tA average contact areas for a unit crack surface in the tangential direction vs. A ...... specimens (S5 and S7) were tested under reversed cyclic loading. ...... The photographs showing cracking during the test were taken using a digital camera ...... (MPa). Failure. Load. (kN). Corresponding. Displacement. (mm). *. 6S. â. 71.
STRENGTH AND DUCTILITY OF HIGH-STRENGTH CONCRETE SHEAR WALLS UNDER REVERSED CYCLIC LOADING
by
Hooshang Dabbagh
A thesis submitted as partial fulfilment of the requirements for the degree of Doctor of Philosophy
School of Civil and Environmental Engineering The University of New South Wales Sydney, Australia
December 2005
CERTIFICATE OF ORIGINALITY
I hereby declare that this submission is my own work and to the best of my knowledge it contains no material previously published or written by another person, nor material which to a substantial extent has been accepted for the award of any other degree or diploma at the University of New South Wales or any other educational institution, except where due acknowledgment is made in the thesis. Any contribution made to the research by others, with whom I have worked at the University of New South Wales or elsewhere, is explicitly acknowledged in the thesis.
I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception or in style, presentation and linguistic expression is acknowledged.
_______________________ Hooshang Dabbagh
Dedicated to my wife, Farnaz, and my children, Koshiar and Katayoun
ABSTRACT
This study concerns the strength and behaviour of low-rise shear walls made from high-strength concrete under reversed cyclic loading. The response of such walls is often strongly governed by the shear effects leading to the shear induced or brittle failure. The brittle nature of high-strength concrete poses further difficulties in obtaining ductile response from shear walls. An experimental program consisting of six high-strength concrete shear walls was carried out. Specimens were tested under inplane axial load and reversed cyclic displacements with the test parameters investigated being longitudinal reinforcement ratio, transverse reinforcement ratio and axial load. Lateral loads, lateral displacements and the strains of reinforcement in edge elements and web wall were measured. The test results showed the presence of axial load has a significant effect on the strength and ductility of the shear walls. The axially loaded wall specimens exhibited a brittle behaviour regardless of reinforcement ratio whereas the specimen with no axial load had a lower strength but higher ductility. It was also found that an increase in the longitudinal reinforcement ratio gave an increase in the failure load while an increase in the transverse reinforcement ratio had no significant effect on the strength but influenced the failure mode. A non-linear finite element program based on the crack membrane model and using smeared-fixed crack approach was developed with a new aggregate interlock model incorporated into the finite element procedure. The finite element model was corroborated by experimental results of shear panels and walls. The finite element analysis of shear wall specimens indicated that while strengths can be predicted reasonably, the stiffness of edge elements has a significant effect on the deformational results for two-dimensional analyses. Therefore, to capture the deformation of walls accurately, three-dimensional finite element analyses are required. The shear wall design provisions given in the current Australian Standard and the Building Code of American Concrete Institute were compared with the experimental results. The comparison showed that the calculated strengths based on the codes are considerably conservative, specially when there exists the axial load.
i
ACKNOWLEDGEMENTS
I would like to express my deepest appreciation and thanks to my supervisor Associate Professor Stephen J. Foster for his continual support, guidance and encouragement throughout the course of this research project and the writing stages of the thesis. I would like to thank Professor B. V. Rangan at the Curtin University of Technology for his interest in this research and serving as co-supervisor. Special thanks to Mark D’urso for his great cooperation and assistance in undertaking the experimental tests. I am thankful to the technical and laboratory assistance provided by the staff of the School of Civil and Environmental Engineering, Heavy Structure Laboratory in Randwick (Ronald Moncay, Frank Scharfe and Saffwan Ramadan) and the Concrete Materials Laboratory in UNSW Kensington campus (Bill Terrey). Special thanks to Chris Gianopoulos in Randwick Heavy Structure Laboratory for his great assistance and managements during the experimental works. In the undertaking this research, I have been supported by a scholarship from the Iranian Ministry of Science, Research and Technology. This support is thankfully acknowledged. The laboratory testing was funded through an Australian Research Discovery Grant (Foster and Rangan 2001-2003) and this support is also gratefully acknowledged. My gratitude is extended to Shamseddin Nejadi, Kak Tien Chong, Adnan Malik, Gregory Lee and Mohammad Bazyar for their friendship and discussion which have made my study at the University of New South Wales such an enjoyable experience. Special thanks to my parents for their love and prayers that made this possible. Finally, very special thanks to my wife, Farnaz, and my children, Koshiar and Katayoun, for their many sacrifices during the development of this research. Without their love, encouragement and support, this thesis would not have been possible to complete.
ii
TABLE OF CONTENTS
ABSTRACT
i
ACKNOWLEDGEMENT
ii
TABLE OF CONTENT
iii
NOMENCLATURE
ix
CHAPTER 1
INTRODUCTION 1.1
General
1-1
1.2
Structural Characteristics of Shear Walls
1-2
1.3
Research Significance
1-6
1.4
Objectives
1-7
1.5
Scope
1-8
1.6
Organisation of Thesis
1-10
CHAPTER 2
LITERATURE REVIEW 2.1
Introduction
2-1
2.2
Constitutive Models
2-1
2.2.1
General
2-1
2.2.2
Concrete
2-2
2.2.3
Reinforcing Steel
2-14
2.2.4
Shear Transfer
2-17
2.3
Finite Element Analysis
2-35
2.3.1
Overview
2-36
2.3.2
Finite Element Analysis of RC Panels and Shear Walls
2-44
iii
2.4
2.5
Experimental Studies
2-52
2.4.1
Barda et al (1977)
2-53
2.4.2
Oesterle et al. (1975, , 1978, , 1984)
2-55
2.4.3
Maier and Thurlimann (1985)
2-57
2.4.4
Lefas et al. (1990) and Lefas and Kotsovos (1990)
2-59
2.4.5
Pilakoutas and Elnashai (1995)
2-61
2.4.6
Gupta and Rangan (1996)
2-63
2.4.7
Kabeyasawa and Hiraishi (1998)
2-66
2.4.8
Palermo and Vecchio (2002)
2-68
2.4.9
Farvashany (2004)
2-71
Summary
2-72
CHAPTER 3
FINITE ELEMENT MODELLING 3.1
Introduction
3-1
3.2
Crack Membrane Model
3-2
3.2.1
Background
3-2
3.2.2
Concepts and Relationships
3-3
3.3
Constitutive Relationships
3-10
3.3.1
Constitutive Models for Concrete
3-10
3.3.2
Constitutive Model for Reinforcing Steel
3-16
3.3.3
Constitutive Models for Interaction between Concrete
3-16
and Steel 3.4
Finite Element Procedure
3-29
3.4.1
Finite Element Formulation
3-30
3.4.2
Implementation
3-36
CHAPTER 4
NUMERICAL EXAMPLES 4.1
Introduction
4-1
4.2
Monotonic Loading
4-2
4.2.1
4-2
Shear Panels
iv
4.3
4.4
4.2.2
Shear Walls
4-10
4.2.3
Shear-Critical Reinforced Concrete Beams
4-13
Cyclic Loading
4-18
4.3.1
Shear Panels
4-18
4.3.2
Shear Walls
4-22
Conclusion
4-25
CHAPTER 5
EXPERIMENTAL PROGRAM 5.1
Introduction
5-1
5.2
Test Parameters
5-1
5.3
Test Specimens
5-2
5.3.1
Dimensions
5-2
5.3.2
Reinforcement Layout
5-4
5.3.3
Material Properties
5-7
5.4
5.5
Construction of Test Specimens
5-14
5.4.1
Formwork
5-15
5.4.2
Assembling the Reinforcement
5-18
5.4.3
Casting
5-18
Instrumentation
5-23
5.5.1
Strain Gauges
5-23
5.5.2
Linear Variable Displacement Transducers (LVTDs)
5-27
5.6
Test Set-up
5-28
5.7
Testing Procedure
5-29
5.7.1
Specimen Installation and Loading
5-34
5.7.2
Data Recording
5-35
CHAPTER 6
EXPERIMENTAL RESULTS 6.1
Introduction
6-1
6.2
Observed Response of Specimens
6-2
6.2.1
6-2
Specimen SW1
v
6.2.2
Specimen SW2
6-12
6.2.3
Specimen SW3
6-23
6.2.4
Specimen SW4
6-32
6.2.5
Specimen SW5
6-41
6.2.6
Specimen SW6
6-47
6.2
Out-of-plane Displacement
6-57
6.4
Summary of Test Results
6-58
CHAPTER 7
ANALYSIS OF EXPERIMENTAL RESULTS 7.1
Introduction
7-1
7.2
Effects of Experimental Parameters
7-1
7.2.1
Transverse and Longitudinal Reinforcement
7-1
7.2.2
Axial Load
7-6
7.3
Test Results vs. Those of Gupta and Rangan (1996)
7-7
7.4
Comparison of the Wall Strengths with Design Code Predictions
7-11
7.5
Simplified Strut-and-Tide Model by Rangan (1997)
7-14
7.6
Finite Element Analysis of Shear Wall Specimens 7.6.1
Two Dimensional Finite Element Analysis
7-15
7.6.2
Three Dimensional Finite Element Analysis
7-22
CHAPTER 8
CONCLUSIONS 8.1
Summary
8-1
8.2
Concluding Remarks
8-3
8.3
Recommendations for Future Study
8-5
REFERENCES
R-1
APPENDIX A
FINITE ELEMENT IMPLEMENTATION A.1 Type of Element
A-1 vi
A.2 Nonlinear Solution Procedure
A-3
A.3 Convergence Criteria
A-6
A.4 Program RECAP
A-7
APPENDIX B
RESPONSE OF LVDTS B.1 Specimen SW1
B-2
B.2 Specimen SW2
B-6
B.3 Specimen SW3
B-11
B.4 Specimen SW4
B-16
B.5 Specimen SW5
B-21
B.6 Specimen SW6
B-26
APPENDIX C
LOADS OF MACALLOY BARS C.1 Specimen SW1
C-2
C.2 Specimen SW2
C-5
C.3 Specimen SW3
C-8
C.4 Specimen SW5
C-11
C.5 Specimen SW6
C-14
APPENDIX D
STRAINS OF EDGE ELEMENT REINFORCEMENT D.1 Specimen SW1
D-2
D.2 Specimen SW2
D-5
D.3 Specimen SW3
D-7
D.4 Specimen SW4
D-10
D.5 Specimen SW5
D-13
D.6 Specimen SW6
D-16
vii
APPENDIX E
STRAINS OF WALL REINFORCEMENT E.1
Specimen SW1
E-2
E.2
Specimen SW2
E-5
E.3
Specimen SW3
E-8
E.4
Specimen SW4
E-11
E.5
Specimen SW5
E-14
E.6
Specimen SW6
E-17
APPENDIX F
DESIGN CODE CLAUSES FOR SHEAR WALLS F.1
F.2
F.3
Australian Standard (AS 3600, 2001)
F-1
F.1.1
Strength in Flexural
F-1
F.1.2
Strength in Shear
F-3
American Concrete Institute Code (ACI 318, 2002)
F-5
F.2.1
Strength in Flexural
F-5
F.2.2
Strength in Shear
F-5
Sample Calculation
F-7
APPENDIX G
STRUT-AND-TIE MODEL
viii
NOMENCLATURE
A
area of finite element
Ag
gross cross-sectional area of shear wall
Al
cross-sectional area of longitudinal reinforcement in shear wall
An
average contact areas for a unit crack surface in the normal direction
As
cross-sectional area of reinforcement
At
average contact areas for a unit crack surface in the tangential direction
Avs
cross-sectional area of shear reinforcement in shear wall
a
effective aggregate size
a1 , a2
Menegotto and Pinto parameters representing the Bauschinger effect on the cyclic response of steel reinforcement
B
strain displacement matrix
bf
width of flange in shear wall
sec D12
secant constitutive matrix in the 1_2 coordinate system
Dc12
constitutive matrix of the cracked concrete in the 1-2 coordinate system
D cr12
constitutive matrix of crack in the orthotropic 1-2 coordinate system
D cts
constitutive matrix of concrete tension stiffening
Dmax
maximum aggregate size
Dsec nt
secant constitutive matrix in the n - t coordinate system
Ds
constitutive matrix of steel reinforcement
D sc12
constitutive matrix of intact concrete between the cracks in the orthotropic 1-2 coordinate system ix
D xy
constitutive matrix of reinforced concrete element in the global X-Y coordinate system
dN
distance from extreme compressive fibre to the neutral axis of wall section
dw
effective transverse length of shear wall
E
modulus of elasticity
Ec
initial elastic modulus of concrete
Ec1 , Ec 2
uniaxial secant moduli of concrete in the orthotropic 1-2 coordinate system
Ects
secant modulus corresponding to concrete tension stiffening
Ectsx , Ectsy
secant moduli due to concrete tension stiffening in X and Y directions
Es
elastic modulus of steel reinforcement
Es sec
secant elastic modulus of steel reinforcement
E sx , E sy
secant elastic moduli of steel reinforcement in X and Y directions, respectively
Ew
strain hardening modulus of elasticity for steel reinforcement
Fc
internal compressive force due to the contribution of concrete in shear wall
Fcr
strength of shear wall corresponding to the onset of cracking
Fs
internal force resulted from all steel bars in shear wall
Fy
yield strength of shear wall
f c*
biaxial compressive strength of concrete
f c'
cylinder compressive strength of concrete
f cc
cube compressive strength of concrete
f cp
concrete peak stress
x
f ct
tensile strength of concrete
f si
stress in each layer of steel reinforcing bars
f su
ultimate stress of reinforcement
f sy
yield stress of reinforcement
G
shear modulus of elasticity
Gc
shear modulus of concrete
Gc12
secant shear moduli of cracked concrete in the 1-2 coordinate system
Gcr
shear modulus of cracked concrete
Gcr12
secant shear modulus of crack in the orthotropic 1, 2-coordinate system
Gf
fracture energy
Gsc12
secant shear modulus of intact concrete between cracks in the orthotropic 1-2 coordinate system
Hw
height of shear wall
K
stiffness of shear wall
K
stiffness matrix
Kc
stiffness matrix for concrete
Ks
stiffness matrix for reinforcement
K ( wcr )
effective ratio of contact area in crack
k
element stiffness matrix in the X-Y coordinate system
k
decay factor in the concrete stress-strain relationship
k3
factor that accounts for the difference in compressive strengths of in-situ concrete with the concrete test cylinder
Lw
length of shear wall
l
length of reinforcement throughout element
xi
lch
characteristic length over which the fracture energy is dissipated
M*
bending moment at the section of shear wall
N*
axial load at the section of shear wall
n
factor in the concrete stress-strain relationship equal to Ec ( Ecp − Ec )
pk
ratio of total volume of the aggregates over the total volume of the concrete
pl
ratio of longitudinal reinforcement in shear wall
pt
ratio of transverse reinforcement in shear wall
pw
ratio of shear reinforcement in shear wall
R
Bauschinger parameter used in Menegotto and Pinto model for steel reinforcement
R0
Menegotto and Pinto parameter representing the Bauschinger effect on the cyclic response of steel reinforcement
r
displacement ratio ( δ cr / wcr )
s
spacing of reinforcement
s rm
average crack spacing measured normal to the cracks
srmx 0 , srmy 0
average crack spacings of uniaxial tension chords in the X and Y directions, respectively
T
transformation matrix
Tε
strain transformation matrix
tf
thickness of flange in shear wall
tw
thickness of web in shear wall
V*
shear force at the section of shear wall
Vu
shear strength of shear wall
Vuc
shear strength contributed by concrete in shear wall
xii
Vus
shear strength contributed by shear reinforcement in shear wall
wcr
crack width
α
parameter to determine the rectangular stress block of compressive concrete
α1 , α 2 , α 3
softening parameters of the concrete response in tension
αd
dowel effect parameter
β
scaling factor applied to the uniaxial compressive stress-strain curve to determine the biaxial compressive stress-strain curve
βs
shear retention factor
Δy
displacement corresponding to the yield strength of shear wall
Δu
displacement corresponding to the ultimate strength of shear wall
Δ εc
total concrete strain increment
Δ ε cr
crack strain increment
Δ ε ic
intact concrete strain increment
Δσcx, Δσcy
X- and Y-component concrete stresses due to tension stiffening
δ cr
tangential displacement of crack surfaces
δs
crack slip
{ε c }
strain vector of the cracked concrete
{ε cr }
strain vector of crack
{ε sc }
strain vector of intact concrete between cracks
ε
strain
ε1u , ε 2u
equivalent uniaxial strains in the orthotropic 1-2 coordinate system
εa
concrete strain just before beginning the unloading
εc
concrete strain at the crushing taken as 0.003
xiii
ε c1 , ε c 2
average strains of concrete in the 1-2 coordinate system
* ε cp
biaxial concrete strain corresponding to the biaxial concrete strength
ε cp
strain corresponding to the concrete peak stress
ε cr1 , ε cr 2
average normal strains due to cracks in the 1-2 coordinate system
εe
elastic component of concrete strain
εm
average strain over the length of the element
ε nn , ε tt
average normal strains in the n ( t ) direction
εo
strain at the intersecting point of asymptotes in Menegotto and Pinto model for steel reinforcement
εp
plastic component of concrete strain
εr
strain at the last strain reversal in Menegotto and Pinto model for steel reinforcement
ε sc1 , ε sc 2
average strains of intact concrete between cracks in the orthotropic 1-2 coordinate system
ε sh
strain corresponding to the onset of strain hardening in reinforcement
ε to
strain corresponding to the concrete tensile strength
εy
yield strain of steel reinforcement
∅
diameter of the reinforcing bar
∅x, ∅y
diameter of the reinforcing bar in X (Y) directions
γ
parameter to determine the rectangular stress block of compressive concrete
γ c12
average shear strain of concrete in the 1-2 coordinate system
γ cr12
average shear strain due to cracks in the 1-2 coordinate system
γ nt
average shear strain in the n - t coordinate system
γ sc12
average shear strain of intact concrete between cracks in the 1-2 coordinate system xiv
η
ratio of the value of concrete strain over the concrete strain corresponding to the concrete peak stress ( ε c ε cp )
λ
ratio of tensile stresses in the concrete due to tension stiffening over the concrete tensile stress
λx , λy
ratios of tensile stresses in the concrete due to tension stiffening in the X and Y directions, respectively, over the concrete tensile stress
μ
roughness coefficient of crack interface
μd
measure of displacement ductility
μf
coefficient of friction in crack interface
μn
normal reduction factor
v12 , v21
Poisson’s ratios
θ
inclination of contact areas in concrete crack
θc
angle between major principal stress and the global X axis
θr
angle between a vector normal to the cracks and the global X axis
ρ
steel reinforcement ratio
ρx, ρy
steel reinforcement ratios in the global X- and Y-directions, respectively
ρv
ratio of reinforcement crossing the crack interface
{σ c }
stress vector of cracked concrete
{σ cr }
stress vector of crack
{σ sc }
stress vector of intact concrete between cracks
σ
stress
σa
concrete stress just before beginning the unloading
σ c1 , σ c 2
average concrete stresses in the orthotropic 1-2 coordinate system
σ cm
average stress in concrete between cracks
xv
σcn,σct
concrete stresses in the n - t coordinate system
σ con
normal contact stress in crack
σ cr
normal stress over crack surface
σ cr1 , σ cr1
average stresses over the crack surfaces in the orthotropic 1-2 coordinate system
σ ctsm
average concrete tensile stiffening stress
σ nn , σ tt
average normal stresses in the n - t coordinate system
σo
stress at the intersecting point of asymptotes in Menegotto and Pinto model for steel reinforcement
σr
stress at the last stress reversal in Menegotto and Pinto model for steel reinforcement
σ sc1 , σ sc 2
average stress of intact concrete between cracks in the orthotropic 1-2 coordinate system
σ sm
average stress in steel bar between cracks
σ s min
minimum stress in steel bar between cracks
σ sr
stress in the steel bar at the cracks
σsx, σsy
steel reinforcement stresses in the global X-Y coordinate system
σx, σy
normal stress in the global X-Y coordinate system
τb
bond shear stress
τ b0
plastic bond strength before yielding of the reinforcing steel
τ b1
plastic bond strength after yielding of the reinforcing steel
τ c12
average concrete shear stress in the orthotropic 1-2 coordinate system
τ cnt
concrete shear stress in the n - t coordinate system
τ co
focal point in the aggregate interlock relationship
τ cr
shear stress over crack surface
xvi
τ cr12
average shear stress over the crack surfaces in the orthotropic 1-2 coordinate system
τ cr max
maximum shear capacity of crack
τ nt
average shear stresse in the n - t coordinate system
τu
ultimate shear capacity due to aggregate interlock
τxy
shear stress in the global X-Y coordinate system
Ω(θ )
distribution function of the crack surface orientations
ξ
normalised strain history parameter in Menegotto and Pinto model for steel reinforcement
ψ
displacement ratio ( δ cr / wcr )
xvii
CHAPTER 1
INTRODUCTION
1.1
General
Reinforced concrete (RC) shear walls are frequently used in multistorey buildings to resist the lateral loads due to wind forces and seismic effects on buildings and the vertical loads due to dead and live loads transmitted by floors. So shear walls are subjected to axial forces, bending moments and shear forces. In practice, there are two different types of shear walls: cantilever and squat shear wall (Figure 1.1). The cantilever shear walls act as a cantilever beam and its design is usually governed by flexural behaviour. Squat shear walls are walls with a ratio of height to length less than 2 and are usually found in low-rise buildings or in the lower storeys of medium to high-rise buildings (Paulay and Priestley, 1992). The response of such walls is often strongly governed by the shear effects leading to shear induced brittle failure and tensile cracks. In Australia and other western countries using cantilever shear walls is a popular practice whilst the squat shear wall is common in Japan, China and Southeast Asia (Mau and Hsu, 1986). The use of shear walls as a lateral load resisting system emerged during the 1950s and 1960s, when the modern approach to earthquake engineering of building commenced. Over the last two decades, the improvements in material technology and in the production of high strength concrete (HSC) have resulted in a number of projects in the
(a)
(b)
Figure 1.1 - (a) Squat shear wall (b) Cantilever shear wall.
