BOND STRESS AND SLIP MODELING€¦ · Element Analysis 3 -7.1 Models of Bond-Slip Element 3.7.2...

BOND STRESS AND SLIP MODELING

IN NONLINEAR FINITE ELEMENT ANALYSIS

OF REINFORCED CONCRETE STRUCTURES

Youai Gan

A Thesis submitted in conformity with the requirements

for the Degree of Master of Applied Science

Graduate Department of Civil Engineering

University o f Toronto

O Copyright by Youai Gan 2000

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ABSTRACT

The stress transfer behavior between reinforcing steel and surrounding

concrete through bond and slip plays an important role in the response of

reinforced concrete structures, especially in their hysteretic response. This

thesis studies the bond-slip relationship under both rnonotonic loading and

cyclic loading. Two types of bond elements, contact elements and linkage

elements, are developed and bond stiffness matrices are derived. Four

bond-slip models are presented based on the findings of previous

experimental studies. The degradation of bond resistance is included in the

bond-slip models. The two bond elements and four bond-slip models are

added into the nonlinear finite element program TRIX99. The finite

element models are then used to study the behavior of reinforced concrete

walls subjected to cyclic lateral load. The accuracy of the models are

assessed by cornparison of the finite element numerical response with

experimental data fiom four reinforced concrete shear walls tested under

cyclic loading.

The NLFEA prograrn with the nonlinear material models for concrete,

reinforcing bar and bond-slip is capable of reproducing the important

features of the measured Iiysteretic response of reinforced concrete walls

with a variety of cyclic loading histones and configurations of walls. The

prograrn is able to successfully predict the load-deflection values of the

reinforced concrete structures under cyclic loading. The proposed finite

element method provides an efficient method for evaluating the rnonotonic

and cyclic response of the reinforced concrete structures.

ACKNOWLEDGMENT

I would like to thank my supervisor, Dr. Frank J. Vecchio, Professor

of Department o f Civil Engineering at the University of Toronto. He

provided the subject for my thesis, the knowledge, the wisdom, the

guidance, the patience and support throughout the course of my research

which allowed me to complete al1 my tasks.

Table of Contents

CHAPTER 1

CHAPTER 2

Introduction

Background

Scope

Plan of Study

Review of previous studies

Introduction

Mechanics of Bond Slip

Factors Mecting Bond Behavior

2.3.1 Monotonic Loading

2.3 -2 Cyclic Loading

2.3.3 Loads

2.3 -4 Failure Modes

2.4 The Experimental Study of Bond Stress

and Slip

2.4.1 Bond Stress and Slip Under Monotonic

Loading

2.4.2 Bond Stress and Slip Under Cyclic

Loading

2 -5 The Study of Bond Stress and Slip by

Theoretical Anal ysis Methods

2.5.1 Constitutive Equation of Bond Stress

and Slip

2.5.2 The Solution of Constitutive Equation of

Bond Stress and Slip

Table of Contents

2.6 Code Development

2.7 The Analysis of Bond Stress and Slip

by Finite Element Method

CHAPTER 3 The Modeling of Bond Stress and Slip

Analysis by Finite le ment Metbod

3.1 Introduction

3 -2 Contact Element

3.3 Linkage Element

3.4 Temperature Changes, Prestrains and Support

Displacements

3.5 Transformation Matrix

3 -6 Material Properties

3 -6 1 Concrete Properties

3.6 -2 Reinforcement Properties

3.7 Bond Stress and Slip Models of Finite

Element Analysis

3 -7.1 Models of Bond-Slip Element

3.7.2 Models of Local Bond Behavior

3.7.2.1 Mehlhorn Model (Model 1)

3.7.2.2 Eligehausen Model (Model 2)

3.7.2.3 Moharned Harji Model (Model 3)

3.7.2.4 Proposed Model (Model 4)

3.7.2.5 Model of Bond Stress-Slip Relationship

with Partial Confining Pressure

Table of Contents

3.7.2.6 Models 2,3, and 4 under Cyclic

Loading

3 -7.2.7 Effect of Hooks in Reinforcement

3.7.2.8 Normal Bond Stiffiiess

CHAPTER 4 Evaluation of Models by Cornparison

Between Finite Element Analysis and

Experimea t 124

4.1 Verification of the Program 124

4 -2 Expenmental Details 125

4.2.1 Dimensions and Reinforcement Amangement

of the Specimens 125

4.2.2 Materiai Properties of the Specimen 126

4.2.3 Loading History of the Specimens 127

4.3 Study of Element Mesh 128

4.4 Comparison of the Analytical and

Experimental Results 129

4 -4.1 Shear Wall- 1 130

4.4.2 Shear Wall-2 134

4.4.3 Shear Wall-3 138

4.4.4 Shear Wall4 141

CHAPTER 5 Discussion and conclusions

5.1 Discussion

5.2 Conclusions

References

Table of Contents

List of Figures Fig. 2-1 Specimen details in pull-out test before the 1970's Fig. 2-2 Specimen details in pull-out test d e r the 1970's Fig. 2-3 (a) Splitting mode of thin cover. Fig. 2-3 (b) Splitting mode of thick cover. Fig. 2-4 Load-slip curves. Fig. 2-5 Details of the beam tested by Lutz. Fig. 2-6 Bond stress distribution in constant moment region near a crack. Fig. 2-7 Longitudinal variation in stresses along steel-concrete interface. Fig. 2-8 Separation between concrete ahd bar near a pnmary crack. Fig. 2-9 Ratio of crack widths at the steel and at the concrete surface. Fig. 2-10 Tension and bond stress distribution for eccentric pull-out tests. Fig. 2-1 1 Bar tension and bond stress distributions in beam at a crack. Fig. 2-12 Effect of concrete strength on eccentric pull-out tests. Fig. 2-13(a) Details of specimens and loads. Fig. 2-13(b) Relationship between bond and steel stress for ail three

specimens. Fig. 2-14 Types of specimens tested by Morita. Fig. 2-15 Bond stress distribution in pull-out specimen type 1. Fig. 2-16 Bond stress distribution in pull-out specimen type 2. Fig. 2-17 Bond stress distribution in pull-out specimen type 3. Fig. 2-1 8 Local bond stress versus local slip relationship in pull-out test. Fig. 2-19 Test specimen used by other researchers for studying bond

behavior. Fig. 2-20 Test specimen used by Jiang for studying bond behavior. Fig. 2-21 Local bond stress and local slip relationship. Fig. 2-22 Average characteristic bond stress versus lug inclination. Fig. 2-23 The bars profiles a to d. Fig. 2-24 Average bond stress-slip characteristic for bars of senes a to d. Fig. 2-25 Effect of concrete strength and cover thickness on slip. Fig. 2-26 Bond slip versus specirnen size. Fig. 2-27 Bond slip of square specimen versus steel stress. Fig. 2-28 Bond stress-slip relationship. Fig. 2-29 Bond stress-slip curve for beam and pull-out specimen. Fig. 2-30 Bond stress distribution in the specimen. Fig. 2-3 1 Bond stress and slip relationship proposed by Hayashi et al. Fig. 2-32 Coefficient of local maximum bond stress deterioration. Fig. 2-33 Bond stress distribution along 25 mm plain round bar. Fig. 2-34 Bond stress distribution for 25 mm cold-worked nbbed bar.

List of Figures

Fig. 2-35 Bond stress distribution for 25 mm hot-rolled ribbed bar. Fig. 2-36 Bond stress-steel stress relationship for plain round bar. Fig. 2-37 Bond stress-steel stress-slip relationship for mild steel bar. Fig. 2-38 Bond stress-steel stress-slip relationship for cold-worked ribbed

bar. Fig. 2-39 Bond stress-steel stress-slip relationship for hot-rolled ribbed bar. Fig. 2-40 Bond stress-slip relationship for cold-worked nbbed bar. Fig. 2-4 1 Bond stress-slip relationship for hot-rolled ribbed bar. Fig. 2-42 Bond stress-slip response for pull-out failure. Fig. 2-43 Bond stress-slip response for splitting failure. Fig. 2-44 Analytical bond model for pull-out failure. Fig. 2-45 Analytical bond model for splitting failure. Fig. 2-46 Experimentally derived local bond stress-slip curves. Fig. 2-47 Idealized local bond stress-slip curve for rnonotonic loading. Fig. 2-48 Effect of lateral pressure on local bond stress-slip relationship. Fig. 2-49 Effect of rate of slip on the local bond stress-slip relationship. Fig. 2-5 1 Specimen tested by Tassios. Fig. 2-52 Bond stress-slip relationship for monotonic loading. Fig. 2-53 Steel stress distributions for monotonic loading. Fig. 2-54 Local bond stress versus local slip curves in rnonotonic loading. Fig. 2-55 Bond stress distributions for various loading levels of cyclic

loading. Fig. 2-56 Bond stress distributions for various cyclic loading levels. Fig. 2-57 Slip distributions for various cyclic loading levels. Fig. 2-58 Local bond stress-local slip relationship under cyclic loading. Fig. 2-59 Relationship of bond stress versus slip under cyclic loading. Fig. 2-60 Local bond stress-slip relationship for repeated loading. Fig. 2-6 1 Local bond stress-slip experimental curves for cyclic loading of

confined specimens by Eligehausen et al. Fig. 2-62 Local bond stress-slip curves for deformed bars with hooks by

experiments. Fig. 2-63 Free body of a reinforced concrete member. Fig. 2-64 Slip distribution along the bar. Fig. 2-65 Bond stress distribution along the bar. Fig. 2-66 Cornparison of bond stress dismbution between Yang and

Samayaji.

List of Figures

Fig. 3-1 Relative displacements A. Fig. 3-2 Linkage element. Fig. 3-3 Relation of local coordinate and global coordinate system. Fig. 3-4 Compressive softening parameter f3 of concrete. Fig. 3-5 Constitutive response of cracked concrete in compression. Fig. 3-6 Constitutive response of cracked concrete in tension. Fig. 3-7 Strength enhancement due to biaxial compression. Fig. 3-8 Stress-strain relation for biaxially compressed concrete. Fig. 3-9 Poisson's ratio of concrete in compression. Fig. 3-1 0 Hysteresis model for concrete. Fig. 3- 1 1 Stress-strain relationship of reinforcement. Fig. 3- 12 Hysteresis model for reinforcement. Fig. 3-13 Relationship of local bond stress and slip. Fig. 3-14 Influence of the direction of casting of concrete corresponding to

the pull-out direction. Fig. 3-15 Influence of the relative rib area of the reinforcement bar

normalized to a,=0.065. Fig. 3-16 Relationship of bond strength and confining pressure on the

contact surface of a steel bar. -

Fig. 3- 17 Relationship of bond strength and gap between steel bar and concrete.

Fig. 3- 18 Angle between the steel bar and the crack of the concrete. Fig. 3-19 Influence of angle between steel bar and the crack and distance

between the linkage element to a nearest crack on f342.

Fig. 3-20 The bond stress and slip relationship of Eligehausen model. Fig. 3-2 1 The bond stress and slip relationship of Model 3 (Harjli). Fig. 3-22 The bond stress and slip relationship of proposed model (Model 4). Fig. 3-23 The bond stress and slip relationship with partially confiniag

pressure. Fig. 3-24 The bond stress and slip relationship under cyclic loading

proposed by Eligehausen et al. Fig. 3-25 Analytical local bond stress and slip model for deformed bars with

hooks.

Fig. 4-1 Finite element mesh of a specimen used b veriQ the TRIX. Fig. 4-2 Distribution of tensile force in bar under load =20 kN

(linkage element)

List of Figures

Fig. 4-3 Distribution of tensile force in bar under load =20 kN (contact element)

Fig. 4-4 Distribution of tensile force in bar under load =70 kN (linkage element)

Fig. 4-5 Distribution of tensile force in bar under load =70 kN (contact element)

Fig. 4-6 Cross-sections of walls (Kuzm~ovic, 1994). Fig. 4-7 Front and side view of specimen (Chio, 1995). Fig. 4-8 Front and side view of reinforcement-Wall-3 (Chio, 1995). Fig. 4-9 Side view of reinforcement-Wall-3 (Chio, 1995). Fig. 4-1 0 Front and side view of reinforcement-Wall4 (Chio, 1995). Fig. 4-1 1 Side view of reinforcement-Wall4 (Chio, 1995). Fig. 4-12 Stress-strain curves of concrete used in Wall-3 and Wall-4. Fig. 4- 1 3 Stress-strain curves for reinforcement steel. Fig. 4- 14 Loading history of the specimen Wall-1 . Fig. 4-15 Loading history of the specimen Wall-2. Fig. 4- 16 Loading history of the specimen Wall-3. Fig. 4-17 Loading history of the specimen Wall-4. Fig. 4- 1 8 Coarse mesh for Wall- 1. Fig. 4- 19 Fine mesh for Wall- 1. Fig. 4-20 Finer mesh for Wall-1. Fig. 4-2 1 Mesh for Wall-l when considering bond-slip between interfaces of

al1 steel bars and concrete of web. Fig. 4-22 Final FE mesh for Wall-1. .

Fig. 4-23 FE mesh for Wall-2 and Wall-3. Fig. 4-24 FE mesh for Wall4 Fig. 4-25 Computed load-displacement response of Wall-1 using coarse

smeared elements. Fig. 4-26 Computed load-displacement response of Wall-1 using fine and

finer smeared elements. Fig. 4-27 Load-displacement of Wall-1 assuming perfect bond. Fig. 4-28 Load-displacement of Wall-l for bond-slip Model 1

(linkage element). Fig. 4-29 Load-displacement of Wall-1 for bond-slip Model 2


(linkage element).

List of Figures

Fig. 4-3 1 Load-displacement of Wall-1 for bond-slip Model 4 (linkage element).

Fig. 4-32 Load-displacement of Wall-1 for bond-slip Model 1 (contact element).


Fig. 4-34 Load-displacement of Wall-1 for bond-slip Madel 3 (contact element).


Fig. 4-36 Load-displacement envelope of Wall-1 computed assuming perfect bond (PB), computed using bona-slip Model 2 with contact elements (CB2), and as measured during test.

Fig. 4-37 Load-displacement of Wall- l fiom test data (Kumanovic, 1994). Fig. 4-38 Load-displacement of Wall- 1 assuming perfect bond. Fig. 4-97 Load-displacement of Wall-2 for bond-slip Model 1

(linkage element). Fig. 4-40 Load-displacement of Wall-2 for bond-slip Mode12


(1 inkage e lement). Fig. 4-42 Load-displacement of Wall-2 for bond-slip Model 4


(contact element). Fig. 4-44 Load-displacement of Wall-2 for bond-slip Model 2



(contact element). Fig. 4-47 Load-displacement envelope of Wall-2 computed assuming

perfect bond (PB), computed using bond-slip Model 2 with contact elements (CB2), and as measured during test.

Fig. 4-48 Load-displacement of Wall-2 fiom test data (Kmanovic, 1994). Fig. 4-49 Load-displacement of Wall3 assuming perfect bond. Fig. 4-50 Load-displacement of Wall3 for bond-slip Model 1

(linkage element).

List of Figures

Fig. 4-5 1 Load-displacement of Wall-3 .for bond-slip Model 2 (linkage element).

Fig. 4-52 Load-displacement of Wall-3 for bond-slip Model 3 (linkage element).

Fig. 4-53 Load-displacement of Wall-3 for bond-slip Mode14 (linkage element).


Fig. 4-55 Load-displacement of Wall-3 for bond-slip Mode12 (contact element).



Fig. 4-58 Load-displacement envelope of Wall-3 computed assuming perfect bond (PB), computed using bond-slip Model 2 with contact elements (CB2), and as measured during test.

Fig. 4-59 Load-displacement of Wall-3 fiom test data (Kuunanovic, 1994). Fig. 4-60 Load-displacement of Wall4 assuming perfect bond. Fig. 4-6 1 Load-displacement of Wall-4 for bond-slip Model 1


(1 inkage element). Fig. 4-63 Load-displacement of Wall4 for bond-slip Model 3


(linkage element). Fig. 4-65 Load-displacement of Wall4 for bond-slip Model 1

(contact element). Fig. 4-66 L~ad-displacement of Wall4 for bond-slip Model 2

(contact element). Fig. 4-67 Load-displacement of Wall4 for bond-slip Model 3

(contact element). Fig. 4-68 Load-displacement of Wall4 for bond-slip Model 4

(contact element). Fig. 4-69 Load-displacement envelope of Wall4 computed assuming

perfect bond (PB), computed using bond-slip Model 2 with contact elements (CB2), and as measured during test.

List of Figures

Fig. 4-70 Load-displacement of Wall4 fiom test data (Kuzmanovic, 1994). Fig. 4-7 1 Load-displacement of Wall-1 computed assuming perfect

bond (PB), and computed using contact elements with bond-slip Model 2 (CB2) under rnonotonic loading.

Fig. 4-72 Load-displacement of Wall-2 computed assuming perfect bond (PB), and computed using contact elements with bond-slip Model 2 (CB2) under rnonotonic loading.

Fig. 4-73 Load-displacement of Wall-3 computed assuming perfect bond (PB), and computed using contact elements with bond-slip Model 2 (CB2) under rnonotonic loading.

Fig. 4-74 Load-displacement of Wall4 computzd assuming perfect bond (PB), and computed using contact elements with bond-slip Model 2 (CB2) under rnonotonic loading.

Fig. 4-75 Bond slip envelope of Wall-1 (Mode! 2, linkage element). Fig. 4-76 Bond slip envelope of Wall4 (Model 3, linkage element). Fig. 4-77 Bond slip envelope of Wall-1 (Model 4, linkage elementj. Fig. 4-78 Bond slip envelope of Wall-1 (Model 2, contact element). Fig. 4-79 Bond slip envelope of Wall-1 (Model3, contact element). Fig. 4-80 Bond slip envelope of Wall-1 (Model 4, contact element). Fig. 4-81 Bond slip envelope of Wall-2 (Model 2, linkage element). Fig. 4-82 Bond slip envelope of Wall-2 (Model 2, contact element). Fig. 4-83 Bond slip envelope of iVall-3 (Model 2, linkage element). Fig. 4-84 Bond slip envelope of Wall-3-(Mode1 2, contact element). Fig. 4-85 Bond slip envelope of Wall4 (Model 2, linkage element). Fig. 4-86 Bond slip envelope of Wall4 (Model 2, contact element). Fig. 4-87 Bond slip at the south bottom of Wall-1 (Model 2, contact

element). Fig. 4-88 Bond slip at the north bottom of Wall-1 (Model 2, contact

element). Fig. 4-89 Bond stress at the south bottom of Wall-1 (Model 2, contact

element). Fig. 4-90 Bond stress at the north bottom of Wall-1 (Model 2, contact

element). Fig. 4-91 Steel stress at the south bottom of Wall-1 (Model 2, contact

element). Fig. 4-92 Steel stress at the north bottom of Wall-1 (Model 2, contact

element) .

List of Figures

Fig. 4-93 Bond slip at the south bottom of Wall-2 (Model 2, contact element).

Fig. 4-94 Bond slip at the north bottom of Wall-2 (Model 2, contact element).

Fig. 4-95 Bond stress at the south bottom of Wall-2 (Model 2, contact element).

Fig. 4-96 Bond stress at the north bottom of Wall-2 (Model 2, contact element).

Fig. 4-97 Steel stress at the south bonorn of Wall-2 (Model 2, contact element).

Fig. 4-98 Steel stress at the north bottom of Wall-2 (Model 2, contact element).

Fig. 4-99 Bond slip at the south bottom of Wall-3 (Model 2, contact element).



Fig. 4-102 Bond stress at the north bottom of Wall-3 (Model 2, contact element).

Fig. 4-1 03 Steel stress at the south bottom of Wall-3 (Model 2, contact element).

Fig. 4-104 Steel stress at the north bottom of Wall-3 (Model 2, contact element).

Fig. 4-105 Bond slip at the south bottom of Wall4 (Model 2, contact element).



Fig. 4- 108 Bond stress at the north bottom of Wall-4 (Model 2, contact element).

Fig. 4-109 Steel stress at the south bottom of Wall4 (Model 2, contact element).

Fig. 4-1 10 Steel stress at the north bottom of Wall4 (Model2, contact element).

Fig. 4-1 1 1 Deflection of Wall-1 after failure (assurning perfect bond). Fig. 4-1 12 Deflection of Wall-1 after failure (linkage element, Mode! 2).

List of Figures

Fig. 4-1 13 Deflection of Wa Fig. 4-1 14 Deflection of Wa Fig. 4- 1 15 Deflection of Wa Fig. 4-1 16 Deflection of Wa Fig. 4- 1 17 Deflection of Wa Fig. 4-1 18 Deflection of Wa Fig. 4-1 19 Deflection of Wa Fig. 4-120 Deflection of Wa Fig. 4-1 2 1 Deflection of Wa Fig. 4-1 22 Deflection of Wa Fig. 4-1 23 Deflection of Wa Fig. 4-1 24 Deflection of Wa Fig. 4-1 25 Deflection of Wa Fig. 4-126 Deflection of Wa Fig. 4-127 Deflection of Wa Fig. 4-128 Deflection of Wa Fig. 4-129 Deflection of Wa Fig. 4-130 Deflection of Wa Fig. 4- 13 1 Deflection of Wa Fig. 4- 132 Deflection of Wa Fig. 4- 133 Deflection of Wa Fig. 4-134 Deflection of Wa Fig. 4-135 Deflection of Wa Fig. 4-136 Deflection of Wa Fig. 4-137 Deflection of Wa Fig. 4-1 38 Deflection of Wa

Fter fai Fter fai Fter fai Fter fai Eter fai Rer fai Rer fai Eter fai !ter fai Rer fai Rer fai Fter fai fier fai Fter fai 3er fai 3er fai 3er fai, 3er fai !ter fai 3er fai 3er fai 3er fai 3er fai 3er fai 3er fai 3er fai

Lure (linkage elernent, Model 3). Lure (linkage element, Model4). Lure (contact element, Model 2). lure (contact element, Model3). lure (contact element, Model 4). Lure (assuming perfect bond). Lure (linkage element, Model 2). lure (linkage element, Mode1 3). lure (linkage element, Model 4). Lure (contact element, Model 2). lure (contact element, Model 3). lure (contact element, Model 4). lure (assum ing perfect bond). .ure (linkage element, Model 2). ure (linkage element, Model 3). ure (linkage element, Model 4). .ure (contact element, Model 2). sre (contact element, Model 3). .ure (contact element, Model 4). ure (assuming perfect bond). ure (linkage element, Model 2). .ure (linkage element, Model 3). ure (linkage element, Model 4). ure (contact element, Model 2). ure (contact element, Model 3). ure (contact element, Model 4).

List of Tables

Table 2-1 Confining pressure in experiments of Figure 2-48 (a) Table 2-2 Confining pressure in experiments of Figure 2-49 (a)

Table 3-1 Ultimate relative displacements Au (mm).

Table 4-1 Matenal data. Table 4-2 Specimen parameters and analytical maximum slips (mm). Table 4-3 Uniaxial compressive strength of the Wall-1 and Wall-2. Table 4-4 Uniaxial compressive strength of the Wall-3 and W a l l 4 Table 4-5 Characteristics of reinforcement steel. Table 4-6 Comparison of ultimate loads acting at top of Wall-1 fiom test

data and FE analysis 0. Table 4-7 Maximum and minimum slips of Wall-1 by FE analysis (mm). Table 4-8 Comparison of ultimate loads for Wall-2 fiom test data and

FE analysis (kN). Table 4-9 Ultimate loads for Wall-3 fiom test data and FE analysis 0. Table 4-10 Comparison of ultimate loads acting for Wall4 from test data

and FE analysis (kN). Table 4-1 1 Comparison of ultimate loads for the four test walls calculated

assuming perfect bond and bond-slip Mode1 2 with contact bond elements under rnonotonic loading 0.

Table 5-1 Comparison of ultimate loads of walls fiom test data and FE analysis 0.

BOND STRESS AND SLIP MODELING IN

NONLINEAR FINITE ELEMENT ANALYSIS OF

REINFORCED CONCRETE STRUCTURES

CHAPTER 1

INTRODUCTION

1.1 Background

Bond stress is the shear stress acting parallel to an embedded bar on the

surface between the reinforcing bar and the concrete. Bond slip is the

relative displacement between the bar and the concrete. Reinforced

concrete depends on the combined action of the concrete and its

embedded reinforcement for satisfactory operation as a construction

material. This action is produced by the interaction between both of its

components, plain concrete and reinforcing bars. The transfer of forces

across the interface between these two materials is completed by bond

action behveen them, so bond plays a very important role in most aspects

of reinforced concrete behavior. To better understand bond behavior, there

have been a number of studies specifically aimed at examining behavior

of bond stress-slip by way of both expenment and theory. Several laws to

describe the behavior have been developed on experimental ground.

