FRAME ANALYSIS OF REINFORCED CONCRETE SHEAR WALLS WITH
OPENINGS
NORLIZAN BINTI WAHID
UNIVERSITI TEKNOLOGI MALAYSIA
iii
To my beloved husband,
mother and family.
Thanks for all the supports..
iv
ACKNOWLEDGEMENT
I wish to express my sincere appreciation to my project supervisor, Assoc.
Prof. Dr. Abdul Kadir Marsono, for encouragement, guidance and constructive
criticisms. Without his support, the project would not have been the same as
presented here.
I am indebted to all the lecturers that had been taught me and their guidance
in the process of gaining knowledge in this field. My sincere appreciation also
extends to all my colleagues and others who have provided assistance at various
occasions. Their views and tips are making life useful.
Finally, thanks to my husband, my mother and all the family members for
their understanding and encouragement.
v
ABSTRAK
Terdapat beberapa jenis kaedah analisis yang tersedia untuk menganalisis
struktur dinding ricih. Analisis kerangka adalah salah satu daripadanya dan telah
digunakan dalam kajian ini untuk mengenalpasti kekuatan muktamad bagi struktur
dinding ricih berpasangan dan kelakuan strukturnya ketika dikenakan daya
mengufuk. Perisian Multiframe versi 5.16 yang berteraskan kepada kaedah analisis
elastik peringkat pertama telah digunakan untuk menganalisis model dinding ricih.
Keputusan analisis menunjukkan bahawa kaedah analisis kerangka elastik peringkat
pertama telah terlebih anggar kekuatan muktamad bagi model dinding ricih dan
kurang tepat untuk menganalisis keadaan ini berbanding dengan Kaedah Analisis
Sambungan Berterusan (CCM) dan Kaedah Analisis Unsur Terhingga Tidak Linear
(NLFEA). Perbezaan keputusan yang diperolehi adalah lebih kurang empat kali
ganda lebih besar daripada keputusan ujian makmal yang dijalankan oleh pengkaji
sebelum ini. Namun begitu kelakuan struktur bagi rasuk penyambung dan dinding
ricih pada keadaan muktamad masih lagi sama seperti yang berlaku pada ujikaji
makmal dan yang diperolehi daripada Kaedah Analisis Unsur Terhingga Tidak
Linear (NLFEA).
vi
ABSTRACT
There are several types of analysis methods available for analysing shear
walls of the building structures. Frame analyses is one of the methods, and were used
in this study to determine the ultimate strength of reinforced concrete coupled shear
wall structures and its structural behaviour under lateral loading. Multiframe Version
5.16 software which based on linear (first-order elastic) analysis method were used to
analyse the shear wall models adopted from previous research. Results show that
linear elastic of frame analysis method was overestimating the ultimate strength of
shear wall models and not reliable enough for this analysis compared to analytical
Continuous Connection Method (CCM) and Non-Linear Finite Element Analysis
(NLFEA) method. The difference was about four times higher than the experimental
results being conducted by the previous researcher. However the structural behaviour
of walls and its coupling beam at ultimate condition was still pose similar behaviour
to the observation made on the experimental test and Non-Linear Finite Element
Analysis (NLFEA).
vii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRAK
v
ABSTRACT
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF SYMBOLS
xiii
LIST OF APPENDICES
xiv
INTRODUCTION
1
1.1
Analysis of Shear Walls
2
1.2
Previous Work
3
1.3
Objectives
3
1.4
Scope of Project
4
LITERATURE REVIEW
5
2.1
Shear Wall Systems
5
2.1.1
Coupled Shear Walls
6
2.1.2
Modes of Failure for Shear Wall
8
2.2
Coupling Beams
9
2.2.1
Flexural mode of failure
10
2.2.2
Pure Shear Mode of Failure
11
viii
2.2.3
2.3
2.4
Flexure plus Shear Mode of Failure
Method of Analysis of Shear Wall Structures
14
2.3.1
Finite Element Analysis (FEA)
15
2.3.2
Continuous Connection Method (CCM)
16
2.3.4
Equivalent Frame Analysis
17
Multiframe Version 5.16
2.4.1
3
5
6
The Shear Wall Models
19
23
RESEARCH METHODOLOGY
26
3.1
Introduction
26
3.2
Modelling and Analysing the Shear Wall Models
27
3.2.1
Installing the Section Properties
27
3.2.2
Building the Frame Models in Multiframe
39
3.2.3
Loading
42
3.2.4
Analysing the Shear Wall Models
43
3.3
4
18
Methods and Conventions Used in
Multiframe Version 5.16
2.5
13
Results Interpretation
44
DATA COLLECTION AND ANALYSIS
46
4.1
Ultimate Load Condition
46
4.2
Deflections
47
4.3
Stresses
53
4.3.1
Resultant Stress
56
4.3.2
Shear Stress
59
DISCUSSION OF RESULTS
61
5.1
Load Displacement Characteristic
61
5.2
Mode of Failure
66
5.3
Concrete Crushed and Crack Observations
67
CONCLUSIONS AND RECOMMENDATIONS
71
6.1
Conclusions
71
6.2
Recommendations
74
ix
REFERENCES
75
APPENDICES
A
Joint Displacements At Ultimate Load
B
Results From Previous Research
By Marsono (2000)
77
90
x
LIST OF TABLES
TABLE NO.
2.1
TITLE
PAGE
Maximum capacities of structural modelling
in Multiframe
18
3.1
Section properties of shear wall members
31
3.2
Material properties
38
3.3
Load cycles and increment of loads
43
3.4
Properties for Model 5 and Model 6
45
4.1
Summary of the deflections measured at ultimate condition
49
4.2
Data for lateral load and displacement of Model 5 and
Model 6
49
Displacement through height of the compression wall
at the ultimate condition
50
Summary of the structural behaviour from the frame
analysis for Model 5 and Model 6
73
4.3
6.1
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
2.1
Shapes of shear walls
6
2.2
a) Typical arrangement of elevation and
b) Plan shapes of shear wall structures
7
Modes of failure of shear walls
a) Mode of failure one, b) Mode of failure two,
c) Mode of failure three
9
2.4
Flexural failure of slender coupling beam
11
2.5(a)
Shear diagonal splitting of coupling beam
12
2.5(b)
Pure shear failure of coupling beam
12
2.5(c)
Shear compressive failure of coupling connection
13
2.6(a)
Shear diagonal splitting and inclined flexural failure
13
2.6(b)
Extended shear diagonal splitting and inclined
flexural failure of coupling beam
14
2.7
Axes and sign convention in Multiframe
20
2.8
Free body diagram of a simple beam
22
2.9
Analysis of simple beam
23
2.10
Scaled models of shear walls.
a) Model 5 and, b) Model 6
25
Reinforcement detail and members’ cross section
for Model 5
29
Reinforcement detail and members’ cross section
for Model 6
30
2.3
3.1
3.2
xii
3.3
Graphical User Interface (GUI) for Section Maker
38
3.4
Modelling the shear walls into frame structures.
a) Model 5 and, b) Model 6
40
Frame models considering the existance of links.
a) Model 5 and, b) Model 6
41
3.6
Graphical User Interface (GUI) for Multiframe4D
42
4.1
Deflected shape for
a) Model 5 and, b) Model 6
48
4.2
Lateral load and displacement of Model 5 and Model 6
51
4.3
Lateral displacements through height of the
compression wall
52
Stresses at members of Model 5.
a) Axial stress, Sx’ (N/mm2), b) Sbz’ top (N/mm2),
c) Sbz’ bottom (N/mm2)
54
Stresses at members of Model 6.
a) Axial stress, Sx’ (N/mm2), b) Sbz’ top (N/mm2),
c) Sbz’ bottom (N/mm2)
55
Resultant stresses and the illustration of its location
on Model 5
57
Resultant stresses and the illustration of its location
on Model 6
58
Shear stresses in members.
a) Model 5 and, b) Model 6
60
5.1
Lateral load and displacement of Model 5
62
5.2
Lateral load and displacement of Model 6
63
5.3
Lateral load and displacement for Load Cycle 1 of
a) Model 5 and, b) Model 6
65
Approximate observation of concrete crushed and
cracks formation of Model 5
69
Approximate observation of concrete crushed and
cracks formation of Model 6
70
3.5
4.4
4.5
4.6
4.7
4.8
5.4
5.5
xiii
LIST OF SYMBOLS
A
-
Area
D, d
-
Depth
δu
-
Ultimate deflection
E
-
Modulus of elasticity
F
-
Vector of applied loads
fc
-
Compression stress
ft
-
Tension stress
fy
-
Yield strength
fcu
-
Characteristic strength
fst
-
Splitting strength
ftc
-
Limiting tensile strength
G
-
Shear modulus
H
-
Height
I
-
Moment of inertia
J
-
Torsion constant
K
-
Stiffness matrix
L, l
-
Length or span
P
-
Load applied
Pu
-
Ultimate load
Sx’
-
Axial stress about local x' axis of member
Sbz’
-
Bending stress about local z' axis of member
x
-
Vector of calculated displacements
xiv
LIST OF APPENDICES
APPENDIX
TITLE
A
Joint Displacements At Ultimate Load State
B
Results From Previous Research
By Marsono (2000)
PAGE
77
90
CHAPTER 1
INTRODUCTION
High rise building is a structure vertically cantilevered from the ground level
subjected to axial loading and lateral forces. It consists of frames, beams, shear walls,
core walls, and slab structures which interact through their connected edges to
distribute lateral and axial load imposed to the building. Lateral forces generated
either due to wind blowing against the building or due to the inertia forces induced
by ground shaking which tend to snap the building in shear and push it over in
bending. These types of forces can be resisted by the use of shear wall system which
is one of the most efficient methods of ensuring the lateral stability of tall buildings.
For building taller than 10-stories, frame action obtained by the interaction of
slabs and columns is not adequate to give the required lateral stiffness (Taranath
1998). It also has become an uneconomical solution for tall buildings. However it
can be improved by strategically placing shear walls as it very effective in
maintaining the lateral stability of tall buildings under severe wind or earthquake
loading.
Coupled shear wall is a continuous wall with vertical rows of opening created
for windows and doors, coupled by beams that interconnecting the wall piers across
the openings. These connecting beams are referred to as coupling beams. It may be
2
shallow or deep beam type constrained by the walls on either side. Coupling beam
has to be ensured adequately strong and stiff under elastic loading, ductile and able to
dissipate energy under inelastic loading to achieve desirable performance of these
structures (Lam et al., 2005) as its behaviour and modes of failure are highly effect
the mode of failure of shear wall.
1.1
Analysis of Shear Walls
There are several types of analysis methods available for analysing shear
walls structure. The analysis can be made in elastic, elasto-plastic and ultimate
condition. However, due to uncertain and in-ability to analyse and interpret the post
elastic behaviour or possibly time constraint, the elastic method is preferred for its
simplicity. This elastic method of analysis consists of Continuous Connection
Method (CCM), Transfer Matrix Method, Wide Column Analogy (WCA) or frame
analysis, Finite Element Method and Discrete Force Method.