Australia, United States, Canada, Europe and Japan that have been completed using such concrete (Nawy, 2001, among many others). The HSC exhibits superior performance allowing shear walls (eg. lift core walls) to be thinner, thereby increasing the amount of rentable floor area. Nevertheless, the brittle nature of HSC being considerably greater than that of normal strength concrete (NSC), poses many difficulties for designers, particularly in obtaining ductile response from shear walls constructed using HSC and subjected to the reversed cyclic loading.
1.2
Structural Characteristics of Shear Walls
To design shear walls against seismic actions, various levels of protection including the preservation of functionality, the different degrees of damage and prevention of loss of life are generally described. In conjunction with these three levels of seismic protection the specific structural properties termed as stiffness, strength and ductility need to be considered. A typical response of a reinforced concrete shear wall subjected to monotonic loading is illustrated in Figure 1.2.
1-2
Δ F
(a) Loading and the ensuing deformation
Load Ductile failure
Fy Idealised response
0.75 Fy Fcr
Brittle failure
K 1
Δy
Δu
Displacement
(b) Load-displacement relationship
Figure 1.2 – Typical response of RC shear wall (Paulay and Priestley, 1992).
The stiffness of a shear wall relates the lateral applied load to the resulting lateral displacement. The stiffness is defined as the slope of idealised linear elastic response, K, where K = Fy / Δ y
(1.1)
as shown in Figure 1.2 where Fy is the load at yield and Δ y is the corresponding
displacement at the point of yielding. For the real load-displacement curve, K can be determined based on the effective secant stiffness at a load of 0.75 Fy (Paulay and
Priestley, 1992). The force Fcr , in Figure 1.2, is the load corresponding to the onset of cracking and Δ u is the displacement corresponding to ultimate failure of the wall.
1-3
Strength is used as a criterion to evaluate the level of protection against the damage due to seismic loading. Since the inelastic response during seismic loading is the main source of concrete damage, the shear wall must have adequate strength to resist internal actions by an elastic seismic response. Therefore, the desired strength can be expressed in terms of the force Fy derived from an elastic analysis and based on the stiffness described previously. The ability of a shear wall to resist collapse associated with sustaining large deformations is known as its ductility. Displacement ductility can be quantified by the ratio of the total imposed displacement Δ u to the displacement at yielding. That is: μ = Δu / Δ y
(1.2)
where μ is the measure of displacement ductility. Since concrete is an inherently brittle material, the reinforcing steel bars must be distributed throughout the concrete shear wall in such a way that the composite can exhibit a ductile response in order that it function under large inelastic deformations caused by severe seismic loading. Therefore, detailing of shear walls becomes an important issue for designers. The failure of a well-detailed cantilever shear wall is usually in a flexural mode similar to that of beams. However, depending on the different parameters such as geometrical dimensions, boundary conditions, the way lateral loads are imposed and the reinforcement detailing, squat shear walls may fail in any of three modes: diagonal tension, diagonal compression or sliding shear (Paulay and Priestley, 1992). The diagonal tension failure mode will occur whenever transverse reinforcement is insufficient to carry shear forces or is insufficiently detailed (Figure 1.3a). When adequate the transverse reinforcement is provided but the wall is subjected to a high shear stress, concrete may crush under diagonal compression (Figure 1.3b). This is common in the squat walls with edge elements. Finally, for walls with sufficiently detailed transverse reinforcement but low quantities of longitudinal reinforcement in the
1-4
(a) Diagonal Tension
(b) Diagonal Compression
(c) Sliding Shaer
Figure 1.3 - Shear failure modes in squat shear walls (Paulay and Priestley, 1992).
web, failure can be due to yielding of longitudinal reinforcement leading to a sliding displacement along base of the wall (Figure 1.3c). This last failure mode is particularly important for walls subjected to cyclic reversals in displacement. The cyclic behaviour of concrete shear walls shows pinched hysteretic loops with significant strength and stiffness degradation as number of cycles increases (Figure 1.4). The reduction in stiffness of the system is as a result of both the concrete cracking and reinforcement yielding. The loss of stiffness results in the hysteresis curve becoming
Load
shallower.
Envelope Curve
Displacement Pinching
Figure 1.4 – Cyclic Behaviour of Shear Walls.
1-5
The hysteresis loops also show a change of slope during the secondary loops resulting in a significant increase of stiffness during reloading, known as pinching effect (Paulay and Priestley, 1992) (Figure 1.4). This is caused by the closing of previously formed cracks during the reloading phase. This effect can significantly reduce the energy absorption at each cycle, represented by the area enclosed within each loop and thus decrease the system damping.
1.3
Research Significance
Fintel (1991) documented the superiority of shear walls, over other load resisting systems, to resist lateral forces resulting from earthquake events. This investigation as well as an extensive number of other publications (as listed by Paulay and Priestley, 1992) show that the ductility of shear walls is of paramount importance. To design a RC shear wall to behave in a ductile manner two issues are critical (Fintel, 1991). First, the wall as well as joints and other members in the building must be appropriately detailed. Second, the strength of wall is required to be governed by the flexural behaviour rather than in shear. In other words, because a shear failure is significantly less ductile compared with flexural failure it should not be permitted to occur. To achieve this, the shear capacity of a wall must be known and be larger than the shear corresponding to its moment capacity. It is also important to be understood what occurs between the beginning of shear cracking and the shear failure. However, the lack of a similar level of confidence for the shear design of walls as is presently available for the flexural design leads one to recognise an immediate need for developing an analytical and experimental understanding of shear response of shear walls (Fintel, 1991). A rational understanding of shear behaviour of concrete shear walls in general, and HSC shear walls in particular, is therefore vital. Extensive information is available with regard to design of normal strength concrete (NSC) walls. In design methods, the flexural strength and the ductility capacity may be calculated using the conventional analysis of reinforced concrete sections subjected to axial force and bending moment (Paulay and Priestley, 1992). Since HSC is more brittle
1-6
than the NSC, it is important that designers can calculate the difference in the safety between walls designed in the NSC and the HSC alternatives. However, there is a lack of the experimental data and proper models necessary for this to be accomplished. More importantly, the current models and design codes developed for the design of NSC shear walls are empirical and cannot be directly transferred to the design of HSC shear walls. To ensure that analytical models predict the behaviour of shear walls accurately, experimental data is required. Experimental data can also provide useful information for better understanding the cyclic response of shear walls. While a large number of NSC shear walls tested under monotonic and cyclic loading have been reported in the literature (Barda et al., 1977, Cardenas et al., 1980, Maier and Thurlimann, 1985, Lefas and Kotsovos, 1990, Lefas et al., 1990, Pliakoutas and Elnashai, 1995a, Palermo, 1998 among many others), only a limited number of research programs on HSC shear walls subjected to monotonic loading (Kabeyasawa et al., 1993, Gupta and Rangan, 1996, Farvashany, 2004) and cyclic loading (Kabeyasawa et al., 1993) have been carried out and many important parameters affecting the cyclic behaviour of HSC walls remain unclear. Hence, further experiments regarding the behaviour of HSC shear walls, especially under cyclic condition, are required. In summary, although shear walls provide good resistance against wind and earthquake loads, the current information about the cyclic response of such walls cast using HSC is not sufficient and further experimental and analytical information is needed so that they can be designed safely and detailed appropriately.
1.4
Objectives
The main aim of this study is the investigation of strength and ductility of HSC shear walls under reversed cyclic loading, so as to provide the analytical models and baseline experimental data needed to safely design shear walls with high strength concrete under seismic loads. The objectives and expected outcomes to reach this are as follows:
1-7
•
To design and undertake an experimental program to investigate the general response, ultimate strength and ductility characteristics of high strength concrete squat shear walls loaded in axial compression and in cyclic shear stresses.
•
To develop a finite element procedure and appropriate constitutive models for predicting the strength and ductility of reinforced high strength concrete shear walls.
•
To verify the applicability of the finite element model by comparing the results of analyses with the experimental observations available in the literature.
•
Analysis of shear walls tested as a part of this research project to determine their strength and ductility and comparing the analytical results with the experimental data.
•
To compare the strength of shear walls tested in the experimental program with the calculated ultimate strengths based on the design codes.
1.5
Scope
This study is concerned with the behaviour and strength of framed HSC squat shear walls subjected to in-plane constant vertical load and reversed cyclic loading (Figure 1.5). The research involves an analytical and an experimental program. The experimental program involves tests on wall specimens that represent approximately one-third scale models of a prototype shear wall in a multistorey building. The compressive strength of the concrete used in the walls is around 80 MPa. To investigate the shear response and strength of HSC shear walls, the specimens were designed to fail in a shear mode. The main parameters included in the study are the vertical load, the longitudinal reinforcement ratio and the transverse reinforcement ratio. Other factors such as overall dimensions of the test specimens and the reinforcement ratio of top, bottom and edge elements were kept constant. 1-8
P
V
Edge Element
Top Element
Wall
Bottom Element
Figure 1.5 - Typical shear wall specimen under loading.
The analytical program includes the development of the crack membrane model as a two-dimensional smeared-fixed crack finite element model. Then, the model is implemented into a computer program, verified against data found through the literature and used to analyse the shear wall specimens with main emphasis on the investigation of the strength and ductility of shear walls. The analysis of other cyclic aspects such as hysteretic characteristics is not included in the analytical program. The investigation of fatigue effects is outside the scope of this work. Also excluded from the scope of this thesis, while not underestimating its importance, is the bond deterioration between concrete and reinforcement.
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1.6
Organisation of Thesis
In Chapter 2, a review of the literature is undertaken on the constitutive models for reinforced concrete, the aggregate interlock effect, the finite element modelling of shear walls, and the experimental works carried out on shear walls. In Chapter 3, constitutive relationships for concrete and reinforcing steel are derived. Then a finite element model for the analysis of reinforced concrete elements in shear walls is developed. In Chapter 4, the finite element model outlined in Chapter 3 is verified by comparing the results obtained in numerical analyses with existing experimental test data over a range of reinforced concrete panels, beams and shear walls. In Chapter 5 a description of the experimental program carried out on 6 HSC squat shear wall specimens including the test parameters, material properties, specimen details, and testing is presented. Chapter 6 contains a summary of the qualitative and quantitative results of the experimental program. The main concentration is on the cyclic response, crack patterns, failure mechanism, and strength and ductility aspects. In Chapter 7, the experimental results of the shear wall specimens given in Chapter 6 are discussed. The wall strengths are also compared to those calculated based on design codes as well as strut-and tie model. Then the results obtained from the analysis of test specimens using the two-dimensional finite element procedure proposed in Chapter 3 as well as a three-dimensional finite element model are compared with the experimental data and discussed. Finally in Chapter 8 the results and outcomes of the thesis are summarised and conclusions and recommendations for future research are given. Other pertinent information is listed in the Appendices A to G at the end of the report.
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CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
In this chapter a review of the previous works carried out to investigate the structural behaviour of reinforced concrete shear walls is presented. The literature review focuses on three areas related to the main objectives of this research including constitutive modelling, the finite element analysis and experimental works. A summary of major outcomes obtained from the literature and the proposed research framework is expressed at the end of this chapter.
2.2
Constitutive Modelling
2.2.1 General The challenge in numerical modelling of reinforced concrete arises from its composite nature. The important aspects such as cracking, crushing, tension stiffening, compression softening, aggregate interlock and bond-slip cause non-linear behaviour of reinforced concrete members. Under cyclic loading, there are further complexities including stiffness degradation in concrete, the Bauschinger effect in reinforcing steel,
bond degradation, crack opening and closing and unloading and reloading process. The accuracy and reliability of numerical modelling of reinforced concrete is governed by the underlying constitutive relationships used in analysis. Over the past decades many constitutive models have been proposed to analyse reinforced concrete structures that can be classified into elasticity-based, plasticitybased, progressive damage-based, micromechanics and endocrinic models (CEB, 1996). An extensive review of these models is available in the literature (ASCE Committee 447, 1982, Chen, 1982, ASCE Committee 447, 1993, CEB, 1996). Amongst these, elasticity-based models that use a Hookean formulation in incremental form are frequently used (ASCE Committee 447, 1982, CEB, 1996). In the following, a review of the constitutive models expressed for the concrete and the reinforcing steel are presented. The emphasising is on those using in the elasticity-based approach.
2.2.2 Concrete Concrete in Compression
One of the first experimental investigations into the behaviour of plain concrete under cyclic loading was conducted by Sinha et al. (1964). The experiment was undertaken on concrete cylinders with compressive strengths from 20 to 28 MPa and subjected to repeated axial compressive loading in order to determine the main factors governing the cyclic response of concrete. To investigate the effects of load history, the load cycles were applied in two different manners including complete and partial unloading. Figure 2.1 shows the response of concrete under cyclic loading. Based on the test results, the following qualitative conclusions were drawn: •
The stress-strain paths caused by cyclic loading do not go beyond an envelope curve regardless of the previous load history. Such envelope curve can be considered unique for the each strength of concrete and expressed by stress-strain curve obtained under monotonic compressive loading to failure.
2-2
•
The unloading and reloading paths present different curves such that a hysteresis feature reflecting the energy dissipation is exhibited in each cycle. Two distinctive pattens represented by two mathematical families of curves are recognised for these curves.
