Post on 19-Feb-2022
The Pennsylvania State University
The Graduate School
Department of Civil and Environmental Engineering
ROLE OF EARLY-AGE CONCRETE PROPERTIES AND CONSTRUCTION
LOADING ON SLAB SERVICEABILITY
A Thesis in
Civil Engineering
by
Je Il Lee
© 2007 Je Il Lee
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
August 2007
The thesis of JE IL LEE was reviewed and approved* by the following:
Andrew Scanlon Professor of Civil and Environmental Engineering Thesis Advisor Chair of Committee
Andrea J. Schokker Associate Professor of Civil and Environmental Engineering
Maria Lopez de Murphy Assistant Professor of Civil and Environmental Engineering
Ali M. Memari Associate Professor of Architectural Engineering
Peggy A. Johnson Professor of Civil and Environmental Engineering Head of the Department of Civil and Environmental Engineering
*Signatures are on file in the Graduate School
iii
ABSTRACT
The slab is modeled using a shell element in the commercial finite element
software package ABAQUS/Standard. To idealize material behavior a user-defined
subroutine (UMAT) is developed. Time-dependent creep and shrinkage effects in
concrete material are also incorporated to the subroutine. Recently proposed creep and
shrinkage models are implemented along with tension stiffening models in a general
purpose computer program for analysis of concrete slabs under sustained time-dependent
loading.
Laboratory tests on nine simply supported one-way reinforced concrete members
subjected to sustained load was performed. Each specimen was subjected to immediate
full live again after six months. Applied load and mid-span deflections were recorded
under immediate live load and sustained load. The test results demonstrated the effect of
shrinkage restraint provided by embedded bars on the flexural cracking of the specimens
under applied load, as well as effects of early age loading on time-dependent response.
Results of an analytical study of reinforced concrete two-way slab systems are
also presented. Numerical results which are obtained using the developed time-dependent
concrete model were compared with available experimental results. The results show
good correlations between analysis and tests in terms of load-deflection and deflection
histories.
A parametric study is carried out in order to investigate the various factors
affecting slab deflections.
iv
TABLE OF CONTENTS
TABLE OF CONTENTS............................................................................................. iv
LIST OF FIGURES .....................................................................................................vii
LIST OF TABLES.......................................................................................................xiv
LIST OF SYMBOLS ...................................................................................................xvi
ACKNOLEDGEMENTS.............................................................................................xxii
Chapter 1 INTRODUCTION......................................................................................1
1.1 Background....................................................................................................3
1.2 Objective and Scope ......................................................................................6
1.3 Literature Review ..........................................................................................8
1.3.1 Material Properties for the Early Age Concrete ..................................9
1.3.1.1 Compressive Strength ...............................................................9
1.3.1.2 Tensile Strength.........................................................................13
1.3.2 Tension Stiffening ...............................................................................14
1.3.3 Creep and Shrinkage of Concrete........................................................15
1.3.4 Concrete Tensile Creep .......................................................................18
1.3.5 Factors Affecting Creep and Shrinkage ..............................................18
1.3.5.1 Cement ......................................................................................19
1.3.5.2 Aggregate ..................................................................................19
1.3.5.3 Admixture..................................................................................20
1.3.5.4 Water-to-Cement Ratio .............................................................21
1.3.5.5 Time ..........................................................................................22
1.3.5.6 Other Factors .............................................................................22
1.3.6 Analysis Approaches ...........................................................................23
1.3.7 Construction Loads..............................................................................25
1.3.8 Experimental Studies...........................................................................29
1.4 Thesis Layout.................................................................................................32
Chapter 2 METHOD OF ANALYSIS........................................................................34
2.1 Introduction....................................................................................................34
2.2 Material Models.............................................................................................35
2.2.1 Concrete Elastic Model .......................................................................35
2.2.2 Tension Stiffening Models ..................................................................37
2.2.3 Equivalent Uniaxial Strain ..................................................................38
v
2.2.4 Cracking Algorithm.............................................................................40
2.2.5 Creep and Shrinkage Algorithm..........................................................44
2.2.6 Strength Development of Concrete .....................................................47
2.2.7 Reinforcing and Post-Tensioning Steel ...............................................49
2.3 Interface of Concrete Model in ABAQUS/Standard .....................................50
2.4 Solution Method ............................................................................................51
2.4.1 Modified Newton-Raphson Method....................................................51
2.4.2 Convergence ........................................................................................53
Chapter 3 EXPERIMENTAL STUDY.......................................................................66
3.1 Introduction....................................................................................................66
3.2 Specimen Design and Preparation .................................................................66
3.3 Material Properties.........................................................................................67
3.4 Test Setup and Procedure ..............................................................................68
3.5 Immediate Deflection due to Application of Live Load................................68
3.6 Long-Term Deflection under Sustained Load ...............................................70
3.7 Summary........................................................................................................70
Chapter 4 VERIFICATION OF DEVELOPED MODEL..........................................90
4.1 Introduction....................................................................................................90
4.2 Scott and Beeby (2005) .................................................................................90
4.3 McNeice Corner Supported Slab (1971) .......................................................92
4.4 Burns and Hemakom (1985)..........................................................................93
4.5 Gilbert and Guo (2005)..................................................................................94
4.6 Analytical Investigation of One Way Slab Specimens..................................95
4.6.1 Calculation of Deflections Using Method Specified in Design Code .......................................................................................................96
4.6.2 Prediction of Cracking Loads..............................................................98
4.6.3 Results of Analysis: Instantaneous Deflections ..................................99
4.6.4 Results of Analysis: Long-Term Deflections ......................................100
4.7 Finite Element Analysis using Developed Concrete Model..........................102
4.7.1 Finite Element Model ..........................................................................102
4.7.2 Immediate Deflections.........................................................................103
4.7.3 Long-Term Deflections .......................................................................104
4.8 Summary........................................................................................................106
Chapter 5 PARAMETRIC STUDY BASED ON THE DEVELOPED MATERIAL MODEL ..........................................................................................148
5.1 Introduction....................................................................................................148
vi
5.2 Slab Design....................................................................................................149
5.3 Finite Element Model ....................................................................................150
5.4 Parameters......................................................................................................151
5.4.1 Load-Time History Model...................................................................152
5.4.2 Slab Thickness.....................................................................................155
5.4.3 Column Stiffness .................................................................................155
5.4.4 Separation of Creep and Shrinkage Effect ..........................................156
5.4.5 Elastic and Nonlinear Analysis ...........................................................157
5.4.6 Extraordinary Superimposed Loading.................................................157
5.4.7 Age of Application of Loading............................................................158
5.5 Long-Term Multiplier....................................................................................159
5.6 Moment Variation..........................................................................................160
5.7 Summary........................................................................................................162
Chapter 6 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS..............197
6.1 Summary........................................................................................................197
6.2 Conclusions....................................................................................................199
6.3 Recommendations..........................................................................................201
Bibliography ................................................................................................................202
Appendix A CREEP AND SHRINKAGE MODELS ................................................209
A.1 ACI 209 Model (1992) .................................................................................209
A.2 CEB-FIP Model (fib, 1999) ..........................................................................212
A.3 GL2000 (Gardner and Lockman, 2001) .......................................................217
Appendix B SHRINKAGE RESTRAINT..................................................................219
vii
LIST OF FIGURES
Figure 2-1: Compressive Stress-Strain Curve of Concrete ........................................57
Figure 2-2: Steel, Concrete, and Bond Stress in a Cracked Reinforced Concrete Prism Member ......................................................................................................58
Figure 2-3: Tension Stiffening Models ......................................................................59
Figure 2-4: Equivalent Uniaxial Stress-Strain Relation .............................................60
Figure 2-5: Time-Dependent Tension Stiffening Model-Damjanic and Owen..........61
Figure 2-6: Stress-Strain Curve of Steel.....................................................................62
Figure 2-7: Stress-Strain Curve of Prestressing Steel.................................................62
Figure 2-8: Incremental Method.................................................................................63
Figure 2-9: Newton-Raphson Method........................................................................64
Figure 2-10: Modified Newton-Raphson Method......................................................65
Figure 3-1: Comparison of Time-Dependent Compressive Strength Between Experiment and Analysis ......................................................................................74
Figure 3-2: Comparison of Time-Dependent Elastic Modulus Between Experiment and Analysis ......................................................................................75
Figure 3-3: Test Setup ................................................................................................75
Figure 3-4: Setup for Live Load .................................................................................76
Figure 3-5: Loading History.......................................................................................76
Figure 3-6: Load-Deflection Response for Loading at 3 Days ..................................77
Figure 3-7: Load-Deflection Response for Loading at 7 Days ..................................77
Figure 3-8: Load-Deflection Response for Loading at 28 Days.................................78
Figure 3-9: Averaged Load-Deflection Response ......................................................78
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Figure 3-10: Load-deflection Response due to First and Second Live Loads of B1D3.....................................................................................................................79
Figure 3-11: Load-deflection Response due to First and Second Live Loads of B2D3.....................................................................................................................79
Figure 3-12: Load-deflection Response due to First and Second Live Loads of B3D3.....................................................................................................................80
Figure 3-13: Load-deflection Response due to First and Second Live Loads of B4D7.....................................................................................................................80
Figure 3-14: Load-deflection Response due to First and Second Live Loads of B5D7.....................................................................................................................81
Figure 3-15: Load-deflection Response due to First and Second Live Loads of B6D7.....................................................................................................................81
Figure 3-16: Load-deflection Response due to First and Second Live Loads of B7D28...................................................................................................................82
Figure 3-17: Load-deflection Response due to First and Second Live Loads of B8D28...................................................................................................................82
Figure 3-18: Load-deflection Response due to First and Second Live Loads of B9D28...................................................................................................................83
Figure 3-19: Deflection History for B1D3.................................................................83
Figure 3-20: Deflection History for B2D3.................................................................84
Figure 3-21: Deflection History for B3D3.................................................................84
Figure 3-22: Deflection History for B4D7.................................................................85
Figure 3-23: Deflection History for B5D7 .................................................................85
Figure 3-24: Deflection History for B6D7.................................................................86
Figure 3-25: Deflection History for B7D28 ...............................................................86
Figure 3-26: Deflection History for B8D28 ...............................................................87
Figure 3-27: Deflection History for B9D28 ...............................................................87
ix
Figure 3-28: The Effect of Age at Loading on Long-Term Deflection......................88
Figure 3-29: Variation of Humidity ...........................................................................89
Figure 3-30: Variation of Temperature ......................................................................89
Figure 4-1: Time-Load History for T16R1..................................................................115
Figure 4-2: Idealization of Axial Member ...................................................................116
Figure 4-3: Assumed Creep Coefficient for Scott and Beeby .....................................117
Figure 4-4: Assumed Tension Stiffening Model .........................................................117
Figure 4-5: Time-Dependent Strain Variations of Concrete........................................118
Figure 4-6: Time-Dependent Average Stress Variations of Concrete .........................118
Figure 4-7: The Geometry of Slab(McNeice, 1967)....................................................119
Figure 4-8: Tension Stiffening Models for McNeice Slab ..........................................120
Figure 4-9: Load-Deflection at Center of McNeice Slab.............................................120
Figure 4-10: Geometry and Tendon Layout of Slab (Burns and Hemakom, 1986) ....121
Figure 4-11: Tendon Profile.........................................................................................122
Figure 4-12: Equivalent Loading and Equivalent Layer Method ................................123
Figure 4-13: Deflection of Panel A..............................................................................124
Figure 4-14: Deflection of Panel B..............................................................................124
Figure 4-15: Deflection of Panel C..............................................................................125
Figure 4-16: Dimension of Slab and Measuring Points...............................................126
Figure 4-17: Reinforcement Layout.............................................................................127
Figure 4-18: The Finite Element Model of Slab..........................................................128
Figure 4-19: Loading History of S3.............................................................................128
Figure 4-20: Assumed Tension Stiffening Model for Gilbert and Guo Slab...............129
Figure 4-21: Creep Coefficient for Gilbert and Guo Slab(Guo and Gilbert, 2002).....129
x
Figure 4-22: Shrinkage Strain for Gilbert and Guo Slab (Guo and Gilbert, 2002) .....130
Figure 4-23: Deflection History for Point 4, 6, 11, and 13..........................................130
Figure 4-24: Deflection History for Point 8 and 9.......................................................131
Figure 4-25: Deflection History for Point 1, 2, 15, and 16..........................................131
Figure 4-26: Deflection History for Point 5 and 12.....................................................132
Figure 4-27: Deflection History for Point 3, 7, 10, and 14..........................................132
Figure 4-28: Schematic Time- Deflection History ......................................................133
Figure 4-29: Prediction of Cracking Load from Load-Deflection Response of Loading at 3 Days.................................................................................................133
Figure 4-30: Prediction of Cracking Load from Load-Deflection Response of Loading at 7 Days.................................................................................................134
Figure 4-31: Prediction of Cracking Load from Load-Deflection Response of Loading at 28 Days...............................................................................................134
Figure 4-32: Comparison Between Experiment and Analytical Results for Loading at 3 Days.................................................................................................135
Figure 4-33: Comparison Between Experiment and Analytical Results for Loading at 7 Days.................................................................................................135
Figure 4-34: Comparison Between Experiment and Analytical Results for Loading at 28 Days...............................................................................................136
Figure 4-35: Long-Term Deflection Based on ACI 318..............................................136
Figure 4-36: Assumed Tension Stiffening Models for Test Slabs...............................137
Figure 4-37: Creep Coefficient Based on ACI 209 .....................................................137
Figure 4-38: Creep Coefficient Based on GL2000 ......................................................138
Figure 4-39: Creep Coefficient Based on CEB-FIP ....................................................138
Figure 4-40: Shrinkage Model Based on ACI 209.....................................................139
Figure 4-41: Shrinkage Model Based on GL2000 .....................................................139
Figure 4-42: Shrinkage Model Based on CEB-FIP....................................................140
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Figure 4-43: Finite Element Model..............................................................................140
Figure 4-44: Comparison Between Analysis and Experiment of Loading at 3 days...141
Figure 4-45: Comparison Between Analysis and Experiment of Loading at 7 days...141
Figure 4-46: Comparison Between Analysis and Experiment of Loading at 28 days .......................................................................................................................142
Figure 4-47: Time-Dependent Deflection of Loading at 3 days Using ACI 209 ........143
Figure 4-48: Time-Dependent Deflection of Loading at 7 days Using ACI 209 ........143
Figure 4-49: Time-Dependent Deflection of Loading at 28 days Using ACI 209 ......144
Figure 4-50: Time-Dependent Deflection of Loading at 3 days Using GL2000.........144
Figure 4-51: Time-Dependent Deflection of Loading at 7 days Using GL2000.........145
Figure 4-52: Time-Dependent Deflection of Loading at 28 days Using GL2000.......145
Figure 4-53: Time-Dependent Deflection of Loading at 3 days Using CEB-FIP .......146
Figure 4-54: Time-Dependent Deflection of Loading at 7 days Using CEB-FIP .......146
Figure 4-55: Time-Dependent Deflection of Loading at 28 days Using CEB-FIP .....147
Figure 5-1: Plan of Flat Plate System.........................................................................172
Figure 5-2: Distribution of Total Moment in the Exterior Panel ...............................173
Figure 5-3: Schematic Reinforcement Lay-out ..........................................................174
Figure 5-4: Arrangement of Reinforcement in Slab...................................................175
Figure 5-5: Mesh of Finite Element Model................................................................175
Figure 5-6: Assumed Tension Stiffening Model........................................................176
Figure 5-7: Creep Coefficient for Loading at 7 Days .................................................176
Figure 5-8: Creep Coefficient for Loading at 14 Days ..............................................177
Figure 5-9: Creep Coefficient for Loading at 21 Days ..............................................177
Figure 5-10: Creep Coefficient for Loading at 28 Days.............................................178
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Figure 5-11: Creep Coefficients along Age of Loading of 8 in Thick Slab...............178
Figure 5-12: Shrinkage Model Based on GL2000 for Parametric Study ...................179
Figure 5-13: Boundary Condition ..............................................................................180
Figure 5-14: Schematic Load-Time History ..............................................................181
Figure 5-15: Simplified Load-Time History in Accordance with ACI 435R ............182
Figure 5-16: Maximum Slab Load Ratio for 2S1R for 7 Days of Construction Cycle (Rosowsky and Stewart, 2001)...................................................................183
Figure 5-17: Maximum Slab Load Ratio for 3S for 7 Days of Construction Cycle (Rosowsky and Stewart, 2001) .............................................................................184
Figure 5-18: The Location of Maximum Deflection of Exterior Panel .....................185
Figure 5-19: Time-Deflection for Given Loading Histories (case: 1, 2, 3, and 4).....186
Figure 5-20: Time-Deflection for Given Slab Thicknesses (case 5, 6, and 7)............186
Figure 5-21: Time-Deflection for Given Column Stiffness (case 8, 9, and 10).........187
Figure 5-22: Separation of Creep and Shrinkage (case: 11, 12, and 13)....................187
Figure 5-23: Comparison Between Elastic Analysis and Nonlinear Analysis (Case: 14 and 15) ..................................................................................................188
Figure 5-24: Extraordinary Loading Condition with Minimum Thicknesses (Case: 16 and 17) ..................................................................................................188
Figure 5-25: Effect of Age of Loading without Shrinkage Restraint (Case: 18, 19, 20, and 21) ............................................................................................................189
Figure 5-26: Effect of Age of Loading with Shrinkage Restraint(Case: 17, 22, 23, and 24) ..................................................................................................................189
Figure 5-27: Loading at 14 Days................................................................................190
Figure 5-28: Loading at 21 Days................................................................................190
Figure 5-29: Loading at 28 Days................................................................................191
Figure 5-30: Long-Term Multiplier Along with Age of Loading ..............................191
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Figure 5-31: The Line of the Column Face and the Center Line of Panel .................192
Figure 5-32: Moment Diagram at the Exterior Column Line ....................................193
Figure 5-33: Moment Diagram at the Intermediate Line ...........................................193
Figure 5-34: Moment Diagram at the Exterior Column Line.....................................194
Figure 5-35: Moment Diagram at the Longitudinal Line...........................................194
Figure 5-36: Time-dependent Moment Diagram at the Exterior Column Line .........195
Figure 5-37: Time-dependent Moment Diagram at the Intermediate Line ................195
Figure 5-38: Time-dependent Moment Diagram at the Interior Column Line ..........196
Figure B-1: Shrinkage Restraint (Gilbert, 1988)........................................................221
xiv
LIST OF TABLES
Table 2-1: Constant for ACI 209 ................................................................................56
Table 2-2: Coefficient for GL2000 Model..................................................................56
Table 2-3: Coefficient for CEB-FIP Model ................................................................56
Table 3-1: Concrete Mix Used....................................................................................72
Table 3-2: Concrete Material Properties.....................................................................73
Table 4-1: Material Properties of Scott and Beeby.....................................................107
Table 4-2: Material Properties of McNeice Slab ........................................................107
Table 4-3: Differences of Deflection Between Analytical and Experimental Results of Burns and Hemakom ...........................................................................108
Table 4-4: Material Properties of Gilbert and Guo Slab.............................................108
Table 4-5: Factor k for Modulus of Rupture...............................................................109
Table 4-6: Differences of Maximum Deflection Between Analytical and Experimental Results ............................................................................................109
Table 4-7: Differences of Irrecoverable Deflection Between Analytical and Experimental Results ............................................................................................110
Table 4-8: Prediction of Long-Term Deflection Based on Equations Specified in ACI 318 ................................................................................................................110
Table 4-9: Long-Term Multiplier from Experiment...................................................111
Table 4-10: Deflection Requirements.........................................................................112
Table 4-11: Input Value of Analytical Model.............................................................112
Table 4-12: Input Values For Creep and Shrinkage ...................................................113
Table 4-13: Differences Between Analytical and Experimental Results....................114
Table 4-14: Differences of Long-term Deflection Between Analytical and Experimental Results after six Months.................................................................114
xv
Table 5-1: Given Design Loads ..................................................................................164
Table 5-2: Given Design Loads Based on Extraordinary Superimposed Dead Load ......................................................................................................................164
Table 5-3: Amount of Reinforcements (Ordinary Loading) : E-W direction............165
Table 5-4: Amount of Reinforcements (Ordinary Loading) : N-S direction .............166
Table 5-5: Amount of Reinforcements(Extraordinary Loading) : E-W direction .....167
Table 5-6: Amount of Reinforcements(Extraordinary Loading) : N-S direction ......168
Table 5-7: Material Properties of Concrete for Parametric Study ..............................168
Table 5-8: Input Values for Creep and Shrinkage in Parametric Study .....................169
Table 5-9: Parameters .................................................................................................170
Table 5-10: Long-Term Multiplier .............................................................................171
Table A-1: Correction Factors for ACI 209 Model ....................................................211
xvi
LIST OF SYMBOLS
Subscripts
c concrete or compressive
d residual in terms of displacement
e effective
el elastic component
p principal direction
pl plastic component
r residual in terms of force
u uniaxial direction
x local x coordinate
y local y coordinate
1 principal 1 coordinate
2 principal 2 coordinate
Superscript
n iteration number in step
Variables
cE elastic modulus of concrete
ciE tangent modulus of elasticity
xvii
cmtE mean modulus of elasticity
iE secant modulus of concrete in uniaxial stress-strain relationship, (i=1,2)
sE elastic modulus of steel
pE elastic modulus of prestressing steel
f force matrix
cf concrete stress
'cf compressive strength of concrete
28ckf characteristic or specified compressive strength of concrete at 28 days
cmf mean compressive strength of concrete
28cmf mean compressive strength of concrete at 28 days
cmtf mean compressive strength at age t
rf modulus of rupture
sf stress of steel
pyf yielding stress of prestressing steel
yf yielding stress of steel
G shear modulus
H humidity
I internal force
crI cracked transformed moment of inertia.
eI effective moment of inertia
xviii
gI gross moment of inertia
l member length
nl clear span length
2l length of the transverse width of strip
aM applied maximum service load moment
crM cracking moment
0M panel moment
P load or external force
LP concentrated live load
R Residual
t time or thickness
oct age of concrete at loading
ost age of concrete at the beginning of shrinkage
u displacement
sv / volume-to-surface ratio
CLw construction live load
constw construction load
Dw dead load
slabw slab self-weight
uw factored load
xix
ε total strain
elε elastic strain component
cε concrete strain
crε creep strain
sε strain of steel
shε shrinkage strain
0ε locus in terms of strain
plε plastic strain component
uiε uniaxial strain component
xε strain component in x direction
yε strain component in y direction
xyγ shear strain component
1ε principal strain component in 1 direction
2ε principal strain component in 2 direction
εΔ incremental total strain
elεΔ incremental elastic strain component
Δ deflection of member or incremental
DΔ instantaneous deflection induced by dead load
irrΔ irrecoverable deflection
LΔ instantaneous deflection induced by live load
xx
DL+Δ instantaneous deflection induced by dead load plus live load
LTΔ long-term deflection
susΔ instantaneous deflection induced by sustained load
plεΔ incremental plastic strain component
uiεΔ incremental uniaxial strain component
crεΔ incremental creep strain
shεΔ incremental shrinkage strain
xεΔ incremental strain component at x direction
yεΔ incremental strain component at y direction
1εΔ incremental principal strain component in 1 direction
2εΔ incremental principal strain component in 2 direction
xyγΔ incremental shear strain component
σ stress
0σ locus in terms of stress
uiσ uniaxial stress component
xσ stress component in x direction
yσ stress component in y direction
xyσ shear stress component
uiσΔ incremental uniaxial stress component
xxi
xσΔ incremental stress component in x direction
yσΔ incremental stress component in y direction
xyσΔ incremental shear stress component
pθ principal direction
),( τφ t creep coefficient
'ρ compression reinforcement ratio
ξ time-dependent factor
λ long-term multiplier
xxii
ACKNOLEDGEMENTS
I would like to present my gratitude to a number of individual who have given me
help and encouragement throughout the process of earning my doctoral degree.
I especially wish to express my appreciation to Dr. Andrew Scanlon, my advisor
and committee chair. Without unconditional help from you, this study would never have
been finished. Thank you for guiding and supporting me through these five years.
I am also grateful to Dr. Andrea J. Schokker, Dr. Maria Lopez de Murphy, and Dr.
Ali M. Memari for providing me knowledge, skill, and their assistance.
I would like to thank all my friends for bearing me during past five years and
standing beside me.
My highest appreciation goes to my family. I thank my beautiful son and wife,
Lucas and Yeonsoo. They always make me happy and have been giving me inspiration
and motivation. Thank you my brothers, Jewon and Jegeun, who have taken care of
mother while I am not with her.
Finally, I want to show my deep appreciation to my mother Myeongrye Jo, and
the late father, Daejong Lee. I know my father is always watching me in the heaven. The
word will never show my appreciation for all your supports, sacrifices and infinite love
from the day I born.
Chapter 1
INTRODUCTION
The design of reinforced and prestressed concrete slab requires a limitation of
deflection and camber. In order to provide the limitation, it is necessary to perform
extensive experiments and develop accurate analysis methods. Because nonlinear
properties of concrete as well as time-dependent effects make the analysis difficult,
practical modeling methods are essential for analysis.
During construction of multistory buildings, shoring and reshoring processes are
employed and construction loads are applied to the slab. The construction load affects
floor and roof slab deflection because the strength of concrete and age of loading vary
according to the construction methods and cycles used. Also, the slab experiences
construction loading due to material storage and construction equipment which may also
cause cracking at early age of concrete. An early loss of stiffness may cause a high short
term deflection. In addition, it is well known that creep and shrinkage effects increase
long-term deflections. The loading history including construction loads is an important
factor which can increase cracking in the concrete slab (ACI Committee 347, 2005; Hurd,
1995).
Concrete shows different material behavior under compression and tension. Under
certain levels of loading, cracks are formed in the concrete slabs. These cracks are the
major factor causing nonlinear material behavior of concrete. In addition, time-dependent
material behavior, creep and shrinkage, are also the major sources of nonlinear behavior
2
of concrete. Nonlinear analysis, therefore, needs to be used to express real behavior of
concrete and reinforcement as closely as possible (Phuvoravan and Sotelino, 2005).