The Non-Linear Finite Element Analysis (NLFEA) can achieve excellent
agreement with the test results (Driver et al., 1998). But this method is time
consuming where relatively simple force distribution output is required. Frame
analysis method offers the advantage of being simple and relatively accurate yet the
results output are still acceptable by engineers. Thus, this study was carried out on
shear wall structures adopted from research models by Marsono (2000) using frame
analysis based software, to evaluate the accuracy of the methods.
3
1.2
Previous Work
Previous researcher, Marsono (2000) has conducted an experimental work on
small scaled model of various types of shear walls structure. Results from the
experiment in the form of stresses and strains, crack distributions and ultimate
strength then used to establish the analytical method (Continuous Connection
Method, CCM) of analysis. The non-linear finite element analysis (NLFEA) was
performed as a tool to affirm the experimental results and the analytical mode of
failure and ultimate strength predictions. The experimental and NLFEA results were
in very close agreement in predicting the ultimate strength and mode of failure of
coupled shear wall structure.
1.3
Objectives
The main objectives of the research are as follows:
a) To carry out a frame analysis on shear wall models using Multiframe Version
5.16 software.
b) To check the reliability of frame analysis method compared to analytical
Continuous Connection Method (CCM) and Non-Linear Finite Element
Analysis (NLFEA) method.
c) To approximately determine the crack formation and crushing of concrete at
shear wall referring to the results obtained from the frame analysis.
4
1.4
Scope of Project
Analysis was carried out on a scaled model of shear walls adopted from
Model Number 5 & 6 of PhD research by Marsono (2000) and no design work on the
structure involved. The software used to analyse the models was Mutiframe version
5.16 which based on linear elastic method of frame analysis. All the material
properties assigned and load applied were also taken from the previous research.
Results obtained from the analysis then being compared to experimental study,
analytical CCM and NLFEA results obtained from previous research.
CHAPTER 2
LITERATURE REVIEW
The following is a review on shear wall structural systems, modes of failure
for shear wall and coupling beam, method of analysis that available for analysing
shear wall structure and background of the Multiframe version 5.16 software.
2.1
Shear Wall Systems
Shear walls have been the most common lateral force resisting elements for
tall building besides frame systems. It is an efficient method of ensuring the lateral
stability of tall buildings and also efficient against torsional effects when combined
together with frame structures. Their stiffness is such that sway movement under
wind load can be minimised.
Structural forms of shear wall are commonly used in buildings of 10 to 30
storeys. Monolithic shear wall can be classified as short, squat or cantilever as in
Figure 2.1 according to their height/depth ratio (Irwin, 1984). The walls may be
planar, flanged or core in shape.
6
D
H
H/D < 1
Short
1 < H/D < 3
Squat
Figure 2.1
2.1.1
H/D > 3
Cantilever
Shapes of shear walls
Coupled Shear Walls
Coupled shear wall is a continuous wall with vertical rows of opening created
by windows and doors, coupled by connecting beams. When two or more shear walls
are interconnected by a system of beams or slabs, the total stiffness of the system
exceeds the summation of the individual wall stiffness because the connecting slab or
beam restraints the individual cantilever action by forcing the system to work as
composite unit. Such an interacting shear wall system can be used economically to
resist lateral loads in buildings up to about 40-stories (Taranath 1998).
Shear wall may come in many forms and there are various types of opening
shape due to architectural and planning requirement as in Figure 2.2. However, due
to ease of analysis, design and construction, regular shapes with openings throughout
the height are preferred by the engineers.
7
Single Band
Asymmetrical
Staggered Band of Openings
Two Bands
Three Bands
Wide Base
(a)
(b)
Figure 2.2
structures
a) Typical arrangement of elevation and b) Plan shapes of shear wall
8
2.1.2
Modes of Failure for Shear Wall
At loaded state, shear wall is subjected to tension and compression along the
height. It may fail in flexure, shear or combined action of flexure and shear.
Primarily, the failure of shear wall structure can be identified by the three main cases
that highly dependent on the effectiveness of coupling beams. The first two modes of
failure are caused by the flexible coupling action in shear wall structures. Referring
to Figure 2.3, the modes of failure of shear walls are;
i) Mode of Failure 1 – Flexural Mode of Failure
This mode of failure is identified by flexural failure of coupling beams. It
happens with relatively shallow coupling beams and reinforced with a small
amount of main bars. The connecting beam will deform in double curvature
developing flexural cracks at wall junction and the walls continue to resist the
loads after the beams have reached their capacities. The walls will reach it
capacity when it crushes at the compressive corners.
ii) Mode of Failure 2 – Shear Diagonal Splitting Mode of Failure
This mode of failure is characterised by the failure of coupling beams in shear
or diagonal splitting mode. Major shear or diagonal splitting concrete cracks
across the compression diagonal developed at the failed beams. It happens on
deep and moderately reinforced coupling beams. The failure of shear walls
occur with the compression wall crushing together with the diagonal splitting
or shear failure of coupling beams completing almost simultaneously.
iii) Mode of Failure 3 – Rigid Actions
The rigid action of coupling beams will cause this mode of failure occurs to
shear walls. Some partial flexural cracks will develop at the coupling beams
while a large number of cracks will develop on the tension side of the wall,
9
and it will bend in a cantilever mode. The wall will fail with the crushing of
the concrete at the highly stressed compression corners.
(a)
Figure 2.3
(b)
(c)
Modes of failure of shear walls. a) Mode of failure one, b) Mode of
failure two and, c) Mode of failure three.
2.2
Coupling Beams
Coupling beams are defined as a connector between two or more vertical
shear walls separated by regular openings created by windows or doors at each storey
level along the shear wall height. It may be shallow or deep beam type, however the
most common type used is a deep beam type. The ACI Code 318-89 defines a deep
beam as a beam in which the ratio of the clear span l to the overall depth d is less
than the limits 1.25 for simple spans and 2.5 continuous spans. While the CIRIA
Guide 2 state deep beam as a beams having an effective span/depth ratio l/d of less
than 2 for single span and less than 2.5 for continuous beams.
10
At lateral loaded state, shear walls undergo different movement between
supporting ends. The coupling beams in between are actually constraint the walls to
deflect similarly and will be subjected to flexural, shear or combination of both types
of deformation, as the beams are required to connect the walls and transfer process
between them.
These coupling beams have great influences on the overall structural
behaviour of the shear walls. Its have to be stiff, strong and ductile enough to ensure
the desirable performance of these structures. Local failure of coupling beams may
lead to a more serious global failure of the whole lateral load resisting system of the
building. The main characteristics of the coupling beams that can be observed at
failure are:
i) Formation of initial flexural cracks at fixed end supports
ii) Formation of major diagonal cracks
iii) Concrete crushing at two ends of diagonal cracks
There are several modes of failure on coupling beams considered as flexural,
shear and combined action of flexure plus shear (Marsono 2000).
2.2.1
Flexural mode of failure
In flexural mode of failure, coupling beams are deformed in double curvature
of bending. The failure will occur when the concrete in compressive zones crushes
11
and the horizontal bars yield. This type of failure occurs in shallow coupling beams
with small flexural capacity as shown in Figure 2.4.
Direction of load
Yield of reinforcement
Flexural cracks
Flexural cracks
Yield of reinforcement
Figure 2.4
2.2.2
Flexural failure of slender coupling beam
Pure Shear Mode of Failure
Pure shear deformation will occur when top and bottom reinforcement along
the beam length is in tension. It can be characterised by the extension of diagonal
cracks to the position of main reinforcement diagonally opposite and by the crushing
of concrete at its end.
12
Direction of load
Major diagonal
splitting crack
Reinforcement
not yield
Figure 2.5(a) Shear diagonal splitting of coupling beam
Direction of load
Flexural of reinforcement
Pure shear cracks
Minor flexural cracks
Figure 2.5(b) Pure shear failure of coupling beam
13
Direction of load
Friction cracks
Bent of reinforcement
Figure 2.5(c) Shear compressive failure of coupling connection
2.2.3
Flexure plus Shear Mode of Failure
The characteristic of failure are the formation of major flexural cracks along
the fixed end and a major diagonal crack on the web section. There are signs of
crushing concrete occurs at two ends of the diagonal that act as a hinge. At the hinge
a large rotational deformation occurs which cause the structure to collapse.
Direction of load
Major shear diagonal crack
Incline flexural cracks
Figure 2.6(a) Shear diagonal splitting and inclined flexural failure
14
Direction of load
Bent of reinforcement
Major diagonal splitting crack
Figure 2.6(b) Extended shear diagonal splitting and inclined flexural failure of
coupling beam.
2.3
Method of Analysis of Shear Wall Structures
Analysis on shear wall structures can be made in elastic, elasto-plastic and
ultimate condition. Due to its simplicity, elastic analysis is still widely in used today
in the design offices.
There are several methods available for the analysis of coupled shear wall as
been introduced in Chapter 1. Three common basic methods that usually been used
are finite element analysis, continuous connection method and equivalent frame
analysis.
15
2.3.1
Finite Element Analysis (FEA)
In finite element method, the main idea is to discrete a complex region
defining a continuum into simple geometric shapes called finite elements. The
material properties and the governing relationships are considered over these
elements and expressed in terms of unknown values at element corners. An assembly
process, duly considering the loading and constraints, results in a set of equations.
Solution of these equations gives us the approximate behaviour of the continuum.
The advantages of finite element analysis includes in which the nonlinearities
behaviour of material or structure can be considered in the analysis. The term
nonlinear is used in structural analysis to describe a situation where the deformation
is not proportional to the applied load. This is may be due to geometric
nonlinearities, material nonlinearities and the contact of bodies with geometric and
material nonlinearities. It also virtually may include various geometrical shapes of
structures.
Factors that usually considered for nonlinear concrete material model used in
the analysis are includes of:
i) Nonlinear behaviour in compression at materials including hardening and
softening
ii) Fracture of concrete in tension based on nonlinear fracture mechanics
iii) Biaxial strength failure criterion
iv) Reduction of the shear stiffness after cracking
16
Non linear finite element analysis (NLFEA) make possible for us to analyze
models’ real-life conditions on the desktop. The analysis can be made in elastic,
elasto-plastic and ultimate conditions. Results obtained could offer very good
alternatives to experimental results. This method is cheaper but time consuming
whereas relatively simple force distribution output is required for design but certainly
not true for research purposes.
2.3.2
Continuous Connection Method (CCM)
Continuous connection method is an analysis where the coupling beams of
shear wall structure are replaced by continuous connected media along its height.
The coupling beams are assumed to deform with a point of contra flexure, normally
at mid-span. The walls are assumed as cantilever system on a rigid foundation and it
neglects the effect of the beam’s axial deformation. The openings are replaced by a
single continuous shear medium.
The method also allows simple evaluation for any load pattern to be included
in the analysis. A simple analytical solution can be derived, including the accuracy of
force and deflection by explicit mathematical relationships which are dependent of
the number of storeys. The analysis can be made in elastic and elasto-plastic
conditions. Elasto-plastic method of analysis based on CCM is done by dividing the
structure into elastic and plastic zone.