•
The locus of points where the unloading and reloading curves of each cycle intersected may be defined as the shakedown limit at which the strains stabilise. Stresses above this shakedown limit produce additional stains showing further damage due to cycling whereas maximum stresses at or below this limit lead to the stabilisation of strains and a closed hysteresis loop is formed in subsequent cycles.
•
The value of shakedown limit depends on the minimum stress in the cycle so that for complete unloading this limit is near the envelope curve while for partial unloading it takes place at a lower value of stress.
Envelope Curve
Figure 2.1 – Experimental stress-strain curves of concrete under complete unloading (Sinha et al., 1964).
2-3
To represent the
concrete response analytically, a polynomial relationship was
adopted for the envelope curve. Also, the unloading and reloading paths were modelled using parabolic and linear equations independent of the previous load history, respectively, although subsequent studies (Karsan and Jirsa, 1969, Bahn and Hsu, 1998) indicated the dependency of unloading and reloading responses on the previous load history. The analytical cyclic response could show the test results qualitatively. Sinha et al. (1964) introduced some main characteristics of the cyclic behaviour of concrete and established a sound basis for the future studies in this area. Based on a fracture mechanics treatment, Shah and Winter (1966a, 1966b) carried out a series of tests on prismatic specimens subjected to cyclic axial compressive loading. Test results indicated that the shakedown limit is approximately equivalent to the critical load at which the number of microcracks in mortar begins to increase sharply, and a continuous pattern of microcracks begins to form. As a result, undamaged portions that carry the load reduce and the stress-strain relationship becomes even more nonlinear. The onset of major microcracking was reported at 70% to 90% of the ultimate load. To gain further insight into the response of plain concrete under different cyclic compressive loading histories, Karsan and Jirsa (1969) performed an experimental study on 46 short rectangular concrete columns with the cylinder compressive strengths from 24 to 35 MPa. The specimens were tested under four different loading regimes including monotonic increasing loading to failure, cycles to envelope curve, cycles to envelope curve adding a specified strain increment during each cycle and cycles between maximum and minimum stress levels. Generally, the envelope curve for the cyclic response of test specimens coincided with the stress-strain curve obtained under monotonic loading to failure regardless the load history. Defining the concept of common point as the intersecting point of unloading and reloading curves at each cycle, it was concluded that the points of intersection due to load cycles to the envelope curve represent an upper limit (shakedown limit) on the common points, whereas the cycles with lower stress levels lead to reducing and stabilising the common points at a lower bound. The lower and upper limits, termed as
2-4
the stability limit and the common point limit, respectively, were reported to be '
'
corresponding to the stress levels of 0.63 f c and 0.76 f c , i.e. 74% and 90% of the specimen ultimate strength (Figure 2.2), showing the concrete response is largely dominated by the effects of microcracking. Researchers found out that unloading and reloading curves are not unique but depend on the previous load history. Introducing the concept of nonrecoverable strain (or plastic strains) as the strain corresponding to a zero stress on the unloading or reloading curves, the shape of these curves was found to be significantly influenced by this factor. Based on the test results, a parabolic relationship was proposed to determine the normalised plastic strains using the normalised strains on the envelope curve at the commencement of unloading. As well, a parabolic relationship was assumed for reloading curve to address the additional strains for stress beyond the common point. Adopting the relationship of Smith and Young (1955) for the envelope curve, the test results could be quite well simulated by analytical model.
Figure 2.2 – Loading and unloading curves (Karsan and Jirsa, 1969).
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While the previous works concerned the plain concrete, Brown and Jirsa (1971) tried to simulate the behaviour a series of 12 reinforced concrete cantilever beams subjected to reversed cyclic loading using the stress-strain curves for concrete and for the reinforcing steel. The concrete model in compression was generally based on studies by Karsan and Jirsa (1969). To address the reversal effects they considered that after cracking no concrete compressive stress is developed until the cracks are completely closed. This was experimentally found to occur whenever the tensile strain of concrete reaches a value of 40% of the tensile strain corresponding to concrete cracking. The model was corroborated against the load-deflection and load-rotation responses obtained from tests and a reasonable agreement was observed. It should be noted that since the experimental responses of RC beams were significantly dominated by the reinforcement response, the concrete model was not as critical. Park et al. (1972) studied experimentally and theoretically the moment-curvature and load-deflection responses for reinforced concrete beams and columns subjected to cyclic loading. The cyclic model used for concrete is shown in Figure 2.3. For unloading it was assumed that 0.75 of previous stress is lost with no reduction in strain followed by a linear path of slope 0.25 Ec to a zero stress level where Ec is the initial elastic modulus of concrete. For reloading the strain had to re-reach the value corresponding to the zero stress level of the last unloading and after that the path was similar to that of unloading, as shown in Figure 2.3. The tensile response of concrete was considered brittle linear elastic with no tensile strength after cracking. Although the concrete model could be simply implemented in a numerical procedure, it suffered three major drawbacks. First, the plastic strains are independent of the loading history. Second, the reloading path terminates at the last unloading stress so that the damage attributed to the cycling loading is ignored. Finally, the tension stiffening effect was not incorporated in the model. Although predicted responses compared well with the test results, the experimental responses seem to be dominated by the reinforcement response, similarly to the specimens tested by Brown and Jirsa (1971), and therefore the concrete model was not as critical.
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Figure 2.3 – Stress-strain curve for concrete under cyclic loading (Park et al., 1972).
Darwin and Pecknold (1976) carried out one of the first finite element analysis of reinforced concrete panels subjected to cycling loading. The model for concrete under cyclic loading (Figure 2.4) was developed based on the experimental results of Karsan and Jirsa (1969). The unloading and reloading curves were considered to terminate and initiate at the plastic strain points, respectively. The reloading curve was represented by a line initiated from the plastic strain, passed through the common point and ending at the envelope curve. The unloading path was modelled using three straight lines with the slopes of initial elastic modulus, the same as the slope of reloading line and zero, respectively. The first and second lines intersected at the turning point determined experimentally. The concrete in tension was assumed as a linear elastic brittle material so that after cracking the tensile tangential stiffness along the principal tensile stress direction was reduced to zero. The concrete model along with that of steel reinforcement was incorporated into a finite element program and corroborated against the response of shear panel W-4 tested by Cervenka and Gerstle (1972). While the predicted results were in agreement with the first cycle, they deviated gradually during the second cycle.
2-7
Figure 2.4 – Stress-strain curve for concrete under cyclic loading (Darwin and Pecknold, 1976).
Concrete in Tension
Similar to the concrete in compression, the tensile behaviour of concrete subjected to cyclic loading can be modelled using two groups of curves including a stress-strain envelope curve and the unloading and reloading curves. Test results have shown that the envelope curve can be approximated by the stress-strain curve of concrete under monotonic loading (Yankelevsky and Reinhardt, 1989). The main characteristics of this curve is as follows (Hillerborg, 1980, ASCE Committee 447, 1982, CEB, 1996): •
The stress-strain curve consists of an ascending branch before ultimate strength and a descending branch after that (Figure 2.5(a)).
•
The stress-strain relationship in ascending part is linear almost up to the peak stress with an elastic modulus equal to the initial tangential modulus of elasticity in compression.
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The descending curve qualitatively includes a very steep branch and then a more
•
flat branch. Since the exact shape of stress-strain descending curve does not have significant influence on the most of practical applications, its selection is primarily a matter of practical convenience. While a linear relationship for ascending branch is commonly considered, for the descending branch a variety of curves such as a linear model (Hillerborg et al., 1976), bilinear model (Petersson, 1981), multilinear model (Gustafsson, 1985) and more complex nonlinear models (Lin and Scordelis, 1975, Cornelissen et al., 1985, Gopalaratnam and Shah, 1985) have been proposed. Figures 2.5 (b) and (c) show the linear and bilinear models, where f ct and ε t 0 are the tensile strength and corresponded strain, respectively; α1 , α 2 and α 3 are softening parameters dependant on fracture energy. These parameters were proposed by Hillerborg et al. (1976) and Petersson (1981), respectively, as follows:
α1 = 0 ; α 2 = α 3 =
2 Ec G f
(2.1)
lch f ct2
and 1 3
2 9
α1 = ; α 2 = α 3 + α1 ; α 3 =
18 Ec G f 5 lch f ct2
(2.2)
where Ec is the initial elastic modulus of concrete, G f is the fracture energy and lch is a characteristic length over which the fracture energy is dissipated.
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fct
fct
fct
α1 fct εto
α3εto
εto
(a)
(b)
εto α 2εto
α3 εto
(c)
Figure 2.5 – Tensile envelope curve of concrete (a) typical response (b) linear descending model (Hillerborg et al., 1976) (c) bilinear descending model (Petersson, 1981).
When the cracking occurs in concrete, the stresses are redistributed throughout the concrete element. The stress redistribution results in non-monotonic variation of stresses, even for monotonic external loading. Therefore, unloading and reloading behaviours are significant not only for cyclic loading but also for monotonic loading and should be incorporated into the concrete model.
Yankelevsky and Reinhardt
(1987b, 1989) conducted a series of tests to investigate the tensile cyclic response of concrete subjected to different types of loading including monotonic displacementcontrol tensile test, complete unloading, reversed loading up to a low level of compressive stress and reversed loading until a greater compressive stress level (type 1 to 4, respectively) (Figure 2.6). It was found that the curvature of the unloading and reloading branches was related to the strain at unloading and the strain at the reloading, respectively.
2-10
Figure 2.6 – Uniaxial cyclic response of concrete in tension for different loading regimes (Yankelevsky and Reinhardt, 1987b, 1989).
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The number of proposed models for unloading and reloading response is limited (Gylltoft, 1984, Reinhardt, 1984, Rots et al., 1985, Yankelevsky and Reinhardt, 1987b, 1989). Prior to the peak stress, both unloading and reloading paths are usually assumed to be the same(linear elastic) (ASCE Committee 447, 1982, CEB, 1996). After the peak stress, however, different relationships have been suggested. Rots et al. (1985) assumed that both unloading and reloading run on a same line starting from the origin of the coordinate system (Figure 2.7(a)). This model approximates the real behaviour roughly and is path-independent. Gylltoft (1984), considering a set of lines parallel to the initial ascending branch for modelling the unloading and reloading behaviour, proposed a better approximation to the real response of concrete (Figure 2.7(b)), although it is also path-independent. Foster and Marti (2003) employed a simple model including the history-dependant aspect to describe unloading and reloading (Figure 2.7(c)). The whole strain was assumed to be composed of an elastic component ε e and a plastic component ε p . The unloading was considered to cross through a sarin of ε p / 2 and reloading path was assumed along the same path.
fct
fct
fct
1
ε ct (a)
ε ct (b)
Ec
Ec ε p /2 εp
ε ct
(c)
Figure 2.7 – Unloading and reloading models: (a) Rots et al. (1985), (b) Gylltoft (1984) (c) Foster and Marti (2003).
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Based on the test results, Yankelevsky and Reinhardt (1989) introduced a focal point model to describe the response of concrete subjected to cyclic tension. Figure 2.8 illustrates the graphical procedures used to form a set of unloading and reloading paths. This model presented a good approximation of cyclic response of concrete and can capture the concrete damage on reloading as well as crack closing. The mathematics to describe this is trivial.
Figure 2.8 – Focal point model of Yankelevsky and Reinhardt (1989).
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2.2.3 Reinforcing Steel The constitutive stress-strain relationship of reinforcing steel bars in the cyclic loading is expressed using two groups of curves. The first group is the backbone envelope curve and the second group includes unloading and reloading curves. One significant consideration to select an appropriate constitutive model for reinforcing steel is its numerical efficiency. Particularly, for a non-linear analysis of large RC structures with a huge amount of numerical processing, the model should be as simple as possible (CEB, 1996). Tests conducted by Belarbi and Hsu (1994) indicated that the monotonic stress-strain curve of steel bars can be used to approximate the backbone envelope curve. This curve generally consists of elastic, plastic, strain hardening and strain softening zones as shown in Figure 2.9(a). In some grades of steel, however, the response may not include a distinctive plastic zone (Figure 2.9(b)), thus the intersection of main stress-strain curve and a line with a slope equal to initial elastic modulus and crossing through the strain of 0.002 is considered as the effective yield point (refer section 5.3.3). In Figure 2.9, f sy and ε y are stress and strain corresponding to yield point, respectively,
ε sh is the strain corresponding to the onset of strain hardening, and f su and ε u are the ultimate stress and corresponded strain, respectively. While several analytical models have been proposed for envelope curve of reinforcement response, more common idealised stress-strain curves have been bilinear (Vecchio, 1989, Belarbi and Hsu, 1994, Elmorsi et al., 1998, Kwak and Kim, 2004a, 2004b, Mansour and Hsu, 2005) and trilinear (Stevens et al., 1987, Vecchio, 1999, Foster and Marti, 2003, Palermo and Vecchio, 2003) as shown in Figure 2.10. Depending on the analysis for monotonic or reversed cyclic loading, different types of unloading and reloading curves may be considered. In the case of monotonic as well as cyclic loading, the unloading and reloading are usually modelled using a similar linear path with a slope equal to initial elastic modulus. This idealisation provides a reasonable approximation for reinforcing steel (ASCE Committee 447, 1982, CEB, 1996). Under a reversed cyclic condition, however, observations have shown that the beyond the initial
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fs
fs strain softening zone
strain softening zone
fsu fsy
fsy
strain hardening zone
strain hardening zone
plastic zone elastic zone
elastic zone
εy
εsh
εs
εu
0.002
εy
εs
(a)
(b)
Figure 2.9 – Typical stress-strain curve for steel reinforcement: (a) with plastic zone (b) without plastic zone.
fs
fs
fsu
fsu
fsy
fsy
εu
εy
εs
εy
(a)
εsh
εu
εs
(b)
Figure 2.10 – Idealised stress-strain relationship for steel reinforcement: (a) bilinear model (b) trilinear model.
yield stress, the unloading path exhibits the Bauschinger effect in the region of reversed loading, which refers to the softening of the steel modulus (Bauschinger, 1887). The models of the cyclic stress-strain response of steel reinforcement may be classified in two major categories including macroscopic models and microscopic models (Popov and Ortiz, 1979, CEB, 1996). The former is based on the measured stress-strain 2-15
relationship while the latter is based on the dislocation theory. Although the microscopic models are obtained from sound theory, they are too complex to use in a nonlinear analysis of RC structures (Balan et al., 1998). The most widely used models to express the hysteretic response of reinforcing steel fall into the first category. Within this group, two main approaches can be distinguished. In the first approach the constitutive relationship is expressed in a form ε = f (σ ) (Ramberg and Osgood, 1943,
Ma et al., 1976) and in the second one it is in the form σ = f (ε ) (Menegotto and Pinto, 1973, Filippou et al., 1983, Chang and Mander, 1994). Since in the finite element method the strains are usually derived first from the strain-displacement relationship, the second approach seems to be more advantageous. The most popular model in this context is the model originally proposed by Mentegotto and Pinto (1973) in which the non-linear relationship is considered as shown in Figure 2.11(a) and expressed by
⎡
σ = σ r + ⎢b ε + ⎢⎣
ε=
⎤ ⎥ (σ o + σ r ) (1 + ε R )1 R ⎥⎦ (1 − b)
ε − εr εo − εr
(2.3)
(2.4)
where σ and ε are the stress and strain of reinforcing steel respectively; σ o and ε o are the coordinates of the point where the asymptotes of the branch under consideration intersect; σ r and ε r are the stress and strain at the point where the last strain reversal with stress of equal sign took place; Es and Ew are the initial and the strain hardening modulus of elasticity respectively as well as the slopes of the asymptotes; b is the strain hardening ratio equal to Ew Es and R is a transition parameter to account for the Bauschinger effect that is given by
R = R0 +
a1 ξ a2 + ξ
(2.5)
where ξ is the normalised strain history parameter that is updated following a strain reversal as shown in Figure 2.11(b); R0 , a1 and a2 are material parameters determined
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Figure 2.11 –Hysteretic model of Menegotto and Pinto (1973): (a) parameters (b) definition of parameter R (ξ ) .
from experiment. For the non-prestressed steel, Menegotto and Pinto
(1973)
recommended R0 = 20 , a1 = 18.5 and a2 = 0.15 .
2.2.4 Shear Transfer When a shear force is transmitted through an uncracked RC member, the concrete response is considered to be linear elastic. As the principal tensile stress at some location develops and reaches the cracking strength of concrete, a crack forms normal to the direction of the principal tensile stress. After cracking, two major mechanisms namely aggregate interlock and dowel action contribute to the shear transfer. Aggregate interlock is essentially a material property depending on aggregate type and shape and the strength of cement paste while the dowel action is a structural property relating to detailing of the reinforcement, the geometry of section and the loads or constraints (CEB, 1996). The contribution of dowel action is less pronounced in comparison to that
2-17
of the aggregate interlock (Walraven, 1980) and to analyse the RC structures with low amount of reinforcement such as shear walls, it may be ignored (Elmorsi et al. 1998).
Aggregate Interlock
Cracked concrete transmits a significant amount of shear force attributed to sliding of the rough crack surfaces along each other and interlocking of aggregate particles. This phenomenon is termed aggregate interlock (Fenwick and Paulay, 1968) and is accompanied with a restraining normal force along the crack due to the reinforcing steel as well as the boundary restraints. Additionally, relative tangential and normal displacements of the two crack surfaces named as the crack slip and the crack width, respectively, occur over the crack interface. Figure 2.12 shows the shear stress τ cr , the normal stress σ cr , the crack slip δ cr and the crack width wcr which occurred over a rough crack. As the crack slip is developed at the crack interface, there is a tendency for widening of the crack known as crack dilatancy resulting in increasing axial stress in the reinforcing bars. The relationship of a planar crack may be written in the form (Bazant and Gambarova, 1980): ⎧dσ cr ⎫ ⎡ D11 ⎬=⎢ ⎨ ⎩ dτ cr ⎭ ⎣ D21
D12 ⎤ ⎧dwcr ⎫ ⎬ ⎨ D22 ⎥⎦ ⎩ dδ cr ⎭
(2.6)
where D11 , D12 , D21 and D22 are crack stiffness coefficients which are dependant on
τ cr , σ cr , δ cr , wcr
and possibly other parameters. The crack stiffness matrix is
asymmetric and non-positive definite, so that crack response tends to be unstable. However, the restraint provided by the reinforcement usually stabilises the response.
