THESIS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Three-Dimensional Modelling of Bond in Reinforced Concrete

Theoretical Model, Experiments and Applications

KARIN LUNDGREN

Division of Concrete Structures

Department of Structural Engineering

Chalmers University of Technology

Göteborg, Sweden 1999

Three-Dimensional Modelling of Bond in Reinforced ConcreteTheoretical Model, Experiments and ApplicationsKARIN LUNDGRENISBN 91-7197-853-4

©KARIN LUNDGREN, 1999

Doktorsavhandlingar vid Chalmers tekniska högskolaNy serie nr 1549ISSN 0346-718X

Publication 99:1Arb nr: 37Division of Concrete StructuresChalmers University of TechnologySE-41296 GöteborgSwedenTelephone + 46 (0)31-772 1000

Cover:Results from analyses of a frame corner with a short splice are visualised. The redcolour indicates cracked concrete; shown enlarged is the splitting crack that is a resultof the bond action at the splice. For more information, see Paper IV, page 14.

Chalmers Reproservice

Göteborg, Sweden 1999

I

Three-Dimensional Modelling of Bond in Reinforced ConcreteTheoretical Model, Experiments and ApplicationsKARIN LUNDGRENDivision of Concrete StructuresDepartment of Structural EngineeringChalmers University of Technology

ABSTRACTThe bond mechanism between deformed bars and concrete is known to be influencedby multiple parameters, such as the strength of the surrounding structure, theoccurrence of splitting cracks in the concrete and the yielding of the reinforcement.However, when reinforced concrete structures are analysed using the finite elementmethod, it is quite common to assume that the bond stress depends solely on the slip.A new theoretical model which is especially suited for detailed three-dimensionalanalyses was developed. In the new model, the splitting stresses of the bond action areincluded; furthermore, the bond stress depends not only on the slip, but also on theradial deformation between the reinforcement bar and the concrete. In addition, thismodel includes the simulation of cyclic loading. Steel-encased pull-out tests subjectedto reversed cyclic loading were carried out. The tangential strain in the steel tubes wasmeasured to investigate how the splitting stresses are affected by cyclic loading.Based on the results of these tests, several improvements of the model were made. Barpull-out tests with differing geometries and with both monotonic and cyclic loadingwere analysed, using the new model for the bond action, and non-linear fracturemechanics for the concrete. The results show that the model is capable of dealing witha variety of failure modes, such as pull-out failure, splitting failure, and the loss ofbond when the reinforcement is yielding, as well as dealing with cyclic loading in aphysically reasonable way.

The new model was used in detailed three-dimensional analyses of frame corners.Until recently, splicing of the reinforcement in frame corners had not been allowed bythe Swedish Road Administration. Since this had led to reinforcement detailing thatwas hard to realise on site, it was of interest to examine how splicing of thereinforcement affects the behaviour of the structure. Tests on frame corners subjectedto closing moments were also carried out. It was found that the analyses coulddescribe the test performance in a reasonable way. The tests and analyses showed thatsplicing the reinforcement in the middle of the corner has advantages over splicesplaced outside the bend of the reinforcement. They also indicated, in agreement withprevious work, that provided the splice length is as long as required in the codes, thereare no disadvantages in splicing the reinforcement within the corner of a framesubjected to closing moment.

Key words: Reinforced concrete, bond, splitting effects, three-dimensional analysis,pull-out tests, cyclic loading, finite element analysis, non-linear fracturemechanics, splicing of reinforcement, frame corners.

II

Tredimensionell modellering av vidhäftning i armerad betongTeoretisk modell, experiment och tillämpningarKARIN LUNDGRENAvdelningen för betongbyggnadInstutionen för konstruktionsteknikChalmers tekniska högskola

SAMMANFATTNING

Vidhäftningsmekanismen mellan kamstänger och betong påverkas av ett antalparametrar, såsom hållfastheten hos den omgivande strukturen, uppkomsten avspjälksprickor i betongen och om armeringen flyter. När armerade betong-konstruktioner analyseras med finita elementmetoden antas dock vanligtvis attvidhäftningen beror enbart på glidningen. En ny teoretisk modell har utvecklats, somär speciellt lämpad för detaljerade tredimensionella analyser. I denna nya modell ärspjälkspänningarna som uppstår på grund av vidhäftningen inkluderade, ochvidhäftningen beror inte enbart på glidningen, utan också på den radielladeformationen mellan armeringsjärnet och betongen. Modellen har även utvecklatsför simulering av cyklisk last. Stålmantlade utdragsförsök med cyklisk belastning harutförts. De tangentiella töjningarna i stålrören mättes för att undersöka hur dencykliska lasten påverkar spjälkspänningarna. Utgående från resultaten i dessa försökgjordes flera förbättringar i modellen. Den nya modellen som beskrivervidhäftningsmekanismen har använts, tillsammans med icke-linjär brottmekanik föratt beskriva betongen, i analyser av utdragsförsök med olika geometrier och med bådemonoton och cyklisk belastning. Resultaten visar att den nya modellen kan hanteraolika brottyper, som utdragsbrott, spjälkbrott, att vidhäftningen minskar närarmeringen flyter, samt att den kan simulera cyklisk last på ett fysikaliskt rimligt sätt.

