisian

unsur terhingga adalah boleh didapati secara komersial untuk penyelidik dan

jurutera. Walau bagaimanapun, ini tidak menepati objektif penyelidikan ini. Dalam

penyelidikan ini unsur baru telah dimajukan, perturan juzuk yang diubah suai telah

dicadangkan, pengiraan algorithmik yang baru telah pun digunakan. Tambahan lagi,

penyiasatan ini menumpukan kepada peningkatan keberkesanan dan model

pengukuhan bangunan dinding ricih konkrit yang sesuai. Model ini meliputi model

fizikal dan juzuk bahan.

(c) Model fizikal

Kebanyakan elemen yang sesuai telah pun diadaptasi dan beberapa unsur baru telah

dimajukan untuk model fizikal sistem struktur dinding ricih. Unsur yang digunakan

untuk tujuan pendiskretan sistem struktur dinding ricih adalah:

• Unsur modified optimal triangle (MOPT) dengan darjah kebebasan penggerudian

untuk mewakili asas dan superstruktur. Unsur ini dibangunkan dalam kajian ini.

• Unsur antara muka untuk menggambarkan sifat perantaraan di antara asas yang

diwakilkan oleh elemen MOPT dengan tiga darjah kebebasan di setiap nod (juzuk

atas) dan media tanah (juzuk bawah) dianggap sebagai masalah tekanan satah /

tegangan satah dengan dua darjah kebebasan di setiap nod. Unsur ini dimajukan

untuk menggabungkan kemasukan dan pemisahan pada interface.

• Unsur super adalah kumpulan unsur yang digunakan untuk menggambarkan

domain sub-struktur yang berbeza.

• Sembilan nod unsur kelompang telah digunakan untuk menggambarkan asas-

superstruktur. Unsur ini adalah berbagai lapis secara semulajadi untuk simulasi

struktur konkrit.

ix

• Pasangan unsur terhingga dan takterhingga digunakan untuk menggambarkan

media tanah yang dekat dan jauh, masing-masing. Unsur ini digunakan apabila

model interaksi struktur-tanah perlu diambil kira.

Semua unsur di atas telah dirumus berdasarkan kepada prinsip mekanik juzuk.

(d) Model juzuk

Undang-undang juzuk untuk bahan-bahan berbeza yang terlibat dalam masalah

sistem dinding ricih-asas-tanah adalah:

• Konkrit; ketidakselanjaran yang timbul akibat daripada kehancuran,

keretakan dan penghasilan plastik pada konkrit dalam satu atau dua arah

telah pun dipertimbangkan.

• Keluli; kekukuhan keluli dianggap berada dalam tahap tegasan satu paksi

dan dimodelkan sebagai bahan dwilelurus dengan penguatan terikan.

• Tanah; bagi menerangkan sifat tak lelurus media tanah, model kenyal tak

lelurus telah digunakan.

Berdasarkan kepada model fizikal dan bahan di atas, dua kod unsure terhingga telah

dibangunkan menerusi kajian ini:

(iii) Analisis Dua Dimensi Struktur Dinding Ricih (2DASW )

Unsur MOPT, unsur super, unsur antara muka dan model kenyal tak lelurus

dengan belbagai kaedah pengiraan telah dimasukkan dalam kod ini.

(iv) Analisis Tiga Dimensi Unsur Terhingga Struktu Dinding Ricih (3DASW)

Kod komputer 3DASW unsur terhingga ini berhubung dengan

ketidakselanjaran juzuk disebabkan oleh konkrit dan keluli. Unsur belbagai

x

lapis untuk rasuk dan kelompang boleh didapati dalam koleksi pengaturcaraan

komputer unsur terhingga ini.

Kedua-dua kod unsur terhingga di atas dilengkapi dengan kemudahan pra-proses,

pos-proses dan animasi grafik. Kod-kod tersebut telah dibangunkan dengan

menggunakan bahasa FORTRAN dan berfungsi di bawah Stesen Kuasa FORTRAN-

95.

Kecekapan dan kejituan kod unsur terhingga telah dicapai menerusi analisis

beberapa contoh tanda aras yang boleh didapati dalam penerbitan. Pendiskretan

unsur terhingga telah menggunakan beberapa conton sebagai tujuan verifikasi.

Kecekapan pengiraan juga menunjukkan jejaring unsur terhingga dengan jaringan

kasar dan belbagai aspek nisbah unsur MOPT menghasilkan keputusan yang baik.

Keputusan tersebut menunjukkan ketepatan dan kepantasan kadar pertemuan elemen

super dan MOPT.