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τcr wcr
σcr
δcr Figure 2.12 – Concrete rough crack.
Early tests conducted to investigate the internal shear force distribution in RC beams did not take into consideration the shear forces transmitted by the interface shear transfer mechanism. Taylor (1959) was the first to undertake tests to study this mechanism and its effect on diagonal cracking. The results of these tests, as well as those of other experimental investigations such as Fenwick (1966), Fenwick and Paulay (1968) and Sharma (1969), revealed the significant contribution of the interface shear transfer mechanism to resist shear forces with between 33 to 50 percent of the total shear applied at a section carried by interface shear. It was also found that the shear stiffness of beams increased with decreasing initial crack width and increasing concrete strength. However, due to limitations of tests undertaken on beams a systematic investigation of the effects of major parameters such as initial crack width and aggregate size, the basic load-displacement relationships could not be determined. During the 1970s a series of direct shear tests were carried out to study the effect of initial crack width, concrete strength, aggregate size, normal restraining stiffness, shear stress intensity and cyclic shear stresses (Loeber, 1970, Taylor, 1970, Houde and Mirza, 1972, White and Holley, 1972, Paulay and Loeber, 1974, Laible et al., 1977). Figure 2.13 shows a typical specimen used in direct shear tests in which a shear force, V, and a normal stress, σ n , are applied. The tests were conducted under different conditions including constant crack width, constant and variable confinement stiffness, constant confinement stress and constant crack dilatancy. The main conclusions drawn from the tests were:
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V
preformed crack
shear plane
σn
σn
V
Figure 2.13 – Typical direct shear test specimen.
•
the aggregate shape and size had little influence on the shear stress-shear displacement relationships.
•
the shear displacement across the crack was largely affected by the crack width.
•
the response of specimens observed prior to the termination of test consisted of the three phases shown in Figure 2.14. In the first phase an initial free crack slip occurred before the aggregates projecting across the crack came into contact with each other or the cement matrix. During the second phase, the interlocking of aggregates was established accompanied with a low shear stiffness. Stage 3, complete interlocking with a linear elastic response and increased shear stiffness.
•
the shear stiffness of specimens increased with increasing normal pressure and decreasing the crack width.
•
when the crack width as well as the shear stress over the crack surfaces increased the shear stiffness of crack gradually decreased.
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τcr termination of test
phase 1
phase 3 phase 2
δ cr Figure 2.14 – Typical shear stress-displacement response of concrete crack.
•
the magnitude of crack width dominated the response of specimens tested under cyclic loading such that the shear displacement accumulated after each load cycle was proportionally related to the crack width.
•
the cyclic shear forces resulted in the degradation of the crack surfaces so that the initial free slip increased with load cycling.
•
the shear stiffness of a crack observed during the first load cycle was about 1/3 to 1/2 of shear stiffness of following cycles. Moreover, the following cycles up to about 17th and similarly between the 18th and 32nd, load repetition were practically coincident (Figure 2.15) (Paulay and Loeber, 1974).
•
upon unloading, residual displacements ranged from 70 to 90 percent of maximum shear displacement corresponding to the onset point of unloading were observed.
•
the interface shear transfer was fully activated at a shear stress and shear displacement of approximately 10% and 70% of the maximum shear stress and shear displacement, respectively, during the cycle considered.
The studies discussed above focused on the shear stiffness rather than shear strength of cracked concrete sections. In studies by Mattock and his co-workers (Hofbeck et al.,
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1969, Mattock and Hawkins, 1972, Mattock et al., 1976) a number of experimental tests were undertaken to investigate the shear strength of cracked reinforced concrete expressed in terms of the ultimate shear stress. These works were used as a base for the design relationships of shear friction described in the ACI 318 building code (ACI Committee 318, 1971, 1977). The specimens were tested either as uncracked or precracked along the shear plane and only the failure load and amount of transverse reinforcement were reported. The test results showed that the strength of the uncracked specimens were higher than those of the initially precracked specimens, although the shear stiffness and upper limit on shear transfer for the both groups were similar (Figure 2.16). Considering that the shear transfer can be described in terms of friction, a shear-frictional equation was proposed by Mattock and Hawkins (1972):
6
Number of cycle 1
2
17
5
Shear Stress (MPa)
to continue 4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
Shear Displacement (mm)
Figure 2.15 – Cyclic response of shear stress due to aggregate interlock ((Paulay and Loeber, 1974).
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12
10
Initially uncracked
τcr (MPa)
8
6
Initially cracked
4
2
0 0
2
4
6
8
10
12
ρvfy (MPa)
Figure 2.16 – Crack shear stress τ cr versus crossing reinforcement ratio ρ v f y (Hofbeck et al., 1969, Mattock and Hawkins, 1972).
τ cr max = ρ v f y μ
(2.7)
where ρ v is the crossing reinforcement ratio, f y is the yield stress of reinforcement and
μ is the coefficient of friction and is dependant on the construction procedure. This study dealt with the ultimate shear friction capacity of cracks and did not investigate the relationship between crack shear stress and crack displacements. Later on, Mattock (2001) re-evaluated the design equations based on the large number of tests carried out on the concrete with strengths ranging from 17 to 100 MPa and obtained a set of modified design equations. Although by the early 1970s some models had been proposed for certain aspects of the phenomenon of aggregate interlock (Hofbeck et al., 1969, Houde and Mirza, 1972, Paulay and Loeber, 1974), comprehensive analytical models of the problem were
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lacking. Shear transfer was introduced into the finite element models of RC structures, by Suidan and Schnobrich (1973) using the relationship:
Gcr = β s Gc
(2.8)
where Gcr and Gc are the shear modulus of cracked and uncracked concrete, respectively, and β s is a shear retention factor ( 0 ≤ β s ≤ 1 ). In the early studies the β s was generally considered constant but later models often adopted a decreasing functions for β s to consider the effect of increasing crack width with increasing applied load (Dei Poli et al., 1990). Further discussion regarding β s is presented in the section 2.3 of this chapter. The use of an empirical shear retention factor, however, does not adequately and comprehensively represent the nature of aggregate interlock and the dependency of shear stiffness on the crack properties. Fardis and Buyukozturk (1979) considered the rough and irregular shape of crack as a superposition of low frequency sinusoidal components identified by the location and size of aggregates and matrix upon high frequency ones comprised of small asperities and protruding particles (Figure 2.17). These components were termed general and local roughness, respectively. The general roughness of a crack interface was assumed to retain its shape during relative movement, whereas the local roughness was considered to be ground and smoothed. Ignoring the effects of local roughness and introducing the shape of the general roughness as functions of the coordinate x (in the tangential direction of the crack interface), the forces developed due to material internal friction at the contact points were determined. Since some of the independent variables such as the detailed shape of the crack are a priory unknown, the model was not a predictive tool but functional model. Applying a multiple regression analysis for the available experimental data and determining these parameters, a model between the response and main variables such as the crack width and the stress resultants at the crack location was developed. The model could qualitively predict the response of concrete crack (Fardis and Buyukozturk, 1979). The limitation of model lies in its attempt to describe a stochastic crack surface profile with a deterministic function.
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y x
Figure 2.17 – General and local roughness of crack interface (Fardis and Buyukozturk, 1979).
During the 1980s, the experimental data collected in the 1970s and the early 1980s (Walraven, 1980) made it possible to develop much more realistic and reliable models. These models may be classified into two categories: empirical models and physical models (Feenstra et al., 1991a). The models of first category include empirical formulations derived from the experimental results, while the models in the latter are based on rational assumptions of the shape of the crack surface. Among the empirical models, the most reliable and comprehensive are those that originate from the rough crack theory (Bazant and Gambarova, 1980, Divakar et al., 1987, Yoshikawa et al., 1989) and the model expressed by Walraven and Reinhardt (1981), whereas those of the physical models are the two-phase model (Walraven, 1980, Walraven, 1981) and the contact density model (Li and Maekawa, 1987, Li et al., 1989). Although the constitutive relationship is assumed to be incrementally linear, due to the complexity of the problem the less general but simpler constitutive laws based on the total deformation approach have been mostly proposed. The major drawback of these models is the path-independency while the inelastic response of aggregate interlock is normally pathdependent. Bazant and Gambarova (1980) introduced the rough crack model considering only the local roughness of crack interface. The number of contact points along the crack interface was assumed to be infinite so that the stress-displacement relationships for a crack could be expressed as continuous. The main assumptions were:
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•
the interface stresses attributed to the wedging effect are assumed to be primarily dependant on the displacement ratio r = δ cr wcr .
•
for large values of the displacement ratio, r , the shear stress τ cr must exhibit an asymptote due to microcracking and crushing in the mortar close to the aggregates.
•
for large values of crack width wcr (e.g. wcr > Dmax / 2 where Dmax is the maximum aggregate size), the stresses vanish because the contact between crack surfaces is lost.
The shear stress τ cr and the normal stress σ cr are expressed as functions of main parameters as follows:
τ cr = f t (r , wcr / Dmax , f c' )
(2.9)
σ cr = f n ( wcr ,τ cr )
(2.10)
By optimising the fits of Paulay and Loeber’s experimental data (1974) the following relationships were identified:
τ cr = τ u r
σ cr = −
a3 + a4 r
3
1 + a4 r 4
a1 (a2 τ cr wcr
)p
(units of mm and MPa)
(2.11)
(units of mm and MPa)
(2.12)
where a1 = 0.000534 ; a2 = 145.0 ; a3 = 2.45 / τ 0 ; a4 = 2.44 (1 − 4 / τ 0 ) ; τ 0 = 0.245 f c' ; 2 τ u = 0.001τ 0 / [ 0.001 + ( wcr / Dmax ) 2 ] ; p = 1.30 [1 − 0.231 /(1 + 0.185 wcr + 5.63 wcr )] and
f c' is the compressive cylindrical strength of concrete in MPa. The response diagram of model is presented in Figure 2.18.
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10
τcr (MPa)
8
wcr = 0.1 mm 0.5 mm 1.0 mm
6 4 2
σcr (MPa)
0 -2
1.0 mm
-4
0.5 mm
-6
0.1 mm
-8 -10 0
0.5
1
1.5
2
δcr (mm) Figure 2.18 – Typical response diagram for f c' = 30 MPa (Bazant and Gambarova, 1980).
Based on the concepts of the rough crack model, the functional relationships among the four major parameters δ cr , wcr , τ cr and σ cr have been expressed in different ways. Divakar et al. (1987) considered the functional relationships:
τ cr = g t (δ cr , σ cr )
(2.13)
wcr = g n (δ cr , σ cr )
(2. 14)
while Yoshikawa et al. (1989) focused on the crack slip and confinement stress as follows:
δ cr = ht (τ cr , wcr )
(2.15)
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σ cr = hn (τ cr , wcr )
(2.16)
Similarly to the original rough crack model by Bazant and Gambarova (1980), in these studies the empirical relationships drawn on the experimental data were proposed to determine the crack response. The rough crack model may be extended to cyclic loads. The CEB (1996) reported a extended cyclic model by Gambarova (1980) drawn on the rough crack model in which the proposed loading and reloading equations were similar to Eqs. 2.14 and 2.15, while different unloading formulations were expressed. Based on extensive and comprehensive experimental tests on the aggregate interlock mechanism in normal weight concrete (Walraven et al., 1979) and data-fit analysis, Walraven and Reinhardt (1981) introduced the following empirical linear formulations (in units of mm and MPa) that fit their experimental data with the greatest accuracy:
τ cr = −
f cc −0.80 −0.707 + [1.80 wcr + (0.234 wcr − 0.20) f cc ]δ cr 30
(2.17)
σ cr = −
f cc −0.63 −0.552 + [1.35 wcr + (0.191 wcr − 0.15) f cc ]δ cr 20
(2.18)
in which
δ cr ≥ 0 ; τ cr ≥ 0 ; σ cr ≤ 0 and f cc is the compressive cube strength of
concrete in MPa. Figure 2.19 illustrates the model. The original model lacked an upper bound for the shear stress. Later, using the same experimental data Vecchio and Collins (1986) proposed the following equation as the maximum shear stress that the crack could carry:
τ cr max =
f c' 24 wcr 0.31 + Dmax + 16
(units of mm and MPa)
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(2.19)
τcr (MPa)
10
wcr = 0.1 mm
0.5 mm
8 6 4
1.0 mm
2 0
σcr (MPa)
-2
1.0 mm
-4 -6 -8
0.5 mm
0.1 mm
-10 0
0.5
1
1.5
2
δcr (mm) Figure 2.19 - Response diagram for f cc = 37.5 MPa (Walraven and Reinhardt, 1981).
Although this model can be simply employed in a finite element model, the initial free slip included in the model leads to some instability into the computational algorithm (Lai, 2001). Moreover, the model has been validated for the crack width up to 1 mm and the model will lose its accuracy when the crack width is too large so that the ratio
τ cr / δ cr (i.e. the crack shear stiffness) as well as σ cr / δ cr degrades dramatically and becomes zero after a crack width limit regardless of the maximum aggregate size. For example for a concrete with the compressive cube strength of 37.5 MPa and maximum aggregate size of 19 mm, the shear stiffness becomes zero at a crack width of about 1.3 mm.
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The two-phase model is a rational and comprehensive analytical model introduced by Walraven (1980) that was developed to model the aggregate interlock in the cracks without reinforcing steel bars. The main assumptions of the model were as follows: •
concrete is assumed as a two-phase material consisting of a very stiff spherical inclusions embedded in a soft matrix.
•
Fuller’s curve is adopted for the grading of the aggregate along a crack plane.
•
the inclusions and the matrix are assumed to be in partial contact attributed to interface displacements. The active contact areas are related to interface displacements using the geometrical relationships and the statistical distribution of aggregates.
•
while the compressive contact strength of matrix is considered as a function of concrete strength, the shear contact strength is assumed as a linear function of the compressive contact strength using a friction coefficient.
Applying the equilibrium conditions in the crack interface, the stress-displacement relationships were given as:
τ cr = σ pu ( An + μ At )
(2.20)
σ cr = −σ pu ( At − μ An )
(2.21)
where An and At are the average contact areas for a unit crack surface in the normal and tangential directions of the crack interface, respectively; σ pu is the maximum compressive strength of matrix and μ is the friction coefficient between inclusions and matrix. The relationships proposed to determine An and At were sophisticated functions of δ cr , wcr , Dmax and relative aggregate volume pk defined as the ratio of total volume of the aggregates over the total volume of the concrete. While a quite good agreement between the predicted values by the model and the experimental data was
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observed by Walraven (1980), its application in the finite element models has been limited due to its complex relationships resulting in some difficulties. The contact density model of Li and Maekawa (1987) and Li et al. (1989) is based on two proposals and three assumptions. The proposals describe the geometry of a crack surface and the direction of contact stress as follows: •
a crack surface contains a set of contact areas (i.e. contact units) with different inclinations θ varied from − π / 2 to π / 2 . A probabilistic contact density function Ω (θ ) can be used to describe the distribution of the crack surface orientations.
•
the direction of each contact stress is proposed to be fixed and normal to the initial contact direction denoted by θ .
The main assumptions are: •
The contact density function Ω(θ ) is independent of the size and the grading of aggregates and given by Ω(θ ) = 0.5 cos θ
•
(2.22)
a simple elastic-perfectly plastic model is assumed for the normal contact stress (σ con ) with regard to the matrix and interface deformations.
•
the effective ratio of contact area K ( wcr ) representing the effect of wcr on the loss of contact area, is given by K (wcr ) = 1 − exp (1 −
0.5 Dmax ) wcr
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(2.23)
Applying the equilibrium conditions, the shear stress τ cr and the normal stress σ cr transferred across the crack are given by
τ cr = ∫
π/ 2
− π/ 2
σ cr = ∫
π/ 2
− π/ 2
σ con . K ( wcr ) . At . Ω(θ ) . sin θ dθ
(2.24)
σ con . K ( wcr ) . At . Ω(θ ) . cos θ dθ
(2.25)
where At is the total area corresponding to the nominal unit area of crack surface. For reinforced concrete with an ordinary reinforcement ratio, where the assumption of smeared crack is valid, a simplified path-independent stress transfer model was formulated (Li et al., 1989, Maekawa et al., 2003):
τ cr = 3.83 3 f c'
ψ2 1 +ψ 2 ⎡π
σ cr = 3.83 3 f c' ⎢ − cot −1ψ − ⎢⎣ 2
(2.26)
ψ2 ⎤ ⎥ 1 +ψ 2 ⎥⎦
(2.27)
where ψ = δ cr / wcr and f c' is in MPa. The response diagram of the simplified model is demonstrated in Figure 2.20. The model is simple and rational and can be easily incorporated into the finite element modelling to analyse the RC structures subjected to either monotonic or cyclic loading (Okamura and Maekawa, 1991, Maekawa et al., 2003). Nevertheless, the model will lose its accuracy if the crack width is too large (typically greater than 1 mm) since the maximum shear transfer in this model is independent of wcr and Dmax and only dependent on f c' (Maekawa et al., 2003).