Constitutive laws between the local bond stress and the local bond slip on

the interface have been formulated for theoretical analysis purposes.

The finite element method has been widely used for analyzing the

response of reinforced and prestressed concrete structures. niere are two

distinct ways of representing cracks in a finite element procedure. The

discrete crack mode1 represents cracks as interslement discontinuities.

There is restriction on the crack propagation direction depending on the

mesh layout. The smeared crack approach represents cracks as a change in

the material property of the element over which the cracks are assumed to

be smeared. In the simplified analysis of reinforced and prestressed

concrete stmctures, complete compatibility between concrete and

reinforcement or prestressed tendons is usually assurned. It means that

perfect bond is presurned. For models with smeared steel, the perfect bond

relationship is the easiest to adopt since it simply involves overlaying the

constitutive matrix of the steel elements with concrete elements. For the

models with discrete reinforcement elements, perfect bond also represents

a very easy solution, since the displacements of the nodal points are the

sarne for both concrete elements and reinforcement elements. Actually,

this assumption is only valid in regions where only low transfer stresses

between the two components exist. In the regions where high transfer

stresses occur between the interfaces of the two components, especially

for the regions near cracks, there are different strains in concrete and

reinforcement. As result of this, relative displacements, which are called

bond slips, occur between concrete and reinforcement. Because bond

stresses are related to bond slips between the two components, the

assumption of petfect bond in a cracked region would require infinitely

high reinforcement strains to explain the crack widths. It would also cause

signi ficant error in the predicted load-deflection response, stresses and

strains of the reinforced structures when the bar slip is large.

Most of the work done in the past to understand bond behavior

focussed on experimental and analytical study of concrete prism

specimens with an embedded steel bar. In most cases, the model was

used to explain the behavior of the specimens fiom which the model was

derived. This does not constitute real verification of the model.

Theoretical methods proposed by some researchers are only applicable

for very simple members such as concrete prism with an embedded steel

bar, and can not be used for reinforced concrete structures. However,

little work has been done towards verimg the bond stress-slip models

obtained fiom experimental investigations for reinforced concrete

structures and studying the effects of bond slip on behavior of reinforced

concrete structures. Such a study is vital in verifjring bond stress-slip

models and investigating behavior of reinforced concrete structures.

1.2 Scope

The scope of this thesis is to study the bond stress-slip models under

both rnonotonic loading and cyclic loading; to verie these models by

comparing analytical response with the experimental results of reinforced

concrete shear walls; to study the effects of bond slip on reinforced

concrete shear walls; and to provide a kind of method of nonlinear finite

element analysis for reinforced concrete structures when bond slip is

considered.

1.3 Plan of Study

Published experimental and theoretical research under both

rnonotonic and cyclic loading are studied in the first part of this thesis

(Chapter 2), to investigate factors which affect bond behavior and to

provide bond-slip models. Chapter 3 studies the modeling of bond stress

and slip analysis by finite element method. First, two types of bond-slip

elernents, contact element and linkage element, are studied and

formulations are presented for use in 'finite element analysis; then four

types of bond-slip models are described and implemented into the

nonlinear finite element program TRIX99. Chapter 4 evaluates the bond-

slip models and elements by cornparison between finite element analysis

results and experimental data of a set of four shear walls tested at the

University of Toronto. Chapter 5 discusses the bond-slip elements and

models presented in chapter 4, and gives the conclusions of this study.

CHAPTER 2

REVIEW OF PREVIOUS STUDIES

2.1 Introduction

Many researchers have studied bond by performing experiments on

strain-gaged reinforcing bars embedded in a variety of concrete

specimens. In these tests, the axial strain in the bars was measured

directly, but bond stresses could only be detennined indirectly from the

slope of the steel strain curve, while the bond slips were determined fkom

displacements obtained by numerical integration of the difference of

strains between steel bar and concrete in the interface.

Study of bond behavior between steel bar and surrounding concrete

began in the 1960s. Most of the studies perfonned in 1960s and 1970s

were experimental investigations under monotonic loading. Study of bond

behavior under cyclic loading has been done since 1980s. Because there

exist many variables that affect the bond stress and bond slip relationship,

and because measurement of bond stress and bond slip is difficult and

sensitive to experimental errors, it is very difficult to get a generally

applicable law from experimental investigations. Some researchers

developed the theoretical method to study bond behavior by analytical

models based on some assumptions since 1980s.

The bond stress between the reinforcing bar and its surrounding

concrete is very complicated. It is dependent on the slip between the steel

bar and the concrete, and the stress in the reinforcing bar as well as many

other factors. Two alternative basic hypotheses have been used in the Fast;

in one bond stress is considered to be a linear function of slip ( Ngo and

Scordelis, 1967), while in the other it is considered to be a nonlinear

relationship between bond stress and slip.

Early experimental studies of bond were concerned with determining

bond failure strengths and the influence of surface deformations on them

by push-in test. Later some researchers found that bond failure occurs at a

higher stress for a push-in test than for the normal pull-out test. The

explanation for this is simple: firstly, the compressive axial stress

developed in the bar during a pushin test causes an increase in bar

diameter because of Poisson's effect, which, in tum, causes an increase in

the radial pressure between bar and concrete. Since friction is an

important element in bond, this increase in pressure leads to an increased

bond strength; secondly, cracking is an important reason to cause

degradation of bond strength. Because there is no cracking in a push-in

test, this leads to the increase of bond strength.

In the ordinary pull-out test, the test specimen is typically a cylinder or

prism with a bar embedded in it. Before the 19709s9 the bar was pulled

fi-om one side while the concrete was held by the reaction pressure on the

same end as shown in Fig. 2-1. Since the bar is in tension and concrete in

compression, differential strains force a relative slip at very low steel

stresses. This is different to what occurs in an actual structure. The most

senous weakness of this pull-out test is that the concrete in compression

eliminates transverse tension cracking. After the 19709s, the method of

pull-out test was improved. A specimen is pulled firom both sides of the

bar (Fig. 2- 2).

Fig. 2-1 Specimen details in pull-out test before the 1970's.

Fig. 2-2 Specimen in pull-out test after the 1970's.

Most researchers believe that there are different relationships for local

bond stresses versus local bond slips at different points of the interface of

the steel bar and the concrete. Since the properties of the interface will not

be different, this change can only corne fiom stress effects. Some

researchers (Morita, 1985; Narnmur and Naaman, 1989; Edward and

Yannopoulos, 1979) thought that the relationship is a material property

and, therefore, independent of location. They thought that there exists a

unique bond slip relationship which depends only on matenal properties

and steel geometry.

2.2 Mechanics of Bond Slip

The strength of the bond between a rebar and the surrounding concrete

is generally made up of three components:

1. Chernical adhesion;

2. Friction;

3. Mechanical interlock between reinforcement and concrete. This

includes the bearing of lugs on concrete and shear strength of concrete

section between lugs.

Bond of plain bars depends primarïly on the first two mechanisms,

although there is some mechanical interlocking due to the roughness of

the bar surface. However, even a low .bar stress causes slip suficient to

break the adhesion immediately adjacent to a crack in the concrete. Over

the slipping length o d y the friction drag remains, and the highest adhesive ,

bond stress can act only close to this slipping portion. Bond resistance is

thus an ultimate bond stress. over a short length where adhesion is about to

fai 1.

Deformed bars change this behavior. Adhesion and fiction still assist,

but the primary resistance has been changed to mechanical interlocking

for superior bond properties. With deformed bars, a pull-out specimen

nearly always fails by splitting; the concrete splits into two or three

segments rather than failing by crushing against the lugs or by shearing on

the cylindrical surface which the lugs-tend to strip out. Splitting results

from the wedging action of the lugs against the concrete. It should be

noted that a split on one face of a concrete member does not represent .

complete failure. When stimps are present, significant

remains.

Clear cover over a reinforcing bar will be significant in

bond resistance

connection with

splitting resistance. Thin cover can be easily split like Fig. 2-3 (a); a thick

cover can greatly delay splitting if bars are not closely spaced laterally. If

a number of bars are closely spaced with thick cover in a beam, a splitting

failure will occur as shown in Fig. 2-3@).

Fig 2 3 (a) Splitting mode of thin cover

Fig. 2-3 (b) Splitting mode of thick cover.

Lutz and Gergely (1967) investigated the mechanics of bond and slip.

For deformed bars, initially, chemical adhesion combined with

mechanical interaction prevents slip. Afier adhesion is destroyed, as slip

occurs, the rib of a bar restrains this movement by bearing against the

concrete between the ribs. Friction, which would occur after the slip of

plain bars, does not occur here because of the presence of the ribs.

Slip of a deformed bar can occur in two ways: (1) the ribs c m push the

concrete away from the bar (wedging action), and (2) the ribs can cmsh

the concrete.

Tests indicate that the movement of the ribs is about the same for al1

ribs with a face angle greater than about 40 degrees. In bars with ribs

having face angles near 90 degrees, the ribs can not push concrete

outward (no wedging action). The friction between the rib face and the

concrete is suficient to prevent relative movement at the interface when

face angles are larger than 40 degrees. The slip is due almost entirely to

tlie crushing of the porous concrete paste in front of the ribs if the rib face

angles are larger than about 40 degrees. Bars with ribs having face angles

less than about 30 degrees exhibit a markedly different load slip

relationship. Here the friction between the rib face and the concrete is not

sufficient to prevent relative movement. Thus, slip is due mainly to the

relative movement between the concrete and the steel bar dong the face of

the rib, and secondarily to some crushing of the concrete.

For the usual case of good fictional properties and a nb face angle

greater than 40 degrees, slip occurs by progressive crushing of the porous

concrete paste structure in front of the rib.

The slip resistance on reloading is considerably higher than the slip

resistance initially, as shown in Figure 2-4.

Lutz et al also experime&ally investigated the cracking effects on the

bond stress distribution in concrete beams. The details of the beam used in

tlie test are shown in Fig. 2-5. Before cracking, the bar adheres chemically

to the concrete. This adhesion acts until slip or movement of the steel

relative to the adjacent concrete occurs. Tests showed that adhesion would

be lost in tension rather than in shear. Before slip and cracking occur,

tension can exist on the interfaçe between the bar and concrete as a result

of the large Poisson's ratio of the steel and concrete (Poisson ratios of 0.3

and 0.15 for the steel and concrete, respectively). Due to the loading, no

bond stresses exist between the concrete and reinforcing bar in the middle

third of the bearn. In the outer thirds of the beam, because there is no slip,

the steel and concrete elongate the sarne amount over any given length,

but due to the difference in the elastic moduli of the two materials, a

larger change in stress occurs in the steel than in the concrete, leading to

bond stresses.

O 2 4 6 8 10

distance dong bar liom loaded end(in. j

Fig. 2-4 load-slip curves (unit of loads: lb).

Fig. 2-5 Details of the beam tested by Lutz.

Mer a flexural crack forms in the constant moment region of the

beam, the stress condition in the vicinity of the crack is very complex.

The bond stress distribution near a crack is sketched in Figure 2 6 . As

shown in Fig. 2-7, the circumferential tensile stresses in the concrete

around the bar are very small prior to flexural cracking. However, near

transverse cracks bonding forces cause large circumferential tensile

stresses. Also, radial tensile stresses, acting normal to the concrete steel

interface, destroy contact near the crack and allow separation and slip

between the bar and the concrete.

Fig. 2-6 Bond stress distribution

in constant moment region near a crack.

Fig. 2-7 Longitudinal variation

in stresses dong steel-concrete interface.

.= 80 -

-ss

crack O 1 2

1 xld

--Radial stress --Circumferent ial stress

--Bond stress

--Longitudinal stress

In the case of plain bars this separation would mean complete loss of

bond in the region next to the crack. The concrete stress at the bar surface

and parallel to the bar would disappear, and the crack width at the bar

would be essentially the same as the width at the surface of the concrete.

However, when the reinforcing bar has transverse nbs, separation does

not produce complete unloading of the concrete adjacent to the bar

inasmuch as the bar nbs prevent much of the opening of the crack at the

bar (Fig. 2-8). Some unloading does occur, allowing the crack to open at

the surface of the bar. This opening is caused partly by the unloading of

the concrete between the crack and the nearest bar rib when the crack .

forms, producing a relative contraction of the concrete, and partly by the

bearing deformation under the ribs. Another factor conbibuting to the

opening is the inclination of the bar ribs, since the separation of the bar

and the concrete is accompanied by a movement along the inclined bar rib

face. The bond stress near a transverse crack is transferred only by bearing

of the concrete against the face of the ribs. With an increase in the load

above the cracking load, additional extemal cracks form. At each of the

surface cracks the steel stress will reach a local peak; between cracks the

steel stress is lower as part of the tension force is carried by the concrete.

The transfer of forces produces bond stresses.

The interna1 transverse cracks tend to occur before longitudinal cracks .

if the steel stress is high relative to the bond stress (in the flexural region),

while splitting would occur fust when the bond stress is large relative to

the steel stress (in the anchorage region). Fig. 2-9 shows the ratio of the

crack width adjacent to the bar to the one at the concrete surfaces. A large

variation in the crack widths occurs fiom the bar to the concrete surface

with increasing steel stress. .

P~~.IMIY or surfaœ crack

Fig. 2-8 Separation between concrete and bar near a primary crack.

Fig. 2-9 Ratio of crack widths at the steel and at the concrete surface.

0.6 -

O

3 0.5 -

f

Ingraffea et al (1984) studied the fi-acture mechanics of bond in

reinforced concrete. They thought there are several possible contributions

to bond behavior. These are: elastic deformation; crushing at points where

concrete bears on steel ribs; secondary radial cracking; and longitudinal

splitting cracking. They researched the radial secondary cracking. This *

cracking was found not to follow the principles of linear elastic fracture

mechanics, and was therefore modeled using a nonlinear discrete-crack

finite element approach. They proposed a new method for finite element

modeling of nonlinear fracture and introduced the "tension-softening

element" for the purpose of modeling bond stress-slip in practical

ws - width of crack at bar surhce WC - wvidth of crack at concrete surface

1 0.3 ,

t O 20 40 60 80 100

*el a r e s at free ends ( ksi )

engineering problems. The characteristics of this element were derived by

studying bond behavior allowed by secondary cracking in the tension pull

specimen.

They concluded that secondary cracking is an important mechanism

allowing bond slip to occur. Secondary and primary cracking are

interrelated and secondary cracks are generally confined to a region close

to the location where a reinforcing bar crosses a primary crack.

2.3 Factors Affecting Bond Behavior

2.3.1 Monotonic Loading

Under static monotonically increasing loads, the main factors that

affect bond behavior are the concrete strength, yield strength of steel bar,

bar size, cover and bar spacing, geometry of bar, surface condition of bar,

transverse reinforcement, and concrete casting position.

2.3 -2 Cycl ic Loading

Al1 parameters that are of importance under rnonotonic loading are

also of importance under cyclic loading. In addition, the bond stress range,

bond slip range, and type of loading are important factors affecting bond

behavior under cyclic loading.

2.3.3 Loads

Loads c m be subdivided into monotonic and cyclic loads. Monotonic

loading implies that slip is always increasing. Cyclic loading implies slip

reverses in direction many times during the load history. Cyclic loading i s

divided into two general categories. The first category is low-cycle

loading or low-cycle, high-stress loading. That is loading containing few

cycles but having a large range of bond stress. The second category is

high-cycle loading or fatigue loading.

The bond behavior under cyclic loading can further be divided

according to the type of stress applied. The first is unidirectional loading,

which implies that the bar stress does not change sense ( tension to

compression) during a loading cycle. The second is stress reversal, where

the bar is subjected alternatively to tension and compression.

2.3.4 Failure Modes

Under rnonotonic loading, two types of bond failures are typical. The

first is direct pull-out of the bar, which occurs when large confinement is

provided to the bar. The second type of failure is a splitting of the

concrete cover when the cover or confinement is insufficient to obtain a

pull-out failure.

2.4 The Experimental Study of Bond Stress and Slip

2.4.1 Bond Stress and Slip Under Monotonic Loading

Peny and Thompson (1966) studied the bond stress distribution in

beams and eccentric pull-out specimens. Bond stress distribution curves

are shown in Fig. 2-10 for eccentric pull-out specimens for different loads.

The point of maximum bond stress moved away from the loaded end as

the force in the bar increased.

For beams, even when shear is zero and moment is constant, large local

bond stresses exist adjacent to each flexural crack. At the crack, most of

the tension is carried by the bars and the steel stress is maximum. Between

cracks, the concrete carries tension and the steel stress drops off. Thus

bond must take stress out of the steel after a crack and put it back in the

steel just before the next crack is reached. Bond stress distribution curves

were plotted in Fig. 2-1 1 for bearns near a crack. It was noted that the

point of maximum bond stress was aiways about 1.5 inches fiom the crack

regardless of the load.

7 0 0 , 1

O 2 4 6 8 10 distance along bar from loaded end (in.)

Fig. 2-10 Tension and bond stress distribution for eccentric

pull-out tests (f,' =2500 psi, unit of loads: Kips).

500 400

.- 300 & zoo 2 100 L C

UJ O w 5-100

-200

-300 400 - -

O 1 2 3 4 5 6 7 8 9 distance along bar from crack-in.

Fig. 2-1 1 Bar tension and bond stress distributions

in beam at a crack (f,'=5000 psi, unit of loads: Kips).

The eEect of concrete strength on the distribution of bond stress in the

eccentric pull-out specimens is s h o w in Fig. 2-12. The point of

maximum bond stress was found to shift toward the unloaded end for the

lower concrete strengths. As the maximum bond stress was reached at any

point along the bar, the bar slipped and caused the bond stress on the side

of the loaded end to reduce gradually to the vzlue of fnctional drag

between the bar and the concrete.

O 1 2 3 4 5 6 7 8 9 distance fiom foaded end - in.

Fig. 2-1 2 Effect of concrete strength on eccentric pull-out tests.

(unit of concrete strength: psi)

The authors cornpared relationships between bond stress and steel

stress for the three types of specimens shown in Fig. 2-13. Three bond

curves were plotted from steel siress distribution curves having

approximately the same steel stress at point zero. The maximum bond

stress in each case was not significantly different, but they occurred at '

different locations. It was seen that the bar cut-off point occurred in a zone

of rapidly changing moment in one bearn, whereas the crack in the other

beam was in a zone of constant moment. The bar cut-off specimen (Beani

a) is different from others because horizontal shearing stresses exist at the

level of the bar, while there are no shearing stresses in others. These

differences probably caused the non-similarity of the bond stress

distribution for the two beam tests. When bars are cut off in a tension

zone, a flexural crack foms prematurely at the discontinuity. The sudden

change in bar stress produces large bond stress locally and causes

splitting.

Fig. 2-13 (a) Details of specimens and loads.

Cut off point, crack Or free end - pullout - beam a - beam b

-200 4 -300 J distance along bar-in.

Fig. 2-13 (b) Relationship between bond

and steel stress for al1 three specimens.

Monta and Fujii (1985) developed a bond stress-slip mode1 used iii

finite element analysis. They conducted extensive tende tests using

deformed bars of 51 mm, 25 mm, 19 mm, 7 mm and 3 mm diameters to

investigate the efTect of bar size upon slip behavior. The following

equation was obtained fiom regression analysis of the test data :

z = 888*(Nd) Os'' @g/cm2) (2- 1

where z is the local bond stress, A is the local bond slip, and d is the

diameter of the steel bar. Using equation (2-l), stress distributions were

calculated numerically. The results indicate that if the distance fiom the

loaded end is normalized with respect to bar diameter, the stress

distributions are almost identical for bar diameters of 3 mm to 51 mm.

This means that the extent of the intemal cracking zone is probably

proportional to the bar diameter.

The other basic aspect of bond behavior is the occwence of radial

forces caused by wedging action of the ribs. Due to this action, interna1

longitudinal cracks occur in the vicinity of the bar. With the increase of

the slip, splitting of the surrounding concrete develops along the bar axis.

The local bond behavior is strongly affected by development of these

longitudinal cracks. The peak stress of local bond is ofien dependent on

the resisting capacity against splitting of entire concrete cover. The

provision of transverse reinforcement is effective only to sustain the bond

up to a large slip.

Morita et al (1985) investigated the effect of location on the

development of bond stress and bond slip (specimen is shown in Fig. 2-

14). Figures 2-15 to 2-17 show the bond stresses along the bar axis at

several loading stages for specimens type 1, type 2 and type 3. The

reduction of bond stress transfer in the neighborhood of primary cracks at

the pull-out end (the region of about 5d) is clearly demonstrated, while at

the push-in end, bond stress transfer near the primary crack is much more

effective throughout the load application. Fig. 2-18 shows the bond stress

versus bond slip relationship at some locations. The closer the location to

the loaded end, the more deteriorated the bond resistance in pull-out tests.

Fig. 2-14 Types of specimen tested by Morita.

(type 1-pull only; type 2-push only, type 3-push and pull)

-c- P=99 ton

+el23 ton - P=149 ton

Free end

O i / 7 t

O 20 40 60 80

didance from pulled end ( cm )

Fig. 2-1 5 Bond stress distribution in pull-out specimen type 1 .

O 20 40 60 80

didance frorn free end (cm)

-+- P=27 ton

-P=51 ton - P=99 ton

Pushed end

Fig. 2-16 Bond stress distribution in push-in specimen type 2.

-t- f i 1 47 ton + P=l23 ton +-ton -x--75 ton *Ml ton

h h end

distance from pulled end ( cm )

Fig. 2-17 Bond stress distribution in combined

pull-out and push-in specimen type 3.

O 0.2 0.4 0.6 0.8 1

local slip ( mm )

Fig. 2- 18 Local bond stress versus local slip relationships in pull-out test.

Nilson (1972) studied the spacing and width of cracks, and the

distribution of concrete stresses in partially cracked members. Usually,

bond stress-slip relationships were studied using pull-out specimens of the

type s h o w in Figure 2-19. This type of test is intended to simulate

conditions in the tension zone of a concrete beam between primary

flexural cracks and beiow the neutral axis. With this type of specimen, it .

is possible to measure the variation in steel strain, and hence to measure

the bond stress distribution, and to measure slips at the two ends.

However, it is difficult to measure local slip, strain distribution in the

surrounding concrete, and the extent of the secoridary cracks. Actually, in

experimental investigations, the exact measurement of concrete strains at

the interface is almost impossible to achieve. Therefore the strains are

measured at a certain distance away fkom the interface. In many studies of

bond slip behavior done using prismatic specimens, concrete strains dong

steel bar are measured at some distance away from the interface to give

the interface slip. The different methods used in measuring the local slip

lead to widely varyiny test results for the bond stress-slip relationship.

Jiang et al (1984) developed a new type of test specimen (Fig.2-20).

They measured directly the interface slip using a microscopic technique.

Since the steel-concrete interface was exposed, the local slip and concrete

strain were measured directly. They found the cracks generally initiated

fiom the location of the first lug to the two ends and extended at an

inclination of about 60 degrees fiom the axis of the bar towards the side of

the specimen with increasing loads. The inclination of the secondary

cracks other than the first mes was generally larger. They assumed that

resistance to relative slip, until the formation of the first secondary crack,

will be largest at the lugs nearest the primary cracks. The peak bond stress

70 (near the fust lug) was empirically related to the steel stress ad at the

end as:

TO =0.034 0, ( 1 - 0 . 0 1 ~ ) ksi (2-2) Jiang et al found that the maximum slip is not at the end of the bar but at the

location where the secondary cracks were ubserved. The relationship

between the bond stress and slip at the steel-concrete interface is not unique

but varies fiom location to location as shown in Fig. 2-2 1.

P * P/2 P/2

Fig. 2- 1 9 Test specimen Fig. 2-20 Test specimen used

used by other researchers by.Jiang for shidying bond behavior.

for studying bond behavior.

1.2 ,

Fig. 2-21 Local bond stress and local slip relationship.

