CHONG Alecs KT - Phd Thesis
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Numerical Modelling of Time-dependent Cracking and Deformation of Reinforced Concrete Structures Kak Tien Chong A thesis submitted as a partial fulfilment of the requirements for the degree of Doctor of Philosophy December 2004 UNSW THE UNIVERSITY OF NEW SOUTH WALES • SYDNEY • AUSTRALIA School of Civil & Environmental Engineering
Transcript of CHONG Alecs KT - Phd Thesis
Numerical Modelling of Time-dependent
Cracking and Deformation of
Reinforced Concrete Structures
Kak Tien Chong
A thesis submitted as a partial fulfilment of the
requirements for the degree of Doctor of Philosophy
December 2004
UNSW THE UNIVERSITY OF NEW SOUTH WALES • SYDNEY • AUSTRALIA School of Civil & Environmental Engineering
CERTIFICATE OF ORIGINALITY
I hereby declare that this submission is my own work and to the best of my knowledge
it contains no materials 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 UNSW or any other educational institution, except where due
acknowledgement is made in the thesis. Any contribution made to the research by
others, with whom I have worked at UNSW 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.
Kak Tien Chong
To those who have the thirst for knowledge
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ABSTRACT
For a structure to remain serviceable, crack widths must be small enough to be
acceptable from an aesthetic point of view, small enough to avoid waterproofing
problems and small enough to prevent the ingress of water that may lead to corrosion of
the reinforcement. Crack control is therefore an important aspect of the design of
reinforced concrete structures at the serviceability limit state. Despite its importance,
code methods for crack control have been developed, in the main, from laboratory
observations of the instantaneous behaviour of reinforced concrete members under load
and fail to account adequately for the time-dependent development of cracking.
In this study numerical models have been developed to investigate time-
dependent cracking of reinforced concrete structures. Two approaches were adopted to
simulate cracking in reinforced concrete members. The first approach is the distributed
cracking approach. In this approach, steel reinforcement is smeared through the
concrete elements and bond-slip between steel and concrete is accounted for indirectly
by including the tension stiffening effect. The second approach is the localized cracking
approach, in which concrete fracture models are used in conjunction with bond-slip
interface elements to model stress transfer between concrete and steel.
Creep of concrete has been incorporated into the models by adopting the principal
of superposition and the time-dependent development of shrinkage strain of concrete is
modelled using an approximating function. Both creep and shrinkage were treated as
inelastic pre-strains and applied to the discretized structure as equivalent nodal forces.
Apart from material non-linearity, non-linearity arising from large deformation was
also accounted for using the updated Lagrangian formulation.
The numerical models were used to simulate a series of laboratory tests for
verification purposes. The models were assessed critically by comparing the numerical
results with the test data and the numerical results are shown to have good correlations
with the test results. In addition, a comparison was undertaken among the numerical
models and the pros and cons of each model were evaluated.
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A series of controlled parametric numerical experiments was devised and carried
out using one of the numerical models. Various parameters were identified and
investigated in the parametric study. The effects of the parameters were thoroughly
examined and the interactions between the parameters were discussed in detail.
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ACKNOWLEDGEMENTS
The work presented in this thesis was undertaken in the School of Civil and
Environmental Engineering at the University of New South Wales.
I wish to express my sincere gratitude to Professor R. Ian Gilbert for giving me
the opportunity to participate in this research project. His patient supervision,
suggestions, critical comments and continuous support throughout the course of this
study are very much appreciated.
I would also like to thank my co-supervisor, Associate Professor Stephen J.
Foster, with whom I had many inspiring and fruitful discussions on numerical methods
and his patient guidance is one of the most important factors promoting the completion
of this study.
This research was funded by Australian Research Council (ARC) Discovery
Grant No. DP0210039 and an Australian Government International Postgraduate
Research Scholarship (IPRS). The ARC and Scholarship supports are gratefully
acknowledged.
I would like to express my deepest gratitude to my family for their love, support
and encouragement while I was thousand of miles away from home. Finally, I wish to
thank my beloved girl friend, Peggy, who has walked me through all the good times
and bad times throughout these years, without whose love and support the completion
of this thesis would not have been possible.
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CONTENT
ABSTRACT iv
ACKNOWLEDGEMENTS vi
NOTATION xii
CHAPTER 1 INTRODUCTION
1.1 Background and Significance 1
1.2 Objective and Scope 3
1.3 Outline of Thesis 5
CHAPTER 2 LITERATURE REVIEW
2.1 Introduction 7
2.2 Instantaneous Behaviour of Concrete 7
2.2.1 Uniaxial Compression 7 2.2.2 Uniaxial Tension 9 2.2.3 Biaxial Loading and Failure Criteria 11
2.3 Time-dependent Behaviour of Concrete 13
2.3.1 Creep 15 2.3.1.1 Factors affecting Creep 16 2.3.1.2 Creep Recovery 17 2.3.1.3 Principle of Superposition 18
2.3.2 Shrinkage 22 2.3.2.1 Chemical Shrinkage 23 2.3.2.2 Drying Shrinkage 24 2.3.2.3 Effects of Shrinkage 26
2.3.3 Interaction of Fracture and Creep 26 2.3.3.1 Influence of Loading Rate on Peak Load 27 2.3.3.2 Load Relaxation at Fracture Zone 28 2.3.3.3 Creep Rupture 28 2.3.3.4 Time-dependent Fracture Models 29
2.4 Behaviour of Reinforcement 31
2.5 Bond between Reinforcement and Concrete 32
2.5.1 Local Bond Stress-slip Relationship 33 2.5.2 Influence of Bond on Cracking 36 2.5.3 Tension Stiffening 37
2.6 Non-linear Modelling of Concrete Structures 38
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2.6.1 Discrete Crack Approach 39 2.6.2 Smeared Crack Approach 41
2.6.2.1 Fixed Crack Model 42 2.6.2.2 Rotating Crack Model 43 2.6.2.3 Multiple Fixed Crack Model 44
2.6.3 Constitutive Models for Concrete 45 2.6.3.1 Elasticity-based Models 46 2.6.3.2 Plasticity-based Models 49 2.6.3.3 Continuous Damage Models 53 2.6.3.4 Microplane Models 55
2.6.4 Fracture Models for Concrete 58 2.6.4.1 Fracture Mechanics 58 2.6.4.2 Fictitious Crack Model 59 2.6.4.3 Crack Band Model 60
2.6.5 Regularization of Spurious Strain Localization 61 2.6.5.1 Non-local Models 62 2.6.5.2 Gradient Models 65 2.6.5.3 Crack Band Formulation as Partial Regularization 66 2.6.5.4 Regularization by Inclusion of Material Viscosity 67
2.6.6 Modelling of Steel Reinforcement 67 2.6.7 Modelling of Steel-Concrete Bond 69
2.6.7.1 Tension Stiffening 69 2.6.7.2 Discrete Bond Modelling 70
2.6.8 Computational Creep Modelling 72
CHAPTER 3 FINITE ELEMENT MODELS FOR REINFORCED CONCRETE
3.1 Introduction 76
3.2 Continuum Modelling 77
3.3 Distributed Cracking Approach 77
3.3.1 Tension Chord Model 79 3.3.2 Cracked Membrane Model 81
3.4 Localized Cracking Approach 83
3.4.1 Crack Band Model 84 3.4.2 Non-local Smeared Crack Model 86
3.4.2.1 Issue Related to Non-local Continuum with Local Strain 87 3.4.2.2 Proposed Non-local Smeared Cracking Formulation 89
3.5 Orthotropic Membrane Formulation 91
3.6 Material Constitutive Models 95
3.6.1 Instantaneous Behaviour of Concrete 95
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3.6.1.1 Stress-strain Relationships for Concrete 96 3.6.1.2 Biaxial Compression State of Stress 98 3.6.1.3 Tension-Compression State of Stress 99 3.6.1.4 Biaxial Tension State of Stress 100
3.6.2 Time-dependent Behaviour of Concrete 101 3.6.3 Shrinkage 102 3.6.4 Creep 103 3.6.5 Solidification Theory for Concrete Creep 103
3.6.5.1 Rate-type Constitutive Model 106 3.6.5.2 Finite Element Implementation of Creep 108
3.6.6 Time-dependent Crack Width 111 3.6.6.1 Cracked Membrane Model 111 3.6.6.2 Crack Band Model 111 3.6.6.3 Non-local Model 112
3.6.7 Stress-strain Relationship for Reinforcing Steel 112 3.6.8 Local Bond-slip Model for Bond Interface Element 113 3.6.9 Concrete Tension Stiffening 116
3.7 Non-linear Finite Element Implementation 118
3.7.1 Spatial Discretization 119 3.7.2 Time Discretization 120 3.7.3 Principal of Virtual Work 121 3.7.4 Incremental Iterative Solution Procedures 123 3.7.5 Geometric Non-linearity 128 3.7.6 Convergence Criteria 129 3.7.7 Computational Solution Algorithm 130
3.8 Finite Element Formulations 133
3.8.1 Four-node Isoparametric Quadrilateral Element 133 3.8.2 Two-node Truss Element 136 3.8.3 Four-node Isoparametric Bond Interface Element 138
CHAPTER 4 EVALUATION OF THE FINITE ELEMENT MODELS
4.1 Introduction 141
4.2 Mesh Sensitivity of the Localized Cracking Models 141
4.3 Creep of Plain Concrete under Variable Stress 147
4.4 Long-term Flexural Cracking Tests 151
4.4.1 Introduction 151 4.4.2 Analysis of Long-term Flexural Cracking Tests and Material Properties 154
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4.4.3 Analysis of Long-term Flexural Cracking Tests using the Distributed Cracking Model – Cracked Membrane Model 156 4.4.3.1 Four-point Bending Beam Tests under Sustained Load 156 4.4.3.2 Uniformly Loaded One-way Slabs under Sustained Load 160 4.4.3.3 Discussion 162
4.4.4 Analysis of Long-term Flexural Cracking Tests using the Localized Cracking Model – Crack Band Model 169 4.4.4.1 Four-point Bending Beam Tests under Sustained Load 170 4.4.4.2 Uniformly Loaded One-way Slabs under Sustained Load 179 4.4.4.3 Discussion 182
4.4.5 Analysis of Long-term Flexural Cracking Tests using the Localized Cracking Model – Non-local Smeared Crack Model 187 4.4.5.1 Four-point Bending Beam Tests under Sustained Load 188 4.4.5.2 Uniformly Loaded One-way Slabs under Sustained Load 193 4.4.5.3 Discussion 196
4.4.6 Summary for Analysis of Long-term Flexural Cracking Tests 201 4.5 Long-term Restrained Deformation Cracking Tests 203
4.5.1 Introduction 203 4.5.2 Analysis of Restrained Deformation Cracking Tests and Material
Properties 205
4.5.3 Comparisons of Numerical and Experimental Results 208 4.5.4 Discussion 217
4.6 Other Numerical Examples 220
4.6.1 Continuous Beams Subjected to Long-term Sustained Load 220 4.6.2 Time-dependent Forces Induced by Supports Settlement of Continuous Beams 225 4.6.3 Slender Columns Subjected to Long-term Eccentric Axial Loads 230
CHAPTER 5 NUMERICAL EXPERIMENTS
5.1 Introduction 237
5.2 Description of Numerical Experiments 239
5.2.1 Beam Specimens 241 5.2.2 Slab Specimens 244 5.2.3 Testing Method 246
5.2.3.1 Test Series A to J: Material and Environmental Parameters 247 5.2.3.2 Test Series K: Amount of Shear Reinforcement 250 5.2.3.3 Test Series L: Impact of 500 MPa Steel Reinforcement 251 5.2.3.4 Test Series M: Load Histories 252
5.3 Presentation and Discussion of Results 256
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5.3.1 Test Series A – Bottom Concrete Cover 257 5.3.2 Test Series B – Diameter of Tensile Reinforcing Steel 261 5.3.3 Test Series C – Quantity of Tensile Reinforcement 264 5.3.4 Test Series D – Quantity of Compressive Reinforcement 267 5.3.5 Test Series E – Tensile Strength of Concrete 270 5.3.6 Test Series F – Bond Strength between Steel and Concrete 274 5.3.7 Test Series G – Concrete Tensile Strength Fluctuation Limit 277 5.3.8 Test Series H – Magnitude of Creep 279 5.3.9 Test Series I – Magnitude of Shrinkage 285 5.3.10 Test Series J – Bond Creep 285 5.3.11 Test Series K – Quantity of Shear Reinforcement 289 5.3.12 Test Series L – Impact of 500 MPa Steel Reinforcement 291 5.3.13 Test Series M – Load Histories 293
5.3.13.1 Comparisons between LH-2 and LH-1 293 5.3.13.2 Comparisons between LH-5 and LH-2 296 5.3.13.3 Comparisons between LH-3 and LH-4 298
5.3.14 Section Geometry and Boundary Conditions 301 5.4 Summary 301
CHAPTER 6 SUMMARY AND CONCLUSIONS
6.1 Summary 305
6.2 Conclusions 308
6.3 Recommendations for Future Research 311
APPENDIX A FE IMPLEMENTATION OF RATE OF CREEP METHOD 313
APPENDIX B ILLUSTRATION OF TREATMENT FOR INELASTIC PRE-STRAIN
BY SIMPLE HAND CALCULATION 318
APPENDIX C CEB-FIP MODEL CODE 1990 – CREEP AND SHRINKAGE
MODELS 324
REFERENCES 327
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NOTATION
A Area; empirical time-dependent parameter.
cA Cross-sectional area of concrete.
cpA , cpB Parameters for time-dependent variation of creep coefficient.
effcA . Effective area of concrete in tension.
eA Surface area of finite element; tangential contact surface area for bond element.
ctfA , ctfB Parameters for time-dependent growth of concrete tensile strength.
sA Cross-sectional area of steel.
scA Cross-sectional area of compressive steel.
shA , shB Parameters for time-dependent development of shrinkage strain.
stA Cross-sectional area of tensile steel.
svA Cross-sectional area per stirrup.
0A Negative infinity area of retardation spectrum.
B Empirical time-dependent parameter.
B Strain-displacement matrix.
C Creep compliance function.
ad Maximum aggregate size.
D Material elasticity matrix.
cD Constitutive matrix for concrete.
ctsD Constitutive matrix for tension stiffening.
12cD Material constitutive matrix in principal directions.
bD Bond constitutive matrix.
eD Elastic stiffness matrix.
epD Elasto-plastic stiffness matrix for plasticity-based model.
sD Constitutive matrix for steel.
secD , secD Secant stiffness matrix.
secntD Secant stiffness matrix in material local axis.
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sec12D Secant stiffness matrix in principal axis.
tanD Material tangent constitutive matrix.
E Modulus of elasticity.
bnE , btE Secant moduli for bond-split and bond-slip, respectively.
sec.bE Secant modulus of bond.
cE Initial modulus of concrete.
14.cE , 28.cE Elastic modulus of concrete at 14 days and 28 days, respectively.
cpkE Secant modulus at peak of concrete stress-strain curve.
ctsxE , ctsyE Secant moduli for tension stiffening in x, y directions, respectively.
cuE Unloading modulus for concrete in compression.
1cE , 2cE Concrete secant moduli in major and minor principal directions, respectively.
sE Initial elastic modulus of reinforcing steel.
secsE Secant modulus of reinforcing steel.
secE Secant modulus.
sxE , syE Secant moduli for steel reinforcement in x, y directions, respectively.
suE Unloading modulus for reinforcing steel.
tuE Unloading modulus for concrete in tension.
wE , uE Hardening moduli for reinforcing steel.
0E Asymptotic modulus of concrete.
µE Elastic modulus of µ-th Kelvin chain unit.
f Yield function for plasticity-based model; damage loading function for continuous damage model; local state variable.
F Function; time-dependent function.
cff Flexural tensile strength of concrete.
cmf Mean compressive strength of concrete.
crf Cracking stress under tension cut-off regime.
ctf Direct tensile strength of concrete.
tctf . Direct tensile strength of concrete at age t days.
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0ctf Concrete tensile strength at zero crack opening rate.
cuf Compressive strength in uniaxial compressive stress-strain curve.
syf Yield stress of steel reinforcement.
swf , suf Hardening stresses of steel reinforcement.
'cf Characteristic compressive strength of concrete.
*cf Biaxial compressive strength of concrete.
f Weighted average state variable for non-local model.
F Structural equivalent pre-strain nodal force vector.
eF Element equivalent pre-strain nodal force vector.
G Shear modulus.
12cG Concrete secant shear modulus in principal directions.
fg Fracture energy density.
fG Fracture energy.
pg Plastic potential function for plasticity-based model.
G Matrix for inclusion of Poisson’s effect to biaxial stress.
h Average width of fracture process zone; volume associated with viscous strain.
ch Crack band width.
sh Notational size of concrete member.
i , j Counters.
J Compliance function.
J Jacobian matrix.
k Decay factor.
K Structural stiffness matrix.
eK Element stiffness matrix.
secK Secant stiffness matrix. tanK Tangent stiffness matrix.
0K Initial tangent stiffness matrix.
chl Material characteristic length.
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fctl Random fluctuation limit of concrete tensile strength.
L Function for retardation spectrum; length of truss element; length of bond element.
l , m , n Orthonormal base vectors.
L Differential operator matrix.
swM , uM Moment due to self-weight and moment capacity, respectively.
N Total number of Kelvin chains; total number of elements; shape function.
N Displacement interpolation matrix.
P External structural nodal force vector.
bp Body force vector.
ep Nodal force vector.
eP External element nodal force vector.
sp Surface traction vector.
2q , 3q , 4q Empirical parameters for solidification theory of creep.
Q Structural internal force vector.
r Distance between target point and source point in non-local analysis.
R Relaxation function; non-local interaction radius.
RH Relative humidity.
R Out-of-balance force vector.
s Slip between concrete and reinforcing steel.
as Axial length of truss element.
is Instantaneous slip.
ts Time-dependent slip.
1s , 2s , 3s Slips defining CEB-FIP bond-slip model.
rms Crack spacing.
rmxs , rmys Crack spacings of an orthogonally reinforced concrete membrane element in x and y directions, respectively.
0rms Maximum crack spacing.
s Location vectors of neighbourhood strains.
t Time or age of concrete.
T Temperature.
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et Thickness of plane stress element.
0t Age at first loading.
't Variable for age at loading.
bT Bond transformation matrix.
BT Diagonal bond transformation matrix.
εT Strain transformation matrix.
u , v Nodal displacements corresponding to x and y directions, respectively.
au Nodal axial displacements.
cu Perimeter of concrete member.
tiu , niu Element nodal displacements parallel to and normal to bond element, respectively.
u Structural nodal displacement vector.
eu Element nodal displacement vector.
u′ Continuous field displacement vector.
V Volume of structure.
eV Volume of finite element.
'V Volume of structure in displaced configuration.
crw Crack opening displacement.
uw Crack opening at which stress transfer at fictitious crack vanishes.
x Location vectors of local strain.
Y Damage energy release rate.
α Weight function; ratio of major and minor principal compressive stresses.
1α , 2α , 3α Tension softening parameters.
'α Normalized weight function.
β Shear retention factor; confinement factor; strength reduction factor.
1β Confinement factor in the major principal direction.
2β Confinement factor in the minor principal direction.
yδ Slip at which reinforcing steel starts to yield.
ε Strain.
crbetw.ε Concrete strain between cracks.
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cε Concrete strain.
ceε Concrete elastic strain.
unce.ε Concrete elastic strain corresponding to unc.σ .
ciε , instε Instantaneous concrete strain.
cpε Creep strain.
cpkε Concrete strain corresponding to peak stress in compressive stress-strain curve.
crε Concrete cracking strain.
uncr.ε Concrete cracking strain corresponding to unc.σ .
unc.ε Concrete strain corresponding to unc.σ on stress-strain curve.
NML εεε ,, Microplane strain components of microplane model.
mε Mean strain in an element.
shε Shrinkage strain.
tpkε Concrete strain corresponding to peak stress in tensile stress-strain curve.
uε Concrete cracking strain when cohesive stress between crack faces vanishes.
1ε , 2ε Concrete strains in major and minor principal directions, respectively.
u1ε , u2ε Equivalent uniaxial concrete strains in major and minor principal directions, respecitively.
eu.1ε , eu.2ε Concrete elastic strain in major and minor principal directions, respectively.
cru.1ε , cru.2ε Concrete cracking strain in major and minor principal directions, respectively.
*cpkε Adjusted concrete strain corresponding to peak stress in biaxial
compressive stress-strain curve. *shε Final shrinkage strain.
crε Concrete non-local cracking strain.
u1ε , u2ε Modified equivalent uniaxial strains due to non-locality in major and minor principal directions, respectively.
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cru.1ε , cru.2ε Concrete non-local cracking strain in major and minor principal directions, respectively.
ε~ Equivalent strain of continuous damage model.
ε Strain vector.
ciε Concrete instantaneous strain vector.
12ε Principal strain vector.
12cε Concrete elastic principal strain vector.
cpε Creep strain vector.
ceε , eε Concrete elastic strain vector.
crε , pε Concrete cracking strain or plastic strain vector.
fε Viscous strain vector.
ntε Strain vector in material local axis.
shε Shrinkage strain vector.
vε Viscoelastic strain vector.
0ε Pre-strain vector.
ε Non-local strain vector.
crε Concrete non-local cracking strain vector.
φ Creep coefficient.
*φ Final creep coefficient.
Φ Microscopic creep compliance function.
µγ Viscoelastic microstrain of µ-th Kelvin chain unit.
γ Viscoelastic microstrain vector.
η Apparent macroscopic viscosity associated with viscous component of creep.
0η Effective viscosity of solidified matter associated with viscous component of creep.
µη Viscosity of µ-th Kelvin chain unit.
κ Hardening or softening parameter for plasticity-based model; history-dependent parameter for continuous damage model.
λ Ratio of crack spacing to maximum crack spacing.
µ Counter for Kelvin or Maxwell chain unit.
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v Poisson’s ratio; volume associated with viscoelastic strain.
12ν , 21ν Poisson’s ratio in major principal direction resulting from stress in minor principal direction and vice versa, respectively.
θ Angle; angle between global x and local n axes; angle between global x and major principal axes.
ρ Ratio of cross sectional area of steel to cross sectional area of concrete.
effρ Ratio of area of tensile steel to effective area of concrete in tension.
xρ , yρ Element reinforcement ratios in x and y directions, respectively.
σ Stress.
cσ Stress in concrete.
cnσ , ctσ , cntτ Concrete stresses in local n (normal to crack), t (crack direction) and cnt (shear) directions, respectively.
ctsmσ Mean concrete tension stiffening stress.
ctsmyctsmx σσ , Mean concrete tension stiffening stresses in the x and y directions, respectively.
1ctsmσ Mean concrete tension stiffening stress in major principal direction.
unc.σ Concrete stress just before commencement of unloading.
1cσ , 2cσ Concrete stress in major and minor principal directions, respectively.
etσ Function of cohesive stress versus cracking strain.
extσ External stress.
intσ Internal stress.
locσ Local stress in concrete.
outσ Out-of-balance stress.
sσ Stress in reinforcing steel.
smσ Mean steel stress.
minsσ Minimum steel stress.
srσ Steel stress at crack or maximum steel stress.
sxσ , syσ Steel stresses in global x, y directions, respectively.
xσ , yσ , xyτ Concrete stresses in global x, y and xy (shear) directions, respectively.
wtσ Function of cohesive stress versus crack opening displacement.
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σ Stress vector.
ntσ Stress vector in material local axis.
12σ Principal stress vector.
bτ Bond stress between concrete and reinforcing steel.
0bτ , 1bτ Bond stresses before yielding and after yielding of steel, respectively.
maxτ Maximum bond stress in local bond stress-slip curve.
fτ Failure bond stress.
µτ Retardation time of µ-th Kelvin chain unit.
ω Damage variable.
ξ , η Natural coordinate system.
ψ Angle of orientation of truss or bond element from the x-axis.
sψ Temperature-dependent shift function.
∅ Diameter of reinforcing bar.
st∅ Diameter of tensile reinforcing bar.
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CHAPTER 1
INTRODUCTION
1.1 Background and Significance
Reinforced concrete is a composite material made up of steel reinforcement embedded
in hardened concrete. The two materials are inter-complimentary. Concrete is ideal for
withstanding compressive forces and steel reinforcement is ideal for carrying tensile
forces, thereby compensating for the low tensile strength of concrete. For the steel
reinforcement to effectively carry the internal tensile forces, the tensile concrete must
crack. Under normal in-service conditions, cracking is inevitable in many reinforced
concrete structures. In most structures, cracking occurs due to the application of
external service loads and due to restrained deformation. Although inclusion of steel
reinforcement does not prevent this type of cracking, it does help distributing cracks
evenly over the cracked regions and therefore effectively controls the development of
cracks. There are several types of cracking that cannot be controlled by reinforcement.
These include cracking originating from the development of internal pressure in
concrete due to corrosion of reinforcement, plastic shrinkage of concrete that occurs in
the first few hours after casting and expansion of concrete associated with chemical
reactions. Cracking caused by these factors can, instead, be prevented by good quality
control during construction of the structure. In this study, only cracking arising from
external loads and restrained deformation (for instance, shrinkage induced movements
and support settlement), which can be effectively controlled by adequate inclusion of
steel reinforcement, is investigated.
The ability of reinforcement to distribute cracks depends greatly on the quality of
bond between the reinforcing steel and the concrete. The composite interaction between
the two materials is established and maintained by bond, which effectively transfers
load between the steel and the concrete. The main mechanism in the development of
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bond is the mechanical interaction between the ribbed or deformed surface of the
reinforcing bar and the concrete. However, other mechanisms such as surface friction
and chemical adhesion also play a role. The quality of bond has a prominent influence
on crack formation and hence affects the spacing between cracks and the crack width.
Cracking results from tension caused by external loads and from tension caused
by restrained deformation due to the effects of shrinkage and temperature. In addition,
the bond in the vicinity of a crack under sustained loads deteriorates due to creep of the
concrete adjacent to the reinforcing steel. This results in an increase in slip between the
steel and concrete with time. The deterioration of bond under long-term loads further
complicates the cracking process, as does the gradual build-up of tension caused by
restraint to shrinkage. As a consequence, cracking is time-dependent, with the extent
and width of cracks gradually increasing with time under sustained loading.
For a structure to remain serviceable, crack widths must be small enough to be
acceptable from an aesthetic point of view, small enough to avoid waterproofing
problems and small enough to prevent the ingress of water that may lead to corrosion of
the reinforcement. Excessively wide cracks can even provoke public concern for the
safety of the structure. Crack control is therefore an important aspect of the design of
reinforced concrete structures at the serviceability limit state and the topic has received
much research attention. The design procedures for crack control can be divided
broadly into two alternative types. The first type is by calculating the crack width
explicitly and comparing the calculated magnitude with the code stipulated crack width
limits. In the second approach, crack control is deemed to be satisfactory as long as
some specific detailing requirements are met, such as permissible maximum distance
between reinforcing bars in the tension region, maximum bar diameters for a specific
steel stress, minimum reinforcement area and so on.
For practising structural engineers, the second approach may seem appealing due
to its simplicity. Nevertheless, the simplified procedures provided in most design codes
are generally less than adequate. In addition, code methods have been developed, in the
main, from laboratory observations of the instantaneous behaviour of reinforced
concrete members under load and fail to account for the time-dependent development
of cracking and the inevitable increase in crack widths with time due primarily to
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shrinkage. As a result, even the use of the first approach (explicit calculation of crack
width) may lead to significant error in the estimation of the final crack width. The
inability to recognize and quantify the non-linear effects of cracking, creep and
shrinkage can lead to excessive deflections and crack widths and miscalculation of
support reactions.
Experimental data relating to the time varying distribution of cracking, in
particular the final crack spacings and crack widths are scarce in the open literature.
The development of a rational and reliable design procedure for control of cracking,
however, requires considerable experimental data for the purpose of investigating and
understanding the critical factors that affect time-dependent cracking.
Two alternatives may be used to yield the required data for this purpose. The
conventional experimental approach involves the fabrication of numerous test
specimens and testing them in the laboratory for a specific objective. Long-term
cracking tests are, however, costly not only in terms of time but also resources of the
laboratory since engagement of a large area in the laboratory over a long period of time
is necessary. Alternatively, numerical techniques such as the finite element method
may be employed to simulate a wide range of virtual long-term cracking tests so as to
facilitate a parametric investigation. This is an efficient and inexpensive method
compared to the conventional experimental approach. Nevertheless, to achieve this the
critical factors that affect accuracy of modelling must be identified. Realistic material
models that take account of cracking, creep and shrinkage of concrete and bond-slip
between steel and concrete are the keys to a reliable numerical simulation of time-
dependent cracking of reinforced concrete structures.
1.2 Objective and Scope
The main objective of the work reported in this thesis is to investigate the time-
dependent behaviour of reinforced concrete structures using the finite element method,
with particular interest in the formation of cracks, both in terms of spacing and width of
the cracks. The following tasks were undertaken to achieve the aforementioned
objective:
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(1) Formulation of a two-dimensional plane stress finite element model and
incorporation of material constitutive models that can realistically characterize
cracking, creep and shrinkage of concrete and bond-slip between concrete and
reinforcing steel, thereby facilitating numerical modelling of reinforced
concrete structures under service load conditions.
(2) Calibration and evaluation of the finite element model using the test data
obtained from the experimental program (Gilbert and Nejadi, 2004; Nejadi and
Gilbert, 2004) conducted in parallel with this study and other test data obtained
in the literature.
(3) Critical assessment of the time-dependent cracking models developed in this
study and comparisons of the pros and cons of each model.
(4) Investigation of the effects of various parameters on time-dependent cracking
of reinforced concrete structures by running a series of numerical parametric
experiments on beam and slab specimens.
The time-dependent finite element models developed in this study account for the
time effects of concrete, namely creep and shrinkage. Although temperature is, in some
respects, considered as one of the factors affecting the time-dependent behaviour of
reinforced concrete structures, it is not within the scope of the work described in this
thesis and therefore thermal effects are not considered. In other words, the ambient
temperature is assumed to remain constant for all simulations carried out in this work.
The finite element models were developed to investigate time-dependent cracking
under service load conditions. Under these conditions, compressive stresses in concrete
are rarely in excess of 40% of the concrete compressive strength and, for this reason, a
linear creep model was implemented into the finite element models. Nevertheless, the
finite element models developed here are capable of computing the instantaneous
behaviour to failure of reinforced concrete structures under the full range of loading, or,
subjected to certain limiting assumptions, the time-dependent behaviour of members
under loads up to failure can also be established.
In addition to material non-linearity, the finite element models can also account
for the complication arising from geometric non-linearity. This greatly facilitates
5
simulation for problems related to creep buckling, which is important in stability
analysis of slender columns.
1.3 Outline of Thesis
Chapter 2 gives a review of the previously published works that are relevant to this
study. This chapter is divided into two parts. The first part is an overview of the
literature dealing with instantaneous and time-dependent material behaviour in general.
The second part deals with previously published approaches for non-linear modelling
of reinforced concrete structures using the finite element method.
In Chapter 3, the formulations of the time-dependent finite element models are
presented. Firstly, various types of concrete cracking models are described. This is then
followed by a description of the material constitutive models employed in the finite
element studies. The finite element implementation of the time-dependent models and
the non-linear solution procedures are also outlined. Finally, the formulations of the
finite elements engaged in this study are shown.
Chapter 4 evaluates the finite element models described in the previous chapter.
A mesh sensitivity test is carried out to verify the objectivity of the results calculated by
the proposed cracking models. The creep model is also evaluated by testing a simple
uniaxially loaded specimen subjected to variable load histories. The accuracy of the
finite element models is examined by comparing the numerical results with existing
laboratory test data, both in terms of formation of cracks and deformation.
Chapter 5 describes a parametric investigation using the proposed finite element
model to study time-dependent cracking. A series of controlled parametric numerical
tests is devised and analysed. The results of the numerical tests are discussed and the
effects of each parameter are examined.
Chapter 6 summarizes the conclusions drawn from the investigation of this work
and a proposal for future research is made.
6
Appendix A presents a finite element model developed in the early stage of this
study. The model employs a relatively simple creep model based on the rate of creep
method. A simulation example is shown and the results are compared with test data.
In Appendix B, a simple hand calculation is shown to demonstrate the method
used in the finite element procedures to account for inelastic pre-strains such as creep
and shrinkage.
Finally, Appendix C presents the creep and shrinkage models of the CEB-FIP
Model Code 1990 (1993).
7
CHAPTER 2
LITERATURE REVIEW
2.1 Introduction
The knowledge for producing concrete and the basic idea of using reinforcement for
building structures has existed for thousands of years. However, due to the
heterogenous nature of concrete and the composite actions of the constituent materials,
research is still undertaken on reinforced concrete structures in order to gain a deeper
understanding of the complex behaviour of the individual and composite materials.
This chapter consists of two major parts. In the first part, a state-of-the-art review
is given on the properties of concrete and steel reinforcement as well as the interaction
between concrete and reinforcement. The material properties and composite action of
the two materials are of vital importance in the modelling and analysis of reinforced
concrete structures. The time-dependent aspect of concrete is discussed and the
influencing factors are identified. The second part is a review of the non-linear
modelling methods for reinforced concrete structures which includes discussions of the
different cracking models, material constitutive models and time-dependent modelling
techniques.
2.2 Instantaneous Behaviour of Concrete
2.2.1 Uniaxial Compression
Concrete is a highly non-linear material in uniaxial compression. The properties of
concrete in uniaxial compression are obtained from cylinder tests or cube tests. A
standard cylinder of 300 mm height and 150 mm diameter is used for cylinder tests
8
under standard conditions in Australia and North America, while tests on 150 mm
cubes are often used in European and Asian countries. Figure 2.1 shows a typical
uniaxial compression stress-strain curve. Concrete is a heterogeneous material, the
shape and the peak of the stress-strain curve varies greatly and are dependent on the
proportions and properties of the constituents, the size and shape of the specimen, the
rate of loading and also the age of the concrete. In Figure 2.1 the response of concrete
may be taken to be linear-elastic up to 30% ~ 40% of the peak stress, cuf (often termed
the compressive strength of concrete).
Beyond this point, concrete behaves in a non-linear manner up to the peak stress.
At levels of stress that are just above the linear-elastic range, microcracking between
the aggregates and mortar becomes apparent and “bond cracks” are formed (Mindess et
al., 1981). As the stress increases further into the range 70% to 90% of the compressive
strength, microcracks start to open and bridge the bond cracks. This leads to the
formation of continuous crack patterns and the stress eventually reaches the maximum
compressive stress.
Immediately after the peak stress, the concrete undergoes strain softening, which
is depicted in Figure 2.1 as the descending branch of the curve. Strain softening of
concrete in compression is a complicated process, it is dependent on the size of the
specimen and the strength of the concrete (Kaufmann, 1998). The softening branch of
the stress-strain curve in compression is steeper for longer specimens than for shorter
specimens and is due mainly to the localization of deformations while other parts of the
specimen are unloading. The shape of the unloading curve depends on the concrete
strength, with higher strength concrete exhibiting a more brittle response (i.e. a steeper
unloading curve). This is attributed to the fact that the specific fracture energy of
concrete in compression does not increase much with the concrete strength. With the
area under the stress-strain curve roughly constant, higher strength concrete must have
a steeper descending curve.
Many uniaxial stress-strain relationships for concrete in compression have been
proposed in the literature, such as the well-known expressions developed by Hognestad
(1951), Desayi and Krishnan (1964), Saenz (1964) and Thorenfeldt et al. (1987).
9
Fig. 2.1 - Uniaxial compression curve.
2.2.2 Uniaxial Tension
The strength of concrete in tension is much lower than the strength in compression. The
ratio of tensile strength to compressive strength is between 0.05 and 0.1, as obtained
experimentally by Johnson (1969). The tensile strength of concrete is normally
evaluated using the split cylinder test (called the Brazilian test) for the indirect tensile
strength, while the modulus of rupture test is used to evaluate the flexural tensile
strength cff . The direct tensile strength ctf is about 0.9 times the indirect split
cylinder strength and about 60% of the modulus of rupture and is often adopted in
situations where the stress field around the locations where cracking occurs is
uniformly distributed. An example is the cracking analysis in finite element modelling.
For flexural cracking, the flexural tensile strength is the quantity required. Normally for
general engineering purposes, the tensile strength may be expressed as a function of the
compressive strength. As recommended by the Australia Standard AS 3600-2001, the
lower characteristic direct tensile strength at age 28 days may be estimated by
'4.0 cct ff = (2.1)
and the lower characteristic flexural tensile strength is given by
'6.0 ccf ff = (2.2)
−fcu
−σc
Uniaxial stress
Longitudinal strain
−ε cpk
Ec
1−ε c
−0.4 fcu
10
where 'cf is the characteristic compressive strength of concrete (obtained from cylinder
tests) and all strength quantities in both Eq. 2.1 and Eq. 2.2 are in MPa.
A full stress-deformation response of concrete in tension cannot be determined
using the aforementioned tests. A direct tension test is required in order to capture the
full deformation of a concrete member in tension. This is a highly sensitive test that
requires the specimen to be sufficiently short, tested using very stiff testing machines,
using precise measuring devices and under displacement controlled conditions. The
typical stress-elongation response of concrete in tension is shown in Figure 2.2. The
pre-peak behaviour of concrete in tension is mostly linear-elastic except for the stresses
near the peak stress. At about 60% to 80% of the peak stress, microcracks begins to
form fairly uniformly throughout the specimen. The overall response of the concrete
member becomes softer and exhibits highly non-linear behaviour. Due to the quasi-
brittle nature of concrete, tensile stress in concrete does not reduce abruptly to zero
after the peak. On the contrary, damage in the specimen starts to localize into a fracture
process zone at the weakest section while the rest of the specimen undergoes unloading.
In the fracture process zone, concrete is able to transfer stress across the crack opening
direction due to bridging of aggregate particles, the tensile stress then drops gradually
with increasing deformation until a complete crack is formed. At this point, no further
tensile stress can be taken by the concrete member and this eventually leads to a
complete tensile failure. This process is known as strain localization and the concrete is
said to undergo tension softening.
However, a direct experimental evaluation of the full stress-strain curve of
concrete in tension has not been possible to date. This is attributed to the localized
nature of fracture in concrete. The elongation of the specimen is contributed to by the
fracture process zone while the other parts of the specimen actually cause a reduction in
length due to elastic unloading. Therefore, a direct measurement of the stress-strain
curve for concrete specimens in tension, even from the same concrete mix, would give
unobjective responses for specimens of different lengths. To overcome this problem,
the prediction of cracking in concrete must not be based solely on a strength criterion
but must also take into account the energy dissipated in cracking of concrete. Hence the
use of fracture mechanics should be considered.
11
(a) (b) (c)
Fig. 2.2 - Uniaxial tensile response: (a) stress-elongation curve and the influence of
specimen size to tension softening; (b) stress-strain curve for the specimen
outside the strain localization zone; (c) post-peak stress-crack opening
displacement curve, fG is known as the fracture energy and is equal to the
area under the softening curve.
2.2.3 Biaxial Loading and Failure Criteria
Concrete subjected to biaxial loading exhibits a different response to that under uniaxial
loading. Since many practical engineering problems may be simplified to a state of
plane stress, much research has been carried out to investigate the behaviour of
concrete in biaxial states of stress for both normal strength and high strength concretes
(Kupfer et al., 1969; Liu et al. 1972; Nelissen, 1972; Tasuji et al.; 1979, van Mier,
1986; Nimura, 1991). Kupfer et al. were probably the first researchers to conduct
reliable biaxial tests on concrete. They tested different concretes with compressive
strengths between 19 MPa and 60 MPa and found that the biaxial strength envelopes
constructed in terms of the ratio of the orthogonally applied stresses and the
compressive strengths, are similar for all the concretes tested. A typical biaxial strength
envelope of Kupfer et al. (1969) is shown in Figure 2.3.
The biaxial state of stress may be divided into three loading combinations,
namely biaxial compression, tension-compression and biaxial tension. Under biaxial
compression, the compressive strength is increased due to the presence of the lateral
compressive stress. The shape of the stress-strain curves is similar to that under uniaxial
fct
σt
−∆ ltpk ∆ l
wcr
wcr
wcr
fct
σt
εtpk ε t
fct
σt
wcr
Gf
Ec 1
Ec 1
12
compression but with a higher compressive strength up to about 1.25 times the uniaxial
compressive strength, depending on the magnitude of the lateral compressive stress.
Kupfer and Gerstle (1973) proposed an equation to model the biaxial compressive
strength envelope in which the maximum compressive strength c2σ may be
obtained by
'22
)1(65.31
cc fα
ασ+
+= (2.3)
where 21 /σσα = ; and 1σ and 2σ are the principal major and minor stresses
respectively. The in-service behaviour of a concrete structure usually involves regions
that lie in the tension-compression and in the biaxial tension states of stress. Under
conditions of tension-compression, which is the second or fourth quadrant in Figure
2.3, concrete exhibits a considerable reduction in compressive strength with a small
increase in transverse tension. Kupfer and Gerstle suggested a straight-line strength
envelope, the tensile strength t1σ decreasing with an increasing compressive stress and
given by
ctc
t ff '2
1 8.01 σσ −= (2.4)
For biaxial tension, Kupfer and Gerstle suggested the use of a constant uniaxial
tensile strength, which is essentially identical to the Rankine failure criterion and this is
generally adopted in modelling cracking in concrete.
13
Fig. 2.3 - Biaxial concrete strength envelope (Kupfer et al., 1969).
2.3 Time-dependent Behaviour of Concrete
The discussion of the behaviour of concrete has so far concentrated on the
instantaneous response. Concrete is indeed a type of viscoelastic cementitious material
that is characterized by time-dependent behaviour. When a concrete specimen is
subjected to a sustained load, it undergoes an immediate deformation. This is followed
by a further deformation with increasing time. The increase in deformation with time is
not negligible and may be several times larger than the instantaneous deformation.
Under constant temperature condition, the additional time-dependent response is
attributed to the effects of both creep and shrinkage of concrete.
Considering a concrete specimen subjected to a uniaxial load at a constant
ambient temperature, the total strain of the specimen at time t may be decomposed into
the following three strain components
)()()()( tttt shcpci εεεε ++= (2.5)
in which )(tciε is the instantaneous strain, )(tcpε is the creep strain and )(tshε is the
shrinkage strain. The decomposition of the total strain into individual components in
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2σ1 / f c '
σ2 / f c '
14
Eq. 2.5 is a simplification. In reality, creep and shrinkage are interdependent
phenomena. However, this assumption is generally acceptable for engineering
purposes.
Figure 2.4 shows the deformation of a specimen loaded uniaxially in compression
at age 0t at a constant stress level. Shrinkage strain develops at the commencement of
drying (here defined as t = 0) while creep strain begins to develop after the specimen is
stressed. For loading within the elastic range of the stress-strain relationship, the
instantaneous strain is equal to the elastic strain. The dashed line in Figure 2.4b denotes
the true elastic strain of the specimen due to the increase in the elastic modulus
with age.
(a)
(b)
Fig. 2.4 - Time-dependent strain development: (a) development of shrinkage strain
after commencement of drying; (b) change in the different strain
components for a specimen after being subjected to a sustained load first
applied at 0t .
εsh
0t t
Shrinkage strain from to t t0
ε
0t t
Shrinkage strain
Nominal instantaneous strain
True instantaneous strain
Creep strain
15
2.3.1 Creep
Creep of concrete was first reported by Hatt (1907). It is a time-dependent deformation
that develops at a decreasing rate under a sustained loading. At low stress levels, creep
originates in the hardened cement paste while aggregates provide only restraint to the
deformation. The cement paste consists of solid cement gel with numerous capillary
pores, which is made of colloidal sheets formed by calcium silicate hydrates and
contain evaporable water. The mechanism of creep in concrete is disputed and no
satisfactory theory is available to describe the formation of creep. However, it is
generally believed that creep is caused by changes in the solid structure, due to the
disordered and unstable nature of the bonds and contacts between the colloidal sheets
(Bažant, 1982).
At high stress levels, interfacial cracks begin to form between the cement paste
and the aggregate and this finally results in a further increase in deformation. The
influence of stress level on creep is illustrated in Figure 2.5. The applied constant stress
is plotted against the corresponding creep strain. At service load conditions, stress in
concrete seldom exceeds 50% of the compressive strength. At these stress levels, the
creep response is approximately linear, and therefore, creep is often assumed to be
proportional to stress.
Fig. 2.5 - Influence of stress level and sustained duration on concrete mechanical
strain (Gilbert, 1988).
1.0 ( - ) = 0t t’
( - ) = infinityt t’
(εci+ εcp)
σc / fc’
Linear rangeCreep limit
Failure limit
0.5
16
Creep may be divided into two components, namely basic creep and drying creep.
Under conditions of hygral equilibrium (no moisture exchange with the ambient
medium and therefore no drying shrinkage), the gradual increase in strain with time in a
loaded specimen is known as basic creep. In a drying specimen, creep occurs
simultaneously with shrinkage, and is significantly higher than basic creep in a sealed
specimen. The extra creep that occurs in excess of the basic creep is termed drying
creep or the Pickett effect. Drying creep is primarily caused by stress-induced
shrinkage (Bažant and Chern, 1985, Bažant, 1988).
2.3.1.1 Factors affecting Creep
The magnitude of creep in concrete structures is influenced by many factors. These
factors can be categorized into two groups: the technological parameters and the
external parameters (CEB, 1997). The former refers to the parameters associated with
the particular concrete mix such as water-cement ratio, mechanical properties and
quantity of aggregates and type of cement. Lorman (1940) suggested that creep is
approximately proportional to the square of the water-cement ratio when all other
factors are kept constant. This is due to the fact that the cement paste content varies for
different water-cement ratios, and the cement paste is the main ingredient that
influences creep deformation of concrete. On the other hand, creep may also be reduced
by using stiffer aggregates and by increasing the aggregate content, thereby inducing a
higher restraining force on the cement paste.
The second group of parameters affecting creep are those associated with the
external conditions, such as age of concrete at first loading, relative humidity, and
shape and size of specimen. The observations of Davis et al. (1934) and Glanville
(1933) show that the rate of creep during the first few weeks under load is much greater
for concrete loaded at an early age than for older concrete. Figure 2.6 shows typical
creep curves for specimens loaded at different ages. It can be observed in Figure 2.6b
that for a specimen loaded at a very old age, the strain level tends to approach the value
0/ Eσ in which 0E is the asymptotic elastic modulus of concrete equivalent to )(∞cE .
The relative humidity has a large influence on creep. It was observed that drying
17
concrete has a higher rate of creep and a higher final creep level than concrete that is
stored in an environment with a higher relative humidity. Creep also depends on the
shape and size of the specimen. Investigation shows that creep decreases with an
increase in size of the specimen. However, the size factor becomes insignificant if the
thickness of the specimen exceeds about 900 mm (Neville et al., 1983). For the
influence of shape, Chivamit (1965) found that the initial rate of creep of a cruciform
section is higher than that of a circular section of the same cross-sectional area. This
may be attributed to the large surface area that exists in the cruciform section, with
creep occuring more rapidly on the drying surface than in the interior of a specimen.
Nevertheless, the influence of shape appears to be significantly smaller than the
influence of size and therefore it is often neglected for engineering purposes.
(a) (b)
Fig. 2.6 - Typical creep curves for a specimen subjected to a sustained stress σ at
various ages 't : (a) creep curves on the normal time scale; (b) creep curves
on the logarithmic time scale.
2.3.1.2 Creep Recovery
In practice, concrete structures are seldom subjected to a single constant load
throughout their service life. Other than the self-weight, service loads generally vary
with time depending upon the function of the structure at different stages in its life.
0t t 1t
2t
3t
σ/E t’c( )
t’ = t 0
t’ = t1
log (t - t’)
σ/E0
(εci+ εcp) (εci+ εcp)
t’ = t2
t’ = t3
18
Consider a specimen subjected to a sustained compressive stress, when the stress
is completely removed, it immediately undergoes an elastic unloading known as the
instantaneous recovery, as depicted in Figure 2.7. If the increase in elastic modulus
with time is ignored, the magnitude of the instantaneous recovery is equal to the
instantaneous strain when the stress is first applied to specimen. The process is
followed by a gradual reduction in strain, which is referred to as creep recovery. Creep
deformation is predominantly an inelastic process. A large amount of creep strain
developed over the loading period is irrecoverable, with only a relatively small amount
being recoverable through creep recovery (generally less than 30% of the total creep
strain).
Fig. 2.7 - Recovery of strain components upon removal of external load.
2.3.1.3 Principle of Superposition
It is well known that under service conditions, the maximum stress in a concrete
structure seldom exceeds 50% of the compressive strength and creep may be assumed
to be proportional to stress. For a concrete member subjected to a constant sustained
uniaxial compressive stress σ, the sum of the instantaneous strain, creep strain and
shrinkage strain may be written as
)()',()( tttJt shεσε += (2.6)
(εci+ εcp)
0t t
Creep recovery
Instantaneous strain
Instantaneous recovery
Irrecoverable residual strain
Creep strain
19
where )',( ttJ is the compliance function (or creep function) which is defined as the
strain at time t produced by a unit sustained stress applied at age 't . The compliance
function may further be expressed in an expanded form as follows
)',()'(
1)',( ttCtE
ttJc
+= (2.7)
where )'(tEc is the elastic modulus of concrete at age 't and )',( ttC is the creep
compliance. The first term in Eq. 2.7 represents the instantaneous deformation. By
introducing the creep coefficient )',( ttφ defined as )'()',()',( tEttCtt c=φ , Eq. 2.7
may be rewritten as
[ ])',(1)'(
1)',( tttE
ttJc
φ+= (2.8)
In a reinforced concrete structure, the concrete stress at any point varies
continuously throughout its service life even though the loads may be kept constant
with time. This is due to the redistribution of stress between the concrete and the steel
reinforcement caused by the gradual development with time of creep and shrinkage
strains in the concrete. By exploiting the advantage of the linear relationship between
creep and stress in the service range, the principle of superposition is generally adopted
to compute the deformation of the structure when subjected to varying stress histories.
The principle of superposition for an aging material was derived by Volterra (1913,
1959) and was first applied to concrete by McHenry (1943). The principle states that
the total strain at time t is calculated by summing the strain increments produced by
each stress increment σd applied at any age 't , and each strain increment is not
affected by the stresses applied earlier or later. The principle of superposition is
illustrated in Figure 2.8. This may be expressed as a Stieltjes integral (Bažant, 1982)
and the uniaxial total uniaxial strain over a period t is given by adding the shrinkage
strain to the integral as
)()'()',()(0
ttdttJt sh
tεσε += ∫ (2.9)
20
(a) (b) (c) Fig. 2.8 - The principle of superposition: (a) constant applied stress 1σ and the
corresponding strain; (b) constant applied stress 2σ and the corresponding
strain; (c) combined stress history of (a) and (b) and the resulting strain
obtained from superposition of strain curve 1 and strain curve 2.
Bazant (1982) pointed out that the Stieltjes integral is advantageous over the
commonly used Riemann integral because it is applicable to both continuous and
discontinuous stress histories. For a continuous stress history, the Riemann integral is
preferred, which is given by
)(''
)'()',()(0
tdtttttJt sh
tεσε +
∂∂
= ∫ (2.10)
0t
1σ
t 1t
σ
t
(εci+ εcp)
0t 1t
strain curve 1
0t
t 1t
σ
t
(εci+ εcp)
0t 1t
σ2 strain curve 2
0t t
1t
σ
t
(εci+ εcp)
0t
1t
σ 1 + σ2
σ1
strain curve 1 + strain curve 2
21
Various creep models are available in the literature, for example ACI model (ACI
Committee 209, 1982), CEB-FIP model (CEB-FIP Model Code, 1978) and BP model
based on the double-power (Bažant and Panula, 1978, 1980, 1982). These models have
a common theoretical ground, that is, concrete is taken as material that undergoes
aging, which means that the material properties are described as a function of the age,
't . A different type of creep model based on the solidification theory for aging creep
was proposed by Bažant and Prasannan (1989a, b). Instead of treating concrete as aging
material, the aging aspect of concrete creep is thought to be a consequence of the
growth of volume fraction of the load-bearing solidified matter due to hydration of
cement. This model was later known as the B3 model (Bažant and Baweja, 1995a, b)
and presented as a RILEM recommendation. One of the main arguments of this model
is to remedy the shortcomings of the aging creep models, which may not necessarily
satisfy the thermodynamics restrictions. From the numerical point of view, the model is
much simpler to implement than the creep models with aging. Recently, a
microprestress-solidification theory was introduced by Bažant et al. (1997), which is an
improved version of the solidification theory. The new model was formulated based on
the physical processes in the microstructures of the hydration products so as to provide
a physically justified explanation for long-term aging and drying creep of concrete.
A different approach to implement the principle of superposition is by expressing
the superposition integral in terms of the relaxation function )',( ttR , which is defined
as the stress caused by a unit constant strain imposed at time 't . The typical relaxation
curves of a relaxation function are shown in Figure 2.9. By this way, the total uniaxial
stress may be written as an integral of the stress increments produced by the stress-
dependent strain increments as
)]'()'([)',()(0
tdtdttRt sh
tεεσ −= ∫ (2.11)
The shrinkage strain increment must be isolated from the total strain increment in
Eq. 2.11 since the development of shrinkage is considered to be independent of the
concrete stress. The creep function and the relaxation function are interchangeable.
22
This may be achieved by solving Eq. 2.9. Bažant and Kim (1979) proposed an
approximated relaxation function from a given creep function as
−
∆+∆−
−−
∆−≈ 1
)',()',(
)1,(115.0
)',(1
)',( 0ttJ
ttJttJttJ
ttR (2.12)
where the unit of time is day, 0∆ is approximately 0.008 day and )'(5.0 tt −=∆ .
Reasonable accuracy was shown by comparing the results of Eq. 2.12 with the
relaxation curves calculated from a direct solution of Eq. 2.9.
(a) (b)
Fig. 2.9 - Typical relaxation curves for a specimen subjected to an imposed strain at
various ages 't : (a) relaxation curves on the normal time scale; (b)
relaxation curves on the logarithmic time scale. (Bažant, 1982).
2.3.2 Shrinkage
The porosity of concrete is determined by the porosity of the cement paste, which is
made up of air voids, capillary pores and gel pores (Bisschop, 2002). Contraction
occurs when the absorbed water in a porous body is removed. This process is known as
shrinkage and it occurs throughout the service life of concrete structures. Shrinkage of
concrete may be defined as the time-dependent change in volume in an unstressed state
at constant temperature. In general, shrinkage may be divided into four different types,
namely plastic shrinkage, thermal shrinkage, chemical shrinkage and drying shrinkage
(Gilbert, 2002). Plastic shrinkage is a consequence of evaporation of water from the
0t t 1t 2t
3t
t’ = t 0
t’ = t1
log (t - t’)
E t’( )
R t t’( , )
t’ = t2 t’ = t3
R t t’( , )
23
surface of wet concrete while still in the plastic state, and this may result in cracking
during the setting process. Thermal shrinkage is the contraction that occurs in setting
concrete as a result of subsequent dissipation of heat generated during hydration of
cement. Plastic shrinkage is prominent during the setting period and thermal shrinkage
is especially important for mass concrete structures, such as dams, where a massive
amount of internal heat is generated during the hydration process. The effects of plastic
shrinkage and thermal shrinkage are not considered in this study.
2.3.2.1 Chemical Shrinkage
Chemical shrinkage refers to the contraction of concrete that results from chemical
reactions within the cement paste. Two major types of chemical shrinkage may be
identified in concrete: autogenous shrinkage and carbonation shrinkage.
Autogenous shrinkage is defined as the bulk volume reduction caused by
chemical reactions during cement hydration, which occurs under isothermal condition
in the absence of hygral exchange with the ambient medium. Hydration of cement
continues long after setting is completed. This continuation of hydration results in
removal of water from the capillary pores and leads to a drying process within the
material. This process is known as self-desiccation and is the source of autogenous
shrinkage. Unlike drying shrinkage, autogenous shrinkage is independent of the size of
the specimen and is treated as an intrinsic characteristic of the material (CEB, 1997).
Except at extremely low water-cement ratios, autogenous shrinkage is relatively small,
only about 5% of the maximum drying shrinkage (Bažant, 1988).
Reaction of calcium hydroxide from the cement paste with the atmospheric
carbon dioxide is the cause of carbonation shrinkage. However, atmospheric carbon
dioxide rarely penetrates through concrete surface for more than a few millimetres, and
the effect of carbonation shrinkage is therefore insignificant compared to drying
shrinkage. Thus, in some cases, autogenous shrinkage and carbonation shrinkage may
well be negligible (Bažant, 1988).
24
2.3.2.2 Drying Shrinkage
Drying shrinkage is the most prominent shrinkage and is defined as the time-dependent
reduction of volume at constant temperature and relative humidity due to loss of water
from concrete stored in unsaturated air. The process of drying shrinkage begins as soon
as the absorbed water is lost to the environment. The mechanisms of drying shrinkage
is so far not fully understood, however, it is believed that the bulk shrinkage of the
cement paste is attributed to phenomena such as capillary stress, disjoining pressure,
movement of interlayer water, and changes in surface free energy (Mindess and Young,
1981; Hansen, 1987; Bisschop, 2002). Drying shrinkage is approximately proportional
to the loss of water from concrete (Carlson, 1937; Pickett, 1946). Nevertheless, the loss
of water with time depends on the size of the specimen. This inevitably makes it
difficult to use the data on the loss of water to predict the final shrinkage. Mensi et al.
(1988) developed a generalized pattern of loss of water with distance from the drying
surfaces of specimens based on the assumption that the rate of diffusion of vapour is
proportional to the square root of the time elapsed. It is, however, far more complicated
for real structures, as the size and shape are non-uniform throughout. Figure 2.10 shows
the relations of the loss of water and the age of concrete for test prisms of various sizes.
Fig. 2.10 - Water loss in specimens of various sizes (L’Hermite, 1978).
1
Age (years)
NB: Size of specimen in mm
700 700 2800xx700 700 1680xx
700 700 840xx
700 700 560xx700 700 280xx
Wat
er lo
ss a
fter m
ixin
g(%
tota
l vol
.)
2 3 4 5 60
5
10
25
The drying shrinkage that occurs in concrete that has been dried in air is not fully
recoverable by rewetting, even if the wetting period is longer than the period of drying.
For most concretes, the irreversible shrinkage can be as large as 30% to 60% of the
ultimate first drying shrinkage (Pickett, 1956; Helmuth and Turk, 1967; L’Hermite,
1960). A possible reason for the irreversible shrinkage is the development of additional
bonds within the gel during the drying process that subsequently reduces the gel pores.
The irreversible shrinkage residual may be reduced if the cement paste in concrete is
hydrated to a considerable extent before drying (Neville, 1995).
Since the main factor causing shrinkage is the evaporable water in cement paste, a
high water-cement ratio in concrete results in a high amount of shrinkage. For a
concrete with water-cement ratio between 0.2 and 0.6, shrinkage of hydrated cement
paste is found to be directly proportional to the water-cement ratio (Brook, 1989;
Neville, 1995). The amount of aggregate also has an important influence on shrinkage.
Aggregate provides restraining actions to the cement paste that undergoes drying
shrinkage. The influence of water-cement ratio and aggregate content on shrinkage is
shown in Figure 2.11.