Usaha telah pun dilakukan bagi menekankan kecekapan perhitungan kod unsur

terhingga 2DASW yang dimajukan dengan menganalisis bangunan dinding ricih

yang sebenar iaitu dengan dan tanpa bukaan. Perbandingan hasil analisis antara

pakej komersial seperti SAP-2000 dan STAAD-PRO dan kod yang dibangunkan

melalui kajian ini telah dibuat. Kontur agihan tegasan, yang dikira melalui STAAD-

PRO, SAP 2000 dan hasil kajian ini yang menggunakan elemen MOPT

menunjukkan hasil yang setanding, tapi unsur MOPT menunjukkan tegasan yang

lebih tepat pada rasuk penyambung.

xi

Lagipun, pembahagian tegasan yang ditaksir melalui kod unsur terhingga yang

dibangunkan berdasarkan kepada rumusan unsr MOPT dengan jaringan kasar, sangat

bersetuju dengan agihan tegasan yang diberikan oleh pakej komersial di mana

jaringan yang baik digunakan. Perbandingan ini juga membuktikan ketepatan

perhitungan rumusan yang dicadangkan untuk elemen MOPT.

Pengesahan elemen 3DASW unsur telah dibuat dengan menganalisis satu siri

struktur konkrit yang dikaji secara eksperimen atau analisis oleh pengkaji terdahulu.

Tindakbalas struktur konkrit menerusi lengkungan muatan-pembiasan, pola retakan

dan ramalan muatan terbaik telah dibuat perbandingan. Hasil menunjukkan bahawa

keputusan yang diramal oleh 3DASW agak bertepatan dengan apa yang telah

dilaporkan sebelum ini.

Kajian yang meluas telah dijalankan berkenaan analisis dimensi sistem dinding ricih-

asas-tanah yang dikenakan dengan beban menegak atau mendatar. Didapati bahawa

sesaran yang diramal oleh kedua-dua analisis interaktif lelulurus dan tak lelulurus

menunjukkan perbezaan ketara pada nilai sesaran dan agihan tegasan pada dinding

ricih. Ini mungkin disebabkan keadaan modulus tangen yang bergantung kepada

tegasan. Sesaran dan tegasan dari analisis interaktif tak selanjar didapati semakin

meningkat apabila dibandingkan dengan analisis interaktif lelurus. Kesan ke atas

interaksi dinding ricih-asas-tanah tidak boleh abaikan.

Aplikasi 3DSWA semakin ditingkatkan dengan melakukan analisis tak kenyal pada

bangunan dinding ricih. Didapati bahawa;

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(i) Corak retakan bermula pada sudut tegangan di dasar dinding tersebut

dan kemudian tersebar ke seluruh lebaran dan ketinggian dinding itu.

(ii) Retakan konkrit mempunyai pengaruh penting kepada penyebaran

semula tegasan normal σy. Ini menghadkan zon tegasan mampatan

dan memindahkan tegasan yang tinggi ke zon tak retak dan tetulang.

(iii) Penentuan awal sasaran sesaran sangat membantu dalam meramalkan

lengkung sesaran-beban yang sempurna.

xiii

ACKNOWLEDGEMENTS

Praise to the Almighty Allah Subhanahu wa Ta’ala, who giving me the strength and

health to complete this stage of study.

I wish to express my deepest gratitude to Assoc. Prof. Dr. Jamaloddin Noorzaei,

thesis supervisor, for his guidance, valuable suggestions, constant encouragement

and his advice during the course of this investigation and throughout the preparation

of this thesis. His willingness to help in every possible way is greatly appreciated.

I am grateful to all my supervisory committee members; Assoc. Prof. Dr. Mohd

Saleh Jaafar and Prof. Dr. Waleed A. M. Thanoon for their advices and suggestions

during this study.

Words fail to express my deep sense of gratitude to my father (Haji Saeed) who

shaped my carrier and blessing to my mother (Hajieh Nosrat) for her continuous

prayer, encouragement and patience. Special thanks to my wife, my sons, my

brothers and my sisters for their constant moral support.