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10
wcr = 0.1 mm
0.5 mm
τcr (MPa)
8
1.0 mm
6 4 2 0
σcr (MPa)
-2 -4 -6
1.0 mm
-8
0.1 mm
0.5 mm
-10 0
0.5
1
1.5
2
δcr (mm)
Figure 2.20 - Response diagram for f c' = 30 MPa (Li et al., 1989).
Feenstra et al.
(1991a, 1991b) carried out a comprehensive comparative study to
evaluate the performance of different aggregate interlock models including the rough crack model of Bazant and Gambarova (1980), the two-phase model of Walraven (1980), the empirical model of Walraven and Reinhardt (1981) and the simplified contact density model of Li et al. (1989). In general, all models showed a reasonable agreement with experimental data, though a large scatter was observed specially at smaller crack widths. They concluded that the contact density model, compared with the other models, seemed to have the edge from a numerical point of view. The aggregate interlock capacity of cracks in high-strength concrete (HSC) is reduced compared with normal strength concrete. For HSC, since the strength of the matrix is more than that of the aggregate particles, the cracks tend to cross the individual pieces of aggregates rather than going around them. Based on an experimental investigation undertaken on RC beams with compressive strength from 21 to 92 MPa,
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Mphonde (1988) concluded that contribution of aggregate interlock decreased considerably as the compressive strength of concrete increased. Walraven (1995) carried out tests on pre-cracked push-off specimens made of concrete with a compressive cube strength of 115 MPa. The tests included both plane and reinforced cracks. It was found that the fracture of the aggregate particles upon crack formation in high-strength concrete resulted in smoother crack surfaces and as a result a significant reduction in the shear capacity of cracks due to aggregate interlock was observed. The reduction for plain cracks was about 65 percent of the value which would be achieved without particle fracture, while that of reinforced cracks was about 25 to 45 percent. Walraven concluded that because of better bond between reinforcing bars and the high strength concrete, the yield stress was reached at a smaller crack width compared with normal strength concrete and therefore the number of contact areas was smaller. Based on the contact density model of Li and Maekawa (1987), Ali and White (1999) developed a method for use in design. The model was used to determine the ultimate shear capacity and validated for a wide range of concrete strengths (20 to 100 MPa). Nevertheless, the model did not provide the full response of the crack interface subjected to the crack displacements. Angelakos et al. (2001) conducted tests on large RC beams with compressive strengths ranging from 20 to 100 MPa to study their shear strengths. The maximum shear stress of a crack was calculated using Eq. 2.22 in which the smoother interface of cracks in HSC was addressed by reducing the effective aggregate size a from the nominal diameter to zero as the concrete strength increases from 60 to 70 MPa. For concrete strengths of 70 MPa or higher, the a was taken as zero.
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2.3
Finite Element Analysis
Since the first use of finite element method for modelling of
simply supported
reinforced concrete beams (Ngo and Scordelis, 1967), the method has been significantly developed and become a powerful and general tool to simulate and study the behaviour of reinforced concrete structures as reported comprehensively by ASCE (1982, 1993) and CEB (1996). Although the progressing in speed and accuracy of analyses in recent years has led to the wide use of the nonlinear finite element analysis softwares in many types of practical applications, the use of them in reinforced concrete remains complex. Besides the significance of the type of finite element and the size of mesh used to idealise the structure, the main challenge in using this method to model RC structures arises from the composite nature of the material. The accuracy and reliability of finite element analysis is largely governed by the abilities of the underlying constitutive relationships to capture different types of nonlinear behaviour such as the nonlinear stress-strain relationships of concrete and reinforcing steel, cracking, crushing, tension stiffening, compression softening, aggregate interlock, dowel action and bond slip. Moreover cyclic loading introduces further complexities such as stiffness and strength degradation in concrete, the Bauschinger effect in reinforcing steel and bond deterioration between concrete and reinforcement. As well, incompatibility of models and approaches is another important problem, so that using some models from one analytical approach to another or combining them with other models may yields errors in the finite element analysis (Vecchio and Palermo, 2001). As a result, to simulate the response of reinforced concrete members and specially shear walls under general loading, the selection of proper and compatible material models and numerical procedures is essential. In the following, a review of the major approaches and the previous works conducted on the nonlinear finite element analysis of reinforced concrete with emphasis on the finite element analysis of RC shear walls and panels is presented.
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2.3.1 Overview The dominate method for the finite element analysis of reinforced concrete consists of developing separate models for the concrete and for the reinforcing steel and then combining those models to make the constitutive matrices at the element level as well as the structure level. The main approaches for modelling of concrete and reinforcing steel in reinforced concrete structures are discussed herein. Concrete Modelling
Reinforced concrete as a composite material exhibits a highly nonlinear behaviour. This nonlinearity arises primarily from the nonlinear behaviour of the constituent materials of steel and concrete, which dominate the pre- and post-cracking responses. A major reason for nonlinearity in reinforced concrete is the tensile cracking which causes a considerable redistribution of stress within the intact concrete as well as the stress transfer from concrete to the reinforcement. The success of a nonlinear finite element analysis for RC structures essentially depends on realistic crack modelling, along with appropriate material representations of reinforcing steel and concrete. In general, two major approaches have been used to model the cracking in concrete structures named as the discrete crack model and the smeared crack model. The former approach models a crack as a geometrical discontinuity in concrete, while the latter treats a cracked concrete as a continuum. In the early discrete crack model that was first introduced by Ngo and Scordelis (1967), cracking was modelled as a separation of nodes along element edges. Post-cracking effects such as tension stiffening and bond-slip can be taken into account using linkage elements or interface elements between the crack surfaces. Although this approach was used in the other early studies on the finite element modelling of RC structures (Nilson, 1968, Franklin, 1970, Mufti et al., 1970), two major drawbacks involved in using the model in the finite element modelling greatly restrict the ease and speed of analysis. First, the crack propagation is limited to follow a predefined path along the element boundaries so that to improve the analytical results, the topology of the finite element mesh is required to be changed continuously throughout the analysis. Second, because
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of considering the additional degree of freedoms for the separated nodes along the new crack faces, the time and cost of computation increase and the efficiency decreases (although with today’s high speed computers this time and cost penalty is less restrictive than it once was). To overcome these issues some modifications were proposed such as remeshing techniques with respect to the potential direction of crack propagation at each crack increment (Cervenka and Gerstle, 1971, Ingraffea and Saouma, 1984, Bocca et al., 1991) and the techniques in which the crack is permitted to extend through the finite elements (Ortiz et al., 1987, Belytschko et al., 1988). While the mentioned drawbacks have made the discrete crack model less popular than other techniques (such as, for example, smeared cracks), for those problems that involve a few dominate cracks or where the local material behaviour is of interest, it is most useful and likely to be the main choice. Nevertheless, the behaviour of many reinforced concrete structures contained closely spaced reinforcement such as shear panels and walls are dominated by distributed parallel cracks. In such cases, the smeared crack model is often considered to be more effective and beneficial than the discrete crack model (ASCE Committee 447, 1982, 1993). In the smeared crack model, introduced by Rashid (1968), once cracking occurs at a integration point in an finite element, the cracks are assumed to be parallel and finely distributed (or smeared) over the entire region that point represents. Thus the cracked concrete can be considered as a continuum and the average stress-strain relationship can be expressed in a continuous manner so that upon cracking the initial isotropic stressstrain relationship can be simply replaced by an orthotropic stress-strain relationship. Consequently, cracking is treated as a reduction of average material stiffness over the area in directions of the major orthotropic axes and the constitutive matrix is correspondingly modified to reflect this. In comparison to the discrete crack model, the smeared crack model neither changes the topology of original finite element mesh nor imposes any constraints on crack propagation directions. Despite this, classical smeared crack approaches suffer from a deficiency when dealing with localised cracking (Bazant, 1983), however, the advantages of the smeared crack approach have made it more widely adopted than the discrete crack approach by researchers to model cracking in the large scale reinforced concrete structures. Throughout the literature, three major
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variants of smeared crack approach have been used; the fixed crack model, rotating crack model and multi-directional fixed crack model. In the fixed crack model that was firstly used by Rashid (1968), the crack takes place normal to the direction of maximum principal stress once it reaches the concrete tensile stress and thereafter the crack orientation is taken as fixed throughout the loading process. The cracked concrete is assumed as an orthotropic material with the orthotropic axes n and t , normal and parallel to the crack direction, respectively. Considering a two dimensional cracked concrete membrane, the orthotropic constitutive relationship may be given by (ASCE Committee 447, 1993) ⎧σ nn ⎫ ⎪ ⎪ sec ⎨ σ tt ⎬ = D nt ⎪τ ⎪ ⎩ nt ⎭
⎧ε nn ⎫ ⎪ ⎪ ⎨ ε tt ⎬ ⎪γ ⎪ ⎩ nt ⎭
(2.28)
where Dsec nt is the secant constitutive matrix; σ nn , σ tt and τ nt are the average stresses and ε nn , ε tt and γ nt are the average strains in the n - t coordinate system, respectively. In the early application of the fixed crack model (Rashid, 1968, Cervenka and Gerstle, 1971) Dsec nt is expressed as
Dsec nt
⎡0 0 0 ⎤ = ⎢⎢0 E 0⎥⎥ ⎢⎣0 0 0⎥⎦
(2.29)
In other words, after cracking the stiffness normal to the crack, the shear stiffness and the Poisson’s ratio for cracked concrete are taken to be zero. Later, because of numerical problems arising from the immediate drop of stiffness, a reduced shear stiffness, β s G was incorporated into the model as follows (Suidan and Schnobrich, 1973) :
sec Dnt
⎡μn E = ⎢⎢ 0 ⎢⎣ 0
0 E 0
0 ⎤ 0 ⎥⎥ β s G ⎥⎦
(2.30)
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where G is the shear stiffness of intact concrete and β s is a shear retention factor and ranged between 0 and 1. While foe many problems the exact amount of β s is not crucial to the solution, a value equals 0.2 is a commonly adopted value. The use of β s may be also thought as a way to account for aggregate interlock and dowel effects. Additionally, to address the strain softening effect in tensile concrete a “normal reduction factor” μ n has been inserted in the constitutive matrix:
sec Dnt
⎡μn E = ⎢⎢ 0 ⎢⎣ 0
0 E 0
0 ⎤ 0 ⎥⎥ β s G ⎥⎦
(2.31)
where μ n is considered as a function of the strain normal to the crack. The fixed crack model has been used to analyse the reinforced concrete structures under both monotonic and cyclic loading. It usually predicts a stiffer response compared with experimental results especially when a structure is anisotropically reinforced (Crisfield and Wills, 1989) because, while the fixed crack model does not allow the crack directions to rotate, in real structures the direction of principal stresses is continuously changed as the loading is increased and consequently a shear stress on the crack surfaces is developed. This matter led to developing the rotating crack model in which the cracks can rotate during loading process and the multi-direction fixed crack model in which new cracks are allowed to develop at the different orientations. The rotating crack model was originally developed by Cope et al. (1980) to take into account crack reorientation. While the basic relationships of rotating crack model are essentially similar to the fixed crack model, the main distinction between them is that in the rotating crack model the axes of orthotropy are always aligned with the direction of major principal stresses as the load increases. Furthermore to make the rotating crack model simpler, the assumption of co-axiality of the principal stresses and the principal strains is usually incorporated into the model (Vecchio, 1989). Consequently, considering the cracked concrete as an orthotropic material, the stress-strain constitutive relationship may be written based on the principal 1-2 axes as
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⎧σ 11 ⎫ ⎪ ⎪ sec ⎨σ 22 ⎬ = D12 ⎪τ ⎪ ⎩ 12 ⎭
⎧ε 11 ⎫ ⎪ ⎪ ⎨ε 22 ⎬ ⎪γ ⎪ ⎩ 12 ⎭
(2.32)
sec where D12 is the secant constitutive matrix; σ 11 and σ 22 are the average stresses and
ε 11 and ε 22 are the average strains in the directions of 1- and 2-principal axes respectively. The co-axiality assumption requires that the shear stress τ 12 and the shear strain γ 12 are always zero and hence the need to employ the shear reduction factor β or a shear stress-strain relationship is removed. In general, the rotating crack model has been criticised for two shortcomings. First, the rotation of crack direction implies that the concrete damage is not permanent and it depends only on the current stress state and therefore it is load history-independent (Bazant, 1983, Noguchi, 1985). Second, with respect to either new crack direction or new axes of orthotropy at the beginning of each load step, a different stress-strain relationship should be used and hence, a whole family of stress-strain curves are required whereas in the fixed crack approach only few curves are used. Nevertheless, the model can be simply implemented into a computer program and many researchers have successfully used the model for analysing the behaviour of RC structures subjected to monotonic loading (Gupta and Akbar, 1983, Milford, 1984, Vecchio, 1989, Hu and Schnobrich, 1990 among others). Stevens et al. (1987) were the first researchers that developed the model to include the reversed cyclic loading. While the model was quite successful at the element level, the analytical procedure at the structure level encountered some numerical problems and their attempt failed. Later Vecchio (1999) and Palermo and Vecchio (2003) successfully employed the rotating crack model to analyse the cyclic response of RC shear walls. Crisfield and Wills (1989) investigated the application of the fixed and rotating crack models to the analysis of a number of RC panels tested by Vecchio and Collins (1982). The main conclusions derived from this investigation can be summarised as: Applying the fixed crack model to the RC members that fail in a ductile manner with response dominated by yielding in the reinforcement steel, the shear retention factor does not affect on the final collapse load, although the load-deflection is dependant on it.
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When the straining pattern changes as the loading is increased, the fixed crack model may result in overestimated failure loads, where the rotating crack model yields a failure load that is less than or equal to that given by the fixed crack model. If the concrete fails in a shear induced mode, the rotating crack model leads to an overestimation of failure loads and often indicates a failure attributed to yielding of the reinforcing steel. The application of a simple, but effective orthogonal fixed crack model is more general than rotating crack model. Nevertheless in absence of such fixed crack model, the rotating crack model appears to offer many advantages. The multi-directional fixed crack model was originally developed by De Borst and Nauta (1985). The model is a refined version of the fixed crack model in which the crack orientation is still assumed constant after cracking takes place. However, new cracks are allowed to develop in different directions if the angle between two consecutively formed cracks exceeds a threshold angle. Based on this concept, the rotating crack model may be conceived as a multi-directional fixed crack model with a zero threshold angle (ASCE Committee 447, 1993). The main concept in this model is that the total concrete strain increment Δ ε c is decomposed into an intact concrete strain increment Δ ε ic and a crack strain increment Δ ε cr which can be expressed as Δ ε c = Δ ε ic + Δ ε cr
(2.33)
where Δ ε cr is the summation of the strain contributions of all cracks at one integration point and is expressed by Δ ε cr = Δ ε cr 1 + Δ ε cr 2 + ... + Δ ε crn
(2.34)
So the cracks can be treated separately and while the effect of new cracks on the total response of reinforced concrete is considered, the influence of the old cracks is taken into account as well. In other words the model is path-dependent and sensitive to load history. The application of the multi-directional fixed crack model is limited compared 2-41
with alternative approaches. This arises from its complex nature as well as the need to a large number of computations to build the total constitutive matrix particularly under cyclic conditions (Crisfield and Wills, 1989). The model was successfully used by Bolander and Wight (1991) for investigating the behaviour of RC shear walls under monotonic loading and Xu (1991) employed the model to simulate the cyclic response of RC members. While the analytical results at element level were generally satisfactory, because of numerical problems the approach was not successful at the structure level.
Reinforcing Steel Modelling
Three major approaches have been used to model the reinforcing steel in the finite element analysis of reinforced concrete including the smeared steel model, the discrete steel model and the embedded steel model (Figure2.21). In the smeared steel approach, the reinforcing steel is assumed to be distributed uniformly over a concrete element at a particular orientation and expressed using the reinforcing steel ratio. Since the reinforcement is considered to be smeared, the complete compatibility between reinforcing steel and concrete is imposed.
The
constitutive matrix of reinforced concrete finite element is considered to be composed of individual contributions of concrete and steel reinforcement. This model is usually helpful for analysing reinforced concrete structures consisting of uniformly distributed reinforcement (eg. shear walls) not having to define separate reinforcing bars (ASCE Committee 447, 1982, Crisfield and Wills, 1989). The discrete steel approach was the first reinforcing steel model used in the finite element analysis of RC structures (Ngo and Scordelis, 1967). In this model the reinforcing steel is modelled as at common separate one-dimensional truss elements connected to the concrete element edges at common nodal points. Bond-slip effects between steel and concrete can be included by inserting the linkage elements that is the important advantage of this model. The main drawback of this model is that the
2-42
direction and location of steel reinforcement are restricted to the boundaries of the concrete elements. In the embedded steel approach, each steel reinforcing bar is considered to be an axial member included into the concrete element in such a way that the displacements of embedded member and those of surrounding concrete element are compatible. Similar to the smeared steel model, the bond between reinforcing steel and concrete is often assumed to be perfect. The one exception to this is the incorporation of a stepped plastic bond slip made into the FE formulation by Foster and Marti (2002, , 2003). The main advantage of embedded reinforcement model is that there is no limitation for representing the direction and location of the steel reinforcement. Using the principle of virtual work the element stiffness matrix, K , can be determined as (Chang et al., 1987) K = Kc + Ks
(2.35)
in which
K s = As Es
∫l B
T
TT T B dl
(2.36)
where K c and K s are the stiffness matrices for concrete and reinforcement, respectively; l , As and Es are the length, the cross-sectional area and the elastic modulus of reinforcement, respectively; B is the strain displacement matrix and T is the strain transformation matrix for reinforcement with respect to global coordinate system.
2-43
(a)
(b)
(c)
Figure 2.21 – Reinforcing steel modelling: (a) smeared steel model (b) discrete steel model (c) embedded steel model.
