FINITE ELEMENT ANALYSIS ON THE DEFECTED REINFORCED
CONCRETE COLUMN
CHONG KEAN YEE
A project report submitted in partial fulfilment
of the requirements for the award of the degree of
Master of Engineering (Civil-Structure)
Faculty of Civil Engineering
Universiti Teknologi Malaysia
JUNE, 2007
iii
To my beloved family
iv
ACKNOWLEDGEMENT
First of all, I would like to express my greatest gratitude to all parties who
have given me the co-operation and help. Without them, I would not be able to
accomplish this Master’s Project. Besides that, I am very thankful to my project
supervisor, Dr. Redzuan Abdullah, for being a wise teacher and an understanding
friend.
I appreciate his guidance, enlightenment and most importantly his
motivation.
Apart from that, sincere appreciation is conveyed to my beloved family and
of course, Miss Goh, Hui Weng. Their invaluable encouragement, supports and
understanding helped me to get through my tough moment.
Last but not least, thanks is extended to all of those who had directly and
indirectly helped in this project.
v
ABSTRACT
In construction industry, misinterpretation of detail drawings is likely to
occur in a tight-scheduled project, leading to the non-conformance with the detail
drawings. This study is conducted on a damaged column of a real construction
project, where the as-built dimension of its stump does not comply with the detail
drawings. The stump is protruded from the wall and is hacked for aesthetic reason,
thus the strength of the column is reduced. The aim for this study is to conduct a
finite element analysis on the reinforced concrete column whose stump is damaged,
to study the behaviour of the column. The strength level and maximum hacking
allowed are determined. Non-linear analyses are performed on the column model
using LUSAS.
The accuracy of the finite element model is verified against
experimental data published. The theoretical results are also used to verify the finite
element model. From the analysis results, the load capacity, deflection and stress
contour of the column with the respected degrees of damage at stump due to hacking
are known.
Subsequently, the failure mode of the column and the maximum
hacking allowed are determined. Besides that, an equation for the particular column
is established to determine the column capacity based on the damage done to the
stump due to hacking. At the end of the study, it is found that the column having its
stump hacked is still able to sustain its design load and maintain its stability.
vi
ABSTRAK
Dalam industri pembinaan, kesilapan membaca lukisan perincian sering
berlaku disebabkan oleh kesuntukan masa pihak bertanggung-jawab.
Hal
menyebabkan kesilapan dalam pembinaan di mana pembinaan tidak sama dengan
lukisan perincian. Kajian ini dilakukan ke atas tiang dengan merujuk kepada projek
pembinaan sebenar, yakni ukuran ‘as-built’ untuk tunggul tiang tidak sama dengan
lukisan perincian. Oleh yang demikian, sebahagian daripada tunggul tiang tersebut
telah dipecahkan, dan menyebabkan kekuatan tiang tersebut telah berkurangan.
Tujuan utama kajian ini adalah untuk menjalankan analisis unsur terhingga ke atas
tiang konkrit bertetulang, bagi mengkaji kelakuan tiang tersebut dan seterusnya
mencari tahap kekuatan serta menentukan tahap pecahan maksimum yang
dibenarkan. Justeru, analisis tidak lelurus dijalankan ke atas model tiang dengan
menggunakan LUSAS. Demi menentukan kejituan analisis unsur terhingga, data
eksperimen makmal dari pihak lain telah dirujuk. Daripada keputusan analisis yang
dijalankan ke atas tiang tersebut, kapasiti beban, pesongan and kontur tegasan telah
diperolehi. Hasil analisis mod kegagalan dan tahap pecahan yang dibenarkan telah
dikenalpasti. Selain itu, satu rumus yang dapat menentukan kapasiti tiang telah
diperolehi. Akhirnya, tiang tersebut didapati masih berupaya untuk menahan beban
rekabentuk.
vii
TABLE OF CONTENTS
CHAPTER
1
2
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xi
LIST OF SYMBOLS
xiv
LIST OF APPENDICES
xvi
INTRODUCTION
1
1.1
Background
1
1.2
Problems Statement
2
1.3
Objectives of the Study
3
1.4
Scopes of the Study
3
LITERATURE REVIEW
4
2.1
Concrete
2.1.1
Stress-strain Relation in Compression of Concrete 4
2.1.2
Elastic Modulus of Concrete
6
viii
2.2
Reinforcing Steel
7
2.3
Reinforced Concrete
8
2.4
Reinforced Concrete Column
9
2.4.1
Types of Column
10
2.4.2
Column Classification and Failure Modes
10
2.4.3
Capacity of the Reinforced Concrete Column
12
2.5
Fracture
13
2.6
Geometrical Non-linearity
15
2.7
Buckling of Slender Column
16
2.8
Finite Element Method
19
2.8.1
Brief History
19
2.8.2
Formulation of the Elements
19
2.8.3
Finite Elements
19
2.8.4
Verification of Results
20
2.8.5
Basic Steps of Finite Element Analysis
20
2.9
LUSAS
21
2.9.1
22
The Iteration Procedures for Non-linear Static
Analysis
2.10
Studies done on Reinforced Concrete Column
23
2.10.1 Slender High-strength Concrete Columns
23
Subjected to Eccentric Loading
2.10.2 A Three Dimensional Finite Element Analysis
27
of Damaged Reinforced Concrete Column
3
RESEARCH METHODOLOGY
31
3.1
Introduction
31
3.2
Development of Finite Element
31
3.2.1
Model Geometry
32
3.2.2
Finite Element Meshing
33
3.2.3
Material Properties
35
3.2.4
Modelling of the Supports and the Load
36
3.2.4.1 Modelling of the Supports and the
37
Load – Cap Type I
ix
3.2.4.2 Modelling of the Supports and the
39
Load – Cap Type II
3.3
3.2.5
Non-linear Analysis Control Setting
41
3.2.6
Verification of the Results and Discussions
41
Modelling of the Undamaged Reinforced Concrete
42
Column
3.4
3.3.1
Column Geometry
43
3.3.2
Finite Element Meshing
45
3.3.3
Material Properties
47
3.3.4
Modelling of the Supports and the Loading
48
3.3.5
Non-linear Analysis Control Setting
49
Modelling of the Damaged Reinforced Concrete
50
Column
3.5
4
5
Verification of the Results and Discussion
53
RESULTS AND DISCUSSION
60
4.1.
Introduction
60
4.2.
Analysis Results
60
4.3.
Failure Mode of the Column
63
4.4.
Parametric study on the Column Capacity
72
4.5.
Column Strength Level
76
4.5.1. Maximum Hacking Allowed
76
4.5.2. Stability of the On-site Column
78
CONCLUSIONS
79
5.1
Conclusions
79
5.2
Recommendations
80
REFERENCES
82
APPENDICES
84
x
LIST OF TABLES
TABLE NO.