The calculation of deflections for two-way slab is complicated even if behavior of
concrete slab is linear elastic. In order to analyze the concrete slab, classical mechanics
based on plate and shell theory, numerical methods such as finite difference method
(FDM) and finite element method (FEM) have been developed. Among these methods,
the finite element method may be the most popular application in a complex concrete slab
system. Although classical plate and shell theory has provided theoretical background for
developing other methods, it has limitations in analyzing the plate and shell when there
are complicated shape of slab, composite materials, loading conditions, and boundary
conditions. On the other hand, as a result of development of micro computer, numerical
method has become more popular. The Finite Element Method has been described in
detail by Chapelle and Bathe (2003), Ugural (1981), and Szilard (1974).
When performing the finite element analysis of concrete slabs, it is necessary to
incorporate nonlinear material properties of concrete and reinforcement into the model. In
addition, proper finite element types should be chosen according to the problem. Under
service load, the most important aspect in concrete material model is a tensile cracking.
Tension stiffening models usually have been introduced to model the stiffness provided
by concrete between cracks. In these models, the concrete cracking is treated as a gradual
reduction of tensile stiffness with increasing load (e.g. Scanlon, 1972; Fields and
Bischoff, 2004; Link et al, 1989).
In the analysis, the tension stiffening effect is normally applied when calculating
the short-term deflection. When the concrete structure is subjected to time-dependent
3
condition, the tension stiffening model needs to be changed into a function of time. This
research introduces a long-term loss of tension stiffening in finite element analysis. In the
level of material model, it is not easy to combine both concrete and reinforce with the
instantaneous and time-dependent material model. Also, proper numerical solution
method is necessary. An important aspect of early age concrete is the development of
compressive and tensile strengths with time. It is reported that shrinkage can cause cracks,
so that the concrete cracking moment can be reduced by shrinkage caused during curing
(Bischoff, 2005, Gilbert, 1992, Scanlon and Murray, 1982). Also, the creep and shrinkage
effect can be increased if the loading starts at early age. Long-term loss of tension
stiffening also needs to be considered when modeling time-dependent effects.
Experimental studies for early age creep showed that the creep is significantly high
(Altoubat and Lang, 2001; Kovler, 1995; Bissonnette and Pigeon, 1995). Therefore, an
analytical model explaining the shrinkage effect during curing needs to be developed.
1.1 Background
Many modeling approaches have been developed in order to analyze concrete
slabs. Smeared crack models and discrete crack models have been developed. Discrete
model approach considers a crack as geometrical discontinuity. The crack occurs
following predefined path by using nodal separation in the finite element analysis. This
method changes the connectivity of nodes continuously. But nodal separation and
predefined crack path are not the nature of finite element method. These weaknesses have
been modified by using graphic-aided algorithms. Discrete model is more proper to
4
idealize the local effect such as punching shear and column-slab connection in the
concrete slab modeling. On the other hand, the smeared approach considers a cracked
concrete as a cracked solid continuum. Smeared crack models use a concept of oriented
damaged elasticity in terms of stress-strain relations. Researchers have suggested
smeared model and proved its accuracy. Especially, in order to calculate the load-
deflection response of concrete slab, a smeared model has provided good results (e.g.
Rots and Blaauwendraad, 1989).
Although commercial finite element program such as ABAQUS (2002) provides a
plasticity concrete models based on plasticity theory, the calculation procedure for creep
and shrinkage have not been implemented with the cracking algorithm for concrete. In
order to calculate the short-term deflections, smeared model and damaged-plasticity
material model for concrete are provided in ABAQUS. Smeared crack model uses a
concept of oriented damaged elasticity in terms of stress-strain relations. The inelastic
compressive stress-strain relation is used to express the isotropic compressive inelasticity
and the tension stiffening effect and tensile stress-strain relation are adopted to account
for tensile cracking. Damaged-plasticity model for concrete was proposed by Lubliner et
al (1989). This model uses fracture-energy-based scalar damaged variables to represent a
damage of concrete. Also, the model introduces elastic and inelastic stiffness degradation
variables which are used to make an inelastic constitutive model of concrete. However,
ABAQUS does not provide a procedure to calculate the creep and shrinkage under
cracked condition. Therefore, it is crucial to combine the procedures to calculate the short
and long-term behavior of concrete slabs together.
5
The inelastic stress-strain relation is used to express the isotropic compressive
inelasticity and the tension stiffening effect. Tensile stress-strain relation is adopted to
account for tensile cracking. The tension stiffening effect is especially important aspect
of tensile behavior of reinforced concrete. It is because concrete can carry tension
between cracks in a reinforced concrete. Scanlon (1971) first introduced the tension
stiffening model in finite element analysis. Tensile stress between cracks in concrete
member is considered as an average tensile stress of concrete. A number of tension
stiffening models have been proposed (e.g. Fields and Bischoff, 2004; Massicotte et al,
1990; Damjanic and Owen, 1984; Lin and Scordelis, 1975; Scanlon and Murray, 1974).
According to Bischoff (2001), the tension stiffening can be reduced due to shrinkage
effect. He addressed that test results can be affected by shrinkage significantly. It is
because reinforcement has compressive stress and concrete has tensile stress as a result of
member shortening caused by shrinkage prior to loading. Fields and Bischoff (2004)
proposed a tension stiffening equation including shrinkage effect. Ostergaard et al (2001)
performed test, and presented result that tensile creep of concrete is much higher when
loading is applied at early age. It is also controversial that the decay of tension stiffening
exists for some time after loading as like the results of Scott and Beeby’s test (2005).
The advantages of smeared model include its efficiency in prediction of load-
deflection response of the concrete slab and simplicity of implementation. Layered model
approaches introduced to embody the smeared model. It allows the stress-strain variation
through thickness and makes possible reinforcement modeled as smeared layer between
concrete layers. It is also an advantage that flexural and membrane action of concrete slab
can be expressed as two dimensional plane stress behavior. Thus, the concrete slab can be
6
idealized as two-dimensional plane finite elements. This shortens the run-time and makes
it easy to develop a material model. Material model is treated as a two-dimensional
biaxial stress-strain relation (Scanlon, 1971; Scanlon and Murray, 1974; Lin and
Scordelis, 1975, Gilbert and Warner, 1978).
Noh et al (2003) presented a finite element analysis using the reinforced concrete
shell element. For the concrete material model an orthotropic model was used. The
concrete model is based on the elastic-plastic damage model for the cyclic and monotonic
loading. Layered shell element based on Reissner-Mindlin shell theory was adopted with
the orthotropic concrete model and a bilinear elastic-plastic reinforcement model.
Phuvoravan and Sotelino (2005) suggested a new finite element for the nonlinear
analysis of reinforced concrete slabs. The developed model was implemented in
ABAQUS using user-defined element (UEL) and user-defined material model (UMAT).
The concrete slab was idealized with four node shell element based on Kirchhoff shell
theory and the reinforcement was modeled with two node beam element. The difference
between existing layered model and proposed method was that the reinforcement was
considered as beam element and connected shell element with rigid link. For the concrete
material model, orthotropic model was adopted and uniaxial stress-strain relation was
used for reinforcement model.
1.2 Objective and Scope
The objective of this study is to develop analytical methods to investigate the
influence of material properties, construction sequence, and time-dependent effect on the
7
performance of concrete flat slab systems. Emphasis will be placed on implementation of
material constitutive model, cracking, and deflection of slabs under service load
conditions. In order to investigate the early age effect of concrete, an experimental
program was performed for one-way reinforced concrete slab. Results of this study will
be used to provide recommendations for improved design of concrete slab systems. This
objective will be achieved within the following scope:
1. Literature review to determine current state-of-the-knowledge including time-
dependent effects, nonlinearity and early age effect of concrete slabs.
2. Evaluations of existing creep, shrinkage, and cracking models obtained from the
literature for implementation in the analytical model.
3. Implementation of selected models in a general purpose program
(ABAQUS/Standard) through user-defined subroutine.
4. Performing the experimental program to investigate the early age effect of
concrete.
5. Validation of the analytical model using available experimental data.
6. Parametric studies to examine the influence of material model parameters,
environmental condition, and construction sequences on slab performance.
7. Development of conclusions and recommendations for design and construction of
concrete slab system.
8
1.3 Literature Review
In order to investigate the structural behavior of concrete slabs, a proper analytical
approach must be chosen and evaluated. Several analytical approaches have been
suggested such as equivalent frame method, classical plate and shell theory, and
numerical analysis. In addition, numerous experimental studies are focused on the
structural behavior of concrete slabs: short-term load-deflection response; creep and
shrinkage effect of concrete; long-term deflection history; effect of construction sequence
and evaluation of construction loads (Ofosu-Asamoah and Gardner, 1997; Rosowsky et al,
1994; Stivaros and Halvorsen, 1990; Jokinen and Scanlon, 1987).
Loading at early age occurs in multistory concrete building construction. The
fleshly poured concrete slab is supported by a system of shores and reshores. During
shoring and reshoring process construction loads are transferred into previously cast
floors which may not have attained the specified concrete strength. These loads also can
be higher than the design service loads. If the construction loads are not evaluated before
the design based on the understanding of the concrete properties at early age, there may
be structural failures or serviceability failures (Hurd, 1995).
The characteristics of the early age concrete are that strengths are still under
development and show low strengths in compression, and tension. A low elastic modulus
and stiffness of loading at early age can cause larger long-term deflections and cracking
than concrete slabs loaded at matured strength and stiffness. Time-dependent creep and
shrinkage of concrete can affect significantly long-term deflections.
9
This chapter presents the literature review to obtain the state-of-art knowledge of
material model for the early age concrete, analytical approaches, experimental studies,
and construction loading of the concrete slabs.
1.3.1 Material Properties for the Early Age Concrete
The strengths of early age concrete are mainly dependent on the rate of strength
development. The early age of concrete may be defined before 28 days after concrete
pouring. Because many construction sequences impose significant construction loads on
the concrete structure even though the concrete strength does not reach its maximum
specified strength, it is essential to know the properties of early age concrete. For
simplicity it may be assume that the strengths of concrete such as flexural strength, shear,
tensile is proportional to the concrete compressive strength at loading age (ACI
Committee 347, 2005).
Cracking and deflections are primarily related to the tensile strength, elastic
modulus and tension stiffening at that age. Prediction of proper concrete strengths at the
loading age is essential in calculation of slab deflections. Time-dependent properties at
early age are also important factors in prediction of the long-term deflections.
1.3.1.1 Compressive Strength
A typical concrete stress-strain relationship depends on various properties
including the strength of concrete, age of concrete, rate of loading, material properties of
10
cement and aggregate and size of specimen. The compressive stress of concrete shows an
approximately linear increase with strain in the range of '45.0~4.0 cf . Once the strain of
concrete exceeds the elastic range, the concrete stress increases nonlinearly and reaches
the specified compressive strength 'cf . After that, the stress decreases nonlinearly and that
is referred to as the softening phenomenon. ACI 318 (2005) code specifies an ultimate
strain of 0.003 for design. In the elastic range concrete is assumed to be an isotropic
linear elastic model. The linear elastic model is valid with the response of concrete
subject to both tensile stress below cracking and compressive stress in the range
of '45.0~4.0 cf . Researchers have extensively investigated nonlinear stress-strain
relations of concrete for decades using mathematical form (Carreira and Chu, 1985;
Popovics, 1970; Hognestad et al, 1955).
The relations for early age concrete may not be investigated thoroughly.
Numerous attempts have been made to investigate the concrete properties at early age.
The development of the compressive strength is varied under different curing temperature
and curing conditions. Klieger (1958) performed experiment about the strength
development under different curing condition. Experiment showed that the compressive
strength can be stronger as the duration of the moist curing is longer. Also, as the initial
temperature and curing temperature are increasing, the compressive strength is lower at 3
months and 1 year. The experiment shows that the development of the compressive
strength showed a nonlinear increase of its value.
Gardner and Poon (1976) tested a series of concrete cylinders to investigate the
compressive strength, tensile strength, and bond strength at early age. Experiment
11
showed that the curing at high temperature increase the rate of strength at early age, but
the final ultimate strength reached lower value.
Gardner (1990) performed experiment to investigate the effect of temperature on
the early age concrete properties. Type I, III, and Type I/Fly ash concrete cylinder and
prism specimens were used to get the compressive, split tensile strength and elastic
modulus. The controlled temperature conditions were 0, 10, 20, and 30 degree in Celsius.
According to experiment, the rate of strength development was related to water-cement
ratio, cement type, and temperature. The development of strength at early age was
retarded at low temperature (0 C) and curing temperature had little effect on the
development of strength of Type III or Type I having 0.35 water-to-cement ratio. Also,
experiment showed that the lower the curing temperature, the higher the rate of strength
development at early age but the lower the strength at early age less than 14 days. In the
research, the final ultimate strength seemed to reach at the expected strength at any
curing temperature regardless of the type of cement. Empirical equations predicting
tensile strength and elastic modulus were suggested as the relation of the power of
compressive strength.
Oluokun (1991) and Oluokun et al (1991) investigated the relationships between
the elastic modulus, Poisson’s ratio and the cylinder strength at early age. The results
were obtained for ages from 6hrs to 28 days. It is concluded that the modulus of elasticity
at appropriate age is proportional to the 0.5 power of the compressive strength. For the
Poisson’ ratio the values was not only insensitive to both the age of concrete and the
concrete mix, but also the value approximately taken as 0.19 did not change with
compressive strength development.
12
Khan et al (1995) performed experiment about the early age compressive stress-
strain characteristics of low (30 MPa), medium (50 MPa) and high-strength (70 MPa)
concrete. The experiment showed that the stress-strain relation for all of the concrete
started to be similar to the relation of 28 days after 24 hrs. Under different curing
conditions, the rate of development of strength follows by the order of temperature-
matched curing, sealed curing, and air-dried curing.
Schutter (1999) suggested the extended compressive stress-strain relation based
on CEB-FIP Model Code 1990 for early age concrete. In the research modifications using
the improved parameters were made. Once the compressive strength, tangent elastic
modulus, and strain at the peak compressive strength for early age concrete were known,
the CEB-FIP model produced the most accurate predictions compared with results from
experiment.
Yi et al (2003) proposed time-dependent stress-strain relation for the compressive
strength of concrete. The proposed model was not only verified against the experimental
results, but also compared with existing empirical equations. The range of specified
compressive strength from 30 MPa to 70 MPa with water-to-cement ratio from 0.89 to
0.30 was investigated. Total 8 different ages, 0.5, 0.75, 1, 2, 3, 7, 14, and 28 days, were
compared with experiment results. Results showed that the proposed model can predict
the compressive stress-strain relation accurately.
13
1.3.1.2 Tensile Strength
Before cracking, tensile stress of concrete is assumed to be linear elastic. After
cracking, the stress decreases according to the tension stiffening equation or softening
equation. Tensile strength is expressed in terms of a specific test method. The direct
tensile, the beam test for modulus of rupture, and split cylinder test are the three kinds of
tests that are frequently have been used. The strain at the maximum tensile stress usually
is assumed to increase up to the modulus of rupture while pure bending condition, and
sometimes can be assumed to be linear or nonlinear decrease. The modulus of rupture or
bending tensile strength rf , direct tensile strength, tf , of concrete is assumed as the
maximum elastic tensile strength of concrete. Therefore, the tensile stress-strain relation
in the elastic range can be assumed as linear elastic response.
Gardner and Poon (1976) showed that the relation that the tensile strength and
bond strength were proportional to the compressive strength. The 0.8 power of cylinder
strength at the appropriate age was suggested. Also, there was no significant effect on the
interrelationship of tensile strength or bond strength and compressive strength according
to different curing at temperature, cement type.
Oluokun (1991) presented a prediction method for tensile strength from the
compressive strength for normal weight concrete. In the research instead of using the 0.5
power relation used in ACI 318 for predicting the splitting tensile strength of concrete the
0.69 power relation is proposed.
Swaddiwudhipong et al (2003) performed the experiment about direct tensile test
and showed the result of tensile strain of concrete at early age. The investigation
14
concluded that the rate of development of tensile strength is lower than the compressive
strength and the average tensile strain at maximum strength is relatively independent
parameter. The average tensile strain at failure does not depend on strength, mix
proportion and age of concrete.
1.3.2 Tension Stiffening
The tension stiffening effect is usually considered to represent the tensile strength
of concrete after cracking in reinforced concrete. The effect contributes to overall
stiffness in cracked reinforced concrete particularly at service load levels. After the
concrete tensile stress reaches the maximum tensile strength, cracking will occur. Once
cracked, the concrete is assumed not to carry any tension at the cracks. But the tension is
transferred by reinforcement into the surrounding concrete. Consequently, the concrete
tensile stress can be assumed as the average tensile stress. The average tensile stress in
the concrete continues to decrease with increasing strain.
The tension stiffening effect was first introduced in finite element analysis by
Scanlon (1971) and various tension stiffening models have been proposed. Lin and
Scordelis (1975) proposed a bilinear type concrete tension stress-strain model to explain
the tension stiffening effect. Damjanic and Owen (1984) also suggested a bilinear type
tension stiffening effect but with sudden drop of tensile stress immediately after concrete
cracking. Fields and Bischoff (2004) proposed tension stiffening equation including
shrinkage before loading. Shrinkage reduced the tension stiffening effect of high strength
reinforced concrete tension member according to experiment. This is because the member
15
is initially shortened due to shrinkage. As result, compressive stress is caused in
reinforcing steel in opposite to concrete. The tensile stress in concrete reduces the axial
force causing cracks. They explained the shrinkage effect on tension stiffening as the
equation fitting experimental data. The tension stiffening equation includes a term for
shrinkage strain.
Scott and Beeby (2005) investigated long-term tension stiffening effects by
laboratory tests. Time-dependent strain variation of concrete and reinforcement due to
creep was investigated. It is reported that concrete tension stiffening was decayed
significantly during experiment. According to tests, decay of tension stiffening may
affect deflection history.
1.3.3 Creep and Shrinkage of Concrete
Time-dependent deformations can be divided into stress-dependent and stress-
independent. Shrinkage is a stress-independent deformation of concrete, and caused by
the loss of moisture of concrete or defined as the time-dependent volume or strain change
of concrete specimen not subjected to an external stress at a constant temperature after
hardening of concrete. On the other hand, creep refers to time- and stress-dependent
variation of strains in hardened concrete subjected to a constant sustained stress. Creep
and shrinkage, as a matter of fact, are interdependent processes. Although they are
affecting each other on their processes, they are treated as independent and assumed to be
additive as independent processes for simplicity. This is because the process between
16
creep and shrinkage is very complicated and hard to identify the process independently
(Bazant, 1988).
Creep and shrinkage is an inelastic behavior of concrete. This phenomenon is
caused by combining complex physical and chemical actions in concrete member.
Shrinkage may be defined as drying shrinkage, autogenous shrinkage, carbonation
shrinkage, and plastic shrinkage. Drying Shrinkage refers to general meaning of
shrinkage phenomenon in concrete (Pickett, 1946). This shrinkage is caused by the loss
of moisture from concrete under drying condition. Strain of drying shrinkage is partially
irreversible. Reversibly, the swelling can occur when concrete is saturated again.
However, the swelling is not only insignificant but insufficient to completely compensate
for shrinkage. Autogenous (hydration or chemical) shrinkage occurs when water is
removed internally by chemical combination during hydration in a moisture-sealed state.
Autogenous shrinkage is quite small for ordinary normal concrete but it is significant for
high-performance concretes. Carbonation shrinkage occurs when concrete is carbonated
in a low, relative humidity environment. Plastic (capillary) shrinkage occurs when water
is lost from concrete while it is in the plastic state (fib, 1999).
Creep of concrete may be divided into two types; basic creep and drying creep.
Basic creep is the time-dependent deformation that occurs when concrete is loaded in a
sealed condition so that moisture cannot escape (ACI Committee 209, 1992). Drying
creep occurs when concrete is loaded in allowing drying. Drying creep is the additional
creep in excess of basic creep and can be considered as stress-induced shrinkage.
According to the concept of drying and basic creep, there are existing differences
between inside and outside of concrete. But both creep strains are considered as one
17
creep strain and assumed to have the same creep strain rate when doing analysis.
Transitional creep strains are used as another nomenclature about creep strain, and those
are related to environmental condition. Transitional creep strains can be divided into
three components; transitional hygral creep, transitional chemical creep, and transitional
thermal creep. Transitional hygral creep refers to wetting creep and drying creep.
Transitional chemical creep strain occurs when concrete is under significant chemical
reactions. Transitional chemical creep strain is caused by hydration, carbonation of
cement paste. Transitional Thermal Creep Strain occurs when temperature is changed
under loading (Bazant, 1988; ACI Committee 209, 2005).
Many different theories have been suggested to explain observed behavior of
creep and shrinkage mechanism. Creep and shrinkage mechanisms could be distinguished
between real and apparent mechanism. Real mechanism could be considered as physical
and chemical properties of materials. This mechanism is independent of size and shape
effect. On the other hand, apparent mechanism is caused by other effect such as
composite action between aggregate and cement paste, moisture gradients, thermal effect
caused by hydration. Creep of concrete is an extremely complex phenomenon, mostly
because concrete is such a complex composite material. Extensive testing has been
performed on numerous varieties of concrete, but the creep mechanism is still not fully
understood today. However, the effects of certain factors have been concluded based on
trends observed in creep testing (Bazant, 1988).
18
1.3.4 Concrete Tensile Creep
Concepts of creep mechanism are usually regarded as compression phenomenon.
Also, tensile creep of concrete can be considered to be as large as the compression creep
(Bazant, 1988; Gilbert, 1988). Experimental studies of tensile creep have been reported
by researchers (Ostergaard et al, 2001; Altoubat and Lange, 2001; Kovler, 1999; Kovler;
1996; Kovler, 1995). It is reported that tensile creep is affected by many factors such as
curing condition, drying condition, loading condition, and stress levels. Although tensile
creep has similar characteristics as compressive creep has, there may be differences
between compression and tensile creep according to the development of micro cracking.
Tensile creep models are not suggested in the design code up to now and compression
creep models are prevalent in the analysis.
1.3.5 Factors Affecting Creep and Shrinkage
Concrete is made of cement, aggregates, water and admixtures. There are
numerous interacting factors affecting creep and shrinkage. The interaction among these
factors is very complicated and difficult to be understood because of complex chemical
and physical interactions. The factors are discussed and categorized to understand the
effect of each factor affecting creep and shrinkage. The factors may be categorized into
internal and external factors. Internal factors can include materials and mix-proportions.
External factors consist of stress level, environmental condition and geometry of the
concrete member, and structure type.
19
1.3.5.1 Cement
Portland cement is usually referred to general meaning of hydraulic cement.
Cement binds aggregates and provides adhesive property. Cement has effect on strength
of concrete and produces heat of hydration. Since the rate of hydration is connected to
development of strength and removal of moisture in concrete, creep and shrinkage are
also affected by cement characteristics. Cement content affects degree of hydration, and
volume change occurs due to hydration of cement paste. Also, cement content affects
compressive strength of concrete. The modulus of elasticity has relationship with strength
of concrete. This means that creep strain is affected by the modulus of elasticity of
concrete (Mehta and Monteiro, 1992).
1.3.5.2 Aggregate
The aggregate as well as cement in concrete mixture occupies most of volume.
About 60 to 80 percent of volume is filled up with the aggregate. The strength of concrete
and dimensional stability and durability of concrete can be affected by the aggregate. For
instance, size, shape, surface texture, and composition of coarse and fine aggregate are
known to affect concrete strength (Mehta and Monteiro, 1992).
The size of aggregate has influence on the proportions of concrete mix. If the ratio
of water to cement is constant, the larger aggregate in mixture, the less compressive
strength of concrete. This is because the large aggregates have smaller surface area than
the small aggregates. As a result, the bond between cement and aggregates is weaker than
the small aggregates (Popovics, 1998).
20
The shape and surface texture of aggregates has effect on the strength of concrete
at early age. The shape of aggregates refers to geometrical properties such as rounded,
angular, elongated, and flaky. Rough textured aggregate may help formation of strong
bond between the cement paste and aggregates, but rough texture or flat shape requires
more water to produce workability (PCA, 1968).
1.3.5.3 Admixture
Admixtures are ingredient of concrete mixture and added to the batch before or
during mixing as well as water, cement, and aggregates. It is difficult to classify
admixtures because there are so many kinds of admixtures and some admixtures have
more than two kinds of effects. However, those can be classified according to chemical
composition and functions on concrete mixture. Mehta and Monteiro (1992) classified
admixture according to their composition, mechanism of action, applications, surface-
active chemicals, set-controlling chemicals, and mineral admixtures.
Air-entraining and water-reducing admixtures can be categorized as surface-
active chemicals. Air-entraining admixtures are used to improve durability under the
weather cycle of freezing and thawing. Those improve workability of concrete mixtures.
Lager amount of air-entraining admixtures can cause delaying in cement hydration and
decrease of concrete strength. Water-reducing admixtures are used to reduce the water
requirement in concrete mixture. An increase in strength can be achieved by reducing
water under condition that cement and slump are constant. In spite of reduction in water
content, it is reported that there is significant drying shrinkage in concrete made of some
21
water-reducing admixtures. Accelerating and retarding admixture can be included in set-
controlling admixtures. Those admixtures control the setting time of concrete and the rate
of strength development at early age. An accelerating admixture is used to accelerate the
setting time and the strength development. Most accelerating admixtures have the effect
on drying shrinkage and those are increase the drying shrinkage. Calcium chloride is the
most common accelerating admixture. On the other hand, it is reported that retarding
admixtures reduce the strength of concrete at early age but shrinkage may not be
predictable (PCA, 1968).
Mineral admixtures refers to pozzolanic and/or cememtitious admixtures. These
admixtures can be obtained from the nature or by-product of the industry. Also, these
admixtures have tendency to increase the volume of fine pores in hydrated cement. Creep
and drying shrinkage in concrete are associated with the water held by small pores. If
concrete has higher pore refinement, the higher drying shrinkage and creep occur (Mehta
and Monteiro, 1992).
1.3.5.4 Water-to-Cement Ratio
Creep and shrinkage effect are directly not influenced by water and cement
content. Water and cement are influencing each other when mixing concrete proportions.