Den nya vidhäftningsmodellen har använts i detaljerade tredimensionella analyser avramhörn. Tidigare har Vägverket inte tillåtit att armeringen skarvas inom ramhörnet.Eftersom det ledde till komplicerade detaljutformningar som var svåra att utföra, vardet av intresse att undersöka hur armeringsskarvar inom hörnområdet påverkar detstrukturella uppförandet. Ramhörn har provats med stängande moment. Det visade sigatt analyserna kunde beskriva försöksresultaten på ett rimligt sätt. Försöken ochanalyserna visade att det är fördelaktigt att skarva armeringen mitt i hörnet, jämförtmed att placera skarven utanför armeringsbocken. De indikerar också, liksom tidigareanalyser och försök, att om skarvlängden är normenlig finns det inga nackdelar medatt skarva armeringen inom hörnområdet i ett hörn belastat med stängande moment.

Nyckelord: Armerad betong, vidhäftning, spjälkande effekter, tredimensionellanalys, utdragsförsök, cyklisk last, finita element-analys, ickelinjärbrottmekanik, skarvning av armering.

III

LIST OF PUBLICATIONS

This thesis is based on the work contained in the following papers, referred to by

Roman numerals in the text:

I ”Modelling Splitting and Fatigue Effects of Bond”, in Fracture Mechanics of

Concrete Structures Proceedings FRAMCOS-3, AEDIFICATIO Publishers,

D-79104 Freiburg, Germany, 1998, pp. 675—685. (Co-author: K. Gylltoft).

II ”Pull-out Tests of Steel-Encased Specimens Subjected to Reversed Cyclic

Loading”, submitted to Materials and Structures.

III ”Bond Modelling in Three-Dimensional Finite Element Analyses”, in July

1999 provisionally accepted for publication in Magazine of Concrete

Research. (Co-author: K. Gylltoft).

IV ”Static Tests and Analyses of Frame Corners Subjected to Closing Moments”,

submitted to Journal of Structural Engineering.

IV

CONTENTS

ABSTRACT I

SAMMANFATTNING II

LIST OF PUBLICATIONS III

CONTENTS IV

PREFACE VI

NOTATIONS VII

1 INTRODUCTION 1

1.1 Background, Aim and Scope 1

1.2 Limitations 1

1.3 Outline of Contents 2

1.4 Original Features 2

2 NON-LINEAR FRACTURE MECHANICS FOR CONCRETE

STRUCTURES 4

2.1 Tensile Behaviour 4

2.1 Compressive Behaviour 7

3 BOND BETWEEN REINFORCEMENT AND CONCRETE 10

3.1 The Bond Mechanism 10

3.1.1 Monotonic loading 10

3.1.2 Cyclic loading 13

3.2 Steel-Encased Pull-Out Tests Subjected to Reversed Cyclic Loading 14

3.3 Theoretical Models of the Bond Mechanism 15

V

4 A NEW BOND MODEL 18

4.1 Presentation of a New Bond Model 18

4.1.1 Elasto-plastic formulation 19

4.1.2 Damaged and undamaged deformation zones 21

4.2 Development of the Bond Model 22

4.2.1 The yield line describing the upper limit 23

4.2.2 Splitting stress in the damaged deformation zone 24

4.2.3 The apex of the yield lines 26

4.2.4 The parameters µ and η within the damaged deformation zone 27

4.3 Calibration of the Model 29

4.4 General Remarks 30

4.4.1 Outer pressure 31

4.4.2 Shrinkage 34

5 FRAME CORNERS SUBJECTED TO CLOSING MOMENTS 35

5.1 Internal Forces in a Corner Subjected to Closing Moment 35

5.2 Frame Corners Subjected to Cyclic Loading 38

5.3 Tests and Analyses of Frame Corners 39

6 CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 42

REFERENCES 44

APPENDIX A DERIVATION OF THE ELASTIC STIFFNESSES IN

THE ELASTIC STIFFNESS MATRIX

VI

PREFACE

In this study, a theoretical model of the bond action in reinforced concrete wasdeveloped and used in finite element analyses of pull-out tests and frame corners.Pull-out tests and tests on frame corners were also carried out. Most of the work wasdone between January 1996 and November 1999. This work is part of a researchproject, "Detailing of frame corners in concrete bridges", which extended from July1996 to June 1999, at the Division of Concrete Structures, Chalmers University ofTechnology. The research project was financed by the Swedish Council for BuildingResearch (BFR), the Development Fund of the Swedish Construction Industry(SBUF), and the Swedish Road Administration (Vägverket). The work has beenfollowed by a reference group consisting of representatives from the building industryand from the Swedish Road Administration. Their interest and valuable comments arehereby acknowledged.

I am most grateful to my supervisor, Professor Kent Gylltoft, for his guidance,support, valuable discussions and encouragement. I also thank Professor BjörnEngström, who was my supervisor during my first period as a doctoral student, for hissupport and valuable discussions, and Professor Emeritus Krister Cederwall, myoffice mate, for his encouragement.

I am also most grateful to all of the doctoral students at the Division of ConcreteStructures for much support, many discussions and the good times we have hadtogether. In particular, I would like to thank Jonas Magnusson and Morgan Johanssonfor many interesting discussions about bond and frame corners. All of the tests werecarried out in the laboratory of the Department of Structural Engineering at ChalmersUniversity of Technology. The laboratory staff is remembered with appreciation fortechnical assistance during the experiments. Lars Wahlström made the photographsfor this thesis, and Wanda Sobko produced some of the figures. I thank also YvonneJuliusson for all assistance, and Lora Sharp-McQueen for improving the language inmost of the thesis. I enjoyed working with Ken Olausson and Carina Haga, who didtheir degree project as a part of the larger research project.