For a one-dimensional model, a parabolic bond stress distribution was

assumed

r = ro [ 1 - ( I -~XL)~] (2-3)

Soretz and ~ ~ i z e n b e i n (1979) studied the influence of the rib

dimension of reinforcing bars to bond behavior. They considered the

influence of the following three parameters related to the formation of

lugs: the height and spacing of the lugs , the inclination of the lugs with

respect to the bar axis, and the cross-section of the lugs. They arriveci at

the following conclusions: Fustly, the bond resistance shows no

significant dependence on the pattern of the ribbed bars with an identical

related nb area. ~econdly, a reduction of the rib height seems to be

advantageous, since the danger of longitudinal cracks in the reinforced

concrete stnicture due to the splitting effect is considerably reduced. An

increase of lug height will cause a decrease of the bendability of the bars,

and the tendency for brittle &acture of the steel bar. To maintain the

necessary bond characteristic, they suggested keeping the lug spacing at

0.3d and the minimum height of lugs at 0.03d (diameter of the reinforcing

bar). Thirdly, when the inclinations of the lugs with respect to the bar axis

are from 45 degrees to 90 degrees, the bond characteristic improves

slightly but that the influence of the rib area is three-fold as important as

that of the lug inclination with increasing inclination of the lugs (Fig. 2-

22). Fourthly, the inclination of the nb flanks towards the surface of the

core from 45 degees to 90 degrees has no significant influence on either

the bond stresses and bond failure due to splitting or on the maximum slip

value (Figs. 2-23, 2-24). Fifthly, changing the rib cross section from a

rectangle to a 45 degrees trapezoid has no significant influence and

changing to a very flat triangle has only a slight influence on the bond

characteristics. The splitting effect is not influenced at all.

4 0 5 0 6 û 7 0 8 0 9 0 lug inclination (deg)

Fig. 2-22 Average characteristic bond stress versus h g inclination.

Y) II) er C II)

Fig. 2-23 The bar profiles a to d.

O "t O 0.1 0:2 0.3 0.4 0.5 0.6 0.7

slip (mm)

profile a

profile b

profile c profile d

Fig. 2-24 Average bond stress-slip characteristic for bars of series a to d.

The crack width specified in various codes is the crack width at the

surface of concrete, and it differs significantly fiom the crack width at the

steel-concrete interface. Experirnental evidence shows that the ratio of the

crack width at the concrete face to that at the steel-concrete interface

varies with the steel stress (shown in Fig. 2-7).

Mirza and Houde (1979) showed that crack formation is inherently

subjected to a greater experimental scatter than other properties of

concrete. The crack spacing can Vary between 1 to 2 times the minimum

crack spacing. If one considers the influence of the crack spacing, it is

normal to expect the crack spacing to be 50 per cent larger or smaller than

the average measured value. The crack spacing was principally govemed

by the concrete cover thickness and was approximately equal to 3c (cover

thickness).

(1) f i 3 0 0 0 psi (2) fc=6000 psi

O I O 20 30 40 50 end sips (0.0001 in.)

Fig. 2-25 Effect of concrete strength and cover thickness on slip.

-t- fs=15 ksi

+ f s--30 ksi

+ f s=45 ksi

O 1 2 3 4 5 6 7 8 square specimen size (in)

Fig. 2-26 Bond slip versus specîmen size.

O 10 20 30 40 50 steel stress (ksi) .

Fig. 2-27 Bond slip of square specimen versus steel stress.

Mina and Houde examined the influence of bar size, concrete

strength and the thickness of the concrete cover. They showed that the slip

is seen to increase linearly with an increase in the steel stress, and to

increase with an increase in concrete strength and an increase in the

specimen dimensions up to a certain size (Figures 2-25 to 2-29). [In

Figures 2-25 and 2-27, S2*2 means that the dimension of a specimen is 2

in x 2 in]. The slip can be explained only by the interna1 cracking of the

first layer of concrete surrounding the bar and by the bending andlor

cracking of the small concrete teeth near bar lugs. For small specimens,

where the crack spacing is small, the slips are small. For large specimens,

the cracking increases with the specimen cross section up to a point where

no transverse cracks are observed. This means that over the length

provided, the force transferred to the concrete does not exceed the tensile

capacity of the concrete section.

The tests done by Mirza and Houde showed that the concrete strength

has an insignificant effect on the observed slip values. The influence of

the steel stress and the section geometry can be expressed by an equation

of the fonn:

S I ip= k 1 f S k Z ( ~ ~ s ) H (2-4) the value of the slip is in 10" inches, and the steel stress f, is in ksi.

O 1 2 3 bond slip (0.001 in)

Fig. 2-28 Bond stress-slip relationship

(x is the distance fiom loaded end, and bond stress ratio is rlf,'ln).

beam

pull

O 2 4 6 8 10 12 bond slip (0.0001 in)

Fig. 2-29 Bond stress-slip curve for beam and pull-out specimen.

The bar slip increased almost linearly with the steel stress and with the

value of the ratio (AC~AS)'" between 45 and 60. Beyond this value, the

larger concrete area imposes a greater. restraint on the steel bar, thereby

causing a decrease in the slip at the steel concrete interface.

Slip has been considered to result from a gradua1 deterioration of the .

concrete in front of the lugs as a result of high bearing and shearing

stresses. Mirza and Houde thought the slip at the steel concrete interface

can be explained only by the bending of the comb-like structures of the

first concrete layers surrounding the bar.

They concluded that the magnitude of force transferred from the steel

to the concrete is dependent on the embedment length, the concrete cross

section and the concrete strength. The bond stress at the steel- concrete

interface is seen to reach the maximum value at a slip of some value.

Before the peak value is reached, the -1ationship between the localbond

stress(r) and the local slip (A) can be expressed by the following

polynomial : 9 2 12 3 15 4 ~=1.95*10~~-2.35*10 A +1.39*10 A -0.33* 10 A (2-5)

Differentiation of both sides of equation (2-5) with respect to A, yields: 12 2 15 3 d~dA=1..95*10~-4.70*10~~+4.17*10 A -1.32*10 A (2-6)

They found fiom tests that the bond stress level was not to be related to

the distance &om the end face before the maximum bond stress was

reached. The maximum bond stress was attained at al1 locations when the

slip value was between 0.001 inch and 0.0012 inch. The bond stress-slip

relationship is thus applicable directly at any point along the bar.

The bond stress slip behavior past the peak point was dependent on the

distance from the end face. For points at some distance fiom the end face

(3 to 4 inches or more) the bond stress was almost constant for al1 slip .

values. For smaller distances, bond stresses decreased progressively with

increasing slip. This loss of bond transfer capacity near by the end faces

can be attributed to the splitting cracks observed.

Hayashi et al (1985) investigated bond behavior near a crack. They

concluded that near cracks, both bond st if iess and maximum bond stress

deteriorate. The local maximum bond stress deteriorates near the crack in

proportion to the distance from the crack. When the distance from the

crack exceeds 4 bar diameten, no bond deterioration is observed. The

thickness of concrete cover does not effects the bond behavior as iong as

the minimum thickness is 2.5 bar diameters.

Figure 2-30 shows the bond stress distribution in the test specimen. The

maximum bond stress is about 40 kgicm2 (about 4 MPa) for al1 specimens.

Figure 2-31 shows the relationship between local bond stress and slip. It

can be seen that the relationship between local bond stress and local slip

varied with distance fiom the loaded end.

80

O S 10 15 20 25

distance from fme end (cm)

Fig. 2-30 Bond stress distribution in the specimen.

Hayashi et al proposed a bond detenoration model for finite element

method analysis of pull-out bond tests and of experiments on reinforced

concrete beams subjected to bending moment and shear force. They

proposed three rnaterial models for bond as s h o w in the following figures.

slip (mm)

a b c

Fig. 2-3 1 Bond stress and slip relationship proposed by Hayashi et al.

(stress unit: kgkm2)

Figure 2-3 l a is derived fiom the relationship between bond stress and

slip at some distance from a crack. Therefore, the bond deterioration near

the crack is not considered. The second model (2-3 1 b) considers the bond

deterioration near cracks. Where the bond stiffness changes, bond

stiffnesses are multiplied by a, which is determined by the response at

the distance fiom the crack. For the third model (2-31c), the maximum

bond stress of the basic bond stress slip relationship is limited to 40

kg/cm2 (4 MPa). The relationship between a and the distance from the

crack proposed here is illustrated in Fig. 2-32. When L is larger than 4 bar

diameter, a is equal to 1.

Kankam (1997) investigated the relationship of bond stress, steel

stress and slip in reinforced concrete for mild steel, cold-worked steel and

hot-rolled steel bars.

a

Fig. 2-32 Coefficient of local maximum bond stress deterioration.

The steel strain distribution curve for the plain round bar showed

very little change in form throughout the range of loading. This implied

that the px-imary mechanism of force transfer fiom the embedded plain

round bar to the surrounding concrete, resulting in the modification of the

steel stress between the two loaded ends, remain unchanged. The

maximum point of each strain distribution curve occurred at the loaded

end of the bar for a plain round bar, indicating a general decrease in slope

towards the central anchor point. For b&h types of ribbed high yield bar, a

typical curve was largely parabolic and characterized by double

curvatures and showed large differences in the magnitude of strain

between the ends and central anchor point. These large sbain differences

in the ribbed bars showed a consistent increase with an increasing appiied

load.

Curves of the slip distribution of the embedded plain round bar and the

liigh-yield ribbed bars were approximately parabolic in the initial stages of

low applied load but later, as the load increased, the tendency was towards

a linear distribution. The magnitude of slip of the nbbed bars was less

than that of the plain round bar. The difference in the magnitude of slip

shown by the high-yield nbbed bars and the plain round bar could be

attributed to the different surface patterns of the bars and the relative

resistance developed against the movement of each reinforcing bar

relative to its surrounding concrete.

Fig. 2-33 Bond stress distribution along 25 mm plain round bar.

Fig. 2-33 shows bond stress distribution curves corresponding to a

number of load increments. It was noted that although the change in the

curves of the bond stress distribution with respect to load increments

within the service range did not confirm to any unique pattern in the plain

bars, the maximum point of each cume always occurred at, or very close

to, the loaded end and the minimum point at the central anchor point. The

magnitude and distribution of the bond stress changed only slightly as the

load increased.

Fig. 2-34 Bond stress distribution for 25 mm cold-worked ribbed bar.

Fig. 2-35 Bond stress distribution for 25 mm hot-rolled nbbed bar.

For the ribbed bars (Figures 2-34 and 2-35), as the tensile load

increased, the form of the distribution curve generally changed slightly.

Nevertheless, it was impossible to describe the curves by any simple

consistent fonn. In general, there was; with increasing load, a consistent

increase in the bond stress at almost al1 points.

2.5 -, Slifl.04 mm Slip=û.OS min

Fig. 2-36 Bond stress-steel stress relationship for plain round bar.

From Fig. 2-36, it c m seen that the bond stresses decreased linearly

with the increase of steel stress when the bond slip was constant.

Fig. 2-37 is a typical curve of the fundamental relationship of bond

stress versus bond slip for different steel stresses for the 25 mm plain

round bar. In the early stages of relatively low slip, the curve showed a

linear increase of the bond stress with the slip, but later the relationship

changed to nonlinear following large increases in the slip. As the

reinforcing bar continued to slip, causing its surface aspenties to corne in

contact with different points of the surrounding concreta, the bond stress

at any given value of the slip was less for greater values of the steel stress.

This decrease in the bond stress would be expected due to the radial

contraction of the bar and cocsequent reduction of the confining pressure

of the surrounding concrete. The following empirical equation for bond

stress-slip relationship was proposed by Kankam:

T=(~Q-k fs)~o-8 (2-7)

b, kl are constants that depend on certain parameters such as concrete

strength, bar size, surface texture of bar, and type of loading; f, is steel

stress.

The authors proposed following values for and kl :

l ~ = 4 l . 7

k* =0.2

O 2 4 6 slip (0.01 ml$

Fig. 2-37. Bond stress-steel stress-slip relationship

for mild steel bar (f, is steel stress, unit: MPa).

For cold-worked and hot-rolled ribbed bars, it was found more

appropriate to examine the relationship between bond stress and slip,

since the steel stress was found to increase from the anchored midpoint to

the pulling ends at different positions along the bar. The bond stress for a

given arnount of slip increased in magnitude with the distance from the '

loaded ends of the specimens. At each position, the bond stress increased

with local slip. Near the center of the embedded bar, the relationship waî

first linear when the local slip was small, but became nonlinear as the slip

increased. At al1 other positions, the local relationship was generally

nonlinear.

Fig. 2-38 Bond stress-steel stress-slip relationship for

cold-worked tibbed bar (f, is steel stress, unit: MPa).

I O

O 1 2 3 4 5 6 7

slip (0.01 mm)

Fig. 2-39 Bond stress-steel stress-slip relationship for

hot-rolled ribbed bar (f, is steel stress, unit: MPa).

The initial dope of the bond stress versus slip curve was diflerent for

each of the positions and, in fact, increased consistently fkom the loaded

ends to the center of the embedded bar. The authors proposed the

following equation of local bond stress and bond slip for nbbed bars:

Cold-worked ribbed bar: T = ( ~ ~ - O . S X ) A ~ - ~ (2-9)

Hot-rolled ribbed bar: T = ( ~ S - O . ~ X ) A ~ - ~ (2- 1 O)

1

O 2 4 6 8

dip (0.Olmm)

Fig. 2-40 Bond stress-slip relationship for cold-worked ribbed bar.

O 1 2 3 4 5 6 7 siip (0.01mm)

Fig. 2-4 1 Bond stress-slip relationship for hot-rolled ribbed bar.

Nikon (1968) studied the spacing, width of cracks, and the distribution

of concrete stress in partially cracked members. In spite of the

considerable scatter of experimental data, Nikon recognized a definite

trend and fitted the following equation by using the least square's

method: 9 2 12 3 ~ 3 . 6 0 6 ~ 1 0 ~ ~ - 5 . 3 5 6 * 1 0 A +1.986*10 A (2-1 1)

where r is the nominal bond stress in psi and A is the local bond slip in

inches. Differentiation of both sides of equation (2-9) with respect to A,

yields: 12 2 dr/d~=3.606*10~-10.712*10~~+5.985*10 A (2-1 2)

which represents the stifiess of the concrete layers transfemng the forces

to the steel bar.

Later, Nikon (1972) reported results obtained using embedded gages to

measure the concrete strain distribution in tension specimens. The

concrete displacement curve was derived directly by intergrating the

concrete strains measiued at points at a distance of 0.5 inch from the bar.

The following bond stress slip relationship was proposed:

t=3 100(1.43~+1 SO)Afc (2- 13)

where the bond stress r<(1.43x+1.50) f, ;

and x is the distance fkom the loaded end in inches, and P, is the concrete

strength in psi .

The typical pull-out tests that many researchers have used are

nonuniform in the local bond stress distribution along the reinforcement.

In these tests, the actual bond stress varies significantly along the

embedment length, and results were reported as the average bond stress

versus slip measured at one end of the specimen. Abrishami and Mitchel .

(1996) proposed a new testing technique that simulates a uniform bond

stress distribution. A suitable combination of pull-out and push-in forces

can simulate a uniform bond stress distribution along the reinforcing bar

embedded in the concrete. This test method bas enabled a more accwate

detennination of the bond stress versus slip response. It simulates a

uniform bond stress by adjusting the top and bottom forces in the

embedded bar such that the strain distribution along the reinforcing bar,

measured by strain gauges, is essentially linear. This linear variation of .

strains results in approximately unifonn bond stress over the embedment

length. Figures 2-42 and 2-43 show the bond stress versus slip response

obtained fiom tests simulating a uniform bond stress distribution for

specimens failing by pull-out and splitting, respectively. They concluded

that the relationship of bond stress versus bond slip response is linear for a

brittle failure due to shearing of the concrete along lugs. On the other

hand, the relationship is nonlinear for a bond splitting failure that exhibits

a more ductile response. Afier splitting cracks form, the predicted bond

stress distribution is almost uniform and does not seem to be affected as

much by the type of specimen loading.

They proposed the analytical relationships (Fig. 2-44 and 2-45) of the .

bond stress versus slip response which capture the key behavioral features

of these two types of bond failure mechanisms, and derived analytical

solutions of both pull-out failures and splitting failures in pull-out tests,

push-in tests, and a combination of pull-out and push-in tests. The

predicted bond stress distribution is dependent on the length of

embedment, bar size, concrete properties, and the size of the specimen. As

the length of the specimen decreases, the bond stress becomes more

unifonn. Because of this, early attempts by the other researchers to

determine bond strength under nearly uniform bond stresses involved

short embedment lengths. However, these short embedment lengths gave

rise to umealistically high bond strength results. On the other hand, the .

use of a longer embedment length in a simple pull-out test gives a large

variation between the maximum and minimum bond stress. Hence, taking

the average bond strength for these specimens is not representative of the

actual bond strength.

9 1

O Y O 1 2 3 4

Average slip (mm)

Fig. 2-42 Bond stress-slip response for pull-out failure.

O Y , O 1 2 3 4

Average dip (mm)

Fig. 2-43 Bond stress-slip response for splitting failure.

Hota et a1 (1997) investigated the interface properties between a straight

steel fiber and two matrix materials. They concluded that the average

interface bond strength and the average interface bond stiffbess for a

matrix material containing polymer (10 percent by weight of cernent)

were approximately twice the values measured using a mortar matrix

without polymer. The energy required for debonding and pull-out was

also approximately doubled by adding 10 percent polymer by weight of

the cernent.

awrage slip

Fig. 2-44 Analytical bond model for pull-out failure.

awrage slip

Fig. 2-45 Analytical bond model for splitting failure.

Ayyub et al (1994) studied bond sîrength of welded wire fabric

(WWF) by pull-out tests and considered the following factors which may

affect the bond strength: penetration and strength of welds between

transverse and longitudinal wires; size of transverse wires; number of

transverse wires, and weld strength; and use of bundled longitudinal

wires. They concluded that the pull-out capacity of the WWF depends .

mainly on the bond strength between the WWF longitudinal wires, the

anchorage effects of the transverse wires, and the shear strength of welds

between the transverse and longitudinal wires. M e r the first slippage of

the longitudinal wires, the load is resisted only by the transverse wires and

the welds between the transverse and longitudinal wires.

The contribution of the adjacent longitudinal wires to the pull-out

resistance was very small, and the majority of the pull-out specimens failed

by bond between the wires and concrete.

Some researchers (Esfahani and Orangun, 1998; Thompson, 1968)

investigated the bond stress between concrete and reinforcing bars in

splices in beams. The test parameters included concrete compressive .

strength, cover to reinforcement, spacing between bars, splice length, and

the type of steel bars. Based on the tests of 22 rectangular simply

supported beams, Esfahani and Rangan proposed the following equations

to calculate the cracking bond strength of short length specimens:

For concrete with compressive strength less than 50 MPa,

7, = 4.9[(C/d +OS)/(C/d +3 -6) fa (2- 14)

For concrete with compressive strength equal to or greater than 50 MPa,

7, = 8.6[(C/d +OS)/(C/d +5.5) f, (2- 15)

In equations (2-14) and (2- 15), C is the minimum cover of concrete.

Equation (2-15) was obtained based on results of specimens made of

concrete with compressive strength of 50 and 75 MPa, and bars with rib

face angle between 40 to 47 degrees. For bars with rib face angle between

23 to 27 degrees, tests showed that the right side of equation (2-15)

should be multiplied approximately by 0.85.

The bond strength of splices was calculated by following expression:

r, = r, [(l+ l/~)/(l.85+0.024M-~)] (0.88+0.12C,J& ) (2-16)

wliere &,, is the minimum of side cover C, , bottom cover C,, and half

spacing between bar center (Cs +d)/2 (Cs is the clear spacing of

reinforcement), Gd is the median value of &, C, and (Cs +d)/2.

M was given by following expression :

M=cosh [0.0022L* (rf, /dlo" ] (2- 17)

where r is a constant which depends on the type of reinforcement.

Eligehausen, Popov and Bertero (1 983) investigated the bond stress-

slip relationship by extensive experimental study. They extended the slip

range up to very large values. Various bond stress-slip diagrarns fiom

experimental results are shown in figure 2-46. A typical diagram can be

idealized as a sequence of linear segments, as shown in Figure 2-47.

Up to a certain value of stress (ri in Figure 2-47), bond is due to

chernical adhesion of the cement paste on the surface of the steel bar and

practically no slip take place; typical values of rl range from 0.5 to 1 .O

MPa. For r>ri adhesion breaks down and bond is provided by fiction and

wedging action between the cernent paste and the rnicroscopic anomalies

(pitting) of the bar surface and also, in the case of deformed bars, by

mechanical interlock of the deformations and the surrounding concrete.

Due to these interlock forces, at a stress level -2 (which is a function of

the tensile strength of concrete as shown in Figure 2-47) bond cracks

form. At approximately the same time, separation of concrete f?om the

reinforcing bar takes place in the region of primary (flexural) cracks. This

separation causes an increase in the circumference of the concrete surface

previously in contact with the bar and, as a result, circumferential tensile

stresses develop. These stresses, in combination with the radial .

component of the force carried by the ribs or indentations, lead to splitting

cracks. At the stress F T ~ (Figure 2-47), these cracks propagate up to the

extemal face of the member and, if there is not enough confinement ,

bond is destroyed and a splitting failure occurs (Figure 2-46 and 2-47). On

the other hand, if the presence of adequate confining reinforcement

inhibits the propagation of the splitting cracks, the bond stress can reach

substantially higher values (t, in Figure 2-47).

7 , 1

O 1 2 3 4 5 6 slip (mm)

a. Local bond stress-slip curves without confinement.

b. Local bond stress-slip curves with confinement.

Fig. 2-46 Experimentally derived local bond stress-slip curves.

Ar A2 A3 4, slip ( mm ) 4

Fig. 2-47 Idealized local bond stress-slip curve for monotonie loading.

The slopes of the consecutive branches of the bond stress-slip diagram

gradually decrease; in other words, the value of the relative slip .

corresponding to a given increment of bond stress increases. Along the

branch defined by the stresses 73 and r,, , a gradua1 deterioration of the

concrete lugs (keys) between adjacent ribs occurs until, at a value of

FT,, , tliese lugs fail in shear .

The descending branch of the bond stress-slip diagram (A>&, )

corresponds to a complete deterioration of concrete between adjacent ribs,

and for A>& , the moderate amount of residual bond stress (r4 ) is due

exclusively to fiction at the cylindncal surface defined by the tips of the

ribs. The stress 4 can remain practically constant for high values of slip,

as shown in the experimental curves of Figure 2-46. The value of the slip

& almost coincides with the spacing of the ribs, since when a nb is

displaced to the position occupied by the adjacent one when loading

started. The only remaining rnechanism of the bond transfer is fiction at

the cyl indrical failure surface.

The bond stress and slip diagram for rnonotonic loading remains

approximately the same for loading in tension, as well as in compression,

provided that an adequate degree of Confinement exists. In the case of

unconfined zones, such as the areas of a beam-column joint which lie

outside the column reinforcement, the bond stress-slip curve is different

for tension and compression. In addition, when the stress in the bar

exceeds its yield strength, the lateral contraction (for tensile loading) or

expansion (for compressive loading) of steel bars will cause a decrease or

increase, respectively, in bond strength. It has been found that this lateral

deformation cannot affect the bond strength more than 2030%, even for

very large steel strains.

Figure 2-48 shows the influence of transverse (confining) pressure,

resulting either h m compressive radial stresses (such as those acting

beam-column joints where beam bars* are anchored or pass through) or

from confinement. As can be seen in the Figure 2-48, both the maximum

bond stress and residual stress due to friction (r4) increase with the

confining pressure, but in a nonlinear fashion. These data indicate the

favorable effect of confinement with regard to bond conditions. Indeed,

the presence of confining reinforcement inhibits a premature, brittle type

of bond failure due to splitting, and in addition it increases bond strength.

O 2 4 6 8 10 12 slip (mm)

5 10

confining pressure (MPa)

Fig. 2-48 Effect of lateral pressure on local bond stress-slip relationship.

Table 2-1. Confining pressure in experiments of Figure 2-48 (a )

curve 1

confining pressure (MPa) O

There are numerical values for the various parameters of the model,

based on the tests by Eligehausen, Popov and Bertero (1983 and 1986) on

concrete specimens (fc=30 MPa) with 25 mm diarneter deformed bars,

having a rib spacing of 10.5 mm and a relative rib area of 0.66. The scatter

in the experimentally derived values of the various parameters was also

pointed out by these investigators, and as a result they decided to suggest

empirical coefficients for correcting the values given in Figure 2-47

wlienever the shape of the deformed bar, the concrete strength and the

spacing of bars are different from those used in their tests. Values of t,,

in the literature Vary from about 10 to 21 MPa, while values of A,, ,

which show considerable scatter, Vary from 0.25 to 2.5 mm.