In practical application, it is not necessary to distinguish between the components
of shrinkage. The concrete shrinkage strain is usually considered to be the sum of the
drying, chemical and thermal components (Gilbert, 2002). No thermal effects are
considered in this work, and so shrinkage is considered as a composite phenomenon of
drying shrinkage and chemical shrinkage.
Fig. 2.11 - Influence of water/cement ratio and aggregate on shrinkage (Ödman,
1968).
0.3 0.4 0.5 0.6 0.7 0.8
Water / cement ratio
Aggregate content by volume (%)
80%
70%
Shrin
kage
( 1
0)
x-6
0
800
400
1600
120060%50
%
26
2.3.2.3 Effects of Shrinkage
Shrinkage usually occurs in different amounts at different locations within a concrete
element depending on the shape of the structure. Shrinkage tends to be largest on the
surface due to rapid moisture loss and lowest in the interior of the concrete furthest
from the drying surface. The high shrinkage on the surface is restrained by the lower
shrinkage in the interior, which induces a differential shrinkage within the member.
This gives rise to the development of tensile stress on the surface and compressive
stress at the interior and may eventually lead to surface cracking. In addition,
differential shrinkage due to unsymmetrical drying may even causes warping in a
concrete member.
Concrete structures are usually made up of plain concrete and reinforcing bars.
The embedded reinforcing bars restrain the concrete from shrinking freely due to bond
action. Consider a singly reinforced section, or an unsymmetrically reinforced section
(amount of tension and compression reinforcements are not equal). Different restraints
are exerted by the top bars, if any, and the bottom bars as shrinkage develops. A
shrinkage induced curvature is produced and this may eventually result in undesirable
shrinkage-induced deflection of the member.
Moreover, most concrete structures consist of statically indeterminate members
and the development of shrinkage provokes redistribution of internal actions that may
lead to cracking. Unsightly wide cracks are commonly observed for members in which
significant restraint is provided to movement caused by shrinkage. In some cases,
cracks are even observed before the application of load.
2.3.3 Interaction of Fracture and Creep
The study of time-dependent fracture of quasi-brittle materials has gained increasing
attention in the last decade. In classical fracture mechanics, the mechanical behaviour
of materials is assumed to be time-independent. In fact, the bond breakage process at
the fracture front is time-dependent, unlike most metals. The viscoelasticity of the creep
outside the fracture process zone and the time-dependent effect in the fracture process
27
zone are not negligible. The influence of creep on fracture is recently evidenced in
some experimental studies of time-dependent fracture under quasi-static loading
conditions (Bažant and Gettu, 1992; Zhou, 1992, 1993; Zhou and Hillerborg, 1992;
Bažant and Xiang, 1997).
2.3.3.1 Influence of Loading Rate on Peak Load
In the three-point bending fracture tests of Bažant and Gettu (1992), the peak load is
higher for a faster loading rate. Figure 2.12a shows Bažant and Gettu’s test results of
two specimens loaded with different crack mouth opening displacement (CMOD) rates.
They also tested specimens of three different sizes in order to investigate the influence
of loading rate on both peak load and size effect. The results are shown in Figure 2.12b,
in which the lines depict the theoretical model of Bažand and Jirásek (1993).
(a) (b)
Fig. 2.12 - (a) Load-CMOD curves for two three-point bend concrete fracture
specimens under different rates of loading (after Bažant and Gettu, 1992).
Dashed lines are the theoretical predictions of Wu and Bažant (1993). (b)
Influence of loading rate and specimen size on the peak load (after Bažant
and Gettu, 1992). Dashed lines are the theoretical predictions of Bažant and
Jirásek, 1993.
28
2.3.3.2 Load Relaxation at Fracture Zone
Another important fracture phenomenon related to the creep effect is load relaxation.
Zhou and Hillerborg (1992) performed tension relaxation tests on notched cylinder
specimens under displacement control at a constant rate. The displacement was
increased right after the peak and held constant for 60 minutes. Load relaxation was
observed at the constant displacement, which is depicted by the vertical stress drop in
Figure 2.13a. After the first relaxation, the displacement was increased again and two
additional relaxations were performed for durations of 30 minutes. The test results are
shown in Figure 2.13.
(a) (b)
Fig. 2.13 - Tensile relaxation tests: (a) stress versus displacement curve; (b) stress
versus time. (after Zhou and Hillerborg, 1992; diagrams extracted from
Bažant and Planas, 1998).
2.3.3.3 Creep Rupture
Zhou and Hillergorg (1992) undertook a series of three-point bending tests on notched
beams, each subjected to sustained constant loading, in order to investigate the effects
of creep on fracture. It was found that the crack gradually grew with time although no
additional load was added. A typical result of the tests and the prediction of their
proposed theoretical model are shown in Figure 2.14. The response is characterized
29
firstly by a decreasing CMOD rate and is followed by a rather constant CMOD rate
over a period of time. The specimen finally failed by creep rupture accompanied by a
large CMOD.
Fig. 2.14 - Results of creep rupture tests of Zhou and Hillerborg, 1992 (diagrams
extracted from Bažant and Planas, 1998).
2.3.3.4 Time-dependent Fracture Models
Since the first publication of work on the time-dependent fracture by Zhou (1992) and
Bažant and Gettu (1992), many researchers have attempted to develop reliable
theoretical models to simulate the observed behaviour. In general, three approaches are
available in the literature.
The first approach is based on the concept of rate-dependent softening. Bažant
(1993) suggested that the bulk creep of the material and the rate-dependence of bond
rupture in the fracture process zone are the factors responsible for time-dependent
fracture of concrete. Bažant derived the rate of bond rupture in the fracture process
zone based on the theory of activation energy (Glasstone et al., 1941) and expressed the
cohesive stress as a function of the crack opening crw and the crack opening rate crw&
at constant temperature (Wu and Bažant, 1993) as follows:
time (s)2001000
0
20
40
60
80
100
CM
OD
(m
)µ
300 400 500 600
ExperimentalTheoretical
30
)/(sinh)]([)(),( 1100 rcrcrwtctcrwtcrcr wwwkfkwwwF &&& −++== σσσ (2.13)
where the superimposed dot denotes the derivative of time, wtσ is a function for the
stress versus crack opening curve obtained from an infinitely slow loading, 0k and 1k
are material constants, 0ctf is the tensile strength at zero opening rate and rw& is an
empirical constant which is called the reference opening velocity. de Borst et al.
(1993b) proposed a simpler equation for the cohesive stress by the use of a viscosity
term m:
+= cr
ctcret f
m εεσσ &1)( (2.14)
where crε is the cracking strain in the fracture process zone and etσ is a function for
the stress versus cracking strain curve.
The second approach is based on the use of rheological models to represent creep
and relaxation in the fracture zone. In the model of Zhou and Hillerborg (1992), a
modified Maxwell chain element is adopted for modelling creep fracture in the
cohesive zone while treating the material outside the cohesive zone as linear elastic.
The cohesive stress is expressed in an incremental form as
wr ddd σσσ += (2.15)
where rdσ is the stress increment due to relaxation and wdσ is the stress increment
due to a crack opening increment crdw and are given by
)1)(( /0 −−= − τασσσ dt
r ed (2.16)
)1(21 / += − τσ dt
uw eFdwd (2.17)
In Eq. 2.16 and Eq. 2.17, σ is the stress at time t, 0σ is the stress obtained from the
static stress-crack opening curve corresponding to crack opening crw , uF is the
unloading slope and α and τ are material constants.
31
The third approach describes time-dependent fracture by combining a time-
independent micromechanical model for tension softening and a time-dependent
rheological model. This approach is aimed at the modelling of very slow or static
fracture such as cracking in mass structures like dams and no rate effects are accounted
for. One of these models was introduced by Sathikumar et al. (1998) and was later
enhanced by Barpi and Valente (2001, 2003) using a fractional order rate law.
In the author’s view, the effect of time-dependent fracture due to creep is only
significant in the case of fracture of plain concrete, especially when the rate of loading
is an important factor. Concrete structures are rarely made up entirely of plain concrete,
they are strengthened with steel reinforcement which induces significant restraint to
cracking. Moreover, creep is accompanied by shrinkage which is a major cause of
crack widening with time. The influence of the rate of loading is important in seismic
and impact analyses, which beyond the scope of the present study. The effects of creep
on cracking in the context of this study are relatively unimportant. Therefore no further
consideration is made related to this phenomenon in the subsequent sections.
2.4 Behaviour of Reinforcement
In general, two types of reinforcing steel may be identified based on the difference in
the stress-strain response (Kaufmann, 1998). The first type is the hot-rolled, low-carbon
or micro-alloyed steel. The tensile stress-strain diagram is shown in Figure 2.15a.
Reinforcing steel of this type is characterized by an initial linear elastic stress-strain
relationship up to the yield stress syf and the stress stays constant over a yield plateau.
This is followed by a strain hardening range and the steel fails completely when the
stress reaches the tensile strength suf . The second type is the cold-worked or high-
carbon steel. This steel type, also, exhibits a linear elastic response at initial loading,
but has no distinct yield point. The stress-strain relationship shifts from linear elastic to
strain hardening through a smooth transition (Figure 2.15b). Due to the absence of an
observable yield point, the 0.2% proof stress is usually taken as the yield stress and, the
yield strain syε is as shown in Figure 2.15b.
32
(a) (b)
Fig. 2.15 - Stress-strain curves for reinforcing steel: (a) hot-rolled, heat-treated, low-
carbon or micro-alloyed steel; (b) cold-worked or high-carbon steel.
In accordance with the Australian and New Zealand standards AS/NZS 4671
(2001), three classes of reinforcing steel are available, namely low ductility, normal
ductility and high ductility for earthquake resistant design, which are denoted by the
letters L, N, and E, respectively. The previous 400Y grade steel has recently been
replaced by 500-grade steel with a higher minimum guaranteed yield strength of 500
MPa, which can be further designated with the appropriate class of ductility mentioned
above. The 500-grade of steel bar often comes with deformed ribbed surface and is
used as main reinforcement. Another generally adopted grade of steel in reinforced
concrete design is the grade 250N plain (or round, bar surface undeformed) steel bar
which has a minimum guaranteed yield strength of 250 MPa. As a result of the low
bond associated with plain round bars, the 250N plain bars can only be used as column
ties and beam stirrups (AS 3600, 2001).
2.5 Bond between Reinforcement and Concrete
The load-carrying capacity of reinforced concrete structures depends on many factors,
including the quality of bond between the reinforcing steel and the concrete. The
composite interaction between the two materials is established and maintained by the
bond stress, which effectively transfers load between the steel and concrete. The main
fsu
σs
εs 0.2%
Es
1
fsy
εsu
fsu
σs
εs
Es
1
fsy
εsuεsyεsy
33
mechanism in the development of bond stress is the mechanical interaction between the
ribbed or deformed surface of the reinforcing bar and the concrete, although other
mechanisms such as surface friction and chemical adhesion also play a role.
2.5.1 Local Bond Stress-slip Relationship
Numerous tests have been performed dedicated to the understanding of the bond
mechanism between reinforcing steel and concrete. In general, bond is treated as an
average effect over the contact surface area of concrete and steel and is commonly
described by a local bond stress-slip relationship. The local bond stress-slip relationship
is obtained by means of pull-out tests on reinforcing bars embedded in concrete and is
characterized by four different stages as shown in Figure 2.16a (fib, 2000). At the
initial stage, the bond stress remains as low as 20% to 80% of the tensile strength of
concrete. No cracking is observed at this stage. Bond mechanism is realized primarily
by chemical adhesion and partly by the micromechanical interaction due to the
roughness of steel surface. As the pull-out deformation proceeds, the relative
displacement between steel and concrete increases and results in a higher bond stress.
The second stage commences when the chemical adhesion is entirely broken
down, in which the bond stress is in excess of 1τ as shown in Figure 2.16a. Large
bearing stresses are exerted by the lugs of the deformed bar and causes formation of
transverse microcracks in the concrete adjacent to the tip of the lugs. These microcracks
are also known as internal bond cracks (Figure 2.16b) and were first observed
experimentally by Goto (1971).
At stage three, the bond stress increases beyond the tensile strength of concrete
and longitudinal splitting cracks begin to form radially around and parallel to the
reinforcing bar as shown in Figure 2.16b. The splitting action is triggered when the
radial stresses induced by the wedge action of the ribs exceed the tensile strength of
concrete (Tepfer, 1973, 1979). This stage ends as soon as the splitting crack reaches the
surface of the concrete member. In the case of low confinement, such as concrete with
light transverse reinforcement, the concrete member usually fails abruptly with a
through splitting crack.
34
(a)
(b)
Fig. 2.16 - Pull-out mechanisms: (a) local bond stress-slip relationship and stages of
debonding (modified from fib, 2000); (b) internal bond cracks and splitting
cracks (Vandewalle, 1992).
At the final stage, the bond stress increases to the peak, beyond which bond stress
gradually decreases with a large increase in slip. Depending on the degree of
confinement provided by the transverse reinforcement, the concrete member may fail
by splitting, pull-out, or both simultaneously. In the case of heavy transverse
reinforcement, the bond failure is caused by pull-out of the bar, whereas for concrete
member with low transverse reinforcement, bond breaks down by splitting failure with
longitudinal cracks penetrating through the concrete cover. In this case, the softening
Slip, s
Bond stress, τb
Stage 1
Stage 3 Stage 4
Pull-out failureStage 2
τ1
Splitting failure
Confinement increases
Abruptsplitting failure
Splitting crack
Pull-out direction
Internal bond crack
35
branch of the bond stress versus slip curve is shifted down by a considerable amount. In
practice, appropriate design of transverse reinforcement and concrete cover must be
enforced in order to prevent the brittle bond failures involved at this stage.
The local bond stress-slip relationships have been investigated by many
reearchers in order to describe the four stages of pull-out mechanisms discussed above
(for example Rehm, 1961; Nilson, 1968; Martin, 1973; Mirza and Houde, 1979; Ciampi
et al., 1981; Shima et al., 1987). The CEB-FIP Model Code 1990 (1993) has adopted
the relation proposed by Ciampi et al. (1981) and provides specific parameters for
confined concrete and unconfined concrete.
Upon yielding of steel reinforcement, the bond between steel and concrete is
significantly reduced due to the lateral contraction of steel bar resulting from the
Poisson’s effect. Shima et al. (1987) included reinforcing steel strain as a parameter in
their proposed relation so as to extend the application to the post-yield range. Huang et
al. (1996) expressed the post-yield bond stress-slip relation with a lower softening
branch for both normal strength concrete and high strength concrete (Figure 2.17) with
the base curve similar to the bond model of the CEB-FIP Model Code 1990 (1993).
The development of splitting crack results in a negative effect on bond transfer,
similar to yielding of the steel bar. It reduces normal forces around the bar and hence
decreases the bond transfer. Gambarova et al. (1982, 1989, 1996, 1997) investigated the
effect of splitting cracks on the bond stress-slip relationship by using pull-out
specimens with preformed splitting cracks.
Fig. 2.17 - Local bond stress-slip relationship of Huang et al. (1996).
τmax
τy
Bond stress, τb
Slip, s
Reinforcing steel in elastic range
Reinforcing steel in plastic range
sy s1
τf
τy.f
s3 sy.f s3 s5 s4
36
The present study focuses on the formation of transverse cracks in concrete
structures at service loads, in which the bond stress levels are normally low. Hence, the
effect of splitting cracks on bond will not be considered here.
2.5.2 Influence of Bond on Cracking
The quality of bond has a marked influence on crack formation in reinforced concrete
structures, both in terms of the spacing between cracks and the crack width. Consider a
reinforced concrete member subjected to an axial tensile force, as shown in Figure
2.18a. The first primary crack is formed when the concrete stress, transferred from the
steel reinforcement via bond action, reaches the tensile strength of the concrete. This
first crack occurs at that section along the member where the tensile strength is
smallest. At the crack, the concrete stress is zero, the steel carries the entire tensile
force, slip takes place between the concrete and the steel and the bond stress is
negligible. Adjacent to the crack, the bond stress τb increases rapidly, the tensile stress
in the concrete σc increases and the tensile stress in the steel σs decreases. At some
distance so from the first crack, the bond stress reduces to zero and the concrete and
steel stresses are unaffected by the crack.
If further tensile force is added, a second primary crack will form when the
concrete tensile strength is exceeded on the next weakest section greater than so from
the first crack. The process continues until the final crack pattern is established. No
further cracking can occur if the distance between the cracks is not large enough to
develop sufficient bond to allow the concrete stress on any section between the cracks
to reach the concrete tensile strength. The final crack spacing is therefore somewhere
between so and 2so. Figure 2.18b shows a magnified view of the concrete-steel
debonding at the primary crack.
37
(a) (b)
Fig. 2.18 - Cracking in a reinforced concrete element: (a) direct tension member and
stress distributions; (b) stress transfer by bond.
2.5.3 Tension Stiffening
One of the most important consequences of the effect of bond in cracked reinforced
concrete is tension stiffening. The conventional approach completely disregards the
effectiveness of the tensile concrete in a cracked reinforced concrete member and the
total applied tension is thought to be resisted only by the steel reinforcement. This
apparently underestimates the total stiffness of the structure and leads to an
unrealistically higher prediction of deformation. As demonstrated in the previous
section, concrete between the cracks is capable of carrying tensile stress due to stress
transfer via the bond between concrete and steel reinforcement. The structural response
of a cracked reinforced concrete tension member is stiffer than that of the naked
reinforcing steel and this stiffening phenomenon is called “tension stiffening”.
The tension stiffening effect may be further illustrated by considering the load
versus average strain response of a reinforced concrete tension chord between two
cracks (see Figure 2.19a). In order to compare with the stiffness of the bare reinforcing
steel, the load on the tension chord is expressed as the force per unit area of steel
reinforcement. The comparison is shown in Figure 2.19c. It is evident that the
contribution of concrete between the cracks increases the stiffness. This can be
observed by comparing the secant modulus of the tension chord smE to the elastic
bond stress τb
steel stress σs
concrete stress σc
Primary crack
Internal crack
38
modulus of the bare reinforcing steel sE at the same stress level 1σ shown in Figure
2.19c. The saw-tooth response in the stress-strain curve represents the formation of
primary cracks in the tension chord.
(a)
(b) (c)
Fig. 2.19 - Concrete tension stiffening: (a) tension chord with embedded reinforcing
steel; (b) bare reinforcing steel; (c) comparison of stiffness of embedded
steel and bare steel.
2.6 Non-linear Modelling of Concrete Structures
Reinforced concrete is well known for its non-linear behaviour. The non-linearity in
reinforced concrete originates from the non-linear stress-strain relationship of plain
concrete. The structural behaviour is further complicated by cracking of concrete which
causes a considerable redistribution of stresses within the intact concrete as well as the
stress transfer from concrete to steel reinforcement.
The finite element method is the most often adopted numerical tool to simulate
the non-linear behaviour of reinforced concrete structures. Rational and reliable
representations of cracking of concrete must be used in conjunction with the finite
element method in order to accurately describe the structural behaviour. Two major
approaches are available in the literature for the modelling of cracking in concrete
σ = /N As
ε
Es 1
σcr
εsεsm
σ1
Esm
1
Embedded steel
Bare steel
39
structures, namely the discrete crack approach and the smeared crack approach. These
two approaches are discussed in the subsequent sections.
2.6.1 Discrete Crack Approach
In the discrete crack model, cracking is simulated as a propagation of discontinuities in
a structure. Two approaches may be identified in the discrete crack model, the inter-
element crack model and the intra-element crack model. In the first approach, crack
propagation is modelled by means of separation of element edges and the cohesive
tractions in the fracture zone are simulated using either linkage elements or interface
elements (as depicted in Figure 2.20a). This approach was employed widely in the early
development of finite element models for reinforced concrete structures (Ngo and
Scordelis, 1967; Nilson, 1969; Mufti et al., 1972; Al-Mahaidi, 1979). However, these
discrete crack models were later replaced by the smeared crack approach which is more
attractive in a variety of computational aspects.
The early discrete crack approaches suffer from two major drawbacks. The first
drawback is the limitation of the crack trajectory, which is constrained to follow the
path along the predefined inter-element boundaries. The second drawback is the
increasing computation cost and the decreasing efficiency due to the additional degree-
of-freedom of the separated nodes along the new crack faces. To remedy the first
drawback, a remeshing technique must be adopted. In this technique, the finite element
mesh topology is modified at each crack increment according to the direction of the
crack propagation. This technique was pioneered by Saouma and Ingraffea (1981) and
Saouma (1981) for linear elastic fracture mechanics problems and was extended to non-
linear mixed mode fracture mechanics of concrete (Ingraffea et al, 1984; Bocca et al.,
1991; and Gerstle and Xie, 1992; Červenka, 1994; Valente, 1995).
The second approach is the intra-element discrete crack model in which the crack
is allowed to propagate through the finite element as shown in Figure 2.20b. Two types
of model are available in this approach, namely the embedded discontinuity model and
the model based on the partition-of-unity concept. The embedded discontinuity model
originally emerged as a tool to deal with strain localization problems such as shear band
40
in metals and has been further extended to cohesive material such as concrete (Ortiz et
al., 1987; Belytschko et al., 1988; Klisinski et al., 1991, Lofti and Shing, 1994; Simo
and Oliver, 1994). The discontinuity at cracks is regarded as a displacement jump in the
element by incorporating additional localization modes to the standard shape function
of the finite element (Sluys and Berends, 1998). The second type of intra-element crack
model is formulated on the basis of the partition-of-unity concept (Duarte and Oden,
1996; Melenk and Babuška, 1996). In this model, discontinuous shape functions are
used and the displacement jump across the crack is represented by extra degrees of
freedom at the existing nodes (Wells and Sluys, 2001).
The discrete crack approach may sound physically appealing since the crack is
represented by a traction free discontinuity and it does not suffer from the spurious
strain localization problems upon mesh refinement which to a degree undermines the
smeared crack approach. The discrete crack approach is most useful in structural
problems with prominent localized cracking. Nevertheless, concrete structures are often
dominated by diffused cracking, which is found especially in reinforced concrete
members with closely spaced reinforcement such as shear walls and the tension side of
a reinforced concrete beam. In such cases, the smeared crack approach is often
considered to be advantageous compared to the discrete crack approach.
(a) (b) (c)
Fig. 2.20 - Concrete cracking models: (a) discrete inter-element crack approach; (b)
discrete intra-element crack approach; (c) smeared crack approach (crosses
in the shaded element are the cracked integration points).
41
2.6.2 Smeared Crack Approach
The smeared crack approach is a concept developed based upon the framework of
continuum mechanics which is described in the notions of stress and strain. This
approach was introduced by Rashid (1968) to overcome the drawbacks of the early
discrete crack models and has become the most widely adopted method for modelling
cracking in structural engineering problems. The cracking in a smeared crack model is
assumed to be smeared over a certain volume of material. Cracking is treated as a
reduction of average material stiffness in the direction of the major principal stresses. In
a finite element model, this is done at the integration point level of the element by
reducing the stiffness according to a particular constitutive relationship (see
Figure 2.20c).
One of the most important advantages of the smeared crack approach is that the
mesh topology of a structure remains unchanged during the nucleation of cracks. This
offers great convenience in numerical implementation. As discussed earlier, cracking in
reinforced concrete members is usually distributed due to the stabilising effect of
reinforcement. Therefore, the distributed nature of cracking in the smeared crack
approach best describes the cracking phenomenon in most concrete structures. In the
case of concrete fracture, the smeared crack approach is justified due to the fact that the
fracture process zone of concrete is densely filled with microcracks within a finite
width of fracture before it finally localizes into a line of discontinuity (Bažant, 1985).
Notwithstanding the advantages in modelling reinforced concrete, the classical
smeared crack approach suffers from a major deficiency when dealing with localized
cracking. In fracture problems, the smeared crack approach tends to localize cracking
into a single row of elements and produces unobjective tensile post peak results upon
refinement of element size. This phenomenon is known as spurious mesh sensitivity
and will be discussed in detail later. On the other hand, the classical smeared crack
approach also suffers from mesh directional bias which tends to predict crack
propagation in alignment with the direction of the mesh. Nevertheless, these
deficiencies may be eliminated by utilising one of a number of regularization models
which will be discussed later.
42
Three major variants of smeared crack approach are available in the literature
depending on the method of development of the crack planes: fixed crack model,
rotating crack model and multi-directional fixed crack model.
2.6.2.1 Fixed Crack Model
The first smeared crack model introduced by Rashid (1968) was based on the fixed
crack concept. In a fixed crack model, the crack initiates normal to the major principal
stress and has a fixed direction throughout the loading the process. Consider a two-
dimensional linear elastic material, the stress may be written as
−−=
xy
y
x
xy
y
x E
γεε
νν
ν
ντσσ
)1(2100
0101
1 2 (2.18)
The material stiffness matrix becomes orthotropic as soon as the maximum
principal stress exceeds the concrete tensile strength. The orthotropic stress-strain
relationship may be written in the normal (n) and tangential (t) directions of the crack,
which gives
ntntnt εDσ sec= (2.19)
where secntD is the secant stiffness matrix in the directions of orthotropy. In the early
versions of the fixed crack model (Rashid, 1968; Červenka, 1970), the stiffness normal
to the crack and the shear stiffness are both set to zero and the Poisson’s effect vanishes
immediately after cracking.
=
00000000
sec EntD (2.20)
43
The sudden drop of stiffness, however, gives rise to numerical difficulties. To
improve this situation, a “shear retention factor” β was introduced (for instance, Suidan
and Schnobrich, 1973) for a smooth transition from uncracked to completely cracked
states. It may also be conceived as a way to account for the aggregate interlock. The
shear component is retained in the stiffness matrix as
=
GEnt
β0000000
secD (2.21)
in which the shear retention factor β is in the range 10 ≤≤ β . To include the strain
softening phenomenon of concrete, a refined model was proposed by Bažant and Oh
(1983). They introduced a “normal reduction factor” µ and incorporated the post-
cracking Poisson’s effect, which leads to the following expression
−−
−−
=
G
EE
EE
nt
β
µνµν
νµµν
νµ
µν
µ
00
011
011
22
22
secD (2.22)
In real structures, the direction of principal stresses is constantly changing during
the loading process. Because of the fixed crack direction, the rotation of the principal
stresses causes the development of shear stress on the crack surface. A large shear
stress may be induced if the rotation of the principal stresses is significant.
Consequently, the fixed crack model often predicts a stiffer response than
experimental results.
2.6.2.2 Rotating Crack Model
The rotating crack model was proposed by Cope et al. (1980), which was motivated by
the experimentally observed phenomenon of the rotating principal stress directions in
concrete structures. The underlying smeared cracking formulation of the rotating crack
44
model is essentially no different to that of the fixed crack model. A crack is formed
normal to the major principal direction upon violation of the tensile strength criterion.
However, in the rotating crack model the crack direction rotates with the principal
stress directions during the entire loading process. Consequently, the orthotropic
constitutive relationship of the cracked concrete changes accordingly with the crack
directions. Eq. 2.19 may now be written in the principal 1-2 axes instead of the local
crack n-t axes as
12sec1212 εDσ = (2.23)
Although questions were raised for the physical justification of the rotating crack
model (Bažant, 1983) due to the fact that cracks in real structures do not rotate but the
principal stress directions, the rotating crack model certainly is an efficient tool for
accurate concrete modelling. In addition, the shear retention factor, which is a sensitive
factor in the fixed crack model, is no longer needed in the rotating crack model. This
simplification greatly facilitates the numerical implementation and is practically
appealing from an engineering point of view.
2.6.2.3 Multiple Fixed Crack Model
The multi-directional fixed crack model is a refined version of the fixed crack model,
which aims at improving the simulation of the change in direction of the principal
stress. The basic concept is identical to the fixed crack model in which rotation of crack
is not permitted. However, new cracks are allowed to develop at different orientations
if the angle between two consecutively formed cracks exceeds a threshold angle.
A systematic incremental strain-decomposition approach was proposed by de
Borst and co-workers (de Borst and Nauta, 1985; de Borst, 1987; Rots, 1988). The total
concrete strain increment ε∆ is decomposed into a concrete elastic strain increment
ceε∆ and a cracking strain increment crε∆ as follows
crce εεε ∆+∆=∆ (2.24)
45
where the cracking strain increment crε∆ is made up of the cracking strains of all
cracks at different orientations, which can be written as
...321 +∆+∆+∆=∆ crcrcrcr εεεε (2.25)
de Borst and co-worker derived a constitutive relation for cracked concrete, in
which the stress increment is related to the strain increment by a reduced elastic
stiffness matrix similar to the elasto-plastic stiffness matrix of plasticity. Since crack
opening and crack closing may happen simultaneously, the computational procedure
for a multiple fixed crack model may get quite complicated and therefore special
computational strategies must be devised to follow the possible bifurcation path
(Bažant and Planas, 1998).
2.6.3 Constitutive Models for Concrete
One very important aspect in smeared crack modelling of concrete is the development
of constitutive models that are capable of describing the behaviour of concrete. In the
broad sense, the constitutive models of concrete may be classified into two categories:
the macroscopic phenomenological models and the micromechanical models (Bažant
and Prat, 1988). The macroscopic models may be further divided into two subgroups
according to the treatment of the stress-strain evolution law. The first group is the
models based on a total formulation, in which the total stress is related to the total strain
by a secant constitutive relation. Examples of total formulation models are the
elasticity-based cauchy elastic models and the continuous damage models. The second
group refers to the models adopting an incremental stress-strain concept via the use of
tangent moduli, which is referred to as the incremental formulation. Plasticity-based
models, elasticity-based hypoelastic models, plastic-fracturing models and endochronic
models are the examples of this subgroup of macroscopic models.
Whilst the macroscopic models describe the material behaviour via a direct stress-
strain constitutive relationship, the micromechanical model is characterized by a
constitutive relationship based upon the mechanical interaction of the cement matrix
46
and aggregates loaded on the microscopic and macroscopic levels. A well-known
example of the micromechanical model is the microplane model.
In the following sections, a brief review and discussion is given of the frequently
adopted constitutive models in concrete modelling, namely the elasticity-based models,
the plasticity-based models, the continuous damage models and the microplane model.
2.6.3.1 Elasticity-based Models
In general, the elasticity-based models can be formulated based on a total formulation
or an incremental formulation. In the elasticity-based models based on the total
formulation, or called the cauchy elastic models (ASCE Task Committee, 1982, Chap.
2), the total stress is a function of the total strain and is given by
)(εσ f= (2.26)
It may be written in a secant constitutive relationship in tensorial notation as
klijklij D εσ sec= (2.27)
where ijσ and klε denote the stress and strain tensors, respectively, and secijklD is a
fourth-order secant stiffness tensor, which is a function of the stress state.
It is evident that the total stress is uniquely related to a single total strain via the
material response function as shown in Eq. 2.26. Due to the deformation path-
independence introduced by the total stress-strain constitutive relationship, these
models are not able to provide a physically sound base for representing the behaviour
of concrete. In the case of loading and unloading stress histories, the total formulation
elasticity-based model requires a different stress-strain relationship in order to handle
the unloading behaviour of concrete. This further gives rise to difficulties in
generalizing the total formulation models to represent behaviour of concrete subjected
to cyclic loading.
47
The elasticity-based models based on the total formulation concept have been
extensively used in simulating the behaviour of reinforced concrete structures under in-
service loading in which compressive loads are relative small and remain elastic.
Concrete prior to cracking can be modelled sufficiently accurate as an isotropic linear
elastic material (for example Ngo and Scordelis, 1967; Rashid, 1968; Bažant and
Gambarova, 1980; Bažant and Oh, 1983). The majority of these models concentrate on
the behaviour of cracking in concrete, including strain-softening and localization in
particular. A secant or an initial stiffness is often utilized to model unloading in
concrete after cracking.
The cauchy elastic model has also been extended to represent non-linearity of
concrete. One of the earliest biaxial models based on the cauchy elastic formulation
was proposed by Kupfer and Gerstle (1973). They formulated an isotropic total stress-
strain model based on biaxial monotonic tests of concrete and established a series of
expressions for the secant shear and bulk moduli. However, good results were only
obtained for comparisons with experimental data at low stress levels.
Despite the criticisms related to the physical interpretation, the total formulation
has been widely adopted in recent years to simulate the non-linear behaviour of
concrete. This is due primarily to the simplicity of implementation of the models,
especially for extending an isotropic linear elastic model for non-linear modelling. The
models have been shown to work well in concrete modelling in many studies (Stevens
et al., 1991; Ramaswamy et al.; 1995, Vecchio, 1989, 2001; Foster and Marti, 2003).
One of the important models in modelling concrete membrane structures is the
modified compression field theory (Vecchio and Collins, 1986), which is formulated
based on the total formulation concept. A concrete compression softening model was
proposed along with the theory, which states that the peak compressive stress for
cracked concrete under compression-tension state of stress decreases with increasing
principal tensile strain. Vecchio and Collins (1989) introduced a strength reduction
factor β , which is defined as the ratio of the reduced peak compressive stress and the
concrete compressive strength, and is given by
48
0.127.085.0
1
2
1≤
−=
εε
β (2.28)
where 1ε and 2ε are the major principal and minor principal strains, respectively.
Vecchio (1989) implemented the modified compression field theory into a secant
stiffness non-linear finite element program and demonstrated good agreement with
experimental results for reinforced concrete membranes.
The second type of elasticity-based models is formulated based on an incremental
stress-strain constitutive relationship, which is also called the hypoelastic models
(Truesdel, 1955). Stress and strain are related by the material tangent stiffness that is
dependent on the current state of stress and may be expressed in tensorial notation as
klijklij dDd εσ tan= (2.29)
where ijdσ and kldε are the incremental stress and strain tensors respectively and
tanijklD is a fourth-order tangent stiffness tensor. From a thermodynamics viewpoint,
these models provide a more physically convincing base than the models based on a
total formulation due to the path-dependent nature of the deformation history. A typical
biaxial model based on the isotropy concept was proposed by Gerstle (1981). In his
model, the response of concrete is divisible into hydrostatic and deviatoric components
and the constitutive relationship is expressed in terms of tangential bulk and shear
moduli. Good correlation was observed with experimental data.
A different approach was developed based on the experimentally observed stress-
induced anisotropy in concrete. As a special case of anisotropy, concrete subjected to a
biaxial state of stress is treated as an orthotropic material and the constitutive relation is
constructed along the axes of orthotropy in the principal stress directions. An important
model of this type was presented by Darwin and Pecknold (1977). In their incremental
orthotropic model, the concept of “equivalent uniaxial strain” was proposed to subtract
the Poisson’s effects in the principal directions so as to represent the degradation in
concrete with an equivalent uniaxial stress-strain relationship. The total equivalent
49
uniaxial strain iuε is calculated through the accumulation of each incremental change
in strain caused by the incremental change in principal stress and is given by
∑=
∆=
incN
i i
iiu E1
σε (2.30)
where incN is the total number of load increments. The incremental constitutive
relations are written in the principal directions of orthotropy as
121212 εDσ dd = (2.31)
in which 12D is the tangent stiffness matrix given by
−+−
=
2121
22
11
212
2(4100
00
11
EEEE
EEEE
ν
νν
νD (2.32)
where ν is an “equivalent” Poisson’s ratio equal to 21νν . 1E and 2E are the tangent
moduli obtained from uniaxial stress-strain curves. The model was demonstrated to be
capable of modelling concrete under monotonic and cyclic loading.
2.6.3.2 Plasticity-based Models
To capture the more general behaviour of concrete over the full loading range, the
plasticity-based models are advantageous over the elasticity-based models. The
parameters required in the plasticity-based models are greatly reduced compared to the
elasticity-based models. The plasticity theory was originally developed to represent the
behaviour of ductile materials such as metals. Extending the theory to represent the
behaviour of concrete, for which the non-linear behaviour is characterized by dense
microcracking in the material, requires significant modifications to the classical
plasticity models.
50
The establishment of a standard plasticity model involves three essential
conditions, a yield surface, a hardening rule and a flow rule. Plastic deformation is
deemed to initiate when the stress of a material reaches the yield surface in stress space.
The evolution of loading surfaces after yield is subsequently governed by the hardening
rule. During plastic deformations, the plastic strain evolution rate is controlled by the
flow rule using a plastic potential function (Chen and Han, 1988; Chen et al., 1992). In
the plasticity theory, inelastic deformation of materials is measured by the amount of
plastic strain developed during the course of loading. Expressing in a differential form,
the total strain rate is divided into the elastic and plastic components
pe εεε &&& += (2.33)
where the superimposed dot denotes the first derivative of time, eε& and pε& are the
elastic and plastic strain rate vectors, respectively. The stress rate is related to the
elastic strain rate by a symmetrical linear elastic constitutive matrix, given by
)( pe εεDσ &&& −= (2.34)
where the bracket term is the elastic strain rate vector obtained from Eq. 2.33. For the
case of isotropic hardening or softening plasticity, the yield surface is described as a
function in stress space given by
0),( =κσf (2.35)
where κ is a scalar value called the hardening or softening parameter that is typically
dependent on the strain history. A stress point is not permissible to lie outside the yield
surface and must remain on the yield surface during the course of plastic flow. Hence, a
condition is introduced, the Prager’s consistency condition, which is given by the
equation:
0),( =κσf& (2.36)
For a given flow rule, the plastic strain rate vector is expressed as the product of a
positive proportionality scalar λ& and a vector m given by
51
mε λ&& =p (2.37)
in which the scalar λ& and the vector m indicate the magnitude and the direction of the
plastic flow, respectively. The vector m is defined as the gradient of the plastic
potential function pg of the flow rule, expressed as
σm
∂
∂= pg
(2.38)
By defining a vector n as the gradient of the yield function
σn
∂∂
=f (2.39)
the consistency equation of Eq. 2.36 can be written as
0T =∂∂
+ κκ&&
fσn (2.40)
where the superscript T denotes the transpose of the vector n. By defining a hardening
or softening modulus h as follows
κκλ&
& ∂∂
−=fh 1 (2.41)
and replacing σ& and κκ &)/( ∂∂ f in Eq. 2.40 using the relations given by Eq. 2.34, Eq.
2.37 and Eq. 2.41 and yield the expression for the proportionality rate constant λ&
mDn
εDn
e
e
h T
T
+=
&&λ (2.42)
Given a yield function and a plastic potential function, the growth of plastic strain can
be determined. Now the stress rate may be related to total strain rate as
εDσ && ep= (2.43)
52
where epD is the elasto-plastic stiffness matrix, which can be perceived as a reduced
elastic stiffness matrix given by
mDn
DnmDDD
e
eeeep
h T
T
+−= (2.44)
The elasto-plastic stiffness matrix presented above is non-symmetrical and was
derived based on a non-associated flow rule. For an associated flow rule, the yield
function coincides with the plastic potential function )( pgf = and hence gives a
symmetrical elasto-plastic stiffness matrix.
Numerous efforts have been dedicated to develop the yield functions, hardening
rules and flow rules in order to accommodate the use of plasticity theory in modelling
of concrete (Chen and Chen, 1975; Buyukozturk, 1977; Murray et al. 1979). Chen and
Chen’s (1975) work is one of the earliest attempts to establish a plasticity-based model
for concrete. They developed an isotropic hardening plasticity model adopting two
similar functions to define the surface in pure compression and tension-compression
stress states. However, the model has been subjected to criticisms on its applicability to
concrete, since the model postulates concrete to be linear elastic even at high
stress levels.
Han and Chen (1985) introduced a non-uniform hardening plasticity model
assuming the associated flow rule. In their model, a failure surface is defined as a
bounding surface to enclose all the loading surfaces. The failure surface remains
unchanged throughout the loading process. During hardening, the loading surface
expands from the shape of the initial yield surface and reaches the final shape
overlapping the failure surface. The model was further improved based on the non-
associated flow rule and demonstrated a good correlation with experimental data.
In a more recent development, Feenstra and de Borst (1996) presented an energy-
based composite plasticity model to describe the behaviour of plain and reinforced
concrete in biaxial stress under monotonic loading conditions. Based on the framework
of incremental plasticity, they employed a Rankine yield criterion and a Drucker-Prager
yield criterion to monitor the respective stresses in tension and compression. The
53
plasticity model is amalgamated with the energy approach based on the crack band
theory (Bažant and Oh, 1983) for loading in both tension and compression. The model
was developed particularly to model tension-compression biaxial stress states in
concrete structures, such as shear wall panels.
2.6.3.3 Continuous Damage Models
Continuum damage mechanics was originally proposed as a simple model to study
creep failure of metal alloys. It provides a means for describing the degradation of
elasticity in material that is caused by microstructural damage (Kachanov, 1958;
Rabotnov, 1969; Lemaitre and Chaboche, 1990). In the 1980’s, continuum damage
mechanics began to attract increasing attention from researchers in the concrete field
and the pioneering application to quasi-brittle material was made by Mazars (1984) and
Mazars and Pijaudier-Cabot (1989). A set of damage variables were introduced to
represent the local loss of material integrity. The formation of a crack is indicated by
the severity of damage in that part of the material domain and propagates in the
direction of the growth of damage in the fracture process zone.
In damage mechanics, the plastic response of concrete is ignored and strains are
assumed to be fully recoverable upon removal of stresses. Based on the concept of total
formulation, the basic structure of the constitutive relationship (de Borst, 2002) is
given by
klmnopmnmijklij qD εωσ ...),,,(sec Ω= (2.45)
where ijσ is the stress tensor, klε is the strain tensor and secijklD is a fourth-order secant
stiffness tensor which may be dependent on several internal variables such as scalar
valued variable mq and tensor-valued variables mnω and mnopΩ . The constitutive
relation in Eq. 2.45 is, at first glance, similar to that in Eq. 2.27 for the elasticity-based
models based on a total formulation. In fact, there exists a major difference between the
two models, that is, the internal variables engaged in the continuous damage model are
54
history-dependent while the elasticity-based models based on a total formulation are
load path-independent as discussed earlier.
To illustrate the theory of damage mechanics, an isotropic scalar damage model is
considered. The total stress-strain relationship is written as
εDσ e)1( ω−= (2.46)
where eD is the elastic stiffness matrix and the scalar ω is a damage variable. The
damage level of a material is measured via the damage variable for which the value
varies from zero to one, indicating the respective state of a fully intact material to a
complete loss of material integrity.
As in plasticity, a damage model is accompanied by a loading and unloading
function f given by
κεκε −= ~),~(f (2.47)
where κ is a history-dependent parameter and ε~ is the equivalent strain. The damage
loading function f and the rate of the history parameter κ must satisfy the Kuhn-Tucker
conditions at all instances, which are given by
0≤f , 0≥κ& and 0=κ&f (2.48)
In a multiaxial generalization, damage growth is deemed to occur when the damage
loading function f and the its first derivative of time f& are equal to zero, that is
0=f and 0=f& (2.49)
The damage variable is obtained through a damage evolution law expressed as a
function of the history parameter κ such that
)(κω F= (2.50)
For modelling the propagation of a crack in concrete, Mazars (1984) expressed the
equivalent strain as a function of the principal strains as follows
55
( )∑=
=3
1
2~i
iεε (2.51)
where iε is the principal strains, and ii εε = for 0>iε and 0=iε for 0≤iε .
de Vree et al. (1995) proposed a different function for the equivalent strain and called it
the modified von Mises definition. This function is given by
22212
21
)1(6
)21()1(
21
)1(21~ JkIk
kI
kk
νννε
++
−
−+
−−
= (2.52)
where ν is the Poisson’s ratio, 1I and 2J are the first invariant of the strain tensor and
the second invariant of the deviatoric strain tensor, respectively. The parameter k is the
ratio of the uniaxial compressive strength to the uniaxial tensile strength ( ctc ff /' ).
The isotropic damage models are sufficient for describing progressive crack
propagation in concrete, however, a more complicated damage model is generally
required for the modelling of reinforced concrete members when damage induced
anisotropy is not negligible. For two-dimensional planar problems, concrete is treated
as an orthotropic material and two damage variables, 1ω and 2ω , are introduced to
represent the loss of material integrity in the cracking opening direction and the shear
direction, respectively (de Borst et al., 1998).
Ragueneau et al. (2000) extended the damage-based model to characterize a
residual hysteretic response for seismic analysis of concrete structures. The main
feature of the model is the coupling of damage mechanics and a sliding effect which
produces the hysteretic behaviour of concrete under cyclic loading.
2.6.3.4 Microplane Models
The constitutive models that have been discussed so far are the macroscopic models. In
contrast to the macroscopic models are the micromechanical models, in which the
constitutive laws are established based upon the microscopic behaviour of the cement
56
and aggregate under loading. The microplane model is a constitutive model of the
microscopic type. The model was developed based on the idea of Taylor (1938), which
was developed in detail for plasticity polycrystalline metals by Batdorf and Budianski
(1949) under the name “slip theory of plasticity”. The concept was extended by Bažant
and Gambarova (1984). They replaced the slip planes by damaged planes and named it
the “microplane model”.
The macroscopic stress and strain describe the constitutive relationship of a
macroscopic model. In the microplane model, the constitutive relationship is defined by
the relationship of the stress and strain acting on the planes at various arbitrary
orientations in the microstructure of the material, which is called the microplane (see
Figure 2.21a). These planes can be imagined as damage planes in the microstructure of
the material. The macroscopic behaviour of the material is essentially a composite
effect imposed by each microplane oriented in various directions.
In the microplane model, the strain on a microplane is obtained as a projection of
the macroscopic strain under a so-called “kinematic constraint”. A set of unit
orthonormal base vectors l, m and n with components il , im and in , respectively, are
introduced on the micro level, in which il and im lying on the microplane and in
normal to the microplane (see Figure 2.21a). By applying the kinematic constraint, the
macroscopic strain tensor ijε is resolved into a normal component Nε and two shear
components Mε and Lε (Bažant et al., 2000) and are given by
ijijN N εε = , ijijM M εε = and ijijL L εε = (2.53)
where
jiij nnN = (2.54a)
)(5.0 ijjiij nmnmM += (2.54b)
)(5.0 ijjiij nlnlL += (2.54c)
57
The lower case subscripts refer to the components of the tensors in Cartesian
coordinates ix , where i = 1, 2, 3. Repetition of lower case subscript, referring to
Cartesian coordinates ix , implies summation over i = 1, 2, 3.
Knowing the strain components on the microplane, the microplane stresses ( Nσ ,
Mσ and Lσ ) may be determined for a given constitutive relationship on the
microplane level. The macroscopic stress is related to the microplane stresses by
applying the principle of virtual work and integrating over the surface Ω of a unit
hemisphere. In a simplified form, Bazant (1984) expressed the macroscopic stress
tensor as
Ω= ∫Ω
dsijij πσ
23 (2.55)
where ijMijLijNij MLNs σσσ ++= . For finite element implementation, Bažant and
Oh (1986) proposed an efficient formula to evaluate the integral by using a system of
21 microplanes per hemisphere (as shown in Figure 2.21b) for each integration point of
the finite elements.
A fourth generation of the microplane model, labelled M4, was presented by
Bažant et al. (2000). From the first generation of the microplane model M1 focussing
on the modelling of tensile fracture to the latest version M4, the microplane model was
extended to successfully handle various load types and has shown good agreements
with the basic experimental data for uniaxial, biaxial and triaxial loadings.
58
(a) (b)
Fig. 2.21 - Microplane model: (a) strain components on a microplane; (b) 21-point
optimal Gaussian integration for each hemisphere (circled points represent
the directions of the microplane normals; a total of 42 circled points on the
whole sphere). (after Bažant et al., 2000).
2.6.4 Fracture Models for Concrete
2.6.4.1 Fracture Mechanics
There are two major approaches in fracture mechanics, namely the linear elastic
fracture mechanics (LEFM) and the non-linear fracture mechanics (NLFM). Griffith
(1921) was the first to develop LEFM and to adopt the energy conservation principle to
describe the formation of a crack. The theory is based on linear-elasticity and states that
the entire body of a material is elastic except in the vanishingly small region at the
crack tip. In reality, an inelastic region does exist in the neighbourhood of a crack tip
for all types of material, in order to apply the theory the inelastic region must be
relatively small in comparison to the size of the body.
LEFM applies perfectly to brittle materials such as glass and brittle ceramic as
they possess a very concentrated fracture process zone at the crack tip. However, for
quasi-brittle materials like concrete, rock and ceramic, the crack tips are surrounded by
a relatively large fracture process zone, which is the main contributing factor to the
tension softening phenomenon. LEFM is therefore strictly applicable to massive
59
structures such as concrete dams. The comparisons of the characteristics of the fracture
process zones for different types of material are shown in Figure 2.22.
Although early attempts had been made to model the non-linear behaviour of
fracture process zone by researchers such as Irwin (1960), Dugdale (1960) and
Barenblatt (1962), fracture mechanics for quasi-brittle material had not been
extensively developed until the 1980’s. Amongst the NLFM theories, the most
significant and most widely adopted models are the fictitious crack model (Hillerborg
et al., 1976) and the crack band model (Bažant and Oh, 1983). The former is an
extension based on the concept of the cohesive crack that was introduced by Dugdale
(1960) and Barenblatt (1962) to model various non-linearities at the crack front. The
fictitious crack model is therefore also called the cohesive crack model or Dugdale-
Barenblatt model. The crack band model for strain softening is based on a similar
concept to the fictitious crack model, in which the fracture process zone is modelled by
a crack band with a finite width instead of a cohesive crack.
Fig. 2.22 - Different types of fracture process zone: (a) brittle materials; (b) ductile
materials; (c) quasi-brittle materials. (Bažant and Planas, 1998).
2.6.4.2 Fictitious Crack Model
In the fictitious crack model, the response of a concrete member in tension is the same
as the process described in Section 2.2.2. The simplification made by Hillerborg was
that, after the peak load, microcracking in the fracture process zone is lumped into a
line crack (fictitious crack) with a finite opening that is able to transfer stress while the
Linear-elastic zone Softening zone Non-linear hardening zone
(a) (b) (c)
60
remaining of the concrete member unloads (also see Figure 2.2 in Section 2.2.2).
Hence, the post-peak total elongation of a concrete specimen in an idealized direct
tension test may be calculated as
crunld wLl +=∆ ε (2.56)
where unldε is the strain in the unloading intact material, L is the total length of the
specimen and crw is the crack opening displacement.
Based on the simplification of the model, Hillerborg assumed that non-linear
fracture in concrete is described in full by a single stress-crack opening displacement
relationship which may be written as )( crwf=σ . The fracture energy fG and the
tensile strength ctf are the two key parameters defining the constitutive relationship
(see Figure 2.2c). The fracture energy is defined as the energy consumed in a unit area
of concrete fracture, it is equivalent to the area under the stress versus crack opening
curve. This may be expressed as
cr
w
f dwGu
∫=0σ (2.57)
where σ is the cohesive stress in the fictitious crack and uw is the crack opening at
which the stress transfer at the fictitious crack vanishes. Thus, given that the fracture
energy and tensile strength are known and with an assumed stress-crack opening
displacement softening function, the deformation of a concrete member in tension may
be predicted using Eq. 2.56.
2.6.4.3 Crack Band Model
When the fictitious crack model was first introduced, Bažant (1976) also presented a
non-linear fracture model for concrete, which forms the basis of the well-known crack
band model that is broadly used in modelling concrete structures in present engineering
61
design. The concept was then refined and introduced in full by Bažant and Oh (1983) to
describe the strain softening phenomenon of concrete.
The crack band model is not dissimilar to the fictitious crack model. The fictitious
crack model involves a distinct line crack, while the crack band model assumes
microcracks to be distributed densely over a certain width over the fracture zone. Thus,
Bažant introduced a characteristic length in addition to the two parameters fG and ctf
discussed previously for the fictitious crack model. In the context of the crack band
model, the characteristic length is the crack band width ch . Instead of using a stress-
cracking opening displacement relation, the crack band model describes the fracture
behaviour of concrete with a tensile stress-strain relationship in which the strain
softening tail of the curve must be modified for different widths of the crack band in
order to guarantee a constant fracture energy dissipation rate in the fracture zone.
2.6.5 Regularization of Spurious Strain Localization
The earliest continuum modelling of cracking in concrete was based on the strength
criterion. In these models, the concrete tensile response is characterized by a linear
elastic pre-peak stress-strain relation and is followed by a sudden drop in stress to zero
upon initiation of cracking. However, these models suffer from the severe deficiency of
mesh size dependence. Numerical results for the same structure vary notably for finite
element discretizations of different mesh sizes especially in the case of localized
cracking with little or no reinforcement. Bažant and Cedolin (1979) pointed out that
objective results could only be obtained based on an energy criterion by considering the
energy release rate fG and, hence, promoted the use of tension softening model for
cracked concrete, that is, by the inclusion of a descending branch of the tensile stress-
strain curve.
Nevertheless, proper regularization tools must be employed in conjunction with
the tension softening model in order to conserve the energy dissipation rate in crack
formation. The direct use of a softening stress-strain curve, like the strength criterion,
can also lead to pathological mesh sensitivity (Bažant, 1976; Crisfield, 1982). This is
62
due to the fact that, when cracking takes place in a smeared crack model, the damage
tends to localize into a single band of element. In addition, energy dissipation per unit
volume in a smeared crack model is governed by the local strain softening constitutive
law. Consequently, the energy dissipation decreases to zero as the mesh size reduces to
an infinitely small volume, which inevitably causes spurious mesh sensitivity.
With the increasing applications of fracture mechanics to continuum modelling of
quasi-brittle materials like concrete, many regularization approaches have been
developed in the last decade. A detailed review of a number of regularization models
for quasi-brittle materials was recently presented by Bažant and Jirásek (2002).
Regularization of strain localization serves a mathematical purpose to prevent the loss
of well-posedness of the incremental boundary value problem, which generates an
infinite number of solutions when the material stability (Drucker’s stability) is lost
(Benallal et al., 1988; de Borst et al., 1993c, 1998). The earliest models of its kind are
the models based on the Cosserat theory (Cosserat and Cosserat, 1909), which was
originally introduced to capture the mechanical behaviour of material due to
microstructural heterogeneity on the mezzo scale. The theory suggests that deformation
in material originates not only from the conventional translational deformation but also
the rotational movement of the material particles. Attempts have been made to
regularize strain localization problems in strain softening materials by using the
Cosserat theory (Mühlhaus and Vardoulakis, 1987; Mühlhaus, 1991; Mühlhaus et al.,
1991). However, it has been replaced by the non-local approach that has gained much
popularity over the past 10 years, or so.
2.6.5.1 Non-local Models
Most engineering problems have been successfully solved by the use of local
continuum models, for which the stress at a point in the continuum is dependent solely
on the strain at the same point. The local continuum models work perfectly well in
cases where the characteristic size of the material heterogeneity is relatively small
compared to the size of the structure. Although strain localization of concrete is a
macroscopic phenomenon, the process of fracture takes place at the scale of the
heterogeneity, which involves dense microcracking in the fracture zone. Consequently,
63
apart from the mathematical shortcoming, the local continuum models, which are
characterized by the local stress and strain of a point, likewise, are not able to
accurately describe the physical process of fracture in concrete.
While a local continuum model describes the local stress and strain, the stress in a
non-local continuum depends not only on the strain at that point but also on the
weighted averages of a state variable in the neighbourhood within a distance from that
point. The spatial interaction of the non-local approach essentially introduces an
internal length scale (de Borst and Mühlhaus, 1992; de Borst et al., 1993a) or a
characteristic length (Bažant and Pijaudier-Cabot, 1988; Bažant and Ožbolt, 1990) into
the constitutive model so as to ensure a constant energy release rate in the fracture
process zone. The notion of non-locality emerged in the late nineteenth century and was
later studied extensively in elastic materials (for example, Eringen, 1965; Kröner, 1967;
Eringen and Edelen, 1972). The non-local concept was first applied to strain softening
damage materials by Bažant et al. (1984).
In general, the non-local approach involves the replacement of a local state
variable f by a weighted average state variable f . The condition for effective non-
local modelling is that the selected state variable must be able to reflect the damage
evolved in the strain softening process. The averaged state variable is obtained by the
use of a non-local averaging operator given by (Jirásek, 1998)
dVffV∫= )(),(')( xsxx α (2.58)
where x and s are the vectors representing the locations of the local variable and the
neighbourhood variables, respectively (Figure 2.23a), V is the volume of the structure
and 'α is a normalized weight function defined as
dVV∫ −
−=
)||()||(
),('sxsx
sxαα
α (2.59)
where α is a scalar weight function shown in Figure 2.23b, which is a function of
|| sx −=r , the distance between point x and point s. The weight function usually takes
the form of the Gaussian distribution function as
64
−=
2
2
2exp)(
chl
rrα (2.60)
where chl is the characteristic length of the non-local continuum. Otherwise, a bell-
shape polynomial function may also be used (Bažant and Ožbolt, 1990). That is
2
2
21)(
Rrr −=α (2.61)
where the pointer bracket denotes a Macauley bracket, in which xx = for 0>x and
0=x for 0≤x , R is the non-local interaction radius which may be related to the
characteristic length chl .
(a) (b)
Fig. 2.23 - Non-local continuum: (a) representative volume for non-local averaging;
(b) weight function and the relation to the characteristic length. (Bažant and
Planas, 1998).
The first non-local model for strain softening is the imbricate continuum model
proposed by Bažant et al. (1984). In this model, the strain was taken to be the state
variable for non-local averaging. This model was found to guarantee mesh insensitivity,
however, the differential equations for the equilibrium and boundary conditions
involved were way too complicated for straightforward numerical implementation
(Bažant, 1990). The consequence of this is the introduction of the concept of the non-
local continuum with local strain in which the strain is kept local but the constitutive
xx
s
y
s
lch
1
α
65
relation for strain softening is dependent on the non-local state variable that causes
strain softening. The non-local damage model (Pijaudier-Cabot and Bažant, 1987;
Bažant and Pijaudier-Cabot, 1988) was developed based on this concept and has proved
to be successful in regularizing strain localization problems. The model was built on the
framework of continuum damage mechanics and non-local averaging was performed
either on the damage energy release rate or the damage variable.
Several variants of the non-local damage models were attempted by using
different non-local variables. For example, Bažant and Lin’s (1988a) non-local smeared
cracking model based on the non-local total strain formulation and used the model to
investigate structural size effect. They showed that the model is capable of rectifying
mesh directional bias that exists in a local smeared crack model. Jirásek and
Zimmermann (1998b) formulated a non-local rotating crack model and evaluated the
stress by multiplying the stiffness matrix with the total strain. The stiffness matrix was
calculated based on the non-local total strain. Jirásek (1998) summarized the possible
non-local formulations with various non-local state variables and compared the
characteristics of each model. The non-local concept has also been extended to the
microplane model (Bažant and Ožbolt, 1990; Ožbolt and Bažant, 1996) and a number
of plasticity-based models (Bažant and Lin 1988b; Nilsson, 1997; Borino et al., 1999).
2.6.5.2 Gradient Models
The mesh sensitivity of strain softening may also be prevented by the use of a gradient
model. The basic idea of a gradient model is that, the stress at a point is not only a
function of the strain but also the first or second spatial derivatives of the strain
(Bažant, 1990). The term gradient originates from the notion of the spatial derivatives
which can be perceived as the gradients of strains with respect to the distance away
from the point under consideration. The gradient model may be derived from the non-
local model by expanding the spatial integral given by Eq. 2.58 into a truncated
Taylor’s series and retaining the even-order derivative (Bažant, 1984; Lasry and
Belytschko, 1988; de Borst, 1997). That is
fcff 2∇+= (2.62)
66
where c is a gradient coefficient depending on the type of the weight function used and
is a function of the internal length scale, 2∇ is the Laplacian operator defined as
∑ ∂∂=∇ i ix/22 , in which )3,2,1( =ixi is the Cartesian coordinate. Due to the
approximated non-local nature of Eq. 2.62, the gradient model is also known as the
“weakly non-local model” (Bažant and Jirásek, 2002). The state variable f of the
gradient models is usually a specific form of strain component depending on the type of
constitutive model where the gradient formulation is built on. For example, in the
framework of continuum damage mechanics, the equivalent strain ε~ (see Eq. 2.47) is
often taken as the gradient dependent state variable (Peerlings et al., 1996). While in a
plasticity model the plastic strain is used for the evaluation of the gradient term (de
Borst and Mühlhaus, 1992; Pamin and de Borst, 1995).