Also, the variation in water and cement content in concrete mixture affects other
proportions. It is difficult to understand what the contribution of each factor is in creep
and shrinkage. For constant water-to-cement ratio, as cement content increases, creep and
shrinkage has tendency to increase (Mehta and Monteiro, 1992).
22
1.3.5.5 Time
Creep and shrinkage effect is the function of time and takes place over long
period. Water movement taken place by capillary tension effect from small pores of
hydrated cement paste to the atmosphere and/or other small pores is the time-dependent
process. 75 to 80 percents of total amount of creep and shrinkage occurs within one year
(Mehta and Monteiro, 1992)
1.3.5.6 Other Factors
Curing condition greatly affects creep and shrinkage effect of concrete.
Depending on curing method and curing history, creep and shrinkage can be varied.
There is significant difference between value obtained in practice under varying humidity
and that obtained in laboratory at constant humidity. For stress level, there is
proportionality between creep strain and stress level. This relation may be valid when
stress level is in the range of elasticity of concrete. In regard to the atmospheric humidity,
while humidity increases, it makes the relative rate of moisture flow from the interior to
the exterior surface of concrete slow down. For a same condition of exposure, it is
reported that the increased humidity in the air reduces the shrinkage and creep. Geometry
and structure type of concrete member can affect creep and shrinkage. Because there is
resistance to water movement from the interior to the exterior of concrete, the rate of
water movement can be governed by the total length of path traveled by water (Mehta
and Monteiro, 1992).
23
1.3.6 Analysis Approaches
In the numerical analysis of reinforced concrete slab a finite element analysis is
usually used. This numerical approach usually provides reasonable accuracy of solution.
In the finite element method, the plate or slab is idealized into a finite number of
elements using triangular or rectangular in shape. Finite elements are connected at their
nodes at which the compatibility and equilibrium conditions are satisfied. For the non-
homogeneous material such as reinforced concrete slab, layered model is adopted with
nonlinear material behavior of concrete (AAlami, 2005; .Wang et al, 2004; Ghosh and
Dey, 1992).
Scanlon and Murray (1974) presented a finite element model in order to simulate
the time-dependent reinforced concrete slab deflections. A layered model was adopted to
idealize a concrete slab. Time-dependent creep and shrinkage strains were considered as
initial strains. For the concrete model, an orthotropic material model was used. In
addition, tension stiffening model was introduced in the finite element analysis.
Gilbert and Warner (1979) also used a layered finite element model with an
orthotropic concrete model to calculate the short-term deflection of slab. In the study,
different tension stiffening models were compared. In order to simulate concrete model,
modified stress-strain diagram for tension steel was used as well as tension stiffening
model using stress-strain diagram for concrete.
Scanlon and Murray (1982) presented methods for calculating deflections of two-
way slabs. In order to take into account restraint stresses due to shrinkage and thermal
effects reducing a modulus of rupture was used instead of using code specified modulus
24
of rupture. In calculation of deflections of two-way slabs the equivalent frame method
and the finite element analysis were presented. Reduction of flexural stiffness due to
cracking was accounted for using effective moment of inertia. Calculation of long-term
deflection could be determined by ACI 209 model.
Graham and Scanlon (1985) investigated the effect of construction loading on
deflection of flat plate slabs using finite element method. Two-way slabs were modeled
using equivalent frame method. For the material model of concrete the modified linear
elastic material properties proposed by Scanlon and Murray (1982) were used. The
reduced flexural stiffness to express a degree of cracking was obtained from a moment-
curvature relationship. The long-term deflections and time-dependent strength of concrete
were calculated by the recommendation of ACI 209 Committee. Construction loading
from three levels of shoring process was adopted to investigate the deflection of slab. The
comparison of model results with field measurement showed that the analysis method
using equivalent frame method with iterative reduced stiffness of slab could produce a
good agreement and the deflection due to construction loads could exceed the deflections
due to service loads. Although the analysis method could predict the deflections of two-
way slab, the modeling method could not consider the deformation of in-plane and the
member forces.
Gardner and Scanlon (1990) addressed that the design estimation of the long-term
deflection of reinforce concrete two-way slab could be different from the field measured
deflections. The reasons of causing discrepancy between calculated and measured
deflections could be explained by numerous effects. The construction loads and schedules
could cause early cracks and reduction of flexural stiffness. Creep and shrinkage could be
25
varied according to environmental conditions and concrete mix and the restraint stresses
could be produced by shrinkage. Also, the analysis method could have errors because the
numerical model could not express the practical problems thoroughly.
1.3.7 Construction Loads
During construction of multistory building with reinforced concrete floor slabs,
shoring and reshoring operations are carried out. The construction is started by setting up
the shoring on the previously cast floors. The second step is to pour fresh concrete on
next floor. The same procedures are practically continued two or three times leaving the
previously installed shore system according to the number of shoring and reshoring
process such as three levels of shores, two levels of shoring and one level of reshoring,
one level of shoring and two levels of reshoring, and so on. During shoring and reshoring
process the weight of freshly poured concrete is transferred by shoring into the previously
cast floors. The construction load may exceed the design loads and the load is usually
expressed as load ratio of construction load to self-weight of slab. It is known that the
distribution of construction loads depends primarily on the shoring/reshoring and the
number of supporting floors.
Grundy and Kabalia (1964) developed a simplified method to estimate the
construction loads in a multistory building. The method is developed based on several
assumptions: elastic behavior of slabs, completely rigid foundation supporting the slabs,
and infinitely rigid shores compared with the slab in vertical displacement. In the
research constant flexural stiffness of slab as well as a flexural stiffness increasing with
26
time was investigated. The flexural stiffness was assumed to be proportional to the
modulus of elasticity. For the three levels of shores the maximum load ratio was 2.36,
while the load ratio was converged for upper levels and the value was 2.0 in both
constant flexural stiffness and increasing flexural stiffness with time.
Agarwal and Gardner (1974) performed the field tests to determine the actual
construction load ratio of multistory flat slab building. The obtained load ratios were
compared with the theoretical load ratios. During building construction three levels of
shore and four levels of reshores were used for the first building and three levels of
shores were used for the second building. They reported the comparisons field
measurement with load ratios by the simplified method by Grundy and Kabalia (1964).
The predicted load ratios were calculated from the one level of shoring and multi levels
of reshoring was used. Results showed that the reshoring process could reduce the
maximum load of supporting slabs significantly. The factored construction load ratio with
number of shores and reshores were suggested. The results showed the accuracy of load
ratios by the simplified method within 10 to 15 percent. Also, the research suggested the
construction ultimate loads on the slabs as simplified mathematical equation.
Lasisi and Ng (1979) presented a modification of Grundy and Kabalia simplified
method. Analysis included the construction live load assuming 50 psf and 10% of self-
weight for the weight of shoring and reshoring. Because the construction live load could
be produced by construction workers and equipments, the peak loads could be produced
by self-weight of slab, formwork, and construction live loads together.
Sbarounis (1984) included the effects of cracking on the previously cast floors
and reported the maximum load ratios could be reduced up to approximately 10 percent.
27
This was because the cracking occurred during construction changed the stiffness of
previously cast slabs and produced different load distribution between these slabs. The
analysis results implied that the cracking during construction might not only cause greater
immediate deflections than predicted from the design, but also the long-term deflection
could be greater.
Gardner (1985) considered the effect of early age strength of concrete in order to
check the safety of structure during construction. The factored construction loads were
compared with the slab strength at certain construction age. The factored construction
loads were calculated from simplified method suggested by Grundy and Kabalia (1964)
and extended by Agarwal and Gardner (1974). The obtained factored construction loads
were compared with the design loads multiplied by the ratio of strength development of
concrete at the age of loading. When the factored construction loads is greater than the
design loads, the construction method using shoring and reshoring could not be used.
This method may be useful when deciding the number of shoring and reshoring process
and construction cycles at the design stage.
Liu et al (1985) developed a three-dimensional finite element model to investigate
the load ratios. In the research the effects of time-dependent material properties of
concrete, foundation rigidity, column axial stiffness, aspect ratio of slab were considered.
The more realistic approaches could be initiated using a finite element model. However,
the model had a limitation that the behavior slabs were considered as linear elastic. Also,
the effects of cracking and time-dependent creep and shrinkage were not considered in
the analysis. In spite of limitations the finite element model could be used in the situation
of various boundary conditions, obtaining slab moments and shore loads, and
28
investigating the influence of shore stiffness. The analysis results showed that the Grundy
and Kabalia method needed to be corrected about 5 to 10% conservatively.
Fu and Gardner (1986) compared construction loads of one level of shoring and
two levels of reshoring (1S2R), one level of shoring and two levels of preshoring (1S2P),
and three levels of shoring (3S). The load ratios were calculated based on Grundy and
Kabalia’s simplified method and the construction cycle were assumed to be a 7-day
casting cycle with stripping after 5 days.
Gardner and Muscati (1989) suggested an algorithm that analyzes the construction
sequences of shoring and reshoring process. To determine the design ultimate
construction loads a simplified method by Grundy and Kabalia and extended by Agawal
and Gardner was used. The empirical equations of the strength of concrete with time
were incorporated in the algorithm. The safety of structure during construction could be
checked by that the ultimate construction loads calculated by the simplified method did
not exceed the design loads.
SEI/ASCE 37-02 (2002) defined minimum design load requirements during
construction for building and other construction. The loads specified involved final loads,
construction loads, material loads, lateral earth pressure, and environmental loads. The
additive load combinations which are not defined in ACI 318 were suggested and the
most critical load combination should be used. The standard provided the combined
uniformly distributed loads for the combined material, personnel, equipment, and other
applicable construction loads on the working surface according to the operational class in
traditional design. The range of construction live loads varied from 20 psf to 70 psf.
29
ACI Committee 347 (2005) recommended that construction loads on formwork be
designed for minimum live load of 50 psf as weights of workers, runaways, screeds, and
other equipment. The minimum live loads of 75 psf are recommended when the
motorized carts are used. Also combined design loads for dead and live loads should be
100 psf and 125 psf when motorized carts are used. For the load factors and strengths of
concrete, ACI 209 and ACI 318 were recommended. Also, it is recommended that
construction loads can be distributed by simplified method.
Stivaros (2005) addressed the estimation of construction load distribution,
strength requirement of early age concrete, and serviceability problems during
construction of multistory buildings. Although Grundy and Kabalia’s simplified method
could be used to calculate the load ratio, caution should be taken because the axial
stiffness of shoring and reshoring could not be ignored in the load distribution. Also, it is
addressed that the time-dependent development of strength needed to be taken into
account and the consideration of load factors not covered in ACI 318-05 for the
construction loads were needed. Finally, the requirement of minimum thickness in
current ACI 318-05 could not be used as safety against excessive deflections and
cracking because the large construction loads could be imposed to the slab at early age.
1.3.8 Experimental Studies
In the construction of multistory building the previously cast slabs experience the
construction load transferred by shoring and reshoring process. The construction
schedules can be shorter due to competitive bids. As the schedules are shorter, relatively
30
large loads can be applied to immature slabs. The loading at early age stage of
construction may cause large immediate and long-term deflections. However, there are a
few experimental researches of long-term deflections considering construction loads.
Washa and Fluck (1952) presented the effect of compressive reinforcement on the
long-term behavior of simply supported reinforced concrete beams. Five different beam
sizes with three different conditions of reinforcement were investigated. As results, the
compressive reinforcement reduced the long-term deflections due to creep and shrinkage
significantly.
Heiman (1974) measured long-term deflections for approximately 8 years of
reinforced concrete buildings. The long-term deflections at mid-spans of interior panels
were measured after construction was completed.
Bakoss et al (1982) tested simply supported and two spans continuous reinforced
concrete beams. The instantaneous and long-term deflections were measured and
compared with values predicted by the design codes and finite element analysis. The
specified compressive strength 30 MPa of normal weight concrete was cast. The creep
coefficient and shrinkage strain were recorded and compared with values specified in the
design codes-British, European, American, and Australian. The cross-section of beam
was 100mm wide 150 mm deep and 12mm diameter deformed rebars were use. The span
lengths of simply supported and each span length of continuous beams were 3750 mm
and 3500 mm respectively. The two simply supported beams were subjected to sustained
load consisting of two point loads, applied at the third points of the span at 28 days after
casting. The two continuous beams were loaded at 23 days after casting. A point load
loaded at the mid-point of each of the two equal spans.
31
Gardner and Fu (1987) investigated the effect of early age construction loads on
the long-term deflections of reinforced concrete flat slabs. In order to simulate the
construction loads the approximately twice the slab dead load. Shrinkage strains were
obtained from 3 x 4 x 15 in and 3 x 2.5 x 15 in prisms. Creep tests were conducted on
both compression and flexure. Concrete cylinders were loaded in accordance with ASTM
in compression. Plain, two singly reinforced, and two doubly reinforced concrete beams
were manufactured to investigate the creep under flexure. The deflections were measured
using dial gages and steel scales.
Gilbert and Guo (2005) investigated immediate and long-term deflections of
seven large-scale of reinforced two-way flat slab structures. Two spans in each
orthogonal direction continuous two-way slabs were cast. A plan dimension of each slab
was 6.2m by 7.2m and two 3m continuous spans in each orthogonal direction. The
thickness of each slab was 90mm. Each slab was supported on nine columns and each
column size was 200 by 200 by 1250 mm. In order to investigate parameters influencing
the long-term deflections, concrete properties, reinforcing spacing and reinforcement
ratios, slab thickness, boundary conditions, and loading history were varied. Slabs were
loaded at 14 and 15 days after casting. Material properties such as compressive strength,
flexural tensile strength, elastic modulus, the creep coefficient, and shrinkage strain were
measured. Time-dependent crack patterns, immediate and long-term deflections were
measured for three years. Results showed that time-dependent cracking significantly
affected the serviceability of flat slabs. The measured long-term deflections were
approximately 5 to 9 times the initial short-term deflections. It was conclude that the
32
current ACI 318 Building Code could not account for this effect appropriately for
deflection calculation and control.
1.4 Thesis Layout
The research carried out is presented in 6 chapters. Chapter 1 introduces the
background of the research and presents the objectives and scope. In addition, literatures
are reviewed to get the state-of-art knowledge of time-dependent effect of concrete,
concrete material model, construction loads in multistory building, analysis of concrete
slab systems and experimental approaches.
In chapter 2, analytical modeling of reinforced concrete slabs is presented. User-
defined material subroutine and layered shell element in ABAQUS/Standard are
introduced. An orthotropic time-dependent concrete model is explained.
Chapter 3 presents the experimental program in order to investigate the loading at
early age on concrete one-way slabs. Procedure and test setup are explained.
Instantaneous and time-dependent deflections are shown.
In chapter 4, verification of developed material model is presented using the
results of experimental programs as well as pre-existing results in the literature.
Chapter 5 deals with parametric study for a flat plate system. Loading history is
considered according to shoring/reshoring method in multistory building. Also, long-term
multiplier and moment diagram of slab based on parametric study are described in this
chapter.
33
Finally, chapter 6 presents the conclusion, summary, and recommendations.
Chapter 2
METHOD OF ANALYSIS
2.1 Introduction
Finite element analysis is used to analyze concrete slab systems. The concept of
finite element method is to divide a complicated structure into a finite number of simple
elements for which the exact or appropriate solution is known. The real structural system
is modeled into a mathematical approximate numerical system (Zienkiewicz and Taylor,
1991; Bathe, 1996). In order to make the mathematical model, simplification,
linearization, idealization, and assumptions are necessary. For the idealization of material,
concrete shows different material behavior under compression and tension. Because
concrete shows nonlinear behavior, a constitutive model is necessary to be proposed and
implemented in the commercial finite element program. The concrete material model is
developed using an orthotropic model which is based on equivalent uniaxial stress-strain
relationship. Also, time dependent material properties are implemented in the user
subroutine in ABAQUS. The creep algorithm is developed based on rate of creep method
(RCM). Incremental creep and shrinkage strains are calculated and implemented based on
initial strain approach in the finite element analysis. Also, nonlinear solution method and
convergence of nonlinear problem in ABAQUS is introduced in this chapter.
35
2.2 Material Models
In order to constitute the element stiffness matrix, material constitutive models are
necessary. In this section material models of concrete and reinforcing steel are described.
For idealization of the concrete model a shell element is used. The shell element provided
in ABAQUS is a three dimensional structural element and every node has 6 degree of
freedom (S4, S8R). However, only a two-dimensional stress-strain relationship is
required to formulate the element stiffness matrix. Plane-stress condition is required.
Although ABAQUS/Standard provides its own material model, time-dependent
effect of concrete such as creep and shrinkage is not defined yet for the concrete material.
Creep and shrinkage effect is defined in the level of material mechanical behavior of
concrete using user-defined material subroutine (UMAT). In this research, orthotropic
concrete model is adopted and implement in the ABAQUS through UMAT.
Also, material behavior of reinforcing steels is presented in this section.
Reinforcing steel is idealized one-dimensional perfect elastic-plastic model which is
based on the plasticity model with isotropic hardening and kinematic hardening rule. The
model is already implemented in the ABAQUS (ABAQUS, 2002).
2.2.1 Concrete Elastic Model
A typical concrete stress-strain relationship depends on various properties
including the strength of concrete, age of concrete, rate of loading, material properties of
cement and aggregate and size of specimen. The compressive stress of concrete shows an
36
approximately linear increase with strain in the range of '45.0~4.0 cf . Once the strain of
concrete exceeds the elastic range, the concrete stress increases nonlinearly and reaches
the specified compressive strength 'cf . After that, the stress decreases nonlinearly and that
is referred to as the softening phenomenon. ACI 318 code specifies an ultimate strain of
0.003 for design. Figure 2-1 shows a typical compressive stress-strain curve. In this
study the stress-strain relation can be input in point-wise fashion.
In the elastic range concrete is assumed to be an isotropic linear elastic model.
The linear elastic model is valid with the response of concrete subject to both tensile
stress below cracking and compressive stress in the range of '45.0~4.0 cf .
The following formula in ACI 318(2005) for Young’s modulus of concrete has
been suggested in Eq. 2.1
where, w is the unit weight of concrete in pound per cubic feet for psi and kilograms per
cubic meter for MPa and 'cf is specified compressive strength of concrete in psi and MPa.
For normal weight concrete the elastic modulus is given by Eq. 2.2
The elastic limit under tensile stress can be assumed to be Eq. 2.3 according to
ACI 318
'5.133 cc fwE = (psi) '5.1043.0 cc fwE = (MPa)
2.1
'57000 cc fE = (psi) '4730 cc fE = (MPa)
2.2
37
2.2.2 Tension Stiffening Models
A prism of reinforced concrete containing one reinforcing bar is shown in
Figure 2-2. A portion of load is transferred to the concrete by bond between cracks. At
cracks, the load is carried by steel only. The distribution of steel, concrete, and bond
stresses are shown in Figure 2-2(b), (c), and (d) respectively. Tension stiffening effect is
a softening of concrete stress-strain functions after cracking. The tension stiffening effect
contributes to overall stiffness in cracked reinforced concrete particularly at service load
levels. After the concrete tensile stress reaches the maximum tensile strength, cracking
occurs. Once cracked, the concrete is assumed not to carry any tension at the cracks. But
the tension is transferred by reinforcement into the surrounding concrete. Consequently,
the concrete tensile stress can be assumed as the average tensile stress. It is noted that the
average tension stiffening effect is only valid in vicinity of reinforcement.
The average tensile stress in the concrete continues to decrease with increasing
strain. The tension stiffening effect was first introduced in finite element analysis by
Scanlon (Scanlon, 1971; Scanlon and Murray, 1974). Tension stiffening effect was
considered as stepwise reduction in tensile stress. After that, various tension stiffening
models have been proposed. For example, Lin and Scordelis (1975) proposed a bilinear
type concrete tension stress-strain model to explain the tension stiffening effect.
Damjanic and Owen (1984) also suggested a bilinear type tension stiffening effect but
'5.7 cr ff = (psi) '62.0 cr ff = (MPa)
2.3
38
with sudden drop of tensile stress immediately after concrete cracking. Bischoff (2004)
proposed nonlinear post-peak tension stiffening effect. Figure 2-3 shows several tension
stiffening models.
2.2.3 Equivalent Uniaxial Strain
In order to employ an orthotropic concrete model, equivalent uniaxial strain is
calculated from the strains. In the model the uniaxial stress-strain relation is defined in
the principal axes of orthotropic (Noh et al, 2003; Phuvoravan and Sotelino, 2005). The
principal axes are updated according to strain state during iteration of analysis. However,
it is assumed that the angle of principal axes is fixed when the strains exceed the strains
corresponding to maximum tensile stress or maximum compressive stress of concrete.
The equivalent uniaxial strains are fictitious strains and assumed that the direction of
uniaxial strain coincides with principal direction. When the uniaxial strains are obtained,
the stiffness of concrete is calculated from the uniaxial stress-strain relationship.
Figure 2-4 shows the uniaxial stress-strain relationship. In the figure concrete equivalent
uniaxial stress is a function of elastic and plastic strain. The concrete stress can be
obtained from the stress-strain relation.
The equivalent uniaxial strains can be written in elastic and plastic strain, which
define the stress-strain relationship and can be additive form given by Eq. 2.4
uipluielui ,, εεε += 2.4
39
in which, u is an equivalent uniaxial and i is the principal direction. The strain uiε is
equivalent uniaxial strain in direction of i . The strain components uiel ,ε and uipl ,ε are
uniaxial elastic and plastic strain in direction i respectively.
The incremental equivalent uniaxial strains are given by Eq. 2.5
In the current (n+1)th iteration , total strains are obtained from the summation of
previous (n)th strain and current incremental strain given by Eq. 2.6
A principal stress is a function of an equivalent uniaxial strain and it can be
obtained from the uniaxial stress-strain relationship given by Eq. 2.7
In order to idealize the irrecoverable strain in a concrete constitutive model, the
locus ( )00 ,σε is introduced by Lee and William (1997). The secant stiffness in the
previous increment of concrete is calculated by Eq. 2.8
uipluielui ,, εεε Δ+Δ=Δ 2.5
nui
nui
nui εεε Δ+=+1
nuiel
nuiel
nuiel ,,1
, εεε Δ+=+
nuipl
nuipl
nuipl ,,1
, εεε Δ+=+
2.6
)( nui
nui f εσ = 2.7
onui
onuin
iEεεσσ
++
= 2.8
40
Similarly for the current increment state the stress and stiffness can be obtained
by Eq. 2.9
The incremental equivalent uniaxial plastic strain can be obtained by Eq. 2.10
The incremental stress can be obtained by Eq. 2.11
The stress in the current iteration can be calculated by Eq. 2.12
The stress in the previous iteration can be calculated by Eq. 2.13
Therefore, the incremental stress can be obtained by Eq. 2.14
2.2.4 Cracking Algorithm
The stress-strain relationship is defined in the principal direction ( 2,1=i ) using
equivalent uniaxial strains in two-dimension plane-stress. In two-dimension three strain
components are decoupled into two principal strains. The equivalent uniaxial strains are
onui
onuin
iEεεσσ
++
= +
++
1
11 2.9
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=Δ + n
ini
nplui EE
1110, σε 2.10
nui
nui
nui σσσ −=Δ +1 2.11
( )1,
111,
11 ++++++ −⋅=⋅= nuipl
nui
ni
nuiel
ni
nui EE εεεσ 2.12
( )nuipl
nui
ni
nuiel
ni
nui EE ,, εεεσ −⋅=⋅= 2.13
[ ]( ) ( )nuipl
nui
ni
nuipl
nui
ni
ni
nui EEE ,,
1 εεεεσ Δ−Δ+−−=Δ + 2.14
41
calculated from given principal strains. Using equivalent uniaxial strain principal stress
can be determined from uniaxial stress-strain relationship according to previously
aforementioned.
For the arbitrary tension stiffening model an equivalent tensile uniaxial behavior
is applied in order to calculate the tensile behavior of concrete. The tensile stress-strain
relationship is provided in point wise fashion. Linear interpolation is used to get a certain
point between two points.
Figure 2-5 shows Damjanic and Owen model for time-dependent reduction of stiffness
of concrete. Let us assume that short-term analysis is completed and concrete section is
cracked at jth increment, the stress-strain is positioned at point ),( jjpl
jel σεε + . The reduced
stiffness of concrete is expressed as a secant modulus of elasticity jE . Also, if there is
increasing of creep and shrinkage strain while creep and shrinkage effect is progressing,
the accumulated creep and shrinkage strains, crε and shε , contribute to total strain with
satisfying the equilibrium of system. At increment kth elastic and plastic strains given by
Eq. 2.15
The reduced secant modulus kE due to creep and shrinkage is obtained by
Eq. 2.16
( )ksh
kcr
kkpl
kel εεεεε +−=+ 2.15
okpl
kel
ok
kEεεε
σσ++
+= 2.16
42
Although the concrete shell element is three-dimensional, only a two-dimensional
plane-stress material constitutive matrix is required to formulate the shell element. In this
study the equivalent uniaxial model, also known as an orthotropic constitutive model is
adopted to express concrete material behavior. The assumption of an orthotropic model is
that the directions of the principal stress and strain are parallel. For plane-stress condition,
there are three strain components and stress components are expressed using matrix form
given by Eq. 2.17 .
The principal direction is calculated by Eq. 2.18.
When the principal direction is decided, the incremental principal strain is
calculated by Eq. 2.19.
Incremental equivalent uniaxial strain can be calculated by Eq. 2.20.
Equivalent uniaxial strain can be calculated by Eq. 2.21.
{ } { } Tnxyyxn γεεε =
{ } { } Tnxyyxn γεεε ΔΔΔ=Δ
{ } { } Tnxyyxn τσσσ =
2.17
yx
xyp εε
γθ
−=2tan 2.18
nxy
y
x
pppp
pppp
n ⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ΔΔΔ
⎥⎥⎦
⎤
⎢⎢⎣
⎡
−=
⎭⎬⎫
⎩⎨⎧ΔΔ
γεε
θθθθθθθθ
εε
sincoscossinsincossincos
22
22
2
1 2.19
nnu
u
⎭⎬⎫
⎩⎨⎧ΔΔ
⎥⎦
⎤⎢⎣
⎡−
=⎭⎬⎫
⎩⎨⎧ΔΔ
2
12
2
1
11
11
εε
νν
νεε
2.20
43
The secant modulus of concrete can be calculated from the uniaxial stress-strain
relationship given by Eq. 2.22.