2.3.2 Finite Element Analysis of RC Panels and Shear Walls The accuracy and reliability of the finite element analysis of reinforced concrete is largely dependant on material constitutive models, the capability to capture the major nonlinear aspects and the numerical solution procedures. Since the cyclic loading introduces further complexities compared with monotonic loading, limited finite element analysis to predict the cyclic response of RC shear walls has been performed. The progress of finite element models describing the cyclic behaviour of RC shear walls and panels may be reviewed in two periods. During the decades 1970s and 1980s, although the numerical problems associated with employing inappropriate constitutive models limited the application of finite element analyses especially at structure level, the investigations led to establishing fundamental concepts, recognising and quantifying the influence of different aspects of concrete and reinforcing steel on cyclic response and developing reliable and stable numerical procedures. In the second period, from 1990 to present, some refined finite element models have been quite successfully developed to predict the cyclic response of RC shear walls and panels.
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The Period from 1970 to 1989 Cervenka and Gerstle (1971, 1972) carried out one of the earliest studies to analyse the behaviour of RC shear panels under monotonic and cyclic loading using the finite element method. They used the smeared-fixed crack approach with only one open crack allowed at a point associated with smeared reinforcing steel. The cracked concrete was modelled as an orthotropic material without stiffness perpendicular to crack and shear stiffness tangential to crack which could carry the stresses only parallel to crack directions. The reinforcing steel was assumed as an elastic-perfectly plastic material while the response of concrete was modelled as being elastic-perfectly plastic and elastic-brittle in compression and tension, respectively. To extend the model toward including cyclic conditions, linear unloading and reloading relationships with the slope equal to initial elastic modulus were applied. While the resulting model could reasonably predict the response of two shear panels tested by Cervenka and Gerstle (1972) under monotonic loading, it failed to simulate that of a shear panel under reversed cyclic loading. This failure is attributed to not including important postcracking aspects such as tension stiffening and aggregate interlock effects into the model and, more importantly, the exclusion of major effects exhibited under cyclic conditions such as the energy dissipation in the concrete and steel during unloading and reloading phases and the degradation of concrete stiffness and strength. With regard to the shear panels tested under monotonic loading, the experimental results were governed by yielding of the reinforcing steel and, therefore, the concrete response was not as critical. Darwin and Pecknold (1974, 1976) were the first to develop a nonlinear finite element model for the analysis of reinforced concrete membranes that included major cyclic aspects. Adopting the smeared-fixed crack approach, the cracked concrete was modelled as an orthotropic material with two orthogonal open cracks allowed at a point. The model included a shear retention factor to represent the aggregate interlock and dowel action but tension stiffening and bond deterioration between concrete and steel reinforcement were ignored. The reinforcing steel was taken as smeared with an elastic-perfectly plastic strain hardening material model, including the Bauschinger effect. To incorporate effects due to the biaxial stress states, Darwin and Pecknold
2-45
introduced the concept “equivalent uniaxial strain” which was later adopted by many researchers. A more realistic stress-strain relationship for concrete proposed by Sanez (1964) was used and some important aspects observed in cyclic response of concrete, including the degradation of concrete stiffness and strength and the energy dissipation at each hysteretic cycle, were included in the model. As the nonlinear solution procedure an incremental-iterative tangent stiffness approach was adopted. The model was corroborated against the shear panel W4 tested by Cervenka and Gerstle (1972) for two and one half cycles. While the analytical results were in a reasonable agreement with the experimental results for the first cycle, they began to deviate during following cycle. Palermo (2001) attributed the deviation of the model from the test data to the numerical procedure used in the finite element analysis rather than the constitutive models. Developing an empirical bond-slip model, Shipman and Gerstle (1979) studied the effect of bond deterioration on the cyclic response of the panel W4 (Cervenka and Gerstle, 1972). Comparing the numerical results with those by Darwin and Pecknold (1974), it was found that the effect due to the degradation of concrete strength and stiffness is more significant than that of the bond deterioration in the cyclic response of the shear panels. To analyse a slender shear wall tested by Cardenas et al. (1980), Aktan and Hanson (1980) proposed a model in which the total response of the wall was decomposed into linear and nonlinear parts. The nonlinear joint element was modelled based on previous extensive experimental results on shear walls. The shear stiffness was also assumed to be a function of normal stress acting on the element after closing the crack. Nevertheless, the experimental moment-curvature curve could not be reproduced well by analytical results, even though the shear pinching characteristics were fairly represented. Furthermore, since it is very difficult to develop the material model for the nonlinear joint element for general structural members, the application of the model is limited. Noguchi (1985) reviewed a large number of finite element models of reinforced concrete members subjected to reversed cyclic loading. He pointed out that the general drawback of these early models was the inability to accurately represent the unloading
2-46
stiffness and the area of hysteresis loops. In other words the degradation of concrete stiffness and the energy dissipation were the challenging issues to be simulated. It is attributed to this fact that almost all studies previous to 1985 emphasised the modelling rather the material nonlinearity. Studies through the 1980s, however, showed that the modelling of the post-cracking effects such as tension stiffening and compression softening are vital to the analyses of reinforced concrete structures (Vecchio and Collins, 1982, 1986, Stevens et al., 1987). To take into account the findings concerning the post-cracking response of reinforced concrete in finite element analysis, Stevens et al. (1987) developed a more comprehensive constitutive model for the concrete based on the modified compression field theory of Vecchio and Collins (1986) and employing a smeared-rotating crack approach for modelling the concrete. Besides considering the effects due to the material nonlinearity and the biaxial stress state in the concrete, post-cracking effects such as tension stiffening, compression softening and bond-slip were incorporated into the finite element model. The model accounts for the cyclic aspects of concrete response containing the degradation of concrete strength and stiffness as well as the energy dissipation during each unloading and reloading cycle.
Adopting a discrete beam
element for reinforcement, the stress-strain relationship of the reinforcing steel was modelled to include the reduced yielding caused by cracking in addition to the strain hardening and the Bauschinger effect. The proposed model was verified against the experimental results of three panels tested by the researchers under cyclic loading. When it was used at the structure level, however, numerical problems arose from the complexity of constitutive relationships and most analyses could not be completed.
The Period from 1990 to Present Xu (1991) developed a finite element model using the multi-directional fixed crack approach as a solution to the drawbacks of the fixed crack and rotating crack models. Considering multiple non-orthogonal cracks at a cracked point, the model could account for the nonlinear response due to each crack and superimpose the effects from all
2-47
cracks. Reinforcing steel was taken as smeared with a constitutive stress-strain relationship including yielding, stress hardening and the Bauschinger effect. Using trilinear unloading and linear reloading relationships, the energy dissipation was included in the cyclic model. In addition the model incorporated the degradation of strength and stiffness of concrete due to cyclic damage. Although the model showed good success at the element level, at the structure level like many previous attempts, the complexity of model led to the numerical problems during the analyses. The progress and improvement of techniques used in finite element modelling and computational procedures as well as the better understanding of main aspects of the cyclic response of reinforced concrete has led to a number of successful cyclic analyses of RC shear walls (Okamura and Maekawa, 1991, Sittipunt and Wood, 1993, Elmorsi et al., 1998, Vecchio, 1999, Palermo and Vecchio, 2003, Kwak and Kim, 2004a, Palermo and Vecchio, 2004). The main features of these models are summarised in Tables 2.1, 2.2 and 2.3. The following conclusions are drawn: The equivalent uniaxial strain approach has been successfully employed to simulate the response of RC shear walls subjected to cyclic loading. The main reason for this success is attributed to the use of empirical relationships expressing the complex behaviour of concrete being directly incorporated into the constitutive relationships (Kwan and Billington, 2001). As extensive cracking in shear walls is usually expected, the smeared crack model has been the dominate approach adopted by researchers. Both fixed crack and rotating crack approaches have been used successfully for modelling of RC structures under reversed cyclic loading. Most researchers have modelled the reinforcing steel as smeared throughout the shear walls. The finite element type used in the studies ranges from 4-node isoparametric membrane elements to 12-node quadrilateral isoparametric elements.
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The tangent stiffness formulation is commonly used for analysis of cyclic response of RC structures as shown in Table 2.1. Although the secant stiffness method has been criticised by some that it cannot be effectively employed to analyse the response to general loading conditions (ASCE Committee 447, 1982), the behaviour of shear walls under reversed cyclic loading has been successfully modelled using this method (Vecchio, 1999, Palermo and Vecchio, 2003, 2004). Post-cracking effects including tension stiffening and compression softening have been included in the finite element models. Assuming perfect bond between reinforcement and concrete, however, the bond-slip effect has been ignored by most researchers except by Okamura and Maekawa (1991). It has been argued that since shear walls usually contain the low amounts of reinforcing steel the effect of bond-slip is not significant. The rotating crack approach eliminates the need to express a separate shear transfer model whereas the studies undertaken based on the fixed crack approach have employed the cyclic shear transfer models. Because typically shear walls have low reinforcement ratios, Okamura and Maekawa (1991) and Elmorsi et al. (1998) neglected the contribution of dowel action and assumed that the cracked shear stiffness is due to only aggregate interlock. In contrast, Sittipunt and Wood (1993) considered the shear stiffness to be attributed to both effects. The major cyclic characteristics of concrete modelling applied by researchers include the unloading and reloading curves, the degradation of concrete strength and stiffness and the crack-closing transition curve. As it is shown in Table 2.2, the cyclic model of concrete employed by Vecchio (1999) compared with other models is simpler not including some cyclic modelling aspects such as strength degradation, nonlinear unloading curve for concrete indicating energy dissipation nor a crack-closing transition curve. Although this model yielded the reasonable prediction for the cyclic behaviour of shear walls dominated by reinforcing steel response, it failed for those that showed pinched hysteretic loops indicating shear dominant behaviour, where the other models were successful.
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Table 2.1 – Comparing the main features of finite element modelling used by various researchers to analyse the cyclic response of RC shear walls Researchers
Approach for Constitutive Relations
Crack Modelling
Steel Modelling
Element Type
Iterative Solution Technique
Okamura and Maekawa (1991)
orthotropic equivalent uniaxial strain
smeared-fixed orthogonal crack
smeared
8-node isoparametric membrane element
tangent stiffness
orthotropic equivalent uniaxial strain
smeared-fixed orthogonal crack
discrete, 2-node truss element
4-node isoparametric membrane element
tangent stiffness
Elmorsi et al. (1998)
orthotropic equivalent uniaxial strain
smeared-fixed orthogonal crack
smeared
12-node quadrilateral membrane element
tangent stiffness
Vecchio (1999)
orthotropic equivalent uniaxial strain
smearedrotating crack
smeared
4-node isoparametric membrane element
secant stiffness
Palermo and Vecchio (2003, , 2004)
orthotropic equivalent uniaxial strain
smearedrotating crack
smeared
4-node isoparametric membrane element
secant stiffness
Kwak and Kim (2004)
orthotropic equivalent uniaxial strain
smearedrotating crack
smeared
4-node isoparametric membrane element
tangent stiffness
Sittipunt and Wood (1993)
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Table 2.2 - Comparing the main features of concrete modelling used by various researchers to analyse the cyclic response of RC shear walls
Post-Cracking Effects
Cyclic Characteristics Researchers
Unloading curve
Reloading curve
Strength degradation
Stiffness degradation
Crack Closing Transition Curve
Tension stiffening
Compression softening
Shear transfer across cracks
Bondslip
Okamura and Maekawa (1991)
Nonlinear in both compression and tension
Linear in both compression and tension
Not included
Included
Linear
Included
Included
Only aggregate interlock
Sittipunt and Wood (1993)
Trilinear in compression, linear in tension
Bilinear in compression, linear in tension
Included only in compression
Included
Nonlinear
Included
Included
Aggregate interlock and dowel action
Joint element between concrete elements Not included
Elmorsi et al. (1998)
Bilinear in compression, linear in tension
Linear in both compression and tension
Included only in compression
Included
Nonlinear
Included
Included
Only aggregate interlock
Not included
Vecchio (1999)
Elastic linear in compression, secant linear in tension
Linear in both compression and tension
Not included
Included
Not included
Considered both cases with and without tension stiffening
Included
Not included because of using rotating crack model
Not included
Palermo and Vecchio (2003, , 2004)
Nonlinear in both compression and tension
Linear in both compression and tension
Included in both compression and tension
Included
Linear
Included
Included
Not included because of using rotating crack mode
Not included
Kwak and Kim (2004)
Nonlinear in compression, linear in tension
Linear in both compression and tension
Not included
Included
Nonlinear
Included
Included
Not included because of using rotating crack mode
Not included
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Table 2.3 – Comparing the main features of reinforcing steel modelling used by researchers to analyse the cyclic response of RC shear walls Researchers
Basic Constitutive
Reduced Yielding
Strain
Bauschinger
Stress-Strain
Stress due to
Hardening
Effect
Curve
Embedding or
Effect
Bond
Okamura and Maekawa (1991)
Bilinear
Sittipunt and Wood (1993)
Linear for elastic and
Not included
Included using a
Included using a
linear relation
proposed model
Included using a
Included using a
nonlinear relation
proposed model
Included using the model
Included using a
Included using the
of Stevens (1987)
linear relation
modelled of Menegotto
Not included
plastic regions and nonlinear for strain hardening
Elmorsi et al. (1998)
Bilinear
and Pinto (1973)
Vecchio (1999)
Trilinear
Palermo and Vecchio (2003, , 2004)
Trilinear
Kwak and Kim (2004)
Bilinear
2.4
Not included
Not included
Included using a
Included using the
linear relation
model of Seckin (1981)
Included using a
Included using the
linear relation
model of Seckin (1981)
Included using the model
Included using a
Included using the
of Belarbi and Hsu
linear relation
modelled of Menegotto and Pinto (1973)
(1994)
Experimental Studies
Prior to the 1970s, although the shear walls had been employed in a large number of buildings, the knowledge about inelastic response of reinforced concrete shear walls subjected to cyclic loading was limited. Many experimental investigations by various research groups were undertaken during the 1970s and later. Over the past 30 years, many tests have been carried out on shear wall specimens, simulating single or
2-52
multi-storey shear walls. The majority of tests have been undertaken on normal strength concrete under monotonic loading. Experimental works on HSC shear walls subjected to cyclic condition are few. With respect to these limitations and to reach some conclusions on the behaviour of shear walls, in general, results of tests conducted on normal strength concrete shear walls are also discussed. However, in order to remain within the scope of the present research, the review is concentrated on behaviour of low-rise shear walls.
2.4.1 Barda et al (1977) Barda et al. (1977) tested eight framed reinforced concrete shear wall specimens (Figure 2.22) subjected to monotonic and reversed cyclic loading without axial load. The compressive strength of the concrete ranged from 21 to 29 MPa. The main parameters of experiment included the height-to-length ratio (0.25 to 1.0) and vertical reinforcement and horizontal reinforcement ratios (0 to 0.5 percent). The main findings from the experimental study were reported as follows: •
The longitudinal reinforcement in the edge elements did not have influence on the shear strength of walls.
•
Increasing the height-to-length ratio for a shear wall, where other properties were maintained as invariable, led to decreasing the shear strengths.
•
The failure load of shear walls under reversed cyclic loading was approximately 10% less than that of similarly detailed shear walls under monotonic loading. The main reasons for this are the existence of cracks in orthogonal directions and increasing degradation of bond between reinforcement and concrete under cyclic loading.
•
Specimens without edge elements failed suddenly compared with the gradual failure of framed shear walls due to the frame action provided by the end elements. 2-53
150 mm
1525 mm
2405 mm
Hv
A
B
457 mm
B
3048 mm
1220 mm
A
610 mm
1220 mm
102 mm
Section A-A
572 mm
102 mm
1904 mm
572 mm
Section B-B
Figure 2.22 – Typical shear wall specimen tested by Barda et al. (1977).
This research was one of the earliest investigations on shear walls and the design equations included in the ACI 318 Code are based on its outcomes. Some important issues such as the effects of axial load and strength of concrete, for example, were not investigated in this study.
2-54
2.4.2 Oesterle et al. (1976, 1978, 1984) As part of an extensive experimental program conducted by the Portland Cement Association (PCA), Oesterle et al. (1976, 1978, 1984) carried out a large number of tests to investigate the inelastic behaviour of reinforced concrete shear walls subjected to monotonic and reversed cycling loading. The test specimens were similar to that tested by Barda et al. (1977) with different wall cross sections including flanged, barbell and rectangular shapes as shown in Figures 2.23 and 2.24. The compressive strength of the concrete ranged from 21 MPa to 55 MPa and the yield stress of reinforcing steel was about 420 MPa. The reinforcement details for each specimen were selected based on the provisions of the 1977 ACI Building Code (1977). The ratio of longitudinal reinforcement was approximately 0.3 percent with the amount of transverse reinforcement varied from 0.3 to 1.38 percent. In addition to the wall cross section, the primary experimental parameters included the axial compression load, the compressive strength of concrete, confinement reinforcement in boundary elements, the amount of transverse shear reinforcement, the level of shear stress, load history and lap splices within plastic regions. Based on the experimental results the following conclusions were drawn: •
The strength and ductility of specimens under reversed cyclic loading are less than those of similar specimens but under monotonic loading.
•
The response of shear wall specimens under reversed cyclic loading was ductile. This aspect was attributed to the special reinforcement details used in this study.
•
Transverse reinforcement greater than the amount recommended by ACI Building Code (1977) did not improve the total shear strength.
•
' The axial load corresponding to an axial compressive stress of 0.1 f c increased
the ductility of the walls. •
The confinement reinforcement in the boundary elements had a considerable effect on the inelastic response of the walls.