One major disadvantage of the original gradient model is the need for the use of
higher order elements to ensure continuity in both the displacement and strain fields
within the element. In the work of Peerlings et al. (1996), an implicit gradient
formulation was developed. The gradient dependent state variable is evaluated by
solving a partial differential equation rather than a straightforward expression of Eq.
2.62. The solution of the partial differential equation was shown mathematically to be a
special case of the fully non-local model, thus alleviating the complication arising from
the use of higher-order elements.
2.6.5.3 Crack Band Formulation as Partial Regularization
The crack band model (Bažant and Oh, 1983) is the simplest regularization method that
is capable of preventing spurious strain localization on mesh refinement. The
formulation of the model is discussed in Section 2.6.4.3. The crack band model is
probably the first fracture model that has successfully promoted the use of an energy
criterion in smeared crack modelling for a strain softening material.
As mentioned earlier, the inclusion of a characteristic length into constitutive
models for strain softening is of crucial importance to produce objective post-peak
results. In the crack band model, the characteristic length is the so-called crack band
67
width, which is the width of the element undergoing fracturing. The basic idea of the
model is to ensure a constant energy dissipation rate by adjusting the softening stress-
strain relationship according to the size of the element and, hence, guaranteeing an
objective tensile response. However, the fact that fracture process takes place
numerically within an element is physically unjustified since the fracture process zone
indeed holds a finite width at the fracture front. Moreover, the crack band model is also
known to suffer from mesh directional bias. This aspect has been studied extensively,
for example, by Rots (1988), Li and Zimmermann (1998) and Jirásek and
Zimmermann (1998a).
2.6.5.4 Regularization by Inclusion of Material Viscosity
The mesh sensitivity arising from strain softening can also be prevented by the
inclusion of material viscosity into the constitutive equation for dynamics problems.
This approach was proposed by Needleman et al. (1988) for damage in metals and has
been applied to strain softening material by Sluys (1992). The inclusion of viscosity
indirectly introduces an internal length scale into the constitutive model and regularizes
the mesh sensitivity problem. However, Bažant and Jirásek (2002) pointed out that the
viscosity-induced regularization is highly dependent on the load duration. The
modelling results may not be objective if the load duration is much larger than the
relaxation or retardation time associated with the type of viscosity employed.
2.6.6 Modelling of Steel Reinforcement
Three distinct approaches are available to represent steel reinforcement in reinforced
concrete modelling, namely distributed steel formulation, embedded steel formulation
and discrete steel approach (ASCE Task Committee, 1982, Chap. 3) (see Figure 2.24).
In the distributed steel formulation, reinforcing steel is assumed to be smeared
over concrete elements at a particular angle of orientation and is often described by the
reinforcement ratio. The total stiffness of a distributed reinforced concrete finite
element consists of the stiffness of the concrete and the stiffness contributed by the
68
smeared steel reinforcement. When the reinforcement is assumed to be smeared, a full
compatibility between steel and concrete is naturally enforced. This type of formulation
is useful for the analysis of reinforced concrete structures with densely distributed
reinforcement, since the exact definition of every single reinforcing bar can be avoided.
In the embedded steel formulation, each reinforcing bar is considered as an axial
member incorporated into the concrete element by the principle of virtual work. The
displacements of the embedded steel are consistent with the displacements of the
concrete element. The major advantage of the embedded steel formulation is that the
reinforcing steel can be defined arbitrarily regardless of the mesh shape and size of the
base concrete element.
The discrete steel approach is based on the use of separate elements to represent
the reinforcing steel. One-dimensional truss elements are commonly adopted since
reinforcing steel is usually assumed to carry axial load only. Steel truss elements are
overlayed onto the boundary of the concrete elements by connecting the nodal points.
This approach greatly facilitates the inclusion of bond-slip effects between steel and
concrete, which may be achieved by inserting bond-slip elements between the concrete
elements and the steel truss elements. A major disadvantage of this approach is that the
mesh boundary of the concrete element must overlap the direction and location of the
steel reinforcement.
(a) (b) (c)
Fig. 2.24 - Modelling of reinforcement: (a) distributed steel approach; (b) embedded
steel; (c) discrete steel approach.
69
2.6.7 Modelling of Steel-Concrete Bond
The bond transfer between steel and concrete is closely related to the formation of
distributed cracking in reinforced concrete structures. This is demonstrated in Section
2.5.2. Consequently, the correct prediction of cracking depends on a realistic modelling
of the steel-concrete bond action. In numerical models, two different approaches are
commonly employed to model bond between steel and concrete. The first includes a
wide range of “smeared crack models”. The inclusion of bond-slip is made possible in
an indirect manner by accounting for the tensile force carried by the intact concrete
between the cracks, namely tension stiffening. The second approach describes the local
bond-slip between steel and concrete using discrete bond elements, which actually
takes into account the relative displacement of the reinforcing steel and the
adjacent concrete.
2.6.7.1 Tension Stiffening
The tension stiffening effect has been incorporated in several different ways. One
approach, developed by Scanlon and Murray (1974), is to adopt a descending branch in
the concrete tensile stress-strain curve. An alternative approach proposed by Gilbert
and Warner (1978) is to modify the stress-strain relationship of the tensile steel to
indirectly model the tension in the concrete and to assume that the concrete possesses
zero stiffness after cracking. These approaches can account for the effects of cracking
on the macroscopic deformation of the member or structure but not at the mezzo or
micro levels.
In the modified compression field theory of Vechhio and Collins (1986), tension
stiffening is modelled using a descending concrete stress-strain curve calibrated using
their test data. In the disturbed cracked field model of Vecchio (2000), an improved
tension stiffening descending function was used, which is made dependent on factors
such as reinforcement ratio, diameter of reinforcing bar, direction of crack and
orientation of reinforcements.
70
Hsu and Zhang (1996) proposed the use of both the average descending curve in
tension of concrete and the average stress-strain curve of reinforcing bars embedded in
concrete to model the effect of tension stiffening. They advocated that the average yield
stress of an embedded reinforcing bar should be lower than the yield stress of a bare
reinforcing bar.
Balakrishnan and Murray (1986, 1988) employed an embedded reinforcement
formulation including the effect of bond-slip by using the concept of virtual work. In
addition, they adopted an approximated triangular concrete stress distribution between
cracks in which the maximum stress was taken to be the concrete tensile strength and,
hence, an average tension stiffening stress of ctf5.0 . In this model, it was assumed that
the tension stiffening stress vanishes completely as soon as the reinforcing bars
begin to yield.
The tension chord model (TCM) proposed by Marti et al. (1998) utilizes a
stepped, rigid-perfectly plastic bond-slip model for the calculation for both crack
spacing and crack widths, together with the average concrete stress between the cracks
and, hence, the tension stiffening effect. Foster and Marti (2002, 2003) implemented
the two-dimensional version of the TCM, the Cracked Membrane Model, into their
finite element program (CMM-FE) and demonstrated promising results for the
modelling of reinforced concrete structures in plane stress.
2.6.7.2 Discrete Bond Modelling
In this approach, the modelling of bond involves the use of discrete bond elements to
simulate the relative displacement between steel and concrete. These bond elements can
be subdivided into two general types, the bond link element and the interface element.
The former was developed by Ngo and Scordelis (1967). Bond is modelled by placing a
bond link element between each individual node along the concrete-steel interface. The
element consists of two orthogonal springs that transmit shear and normal tractions
between the bond-slip surfaces as shown in Figure 2.25a.
71
The second type, the interface element (Figure 2.25b), was introduced by
Goodman et al. (1968) for modelling rock joints. The interface elements are now
commonly used for modelling discrete crack propagation (Červenka, 1994) and bond-
slip phenomena (Mehlhorn and Keuser, 1985; Rots, 1985; Rots, 1988) in reinforced
concrete structures. For modelling of bond, the interface elements are placed
continuously along the concrete-steel interface. In addition, since the element is
formulated based on the continuous relative displacement field between steel and
concrete, it can describe the continuous slip between the two materials more effectively
than the bond link element, which can only model bond-slip at individual nodes.
Recently, an elasto-plastic cyclic bond model was developed (Lundgren, 1999)
and implemented using the interface elements of the finite element program DIANA. A
“damaged and undamaged zones” concept was introduced to describe the reduced bond
friction due to the damage of concrete caused by the displacement of the rib of the
deformed bar during a reverse slip process. The model was used to simulate monotonic
and cyclic pull-out tests and frame corners subjected to closing moments, and showed a
good correlation with test results. This demonstrates that the use of interface elements
with a reliable bond model can in fact produce promising results for phenomenological
modelling of reinforced concrete structures.
(a) (b)
Fig. 2.25 - Discrete bond models: (a) bond link element; (b) bond interface element.
1
4
3
2
2
u4
u3
u8u7
u3
u5
u6
u4
u2 u11
u2
u1rs
θ
rs
θ
72
2.6.8 Computational Creep Modelling
One of the biggest problems in implementing the superposition integral of Eq. 2.9 or
Eq. 2.11 is the necessity for storing the history of stresses or strains. Stresses in a
concrete structure are constantly changing with time due to cracking and redistribution
of stresses between the concrete and the reinforcement. Consequently, a large amount
of computer memory is needed to store every single stress increment throughout the
analysis. The analysis of a structure with a large number of elements will be extremely
expensive computationally.
In order to avoid the explicit storage of stress or strain histories, the superposition
integral may be converted into a series of differential equations which is often known
as the rate-type constitutive equations (Bažant, 1982; Bažant, 1988). The stress or strain
histories are stored internally in the rate-type constitutive equations. To facilitate such a
conversion, the kernel of the superposition integral, the compliance function )',( ttJ or
the relaxation function )',( ttR , must be written in the form of the degenerate kernels as
∑=
−−
−=
N ttetC
ttJ1
/)'(1)'(
1)',(µ
τ
µ
µ (2.63)
and
∑=
−−=N ttetLttR
1
/)'()'()',(µ
τµ
µ (2.64)
where µC and µL are functions of concrete age 't , µτ is called the retardation time
for a compliance function or, the relaxation time for a relaxation function. Expanding
the summations of Eq. 2.63 and Eq. 2.64 gives a series of real exponentials, which is
called the Dirichlet series. Although the approximation of the Dirichlet series is not
exact, its accuracy is regarded as sufficient for the purpose of creep analysis. The rate-
type constitutive equations may be formed by combining the degenerate kernel with the
superposition integral. This is done by substituting Eq. 2.63 into Eq. 2.9 or, Eq. 2.64
into Eq. 2.11. After some mathematical manipulations, it can be shown that the rate-
73
type constitutive equations may be represented by some sort of rheological models
(Bažant and Wu, 1973, 1974).
For the integral-type creep law for aging material using the compliance function,
the rate-type constitutive equation may be described as a series of Kelvin chain units as
shown in Figure 2.26a. The total strain for a uniaxially loaded member may be written
as the sum of the strains µε of the Kelvin chain units and the shrinkage strain and is
given by
∑=
+=N
sh ttt1
)()()(µ
µ εεε (2.65)
Each Kelvin chain unit is characterized by a spring and a dashpot connected in parallel,
in which the properties are described by the spring elastic modulus µE and the dashpot
viscosity µη , respectively. Bažant and Wu (1973) showed that µE and µη can be
related to the parameters of the Dirichlet series shown in Eq. 2.63.
In the case of superposition of stress with relaxation functions for an aging
material, Bažant and Wu (1974) deduced that the Maxwell chain model could best
describe the resulting rate-type constitutive equation from the superposition integral in
Eq. 2.11. The Maxwell chain unit is made up of a spring and a dashpot connected in
series as shown in Figure 2.26b. Considering a uniaxial case, it is evident in Figure
2.26b that the total stress is equal to the sum of the stresses µσ carried in each
Maxwell chain unit. This can be written as
∑=
=N
tt1
)()(µ
µσσ (2.66)
Similar to the Kelvin chain unit, the spring and the dashpot in the Maxwell chain unit
are characterized respectively by an elastic modulus µE and a viscosity µη
corresponding to the µ-th unit of Maxwell chain.
74
(a) (b)
Fig. 2.26 - Rheological models: (a) integral-type creep model with Kelvin chain
models in series; (b) integral-type relaxation model with parallel Maxwell
chain models.
The remaining task left for the computation of the total stress or strain at time t is
to solve the resulting differential equations (or rate-type equations) for µσ or µε . To
make possible an efficient algorithm for solving the differential equations, Bažant and
Wu (1973, 1974) introduced an exponential algorithm based on the quasi-elastic
incremental stress-strain approach (Bažant, 1971, 1972). The algorithm permits the use
of relatively large time steps in a time analysis and shows good numerical stability.
A different form of exponential algorithm was developed by Kabir and Scordelis
(1979). They utilized the Dirichlet series expansion and expressed the temperature-
dependent creep compliance in the following form
[ ]∑=
−−−=N
i
ttTi sieTttC
1
)'()(1),',( ψλα (2.67)
where iα is a scale factor depending on the age at loading, iλ is a constant determining
the shape of the creep curve and sψ is a shift function depending on the temperature T.
µ =1
µ =1µ =2
µ =2
µ =N
µ =N
ε1
ε2
εN
εsh
εsh
ε
σ
σ
σ
σ
µ =3
σ1 σ2 σ3 σN
Kelvinchainunit
Maxwell chain unit
ε
εcp
75
Kabir and Scordelis followed the algebraic manipulation of Zienkiewicz and Watson
(1966) and derived a set recursive equations for calculating the creep strain increment
cpε∆ . The uniaxial version of the equations is given by
[ ]∑=
∆−−=∆N
i
tTicp nninn
eA1
)(1 ψλε (2.68)
where
nnitT
ii taeAA nninn
σψλ ∆+= −−−
∆− )(][ 111
)( for 1>n (2.69)
nnii taAn
σ∆= )( for 1=n (2.70)
The subscript n denotes the number of iteration. The storage of the total stress history is
not needed and only the stress history of the last time step is required for the
computation of the current creep strain increment. This approach has also been used by
Van Zyl and Scordelis (1979), Van Greunen (1979) and Kang and Scordelis (1980) in
time-dependent finite element analysis of concrete structures.
76
CHAPTER 3
FINITE ELEMENT MODELS FOR REINFORCED
CONCRETE
3.1 Introduction
The investigation of behaviour of reinforced concrete structures is often undertaken by
gathering test data from experiments and then following up with a detailed theoretical
study. This method of investigation is often viable for the study of instantaneous
structural behaviour. It is, however, expensive and time consuming for the investigation
of long-term behaviour of structures.
The finite element method provides an extremely powerful numerical tool that is
able to accurately simulate structural behaviour if the appropriate modelling approach
and material laws are effectively utilized. The major aim of this study is to develop a
finite element model which can provide reliable predictions of long-term behaviour of
cracked reinforced concrete structures so as to facilitate a parametric study of cracking
and crack development based upon the data produced by the finite element model. In
this way, the parameters affecting the development of a crack with time may be
identified and quantified.
In this chapter, the detailed formulation of the proposed numerical models of
reinforced concrete structures subjected to sustained service loads are presented,
including the finite element formulation and the iterative numerical solution
procedures.
77
3.2 Continuum Modelling
In this study the smeared crack approach is adopted. The pros and cons of the smeared
crack approach were discussed in Section 2.6.2. Two distinct approaches are employed
to simulate the development of cracks in concrete, namely the distributed cracking
approach and the localized cracking approach.
In the first approach, cracks are stabilized by the reinforcement and are smeared
over a region of cracked concrete (or reinforced concrete) elements. No discrete crack
is observed in the distributed cracking approach and cracking is represented on the
average level by a region of element with reduced stiffness. Hence, this approach may
be regarded as an average model which takes account of the average effects of each
component contributing to the global structural stiffness. There are no distinct cracks,
so the estimation of crack spacing and crack width is problematic.
In contrast, the localized cracking approach aims to capture the phenomenon of
strain localization of concrete. Cracks are modelled as discrete localizations. Fracture
of concrete is one of the main ingredients in this approach. For reinforced concrete, the
transfer of bond between reinforcing steel and concrete are of significant importance to
the formation of a crack as discussed in Chapter 2. The localized cracking approach
fuses non-linear fracture mechanics of concrete and discrete bond action between steel
and concrete in order to capture the formation of cracks at discrete spacings.
3.3 Distributed Cracking Approach
To accurately model crack development in reinforced concrete structures using the
distributed cracking approach, a reliable model that can predict crack spacing and
tension stiffening rationally is the key determinant. The cracked membrane model
(Kaufmann, 1998; Kaufmann and Marti, 1998) is used to model two-dimensional plane
stress reinforced concrete elements and is implemented in the finite element model.
This model is formulated based on the assumptions of the tension chord model (Sigrist,
1995), which was originally developed to analyse problems of cracking and minimum
reinforcement in reinforced concrete members subjected to uniaxial stress.
78
The fundamental concept can be illustrated by taking an orthogonally reinforced
concrete panel subjected to in-plane stresses xσ , yσ and xyτ with parallel cracks
spaced uniformly as shown in Figures 3.1a and 3.1b. The applied forces on the panel
are equal to the sum of the force components contributed by the concrete and the steel
in the global directions. Considering equilibrium of the infinitesimal elements at the
crack shown in Figures 3.1c and 3.1d gives
sxxcntctcnx σρθτθσθσσ +++= )2sin(sincos 22 (3.1a)
syycntctcny σρθτθσθσσ +−+= )2sin(cossin 22 (3.1b)
)2cos()2sin()(5.0 θτθσστ cntctcnxy −−= (3.1c)
where the stresses with subscript c represent the concrete stresses and the subscript s
denotes the reinforcing steel stresses. The element reinforcement ratios in the x and y
directions are given by xρ and yρ , respectively. The t-n axis system denotes the crack
direction (t) and the direction normal to the crack (n) and θ is the angle between x and n
axes (-π/2 ≤ θ ≤ π/2) (see Figure 3.1b).
Fig. 3.1 - Orthogonally reinforced concrete panel subjected to in-plane stresses: (a)
applied stresses; (b) axis notation; (c) and (d) stresses at crack. (after Foster
and Marti, 2003).
t n
θ
τxy
θ
σ cos θcn
ctσ sin θ
cntτ cos θ
τ sin θcnt
1
+
ρ σx sx
yσ
σx
y
σx
xyτ
xy
1
τ
τ sin θτ cos θ
σ cos θct
cnt
cnσ sin θ
θcnt
σy
σx
+
(a)
(b) (d)
(c)
ρ σy sy
(a) (c)
(b) (d)
79
3.3.1 Tension Chord Model
Consider a uniaxial reinforced concrete tension chord element with a single reinforcing
bar of diameter ∅ , as shown in Figure 3.2a. Sigrist (1995) proposed that the bond
between the steel and the concrete could be modelled using a stepped, rigid perfectly
plastic bond shear stress-slip relation (Figure 3.2c). The bond shear stress drops from
0bτ to 1bτ after yielding of the reinforcing steel. From this Sigrist (1995) and Marti et
al. (1998) developed an idealised model known as the “tension chord model”. The
following outlines the derivation of Sigrist’s tension chord model.
For a differential element of the tension chord element (Figure 3.2b) of length dx,
the variation of steel and concrete stresses along the element length are described by
∅−= bs
dxd τσ 4
; )1(
4ρ
ρτσ−∅
= bcdx
d (3.2)
respectively, where ρ is the ratio of the cross sectional area of steel sA to the cross
sectional area of concrete cA of the tension chord and bτ is the bond stress.
For a cracked tension chord, the average stress midway between cracks is
obtained by integrating Eq. 3.2 over one half of the crack spacing )2( rms and noting
that the tensile stress cannot exceed the tensile strength, ctf . Therefore
ct
s
bc fdxrm
≤−∅
= ∫2
0)1(
4 τρ
ρσ (3.3)
When ctc f=σ , the crack spacing is at its maximum and solving Eq. 3.2 gives
ρτρ
00 2
)1(
b
ctrm
fs
−∅= (3.4)
where 0rms is the maximum crack spacing for a tension chord with a fully developed
crack pattern. Marti et al. (1998) showed that the minimum crack spacing is one half of
80
the maximum crack spacing and concluded that the range of possible crack spacings for
a fully developed crack pattern is
0rmrm ss λ= (3.5)
with 0.15.0 ≤≤ λ .
The mean steel stress and maximum steel stress (at the cracks) can be expressed
as a function of the mean (or average) strain mε given that the distribution of the bond
shear stress is known. Hence the stresses in the steel and the concrete between the
cracks can be determined for any known stresses in the steel at the cracks by
considering the equilibrium across the tension chord at the crack and anywhere between
the cracks.
(a)
(b) (c)
Fig. 3.2 - Tension chord model: (a) reinforced concrete tension chord element; (b)
differential element; (c) bond shear stress-slip relationship. (after Marti
et al., 1998).
dxx
∅
Ac
srm
dx
NN
σ σc c+ d σc
τb σ σs s+ d σsτb
τb0
τb
δ δy
τb1
81
Following a similar derivation to that followed for Eq. 3.3, the concrete stress at a
distance x from the crack can be shown to be
xx bc )1(
4)( 0ρρτ
σ−∅
= (3.6)
Knowing the concrete stress and the steel stress at crack srσ , the steel stress at
distance x from the crack can be obtained by considering the equilibrium at the crack
and at the section at distance x from the crack as
xxx bsrcsrs ∅
−=−
−= 04)1()()(τ
σρ
ρσσσ (3.7)
The formulation given above was derived for a reinforced concrete member
subjected to direct tension. To apply the tension chord model to reinforced concrete
flexural members, the reinforcement ratio ρ in the crack spacing equation of Eq. 3.4 is
replaced by the effective reinforcement ratio effρ , which is defined as the ratio of the
area of tension steel stA to the effective area of concrete in tension effcA . . In this study
the effcA . is taken as that suggested in CEB-FIP Model Code 1990 (1993) and is
defined as the product of the section width and a depth equal to 2.5 times the distance
from the tensile face of the section to the centre of the steel, but not greater than one
third of the depth of the tensile zone of the cracked section.
3.3.2 Cracked Membrane Model
Kaufmann and Marti (1998) developed the cracked membrane model (CMM) for the
analysis of reinforced concrete membranes subject to in-plane stresses. The CMM
combines the essence of the modified compression theory (Vecchio and Collins, 1986)
and the tension chord model of Marti et al. (1998). The assumptions of the original
CMM are: the crack faces are stress free and able to rotate normal to the direction of
the principal major strains, giving 0== cntcn τσ . In this study concrete tension
82
softening is accounted for, therefore the crack faces are only free of shear stresses but
the residual tensile cohesive stresses normal to the crack are not negligible.
For equilibrium across the continuum, stresses in the x and y directions can be
written in terms of the principal stresses in the 1-2 coordinate system as
ctsmxsxxccx σσρθσθσσ +++= 22
21 sincos (3.8a)
ctsmysyyccy σσρθσθσσ +++= 22
21 cossin (3.8b)
( ) ( )θσστ 2sin5.0 21 ccxy −= (3.8c)
where ctsmxσ and ctsmyσ are the mean concrete tension stiffening stresses in the x and
y directions, respectively.
By adopting the tension chord model, the crack spacing for a uniaxial element in
the x and y directions, rmxs and rmys , are calculated using Eq. 3.4 and Eq. 3.5 (Figure
3.3). As a simplification of the CMM, it can be shown that the Vecchio and Collins’
(1986) crack spacing equation is justified (Kaufmann and Marti, 1998; Foster and
Marti, 2003) and the mean spacing between cracks may be taken as
1sincos
−
+=
rmyrmxrm ss
sθθ
(3.9)
With the crack spacing known from Eq. 3.9, the instantaneous crack width is
calculated considering the elasticity across the continuum and is given by
][ 12121
c
ctsmrmcr E
swσ
ενε −+= (3.10)
where 1ε and 2ε are the concrete strains in principal directions, 12ν is the Poisson’s
ratio in the 1-direction resulting from the stress applied in the 2-direction and 1ctsmσ is
the mean concrete tension stiffening stress in the major principal direction.
83
Fig. 3.3 - Crack spacing and tension stiffening stresses of an orthogonally reinforced
concrete panel.
3.4 Localized Cracking Approach
Cracking in plain concrete is essentially a process of coalescing microcracks into a line
of discontinuity which is not able to transfer stresses. This process is known as fracture
of concrete and can be described theoretically by non-linear fracture mechanics, in
which the formation of a unit area crack is treated as the consequence of dissipation of
material fracture energy caused by tensile stresses. The details of non-linear fracture
mechanics were discussed in Section 2.6.4.
In the localized cracking approach, concrete and reinforcing steel are modelled as
individual components. Localized cracking is simulated by the use of an appropriate
concrete fracture model and the steel-concrete interaction is achieved by employing the
bond interface element. This aims to provide a realistic description of the stress transfer
between steel and concrete in the vicinity of a crack, which is of paramount importance
in the prediction of the development of cracks at discrete locations (Section 2.5.2). The
use of bond elements, either bond link elements (Ngo and Scordelis, 1967) or bond
x
θ
y
srmx
s rm
srmx
σctsy
σctsx
s rmy
s rmy
s rmy
s rms rm
s rm
σ cts
84
interface elements (Mehlhorn and Keuser, 1985), is not new in the finite element
modelling of reinforced concrete structures. However, to the author’s knowledge, finite
element studies focussing on the development of crack width and crack spacing, by
coupling of bond elements with a reliable localized fracture model, are scarce.
Furthermore, the inclusion of the effects of creep and shrinkage to study the time-
dependent development of crack width and crack spacing is even scarcer.
In the context of a smeared crack modelling, fracture is described as a process of
localization of inelastic strain into a band of finite width. The most formidable task in
formulating a reliable fracture model with the smeared crack approach is to preclude
the infamous mesh sensitivity issues. Appropriate models must be adopted in order to
ensure that results do not vary significantly for different mesh sizes and configurations.
In this study the crack band model (Bažant and Oh, 1983) and the non-local model
(Bažant and Pijaudier-Cabot, 1988; Jirásek and Zimmermann, 1998) are employed to
model fracture of concrete and are presented in the following sections.
3.4.1 Crack Band Model
An introduction to the crack band model (Bažant and Oh, 1983) was given in Section
2.6.4.3. The basic idea of the model is identical to the fictitious crack model (Hillerborg
et al., 1976) in which the fracture front (fracture process zone) is able to transfer
stresses before a crack is completely formed. Three parameters are required to define
the crack band model, namely the fracture energy fG , the material tensile strength ctf
and a characteristic length known as the crack band width ch (the effective width of the
cracking element normal to the crack).
It is assumed that cracking is distributed uniformly over ch . Given that the
average strain within the crack band, which is termed the fracturing strain or the
cracking strain, is known, the crack opening displacement, crw , is given by
ccrcr hw ε= (3.11)
85
where crε is the cracking strain as depicted in Figure 3.4. Bažant also introduced a
parameter, the fracture energy density fg , which is defined as the amount of energy
required to fully damage a unit volume of material. This parameter can be acquired
from the area under the softening stress-strain curve (see Figure 3.4c), which gives an
integral that is similar to Eq. 2.57 for the fictitious crack model. That is
crf dgu
εσε
∫=0
(3.12)
where uε is the cracking strain when the crack is fully opened and the cohesive stresses
between the crack faces completely vanish. By combining Eq. 2.57, Eq. 3.11 and Eq.
3.12, Bažant showed that for a particular fG , fg needs to be adjusted according to ch
in order to keep constant the energy consumed per unit extension of the crack band.
This is given by
cff hGg = (3.13)
In the crack band model, the crack band width is taken as a constitutive material
parameter. Bažant and Oh (1983) examined empirically the effects of the crack band
width with fracture tests for specimens of varying geometries and concluded that the
(a) (b) (c)
Fig. 3.4 - Crack band model: (a) crack path and the fracture process zone; (b)
idealization of constant distribution of cracking strain; (c) tension softening
stress-strain curve.
gf
εcr
hc
ymicrocrack
real crack
fct
σ
εcr cr = w /hcfracture processzone
εu
86
optimum crack band width for concrete is approximately three times the maximum
aggregate size. However, it was found that within the range aca dhd 10≤≤ (where
ad is the maximum aggregate size), the effect of varying the crack band width is not
significant (Bažant 1985).
3.4.2 Non-local Smeared Crack Model
The concept of a non-local continuum with local strain (Bažant, 1990) was introduced
to overcome the deficiencies in the “imbricate continuum model” (Bažant et al., 1984)
as discussed in Section 2.6.5.1. The main feature of the non-local continuum with local
strain is that the strain at a point in a continuum is kept local but the constitutive
relation for strain softening is dependent on the non-local state variable that causes
strain softening. A typical non-local formulation of this type is the non-local damage
model proposed by Bažant and Pijaudier-Cabot (1988) and the constitutive relationship
is given by
[ ] εDεσ eY ))((1 ω−= (3.14)
where the overbar denotes the non-local variable, eD is the elastic stiffness matrix, ω
is the damage variable and Y is the damage energy release rate which is a function of
the non-local strain ε , which is calculated using the non-local averaging operator given
by Eq. 2.58.
Jirásek (1998) described this formulation in the context of the smeared crack
approach. Eq. 3.14 can be written as
εεDσ )(sec= (3.15)
in which secD is the secant stiffness matrix. No damage variable is involved in a
smeared crack model, the secant stiffness directly accounts for the degradation of
material integrity and therefore is taken as a function of the non-local strain. Jirásek and
Zimmermann (1998) adopted this formulation and applied it to their “rotating crack
model with transition to scalar damage”.
87
3.4.2.1 Issue Related to Non-local Continuum with Local Strain
The formulation given by Eq. 3.15 is investigated more closely herein. For simplicity,
consider a one-dimensional case. The material constitutive law is assumed to be linear
elastic and is followed by a linear softening stress-strain relationship, as shown in
Figure 3.5a.
According to Eq. 3.15, the stress tensor σ in a continuum is obtained by
multiplying the secant stiffness matrix secD evaluated from the non-local strain ε by
the local strain tensor ε . Recalling the non-local averaging operator given by
dVV∫= )(),(')( xεsxxε α (3.16)
where x and s are the location vectors of the local strain and the neighbourhood strains,
respectively, V is the volume of the structure and 'α is a bell-shaped normalized weight
function. The value of the weight function is largest at the location of the local strain
(or called the target point) and it decreases for increasing distance away from the target
point (see Figure 2.23b). These points in the neighbourhood are called the source points
in the text that follows.
At a target point in the continuum with tensile strain higher than the tensile strains
in the surrounding neighbourhood source points, spatial averaging will inevitably scale
down the strain at the target point since the normalized weight 'α is smaller at the
source points than at the target point. This decrease of the non-local strain is depicted in
Figure 3.5b by the arrow. In the one-dimensional case, the secant stiffness matrix secD
is equal to secant modulus secE and is obtained by connecting a line through the origin
to the intersecting point of the non-local strain on the stress-strain curve. Knowing the
secant modulus, the stress is computed as the product of secE and ε, as shown in
Figure 3.5c.
It is evident in Figure 3.5c that, for the case of the non-local strain being lower
than the local strain, the formulation given by Eq. 3.15 can potentially lead a stress
level that is higher than the tensile strength of the material due to the high secant
88
modulus evaluated from the non-local strain. In some circumstances the over stress can
be substantial. Though the formulation is not posing problems in structural fracture
analysis, however, in the author’s view, the violation of tensile strength from a
structural perspective is undesirable.
(a)
(b)
(c)
Fig. 3.5 - Non-local model with local strain: (a) bilinear tensile stress-strain curve;
(b) computation of secant modulus from non-local strain; (c) stress as the
product of secant modulus and local strain which is shown to be greater
than the material tensile strength.
σ
ε
Ec 1
fct
Ec 1
εε
Esec 1
σ
ε
fct
Esec
Ec
σ> fct
1
1
εε
σ
ε
fct
89
3.4.2.2 Proposed Non-local Smeared Cracking Formulation
In the light of the aforementioned issue, this section presents a variation of the non-
local formulation that remedies the issue of Eq. 3.15. It is of crucial importance to
retain a principal aspect of the non-local damage model proposed by Bažant and
Pijaudier-Cabot (1988), that is, only the causes of strain softening should be made non-
local, while the elastic related variables are treated locally as in the conventional
smeared crack model.
This can be achieved by applying spatial averaging only to the cracking strain that
exists in the fracture zone, which is a state variable that reflects the degradation of
material integrity. The elastic part of the strain is kept local. With this approach,
avoidance of the instability modes encountered in the imbricate continuum model
(Bažant et al., 1984) is guaranteed. The non-local formulation is then described by the
following relationship
][sec crce εεDσ += (3.17)
where ceε and crε are the local elastic strain and the non-local cracking (or plastic)
strain, respectively. In this formulation, the stress evaluation procedure is not dissimilar
to the traditional method, the only difference is the replacement of the local strain by
the sum of the local elastic part and non-local plastic part of the strain.
To illustrate the methodology, again, the stress-strain relationship shown in
Figure 3.5a, is considered. For the computation of stress, the strain is decomposed into
an elastic component ceε and a plastic component crε as shown in Figure 3.6a. To
facilitate a comparison with the formulation given by Eq. 3.15, the same strain
condition is applied, that is, the tensile strain at the target point is assumed to be higher
than those of the source points in the surrounding neighbourhood. A non-local cracking
strain crε with a lower value than the local cracking strain is obtained by applying the
spatial averaging (indicated by an arrow in Figure 3.6b). The secant modulus secE is
evaluated from the new total strain by summing ceε and crε as depicted in Figure 3.6b.
The stress is computed by multiplying the total strain )( crce εε + by secE .
90
With this approach, the issue of computing stresses higher than the concrete
tensile strength is prevented. This is because the evolution of stress always obeys the
constitutive stress-strain law in which the stress cannot exceed the tensile strength. In
contrast, for the case of Eq. 3.15, the stress-strain law only provides a framework for
the computation of the secant stiffness resulting from the effect of non-locality.
Therefore, it does not guarantee the compliance of tensile strength criterion.
(a)
(b)
Fig. 3.6 - Proposed non-local model: (a) decomposition of strain into elastic and
plastic parts; (b) secant modulus is computed from the sum of elastic strain
and non-local cracking strain and stress is calculated as the product of
secant modulus and new total strain.
σ
εEc
1ε
εcrεce
fct
Esec 1
σ< fctEc
1
εcrεce
ε
fct
σ
ε
91
3.5 Orthotropic Membrane Formulation
The cracking models discussed above are implemented into an iterative secant based
rotating crack model. The base element used in this study for the modelling of plain
concrete is the four-node isoparametric element of Foster and Marti (2002, 2003) with
the concrete taken as orthotropic. The material constitutive matrix 12cD in the principal
directions is given by
−−
=
122112
2221
1121
211212
)1(0000
11
c
cc
cc
cG
EEEE
ννν
ν
ννD (3.18)
where ν is the Poisson’s ratio of concrete; 1cE and 2cE are the concrete secant moduli
in the principal directions and; 12cG is the concrete secant shear modulus. To model
the non-linear behaviour of concrete, 1cE , 2cE and 12cG are updated according to the
stress states of the concrete. For plane stress problems in a global coordinate system,
stress is related to strain by
εDσ = (3.19)
where T][ xyyx τσσ=σ and T][ xyyx γεε=ε .
For known displacements and, hence, strains, the secant moduli are determined
using a modified version of the equivalent uniaxial strain concept of Darwin and
Pecknold (1977). This is done by removing the lateral deformation caused by the
Poisson’s effect and is given by Foster and Marti (2003) as
−
=
2
1
21
12
21122
11
11
1εε
νν
ννεε
u
u (3.20)
where u1ε and u2ε are the equivalent uniaxial strains and 1ε and 2ε are the strains in
the principal 1-2 directions. The secant moduli are obtained from the appropriate
uniaxial stress-strain curve. The biaxial stresses are calculated from
92
=
u
u
c
c
c
cE
E
2
1
2
1
2
10
0εε
σσ
(3.21)
The shear modulus is taken as that derived by Attard et al. (1996) and is given by
[ ])1()1()1(4
1212121
211212 νν
νν−+−
−= ccc EEG (3.22)
For cracked concrete, the Poisson’s ratios are taken as zero and Eq. 3.18 reduces to
=
12
2
1
1200
0000
c
c
c
cG
EE
D (3.23)
Prior to constructing the element stiffness matrix, the material constitutive matrix
(Eq. 3.18 or Eq. 3.23) is transformed to the global coordinate system. That is
εε TDTD 12T
cc = (3.24)
where cD is the concrete constitutive matrix in the global coordinate system and εT is
the strain transformation matrix. The detailed formulation of a two-dimensional planar
element will be discussed in Section 3.8.1.
For the distributed cracking model based on the CMM, the stiffness matrices of
steel reinforcement and concrete tension stiffening are added to the concrete
constitutive matrix as
ctssc DDDD ++= (3.25)
where sD and ctsD are the material constitutive matrices of steel reinforcement and
concrete tension stiffening, respectively, and are given by
=
0000000
syy
sxx
s EE
ρρ
D (3.26)
93
and
=
0000000
ctxy
ctsx
cts EE
D (3.27)
where sxE and syE are the secant moduli for steel reinforcement and ctsxE and ctsyE
are the secant moduli of concrete tension stiffening. All moduli are expressed in the
global x-y coordinate system.
For the non-local model, the column vector of the equivalent uniaxial strain in Eq.
3.21 is replaced by the modified counterpart resulting from the effect of non-locality. It
should be noted that non-locality only takes effect for concrete in tension.
Amalgamating non-locality, Eq. 3.21 is written as
=
u
u
c
cE
E
2
1
2
1
2
10
0εε
σσ
(3.28)
where u1ε and u2ε are the modified equivalent uniaxial strains due to the effect of non-
locality, which are calculated as the sum of the elastic component and non-local plastic
component, that is
iuiu εε = for tpkiu εε ≤ (3.29a)
criueiuiu .. εεε += for tpkiu εε > (3.29b)
where eiu.ε (i = 1, 2) is the local elastic strain, criu.ε is the non-local cracking strain
and tpkε is the strain corresponding to the peak stress in the tensile stress-strain curve.
Prior to decomposition of strain into elastic and plastic components, the local stress
locσ is calculated from the local equivalent uniaxial strain iuε via the material
constitutive law. The local elastic strain and the local cracking strain are calculated as
cloceiu Eσε =. (3.30a)
94
eiuiucriu .. εεε −= (3.30b)
where cE is the initial elastic modulus of concrete. The non-local cracking strain is
computed as (Figure 3.7a)
dArrA
criucriu ∫= ),()(' .. θεαε (3.31)
where the normalized weight function 'α is given by
dArrr
A∫=
)()()('
ααα (3.32)
and α is a polynomial bell-shaped weight function (Figure 3.7b) given by Bažant and
Ožbolt (1990) as
( )221)( Rrr −=α (3.33)
The pointer bracket denotes a Macauley bracket and chlR 9086.0= where chl is the
characteristic length. R is also known as the interaction radius (Jirásek, 1998) as it
represents the maximum distance of influence in the neighbourhood of the target point
(Figure 3.7b).
(a) (b)
Fig. 3.7 - Non-local neighbourhood: (a) spatial averaging on region A; (b) bell-
shaped weight function.
R r
2R
θ
Region A
2R x
α
1 Eq. 3.33
95
3.6 Material Constitutive Models
A powerful numerical tool such as the finite element method requires reliable material
models to perform an accurate numerical analysis. In this section, the material models
adopted in this study are presented. The material models are used in conjunction with
the orthotropic membrane formulation presented in Section 3.5 and implemented with
an iterative non-linear solution procedure discussed in the section that follows.
3.6.1 Instantaneous Behaviour of Concrete
Though this work focuses mainly on the behaviour of reinforced concrete structures at
service loads where the compressive stress in concrete rarely exceeds 40% of the
compressive strength, the full compression behaviour to failure is included for the sake
of completeness of the finite element model. For a two-dimensional concrete model, a
realistic constitutive relationship for the biaxial state of stress is the key to the
successful modelling of concrete. Three biaxial states of stress can be identified,
namely the biaxial compression state, the combined compression and tension state and
the biaxial tension state. It is important to apply reliable models for all three stress
states so as to adequately describe the realistic characteristic of a reinforced concrete
member. The instantaneous concrete model implemented in the finite element model is
based on the concrete model used by Foster and Marti (2002, 2003).
Instead of using a curved concrete strength envelope similar to that proposed by
Kupfer et al. (1969), the strength envelope is approximated by a number of linear
segments (Foster and Marti, 2003). Figure 3.8 shows the comparison of the strength
envelope adopted herein and that of Kupfer et al. (1969). In the author’s view, the
accuracy of a linear-segmented strength envelope is not significantly different from that
of a curved envelope nor does it cause discrepancies in analyses since the experimental
data on which all models of the biaxial strength envelope are based are somewhat
scattered in nature.
96
Fig. 3.8 - Strength of concrete under biaxial state of stress.
3.6.1.1 Stress-strain Relationships for Concrete
The strength of concrete in compression under biaxial states of stress differs from that
in uniaxial compression. The compressive strengths in the principal stress directions are
influenced by the orthogonal interaction of the principal stresses. The biaxial
compressive strength of concrete *cf can be written as the uniaxial compressive
strength 'cf multiplied by a strength factor β as
'*cc ff β= (3.34)
The factor β can be thought as a scaling factor for the concrete compressive
stress-strain curve in accordance with the state of stress. Thus, the concrete strain
corresponds to the peak stress in a compressive stress-strain curve, cpkε , is also
multiplied by the same factor and is given by
cpkcpk εβε =* (3.35)
1.0
(0.6, 1.25)
-σ2c / f ’c
1.00.6
0.6 (1.25, )0.6
(1.15, 1.15)
-σ1c / f ’c
This study(after Foster and Marti, 2003)
Kupfer et al. (1969)
α =0.
48
α = 1.0
97
In this work, the uniaxial compressive stress-strain curve proposed by Thorenfeldt
et al. (1987) is used as the base curve for the strength scaling purposes. The concrete
compressive stress is expressed as
nkccn
nfη
ησ+−
−=1
' (3.36)
in which
cpk
cεε
η = and cpkc
cEE
En
−= (3.37)
where cε is the concrete strain, cpkε is the concrete strain corresponding to the peak
stress on the stress-strain curve, cE is the initial elastic modulus of concrete and cpkE
is the secant modulus at the peak of the stress-strain curve that is cpkccpk fE ε'= . The
negative sign in Eq. 3.36 indicates the stress in compression. The parameter k is a decay
factor that controls the post-peak response and is given by Collins and Porasz (1989) as
0.1=k for cpkc εε ≤ (3.38a)
0.162
67.0'
≥+= cfk for cpkc εε > (3.38b)
The uniaxial compressive base curve and the scaled biaxial compressive stress-strain
curves are shown in Figure 3.9a. The unloading modulus for concrete in compression is
taken as that given by Filippou et al. (1983), that is
)1.015.0( ...
.
+−=
cpk
uncuncunc
unccuE
εε
εε
σ (3.39)
where unc.σ is the concrete stress just before the commencement of unloading and
unc.ε is the strain corresponding to unc.σ on the stress-strain curve.
98
For tensile response of concrete, concrete is taken as a linear elastic material prior
to cracking. After cracking, concrete undergoes tension softening before the bridging
stresses at the crack completely disappear. The bilinear tensile stress versus crack
opening displacement model of Petersson (1981) is adopted herein. The softening
branch is converted to a stress-strain softening curve based on an energy-based scaling
method similar to that of the crack band approach (Bažant and Oh, 1983), as described
in Section 3.4.1. The tensile stress-strain curve is shown in Figure 3.9b. Three softening
parameters, 1α , 2α and 3α , are used to define the softening curve, where
31
1 =α ; 132 92 ααα += ;
hf
GE
ct
fc23 5
18=α (3.40)
where h is an average width over which the fracture energy is dissipated. For reinforced
concrete distributed cracking problems analysed by the CMM, h is the crack spacing,
while for concrete fracture problems, h is the width of the fracture zone. In other words,
h is the crack band width for the crack band model and is the average width of the
strain localization zone for the non-local model. The unloading modulus for concrete in
tension is taken as
uncrunce
unctuE
..
.5.0 εε
σ+
= (3.41)
where unc.σ is the same as that defined for Eq. 3.39 but in tension and, unce.ε and
uncr.ε are the elastic and plastic parts of strain corresponding to unc.σ .
3.6.1.2 Biaxial Compression State of Stress
The biaxial compression state of stress is indicated in the first quadrant of Figure 3.8. It
is evident in the biaxial strength envelope that the compressive strength of a concrete
undergoing biaxial compression is higher than its uniaxial compressive strength. This is
due to the confinement induced by the orthogonal compression struts in the principal
stress directions. In the biaxial compression state of stress, β is a confinement factor
which can be determined from the biaxial strength envelope depending on the ratio of
99
the major and minor principal compressive stresses, cc 21 σσα = (see Figure 3.8). By
combining α and the equations for the linearized strength envelope within the biaxial
compression state of stress, after some simple algebraic manipulation, the confining
strength factor in the minor principal stress direction 2β can be expressed as
4.210.1
2 αβ
−= for 48.00 ≤≤ α (3.42a)
5.51)5.51(15.1 1
2 αβ
++
=−
for 0.148.0 ≤< α (3.42b)
The confining strength factor in the major principal stress direction is then given by
21 βαβ = .
3.6.1.3 Tension-Compression State of Stress
In the tension-compression biaxial state of stress, concrete undergoes tension in the
major principal direction and compression in the minor principal direction. This is
depicted by the second and fourth quadrants of the biaxial strength envelope shown in
Figure 3.8. It is well known that the strength of concrete in compression is substantially
reduced if a large tensile strain is present in the orthogonal direction (for example
Vecchio and Collins, 1986; Miyakawa et al., 1987; Belarbi and Hsu, 1991). This
phenomenon is known as compression softening. Therefore, β is a strength reduction
factor and is obtained from the modified compression field model of Vecchio and
Collins (1986), which is given by
0.134.08.0
11
≤+
=
cpkεε
β (3.43)
where 1ε is the principal tensile strain. Figure 3.9c shows the influence of 1ε to the
scaling of the compressive stress-strain curve.
100
A tension cut-off regime is used in conjunction with the compression softening
model. The tension cut-off is activated as soon as the compressive stress in the normal
direction exceeds 60% of the uniaxial concrete compressive strength as shown in the
biaxial strength envelope (Figure 3.8). The cracking stress crf under the tension cut-
off regime are given by
ctcr ff = for 6.00 '2 ≤≤ cc fσ (3.44a)
)(4.0
2'
' ccc
ctcr f
ff
f σ−= for 0.16.0 '2 ≤< cc fσ (3.44b)
The tension cut-off cracking stress is shown in Figure 3.9b in dashed line.
Under service load conditions, the influence of the tension cut-off is not
prominent as the concrete compressive stress in the normal direction is well below
'6.0 cf , for which the cracking stress is equal the concrete tensile strength.
3.6.1.4 Biaxial Tension State of Stress
The Rankine failure criterion is adopted for concrete under the biaxial tension state of
stress, that is, the cracking is deemed to occur as soon as the tensile strength of concrete
is violated regardless of the biaxial interaction of stresses. Therefore the full trilinear
stress-strain relationship denoted by the solid lines in Figure 3.9b is used for this biaxial
state of stress.
101
(a) (b)
(c)
Fig. 3.9 - Stress-strain relationships for concrete: (a) scaling of biaxial compressive
stress-strain curves; (b) tensile stress-strain curve with bilinear softening;
(c) compressive stress-strain surface under biaxial tension-compression
state of stress.
3.6.2 Time-dependent Behaviour of Concrete
The major factors affecting the time-dependent behaviour of reinforced concrete
structures are creep and shrinkage of the concrete. At any time t after first loading, the
vector of total strain is taken to be the sum of the vectors of instantaneous, creep and
shrinkage strains. That is
-σc
-εc
ε1
fβ c ’
-σc
-εc
Ec 1
fc’
εcpk βεcpk
β < 1.0
β > 1.0
fβ c ’
βεcpk
σc
εc
Ec 1
fct
εtpk α εtpk2
α ctf1
α εtpk3
fcr
Etu 1Ecu 1
102
)()()()( tttt shcpci εεεε ++= (3.45)
where ciε , cpε and shε are the instantaneous, creep and shrinkage strain vectors,
respectively, and T][ xyyx γεε=ε . Before cracking the instantaneous strain is
equal to the concrete elastic strain whereas the post-cracking instantaneous strain
consists of an elastic component and a plastic component. For finite element
implementation, creep and shrinkage strains are treated as inelastic pre-strains updated
with time and applied to the structure as equivalent nodal forces. The details of the
computational procedures will be presented in Section 3.7.
The time-dependent development of shrinkage and concrete tensile strength are
calculated using a function given by
tBAttF+
=)( (3.46)
where A and B are empirically fitted parameters obtained from test control data and
)(tF is the shrinkage strain or concrete tensile strength at time t.
3.6.3 Shrinkage
Shrinkage is defined as the time-dependent and load independent strain resulting from
the reduction in volume of concrete at constant temperature (due mainly to loss of
water resulting from drying and hydration). Shrinkage is taken to be direction
independent and shrinkage shear strain is taken as zero. Thus, the shrinkage strain
vector is written as
T]0)()([)( ishishish ttt εε=ε (3.47)
where )( ish tε is negative and with magnitude calculated from Eq. 3.46 using
appropriate factors for A and B.
103
3.6.4 Creep
Two creep models were developed in this study to simulate the time-dependent
behaviour of concrete. In the early phase of the research, a simple creep model based
on the rate of creep method (Glanville, 1930; Whitney, 1932; Dischinger, 1937; Chong
et al., 2004) was employed. This method gives reasonably good creep prediction if the
stress does not vary too much with time. A more refined model based on the principal
of superposition was developed at a later stage using the theory of solidification for
concrete creep (Bažant and Prasannan, 1989a, b). This model is more versatile than the
rate of creep method and can be used to handle creep problems with varying
stress histories.
For reinforced concrete structures under service load conditions, the concrete
stress rarely exceeds 0.4 times the strength of the concrete. Accordingly, two
assumptions are made in this study: (i) creep is linear with respect to stress; and (ii) the
time-dependent response in tension is identical to that in compression.
In the next section, the creep model based on the solidification theory will be
described and this model is used to analyse reinforced concrete structures throughout
this thesis. The model based on rate of creep model will only be presented in
Appendix A.
3.6.5 Solidification Theory for Concrete Creep
The creep model used in this study is the solidification creep aging model of Bažant
and Prasannan (1989a) using Kelvin chains to describe the viscoelastic component
(Figure 3.10a). In this model, the aging aspect of concrete creep is due to growth, on
the microscale, of the volume fraction of the load-bearing solidified matter and is a
consequence of hydration of the cement particles. An advantage of the solidification
theory is that the elastic properties of concrete are taken as non-aging with the modulus
of elasticity of concrete taken to be an age-independent asymptotic modulus, 0E .
Bažant and Baweja (1995a, b) proposed 28.0 6.1 cEE = , where 28.cE is the elastic
modulus at 28 days. A detailed discussion for the justification of the use of 0E is
104
presented in the works of Bazant and Prasannan (1989a) and Bazant and Baweja
(1995b).
The growth in volume of the solidified matter at time t is divided into the volume
fractions associated with: (i) the viscoelastic strain )(tv and; (ii) the viscous strain )(th
as shown in Figure 3.10b. Using this approach, the creep strain is decomposed as
)()()( ttt fvcp εεε += (3.48)
where vε and fε are the viscoelastic and viscous strains, respectively. The viscoelastic
and viscous strain rates are given by (Bažant and Prasannan, 1989a)
)()()(
tvttv
γε&
& = (3.49a)
)'()'()(0
tdtttγt
σ∫ −Φ= && (3.49b)
)()(
)()()(
0 tt
thttf ηη
σσε ==& (3.49c)
where 't is the variable for concrete age at application of load, )(tγ is the viscoelastic
microstrain, )'( tt −Φ is the microscopic creep compliance function of the solidified
matter associated with the viscoelastic component and 0η and )(tη are the effective
viscosity of the solidified matter and the apparent macroscopic viscosity, respectively,
associated with the viscous component. The viscous component is a linear function of
stress calculated directly from Eq. 3.49c, whereas the viscoelastic component is
evaluated by solving the superposition integral given by Eq. 3.49b. From these
relationships, Bažant and Prasannan (1989a) derived the analytical expression for the
creep compliance of concrete
[ ]
+−++=
'ln)'(1ln)',()',( 4
1.032 t
tqttqttQqttC (3.50)
105
where 2q , 3q and 4q are empirical parameters determined from control tests and
)',( ttQ is a binomial integral. However, as no closed formed solution exists for the
integral )',( ttQ the approximation
)'(1)'(
)',()'(
1)'()',(
trtrf
f ttZtQ
tQttQ
−
+= (3.51)
is substituted (Bažant and Prasannan, 1989a) where
( )[ ]2'log0019.0'log4308.0112.0)'(log tttQ f ++−= (3.52a)
( ) [ ]1.05.0 )'(1ln')',( tttttZ −+= − (3.52b)
8'7.1 12.0 += tr (3.52c)
(a) (b)
Fig 3.10 - Solidification theory for concrete creep: (a) Kelvin chain description for
viscoelastic component; (b) schematic representation of the solidification
creep model. (Bažant and Prasannan, 1989a).
µ =1
µ =2
µ =N
γ1
γ2
γN
γ
σ
σ
σ
σ/E0
εvΦ( )t-t’
γ= Φ σ( ) ( )t-t’ d t’ ∫
εf
εsh
ε
E1
E2
EN
η1
η2
ηN
h t( ) dh t( )
v t( ) dv t( )
σ
η
106
The analytical expression for the creep compliance given by Eq. 3.50 is used in
this study for the determination of the empirical parameters 2q , 3q and 4q for a given
set of experimental creep data. In numerical implementation the explicit calculation of
)',( ttQ is not required.
3.6.5.1 Rate-type Constitutive Model
To facilitate the numerical creep analysis, the integral-type equation based on the
principle of superposition, given by Eq. 2.9, is converted into a rate-type constitutive
equation that allows the stress history to be stored implicitly. A discussion for the rate-
type constitutive equations is given in Section 2.6.8. For the creep model adopted, the
rate-type equation is described using the Kelvin chain model (Figure 3.10a). For an
element in uniaxial state of stress, the relationship between the viscoelastic microstrain
and the applied stress is
σγηγ µµµµ =+ &E (3.53a)
∑=
=Nγγ
1µµ (3.53b)
where µγ , µE and µη are the viscoelastic microstrain, the elastic modulus and the
viscosity of the µ-th Kelvin chain unit, respectively, and N is the total number of Kelvin
chains. For a constant stress σ applied at time 't , the biaxial viscoelastic microstrain
vector )(tγ is obtained by solving Eq. 3.53a for µγ and subsequently substituted into
Eq. 3.53b, that is
∑=
−−−=N tte
Et
1
)'( )1(1)(µ
τ
µ
µσγ (3.54)
where µµµ ητ E= is the retardation time of the µ-th Kelvin chain unit.
107
By comparing Eq. 3.54 and Eq. 3.49b for a constant stress σ applied at time 't ,
the microscopic creep compliance function may be expressed in the form of a Dirichlet
series:
01
)'( )1(1)'( AeE
ttN tt +−=−Φ ∑=
−−
µ
τ
µ
µ (3.55)
where the 0A term is added to include the negative infinity area of the retardation
spectrum (Bažant et al., 1997) in the discretization of the spectrum that follows.
Figure 3.11 shows the numerical integration of the retardation spectrum using the
trapezoidal rule with intervals )(ln µτ∆ . The relationship for the discretization of the
Kelvin chains is given by
)(log10ln)()(ln)( µµµµµ ττττ ∆=∆= LLA (3.56)
where µµ EA 1= and )( µτL describes the retardation spectrum and is given by
Bažant and Xi (1995) as
23
31.0
1.08.2
2)3(
])3(1[])3(9.0[)3(02.0)( qL τ
τ
τττ
+
−−−=
−
23
21.0
8.21.08.2
2)3(
])3(1[)3(01.0])3(9.0[)3(19.0 qτ
τ
τττ
+
−−−−+
−− (3.57)
Bažant and Xi (1995) recommended that for a sufficiently smooth creep curve the
retardation time discretization interval be taken as 1)(log =∆ µτ for each adjacent
Kelvin chain.
Using the log-power law proposed by Bažant and Prasannan (1989a), the
microscopic creep compliance function can be approximated as
[ ]1.02 )'(1ln)'( ttqtt −+=−Φ (3.58)
108
and substituting Eq. 3.58 into Eq. 3.55, the negative infinity area is
[ ] ∑=
−−−−−+=N tte
EttqA
1
/)'(1.020 )1(1)'(1ln
µ
τ
µ
µ (3.59)
Fig. 3.11 - Discretization of a continuous retardation spectrum.
3.6.5.2 Finite Element Implementation of Creep
The exponential algorithm proposed by Bažant (1982, 1988) is adopted for finite
element implementation. The use of an exponential algorithm permits the time step
interval to be greater than the shortest retardation time and ensures numerical stability.
Bažant and Prasannan (1989b) introduced an incremental quasi-elastic stress-strain
relationship for numerical analysis using the solidification creep theory. This algorithm,
however, cannot be incorporated directly into the finite element formulation of this
study. Therefore, modifications were made to facilitate the implementation of the
creep model.
The finite element model adopted in this study is a smeared crack model based on
the total strain formulation. In a total strain formulation, the total stress is related to the
total strain through a single constitutive relationship which is path-independent and the
material response at any instant is a function only of the current state of stress or strain.
ln τ
L( )τµ
Retardation spectrum
A0
lnτ2 lnτ3 lnτ4 lnτ5 lnτ6lnτ1
A1 A2A3
A4
A5
A6
109
Consequently, the components that are required to be determined for the incorporation
of the solidification creep model are the total viscoelastic and viscous strains at a
specific time instance. The formulation is generalized for two-dimensional plane stress
problems and is presented in the next section.