The local constitutive material matrix of concrete can be obtained by Eq. 2.23.
Global stiffness matrix is constituted using local stiffness and transformation
matrices given by Eq. 2.24.
in which, [ ]T is the coordinate transformation matrix (Cook et al, 1989)
Incremental equivalent uniaxial plastic strain is obtained by Eq. 2.25.
The incremental principal plastic strain is calculated by Eq. 2.26.
The incremental plastic strain of each component is calculated by Eq. 2.27.
nu
u
nu
u
nu
u
⎭⎬⎫
⎩⎨⎧ΔΔ
+⎭⎬⎫
⎩⎨⎧
=⎭⎬⎫
⎩⎨⎧
+ 2
1
2
1
12
1
εε
εε
εε
2.21
ui
uicni
fE
εε )(1 =+ , 2,1=i 2.22
[ ]1
222
11
21
)1(0000
11
+
+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
=
n
nL
GEEEE
Cν
νν
ν 2.23
[ ] [ ] [ ] [ ]TCTC LTn
G =+1 2.24
⎟⎟⎠
⎞⎜⎜⎝
⎛−⋅=Δ + n
ini
nplui EE
1110, σε , 2,1=i 2.25
nplu
plu
pl
pl
⎭⎬⎫
⎩⎨⎧ΔΔ
⎥⎦
⎤⎢⎣
⎡−
−=
⎭⎬⎫
⎩⎨⎧ΔΔ
,2
,1
,2
,1
11
εε
νν
εε
2.26
44
The incremental stress is obtained by Eq. 2.28.
2.2.5 Creep and Shrinkage Algorithm
The mechanical constitutive model for concrete considers elastic and inelastic
response. The elastic and inelastic deformation is separated into recoverable and
irrecoverable parts. This separation is based on the assumption that there is additive
relationship between strains. Therefore, the total strains can be written by Eq. 2.29.
where, elε , plε , crε , and shε are elastic, plastic, creep, and shrinkage strain respectively.
Incremental strains can also be written by Eq. 2.30.
Based on orthotropic model creep and shrinkage model is incorporated in the
material behavior of concrete. Time-dependence of creep and shrinkage is incorporated in
the user-subroutine. Creep and shrinkage strains are assumed as initial strains. For creep
strains, incremental creep strain is calculated from rate of creep method (RCM). The
formulations used in this study were proposed by Kawano and Warner (1996). ABAQUS
⎭⎬⎫
⎩⎨⎧ΔΔ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
ΔΔΔ
pl
pl
pppp
pp
pp
nplxy
ply
plx
,2
,122
22
,
,
,
sincossincoscossinsincos
εε
θθθθθθθθ
γεε
2.27
{ } [ ] [ ]( ){ } [ ] { }plnGpl
nG
nGn CCC εεεεσ Δ−Δ−−−=Δ ++ 11 2.28
shcrplel εεεεε +++= 2.29
shcrplel εεεεε Δ+Δ+Δ+Δ=Δ 2.30
45
provides the user subroutine to define a material behavior. RCM can be incorporated in
the user subroutine because only stresses at the previous step are necessary to calculate
the creep strains. For shrinkage strain, the incremental shrinkage strain can be obtained
from various shrinkage functions. For example ACI 209 (1992) provides the hyperbolic
functions according to curing type. In this study creep and shrinkage strains are assumed
independent. Several creep and shrinkage models can be found in Appendix A.
Incremental creep strain is calculated from RCM. In this method, creep strain can
be written by Eq. 2.31.
Integrated by parts and given by Eq. 2.32.
The rate of creep method (Dischinger method) is applied to Eq. 2.32, and from
the assumption of constant stress creep strain can be written by Eq. 2.33.
The incremental creep strain can be written by Eq. 2.34
( ) τττστφε d
ddt
Et
t
ccr
)(,1)(0∫= 2.31
[ ]
ττσττφ
ττσττφσφσφε
ddtd
E
ddtd
Etttt
Et
t
c
t
cccr
)(),(1
)(),(1)0()0,()(),(1)(
0
0
∫
∫
−=
−−=
2.32
[ ]
[ ]),(),()(1
),(),()(1)(
001
00111
tttttE
dtttddt
Et
iiic
n
t
tic
ncri
i
φφσ
ττφφτ
σε
−=
−−=
+
++
∑
∫∑ +
2.33
[ ]),(),()(1)( 001 tttttE
t nnnc
ncr φφσε −=Δ + 2.34
46
The incremental creep strain can be written by Eq. 2.35.
Once creep strain is obtained, it is considered as “initial strain”. The creep and
shrinkage algorithm for the nth iteration is deployed in the orthotropic concrete model.
The summation of plastic strain and elastic strain is obtained by subtracting creep and
shrinkage strains from total strain given by Eq. 2.36.
A secant modulus is obtained from Eq. 2.22 and local material compliance matrix
is calculated by Eq. 2.37.
A global material compliance matrix is obtained by Eq. 2.38.
Incremental creep strain is obtained by Eq. 2.39
Incremental shrinkage strain is calculated by Eq. 2.40
Then the stresses are calculated by Eq. 2.41.
[ ]),(),()(1)( 001 tttttE
t nnnc
ncr φφσε −=Δ + 2.35
{ } { } { })()()()()( nshncrnnplnel ttttt εεεεε +−=+ 2.36
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
−=
GEE
EED L
/1000/1/0//1
22
11
νν
2.37
[ ] [ ] [ ] [ ] TLG TDTD −−= 1
2.38
{ } [ ] [ ] { })(),(),()( 001 nGnnncr tDttttt σφφε −=Δ + 2.39
{ } { } { })()()( 1 nshnshnsh ttt εεε −=Δ + 2.40
47
Once total incremental strain is obtained from equilibrium of system, strains are
updated by Eq. 2.42.
2.2.6 Strength Development of Concrete
Time-dependent development of compressive strength specified in ACI 209
(1992) is obtained by Eq. 2.43
where, t is time in days. Coefficients,α and β , are constants defined by cement types
and curing methods specified in Table 2-1. )82('cf is 28 days compressive strength.
GL2000 model (Gardner and Lockman, 2001) recommends the modulus of
elasticity estimated from the compressive strength and given in Eq. 2.44.
{ } [ ] [ ]( ){ }[ ] { }shcrpl
nG
shcrplnG
nGn
C
CC
εεεε
εεεεσ
Δ−Δ−Δ−Δ−
−−−−=Δ+
+
1
1
2.41
{ } { } { })()()( 1 nnn ttt εεε Δ+=+
{ } { } { })()()( 1 ncrncrncr ttt εεε Δ+=+
{ } { } { })()()( 1 nshnshnsh ttt εεε Δ+=+
2.42
)82(')(' cc ft
ttfβα +
= 2.43
cmtcmt fE 43003500 += (MPa)
cmtcmt fE 52000500000 += (psi) 2.44
48
where, cmtE is mean modulus of elasticity at age, t and cmtf is a mean concrete strength at
age t. A time-dependent mean concrete strength can be calculated from Eq. 2.45.
where is a mean concrete strength at 28 days calculated by Eq. 2.46
Coefficients a and b can be decided by cement type and given in Table 2-2.
In case of CEB-FIP model (fib, 1999), modulus of elasticity can be obtained by Eq. 2.47
where, ciE is a tangent modulus of elasticity at concrete stress is at zero and at a concrete
age of 28 days, coE is 41015.2 × MPa. cmf is mean compressive strength and cmof is 10
MPa. Eα is coefficient defined by type of aggregate.
The development of compressive strength with time can be calculated by
Eq. 2.48 .
where,
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−=
2/1
1/281exp)(
ttstccβ .
284/3
4/3
cmcmt fbta
tf+
= 2.45
51.1 2828 += ckcm ff (MPa) 7001.1 2828 += ckcm ff (psi) 2.46
3/1
⎟⎟⎠
⎞⎜⎜⎝
⎛=
cmo
cmcoEci f
fEE α 2.47
)28()()( cmcccm fttf ⋅= β 2.48
49
)(tfcm is mean compressive strength at a concrete age t, MPa. )28(cmf is mean
compressive strength at a concrete age t, MPa. t is concrete age, days and 1t is 1 day
s is a coefficient decided by the strength class of cement, which is given in Table 2-3.
Time-dependent modulus of elasticity specified may be estimated by Eq. 2.49
where,
( ) 2/1)()( tt ccE ββ =
)(tEci is tangent modulus of elasticity at a concrete age t, MPa. ciE is tangent modulus of
elasticity at a concrete age of 28 days. )(tEβ is a function to describe the development of
modulus of elasticity with time
2.2.7 Reinforcing and Post-Tensioning Steel
For non-prestressed reinforcing steel, the stress-strain relation is usually assumed
to be perfect elastic-plastic, as shown in Figure 2-6. This relation is expressed in
Eq. 2.50.
Post-tensioning steel may be wires, bars or strands. For the stress-strain response
of prestressing steel, there is no significant yield stress. An equivalent yield stress is
defined as the stress at a strain of 1 %. The stress-strain response usually can be
ciEci EttE ⋅= )()( β 2.49
ysss fEf ≤= ε 2.50
50
expressed by the modified Ramberg-Osgood function or other proper functions.
Figure 2-7 shows a typical stress-strain curve of prestressing steel.
2.3 Interface of Concrete Model in ABAQUS/Standard
ABAQUS/Standard provides a user-define material sub-routine (UMAT). This
sub-routine is used to define the material mechanical behavior. In order to implement the
specific material behavior FORTRAN programming is necessary. The interface of user-
define sub-routine is as follows.
SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD, 1 RPL,DDSDDT,DRPLDE,DRPLDT, 2 STRAN,DSTRAN,TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,CMNAME, 3 NDI,NSHR,NTENS,NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT, 4 CELENT,DFGRD0,DFGRD1,NOEL,NPT,LAYER,KSPT,KSTEP,KINC) C INCLUDE 'ABA_PARAM.INC' C CHARACTER*80 CMNAME DIMENSION STRESS(NTENS),STATEV(NSTATV), 1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS), 2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1), 3 PROPS(NPROPS),COORDS(3),DROT(3,3),DFGRD0(3,3),DFGRD1(3,3) user coding to define DDSDDE, STRESS, STATEV, SSE, SPD, SCD and, if necessary, RPL, DDSDDT, DRPLDE, DRPLDT, PNEWDT RETURN END where, STRESS and DDSDDE vectors must be defined. STRESS matrix contains the
stress tensor from previous increment at the beginning of the increment and the
incremental stress tensor is updated at the end of the increment. DDSDDE matrix
51
contains incremental stress-strain variation. This array must be updated at the end of
increment according to stress-strain relationship.
2.4 Solution Method
Solution of nonlinear finite element analysis is based on the iterative methods in
ABAQUS/Standard. In this section modified Newton-Raphson method as well as
convergence of nonlinear problem is introduced. For a given loading condition, loading is
divided into a number of steps, and each step consists of a number of increments. Each
increment must satisfy the equilibrium of system. In order to satisfy equilibrium,
convergence criteria for nonlinear problems are provided.
2.4.1 Modified Newton-Raphson Method
When considering nonlinear problem, the stiffness K is composed of constant
term 0K and nonlinear term NK that depend on deformation. Displacement caused by
load can be expressed in Eq. 2.51
where, )(ufK N = , which means that nonlinear term is a function of displacement.
A nonlinear solution method in ABAQUS/Standard is based on combined
incremental-iterative solution, which is called modified Newton-Raphson method
alternatively. Euler’s method is the simplest incremental method of solving first-order
( ) PuKK N =+0 2.51
52
differential equation. After convergence under load, increase the load to upper level. This
is expressed in Eq. 2.52
where, u is the displacement and PΔ is the load increment.
In general form Eq. 2.52 can be expressed in Eq. 2.53 .
This process is shown on Figure 2-8. The disadvantage of the approximate solution by
incremental method is that the solution drifts further from exact solution with every step.
Newton-Raphson method is based on iterative procedure obtained from a
truncated Taylor series and expressed in Eq. 2.54
By neglecting third and higher order term and set 0=nf , approximate incremental
displacement can be obtained. First iteration can be written in Eq. 2.55.
M
323
212
10
1
2
1
0
PdudPuu
PdudPuu
PdudPu
uuat
uuat
uat
Δ⋅⎟⎠⎞
⎜⎝⎛+=
Δ⋅⎟⎠⎞
⎜⎝⎛+=
Δ⋅⎟⎠⎞
⎜⎝⎛+=
=
=
=
2.52
[ ] 111 ++=
+ Δ⋅+=Δ⋅⎟⎠⎞
⎜⎝⎛+= itii
uuatii PKuP
dudPuu
i
2.53
( ) L+Δ+Δ+≅ 2200
0 21 u
dudfu
dudfffn 2.54
001
0
10
0
uuu
fdudf
u
Δ+=
⋅⎟⎠⎞
⎜⎝⎛−=Δ
−
2.55
53
Second and next iterations can be written in Eq. 2.56
where, if is load imbalance. Iteration will go on until the convergence criteria are
satisfied. The Newton-Raphson method is shown in Figure 2-9 . The disadvantage of this
solution method is that the method provides only a single point solution.
In order to overcome the disadvantage incremental and iterative method could be
combined together. The solution of incremental procedure is considered to be starting
point of solution and iterative procedure improves the convergence of solution. Figure 2-
10 illustrate the modified Newton-Raphson method. Load is divided into a number of
increments. Each increment has its own iterative procedure. This approach can provide
the load-displacement response.
2.4.2 Convergence
The external and internal force must balance each other if the body is in
equilibrium state. The balance is obtained from zero net force at every node. The internal
force acting on node can be calculated by stresses in the elements which are connected to
the node. Let P and I be the external and internal force respectively. The equilibrium
condition is expressed in Eq. 2.57
M112
1
11
1
uuu
fdudfu
Δ+=
⋅⎟⎠⎞
⎜⎝⎛−=Δ
−
2.56
0=− IP 2.57
54
In the Newton-Raphson method, in order to get approximate solution at load 2P , a small
load increment PΔ is added to 1P , in which the load 1P is an applied load and determines
the corresponding displacement 0u . Displacement increment 0uΔ and displacement 1u
for the next step are calculated by Eq. 2.55. Also, the internal force 1I is calculated by
following procedure. First, strain-displacement matrix [B] is obtained using strain-
displacement relation [S-D] and interpolation matrix [N], which is given in Eq. 2.58
Strain matrix { }ε is obtained from [B] and nodal displacement{ }0U , which is given in
Eq. 2.59
Material constitutive matrix [ ]C can be obtained from stress-strain relation. In addition,
incremental strain { }εΔ is calculated from [B] and incremental nodal displacement
{ }0UΔ given in Eq. 2.60
Therefore, incremental stresses are obtained by Eq. 2.61
Stresses are updated from previous stresses and incremental stresses in Eq. 2.62
[ ] [ ]NDSB ][ −= 2.58
{ } { }0][ UB=ε 2.59
{ } { }0][ UB Δ=Δε 2.60
{ } { }pshcrC εεεεσ Δ−Δ−Δ−Δ=Δ ][ 2.61
{ } { } { }001 σσσ Δ+= 2.62
55
Element stiffness matrix is constituted by Eq. 2.63
Current nodal displacement is obtained from previous condition given in Eq. 2.64
Finally, the internal force is obtained by Eq. 2.65
The force residual can be calculated by Eq. 2.66 .
In Newton-Raphson method convergence of the approximate solution is decided
by the force residual R and incremental displacement 0uΔ , which is given in Eq. 2.67
The tolerances for the force residual and incremental displacement in default, rTOL and
dTOL , are 0.5 % and 1% in ABAQUS/Standard respectively. If the convergence is not
satisfied, next iteration is performed.
[ ] [ ] [ ][ ]dVBCBKT
V∫= 2.63
{ } { } { }001 UUU Δ+= 2.64
{ } [ ]{ }11 UKI = 2.65
12 IPR −= 2.66
d
r
TOLuTOLR≤Δ
≤
0 2.67
56
Table 2-1: Constant for ACI 209
Type of Curing Cement type constant α 4.0
I β 0.85 α 2.3 Moist Curing
III β 0.92 α 1.0
I β 0.95 α 0.70
Steam Curing III β 0.98
Table 2-2: Coefficient for GL2000 Model
Type of cement Concretes a b
Type I 2.8 0.77
Type II 3.4 0.72
Type III 1.0 0.92
Table 2-3: Coefficient for CEB-FIP Model
Strength class of cement (MPa) 32.5 32.5 – 42.5 42.5- 52.5
s 0.38 0.25 0.20
57
Ec
f 'c
fc
εc Figure 2-1: Compressive Stress-Strain Curve of Concrete
58
(a) Axially loaded prizm
(b) Variation in steel stress
(c) Variation in concrete stress
(d) Variation in bond stress
PP
Average tensile stress
Figure 2-2: Steel, Concrete, and Bond Stress in a Cracked Reinforced Concrete Prism Member
59
Ec
εct
f cr
f ct
4 Damjanic & Owen
2 nonlinear post-peak
3 bilinear type
5 Constant residual stress
1 stepwise reduction
εcr Figure 2-3: Tension Stiffening Models
60
εpl,ui εel,ui
(ε0 σ0)
n n
εpl,ui εel,uin+1 n+1
σuin+1
σuin
E0 Ej Ej+1
εui
σui
Figure 2-4: Equivalent Uniaxial Stress-Strain Relation
61
ε p l ε e l
(ε 0 σ 0)
j j
ε p l ε e lk k
σk
σj
E 0 E j E k
ε
σ
Figure 2-5: Time-Dependent Tension Stiffening Model-Damjanic and Owen
62
f y
f s
Es
εs
Figure 2-6: Stress-Strain Curve of Steel
f py
Ep
εps
A strain of 1%
f p
Figure 2-7: Stress-Strain Curve of Prestressing Steel
63
P1
P2
Displacement
u
P3
PLoad
ΔP1
ΔP2
ΔP3
u1 u2 u30Δu0 Δu1 Δu2
Figure 2-8: Incremental Method
64
LoadP
Displacement
u
-fo
-f1
u0 u1 u2
Δu0 Δu1
P1
I1
P2
dudf0
df1du
ΔP
Figure 2-9: Newton-Raphson Method
65
Displacement
u
P
P3
P2
P1
Load
Figure 2-10: Modified Newton-Raphson Method
Chapter 3
EXPERIMENTAL STUDY
3.1 Introduction
To evaluate the effects of early age loading on deflection, nine one-way slab
specimens were tested under short term application of live load and sustained dead load.
Mid-span deflection measurements were taken during live load application and removal
as well as during the period of sustained load application.
Three specimens labeled B1D3, B2D3, and B3D3 were removed from the forms
and loaded at 3 days, three labeled B4D7, B5D7, and B6D7 were removed and loaded at
7 days, three labeled B7D28, B8D28, and B9D28 were removed and loaded at 28 days.
This chapter describes the design and preparation of test specimens, material
properties, test set-up and procedure, and results of the deflection measurements.
3.2 Specimen Design and Preparation
All nine test specimens were fabricated with the same dimensions and flexural
reinforcement. The slabs are 12 ft long, 12 in. wide, and 5 in. deep, reinforced with 2 - #3
Grade 60 bottom bars with an effective depth of 4 in. and simple supports located 6 in.
from each end providing a simple span length of 11 ft. The slabs were designed according
to ACI 318 05 (2005) for moment capacity to resist an unfactored dead load due to self
67
weight plus a concentrated live load of 600 lbs at midspan. The design was based on a
specified concrete compressive strength of 4000 psi. Details of the concrete mix are
provided in Table 3-1.
All slab specimens were cast on the same day in forms constructed on the floor of
the testing laboratory. The specimens were cast from the same batch of concrete provided
by a local ready mix supplier. Immediately after finishing the top surfaces, the specimens
were covered with wet burlap and plastic film for curing. Specimens B1D3, B2D3, and
B3D3 were cured for three days and then removed from the forms for testing. Specimens
B4D7, B5D7, and B6D7 were cured for 7 days and the n removed from the forms for
testing, and specimens B7D28, B8D28, and B9D28 were cured for 7 days and then left
exposed to air until removal from forms and testing at 28 days.
3.3 Material Properties
Concrete cylinders (6 in. x 12 in.) were cast from the concrete batch used for the
specimens following ASTM C 31. Six cylinders were made for each of the slab sets (3
day, 7 day, and 28 day loading). For each set, three cylinders were used for split cylinder
tensile tests and the other three were used for compressive strength and elastic modulus
using ASTM test procedures (ASTM C 496, ASTM C 39, and ASTM C 469). Results are
summarized in Table 3-2.
Time-dependent development of compressive strength and elastic modulus up to
28 days compared with models provided by ACI 209 (1992), CEB-FIP (1999), and
Gardner and Lockman GL 2000 (2001) is shown in Figure 3-1 and 3-2. The analytical
68
models show good agreement with test results tending to slightly underestimate values at
3 and 7 days.
3.4 Test Setup and Procedure
Specimens were removed from forms at 3, 7, and 28 days as described above and
set on simple supports on the laboratory floor as shown in Figure 3-3. A dial gage was
installed below each specimen at midspan immediately after the specimen was set on the
supports under self weight providing the datum for all subsequent readings.
Six steel blocks, each weighing an average of 105.1 lbs, were placed at midspan
and dial gage readings were recorded after each block was placed. The blocks were then
removed one by one and dial gage readings were recorded on removal of each block.
Deflection readings were taken periodically over a period of 182 days while each
specimen supported its self weight.
A second application of the concentrated load at midspan was performed at age
156 days and the same procedure as before was used to record applied load and
deflection. Figure 3-4 shows a specimen under the full applied concentrated load. The
loading history is shown in Figure 3-5.
3.5 Immediate Deflection due to Application of Live Load
The load deflection response due to initial application and removal of the live
load is shown for all specimens in Figure 3-6 to 3-8. The average response for each set
69
of three specimens loaded at 3, 7, and 28 days is shown in Figure 3-9. For each loading
age the three specimens in each set show very similar approximately linear response on
initial application of load when the specimens were uncracked.
For loading at 3 and 7 days the specimens show a softening of response at
approximately 400 lbs with a rapid increase in deflection under additional load indicating
the onset of cracking in the midspan region. Significant differences in maximum
deflection under peak load are evident in the plots. For loading age 28 days the softening
of response begins at a lower load in the range of 200 – 300 lbs. This trend is shown
clearly in Figure 3-9. The maximum deflection for the 28 day loading is higher than the
maximum deflection for 3 and 7 day loading.
The difference in response between the 28-day case and the early age loading
cases is attributed to the presence of shrinkage restraint tensile stresses as a result of
drying in the period 7 to 28 days for the 28 day case while early age loading specimens
were loaded immediately after the curing period. The mechanism of shrinkage restraint is
introduced in Appendix B.
Figure 3-10 to 3-18 show the load deflection response on the second application
of live load for each specimen. Time dependent deflections between first and second
application of live load are not included to allow comparison between loading and
unloading on first and second application of live load.-
All plots show the same general trend. Second application of live load closely
follows the unloading curve from first application of live load with a slight increase in
both peak deflection and residual deflection on unloading.
70
3.6 Long-Term Deflection under Sustained Load
Figure 3-19 to 3-27 show the deflection histories for all specimens indicating
increasing deflection with time under sustained load. A comparison of average deflection
versus time for the three sets of specimens is shown in Figure 3-28 which clearly shows
the effect of age at loading on long-term deflection. While the slabs loaded at 28 days
showed higher peak and residual deflection, the slabs loaded at 3 and 7 days show
significantly larger long term deflections.
Variation of temperature and relative humidity with time is shown in Figure 3-29
and 3-30. Local variations in deflection can be attributed to variations in temperature and
relative humidity.
3.7 Summary
Details of the experimental program to evaluate effects of loading age on
immediate and time-deflections of one way slabs have been presented. The results
indicate that tensile stresses due to shrinkage restraint during the drying period reduce the
load at which flexural cracking. The results also show the significant effect of age at
loading on long term deflection under sustained load. Restraint stress should therefore be
considered in the calculation of immediate deflection as suggested by Scanlon and
Murray and among others (Nejadi and Gilbert, 2004; Bischoff, 2001; Gilbert, 1988;
Scanlon and Murray, 1982).
71
In the following chapter test results are compared with analytical results using
simplified beam equation approaches as well as results from finite element analysis.
