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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
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.
vii
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.
viii
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;
xii
(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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