By Eq. 3.49a, the change in recoverable component of creep, that is the
viscoelastic component, can be expressed in the form
½
11)(
+
++
∆=∆
i
iiv v
tγ
ε (3.60)
in which the subscripts i and i + ½ indicate the reference to time it and the time in the
middle of a logarithmic time step ½+it , respectively, where
[ ] 5.00010½ ))(( tttttt iii −−+= ++ (3.61)
and 0t is the age at first loading. By Eq. 3.49b, Eq. 3.53b and Eq. 3.55, the change in
viscoelastic microstrain γ∆ is given by
01
1 )(1
AN
i iiσGγγγ ∆+−=∆ ∑
=+ +
µµµ (3.62)
The viscoelastic microstrain at time 1+it for µ-th Kelvin chain used in this study
is a modified form of that derived by Bažant and Prasannan (1989b) and is
σGσG
γγ ∆−
+−+= ∆−−∆−+
µ
µ
µµµ
λµµ
Ee
Ee yiy
ii
1)1(1
1 (3.63)
where
µ
µ τty ∆
=∆ ; µ
µµ
λy
e y
∆−
=∆−1 ; 1−−=∆ ii σσσ (3.64)
For biaxial stress the Poisson effect is included via the matrix G in Eq. 3.62 and
Eq. 6.63 where
110
+−
−=
)1(2000101
νν
νG (3.65)
The volume of the solidified matter at mid-time of a logarithmic time step, ½+iv , is
then given by
1
2
3
½½
1−
++
+=
tv
ii (3.66)
The change in non-recoverable, viscous, component of creep is evaluated from
Eq. 3.49c. By considering the change over a finite time step we write this as
½
½1)(
+
−+ =∆
∆
i
iiftt
ησGε
(3.67)
where 21½ σσσ ∆+= −− ii . Substituting the apparent macroscopic viscosity, defined
as ½1
4½ +−
+ = ii tqη , into Eq. 3.67, the change in viscous strain can be written as
½
4½1)(
+
−+
∆=∆
i
iif t
tqt
σGε (3.68)
Lastly, the changes in viscoelastic and viscous strain components are added to the
creep strain components obtained from the previous converged time step, that is
)()()( 11 ++ ∆+= iviviv ttt εεε (3.69a)
)()()( 11 ++ ∆+= ififif ttt εεε (3.69b)
The sum of the creep strain components from Eq. 3.69 are then added to the
shrinkage strains to give the total inelastic pre-strains, 0ε . The inelastic pre-strains are
converted to equivalent nodal forces and applied to the nodes of the discretized
structure.
111
3.6.6 Time-dependent Crack Width
3.6.6.1 Cracked Membrane Model
Crack widths at time t are obtained in a similar way as described by Eq. 3.10. By taking
the average strains between two cracks, the crack width equation has a general form
that is given by
][ .1 crbetwrmcr sw εε −= (3.70)
where crbetw.ε is the total concrete strain between the cracks.
In a time-dependent analysis, the concrete tensile stress associated with tension
stiffening induces creep deformation and drying shrinkage causes shortening in
concrete between the cracks. Eq. 3.70 may be elaborated as
−++−= )()()(
)()()( 212
0
11 ttt
Et
tstw shcpctsm
rmcr ενεεσ
ε (3.71)
where 1ctsmσ is the mean tension stiffening stress in the crack opening direction.
The concrete stress associated with tension stiffening and creep contribute to an
expansion in concrete between the cracks. These effects obviously reduce the crack
opening. Meanwhile, drying shrinkage causes a volume reduction in concrete. Since the
distributed crack surfaces act as boundaries that separate concrete into individual
“blocks”, volume reduction of each concrete “block” results in a gradual opening of the
cracks. Due to the fact that the influence of shrinkage in widening the crack is far more
dominant than the influence of concrete tensile stress and creep in closing the crack, the
width of a crack in a concrete structure generally increases with time.
3.6.6.2 Crack Band Model
Cracking of concrete is treated as strain localization in the crack band model. The
effects of creep and shrinkage are only prominent in the intact concrete. Consequently,
112
the computation of time-dependent crack width for the crack band model is not
dissimilar to that for instantaneous loading. The only difference is that the variables
involved in the calculation of crack width are now functions of time. The time-
dependent crack width is given by
[ ])()()()( 1 tththtw ceccrccr εεε −== (3.72)
where crε is the cracking strain of concrete at time t and is calculated by subtracting
the elastic component of strain ceε from the total principal strain 1ε .
3.6.6.3 Non-local Model
One of the characteristics of the non-local model is that strain localization takes place
over a width of a number of elements. The distribution of cracking strain across the
width of the localization is not uniform but is concentrated in the centre of the localized
band and gradually decreases to the edge of the localization.
The crack band model has a well-defined localization boundary and cracking
occurs uniformly over a single band of elements and therefore crack width can be
calculated directly using Eq. 3.72. However, the non-local model is not as
straightforward due to the non-uniform distribution of cracking strain.
In this study a simple method is adopted to determine the crack width in the non-
local model, that is, by measuring the relative displacement of any two nodes that are
located adjacent to and outside of the localization zone. This method is crude and can
only give a rough approximation of the crack width.
3.6.7 Stress-strain Relationship for Reinforcing Steel
The different stress-strain relationships of the two types of steel commonly used in
reinforced concrete were introduced in Section 2.4. In this research, a general trilinear
stress-strain model is adopted to model the reinforcing steel as shown in Figure 3.12a.
113
The reinforcing steel is assumed to be linear elastic before yielding of steel and is
followed by a bilinear strain hardening. At reversal of stress, the unloading modulus
suE is taken as equal to the initial elastic modulus of the steel sE .
The trilinear model provides flexibility in describing a wide range of stress-strain
relationships for reinforcing steel. For reinforced concrete structures under service load
conditions, the stress in the reinforcing steel is generally below the yield stress
therefore, for simplicity, a linear-perfectly plastic stress-strain law is sufficient in a
finite element analysis. This is achieved by setting 0== uw EE .
3.6.8 Local Bond-slip Model for Bond Interface Element
The CEB-FIP Model Code 1990 (1993) bond model is used in this study. The bond
stress bτ between concrete and reinforcing steel under monotonic loading is given by
the following functions at different slips, as
4.0
2
1max
=
ss
b ττ for 10 ss ≤≤ (3.73a)
maxττ =b for 21 sss ≤< (3.73b)
−−
−−=23
2maxmax )(
ssss
fb ττττ for 32 sss ≤< (3.73c)
fb ττ = for ss <3 (3.73d)
The parameters used to define the bond model are given in Table 3.1 and the bond
stress-slip curve is shown in Figure 3.12b.
114
(a) (b)
Fig. 3.12 - (a) Trilinear stress-strain relationship for reinforcing steel; (b) bond stress-
slip relationship (CEB-FIP, 1993).
Table 3.1 - Parameters for bond stress-slip model (CEB-FIP, 1993).
Unconfined concrete1 Confined concrete2
Parameters Good bond conditions
All other bond conditions
Good bond conditions
All other bond conditions
1s 0.6 mm 0.6 mm 1.0 mm 1.0 mm
2s 0.6 mm 0.6 mm 3.0 mm 3.0 mm
3s 1.0 mm 2.5 mm Clear rib spacing
Clear rib spacing
maxτ '0.2 cf '0.1 cf '5.2 cf '25.1 cf
fτ max15.0 τ max15.0 τ max40.0 τ max40.0 τ 1 splitting bond failure; 2 pullout bond failure.
The bond stress-slip model presented above is a time-independent relationship
that does not account for the influence of the creep in concrete. Creep causes a gradual
increase in slip with time and this phenomenon is known as bond creep. The
consequence of bond creep is the reduction of slope of the time-independent bond
stress-slip relationship. The isochrone curves recommended by the CEB-FIP Model
Code 1990 (1993) is adopted to model the increase of slip with time (see Figure 3.13a).
fu
σs
εs
Es
Esu
Ew
Eu
1 1
1 1
fsy
εuεsy
fw
εws
Ebu
Eb.sec
1
1τf
τmax
τb
s1 s2 s3
115
The time-dependent slip ts between steel and concrete under a sustained load is
given by
( ) 08.02401)( tsts it += (3.74)
where t is the load duration in days and is is the instantaneous slip.
For finite element implementation, the slip at time t, )(tst , is calculated from the
relative displacement of the bond interface element nodes connecting the concrete
element and steel element. The time-independent slip is is back calculated from ts via
a relationship obtained from Eq. 3.74 by expressing is in terms of )(tst as
( ) 08.02401
)(
tts
s ti
+= (3.75)
The bond stress corresponding to is is then calculated using the time-independent
bond stress-slip relationships given by Eq. 3.73. In a secant stiffness solution
procedure, the secant modulus of the bond element is computed as )(sec. tsE tbb τ= .
An issue arose when the CEB-FIP bond model was implemented into the finite
element code. The bond model has an infinitely large initial gradient (at zero slip),
which inevitably causes computational difficulty. To prevent the numerical instability
due to the extremely high initial bond stiffness, the initial modulus of the bond model is
replaced by a linear modulus for any slip that is smaller than a threshold slip. The
threshold slip should be a reasonably small value which does not introduce a very large
gradient that may exceed the precision of the compiler. In this study a threshold slip of
10-7 mm is used with a Compaq FORTRAN 90 compiler and is found to work well for
the purpose of this research (see Figure 3.13b).
116
(a) (b)
Fig. 3.13 - (a) Influence of creep on bond stress-slip curve; (b) replacing the infinitely
large initial modulus by a linear initial modulus.
3.6.9 Concrete Tension Stiffening
For the distributed cracking model (see Section 3.3), the tension chord model discussed
in Section 3.3.1 is utilized to model concrete tension stiffening in cracked reinforced
concrete elements. Consider equilibrium across a reinforced concrete element in
tension, the mean concrete tension stiffening stress ctsmσ can be calculated for any
known mean and maximum steel stresses (Foster and Marti, 2003) using
)1()(
ρρσσσ−
−= smsrctsm (3.76)
where srσ is the steel stress at the cracks (maximum steel stress) and smσ is the mean
steel stress in the reinforced concrete tension chord. For the case of a two-dimensional
reinforced concrete membrane element with orthogonal reinforcement, the concrete
tension stiffening stresses are calculated as per the cracked membrane model (see
Figure 3.3).
The maximum variation of steel stress is obtained by integrating the change of
steel stress between the cracks (as given by Eq. 3.2) over one half of the crack spacing.
Therefore srσ can be determined by adding one half of the maximum variation of steel
sthres
CEB-FIP bond model
Linear init ial modulus
s
τb
s
τb
Bond stress-slip curveat time t
Time-independentbond stress-slip curve
117
stress to the mean steel stress, which leads to the following equations for various
loading stages:
∅+= rmb
mssrs
E 0τεσ …… for sysr f≤σ (3.77a)
−
∅+
−
∅−−
∅+=
w
s
b
b
rmbb
w
s
w
s
b
brmbmssy
rmb
sysr
EE
sEE
EEs
Efs
f
1
0
210
1
000
5.0
)(
ττ
τττττ
ετ
σ
…… for srsys f σσ <≤min (3.77b)
∅+
−+= rmb
s
symwsysr
sEf
Ef 1τεσ
…… for minssyf σ< and wsmsy ff <≤ σ (3.77c)
∅+
−−−+= rmb
w
syw
s
symuwsr
sE
ffEf
Ef 1τεσ
…… for minssyf σ< and usmw ff ≤≤ σ (3.77d)
where mε is the average strain of the reinforced concrete element and minsσ is the
minimum steel stress between two cracks of the tension chord. Figure 3.14 shows the
distributions of bond, concrete and steel stresses between two cracks of a cracked
reinforced concrete tension chord at the loading stages corresponding to Eq. 3.77a to
Eq. 3.77d.
118
(a)
(b) (c) (d) (e)
Fig. 3.14 - Distribution of bond stress )( bτ , concrete stress )( cσ and steel stress )( sσ
between cracks: (a) uniaxial reinforced concrete member in tension; (b) Eq.
3.77a; (c) Eq. 3.77b; (d) Eq. 3.77c; (e) Eq. 3.77d.
3.7 Non-linear Finite Element Implementation
The numerical analysis of a reinforced concrete structure is complicated by the non-
linear material stress-strain relationships. This is further aggravated by the nucleation of
cracks when the concrete tensile strength is violated, which results in a massive stress
redistribution within the structure. The reinforced concrete modelling approaches and
the material models presented in the previous sections are implemented using the
displacement-based finite element method. In this section, the fundamentals of the
finite element method are briefly described and the iterative non-linear solution
procedures are presented.
A B C
τb τb τb τb
A B C
σc σc σc
σs
σs min
σs min
σs minσsm
σsm
σsmσsr
σsrσsr
σs σs fu fw fsy
fu fw fsy
fu fw fsy
σc
σs fu fw fsy
σs min
σsmσsr
τb0 τb0 τb0 τb0
τb1 τb1 τb1 τb1
A B C A B CA B C
σctsm σctsm σctsm σctsm
119
3.7.1 Spatial Discretization
In a finite element analysis, a continuum is discretized into a discrete number of
elements and the physical behaviour of the continuum is described by means of
material models for the elements. The elements are interconnected at a discrete number
of nodal points on the element boundaries. For a displacement-based finite element
method, the continuous field displacement vector u′ at any point within an element is
obtained by interpolating the nodal displacement vector eu of that element which is
expressed as
euNu =′ (3.78)
where N is the displacement interpolation matrix containing the shape functions
relating the continuous field to the nodal values.
Knowing the displacements at any point within the element the strain at that point
can be determined by using an appropriate linear operator, where the strain field can be
written as
uLε ′= (3.79)
where ε is a vector of strains and the linear operator matrix L is also known as the
differential operator matrix since ε is a differential function of u′ .
In a displacement-based finite element analysis, the displacements are usually
more readily available at the nodal level. Therefore it is more conveniently to express
the strain field in terms of the nodal displacements. This is done by combining Eq. 3.78
and Eq. 3.79, which gives
euBε = (3.80)
where NLB = and is known as the strain-displacement matrix of the element.
With the strain known at a point, the stress can be calculated for a given material
property. In a time-dependent analysis, it is important to correctly account for the
development of time-dependent inelastic strains in a structure. In this study the time-
120
dependent inelastic strains are creep strain and shrinkage strain of concrete. Therefore,
the stress in the element can be written as
)( 0εεDσ −= (3.81)
where D is the material elasticity matrix and 0ε is a pre-strain vector. For the case of
concrete, 0ε is the sum of creep and shrinkage strains.
3.7.2 Time Discretization
To trace the time-dependent behaviour of reinforced concrete structures due to the
effects of creep and shrinkage in concrete, the time domain needs to be discretized into
a number of finite time steps and the time-dependent analysis is performed based on a
step-by-step integration through the time domain.
Due to the aging nature of concrete, the rate of deformation of a concrete
structure under a sustained load decreases with time. For an efficient and accurate time
analysis, it is desirable to discretize the time domain such that smaller time intervals are
used for deformations at higher rates and larger time intervals for deformations at lower
rates. An effective time discretization should produce a nearly constant change in
displacement over each time interval as depicted in Figure 3.15a. Gilbert (1979)
suggested to discretize the time on a creep-time curve such that the same amount of
specific creep occurs in each time interval (see Figure 3.15b). On the other hand,
Bažant (1979, 1988) recommended the use of about two to three constant time step
intervals per unit logarithmic time scale is sufficient for an effective time analysis since
creep curves are approximately linear on a logarithmic time scale.
In this study the time discretization approach suggested by Bažant (1979, 1988) is
adopted. The rationale of this approach is essentially identical to that given by Gilbert
(1979), as the objective is to keep the change in displacement over the time step as
constant as possible so as to minimize the errors arise during the early stages after
loading when the time-dependent development of strains are rapid.
121
(a) (b)
Fig. 3.15 - Time discretization: (a) time steps with equal increments of displacement;
(b) time discretization based on equally divided creep strains (Gilbert,
1979).
3.7.3 Principal of Virtual Work
According to the principal of virtual work, for any quasi-static and admissible virtual
displacement euδ which takes place relative to an equilibrium configuration, the
internal work done by the stresses of a body must be equal to the external work done by
the external forces acting on that body (Cook et al., 2001). Consider an element
subjected to external forces such as body forces bp , surface tractions sp and
concentrated nodal forces ep , the application of the principal of virtual work leads to
eeA sV bVdAdVdV
eeepupupuσε TTTT δδδδ +′+′= ∫∫∫ (3.82)
where eV is the volume of the element. The term on the left of the equal sign of Eq.
3.82 is the internal work done due to the element stresses. The external work done is
given by the right side, which is due to the body forces, the surface tractions and the
nodal forces.
t0
t
Displacement, u
t1 t2 t3 t4 t5 t6 t0 t
εcp
t1 t2 t3 t4 t5 t6 t7
Equa
l dis
plac
emen
t inc
rem
ents
Equa
l cre
ep st
rain
incr
emen
ts
122
By substituting Eq. 3.78, Eq. 3.80 and Eq. 3.81 into Eq. 3.82, we obtain
dVdVee VeeVe ∫∫ −
0TTTT εDBuuBDBu δδ
++= ∫∫ eA sV be dAdV
eeppNpNu TTTδ (3.83)
Since Eq. 3.83 is valid for any virtual displacement, it can be rewritten as
eeee PFuK += (3.84)
where
dVeVe ∫= BDBK T and dV
eVe ∫= 0T εDBF (3.85a, b)
and
eA sV be dAdVee
ppNpNP ++= ∫∫ TT (3.85c)
in which eK is the element stiffness matrix, eF is the nodal forces originate from the
element pre-strains (for example creep and shrinkage strains) and eP is the total
external nodal forces applied to the element. It should be noted in Eq. 3.85b and Eq.
3.85c that the element pre-strains, the distributed body forces and surface tractions
acting on the element continuous field are converted into sets of equivalent nodal forces
via appropriate interpolations and integrations. In the present study, only the
concentrated nodal forces and element pre-strain are considered. Hence, the time-
dependent deformations resulting from creep and shrinkage of concrete can be treated
in the same way as applying a set of nodal forces to the element.
The concept of applying the principal of virtual work to a single element can be
extended to the global structural level by the full assembly of finite elements into a
structural stiffness matrix. The equivalent equation to Eq. 3.84 at the structural level is
PFuK += (3.86)
123
The displacement vector u contains the displacements corresponding to the degrees of
freedom (dof) at each node in the structure. K, F and P are the structural stiffness
matrix, structural equivalent pre-strain nodal force vector and the external structural
nodal force vector, respectively, and are given by
∑=N
eKK ; ∑=N
eFF and ∑=N
ePP (3.87)
where N is the total number of elements used to model the structure. For a load-
controlled structural analysis, F and P are known in advance and K can be assembled
readily for a given set of material properties. The structural nodal displacements can be
determined by solving the dof×N number of equations (Eq. 3.86) given that the
boundary conditions are properly specified so that the stiffness matrix K is non-
singular, meaning that rigid-body movements of the structure are prevented.
3.7.4 Incremental Iterative Solution Procedures
The equilibrium derivations for finite elements discussed previously is limited to a
linear structural system for which the unknowns in the structural system can be
determined directly by solving the equations. The equilibrium governing equation
given by Eq. 3.86 can be written as
0=− QP (3.88)
where Q is a vector of internal forces of the structure with a volume V which is given
by
dVV∫=−= σBFuKQ T (3.89)
Non-linear structural problems are not possible to be solved directly by the linear
equilibrium conditions given by Eq. 3.88. The solution of non-linear systems require
some iterative solution techniques that can solve the linear equilibrium equations
iteratively until a desirable convergence tolerance is achieved. The most frequently
used iterative solution procedure is the well-known Newton-Raphson (NR) method. In
124
this study the time-dependency of the non-linear structural system is treated as quasi-
static, no time integration scheme is, therefore, involved in the solution procedures. The
time-dependent, non-linear equations are solved in space and time to trace the evolution
of stresses and strains.
Consider a time-dependent non-linear structural system, the equilibrium state
given by Eq. 3.88 cannot be attained. That is, the internal and external forces do not
subtract to zero. In an iterative solution procedure at time tt ∆+ and at iteration j, the
residual forces or the out-of-balance forces R can be written as
)()( jtttt
jtt uQPuR ∆+∆+∆+ −= (3.90)
Assuming a converged solution has been achieved for ju , a truncated Taylor
series expansion and impose the equilibrium condition, that is, the out-of-balance
forces must be zero. This gives
0)(
)()( 1 =∆+= ∆++
∆+j
jj
ttj
ttd
du
uuR
uRuR (3.91)
where jjj uuu −=∆ +1 .
By differentiating Eq. 3.90 with respect to u, the external load vector Ptt ∆+ ,
which is independent of u, is eliminated and therefore gives
dVd
dd
dd
dV
jjj∫−=−=
uuσ
BuuQ
uuR )()()( T (3.92)
By taking an infinitesimal stress σd , the relationship with an infinitesimal
displacement ud can be written as
uBDσ dd tan= (3.93)
where tanD is the material tangent constitutive matrix.
125
The relationship given by Eq. 3.93 is used in Eq. 3.92 and is subsequently,
together with Eq. 3.90, substituted into Eq. 3.91. This leads to the recurrence
relationship of the NR method. That is
)(tanj
ttttjj uQPuK ∆+∆+ −=∆ (3.94)
where tanjK is the tangent stiffness matrix given by
dVV jj ∫= BDBK tanTtan (3.95)
The displacement approximation at iteration 1+j is calculated as
jjj uuu ∆+=+1 . The incremental iterative process carries on until a convergence
criterion is achieved. Figure 3.16a shows a schematic illustration of the NR method.
In the NR method, the tangent stiffness matrix of the discretized structure is
computed for each iteration. This inevitably incurs a high computational cost which is
especially significant for large finite element analyses (with a large number of dof).
Due to the fact that NR is an iterative approximation-correction approach, various
variations of NR have emerged to reduce the computational efforts of assembling the
updated tangent stiffness matrix for each iterative cycle. The frequently utilized
variants of NR are the modified Newton-Raphson (mNR) method and the initial
stiffness Newton-Raphson (iNR) method.
In general, the mNR method involves the update of the tangent stiffness matrix
occasionally during the iterative process. A more effective mNR algorithm is to update
the tangent stiffness matrix upon each convergence of an equilibrium state. Modifying
Eq. 3.94 gives
)(tan0 j
ttttjj uQPuK ∆+∆+
= −=∆ (3.96)
where tan0=jK is the tangent stiffness matrix obtained from the previously converged
time step and is used throughout the current time step.
126
A secant stiffness approach is adopted in this study. Therefore, a secant stiffness
matrix secK is computed instead of the tangent counterpart. Therefore the secant NR
can be expressed as
)(secj
ttttjj uQPuK ∆+∆+ −=∆ (3.97)
and the secant mNR as
)(sec0 j
ttttjj uQPuK ∆+∆+
= −=∆ (3.98)
where secjK is the secant stiffness matrix computed for each iterative cycle and sec
0=jK
is the secant stiffness matrix determined from the previously converged time step.
The secant stiffness approach offers higher numerical stability than the tangent
stiffness approach. This is particularly evident when a softening structure is analysed. A
negative tangent stiffness may be encountered which can subsequently lead to
numerical difficulties. As for the secant stiffness approach, the change of the secant
stiffness is gradual and a positive stiffness is guaranteed. Nevertheless, the stability of a
secant stiffness approach is compromised by a slower rate of convergence.
Whilst an mNR method requires the update of tangent or secant stiffness matrix at
any one of the equilibrium states, the iNR method only requires the formation of the
initial tangent stiffness matrix, that is, the linear elastic stiffness matrix 0K . This leads
to a recurrence relationship as
)(0 jtttt
j uQPuK ∆+∆+ −=∆ (3.99)
Both mNR and iNR reduce the computational efforts per iterative cycle, however,
the convenience is traded off by a lower rate of convergence compared to the regular
NR method. In the author’s view, the stability aspect of a non-linear analysis is far
more important than the speed of the solution techniques, as the latter can always be
overcome by the advancing computer technology while the former is purely numerical,
127
which depends on the iterative numerical algorithm. The iterative procedures of the
tangent mNR, secant NR, secant mNR and iNR are shown schematically in
Figure 3.16.
(a) (b)
(c) (d)
(e)
Fig 3.16 - Non-linear iterative solution procedures: (a) tangent NR method; (b)
tangent mNR method; (c) secant NR method; (d) secant mNR method; (e)
iNR method.
Load
uj
t+ t∆Q uj( )
t+ t∆P
∆uj
1K
tan
Displacement uj +1
tP
t- t∆P
Load
uj
t+ t∆Q uj( )
t+ t∆P
∆uj
Displacement uj +1
tP
t- t∆P
j
1K
tanj=0
Load
uj
t+ t∆Q uj( )
t+ t∆P
1K
sec
Displacement uj +1
tP
t- t∆P
j
∆uj
Load
uj
t+ t∆Q uj( )
t+ t∆P
1
Displacement uj +1
tP
t- t∆P
∆uj
Ksecj=0
Load
uj
t+ t∆Q uj( )
t+ t∆P
∆uj
1K0
Displacement uj +1
tP
t- t∆P
1K0
128
3.7.5 Geometric Non-linearity
The non-linear solution techniques presented so far are limited to structures undergoing
sufficiently small deformation so that the infinitesimal linear strain approximation
remains valid. To account for non-linearity arising from large deformation, the second
Piola-Kirchhoff stress and Green-Lagrange strain shall be considered instead of the
frequently used engineering stress and strain.
The Green-Lagrange strain is the exact strain measure for any size of deformation
for which a quadratic term is present in its mathematical definition. If the quadratic
term in the Green-Lagrange strain is omitted, the result is the engineering strain, which
has a linear relationship with the change in length of the material and is only applicable
when deformation is small. In addition, the second Piola-Kirchhoff stress is the stress
measure for large deformation that is conjugate to the Green-Lagrange strain.
However, the use of a complete different set of stress and strain measures
inevitably requires a restructuring of the numerical procedures for the elements. To
avoid this cumbersome process, an updated Lagrangian formulation (Bathe, 1996;
Zienkiewicz and Taylor, 2000), in which the stress and strain are computed based on
the last computed displaced configuration, is employed. By this way, the deformation
with respect to the deformed shape of the last iterative step becomes arbitrarily small
and hence the use of engineering stress and strain can be retained. Consequently,
relatively few modifications are needed to extend the material non-linear iterative
solution procedures to account for non-linearity arising from large deformation.
The secant NR recurrence relationship given by Eq. 3.97 can now be written with
reference to the last computed displaced configuration which has a volume V’. That is
dVV j
ttttjj ∫ ∆+∆+ −=∆
'Tsec σBPuK (3.100)
where
dVV jj ∫=
'secTsec BDBK (3.101)
129
where secjD is the material secant constitutive matrix evaluated at iteration j based on
the last computed configuration. The iterative process can be modified accordingly to
adopt the mNR and iNR methods as discussed in Section 3.7.4.
In the finite element implementation, the iterative solution procedures are
identical to those for material non-linearity, but the nodal coordinates of the discritized
structure are updated constantly in each iterative cycle. This can be done easily by
adding the total nodal displacements calculated in the last iterative cycle to the
undeformed nodal coordinates of the structure. Stress and strain are evaluated based on
the updated nodal coordinates and the iterative process given by Eq. 3.100 and
Eq. 3.101 continues until a set convergence tolerance is reached.
It should be noted that, the geometric stiffness matrix (Zienkiewicz and Taylor,
2000), which is often found adding to the material stiffness matrix, is deliberately
excluded from Eq. 3.101. Since equilibrium between internal and external forces of a
structure is achieved iteratively in the solution procedure, the exact assembly of the
stiffness matrix is therefore not required. Van Greunen (1979) employed the same
treatment in his numerical study on reinforced concrete slabs and panels and proved
success of the implementation.
3.7.6 Convergence Criteria
The incremental iterative solution techniques discussed above provide only
approximate solutions to non-linear structural analyses. This postulates that the
equilibrium conditions given by Eq. 3.88 can only be met in an approximate manner.
Therefore convergence tests are required to ensure a sufficiently close equilibrium state
is achieved. The test for convergence of the iterative cycles can be performed either on
the norm of the out-of-balance forces or the norm of the displacement increments for
each iteration. That is
maxji rr ϕ< )...,,3,2,1( ij = (3.102)
130
where ϕ is the convergence tolerance, )()( Tjjj uRuRr ⋅= for a force
convergence criterion and jjj uur ∆⋅∆= T for a displacement convergence
criterion.
3.7.7 Computational Solution Algorithm
The mathematical formulations for non-linear finite element analyses presented above
are implemented into a finite element code. The general algorithm for all element types
based on the regular NR method is shown in the flowchart depicted in Figure 3.17. The
finite element program is divided into three main modules, an input data control, an
elements module and an equations solver. The input data control module collects the
geometric data, material properties and configuration of the external forces. These data
are stored temporarily in the computer memory and are used throughout the analysis.
The elements module is responsible for the computation of structural stiffness
matrix, element stresses and the out-of-balance forces. The gauss quadrature integration
scheme is adopted for the integration of the element stiffness matrix (Eq. 3.85a). The
structural stiffness matrix is obtained by summing up the element stiffness matrix over
all elements as given by Eq. 3.87. For the first iterative cycle of the first time step, the
nodal displacements of the discretized structure are zero and the time step is small,
therefore the structural stiffness matrix is determined based on the linear elastic (initial)
material constitutive matrix. At this iterative cycle, the internal forces vector of the
discretized structure is a null vector and the out-of-balance forces (step 14 in flowchart)
are directly taken as the total external forces of the current time step.
After the out-of-balance forces vector is computed, it is transferred to the
equations solver and the displacement increments for current iterative cycle (denoted as
j in step 15 of the flowchart) is calculated by solving the structural equilibrium
equations. The new displacements of the discretized structure are obtained by adding
the displacement increments of the current iterative cycle to the displacements from the
previous iterative cycle or from the previously converged time step.
131
In the step that follows, the convergence is checked if the required tolerance is
reached, which can be based either on the displacement or force convergence criterion
as discussed in Section 3.7.6. If the set tolerance is reached, the results of the current
time step will be printed to a file. Depending on the analysis requirement, the analysis
may proceed to the next time step with or without a further load increment, or the
analysis may terminate. On the other hand, if the convergence criterion is not satisfied,
a new set of stresses and strains of the elements are evaluated from the displacements
calculated in step 15 of the flowchart and hence facilitates the computations of the
structural stiffness matrix and the out-of-balance forces. The iterative process continues
until the set tolerance is attained.
The equivalent nodal forces due to pre-strains are not computed explicitly in the
computational algorithm. Instead, the pre-strains are accounted for in step 9 of the
flowchart before the stresses of the elements are evaluated. The internal forces
computed in step 13 have indirectly included the equivalent pre-strain forces. In
Appendix B, the treatment for pre-strains is illustrated with a simple hand calculation.
132
Fig. 3.17 - Finite element implementation flowchart.
Read elements mesh data and boundary conditions
Read external nodal forces P0
Read material properties
Input data control
1
2
3
Calculate structure nodal displacements u j+1 = u j + ∆ u j
where ∆ u j = K-1 R
Equations solver
Displacement or force convergence test
15
16
Tolerance not reached Tole
ranc
e re
ache
d
Increase time and external forces t = t + ∆ t ; P = k P0
Elements module
Output stresses and strains then proceed to next time step
or STOP program 4
Assemble structural stiffness matrix K = ∫V BT D B dV
Determine element stresses σ = D εi
Calculate structure internal forces Q = ∫V BT σ dV
Calculate out-of-balance forces R = P – Q
Determine material constitutive matrix D 10
17
Determine strains of elements ε = B u
Determine pre-strains ε0 of elements (e.g. creep and shrinkage strains)
Determine instantaneous strains of elements εi = ε - ε0
7
8
9
Geometrically non-linear analysis?
Update nodal coordinates xi = xi + uxi
yi = yi + uyi
Yes No
5
6 11
12
13
14
133
3.8 Finite Element Formulations
Two types of models are proposed in Sections 3.3 and 3.4 to simulate cracking of
reinforced concrete structures. For the distributed crack model, reinforcement is
modelled as an additional smeared stiffness over the concrete plane stress elements.
While for the localized cracking model, concrete and reinforcing steel are modelled as
individual components and the stresses between concrete and steel are transferred via
bond action. Three types of element are used in the finite element analyses in this
study. Plane stress elements are employed for representing reinforced concrete for the
distributed cracking model and plain concrete for the localized cracking model. The
discrete steel representation for the localized cracking model is modelled by means of
truss elements and the bond action is realized by the use of interface elements.
3.8.1 Four-node Isoparametric Quadrilateral Element
A formulation that enables the definition of quadrilateral shapes that can have angles
other than 90° at the corners of the elements is more versatile in generating meshes for
structures of different shapes. The isoparametric elements formulation that is often used
in structural analyses provides this particular facility. The main features of the
isoparametric elements are the use of the natural coordinate system (ξ, η) for numerical
integration and the utilization of shape functions for the interpolations of coordinates
and displacements.
The general four-node isoparametric quadrilateral plane element adopted in this
study is shown in Figure 3.18a. The nodal coordinates indicated in Figure 3.18a are
based on the natural coordinate system where the origin is located at the centre of the
element. In an isoparametric formulation, the term “isoparametric” suggests that the
same set of shape functions is used for the interpolations of coordinates (x, y) and
displacements (u, v) for any point within an element. The coordinates at a point are
given by
∑=
=4
1iii xNx and ∑
==
4
1iii yNy (3.103)
134
and the displacements are given by
∑=
=4
1iiiuNu and ∑
==
4
1iiivNv (3.104)
where i is the numbering of the nodal points shown in Figure 3.18a and iN is the shape
function for node i and is given by
)1)(1(41
1 ηξ −−=N ; )1)(1(41
2 ηξ −+=N ; (3.105a, b)
)1)(1(41
3 ηξ ++=N and )1)(1(41
4 ηξ +−=N (3.105c, d)
The four-node isoparametric quadrilateral element is also known as the bilinear
quadrilateral element as the interpolated field quantities vary linearly with the two-
dimensional Cartesian coordinate system.
The strains within an element can be calculated using the relationship given by
Eq. 3.80. That is
euBε = (3.106)
where Txyyx ][ γεε=ε and T
4411 ]...[ vuvue =u . The strain-
displacement matrix B is given by
[ ]4321 BBBBB = (3.107)
where
=
yixi
yi
xi
iNNN
N
,,
,
,0
0B )4,3,2,1( =i (3.108)
in which the notation xiN , denotes a partial derivative x
Ni∂
∂.
135
Before matrix B can be determined, the shape functions given by Eq. 3.105 are
differentiated with respect to ξ and η. By applying the chain rule, we obtain
=
yi
xi
i
iNN
NN
,
,
,
, Jη
ξ (3.109)
where J is the Jacobian matrix given by
=
ηη
ξξ
,,
,,yxyx
J (3.110)
The derivatives of the shape functions with respect to the x and y can be obtained by
inverting J. That is
−
−=
η
ξ
ξη
ξη
,
,
,,
,,
,
, 1i
i
yi
xiNN
xxyy
NN
J (3.111)
Having known xiN , and yiN , , matrix B can be computed.
For the construction of the element stiffness matrix in global coordinates, the
material constitutive stiffness matrix determined from the material coordinate system
)( 12cD as discussed in Section 3.5 is transformed such that
εε TDTD 12T
cc = (3.112)
where cD is the material matrix in the global coordinate system and εT is the strain
transformation matrix given by
−−−=
22
22
22
22 sccscscscs
cssc
εT (3.113)
136
with θcos=c and θsin=s and θ is the angle between the material coordinate system
(directions of the principal strains) and the global coordinate system. The element
stiffness matrix is obtained by applying the principle of virtual work, that is
dAteA cee ∫= BDBK T (3.114)
where eA and et are the area and the thickness of the plane element, respectively. The
integration of the element stiffness matrix is undertaken adopting a 22× gauss
quadrature rule.
(a) (b)
Fig. 3.18 - (a) Four-node isoparametric quadratic element; (b) two-node truss element.
3.8.2 Two-node Truss Element
In this study reinforcing steel is assumed to carry axial forces only, therefore simple
truss elements are sufficient to serve this purpose. The truss elements are overlaid onto
the concrete plane element discussed in the preceding section. To ensure displacement
continuity of the finite elements, a two-node linear truss element as shown in Figure
3.18b is adopted. By writing the nodal displacements as a nodal axial displacement
)( au , we obtain
ψψ sincos. iiia vuu += (3.115)
(-1,1)
ξξ ξ=1
ξ=-11
2
234 u4 u3
v4 v3
(1, 1) -
(-1, 1) -
(1,1) η
u2u1
u2
v2 v1
v2
u1
v1
u as a
x
y
θψ
x
y1
137
where i (i = 1, 2) denotes the node number of the element and ψ is the angle of
orientation of the truss element from the x-axis. The linear shape functions for the
element are given by
)1(21
1 ξ−=N and )1(21
2 ξ+=N (3.116a, b)
The axial displacement at any point of the element can be interpolated as
[ ] [ ][ ]
++−++−== ∑
= 21
212
1.
)1()1(5.0)1()1(5.0sincosvvuuuNu
iiaia
ξξξξψψ (3.117)
Denoting the axial length as as , the strain in the element is obtained by invoking
the chain rule as
a
as
u∂∂
∂∂
=ξ
ξε (3.118)
The derivative of au is obtained by differentiating Eq. 3.117 with respect to ξ and
gives
[ ]ψψξ
sin)(cos)(21
2121 vvuuua +−++−=
∂∂
(3.119)
Also, given that the total length of the truss element is L, we obtain
Lsa
2=
∂∂ξ (3.120)
Substitute Eq. 3.119 and Eq. 3.120 into Eq. 3.118 gives
euB=ε (3.121)
where the strain-displacement matrix B is given by
[ ]ψψψψ sincossincos1−−=
LB (3.122)
138
The material constitutive matrix for the case of a one-dimensional steel truss
element is a mere 11× matrix equal to the secant modulus secsE of the trilinear stress-
strain model shown in Figure 3.12a. The element stiffness matrix is calculated as
dVEeV se ∫= BBK secT (3.123)
Given that the total cross-sectional area of the reinforcing steel A is known and
hence a relationship ξdALdV 5.0= , the element stiffness matrix can be obtained
explicitly by integrating over the limit 11 ≤≤− ξ as
−−−−
=
2
2
22
22
sec
. ssymcscscsscsccsc
LAEs
eK (3.124)
in which ψcos=c and ψsin=s .
3.8.3 Four-node Isoparametric Bond Interface Element
A four-node interface element (shown in Figure 3.19a) is used to model bond-slip
between the reinforcing steel and the concrete. The relative displacement between node
set 1 (consisting of nodes 1 and 4) and node set 2 (nodes 2 and 3) represents the slip
between the concrete and the steel and is given by
−+ −=∆ tititi uuu and −+ −=∆ ninini uuu )2,1( =i (3.125)
where i is the node set number, the superscripts “+” and “–” denote the upper and lower
faces of the interface element, respectively, and the subscripts t and n represent shear
and normal movement, respectively.
The relative nodal displacements itn.u∆ are linked to the continuous
displacement field tnu∆ by
139
ebi
itnitn N uBTuu =∆=∆ ∑=
2
1. (3.126)
where iN are the linear shape functions given by Eq. 3.116, eu is the nodal
displacement in the global coordinate system and bT is the bond transformation matrix
given by
−
=ψψψψ
cossinsincos
bT (3.127)
where ψ is the angle of the interface element to the global x-axis.
The matrix B in Eq. 3.126 relates the continuous field relative displacements to
the nodal displacements along the interface and is given by
−−
−−=
1221
12210000
0000NNNN
NNNNB (3.128)
The element stiffness matrix is computed by applying the principle of virtual
work and gives
dAeA be BDBK ∫= T (3.129)
where eA is the tangential contact surface area between the interface element and the
adjacent materials and bD is the bond constitutive matrix. For the one dimensional
bond formulation used in this study (see Figure 3.19b), eA is the surface area of the
reinforcing bar encased in the concrete and bD is given by
=
bn
btb E
E0
0D (3.130)
where btE is the bond stress-slip modulus for the current stress state and bnE is the
normal bond stress-split modulus. To maintain compatibility between the reinforcing
140
steel and the concrete in the normal direction, a stiff value for bnE is used. In the
tangential direction the CEB-FIP (1993) model is used to define the bond-stress versus
slip relationship as discussed in Section 3.6.8 (also see Figure 3.12b). For stability of
the solution process the bond stiffness btE is taken as the secant stiffness.
For the four-node linear bond element, integration of Eq. 3.129 is undertaken
explicitly giving
−−−−
−−−−
=
bn
bt
bnbn
btbt
bnbnbn
btbtbt
bnbnbnbn
btbtbtbt
e
EEsym
EEEE
EEEEEE
EEEEEEEE
Lc
202.
02002
020200202
20002020002
6K (3.131)
where c is the sum of the bar circumferences of all bars on a layer and L is the length of
the bond element. The bond element stiffness matrix bK in the global coordinate
system is obtained by
BeBb TKTK T= (3.132)
where = bbbbB TTTTT , which is a diagonal transformation matrix.
(a) (b)
Fig. 3.19 - Bond interface element: (a) four-node isoparametric interface element;
(b) connectivity of interface element to concrete and steel elements.
x
y
ξ
ξ=-1
ξ=1
1
4
2
L
3
ut1
ut2
un1
un2
+
+
+
+
ut1un1
ut2un2 −
−
zero width
Interfaceelement
Steelelement
Concreteelement
Concreteelement
−− ψ
141
CHAPTER 4
EVALUATION OF THE FINITE ELEMENT MODELS
4.1 Introduction
The numerical models described in Chapter 3 were incorporated into a finite element
program RECAP (Foster and Gilbert, 1990; Foster, 1992) which was developed to
model the behaviour of reinforced concrete structures. Prior to applying the finite
element models to investigate the time-dependent behaviour of reinforced concrete
structures, with particular interest in the formation of cracks at service load conditions,
the reliability of the models must be assessed and evaluated. To facilitate this task, the
numerical models are examined and calibrated using the experimental long-term tests
undertaken by Gilbert and Nejadi (2004) and Nejadi and Gilbert (2004), which was a
complementary long-term experimental program carried out in parallel with this
numerical study.
In this chapter, mesh dependency of the fracture models is firstly investigated
with a plain concrete fracture test. A number of creep tests under varying stress
histories are then modelled in order to evaluate the creep model based on the
solidification theory (Bažant and Prasannan, 1989a, b). The finite element models are
also used to simulate the behaviour of beam and slab specimens and other time-
dependent tests on reinforced concrete structures. Comparisons are made between the
numerical and experimental results.
4.2 Mesh Sensitivity of the Localized Cracking Models
An important aspect in finite element modelling for concrete structures is to ensure that
the results of a numerical model are insensitive to different mesh configurations for a
142
specific numerical analysis. As discussed in Section 2.6.5, the numerical results for a
lightly reinforced or unreinforced concrete structure can be highly dependent on the
mesh size if no numerical treatment is employed to regularize spurious strain
localization in the concrete (Bažant and Pijaudier-Cabot, 1988; Bažant and Jirásek,
2002; de Borst et al., 1993a). This is the major cause of mesh sensitivity problems
associated with the modelling of reinforced concrete structures.
The localized cracking models namely the crack band model and the non-local
smeared crack model, are examined herein by analysing a four-point bending plain
concrete fracture specimen using the meshes shown in Figure 4.1. The specimens
spanned 450 mm, had an overall depth of 100 mm and were notched at midspan as
shown in Figure 4.1. The vertically aligned meshes in Figures 4.1a and 4.1b were used
to compare the mesh size dependency. The angled meshes in Figures 4.1c and 4.1d are
oriented at 72° to the longitudinal direction and were used to test the influence of mesh
orientation on crack propagation. The widths of the element above the notch for the
coarse and fine meshes are 10 mm and 5 mm, respectively. The material parameters
used in the four-point bending example were: GPa38=cE , MPa3=ctf and
N/m60=fG . For the crack band model, fracture is expected to localize into a single
band of elements for which the crack band widths were taken as the element width, i.e.
either mm10=ch or mm5=ch . The characteristic length for the non-local smeared
crack model was taken as mm50=chl which gives a width of localization of h = 50
mm approximately. Instead of adjusting the softening branch of tensile stress-strain
curve according to the size of the element as in the crack band model, the tensile stress-
strain curve for the non-local smeared crack model was obtained based on the width of
localization (see Section 3.6.1).
The load versus midspan deflection results for the analyses with various mesh
sizes are shown in Figure 4.2. It is well recognized that the crack band model suffers
from stress-locking (Rots, 1988) if the mesh orientation is not in alignment with the
direction of crack propagation. This is seen in Figure 4.2a for both the coarse and fine
slanted meshes, where the residual loads after cracking maintained at about 1.75 kN
although the cracks had completely opened. Stress-locking is greatly reduced for the
143
(a)
(b)
(c)
(d)
Fig. 4.1 - Finite element meshes: (a) coarse vertical mesh; (b) fine vertical mesh; (c)
coarse slanted mesh; (d) fine slanted mesh.
(a) (b)
Fig. 4.2 - Load versus midspan deflection diagrams: (a) crack band model; (b) non-
local smeared crack model.
0.00.51.01.52.02.53.03.54.04.55.0
0 0.1 0.2 0.3
Midspan deflection (mm)
Load
(kN)
Coarse vertical meshFine vertical meshCoarse slanted meshFine slanted mesh
0.00.51.01.52.02.53.03.54.04.55.0
0 0.1 0.2 0.3Midspan deflection (mm)
Load
(kN)
Coarse vertical meshFine vertical meshCoarse slanted meshFine slanted mesh
Nodes: 314 Elements: 273 100 mm
450 mm
Thickness = 50 mm
Nodes: 696 Elements: 642
Nodes: 325 Elements: 283
Nodes: 725 Elements: 672
144
non-local model. The coarse slanted mesh is seen to exhibit a higher softening tail in
Figure 4.2b, but when the mesh is refined the softening tail reduces to that of the
vertical meshes.
The deformed shapes and strain localization of the slanted meshes are shown in
Figure 4.3 where the darker shading denotes higher strains. Although the crack band
model suffers from stress-locking, it is surprising to observe that the crack propagation
is not excessively sensitive to the mesh orientation as reported by Li and Zimmermann
(1998). The crack path is corrected as the crack propagates through the specimen
(Figure 4.3a). Improved crack path was obtained upon mesh refinement as shown in
(a)
(b)
(c)
(d)
Fig. 4.3 - Deformed meshes and strain localization (plot of principal strain): (a) and
(b) coarse and fine slanted meshes for crack band model, respectively; (c)
and (d) coarse and fine slanted meshes for non-local smeared crack model,
respectively.
145
Figure 4.3b. For the non-local model, as expected, the width of the fracture zone
stretches over a number of elements and propagates vertically through the specimen
without being affected by the alignment of the mesh (Figures 4.3c and 4.3d).
Li and Zimmermann (1998) analysed a notched three-point bending concrete
fracture test with slanted meshes (see Figure 4.4) and reported that crack band models
based on the rotating crack concept suffer from serious mesh directional bias. They
investigated the influence of mesh orientation on crack propagation by setting the
meshes at 65° and 45° to the longitudinal direction and the results are illustrated in
Figures 4.5a and 4.5b. To investigate the crack band model developed in this study,
identical specimens and meshes were analysed. The material parameters were:
GPa20=cE , MPa4.2=ctf and N/m100=fG . The thickness of the specimen has
no influence on the direction of crack propagation but, of course, it does influence the
(a)
(b)
Fig. 4.4 - Three-point bending concrete fracture specimen: (a) slanted mesh at 65° to
the horizontal; (b) slanted mesh at 45° to the horizontal. (after Li and
Zimmermann, 1998).
200 mm
Thickness = 100 mm
40 mm
800 mm 100 mm 100 mm
20 mm
146
(a) (b)
(c)
(d)
Fig. 4.5 - Displaced shapes and crack patterns for three-point bending concrete
fracture specimen: (a) and (b) crack patterns for 65° and 45° (to horizontal)
slanted meshes, respectively (after Li and Zimmermann, 1998); (c) and (d)
crack patterns for 65° and 45° (to horizontal) slanted meshes, respectively,
analysed using the crack band model of this study.
load-deflection response. Therefore the thickness, which is not reported in Li and
Zimmermann’s work, was arbitrarily taken as one half of the depth of the specimen (i.e.
100 mm). The crack patterns from the analyses are shown in Figure 4.5c and 4.5d.
It is obvious in Figure 4.5 that the rotating crack model developed by Li and
Zimmermann (1998) exhibits more severe mesh directional bias than the model of this
study. It is observed that, if the mesh is oriented at a smaller angle to the crack direction
the directional bias is more prominent. This is shown in Figure 4.5c, where the crack
147
path is corrected after the crack has propagated over two elements. If the angle between
the mesh and the crack direction is large, the crack tends to propagate diagonally
through the elements as shown in Figure 4.5d. Though the crack path can be
satisfactorily calculated the model still suffers from stress-locking as discussed earlier.
4.3 Creep of Plain Concrete under Variable Stress
To investigate the capability of the solidification formulation (Bažant and Prasannan,
1989a, b) presented in Chapter 3, the creep tests under varying stress histories
undertaken by Ross (1958) were analysed and were compared with the experimental
results for verification. Ross (1958) loaded cylindrical specimens sized 305mm long
and 117.5 mm diameter with various stress histories. Each specimen was accompanied
by an unstressed control specimen under the same atmospheric conditions in order to
measure the free shrinkage strain. The time-dependent deformation of the specimens
was presented as the sum of elastic and creep strains.
The cylindrical specimens cannot be modelled exactly using the plane stress
elements developed in this study, however, since the specimens were loaded in the
elastic range, the geometry of the specimens in fact has little influence to the
development of strain. A single plane concrete element was used to analyse the
uniaxially loaded creep specimens, as shown in Figure 4.6. The size of the element was
arbitrarily taken to be 100 mm width, 200 mm long and a thickness of 65 mm.
Fig. 4.6 - A single plain concrete element used to model creep under variable stress.
100 mm
200 mm
P/2 P/2
148
The material parameters used for the creep tests were:
GPa95.686.1 28.0 == cEE , MPa41.462 µε=q , MPa6.113 µε=q and
MPa046.34 µε=q . The Dirichlet series was discretized into eight Kelvin chain units
for storage of viscoelastic strain history. The corresponding elastic modulus µE and
retardation time µτ for each Kelvin chain unit are given in Table 4.1 and the negative
infinity area of the continuous retardation spectrum was taken as 0A = 13.2 MPa-1. The
results calculated by the finite element model are compared with Ross’ (1958) creep
tests in Figures 4.7, 4.8 and 4.9 with a good overall agreement. This demonstrates that
the model based on the principle of superposition using the solidification theory
computes generally accurate creep deformation of a concrete structure.
Table 4.1 - Kelvin chain properties for solidification creep model.
µ-th unit Eµ (MPa) τµ (days) 1 0.33892 0.0001 2 0.28833 0.001 3 0.24817 0.01 4 0.21628 0.1 5 0.19098 1 6 0.17089 10 7 0.15494 100 8 0.14229 1000
149
(a)
(b)
Fig. 4.7 - Creep tests of Ross (1958): (a) specimen subjected to sustained constant
load over a period of 46 days; (b) specimen subjected to descending stress
history.
0
100
200
300
400
500
600
0 20 40 60 80 100 120 140 160Age (days)
Elas
tic +
cre
ep s
train
s (x
10-6
)
ExperimentalFEM
0
5
10
15
0 20 40 60 80 100 120 140 160
Stre
ss (M
Pa) 13.8
118.3
5.52.75
0
100
200
300
400
500
600
700
0 20 40 60 80 100 120 140 160Age (days)
Elas
tic +
cre
ep s
train
s (x
10-6
)
ExperimentalFEM
05
101520
0 20 40 60 80 100 120 140 160
Stre
ss (M
Pa)
15.03
150
(a)
(b)
Fig. 4.8 - Creep tests of Ross (1958): (a) specimen subjected to ascending stress
history and a subsequent complete load removal; (b) specimen subjected to
combined loading and unloading stress history.
0
100
200
300
400
500
600
0 20 40 60 80 100 120 140 160Age (days)
Elas
tic +
cre
ep s
train
s (x
10-6
)
ExperimentalFEM
0
5
10
15
0 20 40 60 80 100 120 140 160
Stre
ss (M
Pa)
2.75
8.3
13.813.8
8.3
0
100
200
300
400
500
600
0 20 40 60 80 100 120 140 160Age (days)
Elas
tic +
cre
ep s
train
s (x
10-6
)
ExperimentalFEM
0
5
10
15
0 20 40 60 80 100 120 140 160
Stre
ss (M
Pa)
2.755.5
8.311
13.8
151
Fig. 4.9 - Creep tests of Ross (1958): specimen subjected to prolonged sustained
stress.
4.4 Long-term Flexural Cracking Tests
4.4.1 Introduction
To the author’s knowledge, experimental data available for long-term cracking of
reinforced concrete structures are scarce in the literature. Two series of long-term tests
were conducted by Gilbert and Nejadi, which consist of a series of flexural cracking
tests (Gilbert and Nejadi, 2004) and a series of restrained deformation tests (Nejadi and
Gilbert, 2004). One of the major aims of this complementary experimental program is
to make available the experimental data with specific regard to the time-dependent
development of crack spacing and crack width of reinforced concrete structures.
The flexural cracking tests comprise a total of twelve one-way singly reinforced
simply supported specimens without transverse reinforcement, in which six are beam
specimens and the other six are slab specimens. The tests consist of six pairs of
identical specimens and each pair was subjected to sustained load up to 50%
0
100
200
300
400
500
600
700
0 50 100 150 200 250 300Age (days)
Elas
tic +
cre
ep s
train
s (x
10-6
) ExperimentalFEM
05
101520
0 50 100 150 200 250 300
Stre
ss (M
Pa)
15.03
7.52
152
(designated “a”) and 30% (designated “b”) of its ultimate load (see Tables 4.2 and 4.3).
The specimens were cast and moist cured for a period of 14 days and tested under
sustained loading over a span of 3.5 m for durations up to 400 days.
The beam specimens were subjected to sustained loads at the one-third points
between the supports and the slab specimens were subjected to sustained uniformly
distributed loads over the entire span. The growth of flexural cracking and midspan
deflection for each specimen were monitored with time. The loading configurations and
the dimensions of the specimens are shown in Figure 4.10. The details of the cross-
section and the applied load for each specimen are presented in Tables 4.2 and 4.3.
Companion specimens were also cast at the same time to measure creep and
shrinkage of the concrete. The creep coefficients were calculated from the data
obtained from a creep test which was performed by applying 5 MPa constant stress to
several concrete cylinders. Shrinkage strains were measured from cylindrical specimens
and two rectangular concrete panels after the commencement of drying after 14 days
of age.
Table 4.2 - Details of flexural beam specimens (Gilbert and Nejadi, 2004).
Specimen No. of bars
Bar dia. (mm)
cb (mm)
cs (mm)
s (mm)
Load, P (kN)
B1-a 2 16 40 40 150 18.6 B1-b 2 16 40 40 150 11.8 B2-a 2 16 25 25 180 18.6 B2-b 2 16 25 25 180 11.8 B3-a 3 16 25 25 90 27.0 B3-b 3 16 25 25 90 15.2
Table 4.3 - Details of flexural slab specimens (Gilbert and Nejadi, 2004).
Specimen No. of bars
Bar dia. (mm)
cb (mm)
cs (mm)
s (mm)
Load, w (kN/m)
S1-a 2 12 25 40 308 2.9 S1-b 2 12 25 40 308 1.9 S2-a 3 12 25 40 154 4.9 S2-b 3 12 25 40 154 2.9 S3-a 4 12 25 40 103 5.8 S3-b 4 12 25 40 103 3.9
153
(a)
(b)
(c) (d)
Fig. 4.10 - Flexural cracking tests: (a) beam specimen under four-point sustained
bending; (b) slab specimen subjected to sustained uniformly distributed
load; (c) typical cross-section for beam specimens (section A-A); (d) typical
cross-section for slab specimens (section B-B). (Gilbert and Nejadi, 2004).
A
A
L/3
L = 3500
L/3 L/3
P P
B
L = 3500
w
400
130
stA
sc
b s s
cs
b s
Ast
300
154
4.4.2 Analysis of Long-term Flexural Cracking Tests and Material Properties
All the specimens in this series of tests were cast from the same batch of concrete mix.
Material tests on 300 mm high by 150 mm diameter cylinder gave the mean 28-day
concrete compressive strength as MPa8.24=cmf and the 28-day elastic modulus as
MPa2495228. =cE . The tensile strength was obtained from indirect tension (Brazil)
tests on 150 mm diameter cylinders tested at ages 14, 21 and 28 days and were
MPa0.214. =ctf , MPa6.221. =ctf and MPa8.228. =ctf , respectively. For finite
element modelling, the time-dependent development of tensile strength and shrinkage
are approximated using the parameters A and B given by Eq. 3.46 in Chapter 3. The
parameters for growth of the mean concrete tensile strength were taken as
MPa2.4=ctfA and days15=
ctfB and the growth curve is shown in Figure 4.11a.
The shrinkage parameters were obtained by fitting the experimental shrinkage data and
were µε950=shA and days45=shB (see Figure 4.11b). The concrete fracture
energy fG was taken as mN75 and Poisson’s ratio 2.0=ν .
(a) (b)
Fig. 4.11 - Test data of companion specimens compared with models: (a) growth of
concrete tensile strength; (b) shrinkage strain since commencement of
drying. (Gilbert and Nejadi, 2004).
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0 10 20 30Age (days)
Conc
. ten
sile
str
engt
h f ct
(MPa
)
Experimental data
Model
Afct = 4.2 MPaBfct = 15 days
0
200
400
600
800
1000
0 50 100 150 200Time since commencement of drying (days)
Shrin
kage
str
ain
( µε)
Experimental data
Model
Ash = 950 µεBsh = 45 days
155
For creep modelling using the solidification theory, the asymptotic elastic
modulus of concrete E0 is taken as that recommended by Bazant and Baweja (1995a),
that is, GPa406.1 28.0 == cEE . The empirical material constants 2q , 3q and 4q were
determined by fitting the compliance data obtained from the creep test. The
empirical material constants were MPa5.1822 µε=q , MPa0.13 µε=q and
MPa7.234 µε=q and the approximated compliance curve is shown in Figure 4.12.
The Dirichlet series was discretized into eight Kelvin chain units for storage of
deformation history of viscoelastic strain. The corresponding elastic modulus µE and
retardation time µτ for each Kelvin chain unit are given in Table 4.4 and the negative
infinity area of the continuous retardation spectrum was taken as 0A = 52.8 MPa-1. For
the reinforcing steel, the stress-strain relationship is assumed to be elastic-perfectly
plastic with a yield strength of 500 MPa and an elastic modulus of 200 GPa. In this
numerical study, no correction has been made to account for size effects of creep and
shrinkage of concrete. Considering the cross-section dimensions of the test specimens
compared to the size of the control specimens for which the shrinkage and creep
relationships with time were determined, this is considered to be a reasonable
supposition.
Fig. 4.12 - Compliance curve for flexural cracking tests (Gilbert and Nejadi, 2004).
0
50
100
150
200
0.1 1 10 100 1000Time under load, t-t' (days)
J(t,t
') (1
0-6 /
MPa
)
Experimental data (t'=14 days)
Model (t'=14 days)
156
Table 4.4 - Kelvin chain properties for solidification creep model.
µ-th unit Eµ (MPa) τµ (days) 1 0.08480 0.0001 2 0.07214 0.001 3 0.06209 0.01 4 0.05411 0.1 5 0.04778 1 6 0.04276 10 7 0.03877 100 8 0.03560 1000
To compare the finite element models with the experimental results, the
experimental average crack widths were calculated by summing the widths of all cracks
at the soffit within the constant moment region for the beam specimens or within the
region of moment greater than 90% of the maximum moment for the slab specimens.
The sum of the crack widths was divided by the number of cracks within the region to
give the average crack width. Serviceability of a reinforced concrete structure,
however, depends not on the average crack width but on the widest crack. Therefore,
the comparison of the experimental and numerical maximum crack widths is also a
major interest in this study. Crack widths of the finite element models were computed
using the methods discussed in Section 3.6.6. To illustrate the comparisons of the
results of the finite element models and the experimental data, the detailed numerical
results for four specimens (B1-a, B2-a, S2-a and S3-a) are presented and discussed in
the sections that follow. The comparisons of results for the other 8 specimens are
presented in tabulated form.