Finally, special thanks go to my family, Stefan, Martin and Thomas Lundgren, fortheir love and support.

Göteborg, November 1999

Karin Lundgren

VII

NOTATIONS

CAPITAL LETTERS

A area

A’ area of one rib

D elastic stiffness matrix

D11 stiffness in the elastic stiffness matrix

D12 stiffness in the elastic stiffness matrix

D22 stiffness in the elastic stiffness matrix

Ec modulus of elasticity of concrete

F force

F1 yield line describing the friction

F2 yield line describing the upper limit at a pull-out failure

G plastic potential function

GF fracture energy of concrete

Ld length of damaged zone

LOWER CASE LETTERS

c parameter in yield function F2 (in the second version of the model the stress in

the inclined compressive struts)

d diameter

fcc compressive strength of concrete

fct tensile strength of concrete

l length

lk distance between ribs

r radius

ra inner radius

rb outer radius

t the tractions at the interface

tn normal splitting stress

tn0 apex of the yield lines in the first version of the model

tt bond stress

VIII

u the relative displacements across the interface

un relative normal displacement at the interface

une elastic part of the relative normal displacement at the interface

unp plastic part of the relative normal displacement at the interface

ut slip

ute elastic part of the slip

utp plastic part of the slip

utmax maximum value of the slip which has been obtained

utmin minimum value of the slip which has been obtained

GREEK LOWER CASE LETTERS

η parameter in the plastic potential function G

ηd the parameter η in the damaged deformation zone

ηd0 the lowest value of the parameter ηd in the damaged deformation zone

κ hardening parameter

λ plastic multiplier

µ coefficient of friction

µd the coefficient of friction in the damaged deformation zone

µd0 the lowest value of the coefficient of friction in the damaged deformation zone

µmax maximum coefficient of friction

υ the Poisson ratio

1

1 INTRODUCTION

1.1 Background, Aim and Scope

The bond mechanism between deformed bars and concrete has been investigated by

numerous researchers. While it is known to be influenced by many parameters, the

most important are the confinement of the surrounding structure and yielding of the

reinforcement. However, when reinforced concrete structures are modelled with finite

element analysis, it is quite common to assume that the bond stress depends solely on

the slip. The confinement of the surrounding structure must then be evaluated before

the analysis can be started, in order to choose an appropriate bond-slip correlation as

input. Whether the reinforcement will yield or not must also be known in advance, for

the same reason. The goal of this project was to design a general model of the bond

mechanism for which the same set of input parameters can be used in all cases; here,

the bond-slip is a result of an analysis, rather than input. It was then intended to use

the model in analyses of spliced frame corners.

Until recently, splicing of the reinforcement in frame corners had not been allowed by

the Swedish Road Administration. This had led to complicated reinforcement layouts

that were hard to realise on site. It was therefore of interest to study how splicing the

reinforcement within the corner region affects the behaviour of a structure. The bends

of the reinforcement bars in the corners cause splitting stresses. When the

reinforcement is spliced, additional splitting stresses arising from the anchorage of the

reinforcement could cause a decreased bond capacity. By using detailed three-

dimensional models combined with a suitable model for the bond, these effects could

be taken into account in analyses.

1.2 Limitations

The goal here was to develop a general model of the bond mechanism to be used in

detailed finite element analyses of concrete structures. When such analyses are

conducted, suitable material models for the concrete are of course needed. The

material models used are the ones available in the finite element program DIANA, see

TNO (1998). The results of the analyses showed that sometimes the material model

2

used was not sufficient to describe the behaviour accurately. This applied, for

example, to the analyses in which the concrete was exposed to cyclic loading or to a

triaxial stress state. The improvement of material models is, however, outside the

scope of this thesis.

For the tests and analyses of frame corners to investigate the effect of splices, the

study has been limited to closing moments. The reason for this was that the effect of

opening moments has been studied more extensively by other researchers already.

1.3 Outline of Contents

This thesis consists of four papers and this introductory part. An introduction to

selected topics is given in the first part: Non-linear fracture mechanics is briefly

presented in Chapter 2, the bond mechanism and related models are outlined in

Chapter 3, and the structural behaviour of frame corners is discussed in Chapter 5.

The new work is presented mainly in the papers. The work started with the design of a

new model for the bond mechanism between reinforcement bars and concrete. This

model and analyses of some pull-out tests are described in Paper I. Since there was a

lack of experimental data on how the splitting stresses are affected by cyclic loading,

pull-out tests on steel-encased concrete cylinders were carried out; these are presented

in Paper II. The results of these tests revealed some drawbacks to the model which

was then changed accordingly. The alteration of the model, with reasons for changes,

is presented in Chapter 4. The second version of the model is presented in Paper III,

together with analyses of pull-out tests, specially chosen to describe various types of

failure. Finally, the model was used in three-dimensional analyses of frame corners,

and the results therefrom are compared with results from experiments in Paper IV.

1.4 Original Features

A new theoretical model of the bond mechanism in monotonic and cyclic loading was

developed. The fundamentals of the model are the friction between the reinforcement

bar and the concrete, as well as the limitation of the stresses in the inclined

compressive forces that result from the bond action. This way of describing the bond

mechanism as a combination of basic mechanisms and combining them in an elasto-

plastic model has not, to the author’s knowledge, been tried before. Furthermore,

3

tests, as well as finite element analyses of pull-out tests and frame corners were

conducted. The steel-encased pull-out tests with specimens subjected to cyclic loading

are believed to be unique, since no tests have been found in the literature that show

the effect of the splitting stresses measured during cyclic loading.