72
Table 3-1: Concrete Mix Used
Component Amount / 3yd (Amount/ 3m )
2B Stone* 1872 lb (1109.16 kg)
Sand 1224 lb (725.22 kg)
Type I cement 376 lb (222.78 kg)
GRANCEM** 212 lb (125.61 kg)
Water 19.6 gal (96.88 l)
MBVR*** 10 oz. (386.20 ml)
GELENIUM**** 17.6 oz. (679.71 ml)
*2B Stone = 1” to 1/2” aggregate size **GRANCEM = slag ***MBVR : air entraining agent ****GELENIUM : water reducing admixture
73
Table 3-2: Concrete Material Properties
Details Test Set
Compressive Strength, psi
(MPa)
Direct Tensile Strength, psi
(MPa)
Elastic Modulus, psi
(MPa) 2884
(19.89) 305.84 (2.11)
2994096.52 (20644.30)
2729 (18.82)
302.39 (2.09)
3368358.58 (23224.83) Day 3
2870 (19.79)
353.59 (2.44) -
3562 (24.56)
416.28 (2.87)
4042030.30 (27869.80)
3690 (25.44)
400.10 (2.76)
3921970.00 (27041.98) Day 7
3905 (26.93)
269.06 (1.86) -
4512 (31.11)
448.37 (3.09)
4115521.76 (28376.52)
4796 (33.07)
453.86 (3.13)
4176267.47 (28795.36) Day 28
4969 (34.26)
361.55 (2.49) -
74
0
1000
2000
3000
4000
5000
6000
0 5 10 15 20 25 30
Time(days)
Com
pres
sive
Stre
ngth
(psi
)
ACI 209
GL2000
CEB-FIP
Figure 3-1: Comparison of Time-Dependent Compressive Strength Between Experiment
and Analysis
75
0
500000
1000000
1500000
2000000
2500000
3000000
3500000
4000000
4500000
0 5 10 15 20 25 30Time(days)
Ela
stic
Mod
ulus
(psi
)
ACI 209
GL2000
CEB-FIP
Figure 3-2: Comparison of Time-Dependent Elastic Modulus Between Experiment and
Analysis
CL
Steel Beam Dial Gage
1"
6"
2-#3Bearing Plate(6" x 12")
Figure 3-3: Test Setup
76
Figure 3-4: Setup for Live Load
D
D+L
Age at Loading(3, 7, and 28) Concrete Age (days)
Load
first live load second live load
156 Figure 3-5: Loading History
77
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300
Deflection(in)
Live
Loa
d(lb
)
B1D3
B2D3
B3D3
Figure 3-6: Load-Deflection Response for Loading at 3 Days
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B4D7
B5D7
B6D7
Figure 3-7: Load-Deflection Response for Loading at 7 Days
78
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B7D28
B8D28
B9D28
Figure 3-8: Load-Deflection Response for Loading at 28 Days
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
Day 28
Day 7
Day 3
Figure 3-9: Averaged Load-Deflection Response
79
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B1D3
B1D3_2nd
Figure 3-10: Load-deflection Response due to First and Second Live Loads of B1D3
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B2D3
B2D3_2nd
Figure 3-11: Load-deflection Response due to First and Second Live Loads of B2D3
80
``
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B3D3
B3D3_2nd
Figure 3-12: Load-deflection Response due to First and Second Live Loads of B3D3
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B4D7
B4D7_2nd
Figure 3-13: Load-deflection Response due to First and Second Live Loads of B4D7
81
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B5D7
B5D7_2nd
Figure 3-14: Load-deflection Response due to First and Second Live Loads of B5D7
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300
Deflection(in)
Live
Loa
d(lb
)
B6D7
B6D7_2nd
Figure 3-15: Load-deflection Response due to First and Second Live Loads of B6D7
82
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B7D28
B7D28_2nd
Figure 3-16: Load-deflection Response due to First and Second Live Loads of B7D28
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350Deflection(in)
Live
Loa
d(lb
)
B8D28
B8D28_2nd
Figure 3-17: Load-deflection Response due to First and Second Live Loads of B8D28
83
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
B9D28
B9D28_2nd
Figure 3-18: Load-deflection Response due to First and Second Live Loads of B9D28
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-19: Deflection History for B1D3
84
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-20: Deflection History for B2D3
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-21: Deflection History for B3D3
85
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200
Time(days)
Def
lect
ion(
in)
Figure 3-22: Deflection History for B4D7
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-23: Deflection History for B5D7
86
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-24: Deflection History for B6D7
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-25: Deflection History for B7D28
87
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-26: Deflection History for B8D28
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
Figure 3-27: Deflection History for B9D28
88
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200
Time(days)
Def
lect
ion(
in)
Day 3Day 7Day 28
Figure 3-28: The Effect of Age at Loading on Long-Term Deflection
89
0
10
20
30
40
50
60
70
80
90
100
0 20 40 60 80 100 120 140 160 180 200Time(days)
Hum
idity
(%)
Figure 3-29: Variation of Humidity
0
10
20
30
40
50
60
70
80
90
100
0 20 40 60 80 100 120 140 160 180 200Time(days)
Tem
pera
ture
(F)
Figure 3-30: Variation of Temperature
Chapter 4
VERIFICATION OF DEVELOPED MODEL
4.1 Introduction
In order to verify the developed model, several experiment results are obtained
from the literatures. For immediate deflection the test data reported by McNeice (1971)
and Burns and Hemakon (1985) are analyzed. Also, for long-term behavior the results
from Scott and Beeby (2005) and Gilbert and Guo (2005) are analyzed. Short and long-
term deflections from experiments reported in Chapter 3 are compared with analysis
results in this chapter.
4.2 Scott and Beeby (2005)
Scott and Beeby performed uniaxial tensile tests to evaluate the long-term tension
stiffening effect. Time-dependent decay of tension stiffening was investigated. In the
experiment specimens 1200 mm long by 120mm by 120 mm cross section were tested
with varying concrete strength (30, 70, and 100 MPa) and diameters of reinforcement (12,
16, and 20 mm). Some specimens were loaded on reinforcement directly up to 72 kN, and
sustained up to 3 to 4 months or loaded incrementally 42 kN, 58kN, and 72 kN and each
load was maintained according to each planned time-load history. In this analysis
specimen T16R1 reinforced with a single 16mm diameter bar was analyzed. In order to
91
measure strain, it was reported that total 85 strain gages were attached in specimens along
the length. In this case 43 kN was applied at the beginning of the test, 58 kN was applied
at 41 days, and 74 kN was applied at 69 days and sustained for 28 days. Figure 4-1
shows the time-load history.
The specimen was analyzed using the developed time-dependent orthotropic
concrete model and 4-node shell element with a layered modeling method. A unit length
of axial member is idealized as shown in Figure 4-2. Because it is assumed that the
tension stiffening model is considered as average tensile strength between cracks, the
tensile stress in concrete along the length is assumed to be uniform. Therefore, only unit
length of concrete member is modeled. Concrete points A and C are uncracked and points
B and D are cracked section. These points were obtained during experiment by
researchers who performed the experiment. The points were obtained from the presence
of cracks during experiment.
The assumed tension stiffening model is based on the Damjanic and Owen model.
It is assumed that tensile stress decreases to 50 % of tensile strength of concrete in the
model. For the long-term condition ACI 209 standard condition is used. In the analysis
the elastic modulus of concrete was 22.9 GPa, and the yielding stress of reinforcement
was assumed to be 200 MPa with 20 GPa of modulus of elasticity. The assumed creep
model and tension stiffening model are shown in Figure 4-3 and 4-4 respectively. The
material properties used in the analysis are summarized in Table 4-1.
Figure 4-5 shows a comparison between experiment and analysis results for time-
dependent strain variation of concrete. The analysis result shows good agreement with
92
experimental. Because the tension stiffening effect gives the average value of cracked
concrete between cracks, the analysis results tends to be close to average value of
experiment.
Figure 4-6 shows time-dependent average stress variations of concrete. In general
the analytical results show good agreement with the experimental data.
4.3 McNeice Corner Supported Slab (1971)
The McNeice slab is simply supply supported at the four corners and subjected to
a uniformly distributed load. Figure 4-7 shows the geometry of the McNeice slab and
material properties are provided in Table 4-2. In order to analyze the McNeice slab, a
concrete material model built in ABAQUS and the developed user defined concrete
model were used. To model the slab an 8-node shell element (S8R) was used. Because of
symmetry in the slab, only a quarter of the slab was modeled. For tension stiffening a
bilinear model and the Damjanic and Owen model were input in the user subroutine. In
this study the value for the strain beyond failure at which all tensile strength is lost
is 3100.2 −× . For the Damjanic and Owen model the tensile stress right after cracking is
assumed to equal a half of the modulus of rupture. The assumed tension stiffening model
is shown in Figure 4-8.
Figure 4-9 shows a comparison between experiment and analysis results. The
results indicate that the bilinear model provides a stiffer response than the experiment in
the post-cracking range, while the Damjanic and Owen model tends to underestimate the
93
load-deflection response over most of the post-cracking range. In general there is good
agreement for the tension stiffening models considered.
4.4 Burns and Hemakom (1985)
For the experiment of Burns and Hemakom, a similar modeling technique was
used. The differences here are that 4-node shell elements (S4R) were used to model the
slab. Columns are modeled by using frame elements (FRAME3D). Figure 4-10 shows
the geometry and tendon layout. A series of parabolic tendon profile is used and the
profile is shown in Figure 4-11. For material properties the behavior of slab is assumed
to be elastic. The elastic moduli of concrete and prestressing steel are assumed to be
29.96GPa and 200 GPa respectively.
In this analysis the equivalent loading method performed by Lee (2002) implemented in
SAP 2000 is compared with the equivalent layer method. The prestressing force is
idealized as equivalent loading. On the other hand, in equivalent layer method it is
assumed that the prestressing steel is idealized as a series of horizontal segments under
initial stress of prestressing steel layer for finite element analysis. The concept of
equivalent loading and equivalent layer methods are shown in Figure 4-12 schematically.
The analysis results are shown in Figure 4-13 to 4-15. Both the equivalent load approach
using SAP 2000 and the ABAQUS model based on piece wise horizontal cable segments
give reasonable correlation with the measured deflected shape. The average differences
between ABAQUS and measured and between SAP2000 and measured are 10.98% and
94
15% respectively. The differences are presented in Table 4-3. Note that the deflections in
general are very small(less than 2.5 mm).
4.5 Gilbert and Guo (2005)
Gilbert and Guo performed tests of time-dependent effects on seven large scale
reinforced concrete flat slabs (S1 to S7). The experimental results were also published in
more detail as report (Guo and Gilbert, 2002). In this research slab S3 was analyzed.
Overall plan dimension of Slab S3 was 6.2 x 7.2 m and supported on nine 200 by 200 by
1250 mm high columns. The slab was reinforced with 10mm deformed bars (named Y10).
The clear cover from reinforcement to top and bottom surface was 8 mm respectively.
The thickness of the slab was 90 mm. Figure 4-16 shows the plan view and measurement
points. The reinforcement layout is shown in Figure 4-17.
The slab was idealized using 4-nodes shell elements (S4R) and the columns were
modeled as frame elements (FRAME3D). In total 480 shell elements of 24 by 20
elements in 1 and 2 directions in global coordinate, and 9 frame elements were used. The
boundary condition of column was assumed to be fixed in all 6 degree of freedoms. The
finite element model of Gilbert slab is shown in Figure 4-18.
For loading history the self-weight (2.14 kPa) and additional load (3.10 kPa) were
applied at age 14 days and 28 days respectively. This load was maintained until age 387
days when the additional load was removed. Loading history is shown in Figure 4-19.
95
For material properties of slab the elastic moduli of concrete and steel are
assumed to be 22.92 GPa and 219 GPa respectively. The specified compressive strength
and tensile strength of concrete are assumed to be 18.1MPa and 2.48 MPa. Also yielding
stress of reinforcing steel is assumed to be 650 MPa. Concrete material properties used in
the analysis are summarized in Table 4-4. In the analysis, the Damjanic and Owen
tension stiffening model is adopted and shown in Figure 4-20. For creep and shrinkage,
measured data from the experiment were used. Creep and shrinkage data from
experiment are shown in Figure 4-21 and 4-22 respectively.
The deflections were compared with experiment results. Deflection histories of
the slab S3 at measuring points are shown in Figure 4-23 to 4-27 respectively. The
analysis results show a good agreement with experimental result.
4.6 Analytical Investigation of One Way Slab Specimens
In this section the analysis of nine slabs are performed. The experimental results
reported in Chapter 3 are compared with developed material model as well as two
different effective moment of inertia equations suggested by ACI 318 and Bischoff. Test
results for long term deflections are also compared with calculated values obtained using
the ACI 318 multiplier.
96
4.6.1 Calculation of Deflections Using Method Specified in Design Code
In order to calculate load-deflection relationship, the equations of effective
moment of inertia from ACI 318-05 and Bischoff are used. The immediate deflection due
to live load is calculated by Eq. 4.1.
where, LΔ is deflection due to live load, DL+Δ is the deflection due to live load plus dead
load, and DΔ is deflection due to dead load.
Dead load deflection is calculated by Eq. 4.2
in which, cE is elastic modulus, gI is the gross moment of inertia when the section is not
cracked under self-weight, Dw is distributed dead load in which the dead load is only
self-weight, and l is the member length.
Deflection due to dead load plus concentrated live load is calculated by Eq. 4.3
where, eI is the effective moment of inertia or the gross moment of inertia if the section
is not cracked, eI is the effective moment of inertia. LP is concentrated live load at mid-
span.
The calculated deflections were obtained considering varying effective moment of
inertia using virtual work. Because the section over the length is mostly remained mostly
DDLL Δ−Δ=Δ + 4.1
gc
DD IE
w3845 4l
=Δ 4.2
ec
D
ec
LDL IE
wIE
P3845
48
43 ll+=Δ + 4.3
97
uncracked under concentrated loads, deflections can be overestimated when the effective
moment of inertia is constant over the length. Therefore, it is necessary consider that the
section is uncracked and cracked separately.
According to ACI 318-05 the effective moment of inertia is calculated by Eq. 4.4
where, crM is the cracking moment assuming modulus of rupture equal to cf ′5.7 , aM
is applied maximum service load moment, and crI is the cracked transformed moment of
inertia.
On the other hand, Bischoff (2005) suggested the effective moment of inertia for
members reinforced with steel and fiber reinforced polymer bars. The effective moment
of inertia can be obtained by Eq. 4.5
From short-term deflection relationship of the test result, the modulus of rupture
rf can be calculated back. When the cracking moment is calculated from the load-
deflection relationship, the modulus of rupture is calculated by Eq. 4.6
Using Eq. 4.6, the factor k which decides the relation of modulus of rupture and the
specific compressive strength can be obtained by Eq. 4.7.
gcra
crg
a
cre II
MM
IMM
I ≤⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+⎟⎟
⎠
⎞⎜⎜⎝
⎛=
33
1 4.4
( )( ) gacrgcr
cre I
MMIII
I ≤−−
= 2//11 4.5
g
tcrr I
yMf = 4.6
'c
r
ffk = 4.7
98
The factor k in modulus of rupture specified in ACI 318 is 7.5 for normal weight concrete.
For the long-term deflection due to creep and shrinkage under sustained load, ACI
318-05 provides the multipliers for long-term deflections. The long-term deflection is
determined by multiplying the immediate deflection caused by the sustained load by the
factor, λ is calculated by Eq. 4.8
in which, 'ρ is the compression reinforcement ratio, ξ is the time-dependent factor
which varies from 1.0 to 2.0 according to duration.
Long-term deflection, LTΔ is calculated by Eq. 4.9
where, ot is the age of concrete at loading of dead load, 1t is the age of concrete at
loading of additional sustained load. irrΔ is the irrecoverable deflection after removing
live load. susΔ is the deflection due to additional sustained load and obtained by
DDsussus Δ−Δ=Δ + . In this experiment the additional sustained load is zero. A schematic
immediate and deflection history is shown in Figure 4-28.
4.6.2 Prediction of Cracking Loads
Examination of test results indicated that cracking occurred between load
increments. To determine cracking loads, two tangential lines are drawn and an
intersection point is obtained. From the averaged response the cracking loads can be
'501 ρξλ
+= 4.8
( ) susirrDoLLT tttt Δ⋅+Δ+Δ⋅+Δ=Δ ),(),( 1λλ 4.9
99
estimated using prescribed method. It is noted that the cracking loads obtained from each
load-deflection response are due to live loads. The cracking loads of loading at 3, 7, and
28 days are approximately 400, 410, and 290 lbs respectively. The estimation of cracking
loads is shown in Figure 4-29 to 4-31 respectively. It is also noted that the cracking load
of loading at 28 days is smaller than those of 3 and 7 days although the 28 days strength
of concrete is higher. The calculated factor k is shown in Table 4-5. For age of loading at
3 days the factor k is higher than 7.5 but it was lower for age of loading at 28 days.
4.6.3 Results of Analysis: Instantaneous Deflections
The load-deflection responses are compared between experimental and analytical
results. The comparisons between analytical and experimental results are shown in
Figure 4-32 to 4-34. Also, the differences of maximum deflections between analytical
and experimental results are presented in Table 4-6 and the differences of irrecoverable
deflections are shown in Table 4-7.
For the case of loading at 3 days the estimation of response using Branson’s
equation is closely related to B3D3 which is stiffest response at day 3. On the other hand,
the estimation using Bischoff is closely related to B1D3 which is the most flexible case.
The differences between analytical results and experimental are 22.0% for Branson’s and
26.7 % for Bischoff’s respectively. For unloading part of the response Branson’s equation
underestimates the irrecoverable deflection but Bischoff’s equation predicts the
100
deflection within the range of variation. The differences of irrecoverable deflection are
69.4% and 32.0%.
For loading at 7 days load-deflection response of analytical model shows similar
tendency of that of day 3. Bischoff’s equation overestimates the load-deflection response
about 39.3 % and Branson’s equation under estimate the response of 23.7%. For
irrecoverable deflection due to unloading of live load the differences between analysis
and experiment are 68.4% for Branson’s and 16.2% for Bischoff’s
For loading at 28 days the load-deflection response also shows similar tendency
compared with previous responses. Branson’s equation underestimates the response of
22.0% but Bischoff’s overestimate the response of 34.9%. Branson’s and Bischoff’s
equations predict the load-deflection response in the reasonable range for loading but do
not predict well for the unloading response. The differences of irrecoverable deflection
between analysis and experiment are 60.8 % for Branson’s and 20% for Bischoff’s.
According to analysis results Branson’s equation tends to predict the response to
be stiff. On the other hand, Bischoff’s equation predicts the response to be flexible.
4.6.4 Results of Analysis: Long-Term Deflections
Long-term deflections are calculated in accordance with multiplier for long-term
deflection defined in ACI 318. The long-term multipliers for duration of 30, 60, 90,
120,150, and 180 days are approximately 0.7, 0.9, 1.0, 1.1, 1.15, and 1.2 respectively.
Obtained values are multiplied by initial deflections due to self-weight plus irrecoverable
deflections due to removal of live load. For instance, at 90 days of duration long-term
101
deflection of loading at 3 days, 0.141” is obtained from deflection due to self-weight
0.048” plus irrecoverable deflection 0.093” multiplied by long-term multiplier 1.0. The
prediction of long-term deflection base on ACI 318 is shown in Table 4-8. The results of
analysis show that the long-term multipliers may not be applicable to calculation of long-
term deflections for the early-age concrete. Figure 4-35 shows the comparison of long-
term deflections between results of experiment and analysis.
According to experimental results, the long-term deflections of loading at 3 and 7
days are much higher than prediction using the multiplier specified in ACI 318. Long-
term multiplier may be obtained from experimental results. Long-term deflections are
obtained from measured deflection minus irrecoverable deflection due to removing of
live load. The multiplier is obtained from the ratio of long-term deflection to
instantaneous deflection which is the summation of deflection due to self-weight and
irrecoverable deflection. The calculation of the multiplier is performed using the
averaged deflections of each loading case. Table 4-9 shows the long-term multiplier
which is calculated from experimental results. The long-term multiplier specified in ACI
318 is 2.0 for the duration of 5 years or more. However, the multipliers calculated from
the experiment for the cases of loading at 3 and 7 days are 2.33 and 3.21 at duration of
179 and 175 days. The specified multiplier at the duration of 180 day is 1.2, but obtained
values already go beyond the ultimate value of 2.0. It may be noted that the long-term
multiplier for loading at early-age can be higher than that specified in ACI 318. However,
for the case of loading at 28 days the multiplier agrees reasonably well with specified
value of 1.2 at the duration of 180 days.
102
Table 4-10 presents the maximum permissible deflections for immediate and
long-term deflections. Long-term deflections are obtained from experiment up to 182 day
after casting concrete. Results show that the immediate deflections due to first and second
live loads satisfied the limitations (L/180 and L/360). However, for the long-term
deflections the slabs do not satisfy the limitation of L/480.
4.7 Finite Element Analysis using Developed Concrete Model
In this section the analysis of tested slabs is performed. The analysis results are
compared with experimental results. The load-deflection response and time-deflection
history are obtained from the analysis. In order to perform time-dependent analysis, ACI
209, CEB-FIP, and GL2000 model are compared. Creep models are presented in
Appendix A. Because creep and shrinkage strains are not obtained from test, time-
dependent effect on concrete is calculated using aforementioned models.
Comparison of immediate deflection due to first and second live loading is made
in this section. Shrinkage effect is especially introduced while analyzing the case of
loading at 28 days. The shrinkage reduces the cracking load because shrinkage causes the
tensile stress at the bottom fiber of slab.
4.7.1 Finite Element Model
The slabs are idealized using 4-nodes shell element. Each shell element has one
layer of steel reinforcement of 0.22 sq. in positioned at 4” from top fiber. Total 48 shell
103
elements are used. Self-weight of slabs is idealized into distributed pressure load of 0.414
psi which is equivalent to 143 pcf. Live load is applied at center as concentrated load. For
the tension stiffening model Damjanic and Owen model is used. In the model the tensile
strength is assumed to be from '5.7 cf and drops to 50% of tensile strength. The
assumed tension stiffening model according to age of concrete at loading is shown in
Figure 4-36. The compressive strength, 'cf obtained from cylinder test is used. Input
values are presented in Table 4-11. Also, parameters which are necessary in calculation
of creep and shrinkage are presented in Table 4-12. Creep and shrinkage models based
on ACI209, GL2000, and CEB-FIP are presented in Figure 4-37 to 4-42 respectively.
The boundary condition of slabs is assumed to be simply supported. Figure 4-43
shows finite element model of test slabs. The analysis is divided into 6 steps. First and
second steps are for loading/unloading, third step is for creep and shrinkage analysis,
forth and fifth steps are for loading and unloading of second live load, and sixth step is
for creep and shrinkage analysis.
4.7.2 Immediate Deflections
Load-deflection responses between experiment and analysis using developed
concrete model are compared. The calculated deflections based on finite element model
agree well with the measured deflection. The differences are presented in Table 4-13.
For loading at 3 days the response is within the range of deflections of each slab.
The calculated maximum deflection due to live load is 0.178” and the irrecoverable
104
deflection after removing live load is 0.073”. The differences of maximum and
irrecoverable deflections between analytical and experimental are 7.9 and 27.4 %
respectively. The load-deflection response of loading at 3 days is in Figure 4-44.
For loading at 7 days the maximum and irrecoverable deflections from analysis
shows good agreement with experimental results. The maximum deflection is 0.161”
with difference of 3.9% and the irrecoverable deflection is 0.067” with difference of
3.5%. Figure 4-45 shows the load-deflection response of loading at 7 days.
For loading at 28 days the load-deflection response shows different tendency
compared with cases of loading at 3 and 7 days. The compressive strength of 28 days
apparently is bigger than those of 3 and 7 days. However, the load-deflection response
shows more flexible than those of 3 and 7 days. This may be explained by the effect of
shrinkage. The shrinkage before loading can cause tensile stress due to restraint. In this
study shrinkage strain of 200 and 380 micro strains are assumed and incorporated to the
analytical model. Figure 4-46 shows load-deflection response of loading at 28 days.
According to the response, the response show s good agreement with experiment when
the shrinkage strain is 380 micro strains. The effect of shrinkage restraint may be
calculated in accordance with Appendix B.
4.7.3 Long-Term Deflections
Long-term deflections are obtained using ACI209, GL2000, and CEB-FIP model
because creep and shrinkage strains are not obtained from experiment. All models predict
long-term deflection of loading at 28 days well. However, models show poor correlation
105
for loading at 3 and 7 days. The long-term deflections are calculated up to six months and
compared with experimental results. Differences of long-term deflection between
analytical and experimental results are presented in Table 4-14.
ACI 209 model tends to underestimate creep and shrinkage effect of loading at
early-age concrete. Figure 4-47 to 4-49 show time-dependant deflection using ACI 209
model. For loading at 3 days the long-term deflection after 6 months based on ACI 209
model is 0.202” with difference of 52.0%. In case of loading at 7 days, the long-term
deflection is 0.178” with difference of 57.3%. The results show that the model
underestimates the long-term deflection of loading at early-age. On the other hand, for
loading at 28 days the long-term deflection is 0.178” and difference is 13.1%. The
analytical model predicts the long-term deflection of loading at 28 days reasonably well.
GL2000 model also underestimate the time-dependent deflection of loading for
early-age loading. Time-dependent deflection histories using GL2000 model are shown
in Figure 4-50 to 4-52. The model shows a good agreement with experiment when the
load is applied at 28 days. The long-term deflection of loading at 3 days is 0.283” and
difference is 32.7%. GL2000 model predicts the long-term deflection better than ACI 209
model, but there still exists large difference between analytical and experimental results.
For the case of loading at 7 days similar tendency is observed. The long-term deflection
is 0.246” with difference of 40.8%. For loading at 28 days analytical model predicts the
long-term deflection well. The prediction of long-term deflection is 0.296” with
difference of 11.6%.
106
CEB-FIP model tends to predict time-dependent deflection similar to GL2000
model. The model also underestimates the deflection of loading at early-age, but shows
good correlation for the case of loading at 28 days. Time-dependent deflections are
shown in Figure 4-53 to 4-55. The long-term deflection of loading at 3, 7, and 28 days
after six months are 0.286”, 0.250”, and 0.349” and differences between analytical and
experimental results are 32.0%, 39.9%, and 4.5% respectively.
According to experiment, time-dependent deflections of loading at early-age are
higher than expected values from pre-existing model. All three models show poor
correlation when the application of loading occurs at early-age. It may be necessary to
investigate the creep and shrinkage of early-age loading. During experiment shrinkage
strains and creep coefficients were not obtained from experiment. It is expected that
better correlation could be shown if the long-term properties were measured in the
experiment.
4.8 Summary
The developed material model is verified using pre-existing test data and
experimental data reported in Chapter 3. Long-term multiplier specified in ACI 318 may
not appropriate to calculate long-term deflection of loading at early-age. The developed
time-dependent material model predicts immediate and long-term deflections reasonably.
Shrinkage strain before loading can reduce the tensile strength of concrete. As result, the
load-deflection response is affected by shrinkage restraint significantly.