4.4.3 Analysis of Long-term Flexural Cracking Tests using the Distributed
Cracking Model – Cracked Membrane Model
4.4.3.1 Four-point Bending Beam Tests under Sustained Load
Two types of finite element mesh were used for the beams B1-a and B2-a since the two
beams had different thicknesses of bottom concrete cover. Figure 4.13 shows the finite
157
element meshes defining the two different types of beam specimens. Due to symmetry,
only one-half of each beam was required for the modelling. The mesh for the specimen
with 40 mm cover consists of 199 nodes, 108 concrete elements, 54 reinforced concrete
elements and 2 stiff elastic elements for the supports (Figure 4.13a). The mesh for the
specimen with 25 mm cover is made up of 171 nodes, 108 concrete elements, 27
reinforced concrete elements and 2 stiff elastic elements at the supports (Figure 4.13b).
The shaded mesh represents the concrete elements containing the smeared longitudinal
reinforcement. The details of the reinforcement for all beam specimens are given
in Table 4.5.
Analyses were undertaken for λ = 0.5 and λ = 1 representing analysis based on
the minimum and maximum crack spacings, respectively, which accordingly give the
bounds of the crack opening with time. The numerical crack widths were obtained by
summing the average crack width for all elements (the average crack width for all
integration points within that element) at the soffit within the constant moment region
and divided by the number of element under consideration.
(a)
(b)
Fig. 4.13 - Finite element meshes for beam specimens: (a) beam specimen with 40
mm concrete cover; (b) beam specimen with 25 mm concrete cover.
Stiff elastic elements Plain concrete RC zone
158
Table 4.5 - Reinforcement properties for beam specimens.
Specimen Reinforcement ratio ρx ρy
B1-a 0.01676 0 B1-b 0.01676 0 B2-a 0.02437 0 B2-b 0.02437 0 B3-a 0.03655 0 B3-b 0.03655 0
The results of the calculated time-dependent midspan deflection are plotted in
Figure 4.14 and a good correlation is shown between the numerical results and the
experimental data. The comparisons of the calculated and experimental crack widths in
the constant moment region are shown in Figure 4.15. The calculated time-dependent
crack opening is shown by the shaded region in Figure 4.15 and shows a reasonable
agreement with the experimental results. However, the model underestimates the
maximum crack width for beam B2-a. A second run of the finite element model was
carried out using the crack spacings as observed in the experiment and the results were
found to correlate well with the experimental average crack opening as shown in Figure
4.15. This indicates a strong dependence of the numerical crack width on the crack
spacing for the distributed cracking model.
(a) (b)
Fig. 4.14 - Comparison of FEM and experimental time-dependent midspan
deflections: (a) beam B1-a; (b) beam B2-a.
Beam B2-a
0
2
4
6
8
10
12
14
16
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
ExperimentalFEM ( =1.0)FEM ( =0.5)λ
λ
Beam B1-a
0
2
4
6
8
10
12
14
16
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
ExperimentalFEM ( =1.0)
FEM ( =0.5)λ
λ
159
(a) (b)
Fig. 4.15 - Comparison of FEM and experimental time-dependent crack openings: (a)
beam B1-a; (b) beam B2-a.
The computed crack width envelopes at 200 days after loading are compared with
the experimental results in Figures 4.16a and 4.16b for beam B1-a and beam B2-a,
respectively. The crack widths for the primary cracks within the constant moment
region were plotted against the depth of the specimens. It is seen that the calculated
crack width envelopes have a good agreement with the test data.
(a) (b)
Fig. 4.16 - Comparison of FEM and experimental crack width envelopes across the
depth of specimens at 200 days: (a) beam B1-a; (b) beam B2-a.
0.000.050.100.150.200.250.300.350.400.45
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM (EXP Srm)FEM (TCM Srm)
Beam B1-a maximum
average
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM (EXP Srm)FEM (TCM Srm)
maximum
average
Beam B2-a
Beam B2-a
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4Crack width (mm)
Hei
ght f
rom
bot
tom
(mm
) Experimental
FEM
Beam B1-a
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5Crack width (mm)
Hei
ght f
rom
bot
tom
(mm
) Experimental
FEM
160
4.4.3.2 Uniformly Loaded One-way Slabs under Sustained Load
The longitudinal layout of slab S2-a and slab S3-a (Figure 4.10b) are identical to that of
the beam specimens. The slabs were subjected to uniformly sustained load up to a
period of 400 days. Figure 4.17 shows the finite element mesh for the slabs with the
smeared reinforcement depicted by the shaded area. Table 4.6 shows the details of the
reinforcement for all slab specimens. Only one half of specimens were modelled due to
symmetry. The mesh has 199 nodes, 108 concrete elements, 54 reinforced concrete
elements and 2 extra stiff elastic elements for the steel plate support. The numerical
crack widths were calculated similar to the procedure explained for the beam specimens
but the averaging was undertaken for the region of moment greater than 90% of the
maximum moment, as indicated in Figure 4.17.
Fig. 4.17 - Finite element mesh for slab specimens.
Table 4.6 - Reinforcement properties for slab specimens.
Specimen Reinforcement ratio ρx ρy
S1-a 0.009121 0 S1-b 0.009121 0 S2-a 0.013681 0 S2-b 0.013681 0 S3-a 0.018242 0 S3-b 0.018242 0
> 0.9 Mmax
Stiff elastic elements Plain concrete RC zone
161
Like the beam specimens, the slabs were modelled for λ = 0.5 and λ = 1.0 so that
the maximum and minimum crack widths can be calculated by the finite element
model. The midspan deflection curves with time are shown in Figures 4.18a and 4.18b
for slab S2-a and S3-a, respectively, and the model results show a good correlation with
the test data. In Figure 4.19 the development of crack opening with time of the finite
element model again shows a close agreement with the experimental cracking opening.
The development of crack opening with time, which was computed using the
experimentally observed average crack spacing as an input to the finite element
analysis, is shown in Figure 4.19 denoted by dashed line. The numerical results closely
agree with the experimental results for slab S3-a but are slightly higher than those for
specimen S2-a.
(a) (b)
Fig. 4.18 - Comparison of FEM and experimental time-dependent midspan
deflections: (a) beam S2-a; (b) beam S3-a.
(a) (b)
Fig. 4.19 - Comparison of FEM and experimental time-dependent crack openings: (a)
beam S2-a; (b) beam S3-a.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM (EXP Srm)FEM (TCM Srm)
maximum
average
Slab S2-a
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 100 200 300 400Time (days)
Crac
k wi
dth
(mm
)
ExperimentalFEM (EXP Srm)FEM (TCM Srm)
Slab S3-a
maximum
average
Slab S2-a
0
5
10
15
20
25
30
35
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
ExperimentalFEM ( =1.0)FEM ( =0.5)λ
λ
Slab S3-a
0
5
10
15
20
25
30
35
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
ExperimentalFEM ( =1.0)FEM ( =0.5)λ
λ
162
4.4.3.3 Discussion
In this study the average crack spacing were taken as two third of the maximum crack
spacing as recommended by CEB-FIP Model Code 1990 (1993), which is equivalent to
taking λ = 0.67 in the finite element model. For presentation purposes, the time-
dependent midspan deflections for all other flexural specimens were calculated using
λ = 0.67 and the analysis results are compared with the test data in Table 4.7. Table 4.8
summarizes the average crack spacing calculated by the model and those measured in
the test. The average crack width was calculated, again, with 67.0=λ and the crack
width for the widest cracks with λ = 1.0. The calculated crack openings with time are
presented in Tables 4.9 and 4.10 for comparison with test data.
For the case of the distributed cracking model, the concrete tensile strength is
taken as a constant for the entire structure, thus the concrete elements at the soffit begin
to crack as soon as the cracking load is reached. This inevitably over-softens the
structure and must be compensated by a tension-stiffening model. The cracked
membrane model includes tension stiffening via the tension chord model, however, the
concrete tension stiffening stress between cracks is assumed to have a linear
distribution. This in fact idealizes the actual concrete stress distribution, which
increases non-linearly at a decreasing rate from the crack to the point midway between
adjacent cracks.
A comparison of the linearized and the actual concrete stress distribution is
illustrated in Figure 4.20. Obviously, the linearized stress distribution underestimates
the concrete tension stiffening which may lead to an over-soft structural behaviour. For
the beam specimens that were lightly loaded relative to the cracking load, especially for
beam B1-b, the calculated midspan deflections are higher than the experimental results.
In the author’s view, this is attributed to the limitation discussed above.
163
Fig. 4.20 - Comparison of the concrete stress distribution between cracks for cracked
membrane model and that in real structure.
In the test, it may be the case that the instantaneous load was applied approaching
the cracking load and cracks formed at the weakest locations (lowest concrete tensile
strength) but the crack pattern was yet to develop fully. Secondary cracks might then
begin to form with time due to the effects of creep and shrinkage and therefore the
increase in deflection with time was gradual.
This can be illustrated by examining the experimental crack spacings for the beam
specimens loaded at 50% (designated “a”) and 30% (designated “b”) of the ultimate
loads (see Table 4.8). The specimens subjected to a lower load had wider crack
spacings than those subjected to a higher load. In the distributed cracking model, full
cracking of the specimens occurred immediately after the cracking load was exceeded
and this subsequently leads to a larger deformation. This also explains the less
satisfactory correlation between the calculated and the experimental average crack
spacings for beam B2-b. In summary, the overall correlation for the average crack
spacings is reasonable except for beams B3-a and B3-b, the model calculated a lower
average crack spacing than the test data.
σc
fctActual stress distribution
Cracked membrane model
164
Table 4.7 - Midspan deflections for λ = 0.67 at various times t (days) after loading
(Distributed cracking model – Cracked membrane model).
Midspan deflections (mm) Specimens Instantaneous t = 50 t = 200 t = 380
FEM 1.84 6.94 9.54 10.33 B1-b Exp. 1.98 4.88 6.69 7.44
FEM/Exp. 0.93 1.42 1.43 1.39 FEM 1.89 6.48 9.32 10.19 B2-b Exp. 2.06 5.11 7.06 7.88
FEM/Exp. 0.92 1.27 1.32 1.29 FEM 5.13 10.48 13.50 14.62 B3-a Exp. 5.81 10.08 12.35 13.32
FEM/Exp. 0.88 1.04 1.09 1.10 FEM 1.64 5.63 8.52 9.61 B3-b Exp. 1.97 5.33 7.13 7.90
FEM/Exp. 0.83 1.06 1.19 1.22 FEM 6.08 18.75 24.66 26.66 S1-a Exp. 7.14 18.59 22.87 25.12
FEM/Exp. 0.85 1.01 1.08 1.06 FEM 2.16 15.28 21.14 23.06 S1-b Exp. 2.72 12.62 17.79 19.91
FEM/Exp. 0.79 1.21 1.19 1.16 FEM 3.48 14.56 21.05 23.14 S2-b Exp. 4.43 14.34 19.83 21.93
FEM/Exp. 0.79 1.02 1.06 1.06 FEM 4.61 15.34 21.96 24.26 S3-b Exp. 5.04 15.22 20.65 22.90
FEM/Exp. 0.91 1.01 1.06 1.06
An issue arose when the finite element analysis was undertaken for slab S1-b, the
most lightly reinforced concrete slab specimen subjected to a sustained load of 30% of
its ultimate load. The calculated results show that the specimen was uncracked at
instantaneous loading and this does not agree with the observation in the experiment.
Consequently, a range of concrete tensile strengths was assumed in the finite element
trial simulation in order to predict the gradual cracking in slab S1-b. The results for the
trial simulation are shown in Figure 4.21 and it is seen that the results for
MPa6.114. =ctf correlates the test data most satisfactorily. Therefore, this value of
concrete tensile strength was used for the modelling of slab S1-b.
165
Fig. 4.21 - Finite element trial simulation over a range of concrete tensile strength for
specimen S1-b.
Table 4.8 - Comparison of FEM and experimental average crack spacings for
flexural specimens (Distributed cracking model – Cracked membrane
model).
Specimens FEM (mm) Experimental (mm) FEM/Exp.
B1-a 195 190 1.03 B1-b 195 220 0.89 B2-a 135 170 0.79 B2-b 135 320 0.42 B3-a 87 160 0.55 B3-b 87 170 0.51 S1-a 158 130 1.21 S1-b 158 130 1.21 S2-a 104 120 0.87 S2-b 104 110 0.95 S3-a 78 110 0.71 S3-b 78 130 0.60
The underestimation of the numerical model for the cracking state of slab S1-b is
probably because the applied load was close to the cracking load of the specimen. Since
cracking is a gradual process, slab S1-b could have been mildly cracked when subjected
to the applied load and the cracks remained fine. The finite element model may not be
able to capture this sensitive behaviour properly. In addition, due to the random nature
of cracking and the heterogeneous properties of concrete, there might be some locations
Slab S1-b
02468
1012141618
0 10 20 30 40 50Time (days)
Mid
span
def
lect
ion
(mm
)
ExperimentalFEM (fct=2.0MPa)FEM (fct=1.9MPa)FEM (fct=1.8MPa)FEM (fct=1.7MPa)FEM (fct=1.6MPa)
166
in the specimens where the concrete tensile strengths were much lower than the mean
concrete tensile strength obtained from a laboratory test. Consequently, this could also
be one of the causes resulting in the specimen cracking at a lower load.
As mentioned earlier the crack widths obtained by the distributed cracking model
have a strong dependence on the crack spacing. As expected, for specimen B1-b (beam
that was loaded up to about the cracking moment) the calculated crack widths are
higher than the test data (see Table 4.9) since the model overestimated the deformation
of the specimen. One would expect the same to happen for specimen B2-b (the other
Table 4.9 - Crack widths for beam specimens at various times t (days) after loading
(Distributed cracking model – Cracked membrane model).
Crack widths (mm) Beam specimens t = 7 t = 50 t = 200 t = 380
FEMavg 0.078 0.164 0.218 0.234 Exp.avg 0.046 0.097 0.122 0.137
B1-b
FEM/Exp.avg 1.70 1.69 1.79 1.71 FEMmax 0.149 0.245 0.312 0.332 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 1.96 1.93 2.05 1.87
FEMavg 0.015 0.073 0.117 0.130 Exp.avg 0.042 0.110 0.127 0.152
B2-b
FEM/Exp.avg 0.36 0.66 0.92 0.86 FEMmax 0.023 0.126 0.172 0.185 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.30 0.99 1.13 1.04
FEMavg 0.074 0.108 0.124 0.132 Exp.avg 0.066 0.127 0.149 0.184
B3-a
FEM/Exp.avg 1.12 0.85 0.83 0.72 FEMmax 0.106 0.163 0.177 0.187 Exp.max 0.102 0.254 0.279 0.279 FEM/Exp.max 1.04 0.64 0.63 0.67
FEMavg 0.015 0.042 0.065 0.073 Exp.avg 0.030 0.082 0.102 0.112
B3-b
FEM/Exp.avg 0.50 0.51 0.64 0.65 FEMmax 0.024 0.067 0.100 0.110 Exp.max 0.051 0.127 0.127 0.127 FEM/Exp.max 0.47 0.53 0.79 0.87
167
Table 4.10 - Crack widths for slab specimens at various times t (days) after loading
(Distributed cracking model – Cracked membrane model).
Crack widths (mm) Slab specimens t = 7 t = 50 t = 200 t = 380
FEMavg 0.158 0.223 0.267 0.282 Exp.avg 0.066 0.130 0.155 0.168
S1-a
FEM/Exp.avg 2.39 1.72 1.72 1.68 FEMmax 0.206 0.299 0.365 0.386 Exp.max 0.127 0.203 0.254 0.254 FEM/Exp.max 1.62 1.47 1.44 1.52
FEMavg 0.059 0.151 0.211 0.229 Exp.avg 0.044 0.078 0.105 0.114
S1-b
FEM/Exp.avg 1.34 1.94 2.01 2.01 FEMmax 0.141 0.240 0.305 0.327 Exp.max 0.102 0.127 0.178 0.178 FEM/Exp.max 1.38 1.89 1.71 1.84
FEMavg 0.033 0.083 0.124 0.137 Exp.avg 0.058 0.092 0.117 0.130
S2-b
FEM/Exp.avg 0.57 0.90 1.06 1.05 FEMmax 0.062 0.140 0.188 0.202 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.82 1.10 1.24 1.13
FEMavg 0.035 0.065 0.092 0.101 Exp.avg 0.043 0.094 0.124 0.137
S3-b
FEM/Exp.avg 0.81 0.69 0.74 0.74 FEMmax 0.054 0.101 0.138 0.150 Exp.max 0.076 0.127 0.178 0.203 FEM/Exp.max 0.71 0.80 0.78 0.74
beam that was loaded up to about the cracking moment) as the model also computed a
larger deformation than the observed deformation. However, the crack widths have
been compensated by the smaller crack spacing calculated by the model than that
observed in the experiment, therefore the apparent good correlation obtained by the
model, in the author’s view, was a mere coincidence. The model also calculated larger
crack widths for slabs S1-a and S1-b as shown in Table 4.10, which are due mainly to
the overestimation of crack spacing and the slightly higher deformation calculated by
the model. For other specimens the model results have a reasonable correlation with the
test data.
168
To investigate the correlation between the calculated and the experimental crack
widths, the test data were plotted against the corresponding computed results for 7
days, 50 days, 200 days and 380 days after loading in Figures 4.22a and 4.22b, for both
average and maximum crack widths, respectively. Three reference lines were plotted on
the same diagram to show the lines with ratio of experimental to calculated results r of
1.5, 1 and 1.5-1. The line r = 1 depicts a perfect correlation between the model and the
test results while the lines denoted r = 1.5 and r = 1.5-1 represent a deviation of ±50%
of the calculated results from the test data. It is seen in Figures 4.22a and 4.22b that
most calculated crack widths fall within ±50% of the measured crack widths except for
the data points of beam B1-a, slab S1-a and slab S1-b, which were mentioned
previously in the comparisons of results.
The distributed cracking cracked membrane model, on the whole, gives
reasonable results as shown in the comparisons described above. Nevertheless, the
strong dependence of the numerical results on the crack spacing is the biggest
shortcoming of the distributed cracking model. This may be overcome to some extent
by calibrating the bond shear stress 0bτ , which has a considerable influence on the
calculation of crack spacing.
(a) (b)
Fig. 4.22 - Correlation diagrams for distributed cracking model – cracked membrane
model: (a) average crack widths; (b) maximum crack widths.
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5FEM max. crack width, wm.FEM (mm)
Exp.
max
. cra
ck w
idth
, wm
.Exp
(mm
)
r = wm.Exp / wm.FEM
r = 1.5r = 1
r = 1.5 -1
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4FEM avg. crack width, wa.FEM (mm)
Exp.
avg
. cra
ck w
idth
, wa.
Exp (
mm
)
r = wa.Exp / wa.FEM
r = 1.5r = 1
r = 1.5 -1
169
4.4.4 Analysis of Long-term Flexural Cracking Tests using the Localized
Cracking Model – Crack Band Model
In the preceding section, analyses were performed using the distributing cracking
approach, which can be conceived as an “average model” that gives reasonably good
results in a global sense. In the localized cracking model using the crack band
approach, the cracking mechanism in a reinforced concrete member is decomposed into
finer individual mechanisms such as fracture of concrete and stress transfer between
steel and concrete. To correctly capture the localized fracturing of concrete, a finer
mesh is required in comparison to the distributed cracking model. The size of the mesh
must be sufficiently fine to capture the strain localization at discrete locations with a
finite crack spacing and also to allow for the bond stress to develop over the bond
development length.
To calculate crack widths, the cracked concrete element (at discrete locations) at
the soffit is identified and the element crack width is calculated by averaging the crack
widths computed at the integration points of that element. The average crack width
calculated by the finite element model is determined by summing the crack widths of
the cracked elements and is then divided by the total number of cracked elements.
The treatment of a crack as localized fracture enables the localized cracking
models to trace the locations and propagation of the crack. To improve the simulation
of the random cracking phenomena in real structures, stochastic fluctuations of the
concrete tensile strength are introduced. Clark and Spiers (1978) suggested that the first
major crack to form at about 90% of the mean concrete tensile strength and the last
major crack at about 110% of the mean concrete tensile strength. In the following
examples a ±10% random fluctuation of the mean concrete tensile strength ctf is
assigned to the concrete elements. The random concrete tensile strengths were
generated using the random number generator of the Compaq FORTRAN 90 compiler
used in this study. Different sets of random numbers may be generated using different
“seed” numbers. In the examples that follow the size of each finite element was taken
as three times the maximum aggregate size as recommended by Bažant and Oh (1983).
The parameters of the bond model adopted were as per the recommendation of CEB-
170
FIP Model Code 1990 (1993) for “good bond conditions” for “unconfined
concrete” (also see Table 3.1) and these were: mm6.021 == ss , mm0.13 =s ,
MPa0.10max =τ , MPa5.1=fτ and the unloading modulus was taken as
mmMPa100=uk .
4.4.4.1 Four-point Bending Beam Tests under Sustained Load
The typical finite element mesh used to model all the beam specimens is shown in
Figure 4.23a mm)35( =ch . The bottom concrete cover of the mesh is adjusted
accordingly for each beam specimen. In addition, a fine mesh mm)5.17( =ch and a
slanted mesh mm)35( =ch were used to model beam B1-a, as shown in Figures 4.23b
and 4.23c, respectively, in order to evaluate the mesh size and mesh directional
(a)
(b)
(c)
Fig. 4.23 - Finite element meshes for beam specimens: (a) coarse mesh for all beam
specimens; (b) fine mesh for beam B1-a; (c) slanted mesh for beam B1-a.
171
dependency of the crack band model when considering time effects and with
reinforcing steel. The coarse mesh has 665 nodes, 540 concrete elements, 54 steel
elements, 54 bond-slip elements and 4 stiff elastic support elements. The fine mesh
consists of 2405 nodes, 2168 concrete elements, 108 steel elements, 108 bond-slip
elements and 8 stiff elastic support elements. The slanted mesh is made up of 625
nodes, 510 concrete elements, 51 steel elements and 51 bond-slip elements and 2 stiff
support elements. The steel elements were connected to the concrete element via bond-
slip interface elements. The far left end node of the steel bar element was rigidly
connected to the concrete membrane node to simulate anchorage of the bar. In addition,
to examine the significance of the bond-slip interface element, beam B1-a was also
modelled using the coarse mesh but with the steel truss elements directly connected to
the nodal points of the concrete elements.
The calculated midspan deflection versus time curves are compared with the
experimental results in Figure 4.24. The coarse, fine and slanted meshes gave very
similar deformation with time for beam B1-a but with a very slight overestimation
compared to the experimental results. They are, however, considered to have a good
agreement with the experimental data. It is seen that the calculated time-dependent
midspan deflection for beam B1-a with a perfect steel-concrete bond assumption is not
dissimilar from those calculated using the bond-slip interface elements. Despite this, a
realistic description of steel-concrete bond is crucial and the importance of the bond-
slip interface elements will be seen subsequently when it comes to determining the
(a) (b)
Fig. 4.24 - Comparison of FEM and experimental time-dependent midspan
deflections: (a) beam B1-a; (b) beam B2-a.
Beam B2-a
0
2
4
6
8
10
12
14
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
Beam B1-a
0
2
4
6
8
10
12
14
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
ExperimentalFEM (coarse mesh)FEM (fine mesh)FEM (slanted mesh)FEM (perfect bond)
172
crack opening of the specimens. For beam B2-a the correlation of the calculated
midspan deflection and the experimental results is excellent. The displaced shape of
beam B1-a is shown in Figure 4.25a for which the crack openings are clearly visible at
a number of discrete locations. The bond-slip between steel and concrete is evidenced
by the dislocation of the overlapping nodes of the steel and concrete elements, as
depicted in Figure 4.25b.
(a) (b)
Fig. 4.25 - Cracking of numerical beam B1-a: (a) FEM deflection at 380 days (coarse
mesh: scale × 40); (b) dislocation of nodes due to bond slip.
Figures 4.26a to 4.26l show the crack formation with time for beam B1-a and
compare the numerical results for the coarse and fine meshes at instantaneous loading
and at 380 days. The numerically calculated slips increased with time due to the effect
of bond creep at the interface of reinforcing steel and concrete. At instantaneous
loading, the bond stresses away from the dominantly cracked region (approaching the
support) were nearly zero as no significant bond-slip has taken place due the absence of
cracks. However, bond stresses developed with time in this region (Figures 4.26g and
4.26h) as slippage between the concrete and the steel increased due to shrinkage of
concrete and the subsequent restraint imposed by the steel. This can be well illustrated
by the development of tensile stress in the concrete and increase of compressive stress
in the steel near the support as indicated in Figures 4.26i to 4.26l. Furthermore, the
simulation of fracture of reinforced concrete with the use of bond-slip interface
Crack locations
173
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k) (l)
Fig. 4.26 - Formation of crack for beam B1-a: (a) and (b) instantaneous crack pattern
(cracking strain plot) for coarse and fine meshes, respectively; (c) and (d)
crack pattern (cracking strain plot) for coarse and fine meshes at 380 days,
respectively; (e) and (f) longitudinal slip between steel and concrete; (g)
and (h) longitudinal bond stress; (i) and (j) longitudinal stress of concrete
adjacent to reinforcing steel; (k) and (l) longitudinal reinforcing steel stress.
-200-100
0
100200300
Stee
l str
ess
(MPa
)
380 days
Instantaneous-200-100
0100200300
Stee
l str
ess
(MPa
)
380 days
Instantaneous
0.0
1.0
2.0
3.0
Con
c. s
tres
s (M
Pa)
380 days
Instantaneous
0.0
1.0
2.0
3.0
Con
c. s
tres
s (M
Pa)
380 daysInstantaneous
-5.0-3.0-1.01.03.05.0
Bon
d st
ress
(MPa
)
Instantaneous
380 days-5.0-3.0-1.01.03.05.0
Bon
d st
ress
(MPa
)
Instantaneous
380 days
-0.25-0.15-0.050.050.150.25
Slip
(mm
) Instantaneous
380 days-0.25-0.15-0.050.050.150.25
Slip
(mm
) Instantaneous
380 days
174
elements accounts for the tension stiffening effect in a more realistic manner. This is
evidenced by the development of concrete stresses between the cracks at the constant
moment region as a result of bond action between the concrete and the steel (Figures
4.26i and 4.26j).
Figures 4.27a and 4.27b show the crack patterns for the slanted mesh at
instantaneous loading and at 380 days, respectively. The experimental crack pattern for
beam B1-a at 380 days is shown in Figure 4.27g. It is seen that the computed crack
patterns for the three meshes agree well with that observed in the test. For the slanted
mesh, the cracks propagated in the direction of the mesh orientation. The mesh
directional bias can be alleviated if a finer mesh is used, as demonstrated in Figure 4.3b
for the fracture tests presented in Section 4.2. The computed crack patterns for beam
B1-a assuming perfect bond between steel and concrete (without bond-slip elements)
are shown in Figures 4.27c and 4.27d for instantaneous loading and for 380 days after
loading, respectively. It is apparent that the crack pattern computed without using the
bond-slip interface elements is unobjective. Although localized cracking was
computed, the spacing of the cracks is evidently incorrect compared to the experimental
crack pattern. In addition, no distinct crack opening was obtained at the soffit since
cracking has smeared over the bottom concrete cover of the beam. Figure 4.28
compares the computed crack patterns to the experimental crack pattern for beam B2-a
and, again, a good agreement is obtained.
In fact there is a degree of confusion regarding the significance of the modelling
of bond between concrete and reinforcing steel. It is commonly conceived in finite
element modelling of reinforced concrete structures that, the influence of bond is not
important if the primary interest is to obtain a monotonic load-deflection response of a
reinforced concrete member (Darwin, 1993). Stevens et al. (1991) developed a finite
element code based on an extended version of the modified compression field theory of
Vecchio and Collins (1986) and modelled one of the simply supported reinforced
concrete beams of Bresler and Scordelis (1963). They concluded that the global load-
deflection behaviour of a reinforced concrete member was not sensitive to bond-slip
except for cases where bond failure was critical. Balakrishnan et al. (1988) pointed out
175
(a) (b)
(c) (d)
(e) (f)
(g)
Fig. 4.27 - Crack pattern of beam B1-a: (a) and (b) crack patterns (cracking strain plot)
for slanted mesh at instantaneous loading and at 380 days, respectively; (c)
and (d) crack patterns (cracking strain plot) for coarse mesh without bond-
slip interface elements (perfect bond) at instantaneous loading and at 380
days, respectively; (e) and (f) concrete stress and steel stress distribution
for perfect bond example, respectively; (g) experimental crack pattern at
380 days.
0.0
1.0
2.0
3.0
Con
c. s
tres
s (M
Pa)
380 daysInstantaneous
-200-100
0100200300
Stee
l str
ess
(MPa
)380 days
Instantaneous
176
(a) (b)
(c)
Fig. 4.28 - Crack pattern of beam B2-a: (a) and (b) crack patterns (cracking strain plot)
at instantaneous loading and at 380 days, respectively; (c) experimental
crack pattern at 380 days.
that the inclusion of bond-slip had a great effect on shear critical beams while had a
very little improvement on the load-deflection response on other specimens. Based on
the comments given above, one would naturally reach a conclusion to disregard bond
action in a finite element analysis. However, if crack widths and crack spacings are of
primary interest, the modelling of bond-slip is crucial.
The bond action between steel and concrete can generally be divided into two
major approaches as described in Section 2.6.7. Bond can be accounted for in an
average sense in terms of concrete tension stiffening stress that develops due to stress
transfer via bond or, it can be modelled in a discrete manner by using a specific type of
bond element in conjunction with discrete steel elements. Both the finite element
models developed by Stevens et al. (1991) and Balakrishnan et al. (1988), in fact,
incorporated a concrete tension stiffening model, which infers that the effect of steel-
concrete bond action was considered indirectly in their models. Therefore, it is
incorrect to say that bond action is unimportant in simulating the global behaviour of a
reinforced concrete structure if one is including a tension stiffening model that
significantly affects that behaviour.
177
In addition, it is redundant to model discretely the bond-slip between concrete and
reinforcing steel together with a tension stiffening model as done by Steven et al.
(1991) and Balakrishnan et al. (1988) since this attempts to model the effect of bond
twice. A realistic discrete bond model should be able to capture the tension stiffening
effect through stress transfer between concrete and reinforcing steel (see Figures 4.26i
and 4.26j) and no additional average tension stiffening stress should be superimposed
onto the finite element model. Furthermore, a finite element model employing a tension
stiffening model belongs to the distributed cracking models, for which cracking occurs
in a “smeared” manner throughout the region where tensile stress is higher than the
concrete tensile strength. In these models, the bond-slip element cannot work
effectively since the slip between concrete and steel is small due to the smeared nature
of cracking. Therefore, the discrete bond representation in modelling reinforced
concrete member must be used in conjunction with a concrete fracture model as the
discrete bond model can only be engaged effectively when the cracking of concrete
is localized.
The question remains, why the simulation for beam B1-a without using either the
bond-slip interface elements or a tension stiffening model gives a good correlation with
the experimental deflection as shown in Figure 4.24a? The reinforcing steel elements
were rigidly connected to the nodes of the concrete elements. With the use of a concrete
fracture model, the concrete elements adjacent to the steel elements were free to deform
longitudinally through cracking and shearing along the reinforcing steel. This “apparent
bond-slip” phenomenon assists the transfer of stress from the steel to the concrete and it
can be observed in Figures 4.27e and 4.27f, where the distribution of concrete tension
stiffening stress and the steel stress along the beam are shown. Although the “apparent
bond-slip” cannot correctly simulate the bond mechanisms, it does induce a degree of
concrete tension stiffening in the crack region which prevents the specimen from
becoming over-soft in its structural behaviour. This also explains the good correlation
between the calculated and the experimental deflections for the perfect bond example.
Comparisons are made in Figures 4.29a, 4.29b and 4.29c for the three meshes
between the calculated and measured crack widths at the soffit of beam B1a with
increasing time in the constant moment region. The crack opening with time for beam
178
(a) (b)
(c) (d)
Fig. 4.29 - Comparison of FEM and experimental time-dependent crack openings: (a),
(b) and (c) beam B1-a coarse mesh, fine mesh and slanted mesh,
respectively; (d) beam B2-a coarse mesh.
B2-a is shown in Figure 4.29d. Both maximum and average crack widths are presented
for comparisons between calculated and measured crack widths. It is seen that the
calculated results are not very sensitive to the mesh configurations. An overall good
correlation with the test results is achieved with a deviation less than ±15%.
The comparisons of the calculated and experimental crack envelopes for the
primary cracks at 200 days after loading are shown in Figure 4.30a and 4.30b for beam
B1-a and beam B2-a, respectively. Good agreement is observed for both specimens.
The model also obtained accurate heights of crack penetration up into the beams, which
approximately gives the position of the neutral axis.
0.000.050.100.150.200.250.300.350.400.45
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM (fine mesh)
Beam B1-a maximum
average
0.000.050.100.150.200.250.300.350.400.450.50
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM (coarse mesh)
maximum
average
Beam B1-a
0.000.050.100.150.200.250.300.350.400.45
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM (slanted mesh)
Beam B1-a maximum
average
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
Experimental
FEM (coarse mesh)
maximum
average
Beam B2-a
179
(a) (b)
Fig. 4.30 - Comparison of FEM (coarse mesh) and experimental crack width
envelopes across the depth of specimens at 200 days: (a) beam B1-a; (b)
beam B2-a.
In addition, from the crack patterns presented in Figures 4.26, 4.27 and 4.28, it is
evident that the height of the crack penetration (or position of the neutral axis) has
shifted to a lower position with increasing time. This agrees well with the fact that
neutral axis of a beam moves downwards and the compressive zone gradually becomes
larger due to the effect of creep of concrete with time (Gilbert, 1988).
4.4.4.2 Uniformly Loaded One-way Slabs under Sustained Load
The finite element mesh used to model the slab specimens is shown in Figure 4.31 with
one half of the slab modelled due to symmetry. The finite element mesh consists of 390
nodes, 270 concrete elements, 54 steel elements, 54 bond-slip interface elements and 4
stiff elastic support elements. The steel and concrete were bonded via interface element
with full anchorage provided for the steel node at the far left end of the specimen. The
calculated midspan deflection with time curves are shown in Figures 4.32a and 4.32b
for slab S2-a and slab S3-a, respectively, and both show an excellent correlation with
the experimental data. The calculated crack patterns and cracking strain plots are shown
in Figures 4.33a and 4.33b for slab S2-a and in Figures 4.34a and 4.34b for slab S3-a
both at first loading and at 380 days after loading. Good agreements were obtained
between the numerical results and the experimental crack patterns as presented in
Figures 4.33c and 4.34c.
Beam B2-a
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4Crack width (mm)
Hei
ght f
rom
bot
tom
(mm
) Experimental
FEM
Beam B1-a
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5Crack width (mm)
Hei
ght f
rom
bot
tom
(mm
) Experimental
FEM
180
Fig. 4.31 - Finite element mesh for slab specimens.
(a) (b)
Fig. 4.32 - Comparison of FEM and experimental time-dependent midspan
deflections: (a) slab S2-a; (b) slab S3-a.
(a) (b)
(c)
Fig. 4.33 - Crack pattern of slab S2-a: (a) and (b) crack patterns (cracking strain plot)
at instantaneous loading and at 380 days, respectively; (c) experimental
crack pattern at 380 days.
> 0.9 Mmax
> 0.9 Mmax
Slab S3-a
0
5
10
15
20
25
30
35
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
Slab S2-a
0
5
10
15
20
25
30
35
40
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
181
(a) (b)
(c)
Fig. 4.34 - Crack pattern of slab S3-a: (a) and (b) crack patterns (cracking strain plot)
at instantaneous loading and at 380 days, respectively; (c) experimental
crack pattern at 380 days.
In Figures 4.35a and 4.35b the crack widths calculated by the finite element
model for slab S2-a and slab S3-a are compared with the experimental measurement for
the cracks within the region of moment greater than 90% of the midspan moment. The
model calculated a slightly higher average crack width for both the slab specimens
compared to the observed test results. An excellent correlation is obtained for the
calculated and experimental maximum crack width for slab S2-a. For slab S3-a, the
numerical maximum crack width development correlates well with the test data,
however, the experimental crack width increased abruptly at about 280 days. Overall,
the agreement of the numerical and experimental crack opening with time
is reasonable.
(a) (b)
Fig. 4.35 - Comparison of FEM and experimental time-dependent crack openings: (a)
slab S2-a; (b) slab S3-a.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM
maximum
average
Slab S3-a
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM
maximum
average
Slab S2-a
> 0.9 Mmax
182
4.4.4.3 Discussion
Table 4.11 shows a summary of the comparison of the numerical and experimental
midspan deflections at various times for specimens B1-b, B2-b, B3-a, B3-b, S1-a, S1-b,
S2-b and S3-b. The numerical and experimental average crack spacings for the primary
cracks are summarized in Table 4.12. The the time-dependent crack widths for the
beam and slab specimens are given in Tables 4.13 and 4.14, respectively.
Table 4.11 - Midspan deflections at various times t (days) after loading (Localized
cracking model – Crack band model).
Midspan deflections (mm) Specimens Instantaneous t = 50 t = 200 t = 380
FEM 1.93 5.28 8.45 9.77 B1-b Exp. 1.98 4.88 6.69 7.44
FEM/Exp. 0.97 1.08 1.26 1.31 FEM 2.09 4.98 7.77 8.76 B2-b Exp. 2.06 5.11 7.06 7.88
FEM/Exp. 1.01 0.97 1.10 1.11 FEM 6.05 11.15 13.97 15.03 B3-a Exp. 5.81 10.08 12.35 13.32
FEM/Exp. 1.04 1.11 1.13 1.13 FEM 1.79 5.45 8.61 9.61 B3-b Exp. 1.97 5.33 7.13 7.90
FEM/Exp. 0.91 1.02 1.21 1.22 FEM 8.23 18.85 23.90 26.78 S1-a Exp. 7.14 18.59 22.87 25.12
FEM/Exp. 1.15 1.01 1.05 1.07 FEM 2.46 8.73 17.85 21.28 S1-b Exp. 2.72 12.62 17.79 19.91
FEM/Exp. 0.90 0.69 1.00 1.07 FEM 3.27 14.22 20.70 23.39 S2-b Exp. 4.43 14.34 19.83 21.93
FEM/Exp. 0.74 0.99 1.04 1.07 FEM 6.11 17.19 23.80 26.37 S3-b Exp. 5.04 15.22 20.65 22.90
FEM/Exp. 1.21 1.13 1.15 1.15
183
Table 4.12 - Comparison of FEM and experimental average crack spacings for
flexural specimens (Localized cracking model – Crack band model).
Specimens FEM (mm) Experimental (mm) FEM/Exp.
B1-b 163 220 0.74 B2-b 160 320 0.50 B3-a 147 160 0.92 B3-b 147 170 0.86 S1-a 112 130 0.86 S1-b 114 130 0.88 S2-b 119 110 1.08 S3-b 119 130 0.92
Table 4.13 - Crack widths for beam specimens at various times t (days) after loading
(Localized cracking model – Crack band model).
Crack widths (mm) Beam specimens t = 7 t = 50 t = 200 t = 380
FEMavg 0.027 0.077 0.148 0.172 Exp.avg 0.046 0.097 0.122 0.137
B1-b
FEM/Exp.avg 0.59 0.79 1.21 1.26 FEMmax 0.040 0.126 0.208 0.223 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.53 0.99 1.37 1.25
FEMavg 0.018 0.071 0.131 0.148 Exp.avg 0.042 0.110 0.127 0.152
B2-b
FEM/Exp.avg 0.43 0.65 1.03 0.97 FEMmax 0.031 0.119 0.179 0.191 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.41 0.94 1.18 1.07
FEMavg 0.135 0.172 0.202 0.209 Exp.avg 0.066 0.127 0.149 0.184
B3-a
FEM/Exp.avg 2.05 1.35 1.36 1.14 FEMmax 0.163 0.211 0.250 0.264 Exp.max 0.102 0.254 0.279 0.279 FEM/Exp.max 1.60 0.83 0.90 0.95
FEMavg 0.027 0.048 0.096 0.109 Exp.avg 0.030 0.082 0.102 0.112
B3-b
FEM/Exp.avg 0.90 0.59 0.94 0.97 FEMmax 0.032 0.057 0.116 0.130 Exp.max 0.051 0.127 0.127 0.127 FEM/Exp.max 0.63 0.45 0.91 1.02
184
Table 4.14 - Crack widths for slab specimens at various times t (days) after loading
(Localized cracking model – Crack band model).
Crack widths (mm) Slab specimens t = 7 t = 50 t = 200 t = 380
FEMavg 0.129 0.157 0.183 0.192 Exp.avg 0.066 0.130 0.155 0.168
S1-a
FEM/Exp.avg 1.95 1.21 1.18 1.14 FEMmax 0.178 0.220 0.249 0.254 Exp.max 0.127 0.203 0.254 0.254 FEM/Exp.max 1.40 1.08 0.98 1.00
FEMavg 0.011 0.035 0.133 0.154 Exp.avg 0.044 0.078 0.105 0.114
S1-b
FEM/Exp.avg 0.25 0.45 1.27 1.35 FEMmax 0.022 0.136 0.242 0.250 Exp.max 0.102 0.127 0.178 0.178 FEM/Exp.max 0.22 1.07 1.36 1.40
FEMavg 0.064 0.126 0.165 0.177 Exp.avg 0.058 0.092 0.117 0.130
S2-b
FEM/Exp.avg 1.10 1.37 1.41 1.36 FEMmax 0.030 0.163 0.222 0.234 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.39 1.28 1.46 1.31
FEMavg 0.067 0.100 0.129 0.140 Exp.avg 0.043 0.094 0.124 0.137
S3-b
FEM/Exp.avg 1.56 1.06 1.04 1.02 FEMmax 0.086 0.127 0.161 0.173 Exp.max 0.076 0.127 0.178 0.203 FEM/Exp.max 1.13 1.00 0.90 0.85
For the lightly loaded beam specimens, B1-b and B2-b, the distributed cracking
model computed a generally higher midspan deflection than that measured from the
experiment and the associated limitations were discussed in Section 4.4.3.3. In the
localized cracking model, attempts were made to eliminate these limitations. The
introduction of a random fluctuation of concrete tensile strength provides not only a
closer simulation for a real structure but also facilitates a stable numerical analysis that
prevents bifurcation stemming from the existence of multiple equilibrium paths.
The coupling of a concrete fracture model with the bond-slip interface element
enables a more realistic description of the tension stiffening effect of the cracked
concrete and it also allows for the progressive formation of cracks as load increases.
185
For beam B1-b and beam B2-b, the midspan deflections calculated by this model have
a closer agreement with the test results than those calculated by the distributed cracking
model. Moreover, the issue discussed in Section 4.4.3.3 for slab S1-b still applies for
the current model. Therefore, the concrete tensile strength at 14 days of age was taken
as MPa6.114. =ctf in the analysis in order to obtain a pre-cracked specimen upon
instantaneous loading as observed in the experiment. For slab S1-b, it is seen that the
model underestimated the midspan deflection of the specimen at 50 days after loading
(see Table 4.11), however the calculated results is again in good agreement from 200
days onwards. This is because the model computed less cracking before 50 days and
further cracking occurred with time due mainly to the effect of shrinkage of concrete.
In Table 4.12, it is seen that the computed average crack spacings have an overall
satisfactory correlation with the test results except for beam B2-b, where the model
computed a much lower average crack spacing than that observed in the experiment
mm)320( exp. =rms . In the author’s view, the experimental average crack spacing
reported for beam B2-b (lightly loaded at 30% of the ultimate load with a bottom
concrete cover of 25 mm) is not representative. Perhaps some fine cracks were
overlooked during the experiments. Comparing with the experimental results of beam
B1-b mm)220( exp. =rms which is also loaded at 30% of the ultimate load but with a
thicker bottom concrete cover of 40 mm, the average crack spacing of beam B2-b is
way too large. It is theoretically well established that a specimen with thicker concrete
cover generally has a larger crack spacing, which clearly contradicts the overly large
average crack spacing obtained experimentally for beam B2-b. It is seen in Tables 4.13
and 4.14 that the calculated crack widths are in reasonable agreement with the test
results especially for the final crack widths, except for slab S1-b and slab S2-b, the
model calculated a slightly larger crack width than the test results.
Figures 4.36a and 4.36b show the correlation diagrams for the average crack
widths and the maximum crack widths of all flexural specimens at 7 days, 50 days, 200
days and 380 days after loading. Similar to that for the distributed cracking model, the
0% (denoted by “r = 1”) and ±50% (denoted by “r = 1.5” and “r = 1.5-1”) deviation
lines are plotted on the same diagram for comparison purposes. It is seen that the great
majority of the data points are well within the ±50% deviation lines. Comparing to the
186
distributed cracking model, the localized cracking crack band model obtained a better
correlation with the test data as the data points are less scattered and also, are closer to
the 0% deviation line.
(a) (b)
Fig. 4.36 - Correlation diagrams for localized cracking model – crack band model: (a)
average crack widths; (b) maximum crack widths.
The localized cracking model with the crack band fracture model allows for a
more detailed investigation of time-dependent cracking in reinforced concrete
structures than the distributed cracking model, however, the numerical results of the
model are still sensitive to some parameters, in particular the mean concrete tensile
strength. Although a stochastic fluctuation of concrete tensile strength is used in the
model, the mean concrete tensile strength is still the primary governing factor to the
cracking load of a reinforced concrete structure. An accurate computation of the
cracking load in turn influences the time-dependent behaviour of the structure, as was
discussed in Section 4.4.3.3.
The bond stress-slip model is also a factor that has a great effect on the computed
crack widths. In this study, the bond model of the CEB-FIP Model Code 1990 (1993)
for “good bond conditions” for “unconfined concrete” (Table 3.1) was adopted in the
analyses for all specimens. If a different bond model or the “other bond conditions” of
the CEB-FIP bond model were used in the analyses, a different computation outcome
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5FEM max. crack width, wm.FEM (mm)
Exp.
max
. cra
ck w
idth
, wm
.Exp
(mm
)
r = wm.Exp / wm.FEM
r = 1.5r = 1
r = 1.5 -1
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4FEM avg. crack width, wa.FEM (mm)
Exp.
avg
. cra
ck w
idth
, wa.
Exp (
mm
)
r = wa.Exp / wa.FEM
r = 1.5r = 1
r = 1.5 -1
187
would be expected. The computed crack widths would be narrower if a stiffer bond
model was used or, they would be wider if a less stiff bond model were employed.
It has been shown in Section 4.2 regarding mesh sensitivity, the crack band model
suffers from stress locking if the orientation of the mesh is not aligned with the
direction of crack propagation. Surprisingly, the stress-locking issue is not significant
when modelling a reinforced concrete member. As seen in Figure 4.24, the midspan
deflection with time diagram for beam B1-a using the slanted mesh in fact is very
similar to those of the vertical meshes. In addition, it is also encouraging to obtain a
numerical time-dependent crack opening for the slanted mesh having a very good
correlation with the test data. It can, therefore, be concluded that stress-locking is
merely a plain fracture problem and it becomes insignificant as soon as cracking of
concrete is stabilized by the presence of reinforcement.
4.4.5 Analysis of Long-term Flexural Cracking Tests using the Localized
Cracking Model – Non-local Smeared Crack Model
It was seen in Section 4.2 that the non-local smeared crack model is more advantageous
than the crack band model in modelling fracture of concrete. The model completely
alleviates the mesh sensitivity problem usually associated with a standard smeared
crack model. In addition, the non-local smeared crack model also overcomes the stress-
locking phenomenon that persists in the crack band model. In this section, the non-local
smeared crack model is employed in conjunction with the bond-slip interface element
and steel truss element to simulate time-dependent cracking in reinforced concrete
structures. An evaluation is made to scrutinize the applicability of the non-local model
in reinforced concrete structures.
As in the localized cracking model using the crack band approach, the concrete
tensile strength assigned to each element making up a structure is randomized at a
±10% limit of the mean concrete tensile strength ctf . For the following study, the
characteristic length for the non-local smeared crack model is taken as mm50=chl
and a width of localization of h = 50 mm is used to calculate the tensile stress-strain
188
curve for the concrete. The CEB-FIP (1993) bond stress-slip relationship for “good
bond conditions” and “unconfined concrete” is used, giving mm6.021 == ss ,
mm0.13 =s , MPa0.10max =τ , MPa5.1=fτ and an unloading modulus
mmMPa100=uk .
4.4.5.1 Four-point Bending Beam Tests under Sustained Load
The effectiveness of spatial averaging of the non-local smeared crack model depends
primarily on the total number of integration points contained within the averaging
neighbourhood of the point of interest in a structure. This inevitably calls for the need
of a finer finite element mesh discretization. The finite element meshes used for the
four-point bending specimens with bottom cover 40 mm and 25 mm are shown
respectively in Figures 4.37a and 4.37c, for which the mesh in the constant moment
region is 10 mm wide. A coarse mesh, as shown in Figure 4.37b, was also generated for
beam B1-a to test the mesh size sensitivity. The mesh shown in Figure 4.37a consists of
2826 nodes, 2576 concrete elements, 112 steel truss elements, 112 bond-slip interface
elements and 2 stiff elastic support elements. The coarse mesh shown in Figure 4.37b
consists of 2176 nodes, 1978 concrete elements, 86 steel truss elements, 86 bond-slip
interface elements and 2 stiff elastic support elements. The mesh, for specimens with
25 mm concrete cover (Figure 4.37c), is made up of 2713 nodes, 2464 concrete
elements, 112 steel truss elements, 112 bond-slip elements and 2 stiff elastic elements.
The steel truss elements were overlaid onto the concrete elements via bond-slip
interface elements. A rigid fixity was provided at the node of steel-concrete overlay at
the free end of the specimens near the support to simulate full anchorage.
The experimental and numerical midspan deflections with time after loading are
compared in Figures 4.38a and 4.38b for beams B1-a and B2-a, respectively. The finite
element model is in reasonable agreement with the test data but slightly overestimates
the deflections. In addition, the model is relatively insensitive to mesh size, which can
be seen in Figure 4.38a where the deflection curves for the coarse and fine meshes are
very similar.
189
(a)
(b)
(c)
Fig. 4.37 - Finite element meshes for beam specimens: (a) fine mesh for beam
specimens with 40 mm bottom cover; (b) coarse mesh for beam B1-a; (c)
fine mesh for beam specimens with 25 mm bottom cover.
(a) (b)
Fig. 4.38 - Comparison of FEM and experimental time-dependent midspan
deflections: (a) beam B1-a; (b) beam B2-a.
Beam B1-a
0
2
4
6
8
10
12
14
16
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
ExperimentalFEM (fine mesh)FEM (coarse mesh)
Beam B2-a
0
2
4
6
8
10
12
14
16
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
190
Instead of presenting the crack pattern in a cracking strain plot as for the crack
band model, the crack pattern computed by the non-local smeared crack model is
shown as principal strain contours. Figures 4.39a to 4.39l show the crack patterns, the
longitudinal slips and the stresses for beam B1-a. The results computed for the coarse
and fine meshes are also compared. The model obtained a similar crack pattern for the
coarse and fine meshes and the computed crack patterns compare favourably with the
experimental crack pattern as shown previously in Figure 4.27g. The finite element
computed crack patterns for beam B2-a are presented in Figure 4.40 and are similar to
that observed in the test (Figure 4.28c). It is observed that the non-local smeared crack
model is insensitive to mesh size and the widths of the fracture zones are consistent,
irrespective of the size of the mesh. This is a prominent characteristic of the
non-local model.
However, due to the effect of spatial averaging the model is particularly sensitive
to regions of high tensile stress. This can be seen in Figure 4.39a to 4.39d where
cracking of concrete occurred not only in the transverse direction but also in the
longitudinal direction at the level of the reinforcing steel as a result of the splitting
tension induced by the bond action between steel and concrete. The sensitivity of
spatial averaging in the model also results in a discontinuity in crack propagation from
the soffit across the reinforcement layer causing fracture of concrete not being able to
be localized nicely into a single crack.
Unlike the crack band model, fracture in the non-local smeared crack model
localizes into a number of element widths and this leads to a small slip between the
steel and concrete. Comparing the magnitude of slip in Figures 4.39e and 4.39f to that
in Figures 4.26e 4.26f, the non-local smeared crack model predicted only about 10% of
that computed by the crack band model. The small slip gives rise to a low bond stress
and this influences the distribution of the concrete tension stiffening stress and the steel
stress. In addition, due to the influence of the dispersed fracture at the soffit, the
distribution of concrete tensile stiffening stresses are also in a rather irregular pattern.
191
(a) (b)
(c) (d)
(e) (f)
(g) (h)
(i) (j)
(k) (l)
Fig. 4.39 - Formation of crack for beam B1-a: (a) and (b) instantaneous crack pattern
(principal strain contours) for coarse and fine meshes, respectively; (c) and
(d) crack pattern (principal strain contours) for coarse and fine meshes at
380 days, respectively; (e) and (f) longitudinal slip between steel and
concrete; (g) and (h) longitudinal bond stress; (i) and (j) longitudinal stress
of concrete adjacent to reinforcing steel; (k) and (l) longitudinal reinforcing
steel stress.
-200-100
0100200300
Stee
l str
ess
(MPa
)
380 days
Instantaneous-200-100
0100200300
Stee
l str
ess
(MPa
)
380 days
Instantaneous
0.0
1.0
2.0
3.0
4.0
Con
c. s
tres
s (M
Pa)
380 daysInstantaneous
0.0
1.0
2.0
3.0
4.0
Con
c. s
tres
s (M
Pa)
380 days
Instantaneous
-3.0-2.0-1.00.01.02.03.0
Bon
d st
ress
(MPa
)
Instantaneous
380 days-3.0-2.0-1.00.01.02.03.0
Bon
d st
ress
(MPa
)
Instantaneous380 days
-0.03-0.02-0.01
00.010.020.03
Slip
(mm
) 380 days
Instantaneous-0.03-0.02-0.01
00.010.020.03
Slip
(mm
)
Instantaneous
380 days
192
(a) (b)
Fig. 4.40 - Crack pattern of beam B2-a: (a) and (b) crack patterns (principal strain
contours) at instantaneous loading and at 380 days, respectively.
The variation of the numerical crack width with time within the constant moment
region for beam B1-a are compared to the experimental observations in Figures 4.41a
and 4.41b for the coarse and fine meshes, respectively. Figure 4.41c shows the
(a) (b)
(c)
Fig. 4.41 - Comparison of FEM and experimental time-dependent crack openings: (a)
beam B1-a coarse mesh; (b) beam B1-a fine mesh; (c) beam B2-a fine
mesh.
0.000.050.100.150.200.250.300.350.400.450.50
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM (fine mesh)
maximum
average
Beam B1-a
0.000.050.100.150.200.250.300.350.400.450.50
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
Experimental
FEM (coarse mesh)
maximum
average
Beam B1-a
0.000.050.100.150.200.250.300.350.400.45
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM
maximum
average
Beam B2-a
193
comparison of the numerical and experimental time-dependent crack widths for beam
B2-a within the constant moment region. It is seen that the model calculated a generally
higher instantaneous crack width than those observed in the tests. On the whole, the
model results have an overall good agreement with the test data. The crack width
envelopes across the depth of beams calculated by the model for the constant moment
region at 200 days are compared with the test data in Figures 4.42a and 4.42b for beams
B1-a and B2-a, respectively, and show a satisfactory correlation with the test data.
(a) (b)
Fig. 4.42 - Comparison of FEM (fine mesh) and experimental crack width envelopes
across the depth of specimens at 200 days: (a) beam B1-a; (b) beam B2-a.
4.4.5.2 Uniformly Loaded One-way Slabs under Sustained Load
For the uniformly loaded slab specimens, cracking occurred in a distributed manner
throughout the span of the specimens. To capture properly cracking of the specimens,
fine meshes were used throughout the span of the slabs, as shown in Figure 4.43. A
total of 2091 nodes were used to define the mesh. The mesh consists of 1730 concrete
elements, 173 steel truss elements, 173 bond-slip interface elements and 4 stiff elastic
support elements. As for all the localized cracking examples, steel truss elements were
overlaid onto concrete elements via bond-slip interface elements and a full anchorage
was provided to the steel element node at the free end of the slab.
Beam B1-a
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5Crack width (mm)
Hei
ght f
rom
bot
tom
(mm
) Experimental
FEM
Beam B2-a
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4Crack width (mm)
Hei
ght f
rom
bot
tom
(mm
) Experimental
FEM
194
Fig. 4.43 - Finite element mesh for slab specimens.
The midspan deflection versus time diagrams for slabs S2-a and S3-a are
presented in Figures 4.44a and 4.44b, respectively. In comparison to the test data, the
model obtained results that are in a very close agreement with the experimental results
over the duration of the test. The crack patterns are shown in Figures 4.45a and 4.45b
for slab S2-a and in Figures 4.45c and 4.45d for slab S3-a at instantaneous loading and
at 380 days after first loading. Although strain localization still tends to occur at
locations of high bond stresses, it is seen that the strain localizations for the slab
specimens have better defined individual cracks than those in the beam specimens (see
Figures 4.39a to 4.39d and Figures 4.40a and 4.40b) as the propagation of cracks is
continuous from the soffit through the reinforcement layer. The crack patterns
computed by the model for the two slabs are similar to those observed in the test as
shown in Figures 4.33c and 4.34c for slab S2-a and slab S3-a, respectively. The average
(a) (b)
Fig. 4.44 - Comparison of FEM and experimental time-dependent midspan
deflections: (a) slab S2-a; (b) slab S3-a.
Slab S3-a
0
5
10
15
20
25
30
35
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
Slab S2-a
0
5
10
15
20
25
30
35
40
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
> 0.9 Mmax
195
(a) (b)
(c) (d)
Fig. 4.45 - Crack patterns of slab specimens (principal strain contours): (a) and (b)
slab S2-a at instantaneous loading and at 380 days, respectively; (c) and (d)
slab S3-a at instantaneous loading and at 380 days, respectively.
crack spacings were calculated for slab S2-a and slab S3-a within the region greater
than 90% of the maximum moment and are 110 mm and 92 mm, respectively, which
correlate quite well with the measured average crack spacings given in Table 4.8 (120
mm for slab S2-a and 110 mm for slab S3-a).
The calculated crack widths with time for slab S2-a and slab S3-a are compared
with the test data in Figures 4.46a and 4.46b, respectively. The model results have a
good correlation with the experimental results for the widest crack of both slabs and for
the average crack width of slab S3-a. However, the average crack width calculated by
the model for slab S2-a is higher than the experimental results. Similar to the results for
the beam specimens, the model consistently calculated a high instantaneous crack width
for the two slab specimens.
(a) (b)
Fig. 4.46 - Comparison of FEM and experimental time-dependent crack openings: (a)
slab S2-a; (b) slab S3-a.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM
maximum
average
Slab S3-a
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
ExperimentalFEM
maximum
average
Slab S2-a
196
4.4.5.3 Discussion
The comparisons of the computed and the test results for the other specimens are
summarized in Table 4.15 for time-dependent midspan deflection, Table 4.16 for
average crack spacing and Tables 4.17 and 4.18 for crack widths with time for beam
specimens and slab specimens, respectively.
Table 4.15 - Midspan deflections at various times t (days) after loading (Localized
cracking model – Non-local smeared crack model).