With regard to the influence of confining pressure, they suggested the

relationship shown in Figure 2-48 @), while for the effect of rate of

application of slip, shown in Figure 2-49 (a) for three different rates, the

approximation 2-49 @) was suggested. It is noted that an increase in the

slip rates of 100 times results in increases of r,, and r3 about 15%.

O 2 4 6 8 10 12 dip (mm)

-2 -1 O 1 2 log of rate of slip rs

( b ) Fig. 2-49 Effect of rate of slip on the local bond stress-slip relationship.

Table 2-2. Confming pressure in experiments of Figure 2-49 (a )

0.034 0.02

2.4.2 Bond Stress and Slip Under Cyclic Loading

Bond behavior under cyclic loading is affected by the following factors:

1. Concrete compression strength.

2. Cover thickness and bar spacing.

3. Bar size (bar diameter).

4. Anchorage length.

5. Geometry of bar deformations (ribs).

6. Steel yield strength.

7. Amount and position of transverse steel.

8. Casting position and vibration.

9. Strain (or stress) range.

1 0. Type and rate of loading (strain rate).

1 1. Temperature.

12. Surface condition-coating.

It has been pointed out that the influence of many of foregoing

parameters on bond resistance is only qualitatively understand. Parameters

(1)-(3), (6)-(9, and (12) appear to be the ones mostly affecting bond

under monotonic loading, while (9), -(IO) and the value of maximum

imposed bond stress, in addition to previous parameters, are very

important under cyclic loading conditions.

Tassios and Koroneos (1984) investigated local bond stress-slip

relationship by an optical experimental method (the Moire method) in

rnonotonic and cyclic loading. They proved that the point of maximum

local bond stress moved towards to loaded end with an increase of load

(Fig. 2-52).

Fig. 2-5 1 Specimen tested by Tassios.

Fig. 2-52 Bond stress-slip relationship for rnonotonic loading. (unit of fs: MPa)

Fig. 2-53 Steel stress distributions for rnonotonic loading.

(unit of fs: MPa)

O 10 20 30 40 50 local siip ( mm11 000 )

Fig. 2-54 Local bond stress versus local slip curves

in rnonotonic loading.

Fig. 2-55 Bond stress distributions for various loading

levels of cyclic loading. (unit of fs: MPa)

Fig. 2-56 Bond stress distributions for various cyclic loading levels.

(unit of fs: MPa)

Fig. 2-57 Slip distribution for various cyclic loading levels.

(unit of fs: MPa)

From Fig. 2-54, it can be seen that a common bond stress-slip curve

may be traced with a reasonable scatter of points away fiom the loaded

end (x:L<0.75). On the contrary, for points near the loaded end

(x:L)0.75), a much lower and distinctive local bond stress-slip

relationship is found which gradually tends to zero bond stress levels for

finite slip values. f, in Figures 2-52 to 2-57 is the stress in the steel bar at

the loaded point.

Under cyclic loading, bond deteriorated with increasing number of

loading cycles. Figures 2-55 to 2-57 show the distribution of bond stress

and bond slip along the steel-concrete interface for several levels of

cyclic loading. Residual steel stresses, bond stresses and irreversible slips

are observed at zero external loading. It was noted that the considerable

difference in bond stress distribution for equal external loading depends on

loading history.

The constitutive law of bond stress-slip under cyclic loading was given

experimentally by means of the Moire method in Figure 2-58.

-80 -40 -2Q O 20 40 W

slip ( 0.001 mm )

Fig. 2-58 Local bond stress-lacal slip relationship under cyclic loading .

slip

Fig. 2-59 Relationship of bond stress versus slip under cyclic loading.

Tassios et al (1979 and 1981) studied bond behavior under cyclic

loading. They developed the constitutive curve for local bond stress versus

local slip under cyclic loading shown in Fig. 2-59. Parts of OAM and ON

are the constitutive response under monotonic loading. Unloading from H

follows the bond stress path HCDE, where HC is parallel to OA; CD is at

the negative residual bond strength level -Tri and DE is along the

monotonic compression curve. Unloading fiom E follows the stress path

EFGH, where EF is parallel to OA; FG is at the positive residual bond

strength level 7,; and GH is parallel to OA and passes through the

previously most tensioned point, i.e., point B. Further loading fiom H to 1

moves along the monotonic curve, and unloading fiom 1 follows the stress

path IJKL, where KL is parallel to OA and passes through the previously

most compressed point, i.e., point E. The residual positive and negative

bond strengths were not kept constant during load cycling, but were

considered to deteriorate with increased number of loading cycles. The

authors did not give the numerical value of the residual bond strengths at

either the initial stage or during the cyclic loading.

Ismail et al (1972) studied bond deterioration under low cyclic loads

by pnsm tests and beam tests. They arrived at the following conclusions:

Firstly, the most important factor affecting stress transfer was the peak

stress reached in the preceding cycles. If the peak stress was increased, the

stress transfer at lower stresses was reduced in subsequent cycles. The

most significant reduction was at stress levels well below the peak stress.

Secondly, a small number of repetitions of load cycles with constant peak

stress produced a gradua1 deterioration in stress transfer, but the reduction

was minimal in cornparison with the reduction associated with increases

in peak stress. Thirdly, the bond stresses were higher in those specimens

in which compressive stresses were applied.

Siva Hota et al (1 997) studied the bond stress-slip response between

reinforcing bars and fiber reinforced concrete (FRC) under monotonic and

cyclic loading by pull-out test. They tested the four matrix types: SIFCON

(a fiber reinforced cernent composite with a relatively large fiber volume

fraction between 5 percent to 20 percent), fiber reinforced concrete (FRC,

2 percent fibers), plain concrete (PC), and confined concrete (CC). Tests

showed that al1 SIFCON specimens failed by friction pull-out,

accompanied by large slip, while the concrete surrounding the

reinforcement remained together. On the other hand, the failure of the FRC

specimens started with some fiictional pull-out but eventually ended up

with a splitting type of failure. Al1 of the plain concrete specimens failed

by splitting, while al1 the CC specimens failed in a cone-shaped manner.

The addition of steel fibers to a cernent matrix was shown to improve the

overall bond properties of a reinforcing bar. High fiber content in the

matrix can lead to higher bond strength, higher ductility, reduced post-peak

degradation of the pull-out load versus displacement curve, increased

energy absorption, and ultimately a significantly increased safety level

against collapse and failure. The increase in compressive strength of the

matrix increased the bond strength. Since the tensile strength of concrete is

relatively low, cracking occurs early in concrete. At the onset of cracking,

the role of the steel fibers is to prevent m e r crack opening and to resist

additional tensile forces which the concrete matrix itself can not sustain.

This controls the failure in the concrete matrix itself, thereby presewing

the bond strength between the reinforcing bar and the surrounding concrete

matrix. The higher the volume fiaction of fibers, the higher the bond

strength and the higher the area under the bond stress versus slip response.

The fibers may not much delay the formation of the first crack, but they

keep crack width at small values and prevent the sudden opening of cracks.

The plain concrete specimen with no fibers had very low slip values at

maximum load, while the SIFCON specimen had values up to five times

larger. The presence of the fibers is significant in the increase pull-out

energy and ductility.

The bond stress-slip relationship depends on a number of parameters

such as the concrete compressive strength, concrete cover, anchorage

length, and the rib area at rnonotonic load and first loading cycle. Under

cyclic loading, at low bond stresses, inclined cracks propagate fiom the tip

of the ribs. The transfer of forces across the interface between concrete and

steel bar occurs and is caused by bearing and adhesion. As the loading is

increased, inclined cracks begin to form.

For unidirectional loading, the use of fibers played a major role in

slowing down degradation of the specimen's load canying capacity by

delaying the onset of cracks and slowing down the opening of existing

cracks.

For fully reversed cyclic loading, the degradation of bond strength and

bond stifhess occurs faster than for the unidirectional cyclic loading. The

degradation depends on the maximum slip value at which reversa1 of

loading occurs in either direction. The larger the slip at which reversa1

occurs, the more the damage that results in the reversed loading.

For the same number of cycles, the hlly reversed cyclic loading af5ected

the bond strength and stiffness more severely than the unidirectional cyclic

loading.

Eligehausen, Popov and Bertero (1983) studied the bond stress-slip

relationship under cyclic loading by extensive tests. They showed that the

envelope of bond stress-slip hysteresis loops for repeated loading lies very

close to the curve resulting fiom monotonic loading as shown in Figure 2-

60. It is worth pointing out that even after 20 cycles of consecutive loading

and unloading, the envelope of the .loops remains quite close to the

rnonotonic loading curve, which means that the mechanism of bond

deterioration remains the sarne as that described in the previous section.

O 1 2 3 4 5 6 slip (mm)

Fig 2-60 Local bond stress-slip relationship

for repeated loading.

slip

Fig. 2-6 1 Local bond stress-slip expec-imental curves

for cyclic loading of confined specimens by Eligehausen et al.

Figure 2-61 shows bond stress-slip hysteresis loops derived fiom cyclic

loading tests of specimens with one 25 mm deformed bar and confinement

reinforcement. It is first pointed out that, as in the case of repeated loading,

the residual slip during unloading @ranch EF in Figure 2-61 ) is quite

large, which may be attributed to the fact that the elastic part of slip

consists of the concrete deformation only, which is just a small portion of

the total slip. Whenever the sign of the bond stress reverses, the slope of

the curve remains significant up to a level of stress (ri); this increased

stifhess is due to friction between the bar and the surrounding concrete.

When the frictional resistance is overcome, the bar begins to slip in the

opposite direction (with respect to that of initial loading OAE) until the

ribs of the bar again corne into contact with the surrounding concrete (point

1 in Figure 2-61). It is understood that the foregoing applies when the level

of Loading is such that concrete lugs between the adjacent ribs (see also

Figure 2-61) have been ground, thus creating gaps between the side face of

the ribs and the surrounding concrete.

Reloading in the opposite direction (branch IAIE1 in figure 2-61) is now

taking place at a significantly increased slope and the path followed is

similar to that of monotonie loading. However, if the maximum previously

attained (absolutz) value of bond stress is higher than 70-80% of r,, , the

new envelope (OAIBiCl in Figure 2-61) has reduced ordinates with respect

to original one (OABC). This reduction in available bond resistance is

inore pronounced as the values of slips between which cyclic takes place

increase, and also as the number of cycles increases.

Whenever at a certain point the sign of loading changes (in a

deformation-controlled test, when the sign of the applied slip changes), the

unloading and friction branches (IKLPIiIN) are similar to the previous ones

(EFGHI). Further loading is now taking place dong a new envelope

(OA'B'C') whose ordinates are reduced with respect to the initial one

(OABC). If the level of loading is hi& enough for shear cracks to fonn in

concrete lugs between adjacent ribs, only a portion of these lugs can

contribute to the resistance of the system. Hence, the envelope OA'BfCf

has lower ordinates than the previous one (OA'iBfIC'i). Moreover, if

unloading takes place at a point along the descending branch of the bond

stress-slip curve which corresponds to a pronounced deterioration of

concrete lugs due to shear, the fictional resistance (rfu ) will be higher than

previous value (rr ), since at this stage the interface between the bar and the

surrounding concrete is rougher. This characteristic may be verified if the

corresponding branches of the two loops in Figure 2-61 are cornpared.

It is seen fiom the foregoing discussion of the bond degradation

mechanism under cyclic loading that most of the damage occurs during the

first loading cycle. During subsequent cycles, a gradua1 smoothing of crack

interface occurs, causing a reduction of mechanical interlock and fiction

forces.

Eligehausen et ai (1983) studied experimentally the behavior under

cyclic loading of specimens with 25 mm diarneter deformed bars, having a

clear length of 5d and hooks at their ends. The typical local bond stress-slip

curves for a hooked bar specimen are shown in Fig. 2-62. It is clearly seen

that the available bond resistance under rnonotonic loading remains almost

constant, even for very large values of slip, in contrast to what happens in

bars without hooks. Furthemore, during successive reversed loading cycles

a significant drop of bond resistance is observed for values of the slip lower

than the previously attained peak. M e r this value is reached, bond

resistance is soon recovered, and the corresponding curve at large values of

slip lies quite close to the rnonotonic loading curve, even afier 10 loading

cycles.

Figure 2-61 shows bond stress-slip hysteresis loops derived fiom cyclic -

loading tests of specimens with one 25 mm deformed bar and confinement

-4 O 4 8 12 16 20 Slip ( mm )

Fig. 2-62 Local bond stress-slip curves

for deformed bars with hooks by experiments.

2.5 The Study of Bond Stress and S1ip.b~ Theoretical Analysis Method

2.5.1 Constitutive Equation of Bond Stress and Slip

The increment of the local slip dA within an infinitesimal bar length dx at

the location x can be defined as the difference between the bar strain E,

and the concrete strain E ~ ~ :

ciA/dx=~,-~~~ (2- 1 8)

The differentiation of the equation (2-18) with respect to x gives:

d2A/d2x=de, /dx-d&,/dx (2- 19)

Fig. 2-63 Free body of a reinforced concrete member.

As shown in Figure 2-63, by cutting the segment at x and taking a fiee

body, it gives :

P=Pc+Ps=AcEc~a+ASES~SX = &E,(~,,+nps,) (2-20)

The differentiation of the equation (2-20) with respect to x gives:

&Ec (d&,/dx+npds,/dx)=O (2-2 1 a)

ds,,/dx=-npds,/dx (2-2 1 b)

substituting equation (2-2 1) into equation (2- 1 9) gives:

d2A/d2x=(l +np)d&, /du (2-22)

r,~d=(xd2/4)a, (2-23a)

r,=(d/4) a,=(Esd/4) ds- /dx (2-23b)

substituting equation (2-23) into equation (2-22) gives:

d2A/d2x=[4(i +np)/(&d)] rx (2-24)

Equation (2-24) is a differential equation of local bond slip and local

bond stress. From this equation, it can be seen that some of the factors that

influence the relationship between bond stress and slip are elastic modulus

of concrete and steel bar, percentage of reinforcement and diameter of steel

bar.

2.5.2 The Solution of the Constitutive Equation of Bond Stress and Slip

Because the measurement of local bond stress and local slip along a

stressed reinforcing bar and the surrounding concrete is difficult and very

sensitive to expenmental errors, there has been a wide variation in deduced

bond stress versus bond slip relationships derived from the results of

various experiments. Some researchers developed analytical models based

on various assumptions.

Shah et al (1 98 1) assumed bond stress distribution as:

D 2 ~ d 2 x = ~ e " +Be-' +C (2-25)

where x is the distance fiom the loaded end of the specimen to any section.

From this assumption, it is obvious that the relationship of local bond

stress versus local slip is a function of the distance of the section fiom the

cracked face.

For the pull-out specimen in Figure 2-63, Shah et al solved the

differential equation (2-25) by boundary conditions:

A= ~ e * +Be-' +cx2/2 +Dx +E (2-26)

the five constants A-E are related to cross sectional dimensions A, and 4,

total perimeter of steel bars, constitutive relationships of steel bar and

concrete, the transfer length L, and the length of segment or crack spacing.

The above assumption rnakes the location of the peak bond stress close

to the center of the transfer length. Test results show that peak bond stress

occurs at the section closer to crack face than to the other end of transfer

length. The relationship between bond stress and local slip is not unique

but varies from location to location. Based on these test results, Yang and

Chen (1988) assumed the following relationship for bond stress versus

slip:

T(X) = KA(x) +(ex2 +D) +Ecos(x/~*x&) (2-27)

where K is the bond constant which is the bond stress per unit slip; C, D,

and E are constants to be determined by boundary conditions; L, is the

transfer length, and x is the distance from the loaded end of the specimen

or the cracking spacing to any section.

Yang et al employed the boundary conditions to determined the

theoretical solution of equation (2-24) as following:

A(y)=ZA{ch(cy) - ( ~ ~ ) ~ / 2 ch (c ) +p2[ch(c )-11 [l -cos(yd2)] -1 )

(2-28)

t(y)=ZAK {ch(cy) -ch(c )+[ch(c ) - 1 ] cos(yd2)) (2-29)

where y=x/Lb (2-30a)

a 4 ( l +np)l(dE,) (2-30b)

c = ( a ~ ) ' . ' ~ , (2-3Oc)

p=(2/x)c (2-30d)

ZA={sh(c) -ch(c) +p[ch(c) -11) QU (2-30e)

eo is the steel strain at loaded end of the specimen or cracking face of the

actual structure.

Fig. 2-64 Slip distribution along the bar.

Fig. 2-65 Bond stress distribution along the bar.

Fig. 2-64 and 2-65 show the curves o f A and T of the Yang and Chen

theoretical solution, respectively, with a L t =0.3 mm and c=2,3,4. It can

be seen that the peak values always occur near y=0.7 and move slightly

toward y=l with very weak dependence to an increasing c. Figure 2-66

shows the cornparison of solution of Yang and Chen with the solution of

the Somayaji and Shah method.

Fig. 2-66 Cornparison of bond stress distribution between Yang and Samayaji.

The authors compared the analytical results with those measured in

experiment.

Hamayoun et al (1996) developed an analytical method to predict the

ultimate tensile strength of fiber reinforced concrete when the failure is

govemed by the strength of the fiber concrete interface, and to account for

the interfacial slip between the concrete and the fiber. The model of

analysis is based on the concept of fracture mechanics with the presence of

a traction-free interfacial crack between the fiber and concrete. The model

is developed based on the assumption that both the fiber and the concrete

behave elastically.

For elastic fibers with L/r exceeding 100, the interface crack and the

fiber lengths have no major effect on the pull-out strain energy rate (L is

fiber length, and r is fiber radius); the interface crack becomes significant

if the fiber is ngid (when L/r is equal to or less than 25); and increasing the

fiber spacing increases the strain energy release rate.

2.6 Code Development

The AC1 Building Code (1983) bond strength of reinforcing bars of

No.1 l (36 mm) or smaller sizes in tension is given as:

~ , ~ ~ = ( f ~ & 4 ) [ f;' 0 . 5 / ( ~ . ~ 4 Abfy) PSI . (2-3 1 a)

where 0.04 Abf?=0.004df?=12 in (305 mm). (2-3 1 b)