TITLE
PAGE
2.1
Details of columns in group A, B and C
24
2.2
The properties of the materials
28
3.1
Material properties of concrete
35
3.2
Materials properties of steel
36
3.3
Material properties concrete
47
3.4
Material properties steel
47
3.5
Section properties of the transformed section
55
4.1
Summary of the analysis results
62
4.2
Section properties of the column transformed section
67
4.3
Buckling load
68
4.4
Maximum displacement before buckling
68
4.5
Failure mode of the column (stump 20% - 70% damaged)
69
4.6
Summary of the column failure mode
69
4.7
Parameters in the study
72
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1
The damage to the column
2
2.1
Typical stress-strain curve for concrete in compression
5
2.2
Static modulus of concrete
6
2.3
Typical stress-strain curve for reinforcing steel
7
2.4
Simplified stress-strain curve for reinforcing steel
8
2.5
P-∆ effect on slender column
11
2.6
Failure modes of column
12
2.7
Forces act on the column section
13
2.8
Different Modes of Fracture
14
2.9
P-∆ effect on a column
15
2.10
Buckling of an initially crooked column
17
2.11
Bending moment in the column
18
2.12
Flow chart for the basic steps of finite element analysis
21
2.13
Forms of modified Newton-Raphson iteration
22
2.14
Geometry and details of configurations (a) and (b)
24
2.15
Load arrangement for the test
25
2.16
Instrumentation of slender column
26
2.17
The model of the round column
29
3.1
Detail of column cross section
32
3.2
Column length
33
3.3
2-dimensional bar element with quadratic interpolation
34
3.4
Plane stress element with quadratic interpolation
34
3.5
Elastic-plastic model
35
3.6
Bearing plate
36
xii
3.7
Stresses induced by eccentric point load
37
3.8
Spreading of point load into equivalent stress
38
3.9
Cap models
38
3.10
Redundant portion of cap model
39
3.11
Dimension for triangular cap
40
3.12
Models of cap
40
3.13
Graph of eccentric vertical load versus column mid-height
42
3.14
Detail of column cross section
43
3.15
Detail of column elevation
44
3.16
Geometries defined
45
3.17
Finite Element Meshing
46
3.18
Dimensions for cap model
48
3.19
Model for supports and load
49
3.20
Portion of damages at the stump due to hacking
50
3.21
Transmission length of the reinforcement
51
3.22
Column full model (60% of the stump is damaged)
52
3.23
Finite element model used for verification
53
3.24
Contour of ultimate equivalent stress
54
3.25
Column section
55
3.26
Effective length of the column
56
3.27
Comparison of results for vertical load against vertical
58
displacement
4.1
Graph of eccentric load against vertical displacement
61
4.2
Deflected shape of the column
62
4.3
Stress contour of the column having 0%-damaged stump
64
4.4
Stress contour of the column having 10%-damaged stump
64
4.5
Stress diagram of the critical section
65
4.6
Stress diagram of the critical section
66
4.7
Stress contour of the column having 50%-damaged stump
70
4.8
Elevation detail of the column
71
4.9
Graph of column capacity versus second moment of inertia
73
4.10
Graph of column capacity versus cross sectional area
73
4.11
Damaged stump section for Ae = 90924 mm2
74
xiii
4.12
Isometric view of the damaged stump
75
4.13
Cross section of the stump
77
4.14
Damage done to the column
78
xiv
LIST OF SYMBOLS
Ae
-
Transformed sectional area
As
-
Area of reinforcement
As’
-
Area of compression reinforcement
b
-
Width of column
d
-
Effective depth
d’
-
Depth to the compression reinforcement
E
-
Elasticity
Ec
-
Secant or static modulus of concrete
Es
-
Young’s modulus of steel
e
-
Eccentricity of load
Fcc
-
Concrete compression force
Fsc
-
Reinforcement compression force
Fs
-
Reinforcement tension force
fb
-
Bond stress
fbu
-
Ultimate bond stress
fcu
-
Characteristic strength of concrete
fy
-
Characteristic strength of reinforcement
h
-
Depth of column in the plane under consideration
Ie
-
Transformed section second moment of inertia
lanc
-
Anchorage length
le
-
Effective column height
Mc
-
Moment before column buckle
Mcap
-
Moment capacity
Mo
-
Moment due to eccentric load
N
-
Column design load
Ncap
-
Column capacity
Ncrushing
-
Crushing load of column
xv
P
-
Vertical load to the column
Pc
-
Buckling load
r
-
Radius of gyration
x
-
Depth to the neutral axis
α
-
Modulus ratio
β
-
Coefficient dependant on the bar type
γm
-
Partial safety factor
δ
-
Second order lateral deflection
δo
-
Maximum deviation from the straightness at mid-height
δy
-
Vertical displacement of column
εsc
-
Reinforcement compression strain
εs
-
Reinforcement tension strain
σ
-
Stress
φe
-
Effective bar size
xvi
LIST OF APPENDICES
APPENDIX
A
TITLE
Laboratory test results by Claeson and Gylltoft (1996)
PAGE
84
1
CHAPTER 1
INTRODUCTION
1.1.
Background
In construction industry, structural and architectural elements of a building
are detailed in separate sets of drawings. When the time allocated for a project is
short and the schedule is tight, misinterpretation of the drawings is likely to occur.
As a result, non-conformance with either one of the drawings may happen during
construction stage, leading to a conflict between aesthetic quality and structural
stability.
This study is conducted in reference to a real construction project1 where
non-conformance of architectural and structural drawing has occurred. The site
problem was initiated when a stump was cast higher than finished floor level, due to
the misinterpretation of the drawings during levelling survey work. This resulted in
the protrusion of the stump from acoustic wall surface. Hence, the stump was
hacked to provide a flat surface for the installation of the acoustic wall (see Figure
1.1).
1
The project name is not disclosed due to the request by the project owner.
2
Figure 1.1 : The damage to the column
The strength of the defected column is assumed to have reduced due to
hacking. Because the column is an important structural member of the building, a
study to determine its capacity is proposed.
1.2
Problem Statement
The type of structural defect due to hacking to the column as presented in this
study is not common. Therefore, there is no comprehensive reference available with
regards to the acquisition of the maximum capacity for the column. Also, the current
code of practice (i.e. BS 8110) does not provide any provision on the design of
structural members with openings, hence useful data and references are not available.
For the reasons stated above, analysis is required to understand the structural
behaviour of the defected column and consequently know its load bearing capacity.
The finite element method (FEM) is chosen as the analysis tool in this study, because
3
it has the advantages in the ability of predicting localised and global behaviours of a
structural member.
1.3
Objectives of the study
The objectives of the study are listed as below:
1.
To conduct a study on a reinforced concrete (RC) column using finite
element analysis, before and after the damage due to the over-hacking.
2.
To comprehend the behaviour and to determine the strength level of
the damaged RC column.
3.
To determine the maximum hacking allowed to the RC column before
failure.
1.4
Scopes of the Study
The scopes of the study are listed as below:
1.
The finite element analysis is done by using LUSAS.
2.
The linear and the non-linear analysis is done in 2-dimension.
3.
Material and geometrical non-linearity are included in the analysis.
4.
The study is based on the short-term behaviour of the concrete.
5.
Analysis is conducted on a column according to the as-built details in
the project
4
CHAPTER 2
LITERATURE REVIEW
2.1
Concrete
Concrete is a construction material consisting of fine aggregate, coarse
aggregate, cementitous binder, and other chemical admixtures. It has a very wide
variety of strength, and its mechanical behaviour is varying with respect to its
strength, quality and materials.
2.1.1
Stress-strain Relation in Compression of Concrete
Concrete has an inconsistent stress-strain relation, depending on its respective
strength. However, there is a typical patent of stress-strain relation for the concrete
regardless the concrete strength, as shown in Figure 2.1.
5
Stress
Strain
0
Figure 2.1 : Typical stress-strain curve for concrete in compression
(Arya, 2001)
When the load is applied, the concrete will behave almost elastically,
whereby the strain of the concrete is increasing approximately in a linear manner
accordingly to the stress. Eventually, the relation will be no longer linear and the
concrete tends to behave more and more as a plastic material, which in this state,
recovery of displacement will be incomplete after the removal of the loadings, hence
permanent deformation incurred.
Generally, the concrete is gaining its strength with age, but the rate is varied
depending on the admixture added to the concrete, type of cement used, etc. Usually
the increment of concrete strength is insignificant after the age of 28-day, and
therefore assumption that the concrete strength taken as its strength at the age of 28day is acceptable (Martin et al., 1989).
6
2.1.2 Elastic Modulus of Concrete
The stress-strain relationship for concrete is almost linear provided that the
stress applied is not greater than one third of the ultimate compressive strength. A
number of alternative definitions are able to describe the elasticity of the concrete,
but the most commonly accepted is E = Ec, where Ec is know as secant or static
modulus (see Figure 2.2).
Figure 2.2 : Static modulus of concrete
(Mosley et al., 1999)
BS 1881 has recommended a series of procedure to acquire the static
modulus. In brief, concrete samples in standard cylindrical shape will be loaded just
above one third of its compressive strength, and then cycled back to zero stress in
order to remove the effect of initial ‘bedding-in’ and minor redistribution of stress in
the concrete under the load. Eventually the concrete strain will react almost linearly
to the stress and the average slope of the graph will be the static modulus of
elasticity.
7
2.2
Reinforcing Steel
The reinforcing steel has a wide range of strength. It demonstrates more
consistent properties compared to the concrete, because it is manufactured in a
controlled environment.
The typical stress-strain relations of the reinforcing steel can be described in
the stress-strain curve as shown in Figure 2.3.
stress
(b) High yield steel
0.2% proof stress
(a) mild steel
0.002
strain
Figure 2.3 : Typical stress-strain curve for reinforcing steel
(Mosley et al., 1999)
Graph (a) and graph (b) in Figure 2.3 are indicating the stress-strain relation
of high yield steel and the mild steel respectively. From the graph, it can be seen that
the mild steel behaves as an elastic material until it reaches its yield point, eventually
it will have a sudden increase in strain with minute changes in stress until it reaches
the failure point. The high yield steel on the other hand, does not have a definite
yield point but shows a more gradual change from elastic to plastic behaviour.
Despite of the various strength of the materials, reinforcing steels have a
similar slope in the elastic region with Es = 200 kN/mm2. The specific strength taken
8
for the mild steel is the yield stress. For the high yield steel, the specific strength is
taken as the 0.2% proof stress (see Figure 2.3). BS8110 has recommended that the
stress-strain curve may be simplified as per Figure 2.4. The suggested stress-strain
relation is an elastic-plastic model, which the hardening effect is neglected.
Figure 2.4 : Simplified stress-strain curve for reinforcing steel
(BS8110, 1997)
2.3
Reinforced Concrete
Reinforced concrete is a strong and durable construction material that can be
formed into many varied shapes and sizes ranging from a simple rectangular column
(spanning in 1-dimensional) to a shell (spanning in 3-dimensional). Its utilities and
versatilities are achieved by means of the composite action, with the best feature of
concrete in compression and steel in tension.
The tensile strength of concrete is just about 10% of its compressive strength.