Midspan deflections (mm) Specimens Instantaneous t = 50 t = 200 t = 380
FEM 1.96 4.48 7.05 8.76 B1-b Exp. 1.98 4.88 6.69 7.44
FEM/Exp. 0.99 0.92 1.05 1.18 FEM 2.31 5.18 7.34 8.60 B2-b Exp. 2.06 5.11 7.06 7.88
FEM/Exp. 1.12 1.01 1.04 1.09 FEM 6.24 11.32 14.41 15.88 B3-a Exp. 5.81 10.08 12.35 13.32
FEM/Exp. 1.07 1.12 1.17 1.19 FEM 2.22 6.39 8.76 9.86 B3-b Exp. 1.97 5.33 7.13 7.90
FEM/Exp. 1.13 1.20 1.23 1.25 FEM 10.32 19.40 24.56 26.93 S1-a Exp. 7.14 18.59 22.87 25.12
FEM/Exp. 1.45 1.04 1.07 1.07 FEM 3.49 12.30 16.72 18.72 S1-b Exp. 2.72 12.62 17.79 19.91
FEM/Exp. 1.28 0.97 0.94 0.94 FEM 4.87 14.08 19.27 21.62 S2-b Exp. 4.43 14.34 19.83 21.93
FEM/Exp. 1.10 0.98 0.97 0.99 FEM 4.90 15.12 21.23 23.94 S3-b Exp. 5.04 15.22 20.65 22.90
FEM/Exp. 0.97 0.99 1.03 1.05
197
Table 4.16 - Comparison of FEM and experimental average crack spacings for
flexural specimens (Localized cracking model – Non-local smeared crack
model).
Specimens FEM (mm) Experimental (mm) FEM/Exp.
B1-b 170 220 0.77 B2-b 156 320 0.49 B3-a 153 160 0.96 B3-b 124 170 0.73 S1-a 108 130 0.83 S1-b 106 130 0.82 S2-b 106 110 0.96 S3-b 105 130 0.81
Table 4.17 - Crack widths for beam specimens at various times t (days) after loading
(Localized cracking model – Non-local smeared crack model).
Crack widths (mm) Beam specimens t = 7 t = 50 t = 200 t = 380
FEMavg 0.023 0.043 0.100 0.158 Exp.avg 0.046 0.097 0.122 0.137
B1-b
FEM/Exp.avg 0.50 0.44 0.82 1.15 FEMmax 0.031 0.054 0.132 0.221 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.41 0.43 0.87 1.24
FEMavg 0.043 0.053 0.067 0.094 Exp.avg 0.042 0.110 0.127 0.152
B2-b
FEM/Exp.avg 1.02 0.48 0.53 0.62 FEMmax 0.052 0.072 0.096 0.151 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.68 0.57 0.63 0.85
FEMavg 0.132 0.141 0.155 0.162 Exp.avg 0.066 0.127 0.149 0.184
B3-a
FEM/Exp.avg 2.00 1.11 1.04 0.88 FEMmax 0.236 0.261 0.289 0.304 Exp.max 0.102 0.254 0.279 0.279 FEM/Exp.max 2.31 1.03 1.04 1.09
FEMavg 0.044 0.049 0.057 0.067 Exp.avg 0.030 0.082 0.102 0.112
B3-b
FEM/Exp.avg 1.47 0.60 0.56 0.60 FEMmax 0.069 0.080 0.094 0.109 Exp.max 0.051 0.127 0.127 0.127 FEM/Exp.max 1.35 0.63 0.74 0.86
198
Table 4.18 - Crack widths for slab specimens at various times t (days) after loading
(Localized cracking model – Non-local smeared crack model).
Crack widths (mm) Slab specimens t = 7 t = 50 t = 200 t = 380
FEMavg 0.125 0.181 0.207 0.221 Exp.avg 0.066 0.130 0.155 0.168
S1-a
FEM/Exp.avg 1.89 1.39 1.34 1.32 FEMmax 0.199 0.264 0.296 0.313 Exp.max 0.127 0.203 0.254 0.254 FEM/Exp.max 1.57 1.30 1.17 1.23
FEMavg 0.020 0.059 0.073 0.084 Exp.avg 0.044 0.078 0.105 0.114
S1-b
FEM/Exp.avg 0.45 0.76 0.70 0.74 FEMmax 0.028 0.098 0.125 0.143 Exp.max 0.102 0.127 0.178 0.178 FEM/Exp.max 0.27 0.77 0.70 0.80
FEMavg 0.034 0.072 0.089 0.104 Exp.avg 0.058 0.092 0.117 0.130
S2-b
FEM/Exp.avg 0.59 0.78 0.76 0.80 FEMmax 0.066 0.137 0.171 0.196 Exp.max 0.076 0.127 0.152 0.178 FEM/Exp.max 0.87 1.08 1.13 1.10
FEMavg 0.029 0.067 0.087 0.101 Exp.avg 0.043 0.094 0.124 0.137
S3-b
FEM/Exp.avg 0.67 0.71 0.70 0.74 FEMmax 0.054 0.119 0.155 0.181 Exp.max 0.076 0.127 0.178 0.203 FEM/Exp.max 0.71 0.94 0.87 0.89
In Table 4.15, it is seen that the midspan deflections calculated by the non-local
smeared crack model for the specimens have an overall good correlation with the
measured results. In comparison with the distributed cracking model and the localized
cracking model with the crack band approach, the non-local smeared crack model
obtained results in better agreement with the experimental midspan deflections. This is
especially noticeable for beam specimens subjected to 30% of the ultimate load (beams
designated “b”). The distributed cracking model and the localized cracking crack band
model obtained a final deflection higher than that observed by up to 40% and 30%,
respectively, while the localized cracking non-local smeared crack model predicted
deflections higher than the test data by only up to 25%. In addition, similar to the
199
previous two finite element models, the concrete tensile strength for slab S1-b at age of
loading (age 14 days) was taken as MPa6.114. =ctf in order to obtain a cracked the
specimen at instantaneous loading as observed in the experiment. The midspan
deflection calculated using this concrete tensile strength is in reasonable agreement
with the experimental results.
Table 4.16 summarizes the calculated and experimental average crack spacing
and the comparison shows a reasonable agreement. All calculated crack spacings are
within 30% of the test results except for beam B2-b, where the calculated result is only
half of that observed in the test. A discussion has been given in Section 4.4.4.3
regarding the experimental average crack spacing for beam B2-b. For the same
specimen, the non-local smeared crack model computed an average crack spacing that
is in close agreement with that obtained by the crack band model mm)160( =rms ,
which indicates that, although a different fracture model is used the modelling approach
would give a similar result.
In the comparisons of the time-dependent crack widths in Tables 4.17 and 4.18, it
is seen that the final crack widths calculated by the model have a reasonable correlation
with the experimental results. Nevertheless, the correlation for the calculated and
observed development of crack widths over the test period is merely acceptable. As
presented before for specimens B1-a, B2-a, S2-a and S3-a, the crack width calculated
by the model at first loading is significantly larger than that observed. For the
specimens subjected to 50% of the ultimate load (beam B3-a and slab S1-a) as shown
in Tables 4.17 and 4.18, the same trend of crack opening with time is observed in the
computed results. The instantaneous crack opening is larger than the test data, but is
“corrected” as time approaches 380 days. For the lightly loaded specimens (designated
“b”) except beam B3-b, a different trend in numerical crack width is observed. The
model calculated a gradual increase in crack opening with time, but with an initial
underestimation and a closer agreement for the final crack widths. It is noticed that the
crack patterns computed by the model at low load levels are less well defined. This can
be seen, for example, in the principal strain contours diagrams of beam B1-b as shown
in Figure 4.47. Numerical analysis shows that the beam was cracked at instantaneous
200
(a) (b)
Fig. 4.47 - Crack pattern of beam B1-b: (a) and (b) crack patterns (principal strain
contours) at instantaneous loading and at 380 days, respectively.
loading, however, no obvious crack pattern is observed in Figure 4.47a. A stabilized
crack pattern was only established with time. The crack pattern at 380 days is shown in
Figure 4.47b.
In the author’s view, a number of factors are contributing to the discrepancy of
the agreement between the calculated and experimental time-dependent crack openings.
The nature of the modelling of fracture zones in the non-local smeared crack model
leads to a few undesirable issues in the application to reinforced concrete structures.
The non-uniform distribution of cracking strains across the width of the fracture zone
causes difficulties in the determination of the crack width. The method used in this
study to calculate the crack width is that presented in Section 3.6.6.3, which is simply
to compute the difference in displacements of the two nodes adjacent to the fracture
zone. However, this method only gives an approximate crack width in the fracture
zone. In addition, the issue of the discontinuous fracture across the reinforcement layer
from the soffit, which has been mentioned earlier, further handicaps the determination
of crack width. Therefore, the computation of crack width depends not only on the
magnitude of time-dependent deformation of concrete but also on the selection of the
nodal points adjacent to the fracture zone (or crack). This undoubtedly leads to an
additional variable in the determination of crack width. Another influencing factor is
the effectiveness of the combined usage of the bond-slip interface element together
with the non-local smeared crack model. It has been mentioned earlier that the bond
slip at cracks is particularly low due to the distribution of fracture over a number of
elements, as illustrated in Figure 4.39. It is questionable whether the behaviour of bond-
slip is realistically described in such a condition, which subsequently has a great
influence on the distribution of steel and concrete stresses in the tension region.
201
The correlation diagrams for the localized cracking non-local smeared crack
model are shown in Figures 4.48a and 4.48b for the average crack widths and the
maximum crack widths at 7 days, 50 days, 200 days and 380 days after loading. It is
seen that the model obtained a better correlation with the test data for data points of
larger crack widths, which are made up mostly by data points with a longer period
under load. On the whole, the model results are in a reasonable agreement with the
experimental results.
(a) (b)
Fig. 4.48 - Correlation diagrams for localized cracking model – non-local smeared
crack model: (a) average crack widths; (b) maximum crack widths.
4.4.6 Summary for Analysis of Long-term Flexural Cracking Tests
Comparing the correlation diagrams of the three finite element models (Figures 4.22,
4.36 and 4.48), it is obvious that the localized cracking crack band model has the best
correlation with the test data. The data points obtained from this model, for both
average and maximum crack widths, are consistently close to the line of 0% deviation.
For the localized cracking non-local smeared crack model, the results correlate well
with the test data for data points of large crack widths, while for data points of small
crack widths, the model calculated lower crack widths than the test data. The crack
widths calculated by the distributed cracking cracked membrane model have a uniform
dispersion between the ±50% deviation lines and have a constant tendency of
overestimating the crack widths.
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4FEM avg. crack width, wa.FEM (mm)
Exp.
avg
. cra
ck w
idth
, wa.
Exp (
mm
)
r = wa.Exp / wa.FEM
r = 1.5r = 1
r = 1.5 -1
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5FEM max. crack width, wm.FEM (mm)
Exp.
max
. cra
ck w
idth
, wm
.Exp
(mm
)
r = wm.Exp / wm.FEM
r = 1.5r = 1
r = 1.5 -1
202
In fact, all three models have their respective merits and drawbacks. The
distributed cracking model is the simplest model and requires the least computational
resources as the model simulates reinforced concrete structures in an average manner.
Therefore a coarse finite element mesh is usually sufficient to model most types of
structural problems. Another advantage of the model is the smeared description of
reinforcement, which greatly simplifies the mesh generation process. However, the
shortcoming of the model is that, it only provides average results for a reinforced
concrete structure, such as average stress and average strain, where the determination of
a localized mechanism of the structure is not possible. Moreover, the reliability of the
computed crack openings of a structure depends strongly on the accuracy of the
calculated crack spacing obtained using the tension chord model.
The original motives of a non-local model are to provide a realistic physical
description of the fracture process and to regularize mesh sensitivity of tension
softening materials such as concrete. Despite having a sound mathematical formulation
that can effectively prevent both mesh size and mesh directional sensitivity, the
computational cost is inevitably expensive due to the spatial averaging involved in the
computational algorithm. In addition, a fine finite element mesh with mesh size smaller
than the fracture width is required for an accurate non-local modelling of a structure.
Because of the effect of spatial averaging, the width of a fracture zone generally
stretches over several elements. This reduces the practical applicability of the non-local
model to engineering problems since concrete structures normally contain a large
amount of reinforcement and cracking are densely distributed due to the stabilizing
effect of the reinforcement. If the fracture zone is wider than the crack spacing of a
structure, the computed neighbourhood fracture zones may coalesce and lead to
erroneous numerical results. In addition, the presence of a finite fracture width also
reduces the effectiveness of the bond-slip interface element, as has been shown
previously in the numerical examples. This further complicates the process of
determining the crack openings of a reinforced concrete structure.
In contrast, the crack band model offers several advantages over the non-local
smeared crack model. The crack band model requires only an energy-based adjusted
tensile stress-strain curve and the rest of the computational algorithm is exactly
identical to that of a standard finite element model. To properly capture the localized
203
cracking in a reinforced concrete structure, the crack band model inevitably requires a
finer finite element discretization than the distributed cracking model. Although some
researchers (for example, de Borst, 1997) have pointed out that severe convergence
instability may occur if mesh size for the crack band model is refined, the author has
not encountered any such problems during the numerical studies undertaken in this
study. One very encouraging finding of this study relating to the crack band model is
that, the stress locking issue prevailing in concrete fracture problems appears to be
insignificant in the modelling of reinforced concrete structures. Furthermore,
comparing to the non-local smeared crack model, the crack band model works more
effectively with the bond-slip interface elements in describing the bond-slip
phenomenon, which in turn facilitates a more realistic stress transfer between concrete
and reinforcing steel.
4.5 Long-term Restrained Deformation Cracking Tests
4.5.1 Introduction
This experimental program is the second series of long-term tests performed by Nejadi
and Gilbert (2004). Eight fully restrained reinforced concrete slab specimens were
fabricated and monitored up to a period of 130 days and the growth of cracking caused
by the development of tensile stresses in the restrained specimens due to the effects of
drying shrinkage was recorded. Each specimen was 600 mm wide, 100 mm deep and
2000 mm long and each end of the specimen was rigidly restrained by a 1000 mm by
1000 mm by 600 mm thick concrete block clamping to the reaction floor with high
strength alloy steel rods so as to prevent longitudinal movement. The setup of the
experiment is shown in Figure 4.49. The details of the cross-sections of each specimen
are given in Table 4.19. A letter “R” is added to the original designation given by
Nejadi and Gilbert (2004) for each specimen to differentiate the restrained slab
specimens from the flexural specimens. The specimens were cured in the formwork and
204
Table 4.19 - Details of restrained slab specimens (Nejadi and Gilbert, 2004).
Specimen No. of bars Bar diameter (mm)
cs (mm)
s (mm)
RS1-a 3 12 109 185 RS1-b 3 12 109 185 RS2-a 3 10 110 185 RS2-b 3 10 110 185 RS3-a 2 10 145 300 RS3-b 2 10 145 300 RS4-a 4 10 115 120 RS4-b 4 10 115 120
Fig 4.49 - Restrained deformation cracking tests: (a) typical cross-section through
mid-depth of the specimen; (b) side view; (c) typical cross-section (section
A-A). (Nejadi and Gilbert, 2004).
A
A
1000
1000
2000
2660 1000(a)
600
(b)
Metal sheet to induce first crack
75
600
75
cs
205
kept moist by a covering of wet hessian for 3 days in order to prevent loss of moisture
from the specimens. Drying shrinkage commenced after the removal of the wet hessian.
Companion specimens were cast from the same batch of concrete for measuring
the creep and shrinkage properties of the concrete. The creep coefficients were obtained
from a creep test by loading a 5 MPa sustained stress longitudinally to concrete
cylinders mounted in a standard creep rig at age 3 days. The development of shrinkage
strain was measured from plain concrete specimens with the same cross-section as the
restrained slab specimens and subjected to the same environmental, curing and
drying conditions.
4.5.2 Analysis of Restrained Deformation Cracking Tests and Material
Properties
The crack band model with bond-slip interface element was used to simulate these
tests. For modelling purposes, it was assumed that the end concrete blocks supporting
the slab was rigidly clamped to the floor and the junction connecting the slab to the end
concrete block was fully restrained therefore no movement was possible. A typical
finite element mesh for the tests is shown in Figure 4.50 (specimen RS1-a or b shown).
The mesh consists of 1366 nodes, 1040 concrete elements, 247 steel truss elements and
247 bond-slip interface elements. The steel truss elements were connected to the
concrete elements via bond-slip interface elements. The numbers of node and element
differ slightly for each specimen since they contained different amount of
reinforcement. Instead of exploiting the double symmetric nature for the simulation of
only a quarter of the slab, the entire slab was modelled with a ±10 percent stochastic
fluctuation of the mean concrete tensile strength in order to capture the random
formation of cracks in the reinforced concrete member. The boundary conditions were
introduced such that the slabs were only free to shrink laterally. In addition, the
symmetry line in the longitudinal direction of the slabs was restrained from deforming
laterally by providing longitudinal roller supports to the concrete element nodes on the
symmetry line. The introduction of this boundary condition is to prevent excessive
206
Fig. 4.50 - Typical finite element mesh for restrained slab specimens (slab
RS1-a/b shown).
in-plane rotation of the slab due to non-uniform shrinkage across the cross section
caused by random cracking results from the stochastic concrete tensile strength.
Moreover, this also ensures a better numerical stability for the analyses.
All specimens, except slab RS2-b, were cast from the same batch of concrete.
Since the properties of the two batches of concrete were similar, only one set of the
material parameters was used in the numerical study. The material properties for
the concrete were: MPa3.24=cmf , GPa5.366.1 28.0 == cEE , MPa97.128. =ctf ,
2.0=ν , µε603=shA and days44=shB . The growth of the mean concrete tensile
strength for aging concrete was approximated with MPa1.2=ctfA and
days6.2=ctfB (±10%), as presented in Figure 4.51a. The development shrinkage
strain with time is shown in Figure 4.51b. Similar to that in the flexural cracking tests,
the concrete fracture energy fG was taken as mN75 . The concrete tensile softening
stress-strain curve was scaled with a crack band width of mm36=ch .
The empirical material constants for the solidification creep model were obtained
by fitting the compliance data obtained from the companion creep test.
The solidification creep parameters were: MPa6.2062 µε=q , MPa01.03 µε=q ,
MPa1.174 µε=q and 10 MPa8.58 −=A . The approximated compliance curves are
shown in Figure 4.52. Eight Kelvin chain units were adopted and the elastic moduli µE
Softened element to induce first crack
207
(a) (b)
Fig. 4.51 - Test data of companion specimens compared with models (Nejadi and
Gilbert, 2004): (a) growth of concrete tensile strength; (b) shrinkage strain
since commencement of drying.
and retardation times µτ are those as given in Table 4.20. An elastic-perfectly plastic
stress-strain relationship was assumed for the reinforcing steel with a yield strength of
500 MPa and an elastic modulus of 200 GPa. The parameters defining the bond
model were: mm6.021 == ss , mm0.13 =s , MPa9.9max =τ , MPa5.1=fτ and
mmMPa100=uk .
Fig. 4.52 - Compliance curve for restrained shrinkage tests (Nejadi and Gilbert, 2004).
0.0
0.5
1.0
1.5
2.0
2.5
0 10 20 30Age (days)
Conc
. ten
sile
str
engt
h f ct
(MPa
)
Experimental data
Model
Afct = 2.1 MPaBfct = 2.6 days
0
200
400
600
0 50 100 150Time since commencement of drying (days)
Shrin
kage
str
ain
( µε)
Experimental data
Model
Ash = 603 µεBsh = 44 days
0
50
100
150
200
250
0.1 1 10 100 1000Time under load, t-t' (days)
J(t,t
') (1
0-6 /
MPa
)
Experimental data (t'=3 days)
Model (t'=3 days)
208
Table 4.20 - Kelvin chain properties for solidification creep model.
µ-th unit Eµ (MPa) τµ (days) 1 0.07614 0.0001 2 0.06477 0.001 3 0.05575 0.01 4 0.04859 0.1 5 0.04290 1 6 0.03839 10 7 0.03481 100 8 0.03197 1000
4.5.3 Comparisons of Numerical and Experimental Results
The restrained deformation tests are fundamentally different from the flexural cracking
tests. No external load was applied to the specimens except the tensile reactions
induced at the supports and therefore no explicit displacement could be measured to
characterize the response of the structure. The deformation was solely due to the
development of drying shrinkage of concrete with time, which in turn induced the
tensile stresses in concrete. Cracking occurs as soon as the induced tensile stresses
exceed the concrete tensile strength. In the following presentation of results, a detailed
comparison between the numerical and the experimental results is made only for slabs
RS1-a and RS1-b, the results for all other specimens are presented only for the final age
122 days. The formation of cracks computed by the model and the crack widths at age
122 days for slabs RS1-a and RS1-b are shown in Figures 4.53a to 4.53d. The
development of concrete tensile stress in the specimen between age 6 days and 50 days
are shown in Figures 4.54 and 4.55. The age of the formation of the first crack
calculated by the finite element model has a good correlation with the experimental
observation (at age 7 days).
The shrinkage-induced tension in the slab was relieved immediately after the
formation of the first crack. The slab remained in tension after cracking and the
reinforcing steel carried the entire induced tensile force at the crack. After the first
209
(a)
(b)
(c)
(d)
Fig. 4.53 - FEM crack formation for slabs RS1-a and RS1-b (scale × 40): (a) at age 6
days; (b) at age 40 days; (c) at age 50 days; (d) at age 122 days.
Age = 122 days
Age = 6 days
Age = 40 days
Age = 50 days
0.34 mm 0.17 mm 0.14 mm 0.19 mm
210
Fig. 4.54 - FEM shrinkage induced concrete tensile stresses in slabs RS1-a and RS1-b:
(a) at age 6 days; (b) at age 20 days; (c) at age 30 days.
(a)
(b)
(c)
Age = 6 days
Age = 20 days
Age = 30 days
x
y σc1
Concrete stress (kPa)
211
Fig. 4.55 - FEM shrinkage induced concrete tensile stresses in slabs RS1-a and RS1-b:
(a) at age 40 days; (b) at age 50 days.
cracking, the concrete slab split into two intact pieces and the reinforcing steel at the
crack served as a bridge to transfer the shrinkage-induced tensile force to the concrete
slabs via bond action. As time increases, concrete underwent further shrinkage and
concrete tensile stress began to develop again as shown in Figures 4.54 and 4.55. The
regions in dark red indicate the concrete with high tensile stresses. It can be seen that
the concrete tensile stress concentrated in the region near the support junctions having a
change in width of the slab and also in the regions at a distance away from the first
crack. The stress concentration at the supports was caused by the change in boundary
conditions which can be thought of as a geometric “imperfection”. For the regions at a
distance away from the crack, the stress concentration occurred in the concrete
surrounding the reinforcing steels. At a first glance one may wonder why concrete
(a)
(b)
Age = 40 days
Age = 50 days
x
y σc1
Concrete stress (kPa)
212
tensile stress concentrated only at these regions, and why it was not uniformly
distributed throughout the section when the stresses were fully transferred between
concrete and steel. By examining closely Figures 4.54 and 4.55, the concrete between
the reinforcing steels and at the two sides of the slab shrinks more freely with less
restraint from the steel bars than the concrete surrounding the steel bars and, therefore
the concrete tensile stresses are relatively lower. This is particularly obvious for the
side face concrete of the slab for which the low tensile stress region (denoted by green
to yellow stress contours, for example, as can be seen in Figures 4.54b, 4.54c and
4.55a) stretches a distance up to about 10 elements width away from the midspan crack.
Since tensile stresses are low at the each side of the slab and between the reinforcing
bars, considering the equilibrium at the section where the concrete stress is fully
developed through bond (about 8 elements width away from the midspan crack), the
concrete having a higher restraint from reinforcing steel must have a higher tensile
stress concentration to maintain an equilibrium state in the section.
The computed crack pattern was fully developed at 50 days and no further
cracking was computed up to the age 122 days as the specimen was relieved from the
restraining tension and the development of shrinkage of concrete was not sufficient to
further produce a tensile stress higher than the concrete tensile strength. Figure 4.56
shows the experimental crack patterns and crack widths at age 122 days. Comparing
Figures 4.53d and 4.56, it is seen that the crack pattern and crack widths computed by
the model agrees well with the test results. The comparison of the experimental and the
calculated crack widths, steel stresses and concrete stress at age 122 days are presented
in Table 4.21. A good correlation is obtained except for the steel stress at crack, for
which the model calculated a lower value than that obtained from the experiment.
The comparisons of the numerical and experimental results for all other
specimens are presented in Figures 4.57 to 4.59 and Tables 4.22 to 4.24. In addition to
the comparison of the average crack widths, the sum of all crack widths is also
presented for the comparisons of the numerical and experimental results. The average
crack widths are calculated based on the total number of cracks and the number of
cracks can be quite different for two specimens with the same cross-section. For
example, the experimental average crack widths of slabs RS3-a and RS3-b are very
different (see Table 4.23) because of the difference in number of cracks observed in the
213
test. Therefore, in this case, the sum of all crack widths gives a better understanding of
the overall deformation of the specimens and also provides a more meaningful
comparison with the numerical results. Overall, the calculated results agree well with
the test results.
(a)
(b)
Fig. 4.56 - Experimental crack patterns and crack widths for slabs RS1-a and RS1-b at
age 122 days.
Table 4.21 - Comparison of experimental and FEM results for slab RS1-a and RS1-b
at age 122 days.
Description Slab RS1-a Slab RS1-b FEM Average crack width (mm) 0.22 0.18 0.21 Sum of all crack widths (mm) 0.86 0.68 0.85 Steel stress at crack (MPa) 273 190 147 Steel stress away from crack (MPa) -47.9 -57.9 -57.8 Concrete stress away from crack (MPa) 1.77 1.41 1.40
0.10 mm 0.13 mm
0.34 mm 0.11 mm
0.12 mm
0.21 mm 0.37 mm
0.13 mm
0.15 mm
214
(a)
(b)
(c)
Fig. 4.57 - Crack patterns and crack widths for slabs RS2-a and RS2-b at age 122
days: (a) FEM results; (b) test results for RS2-a; (c) test results for RS2-b.
Table 4.22 - Comparison of experimental and FEM results for slab RS2-a and RS2-b
at age 122 days.
Description Slab RS2-a Slab RS2-b FEM Average crack width (mm) 0.30 0.31 0.35 Sum of all crack widths (mm) 0.90 0.94 1.04 Steel stress at crack (MPa) 250 290 292 Steel stress away from crack (MPa) -41.0 -75.0 -81.8 Concrete stress away from crack (MPa) 1.13 1.46 1.71
0.22 mm 0.25 mm
0.43 mm
0.28 mm
0.21 mm 0.45 mm
0.49 mm 0.27 mm 0.28 mm
215
(a)
(b)
(c)
Fig. 4.58 - Crack patterns and crack widths for slabs RS3-a and RS3-b at age 122
days: (a) FEM results; (b) test results for RS3-a; (c) test results for RS3-b.
Table 4.23 - Comparison of experimental and FEM results for slab RS3-a and RS3-b
at age 122 days.
Description Slab RS3-a Slab RS3-b FEM Average crack width (mm) 0.84 0.50 0.89 Sum of all crack widths (mm) 0.84 1.00 0.89 Steel stress at crack (MPa) 532 467 491 Steel stress away from crack (MPa) -19.2 -33.4 -61.4 Concrete stress away from crack (MPa) 1.45 1.31 1.52
0.89 mm
0.84 mm
0.78 mm 0.22 mm
216
(a)
(b)
(c)
Fig. 4.59 - Crack patterns and crack widths for slabs RS4-a and RS4-b at age 122
days: (a) FEM results; (b) test results for RS4-a; (c) test results for RS4-b.
Table 4.24 - Comparison of experimental and FEM results for slab RS4-a and RS4-b
at age 122 days.
Description Slab RS4-a Slab RS4-b FEM Average crack width (mm) 0.23 0.25 0.22 Sum of all crack widths (mm) 0.93 0.74 0.87 Steel stress at crack (MPa) 270 276 168 Steel stress away from crack (MPa) -45.4 -54.1 -58.2 Concrete stress away from crack (MPa) 1.64 1.71 1.40
0.38 mm
0.16 mm
0.18 mm 0.15 mm
0.18 mm
0.28 mm 0.18 mm
0.29 mm
0.26 mm 0.16 mm 0.32 mm
217
4.5.4 Discussion
Numerical aspects of the simulation of the restrained deformation tests are here
mentioned. Recalling the convergence tests discussed in Chapter 3, Section 3.7.6, a
force or a displacement convergence criterion can be used in a non-linear solution
procedure. Since the restrained deformation specimens were not subjected to externally
applied loads, a displacement convergence criterion must be employed. Furthermore,
the rate of development of both creep and shrinkage of concrete reduces with age of
concrete, the time-dependent deformation of a concrete structure will also reduce with
time. In a time-dependent finite element analysis, time is discretized such that the time
steps are small at early ages and are gradually increased as the rate of deformation of
the structure reduces at an older age. Other than ensuring an effective usage of the
computer resources, the time discretization method also maintains a stable numerical
procedure. It ensures that the increase in deformation of a structure within a particular
time step is not too small to reach the prescribed tolerance for displacement
convergence. In the analyses of the restrained deformation specimens, small time steps
were used at early ages and hence the age of first cracking could be easily traced. As
time increases, the time steps advance with larger time intervals, which causes
difficulties in tracing the age of formation for each crack. For example, taking the
simulation of slab RS1-a or RS1-b (see Figure 4.53), a 10-day time step was used
between age 40 days and age 50 days and three cracks were computed within this time
step. An attempt was made to trace the formation of each individual crack. However,
convergence was difficult due to the use of small time steps. Despite so, the numerical
results have a good agreement with the experimental results.
In the author’s view, even if the model were able to trace one crack at a time, no
further cracking would have been computed after the formation of the second crack,
since the shrinkage-induced tension in the specimen is greatly relieved and the
development of subsequent shrinkage in the aging concrete is too low to induce further
cracking within the duration of the test. This invoked further investigation into the
details of the experimental results. By analysing the concrete surface strains measured
from the experiment, it is believed that, one of the factors causing restrained shrinkage
cracking, which was unaccounted for in most previous studies, is the coalescence of
218
microcracks induced by the internal restraints within the concrete member during the
entire shrinkage process.
To illustrate this phenomenon, the concrete surface strain measurements with
time of two typical specimens are referred to, as shown in Figures 4.60a and 4.60b. The
corresponding locations of the measurements are shown in Figure 4.60c. The curves
with marked indicators in Figures 4.60a and 4.60b are the strain measurements of the
locations containing the cracks of the specimens where the tensile strains are relatively
high. After first cracking, the concrete away from the crack began to shrink, which is
indicated as negative strain measurements in Figures 4.60a and 4.60b. By examining
the test results carefully, it can be noticed that the compressive strains in the concrete
are not uniform throughout the slab length. For concrete adjacent to the crack, this is
because of the low restraint imposed by the reinforcement before the bond was fully
(a) (b)
(c)
Fig. 4.60 - Experimental measurements of concrete surface strain: (a) slab RS1-a; (b)
slab RS2-b; (c) locations of the DEMEC surface strain targets.
-750
-500
-250
0
250
500
750
1000
1250
1500
1750
0 20 40 60 80 100 120 140 160Age (days)
Con
cret
e su
rfac
e st
rain
( µε)
1234567891011
-500
-250
0
250
500
750
1000
1250
1500
0 20 40 60 80 100 120 140 160Age (days)
Con
cret
e su
rfac
e st
rain
( µε)
1234567891011
1 2 3 4 5 6 7 8 9 10 11
DEMEC surface strain target
219
developed. For concrete elsewhere, in the author’s view the non-uniform compressive
concrete strain is caused by the shrinkage-induced microcracking due to the
internal restraints.
It is well recognized that the presence of restraints is the major cause of shrinkage
cracking in reinforced concrete structures. Restraints on shrinkage of concrete can
occur at various scales. The finite element simulation of the restrained deformation
tests has taken into consideration both the external and internal restraints to shrinkage.
These include the external restraint imposed at the end supports and the internal
restraint imposed by the reinforcement. However, the inherent internal restraints on the
mezzo level within concrete, which consist of aggregates restraint and self-restraint
caused by differential shrinkage, were not accounted for or, more precisely, out of the
capability of the present macroscopic modelling method. These internal restraints are
said to be part of the causes of microcracking in concrete, which had been investigated
extensively by, for example, Bažant and Ralfshol (1982) and Bisschop (2002).
In most studies, restrained deformation cracking of reinforced concrete members
is treated in a similar manner to that in a load-induced direct tension member. The only
difference is that, instead of applying external loads, tension in a restrained member
develops through the development of drying shrinkage. For a restrained concrete
member, the first crack forms at the weakest section as shrinkage develops and the
tensile stress in concrete away from the crack is relieved by the formation of the crack.
The tensile stress in the concrete increases as further shrinkage takes place. A new
crack will form if the tensile stress in concrete violates the tensile strength criterion and
the tensile force in the member is once again relieved. This process takes place until the
shrinkage-induced tension becomes insufficient to cause further cracking. This theory
assumes restrained shrinkage cracking as a progressive process where one crack can
only form at a time.
However, the theory above cannot explain the observations in the restrained
deformation tests. For example, the second to the fourth crack of slab RS1-a (see Figure
4.60a) occurred simultaneously at age about 30 days, which obviously disagrees with
the aforementioned theory which assumes crack to form individually at a section at a
time instance. For slab RS2-b (see Figure 4.60b), although the second and third crack
220
did not appear on the same day, the compressive strains over the cracks were constantly
low compared to the strain measurements at other locations. This indicates high tension
was induced simultaneously at the two different locations.
From the experimental observations, it can be postulated that the concrete must
have undergone some sort of deteriorating mechanism and the author believes that
shrinkage microcracking due to internal restraints is the main factor causing
this phenomenon.
4.6 Other Numerical Examples
4.6.1 Continuous Beams Subjected to Long-term Sustained Load
Two simply supported beams and two continuous beams subjected to long-term
sustained loads were tested by Bakoss et al. (1982) to compare the experimental
measurements with the results calculated using empirical design approaches. The
simply supported beams were similar to those tested by Gilbert and Nejadi (2004) for
which the loading points were at the third points of the beams, thus finite element
analysis was only carried out on the continuous beams. The two continuous beams were
identical in both setting and cross-section. The cross-sections of the beams were 100
mm wide and 150 mm deep with two 12 mm diameter bars at an effective depth of 130
mm on the tension sides of the beams, as indicated in Figure 4.61a. A 6 kN point load
was applied at the midspan of each of the two equal 6 m spans at 23 days after casting.
In addition to the beam tests, Bakoss et al. also tested companion specimens cast from
the same batch of concrete in order to measure the creep and shrinkage properties of
the concrete.
The localized cracking crack band model was employed to model the continuous
beam. One half of the beam was modelled with symmetry and the finite element mesh
is shown in Figure 4.61b. The mesh consists of 948 nodes, 714 concrete elements, 119
steel truss elements, 119 bond-slip interface elements and 3 stiff elastic support
221
(a)
(b)
Fig. 4.61 - Details of Bakoss et al.’s (1982) continuous beam: (a) cross-sections of the
beam; (b) finite element mesh.
elements. The steel truss elements were linked to the concrete elements via bond-slip
interface elements. The reinforcing steel at the left end of the beam in Figure 4.61b was
assumed to have full anchorage where no slip is permitted. In the experiment, the
continuous beams were loaded horizontally with the beams suspended sideways
therefore the self-weight of the beams was omitted in the analysis.
Bakoss et al. reported the creep property as creep coefficients with increasing
time for the concrete first loaded at age 23 days. The creep coefficients were converted
into a compliance function using the relationship [ ] 23.)23,(1)23,( cEttJ φ+= . The
concrete parameters used in the analysis were: GPa3.2723. =cE ,
GPa9.496.1 28.0 == cEE , MPa39=cmf , MPa5.228. =ctf , ch = 35 mm, ν = 0.2,
µε850=shA , days150=shB , MPa1252 µε=q , MPa01.03 µε=q ,
MPa264 µε=q and 10 MPa5.35 −=A . The approximated time-dependent shrinkage
strain curve and compliance curve are shown in Figures 4.62a and 4.62b, respectively.
The elastic moduli and retardation times of the Kelvin chain units for the creep model
2∅12 bars15
0
100
130
2∅12 bars
20
Section A-A Section B-B Section C-C
6 kN A
A
B
B
C
C 6000 mm
222
are presented in Table 4.25. By using the equation recommended by AS 3600 (2001),
as given by Eq. 2.1, the mean concrete tensile strength was calculated from the
experimental concrete compressive strength at different ages. The time-dependent
growth of the mean concrete tensile strength was approximated by MPa1.3=ctfA and
days10=ctfB with the concrete tensile strength for each element assigned randomly at
±10% of the mean value. The concrete fracture energy fG was taken as mmN70 in
accordance with CEB-FIP Model Code 1990 (1993). The reinforcing steel was
modelled as elastic-perfectly plastic material with a yield strength of 400 MPa and an
(a) (b)
Fig. 4.62 - Creep and shrinkage measurements of Bakoss et al. (1982) compared with
models: (a) shrinkage strain since commencement of drying; (b)
compliance curve.
Table 4.25 - Kelvin chain properties for solidification creep model.
µ-th unit Eµ (MPa) τµ (days) 1 0.12584 0.0001 2 0.10705 0.001 3 0.09214 0.01 4 0.08030 0.1 5 0.07091 1 6 0.06345 10 7 0.05753 100 8 0.05283 1000
0
50
100
150
200
0.1 1 10 100 1000Time under load, t-t' (days)
J(t,t
') (1
0-6 /
MPa
)
Experimental data (t'=23 days)
Model (t'=23 days)
0
200
400
600
800
0 200 400 600 800Time since commencement of drying (days)
Shrin
kage
str
ain
( µε)
Experimental data
Model
Ash = 850 µεBsh = 150 days
223
elastic modulus of 200 GPa. The parameters for the CEB-FIP 1990 (1993) bond-slip
model were: mm6.021 == ss , mm0.13 =s , MPa5.12max =τ , MPa9.1=fτ and an
unloading modulus of mmMPa100 was assumed.
The midspan deflection calculated by the finite element model is compared with
the experimental average deflection for the two beams in Figure 4.63. The model
calculated a slightly lower time-dependent deflection than the experimental
measurement but with a difference not more than 10%, for which the correlation is
considered reasonable. No further experimental data was reported by Bakoss et al.
except the midspan deflection with time. To demonstrate the formation of crack at
discrete location, the crack patterns computed by the model at instantaneous loading, at
50 days and at 400 days under sustained load are shown respectively in Figures 4.64a,
4.64b and 4.64c. The model computed an average crack spacing of 70 mm. Without the
available experimental data, a comparison is made with the average crack spacings
calculated using the tension chord model and the recommendation of CEB-FIP Model
Code 1990 (1993). For stabilized cracking, the average crack spacing is given by
(CEB-FIP, 1993)
032
rmrm ss = with eff
rmsρ6.30∅
= (4.1)
where 0rms is the maximum crack spacing, ∅ is the bar diameter and effρ is the
effective reinforcement ratio defined as the ratio of the area of reinforcing steel and the
effective area of concrete in tension (see Section 3.3.1). The comparison of the results
is shown in Table 4.26. It is seen that the finite element model computed the largest
average crack spacing while the tension chord model gives the lowest prediction.
224
Fig. 4.63 - Comparison of FEM and experimental midspan deflections versus time.
(a)
(b)
(c)
Fig. 4.64 - Crack patterns computed by FEM (principal cracking strain plot): (a)
instantaneous loading; (b) 50 days after loading; (c) 400 days after loading.
0
2
4
6
8
10
12
14
16
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
225
Table 4.26 - Comparison of average crack spacing calculated by different models.
Model description Average crack spacing FEM 70 mm
Tension chord model 42 mm CEB-FIP Model (1993) 51 mm
In the analysis, the beam was cracked at instantaneous loading. The cracks of the
beam widened with time due to the effects of creep and shrinkage, as can be seen in
Figure 4.64. A summary of the crack opening over 400 days is given in Table 4.27. The
average crack widths were calculated by averaging the widths of the cracks within the
region of moment greater than 80% of the maximum moments. The maximum crack
opening was found to occur at the maximum negative moment region above the middle
support where the bending moment is highest in the beam.
Table 4.27 - Crack widths calculated by FEM at various times t after first loading.
Region Crack width (mm) Instantaneous t = 100 days t = 400 days
Positive moment region Avg. 0.025 0.045 0.062 Max. 0.026 0.055 0.078
Negative moment region Avg. 0.036 0.062 0.083 Max. 0.048 0.0.82 0.106
4.6.2 Time-dependent Forces Induced by Supports Settlement of
Continuous Beams
A series of tests was performed by Ghali et al. (1969) to investigate the time-dependent
reaction forces induced by the differential settlement of supports of continuous
reinforced concrete beams. The test consisted of 4 pairs of two-span continuous beams.
Each pair of the beams was tested in a vertical position so as to disregard the bending
caused by the self-weight. Rollers were inserted at the end of the beams to separate the
beam set and a mid-support “settlement” (or deflection) was introduced by means of
226
two threaded bars tying the beams. The applied settlement was monitored by two dial
gauges and the induced forces were recorded throughout the duration of the test. The
setup and the details of the beams are shown in Figure 4.65. Each pair of the beams was
subjected to the same final deflection of 1.65 mm but with different deflection
increments applied at different ages. The beams were tested up to a period of 300 days.
The details of the age of application of deflections for each beam set are indicated in
Table 4.28. In addition to the support settlement tests, a beam with the same layout and
cross-section was subjected a sustained load of 17.3 kN at the mid-length of the beam
in the absence of the middle support. The beam was also tested in a vertical position
therefore the bending effect due to its self-weight was eliminated.
(b)
(c)
(a)
Fig. 4.65 - Details of Ghali et al.’s (1969) continuous beams: (a) longitudinal layout of
a beam set; (b) section of the test; (c) cross-section of the beam.
914.
4
2133
.6
914.
4
A A
Dial gauge 25.4 25.4
209.6
2∅12.7 2∅12.7
Stirrups ∅6.35 at 152.4
Dial gauge
Threaded bar Calibrated rod
Section A-A
227
Table 4.28 - Details of application of deflections.
Age (days) of application of deflections for increment number
Test No.
1 2 3 4 5
Deflection increment
(mm)
Duration for each deflection increment
(minute)
1 9 - - - - 1.65 30 2 12 12⅛ 12½ 13¼ 14¼ 0.33 10 3 12 13 15¼ 20 26⅓ 0.33 10 4 11½ 15 27¼ 41¼ 72¼ 0.33 10
Localized cracking is not the major interest in this investigation and in addition,
the beams contained both longitudinal and transverse reinforcements, it is more
appropriate to adopt a distributed cracking model, in this study it is the cracked
membrane model. By exploiting the symmetric nature of the beam, only one half of the
beam is required for the simulation. The mesh of the beams is shown Figure 4.66 with
the material regions indicated. The mesh is made up of 374 nodes and contains 320
reinforced concrete elements and 3 stiff elastic support elements. Table 4.29 shows the
reinforcement properties of the reinforced concrete zones indicated in Figure 4.66.
Fig. 4.66 - Finite element mesh for Ghali et al.’s (1969) beams.
Table 4.29 - Reinforcement properties for Ghali et al’s (1969) beams.
RC zone Reinforcement ratio ρx ρy
1 0 0.004091 2 0.049090 0.004091
RC zone 1 RC zone 2 Stiff elastic elements
12
228
The time-dependent properties of the concrete were not reported by Ghali et al.
(1969). To facilitate the numerical analysis, the compliance function and shrinkage
function recommended by CEB-FIP Model Code 1990 (1993) were adopted and were
incorporated into the finite element model by using the appropriate creep and shrinkage
parameters. A relative humidity of 65% was used in the calculation of the compliance
function and shrinkage function. The CEP-FIP 1990 creep and shrinkage models are
presented in Appendix C. Ghali et al. tested a number of concrete cylinders to obtain
the compressive strengths at ages range from 7 days to 190 days and approximated the
growth of concrete compressive strength by a function of age t given by
75.0792.37'
+=
tfc (4.2)
The growth of concrete tensile strength can be calculated using the
recommendation in AS 3600 (2001) (Eq. 2.1, Chapter 2) based on the compressive
strengths at various ages. The concrete parameters used in the analysis
were: GPa3.466.1 28.0 == cEE , MPa38=cmf , MPa5.228. =ctf , MPa8.2=ctfA
days5.3=ctfB , 2.0=ν , µε450=shA , days60=shB , MPa3.1422 µε=q ,
MPa4.63 µε=q , MPa3.164 µε=q and 10 MPa4.40 −=A . The elastic moduli and
corresponding retardations times of the Kelvin chain units are tabulated in Table 4.30.
The concrete fracture energy fG was taken as mmN75 . For the reinforcing steel, an
elastic-perfectly plastic stress-strain relationship with an elastic modulus of 200 GPa
and a yield strength of 400 MPa was employed.
Figures 4.67a to 4.67d show the comparisons of the numerical and experimental
results for the reactions at mid-supports with time induced by the applied deflections. It
is seen that the model calculated a more rapid change in reaction than the test results.
This is attributed to the effects of creep and shrinkage of concrete. A better correlation
could be obtained if the experimental creep and shrinkage data were available.
Nevertheless, the results calculated by the model are overall in a good agreement with
the test data. In Figure 4.68, the calculated midspan deflection for the beam subjected
to a sustained load is compared with the test results with a good correlation obtained.
229
Table 4.30 - Kelvin chain properties for solidification creep model.
µ-th unit Eµ (MPa) τµ (days) 1 0.11054 0.0001 2 0.09404 0.001 3 0.08094 0.01 4 0.07054 0.1 5 0.06229 1 6 0.05574 10 7 0.05054 100 8 0.04641 1000
(a) (b)
(c) (d)
Fig. 4.67 - Comparison of FEM and experimental time-dependent reactions at mid-
supports of Ghali et al.’s (1969) controlled deflection specimens: (a) to (d)
test 1 to test 4, respectively.
0.02.04.06.08.0
10.012.014.016.018.020.0
0 50 100 150 200 250Age (days)
Rea
ctio
n at
mid
-sup
port
(kN)
Experimental
FEM0.02.04.06.08.0
10.012.014.016.018.0
0 50 100 150 200 250Age (days)
Rea
ctio
n at
mid
-sup
port
(kN)
Experimental
FEM
0.02.04.06.08.0
10.012.014.016.018.0
0 50 100 150 200 250Age (days)
Rea
ctio
n at
mid
-sup
port
(kN)
Experimental
FEM0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
0 50 100 150 200 250Age (days)
Rea
ctio
n at
mid
-sup
port
(kN)
Experimental
FEM
230
Fig. 4.68 - Comparison of FEM and experimental time-dependent midspan deflection
for the beam subjected to a sustained load of Ghali et al. (1969).
4.6.3 Slender Columns Subjected to Long-term Eccentric Axial Loads
This section deals with a slightly different type of structural problem where non-
linearity does not arise solely from the material properties of reinforced concrete but
also from the excessive deformation of the structure due to the effects of creep. This
geometric non-linearity may lead to creep buckling and is common in slender
reinforced concrete columns subjected to sustained eccentric axial compression.
Bradford (2005) tested a series of eccentrically loaded slender reinforced concrete
columns and used the test results to investigate the behaviour of slender columns
analytically. Five identical columns were tested with various degree of end eccentricity.
All columns were 5 m long and the eccentric compressive loads were applied at the two
ends of the columns. The test arrangement and the cross-section details of the
specimens are shown in Figure 4.69. Three dial gauges were placed on the side face of
the column, as shown in Figure 4.69a, in order to measure the sideways deflections.
One dial gauge was placed at the mid-height whilst the other two were positioned at the
quarter points. The details of the compressive loads and the end eccentricities eT and eB
as shown in Figure 4.69a are given in Table 4.31.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 50 100 150 200 250Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
231
(a)
(b)
Fig. 4.69 - Details of Bradford’s (2005) slender columns: (a) layout of the test; (b)
cross-section of the column.
A A
5000
1250
1250
1250
1250
eT
B e
Strong wall
Dial gauge
Dial gauge
Dial gauge
Steel channel section
Eccentric loading
Tensioning cable
Test column
Hydraulic jack
Load cell
I-section loading arm
150
Section A-A
Stirrups Ø10 at 150
2N12
2N12
Clear cover 15 mm
232
Table 4.31 - Details of the loading conditions for Bradford’s (2005) columns.
Specimen C1 C2 C3 C4 C5 eT (mm) 50 50 50 50 50 eB (mm) 50 25 0 -25 -50
Load (kN) 70.0 70.0 80.0 80.0 85.0
The distributed cracking cracked membrane model was used to simulate these
tests. The finite element mesh for the columns is shown in Figure 4.70 and the
reinforcement details for the mesh are given in Table 4.32. The 200 rows by 6 columns
mesh consists of 1407 nodes and 1200 reinforced concrete elements.
The properties of concrete were taken as those reported by Bradford. The tensile
strength of concrete, which was not reported by Bradford, was taken as that calculated
using the equation given in AS 3600 (2001). The parameters for creep and shrinkage
were determined by fitting the test data obtained from the creep and shrinkage tests
conducted in conjunction with the slender column tests. Figures 4.71a and 4.71b show
(a) (b) (c) (d) (e)
Fig. 4.70 - Finite element mesh for Bradford’s (2005) columns: (a) column C1; (b)
column C2; (c) column C3; (d) column C4; (e) column C5.
5000
(2
00 ro
ws o
f ele
men
ts)
50
50
50
25
50 50
25
50
50
1 1 1 1
2 2 2
RC zone 1 RC zone 2
233
Table 4.32 - Reinforcement properties for Bradford’s (2005) columns.
RC zone Reinforcement ratio ρx ρy 1 0.00711 0.03492 2 0.00711 0
the approximated curves for shrinkage strain and for compliance data, respectively.
Since the age of loading for the columns were not reported explicitly by Bradford, it
was taken as 12 days in this study and it is the loading age for the creep test.
The concrete parameters used in the simulation were: GPa4.356.1 28.0 == cEE ,
MPa3.29=cmf , MPa2.228. =ctf , MPa2.2=ctfA days12=
ctfB , 2.0=ν ,
µε420=shA , days90=shB , MPa2.802 µε=q , MPa5.23 µε=q ,
MPa5.384 µε=q and 10 MPa8.22 −=A . The elastic moduli and retardations times
for the Kelvin chain units are shown in Table 4.33. Similar to the previous examples,
the concrete fracture energy fG was taken as mmN75 and the stress-strain
relationship for reinforcing steel was taken to be elastic-perfectly plastic with an elastic
modulus of 200 GPa and a yield strength of 500 MPa.
(a) (b)
Fig. 4.71 - Comparison of experimental shrinkage and creep measurements (Bradford,
2005) with approximated models: (a) shrinkage strain; (b) compliance
curve.
0
50
100
150
200
250
0.1 1 10 100 1000Time under load, t-t' (days)
J(t,t
') (1
0-6 /
MPa
)
Experimental data (t'=12 days)
Model (t'=12 days)
0
100
200
300
400
0 50 100 150 200 250 300 350Time since commencement of drying (days)
Shrin
kage
str
ain
( µε)
Experimental data
Model
Ash = 420 µεBsh = 90 days
234
Table 4.33 - Kelvin chain properties for solidification creep model.
µ-th unit Eµ (MPa) τµ (days) 1 0.19612 0.0001 2 0.16685 0.001 3 0.14361 0.01 4 0.12516 0.1 5 0.11051 1 6 0.09889 10 7 0.08966 100 8 0.08234 1000
The deflected shapes calculated by the finite element model are shown in Figure
4.72. It is seen that the column with the largest equal end eccentricities (specimen C1)
has the largest deflection, whilst column C5, which was subjected to the largest unequal
end eccentricities, has the smallest deflection. The calculated deflections at mid-height
and top and bottom quarter points with time are compared with the experimental results
in Figure 4.73. Although the calculated results for column C1 are slightly lower than
that observed in the test, the overall correlation between the model results and the test
data is reasonable.
(a) (b) (c) (d) (e)
Fig. 4.72 - Deflected shapes computed by the model (scale ×30): (a) column C1; (b)
column C2; (c) column C3; (d) column C4; (e) column C5.
235
(a) (b)
(c) (d)
(e)
Fig. 4.73 - Comparison of FEM and experimental deflections versus time: (a) column
C1; (b) column C2; (c) column C3; (d) column C4; (e) column C5.
Comparing the calculated central deflection for column C5 with the test data, the
model seems to have computed the column to deflect to the wrong side since the
calculated and experimental central deflections are of different signs as seen in Figure
4.73e. A close examination reveals that the computation was theoretically sensible.
0.05.0
10.015.020.025.030.035.040.045.050.0
0 100 200 300Time since loading (days)
Defle
ctio
ns (m
m)
Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)
Column C1
0.0
5.0
10.0
15.0
20.0
25.0
30.0
0 100 200 300Time since loading (days)
Defle
ctio
ns (m
m)
Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)
Column C2
0.0
5.0
10.0
15.0
20.0
25.0
0 100 200 300Time since loading (days)
Defle
ctio
ns (m
m)
Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)
Column C3
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0 100 200 300Time since loading (days)
Defle
ctio
ns (m
m)
Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)
Column C4
-6.0-5.0-4.0-3.0-2.0-1.00.01.02.03.04.05.0
0 100 200 300Time since loading (days)
Defle
ctio
ns (m
m)
Exp (top) FEM (top)Exp (mid) FEM (mid)Exp (bot) FEM (bot)
Column C5
236
Although column C5 was subjected to end eccentricities of ±50 mm, the magnitudes of
the end moments were not identical. The column had a larger moment at the bottom
end due to the self-weight of the column. This can be seen in the test results for the
quarter points shown in Figure 4.73e, the deflection of the bottom quarter point is
slightly larger than that of the top quarter point. Due to the larger moment at the bottom
end, the central deflection should also deflect to the same direction as the bottom
quarter point, as computed by the model. The author believes that the difference
between the results is due to vagaries of experiments or more undesirably, errors arising
when measuring the test data.
237
CHAPTER 5
NUMERICAL EXPERIMENTS
5.1 Introduction
Numerous studies were conducted in the 1950s to 1970s to gain better understanding of
the cracking mechanisms of reinforced concrete structures (for example, Clark, 1956,
Broms., 1965, Base et al., 1966, Ferry-Borges, 1966, Gergely et al., 1968 and Nawy et
al., 1970). However, these studies are limited to instantaneous cracking. Cracking of
reinforced concrete structures under long-term loads has received relatively less
attention. Although there are studies dedicated to the investigation of the time-
dependent behaviour of reinforced concrete, their objectives are usually concerned with
the variation of deflection with time. Relatively little attention has been given to the
study of the time-dependent change in crack widths and crack spacings in reinforced
concrete structures. The reason for this is that time-dependent tests are not only time
consuming but also require a large amount of laboratory resources. Cracking of
reinforced concrete is well known for its semi-random nature. A large number of
specimens are needed for experimentation in order to obtain statistically representative
results. Therefore, the need for a large laboratory area over a lengthy period is
unavoidable and expensive.
In this chapter, a number of series of numerical experiments were devised and
analysed using the numerical model described in Chapter 3 in order to investigate time-
dependent cracking of reinforced concrete structural elements. The primary aims of the
investigation are to identify the major parameters influencing time-dependent flexural
cracking and to acquire a better qualitative understanding of the interactions of the
parameters. As a result, the localized cracking Crack Band Model with the use of bond-
slip interface elements (see Section 3.4 for formulation and Chapter 4 for evaluation) is
better suited for this parametric investigation compared to the localized cracking non-
238
local model due to its capability of modelling the concrete-steel stress transfer via bond
and the subsequent formation of cracks in a reinforced concrete structure, while
remaining more computationally effective.
This investigation essentially focuses on the following aspects of cracking in
reinforced concrete structures:
(1) Short-term and long-term crack spacings;
(2) Short-term and long-term crack widths;
(3) The ratio of long-term to short-term crack width;
The factors affecting the time-dependent behaviour of reinforced concrete
structures can be classified primarily into three categories: (i) factors associated with
material properties; (ii) factors related to environmental conditions (such as relative
humidity); and (iii) factors related to the structural type, including geometry and
boundary conditions (Gilbert, 1979). In this study the following parameters have been
selected as variables to be investigated:
(A) Bottom concrete cover bc ;
(B) Diameter of tensile reinforcing steel st∅ ;
(C) Quantity of tensile reinforcement stA ;
(D) Quantity of compressive reinforcement scA ;
(E) Magnitude of mean tensile strength of concrete ctf ;
(F) Bond strength between steel and concrete maxτ ;
(G) Random fluctuation limit of concrete tensile strength fctl ;
(H) Magnitude of creep;
(I) Magnitude of shrinkage;
(J) Bond creep;
(K) Quantity of shear reinforcement svA ;
239
(L) Yield stress of reinforcing steel syf ;
(M) Load histories;
(N) Geometry of sections;
(O) Boundary conditions.
The numerical experiments were designed so that a comprehensive parametric
investigation of cracking could be undertaken in reinforced concrete beams and slabs
under service loads. In each series of tests, one parameter is selected as the variable
while the others are kept constant or otherwise manipulated in order that their effects
are not significant on the crack width and crack spacing. In this way the effect of the
selected variable can be thoroughly examined.
5.2 Description of Numerical Experiments
Beam and slab specimens were used in the numerical experiments with different
section geometries. Both the beam and slab specimens were subjected to two types of
boundary conditions so as to examine the effects of boundary conditions on time-
dependent cracking. A simply supported case and a continuous case, as shown in
Figure 5.1, were considered in this study.
All specimens were first loaded at age 14 days and, unless stated otherwise, the
benchmark material properties for each specimen were taken as follows:
MPa25=cmf , MPa2=ctf , GPa406.1 28.0 == cEE , 2.0=ν and N/m70=fG .
A ±10% random fluctuation of the mean concrete tensile strength was assigned to the
concrete elements. Shrinkage was assumed to commence at age 14 days and the
shrinkage parameters were taken as µε600=shA and days45=shB , which give a
final shrinkage strain µεε 600* =sh . The solidification creep parameters were obtained
by fitting the creep curves of CEB-FIP Model Code 1990 and were
MPa1.1212 µε=q , MPa7.63 µε=q , MPa8.204 µε=q , which correspondingly
240
(a)
(b)
Fig. 5.1 - Boundary conditions for beam and slab specimens: (a) simply supported;
(b) continuous.
give a final creep coefficient 2.2* =φ . The negative infinity area of the continuous
retardation spectrum was 0A = 34.3 MPa-1. Eight Kelvin chain units were used to store
viscoelastic strain history and the corresponding elastic modulus µE and retardation
time µτ for each Kelvin chain unit are shown in Table 5.1. Reinforcing steel was taken
as an elastic-perfectly plastic material with a yield strength MPa500=syf and an
elastic modulus GPa200=sE . The parameters for the bond model were:
mm6.021 == ss , mm0.13 =s , MPa0.10max =τ , MPa5.1=fτ and the unloading
modulus was taken as mmMPa100=uk .
L/3 L/3 L/3
L = 5000
P P
L/3 L/3 L/3
L = 5000
P P
Tensile steel
Compressive steel
241
Table 5.1 - Kelvin chain properties for solidification creep model.