Several problems may arise when obtaining the solution to the equation if
unusual base forms, irregularities of openings, such that new boundary conditions
that has to be applied to the equation.
17
2.3.4
Equivalent Frame Analysis
Frame analysis may also be called wide frame analogy. It is a simple method
and can be used in plan frame programs. This method treats the walls and lintel
beams as discrete frame members. Walls and connecting beams are replaced by the
line element of stiffness equal to those of the units they replaced.
The method of analysis is based on the assumption that a linear relationship
exists between the applied actions and the resulting displacements. This assumption
requires, first, the material of the frame shall behave in Hookean manner at all points
and through out the range of loading considered. Second, it assumes that the changes
in the geometry of the structure are small enough to be neglected when the internal
actions are calculated (Hall and Woodhead, 1967).
Two basic procedures in frame analysis are flexibility method and deflection
or stiffness method. In the first approach, certain actions are temporarily removed;
these actions are the unknowns in the compatibility equations which lead to the
complementary solution. In the second approach, certain displacements are prevented
or removed. The equilibrium equations are written in terms of these unknown to be
sought displacements.
Stiffness method is the basic method used by Multiframe to analyse
structures. It can be performed in a linear (first-order elastic) analysis or geometrical
non-linear (second-order elastic) analysis. However, this second-order elastic
analysis has not yet included in the version 5.16 of Multiframe, it only included in
the latest version 9.5 of Multiframe. Since this study was using Multiframe version
5.16, so the shear wall models were actually analysed using first-order elastic
analysis of stiffness method. In this method, the structures assumed to behaves
linearly elastic so that the principles of superposition applied.
18
Occasionally this method gave the wrong impression on the behaviour of
shear wall structure under loading (Kwan 1993), however due to uncertain and inability to understand the post-elastic behaviour, time constraint and also its
simplicity, the results output are still acceptable by engineers. Thus this study was
carried out to view the reliability of the method compared to the analytical and
NLFEA method.
2.4
Multiframe Version 5.16
Multiframe version 5.16 is a commercially available software package for
structural frame analysis. It provides the GUI (Graphical User Interface) facility for
the ease of structural modelling. Detail step of modelling the shear walls in this
software will be explain in the next chapter. The absolute maximum capacities for
structural modelling in Multiframe are as shown in Table 2.1
Table 2.1
Maximum capacities of structural modelling in Multiframe
Number of joints
No limit
Number of members
No limit
Number of restraints and prescribed displacements
No limit
Number of springs
No limit
Number of load cases
500
Number of joint loads
No limit
Number of member loads
No limit
Number of thermal loads
No limit
Number of members connected at one joint
18
19
In practice, some of the above limits may be reduced by the amount of
memory available at the time the program is running. Memory can be increased by
modifying virtual memory settings if required. The amount of memory required is
independent of the order in which the joints are numbered. Multiframe will
automatically optimise the internal numbering of the joints for the best use of
memory available.
The actual size of the structure that can be solved will depend on the number
of load cases and the geometric configuration of the structure. The more load cases
being used the smaller structure that will be able to analyse.
2.4.1
Methods and Conventions Used in Multiframe Version 5.16
In version 5.16, Multiframe carries out a first order, linear elastic analysis to
determine forces and deflections. It uses the matrix stiffness method for solving a
system of simultaneous equations to determine these forces and deflections in a
structure.
The matrix stiffness method forms a stiffness matrix for each member of the
structure and given a list of applied loading, solves a system of linear simultaneous
equations to compute the deflections in the structure. The internal forces and
reactions are then computed from these deflections. Deformations due to shear action
in deep beams or warping deformation due to torsion does not take into account in
Multiframe.
Two coordinate systems for defining geometry and loading are use in
Multiframe. The global coordinate system is a right handed x, y, z system with y
20
always running vertically and x and z running horizontally. Gravity loads due to self
weight are always applied in the negative y direction. To distinguish between local
and global axes, Multiframe uses the ‘'’ suffix to indicate a local axis as in Figure 2.7
below.
Figure 2.7
Axes and sign convention in Multiframe.
There are six degrees of freedom at each joint uses in Multiframe when
performing its calculations. These comprise three displacements along the axes and
three rotations about the axes at each joint. The local element stiffness matrix K used
by Multiframe is as follows:
21
Where each element behaves according to the equation:
F=Kx
F={Px1, Py1, Pz1, Mx1, My1, Mz1, Px2, Py2, Pz2, Mx2, My2, Mz2}
x={dx1, dy1, dz1, Øx1, Øy1, Øz1, dx2, dy2, dz2, Øx2, Øy2, Øz2}
Where F is the vector of applied loads, K is the stiffness matrix above and x is a
vector of calculated displacements. All relative to the local member coordinate
system.
22
Member actions are computes relative to the local member coordinate system.
When calculating an action at an intermediate point along a member, Multiframe
checks the free body diagram of the member to the left of the point of interest and
uses the balance of forces at this point for the sign of the computed action. Referring
to Figure 2.8, consider the shear force at a point on a simple beam subject to a central
point load. At the left hand portion of a beam, the sum of the shear forces is positive
as shown in the figure.
Figure 2.8
Free body diagram of a simple beam.
For the common case of a beam with joint 1 at the left hand end and joint 2 at
the right hand end, a load acting downwards will be negative in magnitude and the
forces will be as shown in Figure 2.9.
23
Load
Restraint
Restraint
Bending Moment
Shear Force
Deflection
Figure 2.9
2.5
Analysis of simple beam.
The Shear Wall Models
Previous researcher, Marsono (2000) has conducted an experimental work on
a series of approximately 1:25 scale models as representatives of full scale actual 1215 storeys shear wall structures for detail study of the structural behaviour. The study
covers the behaviour of the structures with respect to crack formation, deformation
and strain development up to failure and identification of the model failure.
In this research, results from these experiments were used to establish the
analytical method (Continuous Connection Method, CCM) of analysis. The nonlinear finite element analysis (NLFEA) was performed as a tool to affirm the
24
experimental results and the analytical mode of failure and ultimate strength
predictions. The experimental and NLFEA results were in very close agreement in
predicting the ultimate strength and mode of failure of coupled shear wall structure.
In the study, Marsono (2000) has classified the models into four series,
depending on the arrangements of its openings:
i) 2 numbers of coupled shear walls with single band of openings.
ii) 4 numbers of coupled shear walls with two bands of openings.
iii) 3 numbers of coupled shear walls with staggered openings.
iv) 2 numbers of flanged shear walls with staggered openings.
Series two consists of shear wall models with two bands of openings,
comprising of symmetrical wall sections, Model 3 and asymmetrical wall section
Model 4, Model 5 and Model 6. The behaviour of the coupling beams is expected to
be similar throughout the height of the structure.
For the purpose of this study, two models (Model 5 and Model 6) were
adopted from the second series of the Marsono’s (2000) test specimens as shown in
Figure 2.10. The models are vertical planar shear wall structures 40mm in thickness
and constrained at a rigid base of thickness 200mm to 300mm and 300mm deep. The
thickness of all the coupling beams was 30mm.
Model 5 and Model 6 were actually two same sizes of asymmetrical walls
with two bands of openings. The difference between them is that in Model 5 the
lateral point load is applied to the wider wall (280mm) where as in Model 6, the load
was applied to the narrower wall (180mm).
25
(a)
(b)
Figure 2.10
Scaled models of shear walls a) Model 5, b) Model 6
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
The research methodology was start with problem identification on reinforced
concrete shear wall and setting up the objectives and scope of study. Then all the
related background information were collected and studied for the literature review
for knowledge updating.
The major parts of this study are structural modelling and computational
analysis using frame analysis method in Multiframe version 5.16. The results
obtained then being assessed and interpreted and compared to one which obtained
from the previous research by Marsono (2000). From here the reliability of frame
analysis can be determined compared to NLFEA and analytical CCM.
27
3.2
Modelling and Analysing the Shear Wall Models
Generally there are several steps in modelling and analysing the shear walls.
First is by installing the section properties for every part of the shear walls using
Section Maker in Multiframe. Followed by building the frame models for every shear
walls in Multiframe 4D and all the section properties and restraints are assigned to
the respective part of the structure. Then applying the structure with load and
analysed it to obtain all the results.
3.2.1
Installing the Section Properties
To compute deflections and stresses in the structure, it is necessary to know
all the material properties and dimensions of the section used in the structure. In
Multiframe, as it uses frame analysis method to analyse structures, all the walls and
beams are being treated as a line element members. The line element has a same
stiffness with the unit they replaced and will represent the actual behaviour of the
structure. This same stiffness can be achieved by assigning the actual material and
section properties to the line element.
Multiframe has a built-in table of the most commonly used structural sections
from which we can select the desired section type. If the structural section required is
not contained in the Section Library, and is not one of the standard shapes supported,
new section can be define into Section Library using Section Maker, facility
provided by Multiframe for defining material and section properties use in modelling
frame structures.
28
For this study, all the section properties were newly define in the Section
Maker since it do not use the standard section provided in the Section Library. All
the steel reinforcement was being assigned compositely with the concrete member
referring to the reinforcement detail and cross section of every part of the structure as
in Figure 3.1 and Figure 3.2.
As mentioned in the previous chapter, two models of shear walls were
adopted from the previous research by Marsono (2000) for the purpose of this study.
The models are Model 5 and Model 6, shear wall models with two bands of
openings. Its two same sizes of asymmetrical walls with two bands of 100mm
openings, two 280mm wide walls and one 180mm narrow wall. The difference
between them is that in Model 5, the lateral point load is applied to the wider wall
where as in Model 6, the load was applied to the narrower wall.
29
Wall 1
Wall 2
Wall 2
R6
R6
R8
R8
Figure 3.1
Reinforcement detail and members’ cross section for Model 5
30
Wall 2
Wall 2
Wall 1
R6
R6
R8
R8
Figure 3.2
Reinforcement detail and members’ cross section for Model 6
31
For the purpose of modelling the narrower wall (180mm) was named by
‘Wall 1’ and the wider wall (280mm) named by ‘Wall 2’ for every models. These
walls contain same vertical steel reinforcement throughout the height. However, the
horizontal reinforcements are different at certain part. There are two different steel
which are 6mm diameter steel used at middle span of the wall and 8mm diameter
steel used at the area of coupling beams. There are three different section properties
for every walls, two for cross section of wall that contain 6mm and 8mm diameter
steel respectively and the other for the area where no horizontal steel.
All the coupling beams were named by ‘Beam’ and the base of the wall as
‘Base’. Since the existence of the horizontal bars have to be included in the analysis,
every part of the structure is to be split into two section properties and it was
differentiated by the name of ‘with links’ or ‘without links’. These section properties
will be shown in Table 3.1.
Table 3.1
Section properties of shear wall members.