The corresponding equation proposed by AC1 cornmittee 408 is

~~~~=(fsd/4)[(Kf~"-~)/(5 500&)(60,000/fy) psi (2-3 2a)

where (5500Ab )/(~f,'~~)*(60,000/f,)>=12 in (305 mm) (2-32b)

K=(C+Kw)<=3 d (2-32~)

KpAtrfY/( 1 5 00S)<=d (2-32d)

S(stimip spacing)<=Ld/2 (2-3 2e)

Mina (1987) investigated the variability of bond strength of bottom-

tension reinforcement in concrete beams. He used the Monte Carlo

technique to generate the variability of the ultimate strength of bond

between concrete and reinforcement zind to estimate statistical properties

of the bond strength by analyzing the simulated sample by considering

following parameters : ratio C/d , where C is the smaller of the bottom or .

side cover to the center of flexural tension reinforcement, and d is the

diameter of longitudinal bar; ratio of transverse reinforcement A&f/sd ;

concrete strength; and grade of flexural steel. The author proposed the

fol lowing equation for calculating bond strength :

rh=(fyid/4) { [ 1 .2+3 C/d+(Ayfyt)/(5 00sd)]/[d(fyI/(.62 5 fspr)O)] } psi

(2-33a)

where (AUf,J/(500sd)~=3, (2-33b)

C/d<=2.5 (2-33d)

Cs/(Cbd)<=3 (2-33e)

fYl and fF are the yield stresses of the longitudinal and transversal

rein forcement, respectively, and f, is the splitting strength of concrete.

Due to the nonuniformity in the bond stress distribution, no theoretical

method currently exists for evaluating bond strength of a reinforcing bar

embedded in concrete. Hence an "average bond stress" or "average bond

strength" is used in codes of practice. Several studies have show that

there is a significant variation of the actual bond stress distribution, with

the maximum bond stress in some cases being much greater than the

average bond stress. In addition, it has been shown that the bond stress

distribution varies greatly as slip develops.

Mirza estimated the ultimate bond strength for varied material strengths

and steel properties by a Monte Car10 analysis, and compared the

theoretical strengths with the results fiom AC1 408 method (equation 2-32)

and AC1 Building Code method (equation 2-3 1). It was found that AC1

408 method produced bond strengths having less variation with the

theoretical value than did the AC1 building Code method.

2.7 The

Ngo

Analysis of Bond Stress and Slip by Finite Element Method

and Scordelis (1967) were the fust to use the finite element

method to analyze reinforced concrete structure accounting for bond-slip

effects using a linear relationship between bond stress and bond slip. Later

in 1968, Nilson studied the nonlinear analysis of reinforced concrete by

finite element method. Nilson considered nonlinear bond stress-slip

relationships, nonlinear material properties, and the influence of

progressive cracking. A third degree polynomial was employed as the

relationship of bond stress versus slip. That is:

6 2 9 3 s=3606*103~-5356*10 A +1986*10 A (2-34)

Shava et al (1989) developed a elastic-plastic cracking constitutive

model for the analysis of reinforced concrete by considering the

nonlinearities due to tensile cracking, aggregate interlock, plasticity in

compression, yielding of steel, bond stress and slip, and tension stiffening.

Mehlhom et al (1985) developed one- and two-dimensional

isopararnetric contact elements and used these elements to analyze the

reinforced structures.

Shiro and Morita (1987) thought that the bond model used in linkage

elements requires quite cumbersome calculations to determine the global

behavior and did not include the mechanism of stress transfer by wedging

action. They assumed that the concrete deformation due to interna1

secondary cracking occurs at the interface and it was involved as a part of

the local slip of the bond model. They proposed a combined link element

which consists of the slip link element ( linkage element) and wedge link

element to account for wedging action.

Keuser et al (1987) investigated the properties of the elements and the

quality of the results which are affected by the displacement function of

the elements, the density of the element mesh, and bond stress-slip

relat ionship.

The quality of the results depends strongly on the design of the element

mesh and the stiffnesses of individual elements. For the evaluation of an

element stiffness matrix, two basic assumptions are made; one conceming

the displacement function and other regarding material behavior.

Differences between an exact solution and results of an finite element

analysis are caused by the fact that finite elements actually model only part

of the solution with the same or a lower order than the displacement

function used for the finite elements. Higher order effects are only

approximated.

The influence of the displacement function is investigated by an energy

considerat ion.

For a loading structure, the intemal energy in the contact area of steel

and concrete is:

In finite element analysis, the interna1 energy is:

The energy values for elements with different displacement functions

and for various numbers of degrees of fieedom were calculated using

equation (2-36) . These results were compared to theoretical values which

were obtained by equation (2-35).

The authors studied the results of five prescribed linear or cuwed

functions and concluded that:

1. The bond linkage element with its constant displacement function is

not very well suited to mode1 non-constant slip curves. The

approximation of antisymmetric portions is especially poor. The

results can be improved by increasing the number of degrees of

fieedom, but the convergence is still rather slow because the bond

stress and slip curves are approximated by stepwise functions.

2. The contact element with a linear displacement function models

linear slip curves exactly and even gives good approximations of the

nonlinear slip curves with only a few elements.

3. Results for the contact element can be improved by use of quadratic

or higher order displacement functions, but the difference is much

smaller than between the use of a constant displacement function

and a linear one.

Allwood and Bajman (1996) also modeled nonlinear bond stress-slip

behavior for the finite element method. They ignored the adhesion effect,

and assumed that the contribution of friction and interlock is proportional

to radial pressure. The radial pressure between bar and concrete cornes

fiom the total effect of:

1. Pressure on the bar generated by the shrinkage of concrete during

setting.

2. Changes in diameter of the bar due to Poisson's ratio as axial stresses '

are developed in a bar.

3. Lateral confming stress in the surrounding concrete.

They developed a new approach for bond stress and slip analysis within

the finite element method. The b a i s of the method is to analyze the

concrete and steel separately and then to bring the two solutions together

by a rapidly converging iterative process that adjusts the bond stresses

linking the two components together.

Pochanart and Harrnon (1989) studied the bond stress-slip model

including fatigue under cyclic loading. By the load-controlled test and slip-

controlled test, they proposed an analytical model of bond stress-slip for

rnonotonic load and cyclic load, including a reduced bond stress-slip.

envelope .

CHAPTER 3

THE MODELING OF BOND STRESS-SLIP

BY FINITE ELEMENT METWOD

3.1 htrodunt' ion

Accurate prediction of the nonlinear response of reinforced concrete to

loads using the finite element method is dependent on the knowledge of

several complex phenomena such as the behavior of concrete and steel,

relationship of bond stress versus slip at the steel-concrete interface,

aggregate interlock at the cracks and time-dependent phenomena such as

creep and shrllikage.

There are two basic approaches on which the bond stress-slip modcl are

applied: (1) rnicroscopic analysis of the stress state in the neighborhood of

the reinforcement (Gajer and Dux in 1990, Ingraffea et al in 1984), and (2)

macroscopic analysis of the global behavior of a member or a structural

assemblage (Ngo and Scordels, 1967; Nilson, 1968; Keuser and Mehlhorn,

1987).

In rnicroscopic analysis, the stress transfer mechanism by bond should

be treated as the local contact in front of bar ribs, not as the one

dimensional bond-slip at the interface. The interface slip is due to cmshing

of concrete within the limited zone in fiont of ribs, and can be derived fiom

material properties and the configuration of the bar. It is essential to mode1

appropriately the radial action of bond, as well as shear transfer parallel to

the axis of the bar. Stress States in the concrete, especially in the

neighborhood of the reinforcement, should be evaluated accurately, and

initiation and propagation of interna1 cracks, both longitudinal splitting and

cone-shaped cracks, should be followed analytically.

Thus, the local bond stress versus the local slip relationship cm be

denved as the result of the analysis. Therefore, the bond stress-slip

relationship depends on the local stress state and in general varies from

location to location.

In macroscopic analysis, the bond stress versus the slip relationship may

be directly modeled by determining the constitutive law of a one- or two-

dimensional bond element using experimental data or the results of

microscopie analysis. The stress state of the concrete near the bar is not

more realistic, but it is a fictitious one to sirnplifi the behavior.

Reinforcement can be introduced in a finite element mesh in three ways.

In the first method, steel is represented as discrete bar or beam elements

connecting the nodes of the finite element mesh. In the second method (the

embedded steel element), steel is placed anywhere in the finite element

mesh by embedding the steel bar element within the concrete element and

enforcing displacement compatibility through interpolation and

transformation. In the third method (the smeared steel element), a

composite steel concrete material matrix is employed, and this requires a

minimum number of elements and nodes. Third method can be employed

advantageously in structures where steel is distributed throughout. The

drawback of the latter two representations is that bond stress and slip can

not be modeled. The discrete bar elernents can mode1 bond stress and slip

through the provision of linkage elements or special bond-slip elements.

The stress strain curve usually employed for steel is either elastic-perfectly

plastic or elastic-hardening .

The nonlinearity due to bond stress-slip is usually modeled either by

using linkage elements or by special contact elernents. The parameters for

these elements are obtained fiom experimental bond stress-slip

relationships.

Tension stiffening of concrete can be modeled in two ways. In the fust

method, it is assumed that the tensile stress in concrete reduces gradua-lly

to zero &er tensile cracking. In the second method, a modified stress

strain curve for concrete is used. The dowel action of steel can be modeled

by using a normal stifhess for bond-slip elements. Aggregate interlock is

usually modeled by assuming a positive shear modulus after cracking.

In the finite element method, the elements are connected to each other

at the nodal points. The adjacent elements have identical displacements at

the common nodes; referred to as the displacement compatibility. If the

bond slip between concrete and reinforcement is taken into account, the

condition of displacement compatibility will not be satisfied. Special

interface elements will have to be used in conjunction with the discrete

concrete elements and reinforcement elements, while constitutive laws will

be required to mode1 bond stress and slip between these two components.

In finite element analysis, the linkage element, developed by Ngo and

Scordelis in 1967, has been used most commonly for modeling bond-slip

behavior. This element connects one node of a concrete element with one

node of an adjacent bar element. The linkage element has no physical

dimensions, so two connected nodes have identical coordinates before slip

occurs in the interface between steel bar and concrete. For plane stress

problems, a linkage element consists of two springs, one parallel and the

another one normal to longitudinal axis of the reinforcing bar.

Compared with the linkage element, the contact element (also called

bond-zone element), which was developed by de Groot et al in 1981, is

completely different. The most important differences are that contact

element has the dimension along the steel-concrete interface (it does not

have physical dimension in other two directions) and it provides a

continuous contact surface between - steel bar and concrete. The contact

surface between the steel bar and the concrete in the immediate vicinity of

the steel bar is modeled by a bond stress-slip law which considers the

special properties of the bond zone.

3.2 Contact Element

The bond between steel and concrete is modeled by a double node contact

element. The order and type of the contact elements should be compatible

with the order and type of the steel bar and concrete elements. For example,

if the concrete element is a two dimensional isoparametric element with nine

nodes and the steel bar element is a one-dimensional isoparametric element

with three nodes, a one dimensional parametic contact element with three

double nodes is required to connect it to the concrete and steel elements.

in the contact interface, the two elements connected by a contact

element have independent element nodes, and double nodes of the contact

element also have independent element nodes. One group of nodes of the

contact element is connected to the steel bar element, and another group of

nodes of the contact element is comected to the concrete element. In the

unloaded state, these double nodes have identicai coordinates. In the loaded

state, there are displacements between the double nodes so the double

nodes have the different final coordinates.

In this thesis, the contact element considered is the two double-noded

element with a linear displacement function for the plane problems. Two

nodes of a concrete element are connected to the two nodes of a steel tniss

bar element by the two double nodes of a contact element.

It is assumed that the relative displacernent of any point in a contact

elernent after loading in local coordinate system is :

matrix form

The relationship between nodal relative displacements A and nodal

displacements { 6 } is :

deformed element

u n d e f k e d i element .

Fig.3-1 Relative displacements A.

The joint coordinates in the local coordinate system to element k are:

Double nodes local coordinate r

i and j

mandn

The relative displacements of two 7 -

. O

lk

double nodes i , j and m, n are:

Substituting Eq. 3-4 into Eq. 3-2, the following relationship between the

displacements of any point and the relative displacements of two double

nodes in a contact element is got:

{ f ) =N (a)=PJl [Al -' {A} (3-5)

Substituting Eq. 3-2 into Eq. 3-5, the relationship between the

displacements of any point and the displacements of two double nodes in a

contact element is obtained:

{ f ) =N [AI -' ICI( 8 1 (3-6)

In the local coordinate system, the relationship between contact stress

and relative displacements in a contact element is

J

(3-7)

where [G] is the constitutive law matrix between contact stresses and

relative displacements in the contact interface in the local coordinate system.

($1 is the initial stress vector in a contact interface in the local coordinate

system.

I f one considen the special case in which the contact behavior in the two

is independent of each other, the constitutive law coordinate directions

matrix will be - O

Gtt-

Thus the strain energy per unit volume is :

U=% (oT {O )=% ( f ) ([G] {f )+{ao ))

The potential energy expression of element k is :

nplc=u-(p)rnfcT {8)l~=xI V O ~ ( { ~ } ~ [ ~ ] { f } l c +{f } k T {go ) k)d~tc-(p)rnk~ (5) ~t

(3-9)

Substituting Eq. (3-6) into Eq. (3-9), one obtains:

n P = j W~ r r 6 1 * [cT([AI - ' ) T ~ T [ ~ ~ ~ ~ [ ~ ~ -l [CI ( 6 )

+@) k T r ~ ~ T ( [ ~ ~ k - l ) T ~ T { ~ G 1 d dvk 4p)&1 IC (3- 1 O)

The sum of the potential energy for z structure with n contact elements is :

T O + 2 (6) i T [ ~ ] T 1 ([A] i 'OTIN] {a ) i dvi 2 (O) i T{~}m i

1=1 vol

r = l

T O + 2 I = I {d)iT[clT Id ([A] i- ')T[~] {a ) i dvi- 2 (d) i T ( p ) i

i=l

wliere {p), i and {p) i are the nodal load vectors for the contact element i in

the local and global coordinate system. (P) is the nodal loading vector of

the whole structure in global coordinate system. {dli is the nodal

displacement vector of the element in the global coordinate system.

(d) i =

and {D) is the nodal displacement

global coordinate system. -

Dl

..... D2

{Dl= DI Di+ 1

...... Dm1 D n -

vector of the whole structure in the

NOW let [BI i = N [A], [Cl -1 T Then @3] iT=[clT([~] i ) mT

The element stifhess matrix

The initial load vector is

{r) *=I, ~ ] k = { ~ o ~ I c d ~ k

The load vector is :

W k = w mk + { ~ } m k

of the element K in the local coordin

Then the element st if iess equation of element k in the local coordinate

system is:

[SI mi^ { 8) L= (RI L: (3- 19)

Equation (3-1 1) becomes

we get the structural s t i f iess equation as following from (3-1 8):

[SI i and [SI are the element stifiess matrix of the element i and the

stiffness matrix of the whole structure in the global coordinate system.

From above equation, it c m be shown that:

3.3 Linkage Element

A linkage element is required at each node to connect between steel bar

elements and concrete elements. The element has no physical dimensions.

It connects two nodes with identical coordinates and can be conceptually

thought of as consisting of two linear springs. One is parallel to the steel

bar axis and another is normal to bar axis. Both nodes occupy the sarne

coordinate in space before loading, but they undergo relative displacement

by the deformation of the linkage springs, resulting in different coordinates

after loading. The component of linkage force in the direction of the bar

axis gives the bond force, and the normal component of linkage force gives

the radial splitting force.

concrete

I concrete

reinforcing bar

Fig. 3-2 Linkage element.

The relative displacements at nodes i j are :

The radial and tangential stresses are given by:

or fd=[Kl{ A1 (3-33)

Thus the strain energy per unit volume is :

u=% { G } ~ {A )=i/2{s)T[c]T[~~ [CI {ô) (3 -34)

The potential energy expression of element k is :

npk=u-(p}mkT (6 }k=x {ô}? [c]~[K][cI (6)t -{p)mkT ( 6 ) k (3-3 5 )

The sum of the potential energy for a structure with n linkage elements is:

where {dli and {D) are the nodal displacement vectors of element i in the

local coordinate system, and of the overall structure in the global system,

respectively. {ph) and (pl i are the nodal loading vectors of linkage

element i in the local coordinate system and global coordinate system,

respectively .

Letting XiddDi =O for i=1,2 . . . . n.

One obtains the stifbess matrix of element k in the local coordinate

system as following:

and the stifbess matrix of element k in the global system is:

[S]k= [TI k [clT[K] [CI [ ~ ] k = [ ~ ] t [ ~ ] r n [ ~ ] k (3-38)

and the stifiess equation of element k is :

The stiffiless matrix of the whole structure is:

The stiffhess equation of the whole stnicture is:

[SI {DJ=fFf (3-41)

Substituting the matrices [KI, [Cl and [Tl (see section 3.5

Transformation Mamx) into above equations, we get:

[SI, = O -Kt O

3.4 Temperature Changes, Prestrains and Support Displacements

The effects of temperature changes, prestrains and support

displacements are considered as following:

Firstly, one calculates the strains (initial strains) of the concrete element

and the steel element at the interface due to temperature changes,

prestrains and support displacements. Then one calculates the nodal initial

dis?lacements due to the initial strains. Assuming the initial nodal

displacements are (Do), the stifiess equation of the whole structure can

3.5 Transformation Matrix

As shown in Figure 3-2, the relationship between the displacements

{6Ic the nodal forces (RIk in local coordinate system, and between the

displacements (dlr and the nodal forces {FIi, in global coordinate system ,

are :

{6)k=[T]k {d)k (3-47a)

{R)k =[T]k {F)k (3-47b)

so { ~ I ~ = [ T I ~ Tm1, (3-48a)

and {F) l~=[T]i, {R) k (348b)

where [Tlk is the transformation matrix of elemtnt k.

I

Fig. 3-3 Relation of local coordinate

and global coordinate system.

3.6 Material Properties

The Modified Compression Field Theory (MCFT) was proposed by

Vecchio and Collins (1986) for predicting stresses, strains and deformation

response of reinforced concrete subjected to in-plane shear and normal

stresses. The theory presented new constitutive laws for modeling the

material response of concrete and steel reinforcement, based on the results

of 30 reinforced concrete panels tested in pure shear or in combinations of

shear and normal loads. The theory has yielded excellent agreement with

many experimental results tested by many researchers. Vecchio (1989)

developed a nonlinear finite element program (TRIX) for plane stress

analysis of reinforced concrete membranes according to the formulations

of the MCFT. In the program, cracked reinforced concrete was treated as

an orthotropic material using a smeared, rotating crack modeling approach.

The solution procedure used was based on a secant-stifThess formulation,

giving good numerical stability. Vecchio and other researchers improved

the mode1 of material properties after that and produced the 1999 version

TRIX99. In tliis thesis, the bond element was implemented into the

TRIX99 nonlinear finite element program for considering the slip between

steet reinforcement and concrete.

3.6.1 Concrete Properties

For cracked concrete subjected to a tension-compression stress state,

the constitutive law adopted was according to the MCFT. The strength

reduction factor B for concrete in compression, illustrated in Fig. 3-4, is

p= ll(0.85- 0 . 2 7 ~ ~ 1 ~ ~ ) <=1 .O (3-50)

where E, and ~2 are the strains in the principal tensile and principal

compressive directions, respectively. Thus, the stress-strain relationship of

concrete in compression, illustrated in Fig. 3-5, is

fc2= fp [2 (&hp )- (&dgp l2 1 for O> EZ >sp (3-5 1)

f c 2 = f p [ ~ - { ( & 2 - ~ p ) / ( 2 ~ ~ - & p ) } 2 ] for E ~ > E ~ > ~ E ; (3-52)

where E, ' is peak compressive strain of uncracked concrete.

B

Fig. 3-4 Compressive softening parameter P of concrete.

G2

ognestad . - - -. . . -. . - - . . - 0 . . . - 8 - o . . . *.

O - Parabola . - .

Fig 3-5 Constitutive response of cracked concrete in compression.

The peak compressive stress of the cracked concrete, f,, is:

fp= p fcf (3-53)

occurring at the peak strain

g p = P E: (3 -54)

The stress-strain relationship for concrete in tension, s h o w in Fig. 3-6,

is given by

fcl =Em CI for 0-1 (3-55)

fcl = f , '/[li- ( ~ o o E ~ ) ~ - ~ ] for EI (3-56)

For concrete in a biaxial compression state, strength enhancement was

modeled using a relationship approximating the Kupfer et a1 (1969) model.

The strength enhancement factor for concrete in the 3-direction, due to the

stress fci acting in the 1- direction, is

Kc2 = 1 +O .92(fcl/fi) - 0.76(f~~/fç')~ (3-57)

Cut-off due to local stress conditions at crack

El

Fig 3-6 Constitutive response of cracked concrete in tension.

The peak stress, f and strain at peak stress, EN, are

f p2 = Ks2 f: (3-58)

Ep2 = I(c2 f: (3-59)

The strength enhancement factor for concrete in the 1-direction, due to the

stress f,- acting in the 3- direction, is

Kcl = 1 +O .92(fC2/fcr) - 0. 76(fc2/fc')2 (3-60)

The peak stress, fPi, and strain at peak stress, are

f p i = Ki fc' (3-6 1 )

€pl = LI fc' (3-62)

Fig. 3-7 Strength enhancement due to biaxial compression.

Eo Ep 2%

Fig 3-8 Stress-strain relation for biaxially compressed concrete.

The constitutive law for biaxially compressed concrete is based on the

mode1 of Kent and Park (1971), which was modified later by Scott et al.

j1982), shown in Fig. 3-8. The formulation of constitutive law for biaxial

concrete is

f=z= fp2 [2 (E&Z )- ( ~ 2 / & ~ 2 l2 ] for O> €2 > EN (3 -63)

fc2= fp2 [ 1 + &(tz2 - zp2 )] <=O. 2f p2 for EZ < (3-64)

where &= O S / [ (3+0.29fc' )/(145f,' -1000) ] + fc1/450+ (365)

The stress-strain relation for concrete confined by rectangular hoops is :

fc= fp [2 ( d s p )- ( d & p l2 1 for O> E, > sp (3-66)

f,= fp [ 1 + &(gC - gp )] <=O .2f p for cc < E, (3-67)

where

Kc = 1 +QSffi/f,' (3-68)

The peak stress f, and strain at peak stress %are

f p = &fC' (3-69)

sp= I(c f l (3-70)

&= 0.5/[ (3+0.29f,' )/(145f,' - 1 OOO)+ O .75*~,*(h' /~t , )~-~- (3-7 1)

where fyh is the yield strength of hoop reinforcement (MPa), Qs is the ratio

of volume of hoop reinforcement to volume of concrete core measured to

outside of the hoops, h' is the width of concrete core measured to the

outside of the peripheral hoop (mm), and sh is the center-to-center spacing

of hoop sets (mm) .

Concrete in compression exhibits a lateral expansion charactenzed by a

progressively increasing Poisson ratio. The mode1 used for the Poisson

effect, which was proposed by Vecchio [see Fig. 3-91, is

~ 1 2 = Vo for O> ~2 > 6 2 (3-72)

~ 1 2 = v g [ 1 + 1 . 5 ( 2 ~ ~ / ~ - 1 ) ~ ] for ~ 2 ~ 0 . 5 ~ ~ (3-73)

v21 = vo for O> > 0 . 5 ~ ~ (3-74)

v2l = vo [1+ 1.5(2~, / cp -112] for < 0 . 5 ~ ~ (3 -7 5)

where both viz and vzl do not exceed 0.5; vo is the initial value of the

Poisson ratio; Vij is the expansion in the i-direction due to the concrete

stress in the j-direction, fcj. For concrete in tension, pnor to cracking, the

Poisson ratio was considered constant at vo. M e r cracking, the Poisson

ratio was equated to zero for expansion normal to the tensile direction

only. For example, if 1-direction is the tensile direction, then v2,=0, v+O.

Fig. 3-9 Poisson's ratio of concrete in compression.

( a ) Compression response.

( b ) Tension response.

Fig. 3-10 Hysteresis model for concrete.

For the stress-strain relationship under cyclic loading, a plastic offset

formulation was used [Vecchio 19991. The hysteresis model for concrete is

shown in Fig. 3-10.

On a reloading cycle where the concrete plastic strain in effect is E$,

the concrete compressive stress is calculated as:

fc(&)=0 for E, >ccP or E= >O (3-76a)

fc(&c)= (8c-~cP)fm/(&m-&~ for EcP > Ec >Ecm (3-76b)

f, (& )=fb& ) for E, CE, (3-76c)

where E, is the maximum compression strain attained during previous