107
Table 4-1: Material Properties of Scott and Beeby
'cf
(MPa) cE
(GPa) crf
(MPa) ν
Input 23.5 22.9 2.10 0.15
Table 4-2: Material Properties of McNeice Slab
'cf
(MPa) cE
(GPa) crf
(MPa) ν
Input 37.92 28.6 3.17 0.15
108
Table 4-3: Differences of Deflection Between Analytical and Experimental Results of Burns and Hemakom
Point Measured (mm)
ABAQUS(mm) Diff (%) SAP2000
(mm) Diff (%)
10 -0.066 -0.066 0.18 -0.050 24.56 11 -0.079 -0.093 18.13 -0.070 11.03 12 -0.040 -0.054 35.20 -0.038 4.05 18 -0.046 -0.027 40.67 -0.033 28.26 19 -0.060 -0.061 1.76 -0.054 9.58 20 -0.037 -0.037 1.29 -0.036 1.81 21 -0.043 -0.043 0.35 -0.050 15.44 22 -0.044 -0.038 13.15 -0.038 13.11 23 -0.070 -0.071 1.54 -0.062 10.79 24 -0.032 -0.032 0.42 -0.027 16.25 30 -0.053 -0.044 16.88 -0.044 17.53 31 -0.063 -0.076 20.27 -0.082 30.13 32 -0.052 -0.054 3.91 -0.054 2.96 33 -0.061 -0.060 0.86 -0.077 26.26 34 -0.064 -0.055 13.46 -0.056 12.63 35 -0.091 -0.084 7.88 -0.093 1.97 36 -0.051 -0.046 10.63 -0.036 28.69
Average 10.98 15.00
Table 4-4: Material Properties of Gilbert and Guo Slab
'cf (MPa)
cE (GPa)
crf (MPa)
ν
Input 18.1 22.62 2.48 0.20
109
Table 4-5: Factor k for Modulus of Rupture
Age at loading (day)
Self-weight (pcf)
Live load at cracking
(lbs)
'cf (psi)
ty (in)
gI (in4)
crM (lb-in)
k
3 400 2828 2.45 129.23 24014 8.6 7 410 3719 2.46 128.30 24344 7.7
28 143
290 4759 2.47 128.15 20384 5.7
Table 4-6: Differences of Maximum Deflection Between Analytical and Experimental Results
Day at loading
Slab No.
maximumdeflection
(in)
Branson (in) Diff
(%) Bischoff
(in) Diff (%)
B1D3 0.241 B2D3 0.195 B3D3 0.140 3
Ave. 0.192
0.150 22.0 0.243 26.7
B4D7 0.141 B5D7 0.177 B6D7 0.146 7
Ave. 0.155
0.118 23.7 0.215 39.3
B7D28 0.218 B8D28 0.270 B9D28 0.246 28
Ave. 0.245
0.191 22.0 0.330 34.9
110
Table 4-7: Differences of Irrecoverable Deflection Between Analytical and Experimental Results
Day at loading
Slab No.
irrecoverable deflection
(in)
Branson(in) Diff
(%) Bischoff
(in) Diff (%)
B1D3 0.122 B2D3 0.096 B3D3 0.061
3
Ave. 0.093
0.028 69.4 0.063 32.0
B4D7 0.059 B5D7 0.084 B6D7 0.065
7
Ave. 0.069
0.022 68.4 0.058 16.2
B7D28 0.116 B8D28 0.144 B9D28 0.130
28
Ave. 0.130
0.051 60.8 0.104 20.0
Table 4-8: Prediction of Long-Term Deflection Based on Equations Specified in ACI 318
Day 3 Day 7 Day 28 t λ
DΔ irrΔ TΔ DΔ irrΔ TΔ DΔ irrΔ TΔ 0 0.00 0.000 0.000 0.000
30 0.70 0.099 0.076 0.117 60 0.90 0.127 0.097 0.151 90 1.00 0.141 0.108 0.167 120 1.10 0.155 0.119 0.184 150 1.15 0.162 0.124 0.193 180 1.20
0.048 0.093
0.169
0.039 0.069
0.130
0.037 0.130
0.201 DΔ :deflection due to self-weight, theoretically obtained
irrΔ :irrecoverable deflection after removing live load ( )irrDT Δ+Δ⋅=Δ λ , time-dependent deflection
t : duration
111
Table 4-9: Long-Term Multiplier from Experiment
(a) Loading at 3 days
Day 3 t
DΔ irrΔ TΔ λ 0 0.000 0.00
30 0.180 1.28 61 0.231 1.64 90 0.232 1.65 123 0.269 1.91 151 0.281 1.99 179
0.048 0.093
0.327 2.33 (b) Loading at 7 days
Day 7 t
DΔ irrΔ TΔ λ 0 0.000 0.00
30 0.195 1.81 61 0.249 2.31 91 0.245 2.27 119 0.288 2.67 149 0.315 2.91 175
0.039 0.069
0.347 3.21 (c) Loading at 28 days
Day 28 t
DΔ irrΔ TΔ λ 0 0.000 0.00
31 0.086 0.51 60 0.098 0.59 91 0.124 0.74 119 0.166 0.99 154
0.037 0.130
0.204 1.22 TΔ : obtained from time-dependent measured deflection minus irrecoverable deflection
)/( irrDT Δ+ΔΔ=λ
112
Table 4-10: Deflection Requirements
Day at Loading
Slab No. LΔ , 1st LΔ , 2nd TΔ LTΔ L/180 L/360 L/480 L/240
B1D3 0.241 0.140 0.339 0.580B2D3 0.195 0.125 0.328 0.523B3D3 0.140 0.102 0.315 0.455
Day 3
Ave. 0.192 0.122 0.327 0.519B4D7 0.141 0.112 0.334 0.475B5D7 0.177 0.130 0.377 0.554B6D7 0.146 0.107 0.329 0.475
Day 7
Ave. 0.155 0.116 0.347 0.501B7D28 0.218 0.128 0.190 0.408B8D28 0.270 0.154 0.210 0.480B9D28 0.246 0.137 0.213 0.459
Day 28
Ave. 0.245 0.140 0.204 0.449
0.733 0.367 0.275 0.550
Table 4-11: Input Value of Analytical Model
Details
Test Set
*Compressive Strength, psi
*Elastic Modulus, psi
**Tensile Strength, psi
Day 3 2828 3181227.552 398.60 Day 7 3719 3982000.148 457.38 Day 28 4759 4145894.615 517.39
*compressive strength and elastic modulus are from cylinder test **tensile strength is obtained using modulus of rupture specified in ACI 318
113
Table 4-12: Input Values For Creep and Shrinkage
Models Parameters ACI 209 GL2000 CEB-FIP
Relative Humidity, H (%) 56
Volume-to-Surface Ratio, sv / ( in )
1.72 -
notional size of member, h = uAc /2 (in) - - 3.53
Slump, s ( in ) 2.7 (assumed) - -
Fine aggregate F (%) 40 - -
Air content, A (%) 6 (assumed) - -
Age of concrete at loading, oct (days)
3, 7, and 28 days
Cement content ( 3/ ydlb ) 376 - -
28 days specified concrete strength, 28ckf
(psi) - 4700
Age of concrete at the beginning of shrinkage,
ost (days) loading at 3 days: 3 days
loading at 7 and 28 days: 7 days`
Cement type Type I (normal hardening concrete)
114
Table 4-13: Differences Between Analytical and Experimental Results
Day at loading
Slab No.
maxΔ (in)
irrΔ (in)
ABAQUSmaxΔ
(in)
diff (%)
ABAQUS irrΔ
(in)
diff (%)
B1D3 0.241 0.122 B2D3 0.195 0.096 B3D3 0.140 0.061 3
Ave. 0.192 0.093
0.178 7.9 0.073 27.4
B4D7 0.141 0.059 B5D7 0.177 0.084 B6D7 0.146 0.065 7
Ave. 0.155 0.069
0.161 3.9 0.067 3.5
B7D28 0.218 0.116 B8D28 0.270 0.144 B9D28 0.246 0.130 28
Ave. 0.245 0.130
0.247 0.9 0.140 7.1
Table 4-14: Differences of Long-term Deflection Between Analytical and Experimental Results after six Months
Day at loading
Slab No.
Deflection(in)
ACI209(in) Diff
(%)GL2000
(in) Diff (%)
CEB-FIP
Diff(%)
B1D3 0.461 B2D3 0.424 B3D3 0.376
3
Ave. 0.420
0.202 52.0 0.283 32.7 0.286 32.0
B4D7 0.393 B5D7 0.461 B6D7 0.394
7
Ave. 0.416
0.178 57.3 0.246 40.8 0.250 39.9
B7D28 0.306 B8D28 0.354 B9D28 0.343
28
Ave. 0.334
0.290 13.1 0.296 11.6 0.349 4.5
115
74
58
43
P(kN)
41 69 97time(day)
Figure 4-1: Time-Load History for T16R1
116
P/2 P/2 (applied at node)
rebar layer
concrete layer
shell element
(a) Cross section
(b) Unit length
P
P
A
B
C
D
A and C are uncracked section pointsB and D are cracked section points
Figure 4-2: Idealization of Axial Member
117
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0 20 40 60 80 100Time(days)
Cre
ep C
oeffi
cien
t
Figure 4-3: Assumed Creep Coefficient for Scott and Beeby
0
0.5
1
1.5
2
0 0.0005 0.001 0.0015 0.002Strain(mm/mm)
Stre
ss(M
Pa)
Figure 4-4: Assumed Tension Stiffening Model
118
0
400
800
1200
1600
2000
0 20 40 60 80 100Time(days)
Con
cret
e Te
nsile
Stra
in(m
icro
stra
in)
ABCDDamjanic & Owen
Figure 4-5: Time-Dependent Strain Variations of Concrete
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 20 40 60 80 100Time(days)
Ave
rage
Con
cret
e S
tress
(MP
a)
Experiment
Damjanic & Owen
Figure 4-6: Time-Dependent Average Stress Variations of Concrete
119
33.3 44.5
Point Load
8 nodes shell Element(S8R)
914.4
CL
Y
CL X
unit (mm)
Figure 4-7: The Geometry of Slab(McNeice, 1967)
120
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
0 0.0005 0.001 0.0015 0.002 0.0025
Strain(mm/mm)
Stre
ss(M
Pa)
Damjanic & OwenBilinear
Figure 4-8: Tension Stiffening Models for McNeice Slab
0
2
4
6
8
10
12
14
16
18
20
0 2 4 6 8 10 12Deflection(mm)
Load
(kN
)
UMAT(bilinear)UMAT(Damjanic & Owen)ABAQUS(bilinear)experiment
Figure 4-9: Load-Deflection at Center of McNeice Slab
121
3048762
762
3048
313048
30 32 33
3048 102
3634 35
3048
193048
18 20 21
102
2422 23
1210 11 Panel A
Panel B
Panel C
Plan
70 536
Elevation
Figure 4-10: Geometry and Tendon Layout of Slab (Burns and Hemakom, 1986)
122
-40-20
02040
0 2000 4000 6000 8000 10000
Distance (mm)
ecce
ntric
ity(m
m)
Figure 4-11: Tendon Profile
123
Equivalent concrete layer
Equivalent reinforcement layer
equivalent reinforcement layer(initial stress condition as prestressing force)
P
Psinθ
θ
Pcosθ
w=8Pe/l2
Psinθ
P
Psinθ
Equivalent reinforcement layer
Finite Element
e
Psinθ
Equivalent Load
Pcosθ
Figure 4-12: Equivalent Loading and Equivalent Layer Method
124
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
10 11 12Point
Def
lect
ion
(in)
Measured ABAQUS SAP 2000
Figure 4-13: Deflection of Panel A
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
18 19 20 21 22 23 24Point
Def
lect
ion
(in)
Measured ABAQUS SAP 2000
Figure 4-14: Deflection of Panel B
125
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
30 31 32 33 34 35 36Point
Def
lect
ion
(in)
Measured ABAQUS SAP 2000
Figure 4-15: Deflection of Panel C
126
C1 C2 C3
C6C5C4
C7 C8 C9
6200
3000
3000
7200
3000 3000
1 4 8 11 15
3 10
5 12
2 6 9 13 16
7 14
(Unit: mm)
Figure 4-16: Dimension of Slab and Measuring Points
127
1200
13Y
10@
140
TOP
13Y
10@
140
TOP
1000
1200
1500 1200 1800
8Y10
@14
0 TO
P
6Y10@250 TOP
48Y
10@
140
TOP
8Y10@250 TOP
11
1800
1200
1000
11Y10@140TOP
6
13Y10@140 TOP
13
29Y
10@
220
TOP
1500
6200
33Y10@220 BOT
Figure 4-17: Reinforcement Layout
128
Figure 4-18: The Finite Element Model of Slab
Load(kPa)
2.16
5.26
5993872814Concrete Age (days)
Figure 4-19: Loading History of S3
129
0.00
0.50
1.00
1.50
2.00
2.50
3.00
0 0.0005 0.001 0.0015 0.002 0.0025Strain(mm/mm)
Stre
ss(M
Pa)
Figure 4-20: Assumed Tension Stiffening Model for Gilbert and Guo Slab
0
0.5
1
1.5
2
2.5
0 50 100 150 200 250 300 350 400 450Time(days)
Cre
ep C
oeffi
cien
t
Loading at 14 days
Loading at 28 days
: 0.6562 ln(t)-1.6692
: 0.8371 ln(t)-2.6092
Figure 4-21: Creep Coefficient for Gilbert and Guo Slab(Guo and Gilbert, 2002)
130
0
100
200
300
400
500
600
0 50 100 150 200 250 300 350 400 450Time(days)
Shr
inka
ge S
train
(mic
ro s
train
)
Shrinkage strain :153.51 ln(t)-396.36
Figure 4-22: Shrinkage Strain for Gilbert and Guo Slab (Guo and Gilbert, 2002)
0
2
4
6
8
10
12
14
16
0 100 200 300 400 500 600Time (days)
Def
lect
ion
(mm
)
analysis461113
Figure 4-23: Deflection History for Point 4, 6, 11, and 13
131
0
2
4
6
8
10
12
14
0 100 200 300 400 500 600Time (days)
Def
lect
ion
(mm
)
analysis
8
9
Figure 4-24: Deflection History for Point 8 and 9
0
1
2
3
4
5
6
7
8
9
0 100 200 300 400 500 600Time (days)
Def
lect
ion
(mm
)
analysis
1
2
15
16
Figure 4-25: Deflection History for Point 1, 2, 15, and 16
132
0
2
4
6
8
10
12
0 100 200 300 400 500 600Time (days)
Def
lect
ion
(mm
)
analysis
5
12
Figure 4-26: Deflection History for Point 5 and 12
0
1
2
3
4
5
6
7
8
9
0 100 200 300 400 500 600Time (days)
Def
lect
ion
(mm
)
analysis371014
Figure 4-27: Deflection History for Point 3, 7, 10, and 14
133
ΔD
ΔLΔirr
ΔL
Concrete Age (days)
Deflection
Figure 4-28: Schematic Time- Deflection History
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250Deflection(in)
Live
Loa
d(lb
)
Figure 4-29: Prediction of Cracking Load from Load-Deflection Response of Loading at 3 Days
134
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200
Deflection(in)
Live
Loa
d(lb
)
Figure 4-30: Prediction of Cracking Load from Load-Deflection Response of Loading at 7Days
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300
Deflection(in)
Live
Loa
d(lb
)
Figure 4-31: Prediction of Cracking Load from Load-Deflection Response of Loading at 28 Days
135
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
Branson
B1D3
B2D3
B3D3
Bischoff
Figure 4-32: Comparison Between Experiment and Analytical Results for Loading at 3
Days
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250Deflection(in)
Live
Loa
d(lb
)
Branson
B4D7
B5D7
B6D7
Bischoff
Figure 4-33: Comparison Between Experiment and Analytical Results for Loading at 7 Days
136
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350Deflection(in)
Live
Loa
d(lb
)
Branson
B7D28
B8D28
B9D28
Bischoff
Figure 4-34: Comparison Between Experiment and Analytical Results for Loading at 28
Days
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.350
0.400
0 50 100 150 200 250Age of concrete (days)
Def
lect
ion
(in)
Day 3Day 7Day 28Day 3_ACIDay 7_ACIDay 28_ACI
Figure 4-35: Long-Term Deflection Based on ACI 318
137
0
100
200
300
400
500
600
0 0.0005 0.001 0.0015 0.002 0.0025 0.003Strain(in/in)
Stre
ss(p
si)
Day 3
Day 7
Day 28
Figure 4-36: Assumed Tension Stiffening Models for Test Slabs
0.00
0.50
1.00
1.50
2.00
2.50
1 10 100 1000 10000Time(days)
Cre
ep C
oeffi
cien
t
Loading at 3 days
Loading at 7 days
Loading at 28 days
Figure 4-37: Creep Coefficient Based on ACI 209
138
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
1 10 100 1000 10000
Time(days)
Cre
ep C
oeffi
cien
t
Loading at 3 days
Loading at 7 days
Loading at 28 days
Figure 4-38: Creep Coefficient Based on GL2000
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
1 10 100 1000 10000Time(days)
Cre
ep C
oeffi
cien
t
Loading at 3 days
Loading at 7 days
Loading at 28 days
Figure 4-39: Creep Coefficient Based on CEB-FIP
139
0
50
100
150
200
250
300
350
400
450
500
1 10 100 1000 10000Time(days)
Stra
in (m
icro
stra
in)
Cured for 3 days(Loadingat 3 days)Cured for 7 days(Loadingat 7 and 28 days)
Figure 4-40: Shrinkage Model Based on ACI 209
0
100
200
300
400
500
600
700
800
1 10 100 1000 10000Time(days)
Stra
in (m
icro
stra
in)
Cured for 3 days(Loadingat 3 days)Cured for 7 days(Loadingat 7 and 28 days)
Figure 4-41: Shrinkage Model Based on GL2000
140
0
100
200
300
400
500
600
700
1 10 100 1000 10000Time(days)
Stra
in (m
icro
stra
in)
Cured for 3 days(Loadingat 3 days)Cured for 7 days(Loadingat 7 and 28 days)
Figure 4-42: Shrinkage Model Based on CEB-FIP
Figure 4-43: Finite Element Model
141
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300Deflection(in)
Live
Loa
d(lb
)
ABAQUS
Day3 (average)
Figure 4-44: Comparison Between Analysis and Experiment of Loading at 3 days
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300
Deflection(in)
Live
Loa
d(lb
)
ABAQUS
Day 7(averaged)
Figure 4-45: Comparison Between Analysis and Experiment of Loading at 7 days
142
0.0
100.0
200.0
300.0
400.0
500.0
600.0
700.0
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350Deflection(in)
Live
Loa
d(lb
)
ABAQUS( 0 mcro)
ABAQUS(200 micro)
ABAQUS(380 micro)
Day28(average)
Figure 4-46: Comparison Between Analysis and Experiment of Loading at 28 days
143
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
ACI209B1D3B2D3B3D3
Figure 4-47: Time-Dependent Deflection of Loading at 3 days Using ACI 209
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200
Time(days)
Def
lect
ion(
in)
ACI 209B4D7B5D7B6D7
Figure 4-48: Time-Dependent Deflection of Loading at 7 days Using ACI 209
144
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
ACI 209B7D28B8D28B9D28
Figure 4-49: Time-Dependent Deflection of Loading at 28 days Using ACI 209
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
GL2000B1D3B2D3B3D3
Figure 4-50: Time-Dependent Deflection of Loading at 3 days Using GL2000
145
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200
Time(days)
Def
lect
ion(
in)
GL2000
B4D7
B5D7B6D7
Figure 4-51: Time-Dependent Deflection of Loading at 7 days Using GL2000
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200
Time(days)
Def
lect
ion(
in)
GL2000
B7D28
B8D28
B9D28
Figure 4-52: Time-Dependent Deflection of Loading at 28 days Using GL2000
146
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
CEB-FIPB1D3B2D3B3D3
Figure 4-53: Time-Dependent Deflection of Loading at 3 days Using CEB-FIP
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
CEB-FIPB4D7B5D7B6D7
Figure 4-54: Time-Dependent Deflection of Loading at 7 days Using CEB-FIP
147
0
0.1
0.2
0.3
0.4
0.5
0.6
0 50 100 150 200Time(days)
Def
lect
ion(
in)
CBE-FIPB7D38B8D28B9D28
Figure 4-55: Time-Dependent Deflection of Loading at 28 days Using CEB-FIP
Chapter 5
PARAMETRIC STUDY BASED ON THE DEVELOPED MATERIAL MODEL
5.1 Introduction
The deflections of reinforced concrete slabs are affected by a large number of
factors. These factors include the material properties, loading condition, geometry of slab
and boundary conditions. The purpose of the parametric study presented in this chapter is
to investigate the effect of loading history, geometry of slab, time-dependent behavior,
cracking, and column stiffness on slab deflections. Also, long-term multiplier and the
moment diagrams in the column face and the center line is investigated. Long-term
multipliers considering shrinkage restraint are obtained.
The study is restricted to a square exterior panel of a flat plat floor system and the
following parameters are considered.
1. Load-time history model
2. Thickness of slab
3. Column stiffness
4. Creep and shrinkage effect
5. Comparison between Elastic and Nonlinear analysis
6. Extraordinary superimposed load
7. Age of loading
8. Shrinkage restraint
149
5.2 Slab Design
A flat plate system of three by three panels in plan is designed according to the
Direct Design method specified in ACI 318-02(2002). The clear height between the
floors is 12 ft, and the flat plate has no edge beams. The panel dimensions as well as the
size of the supporting columns are shown in Figure 5-1.Two different loading conditions
are assumed: ordinary loading summarized in Table 5-1 and extraordinary loading
reported by Bondy (2005) presented in Table 5-2.
For ordinary loading condition, in order to decide the thickness of slab ACI 318
minimum thickness of slab without interior beams is used. The minimum thickness is
calculated as 30
nl in inch unit with 20.17ft of clear span, nl . Therefore, minimum
thickness for the given geometry of slab is 8.0 inches. Thicknesses of %100.8 ± (7”, 8”,
and 9”) are chosen to perform the parametric study. It is also assumed that 4000 psi
normal weight concrete and Grade 60 reinforcement are chosen. The design dead loads
by self-weight for thicknesses of 7”, 8”, and 9” are 87.5, 100, and 112.5 psf respectively.
The superimposed dead load is 15 psf. Load from partition and finishes is assumed to be
20 psf. The design live load of 70 psf is assumed.
For extraordinary loading condition, the minimum thickness is not only obtained
by ACI 318 but also obtained by equation suggested by Scanlon and Lee (2006). The
obtained thickness for parametric study is 11”. Also, minimum thickness by ACI 318 of
8” is used to perform the parametric study. It is noted that the minimum thickness
150
specified in ACI 318 is independent of loading condition. The design loads for this case
are superimposed load of 200 psf and live load of 150 psf.
The required reinforcements are obtained from direct design method of the ACI
318-05 Building Code. The total panel moment 0M is obtained from Eq. 5.1
where, uw represent the factored load per unit area on the panel, 2l is the transverse
width of strip, and nl is the clear span between columns faces in the longitudinal direction.
The total moment is distributed between column and middle strip and positive and
negative moment as shown in Figure 5-2. Using the distributed positive and negative
moment of each column and middle strips, the required reinforcement is obtained. The
required amount of reinforcement per unit length in each direction is given in Table 5-3
to Table 5-6 including design moments. The layout of reinforcement is schematically
illustrated in the Figure 5-3. Arrangement of reinforcement in slab is shown in Figure 5-
4. The design is based on flexural behavior, and it is assumed that punching shear is not a
critical condition in the design.
5.3 Finite Element Model
The finite element model of exterior slab is idealized using shell elements and
frame elements in three dimensional space. Four-node layered shell elements are used to
model the slab and reinforcement. The columns are idealized using three dimensional
8
22
0nu llw
M = 5.1
151
frame elements with 6-degrees of freedom. The mesh of model is shown in Figure 5-5.
In total, 576 of shell elements and 96 of frame elements are used respectively. The shell
element aspect ratio is 1.0.
A specified compressive strength of concrete of 4000 psi and steel yield stress of
60 ksi are assumed in the analysis. The Damjanic & Owen model is adopted for tension
stiffening. The tensile strength of concrete is defined by the modulus of rupture specified
in ACI 318. The tension stiffening factor is assumed to be 20 in this study. Material
properties of concrete for parametric study are summarized in Table 5-7 and the assumed
Damjanic and Owen model is shown in Figure 5-6. Creep and shrinkage model is based
on the GL2000 model as described in Appendix A. The assumptions are that the relative
humidity is 40%, slab is moist cured for 7 days, and Type I cement is used as
summarized in Table 5-8. Creep and shrinkage plots are shown in Figure 5-7 to 5-12.
For the boundary condition at the end of column all 6-degrees of freedom are constrained.
Figure 5-13 shows the boundary condition for the slab.
5.4 Parameters
The parametric study is performed considering the factors listed in Section 5.1.
Loading histories are considered according to shoring and reshoring methods in the
construction of multistory buildings. During construction adding and removing of
formwork are decided according to the selected construction method and cycles which
depend on job plans of reuse of materials and the rate of strength gain in the structure
152
(Hurd, 1995). The serviceability of a floor system is maintained by controlling the
deflection and cracking of a floor system. The thickness can affect the flexural stiffness
of slab, so that deflection can be controlled by the thickness. It is known that the column
stiffness also contributes to the deflection calculation when the equivalent frame method
approach is used to calculate the deflection of slab. In addition, the column (or wall)
stiffness affects the in-plane action of the slab due to shrinkage restraint. While varying
the parameters, short and long term deflections of slab are evaluated. As well, creep and
shrinkage effect are separated in order to investigate the sensitivity of each effect.
5.4.1 Load-Time History Model
In the construction of multistory buildings with reinforced concrete slabs, shoring
and reshoring procedures are usually employed. The sequence consists of steps of setting
up shoring on the most recently poured floor, forming the next floor, setting of
reinforcement, and placing concrete in the forms. Because the maturity of the floor below
the floor concrete being placed is only 3 to 14 days old, the concrete may not have
attained sufficient strength to carry loads as great as those imposed during construction
(Lie et al, 1989). It is common to leave formwork support in place between floors and
one or more floors below the recently placed floor. In order to reduce the imposed load
during removal process of formwork, shoring and reshoring processes are employed.
Typically one or two levels of shoring and one or more levels of reshoring are involved in
the process. Once the shores are removed from beneath a floor slab while allowing the
floor to deflect and carry its own weight, reshores are installed in order to allow the loads
153
to be shared by previously cast floors during concrete pouring (Chen and Mosallam,
1991).