In the design of reinforced concrete the tensile strength of concrete is neglected thus
9
the tensile force is assumed to be resisted by the reinforcing steel entirely. The
tensile stress is transferred to reinforcing steel from concrete through the bonding
formed between the interface of the steel and concrete, therefore insufficient bond
will cause the reinforcement to slip within the concrete.
Reinforcing steel can only develop its strength in concrete provided that it is
anchored well to the concrete. BS 8110 has recommended formula in seeking the
anchorage bonding stress, quoted.
fb =
Fs
πφe l anc
(eq. 2.1)
The ultimate bond stress may be obtained by,
f bu = β
f cu
(eq. 2.2)
When the thermal strain is considered, the differential movement between the
reinforcing steel and the concrete will still be insignificant. This is because both the
materials have a near value of thermal expansion co-efficient.
2.4
Reinforced Concrete Column
A column in a structure transfers loads from beams and slabs down to
foundations. Therefore, columns are primarily compression members, although they
may also have to resist bending forces due to the eccentricity. Design of the column
is governed by the ultimate limit state, and the service limit state is seldom to be
considered (Mosley et al., 1999).
10
2.4.1 Types of Column
There are two types of column, namely braced and unbraced. A braced
column is the column that does not resist lateral load, because the load is resisted by
the bracing members (i.e. shear wall). An unbraced column is the column that is
subjected to lateral loads. The most critical arrangement of load is usually that
causes the largest moment and axial load.
2.4.2 Column Classification and Failure Modes
A column can be classified as short or slender by a ratio of effective height, le,
in the bending axis considered, to the column depth in the respective axis, h, (le/h).
Based on the ratio, BS 8110 has recommended that a slender column can be
determined by equation 2.3 (for braced structure) or equation 2.3 (for unbraced
structure). By knowing the column is short or slender, the failure mode of the
column can be predicted.
Braced columns:
le
≥ 15
h
Unbraced columns:
le
≥ 10
h
(eq. 2.3)
(eq. 2.4)
A short column is unlikely to buckle, hence it will fail when the axial load
exceeded its material strength. This will cause the column to bulge and finally to
crush.
11
Buckling is a failure mode characterised by a sudden failure of a structural
member. Buckling may cause a slender column to deflect sideways and thus induces
an additional moment, M = P δ, as illustrated (P-∆ effect) in Figure 2.5.
P
δ
P
Figure 2.5 : P-∆ effect on slender column
(Mosley et al., 1999)
In brief, the column may fail in 3 modes (Mosley et al., 1999), namely,
1.
Crushing - Material failure with negligible lateral deflection, which
usually occurs in short columns. However, it is possible to occur
when there are large end moments acted on a column with an
intermediate slenderness ratio.
2.
Intermediate - Material failure intensified by the lateral deflection and
the additional moment. This type of failure is typical of intermediate
columns.
3.
Buckling - Instability failure which occurs with slender columns and
it is liable to be preceded by excessive deflections.
12
Figure 2.6 illustrates the failure mode of columns
(a)
(b)
Figure 2.6 : Failure modes of columns a) column failure in crushing b) column
failure in buckling
2.4.3
Capacity of the Reinforced Concrete Column
In the practical case of cast in-place concrete structure, moment exists in the
short column that supports an approximately symmetrical arrangement of beams. The
moment for the case is small, but cannot be neglected.
Thus, BS 8110 has
recommended that the capacity of the column, N should be expressed as,
N = 0.35 f cu bh + (0.70 f y − 0.35 f cu )Asc
(eq. 2.5)
The capacity of columns that are subjected to moments and axial forces can
be acquired by performing analysis on its cross section (see Figure 2.7).
13
Figure 2.7 : Forces act on the column section
(Mosley et al., 1999)
From Figure 2.7, the basic equation for axial force capacity, N, and moment
capacity, M, can be derived, as follows:
N = Fcc + Fsc + Fs
= 0.405 f cu bx + f sc A' s + f s As
h⎞
⎛ h 0.9 x ⎞
⎛h
⎞
⎛
M = Fcc ⎜ −
⎟ + Fsc ⎜ − d ' ⎟ − Fs ⎜ d − ⎟
2 ⎠
2⎠
⎝2
⎝2
⎠
⎝
2.5
(eq. 2.6)
(eq. 2.7)
Fracture
The understanding on fracture is important to predict the occurrence of crack
on the concrete. It is a complex process that involves the nucleation and growth of
micro and macro voids or cracks, mechanisms of dislocations, flip bands,
propagation of micro-cracks, and the geometry of the concrete.
14
Concrete materials degrade by initiation and propagation of microcracks in
the heterogeneous microstructure. At regions of high stresses, the concrete will
suffer from microscopic damage, thus forming microcracks. Therefore, a process
zone will be formed in the concrete.
It is the zone where the interaction and
coalescence of microcracks taking place, and eventually leads to localization of a
macroscopic discontinuity.
The process zone formed in the concrete is of significant dimension,
compared to the member geometry. Hence, it cannot be predicted with the classical
linear elastic fracture mechanics. Over the past few decades, theoretical approaches
that are both physically sound and practically applicable have been developed for
fracture analysis in concrete and concrete structures.
Fractures can be taking place in three basic modes, as shown in Figure 2.8.
Figure 2.8 : Different Modes of Fracture
In mode I, the forces are perpendicular to the crack, pulling it to open. Thus,
the mode I fracture is also called as the opening mode. The flexural crack at the
bottom of beams at mid-span is an example of the fracture mode. In mode II, the
forces are parallel to the crack. In the same direction, one force is pushing the top
half of the crack backward and the other is pulling the bottom half of the crack
15
forward. Hence, the crack is sliding along itself and a shear crack is formed.
In mode III, the forces are perpendicular to the crack. The forces are moving
in opposite directions of left and right, in order to grow the crack in front-back
direction. The mode of fracture is called out-of-plane-shear, because the material
will be separate and slide along itself, moving out of its original plane.
2.6
Geometrical Non-linearity
Geometric nonlinearities arise from significant changes in the structural
configuration during loading. The geometrical non-linearity is the main factor that
causes the second order effect (P-∆ effect).
P
δ
Figure 2.9 : P-∆ effect on a column
(LUSAS Modeller User Manual)
16
A loaded column as illustrated in Figure 2.9 is referred. The linear solution
would fail to consider the bending moment due to the progressive eccentricity, δ, of
the vertical load, P. Depending on how large the deflections are, serious errors could
be introduced if the effects of nonlinear in geometry are neglected. Hence,
consideration in geometrical non-linearity is required for knowing the actual
response of the column to the load.
2.7
Buckling of Slender Column
An assumption can be made on the short column, which the failure of the
column will be due to crushing. However, exemption might occur, and the column
that is not satisfying the condition is classified as a slender column.
When loaded at the centroid, the collapsing load for a slender column can be
determined by the Euler Buckling Load, Pc
Pc =
π 2 EA
⎛ kL ⎞
⎜ ⎟
⎝ r ⎠
(eq. 2.8)
2
In reality, loads are seldom loaded on the column centroid, and a moment is
likely to present. The moment may react due to the small eccentricity or because of
the initial crookedness of the column. Suggested by Meyer (1996), the initial column
shape before the application of load can be approximated by the sine function
y o = δ o sin
πx
L
(eq. 2.9)
17
Where δo is the maximum deviation from the straightness at mid-height (see
Figure 2.10). Also, the maximum displacement due to buckling will be at the midheight, and can be expressed as
y max =
δo
P
1−
Pc
(eq. 2.10)
Figure 2.10 : Buckling of an initially crooked column
(Meyer, 1996)
The P-∆ effect must be considered in the buckling. Therefore, when the
column reaches its instability state, the bending moment taking place in the column
will be as illustrated in Figure 2.11.
18
Figure 2.11 : Bending moment in the column
Thus, the maximum moment before the column buckle reads
Mc =
1
P
1−
Pc
Mo
(eq. 2.11)
Where Mo = P e is the moment due to the load eccentricity, e, and the factor
1/[1-(P/Pc)] is the moment magnification factor. From the equation of the maximum
moment, the buckling moment will grow unbounded when the column load reaches
the buckling load.
19
2.8
Finite Element Method
2.8.1
Brief History
Finite element is a numerical method to solve the engineering problems. It is
a very powerful method in performing linear and non-linear problem in structural
analyses. Basic ideas of finite element methods originated from advances in air-craft
structural analysis. At 1956, Turner et al. had presented their findings on the stiffness
matrices for beam, truss and other elements. The very first attempt to analyse the
reinforced concrete by finite element was done by Ngo and Scordelis (1967). Later,
mathematical foundations were laid in the 1970s, and the knowledge of finite
element method is progressively developed.
2.8.2
Formulation of the Elements
There are three approaches that are usually applied in formulating the
elements, namely the direct formulation, minimum total weight potential energy
formulation, and weighted residual formulation (Moaveni, 2003).
2.8.3
Finite Elements
A finite element is a subregion of a discretized continuum. It is of a finite
size (not infinitesimal) and usually has a simpler geometry than that of the continuum
(Boukais, 1989).