2-55
•
Failure of shear walls under reversed cyclic loading is limited by web crushing.
•
Web crushing depended on the amount of shear stress and the level of
203 mm
deformation.
2360 mm
4570 mm
1910 mm
3050 mm
A
610 mm
5383 mm
A
1220 mm
102 mm
1220 mm
Section A-A
Figure 2.23 – Typical shear wall specimen tested by Oesterle et al. (1984).
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102 mm
(a)
305 mm
102 mm
1910 mm
(b)
(c) 102 mm
1706 mm
305 mm
910 mm
1300 mm
102 mm
305 mm
102 mm
Figure 2.24 – The cross-section of specimens tested by Oesterle et al. (1984): (a) rectangular, (b) barbel, (c) flanged .
2.4.3 Maier and Thurlimann (1985) Maier and Thurlimann (1985) undertook tests on 10 low-rise shear wall specimens, of dimensions shown in Figure 2.25, to investigate strength and behaviour under vertical and horizontal loads. Eight specimens were tested under monotonic loading and two specimens (S5 and S7) were tested under reversed cyclic loading. The height-to-length ratio of the specimens was about 1.0 and the ratios of reinforcement were varied as 0.0, 0.57 and 1.0 percent in the horizontal direction and 0.57, 1.0 and 2.5 percent in the vertical direction. The compressive strength of concrete was 29 to 37 MPa and the yield stress of reinforcement was about 500 MPa. The test variables included the shape of wall cross section, openings in compression zone, arrangement and amount of
2-57
reinforcement and amount of axial load. The main conclusions drawn on the experimental results are summarised as follows: •
The horizontal resistance is a function of the cross-sectional geometry, the percentage of longitudinal reinforcement and the axial load. Increasing the last two parameters and adding boundary elements improve the strength of shear walls.
•
The amount of transverse reinforcement has a negligible effect on the ultimate
B
B
150 mm
1700 mm
100 mm
700 mm
100 mm
Section A-A
400 mm 150 mm
A
380 mm
1820 mm
A
1200 mm
240 mm
load but improves the ductility of shear walls.
100 mm
260
1180 mm
260
Section B-B
Figure 2.25 – Typical shear wall specimen tested by Maier and Thurlimann (1985).
2-58
•
A high axial load or a high percentage of longitudinal reinforcement decreases the ductility of shear walls and results in brittle fracture.
•
Under lateral cyclic loading and low axial load, no reduction in ultimate strength and no significant change in ductility were observed as compared with similar specimen but under monotonic loading. However at higher axial load, the failure load is decreased due to the load reversal.
•
In general, the observed failure mode was lateral splitting of the concrete and spalling of concrete cover in the compression corner of the specimens.
•
Under high axial load, a horizontal fracture zone over the length of specimens was observed.
While the study provided useful information regarding the behaviour of squat shear walls cast with the normal strength concrete, the effects due to the use of high strength concrete remained to investigate.
2.4.4 Lefas et al. (1990) and Lefas and Kotsovos (1990) Lefas et al. (1990) undertook an experimental program on the behaviour of shear walls loaded under either monotonic or cyclic loading. It the first part of the experiment programme, thirteen shear walls, shown in Figure 2.26, were tested under a combination of the constant axial load and the horizontal monotonic loading. The experimental parameters investigated were the height-to-width ratio, the axial load, the concrete strength and the ratio of transverse reinforcement. Two groups of shear walls with the height-to-width ratios equal to 1 and 2 were tested. The wall cross-sections were rectangular with additional reinforcement confined by stirrups used at the edge boundaries. The compressive strength of the concrete was varied from 30 MPa to 55 MPa and 500 MPa grade reinforcement was used. The ratio of longitudinal reinforcement for all specimens was constant and equal to 2.4 percent. The horizontal
2-59
reinforcement ratio ranged from 0.4 percent to 1.1 percent. The main findings from the monotonic test were: •
The transverse reinforcement did not significantly influence the shear strength of the walls. The strength and deformation characteristics of specimens were independent of
B
B
65 mm
300 mm
1750 mm
A
1300 mm
150 mm
the compressive strength of concrete within the range of 30 to 55 MPa.
1150 mm
A
200 mm
Section A-A 200 mm
•
250 mm
250 mm
650 mm
Section B-B
Figure 2.26 – Typical shear wall specimen tested by Lefas et al. (1990).
2-60
•
While the axial load decreased either vertical or horizontal deformation it caused an increase in both the horizontal failure load and the lateral stiffness characteristics. For a higher ratio of height-to-width this increase became further pronounced.
In the second part of the investigation, Lefas and Kotsovos (1990) tested one shear wall under monotonic loading and three specimens in the different regimes of reversed cyclic loading. The test specimens were similar to those tested previously and shown in Figure 2.5. The longitudinal and transverse reinforcement ratios for all specimens were equal to 1.5 percent and 0.35 percent, respectively. The following conclusions were drawn on the experimental results: •
The shear wall specimens showed a flexural response before the failure with considerable energy dissipation.
•
The strength and ductility of the specimens were independent of the cyclic loading regime.
This study led to further information about the affects of some major parameters on the response of shear walls subjected to monotonic loading. It also showed the independency of cyclic response of shear walls to the reversed cyclic loading regime. All specimens failed in a flexural mode and, therefore, the response of shear walls failing in shear was not investigated.
2.4.5 Pilakoutas and Elnashai (1995) An experimental research program was conducted by Pilakoutas and Elnashai (1995a, 1995b) to study the behaviour of six RC shear walls under severe cyclic loading to destruction. The shear wall specimens had a rectangular cross section with a height-towidth ratio equal to 2 as shown in Figure 2.27. The walls were built using concrete with a compressive strength between 32 and 46 MPa. The yield stress of reinforcement was approximately 500 MPa. The cyclic loading was imposed on the specimens at a slow 2-61
rate in displacement increments of 2 mm until failure occurred. At each displacement level two full excursions performed. All specimens had no imposed axial load. The main test parameters included the ratios of transverse and longitudinal reinforcement in the web wall and the longitudinal and tide reinforcement ratios as well as the width of the boundary elements. The following conclusions were drawn on the experimental results: •
The failure mode was primarily dependant on the amount and distribution of the wall transverse reinforcement.
•
The shear force is transmitted through the concrete in compression and the link reinforcement in the tensile zone. The walls failed after the yielding of link reinforcement and when the shear capacity of compressive concrete was exceeded.
•
The wall transverse reinforcement beyond the amount needed to carry the maximum lateral load did not have significant effects on the strength and ductility aspects of specimens.
•
Although the shear deformation was a considerable component of total deformation, the overall energy dissipation was significantly attributed to the flexural action.
•
The walls tested without applied axial load exhibited a longitudinal extension due to cyclic loading, particularly in the lower levels of the walls.
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600 mm
A
1200 mm
Stiff Top Beam
60 mm
Stiff Bottom Foundation
A
Section A-A
Figure 2.27 – Typical shear wall specimen tested by Pliakoutas and Elnashai (1995a).
2.4.6 Gupta and Rangan (1996) Gupta and Rangan (1996) conducted tests on eight HSC shear wall specimens subjected to constant vertical and monotonically increasing horizontal loads. The test specimens were similar to those tested by Maier and Thurlimann (1985), as shown in Figure 2.28, but built by using high strength concrete. The compressive strength of concrete ranged from 60 MPa to 80 MPa and the nominal yield stress of the reinforcement was 400 MPa. The major test parameters of the study were the ratio of horizontal reinforcement, the ratio of vertical reinforcement and axial load.
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200 mm
B
B
400 mm
1600 mm
A
1000 mm
1300 mm
100 mm
1800 mm
A
575 mm
100 mm
375 mm
75 mm
Section A-A
100 mm
400 mm
1000 mm
400 mm
Section B-B
Figure 2.28 - Typical test specimen tested by Gupta and Rangan (1996).
2-64
For each specimen the full axial load was initially applied and then the horizontal load was monotically applied in increments of 50 kN before cracking and 10 kN after cracking up to failure. A summary of test results is given in Table 2.4. Based on the study the following conclusions were drawn: •
The lateral failure load was increased and the ductility was reduced with an increase in the vertical load.
•
An increase in longitudinal reinforcement ratio led to an increase in the horizontal failure load. This increase was greater when the vertical load was zero.
•
Increasing the transverse reinforcement ratio gave only a small increase in the lateral failure load.
Table 2.4 - Summary of test results reported by Gupta and Rangan (1996) Specimen
Equivalent ratio of transverse reinforcement (percent)
Equivalent ratio of longitudinal reinforcement (percent)
Axial load
Failure transverse load
(kN)
(kN)
S-1
0.5
1.0
0.0
428
shear
S-2
0.5
1.0
610
720
shear
S-3
0.5
1.0
1230
851
shear
S-4
0.5
1.5
0.0
600
shear
S-5
0.5
1.5
610
790
shear
S-6
0.5
1.5
1230
970
shear
S-7
1.0
1.0
610
800
shear
S-F
0.5
1.0
310
487
flexural
2-65
Failure mode
The conclusions from the study are similar to those for tests carried out on specimens cast with normal strength concrete. Although this study did not involve tests with cyclic loading, it is one of few available on HSC shear walls.
2.4.7 Kabeyasawa and Hiraishi (1998) Kabeyasawa and Hiraishi (1998) reported the results of an extensive experimental research program undertaken on 21 HSC shear walls as a part of a five-year national research project in Japan. The specimens were approximately one-quarter scale with the same cross sectional dimensions (Figure 2.29). Four series of tests were conducted as follows: •
RC flexural tests: six specimens designed to yield first in flexure and then reach different ultimate deformations.
•
RC preliminary shear tests: two specimens subjected to non-symmetric lateral loading designed to fail in shear mode prior to yielding the longitudinal reinforcement.
•
RC shear tests: eight specimens subjected cantilever loading designed to fail in shear mode prior to yielding the longitudinal reinforcement.
•
RC bi-directional tests: five specimens tested to investigate the deformation capacity of HSC flexural walls subjected to bi-directional lateral earthquake loads.
The concrete strength and the yield stress of reinforcing steel used in walls varied from 60 to 137 MPa and from 753 to 1420 MPa, respectively. The concrete strength, the reinforcing steel strength, transverse and longitudinal reinforcement ratios and axial and horizontal load conditions were systematically varied for the investigation of strength and ductility characteristics in both flexural or shear failure modes. It was concluded from the experimental results that:
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•
The high strength concrete can effectively be used to build ductile shear walls failing in a flexural mode, though the pinching effect observed in the hysteresis aspect should be considered in design and analysis.
•
The flexural strength and deformations could be evaluated using the conventional flexural theory. The strength of walls failing in shear with yielding of web reinforcement could fairly be estimated using the arch and truss model by modifying the effective factors for HSC. For walls failing in shear without yielding of web reinforcement, however, the model overestimated the shear strength.
B
3000 mm
B
200 mm
600 mm
4200 mm
600 mm
A
A
2000 mm
500 mm
Section A-A 80 mm
150 mm 150 mm
•
200
1300 mm
200 mm
Section B-B
Figure 2.29 – Typical shear wall specimen tested by Kabeyasawa et al. (1993).
2-67
2.4.8 Palermo and Vecchio (2002) Palermo and Vecchio (2002) tested two large scaled squat flanged shear walls, DP1 and DP2, subjected to lateral displacement reversals. The experimental program was designed to investigate the behaviour of shear walls under cyclic displacements and the main test parameter included in the study was axial load. The geometrical properties of
3280 mm
A
95 mm
B
75 mm
B 4415 mm
Section A-A
462.5 mm
685 mm
3045 mm
462.5 mm
A
620 mm
4000 mm
685 mm
2020 mm
640 mm
the specimens were similar as shown in Figure 2.30.
3075 mm
Section B-B
Figure 2.30 - Typical test specimens by Palermo and Vecchio (2002).
2-68
The compressive strength of concrete used in wall and end flanges was 20 MPa and the yielding and ultimate stresses of reinforcement were 605 MPa and 652 MPa, respectively. The ratios of longitudinal and transverse reinforcement were 0.73 percent and 0.79 percent, respectively. Specimen DP1 had a constant axial load of 960 kN, while specimen DP2 had no axial load. After applying the axial load the shear walls were displaced laterally in 1 mm increments. The test results showed that the axial load had an important effect on the response of shear walls under reversed cyclic loading. As it can be observed in Figure 2.31 and Table 2.5, the maximum load resisted by DP1 was about 40 percent more than that of DP2. Also DP1 was more ductile and dissipated more energy than DP2. Furthermore, the hysteresis loops of DP2 showed more pinching. It is important to note that the squat shear walls specimens presented greatly pinched hysteresis response with small energy dissipation and their behaviour was extensively dominated by shear induce mechanisms. The failure modes of the two specimens were different. Whereas DP1 failed because of crushing of concrete over a widespread zone of web wall, the failure of DP2 was due to a sliding shear plane extending the entire length of the web wall. The researchers concluded that the stiffness of the flange wall had key role in the failure mode so that stiff flanges resulted in forming vertical slip planes in DP1. In DP2, however, probably weaker concrete in the top zones caused a shear sliding plane to form near the top slab.
Table 2.5 - Summarised Test Results of Palermo and Vecchio (2002) Axial load
Positive Direction
Negative Direction
(kN)
Ultimate load (kN)
Corresponding Displacement (mm)
Ultimate load (kN)
DP1
940
1298
11.14
-1255
-11.09
DP2
0
904
9.15
-879
-9.08
Specimen
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Corresponding displacement (mm)
Failure mode
Concrete Crushing Sliding Shear
(a)
(b)
Figure 2.31 - Load-Deformation response: (a) Specimen DP1; and (b) Specimen DP2 (Palermo and Vecchio, 2003, 2004).
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2.4.9 Farvashany (2004) To investigate the questions raised in the earlier research project conducted at the Curtin University of Technology (Gupta and Rangan, 1996, 1998), Farvashany (2004) carried out tests on seven HSC shear walls subjected to monotonic loading. The specimens were similar to those of Gupta and Rangan (1996) but with an increased height-to-width ratio equal to 1.25. The dimensions of the specimens are shown in Figure 2.32 with the walls detailed to fail in shear. The compressive strength of concrete ranged from 83 to 100 MPa and the yield stress of reinforcement used in the walls was about 500 MPa. The main test parameters of the study were axial load, compressive strength of concrete and longitudinal and transverse reinforcement ratios. Fro the experimental results Farvashany concluded that: •
An increase in the axial load gave an increase in the horizontal failure load but a decrease in lateral deformation at the failure load.
•
An increase in the ratio of longitudinal reinforcement in the wall resulted in an increasing in the lateral load at failure.
•
The increase in transverse reinforcement ratio did not have considerable effect on the failure load but it increased the ductility.
•
The failure load of specimens increased with increasing the concrete compressive strength.
This study, accompanied with that of Gupta and Rangan (1996), provides a considerable data base to validate the analytical models and resulted in valuable information regarding to the behaviour of HSC low-rise shear walls under monotonic loading.
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200 mm
1300 mm
1600 mm
90 mm
B
375 mm
1100 mm
A
75 mm
300 mm
B
575 mm
A
100 mm
1800 mm
100 mm
375 mm
75 mm
Section A-A
90 mm
460 mm
880 mm
460 mm
Section B-B
Figure 2.32 – Typical shear wall specimen tested by Farvashany (2004).
2.5
Summary
The analytical, numerical and experimental researches related to the behaviour of shear walls under monotonic and cyclic loading have been reviewed. Based on the elasticitybased equivalent uniaxial stains approach, a variety of constitutive models for concrete, steel reinforcement and the interaction between them have been introduced. These models can be successfully implemented into a finite element procedure to analysis of reinforced concrete structures, if they capture major aspects of their behaviour such as non-linear stress-strain relationships of concrete and steel, concrete cracking, 2-72
compression softening, tension stiffening and so on. Finite element methods based on the smeared crack model have been commonly used to analyse shear walls. Two major approaches in this area are fixed and rotating crack model. While the former is a more general model and has been used by many researches for analysis of shear walls, the latter is simpler and is more easily implemented. The experiments carried out on normal strength concrete shear walls have shown the effects of major parameters such as longitudinal and transverse reinforcement ratios, axial load, the ratio of height-to-length, edge elements and loading regime on the behaviour of walls. Some of these parameters have been also investigated by a few researchers that conducted tests on high-strength concrete shear walls under monotonic loading. However, experimental data on the behaviour of high-strength concrete shear walls under reversed cyclic loading is very limited and the study on this area is essentially required.
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CHAPTER 3
FINITE ELEMENT MODELLING
3.1
Introduction
The non-linear finite element analysis of RC structures involves the simulation of their response under loading and ambient conditions. The significant advantage of this method is that it can be applied to a wide range of problems. The reliability of a simulated response is largely dependant on the nonlinear constitutive material models used as well as the finite element procedure adopted for the analysis. The constitutive models should not only permit consistent interpretations of past behaviour of RC members but also predict accurately their future response. The separate constitutive models describing the different phenomena should be combined and integrated into a consistent theoretical framework for the treatment of the entire response. Starting from a description of the crack membrane model (CMM) as the fundamental theory of analysis, this chapter presents several constitutive material models used in this study. Then, the finite element implementation of constitutive models and the numerical algorithms used in non-linear analysis are described.