µ-th unit Eµ (MPa) τµ (days) 1 0.12989 0.0001 2 0.11050 0.001 3 0.09511 0.01 4 0.08289 0.1 5 0.07319 1 6 0.06549 10 7 0.05938 100 8 0.05453 1000
5.2.1 Beam Specimens
The details of a typical beam section are shown in Figure 5.2. The beams were 300 mm
wide and the effective depth to the tensile reinforcement was consistently 450 mm in
both the positive and negative moment regions. The top concrete cover was kept the
same as the bottom concrete cover for all specimens.
A continuous beam can withstand much higher loads than a simply supported
beam of the same clear span due to the additional moment resistance at the supports.
The capability of taking high loads inevitably leads to a high shear force in the shear
span of the specimens (distance from support to the position of the load). Since the
major purpose of this study is to investigate time-dependent flexural cracking, it is
undesirable for the specimens to fail prematurely in shear. According to the design
Fig. 5.2 - Typical cross-section for beam specimens.
300
450
bb
Asc
Ast
242
procedures of AS 3600 (2001), calculation shows that some specimens will experience
shear failure rather than flexural failure if shear reinforcement is not provided.
Therefore, for the sake of preventing premature shear failure in the numerical beam
experiments, unless stated otherwise, all continuous beams are reinforced with 2-legged
N10 stirrups at 120 mm spacing over the shear span. The concrete cover bc (Figure
5.2) is measured from the top or bottom edge of the beam to the nearest side of the
longitudinal reinforcement (and not to the stirrup). Although in real structures cover is
usually measured to the stirrup, it does not matter in the current two-dimensional
modelling since each stirrup acts essentially as a vertical tie in the beam to prevent
excessive shear cracking.
The finite element meshes for the simply supported and continuous beam
specimens are shown in Figures 5.3 and 5.4, respectively. The tensile reinforcement is
shown as solid lines and the compressive reinforcement is shown as dashed lines in
each figure. All meshes were generated such that the crack band width is ch = 40 mm.
(a)
(b)
Fig. 5.3 - Finite element meshes for simply supported beam specimens: (a) beam
with tensile reinforcement; (b) beam with both tensile and compressive
reinforcement.
Tensile reinforcement Compressive reinforcement
Asc
Ast
Ast
243
The longitudinal steel elements were connected to the concrete elements via bond-slip
interface elements while the transverse steel elements were overlaid directly onto the
concrete elements.
For the singly reinforced simply supported beams (see Figure 5.3), the mesh
consists of 941 nodes and is made up of 792 concrete elements, 66 steel elements, 66
bond-slip elements and 4 stiff elastic support elements. The simply supported beam
with compressive reinforcement has an additional 65 nodes, 66 steel elements and 66
bond-slip interface elements. The nodes of the steel elements at the outer edge of the
beam were rigidly connected to the nodes of the concrete elements in order to simulate
full anchorage of the reinforcement.
(a)
(b)
Fig. 5.4 - Finite element meshes for continuous beam specimens: (a) beam with
tensile reinforcement; (b) beam with both tensile and compressive
reinforcement.
Tensile reinforcement Compressive reinforcement
Legend for longitudinal reinforcement:
Nodal enslavement in horizontal direction
Ast-
Asc- Ast
Asc
Ast
Ast-
244
The mesh of the continuous beam containing tensile reinforcement only is made
up of 962 nodes, 792 concrete elements, 91 longitudinal steel elements, 91 interface
elements and 204 transverse steel elements. The detailing of the longitudinal
reinforcing steel was done in accordance with the guidelines in AS 3600 (2001), so that
the tensile reinforcement, both top and bottom, was extended a distance equal to the
total depth of the section past the point of inflection. The major aim of this study is to
investigate time-dependent flexural cracking of reinforced concrete members. Instead
of providing roller supports at the vertical row of nodes at the right boundary, the mesh
was enslaved in the horizontal direction so as to maintain a zero slope at the midspan
and to prevent boundary-induced membrane actions due to arching effect.
5.2.2 Slab Specimens
The slab section used in the numerical experiments was 1000 mm wide, with an
effective depth of 200 mm. Like the beam specimens, the top and bottom reinforcement
had the same concrete cover. Figure 5.5 shows the typical cross-section of a slab
specimen containing compressive reinforcement. Both simply supported and
continuous slabs were analysed in this study and the arrangement of these two types of
slabs has been shown previously in Figure 5.1.
Fig. 5.5 - Typical cross-section for slab specimens.
The same crack band width ( ch = 40 mm) as for the beam meshes was used to
generate the finite element meshes for the slab specimens. Figures 5.6 and 5.7 show the
finite element meshes for the simply supported and continuous slabs, respectively. The
bb
200
Asc
Ast
245
mesh for the singly reinforced simply supported slab (Figure 5.6a) contains 539 nodes,
396 concrete elements, 66 steel elements, 66 bond-slip interface elements and 4 stiff
elastic support elements. The bond-slip interface elements were placed between the
concrete and steel elements for transfer of stress via bond. The overlapping nodes of the
concrete and steel elements at the left edge of the mesh were rigidly connected so as to
simulate full anchorage of the reinforcement. For the doubly reinforced simply
supported slab (Figure 5.6b), there is an extra row of steel and bond-slip interface
elements in the compression zone, which leads to an additional 65 nodes, 66 steel
elements and 66 bond-slip interface elements.
(a)
(b)
Fig. 5.6 - Finite element meshes for simply supported slab specimens: (a) slab with
tensile reinforcement; (b) slab with both tensile and compressive
reinforcement.
The slab specimen has a much larger cross-sectional area than the beam specimen
and hence has a higher shear strength. A check in accordance with AS 3600 showed
that no shear reinforcement was required for the continuous slab specimens.
Furthermore, an additional development length equal to the total depth of the section
was provided for both the top and bottom tensile reinforcements at the point of
inflection. The finite element mesh for the continuous slab specimens containing tensile
reinforcement only (Figure 5.7a) is made up of 548 nodes, 396 concrete elements, 79
steel elements and 79 bond-slip interface elements and the mesh for continuous slabs
Tensile reinforcement Compressive reinforcement
Ast
Asc
Ast
246
with both tensile and compressive reinforcement (Figure 5.7b) has 599 nodes, 396
concrete elements, 132 steel elements and 132 bond-slip interface elements. As for the
previous finite element meshes, the steel element nodes were connected to the concrete
nodes via bond-slip interface elements. For the same reasons as discussed for the
continuous beam specimens, the nodes at the right edge of the continuous slab
specimens were enslaved in the horizontal direction.
(a)
(b)
Fig. 5.7 - Finite element meshes for continuous slab specimens: (a) slab with tensile
reinforcement; (b) slab with tensile and compressive reinforcements.
5.2.3 Testing Method
This section describes the series of tests (designated test series A to M). In each series a
parameter was selected as the experimental variable, while other parameters were held
constant. The number and designation of the tests on each type of specimen are
presented in Table 5.2.
Nodal enslavement in horizontal direction
Tensile reinforcement Compressive reinforcement
Legend for longitudinal reinforcement:
Ast-
Ast
Ast
Asc
Asc-
Ast-
247
Table 5.2 - Tests conducted on specimens.
No. Specimen Test Series Total No. of tests 1 Simply supported beam specimens A to J, L and M 12 tests 2 Continuous beam specimens A to K 11 tests 3 Simply supported slab specimens A to J, L and M 12 tests 4 Continuous slab specimens A to J 10 tests
5.2.3.1 Test Series A to J: Material and Environmental Parameters
The details of the combinations of variables are given in Tables 5.3 and 5.4 for beam
and slab specimens, respectively. The convention for the test numbering is such that the
first letter denotes the member type (‘B’ for beam or ‘S’ for slab), the second letter
denotes the test series (A to M) followed by the test number in that particular series.
The test designated B-A0 and S-A0 are control specimens. The shaded columns in
Tables 5.3 and 5.4 represent the experimental variables for the test series. It should be
noted that the reinforcement details shown in both Tables 5.3 and 5.4 are for positive
moment regions only and the reinforcement details in negative moment regions for
continuous beams and slabs are shown in Tables 5.5 and 5.6, respectively. All
specimens in test series A to J were subjected to a sustained load at age 14 days that
produced 50% of the moment capacity uM at the critical section. The details of the
loads P (see Figure 5.1) required to achieve the desired moments are given in Tables
5.11 to 5.14. In test series E, the tensile strength of concrete is selected as the
parametric variable. The concrete tensile strength was varied with the tension softening
parameters (see Section 3.6.1 of Chapter 3) held constant and the bond strengths of the
specimens were adjusted according to the tensile strength using a relationship obtained
by combining Eq. 2.1 of Chapter 2 and the definition of maxτ given in Table 3.1 of
Chapter 3, that is, ctf5max =τ .
248
Table 5.3 - Combinations of variables for beam specimens (simply supported and
continuous specimens).
Test No.
cb (mm)
∅st†
(mm) Ast
†
(mm2) Asc
† (mm2)
fct
(MPa) τmax
(MPa) lfct
(%) φ *†† εsh
*
(µε) Bond
creep?
B-A0 20 2N20 620 0 2 10 ±10 2.2 600 Yes B-A1 35 2N20 620 0 2 10 ±10 2.2 600 Yes B-A2 50 2N20 620 0 2 10 ±10 2.2 600 Yes B-A3 70 2N20 620 0 2 10 ±10 2.2 600 Yes B-B1 20 2N16 400 0 2 10 ±10 2.2 600 Yes B-B2 20 2N24 900 0 2 10 ±10 2.2 600 Yes B-C1 20 3N20 930 0 2 10 ±10 2.2 600 Yes B-C2 20 4N20 1240 0 2 10 ±10 2.2 600 Yes B-D1 20 2N20 620 330 (3N12) 2 10 ±10 2.2 600 Yes B-D2 20 2N20 620 620 (2N20) 2 10 ±10 2.2 600 Yes B-E1 20 2N20 620 0 1 5 ±10 2.2 600 Yes B-E2 20 2N20 620 0 1.5 7.5 ±10 2.2 600 Yes B-E3 20 2N20 620 0 2.5 12.5 ±10 2.2 600 Yes B-E4 20 2N20 620 0 3 15 ±10 2.2 600 Yes B-F1 20 2N20 620 0 2 3 ±10 2.2 600 Yes B-F2 20 2N20 620 0 2 5 ±10 2.2 600 Yes B-F3 20 2N20 620 0 2 15 ±10 2.2 600 Yes B-G1 20 2N20 620 0 2 10 ±1 2.2 600 Yes B-G2 20 2N20 620 0 2 10 ±5 2.2 600 Yes B-G3 20 2N20 620 0 2 10 ±15 2.2 600 Yes B-G4 20 2N20 620 0 2 10 ±20 2.2 600 Yes B-H1 20 2N20 620 0 2 10 ±10 0.3 0 Yes B-H2 20 3N20 620 0 2 10 ±10 3.9 0 Yes B-H3 20 2N20 620 0 2 10 ±10 0.3 600 Yes B-H4 20 3N20 620 0 2 10 ±10 3.9 600 Yes B-I1 20 2N20 620 0 2 10 ±10 2.2 0 Yes B-I2 20 2N20 620 0 2 10 ±10 2.2 300 Yes B-I3 20 2N20 620 0 2 10 ±10 2.2 900 Yes B-J1 20 2N20 620 0 2 10 ±10 2.2 600 No B-J2 20 3N20 930 0 2 10 ±10 2.2 600 No B-J3 20 4N20 1240 0 2 10 ±10 2.2 600 No
† Reinforcement details for positive moment regions only. †† 3.0* =φ and 9.3* =φ were achieved by setting 4q as 0 and 40 µε/MPa,
respectively.
249
Table 5.4 - Combinations of variables for slab specimens (simply supported and
continuous specimens).
Test No.
cb (mm)
∅st†
(mm) Ast
†
(mm2) Asc
† (mm2)
fct
(MPa) τmax
(MPa) lfct
(%) φ *†† εsh
*
(µε) Bond
creep?
S-A0 20 5N16 1000 0 2 10 ±10 2.2 600 Yes S-A1 35 5N16 1000 0 2 10 ±10 2.2 600 Yes S-A2 50 5N16 1000 0 2 10 ±10 2.2 600 Yes S-A3 70 5N16 1000 0 2 10 ±10 2.2 600 Yes S-B1 20 5N12 550 0 2 10 ±10 2.2 600 Yes S-B2 20 5N20 1550 0 2 10 ±10 2.2 600 Yes S-C1 20 3N16 600 0 2 10 ±10 2.2 600 Yes S-C2 20 8N16 1600 0 2 10 ±10 2.2 600 Yes S-D1 20 5N16 1000 600 (3N16) 2 10 ±10 2.2 600 Yes S-D2 20 5N16 1000 1000 (5N16) 2 10 ±10 2.2 600 Yes S-E1 20 5N16 1000 0 1 5 ±10 2.2 600 Yes S-E2 20 5N16 1000 0 1.5 7.5 ±10 2.2 600 Yes S-E3 20 5N16 1000 0 2.5 12.5 ±10 2.2 600 Yes S-E4 20 5N16 1000 0 3 15 ±10 2.2 600 Yes S-F1 20 5N16 1000 0 2 3 ±10 2.2 600 Yes S-F2 20 5N16 1000 0 2 5 ±10 2.2 600 Yes S-F3 20 5N16 1000 0 2 15 ±10 2.2 600 Yes S-G1 20 5N16 1000 0 2 10 ±1 2.2 600 Yes S-G2 20 5N16 1000 0 2 10 ±5 2.2 600 Yes S-G3 20 5N16 1000 0 2 10 ±15 2.2 600 Yes S-G4 20 5N16 1000 0 2 10 ±20 2.2 600 Yes S-H1 20 5N16 1000 0 2 10 ±10 0.3 0 Yes S-H2 20 5N16 1000 0 2 10 ±10 3.9 0 Yes S-H3 20 5N16 1000 0 2 10 ±10 0.3 600 Yes S-H4 20 5N16 1000 0 2 10 ±10 3.9 600 Yes S-I1 20 5N16 1000 0 2 10 ±10 2.2 0 Yes S-I2 20 5N16 1000 0 2 10 ±10 2.2 300 Yes S-I3 20 5N16 1000 0 2 10 ±10 2.2 900 Yes S-J1 20 3N16 600 0 2 10 ±10 2.2 600 No S-J2 20 5N16 1000 0 2 10 ±10 2.2 600 No S-J3 20 8N16 1600 0 2 10 ±10 2.2 600 No
† Reinforcement details for positive moment regions only. †† 3.0* =φ and 9.3* =φ were achieved by setting 4q as 0 and 40 µε/MPa,
respectively.
250
Table 5.5 - Reinforcement details for continuous beam specimens.
Test No. Positive moment region Negative moment region Ast (mm2) Asc (mm2) Ast
- (mm2) Asc- (mm2)
B-A0 to A3; B-E1 to E4; B-F1 to F3; B-G1 to G4; B-H1 to H4; B-I1 to I3; B-J1; B-K1 to K3.
620 (2N20) 0 1350 (3N24) 0
B-B1 400 (2N16) 0 800 (4N16) 0 B-B2 900 (2N24) 0 2040 (2N36) 0 B-C1; B-J2. 930 (3N20) 0 2250 (5N24) 0 B-C2; B-J3. 1240 (4N20) 0 3060 (3N36) 0 B-D1 620 (2N20) 330 (3N12) 1350 (3N24) 620 (2N20) B-D2 620 (2N20) 620 (2N20) 1350 (3N24) 1350 (3N24)
Table 5.6 - Reinforcement details for continuous slab specimens.
Test No. Positive moment region Negative moment region Ast (mm2) Asc (mm2) Ast
- (mm2) Asc- (mm2)
S-A0 to A3; S-E1 to E4; S-F1 to F3; S-G1 to G4; S-H1 to H4; S-I1 to I3; S-J2.
1000 (5N16) 0 2200 (11N16) 0
S-B1 550 (5N12) 0 1100 (10N12) 0 S-B2 1550 (5N20) 0 3720 (12N20) 0 S-C1; S-J1. 600 (3N16) 0 1200 (6N16) 0 S-C2: S-J3. 1600 (8N16) 0 3720 (12N20) 0 S-D1 1000 (5N16) 600 (3N16) 2200 (11N16) 1000 (5N16) S-D2 1000 (5N16) 1000 (5N16) 2200 (11N16) 2200 (11N16)
5.2.3.2 Test Series K: Amount of Shear Reinforcement
This series of tests aims to investigate the influence of the amount of shear
reinforcement on time-dependent cracking in reinforced concrete beams. The tests were
conducted on the continuous beam specimens only, which are subjected to high shear
stresses over the shear span. The continuous beam of test No. B-A0 (see Tables 5.3 and
5.5) was selected as the base specimen and the specimens were analysed with various
amounts of shear reinforcement. The specimens were loaded to 50% of their moment
capacity uM at the critical section at age 14 days. Thereafter the load was kept
constant throughout the test (refer to Table 5.13 for magnitude of load P). The details
251
of the shear reinforcement are shown in Table 5.7 and the following parameters were
kept unchanged: bc = 20 mm, ctf = 2 MPa, maxτ = 10 MPa, fctl = ±10% and bond
creep, creep )2.2( * =φ and shrinkage )600( * µεε =sh were accounted for.
Table 5.7 - Details of shear reinforcement for test series K (continuous beams).
Test No. Stirrup at spacing s (mm) Asv/s (mm) B-K1 No stirrup 0 B-K2 2-legged N10 at 240 0.6667 B-K3 2-legged N10 at 120 1.3333
5.2.3.3 Test Series L: Impact of 500 MPa Steel Reinforcement
To investigate the impact of the introduction of the 500 MPa high yield reinforcing
steel to replace the previously used 400 MPa reinforcing steel, the specimens were
designed as under-reinforced members using both the old and the new types of steel.
The beam specimens were designed to resist ultimate moments of 130 kNm and 250
kNm and the slabs specimens to resist ultimates moments of 115 kNm and 145 kNm.
Only simply supported specimens were tested in this series of test. The specimens
tested were subjected to a load history identical to that for test series A to K, that is, a
moment of 50% of the moment capacity uM of the critical sections was applied at age
14 days and held constant throughout the test. The loads P to produce these moments
are given in Tables 5.11 and 5.12. The details of this test series are shown in Tables 5.8
and 5.9 for beam and slab specimens, respectively, and the following parameters were
held constant: bc = 20 mm, scA = 0, ctf = 2 MPa, maxτ = 10 MPa and fctl = ±10%.
Bond creep, creep )2.2( * =φ and shrinkage )600( * µεε =sh were included.
Furthermore, the same bar size was used in each specimen.
252
Table 5.8 - Combinations of variables for test series L (simply supported beams).
Test No. Mu (kNm) fsy (MPa) ∅st (mm) Ast (mm2) B-L1 130 400 4N16 800 B-L2 130 500 3N16 600 B-L3 250 400 5N20 1550 B-L4 250 500 4N20 1240
Table 5.9 - Combinations of variables for test series L (simply supported slabs).
Test No. Mu (kNm) fsy (MPa) ∅st (mm) Ast (mm2) S-L1 115 400 5N20 1550
S-L2 115 500 4N20 1240
S-L3 145 400 10N16 2000
S-L4 145 500 8N16 1600
5.2.3.4 Test Series M: Load Histories
Five types of load histories as shown in Figures 5.8a to 5.8e were adopted in the
numerical experiments. Each of the loading histories was applied up to age 1000 days.
Load history LH-1 as presented in Figure 5.8a was applied to test series A to K as
described in the previous sections. The remaining load histories were applied only to
simply supported beam and slab specimens with the following material parameters held
constant: mm20=bc , scA = 0, ctf = 2 MPa, MPa10max =τ , %10±=fctl and
inclusive of bond creep, creep )2.2( * =φ and shrinkage )600( * µεε =sh . The
reinforcement details of these specimens are as shown in Table 5.10.
Load history LH-2 is similar to LH-1 but with a lower sustained moment equal to
35% of the moment capacity uM of the specimens. Load histories LH-3 and LH-4 are
designed to examine the effects of unloading and age of unloading on time-dependent
cracking. The specimens were loaded up to uM5.0 at age 14 days and then unloaded
(with specimens only subjected to moments due to their self-weights swM ) at age 50
days and 200 days for load histories LH-3 and LH-4, respectively. For load history
253
Fig. 5.8 - Load histories for simply supported beam and slab specimens: (a) LH-1;
(b) LH-2; (c) LH-3; (d) LH-4; (e) LH-5.
LH-5, the specimens were first loaded up to uM5.0 and then immediately unloaded to
uM35.0 . The load was then held constant up to age 1000 days. The magnitudes of the
load P to achieve the desired moments are given in Tables 5.11 and 5.12.
0.5Mu
Moment
Time (days)14 1000
LH-1
(a)
Moment
Time (days)14
0.35Mu
1000
LH-2
(b)
0.5Mu
Moment
Time (days)14 50 1000
Msw
LH-3
(c)
Moment
Time (days)14 200 1000
0.5Mu
Msw
LH-4
(d)
0.5Mu
Moment
Time (days)14
0.35Mu
1000
LH-5
(e)
254
Table 5.10 - Reinforcement details for test series M (simply supported specimens).
Test No. ∅st (mm) Ast (mm2)
BEAM specimens B-M1 2N20 620
B-M2 3N20 930
B-M3 4N20 1240
SLAB specimens S-M1 3N16 600
S-M2 5N16 1000
S-M3 8N16 1600
Table 5.11 - Magnitudes of external loads for simply supported beams.
Test No. Reinforcement area† Ast (mm2)
Moment M (kNm)
Load P (kN)
SIMPLY SUPPORTED BEAMS B-A0 to A3; B-D1 to D2; B-E1 to E4; B-F1 to F3; B-G1 to G4; B-H1 to H4; B-I1 to I3; B-J1.
620 (2N20) 0.50Mu 32.9
B-B1 400 (2N16) 0.50Mu 19.4 B-B2 900 (2N24) 0.50Mu 49.3 B-C1; B-J2. 930 (3N20) 0.50Mu 51.0 B-C2; B-J3. 1240 (4N20) 0.50Mu 67.9 B-L1 800 (4N16) 0.50Mu 32.0 B-L2 600 (3N16) 0.50Mu 32.0 B-L3 1550 (5N20) 0.50Mu 67.9 B-L4 1240 (4N20) 0.50Mu 67.9 B-M1 620 (2N20) 0.50Mu 32.9 0.35Mu 21.1 B-M2 930 (3N20) 0.50Mu 51.0 0.35Mu 33.7 B-M3 1240 (4N20) 0.50Mu 67.9
0.35Mu 45.5 † Reinforcement details for positive moment regions only.
255
Table 5.12 - Magnitudes of external loads for simply supported slabs.
Test No. Reinforcement area† Ast (mm2)
Moment M (kNm)
Load P (kN)
SIMPLY SUPPORTED SLABS S-A0 to A3; S-D1 to D2; S-E1 to E4; S-F1 to F3; S-G1 to G4; S-H1 to H4; S-I1 to I3; S-J2.
1000 (5N16) 0.50Mu 18.1
S-B1 550 (5N12) 0.50Mu 6.5 S-B2 1550 (5N20) 0.50Mu 31.1 S-C1; S-J1. 600 (3N16) 0.50Mu 7.3 S-C2; S-J3. 1600 (8N16) 0.50Mu 33.3 S-L1 1550 (5N20) 0.50Mu 23.4 S-L2 1240 (4N20) 0.50Mu 23.4 S-L3 2000(10N16) 0.50Mu 32.4 S-L4 1600 (8N16) 0.50Mu 32.4 S-M1 600 (3N16) 0.50Mu 7.3 0.35Mu 2.1 S-M2 1000 (5N16) 0.50Mu 18.1 0.35Mu 9.7 S-M3 1600 (8N16) 0.50Mu 33.3
0.35Mu 20.3 † Reinforcement details for positive moment regions only.
Table 5.13 - Magnitudes of external loads for continuous beams.
Test No. Reinforcement area† Ast (mm2)
Moment M (kNm)
Load P (kN)
CONTINUOUS BEAMS B-A0 to A3; B-D1 to D2; B-E1 to E4; B-F1 to F3; B-G1 to G4; B-H1 to H4; B-I1 to I3; B-J1; B-K1 to K3.
620 (2N20) 0.5Mu 112.0
B-B1 400 (2N16) 0.5Mu 71.5 B-B2 900 (2N24) 0.5Mu 161.0 B-C1; B-J2. 930 (3N20) 0.5Mu 166.1 B-C2; B-J3. 1240 (4N20) 0.5Mu 216.8
† Reinforcement details for positive moment regions only.
256
Table 5.14 - Magnitudes of external loads for continuous slabs.
Test No. Reinforcement area† Ast (mm2)
Moment M (kNm)
Load P (kN)
CONTINUOUS SLABS S-A0 to A3; S-D1 to D2; S-E1 to E4; S-F1 to F3; S-G1 to G4; S-H1 to H4; S-I1 to I3; S-J2.
1000 (5N16) 0.5Mu 73.6
S-B1 550 (5N12) 0.5Mu 36.8 S-B2 1550 (5N20) 0.5Mu 115.5 S-C1; S-J1. 600 (3N16) 0.5Mu 41.0 S-C2; S-J3. 1600 (8N16) 0.5Mu 119.1
† Reinforcement details for positive moment regions only.
5.3 Presentation and Discussion of Results
The presentation of the results for the numerical experiments is organized in the same
sequence as the test series. The numerical test results for all specimens, including the
simply supported beams, the simply supported slabs, the continuous beams and the
continuous slabs, are shown in groups so that a direct comparison of the results for all
types of specimens can be made conveniently.
The numerical test results are shown in terms of widths and spacings of cracks
within the constant moment region (region between the two point loads, see Figure
5.1). Average crack widths were calculated by averaging the widths of the primary
cracks (i.e. crack widths of cracked elements) within the constant moment region.
Crack widths are determined at the soffit of the specimen. In the presentation of the
results, only average and maximum crack widths at age 1000 days (final crack widths),
are shown. In addition, the ratios of final crack width to crack width at instantaneous
loading (initial crack width) are presented to show the change in width of cracks
with time.
257
5.3.1 Test Series A – Bottom Concrete Cover
Figure 5.9 shows the final crack widths and the crack spacings for the specimens
analysed taking bottom concrete cover as the variable parameter. It is evident that the
thickness of the bottom cover has a pronounced effect on crack spacing and crack width
irrespective of the type of structure and the boundary conditions. The cracks are wider
as the thickness of the bottom cover increases. It is seen that the beam and slab
specimens have very similar trends for both final crack width and crack spacing at
various bottom covers. The boundary conditions of the specimens do not seem to have
a prominent effect on crack width and crack spacing.
In a flexural member, the strain is linearly distributed across the section. For the
specimens analysed in this series, the effective depths were held constant both in the
positive and negative (if any) moment regions. Consequently, the sectional strain
distribution between the extreme compressive fibre and the tensile steel level must be
the same for all specimens of the same type. The strain at the extreme tensile fibre
depends on the bottom concrete cover of the specimens and so, clearly, the thicker the
bottom cover is, the larger the strain at the extreme tensile fibre. Therefore the crack
width for sections with thicker bottom cover is always larger since crack width is
directly proportional to the strain in the crack opening direction (see Eq. 3.10 of
Chapter 3).
In addition, the large crack width for sections with thick bottom cover is because
the crack spacings are larger. As seen in Eq. 3.10, it is clear that crack width is
proportional to crack spacing. It is, however, useful to examine the relations between
concrete cover and crack spacing. Taking a reinforced concrete tension chord similar to
that discussed in Section 3.3.1, the rate of stress transfer between concrete and steel
between two cracks depends on the bond stress, the amount of reinforcement and the
area of concrete. From Eq. 3.2 of Chapter 3, which is derived from equilibrium across
the tension chord, it can be seen that the stress transfer rate between concrete and steel
is lower if the concrete area of the tension chord is large. In other words, a tension
chord with low reinforcement ratio has a larger development length of bond (or weaker
bond), which means a longer distance is required to form the next crack and hence a
258
Fig. 5.9 - Final crack widths and crack spacings for test series A: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 20 40 60 80Bottom concrete cover, cb (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(e) (f)
0
50
100
150
200
250
300
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Simply supported slabs Simply supported slabs
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 20 40 60 80Bottom concrete cover, cb (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
300
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
300
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 20 40 60 80Bottom concrete cover, cb (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
Simply supported beams Simply supported beams
(a) (b)
0
50
100
150
200
250
300
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0 20 40 60 80Bottom concrete cover, cb (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(g) (h)
Continuous slabs Continuous slabs
259
larger crack spacing. Moreover, weaker bond also indicates larger slips in the vicinity
of the cracks which causes the cracks to open more widely.
In the case of a flexural member, the tension chord analogy can be applied to the
tensile fibre containing the tensile reinforcement. For a fixed quantity of tensile
reinforcement, the spacing of cracks is larger for a bigger effective area of concrete in
tension surrounding the reinforcing steel since the effective reinforcement ratio (area of
tensile steel divided by effective area of concrete in tension) in the tension region is
low. In both the provisions of CEB-FIP Model Code 1990 (1993) and Eurocode 2
(1992), the effective tensile area of concrete is taken as the width of the section times a
depth equal to 2.5 times the distance from the centroid of the reinforcement to the
tensile face of the section. Therefore, the effective tensile area of concrete is larger with
thicker bottom cover and as a result, both crack spacing and crack width will
be larger.
In addition, the crack spacings and crack widths appear to increase at a decreasing
rate as the bottom cover increases. It seems that the crack spacing is approaching an
upper bound limit no matter how thick the concrete cover is. After a detailed
investigation of the finite element results, it is found that the bottom cover ceases to
affect the effective area of concrete in tension after a threshold bottom cover thickness
is reached. This can be explained by examining the stress distribution of concrete
between any two adjacent cracks in the constant moment region, as shown in Figure
5.10. Concrete stress developed around the surface of the reinforcing steel (at 20 mm
(a) (b)
Fig. 5.10 - Typical stress distribution of section between cracks for the simply
supported beam specimen with: (a) mm20=bc ; (b) mm70=bc .
0
100
200
300
400
500
-8 -6 -4 -2 0 2Concrete stress betw cracks (MPa)
Dis
tanc
e fr
om b
otto
m (m
m)
Instantaneous
t=1000 days
Effective depth of concrete in tension
0
100
200
300
400
500
-8 -6 -4 -2 0 2Concrete stress betw cracks (MPa)
Dis
tanc
e fr
om b
otto
m (m
m)
Instantaneous
t=1000 days
Effective depth of concrete in tension
cb = 20 mm cb = 70 mm
260
and at 70 mm above the soffit of the specimens) through stress transfer via bond. It can
be seen in Figure 5.10a that the bottom cover has limited the potential development of
stress in concrete, which basically results in a smaller effective area of concrete in
tension. For the case of mm70=bc as shown in Figure 5.10b, the bottom cover is
sufficiently thick to allow the full development of tensile stress in concrete. This means
the bottom cover has reached a value where further increase will no longer affect the
effective tensile area of concrete and consequently the crack spacing.
The ratios of final to initial crack width are plotted against bottom cover thickness
for all specimens and are shown in Figure 5.11. It is seen that the time-dependent
percentage increase in crack width is not affected by the thickness of bottom cover. In
addition, the continuous specimens have a slightly larger percentage increase in crack
width with time compared to the simply supported specimens.
Fig. 5.11 - Ratios of final to initial crack width for test series A: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
0.00
1.00
2.00
3.00
4.00
5.00
6.00
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
Simply supported beams Continuous beams
(a) (b)
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 20 40 60 80Bottom concrete cover, cb (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
261
In Figure 5.11b, there is a data point for the widest crack that is particularly
higher than the rest of the data points. This is because the crack that has the largest
width at age 1000 days was not the widest crack at first loading. With time, the crack
opened extensively and became the widest crack.
5.3.2 Test Series B – Diameter of Tensile Reinforcing Steel
In this series of test, the effects of the diameter of tensile reinforcing bars are
investigated. All specimens had the same bottom cover, the same spacing between bars
and the same effective depth. Although the spacing between reinforcing bars cannot be
modelled by a two-dimensional finite element model, it can be indirectly simulated by
using the same number of bars for each specimen and assume that the bars are spaced
sufficiently far to develop full bonding between steel and concrete. In addition, the
specimens were loaded such that the reinforcing steels are resisting the same level
of stress.
The results for final crack width and crack spacing are plotted against bar
diameter and are shown in Figure 5.12. Crack widths are smaller if reinforcing bars of
larger size are used and this applies to all specimen types and boundary conditions. The
crack spacings are also in the same trend as the crack widths. At first glance, one may
be surprised by this result. It seems at odds with the detailing rules for crack control
and conventional wisdom, which is to avoid the use of a small number of large
diameter bars and instead, use a larger number of small diameters bars. In this test,
however, the same number of bars has been used in each specimen. Examining the
results closely, it can be noticed that the total contact surface of the bars and concrete is
in fact the primary factor affecting the distribution of cracks and crack widths. Better
bonding can be achieved by providing larger contact surface area of concrete and steel.
With good bonding, the load transfer rate between concrete and steel will be higher and
therefore cracks form at a closer spacing and are smaller in width. Since the number of
bars were the same for each specimen, the bar diameter becomes the only variable
262
Fig. 5.12 - Final crack widths and crack spacings for test series B: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0
50
100
150
200
250
300
10 15 20 25Bar diameter, Øst (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.450.50
10 15 20 25Bar diameter, Øst (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(g) (h)
Continuous slabs Continuous slabs
0.000.050.100.150.200.250.300.350.400.450.50
10 15 20 25Bar diameter, Øst (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(e) (f)
0
50
100
150
200
250
300
10 15 20 25Bar diameter, Øst (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Simply supported slabs Simply supported slabs
0.000.050.100.150.200.250.300.350.400.450.50
10 15 20 25 30Bar diameter, Øst (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
300
10 15 20 25 30Bar diameter, Øst (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
300
10 15 20 25 30Bar diameter, Øst (mm)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.450.50
10 15 20 25 30Bar diameter, Øst (mm)
Fina
l cra
ck w
idth
(mm
)
AvgMax
Simply supported beams Simply supported beams
(a) (b)
263
affecting the concrete-steel contact surface area. Consequently, the bond between
concrete and steel is better for reinforcing steel with larger diameter. This clearly
explains why the spacing and the width of cracks are smaller for specimens containing
larger bars.
Some design codes stipulate a maximum permissible spacing of tensile
reinforcement to control cracking in reinforced concrete structures. As a consequence,
it is commonly thought that the spacing of reinforcing steel is one of the factors
affecting crack spacing and crack width. However, in the author’s view the concrete-
steel contact surface area is the primary determinant. As long as the reinforcing steels
are spaced sufficiently far apart for the development of optimal bond, the bar spacing is
important only in terms of limiting the number of large tensile bars detailed in a
section. By limiting the maximum bar spacing, structural designers must resort to using
reinforcing steels of smaller sizes in order to provide the required amount of
reinforcement. This indirectly provides more concrete-steel contact surface area and
results in better bonding to distribute cracks.
The ratio of final to initial crack width is shown in Figure 5.13. The bar diameter
appears to have no influence on the crack width ratio. The amount of time-dependent
crack opening is generally larger for the continuous specimens than for the simply
supported specimens. However, the difference is only slight. For the simply supported
slab specimens, it is seen that the crack width ratio calculated from average crack
widths for the specimens containing 5N12 reinforcement (test S-B1) is significantly
higher than other data points (see Figure 5.13c). This is because this slab specimen was
quite lightly reinforced (reinforcement ratio of 0.275%) and the uM5.0 external
moment applied to the specimen only just exceeded the cracking moment. At first
loading the cracks were quite fine. The cracks opened extensively with time and
consequently a high crack width ratio.
264
Fig. 5.13 - Ratios of final to initial crack width for test series B: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
5.3.3 Test Series C – Quantity of Tensile Reinforcement
The effects of the amount of tensile reinforcement are investigated in this series of
tests. Only one single bar size was used for each type of specimen and various
reinforcement ratios were achieved by varying the number of reinforcing bars. The
results for final crack width and crack spacing are plotted against reinforcement ratio
and are shown in Figure 5.14. It is observed that both final crack width and crack
spacing decrease with increasing reinforcement ratio. The results for the simply
supported and continuous beam specimens have very similar trends.
The reduction in the crack width and crack spacing with increasing reinforcement
ratio is in fact, again, due to the increasing contact surface area between the concrete
and the reinforcing steel. Since the bar size was held constant, the number of bars
0.00
0.50
1.00
1.50
2.00
2.50
10 15 20 25 30Bar diameter, Øst (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
10 15 20 25 30Bar diameter, Øst (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(a) (b)
Simply supported beams Continuous beams
0.000.501.001.502.002.503.003.504.00
10 15 20 25Bar diameter, Øst (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.000.501.001.502.002.503.003.504.00
10 15 20 25Bar diameter, Øst (mm)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
265
Fig. 5.14 - Final crack widths and crack spacings for test series C: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
050
100150200250300350400
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
AvgMax
(g) (h)
Continuous slabs Continuous slabs
050
100150200250300350400
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
AvgMax
(e) (f)
Simply supported slabs Simply supported slabs
0.000.050.100.150.200.250.300.350.400.45
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
AvgMax
Simply supported beams Simply supported beams
(a) (b)
266
is the main factor that governs the total contact surface area between concrete and steel.
As discussed earlier in Section 5.3.2, cracks are better distributed with larger concrete-
steel contact surface area. When the reinforcement ratio increases, with the same bar
size, the total concrete-steel contact surface area also increases proportionally, and
therefore crack widths and crack spacings are smaller.
Figure 5.15 shows the ratios of final to initial crack width for all specimens of test
series C. The percentage increase in crack width with time is in the range of 40% to
90% of the instantaneous crack width and it is seen that the quantity of reinforcement
does not significantly affect this crack width ratio.
Fig. 5.15 - Ratios of final to initial crack width for test series C: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
0.000.200.400.600.801.001.201.401.601.802.00
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.000.200.400.600.801.001.201.401.601.802.00
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(a) (b)
Continuous beams Simply supported beams
0.00
0.50
1.00
1.50
2.00
2.50
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
267
5.3.4 Test Series D – Quantity of Compressive Reinforcement
The results for final crack width and crack spacing of this test series are shown in
Figure 5.16. The amount of compressive reinforcement is presented as ratios of
compressive reinforcement area to tensile reinforcement area. From many previous
studies on time-dependent behaviour of reinforced concrete structures (for example,
Gilbert, 1979), it is well known that the use of compressive reinforcement is important
in controlling long-term deflection in flexural members. However, it is seen from the
results that the quantity of compressive reinforcement has no pronounced effect on
crack width and crack spacing.
For a reinforced concrete structure subjected to long-term loads, the compressive
reinforcement restrains the concrete from undergoing excessive creep deformation in
the compression zone of the section. This limits the increase of curvature with time and
consequently reduces the long-term deflection. In addition, the presence of compressive
reinforcement enables a further relief of the compressive stress in the concrete as creep
develops with time. This is because a significant portion of the compressive force is
gradually transferred from the concrete to the compressive reinforcement. Therefore
after a period of long-term loading, the depth of the neutral axis for a cracked member
containing compressive reinforcement is smaller than that without or with less
compressive reinforcement as illustrated in Figure 5.17.
Despite the reduction in creep deformation in the compression zone as well as the
smaller final depth of neutral axis, the resultant influence to the strain in the tensile
reinforcement is insignificant. This is also shown schematically in Figure 5.17. As a
result, although the presence of compressive reinforcement is important to the long-
term deflection of reinforced concrete structures, it is not important with regard to the
spacing and the opening of cracks.
Figure 5.18 shows the ratios of the final to initial crack width of the specimens.
The ratios are in the same range as those of the previously presented test series.
Moreover, the percentage increase of crack width under sustained loads is not sensitive
to the quantity of compressive reinforcement.
268
Fig. 5.16 - Final crack widths and crack spacings for test series D: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0
50
100
150
200
250
300
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.450.50
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Fina
l cra
ck w
idth
(mm
)
AvgMax
(g) (h)
Continuous slabs Continuous slabs
0.000.050.100.150.200.250.300.350.400.450.50
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Fina
l cra
ck w
idth
(mm
)
AvgMax
(e) (f)
0
50
100
150
200
250
300
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Simply supported slabs Simply supported slabs
0.000.050.100.150.200.250.300.350.400.45
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
300
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
300
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Fina
l cra
ck w
idth
(mm
)
AvgMax
Simply supported beams Simply supported beams
(a) (b)
269
(a) (b)
Fig. 5.17 - Schematic description of the long-term effects of compressive
reinforcement on tensile steel strain: (a) section without compressive
reinforcement; (b) section with compressive reinforcement.
Fig. 5.18 - Ratios of final to initial crack width for test series D: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
dNA’ 1 κ’
εst
dNA’ 1
κ’
εst
εsc ∆εsc
Instantaneous strain
Final strain
Instantaneous steel strain
Steel strain increment over time
0.00
0.50
1.00
1.50
2.00
2.50
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(a) (b)
Simply supported beams Continuous beams
0.00
0.50
1.00
1.50
2.00
2.50
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0 0.5 1 1.5Compression steel quantity, Asc/Ast
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
270
5.3.5 Test Series E – Tensile Strength of Concrete
The bond strength between concrete and steel is closely related to the properties of
concrete, such as compressive strength and tensile strength. This is evident in many
bond models, which were formulated as a function of either the compressive strength or
the tensile strength of concrete (for instance, Dörr, 1980 and Ciampi et al., 1981). For
the same type of deformed bar the bond strength increases with increasing concrete
tensile strength (or concrete compressive strength). To achieve a more realistic study of
the influence of concrete tensile strength on crack width and crack spacing, the bond
strength of each specimen was adjusted accordingly with the concrete tensile strengths
using the method described in Section 5.2.3.1.
The final crack width and crack spacing versus concrete tensile strength diagrams
are shown in Figure 5.19. It should be pointed out that the absence of results for the
continuous slab specimens in Figures 5.19g and 5.19h at MPa1=ctf is due to the fact
that the concrete tensile strength is too low and results in a large shear crack occurring
in the negative moment region, which leads to a premature shear failure.
In Figure 5.19, the crack spacings at first loading tend to increase with increasing
concrete tensile strength. At age 1000 days the crack spacings reduce to a constant final
value for concrete tensile strengths between 2.0 MPa and 3.0 MPa. The influence of
concrete tensile strength on final crack width is, however, less significant than its
influence on crack spacing except for the continuous slab specimens (Figure 5.19g).
The crack widths have a slight ascending trend for concrete tensile strengths between
1.0 MPa and 1.5 MPa and remain roughly constant or decrease slightly with further
increase in tensile strength. The continuous slab specimens have the most obvious
descending final crack widths at high concrete tensile strengths. On the whole, the
effects of concrete tensile strength on final crack spacings and final crack widths are
not very significant.
Both concrete tensile strength and bond strength are important factors affecting
the formation of cracks in a reinforced concrete structure. Bond strength is a governing
factor of the rate of stress transfer between concrete and steel. With higher bond
271
strength, the stress transfer rate is higher and therefore the distance required to reach
the concrete tensile strength is shorter, which means a smaller crack spacing. On the
other hand, for a given bond characteristic (or stress transfer rate), higher concrete
tensile strength results in larger crack spacing since a relatively long distance is
required for the concrete stress to develop to the tensile strength from an existing crack.
In this series of tests, bond strength is taken to be directly proportional to the concrete
tensile strength of the specimens. Consequently, the effect of bond strength to a large
extent cancels out the effect of concrete tensile strength and results in an almost
constant final crack spacing especially at tensile strengths between 1.5 MPa and 3.0
MPa. As a consequence, the crack widths are also nearly constant for various concrete
tensile strengths except for those in the continuous slab specimens.
The decrease of crack widths with increasing concrete tensile strengths as
observed in Figure 5.19 is attributed to the fact that the concrete was too strong to crack
extensively at instantaneous loading. As the concrete tensile strength increases, a point
is reached that the specimens only just crack at the service load levels. Although
cracking was successfully initiated in these specimens, the initial crack width was
small, but widened significant with time. This can also be seen in Figure 5.20 where the
ratios of final to initial crack width are almost constant at low concrete tensile strengths
but increase significantly after the concrete tensile strength exceeds about 2.5 MPa.
This indicates that the initial crack widths are small compared to the final crack widths.
Although cracks in the specimens with high concrete tensile strength opened with time,
they did not open as much as the cracks in those specimens with lower tensile strengths
which had cracked more extensively at instantaneous loading. Therefore, the final crack
width decreases with increasing tensile strengths.
The results presented herein for this series of tests were obtained using a
particular relationship between the bond strength and the concrete tensile strength. It
should be pointed out that if a different relationship is used the results may vary
accordingly. However, due to the combined effects of bond strength and concrete
tensile strength, the author believes that the final outcome would be similar to that
obtained in this investigation, for which the overall effects of variations in the concrete
272
Fig. 5.19 - Final crack widths and crack spacings for test series E: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0
50
100
150
200
250
300
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.450.50
0 1 2 3 4Concrete tensile strength, fct (MPa)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(g) (h)
Continuous slabs Continuous slabs
0
50
100
150
200
250
300
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.450.50
0 1 2 3 4Concrete tensile strength, fct (MPa)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(e) (f)
Simply supported slabs Simply supported slabs
0.000.050.100.150.200.250.300.350.400.450.50
0 1 2 3 4Concrete tensile strength, fct (MPa)
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
300
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
300
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.450.50
0 1 2 3 4Concrete tensile strength, fct (MPa)
Fina
l cra
ck w
idth
(mm
)
AvgMax
Simply supported beams Simply supported beams
(a) (b)
273
Fig. 5.20 - Ratios of final to initial crack width for test series E: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
tensile strength are not significant to crack spacings and crack widths. This is evident in
the crack spacing equations proposed in many previous studies, for example CEB-FIP
Model Code 1990 (1993) and Eurocode 2 (1992), where concrete tensile strength is not
taken as a critical factor affecting the spacing of cracks. In addition, although the
tension chord model crack spacing equation used in this study (see Eq. 3.4 of Chapter
3) has included concrete tensile strength as one of the governing parameters, the use of
a direct proportionality between bond stress and tensile strength in fact cancels out the
influence of concrete tensile strength on crack spacings, and hence crack widths.
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(a) (b)
Simply supported beams Continuous beams
0.000.501.001.502.002.503.003.504.004.50
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.000.501.001.502.002.503.003.504.00
0 1 2 3 4Concrete tensile strength, fct (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
274
5.3.6 Test Series F – Bond Strength between Steel and Concrete
Figure 5.21 shows the final crack widths and crack spacings plotting against the
maximum bond capacity of the specimens. The importance of bond between concrete
and steel to the formation of cracks has been extensively emphasized throughout this
thesis. Therefore, bond strength is well expected to have a significant effect on crack
width and crack spacing. Although it is apparent in Figure 5.21 that crack width and
crack spacing reduce with increasing maximum bond capacity, it should be noted that it
is not the maximum bond capacity that affects the formation of cracks but the
characteristics of the of the bond model.
The bond-slip model of CEB-FIP Model Code 1990 (1993) has an infinite initial
modulus and the gradient of the bond-slip curve reduces with increasing slip. Upon
reaching the bond strength, the bond stress descends to a residual frictional bond stress
corresponding to a pull-out failure. Under service load conditions, bond stresses are
well below the maximum bond capacity. Figure 5.22 shows the bond stress versus slip
curves for the CEB-FIP bond models used in this series of tests. It can be seen that
bond is stiffer (higher bond stress at the same slip) for the curves with higher bond
strength. A stiff bond model has better stress transfer capability and consequently
reduces the spacing and width of cracks.
Another interesting observation is the difference between maximum and average
crack widths at various bond strengths. The difference between maximum and average
crack widths is large at low bond strength but reduces as bond strength increases. For
specimens with low bond strength, the widest crack has a noticeably larger width
compared to other cracks of the same specimens and it tends to occur below the third
point loads where the specimens undergo a sudden change in curvature. This is due to
the poor distribution of cracking resulting from low bond strength. The large crack
spacing causes the cracks to open more widely at locations with a sudden change in
curvature such as the third points of the specimens. On the other hand, with high bond
strength the cracks are spaced more closely and the opening of cracks due to change in
curvature can be easily distributed between the nearby cracks. Consequently, the
difference between maximum and average crack width is much smaller.
275
Fig. 5.21 - Final crack widths and crack spacings for test series F: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Fina
l cra
ck w
idth
(mm
) AvgMax
(e) (f)
050
100150200250300350400
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Simply supported slabs Simply supported slabs
050
100150200250300350400
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Fina
l cra
ck w
idth
(mm
) AvgMax
(g) (h)
Continuous slabs Continuous slabs
0
50
100
150
200
250
300
350
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Fina
l cra
ck w
idth
(mm
) AvgMax
Simply supported beams Simply supported beams
(a) (b)
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Fina
l cra
ck w
idth
(mm
) AvgMax
0
50
100
150
200
250
300
350
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
276
Fig. 5.22 - CEB-FIP bond models used in test series F.
Figure 5.23 shows the ratios of final to initial crack width plotted against bond
strength. Bond strength appears to have little influence on the percentage increase of
crack width with time. The slightly higher crack width ratios for the continuous beam
Fig. 5.23 - Ratios of final to initial crack width for test series F: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
02468
10121416
0 0.5 1 1.5Slip (mm)
Bond
stre
ss (M
Pa)
τmax = 3 MPa
τmax = 5 MPa
τmax = 10 MPa
τmax = 15 MPa
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(a) (b)
Simply supported beams Continuous beams
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 5 10 15 20Maximum bond capacity, τmax (MPa)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
277
specimens with MPa3max =τ and MPa5max =τ are due to the switch of the largest
cracks during the sustained loading period, which is of the same reason as discussed in
the last paragraph of Section 5.3.1 for test series A.
5.3.7 Test Series G – Concrete Tensile Strength Fluctuation Limit
The randomly generated concrete tensile strengths were incorporated in the finite
element model for two reasons. The first reason is to model the stochastic nature of
concrete tensile strength. The second is to avoid bifurcation of the numerical solutions.
The interests of this test series are twofold: to investigate the effects of the variability of
concrete tensile strength for a given mean value and, to examine the effects of the
fluctuation limit on the numerical outcomes.
The results for final crack width and crack spacing are plotted versus concrete
tensile strength fluctuation limit in Figure 5.24. It is seen that the fluctuation limit does
not have significant influence on crack width or crack spacing. At low fluctuation
limits, the crack widths and crack spacings tend to be slightly smaller than those at high
fluctuation limits. This is probably because of the tensile stresses just prior to cracking
in the concrete elements at the soffits were so close that more cracks tended to form at
the same instance. Similarly for the case using a single concrete tensile strength
%)0( =fctl , one would expect the concrete elements at the soffit to crack at the same
time. In addition, it is seen that the effects of fluctuation of concrete tensile strength
vanishes at fluctuation limits beyond 10%.
The ratios of final to initial crack width are presented in Figure 5.25. The ratios
for this test series fall between 1.5 and 2, which are similar to those for the earlier test
series. Furthermore, the percentage increase in crack width with time is unaffected by
the concrete tensile strength fluctuation limit.
278
Fig. 5.24 - Final crack widths and crack spacings for test series G: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0
50
100
150
200
250
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.450.50
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(g) (h)
Continuous slabs Continuous slabs
0.000.050.100.150.200.250.300.350.400.450.50
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Fina
l cra
ck w
idth
(mm
)
AvgMax
(e) (f)
0
50
100
150
200
250
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Simply supported slabs Simply supported slabs
0.000.050.100.150.200.250.300.350.400.45
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Fina
l cra
ck w
idth
(mm
)
AvgMax
Simply supported beams Simply supported beams
(a) (b)
279
Fig. 5.25 - Ratios of final to initial crack width for test series G: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
5.3.8 Test Series H – Magnitude of Creep
The final crack width and crack spacing of the specimens are plotted against final creep
coefficient in Figure 5.26. The ratios of final to initial crack width are shown in Figure
5.27. The results show that cracks are slightly larger in width with increasing creep but
the crack spacings are not sensitive to the amount of creep. The final crack widths of
the specimens with a final shrinkage strain of 600 µε have a similar trend as those
without shrinkage. It should be pointed out that the crack patterns within the constant
moment regions of the specimens in this test series have fully developed at
instantaneous loading and therefore crack spacings do not reduce further under the
effects of shrinkage. For this reason, the crack spacings of the specimens with
shrinkage )600( * µεε =sh and without shrinkage )0( * µεε =sh are not dissimilar as
0.00
0.50
1.00
1.50
2.00
2.50
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(a) (b)
Simply supported beams Continuous beams
0.00
0.50
1.00
1.50
2.00
2.50
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0 5 10 15 20 25Concrete tensile strength limit, lfct (%)
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
280
Fig. 5.26 - Final crack widths and crack spacings for test series H: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0.000.050.100.150.200.250.300.350.400.45
0 1 2 3 4 5Final creep coefficient, φ*
Fina
l cra
ck w
idth
(mm
)
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
0
50
100
150
200
250
0 1 2 3 4 5Final creep coefficient, φ*
Crac
k sp
acin
g (m
m)
Instantaneoust=1000 days
Simply supported beams Simply supported beams
(a) (b)
0
50
100
150
200
250
0 1 2 3 4 5Final creep coefficient, φ*
Crac
k sp
acin
g (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0 1 2 3 4 5Final creep coefficient, φ*
Fina
l cra
ck w
idth
(mm
)
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
300
0 1 2 3 4 5Final creep coefficient, φ*
Crac
k sp
acin
g (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0 1 2 3 4 5Final creep coefficient, φ*
Fina
l cra
ck w
idth
(mm
)
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
(e) (f)
Simply supported slabs Simply supported slabs
0
50
100
150
200
250
300
0 1 2 3 4 5Final creep coefficient, φ*
Crac
k sp
acin
g (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0 1 2 3 4 5Final creep coefficient, φ*
Fina
l cra
ck w
idth
(mm
)
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
(g) (h)
Continuous slabs Continuous slabs
281
Fig. 5.27 - Ratios of final to initial crack width for test series H: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
shown in Figures 5.26b, 5.26d, 5.26f and 5.26h. In addition, the crack width ratios also
have the same trend as the crack widths, which tend to be higher at larger final creep
coefficients.
Initially, the author thought that creep of concrete might be beneficial in reducing
crack width of the specimens. The reason for this is that, the concrete between cracks
undergoes tension stiffening and it expands longitudinally in the direction of
reinforcing steel towards the cracks due to elasticity and creep. Higher creep increases
the time-dependent expansion of the concrete between cracks and causes the cracks to
close. To illustrate this, the results of the continuous beam specimen with µεε 0* =sh
for 3.0* =φ and 9.3* =φ are presented. The typical stress and strain distributions at a
section between two adjacent cracks within the constant moment region are shown in
Figures 5.28. Comparing Figures 5.28a and 5.28c, the concrete strain surrounding the
0.000.200.400.600.801.001.201.401.601.802.00
0 1 2 3 4 5Final creep coefficient, φ*
Crac
k wi
dth
ratio
, wcr
.f/wcr
.i
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
0.000.200.400.600.801.001.201.401.601.802.00
0 1 2 3 4 5Final creep coefficient, φ*
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
(a) (b)
Simply supported beams Continuous beams
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4 5Final creep coefficient, φ*
Cra
ck w
idth
ratio
, wcr
.f/wcr
.i
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
0.00
0.50
1.00
1.50
2.00
2.50
0 1 2 3 4 5Final creep coefficient, φ*
Crac
k wi
dth
ratio
, wcr
.f/w
cr.i
AvgMaxAvgMax
(εsh*=0 µε)(εsh*=0 µε)(εsh*=600 µε)(εsh*=600 µε)
(c) (d)
Simply supported slabs Continuous slabs
282
reinforcing steel (indicated by dashed circle) for 9.3* =φ is clearly larger than that for
3.0* =φ , although the concrete tension stiffening stresses for both specimens are of the
same magnitude (about 1.6 MPa as shown in Figures 5.28b and 5.28d). Nevertheless,
the test results are not as expected and crack widths tend to be larger for concrete with
higher final creep coefficients.
After a detailed investigation, it was found that creep in fact has a noticeable
effect on the time-dependent change in steel stress. Figures 5.29a and 5.29b compare
the time-dependent increase in steel stress between 9.3* =φ and 3.0* =φ for the
continuous beam specimens without the effect of shrinkage. It is seen that concrete with
larger creep actually causes larger increase in steel stress and results in widening of
cracks with time. The increase in crack width caused by the increase in steel stress is
(a) (b)
(c) (d)
Fig. 5.28 - Typical stress and strain distributions between cracks for continuous beam
specimens: (a) strain diagram for 9.3* =φ ; (b) stress diagram for 9.3* =φ ;
(c) strain diagram for 3.0* =φ ; (d) stress diagram for 3.0* =φ .
0
100
200
300
400
500
-10 -5 0 5Concrete stress betw cracks (MPa)
Dis
tanc
e fr
om b
otto
m (m
m)
Instantaneous
t=1000 days (εsh*=0 µε)0
100
200
300
400
500
-1500 -1000 -500 0 500Concrete strain betw cracks (µε )
Dis
tanc
e fr
om b
otto
m (m
m)
Instantaneous
t=1000 days
smaller expansion in concrete betw cracks
(εsh*=0 µε)
3.0* =φ 3.0* =φ
0
100
200
300
400
500
-10 -5 0 5Concrete stress betw cracks (MPa)
Dis
tanc
e fr
om b
otto
m (m
m)
Instantaneous
t=1000 days (εsh*=0 µε)
0
100
200
300
400
500
-1500 -1000 -500 0 500Concrete strain betw cracks (µε )
Dis
tanc
e fr
om b
otto
m (m
m)
Instantaneous
t=1000 days
larger expansion in concrete betw cracks
(εsh*=0 µε)
9.3* =φ 9.3* =φ
283
(a) (b)
Fig. 5.29 - Longitudinal distribution of stress for bottom reinforcing steel of
continuous beam specimens for: (a) 9.3* =φ ; (b) 3.0* =φ .
larger than the crack closing effect resulting from creep induced expansion of concrete
between cracks. Consequently, the final outcome is the increase in crack width with
increasing final creep.
Another important observation from this test series is the extent of cracking with
time in the presence of shrinkage. Figures 5.30 and 5.31 show the crack patterns for
beam and slab specimens (all specimens shown have a final shrinkage strain
µεε 600* =sh ), respectively, and compare the effects for 3.0* =φ and 9.3* =φ . The
crack patterns at instantaneous loading are the same for both 3.0* =φ and 9.3* =φ
since creep under short-term loading is insignificant. It can be seen that the simply
supported specimens with very low creep (Figures 5.30c and 5.31c) underwent much
more severe time-dependent cracking in the shear span than those with higher creep
(Figures 5.30e and 5.31e). This is due to the fact that creep relieves the shrinkage-
induced tension in concrete caused by the restraint produced by the reinforcing steel. In
absence of creep (or with very low creep), the shrinkage-induced tension in concrete at
the level of the reinforcing steel develops and eventually causes further cracking in the
shear span. However, this phenomenon is not conspicuous in the continuous specimens
due to the nature of the moment distribution in which there is a transition of sagging to
hogging moments.
Although higher creep causes larger time-dependent opening of cracks, its
presence is of crucial importance in preventing extensive time-dependent cracking in
reinforced concrete structures.