4

2 NON-LINEAR FRACTURE MECHANICS FOR

CONCRETE STRUCTURES

2.1 Tensile Behaviour

Since the fictitious crack model was presented by Hillerborg et al. (1976), and the

crack band theory by Bažant and Oh (1983), non-linear fracture mechanics for

concrete structures has been extended and used by many researchers. A brief

overview of the subject is given here. For more information, see for example

Jirásek (1999).

The two basic ideas of non-linear fracture mechanics are that some tensile stress can

continue to be transferred after microcracking has started, and that this tensile stress

depends on the crack opening, which is a displacement, rather than on the strain (as it

does in the elastic region), see Figure 1. The area under the tensile stress versus crack

opening curve equals an energy which is denoted the fracture energy, GF. This is

assumed to be a material parameter.

w

σ

ε

ε

σ

ε

ft

w

.L

L∆

Unloading response

at maximum load

L+εL+w

σ

σ

w

wwu

= f ( w )GF+

Figure 1 Mean stress-displacement relation for a uniaxial tensile test specimen,

subdivided into a general stress-strain relation and a stress-displacement

relation for the additional localised deformations.

5

From the first models, that used discrete crack elements, the smeared approach was

devised. This means that the deformation of one crack is smeared out over a

characteristic length. When modelling plain concrete, or when slip is allowed between

the reinforcement and the concrete, this characteristic length is approximately the size

of one element. This means that the tensile stress versus strain used will depend on the

size of the element. For axisymmetric analyses, the characteristic length depends on

the number of radial cracks assumed. The more radial cracks that are assumed, the

smaller the characteristic length will be, see Figure 2. When modelling reinforced

concrete and assuming complete interaction between the steel and the concrete, the

deformation of one crack is smeared out over the mean crack distance.

In the first models that used the smeared approach, the direction of the cracks was

fixed. Special input was required in order to determine how large the shear stresses

were that could continue to be transferred across a crack. Several cracks could

develop within the same element. There was, however, a certain threshold angle, that

specified the minimum angle between two cracks. The transfer of shear stresses across

a crack, combined with this threshold angle, allowed the tensile stresses in the

material to exceed the tensile strength, as long as the direction of the tensile stress was

close enough to an already formed crack. In particular, when the direction of the

principal stress changes after cracking, there can be large tensile stresses.

characteristic length

Figure 2 Characteristic length in axisymmetric models.

6

To avoid these large stresses, rotating crack models were developed. In these models,

the direction of a crack is not fixed, but rotates with the direction of the maximum

tensile strain. Generally, coaxiality between principal stresses and principal strains is

assumed. The special input for the shear stresses across the crack is no longer needed,

since these stresses become zero by definition. The behaviour of the rotating crack

models is rather close to elasto-plastic models that have been worked out and used, for

example the Rankine criterion that limits the maximum tensile stress.

After the smeared approach, the concept of embedded crack models was evolved, see

for example Åkesson (1996). Here, the crack is modelled as a strain localisation

within an element. This approach has the benefit of not needing any characteristic

length as input. However, since no three-dimensional model was available when this

project started, the smeared approach was chosen for the analyses.

As mentioned, the smeared approach needs a characteristic length as input. There are

some problems in choosing the characteristic length that arise almost immediately

when modelling reinforced concrete structures. Some examples that have appeared

during this work are discussed here. Since slip was allowed between the

reinforcement and the concrete in the analyses carried out, this characteristic length

should be related to the size of one element. However, this is a problem when the

dimensions of the elements are not the same in all directions. If the crack pattern is

known before the analysis is carried out, the most accurate assumption would be to

use the size of the element perpendicular to the crack plane, see Figure 3. If, however,

the crack pattern is not known in advance, or when cracks appear in more than one

direction in an element, a mean value is usually used. This means that the ductility of

the concrete in one direction is overestimated (the length of the elements), and in

another direction underestimated (the width of the elements).

characteristic length

Figure 3 Characteristic length in oblong elements.

7

The easiest and simplest solution to this problem is of course to use meshes in which

the elements have about the same size in all directions. However, there can also be

problems in doing this. In the three-dimensional analyses of frame corners presented

in Paper IV, the mesh had to be adjusted to fit around the main reinforcement bar.

This means that the smallest dimension of an element had to be as small as about

4 mm. If this size had also been chosen for the dimension in the direction along the

reinforcement bar, the number of elements needed to model the corner region would

have become very large, and the time required for the analysis would not have been

reasonable. Furthermore, another problem was that slip between the main

reinforcement and the concrete was accounted for, while the transverse reinforcement

was modelled with complete interaction. These problems were solved (by good

fortune more than skill); the characteristic length was chosen as the length of the

elements along the main reinforcement bars and the splitting cracks localised in two

elements instead of in one, see Figure 11 in Paper III and Figure 14 in Paper IV. Thus,

the characteristic length chosen was rather realistic for cracks in both directions.

2.1 Compressive Behaviour

Since cracks are easy to spot, localisation of the deformations in a tensile failure of

concrete is not difficult to understand. However, there is also localisation of the

deformations in a compressive failure. Van Mier (1984) showed that the compression

softening behaviour is related to the boundary conditions and the size of the specimen.