20
2.8.4
Verification of Results
As discussed by Moaveni (2003), following to the rapid growth of finite
element in recent years, finite element analysis software has become a common tool
in the hands of design engineers. The results of the finite element analysis have to be
verified so that it does not contain errors as listed below.
1.
Wrong input data, such as physical properties and dimensions
2.
Selecting inappropriate types of elements
3.
Poor element shape and size after meshing
4.
Applying wrong boundary conditions and loads
One must always find ways to check the results. While experimental testing
of the model may be the best way to do so, but it may be expensive and time
consuming. Therefore, it is always a good practice to start by applying equilibrium
conditions and energy balance to different portions of a model to ensure that the
physical laws are not violated.
2.8.5
Basic Steps of Finite Element Analysis
Basic steps for the finite element analyses are illustrated as a flow chart in
Figure 2.12.
21
Pre-processing Phase
1. Create and discretize the solution domain into finite elements; that
is, subdivide the problem into nodes and elements.
2. Assume a shape function to represent the physical behaviour of an
element; that is, a continuous function is assumed to represent the
approximate solution of an element.
3. Develop equations for an element.
4. Assemble the elements to present the entire problem. Construct the
global stiffness matrix.
5. Apply boundary conditions, initial conditions and loading.
Solution Phase
6. Solve a set of linear or non-linear algebraic equations
simultaneously to obtain nodal results, such as displacement values
at different nodes.
Post-processing Phase
7. Obtain other important information. (i.e. principle stresses)
Figure 2.12 : Flow chart for the basic steps of finite element analysis
(Moaveni, Saeed, 2003)
2.9
LUSAS
LUSAS is an associative feature-based modeller. The model geometry is
entered in terms of features which are sub-divided (discretized) into finite element in
order to perform the analysis.
22
2.9.1 The Iteration Procedures for Non-linear Static Analysis
LUSAS applies the modified Newton-Raphson iteration in the non-linear
static analysis.
Newton-Raphson iteration is stable and converges quadratically, provided an
initial estimation which is close enough to the solution is available. Besides that, the
tangent stiffness matrix needs to be inverted during each iteration. Because of the
complex iteration procedure, it may fail to converge when extreme material
nonlinearities are present in a structure.
Hence, modification was done on Newton-Raphson iteration.
With the
modified Newton iteration, the current tangent stiffness matrix is replaced with a
previous stiffness matrix. This reduces the number of iteration as the factorisation of
the tangent stiffness matrix is not required for every iteration. Three common forms
of modified Newton-Raphson iteration are illustrated in Figure 2.13.
23
Figure 2.13 : Forms of modified Newton-Raphson iteration
(LUSAS Modeller User Manual)
2.10
Studies done on Reinforced Concrete Column
There are few studies conducted by researchers to comprehend the behaviour
of the reinforced concrete column. Reviews are done by studying the article and
material from the credible sources.
2.10.1 Slender High-strength Concrete Columns Subjected to Eccentric
Loading
An experimental study is done by Claeson and Gylltoft (1998). The research
is to study the behaviour of slender reinforced concrete columns. In the research, 12
full-scale reinforced concrete columns were tested to failure under the eccentric load.
The research was then followed by performing numerical simulations to verify with
the test results.
24
The detailing and configuration of the column cross section is as indicated in
Figure 2.14 and as tabulated in Table 2.1.
Figure 2.14 : Geometry and details of configurations (a) and (b)
(Claeson and Gylltoft, 1998)
Table 2.1 : Details of columns in group A, B and C
(Claeson and Gylltoft, 1998)
Column
Length
group
(mm)
A
2400
B
C
Configuration
Eccentricity
Stirrup spacing
(mm)
(mm)
a
20
100/180
3000
b
20
130/240
4000
b
20
130/240
There were 4 test sample prepared for each group of column. In each group,
2 columns were cast by normal-strength concrete and the other 2 columns were cast
by high-strength concrete. The materials for the columns were tested for acquiring
the data on its mechanical properties (i.e. elasticity, strength, etc.).
Before the test, both the ends of the column were attached to a bearing plate
to produce a good pinned support to the column. Figure 2.15 is illustrating the load
arrangement for the test. It can be seen that the column are supported and loaded
25
with an eccentricity of 20mm.
Figure2.15 : Load arrangement for the test
(Claeson and Gylltoft, 1998)
Sensors consist of different types of gauges were attached to the test sample
at different height level, as indicated in Figure 2.16. The gauges were used to
measure the lateral displacement of the column, as response to the vertical eccentric
load.
26
Figure 2.16 : Instrumentation of slender column
(Claeson and Gylltoft, 1996)
Results acquired from the laboratory test are attached in Appendix A.
The research is continued with the finite element modelling. The objective
was to develop a non-linear finite element model that could simulate the failure
mechanism of the column. ABAQUS was used to perform the analysis. A 3-noded
3-dimensional hybrid beam element was developed for the model of the columns.
The reason for this is because it enabled the analysis to run in a reasonable amount of
time. Also, a similar study done by Claeson (1995) has concluded that, there is a
good agreement between the analysis using beam elements and one using solid
elements.
During the modelling, the program ABAQUS combines the standard
elements of plain concrete with a special option, called the rebar. The option
27
strengthens the concrete in the direction chosen, thereby simulating the behaviour of
a reinforcement bar. By the approach, the material behaviour of the plain concrete is
taken into account independently of the reinforcement.
In order to simulate the behaviour of the materials in plastic state, constitutive
models for the concrete and the reinforcement were developed. The concrete model
was developed by the smeared crack approach, and the reinforcement was modelled
by a linear elastic-plastic model.
The research has concluded that the high-strength concrete columns fail in a
brittle manner, and the spacing of the links does not affect the column ultimate
strength. Secondly, the failure mode of the column is depending on the eccentricity
of the vertical load. When the eccentricity of the load is low, the material crushing
strength played a dominant role. When the load eccentricity is increased, the yielding
of the compressive reinforcement determined the column capacity. Thirdly, the
maximum lateral deflections at the mid-height of the columns were almost the same.
Finally, the column strength is very much depending on the eccentricity of the
vertical load.
2.10.2 A Three Dimensional Finite Element Analysis of Damaged Reinforced
Concrete Column
A numerical study is done by Boukais (1989). The investigation was to
comprehend the structural behaviour of damaged reinforced concrete columns due to
spalling. A stress analysis on the column cross section was conducted. The purpose
was to understand the stress distribution of the column section which is neighbouring
to the cut-out due to the spalling. The research is a pure computer analysis study by
using PAFEC finite element package. In the study, three parameters were involved,
28
namely the height, width and depth of the cut-out. Among the parameters, only the
height of the cut-out is the variable.
The study was conducted in three parts. The first part is the development of a
3-dimensional model. The study was done based on a circular column of a bridge.
The dimension of the column is 5000mm height and 500mm in radius. An initial
loading of 20 N/mm2 is assigned to the top surface of the column. The properties of
the materials are listed in Table 2.2.
Table 2.2 : The properties of the materials
(Boukais, 1989)
Young’s
Material
Modulus, E
2
(N/mm )
Poisson’s
Mass Density
Diameter
Ratio, υ
(N/mm3)
(mm)
Steel
209 x 103
0.3
78 x 106
20
Concrete
30 x 103
0.2
24 x 106
1000
In the modelling, the reinforcement was modelled as a beam element, and the
concrete was modelled as a solid element. Perfect bond between the reinforcement
and concrete was assumed. Hence, the nodes of the solid element and the beam
element were superposed. Besides that, the study recommended the use of fine mesh
in zones where stresses are likely to vary rapidly. Furthermore, the use of elements
with higher node number is advised. The model of the circular column is illustrated
in Figure 2.17.
29
The cut-out
portion
Concrete
modeled by
solid element
The exposed
reinforcement
by beam
element
Depth
Height
(a)
(b)
Figure 2.17 : The model of the circular column a) side view b) dimension of the
cut-out portion (Boukais, 1989)
The second part of the study is to perform the analysis on the column with
cut-out. From the results obtained, the general form of load distribution and stress
diffusion in the region of the cut-out were studied. Hence, the critical zone in the
concrete can be determined.
30
Finally, the third part of the study is to analyse the buckling of the
reinforcement. When the concrete cover is corroded, the reinforcement will be
exposed. Hence, composite reaction between the concrete and the reinforcement will
be voided. Studies were conducted to understand the level of strength that the
exposed reinforcement was able to provide, against the compressive load to the
column.
The aim of the analysis was to predict the allowable length of the
reinforcement, corresponding to the height of the cut-out before the reinforcement
started to buckle.