MODEL 5
Section Properties
R6
180
Wall 1 (without horizontal steel)
40
32
R6
Wall 1 (with horizontal steel R6)
R6
180
40
R6
Wall 1 (with horizontal steel R8)
180
R8
40
R6
Wall 2 (without horizontal steel)
280
40
33
R6
Wall 2 (with horizontal steel R6)
R6
280
40
R6
Wall 2 (with horizontal steel R8)
R8
280
40
Beam (without link)
R8
50
30
Beam (with link)
50
R8
R6
30
34
T16
Base (without link)
300
T16
200
T16
Base (with link)
R10
300
T16
200
MODEL 6
Section Properties
R6
180
Wall 1 (without horizontal steel)
40
35
R6
Wall 1 (with horizontal steel R6)
R6
180
40
R6
Wall 1 (with horizontal steel R8)
180
R8
40
R6
Wall 2 (without horizontal steel)
280
40
36
R6
Wall 2 (with horizontal steel R6)
R6
280
40
R6
Wall 2 (with horizontal steel R8)
R8
280
40
Beam (without link)
R8
50
30
Beam (with link)
50
R8
R6
30
37
T16
Base (without link)
300
T16
200
T16
Base (with link)
R10
300
T16
200
The type of material for each shape of section can be chose from the range of
material properties stored in the Sections Library. If different value of material
properties has to be use, it can be change at the Edit Material section. As for this
study, the material properties are not taken from the properties stored in Section
Library. All the material properties assigned to the models were obtained from the
laboratory test that has been carried out during experimental study as in Table 3.2.
38
Table 3.2
Concrete
Material properties.
Steel
Splitting Modulus of
Cross
Yield Modulus of
Characteristic
Strength, Elasticity, Diameter Sectional Strength, Elasticity,
Strength, fcu
fst
Ec
Area
fy
Es
2
2
2
2
2
N/mm
N/mm
kN/mm
mm
mm
N/mm
kN/mm3
6
28.27
280.2
208.8
52.75
3.42
28.9
8
50.27
416.4
216.8
MODEL 5
10
78.54
381.9
212
16
201.06
542.2
215
6
28.27
280.2
208.8
52.18
4.22
23.99
8
50.27
416.4
216.8
MODEL 6
10
78.54
381.9
212
16
201.06
542.2
215
For the ease of installing new section properties in Section Library,
Multiframe provides the Section Maker in Graphical User Interface (GUI) form as in
Figure 3.3 below.
Figure 3.3
Graphical User Interface (GUI) for Section Maker
39
3.2.2
Building the Frame Models in Multiframe
The main concept of frame analysis is to replace all structural members as a
line element of stiffness equal to those of the units they replaced, located at its
centroidal axis. These line elements will form a one-dimensional frame model
represent the actual structure and being analysed to obtain deflection and stresses of
the model due to applied load. Figure 3.4 shows how the line replacement for Model
5 and Model 6.
Section properties that has been installed in section maker was in 1 axis
(cross section for structural member). The existence of links has been considered by
representing the frame members with different section properties as explained in
Figure 3.5.
All the above step of modelling can be done using Graphical User Interface
(GUI) in Multiframe4D provided by Multiframe. The interface of Multiframe 4D is
as shown in Figure 3.6.
40
Wall 1 Wall 2
Wall 2
Coupling
Beam
MODEL 5
Base
Actual shear wall model
Line element in frame model
(a)
Wall 2
Wall 2
Wall 1
Coupling
Beam
MODEL 6
Base
Actual shear wall model
Line element in frame model
(b)
Figure 3.4
b) Model 6
Modelling the shear walls into frame structures. a) Model 5 and
41
Actual model with
reinforcement detail
Idealization into frame model
(a)
Idealization into frame model
Actual model with
reinforcement detail
(b)
Figure 3.5
b) Model 6
Frame models considering the existance of links.
a) Model 5 and,
42
Figure 3.6
3.2.3
Graphical User Interface (GUI) for Multiframe4D
Loading
The models were applied with horizontal static point load at the top right
most part of the structure. The increment of loads was done manually in Multiframe
4D. It was applied according to the three load cycles applied on the previous research
by Marsono (2000) as in Table 3.3. In each cycle, the load was applied in several
steps of small increment to predetermined levels.
From previous research, first cycle was considered as elastic state of loading.
The models was loaded to approximately 10% of it analytically predicted ultimate
load or when a very fine hair line crack started to appear in any part of the model,
whichever first. Then it was unloaded back to zero.
43
The second load cycle was considered as elasto-plastic or service load cycle.
The load was incremented to approximately 30% of its predicted ultimate load or
when the crack widths on any part of the model reached 0.03mm. The model was
then unloaded back to zero.
The final load cycle was to determine its ultimate capacity. The load was
increased until the models lost its ability to sustain load anymore. At this condition,
the models were considered to have reached its ultimate load carrying capacity.
Table 3.3
Load cycles and increment of loads.
Load
Load Increment (kN)
Cycle
1
2
3
3.2.4
Model 5
Model 6
0, 0.5, 1, 1.5, 2, 2.5, 3, 2.5, 2, 1.5, 0, 0.5, 1, 1.5, 2, 2.5, 3, 2.5, 2, 1.5,
1, 0
1, 0
0, 1.5, 3, 4.5, 6, 7.5, 9, 10, 9, 7.5, 0, 1.5, 3, 4.5, 6, 7.5, 9, 10, 9, 7.5, 6,
6, 4.5, 3, 1.5, 0
4.5, 3, 1.5, 0
5, 10, 15, 20, 25, 30, 40, 50, 60, 5, 10, 15, 20, 25, 30, 40, 50, 60, 70,
70, 80, 90, 100, 110, 120, 122.6
80, 90, 100, 110, 120, 126.6
Analysing the Shear Wall Models
After completing the modelling process and applying the loads, the models
were analysed using static analysis or linear (first order elastic) type of analysis
provided in Multiframe version 5.16. Non-linear (second-order elastic) type of
analysis was known can offer a better results for the ultimate conditions of structure,
however it only included in the latest version 9.5 of Multiframe. Thus, a stiffer
structure were expected from the results compared to one obtained from
experimental test and NLFEA.
44
3.3
Results Interpretation
In Multiframe 4D, a graphical display of forces, actions and deflections
within the structures being analysed are shown in the Plot window. From here, the
area with high stresses or big deflection can be easily assessed graphically. These
results can also be viewed in numerical form. In the Result window, displacements of
the joints in the structure and the joint reactions as computed in the analysis will be
displayed in tables.
Since the analysis cannot determine the ultimate condition of the structure by
itself, so assumption has to be made for this condition. For the shear wall models,
failure occurs when one or more of the wall crushes in compression and stresses at
the compression corner exceeded characteristic strength, fcu of concrete. Crushes in
concrete occurs when compression strength higher than crushing strength of
concrete, taken as 0.8fcu.
The existence of crack in concrete cannot be obtained directly from the
analysis results because it only gives the value of stresses on the structure. Thus, the
determination was made by comparing tensile stress acting on the structure with the
limiting tensile strength, ftc of concrete. Cracks occur when the tensile stress reaches
this limiting tensile strength. The value of ftc here is taken as fcu/21 same as the value
taken by Marsono (2000) in his research.
Table 3.4 shows the value of crushing strength and limiting tensile strength of
concrete as reference properties to determine the ultimate condition of the models
and investigate the mode of failure of the structures. Referring to these reference
properties, assumption was made on the location of concrete crushed and cracks
formation and was approximately being sketched.
45
Table 3.4
Properties for Model 5 and Model 6.
Characteristic Limiting Tensile
Crushing
= fcu/21
= 0.8fcu
Strength, fcu Strength, ftc
Strength
MODEL 5
(N/mm2)
52.75
(N/mm2)
2.51
(N/mm2)
42.2
MODEL 6
52.18
2.48
41.74
CHAPTER 4
DATA COLLECTION AND ANALYSIS
This chapter consists of the results from frame analysis. The analysis was
carried out to study the structural behaviour of coupled shear wall structures under
lateral loading at the ultimate condition. The study is focused on the determination of
the ultimate load, deflections characteristic and stresses in the structures.
4.1
Ultimate Load Condition
The coupled shear wall models considered failed once the failure
characteristic occurs. It is when one or more of the wall crushes in compression. To
determine this condition, the resultant of compression stresses on the models was
compared to the characteristic strength, fcu of concrete. Load acting in sync with the
stresses that exceeded fcu of concrete was considered as the structure has reached
ultimate failure.
The ultimate load for Model 5 was determined as 122.6 kN with the
maximum compression stress at the corner of compression wall greater than fcu for
47
Model 5, 52.75 N/mm2. For Model 6, the ultimate load was found to be 126.6 kN
with the maximum compression stress greater than fcu, which was 52.18 N/mm2 at
the same location of Model 5.
The ultimate load for Model 6 was bigger than Model 5 because the
compression wall for Model 6 was bigger than Model 5 and load was applied at the
smaller tension wall. Since the behaviour of walls at ultimate limit state is governed
by compression zone of walls, for the same dimension of asymmetrical shear wall
structures, the model with bigger compression wall is actually stronger than the other
model.
4.2
Deflections
In this study, deflections were measured at top left most of the models, which
were joint 251 for Model 5 and Model 6. Figure 4.1 shows the significant joint
numbers for Model 5 and Model 6 and its deflected shape.
The maximum deflection for Model 5 under ultimate load was found to be
6.54 mm slightly smaller than deflection of Model 6 which was 7.93 mm. Table 4.1
shows the summary of the horizontal and vertical displacements measured for every
model. Results of displacements with respect to load applied are shown in Table 4.2
and Table 4.3 and being plotted as in Figure 4.2 and Figure 4.3.
48
258
266
251
220
229
183
190
213
174
145
152
105
116
64
76
137
95
57
1
9
16
25
39
(a)
259
266
221
229
251
208
173
183
191
140
148
133
1
91
103
113
56
63
71
9
18
25
39
(b)
Figure 4.1
Deflected shape for a) Model 5 and, b) Model 6.
49
Table 4.1
Summary of the deflections measured at ultimate condition.
Ultimate
Load
(kN)
Deflection
Vertical
Horizontal
Compression Wall Tension Wall
(mm)
(mm)
(mm)
MODEL 5
122.6
6.54
0.96
1.43
MODEL 6
126.6
7.93
0.83
2.09
Table 4.2
Data for lateral load and displacement of Model 5 and Model 6.
MODEL 5
MODEL 6
1
Load
(kN)
0
Deflection
(mm)
0
Load
(kN)
0
Deflection
(mm)
0
2
5.0
0.27
5.0
0.31
3
10.0
0.53
10.0
0.63
4
15.0
0.80
15.0
0.94
5
20.0
1.07
20.0
1.25
6
25.0
1.33
25.0
1.57
7
30.0
1.60
30.0
1.88
8
40.0
2.14
40.0
2.51
9
50.0
2.67
50.0
3.13
10
60.0
3.20
60.0
3.76
11
70.0
3.74
70.0
4.39
12
80.0
4.27
80.0
5.01
13
90.0
4.80
90.0
5.64
14
100.0
5.34
100.0
6.27
15
110.0
5.87
110.0
6.89
16
120.0
6.40
120.0
7.52
17
122.6
6.54
126.6
7.93
50
Table 4.3
Displacement through height of the compression wall at the ultimate
condition.