An explanation could be that the lateral deformations are partly restrained at the

supports, even though brushes were used to reduce the frictional restraint at the end-

zones. However, these effects are most likely partly due to localisation of the

deformations in a compressive failure, see Figure 4. This has been confirmed in a

Round Robin Test, see van Mier et al. (1997). Markeset (1993) has presented a model

for this, see Figure 5. One of the parameters of the model was the length of the

damaged zone, Ld, shown in the figure. It was assumed to be about 2.5 times the

smallest lateral dimension for centric compressed specimens. When strain gradients

were present, it was assumed to depend on the depth of the damaged zone.

Reinforcement probably affects the length of the damaged zone also.

8

0

0.2

0.4

0.6

0.8

1.0

1.2

0 5 10

height 50 mmheight 100 mmheight 200 mm

Strain [‰](a)

stress MaximumStress

0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6

stress MaximumStress

Displacement [mm](b)

height 50 mmheight 100 mmheight 200 mm

Figure 4 Results from uniaxial compressive tests by van Mier (1984): (a) Stress

versus strain, and (b) Post-peak stress versus displacement for various

specimen heights.

L Ld

σc σc

εc

Win

Gl

w

σc

Ws

εc

σc

Figure 5 Illustration of the model developed by Markeset (1993) for a specimen

loaded in uniaxial compression.

9

The model by Markeset (1993) can serve as a tool for analyses of beams and columns

with uniaxial compression. However, there is at present no convenient way to take the

effect of localisation into account in a generalised material model suited for finite

element analysis, especially not for a general case with triaxial stress states. One

problem is that the number of elements in which the compressive region will localise

is not known when the analysis is started. While in tension, it seems reasonable to

assume that a crack will localise in one element, an assumption that is not so obvious

for compression. In the analyses presented in this thesis, simple stress versus strain

relations for the compressive behaviour were used without taking into account the size

of the elements.

10

3 BOND BETWEEN REINFORCEMENT AND

CONCRETE

3.1 The Bond Mechanism

The bond mechanism is the interaction between reinforcement and concrete. It is this

transfer of stresses that makes it possible to combine the compressive strength of the

concrete and the tensile capacity of the reinforcement in reinforced concrete

structures. Thus, the bond mechanism has a strong influence on the fundamental

behaviour of a structure, for example in crack development and spacing, crack width,

and ductility.

3.1.1 Monotonic loading

The bond mechanism is considered to be a result of three different mechanisms:

chemical adhesion, friction, and mechanical interlocking between the ribs of the

reinforcement bars and the concrete, see Figure 6. This statement can be found in, for

example, ACI (1992). However, the mechanical interlocking can be viewed as

friction, depending on the level at which the mechanism is considered. The bond

resistance resulting from the chemical adhesion is small; it is lost almost immediately

when slipping between the reinforcement and the concrete starts, ACI (1992),

CEB (1982). The inclined forces resulting from the bearing action of the ribs make it

possible, however, to continue to transfer forces between the reinforcement and the

concrete. This implies that bond action generates inclined forces which radiate

outwards in the concrete. The inclined stress is often divided into a longitudinal

component, denoted the bond stress, and a radial component, denoted normal stress or

splitting stress, see Figure 7.

Bearing(c)Friction(b)Adhesion(a)

Figure 6 Idealised force transfer mechanisms, modified from ACI (1992).

11

Stress on the concreteand its components

(b)

P

Stress on thereinforcing bar

(a)

Figure 7 Bond and splitting stresses between a deformed bar and the surrounding

concrete. From Magnusson (1997).

The inclined forces are balanced by ring tensile stresses in the surrounding concrete,

as explained by Tepfers (1973), see Figure 8. If the tensile stress becomes large

enough, longitudinal splitting cracks will form in the concrete. Another type of crack

that is directly related to the bond action are the transverse microcracks which

originate at the tips of the ribs, Goto (1971), see Figure 9. These cracks are due to the

local pressure in front of the ribs, which gives rise to tensile stresses at the tips of the

ribs. These transverse microcracks are also called bond cracks.

Splitting crack

Figure 8 Ring tensile stresses in the anchorage zone, according to Tepfers (1973).

12

N

Adhesion and friction

Support of the ribs

Longitudinalsplitting

Largedeformation

Transverse crack

Figure 9 Deformation zones and cracking caused by bond, modified by

Magnusson (1997) from Vandewalle (1992).

It should be noted that the presence of the normal stresses is a condition for

transferring bond stresses after the chemical adhesion is lost. When, for some reason,

the normal stresses are lost, bond stresses cannot be transferred. This is what happens

if the concrete around the reinforcement bar is penetrated by longitudinal splitting

cracks, and there is no transverse reinforcement that can continue to carry the forces.

This type of failure is called splitting failure. The same thing happens if the

reinforcement bar starts yielding. Due to the Poisson effect, the contraction of the

steel bar increases drastically at yielding. Thus, the normal stress between the

concrete and the steel is reduced so that only low bond stress can be transferred.

When the concrete surrounding the reinforcement bar is well-confined, meaning that

it can withstand the normal splitting stresses, and the reinforcement does not start

yielding, a pull-out failure is obtained. When this happens, the failure is characterised

by shear cracking between two adjacent ribs. This is the upper limit of the bond

capacity.

A common way to describe the bond behaviour is by relating the bond stress to the

slip, that is the relative difference in movement between the reinforcement bar and the

concrete. As made clear above, the bond versus slip relationship is not a material

parameter; it is closely related to the structure. It also depends on several parameters

such as casting position, vibration of the concrete and loading rate. Examples of

schematic bond-slip relationships are shown in Figure 10.

13

Bondstress

Slip

(a)(b)

Figure 10 Schematic bond-slip relationship: (a) pull-out failure; (b) splitting failure,

or loss of bond due to yielding of the reinforcement.