loading, f, is the stress corresponding to E, , and fk(cC) is the stress

calculated from the base curve for a strain cc. If E, < E ~ , then E, and fc,

are updated to E,' and fm' respectively.

At each load stage, the instantaneous plastic strain gCP' is calculated as

follows:

scP' = E,-E~ [0.87(cj~J - 0.29 (dcJ2 ] Q 1 . 5 ~ ~ (3-77a)

ccP' = cc- 0.00 1305(&, /0.002) cc< 1 . 5 ~ ~ (3-77b)

where cp is the strain corresponding to the peak stress in the base curve. If

the instantaneous plastic strain ccP' exceeds the plastic offset G ~ , then the

latter is updated accordingly.

On a reloading cycle, when the active plastic strain is E?, the concrete

tensile stress is calculated by

fc(ec)= ( E ~ - E ~ ~ ) ~ ~ ( ~ ~ - E ~ ~ for tzCP < zC < tzEI. (3-78a)

fc (CC )=fbt(sc ) for ec >cm (3-78b)

where cm is the maximum tensile strain attained during previous loading,

f, is is the stress corresponding to cm , and fb<(ec ) is the stress calculated

fiom the base curve for a strain .

On the unloading cycle i where the concrete plastic strain in effect is E?',

the concrete compressive stress is calculated as:

N=Ec (ccp' - E ~ * ) / [ ~ ~ ~ + E ~ ( E ~ - cm3)] (3-79a)

For WN-30:

fco=EC(~, - - B cm *lN-'

fcl=fms+Ec(~c emY)+fd

For N<=l and N>=20:

fc l=Ec(~~ ES') For any value of N:

fc2=fci-i+Ec(~c - &ci-1)

fci= the most negative value ( fcl, f ; )

On the unloading cycle i where the concrete plastic strain in effect is G~',

the concrete tensile stress is calculated as:

N=Ec (smY - Ec(~tmw - E~*') - fa*] (3-8 1 a)

For 1 <N<20: p' N-1 f c ~ = E & ~ ~ - E~)~/N(&~~ - 4 ) (3-8 1 b)

7

fci=fb* -EC(ct, - &=)+fd (3-8 1 C)

For N<=l and N>=20:

fcl=Ec(~c - acp') (3-8 1 d)

For any value of N:

fd=fci- 1 +E,(E, - E ~ - ,) (3-8 1 e)

fci= the maximum value ( cl, G) (3-82)

3.6 -2 Reinforcement Properties

The stress-strain relationship of steel bars is the same for tension and

compression. The rnonotonic stress-strain curve for reinforcement consists

of three regions: the linear region, the yieid plateau, and the strain

hardening region. A trilinear curve was used for modeling stress-strain

relationship of reinforcement as shown in Fig. 3-1 1.

fs=EreS for % c c y (3-83)

fs=fy for (3-84)

fs=f, +ESh(&. - E& )<fu for es>&h (3-8 5)

where f, is the yield stress of reinforcement; Es is the elastic modulus of

reinforceinent; Es is the strain-hardening modulus; and E* is the strain at

the commencement of strain hardening.

Fig 3 - 1 1 Stress-strain relationship of reinforcement.

The reloading and hysteretic response of the reinforcement is modeled

after Seckin (1971) with some minor simplifications. The first loading

circle is the rnonotonic response. The unloading relationship is

f s (~ i ) =f s i-I+w~i - ci-,) (3-86)

where E, is the unloading modulus.

E, = Es for - (gm O go)

E,=Es[l .OS-O.OS(&m-~)/&y] for E y < ( ~ m œ ~ ) < 4 ~ y

E, = 0.85Es foi ( E ~ - E ~ ) > ~ E ~

The stresses upon reloading are

&(&i )=Er( ci- EO ) +( Em -ET ) ( ci- EO ) N / ~ ( &mg EO ) N-l]

where

N=( E, -5 ) ( Em- QI )/[fk EX Ei- &O )]

Fig. 3-12. Hysteresis mode1 for reinforcement.

In a positive cycle, cm is the maximum positive strain attained during

previous cycles, f, is the stress corresponding to E, as determined from the

hysteresis curve, and & is the tangent stiffness at E, . The parameter is

the plastic offset strain corresponding to the zero stress point for the

present cycle; it is redefined whenever the stress passes through zero.

In a negative cycle, the sarne formulations apply except that E, is the

maximum negative strain previously attained. The stress f, and stifhess

E, are evaluated accordingly.

3.7 Bond Stress-Slip Models for Finite Element Analysis

3 -7.1 Models of Bond-Slip Element

Perfect bond was assumed in lower strain regions. Bond slip was

considered in high strain regions. For the plane stress problem, the

concrete element used was either a rectangular or a triangular constant

strain element, and the steel element was a tmss bar element. For the

reinforced concrete shear wdls which were analyzed in this paper, in the

top beam and top part of the web, the slips were very small so the smeared

matrix constant strain rectangular element with four nodes was adopted.

High strains and large slips occurred in the bottom part of the web, so

discrete steel truss elements and concrete elements as well as bond-slip

elernents in the interface between steel and concrete were used. In the part

of the bottom beam which was comected with the web, bond slip was

perhaps significant in the stages of large displacements, so bond-slip

elements were employed in this part of the bottom beam.

An incremental load method was adopted for the nonlinear analysis

procedure. An initially uncracked element is loaded incrementally until the

principal tensile stress exceeds the tensile strength of the concrete at one or

more locations. At each loading step, concrete cracking and its propagation

are checked, then material properties and stifiess coefficients of each

element are calculated according to the current stress-strain states. If the

average value of the principal tensile stress in a element exceeds the tensile

strength, then a crack is established in this element, and the concrete

element stifhess of the element in the crack direction is modified

according to the stress-strain curve of the concrete in tension.

3 -7 -2 Models of Local Bond Behavior

3.7.2.1 Mehlhorn Model (Model 1)

This mode1 was proposed by Mehlhom et al. The relationship of local

bond stress and local slip is s h o w in Figure 3-13 . The cwves are defined

as :

D

Fig. 3- 13 Relationship of local bond stress and slip.

(AB and CD are unloading and reloading curve)

At a particular stage of monotonic loading, the bond stress is calculated as

s = p fct [ 5 NA^) - 4.5 1 d A / A , ~ +1.4 ( A / A ~ ) ~ ] for 1 A/ <Ai

(3-90a)

r =1.9 P fct for AU> A>=Al (3-90b)

r -1.9 P fct for -Au >A<=Al (3-90c)

r =O for 1 > = A ~ (3-90d)

P= Pi f32 P3 P 4 (3-90e)

In the case of cyclic loading, the first loading cycle is the rnonotonic

response. During an unloading and reloading cycle, the bond stress is

calculated as :

t =O for O<A< A~ or Ap <A<O (3-9 1 a)

r = (A-A~) T&(A,+-A~ for Ape cA<Amc (3-9 1 b)

r =(A-Ap) qJ(A,,,--Ap) for Am- <A < Ap (3-9 1 c)

r =f(A) for A >= Ae or A <= Am- (3-9 1 d)

where fci is the concrete tensile strength; f(A) is the bond stress-slip curve

of the rnonotonic loading given in Eq. 3-78; & and A,,,. are the maximum

and minimum slips attained during al1 previous loading stages dong the

concrete-steel interface, respectively; and r , and r ,. are the bond

stresses corresponding to bond slips A+ and &. , respectively . APC and

A~ are the plastic slips at the bond slips A= A,+ and A= &- respectively.

They are calculated as

A~ =Am+ -im+ /Em (3-92a)

Ap =Am- - rm- /EbO (3092b)

where EbO is the inclination slope of the unloading or reloading curve,

equal to the inclination slope of the rnonotonic loading curve at slip A=O. It

cm be calculated as

EbO= dddA 1 A=o=S P fct /AI (3-93)

where Al is the limit value for the slip. When the slip is greater than Al

,the bond stress remains constant until total bond failure occurs at slip=Au.

AI depends on the lateral pressure to the axis of the reinforcing bar. It can

be expressed in the following formula:

Al = 0.06 mm+0.004q (Wmm2) (3-94)

where q is the confining pressure on the reinforcing bar in the contact

surface.

The factor Pi considers the size distribution of the aggregates (pl ranges between 0.8 and 1.25) and the position of the concrete during

casting. The influence of the concrete casting is shown in Fig. 3-14

according Martin and Doerr (1 98 1) tests. When the directions of concrete

casting and of bar slip are identical, q=o0. When the directions of concrete

casting and of bar slip oppose each other, cp=180? When the directions of

concrete casting and of bar slip are perpendicular each other, q1=90~.

Fig. 3- 14 Influence of the direction of casting

of concrete corresponding to the pull-out direction.

The factor B2 considers the geometry of the bar surface, expressed by

the relative rib area, which is defined as the quotient of the rib area and the

bar surface between two ribs. The relative nb area is calculated as

a,=k*Fr*sin<p /(scdsi)+ih/st (3-95)

where k is the number of lug series ,

Fr is the area of the longitudinal section of one lug,

<p is the h g inclination towards the bar axis,

d is the nominal diameter of the steel bar,

si is the distance between lugs,

i is the number of longitudinal ribs,

h is the height of longitudinal rib, and

st is the pitch of twist of twisted bars.

Because the local bond stress-slip relationship is based on Doerr's tests

for ribbed reinforcement of 16 mm diameter, the influence of the geometry

of the bar surface is related to the 16 mm diameter bar. For this bar, the

relative rib area is ar=0.065. Marin (1981) studied the bond behavior of

reinforced concrete with varying geometry of the surface of the tested bar.

For relative displacements between 0.01 mm to 0.50 mm, Marin proposed

a bond stress-slip relationship normalized with respect to the compressive

concrete strength. The mean values related to the relative rib area &+.O65

can be represented by the cuve given in Fig. 3-1 5 .

Fig 3-1 5 Influence of the relative nb area of

the reinforcement bar normalized to ar=0.065

Related to the relative rib area, the ultimate relative displacement Au is

given in Table 3-1 for various concrete compressive strengths.

Table 3-1. Ultimate relative displacements Au(mm)

The factor B3 considers the transverse pressure or gaping based on the

expenmental research of Doerr (1978) and Eligehausen et al (1 983). Doerr

investigated this influence under rotationally symmetric pressure p leading

to the pressure q in the contact surface. With increasing confining pressure,

the bond strength increases. Eligehausen et a1 (1983) studied the influence

of confined pressure in only one direction. The results are summarized in

Fig. 3-16.

Fig 3-1 6 Relationship of bond strength and confining pressure

on the contact surface of a steel bar (q,=q for rotational

symmemc pressure, q,= uniaxial pressure).

There are no experimental data available to consider the influence of

lateral tension on the bond strength. It is obvious that lateral tension

applied to the reinforcing bar will decrease the bond strength. Mehlhorn et

al (1985) recommended that the factor P3 depends on the gap size At as

shown in Fig. 3-17. The h is the height of the lug.

The lateral pressure or tension on a reinforcing bar, produced by

concrete pressure or tension stress around the bar, is variable during the

loading procedure. Hence, it is not convenient to use the above curves. The

following formula can be obtained by the least squares method.

The increase of bond strength due to uniaxial confined pressure on the

contact surface of reinforcing bars is given by:

The decrease of bond strength due to gap At between concrete and

reinforcing bars is:

Fig. 3- 1 7 Relationship of bond strength

and gap between steel bar and concrete.

The factor P4 considers local damage of the concrete in the bond zone

caused by the development of secondary cracks. The influence of the

strain cs of the steel bar and the ratio of the yield stress f, to ultimate

strength f, of the steel bar are considered to express the damage of the

bond near the crack. The directions of the steel bar and the crack are also

considered by the test result. Multiplying these two influencing factors

leads to Eq. 3-84 to consider the effects of bond damage.

p 4 1 = 1- (f&ni) (&o/Esy)

P42 = sinû+(l-sine) d(4d) for x<4d

p 4 2 =1 for x>4d

P 4 =P41 p42 (3-99)

where 0 is the angle between the steel bar and the nearest crack of the bar

as shown in figure 3-18 , and x is the distance between the bond element to

the nearest crack.

7- steel bar

cracks

Fig. 3-18 Angle between the steel bar and the crack of the concrete.

P42

Fig. 3-1 9 Influence of angle between steel b;u and the crack

and distance between the linkage element to a nearest crack on Bq2 .

For uncracked concrete, Pd2 =l. For cracked concrete, in a theoretical

analysis, it 1s impossible to determine the positions of cracks accurately, so

Eq. 3-86 can not be used to calculate Pd2. In the analyses presented in this

thesis, it is assumed that P42 =l.

The secant method is used for considering nonlinear characteristics of

bond stress-slip relationship. The stifiess of a bond element is calculated

as following:

a) On the initial loading increment

For linkage elements:

Km=Ak*Ew

For contact elements:

K p E m

b) On the other loading increments

For linkage elements:

K,=Ak dddA (3-101a)

For contact elements:

Km= dddA (3-101b)

Where Ai, is the contributary area of bond strength of element k. This

area for each linkage element, at a concrete node, is computed based on the

tributary surface area of the steel bars associated with each element

connected at that node. .

3 X2.2 Eligehausen Model (Model 2)

This model is very different fiom Mehlhorn model. It allows for much

larger slips than does Mehlhorn model, and it utilizes larger bond

strengths. Eligehausen, Popov and Bertero (1 983), basing their

formulations on the findings of an extensive experimental program

descnbed previously, suggested a non-linear curve for the bond stress-slip

relationship. Other researchers complemented the model. Soroushian and

Chio (1989) obtained a relationship between the peak bond stress and the

bar size. Pochanart and Harmon (1989) examined the effect of bar surface

geometry and developed a relationship between the surface geometry of

the bar and the peak fictional bond stress. The Eligehausen model can be

defined by the following expressions:

Under monotonically increasing loading, The local bond stress and slip

relationship with suficient confined pressure to the reinforcing bars

(causing pull-out failure) is shown in Figure 3-20.

The bond stress and slip relationship can be defined by the following

expressions:

7=71 for A<=Al (3-102)

where rl is the maximum bond stress value, and Al is the bond slip

corresponding to the bond stress r=zlm.

Fig. 3-20 The bond stress and slip relationship of Eiigehausen Model.

This branch is followed by a plateau branch( r=tt=tl ) corresponding to a

slip field Ai<=A<=A2 .

T=TZ for Al<=A<=A2 (3-103)

The third branch is linearly decreasing up to the value r=ti , which is

achieved at A=A3 .

F r 2 -(A- A2)K A39 A2 ) ( 1 2 - ri) for A2<=A<=A3

The fourth branch is a horizontal branch where the bond stress remains

constant and equal to its minimum value T = T ~ ( the ultimate fiictional bond

resistance).

T-q- for b A 3 (3-105)

where

ri =tz =(20-d/4) (f,'/30)~-' ( m a ) (3-1 06)

~~=(5.5-0.07S/H)(f,'/27.6)~-~ (MPâ) (3- 107)

Ai= (f,'/3 o)'.~ ( mm ) (3-108)

A2== 3.0 mm (3- 109)

A3= S (3-1 10)

a=0.4 (3-1 11)

where S and H are the clear spacing and height of lugs on the bar,

respectively .

For splitting failure, the local bond stress and slip c w e s are given by

the dashed line in Figure. 3-20. The curves are expressed by following

expressions:

T 1, =.rzs =O. 74 8 (f,'~/d)*.~ <=r 1 (MPa) (3-1 12)

T fs =O. 2 3 4 (f;'~/d)O.~ <=Q W a ) (3-1 13)

AIs=Ale (Va) in(~ld~1) - 0.5 (Va) ln(tlsk1) - (fJ30) e (mm) (3-1 14)

A2s =A2 (3-1 15)

A3s =A3 (3-1 16)

where c is the smaller of the concrete-cover thickness, and one-half the bar

spacing. 71, is the maximum bond stress value in splitting failure, and AIS is

the corresponding bond slip to bond stress r=ti,. AZs is the maximum slip

of the plateau branch ( ~ = t ~ ~ = t ~ ~ ). tfs is the ultimate fiictional bond

resistance in splitting failure, A3, is the corresponding bond slip to bond

stress FT~ , .

3.7.2.3 Mohamed Harjli Model (Model 3)

Mohamed Harjli et al (1995) proposed the same shape of bond stress and

slip curve for pull-out failure but with a different value of a and with

slightly different values of .ri and TC. The expressions are as follows : r 0.5

71 =r2=2.575 (f, ) (MPa (3-1 17)

T~ =0.3ki (3-1 18)

Al= 0.75A- (3-1 19)

A2= 1.75A- (3-120)

A3= S (3-121)

a=O .3 (3- 122)

A,, is the slip at which the peak bond resistance Tm&, is reached and

A,, =0.189S+0.18 (3-123)

or more simply,

Ama. =0.2s

For splitting failure, the local bond stress and slip curves are given by

the dashed line in Figure 3-21. The curve is expressed by following

expressions:

T=T I @A I for A<=Als (3-124)

T=O for A>Als (3-125) r 0.5

ri, =(0.249+0.29 1dd) (f, ) <=q (MPa) (3-1 26)

Als=A1e (Va) in ( t l d~1 ) - - 0.75A,, e (Va) i n ( ~ 1 d ~ i ) (3-127)

T ~ ~ = T ~ ~ = O (3-1 28)

where ri, is the maximum bond stress value in splitting failure, and AIS

is the bond slip corresponding to bond stress -1, .

Fig. 3-21 The bond stress and slip relationship of Model 3 (Harjli).

3 X2.4 Proposed Model (Model 4)

This model is proposed by the author of this thesis according to scme

test results fiom published papers. This model is modified version of the

Eligehausen Model (Model 2). The local bond stress and slip relationship

is same as with Model 2 for pull-out failure, and some modifications are

made for splitting failure.

For splitting failure, the local bond stress and slip curves are aven in

the dashed line in Figure 3-22. They are defined by the following

expressions:

~ - t 1 d u A l SIa for Ac=Ais (3- 129)

F ~ I S -@- AIS)/( A ~ s - AIS ) (% - T ~ s ) for Al,<=A<=A2. (3- 130)

I-Tfs for 1A>A2, (3-13 1)

Fig. 3-22 The bond stress and slip relationship

of proposed model (Model 4).

sk , ns ,and A2 can be calculated usinp following formulations :

71s =T~*. 1 52lS (3-1 34)

A2,=2.0 mm (3-1 35)

3.7.2.5 Model of Bond Stress and Slip Relationship With Partial Confining

Pressure

According to the CEB (1993) code, largely based on the Eligehausen,

Popov and Bertero (1983) model described previously, the parameters for

confined concrete are applicable whenever the transverse pressure p>=7.5

MPa or closely spaced transverse reinforcement is present, satisfjing the

conditions LAmv>=nAS. Where LA, is the cross area of st imps over a

length equal to the anchorage length; n is the number of the longitudinal

bars enclosed by the s h p s ; and A, is the area of one longitudinal bar. For

O< LAJIIA~ 4.0 or O c p < 7.5 MPa, a linear interpolation between the

values of confined and unconfined concrete are be used to derive the mode1

parameters.

Fig. 3-23 The bond stress and slip relationship

with partially confining pressure.

The local bond stress and slip relationship is shown in Figure 3-23. It is

defined as:

~lsp=~ls+(~~-~ls)P

~2sp=~2sf (~3-72s)P

rrsp=~â+(~rrrJP

A1sp=A1s +(Ar Al n ) P

A2sp=AzS+(A3A2s)p

A3Sp=A3

where p is the confining pressure factor (O<=P<=l .O) .

3.7.2.6 Models 2 , 3 and 4 Under Cyclic Loading

On the basis of their cyclic loading tests, described in the previous

section, Eligehausen et al proposed the analytical mode1 shown in Figure 3-

24 together with corresponding experimental curves.

The f ~ s t loading cycle is the rnonotonic response. The secant method is

used in the nonlinear finite element analysis. The st if iess of a bond

element is calculated as :

On the initial loading increment:

For the linkage elements:

G,=AkGo

For the contact elements:

G,=Go

On other loading increments:

For the linkage elements:

G,=Akdh

For the contact elements:

Gn=dA

The initial bond stiffness is attained at A/A1=l/l 00. Then

Go=dr/dA=tia* 100'" (3- 144)

Unloading is considered to take place at a constant slope of bond

stifkess. The same slope of bond stifhess is retained during reloading

towards the envelope.

n i e unloading curve at point A is curve AD and the subsequent

horizontal branch DE. Denoting rPm as the maximum bond stress value

which is reached at the end of the previous loading phase, the horizontal

branch subsequent DE is afTected by the ordinate run=0.25zp,.

Fig. 3-24 The bond stress and slip relationship under cyclic

loading proposed by Eligehausen et al .

The reduction of bond resistance with increasing slip and number of

loading cycles is calculated using damage index D, which is equal to zero

for no damage and to one when bond breaks down completely ( ~ 0 ) . The

index D is a hnction of the ratio Efi , where E is the hysteretic energy

dissipation at the stage of unloading, and & is the energy corresponding to

the area under the rnonotonic bond stress and slip curve up to the value A3.

In calculating the value of E, only 50% of the energy due to friction is taken

into account, and the remaining 50% is assumed to be spent in overcoming

the friction resistance without causing any bond degradation.

During cycle i, the stifhess Ei can be calculated using the following

equations:

Ei=O (3-145)

Ei=Ei.i+AEi (3- 146)

AEi =O for (3-147)

AEi = rimA1 (Aim /Al ) ('*) /(1 +a)+0.5 *0.25timA im

for &.8maxcAun<=A1 (3- 148)

AEi = ~ ~ ~ A ~ / ( l + a ) + 0.5(rli +rh)(A -Al) +0.Sf 0.2SimA U.

for AI< A, <=A2 (3-149)

AEi =ri iA /(l +a)+0.5(rl i+tti)(A3Al) +O .5(~~+7, ) ( A h -&)

+ O S *0.25zkA, for A2 <Ah <=A3 (3- 1 50)

iAl/(l +a)+O. 5 ( ~ 1i+~2i)(A3d1)+0 ~ ( T ~ ~ + T ~ ) ( A ~ - A ~ )

+tli(Aun-A3) +0.5 *0 .25~~A, for A,>A3 (3-1 5 1)

For pull-out failure, & is given by:

b = r 1 Ai/(l +a)+r (A3A1)+0 S( t l+~f)(A3-A2) (3-152)

For splitting failure, & is given by:

&=TI Ad(1 +a)+0.5(~ ls+~2s)(A2s-Als)+0.5(~ 1 &tfs)(A~s-A2s)

(3-1 53)

Di=l-e a (3- 1 54)

a=-l .z(E~/Eo)'.' (3455)

where Aim is the maximum absolute value of the bond slip reached during

the first half-cycle loading of cycle i if the present loading stage is in the

second half-cycle loading of cycle i , and is the maximum absolute value of

the bond slip in the second half-cycle loading of cycle i-l (previous loading

cycle ) if the present loading stage is in the first half-cycle loading of cycle i

. th is the absolute value of the bond stress corresponding to bond-slip

Ah.

For pull-out failure, t l i , rzi and ru are bond stresses corresponding to

bond slips Ai , A2 and A3 in the cycle i , respectively. They are calculated

as:

TI^ =72i = ~ l (1-Di ) (3-1 56)

Tfi =tf (1-Di ) (3-1 57)

For splitting failure without confining pressure, 7 i i , Tl i and ra are bond

stresses corresponding to bond slips Ais , AZr and A3, in the cycle i ,

respectively. They are calculated as:

~ l i =Tls (l-Di ) (3.158)

~ 2 i =TZ= (1 -Di ) (3- 159)

tfi =rfs (1-Di ) (3-160)

For splitting failure with partial confining pressure, Tl i , rzi and TG are

bond stresses corresponding to bond slips AI, , A2, and A3, in the cycle i ,

respectively. They are calculated as: *

rii (1-Di ) (3-161)

~ 2 i =Tzse (1 -Di ) (3.162)

tfi = ~ f s (1 -Di ) (3- 163)

Dunng the unloading in cycle i , bond stress is calculated as :

Ti =O .2Srim for O < = ) A ~ I < ~ A ~ ~ I (3-164)

ri = ( A i - A i ? r i d ( A i , - ~ ~ P > for 1 ~ ~ ~ 1 ~ = ~ A ~ ~ ~ ~ A ~ ~ ~

(3-1 65)

r i =f(Ai) for IhiI > = I h i m I (3-166)

where Aim is the maximum (absolute) bond slip in the loading (or

reloading) stage of cycle i , and r, is the bond stress corresponding to

bond slip A,. A: is the plastic bond slip at bond slip Ah . It is defined

as:

A r = Aim TU^ / Go (3-167)

On reloading in cycle i , the bond stress is calculated as :

r , =f(Ai)>=0.25~U, for ri>=O (3-168a)

r i =f(Ai)<=-O.25rh for T ~ < O (3- 168b)

where r, is the bond stress corresponding to the maximum absolute value

of bond slip at the previous loading (or reloading) stage. In equations (3-

152) and (3-154), f(Ai) 1s the bond stress calculated from the base curve

considering the bond degradation of cyclic loading of cycles i.

3.7 -2 -7 Effect of Hooks in Rein forcement

From the experimental results of Eligehausen, Bertero and Propov (1 983),

the local bond stress and slip curves for bars with hooks under cyclic loading

are shown in Figure 3-25. They are given by the following expressions:

For monotonic loading:

r = 2 2 . 0 ( ~ / ~ ~ ) " a ) for IAI<=A, (3-169)

~ 2 2 . 0 (h4Pa) for I A ~ ; A * (3- 170)

On a unloading and reloading cycle, bond stress is calculated as :

t =O for OcAc A"+ or Ap <Ac0 (3- 17 1)

r =(A-~p) 7mJ(Am+-A9 for A~ <A<Am+ (3- 172)

( A - ) ( A - A ) for A, <A < A~ (3- 173)

r =f(A) for A >= Am+ or A >= Am+ (3- 1 74)

where f(A) is the bond stress aiid slip curve of rnonotonic loading given

in (3-155) and (3-156). Am+ and A,.are the maximum and minimum slips

attained during al1 previous loading stages along the concrete-steel

interface respectively; r ,+ and r ,. are the bond stresses corresponding to

Am+ and Am. respectively. A~ and Ap are the plastic slips at the bond slip

A= Am+ and A= Am. respectively. They are calculated as

A~ =Am+ /Go

Ap =Am- 0 ~ ~ - /Go

Fig. 3-25 Analytical local-bond stress and slip mode1

for deforrned bars with hooks.

Vi 2 O ' C Y)

w -5 E O p -10

-1 5

-20

-25

3.7.2.8 The Normal Bond Stiflhess

1

Because there is a lack of experimental research regarding the local

bond stifiess between a steel bar and concrete in the direction normal to

the interface, the assumption made by M. Keuser et al (1983) is used in

this paper. The normal bond stifniess is assumed to equal 100 times the

tangent bond stifhess.

Gp100 Gr OC Ku=lOO Km (3-1 77)

-10 -8 -6 -4 -2 O 2 4 6 8 10

slip (mm)

CHAPTER 4

EVALUATION OF MODELS BY COMPARISON BETWEEN

FINITE ELEMENT. ANALYSIS AND EXPERIMENT

4.1 Verification of the Program

The expenmental results fkom bond specimens, given in the paper by

Keuser et al (1987), are used to ver@ the program. The specimen and

finite element mesh are shown in Figure 4-1. Linkage elements and contact

elements were altematively used at the interface between the concrete and

the steel bar. The specimen is a concrete cylinder of 75 mm diameter, with

a 16 mm deformed bar. Material data are given in Table 4-1. It should be

noted that this is a 3-D problem actually, so it is certain to cause some

degree of error due to simplification to the plane stress problem. The

parameters and maximum bond slips (for load= 70 kN) of the four bond

slip models are given in Table 4-2.

The distributions of steel force for low stress and high stress conditions

are plotted in Figures 4-2 to 4-5. From the Figures, it can be seen that

analytical results of two kinds of elements (linkage and contact elements) are

aimost identical. The analytical results of al1 models agree very well with

experimental results for low steel stress (load=20 kN), but the results fkom

Models 2 and 4 agree better than those fiom Models 1 and 3. For the high

steel stress conditions (load=70 kN), the analytical results of Model 1 differ

from the test results signifi cantly; the farther away fkom the loaded end, the