The accomplishment of the study enabled the following conclusions to be
made. Firstly, 3-dimensional finite element analysis consumes considerable amount
of resources (i.e. hard disk memories, etc.). Secondly, superposition of the nodes of
the solid element and the beam element is able to provide a realistic prediction on the
composite action between the concrete and the reinforcement. Thirdly, the program
PAFEC can only deal with materials possessing the same stress-strain relation in both
tension and compression. Hence, it is advisable to only consider the compression
strength of the concrete, when the program is used. Fourth, finer mesh should be
used where changes in section geometry occur, because this is likely to be the zone
where the stresses varying rapidly. Finally, when the cut-out due to the spalling is
having a depth-to-height ratio lesser than 1.75, some areas in the concrete will be
doubly stressed.
31
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
The research methodology in general can be divided into two main parts,
namely the development of model and verification of analysis results.
3.2
Development of Preliminary Finite Element Model
Based on the test configuration by Claeson and Gylltoft (1998), a finite
element model is developed to ensure its accuracy and the reliability.
LUSAS Example has suggested that a reinforced concrete member can be
modelled by plane stress modelling.
Thus, the modelling method suggested is
referred and some modifications are made. A perfect bond between concrete and
reinforcement is assumed.
Hence, the nodes of the reinforcement model are
32
superposed with the nodes of the concrete model.
3.2.1 Model Geometry
The geometry of the model is based on the details of the tested column, which
is illustrated in Figure 3.1 and Figure 3.2.
Figure 3.1 : Detail of column cross section
(Claeson and Gylltoft, 1998)
33
4000mm
Figure 3.2 : Column length
(Claeson and Gylltoft, 1998)
3.2.2 Finite Element Meshing
The 2-dimensional bar element (see Figure 3.3) with quadratic interpolation
is selected for meshing the reinforcement model. Number of division for the line
meshes is set to be 4, for controlling the aspect ratio of the concrete model.
34
Figure 3.3 : 2-dimensional bar element with quadratic interpolation
(LUSAS Theory Manual)
The plane stress elements with quadratic interpolation are selected for the
surface mesh of the concrete model (see Figure 3.4). Besides that, the surface mesh
is set to be a regular mesh with automatic divisions.
Figure 3.4 : Plane stress element with quadratic interpolation
(LUSAS Theory Manual)
LUSAS Theory Manual emphasized that the aspect ratio of the surface
meshes should not be more than 10. Hence, the line mesh divisions are used to
control the aspect ratio of the surface mesh, because the nodes of the line and surface
meshes are superposed.
35
3.2.3 Material Properties
The concrete is defined as an isotropic material in the modelling. In LUSAS,
few material models are available for simulating the behaviour of the concrete in the
plastic stage. This is because the column is principally loaded in compression, and is
likely to crush. Tabulated in Table 3.1 are the properties of the concrete.
Table 3.1 : Material properties of concrete
Properties
Value
Secant Modulus (Elasticity)
Compression strength
Poisson’s Value
Tensile Strength
Strain at peak compressive stress
Ultimate compressive strain
Fracture energy
28.75 kN/mm2
37 N/mm2
0.2
3.7 N/mm2
0.0022
0.0035
0.13 N/mm2
The reinforcing steel is also been defined as an isotropic material. The elasticplastic model of stress-strain curve is selected for the steel, by neglecting the
hardening effect of the material (see Figure 3.5).
Figure 3.5 : Elastic-plastic model
36
Von Misses stress resultant model is assigned for simulating the behaviour of
the reinforcement model in plastic stage. Tabulated in Table 3.2 are the properties of
the reinforcing steel.
Table 3.2 : Materials properties of steel
Properties
Young’s Modulus
Yielding stress
Poisson’s Value
Value
207 kN/mm2
636 N/mm2
0.3
3.2.4 Modelling of the Supports and the Load
In the laboratory test done by Claeson and Gylltoft (1998), both ends of the
column are attached to a plate with a bearing. The bearing on the plate is having an
eccentricity of 20mm (see Figure 3.6). This means that when compressed, the
Figure 3.6 : Bearing plate
(Claeson and Gylltoft, 1998)
37
column is suffering from a combination of compressive force and buckling moment.
Eventually, the stress transferred to both ends of the column will not be uniform, and
it is rather trapezoidal in shape (see Figure 3.7). To simulate the actual condition as
explained, several types of model for the support and eccentric load are done.
Figure 3.7 : Stresses induced by eccentric point load
(Wang et.al., 2007)
3.2.4.1 Modelling of Supports and Load – Cap Type I
In type-I cap model, a rectangular cap is modelled to the top and the bottom
of the column model. The cap model is intended for spreading the point load into the
trapezoidal distributed stress. An excessively high value of strength and elasticity is
assigned to the cap model, so that it behaves almost as a rigid component. The
reason for this is to minimize any effect given by the cap model, against the
acquisition of a good result.
The rectangular cap model is expected to spread the point load with an angle
not more than 45o. The dimension of the cap is as illustrated in Figure 3.8.
38
Figure 3.8 : Spreading of point load into equivalent stress
Subsequently, an initial point load of 10 kN is loaded on the cap located at the
top of the column with an eccentricity of 20mm (see Figure 3.8).
Figure 3.9
illustrates the completed models of the cap.
The eccentric
point load
Fixed in
x-direction
The cap
Pinned
support
(a)
(b)
Figure 3.9 : Cap models a) at the bottom of the column b) at the top of the column
39
3.2.4.2 Modelling of Supports and Load – Cap Type II
A triangular cap is modelled to the top and the bottom of the column model
for the type-II cap model. In this case, the whole cap model is utilised to spread the
point load into the stress. This is because the present of the redundant portion (see
Figure 3.14) of the rectangular cap might alter the path of the stress distribution.
Point load
20mm
Redundant portion
of the cap model
Figure 3.10 : Redundant portion of cap model
In order to ensure the cap model is not giving significant effect to the
analysis results, it is modelled to be nearly a rigid component.
By doing so,
excessively high value of elasticity and strength is assigned to the cap model. Figure
3.11 indicates the dimension for the triangular cap model.
40
Figure 3.11 : Dimension for triangular cap
An initial load of 10 kN is assigned on the tip of the cap located at the top of
the column. In addition, a pinned support is assigned to the tip of the cap at the
bottom of the column. Figure 3.12 is illustrating the completed cap models.
Eccentric
point load
Fixed in
x-direction
Cap model
Pinned
support
(a)
(b)
Figure 3.12 : Models of cap a) at bottom of column b) at top of column
41
3.2.5
Non-linear Analysis Control Setting
The option “fine integration for stiffness and mass” is selected for the
analysis solution. This function is monitoring the iterative analysis to ensure no
element mechanisms are induced when the material yields. Besides that, the “Total
Lagrangian Geometrical Non-linearity” is activated.
This is to ensure that the
geometrical non-linearity is put into consideration during the analysis.
3.2.6
Verification of Preliminary Model Results and Discussions
The verification of the modelling is done by comparing the results from the
laboratory test with the finite element analysis. The aim for the verification is to
ensure that no mistake is made during the modelling. Besides that, verification of the
results ensures the reliability of the finite element analyses.
The plot shown in Figure 3.13 compares the results from the laboratory test
and the finite element analysis.
42
1000
900
Laboratory
Cap type I
800
Cap type II
Load
Vertical Load (kN)
700
600
500
L/2
400
Displacement
300
L/2
200
100
0
0
5
10
15
20
25
30
35
40
Column Mid-height Horizontal Displacement (mm)
Figure 3.13 : Graph of eccentric vertical load versus column mid-height lateral
displacement
From the graph, it can be seen that the analysis results are near to the
laboratory test result before the model failed. However, the case II modelling can
provide a more precise result until the model fails. To conclude, modelling of
reinforced concrete column should be done as the type II model.
3.3
Modelling of the Undamaged Reinforced Concrete Column
The finite element modelling method as discussed previously is expanded to
account for the damaged stump of the reinforced concrete (RC) column. Assumption
is made that the concrete and reinforcement is bonded perfectly, hence the nodes of
the bar elements and the plane stress elements are superposed.
43
3.3.1 Column Geometry
The geometry of the model is based on the details of the RC column, which is
illustrated in Figure 3.14 and Figure 3.15.
Figure 3.14 : Detail of column cross section
44
1F Level
Ground Level
Figure 3.15 : Detail of column elevation
The finite element model of the column is shown in Figure 3.16.
45
Figure 3.16 : Geometries defined
3.3.2 Finite Element Meshing
2-dimensional bar elements are selected for meshing the reinforcement
model, and plane stress elements are selected for meshing the concrete model (see
Figure 3.17). As illustrated in Figure 3.17, the surfaces neighbouring to the columnstump connection is meshed with finer and irregular meshes. This is because stresses
at the highlighted zone are likely to vary rapidly.
46
Finer mesh assigned
Figure 3.17 : Finite element meshing
47
3.3.3
Material Properties
The concrete is defined as an isotropic material in the modelling. In addition,
Cracking Concrete with Crushing concrete model is selected. Tabulated in Table 3.3
are the properties of the concrete from various sources.