3.1.2 Cyclic loading

A typical response for bond in cyclic loading is shown in a bond versus slip diagram

in Figure 11. The monotonic curve is followed for the first loading until point A in the

figure. Thereafter occurs a steep unloading to point B, and then an almost constant,

low bond stress until the original monotonic curve is reached at point C. As for

monotonic loading, the response depends on the structure, and the influencing

parameters are the same. Moreover, the response is also influenced by the type of

cyclic loading. According to ACI (1992), load cycles with reversed loading cause a

greater degradation of bond strength and stiffness than the same number of load

cycles with unidirectional loading. The peak value of the slip is a critical factor.

Additional cycles between slip values smaller than earlier ones do not significantly

influence the bond behaviour, according to Eligehausen et al. (1983), Balázs (1991)

and ACI (1992).

B

ABondstress

SlipC

Figure 11 Typical bond versus slip for cyclic loading.

14

3.2 Steel-Encased Pull-Out Tests Subjected to Reversed Cyclic

Loading

As discussed in Section 3.1.1, anchoring deformed bars in concrete gives rise not only

to bond stresses but also to splitting stresses. Although many experiments have been

conducted to study the bond stresses, the splitting stresses are less investigated.

Tepfers and Olsson (1992) have done “ring tests” in which a reinforcement bar was

pulled out of a concrete cylinder surrounded by a thin steel tube. By measuring the

tangential strains in the steel tube, the splitting stresses could be evaluated. A few

other researchers have also carried out tests to find solutions to the problems of

measuring the splitting stresses, for example Malvar (1992). The effect on bond of

cyclic loading has been investigated by, among others, Eligehausen et al. (1983) and

Balázs and Koch (1995), who have conducted large programmes of pull-out tests with

cyclically loaded specimens. However, no tests were found in the literature that show

the effect of the splitting stresses measured during cyclic loading.

Therefore, steel-encased pull-out tests subjected to reversed cyclic loading were

carried out, see Paper II or Lundgren (1998). The main purpose of these tests was to

give reference information for calibrating models of the bond mechanism, to improve

knowledge of the splitting stresses, and to investigate how they are affected by

reversed cyclic loading. Hence, a reinforcement bar was pulled out of a concrete

cylinder surrounded by a thin steel tube. The effect of the splitting stresses during

cyclic loading could be studied by measuring the tangential strains in the steel tube,

together with the applied load and slip. In five tests, specimens were loaded by

monotonically increasing the load, while nine other tests subjected specimens to

reversed cyclic loading. All of the tests resulted in pull-out failures. The results from

the monotonic tests indicate that the splitting stresses decreased after the maximum

load had been obtained, although not as much as the load decreased. The results from

the cyclic tests show a typical response for bond in cyclic loading. When there was

almost no bond capacity left, the measured strain in the steel tubes stabilised and

remained more or less unaffected by the last load cycles. The test results provided

valuable information which influenced not only the calibration of the bond model but

also the formulation of the model; more detail is given in Section 4.2.

15

3.3 Theoretical Models of the Bond Mechanism

When reinforced concrete structures are analysed, complete interaction between the

reinforcement and the concrete is perhaps the most frequent assumption. This

assumption is used in almost all hand calculations, for example in the analytical

models for bending moment in the ultimate limit state. In finite element analyses also,

this is a rather commonly used assumption; especially when the overall behaviour of a

larger structure is examined, this assumption is often sufficient for the level of

modelling desired.

Nevertheless, for more detailed analyses of smaller parts of a structure, especially if

one is interested in following the crack development more thoroughly, the bond

mechanism needs to be taken into account. The most usual way to do this is to employ

bond versus slip relations as input. Several researchers have examined the bond

mechanism and suggested various bond versus slip curves to be used in analyses, for

example Tassios (1979), and Eligehausen et al. (1983) include both monotonic and

cyclic loading. However, as discussed in Section 3.1.1, the bond versus slip depends

on the structure. As long as this is kept in mind, a reasonable bond versus slip relation

can be assumed by taking parameters such as the actual concrete cover, the amount of

transverse reinforcement etc. into account. If one wishes to study crack development

in structural members for example, then this way of taking the bond mechanism into

account offers a sufficient level of accuracy and detail.

However, for more detailed analyses of parts of a structural member where the bond

mechanism plays a decisive role for the behaviour, a more refined model for the bond

is needed. This is needed mainly for analyses of anchorage regions, such as in splices

and anchorage of the reinforcement at end supports, but also for analysis of the

rotational capacity, where the bond plays a crucial part. A requirement for this type of

model is that the bond mechanism be described in such a way that the bond versus

slip achieved in a structure is a result of the analysis, rather than input. Another

requirement is that the model includes not only the bond stresses, but also the splitting

stresses that result from the anchorage.

The model by Gylltoft (1983) included the effect of normal stresses, which allows an

outer pressure to increase the capacity. Furthermore, the model could deal with cyclic

loading; fracture mechanics was used to describe the damage. However, the model did

16

not include any active normal splitting stresses that result from the anchorage, and

bond versus slip was used for the input.