larger the difference. Model 1 gives much larger slips than do the other

models, and it under-estimates the bond strength. The analytical results of

Models 2 and 4 match very well with the test results. The results of the finite

element calculation generally show satisfactory agreement with experimental

data for Models 2,3, and 4.

Table 4- 1 Material data.

Concrete 1 Steel Bar 1

I

Table 4-2 Specimen parameters and analytical maximum slips (mm ).

E'=37.2 MPa

Gt=2.7 MPa

fo ,420 MPa

Eh=2050 MPa

Element type

4 -2 Experiment Details

4.2.1 Dimensions and Reinforcement Arrangement of the Specimens

A set of four specimens of low-rise reinforced concrete walls was

designed and tested by Eun Hee Choi and Sasha Kurmanovic under

guidance of Professor J.F. Bonacci at the University of Toronto. The main

difference between the four walls was in the design of boundary elements.

Wall- 1 had uniformly distributed web reinforcament without boundary

elements. Concentrated vertical reinforcement was added near the vertical

edges of Wall-2 and Wall-3, but only Wall-3 had transverse (horizontal)

hoops enclosing the concentrated vertical reinforcement. Wall- 1, -2, and -3

link

contact

Mode1 1

Au=0.530

A,-4.290

A-- 4 . 2 8 9

Mode12 1 Mode1 3 Mode1 4 1 A1=0.178

A,, =O. 124

A,, 4 .120

A 104

A-4.097

=0.093

- -

Ai= O. 178

&=O. 124

&, 4 .120

had a rectangular cross section for the wall web. Wall4 had a small flange

on each side; the area of concentrated reinforcement was the same as in the

Wall-3 (showri in Figure 4.6).

A top and a bottom beam were designed for each wall. The vertical

reinforcement in the walls was anchored to the top and bottom beams to

achieve the full yielding capacity. The bottom beam was 3500 mm long,

500 mm deep, and 500 mm thick, and the top beam was 2325 mm long,

300 mm deep, and 400 mm thick.

The dimensions and reinforcing bars -of the four specimens are s h o w in

Figures 4.7 to 4.1 1 .

4.2.2 Material Properties of Specimens

In order to produce a strong base beam and relatively weak wall, two

different types of pre-mixed concrete were used : high strength concrete

for the base beams and normal strength concrete for the wall and the top

beam. Cylinder samples for each type of concrete were taken, cured,

ground and tested. Uniaxial compression tests were performed, and the

results are summarized in Table 4-3, Table 4-4 and Figure 4-12.

Table 4-3 Uniaxial compressive strength of the Wall-1 and Wall-2

Base beam

Top beam and wall

Test day

Moist cwed

75 MPa

33 MPa

Test day

Air cured

----- 36 MPa

7 days

Moist cured

50 MPa

23 MPa

28days

Moist cured

O--e-

32 MPa

Table 4-4 Uniaxial compressive strength of the Wall-3 and Wall4

7 days 14days 1 7 days after Test day

Four different types of bars were used to meet the guidelines of the AC1

and CSA Standard. A list of the bars and their characteristics are presented

in Table 4-5 and Figure 4-13.

Base beam

Top beam and

Table 4-5 Characteristics of reinforcement steel.

48 MPa -II-- 66 MPa

26 MPa

4.2.3 Loading History of Test Specimens

Most specimens tested under lateral loading are subjected to a point

Types of bars

D4

load at one end of the top beam, and supported at the opposite corner of the

----

base beam. Therefore, the test specimens c m be interpreted as being

39 MPa

Area (mm2)

25.7

loaded diagonally. The method of loading adopted in this experiment is

different fiom the method commonly used in other studies. A steel plate

Yield stress (MPa)

593

with stiffeners and an adapter-plate for an actuator were attached to the top

Ultimate stress (MPa)

643

beam in this series of tests. The lateral force was applied at the adapter-

plate that was located in the middle of the steel plate and uniformly

distributed through the steel plate.

The loading history is outlined as below:

1. A 140kN lateral force was applied for one complete reverse cycle.

2. For the second cycle, load was applied until top beam displaced Imm.

3. Cycle 2 was repeated.

4. The wall was loaded until fust yielding of reinforcement was detected in

the load-deflection response. Then it was unloaded and reloaded in the

reverse direction for the same deflection. It was unloaded to finish this

cycle.

5 . Cycle 4 was repeated.

6. The sixth cycle proceeded in the sarne rnanner as cycle 4, but the top

deflection imposed was about two times larger than the deflection of

cycles 4 and 5.


8. The load was applied in a similar manner as cycle 6, but the top

deflection imposed was about two times larger as the deflection of cycle

6.


10. Load was applied until the wall failed.

The loading history of the four specimens is plotted in Figure 4-14 to

Figure 4-17.

4.3 Study of the Element Mesh

To study the effect of element meshes on the FE results, three kinds of

smeared element meshes (shown in Figures 4-18 , 4-19 and 4-20 ) were

used in the analysis for Wall-1. The top load-deflection curve of Wall-1 is

given in Fi-gure 4-25 for a coarse mesh (Figure 4-18) and in Figure 4-26

for a fine mesh (Figure 4-19). The loads calculated using the coarse mesh

were 10 -1 5 percent larger thm with the fine mesh; hence, the coarse mesh

is too course to attain the sufficiently accurate result. The results with the

fine mesh show only about a 1 percent -difference fiom the those obtained

with a finer mesh (Figure 4-20); hence, it is not necessary to use the finer

mesh to achieve so little improvement in accuracy at the expense of

computing time two to three times greater.

The element mesh of Wall- 1, shown in Figure 4-2 1, was used to study

the bond element response in the finite element model. From the analytical

results, the following can be concluded. Compared to the slip between the

vertical bars and surrounding concrete in the lower half of the web, the

slips between the horizontal bars and surrounding concrete and the slips

between the vertical bars and surrounding concrete in the upper half of the

web are negligible. Hence, in this thesis, the bond elements are employed

only between the vertical bars and surroundiog coiicrete in the lower half

of the web (Figs. 4-22 to 4-24).

4.4 Cornparison of the Analytical and Experimental Results

In order to verify the nonlinear bond stress and slip models described

above, the experimental results f b m the four shear walls tested at

University of Toronto are compared to analytical results. The walls are

fixed against translation in both the horizontal and vertical directions at the

base of the bottom beam. The measured deflection history is imposed

uniformly at the top of the walls.

The success of the analytical models is evaluated by comparing the

measured and computed load-deflection-response of the top of the wall.

4.4.1 Shear Wall-1

The elernent mesh used is shown in Figure 4-22, and the ultimate loads

acting at the top of the wall are given in Table 4-6 for the FE analysis and

test data. From the ultimate loads of Table 4-6, it can be seen that there is

good correlation between the results calculated using TFUX and

experimental results, and the differences varying fiom 0.2% to 3 -7% . The

contact elements give marginally better results than do the linkage

elements, but the merence between results attained from the two types of

elements is small.

Table 4-6. Cornparison of ultimate loads acting at top of Wall-1

fiom test data and FE analysis ( kN )

1 FE analysis 1 Test data

* assumes perfect bond between the concrete and the steel bars.

Element Type

linkage

contact

The computed load-deflection responses of Wall-1 (at the top of the

wall) are plotted in Figures 4-27 to 4-35 for the five bond slip models, and

two types of elements. The load-deflection envelope computed assuming

perfect bond, and computed using bond-slip Model 2 with contact

elements, and test response are plotted in Figure 4-36. The expenmentally

determined response is given in Figure 4-37. Although Model 1 gives a

ultimate load that is very close to test result, the curves of the load-

Model 1

409

403

Model 2

408

404

Model 3

385 - 399

Mode14

403

403

No slip*

425 400

deflection are significantly different each other. The convergence of Model

1 is very poor for large slips/deflections (top deflections larger than 4 mm

for the linkage elements and 6 mm for the contact elements). This mode1

did not converge for deflections greater than about 6 mm for the linkage

elements and 10 mm for the contact elements, so the ultimate loads and

other results obtained fiom Model 1 are not reliable. For other three

Models, the results fiom the two types of bond elements do not have

apparent differences. There are not significant differences between the

perfect bond Model and Models 2,3 and 4 when top deflections of the wall

are smailer than 10 mm, but Models 2 and 4 give the better results when

top deflections of the wall are larger than 10 mm. Al1 bond-slip Models

give larger loads than those measured from test for top deflections of the

wall from about 10 mm to 16 mm, but the wall is failing in the analytical

models for a top deflection over 20 mm.

The calculated hysteretic response exhibits the same trend as the

measured data and successfully represents the amplitude of the measured

data except for the last cycle. The only notable discrepancies between

calculated and measured behavior are the degree of pinching evident in the

load deflection hysteresis and failure cycle.

The bond slip envelopes of the Wall-1 are plotted in Figures 4-75 to 4-

80 for Models 2 to 4. The bond slip at a node is defined by the

displacement of the steel tmss element node less that of the concrete

element node which is connected to ûuss element node by the bond

element. The maximum slips and minimum slips are given in Table 4-7.

During the first three cycles (when the peak deflections at the top of the

wall are not more than 1 mm), the bond slips are very small, and

significant bond stresses exist in the interfaces between the steel bars of

the bottom web and the surrounding concrete. In these loading stages, the

shear wall works as if there is perfect bond between the steel bars and the

surrounding concrete. The large active slips occur at the top of the bottom

beam, and large negative slips occur in the tensile side of a zone near 40%

height of the web wall. When the outside concrete of the bottom web is

uncracked or when cracks are very small, the maximum slips occur at the

interfaces between the outside tensile steel bars and the surrounding

concrete. As the cracks of the outside concrete at the web bottom become

large, the maximum slips move toward the interfaces between inside

tensile steel bars and the surrounding concrete. In the tensile zone near

40% height of the web wall, where the concrete cracks are large, the

displacement of a concrete node is larger than that of the corresponding

steel bar node causing large negative slip. In this area, the peak values of

maximum and minimum slips calculated using contact elements are larger

than those attained using linkage elements. nie maximum and minimum

slips calculated using contact elements are much smaller than those

aîtained using linkage elements in the cycle 1 to 7. This means that the

maximum and minimum slips caiculated using contact elements are

smaller than those attained using linkage elements when the top deflections

are small, and the maximum and minimum slips calculated using contact

elernents are larger than those attained using linkage elements when the top

deflections are large. The difference bekeen the maximum and minimum

slips obtained fkom Models 2 and 4 is not much when linkage elements are

used, but the difference is large when contact elements are used. The slip

envelopes of Mode1 2 are the same for linkage and contact elements.

Table 4-7 Maximum and minimum slips of Wall-1

by FE analysis (mm)

1 Bond Model ( Model 2 1 Model 3 1 Model 4 1

Figures 4-87 to 4-92 give the bond slip, bond stress and steel stress at

the outside bottom of the web of Wall-1 for Model 2 using contact

elements. Very small slips are seen in the first three cycles. From cycles 4

to 7 (the peak top deflections of the wall are 2 mm to 4 mm), there are

apparent bond slips. The bond slips are large fkom cycles 7 to 9 (the peak

top deflection of the wall is 10 mm). The peak values of the bond stresses

do not change much fiom cycles 4 to 9. Actually, the peak values of the

bond slips in these cycles lie between Al and A2 , and the bond strength in

these peak values is constant (the horizontal branch FTI of bond stress-slip

curve) for Model 2. In the results coming from Models 3 and 4 , the peak

values of bond stresses decrease gradually from cycle 7 due to peak bond

slips that are larger than Al .

The steel stresses in the three outside vertical bars at the bottom reach

the yield stress during the 6th cycle (the peak top deflection is 4 mm).

min

-0.45

-0.45

max

Linkage elemenq 0.53

Dunng the 8th cycle (the peak top deflection was 10 mm), the steel stresses

in the outside four vertical bars exceeded the yield stress and the stresses in

other vertical steel bars ranged between 40% and 92% of yield stress at the

bottom of the web except for three vertical bars in the middle of the web.

max

0.84

2.94 Contact element

The steel stresses in the outside vertical bars at the bottom web dropped

133

max

0.53

0.64

min

-2.42

-1.90 0.53

min

-0.59

-1.36

significantly before failure due to bond damage, and the steel stresses in

the inside vertical bars near the horizontal middle of the web were very

large (the largest steel stress is about 1:2 f,). When the wall was about to

fail, the crack width was very large (the largest crack was more than 10

mm). There were large slips (over A2 ) in a zone of near 40% height of the

web at the tensile side, bond stresses were very small, and the steel stresses

dropped suddenly. The wall- then failed.

The maximum concrete compressive stress, at the compressive side of

the wall, was fd= f l when the failure was about to occur. The deflection of

Wall-1 after failure are shown in Figure 4-1 1 1 to 4-1 17 for assuming

perfect bond and bond-slip Models 2 to 4 and two bond elements. Wall-1

failed due to two factors: large steel bar slips (at the zone near the base of

the web when assuming perfect bond; at the zone near 40% height of the

web wall for Models 2 , 4 ; and at the zone near one-third height of the web

wall for Model 3 in tensile side), and concrete crushing at the compression

face. The computed failure mode was a combination of bond slip and

concrete crushing failure.

4.3.2 Shear Wall-2

The element mesh used to mode1 Wall-2 is shown in Figure 4-23. The

ultimate loads acting at the top of the wall are given in Table 4-8 for the FE

analysis and the test data. The analytical predictions fiom the finite

element method fit very well with the experimental results except for

Model 1; the differences range fiom 0.3% to 5.5% . The ultimate loads

predicted by Model 4 have only a 0.3% difference for contact elements,

and a 0.8% difference for linkage elements when with the compared test

data. The load-deflection response cornputed using contact elements is

closer to experimental results than thaf obtained using linkage elements,

but the difference in results attained using the two kinds of elements

small.

Table 4-8. Cornparison of ultimate loads for Wall-2

fkom test data and FE analysis ( kN )

- - . -

* Assumes perfect bond between the concrete and the steel bars.

The computed load-deflection of Wall-2 (at the top of the wall) is

plotted in Figures 4-38 to 4-46 for the five bond models and two types of

elements. Load-deflection envelope computed assuming perfect bond, and

computed using bond-slip Model 2 with contact elements is compared to

the test data in Figure 4-47. The experimentally observed response is given

in Figure 4-48. The convergence of the Model 1 is very poor for large

slips/deflections (deflections larger than 4 mm for linkage elements and 6

mm for contact elements), hence, the ultimate loads and other results

obtained f'kom Model 1 are not reliable. For the other three models, the

results fiom the two types of bond elements show no obvious difference.

The loads calculated assuming perfect bond (i.e., no bond slip) are larger

by about 17% compared to those attained fiom the test. The wall fails

Test data

640

FE analysis

Element Type

1 in kage

contact

linkagekest

contadtest

Model 1

265

600

0.4 14

0.93 8

Model 2

645

642

1.008

1.003

Mode13

605

623

0.945

0.973

Model 4

624

633

0.974

0.988

No slip*

747

1.167

earlier in the analytical models than in the test. The load deflection curves

derived fiom Model 2 agree best with the experimental curves.

The analytical hysteretic loops follow the same trend as the observed

experimental data and successfully represented the amplitude of the

measured data except for the last cycle. The only notable discrepancies

between calculated and observed behavior are the degree of pinching

evident in the load-deflection hysteresis and failure cycle.

The bond slip envelopes for Wall-2 are plotted in Figure 4-8 1 and 4-82

for Model 2. It is evident in the figures that, in the first three cycles, the

slips are very small and the shear wall works as if there is perfect bond

between steel bars and surrounding concrete. The maximum slips are 0.82

mm and 0.87 mm, and minimum slips are -2.88 mm and -2.93 mm for

linkage and contact elements, respectively. The large active slips occur at

the top of the boaom bearn and large negative slips occur in the web zone

near one-third height of the wall up the bottom, due to large concrete

cracks (the largest crack is about 6 mm). When the outside concrete of the

bottom web is uncracked or when cracks are small, the maximum slips

occur at the interfaces between outside tensile steel bars and surrounding

concrete at the bottom of the web. As the cracks of the outside concrete of

the bottom web become large, the maximum slips move toward the

interfaces between the inside tensile steel bars and the surrounding

concrete at bottom of the web. The slip envelopes of Model 2 are the same

for the linkage and contact elements.


the outside bottom of the web of Wall-2 for Model 2 using contact

elements. In the fourth and fifth cycles (the peak top deflection of the wall

is 3.5 mm), there are obvious bond slips. The bond slips are large in cycles

6 and 8 (the peak top deflection of the wall is about 7 mm). The peak

values of the bond stresses do not change much fiom cycles 4 to 8, except

during cycle 7 when the peak values of the bond slips in these cycles lie

between Ai and A2 , and the bond strength in these peak values is constant

(the horizontal branch of bond stress and slip c w e ) for Model 2. In

cycle 7 , the maximum (or minimum) slip is less than that of the previous

cycle, so bond stresses are very small. In cycle 8, the bond stresses in most

elements of the web are smaller than in cycle 6 for the large bond slips. In

the results obtained fkom Models 3 and 4 , the peak values of the bond

stresses decrease greatly in cycle 8 due to peak bond slips larger than Ai .

For Model 2, the steel stresses of the three outside vertical bars at the

bottom of the web reach the yield stress during cycles 6 and 8. For Models

3 and 4, the steel stresses of the outside vertical bars at the bottom drop

due to large slips (the peak slips are larger than AI). When the wall is on

the verge of failing, the steel stresses of the inside vertical bars are very

large (the Iargest steel stress was about 1.1 fy), and there are large slips in

the section at one-third height of the web (over A2 ). Bond stresses are very

small, the steel stresses drop suddenly, and then the wall fails .

In al1 loading stages, the compressive stresses in the concrete are less

than the compressive strength of the concrete. The deflections of the Wall-

2 after failure are shown in Figures 4-1 18 to 4-124 for the various bond

models and bond elements. The computed failure of Wall-2 is due tu large

slips and concrete shear damage at the base of the web when assuming

perfect bond, and at a section at about one-third-height of the web for al1

other bond-slip models.

4.3.3 Shear Wall-3

The element mesh

ultimate loads acting

various analyses and

used to mode1 Wall-3 is show in Figure 4-23. The

at top of the wall are given in Table 4-9 for the

test data. From Table 4-9, it can be seen that the

ultimate loads calculated using al1 bond-slip models, except Model 1, are

reasonably similar to the experimental results, with the differences ranging

fiom 0.0% to 5.5% . Model 2 gives ultirnate loads which agree to within

0.0% for the contact elements and 1.9% for the linkage elements. In

general, the contact elements give better ultimate loads than do the linkage

elements. The ultimate load calculated assuming perfect bond is about 23%

larger than that obtained in the experiment.

Table 4-9. UItimate loads for Wall-3


- - -- - - -

* assumes perfect bond between the concrete and the steel bars.

The computed load-deflection responses of Wall-3 (at the top of the

wall) are plotted in Figures 4-49 to 4-57 for the five bond-slip models and

the two types of element. The load-deflection envelopes computed

assuming perfect bond, and computed using bond-slip Model 2 with

contact elements, are plotted against the test result in Figure 4-58. Plotted

Test data

640

FE analysis

Element Type

linkage

contact

linkagehest

contacthest

Model 1

550

562

0.859

0.878

Model 2

64 1

652

1.000

1.019

Model 3

605

611

0.945

0.955

Model 4

627

632

0.980

0.988

No slip*

787

1.230

in Figure 4-59 is the experimental load-deflection response. The

convergence of Model 1 is very poor for large slipddeflections (deflections

larger than 3 mm), so ultimate loads and other results coming fiom Model

1 are not reliable. For the other three models, the results fiom the two types

of bond elements have some apparent differences. The predictions usine

contact elements give better results; the analytical results agree very well

with test data except for the failure cycle. The loads calculated assuming

perfect bond are about 20% larger than those attained fiom the test. The

wall fails earlier in the analytical models than in the test. The load

deflection curves for Wall-3 derived fiom Model 2 best agree with the

experimental curves.



data except for the last cycle. The only notable discrepancy between the

calculated and measured behavior is the degree of pinching evident in the

load deflection hysteresis.

The bond slip envelopes for Wall-3 are plotted in Figures 4-83 and 4-84

for Model 2 . It is seen in these figures that slips are very small and the


surrounding concrete in the first three cycles. The maximum slips are about

0.24 mm and 1.08 mm, and minimum slips are -3.16 mm and -2.73 mm

for linkage elements and contact elements, respectively. The largest active

slips occur at the top of the bottom bearn and the largest negative slips

occur at the one-third height of the web wall due to large cracks (the widest

crack was 5.6 mm) at the tension side of the web.


the outside bottom of the web of Wall-3 for Model 2 with contact

elements. During cycles 4 and 5 (the peak top deflection of the wall is 3

mm), there is some apparent bond slip. The bond slips are large during

cycles 6 to 8. The peak values of the bond stresses do not change f?om

cycles 6 to 8, while the peak values of the bond slips in these cycles lie

between Ai and A2 , and the bond strength in these peak values is constant

(the horizontal branch F r , of bond stress-slip cuve) for Model 2. In the

8th cycle, the bond stresses in most elements in the web are very small due

to the large bond slips. In the results obtained fkom Models 3 and 4 , the

peak values of bond stresses decrease gradually fiom cycle 6 due to peak

bond slips being larger than AI .

The steel stresses in the outside vertical bars at the bottom web reach

the yield stress in the sixth cycle, and exceed it during the seventh cycle.

The steel stresses of the outside vertical bars at the bottom decrease greatly

when the wall is at the point of failure in the 8th cycle. For Models 3 and 4,

the steel stresses of the outside vertical ban at the bottom cirop earlier and

to a greater extent. When the wall is failing, large slips occur in the zone

near the one-third height of the web. Bond stresses are very small, and the

steel stresses &op suddenly.

At al1 loading stages, the compressive stress in the concrete is less than

the concrete compressive strength. The cracks and defiections of Wall-3

are shown in Figures 4-126 to 4-130 assurning perfect bond and bond-slip

Models 2 to 4, and two bond elements. The computed failure of Wall-3 is

due to large bond slips and concrete shear damage at the base of the web

when assuming perfect bond, and at about one-third height of the web for

al1 other bond-slip models.

4.3.4 Shear Wall4

The element mesh used to mode1 Wall4 is shown in Figure 4-24. The

ultimate loads acting at the top of the wall are given in Table 4-10 for

analytical and test results. From the ultimate loads of Table 4-10, it can be

seen that the correlation between the ultimate loads calculated using TRIX

and experimental results is good; the differences range fiom 1.7% to 3.2%

for al1 models except Model 1. Models 2 and 4 give ultimate loads that

agree exceptionally well with the test data; differences of 1.8% and 1.7%

for the linkage elements, and 2.6% and 2.4% for the contact elements,

respectively, for Models 2 and 4. The ultimate load calculated assuming

perfect bond is about 26% larger than the experimental result.

Table 4-10. Comparison of ultimate loads for Wall-4


1 FE analysis ITest data 1 Elernrnt ~ ~ ~ e ( Model 1 1 Model 2

linkage

contact

Mode13 1 Model 4 1 No slip* 1 I

* assumes perféct bond between thé concrete and the steel bars.

The load-deflection responses of Wall4 are plotted in Figures 4-60 to

4-68 for the five bond Models and the two types of elements. The load-

deflection envelopes computed assuming perfect bond, computed using

bond-slip Model 2 with contact elements, and from the test are plotted in

Figure 4-69. The expenmental result is given in Figure 4-70. The

convergence of Model 1 is very poor for large slips/deflections (deflections

larger than 3 mm), so ultimate loads and other results obtained from Model

1 are not reliable. For the other three bond-slip models, the results corn the