I also thank Jamal Hakim, Ali Naghshineh, Shibli Khan, Buki and all my fellows in

UPM, for their continues help and supports.

xiv

PPROVAL I certify that an Examination Committee has met on 24th April 2008 to conduct the final examination of Masoud Paknahad on his Doctor of Philosophy thesis entitled “Development of Finite Element Codes for Shear Wall Analysis” in accordance with University Pertanian Malaysia (Higher Degree) Act 1980 and University Pertanian Malaysia (Higher Degree) Regulations 1981. The Committee recommends that the candidate be awarded the relevant degree. Members of the Examination Committee are as follows: Rasnasamy Muniandy, PhD Associate Professor Faculty of Engineering University Putra Malaysia (Chairman) Abang Abdullah Abang Ali, PhD Professor Faculty of Engineering University Putra Malaysia (Internal Examiner) Bujang Kim Huat, PhD Professor Faculty of Engineering University Putra Malaysia (Internal Examiner) Wan Hamidon Wan Badaruzzaman, PhD Professor Faculty of Engineering University Kebangsan Malaysia (External Examiner)

___________________________________

HASANAH MOHD. GHAZALI, PhD Professor and Deputy Dean School of Graduate Studies University Putra Malaysia Date: 14 August 2008

xv

This thesis was submitted to the Senate of University Putra Malaysia and has been accepted as fulfilment of the requirement for the degree of Doctor of Philosophy. The members of the Supervisory Committee were as follows: Jamaloddine Noorzaei, PhD Associate Professor Faculty of Engineering University Putra Malaysia (Chairman) Mohd Saleh Jaafar, PhD Associate Professor Faculty of Engineering University Putra Malaysia (Member) Waleed A. M. Thanoon, PhD Professor Faculty of Engineering University Technology Petronas (Member) _____________________________ AINI IDERIS, PhD Professor and Dean School of Graduate Studies University Putra Malaysia Date: 14 August 2008

xvi

DECLARATION

I hereby declare that the thesis is based on my original work except for quotations and citations which have been duly acknowledged. I also declare that it has not been previously or concurrently submitted for any other degree at UPM or other institutions.

______________________ MASOUD PAKNAHAD

Date: 10 August 2008

xvii

TABLE OF CONTENTS Page DEDICATION ii ABSTRACT iii ABSTRAK viii ACKNOWLEDGEMENTS xiv APPROVAL xvi DECLARATION xvii LIST OF TABLES xxii LIST OF FIGURES xxiii

CHAPTER

1 INTRODUCTION 1 1.1 General 1 1.2 Historical Development of Tall Buildings 2 1.3 Methods of Analysis of Shear Wall Buildings 4 1.3.1 Continuous Connection Method 4 1.3.2 Frame method 5 1.3.3 Finite Element Method 5 1.4 Structural behaviour, Analysis and Loads of

Shear Wall Buildings 6

1.4.1 Behaviour of Shear Wall Buildings 6 1.4.2 Dynamic Analysis of Shear Wall Buildings 8 1.4.3 Loads on Shear Wall Buildings 9 1.5 Necessities of New Elements in Discretisation of

Shear Wall Foundation-Soil Interaction 13

1.6 Commercial Packages and Soil Foundation Shear Wall Interaction

14

1.7 Problem Statements 15 1.8 Objectives of the Present Study 17 1.9 Scope and Limitation of the present study 18 1.10 Layout of the Thesis 19 2

REVIEW OF LITERATURE

22

2.1 Introduction 22 2.2 Analysis of Shear Wall Structures 23 2.3 Shear Wall Structures-Foundation-Soil

Interaction 31

2.4 Modelling of the Far-field in Problem of Soil Structure Interaction 40

2.5

Interface Modelling Characteristics Foundation-Soil Media

42

2.6 Matrix Condensation Techniques Related to Huge Structural System 42

2.7 Finite Element with Drilling Degrees of Freedom 45

2.8 Material Stress-strain Relationship of the Materials in Shear wall and coupled shear wall (Concrete and Steel)

48

xviii

2.9 Discussion 51 2.10 Justification of Choosing the Problem to be

Investigated 53

3

FORMULATION OF NEW FINITE ELEMENTS AND DEVELOPMENT OF COMPUTER PROGRAM

55

3.1 Introduction 55 3.2 Finite Element of the New Elements 58 3.2.1 Introduction of Finite Element

Formulation Based on Virtual Work 58

3.3 Development of the Super-Element 59 3.3.1 Generation of Super Element 61 3.3.2 Computational Procedures in Generation

and Implementation of Super Element 63

3.4 Development of Modified Optimal Triangle Element

65

3.4.1 Formulation of Optimal Membrane --------------- Triangle Element 66

3.4.2 Element Description 67 3.4.3 Natural Strains 68 3.4.4 The basic stiffness 68 3.4.5 The Higher Order Stiffness 69 3.4.6 Modified Formulation of Higher --