Some models that include the active splitting stresses, while still using a form of bond

versus slip as input, have been developed, see for example Mainz (1993). Also, some

attempts to model the bond mechanism in a more thorough way by including the ribs

of the reinforcement in the geometrical model have been done, for example by

Reinhardt et al. (1984). The model by den Uijl and Bigaj (1996), see also

Bigaj (1999), includes the splitting stress; the bond stress is related not only to the

slip, but also to the radial deformation between the reinforcement bar and the

concrete. The model can therefore describe the loss of bond if the reinforcement

yields. This is an analytical model, for which the effect of the confinement is obtained

from analyses of a thick-walled cylinder. The results of the model show good

agreement with test results. This model can serve as a valuable tool for getting

information about what bond versus slip should be used as input in an analysis of a

structure. However, it does not seem possible to implement it in a more general way,

for example in a finite element program. Hence, if a part of a structure is to be

modelled, some results of the analysis need to be known in advance, such as whether

splitting failure will occur.

The model by Åkesson (1993) and the one by Cox (1994) represent a new kind of

model. In these models, the splitting stresses are included, and the bond stress

depends not only on the slip but also on the radial deformation between the

reinforcement bar and the concrete. This makes it possible to include the effect of the

confinement of the surrounding structure. Both models use elasto-plastic theory, as

shown by the yield lines in Figure 12.

splittingstress

bondstress at max.

initial

bondstress

splittingstress(b)(a)

Figure 12 Yield lines of the models of (a) Åkesson (1993) and (b) Cox (1994).

17

In Åkesson’s model, the yield line describes the friction with adhesion included. The

adhesion is assumed to decrease to zero for relatively small slip. This model was

devised for studies of the release of prestressed strands in hollow core slabs. It was

therefore intended to be used only for monotonic loads.

To limit the bond capacity, Åkesson made the elastic stiffness describing the relation

between the normal stress and the slip non-linear, with a maximum followed by

decreasing normal stress. This gives reasonable results for monotonic loading. For

cyclic loading, however, it can give unexpected results, such as that the normal stress

increases at unloading. Note, however, that the model was not intended for cyclic

loading.

Another drawback to Åkesson’s model is that there is no upper limit of the bond stress

prescribed by the yield lines; as can be seen in Figure 12 (a), the bond stress can

become infinitely high as long as enough normal stress is present. This does not agree

with the experimental results of, for example, Robins and Standish (1984). Their tests

showed that lateral confinement changed the failure mode from splitting failure to

pull-out failure. Yet, further increase of the lateral confinement had no effect on the

bond capacity. However, outer pressure was outside the scope of the model.

The model by Cox (1994) does not have this drawback; as can be seen in

Figure 12 (b), the bond stress curves towards an upper limit when the normal stress

increases. The initial increase followed by a decrease in bond stress (compare with the

bond versus slip curves in Figure 10) is obtained in this model by letting the yield

surface harden, as shown in Figure 12 (b), and thereafter soften almost to the initial

yield line again. This model is probably a more general model of the bond mechanism

than the model by Åkesson. Still, it has not been shown to describe the loss of bond

when the reinforcement yields. Furthermore, it seemed entirely possible that the

physical behaviour could be described in a more fundamental way.

18

4 A NEW BOND MODEL

4.1 Presentation of a New Bond Model

A new bond model which includes the splitting stresses was developed. With one set

of input parameters, this new model produces different bond-slip curves, determined

by the confinement of the surrounding structure and whether or not the reinforcement

is yielding. The effect of cyclic loading with varying slip direction is also important

for the bond resistance, which is why this effect was included in the model. The

model was implemented in the finite element program DIANA, for more detail see

Lundgren (1999a). In DIANA, there are interface elements available, which describe

a relation between the tractions t and the relative displacements u at the interface.

These elements are used at the surface between the reinforcement bars and the

concrete. The physical interpretations of the variables tn, tt, un and ut are shown in

Figure 13. The interface elements have, initially, a thickness of zero.

reinforcementbar

uttn

tt

un

���

�=�

���

�=

��

��

�=�

���

�=

sliplayer at thent displaceme normal relative

stress bondstress splitting normal

t

n

t

n

uu

tt

u

t

Figure 13 Physical interpretation of the variables tn, tt, un and ut.

19

4.1.1 Elasto-plastic formulation

The new bond model is a frictional one, using elasto-plastic theory to describe the

relations between the stresses and the deformations. Thus, the model has yield lines,

flow rules, and hardening laws. The relation between the tractions t and the relative

displacements u is in the elastic range:

���

�

����

���

�

�

=��

��

�

t

n

t

t

t

n

uu

D

Duu

Dtt

0 22

1211 (1)

where D12 is normally negative, meaning that slip in either direction will cause

negative tn, i.e. compressive forces radiating outwards. The yield lines are described

by two yield functions, one for the friction, F1, assuming that the adhesion is

negligible:

0=+= nt1 ttF µ . (2)

The other yield line, F2, describes the upper limit for a pull-out failure. This is

determined from the stress in the inclined compressive struts that result from the bond

action, see Figure 14.

0222 =⋅++= nnt tcttF (3)

-tndl

r·dϕ

c

α

tt

dl·sinα Equilibrium gives:

0

sin

sin

22

22

22

=⋅++

+

−=

⋅⋅=⋅⋅+

nnt

nt

n

nt

tctt

tt

t

rddlcrddltt

α

ϕαϕ

Figure 14 The stress in the inclined compressive struts determines the upper limit.

20

For plastic loading along the yield line describing the upper limit, F2, an associated

flow rule is assumed. For the yield line describing the friction, F1, a non-associated

flow rule is assumed, for which the plastic part of the deformations is

0 , =+== ntt

t ttuu

GGdd η∂∂λ

tup (4)

where dλ is the incremental plastic multiplier. The yield lines, together with the

direction of the plastic part of the deformations, are shown in Figure 15. At the

corners, a combination of the two flow rules is used; this is known as the Koiter rule.