The construction loads applied to the floors, shores, and reshores are decided
according to the number of levels of shoring, reshoring and sequence of stripping the
shores and reshores. It is economical when a small number of floors of shoring and
reshoring are used. However, it may be impossible to select a single general shoring and
reshoring process in the construction of multistory building because the procedures differ
from one construction project to another (Hurd, 1995; Rosowsky and Stewart, 2001).
Although a large number of shoring and reshoring procedures and combinations have
been used in the construction of multistory building, the most widely used are three levels
of shoring (3S) and two levels of shoring and one level of reshoring (2S1R) (Rosowsky
and Stewart, 2001).
The schematic loading history during construction and in service of slab in a
multistory building is shown in Figure 5-14. During construction, the load on the slab
increases as new floors are placed above. The construction load increases or decreases
along with shoring and reshoring procedure. When the shoring and reshoring is removed,
the floor supports its own weight and additional superimposed dead load as well as live
loads in service life of slab. A simple procedure to determine slab loads during
construction was suggested by Grundy and Kabalia (1963). The recent construction
model by Chen and Mosallam (1991) may be the most realistic and adopted by Rosowsky
and Stewart (2001) recently. In this research the simplified loading history recommended
by Graham and Scanlon (1986) and construction loading history model adopted from
Chen and Mosallam (1991) are adopted. Figure 5-15 shows the simplified load-time
154
history suggested by Graham and Scanlon. Figure 5-16 and Figure 5-17 present the
construction load model for 3S and 2S1R presented by Rosowsky and Stewart.
In the parametric study two kinds of simplified loading histories are assumed.
First case is that simplified loading history consists of a load of 2.59D applied at 28 days
because it is assumed that construction cycle of multistory building is 7 days. The
maximum construction load is calculated by Eq. 5.2 which is specified in ACI 435R
(ACI Committee 435, 1995).
where,
k1 =k2 =1.1
R = 2.0, N = number of shoring and reshoring levels,
CLw = construction live load, 50 psf (SEI/ASCE 37-02, 2002)
The load is then reduced to 1.35D and remains constant thereafter. The second simplified
loading history consists of a load of 2.03D calculated from dead loads plus live loads.
Then the load is reduced to 1.35D and then held constant.
In the case of 2S1R, a load of 0.93DL is applied at 7 days and remains constant
up to 14 days at which a load of 1.84DL applied instantaneously. The load of 1.84DL
holds constant to 21 days at which the reduced to 1.36DL and remains to 28 days. The
sustained load of 1.35 is then remains constant for 5 years.
For the case of 3S, a load of 1.19DL is applied instantaneously at 7 days and
holds constant to 14 days at which a value of 1.36DL is applied and holds constant to 21
Nw
Rwkkw CLslabconst += 21 5.2
155
days. The load of 1.45DL is applied at 21 days and remains constant to 28 days at which
the sustained load of 1.35DL applied immediately and then remains to 5 years.
According to analysis the maximum load occurs at 0.458L from the corner of free
edge. Figure 5-18 shows the location of maximum deflection. Load-time history is
obtained from this location for parametric study. Load-time history according to
construction loading method is shown in Figure 5-19.
5.4.2 Slab Thickness
Slab thicknesses are determined from the design of slab for exterior panel. 8” of
minimum thickness of slab for exterior panel without interior beam is decided. 10% of
variations are chosen for parametric study. Therefore, 7, 8, and 9 inches of thicknesses
are used in ordinary loading condition. For the given thicknesses, a simplified loading
history (D+L) based on ordinary loading condition is used.
Figure 5-20 shows time-deflection response for given thicknesses. Although the
sustained load of 9” thick slab is biggest, final deflection after 5 years is smallest. Long-
term deflection also is less affected by creep and shrinkage. This is because the flexural
stiffness is greater than the other two cases.
5.4.3 Column Stiffness
The column stiffness affects the restraint of shrinkage in the slab. In order to
express the in-plane boundary conditions of the slab, the column stiffness is varied. The
156
column stiffness is controlled using the modulus of elasticity. In this research the
behavior of column is assumed to be elastic. The elastic modulus is assumed to be same
as for the slab (equal to 6106.3 × psi for a specified compressive strength of 4000 psi).
The column modulus of elasticity for the column is varied from one half to four times the
design value to represent a range of column stiffness values.
Figure 5-21 shows the time-deflection history for varying column stiffness. The
result shows that the column stiffness affects immediate deflection significantly. As the
column stiffness increase, the immediate deflections are getting smaller. This is because
the rotation at connection of slab and column become flexible as the column stiffness
decreases. The long-term deflections are also affected by column stiffness. If the flexural
stiffness of column is strong, the long-term deflection increases due to in-plane shrinkage
restraint.
5.4.4 Separation of Creep and Shrinkage Effect
For creep and shrinkage, model GL2000 (Gardner and Lockman, 2001) is adopted
in this study. It is assumed that relative humidity is 40%, Type I cement is used, 28 days
specified compressive strength of 4000 psi, age of concrete at loading is 7 days, age of
concrete at the beginning of shrinkage is 7 days, and moist curing method is used. The
simplified loading history (D+L) based on ordinary loading condition is used. Figure 5-
22 shows that the effect of creep is larger than that of shrinkage. It is noted that the effect
of shrinkage rapidly increases after 800 days.
157
5.4.5 Elastic and Nonlinear Analysis
In order to investigate the effect of material nonlinearity the parametric study is
performed under assumption of elastic and nonlinear behavior of slab. Elastic behavior is
based on no cracking under short and long-term loading. On the other hand, nonlinear
analysis allows slab to have the effect of cracks and stiffness degradation. As a result, the
deflection of slab is higher than that of slab under elastic behavior. Figure 5-23 shows
the comparison between elastic analysis and nonlinear analysis in terms of deflection
history. The graph shows that deflection of nonlinear analysis is much higher than that of
elastic analysis as expected.
5.4.6 Extraordinary Superimposed Loading
Current ACI 318 code specifies minimum thicknesses for two-way slabs as a form
of the longest span divided by a coefficient which varies according to the existence of
drop panels, interior or exterior panels. For exterior panel, the coefficient varies
according to the presence of an edge beam. The minimum thickness is independent of
loading. According to Bondy (2005) the podium slab which is commonly constructed in
California can experience superimposed loading ranging from 100 to 200 psf, and 80
to150 psf of live load. Therefore, the minimum thickness specified in ACI 318 may not
be appropriate to establish the design thickness. Scanlon and Lee (2006) suggested
equation which can predict minimum thickness according to various given conditions
such as loading condition, boundary condition, and geometry of slab. In the present study,
158
minimum thicknesses are obtained from ACI 318 and Scanlon and Lee’s equation. The
obtained thicknesses are 8” and 11” respectively. Simplified loading history (D+L) with
live load of 150psf and sustained load of 200 psf is used. Figure 5-24 shows deflection
history. The deflection history of minimum thickness by ACI 318 is much higher than
that of minimum thickness by Scanlon and Lee as expected.
5.4.7 Age of Application of Loading
The parametric study is performed to investigate the effect of age of loading. The
analysis is performed while varying age of loading at 7, 14, 21, and 28 days. Also, the
shrinkage restraint occurring before loading is investigated. Slab thickness of 8” is used
to perform the parametric study. Simplified loading (D+L) with ordinary loading
condition is assumed. At 28 days maximum load which consists of dead loads plus live
load is applied followed by the sustained load. In the study, it is assumed that shrinkage
starts at 7 days. Therefore, there is a time gap between the day shrinkage begins and
initial loading is applied. Figure 5-25 shows deflection history without shrinkage
restraint, and deflection history with shrinkage restraint is presented in Figure 5-26.
Individual comparisons between with and without shrinkage restraint for different ages of
loading are presented in Figure 5-27 to 5-29.
159
5.5 Long-Term Multiplier
ACI 318 specifies a long-term multiplier to calculate long-term deflection which
is determined by multiplying the immediate deflection caused by the sustained load by
the multiplier. Additional long-term deflection of one-way and two-way slab system is
calculated in accordance with 9.5.2.5. However, the multiplier specified in ACI318 does
not include the effect of age of loading and shrinkage restraint before loading.
Results of the parametric study based on the simplified loading history are used to
calculate the long-term multiplier as a function of age of application of loading. Because
deflection histories are obtained from parametric study, the effective long-term multiplier
is simply obtained by Eq. 5.3.
where,
tλ =long-term multiplier
tΔ =long-term deflection
iΔ =instantaneous deflection due to sustained load
Long-term multiplier is calculated at age 5 years in the parametric study.
The values of long-term multiplier are summarized in Table 5-10. The values are
calculated with and without shrinkage restraint occurred before loading. Figure 5-30
shows long-term multiplier after 5 years along with application of loading. The long-term
i
itt Δ
Δ−Δ=λ 5.3
160
multiplier tλ is calculated for every year up to 5 years for loading age of 7, 14, 21, and 28
days.
The values of long-term multiplier without shrinkage restraint indicate that long-
term multiplier tends to be converged as loading is increasing. The effect of age of
loading becomes insignificant when the loading is applied after 14 days. Similar tendency
is observed when the shrinkage restraint is introduced in the analysis. The effect of
loading age is not significant after 14 days. Although Shrinkage restraint makes the
immediate deflection increase, the values of long-term multiplier are smaller than those
of long-term multiplier without shrinkage restraint. This is because of the increase of
immediate deflection. Immediate deflection is increased due to reduced tensile strength
by shrinkage, but the increased immediate deflection reduced the values of multiplier in
the calculation. This means the increase of long-term deflection is not as high as that of
immediate deflection. From the results, it may be concluded that the long-term multiplier
is affected by age of loading when the loading is applied at less than 14 days.
5.6 Moment Variation
In order to demonstrate moment variation at the middle and column strip, the E-W
direction moment ( 11M ) is obtained from the exterior column line, intermediate line,
interior column line, and longitudinal line. Figure 5-31 shows nodes from which the
values of moment are obtained. The origin of slab is lower left corner. The moment
variation is obtained from elastic analysis and nonlinear analysis. Elastic analysis and
nonlinear analysis represent the presence of cracks in the slab. Also, in order to
161
investigate the effect of shrinkage, the moment variations are compared with and without
shrinkage restraint occurred before loading is presented. It is known that the shrinkage
induces in-plane action in the slab, so that the moment variation may be affected by that.
Under total load (D+L), the moment diagrams of uncracked, cracked, and cracked with
shrinkage restraint are presented in Figure 5-32 to 5-35.
Also, unfactored moments based on direct design method are presented in Figures.
In the moment diagram at the exterior column line, the negative moment occurs in the
vicinity of exterior and interior column. The positive moment at the middle strip is near
zero. Cracking and shrinkage restraint increases negative moment in the vicinity of
column, but there is not much change in the middle strip. In the moment diagram at the
intermediate line, the positive moment at the middle strip is lower than at the interior and
exterior column strip. The positive moment in the interior column strip is increased
slightly by cracking and shrinkage restraint, but the positive moment is decrease in the
middle and exterior column strip. In the moment diagram at the interior column line, the
negative moment is decreased by cracking and shrinkage restraint at the middle and
column strips generally. In the longitudinal direction, the moment in the vicinity of the
exterior column is increased but the moment in the vicinity of interior column is
decreased due to cracking and shrinkage restraint.
In order to investigate time-dependent effect on the values of moment, time-dependent
moment diagrams for sustained load are plotted while changing the time frame for 28
days and 5 years. Time-dependent moment diagrams are shown in Figure 5-36 to 5-38.
162
The time-dependent effect increases the negative moments, but decreased the positive
moment at the exterior column line, intermediate, and interior column line.
It is noted that the discontinuities in the moment diagram are due to different
amount of reinforcement in the column and middle strip, so that the values of moment
can be different at the integration points of shell element.
5.7 Summary
A parametric study of several factors which affect deflections of flat plate floor
system is described in this chapter. The parameters considered are summarized in
Table 5-9.The study is performed for a square exterior panel with uniformly distributed
loading. Although the construction load history is complex according to construction
method, the construction load models are simplified in order to estimate the long-term
deflection of slab system. The sensitivity of slab thickness is investigated while varying
the thickness. Also in order to investigate in-plane action due to shrinkage, the column
stiffness is varied. The long-term deflection is investigated while separating or combining
creep and shrinkage effect. Finally, the effect of age of loading is investigated while
considering shrinkage restraint.
Long-term multiplier was calculated based on parametric study. The values are
obtained with and without shrinkage restraint before loading. The shrinkage restraint
increased immediate and long-term deflections. The moment diagram along loading stage
was investigated. The moment diagram was obtained from linear elastic and nonlinear
163
analysis, and compared. As well, the effect of shrinkage on the moment diagram was
investigated.
164
Table 5-1: Given Design Loads
thickness (inches)
self-weight (psf)
superimposed dead load
(psf)
partition and finishes
(psf)
live load (psf)
7.0 87.5 8.0 100.0 9.0 112.5
15 20 70
Table 5-2: Given Design Loads Based on Extraordinary Superimposed Dead Load
thickness (inches)
self-weight (psf)
superimposed dead load
(psf)
live load (psf)
8.0 100.0 11.0 137.5 200 150
165
Table 5-3: Amount of Reinforcements (Ordinary Loading) : E-W direction
(a) Negative Moments at Exterior End of End Span
Strip Strip Length (in) Index
Slab thickness
(in)
Design Moment
(k-ft)
Provided sA(in2/in)
7 -37.66 0.023 8 -39.84 0.023
Edge Column
Strip 66 REETX
9 -42.02 0.019 7 0. 0.014 8 0. 0.016 Middle
Strip 132 RMETX 9 0. 0.016 7 -75.31 0.023 8 -79.68 0.021 Column
Strip 132 RIETX 9 -84.04 0.019
(b) Positive Moments in End Span
Strip Strip Length (in) Index
Slab thickness
(in)
Design Moment
(k-ft)
Provided sA(in2/in)
7 45.19 0.028 8 47.81 0.028
Edge Column
Strip 66 REMBX
9 50.42 0.023 7 60.25 0.019 8 63.74 0.016 Middle
Strip 132 RMMBX9 67.23 0.016 7 90.38 0.028 8 95.61 0.026 Column
Strip 132 RIMBX 9 100.84 0.023
(c) Negative Moments at Interior End of End Span
Strip Strip Length (in) Index
Slab thickness (in)
Design Moment
(k-ft)
Provided sA(in2/in)
7 -76.04 0.047 8 -80.44 0.042
Edge Column
Strip 66 REITX
9 -84.85 0.038 7 -50.69 0.016 8 -53.63 0.016 Middle
Strip 132 RMITX 9 -56.56 0.016 7 -152.08 0.049 8 -160.88 0.040 Column
Strip 132 RIITX 9 -169.69 0.038
166
Table 5-4: Amount of Reinforcements (Ordinary Loading) : N-S direction
(a) Negative Moments at Exterior End of End Span
Strip Strip Length (in) Index
Slab thickness
(in)
Design Moment
(k-ft)
Provided sA(in2/in)
7 -37.66 0.028 8 -39.84 0.023
Edge Column
Strip 66 REETX
9 -42.02 0.023 7 0. 0.014 8 0. 0.016 Middle
Strip 132 RMETX 9 0. 0.016 7 -75.31 0.026 8 -79.68 0.023 Column
Strip 132 RIETX 9 -84.04 0.021
(b) Positive Moments in End Span
Strip Strip Length (in) Index
Slab thickness
(in)
Design Moment
(k-ft)
Provided sA(in2/in)
7 45.19 0.033 8 47.81 0.028
Edge Column
Strip 66 REMBY
9 50.42 0.028 7 60.25 0.021 8 63.74 0.019 Middle
Strip 132 RMMBY9 67.23 0.016 7 90.38 0.031 8 95.61 0.028 Column
Strip 132 RIMBY 9 100.84 0.026
(c) Negative Moments at Interior End of End Span
Strip Strip Length (in) Index
Slab thickness
(in)
Design Moment
(k-ft)
Provided sA(in2/in)
7 -76.04 0.056 8 -80.44 0.047
Edge Column
Strip 66
REITY
9 -84.85 0.042 7 -50.69 0.019 8 -53.63 0.016 Middle
Strip 132 RMITY
9 -56.56 0.016 7 -152.08 0.054 8 -160.88 0.045 Column
Strip 132 RIITY
9 -169.69 0.040
167
Table 5-5: Amount of Reinforcements(Extraordinary Loading) : E-W direction
(a) Negative Moments at Exterior End of End Span
Strip Strip
Length (in)
Index Slab
thickness (in)
Design Moment
(k-ft)
Provided sA(in2/in)
8 -87.24 0.047 Edge Column Strip 66 REETX
11 -93.78 0.033 8 0.00 0.016 Middle Strip 132 RMETX
11 0.00 0.021 8 -174.47 0.049 Column Strip 132 RIETX
11 -187.56 0.033 (b) Positive Moments in End Span
Strip Strip
Length (in)
Index Slab
thickness (in)
Design Moment
(k-ft)
Provided sA(in2/in)
8 104.68 0.056 Edge Column Strip 66 REMBX 11 112.53 0.042
8 139.58 0.038 Middle Strip 132 RMMBX 11 150.05 0.028 8 209.37 0.056 Column Strip 132 RIMBX 11 225.07 0.040
(c) Negative Moments at Interior End of End Span
Strip Strip
Length (in)
Index Slab
thickness (in)
Design Moment
(k-ft)
Provided sA(in2/in)
8 -176.15 0.099 Edge Column Strip 66 REITX 11 -189.36 0.070
8 -117.43 0.033 Middle Strip 132 RMITX 11 -126.24 0.023 Column Strip 132 RIITX 8 -325.30 0.099
168
Table 5-6: Amount of Reinforcements(Extraordinary Loading) : N-S direction
(a) Negative Moments at Exterior End of End Span
Strip Strip
Length (in)
Index Slab
thickness (in)
Design Moment
(k-ft)
Provided sA(in2/in)
8 -87.24 0.053 Edge Column Strip 66 REETY 11 -93.78 0.038
8 0.00 0.016 Middle Strip 132 RMETY 11 0.00 0.021 8 -174.47 0.049 Column Strip 132 RIETY 11 -187.56 0.035
(b) Positive Moments in End Span
Strip Strip
Length (in)
Index Slab
thickness (in)
Design Moment
(k-ft)
Provided sA(in2/in)
8 104.68 0.061 Edge Column Strip 66 REMBY 11 112.53 0.047
8 139.58 0.040 Middle Strip 132 RMMBY 11 150.05 0.028 8 209.37 0.061 Column Strip 132 RIMBY 11 225.07 0.045
(c) Negative Moments at Interior End of End Span
Strip Strip
Length (in)
Index Slab
thickness (in)
Design Moment
(k-ft)
Provided sA(in2/in)
8 -176.15 0.113 Edge Column Strip 66 REITY 11 -189.36 0.075
8 -117.43 0.033 Middle Strip 132 RMITY 11 -126.24 0.023 Column Strip 132 RIITY 8 -325.30 0.113
Table 5-7: Material Properties of Concrete for Parametric Study
'cf
(psi) cE
(psi) crf
(psi) ν
Input 4000 6106.3 × 474.3 0.15
169
Table 5-8: Input Values for Creep and Shrinkage in Parametric Study
Models Parameters GL2000
Relative Humidity, H (%) 40
t=7” t=8” t=9” t=11” Volume-to-Surface Ratio, sv / ( in ) 3.32 3.77 4.21 5.08
Age of concrete at loading, oct (days)
7,14, 21, and 28 days
28 days specified concrete strength, 28ckf
(psi)
4000
Age of concrete at the beginning of shrinkage, ost (days) 7 days
Cement type Type I (normal hardening concrete)
170
Table 5-9: Parameters
Case No. of Variables Variables Constant
1 LH1 Simplified loading(D+L)
2 LH2 Simplified loading(ACI 435R)
3 LH3 2R1S 4 LH4 3S
h=8” Column E=average
5 TH1 h=7” 6 TH2 h=8” 7 TH3 h=9”
Simplified Loading(D+L) Column E=average
8 CM1 E=small 9 CM2 E=average
10 CM3 E=high
h=8” Simplified Loading(D+L)
11 LE1 creep only 12 LE2 shrinkage only 13 LE3 creep & shrinkage
h=8” Simplified Loading(D+L)
Column E=average
14 AN1 Elastic Analysis
15 AN2 Nonlinear Analysis
h=8” Simplified Loading(D+L)
Column E=average
16 MT1 minimum thickness by ACI 318, h=8”
17 MT2 minimum thickness Scanlon & Lee, h=11”
Extraordinary superimposed Loading
simplified Loading(D+L) Column E=average
18 LA1 loading at 7 19 LA2 loading at 14 20 LA3 loading at 21 21 LA4 loading at 28
W/O shrinkage restraint h=8”
Simplified Loading (D+L) Column E=average
22 LAS1 loading at 14 23 LAS2 loading at 21 24 LAS3 loading at 28
W/ shrinkage restraint h=8”
Simplified Loading (D+L) Column E=average
171
Table 5-10: Long-Term Multiplier
(a) Without Shrinkage Restraint
year
age of loading
1 2 3 4 5
7 1.12 1.35 1.49 1.59 1.65 14 0.85 1.08 1.26 1.38 1.46 21 0.83 1.08 1.25 1.39 1.48 28 0.81 1.05 1.23 1.38 1.48
(b) With Shrinkage Restraint
year
age of loading
1 2 3 4 5
7 1.12 1.35 1.49 1.59 1.65 14 0.74 0.97 1.11 1.20 1.28 21 0.64 0.89 1.03 1.13 1.21 28 0.60 0.84 0.99 1.09 1.17
172
22' 22' 22'
22'
22'
22'
A B C D
1
2
3
4
exterior panel 22" x 22"
column
22" x 22" column
7"8" thickness9"
Figure 5-1: Plan of Flat Plate System
173
M0
Positive Moment0.52 M0
Negative Moment at Interiro end
-0.70 M0
Negative Moment at exterior end
-0.26 M0
Exterior end column strip
negative moment
1.0 x -0.26 M0
Exterior end middle strip
negative moment
0. x -0.26 M0
Column strip Postive Moment0.6 x 0.52 M0
Middle Strip postive moment0.4 x 0.52 M0
Interior end column strip
negative moment
0.75 x -0.70 M0
Interior end middle strip
negative moment0.25 x -0.70 M0
(from edge column strip to middle strip)
0.125 x -0.70 M0
Figure 5-2: Distribution of Total Moment in the Exterior Panel
174
exteriorcolumn
strip
middlestrip
interiorcolumn
strip
exteriorcolumn
strip
middlestrip
interiorcolumn
strip
REETX REITXREMBX
RMMBX
RIMBX
RMETX
RIETX
RMITX
RIITX
(a) E-W Direction
REETY
REMBY
REITY
RMETY
RMMBY
RMITY
RIETY
RIMBY
RIITY
(b) N-S Direction
Figure 5-3: Schematic Reinforcement Lay-out
175
E-W reinforcement
N-S reinforcement
t=7", 8", 9", and 11"
clear cover=3/4" Used rebar: #5 (diameter=0.625”)
thickness(in) 7 8 9 11 E-W N-S E-W N-S E-W N-S E-W N-S depth(in) 6.19 5.56 6.94 6.31 7.94 7.31 9.94 9.31 Figure 5-4: Arrangement of Reinforcement in Slab
Figure 5-5: Mesh of Finite Element Model
176
0
100
200
300
400
500
0 0.0005 0.001 0.0015 0.002 0.0025 0.003Strain(in/in)
Stre
ss(p
si)
Figure 5-6: Assumed Tension Stiffening Model
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
1 10 100 1000 10000
Time(days)
Cre
ep C
oeffi
cien
t
V/S=3.32 (t=7 in)
V/S=3.77 (t=8 in)
V/S=4.21 (t=9 in)V/S=5.08 (t=11 in)
Figure 5-7: Creep Coefficient for Loading at 7 Days
177
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
1 10 100 1000 10000
Time(days)
Cre
ep C
oeffi
cien
t
V/S=3.32 (t=7 in)
V/S=3.77 (t=8 in)
V/S=4.21 (t=9 in)V/S=5.08 (t=11 in)
Figure 5-8: Creep Coefficient for Loading at 14 Days
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
1 10 100 1000 10000
Time(days)
Cre
ep C
oeffi
cien
t
V/S=3.32 (t=7 in)
V/S=3.77 (t=8 in)
V/S=4.21 (t=9 in)V/S=5.08 (t=11 in)
Figure 5-9: Creep Coefficient for Loading at 21 Days
178
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
1 10 100 1000 10000
Time(days)
Cre
ep C
oeffi
cien
t
V/S=3.32 (t=7 in)
V/S=3.77 (t=8 in)
V/S=4.21 (t=9 in)V/S=5.08 (t=11 in)
Figure 5-10: Creep Coefficient for Loading at 28 Days
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
1 10 100 1000 10000
Time(days)
Cre
ep c
oeffi
cien
t
7 days
14 days
21 days
28 days
Figure 5-11: Creep Coefficients along Age of Loading of 8 in Thick Slab
179
0
100
200
300
400
500
600
700
800
1 10 100 1000 10000
Time(days)
Stra
in (m
icro
stra
in)
V/S=3.32 (t=7 in)
V/S=3.77 (t=8 in)
V/S=4.21 (t=9 in)
V/S=5.08 (t=11 in)
Figure 5-12: Shrinkage Model Based on GL2000 for Parametric Study
180
B E
GD C H
NM
O
I J
L
C/2=11"
A
24'
F
KP1
2
1
3
Q
S
R
T
Figure 5-13: Boundary Condition
BOUNDARY CONDTIONS:
HI, HK= 02
=∂∂
∂=
∂∂
=∂∂
yxw
yw
xw
FGIJ= 02
=∂∂
∂=
∂∂
=yxw
xwu
JKNO= 02
=∂∂
∂=
∂∂
=yxw
ywv
P, Q, R, S=Fixed (6 DOF)
181
ConstructionInstallation of Non-structural Elements
Load
Timet1 t2 t3
wsust
wL
Figure 5-14: Schematic Load-Time History
182
(a) Simplified loading history (ACI 435R)
Time(days) 28 28 1825 Load ratio 2.59* 1.35** 1.35**
Applied Load (psf) 259 135 135
DL=100 psf LL=70 psf SDL=15 psf Partition and finishes=20 psf *maximum load during construction, constw :
slab
slab
CLslabconst
w
psfw
Nw
Rwkkw
59.23
50)0.2)(1.1)(1.1(
21
=
+=
+=
(from ACI 435R)
where, CLw construction live load, 50 psf (from ASCE) **sustained load=DL + SDL + Partition and finishes= 135 psf=1.35DL (b) Simplified loading history (D+L)
Time(days) 28 28 1825 Load ratio 2.03 1.35 1.35
Applied Load (psf) 203 135 135
Load
Timet1 t2 t3
wsust
wL
wconst
Figure 5-15: Simplified Load-Time History in Accordance with ACI 435R
183
Time(days) 7 14 14 21 21 28 28 1825 Load ratio 0.93 0.93 1.84 1.84 1.36 1.36 1.35* 1.35*
Applied Load (psf) 93 93 184 184 136 136 135 135
DL=100 psf LL=70 psf SDL=15 psf Partition and finishes=20 psf *sustained load=DL + SDL + Partition and finishes= 135 psf=1.35DL
0
0.5
1
1.5
2
2.5
0 7 14 21 28 35Time since floor placement (days)
Max
imum
sla
b lo
ad ra
tio
3,4 5
6
7,8
9 10
11,1213
14
0.93
1.27
1.841.76
1.00
1.36
1.27
1.00
Figure 5-16: Maximum Slab Load Ratio for 2S1R for 7 Days of Construction Cycle (Rosowsky and Stewart, 2001)
184
Time(days) 7 14 14 21 21 28 28 1825 Load ratio 1.19 1.19 1.36 1.36 1.45 1.45 1.35* 1.35*
Applied Load (psf) 119 119 136 136 145 145 135 135
DL=100 psf LL=70 psf SDL=15 psf Partition and finishes=20 psf *sustained load=DL + SDL + Partition and finishes= 135 psf=1.35DL
0
0.5
1
1.5
2
2.5
0 7 14 21 28 35
Time since floor placement (days)
Max
imum
sla
b lo
ad ra
tio
4
5 6,7
89,10
11
12
0.89
1.19
1.36
1.02 1.09 1.00
1.10
1.45
Figure 5-17: Maximum Slab Load Ratio for 3S for 7 Days of Construction Cycle (Rosowsky and Stewart, 2001)
185
L=22 ft
Figure 5-18: The Location of Maximum Deflection of Exterior Panel
0.458L
186
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 10 100 1000 10000Time (days)
Def
lect
ion
(in)
Simplified Loading(ACI 435R)2S1R3SSimplified Loading(D+L)
Figure 5-19: Time-Deflection for Given Loading Histories (case: 1, 2, 3, and 4)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1 10 100 1000 10000Time(days)
Def
lect
ion(
in)
h=7"
h=8"
h=9"
Figure 5-20: Time-Deflection for Given Slab Thicknesses (case 5, 6, and 7)
187
0
0.2
0.4
0.6
0.8
1
1.2
1 10 100 1000 10000Time(days)
Def
lect
ion(
in)
Ecol=0.5Eslab
Ecol=Eslab
Ecol=4Eslab
Figure 5-21: Time-Deflection for Given Column Stiffness (case 8, 9, and 10)
0
0.2
0.4
0.6
0.8
1
1.2
1 10 100 1000 10000Time(days)
Def
lect
ion
(in)
Creep+ShrinkageCreep onlyShrinkage only
Figure 5-22: Separation of Creep and Shrinkage (case: 11, 12, and 13)
188
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 10 100 1000 10000Time (days)
Def
lect
ion
(in)
Nonlinear
Elastic
Figure 5-23: Comparison Between Elastic Analysis and Nonlinear Analysis (Case: 14 and 15)
0
0.5
1
1.5
2
2.5
3
1 10 100 1000 10000Time (days)
Def
lect
ion
(in)
ACI 318 (t=8 in)
Scanlon & Lee (t=11 in)
Figure 5-24: Extraordinary Loading Condition with Minimum Thicknesses (Case: 16 and 17)
189
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1 10 100 1000 10000Time (days)
Def
lect
ion
(in)
7 days
14 days(W/O Shrinage Restraint)
21 days(W/O Shrinkage Restraint)28 days(W/O Shrinkage Restraint)
Figure 5-25: Effect of Age of Loading without Shrinkage Restraint (Case: 18, 19, 20, and 21)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1 10 100 1000 10000
Time (days)
Def
lect
ion
(in)
7 days14 days(W/ Shrinkage Restraint)
21 days(W/ Shrinkage Restraint)28 day(W/Shrinkage Restraint)
Figure 5-26: Effect of Age of Loading with Shrinkage Restraint(Case: 17, 22, 23, and 24)
190
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 10 100 1000 10000Time (days)
Def
lect
ion
(in)
14 days
14 days with shrinkage restraint
Figure 5-27: Loading at 14 Days
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 10 100 1000 10000Time (days)
Def
lect
ion
(in)
21days
21 days with shrinkage restraint
Figure 5-28: Loading at 21 Days
191
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1 10 100 1000 10000
Time (days)
Def
lect
ion
(in)
28days
28 days with shrinkage restraint
Figure 5-29: Loading at 28 Days
0.00
0.50
1.00
1.50
2.00
2.50
0 5 10 15 20 25 30
Age of Application of Loading(days)
Long
-term
Mul
tiplie
r afte
r 5 y
rs
Multipler W/O shrinkage restraint
Multiplier W/ shrinkage restraint
Figure 5-30: Long-Term Multiplier Along with Age of Loading
192
Figure 5-31: The Line of the Column Face and the Center Line of Panel
-M11
Origin
E
The longitudinal line
-M11
The intermediate line
The interior column line
W
The exterior line
193
-15000
-10000
-5000
0
5000
10000
15000
0 50 100 150 200Distance (in)
M11
(in lb
/in)
D+L (uncracked)D+L (cracked)D+L (cracked + sh)Unfactored Moment
Figure 5-32: Moment Diagram at the Exterior Column Line
-9000
-8000
-7000
-6000
-5000
-4000
-3000
0 50 100 150 200 250Distance (in)
M11
(in lb
/in)
D+L (uncracked)D+L (cracked)D+L (cracked + sh)Unfactored Moment
Figure 5-33: Moment Diagram at the Intermediate Line
194
-1000
1000
3000
5000
7000
9000
11000
13000
15000
0 50 100 150 200Distance (in)
M11
(in lb
/in)
D+L (uncracked)
D+L (cracked)
D+L (cracked + sh)
Unfactored Moment
Figure 5-34: Moment Diagram at the Exterior Column Line
-15000
-10000
-5000
0
5000
10000
15000
20000
25000
30000
0 50 100 150 200Distance (in)
M11
(in lb
/in)
D+L (uncracked)
D+L (cracked)D+L (cracked + sh)
Unfactored Moment
Figure 5-35: Moment Diagram at the Longitudinal Line
195
-10000
-8000
-6000
-4000
-2000
0
2000
4000
6000
8000
0 50 100 150 200Distance (in)
M11
(in lb
/in)
D (28 days)D (5 years)
Figure 5-36: Time-dependent Moment Diagram at the Exterior Column Line
-9000
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
0 50 100 150 200Distance (in)
M11
(in lb
/in)
D (28 days)D (5 years)
Figure 5-37: Time-dependent Moment Diagram at the Intermediate Line
196
-9000
-4000
1000
6000
11000
16000
0 50 100 150 200Distance (in)
M11
(in lb
/in)
D (28 days)D (5 years)
Figure 5-38: Time-dependent Moment Diagram at the Interior Column Line
Chapter 6
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
6.1 Summary
A combined experimental and numerical investigation of reinforced concrete
slabs at service load level has been presented. Time-dependent concrete properties are
incorporated in the commercial finite element program, ABAQUS through user-defined
subroutines. The developed model was verified against experiment results obtained in the
present study as well as those reported in the literature.