Height
Displacement (mm)
(mm)
MODEL 5
MODEL 6
300
0.88
1.08
600
1.78
2.19
900
2.86
3.50
1,200
4.07
4.95
1,500
5.33
6.46
1,800
6.54
7.93
160
Model 5
Pu = 122.6 kN
δu = 6.54 mm
140
Model 6
Pu = 126.6 kN
δu = 7.93 mm
Lateral Load (kN)
120
100
80
60
40
20
0
0
1
2
3
4
5
6
7
8
9
10
Lateral Displacement (mm)
Figure 4.2
Lateral load and displacement of Model 5 and Model 6.
51
2000
Model 5
Pu = 122.6 kN
1800
Model 6
Pu = 126.6 kN
Height of Wall (mm)
1600
1400
1200
1000
800
600
400
200
0
0
1
2
3
4
5
6
7
8
9
Lateral Displacement (mm)
Figure 4.3
Lateral displacements through height of the compression wall.
52
53
The load-displacement graph is linear for both models from start until it
reaches the ultimate load because the models were analysed using first-order elastic
method of frame analysis. The material and section properties were assumed to be
elastic along the analysis. The geometric and material nonlinearities were also not
considered in this analysis.
The displacement in Figure 4.2 is actually for Load Cycle 3. Displacements
for Load Cycle 1 and Load Cycle 2 are not shown here because the value is actually
same with those obtained from Load Cycle 3. The load-displacement graph for Load
Cycle 1 and 2 are also linear and will follow the same path as for Load Cycle 3 since
the models were analysed using linear elastic analysis. The graph will only be shown
in the next chapter as comparison with the deflection obtained from experimental
study and NLFEA.
4.3
Stresses
Figure 4.4 to Figure 4.5 show the computed axial stresses, Sx’ and bending
stresses, Sbz’ of the models at ultimate load. In frame analysis, since beams and walls
were considered as line elements, determining the bending stress at specific location
would be complicated. However, Multiframe has simplified this problem by
categorizing the bending stress into two. First is Sbz’ top, bending stress about the
local z' axis at the top of each member and second is Sbz’ bottom, bending stress
about the local z' axis at the bottom of each member.
34C
8C
57C
97C
58C
58T
85C
57T
85T
9C
102C
102T
7C
7T
102C
131C
131T
118C
13C
97T
138C
119T
102T
138C
136C 138T
119C
136T
118T
11T
137C
138C
118C
18C
15T
118T 107C
137T 136C
109C
26C
19T
109T
100C
134T
101T
149T
138T
137C
117T
134C 138T
107T
154C
101C
12T
136T
149C
100T
154T
27C
23T
48T
26C
28C
(a)
Figure 4.4
47C
26T
(b)
Stresses at members of Model 5.
a) Axial stress, Sx’ (N/mm2),
28T
(c)
b) Sbz’ top (N/mm2),
c) Sbz’ bottom (N/mm2).
Note : T = Tension, C = Compression.
54
41C
13C
76C
75T
86C
75C
72T
72C
76T
123C
6C
7C
124C
123T
6C
9T
136T
14T
108C
108T
136C
10C
86T
116C
124T
108C
106T
133C
106C 116T
136C
136T
112C
135C
108T
111C
112T
14C
19T
18C
135T 124C
24T
125T
111T 117C
125C 133T 115C
115T 141C
126C
124T
129C
117T
138C
6T
129T
22C
138T
126T
141T
31T
45T
30C
33C
(a)
Figure 4.5
45C
30T
(b)
Stresses at members of Model 6.
a) Axial stress, Sx’ (N/mm2),
33T
(c)
b) Sbz’ top (N/mm2),
c) Sbz’ bottom (N/mm2).
Note : T = Tension, C = Compression.
55
56
Only stresses values that can be considered as critical are state in Figure 4.4
and Figure 4.5. To simplify the direction of the stresses, tension stress was marked as
‘T’ and compression stress was marked as ‘C’.
It can be seen that all the coupling beams for both models were undergone
high bending stress with average of stress 100 N/mm2 and above for compression
and tension. The axial stresses in compression wall for Model 5 are slightly higher
than Model 6 with the maximum compression stress of 27 N/mm2 and 22 N/mm2
respectively. But the stresses in tension wall of Model 5 are smaller than Model 6
with the maximum tension stress of 23 N/mm2 and 31 N/mm2 respectively. These are
because Model 5 carried the compression force with smaller section of wall while the
tension force was carried by the bigger section of wall. Thus, Model 5 is actually
weaker in compression but stronger in tension. That is why the ultimate load for
Model 6 is higher than Model 5. Because of the criteria for the models to fail was
when the stress at compression corner exceed characteristic strength, Model 6 with
higher capacity in compression corner require a bigger load to make it fail.
4.3.1
Resultant Stress
The resultant of axial and bending stress is also shown differently for top and
bottom part of members as in Figure 4.6 and Figure 4.7. For the ease of analysing,
the stresses are combined and marked at the full figure of the models.
65C
50T
131C
65C
51T
105C
120C
50T
146C
105C
T
C
C
T
T
T
C
C
C
T
T
T
C
C
T
T
C
C
T
T
C
C
T
T
C
C
T
T
C
C
C
T
63T
140C
99T
118C
123T
99T
136C
119T
119C
137T
116C
119T
118T
137C
1188C
138T
106C
110T
129T
134C
107C
135T
T
139T
T C
T
C
T
C
T
C
T
C
T
T C
T
139T
133C
101C
T C
T
T C
T
136C 138T
99C
161T
23T
T C
T
T C
T
100T
166T
53C
T C
109T
141C
102T
T
T
136C
117T
135C
T C
25C
2C
Sx’ + Sbz’ top
34C
71T
Sx’ + Sbz bottom
Figure 4.6
Resultant stresses and the illustration of its location on Model 5
Note : T = Tension, C = Compression.
57
127C
87C
113C
88C
62T
T C
31T
63T
130C
123C
128C
101T
117T 135C
133C
52C
T
C
T
T
T
C
T
T
T
C
T
T
T
C
T
T
T
C
T
C
T
C
C
C
C
112T 117C
C
C
C
C
C
T
110T
110C
135T
124C
115T
127T
T
109T
136T
112C
127C
T
C
T
115C
136C
108T
126T
C
C
C
T
118T
106C
132C
136T 123C
T
T
C
45T
104C
137T
T C
C
T
113T
116C
125T
135C
130C
144T
32C
8T
Sx’ + Sbz’ top
T
T C
T
T C
T
T C
T
116T
133C
124T
15C
T C
146T
35T
T C
T
T C
T
76T
Sx’ + Sbz bottom
Figure 4.7
Resultant stresses and the illustration of its location on Model 6
Note : T = Tension, C = Compression.
58
59
In Figure 4.6 and Figure 4.7, the ‘T’ signs in blue colour indicate the critical
tension stress which has exceeded the limiting tensile strength, ftc of concrete. So it is
also indicate the locations where the cracks occurred on the shear wall models. While
the ‘C’ signs in concrete indicate the critical compression stress that has exceeded the
crushing strength of concrete. It shows where the existence of concrete crushed
which mostly at the opposite diagonal side of the coupling beams tension corner and
at the bottom compression corner of compression walls.
4.3.2
Shear Stress
Figure 4.8 shows the shear stress on members for Model 5 and Model 6. It
can be seen that there are high shear stress at all the coupling beams, greater than the
limit that had been stated in BS8110 (1997) which was 5 N/mm2. This indicates that
all the coupling beams are also failed in shear. Only certain parts of the walls for
both models have high shear stress that greater than 5N/mm2, mainly at the middle
wall.
60
9.9
15.8
17.7
23.2
20.4
5.5 23.6
20.3
5.2 23.7
5.2 23.4
18.7
17.4
5.2 26.1
5.1
(a)
12.8
13.4
21.0
5.1 19.1
23.1
5.3 18.2
22.8
5.0 19.0
21.2
5.1 19.8
21.7
5.4 23.7
(b)
Figure 4.8
Shear stresses in members. a) Model 5 and, b) Model 6
CHAPTER 5
DISCUSSION OF RESULTS
In this chapter, results obtained from the frame analysis will be discussed and
compared to the results obtained from previous research by Marsono (2000). Here
the performance of frame analysis using linear elastic method in analysing coupled
shear wall structure compared to Non-Linear Finite Element Analysis (NLFEA) and
analytical Continuous Connection Method (CCM) will be observed.
5.1
Load Displacement Characteristic
Due to the fact that Multiframe version 5.16 uses linear elastic method for
frame analysis, very stiff structures were expected. The load displacement graph
from Multiframe is linear from the beginning until it reaches the ultimate load as in
Figure 5.1 and Figure 5.2. This is because materials were assumed to be elastic along
the analysis and the geometrical nonlinearities were not considered in the calculation.
140
Multiframe
Pu = 122.6 kN
δu = 6.54 mm
120
Lateral Load (kN)
100
80
60
CCM
Pu = 20.8 kN
δu = 0.81 mm
40
NLFEA
Pu = 29.2 kN
δu = 6.7 mm
Lab Test
Pu = 24.5 kN
δu = 12.81 mm
20
0
0
2
4
6
8
10
12
14
16
Lateral Displacement (mm)
Figure 5.1
Lateral load and displacement of Model 5
62
140
Multiframe
Pu = 126.6 kN
δu = 7.93 mm
120
Lateral Load (kN)
100
80
60
40
Lab Test
Pu = 27.5 kN
δu = 17.55 mm
NLFEA
Pu = 31.1 kN
δu = 6.81 mm
CCM
Pu = 24.5 kN
δu = 0.97 mm
20
0
0
2
4
6
8
10
12
14
16
18
20
Lateral Displacement (mm)
Figure 5.2
Lateral load and displacement of Model 6
63
64
From the load displacement graphs, there can be seen that ultimate load
obtained from Multiframe is about four times bigger than the experimental test and
other method of analysis. However the displacement obtained are only about half of
the actual displacement from experimental test. This concludes that the frame
analysis using linear elastic method has overestimated the ultimate capacity of the
models as much as 400%.
Although the displacement from NLFEA are close to those from Multiframe,
but the ultimate load obtained from NLFEA are much closer to the experimental test.
This shows that linear elastic frame analysis is not reliable enough to analysed the
ultimate condition of structures.
In NLFEA, various aspects of nonlinearities for materials and geometrical
properties are taken into account in analysis. So the analysis actually done close to
real-life conditions of the structure. But in linear elastic frame analysis, there are too
much assumptions made for the analysis. The material and geometrical properties are
assumed to be constant along the analysis, the crack properties are also not
considered as in NLFEA, and the structural members are just simplified into line
element. Thus makes the frame analysis not comparable enough to NLFEA in
analysing structures to its ultimate condition.