-100
0
100
200
300
0 1000 2000 3000Distance from left edge (mm)
Stee
l str
ess
(MPa
) Instantaneous
t=1000 days(εsh*=0 µε)
-100
0
100
200
300
0 1000 2000 3000Distance from left edge (mm)
Stee
l str
ess
(MPa
) Instantaneous
t=1000 days(εsh*=0 µε)
9.3* =φ 3.0* =φ
284
Fig. 5.30 - Crack patterns (cracking strains) for beams: (a) and (b) instantaneous crack
patterns for simply supported and continuous beams, respectively; (c) and
(d) final crack patterns calculated using 3.0* =φ for simply supported and continuous beams, respectively; (e) and (f) final crack patterns calculated
using 9.3* =φ for simply supported and continuous beams, respectively. Fig. 5.31 - Crack patterns (cracking strains) for slabs: (a) and (b) instantaneous crack
patterns for simply supported and continuous slabs, respectively; (c) and
(d) final crack patterns calculated using 3.0* =φ for simply supported and continuous slabs, respectively; (e) and (f) final crack patterns calculated
using 9.3* =φ for simply supported and continuous slabs, respectively.
(c) (d)
(e) (f)
(a) (b)
(c) (d)
(e) (f)
(a) (b)
285
5.3.9 Test Series I – Magnitude of Shrinkage
It is well known that shrinkage is the most important factor affecting the time-
dependent growth of cracks in reinforced concrete structures. This is further reassured
in this series of tests. Figure 5.32 shows the final crack widths and crack spacings for
the specimens plotted against final shrinkage strain. The effects of shrinkage on crack
width are apparent. Since the crack patterns of the specimens have fully developed at
instantaneous loading, the crack spacings are not very sensitive to the amount of
shrinkage except for the specimens with the largest final shrinkage strain
)900( * µεε =sh , where a reduction in crack spacing is observed at age 1000 days. It is
also seen that the width of cracks increases linearly with the final shrinkage strains.
The mechanisms of the widening of cracks with time by shrinkage are
straightforward. The specimens of this test series were pre-cracked at instantaneous
loading. As the concrete between each adjacent crack shrinks the crack widths increase.
The amount of time-dependent widening of cracks depends on the magnitude and rate
of shrinkage. The ratios of final to initial crack width versus final shrinkage strain are
shown in Figure 5.33. The percentage increase in crack width with time is larger if the
concrete has a higher final shrinkage strain and the relationship also appears to
be linear.
5.3.10 Test Series J – Bond Creep
The effects of bond creep were investigated herein with three different reinforcement
ratios for each specimen. The specimens were analysed with and without bond creep.
The results for analyses with bond creep are identical to those shown in Figure 5.14 of
Section 5.33. The final crack widths and crack spacings for the specimens both
inclusive and exclusive of bond creep are plotted versus reinforcement ratio in Figure
5.34. The results for the specimens with bond creep are indicated as control specimens
in the figures.
286
Fig. 5.32 - Final crack widths and crack spacings for test series I: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0.000.050.100.150.200.250.300.350.400.45
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
300
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
Continuous beams Continuous beams
(c) (d)
0
50
100
150
200
250
300
350
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Fina
l cra
ck w
idth
(mm
)
AvgMax
(g) (h)
Continuous slabs Continuous slabs
0
50
100
150
200
250
300
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
0.000.050.100.150.200.250.300.350.400.45
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Fina
l cra
ck w
idth
(mm
)
AvgMax
Simply supported beams Simply supported beams
(a) (b)
0.000.050.100.150.200.250.300.350.400.45
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Fina
l cra
ck w
idth
(mm
)
AvgMax
0
50
100
150
200
250
300
350
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck s
paci
ng (m
m)
Instantaneoust=1000 days
(e) (f)
Simply supported slabs Simply supported slabs
287
Fig. 5.33 - Ratios of final to initial crack width for test series I: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
The final crack widths for the specimens without bond creep in Figure 5.34 have
the same trends as those with bond creep, but with a somewhat reduced magnitude.
This indicates that inclusion of bond creep increases the crack widths. The crack
spacings for the specimens without bond creep are generally lower compared to those
with bond creep. This is due to the fact that bond without taking account of bond creep
is much stiffer and has better stress transfer between concrete and steel, which
consequently results in smaller crack spacing.
Figure 5.35 shows the ratios of final to initial crack width of the test specimens.
The crack width ratios for specimens with bond creep are slightly higher than those
without bond creep, which indicates bond creep causes a further increase in crack
opening with time.
0.000.200.400.600.801.001.201.401.601.802.00
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck w
idth
s ra
tio, w
cr.f/
w cr.
i
AvgMax
0.000.200.400.600.801.001.201.401.601.802.00
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck w
idth
s ra
tio, w
cr.f/
w cr.
i
AvgMax
(a) (b)
Simply supported beams Continuous beams
0.00
0.50
1.00
1.50
2.00
2.50
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck w
idth
ratio
, wcr
.f/w c
r.i
AvgMax
0.00
0.50
1.00
1.50
2.00
2.50
0 200 400 600 800 1000Final shrinkage strain, ε sh* (µε )
Cra
ck w
idth
ratio
, wcr
.f/w c
r.i
AvgMax
(c) (d)
Simply supported slabs Continuous slabs
288
Fig. 5.34 - Final crack widths and crack spacings for test series J: (a) and (b) simply
supported beams; (c) and (d) continuous beams; (e) and (f) simply
supported slabs; (g) and (h) continuous slabs.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
(e) (f)
050
100150200250300350400
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Inst. (no bondcrp)t=1000days (no bondcrp)Inst. (control)t=1000days (control)
Simply supported slabs Simply supported slabs
050
100150200250300350400
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Inst. (no bondcrp)t=1000days (no bondcrp)Inst. (control)t=1000days (control)
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
(g) (h)
Continuous slabs Continuous slabs
0
50
100
150
200
250
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Inst. (no bondcrp)t=1000days (no bondcrp)Inst. (control)t=1000days (control)
0.000.050.100.150.200.250.300.350.400.45
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
Simply supported beams Simply supported beams
(a) (b)
0.000.050.100.150.200.250.300.350.400.45
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Fina
l cra
ck w
idth
(mm
)
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
0
50
100
150
200
250
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck s
paci
ng (m
m)
Inst. (no bondcrp)t=1000days (no bondcrp)Inst. (control)t=1000days (control)
Continuous beams Continuous beams
(c) (d)
289
Fig. 5.35 - Ratios of final to initial crack width for test series J: (a) simply supported
beams; (b) continuous beams; (c) simply supported slabs; (d) continuous
slabs.
5.3.11 Test Series K – Quantity of Shear Reinforcement
As mentioned previously in Section 5.2.1, the shear stresses in the shear span of the
continuous beam specimens are high and therefore shear reinforcement must be
provided to prevent premature shear failure in the specimens. The idealisation of
reinforcement as one-dimensional truss elements in the finite element model does not
account for the stress concentration effects caused by the geometry of stirrups.
However, the primary interest of this study is to investigate time-dependent cracking
related to longitudinal reinforcement and the inclusion of shear reinforcement is merely
for the sake of preventing unwanted shear failure in the specimens. Therefore, instead
of attempting to model the precise physical influence of stirrups in a real beam, the
actual purpose of this test series is to examine the influence of the amount of shear
resisting truss elements on the results of the numerical model.
0.00
0.50
1.00
1.50
2.00
2.50
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
0.00
0.50
1.00
1.50
2.00
2.50
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
(a) (b)
Simply supported beams Continuous beams
0.00
0.50
1.00
1.50
2.00
2.50
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
0.00
0.50
1.00
1.50
2.00
2.50
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
ck w
idth
ratio
, wcr
.f/w
cr.i
Avg (no bondcrp)Max (no bondcrp)Avg (control)Max (control)
(c) (d)
Simply supported slabs Continuous slabs
290
The results of final crack width, crack spacing and final to initial crack width ratio
are plotted against quantity of shear reinforcement are shown in Figures 5.36a, 5.36b
and 5.36c, respectively. The amount of shear reinforcement has no effect on the
development of flexural cracks, either in terms of width, spacing or increase of width
with time. The outcome of this test series does not agree with the experimental
observation of Divakar and Dilger (1987), from which they concluded that cracking
tends to occur at the location of stirrups. However, for the purpose of this study, it may
be concluded that the inclusion of shear reinforcement does not induce any
unanticipated effects on the numerical results. This is important in the sense of
facilitating an objective comparison of results between the simply support beam and
continuous beam specimens.
Fig. 5.36 - Results of continuous beam specimens for test series K: (a) and (b) final
crack width and cracking spacing, respectively; (c) ratio of final to initial
crack width.
0.000.200.400.600.801.001.201.401.601.802.00
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020406080
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Instantaneoust=1000 days
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291
5.3.12 Test Series L – Impact of 500 MPa Steel Reinforcement
Figure 5.37 shows the results for final crack width, crack spacing and final to initial
crack width ratio plotted against moment capacity of the specimens. It is seen that the
final crack widths for specimens using the reinforcement with MPa500=syf are
generally larger than those containing reinforcement with MPa400=syf .
For under-reinforced flexural members, the reinforcing steel reaches the yield
stress before the concrete fails in crushing. Consequently, the moment capacity of a
section is proportional to the strength of the reinforcement. Comparing the two types of
reinforcement, the use of 500 MPa steel inevitably reduces the quantity of steel
required for a desired strength. Using the same bar size, the required number of bars in
a specimen containing 500 MPa steel must be fewer than that using 400 MPa steel. The
consequence of this is a smaller concrete-steel contact surface area, which means
weaker bond. Weak bond results in large crack spacing and wider cracks. This clearly
explains why the use of 500 MPa steel leads to larger crack widths and crack spacing as
observed in Figure 5.37.
In addition, under the same loads, smaller quantities of steel also result in higher
tensile steel stresses. This is exactly the case for the specimens containing 500 MPa
steel. The higher steel stress in the 500 MPa steel inevitably results in wider cracks.
This is evidenced in Figures 5.37a and 5.37b for the beam specimens with
kNm130=uM . The crack spacings are the same for both beams containing 400 MPa
and 500 MPa steel, however, crack widths are significantly different. This is because of
the significantly higher tensile stress in the 500 MPa steel than in the 400 MPa steel at
service loads.
In Figures 5.37e and 5.37f, it is seen that the percentage increase in crack width
for the specimens with 400 MPa steel are in general higher than those with 500 MPa
steel. This is, again, due to the difference of the tensile steel stresses in these two types
of steel reinforcement. The lower tensile steel stress in the 400 MPa steel results in less
292
Fig. 5.37 - Results for test series L: (a) and (c) final crack widths for simply
supported beams and slabs, respectively; (b) and (d) crack spacings for
simply supported beams and slabs, respectively; (e) and (f) ratios of final to
initial crack width for simply supported beams and slabs, respectively.
expansion in the concrete between two adjacent cracks due to creep and elasticity (also
see the discussion in Section 5.3.8 for test series H). With less expansion, the cracks of
the specimens with 400 MPa steel opened more widely with time due to shrinkage.
It is evident that the use of 500 MPa steel reinforcement changes the way
reinforced concrete structures behave at service load conditions. The consequent
020406080
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Simply supported beams Simply supported slabs
293
reduction of tensile reinforcement adversely increases the tensile steel stress. In
addition, the reduction of steel quantity can easily lead to reduced bond between
concrete and steel, particularly if the detailing of reinforcement is not done with care.
This eventually results in larger crack spacings. The consequence of these is wider
cracks and an increased likelihood of serviceability problem.
5.3.13 Test Series M – Load Histories
5.3.13.1 Comparisons between LH-2 and LH-1
In this part of the test series, the effects of the sustained load levels are investigated.
The final crack width, crack spacing and ratios of final to initial crack width for two
different sustained load levels are presented in Figure 5.38. The specimens were
subjected to either load histories LH-2 (sustained moment uM35.0 ) or LH-1 (sustained
moment uM5.0 ). It should be noted in Figures 5.38c, 5.38d and 5.38f that the data
points for the lightly reinforced slab containing 3N16 reinforcing bars (ρ = 0.003)
subjected LH-2 are absent because the uM35.0 applied moment was too low to initiate
cracking in the slab specimen.
As expected, the final crack widths are, on the whole, smaller for the specimens
subjected to lower sustained loads. However, it is surprising to see that the final crack
widths caused by a sustained moment of uM35.0 are not very much smaller than those
caused by uM5.0 . The specimens subjected to the lower sustained moments generally
have a larger crack spacing, due to the fact that a stabilized crack pattern does not form
initially under the smaller loads. The large crack spacings cancel out the effects of low
tensile stress in the steel due to the smaller sustained moment uM35.0 . Therefore the
crack widths are not much smaller than those caused by uM5.0 .
294
Fig. 5.38 - Test series M – comparisons of results for load histories LH-2 (sustained
moment 0.35Mu) and LH-1 (sustained moment 0.5Mu): (a) and (c) final
crack widths for simply supported beams and slabs, respectively; (b) and
(d) crack spacings for simply supported beams and slabs, respectively; (e)
and (f) ratios of final to initial crack width for simply supported beams and
slabs, respectively.
An interesting feature is observed in Figures 5.38e and 5.38f, which show the
final to initial crack width ratios for the specimens. The percentage increase in crack
width for the specimens subjected to uM35.0 are noticeably higher than that for
050
100150200250300350400
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
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Inst. (LH-2)t=1000days (LH-2)Inst. (LH-1)t=1000days (LH-1)
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Simply supported slabs Simply supported slabs
(c) (d)
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, wcr
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(e) (f)
Simply supported beams Simply supported slabs
295
specimens subjected to uM5.0 . This indicates that the crack widths for the specimens
loaded at uM35.0 were initially small at instantaneous loading but increased
significantly with time.
Figures 5.39a and 5.39b show the typical stress distributions on the sections
containing cracks resulting from load histories LH-1 and LH-2, respectively, for the
simply supported beams. For the beam subjected to sustained moment uM5.0 , the
concrete stress at the crack (indicated by a dashed circle in Figure 5.39a) has
completely reduced to zero as expected. However, for the beam subjected to lower
sustained moment it is seen that some residual concrete tensile stresses are still present
at the crack (dashed circle in Figure 5.39b). These residual stresses are the cohesive
stresses in the concrete at the crack front (fracture process zone), which indicates that a
portion of the crack is still undergoing the process of fracture (tension softening). The
implication of this is that, the beam specimen subjected to sustained moment uM35.0
is yet to be thoroughly cracked. In other words, the crack faces were not completely
separated. With time, the cohesive stresses started to dissipate as the cracks widened
due to shrinkage of concrete. This can be seen in Figure 5.39b where the crack is
completely stress-free at age 1000 days.
The phenomenon described above can explain why the lightly loaded specimens
have a higher ratio of final to initial crack width. In the lightly loaded specimens,
although cracking has been initiated upon loading, the cracks were unable to open
freely due to the presence of residual stress in the cracks. Shrinkage of concrete
overcame the residual stress and widened the cracks as they developed with time.
This observation is not merely academic but actually happens to reinforced
concrete structures in service. This observation has important implications to long-term
serviceability of reinforced concrete structures. Fine cracks in a lightly loaded structure
that seem to be acceptable under short-term loading can turn out to be large unsightly
cracks, or more seriously, they can affect the long-term durability of the structure.
296
(a) (b)
Fig. 5.39 - Typical stress distributions of section containing cracks for the simply
supported beam specimen subjected to: (a) LH-1 (sustained moment
uM5.0 ); (b) LH-2 (sustained moment uM35.0 ).
5.3.13.2 Comparisons between LH-5 and LH-2
This part of the test is complementary to the previous part that has been presented in
Section 5.3.13.1. The specimens were tested under load history LH-5, for which the
specimens were loaded to uM5.0 and followed by an immediate unloading to
uM35.0 . The results for this loading history are compared with the results for load
history LH-2 (sustained moment uM35.0 ) in Figure 5.40, which are presented in terms
of final crack width, crack spacing and final to initial crack width ratio. It should be
noted that crack width ratio was calculated by dividing the crack width at age 1000
days by the crack width right after unloading to uM35.0 .
Although the magnitudes of the long-term moment are the same for both load
histories LH-5 and LH-2, the final crack widths and crack spacings for the specimens
subjected to LH-5 are smaller than those subjected to LH-2. The difference between the
two load histories is that, under load history LH-5 the specimens were fully cracked
(cracks are stress-free) at instantaneous loading and the stabilized crack patterns were
established. These also explain why the cracks of the specimens subjected to LH-5
were more closely spaced and have smaller widths. Although the instantaneous crack
0
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-10 -8 -6 -4 -2 0 2Concrete stress (MPa)
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LH-1 LH-2
297
Fig. 5.40 - Test series M – comparisons of results for load histories LH-5 (initial
moment 0.5Mu then to 0.35Mu) and LH-2 (sustained moment 0.35Mu): (a)
and (c) final crack widths for simply supported beams and slabs,
respectively; (b) and (d) crack spacings for simply supported beams and
slabs, respectively; (e) and (f) ratios of final to initial crack width for
simply supported beams and slabs, respectively.
widths may not be as small as those for specimens subjected to LH-2, the long-term
crack widths are in fact smaller and are more acceptable from a serviceability point
of view.
050
100150200250300350400
0.002 0.004 0.006 0.008 0.01Reinforcement ratio, ρ
Cra
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paci
ng (m
m)
Inst. (LH-5)t=1000days (LH-5)Inst. (LH-2)t=1000days (LH-2)
0.000.050.100.150.200.250.300.350.400.45
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Fina
l cra
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(mm
)
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(c) (d)
Simply supported slabs Simply supported slabs
0
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, wcr
.f/w
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(e) (f)
Simply supported beams Simply supported slabs
298
In addition, the lower final to initial crack width ratio for the specimens under
load history LH-5 is less “deceptive” than that due to LH-2. This is more advantageous
in practical situations in that the users of the structures frequently become unduly
concerned if cracks gradually appear with time and subsequently continue to widen.
5.3.13.3 Comparisons between LH-3 and LH-4
In this part of the test series, the effects of the age of unloading were investigated. All
specimens were initially loaded to uM5.0 and held constant with time. After a certain
period, the external loads were completely removed. To achieve a more realistic
simulation as in real structures, the self-weight of the specimens remained in place after
the external loads were removed. The specimens were unloaded either at age 50 days or
at age 200 days (designated as load histories LH-3 and LH-4, respectively).
Figure 5.41 shows the final crack width, the crack spacing and the ratio of final to
initial crack width, for both load histories LH-3 and LH-4. Since the loads are the same
for the both load histories at instantaneous loading, the crack spacings are also not
different. Moreover, no further change of crack patterns is observed after the loads
were removed. In Figures 5.41a and 5.41c, it is seen that the final crack widths for the
specimens subjected to LH-3 are generally smaller than those subjected to LH-4, but
the difference is only slight.
In most sustained loading cases, the time-dependent opening of cracks is largely
dependent on the ability of a concrete to shrink. However, when the loads vary with
time, the tensile stress in the reinforcing steel changes and this affects the width of
cracks. Since all specimens were assumed to have the same shrinkage properties, with
the same crack spacing the cracks should open by the same amount under the effects of
shrinkage. The larger crack widths observed in Figures 5.41a and 5.41c for the
specimens unloaded at age 200 days are due to the fact that aging concrete is less able
to undergo creep recovery. In addition, the specimens unloaded at age 200 were also
subjected to an additional 150 days (from age 50 days to age 200 days) of sustained
299
Fig. 5.41 - Test series M – comparisons of results for load histories LH-3 (initial
moment 0.5Mu and unloaded at age 50 days) and LH-4 (initial moment
0.5Mu and unloaded at age 200 days): (a) and (c) final crack widths for
simply supported beams and slabs, respectively; (b) and (d) crack spacings
for simply supported beams and slabs, respectively; (e) and (f) ratios of
final to initial crack width for simply supported beams and slabs,
respectively.
loads which inevitably causes a higher tensile steel stress with time than those
subjected to a shorter period of loading. Figure 5.42 compares the average tensile steel
0.000.050.100.150.200.250.300.350.40
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(c) (d)
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Simply supported slabs Simply supported slabs
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0.000.020.040.060.080.100.120.140.160.18
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(mm
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(a) (b)
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0.000.100.200.300.400.500.600.700.80
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, wcr
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300
stresses of the beam specimens containing 3N20 steel bars (ρ = 0.0069) subjected to
load histories LH-3 and LH-4. The additional increase in steel stress due to a longer
period of sustained loading and the reduced ability of concrete to undergo creep
recovery at an older age result in higher final stress in steel and therefore wider cracks.
In Figures 5.41e and 5.41f, it is seen that the ratios of final to initial crack width
for the specimens subjected to unloading at age 50 days are slightly lower compared to
those subjected to unloading at age 200 days. This is to be expected since the
specimens containing the same quantity of steel have the same instantaneous crack
widths but the final crack widths of the specimens unloaded at age 50 days are smaller,
which results in a smaller crack width ratio.
Another observation in Figures 5.41e and 5.41f is the reduction of crack width
ratio with increasing reinforcement ratio. The results may seem intriguing at first
glance since the crack width ratios for specimens subjected to sustained moments in the
previous test series are consistently insensitive to the quantity of reinforcement. In fact,
this feature originates from the different magnitudes of unloading for each specimen.
For example, the most heavily reinforced specimens were subjected to the largest initial
moment equivalent to uM5.0 . Having the same self-weights as other specimens, the
most heavily reinforced specimens undergo the largest unloading which subsequently
causes the crack widths to have the largest decrease, too.
Fig. 5.42 - Comparison of steel stress of beam specimens containing 3N20 steel
reinforcement subjected to load histories LH-3 and LH-4.
0
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300
0 200 400 600 800 1000Age (days)
Ave
rage
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el s
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(MPa
)
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301
For the same reason, the slab specimens have the same trend of crack width ratios
as the beam specimens, but with a higher range. The ratio for the slabs lies between 0.7
and 1.4, while the crack width ratios for beam specimens are between 0.4 and 0.65. In
addition, the higher crack width ratios are due to the larger self-weights of the slab
specimens, which effectively retains a larger width of crack after unloading.
5.3.14 Section Geometry and Boundary Conditions
Examining the results of all test series, it is observed that the type of specimen (beam or
slab in this study) has no effect on crack spacing and crack width. The boundary
conditions have a slight influence on the time-dependent growth of cracks. In general,
the continuous specimens have slightly higher final to initial crack width ratios. This is
probably due to the large bending moment in the negative moment region, which
causes cracking to occur above the supports before cracking in the constant positive
moment region. The delay of cracking in the constant positive moment region may
have caused a relatively smaller instantaneous crack width than that in the simply
supported case. With time, these cracks opened widely and achieved similar crack
widths. Nevertheless, the difference of the crack width ratios between simply supported
and continuous specimens is very small.
5.4 Summary
Various parameters have been investigated in this parametric study. The effects of each
parameter were examined thoroughly and the test results have been evaluated and
discussed. In addition to the investigation of the parameters, the versatility of the
proposed finite element model has also been demonstrated. The finite element model
has been shown to be able to produce a large amount of data for the study of time-
dependent cracking of reinforced concrete structures, and this can be used as an
alternative to conventional laboratory experimentations.
The parameters tested in this study are broadly divided into three groups. The first
group includes the parameters which affect both the instantaneous crack spacing and
302
crack width and the time-dependent growth of cracks. The second group consists of the
parameters that do not affect the formation of cracks at instantaneous loading but are
important in the time-dependent widening of cracks. The final group are made up of the
parameters that have no significant effect on either instantaneous or time-dependent
cracking of reinforced concrete structures.
Parameters such as bottom concrete cover, bar diameter, quantity of tensile
reinforcement, concrete tensile strength and bond strength are in the first group. These
parameters are important in the formation of cracks under short-term loads. A common
feature of these parameters is that the ratios of final to initial crack widths are relatively
insensitive to the variation of these parameters. In the broad sense, these parameters are
interrelated in a way that they have some common interactions. They affect the
formation of cracks by either changing the effective reinforcement ratio (area of tensile
steel divided by effective area of concrete in tension) or the bond characteristic between
steel and concrete.
Bottom concrete cover and the quantity of tensile reinforcement are the factors
affecting the effective reinforcement ratio which subsequently govern the crack spacing
and crack width. A threshold thickness of bottom cover was identified beyond which
the bottom cover ceases to affect the distribution of cracks in a structure.
Bar diameter, quantity of tensile reinforcement and bond strength all influence the
final bond characteristics of the structure. Bond quality can be improved by using
stronger concrete which provides good confinement to the reinforcing steel and
consequently improves local bond characteristics between concrete and steel. An
alternative is to adjust the bar size or the number of bars, or both, so as to achieve a
larger concrete-steel contact surface area. In addition, this investigation also reveals
that spacing of reinforcing steel, which is often quoted in codes of practice for crack
control, is not a critical factor that affects crack spacing and crack width. The more
important factor is the concrete-steel contact surface area which directly affects the rate
of stress transfer between steel and concrete.
The concrete tensile strength parameter is a somewhat special factor which affects
the local bond characteristics of the reinforcing steel and also limits the extent of the
bond development length. The effects of concrete tensile strength in these two aspects
303
are inter-related and rather obscure. It is well known that bond strength increases with
increasing concrete tensile strength, and this affects the stress transfer between concrete
and steel. When these two aspects are considered simultaneously, the effects of
concrete tensile strength become less important as demonstrated in Section 5.3.5.
However, if a different relationship between bond and concrete tensile strength is used,
the outcome of the tests may be different. Therefore, more research on the interaction
of concrete tensile strength and bond is required before a final conclusion can be made.
Creep, shrinkage, bond creep and load history are categorized into the second
parameter group. These parameters vary constantly with time. Undoubtedly, shrinkage
is the most important time-dependent factor that affects the opening of cracks with
time, while creep is a less influential one. Nevertheless, the importance of creep cannot
be completely overlooked, since creep has a prominent effect on the internal
redistribution of stress in a structure which may subsequently lead to an increase in
tensile stress in the steel. Although it is seen in the parametric study that the effects of
creep are less significant on crack opening, care should still be taken when considering
the effects of creep on time-dependent cracking of reinforced concrete structures.
Load history has an important influence on crack width in the sense that the
tensile steel stress changes according to the change in loads. Since the crack opening
mechanisms depend largely on shrinkage (load-independent) and tensile stress in steel
(due mainly to change in load), it is concluded that the final crack widths in a
reinforced concrete structure are not particularly sensitive to the age of change of loads
as demonstrated in Section 5.3.13.3.
In addition, this investigation also shows the benefits of establishing a stabilized
crack pattern at instantaneous loading. The instantaneous crack widths of a stabilized
crack pattern are usually larger than those of a pre-stabilized crack pattern developed
under a smaller load. However, in the course of time, the smaller crack spacing of the
stabilized crack pattern is more beneficial in reducing the long-term width of cracks.
Furthermore, the fine instantaneous cracks which result from a relatively small initial
load may become surprisingly wide over time, and this can have devastating effects on
the serviceability of the structure.
304
The third parameter group includes parameters such as compressive
reinforcement, concrete tensile strength fluctuation limit, structure type and boundary
conditions. The effects of these parameters are less important to crack spacing and
crack width. Although concrete tensile strength fluctuation limit does not have
significant effects on crack spacing and crack width, it is important in the initiation of
cracking in concrete and the prevention of bifurcation of the numerical solution. The
shear reinforcement parameter is left out of any of the parameter group since shear
reinforcement was included only to prevent premature shear failure.
Lastly, an investigation was conducted to examine the effects of increasing the
steel yield stress from 400 MPa to 500 MPa. The results of the investigation show that
the consequent reduction of the required quantity of steel in structural members results
in higher tensile steel stress. As a result, the crack spacings are larger and the cracks are
wider. This study demonstrates the adverse effects of the use of 500 MPa steel
reinforcement and urges an appropriate review of the crack control procedures in the
current Australian Standards, AS 3600 (2001).
305
CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 Summary
The analysis of a reinforced concrete structure is complicated by the non-linear
behaviour arising from cracking of the concrete. A time-dependent analysis is further
complicated by time effects resulting from the load history and creep and shrinkage of
concrete. In this thesis, several two-dimensional numerical models for time-dependent
analysis of reinforced concrete structures have been presented, with particular emphasis
on the formation of cracks at service load conditions. Three numerical models, namely
the distributed cracking Cracked Membrane Model, the localized cracking Crack Band
Model and the localized cracking Non-local Smeared Crack Model, were developed to
trace the gradual cracking process in a reinforced concrete structure subjected to the
time-dependent effects of creep and shrinkage. In addition to the non-linearities
resulting from cracking and time effects, non-linearity due to large deformation is also
accounted for in the numerical models.
In the first part of Chapter 2, the properties of concrete and steel reinforcement
and the interaction between these two materials are described. An overview is given of
the instantaneous and time-dependent properties of concrete including the well-known
behaviour of uniaxially and biaxially loaded specimens, creep and shrinkage and the
time-dependent fracture of concrete. The second part of Chapter 2 presents the state-of-
the-art of non-linear modelling of reinforced concrete structures. Various methods for
modelling cracking in concrete are described and the pros and cons of each model are
discussed. The commonly employed constitutive models for concrete are also
presented. An introduction is given of issues related to strain localization in a standard
smeared crack model and various regularization techniques developed to overcome
these difficulties are described. In addition, the representation of reinforcing steel in the
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context of a finite element formulation is presented. Lastly, the numerical treatment for
a creep analysis based the principle of superposition is described.
The formulations of the numerical models are presented in Chapter 3. The
numerical models have been developed based on a smeared crack approach for which
cracking is modelled as a reduction of material stiffness in the crack opening direction.
Two modelling approaches have been adopted to simulate the behaviour of reinforced
concrete structures. The first type is the distributed cracking approach. In this approach,
steel reinforcement is smeared through the concrete elements and the effect of bond-
slip between steel and concrete is accounted for indirectly as tension stiffening in a
cracked reinforced concrete member. One of the characteristics of the distributed
cracking model is that cracking occurs in a “smeared” manner where no visible
localized crack is computed. For an effective modelling of tension stiffening, the
cracked membrane model of Kaufmann (1998) is employed. Based upon a stepped,
rigid perfectly plastic bond stress-slip relationship as adopted in the tension chord
model of Sigrist (1995), the crack spacing and the stress distribution in steel and in
concrete for a cracked reinforced concrete member can be determined. In this way,
tension stiffening is treated in a rational and tractable manner.
The second approach described in Chapter 3 for modelling cracking is the
localized cracking approach. In this approach, cracking in concrete is treated as
localized fracture and reinforcing steel is modelled as discrete steel elements connected
to concrete using bond-slip interface elements. The localized cracking approach
facilitates a more refined simulation of the mechanism of cracking in a reinforced
concrete structure. Tension stiffening is modelled as a result of stress transfer between
the concrete and the steel via bond action. In this study two different fracture models
were employed, namely the crack band model (Bažant and Oh, 1983) and the non-local
smeared crack model. For the non-local smeared crack model, a new formulation is
proposed to avoid the unrealistic concrete stress in excess of the concrete tensile
strength computed by Jirásek and Zimmermann (1998) in their model based on the non-
local model with local strain formulation (see Section 3.4.2).
In this study, plain concrete is treated as orthotropic material and the biaxial
stresses are obtained using a modified form of the equivalent uniaxial strain concept of
307
Darwin and Pecknold (1977). Although the work in this thesis focuses on time-
dependent behaviour of reinforced concrete structures at service loads, the stress-strain
relationships for biaxial loading to failure are incorporated in the numerical models so
that a short-term failure analysis can also be performed. Creep is modelled using the
solidification theory of Bažant and Prasannan (1989a, b) and shrinkage is approximated
by a growth function. Both these time effects are treated as inelastic pre-strains updated
with time and applied to the structure as equivalent nodal forces. A time-dependent
bond model is implemented with the bond-slip interface elements and is used in the
localized cracking models and therefore the deterioration of bond due to creep under
sustained loads is accounted for. The time-dependent analysis of a reinforced concrete
structure is treated in a quasi-static manner such that the time domain is discretized into
a number of finite time steps where the time-dependent cracking and deformation of the
structure are determined through a step-by-step integration through the time domain. In
addition, for large deformation problems such as creep buckling of slender columns, an
updated Lagrangian formulation is adopted. Using this formulation the stress and strain
of a structure are computed based on the last computed displaced configuration and
therefore the engineering stress and strain notion can be retained.
The numerical models developed in Chapter 3 are verified in Chapter 4. The
major task is to evaluate the numerical models in computing crack spacing and crack
width of reinforced concrete members. The long-term flexural cracking specimens and
the restrained deformation slabs tested by Gilbert and Nejadi (2004) and Nejadi and
Gilbert (2004) were simulated using the numerical models. Despite the random nature
of cracking in the reinforced concrete members, the calculated results are in good
agreement with the test data. In addition, a comparison of accuracy of the computed
results is made among the numerical models and a discussion is given on the
advantages and disadvantages of each model. Other types of structures such as
continuous beams with long-term settling supports and creep buckling of slender
columns have also been analysed. The good correlations between the calculated results
and the test data demonstrate the ability of the numerical models in simulating time-
dependent behaviour of reinforced concrete structures under varying load histories and
under the effect of geometric non-linearity.
308
In Chapter 5, a parametric study is conducted using the localized cracking Crack
Band Model in order to investigate the qualitative interactions of the parameters
affecting time-dependent cracking of reinforced concrete structures. A series of
numerical experiments is devised based on various parameters associated with material
properties, environmental conditions and structural type. The effects of the parameters
are examined and discussed based on the observations from the numerical calculations.
In addition, the impact of the introduction of 500 MPa steel to replace the previously
used 400 MPa steel in Australia is also examined and evaluated using the
numerical models.
6.2 Conclusions
The major objective of this thesis is to investigate time-dependent cracking in
reinforced concrete structures with particular focus on the qualitative understanding of
the time-dependent cracking process and the quantitative evaluation of crack spacing
and crack width due to time effects. In Chapter 4, the numerical models developed in
this study have been demonstrated to compute reasonably accurate quantitative results
and, in Chapter 5, the models have been employed to investigate the process of time-
dependent cracking in reinforced concrete structures through a series of numerical
experiments. Based upon the work presented in this thesis, the following conclusions
have been reached.
A realistic description of bond between concrete and steel is of vital importance
for the accurate computation of crack spacing and crack width in a reinforced concrete
structure. Although bond-slip is not modelled explicitly as a physical interaction
between the concrete and the steel elements in the distributed cracking approach, the
bond condition assumed in the tension chord model is crucial for the derivation of the
crack spacing equation and hence the determination of crack width. For the localized
cracking model, the influence of bond is even more apparent. As can be seen from the
results of the parametric study, the characteristic of bond between concrete and steel
has a marked effect on the crack spacing and crack width.
309
The distributed cracking approach using the Cracked Membrane Model provides
a computationally economical tool that can be used to accurately model the time-
dependent behaviour of reinforced concrete structures since it requires relatively few
elements in a numerical analysis. However, due to the strong dependence of the model
on crack spacing for the calculation of crack width, the tension chord model, which is
used to calculate crack spacing, needs to be further calibrated with more experimental
data from various types of structures.
For the modelling of a reinforced concrete structure using the localized cracking
approach, a concrete fracture model must be complemented by the use of an
appropriate bond element (for example the bond-slip interface element employed this
study) that can simulate the slip between concrete and steel in the vicinity of a crack,
thereby facilitating the opening of a crack. On the other hand, the use of a bond element
without a reliable concrete fracture model (for example, cracking model with a sudden
drop in stress to zero) can render the computation outcomes unobjective. The model
may still be able to compute localized cracking without the use of a concrete fracture
model. However, since the fracture energy dissipation in the fracture process zone of
the concrete is not properly accounted for, the computed results will be highly sensitive
to the finite element mesh configuration.
The non-local model is widely known as one of the most effective tools for
alleviating mesh sensitivity in a standard smeared crack model. However, most of the
previous research relating to non-local models focuses mainly on fracture of plain
concrete. The applicability of such models to reinforced concrete structures is often
overlooked. This study has revealed the drawback of the non-local model in the context
of reinforced concrete structures. The fracture zone of a non-local model stretches over
several elements and the non-linear distribution of cracking strain within the fracture
zone inevitably handicaps the computation of crack width. Moreover, the sensitivity of
spatial averaging to high tensile stress regions in the model also prevents a crack from
forming nicely into a single discontinuity. As a result the computed crack pattern in
some cases can be rather dispersed and the localized cracks are difficult to identify.
In contrast, the crack band model, which is less sophisticated than the non-local
model, can better describe localized cracking in reinforced concrete structures. In
310
addition, the bond-slip interface element also works more effectively with the crack
band model as shown in Section 4.4.4. An interesting finding for the crack band model
is the stress-locking phenomenon observed in concrete fracture tests becomes
insignificant when modelling a reinforced concrete structure. This indicates that stress-
locking in the crack band model is only a plain concrete fracture problem and the
problem is eliminated by the stabilizing effect of reinforcing steel.
From the observation of the parametric investigation presented in Chapter 5, it is
concluded that the contact surface area between the tensile steel and the concrete in a
reinforced concrete member is an important factor affecting the overall bond
characteristic in the tension zone of the member. Larger concrete-steel contact surface
area can improve the bond characteristic of a member. The permissible maximum bar
spacing stipulated in the simplified crack control procedures of many design codes is in
fact not a critical factor for crack spacing and crack width. In the author’s opinion, the
objective of the limitation on tension bar spacing is to infer the use of more reinforcing
bars of smaller size so as to provide a larger concrete-steel contact surface area and thus
reduce crack spacing and crack width. Therefore, the author recommends the use of a
more explicit requirement such as concrete-steel contact surface area per unit length for
detailing of tensile reinforcing bars as a substitute for the currently stipulated maximum
bar spacing in design codes of practice.
The effective reinforcement ratio (i.e. the area of tensile steel divided by the
effective area of concrete in tension) is one of the important factors that affects the
stress transfer rate between the steel and the concrete in a cracked reinforced concrete
member. This in turn has a marked effect on crack spacing. In addition to quantities of
tensile reinforcement, bottom concrete cover is an important parameter in determining
the effective concrete ratio. For a reinforced concrete member with a thicker concrete
cover, the crack spacing is relatively larger since the effective reinforcement ratio is
lower than a member with a smaller concrete cover.
Shrinkage, creep, bond creep and load history are the main factors causing time-
dependent crack opening in reinforced concrete members. This study has confirmed the
well-known influence of shrinkage on crack width. Deterioration of bond under
sustained load results in increase in slip between concrete and steel with time. The
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effect of creep is less prominent on crack width. However, due to its slight influence on
the increase in tensile steel stress with time, the effect of creep cannot be totally
overlooked since large tensile steel stress can also lead to unsightly wide cracks. On the
other hand, the load history of a structure is important due to the fact that it changes the
stress in the tensile steel, which effectively influences the crack width.
This study has demonstrated that the replacement of 400 MPa steel reinforcement
by the new 500 MPa steel reinforcement in AS 3600 (2001) can adversely lead to
reduction in the required quantity of tensile reinforcement in a member, which results
in undesirable increase in tensile steel stress and a possible tendency of using fewer
bars of larger diameters. Consequently, a revision of the current crack control
procedures in AS 3600 (2001) should be undertaken immediately.
6.3 Recommendations for Future Research
It has been demonstrated in this thesis that the numerical models developed in
Chapter 3 and verified in Chapter 4 can accurately model time-dependent behaviour
and trace time-dependent development of cracks of reinforced concrete structures. In
Chapter 5, the model has been employed to investigate time-dependent cracking of
reinforced concrete members and has provided an insight into the influence and
interaction of the parameters affecting time-dependent cracking. However, this does not
mean that the task is complete or the problem is solved, a great deal of work remains to
be done. Some of the recommended future work is outlined below:
• Development of a three-dimensional finite element model for concrete, steel and
bond in order to study the three-dimensional bond mechanism and the interaction of
bond and effective tensile area of concrete in a cracked reinforced concrete
member. This may provide a better understanding of the formation of cracks and
the effects of bond.
• An intra-element discrete crack model to simulate cracking in reinforced concrete
structures should be developed. The intra-element crack model would simulate a
312
crack as a complete stress-free discontinuity and would not require a remeshing
technique to trace the crack trajectory.
• Modification of the algorithm for the generation of stochastic concrete tensile
strengths based on the normal distribution in the localized cracking model.
• Extension of the finite element model for moisture migration analysis using the
non-linear diffusion theory so as to obtain shrinkage at any point in a continuum
taking into consideration boundary and size effects.
• A separate treatment for drying creep by employing a unified approach for long-
term aging and drying of concrete such as the microprestress-solidification theory
(Bažant et al., 1997), which is an improved version of the solidification theory of
creep adopted in this study.
• Further calibration of the crack spacing equation of the tension chord model with
more experimental data from different types of structures.
• The use of different time-dependent bond models and investigation of the effect of
each model on crack opening with time.
• Incorporation of an automatic time-stepping algorithm in the time-dependent finite
element model so that the size of the time step can be adjusted automatically
according to the state of deformation of the structure.
• Inclusion of thermal effects as a component in the time-dependent analysis of a
reinforced concrete structure.
313
APPENDIX A: FE Implementation of Rate of Creep Method
The rate of creep method (RCM) has been used to model the time-dependent
development of creep strain (Gilbert, 1988; Chong et al., 2004). The RCM requires as
input a single creep coefficient versus time curve associated with the initial application
of load. For all subsequent loadings, the creep coefficient versus time curve is assumed
to be affine with that at first loading. That is, the rate of change of creep is assumed to
be independent of the age at first loading. This, of course, is approximate and
introduces some errors, particularly when the change of concrete stress with time is
significant. However, the advantage is that the storage of the time-dependent stress
history is not required.
Similar to the solidification theory of creep presented in Chapter 3, two
assumptions are made: (i) creep is linear with respect to stress; and (ii) the time-
dependent response in tension is identical to that in compression. For the
implementation of the RCM into the finite element model, time is discretized into small
intervals and loads are taken to remain constant during each time increment. The creep
strain at the current time is obtained by summing the increments of creep obtained from
the previous time intervals, that is
∑ ∆= φσε)(
)()(0tE
ttc
cp (A.1)
where φ∆ is the change in creep coefficient during a particular time interval. The
change of creep strain at time t is
)()()( ttt cecp φεε ∆=∆ (A.2)
where )(tεce is the concrete elastic strain at time t. For the finite element
implementation, Eq. A.2 is written as
)()()( 1212 iicicp ttt φ∆=∆ εε (A.3)
314
where )(12 icp tε∆ is the change in creep strain in the principal strain directions and
)(12 ic tε is the concrete elastic principal strain vector. Note that the change in creep
coefficient )( itφ∆ is a scalar.
Before proceeding to the computation of creep strain of the current time step, the
concrete global elastic strain vector T])()()([)( icxyicyicxice tγtεtεt =ε as
determined from the last time step is stored. The elastic strain vector in global
directions is then transformed through an angle θ using the strain transformation
matrix εT to the elastic strain vector in principal directions giving
)()(12 iceic tt εTε ε= (A.4)
where T2112 ]0)()([)( icicic tεtεt =ε is the principal concrete elastic strain. The
change of creep strain in the current time step is calculated from Eq. A.3, in which the
change of creep coefficient is given by
)()()( 1−−=∆ iii ttt φφφ (A.5)
The change of creep strain )(12 icp tε∆ is transformed through θ to the global
directions restoring strains to the global directions. The iterative procedures for the
calculation of the change of creep strain over a time step is performed for each
integration point of each element and is shown in Figure A.1.
Lastly, the total creep strain is calculated by adding the change of creep strain
)( icp tε∆ in the global directions to the stored total creep strain from the last time step
at time 1−it . That is
)()()( 1−+∆= icpicpicp ttt εεε (A.6)
The current material stress state is calculated from
)]()([)( 0 iici ttt εεDσ −= (A.7)
315
in which the inelastic pre-strain vector 0ε is the sum of the creep strain component
calculated from Eq. A.6 and a shrinkage strain component (see Section 3.6.3 of Chapter
3). In the finite element implementation, the inelastic pre-strain vector is converted into
a set of equivalent nodal forces and applied to the discretized structures. The details of
the finite element implementation have been discussed in Section 3.7 of Chapter 3).
Fig. A.1 - Computation of creep strain from elastic strain.
To verify the numerical treatment of creep using RCM, specimen B2-a tested by
Gilbert and Nejadi (2004) was selected and analysed using the localized cracking
cracked band model (see Section 3.4.1 of Chapter 3). The finite element mesh for the
beam is shown in Figure A.2 and the mesh is made up of 665 nodes, 540 concrete
elements, 54 steel elements and 54 bond-slip elements. The steel elements were
connected to the concrete elements via bond-slip interface elements and the free edge
node of the steel bar was rigidly connected to the concrete element node to simulate
anchorage of the bar. A ±10% random fluctuation of the mean concrete tensile strength
Calculate creep strain from instantaneous strain
Transform to
Principal directions
εcx
εcy γcxy y
x
εc2 εc1
2 1
Global directions
∆εcpx
∆εcpy ∆γcpxy
y x
∆εcp2 ∆εcp1
2 1 Transform to
316
Fig. A.2 - Finite element mesh for beam B2-a.
was assigned to the concrete elements. The material properties for the concrete were:
MPa25=cmf , GPa25=cE , ( )10%MPa0.2 ±=ctf , N/m75=fG , mm35=ch ,
2.0=ν . For the time-dependent properties of concrete, the development of shrinkage
strain and creep coefficient were approximated using Eq. 3.46 in Chapter 3 giving
950=shA , days45=shB , 5.1=cpA and days24=cpB , and the comparison between
the approximated models with test data is shown in Figure A.3. For the reinforcing
steel, a nominal yield strength of 500 MPa and elastic modulus of 200 GPa
were employed. The bond-slip parameters were: mm6.021 == ss , mm0.13 =s ,
MPa0.10max =τ , MPa5.1=fτ and mmMPa100=uk .
(a) (b)
Fig. A.3 - Creep and shrinkage measurements of Gilbert and Nejadi (2004) compared
with approximated models: (a) creep coefficient versus time since first
loading; (b) shrinkage strain versus time since commencement of drying.
0
200
400
600
800
1000
0 50 100 150 200Time since commencement of drying (days)
Shrin
kage
str
ain
( µε)
Experimental data
Model
Ash = 950 µεBsh = 45 days
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0 50 100 150 200Time since first loading (days)
Cree
p co
efic
ient
Experimental data
Model
Acp = 1.5Bcp = 24 days
317
The calculated mid-span deflection versus time curve is compared with the
experimental results in Figure A.4a with a good correlation. In the constant moment
region a comparison is made in Figure A.4b between the calculated and measured crack
widths at the soffit of the beam with increasing time. The comparison is presented for
the variation of crack width with time for the widest crack and for the average of crack
width as observed in the test and as predicted by the model. The agreement is
reasonable, but the model calculated a slightly lower maximum crack width and a
higher average crack width compared to the observed results. Figures A.5a and A.5b
show the computed crack patterns at instantaneous loading and at 380 days,
respectively. It is seen that the computed crack spacing agrees well with the test results
as shown in Figure 4.28 of Chapter 4.
(a) (b)
Fig. A.4 - Comparison of calculated time-dependent behaviour and test data: (a)
midspan deflection with time; (b) crack widths with time.
(a) (b)
Fig. A.5 - Crack pattern of beam B2-a: (a) and (b) crack patterns (cracking strain plot)
at instantaneous loading and at 380 days, respectively.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 100 200 300 400Time (days)
Cra
ck w
idth
(mm
)
Experimental
FEM
maximum
average
Beam B2-aBeam B2-a
0
2
4
6
8
10
12
14
0 100 200 300 400Time (days)
Mid
span
def
lect
ion
(mm
)
Experimental
FEM
318
APPENDIX B: Illustration of Treatment for Inelastic Pre-
strain by Simple Hand Calculation
The behaviour of a material is often characterized by the stress-strain relationship of the
material, in which the state of stress is related through a relationship to the state of
strain of a structure. However, in a time-dependent analysis, when the effects of creep
and shrinkage are prominent, the state of stress in a structure cannot be related directly
by a stress-strain law to the state of strain as the strain now contains the creep and
shrinkage components.
This section aims at describing the algorithm for including the time-dependent
components, namely creep and shrinkage of concrete, into a finite element model by
presenting a simple illustrative example which can be analysed by hand calculation. To
facilitate this task, the finite element implementation flowchart presented in Figure 3.17
of Chapter 3 is simplified and is shown in Figure B.1. The iterative procedure for a
finite element stiffness analysis is reduced to an iterative stress-strain analysis where
the computation of stiffness matrix is not required. For simplicity, no attempt is made
to include the material non-linearity arising from non-linear stress-strain relationship of
concrete and this assumption is justified for a concrete structure under service
load conditions. In addition, the rate of creep method presented in Appendix A is
employed for the computation of creep strain in this section.
In Figure B.1 the calculation begins with a short-term analysis and time is set to
0=t . In the first iterative loop, the creep strain cpε , shrinkage strain shε and the total
strain ε are all zero. Therefore, the instantaneous strain and the internal stress intσ are
also zero and the out-of-balance stress outσ in step 6 of the flowchart is directly equal
to the applied external stress extσ . Having known outσ , the strain increment ε∆ can
be calculated from a stress-strain relationship and the total strain is determined by
adding ε∆ to the total strain ε obtained from the previous iterative loop. The next step
is to check convergence for the out-of-balance stress and to decide whether more
iteration is required or to proceed to the next time step. If the prescribed convergence
319
tolerance is not satisfied, the process is transferred back to step 3 of the flowchart. The
total pre-strain 0ε is calculated by summing creep strain cpε and shrinkage strain shε
obtained based on the current time and the current state of stress. The instantaneous
strain instε is computed by subtracting the pre-strain from the total strain and the
internal stress of the structure is calculated using the stress-strain relationship as shown
in step 5. Out-of-balance stress outσ is recalculated and the iterative process from step
3 to step 9 continues until the prescribed convergence tolerance is reached. After the
equilibrium state is sought, calculation can proceed to the next time step or it can be
terminated as indicated in step 10.
Fig. B.1 - Calculation flowchart for time-dependent analysis.
Increase time instance t = t + ∆ t
Apply external stress σext and initialise short-term analysis
t = 0; ∆ t = 0
Calculate out-of-balance stress σout = σext – σint
Proceed to next time step or
STOP calculation
1
2
6
10
Check convergence tolerance σout / σout.max < % tol. ?
Tolerance not reached
9
Tole
ranc
e re
ache
d
Determine strain increment ∆ε = σout / Ec
7
Determine pre-strains due to creep and shrinkage
ε 0 = εcp + εsh 3
Determine instantaneous strain εinst = ε – ε 0
4
Calculate internal stress σint = εinst Ec
5
Calculate total strain ε = ε + ∆ε
8
320
To illustrate the algorithm, a plain concrete column subjected to sustained axial
compression as shown in Figure B.2 is considered. The dimension of the column is 300
mm by 300 mm by 6000 mm long. The axial load and the properties of concrete are
shown in Figure B.2 and in Table B.1. In this example the instantaneous analysis at
days01 =t is followed by two time steps days101 =∆t and days402 =∆t giving
days102 =t and days503 =t . The convergence tolerance is taken as 1%.
(a)
Fig. B.2 - Details of the plain concrete column subjected to sustained axial
compression: (a) longitudinal loading configuration; (b) section A-A.
Table B.1 - Time-dependent concrete properties.
Time, t Concrete parameter 0 days 10 days 50 days
Creep coefficient, φ 0 0.5 1.5
Shrinkage strain, εsh (×10-6) 0 -100 -300
300
300
Section A-A (b)
Ac = 90000 mm2 Ec = 25000 MPa
Ac
A A
P = 1000 kN
321
Initialise short-term analysis
Step 1: MPa11.11mm90000
kN10002
−=−
==c
ext APσ
Step 2: day0=t
Time t = 0 day – iteration 1
Step 3: 00 =ε
Step 4: 0=instε
Step 5: 0=intσ
Step 6: MPa11.110MPa11.11 −=−−=outσ
Step 7: 61044.444MPa25000MPa11.11 −×−=
−=∆ε
Step 8: 66 1044.444)1044.444(0 −− ×−=×−+=ε
Step 9: %100max. =outout σσ > % tolerance (convergence not satisfied)
Time t = 0 day – iteration 2
Step 3: 00 =ε
Step 4: 66 1044.44401044.444 −− ×−=−×−=instε
Step 5: MPa11.11MPa250001044.444 6 −=××−= −intσ
Step 6: MPa0MPa)11.11(MPa11.11 =−−−=outσ
Step 7: 0=∆ε
Step 8: 66 1044.44401044.444 −− ×−=+×−=ε
Step 9: %0max. =outout σσ < % tolerance (convergence satisfied)
Result for t = 0 day: 61044.444 −×−=ε
Step 10: Proceed to the next time step
Step 2: days10days10day01 =+=∆+= ttt
Time t = 10 days – iteration 1
Step 3: 660 1022.322)10100(5.0
25000MPa11.11 −− ×−=×−+×
−=ε
322
Step 4: 666 1022.122)1022.322(1044.444 −−− ×−=×−−×−=instε
Step 5: MPa06.3MPa250001022.122 6 −=××−= −intσ
Step 6: MPa06.8MPa)06.3(MPa11.11 −=−−−=outσ
Step 7: 61022.322MPa25000MPa06.8 −×−=
−=∆ε
Step 8: 666 1067.766)1022.322(1044.444 −−− ×−=×−+×−=ε
Step 9: %100max. =outout σσ > % tolerance (convergence not satisfied)
Time t = 10 days – iteration 2
Step 3: 660 1022.322)10100(5.0
25000MPa11.11 −− ×−=×−+×
−=ε
Step 4: 666 1044.444)1022.322(1067.766 −−− ×−=×−−×−=instε
Step 5: MPa11.11MPa250001044.444 6 −=××−= −intσ
Step 6: MPa0MPa)11.11(MPa11.11 =−−−=outσ
Step 7: 0=∆ε
Step 8: 66 1067.76601067.766 −− ×−=+×−=ε
Step 9: %0max. =outout σσ < % tolerance (convergence satisfied)
Result for t = 10 days: 61067.766 −×−=ε
Step 10: Proceed to the next time step
Step 2: days50days40days102 =+=∆+= ttt
Time t = 50 days – iteration 1
Step 3: 660 1067.966)10300(5.1
25000MPa11.11 −− ×−=×−+×
−=ε
Step 4: 666 1000.200)1067.966(1067.766 −−− ×=×−−×−=instε
Step 5: MPa00.5MPa250001000.200 6 =××= −intσ
Step 6: MPa11.16MPa00.5MPa11.11 −=−−=outσ
Step 7: 61044.644MPa25000MPa11.16 −×−=
−=∆ε
Step 8: 666 1011.1411)1044.644(1067.766 −−− ×−=×−+×−=ε
323
Step 9: %100max. =outout σσ > % tolerance (convergence not satisfied)
Time t = 50 days – iteration 2
Step 3: 660 1067.966)10300(5.1
25000MPa11.11 −− ×−=×−+×
−=ε
Step 4: 666 1044.444)1067.966(1011.1411 −−− ×=×−−×−=instε
Step 5: MPa11.11MPa250001044.444 6 −=××−= −intσ
Step 6: MPa0MPa)11.11(MPa11.11 =−−−=outσ
Step 7: 0=∆ε
Step 8: 66 1011.141101011.1411 −− ×−=+×−=ε
Step 9: %0max. =outout σσ < % tolerance (convergence satisfied)
Result for t = 50 days: 61011.1411 −×−=ε
Step 10: Stop calculation
The same example is analysed using a direct approach in which the total
deformation is calculated as the sum of elastic strain, creep strain and shrinkage strain
and the results are shown in Table B.2. It is seen that the approach presented in Figure
B.1 and the direct approach both give the same answers. Although the approach shown
Figure B.1 may seem cumbersome, the algorithm is useful for the analysis of large non-
linear problems using the finite element method. Furthermore, with the same algorithm,
the analysis method of Figure B.1 or, a more complete version as shown in Figure 3.17
of Chapter 3, can be extended to include pre-strain in reinforcing steel and expansion of
concrete due to heat effects.
Table B.2 - Calculation of deformation using the rate of creep method.
Time t
Instantaneous strain
Creep coef.
Creep strain
Shrinkage strain
Total Strain
(days) εinst φ(t) εcp(t) εsh(t) ε(t) = εinst + εcp(t) + εsh(t) 0 -444.44 × 10-6 0 0 0.5 -444.44 × 10-6 10 -444.44 × 10-6 0.5 -222.22 × 10-6 -100 × 10-6 -766.67 × 10-6 50 -444.44 × 10-6 1.5 -666.67 × 10-6 -300 × 10-6 -1411.11 × 10-6
324
APPENDIX C: CEB-FIP Model Code 1990 – Creep and
Shrinkage Models
In the CEB-FIP Model Code 1990 (1993), the models for creep and shrinkage are
provided for the prediction of the mean behaviour of a concrete cross-section. The
models are valid for structural concrete with compressive strengths of 12 MPa to
80 MPa subjected to a compressive stress less than 40% of the compressive strength
and exposed to mean relative humidity in the range of 40% to 100% at mean
temperature of between 5°C and 30°C.
Creep Model
The compliance function )',( ttJ , which is defined as the strain at time t produced by a
unit stress applied at 't and is given by
28.
)',()'(
1)',(cc Ett
tEttJ φ
+= (C.1)
where 't is the variable for age of loading, )',( ttφ is the creep coefficient, 28.cE is the
elastic modulus of concrete at age 28 days in MPa and )'(tEc is the elastic modulus of
concrete (MPa) at the age of loading which can be estimated by
5.05.0
28.28exp)(
−=t
ssEtE cccc (C.2)
where time t is in days and cs is a coefficient depending on the type of cement and is
taken as cs = 0.2 for rapid hardening high strength cement, cs = 0.25 for normal and
rapid hardening cements and cs = 0.38 for slow hardening cement. The creep
coefficient )',( ttφ at time t (days) for a concrete loaded at age 't (days) in Eq. C.1 is
given by
325
)'()'()()',( 21 tttftt cpcmRH −= βββφφ (C.3)
in which
31)01.0(46.0
01.011s
RHhRH−
+=φ (C.4)
5.01)1.0(
3.5)(cm
cmf
f =β (C.5)
2.02'1.0
1)'(t
t+
=β (C.6)
3.0
)'(')'(
−+
−=−
tttttt
Hcp β
β (C.7)
with
( ) 1500250012.015.1 18 ≤++= RHhsHβ (C.8)
where RH is the relative humidity of the ambient environment in percentage, cmf is the
mean compressive strength of concrete in MPa and sh is a notational size of a member
in mm which is defined as
ccs uAh 2= (C.9)
where cA is the cross-section area of the member and cu is the perimeter of the
member exposed to the air.
326
Shrinkage Model
The shrinkage strain at time t (days) for a concrete member commences drying at time
sht (days) can be calculated from
)()(),( shshRHcmshsshsh ttftt −= ββεε (C.10)
where
3)01.0(155.1 RHRH +−=β for %99%40 <≤ RH (C.11)
25.0=RHβ for %99≥RH (C.12)
5.0
2 )()01.0(350)(
−+
−=−
shs
shshsh
tth
ttttβ (C.13)
610)1.09(10160)( −×−+= cmshccmshs ff βε (C.14)
where RH, cmf and sh are identical to those defined for the creep model presented
previously and shcβ is a coefficient accounting for the type of cement and is taken as
4=shcβ for slow hardening cement, 5=shcβ for normal or rapid hardening cement
and 8=shcβ for rapid hardening high strength cement.
327
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