Time dependent concrete material model is developed based on an orthotropic
model. The time-dependent effect of concrete is incorporated in the model to consider
creep and shrinkage. The developed algorithm is implemented in the commercial finite
element program ABAQUS using user-defined material FORTRAN subroutine. Creep
and shrinkage algorithm based on ACI 209, GL2000, and CEB-FIP models is applied in
the model. Tension stiffening model is adopted to simulate post-cracking behavior of
reinforced concrete. In the model concrete and reinforcement are assumed to be
equivalent layers.
Test on one-way slab specimens was performed to investigate early-age loading
which has an effect on immediate and long-term deflection of slab. Early-age loading can
occur during construction and the loading may cause unexpected cracking in the slab
system as well as increased time dependent deformation due to creep. The slab could
198
experience excessive deflection during service life of structure. In order to simulate the
construction load, a simplified loading method was assumed in the experiment. The total
load (D+L) was applied at 3, 7, and 28 days respectively. After removing live load, the
sustained loading due to self weight was maintained over time. Total nine slabs designed
in accordance with ACI 318 were used in the experiment. In order to get material
properties, cylinder test for compressive strength, splitting tensile strength and modulus
of elasticity was obtained. The development of compressive strength with time shows
good agreement with predictions using existing models. According to load-deflection
response and time-deflection histories of tested slabs, although the immediate deflection
of 28 days was higher than that of loading at 3 and 7 days due to shrinkage restraint
before loading, early-age loading at 3 and 7 days produced higher long-term deflection
than that of loading at 28 days.
The developed concrete model was verified against existing test data of Scott and
Beeby (2005), McNeice (1971), Burns and Hemakom (1985), Gilbert and Guo (2005),
and the present experimental study. Immediate and long-term deflection of analysis
results were compared with experimental results. The analysis results showed good
correlation with experimental results for immediate deflection, and for long-term
deflections when project specific creep and shrinkage data were available. However, the
long-term deflection based on creep and shrinkage models selected for the study showed
poor correlation for the case of loading at 3 and 7 days.
A parametric study was performed to investigate the effect of various factors such
as loading history, thickness of slab, column stiffness, separation of creep and shrinkage
effect, comparison between elastic and nonlinear analysis, extraordinary superimposed
199
load, age of loading, and shrinkage restraint. The study was limited to the exterior panel
of multi-panel system, which was designed in accordance with the direct design method
specified in ACI 318. In the study Damjanic and Owen tension stiffening model and
GL2000 creep and shrinkage models were used. Long-term multipliers and the moment
diagrams at the exterior column and intermediate and interior column line were obtained
and compared based on the results of the parametric study.
6.2 Conclusions
Based on the results of this combined experimental and Analytical study the
following conclusions can be drawn.
1. Slab specimens subjected to drying shrinkage crack at a lower applied
load than those loaded immediately after the end of curing, even when
loaded at a later age when the concrete strength has increased. Tensile
stresses induced by shrinkage restraint due to embedded bars are
considered to be the primary contributing factor. The effect can be
expected to be even more significant for full-scale structures because of
other sources of in-plane restraint such as stiff walls and columns.
Shrinkage restraint should therefore be considered when calculating
deflection of lightly reinforced members.
2. Members loaded at early age (3 and 7 days) show significantly larger
long time deflections than those loaded at 28 days. Higher long time
200
multipliers than currently specified in ACI 318 should therefore be
considered for members loaded at early age.
3. Branson’s effective moment of inertia expression tended to
underestimate immediate deflections while Bischoff’s expression
tended to overestimate immediate deflections. However Bischoff’s
expression provided significantly better comparisons with measured
residual deflections on unloading.
4. The time dependent algorithm based on the rate of creep method
implemented in ABAQUS provided good correlation with test data
when measured creep and shrinkage data were available. Comparisons
with test data on axially loaded tension prisms indicated that the
algorithm is capable of modeling time-dependent behavior in tension
zones.
5. Results of the parametric study demonstrated the importance of
construction loading effects on deflection history for reinforced
concrete slabs.
6. Slab thickness should take into account the design loading particularly
in cases where design loads are higher than normal for building floor
systems.
201
6.3 Recommendations
Based on the results of this study the following recommendations are made.
1. Calculation of immediate deflection of slab systems should be made on
the basis of the Bischoff expression for effective moment of inertia
taking into account shrinkage restraint and construction loading.
2. Long term multiplier applied to immediate deflection should be
increased to account for early age loading for slabs subjected to
significant construction loading.
3. The analytical model developed for this study can be used to conduct
additional parametric studies for other slab systems, and for use in
simulations to investigate uncertainties in predicting slab deflections..
Available models for creep and shrinkage should be investigated to further
evaluate their applicability to early age loading.
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Appendix A
CREEP AND SHRINKAGE MODELS
A.1 ACI 209 Model (1992)
ACI model is based on the ACI committee 209 report. The following equations
are general form of creep and shrinkage equation (ACI 209, 1992).
-For shrinkage
-For creep
where,
fd , = in days
ψα , = constant for a given member shape and size
SHε = the ultimate shrinkage strain
uφ = the ultimate creep coefficient
t = the time after loading in shrinkage and time from the end of the initial curing
in creep
ASTM C 512 recommends the tests for obtaining values of ,,,,, fd u αφψ and SHε .
Normal range of constants is as follows.
SHsh tdtt εε α
α
⎥⎦
⎤⎢⎣
⎡+
=)( A.1
utftt φφ ψ
ψ
⎥⎦
⎤⎢⎣
⎡+
=)( A.2
210
ψ = 0.40 to 0.80
d = 6 to 30 days
uφ = 1.30 to 4.15
α = 0.90 to 1.10
f = 20 to 130 days
SHε = 610415 −× to 6101070 −× in./in.
ACI 209 model was developed based on normal weight, sand lightweight and all
lightweight concretes including curing condition and cement type. Concrete having a
compressive strength of approximately 5000 psi or less can be predicted according to
ACI 209 model.
Following equations are recommended equations for an unrestrained shrinkage
strain and a creep coefficient including ultimate values.
Shrinkage
Shrinkage after age 7 days for moist-cured concrete
Shrinkage after age 1~3 days for steam cured concrete
where SHε is Ultimate shrinkage strain, shγ610780 −× , sb
sac
sf
ss
sv
shsh KKKKKK=γ
SHsh tdaystt εε ⎥
⎦
⎤⎢⎣
⎡+
=35
)( A.3
SHsh tdaystt εε ⎥
⎦
⎤⎢⎣
⎡+
=55
)( A.4
211
Creep
where uφ is Ultimate creep coefficient, 2.35 crγ , cto
cac
cf
cs
cv
chcr KKKKKK=γ
The correction factors, shγ and crγ are used for conditions other than the standard
concrete composition, and summarized in Table A-1
uo
oo ttdays
tttt φφ ⎥
⎦
⎤⎢⎣
⎡−+
−= 6.0
6.0
)(10)(
),( A.5
Table A-1: Correction Factors for ACI 209 Model
Standard Condition
Creep correction factor Shrinkage correction factor
Relative Humidity,
H (%) %40=H
0.10067.027.1 ≤−= HK ch
%40>H
%8040 ≤< H 0.101.04.1 ≤−= HK s
h
%10080 ≤< H HK s
h 03.00.3 −= Volume-to-
Surface Ratio, sv / ( in )
insv 5.1/ = ]13.11)[3/2( /54.0 svcv eK −+=
svsv eK /12.02.1 −=
Slump, s ( in ) ins 7.2= sK cs 067.082.0 += sK s
s 041.089.0 +=
Fine aggregate Percentage,
F (%) %50=F FK c
f 0024.088.0 +=
%50≤F FK s
f 014.03.0 += %50≥F
FK sf 002.09.0 +=
Air content, A (%)
%6=A 0.109.046.0 ≥+= AK cac AK s
ac 008.095.0 +=
Age at loading,)(dayst
Steam cured: dayst 3=
Moist cured: dayst 7=
1~3 days Steam cured concrete
094.013.1 −= tK cto
7 days Moist cured concrete
118.025.1 −= tK cto
Cement content ( 3/ ydlb )
3/695 ydlbB = BK sb 00036.075.0 +=
212
A.2 CEB-FIP Model (fib, 1999)
The International Federation for Structural Concrete (fib, 1999) is recently
updated from CEB-FIP Model Code 90, and it includes the behavior of high-performance
concrete. Total shrinkage is decomposed into autogenous shrinkage and the drying
shrinkage component. Creep equation has been adjusted in order to take into account the
particular characteristics of high-performance concretes.
Shrinkage
Autogenous shrinkage, also called self-desiccation shrinkage or chemical
shrinkage and drying shrinkage are the most important type of shrinkage. The total
shrinkage may be calculated from following equations.
where,
),( scs ttε = total shrinkage at time t
)(tcasε = autogenous shrinkage at time t
),( scds ttε = drying shrinkage at time t
Autogenous shrinkage )(tcasε is estimated from following equation.
where,
)( cmcaso fε = notional autogenous shrinkage coefficient,
),()(),( scdscasscs ttttt εεε += A.6
)()()( tft ascmcasocas βεε ⋅= A.7
213
65.2
0
0 10/6
/)( −⋅⎟⎟
⎠
⎞⎜⎜⎝
⎛+
−=cmcm
cmcmascmcaso ff
fff αε
)(tasβ = function to describe the time development of autogenous shrinkage,
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−−=
5.0
1
2.0exp1)(tttasβ
cmf = mean compressive strength (MPa)
0cmf = 10 MPa
asα = coefficient which depends on the type of cement:
800=asα for slowly hardening cements
700=asα for normal or rapidly hardening cements
600=asα for rapidly hardening high-strength cements
t = concrete age (days)
1t = 1 day
Autogenous shrinkage is independent of the ambient humidity and of member
size and that it develops more rapidly than drying shrinkage.
For drying shrinkage ),( scds ttε the subsequent equations may be applied.
where,
)( cmcdso fε = notional drying shrinkage coefficient,
621 10exp)110220()( −⋅⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅⋅+=
cmo
cmdsdscmcdso f
ff ααε
)()()(),( sdsRHcmcdsoscds ttRHftt −⋅⋅= ββεε A.8
214
)(RHRHβ = coefficient to take into account the effect of relative humidity on
drying shrinkage,
1
3
%99155.1)( so
RH RHforRHRHRH ββ ⋅<
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
1%9925.0)( sRH RHforRH ββ ⋅≥=
)( sds tt −β = function to describe the time development of drying shrinkage,
5.0
12
1
/)()/(350/)(
)( ⎥⎦
⎤⎢⎣
⎡−+
−=−
ttthhttt
ttso
ssdsβ
t = concrete age (days)
st = concrete age at the onset of drying (days)
)( stt − = duration of drying (days)
1.0
15.3
⎟⎟⎠
⎞⎜⎜⎝
⎛=
cm
cmos f
fβ
1dsα = coefficient of type of cement:
31 =dsα for slowly hardening cements
41 =dsα for normal or rapidly hardening cements
61 =dsα for rapidly hardening high-strength cements),
2dsα = coefficient of type of cement:
13.02 =dsα for slowly hardening cements
11.02 =dsα for normal or rapidly hardening cements
12.02 =dsα for rapidly hardening high-strength cements)
215
1sβ = coefficient to take into account self-desiccation in high-performance
concretes
RH = ambient relative humidity (%), ( oRH = 100 %)
h = uAc /2 notional size of member ( mm )
cA = the cross-section ( 2mm )
u = the perimeter of the member in contact with the atmosphere ( mm )
oh = 100 mm .
Creep
The creep function ),( 0ttφ of a concrete at an age t which has been subject to a
constant sustained load at an age 0t is in the below.
where,
oφ = notional creep coefficient,
)()( ocmRHo tf ββφφ ⋅⋅=
213 /1.0
/11 ααφ ⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅
⋅
−+=
o
oRH hh
RHRH
cmocmcm ff
f/3.5)( =β
2.01 )/(1.0
1)(tt
to
o +=β
),(),( ocoo tttt βϕφ ⋅= A.9
216
5.01)/(2
92.1
,1,, ≥
⎥⎥⎦
⎤
⎢⎢⎣
⎡+
+⋅=
α
TToToo tt
tt
),( oc ttβ = coefficient to describe the development of creep with time after loading,
3.0
1
1
/)(/)(
),( ⎥⎦
⎤⎢⎣
⎡−+
−=
tttttt
ttoH
ooc β
β
33
18
15002502.11150 ααβ ≤+⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⋅+=
ooH RH
RHhh
RH = relative humidity of the ambient environmental in (%), 0RH = 100 %
h = uAc /2 , notional size of member ( mm )
cA = the cross-section ( 2mm )
u = the perimeter of the member in contact with the atmosphere ( mm )
oh = 100 mm
cmf = mean compressive strength (MPa)
0cmf = 10 MPa
t = age of concrete in days at the moment considered
Tot , = age of concrete at loading adjusted according to the concrete temperature,
for T=20℃, Tot , corresponds to ot
ot = age of concrete at loading in days
Tt ,1 = 1 day
1t = 1 day
α = constant which depends on the type of cement
217
1−=α for slowly hardening cements
0=α for normal or rapidly hardening cements
1=α for rapidly hardening high-strength cements
3,2,1α = coefficient which depend on the mean compressive strength of concrete,
7.0
1 /5.3
⎥⎦
⎤⎢⎣
⎡=
cmocm ffα
A.3 GL2000 (Gardner and Lockman, 2001)
Shrinkage
where,
)18.11()( 4hh −=β
65.0
28
1043501000 −⋅⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⋅=
cmSH f
Kε
5.0
2)/(97)( ⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅+−
−=
SVtttt
tos
osβ
7001.1 2828 += ckcm ff (psi)
cmtcmt fE 52000500000 += (psi)
K =Factors determined by cement type
1=K , Type I
70.0=K , Type II
)()( thSHsh ββεε ⋅⋅= A.10
218
15.1=K , Type III
V/S = Volume to surface ratio
h =Humidity expressed as decimal
28cmf =28 days mean concrete strength, psi
28ckf =28 days specified concrete strength
cmtf = Mean concrete strength at age t, psi
cmtE =mean modulus of elasticity at age t, psi
oct = Age of concrete at loading, days
ost = Age of concrete at the beginning of shrinkage
Creep
where,
⎥⎥⎦
⎤
⎢⎢⎣
⎡−+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−
−⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−
−= )()086.11(5.2
77
14)()(
2)( 25.05.0
3.0
3.0
28 thtt
ttttt
ttt
os
os
ocos
osos βφφ
5.05.0
2)/(971)(
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛⋅+−
−−=
SVtttt
tos
ososφ
If osoc tt = , 1)( =ctφ when osoc tt >
oct = Age of concrete at loading, days
ost = Age of concrete at the beginning of shrinkage
( )2828
11),( φ+=cm
oc EttJ A.11
Appendix B
SHRINKAGE RESTRAINT
Concrete shrinkage induces compressive stresses in the steel which are
equilibrated by tensile stresses in the concrete. The induced tensile stresses can reduce the
specified tensile strength of concrete. When the loading is applied to concrete member,
the tensile stresses reach at the tensile strength of concrete quickly. Residual tensile stress
induced by shrinkage can be calculated using compatibility and equilibrium of beam
section. .The effect of shrinkage restraint on cracking is well known (Nejadi and Gilbert,
2004; Bischoff, 2001; Gilbert, 1988; Scanlon and Murray, 1982)
In an uncracked concrete section with a tensile reinforcement shown in Figure B-
1, shrinkage develops the compressive stress in the reinforcement during dehydration of
concrete. The compressive force in the reinforcement is equilibrated by the tensile force
acting on the concrete section. The compressive force acting on the reinforcement is
calculated by Eq. B.1
where sE , sA , and sε are the elastic modulus, area of cross section, and the strain of
reinforcement respectively. The equal amount of opposite tensile force cF is created on
concrete section and expressed in Eq. B.2
ssss AEF ε= B.1
cs FF −= B.2
220
The stress in the concrete at the level of reinforcement can be calculated by
Eq. B.3
The concrete strain at the level of reinforcement caused by stress can be obtained
by Eq. B.4
The total concrete stain at the level of reinforcement can be calculated by
summation of shrinkage strain and strain caused by the stress and expressed in Eq. B.5
From the compatibility condition in the strain diagram, sc εε = condition is
satisfied. Therefore, the tensile force acting on concrete section is obtained by Eq. B.6
By arrangement, Eq. B.7 is obtained.
The tensile stress at the bottom fiber can be obtained by Eq. B.8
g
cbc
g
csc I
dyFAF
f2
,)( −
+= B.3
c
scsc E
f ,, =ε B.4
shscc εεε += , B.5
shg
cb
cc
c
ss
c
Idy
AEF
AEF
ε+⎟⎟⎠
⎞⎜⎜⎝
⎛ −+=−
2)(1 B.6
⎟⎟⎠
⎞⎜⎜⎝
⎛ −++
⋅−=
g
cb
cs
cshc
Idy
AnA
EF2)(11
ε B.7
g
bcbc
g
cres I
ydyFAF
f)( −
+= B.8
221
The effective tensile strength of concrete may be calculate by Eq. B.9
where, resf is residual stress induced by shrinkage.
The derived equation is valid only on the simply supported one-way slab. It is
difficult to derive the equation which is valid on two-way slab system because there are
many factors to be considered such as boundary condition and geometry of slab. It may
be proper to use numerical analysis method to calculate the effect of shrinkage.
resre fff −= B.9
dx
εsh
εs
dx
(-)
(+)
(b)section (c)strain (d)concrete stress
Fc= -Fs
fres
εc,sdc
yt
yb
(a)symply supported beam
Figure B-1: Shrinkage Restraint (Gilbert, 1988)
VITA
JE IL LEE
Je Il Lee was born in Incheon, Korea on August 01, 1974. He majored civil
engineering and earned a Bachelor of Engineering degree in February 1999 and conferred
Master of Science of Engineering in civil engineering in February, 2001 from Inha
University, Incheon, Korea
After earning his Master of Science degree, Lee began a career and as a structural
engineer for one year. He came to the United States to pursue doctoral degree in Civil
Engineering at Pennsylvania State University. He earned his doctoral degree in August
2007. The focus of his doctoral research was numerical analysis of steel and concrete
structure under construction load.
Lee is certified as an Engineer Civil Engineering in Korea. He is also a certified
Engineering Intern in the state of Ohio. He holds membership in the American Society of
Civil Engineers (ASCE).