In early stage, the results obtained from Multiframe looked close to those
obtained from CCM. It is because the deflection calculation in CCM was also based
on elastic method of analysis. Both methods overestimated the capacity of the
structure nearly at the same value of load-displacement. But due to the method of
analysis of CCM that determining the mode of failure of the coupling beams and the
walls before analysing the structures, makes it able to obtain the ultimate load that
close to experimental test.
65
Figure 5.3 shows the load-displacement graph for Load Cycle 1 which
considered as the elastic stage of loading in previous research. Even in the elastic
stage of loadings, the frame analysis results still differ significantly from the
experimental results. It is because frame analysis has overestimated the stiffness of
the models and since the coupled shear wall was modelled as before the load applied.
If the condition of section properties reduction as well as material strength
degradation is included in the analysis, better results would be achieved.
3.5
Lab Test
Multiframe
3
Lateral Load (kN)
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Lateral Displacement (mm)
(a)
3.5
Multiframe
Lab Test
3
Lateral Load (kN)
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Lateral Displacement (mm)
Figure 5.3
b) Model 6
(b)
Lateral load and displacement for Load Cycle 1 of a) Model 5 and,
66
5.2
Mode of Failure
At the ultimate condition for both models, there are indications of a
symmetrical behaviour at the opposite corners of all the coupling beams. Stresses at
two diagonal compression corners are found greater than the crushing strength,
indicate that the concrete has crushed. While at the opposite corners, the tension
stresses are also found high, showing the existence of crack formations.
Shear stresses at all the coupling beams for Model 5 and Model 6 are found
high and greater than 5N/mm2, means all the coupling beams were failed in shear.
Thus, based on the above conditions, all coupling beams are considered failed with
flexure plus shear (diagonal-splitting) mode of failure as in Figure 5.4 and Figure
5.5.
The mechanism of failure for the models were started with the failure of all
coupling beams then followed by the failure of shear walls. Thus the mode of failure
for shear walls were actually characterised by the mode of failure of coupling beams.
For Model 5 and Model 6, both were undergone mode of failure category 2 which
characterised by the failure of coupling beams in shear or diagonal splitting mode.
This mode of failure was caused by the flexible coupling action in shear wall
structures. The shear walls for both models were failed in combined action of flexure
and shear.
The frame analysis has shown that the attitude of mode of failure can be
determined from the analysis.
67
5.3
Concrete Crushed and Crack Observations
Observations made from the study shows that the appearance of concrete
crushed and cracks formation in coupling beams was starts at early stage of loading.
The cracks occurred in coupling beams of Model 5 at load 2kN and Model 6 at
2.5kN. While the concrete crushed occurs at coupling beams of Model 5 and Model
6 at load 44.3kN and 39kN respectively. The appearance of cracks formation at shear
wall also starts at the early stage of loading but slightly higher than the loads which
cracks started for coupling beams. Cracks were start to occur at shear wall of Model
5 and Model 6 at load 4.5kN. While the concrete of shear walls started to crush at
load 98.5kN for Model 5 and 101.3kN for Model 6.
The observations of cracks at very small load applied may not really accurate
since reinforcement not being modelled as single element. In reinforced concrete
structure, tension stress is mainly carried by the steel reinforcement and cracks can
also be occurred when the steel yield. However the yielding of steel reinforcement in
this study cannot be detected since it was modelled compositely in the structure. The
cracks were only assumed to be occurred when the combined tension stress exceeds
limiting tensile strength, ftc of concrete.
From the stress resultant that had been determined in previous chapter,
assumption was made on the location of concrete crushed and cracks formations on
the models. Then the assumptions were approximately sketched as in Figure 5.4 and
Figure 5.5. The figures of concrete crushed and cracks formations on the models are
not obtained from the Multiframe analysis. It was only sketched assumption for the
ease of results simplifications.
The figures show that there is a sign of concrete crushed at the bottom left
corner of compression wall for both models. The tension wall for both models was
undergone severe cracks formation from the bottom up until level 5. Cracks were
68
also occurred at the right side of the middle wall. Major crack was observed at the
bottom tension corner of tension wall for Model 5 and Model 6 with tension stress of
71N/mm2 and 76N/mm2 respectively.
All the coupling beams were observed may be to fail with flexure plus shear
(diagonal splitting) mode of failure as in Figure 5.4 and Figure 5.5. There was a
symmetrical behaviour at the opposite corners of all the coupling beams. At the
compression corners, the concrete was observed to be crushed. While at the opposite
corners, there were indication of the existance of cracks.
All the observations on concrete crushed and cracks formations were nearly
same with what had been observed in the experimental study. The signs of mode of
failures for coupling beams and shear walls also similar with the previous study.
T
C
C
T
T
T
Flexure plus shear (diagonal
splitting) mode of failure
C
C
C
C
C
C
C
C
C
C
C
C
C
T C
T
T
T C
T
T C
T
T C
T
T
C
T
C
T
C
T
C
T
T C
T
T C
T
T C
T
T C
T
T
T
T
T
T
T
T
T
T
T
C
T
Figure 5.4
Concrete crushed
fc = 52.8N/mm2
Major crack, ft = 71N/mm2
Approximate observation of concrete crushed and cracks formation of Model 5.
Note : T = Tension, C = Compression
T C
T
C
T C
C
T
T
T
T
C
Flexure plus shear (diagonal
splitting) mode of failure
C
C
C
T
T
T
C
C
C
T
T
T
C
C
C
T
T
T
C
C
C
T
T
T
T
C
C
C
T
T C
T
T C
T C
C
T
T
C
T
T C
T
T C
T
T C
T
T
T
C
T
Figure 5.5
Concrete crushed
fc = 52.2N/mm2
Major crack, ft = 76N/mm2
Approximate observation of concrete crushed and cracks formation of Model 6
Note : T = Tension, C = Compression
CHAPTER 6
CONCLUSIONS AND RECOMMENDATIONS
6.1
Conclusions
A study was carried out on Model 5 and Model 6 adopted from previous
research done by Marsono (2000). From the analysis of frame, the ultimate load of
the models had been obtained and the structural behaviour of the structure under
ultimate conditions had been observed.
The method had overestimated the ultimate load capacity of the models about
four times bigger than NLFEA, analytical CCM method and the actual load from
experimental test. This can be considered as unacceptable and the linear elastic frame
analysis can be conclude as not reliable enough to analyse the ultimate conditions of
structure although it has been used by engineers for more than four decades. The
overestimation happens because various aspects of deterioration for materials and
geometrical size reduction are not taken into account in analysis as in NLFEA. Too
much simplifications for the analysis such as material and geometrical properties are
assume to be constant along the analysis and members are simplified into line
element may cause the false representation of the results.
72
All the coupling beams were observed to fail with flexure plus shear
(diagonal splitting) mode of failure with high stresses in compression corners
exceeded the crushing strength and high tension stress at the opposite corners
exceeded the limiting tensile strength. Shear stresses were also found high in all the
coupling beams, greater than 5N/mm2.
The mechanism of failure was started with the failure of coupling beams
since concrete crushed and cracks formations were observed at coupling beams at the
early stage of loading. Then the mechanism followed by the failure of walls until
both of the models reach its ultimate condition.
The ultimate load for Model 5 was determined as 122.6kN, smaller than
Model 6, 126.6kN because the compression wall for Model 5 was smaller than
Model 5 and load was applied at the bigger tension wall. The behaviour of walls at
ultimate limit state is governed by compression zone of walls, thus for the same
dimension of asymmetrical shear wall structures, the model with bigger compression
wall is stronger than the other model.
Crushing of concrete and cracks formations had been observed and
assumption was made on its locations. Then the assumption was approximately
sketched for a better visualization of the results. From the sketches, it can be
conclude that the structural behaviour of walls and its coupling beams under ultimate
conditions was similar to those had been observed from the experimental test and
NLFEA.
Results obtained from the analysis and structural behaviour of the models at
ultimate condition has been summarized in Table 6.1.
Table 6.1
Structure
MODEL 5
MODEL 6
Ultimate Load
(kN)
122.6
126.6
Summary of the structural behaviour from the frame analysis for Model 5 and Model 6.
Deflections (mm)
Horizontal
6.54
7.93
Vertical
0.96
0.83
Structural Behaviours
Coupling Beams
Concrete crushed at
compression
corners and cracks
formed at the
opposite corners.
Failed with flexure
plus shear
(diagonal splitting)
mode of failure
Concrete crushed at
compression
corners and cracks
formed at the
opposite corners.
Failed with flexure
plus shear
(diagonal splitting)
mode of failure
Compression Wall
Middle Wall
Tension Wall
Crushed of concrete at
the bottom
compression corner.
No concrete crushing
and cracks formation
at compression
corner. Cracks only
observed at the
tension side of wall.
Cracks formed at
both side of wall and
major crack occurs at
the bottom tension
corner.
Crushed of concrete at
the bottom
compression corner.
No concrete crushing
and cracks formation
at compression
corner. Cracks only
observed at the
tension side of wall.
Cracks formed at
both side of wall and
major crack occurs at
the bottom tension
corner.
73
74
6.2
Recommendations
Since the linear elastic method of frame analysis has overestimate the
ultimate capacity of structures, it only suitable to analyse structure at elastic stage
conditions. At ultimate state of frame analysis, the materials were naturally at elastic
conditions and nonlinearities of other properties are not represented. So the results
still can be considered as acceptable only on mode of failure, behaviour at failure,
but certainly not an ultimate load and deflection.
The analysis results may be improved if the latest version 9.5 of Multiframe
was being used to analyse the shear wall models but it will still does not include the
effect of material strength reduction. The second-order elastic (non-linear) analysis
as an option to analyse model may improve the results of analysis very slightly. The
facilities may consider second order elastic nonlinearities which are due to the P-δ,
P-Δ, and flexural shortening effects. It also accounts for the influence of any tension
or compression only members within a structure.
The erroneous difference of analysis results of ultimate load and deflection
may also can be reduced if more detailed model of the shear walls was build in
Multiframe. This can be done by considering all the steel reinforcements and links as
individual element in the modelling. So the exact behaviour of steel can be obtained
and the yielding of steel can be observed accurately. But it also may improve the
results very slightly unless the materials strength degradation properties can be
included in the analysis as in NLFEA.
75
References
British Standard Institutions. (1997). Structural Use of Concrete. London, BS 8110.
Driver, R. G., Kulak, G. L., Elwi, A. E. and Laurie Kennedy, D. J. (1998). FE and
Simplified Models of Steels Plate Shear Wall. Journal of Structural
Engineering. 124(2): 121-130
Hall, A. S. and Woodhead, R. W. (1967). Frame Analysis. 2nd ed. New York: John
Wiley & Sons.
Irwin, A. W. (1984). CIRIA Report 102. Design of Shear Wall Buildings. London:
CIRIA Publication.
Kemp, A. R. (2002). A Mixed Flexibility Approach for Simplifying Elastic and
Inelastic Structural Analysis of Frames. Journal of Construction Steel
Research. 58: 1297-1313
Kwan, A. K. H. (1993). Improved Wide Column Frame Analogy for Shear/Core
Wall Analysis. Journal of Structural Engineering. 119(2): 420-437
Lam, W. Y., Su, R. K. L. and Pam, H.J. (2005). Experimental Study on Embedded
Steel Plate Composite Coupling Beams. Journal of Structural Engineering.