Table 3.3 : Material properties of concrete
Properties
Secant Modulus (Elasticity)
Compression strength
Poisson’s Value
Value
References/ Sources
20.13 kN/mm2
30.67 N/mm2
0.2
BS 8110: Part 2: 1985
Cube tests result
LUSAS
Modeller
User
Manual
Mosley, W.H. et. al. (1999)
LUSAS
Modeller
User
Manual
BS 8110: Part 2: 1985
LUSAS
Modeller
User
Manual
Tensile Strength
Strain at peak compressive stress
3.07 N/mm2
0.0022
Ultimate compressive strain
Fracture energy
0.0035
0.13 N/mm2
The reinforcing steel is defined as an isotropic material. The behaviour of the
steel in the plastic stage is determined by the Von Misses type of stress resultant
model. In order to simplify the stress-strain curve for the steel, the hardening effect
of the steel is neglected. Tabulated in Table 3.4 are the material properties for the
reinforcing steel.
Table 3.4 : Material properties of steel
Properties
Young Modulus
Yielding stress
Poisson’s Value
Value
References/ Sources
200 N/mm2
555 N/mm2
0.3
BS 8110: Part 1: 1985
Mill cert
LUSAS Modeller User Manual
48
3.3.4 Modelling of the Supports and the Loading
The modelling of the supports and the load is referred to the Type II model as
discussed previously. At the top of the column, the cap for converting the eccentric
point load into the trapezoidal stresses is modelled according to the dimensions
indicated in Figure 3.18.
Figure 3.18 : Dimensions for cap model
An initial point load of 100 kN is assigned to the tip of the cap. The point
load is eccentric because it is recommended by BS 8110, to signify the potential
effect of buckling to the column.
According to the standard, the eccentricity
suggested is 5% of the column width. Thus, the point load is loaded on the column
with an eccentricity of 25mm.
In order to model the boundary condition of the model, a roller fixed in xdirection is modelled at the top of the column model. The aim is to simulate the
constraint given by the beam at the first floor level. Besides that, the pinned support
is assigned to the line at the bottom of the stump model. The pin supported line is to
represent the constraint given by the pile cap-stump connection. Pinned support is
sufficient to give full constraint to the base of the model, because the nodes of plane
stress elements posses only 2 nodal degree of freedom.
49
Figure 3.19 illustrates the models for the load and supports.
The eccentric
load
The cap model
Roller fixed in
x-direction
(a)
Pinned support
assign to the line
(b)
Figure 3.19 : Model for supports and load a) at top of column b) at bottom of
column
3.3.5 Non-linear Analysis Control Setting
The options “fine integration for stiffness and mass” and “Total Lagrangian
Geometrical Non-linearity” which are discussed previously, is selected for the
analysis solution.
50
3.4
Modelling of the Damaged Reinforced Concrete Column
The model of the damaged column is done to comprehend its behaviour in
response to the eccentric load. The damage due to the hacking work is modelled by
cutting away some portion of section from the undamaged stump model. Figure 3.20
shows the cross section of the stump in various degree of damage.
Damaged
portion
(a)
(c)
(b)
Damaged
portion
(d)
51
(e)
Damaged
portion
(f)
Figure 3.20 : Portion of damages at the stump due to hacking
a) 70%
b) 60%
c) 50% d) 30% e) 20% f) 10%
The modelling method for the damaged column is same as the undamage
column modelling, which is explained previously.
However, the reinforcement
model will stop at 600mm before the cut out (see Figure 3.21).
Well developed
bonding strength
The reinforcement
is developing
bonding strength
Anchorage
bond length
of 600 mm
Cut out at the
stump
Figure 3.21 : Transmission length of the reinforcement
The reinforcement requires transmission length to develop the interfacial
bond with the concrete. In BS8110, it is called the anchorage bond length, and the
value suggested is 580mm. The value is rounded up as 600mm in the modelling.
52
The basic principle for the modelling is that a perfect bond is assumed
between the concrete and the reinforcement.
Therefore only the fully bonded
reinforcement is modelled. Because the interfacial bond is still developing at the
portion of the reinforcement 600mm before the cut out, therefore it is not included
into the model.
Illustrated in Figure 3.22 is a sample of the developed model for the damaged
RC column.
Figure 3.22 : Column full model (60% of the stump is damaged)
53
3.5
Verification of the Results and Discussion
An examination on the behaviour of the RC column model is conducted to
ensure that the model for the column behaves as in the actual. Thus, a column model
is developed without the stump portion. The height of the column is set to be 10 m.
The bottom of the model is fixed, and a roller (fixed in x-direction) is assigned to the
top of the model. An initial load of 100 kN is loaded to the central of the column, in
the form of uniformly distributed load (UDL) (with magnitude of 200 N/mm).
Figure 3.23 illustrates the model developed.
Uniformly
distributed stress
(red arrows)
Roller fixed in xdirection (blue arrows)
Pinned supports
Figure 3.23 : Finite element model used for verification
54
The analytical results are required to verify the model, thus classical analysis
is done on the column. Firstly, the transformed (equivalent to the concrete) section
properties of the column are determined. By studying the contour of ultimate
equivalent (Von Mises) stress in the concrete (see Figure 3.24), it is known that the
column section is uncracked.
Figure 3.24 : Contour of ultimate equivalent stress
55
The section properties for the uncracked transformed section (see Figure
3.25) are tabulated in Table 3.5.
As
αs As
Centroidal Axis
As
(a)
αs As
(b)
Figure 3.25 : Column section a) cross section b) transformed section
Table 3.5 : Section properties for the transformed section
Properties
Value
Modulus ratio
9.9354
Second moment of inertia, I
5.7633 x 109 mm4
Cross sectional area, A
2.8744x105 mm2
Radius of gyration, r
141.6 mm
The boundary condition of the model can be simplified as in Figure 3.26,
hence the effective length factor, k = 0.7.
56
Figure 3.26 : Effective length of the column
(Meyer, 1996)
The buckling load of the column can be calculated as,
Pc =
Pc =
π 2 EA
⎛ kL ⎞
⎜ ⎟
⎝ r ⎠
(eq. 3.1)
2
π 2 × (20.13 × 10 3 )× (2.8744 × 10 5 )
⎛ 0.7 × 10000 ⎞
⎜
⎟
⎝ 141.6 ⎠
2
Pc = 2.3368 × 10 7 N
Hence,
Pc = 23368 kN
The crushing load of the column can be calculated as,
Ncrushing = fcuAc + fyAs
Ncrushing = 30.67 x 250000 + 555 x 3768
(eq. 3.2)
57
Ncrushing = 9758740 N
Hence,
Ncrushing = 9759 kN
The design load of the column can be calculated as,
N = 0.35 f cu bh + (0.70 f y − 0.35 f cu )Asc
(eq. 3.3)
N = 0.35 x 30.67 x 500 x 500 + (0.7 x 555 – 0.35 x 30.67) x 6280
N = 5055992.34 N
Hence,
N = 5056 kN
By neglecting the slendering and the buckling effect of the column, a loaddisplacement relation of a (ideal elastic) column can be derived from equation 3.4 as,
σ=Eε
(eq. 3.4)
Where,
σ=
ε=
P
A
δy
le
Hence,
δy
P
=E
A
le
(eq. 3.5)
Therefore, the function can be obtained by applying equation 3.5.
P
200 ⎞
⎛
250000 + ⎜ 3768 ×
⎟
20.13 ⎠
⎝
= 20.13
∆l
10000
58
Hence,
P = 578.61 δy
(eq. 3.6)
Where P in kN and δy in mm.
The results of the analytical analysis are compared with the finite element
analysis results, as in Figure 3.27.
Figure 3.27 : Comparison of results for vertical load against vertical displacement
The graph illustrated in Figure 3.27 shows that during the elastic stage, the
finite element graph is lapping with the linear graph of the ideal elastic column. As
the load increases, the material non-linearity is taking place. This explains why the
finite element graph gradient is gradually reducing.
When the load is increased to 8804 kN, the column model experienced a
sudden failure. This happens because the column buckling load is much greater than
59
the crushing load, thus the column model is failed in crushing. When the concrete
crushed, no ductile behaviour will be demonstrated, hence this explains the sudden
failure of the column.
In Figure 3.27, a good agreement is observed between the finite element
results and the analytical results. Hence, conclusion can be made that the results
obtained from the finite element analysis in this study is reliable.
60
CHAPTER 4
RESULTS AND DISCUSSIONS
4.1
Introduction
Verifications of the finite element analysis results were done during the
modelling, to ensure the results obtained are reliable. In this part of the investigation,
a preliminary study is conducted for having an early understanding on the column
behaviour.
Subsequently, the results are inferred, leading to the outcomes that
achieve the aims of the study.
4.2
Analysis Results
The graph of eccentric load against vertical displacement of the column is
plotted based on the analysis results acquired (see Figure 4.1). From the figure, it
can be seen that the columns are suffered from a sudden failure. The failure of the
columns initiated just after the material non-linearity takes place, and no ductile
behaviour demonstrated after since. Besides that, as the damage to the column stump
61
increases, the capacity of the column decreases as well. From the graph, the vertical
displacement of the column is increasing from 0% to 10% damage done to the stump.
However, when the damage done to the stump is equal or more than 10%, the vertical
displacement of the column is decreasing.