Order Stiffness Matrix 71

3.4.7 Computational Procedure and Development of a Finite Element (FE) Code for Modified Optimal Triangle (MOPT) Element

72

3.5 development and Formulation of the Interface Element 77

3.5.1 Computational Procedure of Interface Finite Element 81

3.6 Nine node Lagrangian Shell Element 82 3.6.1 Nonlinear Modelling for Reinforced

Concrete 85

3.6.2 Geometric Nonlinearity 85 3.6.3 Material Nonlinearity 86 3.6.4 Smeared Crack Mechanics and Cracked ----

------ Concrete 95

3.6.5 Strain Softening in Compression 98 3.6.6 Stress Reversal and Concrete States 99 3.6.7 Cracking and Tension Stiffening model 100 3.6.8 Cracked Shear Stiffness 103 3.6.9 Modelling of Reinforcing Steel 104 3.6.10 Computational Procedures for Nonlinear

Analysis of Reinforced Concrete Structures

105

3.7 Stress-strain Relationship for Soil 110 3.7.1 Nonlinear Elastic Model 111 3.7.2 Nonlinear Elastic Computational Procedure 115

xix

for Soil 3.8 Additional Finite and Infinite Element 117 3.9 Development of Computer Programs 118 3.10 Pre Processing, Post Processing and Animation

Facilities 124

3.10.1 Pre-Processor 124 3.10.2 Post-Processor 125 3.11 Three Dimensional Analysis of Shear Wall

Building (3DASW) 125

3.12 Concluding Remarks 126

4 VALIDATION OF THE DEVELOPED FINITE ELEMENT CODE

129

4.1 Introduction 129 4.2 Testing and Verification of Super Element 130 4.2.1 Analysis of Solid Shear Wall 130 4.3 Testing and Verification of the Modified Optimal

Element (MOPT) 133

4.3.1 Example I -Cantilever Beam Under End Moment

134

4.3.2 Example II - The Shear-Loaded Short Cantilever

135

4.3.3 Example III - Cook’s Problem 138 4.3.4 Example IV - Solid shear wall with MOPT 139

4.4 Validation of Inelastic Formulation of Reinforced Concrete Structures 141

4.4.1 Example I - Pure Bending of Plate Strip 141 4.4.2 Example II -Simply Supported Square Plate

Subjected to Uniform Vertical Load 143

4.4.3 Example III - Reinforced Concrete Beams 148 4.4.4 Example IV - Mcneice Slab 153 4.4.5 Example V - Cylindrical Shell 158 4.5 Concluding Remarks 162 5

ANALYSIS OF SHEAR WALL BUILDINGS

164

5.1 Introduction 164 5.2 Analysis of an Eight Storey Shear Wall Building

with Opening 164

5.2.1 Non-Interactive Analysis 164 5.2.2 Linear Interactive Analysis via MOPT and

--- -- Conventional Finite Elements 167

5.2.3 Linear Interactive Analysis with MOPT -----------and Super Element 171

5.3 Interactive Analysis of Five Storey Solid Shear Wall

174

5.3.1 Results and Discussion 176 5.3.2 Nonlinear Interactive Analysis 182 5.4 A Comparative Study of Shear Wall with

Opening Using MOPT and Commercial Packages

184

xx

5.4.1 Discussion and Comparison 185 5.5 Concluding Remarks 188

6

INELASTIC R ANALYSIS OF SHEAR WALL

190

6.1 Introduction 190 6.2 Inelastic Response of Reinforced Concrete Shear

Wall (Large Scale Test) 190

6.2.2 Geometry, Material Detail and Finite--------------- Element Model of Shear Walls

191

6.2.2 Results and Discussions 193 6.3 Inelastic Analysis of Shear Wall 199 6.3.1 Results and Discussion 202 6.4 Concluding Remarks 209