131(8): 1294-1302.
Marsono, A. K. (2000). Reinforced Concrete Shear Walls With Regular and
Staggered Openings. University of Dundee: Ph.D Thesis.
76
MULTIFRAME. 2006. Multiframe User Manual. Australia: Formation Design
Systems Pty Ltd.
Ove Arup and Partners (1977). CIRIA Guide 2. The Design of Deep Beams in
Reinforced Concrete. London: CIRIA Publication.
Taranath, B. S. (1998). Steel, Concrete and Composite Design of Tall Buildings. 2nd
ed. New York: McGraw-Hill.
77
APPENDIX A
JOINT DISPLACEMENTS AT ULTIMATE LOAD STATE
a) Joint Displacements of Model 5
b) Joint Displacements of Model 6
78
a) Joint Displacements of Model 5 at Load = 122.6kN
Joint
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
dx
0
0
0
0
-0.002
-0.002
-0.006
-0.006
-0.006
-0.008
-0.008
-0.011
-0.011
-0.013
-0.013
-0.014
-0.015
-0.015
-0.015
-0.015
-0.015
-0.015
-0.016
-0.016
-0.016
-0.014
-0.014
-0.011
-0.011
-0.008
-0.008
-0.005
-0.005
-0.002
-0.002
0
0
0
0
-0.428
-0.414
-0.435
-0.448
-0.346
-0.359
-0.605
-0.59
-0.602
dy
-0.038
-0.013
-0.01
0
0.029
0.025
0.076
0.071
0.085
0.132
0.139
0.214
0.206
0.289
0.298
0.338
0.386
0.377
0.455
0.462
0.518
0.514
0.545
0.546
0.548
0.537
0.533
0.472
0.48
0.384
0.372
0.258
0.244
0.099
0.114
0
-0.038
-0.054
-0.153
-0.163
-0.16
0.29
0.289
0.756
0.759
-0.231
-0.227
0.276
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.022
0.022
0.022
0.022
0.024
0.023
0.031
0.03
0.033
0.039
0.04
0.046
0.046
0.05
0.05
0.051
0.048
0.049
0.04
0.039
0.025
0.026
0.008
0.007
0.001
-0.02
-0.021
-0.047
-0.045
-0.065
-0.066
-0.079
-0.079
-0.086
-0.086
-0.088
-0.088
-0.088
-0.088
0.132
0.131
0.124
0.124
0.127
0.128
0.141
0.141
0.133
79
Joint
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
dx
-0.615
-0.522
-0.537
-0.859
-0.839
-0.838
-0.784
-0.898
-0.878
-0.878
-0.878
-0.877
-0.877
-0.876
-0.876
-0.876
-0.857
-0.895
-0.871
-0.87
-0.851
-0.852
-0.834
-0.833
-0.806
-0.919
-0.915
-0.827
-0.871
-0.849
-1.198
-1.18
-1.171
-1.189
-1.165
-1.146
-1.408
-1.427
-1.416
-1.397
-1.381
-1.4
-1.75
-1.725
-1.717
-1.709
-1.775
-1.801
-1.775
-1.775
-1.772
dy
0.276
0.814
0.817
-0.323
-0.319
0.258
0.892
-0.337
-0.331
-0.095
-0.109
-0.094
-0.089
-0.076
-0.09
0.256
0.257
0.255
0.588
0.6
0.545
0.56
0.511
0.526
0.895
-0.34
0.254
0.902
0.909
0.907
-0.417
-0.414
0.242
0.241
0.973
0.97
-0.47
-0.473
0.232
0.232
1.017
1.019
-0.55
-0.547
0.219
1.08
-0.556
-0.561
-0.264
-0.281
-0.246
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.133
0.144
0.145
0.142
0.142
0.137
0.154
0.144
0.141
0.123
0.141
-0.054
-0.054
0.136
0.118
0.137
0.137
0.14
0.135
0.109
-0.145
-0.145
0.126
0.153
0.155
0.146
0.141
0.155
0.158
0.157
0.172
0.171
0.169
0.17
0.177
0.177
0.181
0.181
0.181
0.181
0.185
0.186
0.18
0.181
0.184
0.19
0.18
0.183
0.153
0.172
-0.036
80
Joint
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
dx
-1.772
-1.77
-1.77
-1.743
-1.794
-1.769
-1.767
-1.767
-1.766
-1.766
-1.764
-1.764
-1.736
-1.826
-1.82
-1.789
-1.763
-1.816
-2.173
-2.151
-2.169
-2.147
-2.167
-2.146
-2.427
-2.449
-2.445
-2.422
-2.418
-2.441
-2.832
-2.803
-2.799
-2.829
-2.825
-2.795
-2.891
-2.862
-2.861
-2.861
-2.86
-2.86
-2.86
-2.859
-2.889
-2.859
-2.858
-2.858
-2.857
-2.857
-2.856
dy
-0.242
-0.225
-0.208
0.219
0.217
0.217
0.653
0.637
0.647
0.637
0.633
0.65
1.082
-0.564
0.216
1.092
1.088
1.094
-0.624
-0.622
0.206
0.206
1.143
1.142
-0.666
-0.668
0.198
0.199
1.178
1.18
-0.728
-0.726
0.188
0.188
1.23
1.228
-0.737
-0.733
-0.392
-0.412
-0.36
-0.358
-0.328
-0.309
0.186
0.187
0.697
0.678
0.702
0.709
0.715
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
-0.036
0.149
0.168
0.183
0.186
0.183
0.141
0.164
-0.091
-0.09
0.148
0.171
0.19
0.184
0.187
0.192
0.19
0.193
0.209
0.209
0.209
0.209
0.208
0.208
0.217
0.217
0.217
0.216
0.214
0.214
0.211
0.212
0.214
0.213
0.216
0.216
0.212
0.21
0.181
0.201
-0.026
-0.026
0.176
0.196
0.215
0.212
0.17
0.194
-0.067
-0.067
0.173
81
Joint
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
dx
-2.856
-2.855
-2.921
-2.919
-2.916
-2.885
-3.291
-3.316
-3.29
-3.314
-3.309
-3.285
-3.599
-3.624
-3.596
-3.621
-3.587
-3.612
-4.042
-4.01
-4.01
-4.001
-4.106
-4.074
-4.074
-4.074
-4.074
-4.074
-4.074
-4.074
-4.107
-4.042
-4.074
-4.073
-4.073
-4.071
-4.071
-4.069
-4.069
-4.068
-4.034
-4.101
-4.138
-4.139
-4.134
-4.557
-4.531
-4.535
-4.561
-4.563
-4.537
dy
0.734
1.234
-0.739
0.186
1.238
1.237
-0.779
-0.781
0.178
0.178
1.274
1.273
-0.81
-0.812
0.172
0.171
1.299
1.301
-0.854
-0.853
0.163
1.335
-0.86
-0.858
-0.51
-0.488
-0.443
-0.442
-0.399
-0.377
0.162
0.163
0.162
0.698
0.719
0.74
0.745
0.768
0.789
1.34
1.337
1.342
-0.861
0.161
1.343
-0.885
-0.884
0.155
0.155
1.365
1.364
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.197
0.216
0.214
0.216
0.218
0.217
0.235
0.235
0.234
0.235
0.231
0.231
0.24
0.239
0.239
0.239
0.236
0.237
0.228
0.229
0.232
0.237
0.229
0.227
0.218
0.198
-0.01
-0.01
0.194
0.214
0.232
0.232
0.23
0.212
0.188
-0.046
-0.047
0.194
0.218
0.237
0.237
0.238
0.23
0.233
0.239
0.246
0.246
0.247
0.248
0.252
0.251
82
Joint
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
dx
-4.876
-4.85
-4.882
-4.856
-4.865
-4.892
-5.27
-5.302
-5.281
-5.315
-5.366
-5.334
-5.336
-5.336
-5.34
-5.34
-5.344
-5.344
-5.347
-5.314
-5.38
-5.354
-5.355
-5.367
-5.367
-5.38
-5.38
-5.387
-5.351
-5.398
-5.412
-5.46
-5.424
-5.782
-5.807
-5.831
-5.805
-5.896
-5.925
-6.086
-6.111
-6.142
-6.117
-6.248
-6.276
-6.483
-6.513
-6.523
-6.718
-6.543
-6.551
dy
-0.903
-0.902
0.15
0.15
1.381
1.382
-0.926
-0.927
0.144
1.403
-0.93
-0.929
-0.579
-0.556
-0.494
-0.491
-0.407
-0.429
0.143
0.143
0.143
0.686
0.708
0.741
0.738
0.804
0.78
1.406
1.404
-0.931
0.142
1.407
1.407
-0.939
-0.939
0.139
0.139
1.416
1.416
-0.945
-0.946
0.137
0.137
1.423
1.423
-0.954
-0.954
0.133
1.432
-0.955
-0.604
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.246
0.247
0.248
0.248
0.257
0.257
0.231
0.23
0.236
0.258
0.229
0.228
0.22
0.202
0.024
0.024
0.219
0.201
0.233
0.235
0.235
0.215
0.192
-0.03
-0.03
0.239
0.215
0.258
0.258
0.229
0.235
0.26
0.259
0.237
0.237
0.242
0.242
0.271
0.271
0.234
0.233
0.239
0.239
0.272
0.272
0.218
0.217
0.224
0.265
0.215
0.197
83
Joint
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
dx
-6.55
-6.562
-6.562
-6.573
-6.574
-6.585
-6.554
-6.636
-6.639
-6.687
-6.69
-6.741
-6.738
-6.792
-6.755
dy
-0.625
-0.499
-0.51
-0.406
-0.385
0.133
0.133
0.638
0.658
0.721
0.726
0.827
0.804
1.433
1.432
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.207
0.096
0.096
0.195
0.205
0.221
0.223
0.197
0.182
0.042
0.043
0.237
0.22
0.263
0.264
84
b) Joint Displacements of Model 6 at Load = 126.6kN
Joint
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
dx
0
0
0
0
-0.003
-0.003
-0.008
-0.008
-0.011
-0.012
-0.011
-0.014
-0.014
-0.016
-0.016
-0.018
-0.018
-0.019
-0.018
-0.018
-0.017
-0.017
-0.016
-0.016
-0.015
-0.015
-0.015
-0.012
-0.012
-0.008
-0.009
-0.005
-0.005
-0.002
-0.002
0
0
0
0
-0.531
-0.515
-0.538
-0.555
-0.446
-0.464
-0.743
-0.725
-0.765
dy
-0.06
-0.021
-0.015
0
0.046
0.04
0.117
0.109
0.173
0.208
0.198
0.316
0.305
0.417
0.428
0.537
0.526
0.583
0.623
0.632
0.693
0.689
0.714
0.714
0.698
0.686
0.691
0.613
0.602
0.473
0.488
0.327