(b)
(a)
(c)
(d)
(e)
(f)
(g)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure 4.1 : Graph of eccentric load against vertical displacement
From the analysis results, the lateral deflected shape of the column in ultimate
state, with respect to the degree of damage done to its stump is determined ( see
Figure 4.2). From the figure, it can be observed that lateral deflection of the columns
with the 0%-damaged and 10%-damaged are relatively small. In addition, all of the
columns are having the maximum lateral deflection at the height in the range from
2000mm to 4000mm. The graph also shown that with the increasing damage to the
stump until 60%, the column lateral deflection is increasing as well. After that, the
lateral deflection of the column is decreased until 70% of the stump is damaged.
62
7000
5000
0% Damaged
10% Damaged
4000
20% Damaged
30% Damaged
3000
50% Damaged
Column Height (mm)
6000
2000
60% Damaged
70% Damaged
1000
0
-16
-14
-12
-10
-8
-6
-4
Lateral Deflection (mm)
-2
0
2
Figure 4.2 : Deflected shape of the column
The information acquired from Figure 4.1 and Figure 4.2 is then summarised
and tabulated in Table 4.1.
Table 4.1: Summary of the analysis results
Degree of Damage
Load
Maximum Vertical
Maximum Lateral
to the Stump
Capacity
Displacement
Displacement
(%)
(kN)
(mm)
(mm)
0
7611
8.23
0.01
10
7470
8.76
0.04
20
6583
7.51
2.70
30
5662
6.93
4.90
50
3394
5.01
5.01
60
2740
4.78
9.40
70
1768
3.66
3.40
63
4.3
Failure Mode of the Column
This investigation is to confirm the failure mode of the RC column with
respect to its degree of damage.
From the literature, the failure modes of a column (Mosley et al., 1999) can
be defined as:
1.
Crushing - Material failure with negligible lateral deflection, which
usually occurs in short columns. However, it is possible to occur
when there are large end moments acted on a column with an
intermediate slenderness ratio.
2.
Intermediate - Material failure intensified by the lateral deflection and
the additional moment. This type of failure is typical of intermediate
columns.
3.
Buckling - Instability failure which occurs with slender columns and
it is liable to be preceded by excessive deflections.
Among the three modes stated above, the crushing mode is the only failure
mode with the whole column section crushed.
Hence the first part of the
investigation is to determine the columns that failed in the crushing mode.
From Table 4.1, it can be observed that the columns with the 0%-damaged
and 10% damaged stump are having relatively negligible lateral deflection and
obvious shortening. Hence, the contour of equivalent (Von Mises) stress in the
columns is studied (see Figure 4.3 and Figure 4.4).
64
The critical
section
Figure 4.3 : Stress contour of the column having 0%-damaged stump
The critical
section
Figure 4.4 : Stress contour of the column having 10%-damaged stump
65
From the stress contour as illustrated in Figure 4.3 and Figure 4.4, the section
with the highest stress intensity is determined as the critical section. It can be
observed that the critical section for both the columns is located at the column
portion. Subsequently, the stress diagram from the critical section is plotted (see
Figure 4.5).
30
2
Stress in concrete (N/mm )
35
25
20
0% Damaged
15
10% Damaged
10
Concrete Crushing
Strength
5
0
0
100
200
300
400
500
Distance from the left surface of the columns (mm)
Figure 4.5 : Stress diagram of the critical section
Figure 4.5 indicates that the ultimate stress in the critical section of the
columns with the 0%-damaged and 10%-damaged stump is uniform and approaching
the concrete crushing strength. Hence, the whole column section is failed in crushing.
The second part of the investigation is to determine the column that failed in
the intermediate mode and the buckling mode.
As similar to the previous
investigation, the ultimate equivalent (Von Mises) stress contour of the columns
having its stump equal or more than 20% damaged are studied to determine the
66
critical section. The section with the highest stress intensity is identified as the
critical section. Eventually, the stress diagram of the critical sections is plotted in
Figure 4.6.
32
30
28
Stress in concrete (N/mm2)
26
24
22
20
18
16
20% of Damage
14
30% of Damage
12
50% of Damage
10
8
60% of Damage
6
70% of Damage
4
2
0
0
100
200
300
400
500
600
700
Distance from the left surface of undamaged stump (mm)
Figure 4.6 : Stress diagram of the critical section
It can be observed from Figure 4.6 that the ultimate stress in the critical
sections is increasingly uneven. The stress diagram has shown that the stresses at the
right side are decreasing, although the stresses at the left side are still approaching
the material crushing strength. As the damage to the stump is equal or more than
50%, it can be seen that the ultimate stress in the critical section is significantly
uneven. The stresses at the left side are approaching the material crushing strength,
but the stresses at the right side are approaching zero. From the observation, the
columns having the stump equal or more than 20% damaged will not fail in crushing
mode. Instead, the columns will only fail in intermediate or buckling mode.
67
In order to differentiate the failure mode among the columns, the literatures
are referred. It is found that the lateral displacement of the column can be used to
determine whether the eccentric loaded column is failed in buckling. By doing so,
the maximum displacement due to buckling (see equation 2.10) is used to compare
with the maximum lateral displacement of the column. If the value of the column
maximum lateral deflection is not closed to the maximum deflection due to buckling,
then the column is failed in intermediate mode.
Firstly, the buckling load of the columns is determined based on equation 2.8.
The section properties of the column that are required to obtain the buckling load is
tabulated in Table 4.2.
Table 4.2 : Section properties of the column transformed section
Stump
Column
Column
ultimate load,
P
(kN)
Cross section
area, Ae
(mm2)
Second
moment of
inertia, Ie
(mm4)
Cross section
area, Ae
(mm2)
Second
moment of
inertia, Ie
(mm4)
6583
435676
1.160 x 1010
287437
6.318 x 109
5662
380437
7.645 x 109
287437
6.318 x 109
3394
276197
2.771 x 109
287437
6.318 x 109
2740
220958
1.378 x 109
287437
6.318 x 109
1768
171958
6.000 x 108
287437
6.318 x 109
The section properties of either the stump or column portion are chosen to
acquire the minimum buckling load. Table 4.3 shows the value chosen and the
buckling load determined.
68
Table 4.3 : Buckling load
Degree of
Cross
section
area, Ae
(mm2)
Second
moment of
inertia, Ie
(mm4)
Radius of
Buckling
gyration, r
load, Pc
(%)
Column
ultimate
load, P
(kN)
(mm)
(kN)
20
6583
287437
6.318 x 109
148.26
67731
30
5662
287437
6.318 x 109
148.26
67731
50
3394
276197
2.771 x 109
100.16
29703
9
damage
60
2740
220958
1.378 x 10
78.97
14772
70
1768
171958
6.000 x 108
59.07
6432
Hence, the maximum displacement due to buckling is determined (see Table
4.4).
Table 4.4 : Maximum displacement before buckling
Degree of
Eccentricity,
damage
e = δo
(%)
Buckling
Max. disp. before
load, Pc
buckling, ymax
(mm)
Column
ultimate load,
P
(kN)
(kN)
(mm)
20
25
6583
67731
28
30
25
5662
67731
27
50
25
3394
29703
28
60
25
2740
14772
31
70
25
1768
6432
34
Finally, comparison can be made between the column maximum lateral
displacement and the maximum displacement due to buckling (see Table 4.5). It can
be seen that the column maximum lateral displacement is far lesser than the
maximum displacement due to buckling.
intermediate mode.
Thus, the columns are failed in
69
Table 4.5 : Failure mode of the column (stump 20% - 70% damaged)
Maximum Lateral
Max. disp. before
Failure mode
Displacement
buckling, ymax
(Intermediate /
(mm)
(mm)
Buckling)
20
2.70
28
Intermediate
30
4.90
27
Intermediate
50
5.01
28
Intermediate
60
9.40
31
Intermediate
70
3.40
34
Intermediate
Degree of damage
(%)
The failure mode of the columns is summarized in Table 4.6.
Table 4.6 : Summary of the column failure mode
Degree of damage
(%)
Failure mode
0
Crushing
10
Crushing
20
Intermediate
30
Intermediate
50
Intermediate
60
Intermediate
70
Intermediate
From the investigation done, it can be observed that the damage done to the
stump is reducing the stump cross section area and second moment of inertia. It is a
fact that the cross section area of the column determines its compressive strength,
and the second moment of inertia determines the moment capacity of the column.
Thus, the load capacity of the column is decreased followed by the increased degree
of damage to the stump (see Table 4.2).
70
As referred to equation 2.8, it can be seen that the cross section area and
second moment of inertia are the parameter in determining the column buckling load.
However, these parameters are reducing with the increased degree of damage done to
the column stump. As indicated in Table 4.3, the column buckling load is eventually
reduced and the column is increasingly vulnerable to buckling.
Based on the discussion, it is known that the cross section area and second
moment of inertia of a column determine the structural strength and stability of a
column.