7 CONCLUSION AND SCOPE FOR FUTURE RESEARCH

210

7.1 Development and formulation of the new elements

210

7.2 Development of the finite element programs 211 7.3 Computational Efficiency and Application of

New Finite Element program 211

7.3.1 Computational Efficiency 211 7.3.2 Interactive Linear and Nonlinear Analysis

of Shear Wall 212

7.3.3 Inelastic Analysis of Reinforced Concrete Shear Wall 213

7.4 Scope for Further Research 214 REFERENCES

215

xxi

LIST OF TABLES

Table

Page

2.1 Equivalent Static Formula Practiced in Different Seismic Code of Practice

12

2.1 Displacement of Footing (mm) (Pandey, 1994)

35

2.2 Forces on Footing (Pandey, 1994)

36

2.3 Comparison of Analysis

44

3.1 Dimensionless Parameter of OPT Element

70

3.2 Definition and Mathematical Expression for Different Finite Element Parameters

84

3.3 Stress-strain Relation of Elastic, Cracked and Crushed Concrete

97

3.4 Tension Stiffening Factor

103

3.5 Shape Function for Elements Used in Idealization of Shear Wall and Soil

118

4.1 Identifier of Triangle Element Instances

133

4.2 Steel and Concrete Property

150

4.3 Steel and Concrete Property

153

4.4 Steel and Concrete Properties (Bouma et al. 1961)

160

5.1 Soil Properties for Different Type of Soils

167

5.2 Soil Properties for Different Type of Soils

175

5.3 Horizontal and Vertical Displacement at the TOP of Shear Wall

176

5.4 Hyperbolic Parameters of Soil for Duncan Model

182

5.5 Comparison of the Lateral Deflection at Different Storey Level

186

6.1 Material Properties, Failure Force and Failure Displacement for Different Shear WALL (Ioannis et al. - 1990)

191

6.2 Material properties and Reinforcement Detail of Solid Shear Wall

200

6.3 Incremental Applied Load 201

xxii

LIST OF FIGURES

Figure Page

1.1 Arrangements of Shear Wall in Plan (Toutanji, 1990) 2

1.2 Analysis Processes of Shear Wall Buildings 6

1.3 Structural Concept of Tall Building 7

1.4 Behaviour of Shear Wall Building 8

1.5 Various Types of Loads on Shear Wall Buildings 10

1.6 Membrane elements with drilling degree of freedom 13

2.1 Different Families of the Finite Element

26

2.2 (a) Coupled Shear Wall (b) Equivalent Continuous Beam

27

2.3 Idealized Coupling Beam

29

2.4 (a) Shear Wall, (b) Finite Element Model, (c) Beam Idealization

29

2.5 Modelling Procedure of Shear Wall with Opening (kim, and lee, 2003)

30

2.6 Schematic Interactive Analysis of Shear Wall Building

32

2.7 Element Description

34

2.8 Finite Element Modelling of the Structure of Ten Storey Building

35

2.9 Three Nodded Isoparametric Soil Interface (Badie, 1997a)

37

2.10 Coupled Shear Wall on Flexible Foundation (Nadji, and Johnson, 1998)

38

2.11 Settlement Along Centre Line (1) Without Considering Water (2) Considering Water (3) Measured

39

2.12 Flow Chart of Program for Combination Method of Analytic Solution with FEM. (Zhao et al. 2006)

40

3.1 The Overall Schematic Presentation of the Present Study

57

3.2 Illustration of a Substructure Scheme, and Generation of Super Element

60

3.3 Implementation of Super Element to Shear Wall Building

62

xxiii

3.4 Flowchart of Generation and Implementation of Super Element

64

3.5 Membrane Element with Drilling Degree of Freedom

66

3.7 Triangle Geometry

67

3.8 Generation of Four Node Element From 2 MOPT Elements

74

3.9 Generation of Four Node Element From 4 MOPT Elements With Half Thickness

74

3.10 Outline of Implementation of MOPT

75

3.11 Flowchart of Implementation of MOPT

76

3.12 Interface Element sandwiched Between Two finite elements

78

3.13 Flowchart of Implementation of Interface Element

81

3.14 Geometry of Nine Node Element

83

3.15 Experimental Stress-strain Curve for Biaxial Compression (Kupfer and Gerstle – 1973)

87

3.16 Experimental Stress-strain Curves for Tension- Compression (Kupfer and Gerstle – 1973)

88

3.17 Experimental Biaxial Strength Envelope of Plain Concrete (Kupfer and Gerstle – 1973)

88

3.18 Equivalent Uniaxial Stress-strain Curve

92

3.19 Biaxial Strength Envelope Used in the Present Study

93

3.20 Smeared Crack Model

97

3.21 Strain Softening Beyond Maximum Compressive Stress

99

3.22 Load Reversal and Different State in Concrete

100

3.23 Stresses in the Concrete and Steel Due to Unloading

102

3.24 Modified Stress-strain Curve for Steel Including Tension Stiffening Effect

102

3.25 Multi-linear Stress-Strain Curve for Steel

105

3.26 Flowchart of Computational Procedures of Nonlinear Reinforced Concrete Using Nine Node Shell Element

109

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