ROLE OF EARLY-AGE CONCRETE PROPERTIES AND CONSTRUCTION ...

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

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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.

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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

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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

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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

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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 3-16: Load-deflection Response due to First and Second Live Loads of B7D28...................................................................................................................82



Figure 3-19: Deflection History for B1D3.................................................................83




Figure 3-23: Deflection History for B5D7 .................................................................85


Figure 3-25: Deflection History for B7D28 ...............................................................86



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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

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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 4-26: Deflection History for Point 5 and 12.....................................................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 5-30: 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

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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


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


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


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


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


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


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


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


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


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


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)


0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150 200Time(days)

Def

lect

ion(

in)


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)


0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150 200Time(days)

Def

lect

ion(

in)


86

0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150 200Time(days)

Def

lect

ion(

in)


0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150 200Time(days)

Def

lect

ion(

in)


87

0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150 200Time(days)

Def

lect

ion(

in)


0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150 200Time(days)

Def

lect

ion(

in)


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)


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)


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


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


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)


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)


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)


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)


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


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


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


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


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


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


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


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


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



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



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



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


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


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


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


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


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


Column E=average

14 AN1 Elastic Analysis

15 AN2 Nonlinear Analysis


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)


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)


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)


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



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



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)


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)


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).

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