Finally, the third part of the investigation is to identify the cause to the
column sudden failure. The previous discussion has concluded that the columns with
the stump having 0% and 10% damage due to hacking are failed in crushing. Hence,
the failure mode of the column explained the reason for the sudden failure.
The ultimate stress contour of the columns failed in intermediate mode is
studied (see Figure 4.7).
High stress intensity
at the corner of the
cut out
Figure 4.7 : Stress contour of the column having 50%-damaged stump
71
Figure 4.7 indicates partly of the stress contour for the column with the 50%damaged stump. From the contour, it can be seen that a high intensity of stress is
concentrated at the corner of the cut out, which is formed due to hacking. In fact, the
same phenomenon is observed from all the columns that fail in intermediate mode.
The elevation detail of the column (see Figure 4.8) indicates that the portion
of concrete is unreinforced.
Corner of the cut
out unreinforced
Figure 4.8 : Elevation detail of the column
The concrete neighbouring to the corner of the cut out is unreinforced and it
also does not possess ductile property. Thus, the concrete at the particular zone will
crush in a sudden when the stress reaches its material crushing strength. Eventually,
the column is suffered from a sudden failure, initiated by the concrete crushing from
the corner of the cut out.
72
4.4
Parametric study on the Column Capacity
In this part of the study, a parametric study is conducted on the column that is
not failed in crushing (damage to the stump is equal or more than 20%). The purpose
for this is to establish a function that is able to relate the column capacity with the
section property of the stump. Parameters selected are the column capacity and the
respected section properties of the damaged stump transformed section (see Table
4.7).
Table 4.7 : Parameters in the study
Column
capacity
(kN)
6583
Cross section area of the stump
transformed section, Ae
(mm2)
435676
Second moment of inertia of the
stump transformed section, Ie
(mm4)
1.160 x 1010
5662
380437
7.645 x 109
3394
276197
2.771 x 109
2740
220958
1.378 x 109
1768
171958
6.000 x 108
Two graphs are plotted (see Figure 4.9 and Figure 4.10) based on the data
listed in Table 4.7.
73
8000
Column Capacity, Ncap (kN)
7000
6000
5000
4000
Ncap = 0.0037Ie3 - 1.0276Ie2 + 122.28Ie + 1131.3
3000
2000
1000
0
0
20
40
60
80
100
120
8
140
4
Second Moment of Inertia of the Stump Transformed Section x 10 (mm )
Figure 4.9 : Graph of column capacity versus second moment of inertia
8000
Column Capacity, Ncap (kN)
7000
6000
5000
4000
Ncap = 0.021Ae – 1909.4
= 0.9987
R20.021I
Ncap =
e - 1909.4
3000
2000
90924
1000
0
0
50000
100000
150000
200000
250000
300000
350000
400000
450000
2
Area of the Stump Transformed Section, Ae (mm )
Figure 4.10 : Graph of column capacity versus cross sectional area
500000
74
Figure 4.9 indicates that the relation between the column strength and the
transformed section second moment can be expressed by a third order polynomial.
However, Figure 4.10 indicates that the column capacity is related to the cross
section area of the transformed section in a linear function. Comparatively, the linear
function derived from the graph in Figure 4.10 is more simple and easy to use, thus
the parametric study is concentrated on the relation between the column capacity and
the cross section area of the damaged stump transformed section.
From the graph in Figure 4.10, the interception with the x-axis reads, Ae =
90924 mm2. When the cross section area of the stump transformed section, Ae =
90924 mm2, the detail of the stump cross section is as detailed in Figure 4.11.
Portion of
concrete
hacked
Figure 4.11 : Damaged stump section for Ae = 90924 mm2
The isometric view of the damaged stump is illustrated in Figure 4.12.
75
Column
portion
Stump
portion
Figure 4.12 : Isometric view of the damaged stump
From Figure 4.12, it can be seen that at this rate of damage to the stump due
to hacking, the portion of the stump section underneath the column is totally
damaged. Thus, the column will be unsupported and its capacity, Ncap = 0kN.
From the discussion, it can be seen that the linear graph from Figure 4.10 is
having an R-square value closed to 1 (R2 = 0.9987). This means that the graph can
represent very well the relation between the column capacity and the cross section
area of the damaged stump transformed section. Besides that, the interception of the
graph to the x-axis is verified. To conclude, equation 4.1 is able to predict the
column capacity based on the cross section area of the damaged stump transformed
section. However, the use of equation 4.1 is only valid when the damage to the
stump is equal or more than 20%, and the unit for Ncap is in kN and Ae is in mm2.
Ncap = 0.021 Ae - 1909.9
(eq. 4.1)
76
4.5
Column Strength Level
4.5.1
Maximum Hacking Allowed
BS 8110 has recommended that the column capacity can be obtained as
N =0.35fcubh+(0.70fy – 0.35 fcu) As
(eq. 4.2)
Therefore, the column design strength is calculated as,
N = 0.35 x 30.67 x 500 x 500 + (0.7 x 555 – 0.35 x 30.67 ) x 3768
= 4107045 N
= 4107 kN
The required transformed section for the column to maintain the capacity, N,
can be calculated by using equation 4.1.
N = 0.021 Ae - 1909.9
4107 = 0.021 Ae - 1909.9
Ae = 286519 mm2
When Ae = 286519 mm2, the cross section of the stump is illustrated in Figure
4.13.
77
Figure 4.13 : Cross section of the stump
The degree of damage in this study is based on the percentage of the damaged
concrete cross section area at the stump, compared to the concrete cross section area
at the undamaged stump, thus it can be calculated by using equation 4.3. The
maximum degree of damage to the stump due to hacking before the column fail on
its design load can be calculated as,
Degree of damage (%) =
=
Aconcrete,damaged
Aconcrete,undamaged
× 100%
(eq. 4.3)
[(700 − 365) × 700] × 100%
(700 × 700)
= 48 %
In short, the degree of damage to the stump due to hacking must not exceed
48%.
78
4.5.2 Stability of the Column on-site
The picture of the damaged condition of the stump is illustrated in Figure
4.14.
Figure 4.14 : Damage done to the column
The model of the column having the 20%-damaged stump is actually the
model for the on-site column. Based on the previous discussion, the column will fail
in intermediate mode with a sudden collapse at the ultimate limit state. However, the
finite element analysis gives the result that the column is actually having a load
capacity of 6583 kN. Compared to the design load recommended by BS 8110 of
4107 kN, the column is still having an acceptable margin of safety.
To conclude, the column on site is able to sustain its design load and maintain
its stability. However, extra care should be given by the parties who are using the
building, to not to overload the column by means of heavy renovation work and etc.
Because when the column reaches the ultimate limit state, it will collapse in a sudden
and thus jeopardize the structural stability and integrity of the whole building.
79
CHAPTER 5
CONCLUSIONS
5.1
Conclusions
In this study, the behaviour of the column having its stump damaged due to
hacking was examined using finite element model. The finite element model was
first developed based on the test detail available in the literatures.
Upon the
calibration of the model against the test data, it was further expanded to study the
column having its stump damaged. From the results of the study the following
conclusions can be deduced:
1.
The plane stress model can be used to simulate the behaviour of the
reinforced concrete column in uniaxial bending.
2.
Superposition of the nodes of the plane stress and bar elements can
well simulate the full interfacial bond between the concrete and
reinforcement.
3.
The failure mode of the column can be known by studying the stress
contour of the critical section, and by comparing its maximum lateral
deflection to the deflection before buckling.
80
4.
A linear function that relates the remaining capacity and effective area
for the column in this study has been developed.
5.
The column having its stump damaged up to 10% will fail in crushing
mode and suffers from a sudden collapse.
6.
Damage to the column stump equal or more than 20% will cause the
column to fail in intermediate mode. The sudden collapse of the
column is initiated from the concrete crushing at the corner of the cut
out, which is formed due to the hacking.
7.
The removal of the stump section by means of hacking should not be
more than 48%, to enable the column to take the design load.
8.
The actual column capacity with the stump over-hacked as found on
site still higher than the design load, with an acceptable safety margin.
However, extra care must be given to not to overload the column
because as from the analysis, it will fail in intermediate mode with a
sudden collapse.
5.2
Recommendations
The present study opened a number of suggestions for future work
concerning the over-hacked column, listed as:
1.
Further study on the long-term behaviour of the reinforced concrete
column, mainly by the effect of creep and shrinkage of the concrete.
2.
Further investigation on the biaxial bending behaviour of the damaged
column.
81
3.
Further study on the effective length of the damaged column, to
determine the buckling capacity of the column in precise.
4.
Study on the buckling behaviour and capacity for columns or struts
which is possessing non-uniform cross sectional properties along its
height.
82
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British Standards Institution. Structural Use of Concrete. London. BS 8110: Part 2.
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Chandrupatla, Tirupathi R. and Belegundu, Ashok D. (2002). Introduction to Finite
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83
Wang Chu-Kia, Salmon, Charles G. and Pincheira, José A. (2007). Reinforced
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84
APPENDIX A :Laboratory test results by Claeson and Gylltoft (1996)