3.2
Crack Membrane Model
3.2.1 Background The compression field theory (CFT) (Mitchell and Collins, 1974, Collins, 1978, Collins and Mitchell, 1981) formulated equilibrium, compatibility and stress-strain relationships in terms of average stresses and averages strains assuming the cracks as uniformly distributed and smeared across an element. The CFT was based on a number of basic concepts and described using Mohr’s circles for the transformations of strains and stresses with the full response of a RC membrane determined by adding the stress-strain relationships in principal directions of the concrete to that of the reinforcing steel. The basic assumption behind the CFT is that the directions of the principal average total strains are the same as the directions of principal average concrete stresses and, thus, the cracks are taken as rotating and stress-free. The main drawback of the CFT was that the tension stiffening effects were neglected so that the predicted response of a RC structure is too soft. To address this weakness, as well as to determine a better constitutive relationship for concrete, Vecchio and Collins (Vecchio and Collins, 1982, 1986) developed the modified compression field theory (MCFT) by introducing empirical relationships between average total strains and the average tensile and compressive stresses in the concrete to account for tension stiffening and compression softening effects, respectively. The MCFT contributed significantly to the progress in the research areas of structural concrete in the 1980s and the 1990s and was accepted by many analysts. Although the MCFT led to more realistic results, the link to limit analysis was lost as equilibrium was expressed in the form of average stresses across an element (Kaufmann and Matri, 1998). Also, the empirical tension stiffening relationship for concrete has and continues to undergo debate with various researchers proposing various models and variations to that originally formulated (Collins and Mitchell, 1991, Okamura and Maekawa, 1991, Belarbi and Hsu, 1994, Bentz, 1999).
3-2
Kaufmann and Marti (1998) proposed an alternative tension stiffening model by developing a two-dimensional representation of the tension chord model (TCM) of Marti et al. (1998) and combining it with the basic components of the MCFT. The new model was named the crack membrane model (CMM) for the analysis of RC membranes. The CMM maintains equilibrium at the crack faces and, thus, there is no need to describe the equilibrium in terms of average stresses across an element. In this way the link to limit analysis is maintained. Moreover the tension stiffening effect is developed as a rational model based on a simple bond shear stress-slip constitutive relationship. Assuming the crack faces to be stress free and the concrete average principal stresses and the average principal strains being coincident, the model was implemented as a rotating crack procedure. Although a generalised CMM involving shear and normal stresses acting on the cracks was expressed (Kaufmann and Marti, 1998), no progress in this direction has yet been implemented (Marti, 2005). In this study, the CMM is considered in its more general form and developed as a fixed crack model.
3.2.2 Concepts and Relationships Considering a plane concrete element subjected to in-plane stresses, uniformly intersected by a system of parallel cracks and orthogonally reinforced by a regular grid of steel bars, as shown in Figure 3.1, from equilibrium at the cracks, it can be shown that
σ x = σ cn cos 2 θ r + σ ct sin 2 θ r + τ cnt sin (2θ r ) + ρ xσ sx
(3.1a)
σ y = σ cn sin 2 θ r + σ ct cos 2 θ r − τ cnt sin(2θ r ) + ρ yσ sy
(3.1b)
τ xy = 0.5(σ cn − σ ct )sin (2θ r ) − τ cnt cos(2θ r )
(3.1c)
where θr is the angle between a vector normal to the cracks and the global X-axis (-π/2 ≤ θr ≤ π/2); σx, σy and τxy are the in-plane normal and shear stresses in the global XY-coordinate system, respectively; σcn and σct are the concrete stresses normal and
3-3
parallel to the direction of cracking, respectively; τcnt is the corresponding shear stress;
ρx and ρy are the steel reinforcement ratios in the global X- and Y-directions and σsx and σsy are the stresses in the X- and Y-reinforcement, respectively. It is assumed that the element is sufficiently large compared to the crack and bar spacings and that the internal
forces are uniformly distributed over a distance of several cracks and bar spacings. The average tensile stresses in the concrete between cracks in the X and Y directions, as per the tension chord model shown in Figure 3.2, can be written as
σ ctsy =
λ x f ct 2
λ y f ct 2
⋅ g (τ b )
(3.2a)
⋅ g (τ b )
(3.2b)
σy
(a)
σx
σx
σx
θr
τxy
τcnt cos θ r
+
τcnt sin θr σct sin θ r
σy
ρ y σsy
(d)
n
t
(b)
θr
ρ x σsx
σcncos θ r
(c)
τxy
1
σ ctsx =
σct cos θ r τ cnt cos θr θ r τxy
+
σcn sin θr τcnt sin θr
σy 1
Figure 3.1- Orthogonally reinforced membrane subject to plane stress: (a) applied stresses; (b) axis notation; (c) and (d) stresses at a crack (Kaufmann and Marti, 1998). 3-4
m
sr
m
(a)
srmy
sr
m
sr
cft
(b) Y
λ x f ct
s rmy s rmy s rmy
λ y fct
X
srmx
θr srmx
Tension Stiffening Stresses
srmx
Figure 3.2- Tension stiffening stresses: (a) in material axis directions and (b) in orthogonal tension chord of cracked membrane (Kaufmann and Marti, 1998).
where f ct is the concrete tensile strength, λ x f ct and λ y f ct are the maximum tensile stresses in the concrete due to tension stiffening in the X and Y directions, respectively, and g (τ b ) is a function of bond shear stress τ b and is discussed further in the next parts. The tension stiffening factors, λ x and λ y , are determined based on the crack spacings as shown in Figure 3.2 and are given by
3-5
λx =
λy =
Δσ cx f ct
Δσ cy f ct
=
s rm s rmx0 cos θ r
(3.3a)
=
s rm s rmy0 cos θ r
(3.3b)
where Δσcx and Δσcy are the X- and Y-component stresses due to tension stiffening, s rm is the average crack spacing measured normal to the cracks and s rmx0 and s rmy0 are average crack spacings of uniaxial tension chords in the X and Y directions, respectively. The crack spacings in a fully developed crack pattern in the X and Y directions, s rmx and s rmy , are limited by (Marti et al., 1998)
s rmx0 ≤ s rmx ≤ s rmx0 2
s rmy0 2
(3.4a)
≤ s rmy ≤ s rmy0
(3.4b)
leading to 0.5 ≤ λ x ≤ 1 and 0.5 ≤ λ y ≤ 1 . Modelling steel-concrete bond shear stress as a stepped, rigid-perfectly plastic constitutive relationship (Figure 3.3(a)), the TCM gives the crack spacings for uniaxial tension chords in X and Y directions as (Marti et al., 1998) f ∅ (1 − ρ x ) srmx 0 = ct x ρx 2τ b 0
srmy 0 =
(
f ct ∅ y 1 − ρ y 2τ b0
(3.5a)
)
(3.5b)
ρy
3-6
(a)
τb τ b0 τ b1 δy
δ 1
2
cos θc
1
fct
θc Δσcx
θc
(b)
Δσcy sin θc
Figure 3.3- Tension chord model: (a) bond stress versus slip relation ship and (b) tension stiffening stress components (Foster and Marti, 2003).
where ∅x and ∅y are the diameters of the reinforcing bars and ρ x and ρ y are the ratios of reinforcement in X and Y directions, respectively, and τ b0 is the plastic bond strength before yielding of the reinforcing steel and reduces to τ b1 after yielding. Considering the fixed crack approach, the crack is geometrically fixed once first cracking takes place and then the cracked concrete is treated as an orthotropic material with the fixed 1- and 2- orthotropic axes that are, respectively, normal and tangential to the direction of the crack (Figure 3.4). Applying equilibrium of forces at the point of cracking (Figure 3.3(b)) gives Δ σ cx cos 2 θ c + Δ σ cy sin 2 θ c = f ct
(3.6)
3-7
Y
2
Reinforcement
1
θr X
Figure 3.4– Orthotropic and global axes in a cracked concrete element.
where θ c is the angle between the global X axis and the major principal stress, measured midway between the cracks. At the point of cracking θ c = θ r where θ r is the angle of principal stresses at the cracks. Substituting Eq. (3) into Eq. (6) results in the Vecchio and Collins (1986) crack spacing equation:
⎡ cos θ r sin θ r + s rm = ⎢ s rmy0 ⎢⎣ s rmx0
⎤ ⎥ ⎥⎦
−1
(3.7)
Foster and Marti (2002) showed that while the crack spacing can be rationally and fully determined using Mohr’s failure criteria, Eq.(3.7) is a reasonable approximation to the exact solution. To analyse the post-cracking behaviour, the crack orientation and crack spacing are assumed fixed and equal to their values at the moment of cracking. For fully developed cracks spaced at s rm across a continuum, and assuming no transmission of normal stresses in the concrete across the cracks, the crack width is given by (Foster and Marti, 2002) as
wcr = s rm (ε 1 + ν 12 ε 2 − λ f ct 2 E c )
3-8
(3.8)
where ε1 and ε2 are the average strains for element in 1- and 2- directions respectively,
ν12 is Poisson’s ratio for expansion in the 1-direction resulting from stress in the 2direction, λ is the uniaxial tension stiffening factor (0.5 ≤ λ ≤ 1.0) and Ec is the initial modulus of elasticity for concrete. If the dowel action across the cracks is ignored and cracks are assumed to be continuous, the stresses in the solid concrete between the cracks could be considered the same as stresses on the cracks (Bazant and Gambarova, 1980). By substitution of Eq. (3.2) into Eq. (3.1) and writing in terms of the orthotropic 1-2 coordinate system, the average stresses in an element due to any boundary tractions are given by
σ x = σ c1 cos 2 θ r + σ c 2 sin 2 θ r + τ c12 sin (2θ r ) + ρ xσ sx +
σ y = σ c1 sin 2 θ r + σ c 2 cos 2 θ r − τ c12 sin (2θ r ) + ρ yσ sy +
τ xy =
(σ c1 − σ c 2 ) 2
λ x f ct 2
λ y f ct 2
g (τ b )
(3.9a)
g (τ b )
(3.9b)
sin (2θ r ) − τ c12 cos(2θ r )
(3.9c)
where σ c1 , σ c 2 and τ c12 are average tensile, compressive and shear stresses in the concrete in the orthotropic 1-2-axis system, respectively. It is to be noted that prior to cracking the concrete is taken to carry tensile stresses in a linear elastic manner. After cracking, concrete tensile stresses result from two independent mechanisms, namely tension softening and tension stiffening. Whilst taking the former into account is especially important in analysing the concrete members with little or no reinforcing steel (Vecchio, 2000b), the latter has a significant effect on the general response of reinforced concrete members. It is to the advantage of the CMM that both effects are incorporated into Eq.(3.9) separately and, thus, the model can be applied for a wide range of problems without modification including problems involving fracture.
3-9
3.3
Constitutive Relationships
The accuracy and reliability of nonlinear finite element analysis of reinforced concrete structures depends on the underlying constitutive relationships applied in each finite element domain. To simulate the nonlinear response of cracked reinforced concrete; a combined model based on separate material models for the concrete state, reinforcing steel state and the state of interaction between concrete and reinforcement need to be considered (Foster et al., 1996).
3.3.1 Constitutive Models for Concrete To establish the constitutive models for concrete, a diverse number of approaches have been used and can be classified as the elasticity-based approach, plasticity-based approach, damaged-based approach, micro modelling and endochronic theory (CEB, 1996). Among these the elasticity-based approach is more popular because of its simplicity, the capability of modelling concrete under various states of loading (monotonic, cyclic, etc.) and giving reasonably good results (ASCE Committee 447, 1982, Bahlis and Mirza, 1987). The elasticity-based approach includes a variety of models, the most common being the orthotropic, equivalent uniaxial, model. Many researchers have employed the non-linear elasticity approach to simulate the behaviour of RC shear walls under cyclic loading (Okamura and Maekawa, 1991, Sittipunt and Wood, 1995, Elmorsi et al., 1998, Vecchio, 1999, Kwak and Kim, 2004, Palermo and Vecchio, 2004). A common version of this model is based on the concept of “equivalent uniaxial strains” introduced by Darwin and Pecknold (1976) , with the constitutive relationships for concrete founded on the uniaxial stress-strain curves modified to take into account the effects of stresses in other directions. Adopting orthotropic, equivalent uniaxial strain approach in this research, the constitutive relationships for concrete in tension and compression are expressed using two sets of curves. The behaviour of concrete subjected to biaxial states of stress is taken into account by modifying the uniaxial response of the concrete.
3-10
Concrete in Tension
Before cracking, concrete in tension is considered as a linear elastic material. Adopting the bilinear softening model of Petersson (1981) for the post-cracking response, the envelope curve of concrete in tension is expressed by the monotonic stress-strain curve as shown in Figure 3.5(a). The main softening parameters of the model are given by
α1 =
1 ; 3
2 9
α 2 = α 3 + α1 ;
α3 =
18 Ec G f 5 l ch f ct2
(3.10)
where Ec is the initial elastic modulus of concrete, G f is the fracture energy and lch is a characteristic length over which the fracture energy is dissipated. For RC elements analysed using the smeared crack approach, this length is taken as the crack spacing. The unloading stiffness modulus for concrete in tension is expressed by
Ed =
σa εe + ε p 2
(3.11)
σc1
σc1
f ct α1f ct
f ct σa
Ec
Ed Ec
1
εtp α2ε tp
α3ε tp
εc1
1
1
εp/2 εp/2 εe
(a)
εa
εc1
(b)
Figure 3.5- Concrete in uniaxial tension: (a) envelope stress-strain curve (b) unloading and reloading curves (Foster and Marti, 2003). 3-11
where σ a is the concrete stress just before beginning the unloading and ε e and ε p are the elastic and plastic components of strain ε a corresponding to σ a and are given by
ε e = σ a Ec
(3.12)
ε p = εa − εe
Concrete in Compression
In this study, the uniaxial stress-strain curve proposed by Thorenfeldt et al. (1987) is adopted (Figure 3.6). This model is a convenient expression, which accurately describes the shape of the ascending and descending branches of high-strength concrete cylinder stress-strain curve (Collins et al., 1993). The curve forms the envelope for concrete in compression with the compressive stress expressed as
σ c = − f cp
nη
(3.13)
n − 1 + η nk
(
)
where η = ε c ε cp and n = Ec Ec − Ecp . In Eq. 3.13, ε c is the concrete strain, ε cp is the strain corresponding to the peak stress f cp on the curve that is equivalent to the compressive strength of concrete, Ec is the
initial elastic modulus of the concrete, Ecp = f cp ε cp and k is a decay factor that controls the post-peak response. Collins and Porasz (1989) calibrated the decay factor for conventional and high strength concrete and proposed it as follows:
ε ≤ ε cp ............ k = 1
(3.14a)
ε > ε cp ............ k = 0.67 + f cp 62 ≥ 1.0
(3.14b)
3-12
where f cp is in MPa.
σc f c'
σa Ed
Ec 1
1
ε cp
εa
εc
Figure 3.6 – Envelope stress-strain curve for concrete under uniaxial compression.
For unloading of concrete in compression, the stiffness modulus E d defined by Filippou et al. (1983) is used; that is
Ed =
σa
ε a − ε a (0.1 + 0.15 ε a ε cp )
(3.15)
where σ a and ε a define the point of unloading (refer Figure 3.6).
Biaxial Behaviour of Concrete
The strength and stress-strain response of concrete subjected to biaxial states of stress are different compared with uniaxial conditions and vary as the functions of the combinations of stresses. Appropriate modelling of concrete in the two-dimension using
3-13
the orthotropic equivalent uniaxial approach needs modification of uniaxial concrete response in tension and in compression so that the biaxial behaviour of concrete can be captured. Figure 3.7 shows the biaxial strength envelope for concrete used by Foster and Marti (2003) and is used in this study. Commonly for a two-dimensional concrete model three biaxial states of stress are considered: biaxial compression (C-C), biaxial compression and tension state (C-T) and the biaxial tension (T-T). In biaxial compression the strength of concrete is greater than the uniaxial compressive strength (Kupfer and Gerstle, 1969, Liu et al., 1972) due to the confinement effect. Under combinations of compression and tension the effects of tensile cracking causes the compressive strength of concrete to be reduced and this is known as the compression softening effect (Robinson and Demorieux, 1977, Vecchio and Collins, 1982, 1986, Miyakawa et al., 1987, Belarbi and Hsu, 1995). Finally, when concrete is subjected to biaxial tension the strength is close to that in uniaxial tension (Kupfer and Gerstle, 1969). The biaxial compressive strength of concrete f c* may be written as the in-situ uniaxial compressive strength f cp modified by a strength factor β as
−σ2c /fcp
(0.6, 1.25) (1.15, 1.15)
1.0 0.6
(1.25, 0.6)
1.0 0.6
3-14
−σ1c /fcp
Figure 3.7- Strength envelope curve for concrete in biaxial stress (Foster and Marti, 2003).
f c* = β f cp
(3.16)
In the Eq. 3.16, the coefficient β is considered as a scaling factor applied to the * compressive stress-strain curve depending on the state of stress. The strain ε cp
corresponding to f c* may be considered the same as the uniaxial peak strain ε c (Vecchio and Collins, 1986) or modified using the same factor β as follows (Vecchio and Collins, 1982, Belarbi, 1991): * ε cp = β ε cp
(3.17)
The latter approach is used in this study (Figure 3.8). For the C-C state, β is a confinement factor that can be determined from the biaxial strength envelope (Figure 3.7). For the T-C state, β is a strength reduction factor obtained from the modified compression field theory (Vecchio and Collins, 1986) and is expressed as
β=
1
ε 0.8 + 0.34 c1 ε cp
≤ 1.0
(3.18)
where ε c1 is the tensile strain in the material 1-direction. The biaxial tensile strength of concrete f ct* for the C-T and T-T states, based on the biaxial strength envelope (Figure 3.7), is given by
f ct* = f ct
if
3-15
σ c2 f c'
≤ 0.6
(3.19a)
σ f ct* = 2.5 f ct (1 − c 2 ) f c'
if
σ 0.6 < c 2 ≤1.0 f c'
(3.19b)
−σ c β fcp fcp
β >1
β fcp
β =1 1
βε cp
Ed
ε cp βε cp
β