two types of bond elements are not much diEerent, but the analytical

response obtained using contact element fits better with the expenmental

results. The analytical results using Models 2 to 4 agree very well with test

data except during the failure cycle. The loads calculated assuming perfect

bond are about 20025% larger than those attained from the test. The wall

fails earlier in the analytical models than in the test. The load deflection

cuves derived fiom ~ o d e i s 2 and 4 agree best with the expenmental

Cumes.



data except for the last cycle. The only notable discrepancy between

calculated and measured behavior is the degree of pinching evident in the

load deflection hysteresis.

The bond slip envelopes of Wall4 ire plotted in Figure 4-85 and 4-86

for Model 2. It is seen in these figures that the maximum slips are about

0.5 mm and 1-84 mm, and the minimum slips are -0.74 mm and -0.7 1 mm

for linkage elements and contact elements, respectively. The large active

slips occur at the top of the bottom bearn, and large negative slips occur

near the middle-height of the web due to large cracks (the widest crack is

5.7 mm) in the tensile side of the web.

Figures 4- 105 to 4-1 10 give the bond slip, bond stress and steel stress at

the outside bottom of the web of Wall-4 calculated using Model 2 with

contact elements. During the fust cycle, the slips are very small and the


132

surrounding concrete. During cycles 2 and 3 (the peak top deflections of

the wall are 1 mm), there are apparent bond slips. During cycles 4 and 5

(the peak top deflections of the wall are between 4 mm and 5 mm), there

are large bond slips. The peak values of the bond stresses do not change

from cycle 4 until the wall is about to fail, while the peak values of the

bond slips in these cycles lie between Ai and A2 . n e bond strength in

these peak values is constant (the horizontal brmch r=rl of bond stress-slip

curve) for Model 2. When the wall is about to fail in the 8th cycle, the

bond stresses in most elements of the .web are very small for the large

bond slips. In the results obtained fkom Models 3 and 4, the peak values of

bond stresses decrease gradually from cycle 6 due to peak bond slips that

are larger than Ai . The stresses in the outside vertical bars at the bottom tension side

reach the yield stress during the first half-cycle of cycle 6, and exceed it in

the second half-cycle of cycle 6 and first half-cycle of cycle 7. The steel

stresses at the bottom of outside vertical bars decrease greatly in the second

half-cycle of cycle 7. For Models 3 and 4, the steel stresses at the bottom

of outside vertical bars drop more rapidly. When the wall is near failing,

large slips occur at the mid-height of the web for Models 2 and 4 using

linkage elements, and at the bottom of the web for d l models using contact

elements and Model 3 (over A2 ) using linkage element. The bond stresses

are very small, and the steel stresses drop suddenly. The wall fails due to

large slips and shear damage.

The maximum compressive stress in the concrete is less than the

compressive strength during al1 loading stages. The computed deflections

of Wall-4 after failure are shown in Figures 4-132 to 4-138 for the various

bond models and bond elernents. The computed failure of Wall4 is due to

large slips and concrete shear damage- at the mid-height of the web for

Models 2 and 4 using linkage elements, and at the bottom of the web for

the other bond-slip Models and bond elements. If assuming perfect bond,

Wall4 fails due to concrete cnishing at the compression side, at the base of

the web.

The computed load-deflection responses of the four test specimens are

plotted in Figure 4-71 to 4-74 for Model 2 with contact bond elements and

assuming perfect bond under monotonic loading. The ultimate loads acting

at the top of the walls are given in Table 4-1 1. When displacements are

small, the differences between the results obtained from the two models

(assuming perfect bond, and Model 2 with contact elements) are small

because of the small bond slips. When displacements are larger, the

differences in the results are greater because of larger bond slips. When the

walls are near failing, the differences in the results fiom the two models

become smaller. The ultimate loads under rnonotonic loading are larger

than those under cyclic loading for both models, but the differences are

small. The ratios of the ultimate loads calculated assuming perfect bond to

those calculated using bond-slip Model 2, under rnonotonic loading, are

slightly smaller than those calculated under cyclic loading.

Table 4-1 1. Compatison o f ultimate loads for the four test walls

calculated assuming perfect bond and- bond-slip Model 2 with

Fig. 4-1 Finite element mesh of a specimen

used to verify the TRIX.

contact bond elements under monotonie loading( kN )

Perfect bond

Mode1 2 + contact

PerfectModel2

Wall- 1

430

408

1 .O54

Wall-2

76 1

647

1.176

Wall-3

802

659

1.217

Wall4

867

703

1.233

Fig. 4-2 Distribution of tensile force in bar under Load =20 kN (linkage element)

- test data

+mode1 1

,+ mode12

mode13

-mode1 4

-îest data

-m-- mode11

,-mode1 2

m o d e 1 3

m o d e 1 4

Fig. 4-3 Distribution of tensile force in bar under Load =20 kN (contact element )

t e s t data

+ model 1

+mode1 2

+ mode13 -x- mode14

Fig. 4-4 Distribution of tensile force in bar under Load =70 kN (linkage element )

- test data

+mode1 1

+mode1 2

m o d e 1 3

-x- mode14

Fig. 4-5 Distribution of tensile force in bar under Load =70 kN (contact element )

- -

Fig. 4-6

Wall- 1

Cross-sections of walls (Kuzmanovic, 1994)

Fig. 4-9 Side view of reinforcement-Wall- (Chio, 1995)

Fig. 4-1 1 Side view of reinforcement-Wall-4 (Chio, 1 995)

W a l l -Wall

W a l l *base -bas8 -base

Fig 4- 12 Stress-strain curves of concrete used in Wall-3 and Wall-4.

Fig. 4- 13 Stress-strain curves for reinforcement steel.

O 1 2 3 4 5 6 7 8 9 10

cycle

Fig. 4- 14 Loading history of the specimen Wall-1

O 1 2 3 4 5 6 7 8 9 1 O

Cycle

Fig. 4-1 5 Loading historv of the s~ecimen Wall-2

Fig. 4-16 Loading history of the specimen Wall-3

Cycle

Fig. 4-1 7 Loading history of the specimen Wall4

Fig. 4-1 8 Coarse rnesh for Wall-1

Number of nodes : 103

Number of elements : 78

Dimensions of web element: 22Smm x225mm

Fig. 4-19 Fine mesh for Wall-1



Dimensions of web element : 1 12.5mmx 1 1 ZSmm

Fig. 4-20 Finer mesh for Wall-l



Dimensions of web element: 75mm x75mm

Fig. 4-21 Mesh for Wall-1 when considenng bond-slip

between interfaces of al1 steel bars and concrete of web.


Number of elements (linkage element model): 703

Number of elements (contact element model): 688

Dimensions of concrete web element : 1 LMmm x 1 l2.Smm

Length of truss element at the bottom steel hooks : 50 mm (about 5d)

* Thick Iine represents steel tmss elements and bond slip elements.

Fig. 4-22 Final FE mesh for Wall-1

1. For perfect bond model



2. For other bond-slip models

Nurnber of nodes : 364

- Numberofekment~link;rgeeleme~t mo$eQ:39? - - - - - - -


Dimensions of concrete web element: 1 12.5mmx 1 12.5mm

Length of truss element at the bottom steel hooks : 50 mm (about 5d)

* Thick line represents steel truss elements and bond slip elements

Fig. 4-23 FE mesh for Wall-2 and Wall-3

For perfect bond model

Number of nodes : 3 16


For other bond-slip models




Length of m i s s element at the bottom steel hooks : 50 mm (about 5d)

* Thick line represents steel tniss elements and bond slip elements

Fig. 4-24 FE mesh for Wall-4

For perfect bond model



For other bond slip models




Length of truss element at the bottom steel hooks : 50 mm (about Sd)

* Thick line reDresents steel tmss elements and bond s l i ~ elements

-26 -24 -22-20 -18 -16 -14 -12-10 4 -6 4 -2 O 2 4 6 8 10 12

Displacement (mm)

f i n msh

Fig. 4-26 Computed load-displacement response of Wall-1 using fine and finer smeared elements

û'splacememt (mm)

Fig. 4-27 Load-displacement of Wall- 1 assuming perfect bond

-7 -6 -5 4 -3 -2 -1 O 1 2 3 4 5

R'spiacernent (mm)

Fig. 4-28 Load-displacement of Wall-1 for bond-slip Mode1 1 (linkage elements)

-26 -24 -Z -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 O 2 4 6 8 10

Dispiacement (mm)

Fig. 4-29 Load-displacement of Wall-1 for bond-slip Model 2 (linkage elements)

-26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 4 -2 O 2 4 6 8 10 12 Displacement (mm)


-26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 O 2 4 6 8 10

Displacement (mm)

Fig. 4-3 1 Load-displacement of Wall-1 for bond-slip Model 4 (linkage elements)

1

-12 -10 -8 -6 4 -2 O 2 4 6 8 10 12 Displacement (mm)

Fig. 4-32 L.oad-displacement of Wall-1 for bond-slip Model 1 (contact elements)

R'splacement (mm)

Fig. 4-33 Load-displacement of Wall-l for bond-slip Mode1 2 (contact elements)

-26 -24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 O 2 4 6 8 10 12

Dis placement (mm)

Fig. 4-34 Load-displacement of Wall-1 for bond-slip M ~ d e l 3 (contact elements)

-26 -24 -22 -20 -18 -16 -14 -12 -10 -8 6 4 -2 O 2 4 6 8 I O 12

Displacement (mm)

Fig. 4-35 Load-displacement of Wall-1 for bond-slip Model 4 (contact elements)

-28 -24 -20 -16 -12 -8 -4 O

Displacement ( mm )

Fig. 4-36 Load-displacement envelope of Wall-1 computed assuming perfect bond (PB), computed using bond-slip Model 2 with contact

elements (CB2), and as measured during test.

ûispiacement (mm)

Fig. 4-38 Load-displacement of Wall-2 assuming perfect bond

-8 -6 4 -2 O 2 4 6 8 Oisplacement (mm)

Fig. 4-39 Load-displacement of WalI-2 for bond-slip Model 1 (linkage elements)

Displacement (mm)

Fig. 4-40 Load-displacement of Wall-2 for bond-slip Model 2 (iinkage elements)

-ô -6 4 -2 O 2 4 6 8

Dispiacement (mm)


-2 O 2

ûspiacement (mm)


-2 O 2

R'splacement (mm)


-2 O 2 asplacement (mm)


-2 O 2 Displaœment (mm)


-8 -6 -4 -2 O 2 4 6 8

R'spiacement (mm)


-16 -12 -8 4 O 4 8 12 t6 20

Displacement ( mm )

- PB + CB2

+ Test

Fig. 4-47 Load-displacement envelope of Wall-2 computed assuming perfect bond (PB), computed using bond-slip Model 2

with contact elements (CB2), and as rneasured during test.

-16 -14 -12 -10 -8 -6 -4 -2 O

Displacement (mm)

Fig 4-49 Load-displacement of Wall-3 assuming perfect bond

-6 -4 -2 O Displacement (mm)

Fig 4-50 Load-displacement of Wall-3 for bond-slip Mode1 1 (linkage elements)

-1 2 -10 -6 -4 -2 O 2 4 6 8

Dispîacement (mm)

Fig. 4-5 1 Load-displacement of Wall-3 for bond-slip Model 2

-1 0 -8 -6 4 -2 O 2 4 6 8 Dis placement (mm)


-1 2 -1 0 -8 4 4 -2 O 2 4 6

Displacement (mm)

Fig. 4-53 Load-displacement of Wall-3 for bond-slip Mode14 (linkage elements)

-1 2 -1 O -8 -6 -4 -2 O 2 4 6 8 Displacem ent (mm)

Fig 4-54 Load-displacement of Wall-3 for bond-slip Mode1 1 (contact elements)

Displacement (mm)

Fig. 4-55 Load-displacernent of ~ a l l - 3 for bond-slip Model 2 (contact elements)

-12 -10 -8 -6 -4 -2 O 2 4 6 8

DispiacemeM (mm)



- PB

+C82

+ Test

Displacement ( mm )

Fig 4-58 Load-displacement envelope of Wall-3 computed assuming perfect bond (PB), computed using bond-slip Model 2

with contact elements (CB2), and as determined fkom the test.

-10 -8 -6 4 -2 O 2 4 6 8 10 12 14 16

Dispiacement (mm)

Fig 4-60 Load-displacement of Wall-4 assuming perfect bond

-10 -8 ô 4 -2 O 2 4 6 8 10 12 14 16 Displacement (mm)

Fig. 4-6 1 Load-displacement of Wall4 for bond-slip Mode1 1 (linkage elements)

-10 -8 -6 4 -2 O 2 4 6 0 10 12 14

Displacement (mm)

Fig. 4-62 Load-displacement o f Wall4 for bond-slip Model 2 pnkage elements)

-10 -8 -6 4 -2 O 2 4 6 8 10 12 14 16 û'splacement (mm)

Fig. 4-63 Load-displacement of Wall4 for bond-slip Model 3 (linkage elements)

-10 -8 -6 4 -2 O 2 4 6 8 10 12 14

Displacement (mm)

Fig. 4-64 Load-displacement of Wall4 for bond-slip Model 4 (linkage elements)

-10 4 4 4 -2 O 2 4 6 8 10 12 14 16 Dispiacemecit (mm)


-12 -10 -8 -6 -4 -2 0 2 4 6 8 10 l2 l4 l6

R'splacement (mm)

Fig. 4-66 Load-displacement of Wall4 for bond-slip Mode1 2 (contact elements)

Displacernent (mm)

Fig. 4-67 Load-displacement of Wall-4 for bond-slip Mode1 3 (contact elements)

-10 -8 4 4 -2 O 2 4 6 8 10 12 14 16 Oisplacement (mm)

Fig. 4-68 Load-displacement of Wall4 for bond-slip Model 4 (contact elements)

Fig. 4-69 Load-deflection envelope for Wall4 computed assuming perfect bond(PB) , cornputed using bond-slip Model 2 witb contact elements (CB2), and as measured dunng the test.

O 5 10 - 15 Displacement ( mm )

Fig. 4-7 1 Load-displacement of Wall- 1 calculated assuming perfect bond (PB) and calculated using contact elements

with bond Model 2 (CB2) under rnonotonic loading.

Fig. 4-72 Load-displacement of Wall-2 calculated assuming perfect bond (PB) and calculated using contact elernents


Fig. 4-73 Load-displacement of Wall4 calculated assurning perfect bond (PB) and calculated using contact elements


O 5 10 15 Displacement ( mm )

Fig. 4-74 Load-displacement of Wall4 calculated assuming perfect bond (PB) and calculated using contact elements with bond Model 2 (CBZ) under rnonotonic loading.

4 5

Cycle

Fig. 4-75 Bond slip envelope of Wall-1 ( Model 2, linkage elements)

L 1

(

u O 1 2 3 4 5 6 7 8 9 10

Cycle

Fig. 4-76 Bond slip envelope of Wall 1 ( Model 3, linkage elements)

4 5 6 Cycle


Fig. 4-7 8 Bond slip envelope of Wall-1 ( Model 2, contact elements)

4 5 6 Cycle

Fig. 4-79 Bond slip envelope of Wall-1 ( Model 3, contact elements)

4 5 6 Cycle


O 1 2 3

Fig. 4-81 Bond slip envelope

4 5

Cycle

of Wall-2 (

6

Model

7

2, linkage

8 9

elements)

4 S

Cycle


Cycle


3 4 Cycle


3 4 Cycle

Fig. 4-85 Bond slip envelope of Wall4 ( Model 2, linkage elements)

2

1.6

1.2 A

E 0.8 E u

.- 0 0.4 V)

O

-0.4

-0.8 -

1

t I 1

O 1 2 3 4 5 6 7 8 Cyc le


4 5 6 Cycle

Fig. 4-87 Bond slip at the south bottom of Wall-1 (Model 2, contact elements)

Fig. 4-88 Bond slip at the north bottom of Wall-1 (Model 2, contact elements)

Cycle

Fig. 4-89 Bond stress at the south bottom of Wall-1 (Model 2, contact elements)

4 5 6

Cycle

Fig. 4-90 Bond stress at the north bottom of the Wall-1 (Model 2, contact elements).

Fig. 4-91 Steel stress at the south bottom of Wall-1 (Model 2, contact elements)

O 1 2 3 4 5 6 7 8 9 10 Cycle

Fig. 4-92 Steel stress at the north bottom of Wall-1 (Model 2, contact elements)

O 1

Fig. 4-93

4 5

Cycle

Bond slip at the south bottom of Wall-2 (Model 2, contact elements)

Fig. 4-94

4 5

Cycle

Bond slip at the north bottom (Model 2, contact elements)

- -

4 5

Cycle


ô ' 1

O 1 2 3 4 5 6 7 8 9

Cple

Fig. 4-96 Bond stress at the north bottom of Wall-2 (Model 2, contact elements)

-500 ' I O 1 2 3 4 5 6 7 8 9

Cycle

Fig. 4-97 Steel stress at the south bottom of Wall-2 (Model 2, contact elements)

4 5

Cycle

Fig. 4-98 Steel stress at the north bottom of Wall-2 (Model 2, contact elernents)

4

Cycle

Fig. 4-99 Bond slip at the south bottom of Wall-3 (Model2, contact elements)

4

Cycle

Fig. 4-100 Bond slip at the north bottom of Wall-3 (Mode1 2, contact elements)

4

Cycle


-5 J

O 1 2 3 4 5 6 7 8

Cycle

Fig. 4-102 Bond stress at the north bottom of Wall-3 (Model 2, contact elements)

O 1 2 3 4 5 6 7 8

Cycle

Fig. 4-1 03 Steel stress at the south bottom of Wall-3 (Model 2, contact elements)

Fig. 4-104 Steel stress at the north bottom of Wall-3 (Model 2, contact elements)

Cycle

Fig. 4-105 Bond slip at the south bottom of Wall4 (Model 2, contact elements)

O 1 2 3 4 5 6 7

Cycle

Fig. 4-106 Bond slip at the north bottom of Wall4 (Model 2, contact elements)

Cycle

Fig. 4-107 Bond stress at the south bottom of Wall-4 (Mode1 2, contact elements)

Fig. 4-108 Bond stress at the north bottom of Wall4 (Model2, contact elements)

4

Cycle

Fig. 4-109 Steel stress at the south bottom of Wall-4 (Mode1 2, contact elements)

4

Cycle

Fig. 4-1 10 Steel stress at the north bottom of Wall4 (Mode1 2, contact elements)

Fig. 4- 1 1 1 Deflection of Wall-1 after failure (assuming perfect bond)

Fig. 4- 1 12 Deflections of Wall-1 after failure (linkage elements, Mode1 2)

Fig. 4-1 13 Deflections of Wall-1 after failure (linkage elements, Mode1 3)

Fig. 4- 1 14 Defiections of Wall- l afier failure (linkage elements, Mode1 4)

Fig. 4-1 15 Deflections of Wall-1 afier failure (contact elements, Mode1 2)

Fig. 4- 1 16 Deflections of Wall- 1 after failure (contact elements, Mode1 3)

Fig. 4- 1 17 Deflections of Wall- 1 afier failure (contact elements, Mode1 4)

Fig. 4-1 18 Deflection of Wall-2 after failure (assuming perfect bond)

Fig. 4-1 19 Deflections of Wall-2 afier failure (linkage elements, Mode1 2)

Fig. 4-120 Deflections of Wall-2 afier failure (linkage elements, Mode1 3)

Fig. 4-1 2 1 Deflections of Wall-2 after failure (linkage elements, Mode1 4)

Fig. 4-1 22 Deflections of Wall-2 after failure (contact elements, Mode1 2)

Fig. 4-123 Deflections of Wall-2 after failure (contact elements, Mode1 3)

Fig. 4- 124 Deflections o f Wall-2 after failure (contact elements, Model4)

Fig. 4- 125 Deflection of Wall-3 after failure (assuming perfect bond)

Fig. 4-1 26 Deflections of Wall-3 after failure (linkage elements, Model2)

Fig. 4-127 Deflections of Wall-3 afier failure (linkage elements, Mode1 3)

Fig. 4-1 29 Deflections of Wall-3 afier failure (contact elements, Model2)

Fig. 4-130 Deflections of Wall-3 after failure (contact elements, Mode1 3)

Fig. 4- 13 1 Deflections of Wall-3 afier failure (contact elements, Mode1 4)

Fig. 4- 1 3 2 Deflection of Wall-4 after failure (assuming perfect bond)

Fig. 4-1 33 Deflection of Wall-4 after failure (linkage element, Mode1 2)

Fig. 4-1 35 Deflection o f Wall-4 after failure (linkage element, Mode1 4)

Fig. 1- 136 Deflections of Wall-4 after failure (contact elements, Mode1 2)

Fig. 4-137 Deflections of Wall-4 afier failure (contact elements, Mode1 3)

Fig. 4-1 38 Deflections of Wall-4 after failure (contact elements, Mode1 4)

CNAPTER 5

DISCUSSION AND CONCLUSIONS

5.1 Discussion

For the four examined walls, during the first two cycles (peak

deflections not more than 1 mm), bar slips were negligible and the shear

walls behaved as if there was perfect bond between the steel bars and the

surrounding concrete.

Large positive slips occwed above the bottom beam and large negative

slips occurred at the section from one-third to one-half the height of the

web due to large cracks in the tension zone of the web. When the outside

concrete of the bottom web was uncracked or when cracks were srnall, the

maximum slips occurred at the interfaces between outside tension steel

bars and the surrounding concrete near the bottom regions of the web. As

the cracks on the outside concrete of the bottom web became larger, the

maximum slips moved toward the interfaces between inside tension steel

bars and the surrounding concrete at the base of the web.

The differences between the perfect bond mode1 and the experimental

data increased with the increase in stifniess of the wdl. The larger the

stiffness of the wall, the more significant the difference.

The damage index D of the bond resistance due to cyclic loading was

small. It was very close to zero for many of the elements, and about 8% to

20% only for a small number of the elements. It should be emphasized that

the damage index D does not compute accurately in the models used in this

paper. Due to the lack of experimental data on the reduction of bond

resistance under unidirectional cyclic loading, it is assumed that the

damage of bond resistance occurs under reversed cyclic loading only.

Actually, the damage of bond resistance also occurs under unidirectional

cyclic loading. Although the loading history of al1 four walls is reversed

cyclic loading, the slips are unidirectionai cyclic for a certain number of

elements; for example, the elements in the horizontal middle of the web.

This factor is very important for predicting the load-deflection response

accurately when top deflections are large, because the bond resistance in

the outside web of the tension side is very small in this situation. To

analyze the reinforced concrete structures more accurately for large slips

under cyclic loading, it is recommended that experimental research be

undertaken in order to obtain test data under unidirectional cyclic loading.

It is noted that Wall-2 and Wall-3 achieved the sarne ultimate load

during testing, and Wall4 with a barbell-shaped cross section had a

ultimate load slightly higher than that of a rectangular section with the

same amount and detailing of web and boundary reinforcement. Choi

concluded fiom the above data that the horizontal hoops did not affect the

shear strength of the wall, and the flange of the cross section only aflected

the shear strength slightly. It is this author's contention that his conclusions

were incorrect. Because the loading histories were different, the responses

of Wall-2, Wall-3 and Wall4 are not comparable. Actually, the effect of

the reinforcement hoops is considered in the theoretical models by

enhancement of the bond resistance and concrete strength in the area of the

hoops, and the ultimate loads predicted by finite analysis were very close

to those observed in the experiments. This indicates that the hoops

influenced the response of the walls. As for the function of the web

flanges, they affected the response greatly when uncracked, and the effect

become smaller when cracked. However, they still had the efFect of

undertaking compression at the compressive side of the wall.

For al1 four shear walls modeled by bond-slip Models 2 to 4, the

analytically computed hysteretic responses showed the same trends as

measured experimentally, and the computed ultimate lateral loads

correlated very closely to those experimentally recorded, as shown in

Table 5-1. The stifniess of the walls is diminished by increasing the

number of cycles of reversed cyclic bading. Residual deflections upon

unloading are significant and cumulative. Excellent agreement of the

Models 2, 3 and 4 with test results is observed for al1 experimental walls.

There are only two apparent differences between analytical responses with

experimental results. Firstly; the experimental load-deflection curves show

more pinching in the hysteretic loops than those of finite element analysis.

Secondly, the al1 four walls failed one cycle earlier in theoretical analysis

than in experiment. These may be related to two factors. First, the

hysteretic models used for the compression and tension response of

concrete are preliminary, and currently under further development.

Secondly, the hysteretic models of the bond stress-slip relationship may be

inaccurate. It is noted that the bond stress-slip relationship shows wide

scatter in experimental investigations, even under rnonotonic loading. Of

course, the hysteretic mode1 of the bond stress-slip relationship shows

even greater scatter under cyclic loading. It is very dificult to predict bond

stresses and slips of a reinforced concrete structure accurately at present.

The darnage of bond resistance is underestimated due to ignoring the

reduction of bond resistance under unidirectional cyclic slips. It results in a

computed response that is stiffer than seen in the actual structure, therefore

the ultimate loads occur earlier and the structures fail earlier than obsewed

in the experiment.

Table 5-1 Cornparison of ultimate loads of walls


1 FE Analysis 1 Test data 1

Model 2 gives the closest fit to the experimental data. Model 4 also

gives excellent agreement with the data observed in experiment. The

analyses using Model 1 are convergent only when the slips are very small;

this model is not recommended for theoretical modeling of the bond stress-

slip relationship.

The analytical responses computed using contact elements match

slightly better with the experimental data than those cdculated using

linkage elements. The linkage element is not well behaved because its

constant displacement function can not model the non-constant slip field.

The contact element, with a linear displacement function, c m model the

Wall

Wall-1

Wall-2

Wall-3

Wall4

Total

ink ka& Element

load

FE/Test

load

FE/Test

load

FE/Test

load

FE/Test

FE/Test

Model 2

408

1.020

643

1 .O08

64 1

1 .O02

692

1.018

1.012

400

640

640

680

Contact Element

Model 4-

403

1.008

624

0.974

627

0.980

69 1

1.017

0.995

Model 2

404

1 .O10

64 1

1.003

652

1.019

698

1 .O26

Mode14

403

1.008

633

0.989

632

0.988

696

1 .O24

1.015 1 1.002

linear slip field exactly. Also, it gives a better approximation of the

nonlinear slip field than does the lïnkage element.

5.2. Conclusions

This thesis studied the bond stress-slip relationship under both

rnonotonic loading and cyclic loading. It presented formulations for four

types of bond-slip models and two types of bond-slip elements for use in

finite element analysis. These formulations were implemented into the

finite element program TRIX99. The thesis compared results £kom

different analytical models and bond elements for a series of four

reinforced concrete shear walls. The bond-slip Model 2 (by Eligehausen et

al) and Model 4 (proposed by the author of this thesis), in conjunction

with contact elements, is recommended for use in analysis of reinforced

concrete structures.

The thesis documents a reliable analytical tool. The results of the

validation studies demonstrate that the analytical models are capable of

reproducing most of the important aspects of the measured cyclic

responses of reinforced concrete walls with a variety of cyclic loading

histones and wall configurations. The degradation of bond resistance is

included in the finite element program.

The analytical results obtained using the modified program T m 9 9

show excellent correlation with the experimental measurements of the

reinforced concrete walls with different cross sections and reinforcement

arrangements. The prograrn is able to successfully compute the load-

deflection values of the walls under cyclic loading. The proposed finite

element formulation provides an efficient method for evaluating the

rnonotonic and cyclic response of the reinforced concrete structures.

Although this study is restricted to 2-D prablems, the bond-slip element

can easily be adapted to 3-D problems and added to 3-D finite element

programs.

Due to the lack of experimental data on t!!e reduction of bond resistance

under unidirectional cyclic loading, the darnage of bond resistance is

considered only under reversed cyclic loading in this thesis. To analyze the

reinforced concrete structures more accurately for large slips under cyclic

loading, more experimental research is needed in order to obtain the

unidirectional experimental data.

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