0.309
0.125
0.144
0
-0.068
-0.048
-0.193
-0.066
-0.063
0.601
0.601
1.031
1.035
-0.131
-0.128
0.606
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.034
0.034
0.034
0.034
0.036
0.036
0.045
0.045
0.056
0.058
0.058
0.064
0.063
0.064
0.064
0.06
0.06
0.056
0.048
0.047
0.025
0.026
-0.001
0.001
-0.024
-0.031
-0.029
-0.06
-0.062
-0.085
-0.084
-0.1
-0.101
-0.109
-0.109
-0.111
-0.111
-0.111
-0.111
0.156
0.155
0.157
0.157
0.176
0.177
0.17
0.17
0.167
85
Joint
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
dx
-0.748
-0.684
-0.704
-1.032
-1.057
-1.067
-1.043
-1.082
-1.083
-1.083
-1.087
-1.086
-1.089
-1.09
-1.091
-1.087
-1.087
-1.077
-1.078
-1.067
-1.068
-1.04
-1.065
-1.015
-1.134
-1.108
-1.139
-1.114
-1.091
-1.116
-1.479
-1.457
-1.453
-1.475
-1.468
-1.446
-1.733
-1.756
-1.752
-1.729
-1.73
-1.754
-2.186
-2.123
-2.155
-2.118
-2.12
-2.218
-2.185
-2.185
-2.183
dy
0.606
1.122
1.126
-0.217
-0.22
0.612
0.612
-0.227
0.214
0.231
0.205
0.217
0.187
0.203
0.613
1.039
1.023
0.992
1.007
0.981
0.964
1.252
1.262
1.247
-0.236
-0.233
0.614
0.614
1.27
1.274
-0.309
-0.307
0.622
0.622
1.372
1.369
-0.36
-0.362
0.628
0.628
1.441
1.444
-0.441
-0.433
-0.435
0.636
1.539
-0.446
0.082
0.102
0.111
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.167
0.191
0.191
0.181
0.181
0.169
0.169
0.181
0.18
0.155
-0.117
-0.117
0.143
0.167
0.169
0.141
0.167
-0.145
-0.145
0.178
0.152
0.182
0.179
0.183
0.185
0.184
0.174
0.172
0.183
0.185
0.208
0.208
0.207
0.207
0.215
0.214
0.219
0.22
0.22
0.22
0.224
0.224
0.227
0.227
0.227
0.222
0.22
0.229
0.205
0.182
-0.082
86
Joint
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
dx
-2.183
-2.181
-2.181
-2.18
-2.149
-2.211
-2.18
-2.18
-2.181
-2.181
-2.181
-2.181
-2.151
-2.182
-2.25
-2.242
-2.243
-2.212
-2.645
-2.671
-2.662
-2.636
-2.661
-2.635
-3
-2.973
-2.994
-2.967
-2.993
-2.966
-3.463
-3.427
-3.42
-3.499
-3.498
-3.498
-3.495
-3.495
-3.493
-3.493
-3.491
-3.456
-3.49
-3.49
-3.489
-3.489
-3.488
-3.488
-3.487
-3.417
-3.452
dy
0.102
0.109
0.128
0.637
0.637
0.638
1.168
1.148
1.179
1.185
1.219
1.198
1.543
1.551
-0.448
0.638
1.561
1.558
-0.504
-0.506
0.645
0.645
1.637
1.635
-0.548
-0.546
0.65
0.65
1.694
1.691
-0.606
-0.604
0.657
-0.611
0.014
-0.009
0.028
0.035
0.07
0.047
0.658
0.658
1.276
1.253
1.322
1.32
1.392
1.368
1.777
1.768
1.771
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
-0.082
0.176
0.2
0.221
0.222
0.224
0.179
0.201
-0.061
-0.061
0.207
0.185
0.219
0.218
0.23
0.226
0.223
0.221
0.249
0.25
0.252
0.251
0.252
0.251
0.258
0.258
0.26
0.26
0.26
0.26
0.261
0.261
0.257
0.26
0.211
0.237
-0.072
-0.072
0.231
0.206
0.255
0.256
0.214
0.235
-0.017
-0.017
0.241
0.219
0.251
0.253
0.252
87
Joint
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
dx
-3.572
-3.536
-3.527
-3.563
-3.558
-3.523
-4.047
-4.018
-4.037
-4.008
-3.999
-4.029
-4.381
-4.41
-4.375
-4.405
-4.396
-4.366
-4.874
-4.872
-4.861
-4.991
-4.952
-4.913
-4.952
-4.952
-4.951
-4.951
-4.95
-4.95
-4.911
-4.989
-4.95
-4.947
-4.947
-4.943
-4.944
-4.94
-4.94
-4.977
-4.938
-4.9
-5.031
-5.028
-5.016
-5.532
-5.502
-5.538
-5.506
-5.529
-5.497
dy
-0.616
-0.614
0.659
0.659
1.784
1.782
-0.657
-0.655
0.663
0.663
1.838
1.84
-0.685
-0.687
0.666
0.667
1.88
1.879
-0.726
0.671
1.934
-0.733
-0.731
-0.728
-0.058
-0.083
-0.031
-0.025
0.002
0.027
0.671
0.671
0.671
1.349
1.323
1.42
1.418
1.491
1.517
1.944
1.941
1.937
-0.734
0.671
1.945
-0.759
-0.758
0.672
0.672
1.981
1.98
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.263
0.262
0.258
0.259
0.255
0.254
0.279
0.278
0.281
0.281
0.28
0.28
0.283
0.283
0.287
0.287
0.286
0.286
0.281
0.28
0.278
0.281
0.279
0.28
0.23
0.256
-0.055
-0.055
0.228
0.254
0.279
0.28
0.278
0.239
0.259
0.018
0.018
0.245
0.266
0.278
0.275
0.277
0.281
0.281
0.28
0.291
0.291
0.3
0.299
0.305
0.304
88
Joint
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
dx
-5.909
-5.879
-5.895
-5.927
-5.929
-5.896
-6.46
-6.382
-6.421
-6.414
-6.445
-6.5
-6.467
-6.467
-6.477
-6.477
-6.488
-6.487
-6.454
-6.494
-6.534
-6.502
-6.502
-6.515
-6.514
-6.527
-6.527
-6.532
-6.489
-6.539
-6.574
-6.576
-6.621
-7.035
-7.005
-7.059
-7.091
-7.169
-7.206
-7.401
-7.372
-7.445
-7.476
-7.621
-7.657
-7.896
-7.858
-7.95
-8.216
-7.934
-7.958
dy
-0.777
-0.776
0.673
0.673
2.007
2.006
-0.803
-0.8
-0.801
0.674
2.042
-0.804
-0.148
-0.122
-0.071
-0.073
0.006
-0.021
0.674
0.674
0.674
1.346
1.372
1.449
1.446
1.535
1.566
2.046
2.043
-0.804
0.674
2.047
2.048
-0.813
-0.813
0.674
0.674
2.062
2.063
-0.82
-0.82
0.673
0.673
2.073
2.074
-0.829
-0.829
0.673
2.088
-0.83
-0.206
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.291
0.292
0.302
0.301
0.314
0.314
0.281
0.283
0.282
0.289
0.313
0.282
0.259
0.236
-0.019
-0.019
0.265
0.241
0.288
0.286
0.288
0.267
0.247
0.029
0.028
0.28
0.302
0.312
0.313
0.282
0.289
0.316
0.318
0.285
0.285
0.3
0.3
0.347
0.347
0.282
0.283
0.297
0.296
0.35
0.349
0.272
0.273
0.278
0.327
0.271
0.245
89
Joint
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
dx
-7.959
-7.981
-7.98
-8.002
-8.003
-7.989
-8.027
-8.106
-8.11
-8.18
-8.183
-8.256
-8.253
-8.307
-8.261
dy
-0.181
-0.08
-0.088
0.014
0.04
0.673
0.673
1.305
1.331
1.433
1.443
1.597
1.566
2.09
2.088
dz
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øx
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øy
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Øz
0.231
0.075
0.075
0.234
0.249
0.277
0.275
0.248
0.235
0.098
0.099
0.307
0.291
0.322
0.325
90
APPENDIX B
RESULTS FROM PREVIOUS RESEARCH BY MARSONO (2000)
a) Results for Model 5
b) Results for Model 6
a) Results for Model 5
Load
0
0.5
1.0
1.5
2.0
2.5
3.0
2.5
2.0
1.5
1.0
0.5
0
Cycle 1
Displacement
0
0.03
0.10
0.16
0.23
0.29
0.35
0.33
0.28
0.22
0.16
0.09
0.03
Experimental Test
Cycle 2
Load
Displacement
0
0.03
1.5
0.16
3.0
0.33
4.5
0.53
6.0
0.90
7.5
1.41
9.0
2.02
10.0
2.42
9.0
2.47
7.5
2.31
6.0
2.12
4.5
1.87
3.0
1.60
1.5
1.32
0
0.93
NLFEA
Load
0
3.0
6.0
8.0
10.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
24.5
Cycle 3
Displacement
0.93
1.05
1.20
1.51
1.97
2.79
3.15
3.71
4.10
4.52
5.02
5.47
6.21
7.16
7.85
8.65
9.48
10.90
13.74
Analytical CCM
Load
Displacement
Displacement
Load
0.05
0.10
0.18
0.29
0.46
0.71
1.09
1.65
2.34
3.15
4.13
5.31
6.79
8.77
11.70
16.00
22.30
29.20
29.20
29.20
0.93
0.94
0.94
0.95
0.96
0.97
0.99
1.03
1.07
1.15
1.26
1.42
1.68
2.06
2.63
3.48
4.77
6.70
6.70
6.71
0.81
20.8
91
a) Results for Model 6
Load
0
0.5
1.0
1.5
2.0
2.5
3.0
2.5
2.0
1.5
1.0
0.5
0
Cycle 1
Displacement
0
0.07
0.16
0.27
0.33
0.46
0.57
0.54
0.48
0.40
0.33
0.24
0.15
Experimental Test
Cycle 2
Load
Displacement
0
0.15
1.5
0.28
3.0
0.55
4.5
0.98
6.0
1.49
7.5
2.05
9.0
2.62
10.0
2.99
9.0
2.93
7.5
2.73
6.0
2.49
4.5
2.21
3.0
1.88
1.5
1.53
0
1.00
NLFEA
Load
0
3
6
8
10
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
27.5
Cycle 3
Displacement
1.00
1.79
2.42
2.83
3.33
4.02
4.52
4.89
5.28
5.76
6.28
6.82
7.10
7.67
8.16
8.70
9.26
10.11
11.09
12.69
14.97
17.55
Analytical CCM
Load
Displacement
Displacement
Load
0.05
0.10
0.18
0.29
0.46
0.71
1.09
1.63
2.35
3.25
4.25
5.49
7.12
9.40
12.60
17.20
23.80
31.10
1.00
1.01
1.01
1.02
1.03
1.04
1.06
1.10
1.15
1.22
1.33
1.50
1.75
2.13
2.70
3.57
4.86
6.81
0.97
24.5
92