For the hardening law of the model, a hardening parameter κ is established. It is

defined by

22 pt

pn dudud +=κ . (5)

The variables µ and c in the yield functions are assumed to be functions of κ.

tup

∂∂

= 2Fdd λ

tn

tt

F1

G

F2t

up

∂∂= Gdd λ

ttup

∂∂+

∂∂= 2

21FdGdd λλ

µη

11

Figure 15 The yield lines. The plastic part of the deformations, dup, is given by an

associated flow rule at the yield line describing the upper limit, F2, and a

non-associated flow rule at the yield line describing the friction, F1.

21

4.1.2 Damaged and undamaged deformation zones

A typical response for bond with varying slip direction is a steep unloading and then

an almost constant low bond stress until the original monotonic curve is reached; this

is described in Section 3.1.2. To make the model describe this typical response, a new

concept, called damaged and undamaged deformation zones, is used. The idea is that

when the reinforcement slips in the concrete, the friction will be damaged (reduced) in

the range of the passed slip. This is a simplified way to describe the damage of the

cracked and crushed concrete. In Figure 16 (b), the reinforcement is back in its

original position after slipping in both directions. The concrete, consequently, is

crushed in the range of the passed slip. While this crushed concrete still has some

capacity to carry compressive load, it has no capacity at all in tension. The friction is

therefore assumed to vary in the damaged zone, depending on whether loading is

applied in the direction away from, or towards, the original position, as shown in

Figure 16 (c) and (d). It is assumed to drop immediately to a low value, µd0, at load

reversal, and to keep this value until the original position is reached. For further

loading, away from the original position, the friction is assumed to increase gradually

until the undamaged zone is reached and the normal value of µ is used again. To

describe this gradual increase, an equation of the second degree has been chosen.

The parameter η also has a lower value in the damaged deformation zone, varying in

the same way as just described for the coefficient of friction. This lower value

corresponds physically to the fact that the increase in the stresses is lower in the

damaged than in the undamaged deformation zone.

22

“crushedconcrete”

utmax

concrete

reinforcement bar

(b)

utmin

tt

ut

damageddeformationzone

undamageddeformationzone

undamageddeformationzone

(a)

utmax

utmin

µ, η

ututmaxutmin

µd0, η d0

(c)

µ, η

ututmaxutmin

µd0, η d0

(d)

loadingdirection

loadingdirection

Figure 16 (a) One load cycle with varying slip directions. (b) The reinforcement bar

is back in its original position, after slipping in both directions. Maximum

and minimum values of the slip are marked. (c) and (d) The parameters µ

and η vary within the damaged deformation zone depending on whether

the loading is directed towards or away from the original position.

4.2 Development of the Bond Model

The bond model described in the previous section is the same as that presented in

Paper III. In Paper I, an earlier version of this model is described. The two versions

are slightly different: the one in Paper III can be viewed as an improvement of the

first one. The main reason for the changes was the results from the steel-encased pull-

out tests that are reported in Paper II. Sections 4.2.1 through 4.2.4 cover the

differences between the two versions of the models with reasons for the changes.

23

4.2.1 The yield line describing the upper limit

In both versions of the model, the yield lines are two yield functions, one describing

the friction and the other describing the upper limit of a pull-out failure. In the first

version of the model, little attention was paid to the formulation of the upper limit.

Only one example was considered: the theoretical one with zero bond stress, which

leads to a limit of the splitting stress about the same as the compressive strength of the

concrete. By examining the results from pull-out tests, a reasonably large bond

capacity was then obtained simply by setting an upper limit with straight lines, as

shown in Figure 17.

In the second version of the model, the combinations of splitting stresses and bond

stresses were recognised as inclined compressive struts. By letting the stress in these

compressive struts be limiting, a new expression was derived for the upper limit, see

Figure 14. This new expression is believed to be better than the first one, since it

corresponds more closely to the physical reality. When results from analyses were

compared with results from the monotonically loaded steel-encased pull-out tests, it

also appeared that the second version of the model gave improved results. The main

drawback to the first version of the model was that the tangential strains in the steel

tube were too small in the analyses, when compared with the measured ones. With the

second version of the model, larger strains were obtained for the analyses. The reason

for this can be seen directly in Figure 17, where the second expression for the upper

limit gives greater splitting stress than the first one for the same bond stress. This is so

when the coefficient of friction is between zero and one, as it is when the maximum

capacity at a pull-out failure is obtained. When the coefficient of friction is larger than

one, it is the other way around; i.e. the second expression for the upper limit gives a

lower splitting stress than the first one for the same bond stress. Since the largest

value of the coefficient of friction was 1.0 in the calibration of the second version,

however, this example is not valid here.

24

tn

tt

F2 second version

F2 first version

Figure 17 Comparison of the yield lines for the two versions of the model.

4.2.2 Splitting stress in the damaged deformation zone

In the first version of the bond model, it was assumed that the splitting stress

decreased during unloading until the bond stress was zero, and then increased again

when bond stress in the opposite direction was obtained. The results from the

cyclically loaded steel-encased pull-out tests showed, however, that this was not so.

As can be seen in Figure 18, the tangential strain in the steel tube decreased during

unloading, on the other hand, it continued to decrease also when there was a small

bond capacity in the opposite direction. The tangential steel strain did not start to

increase again until the reinforcement had returned to its original position, most

clearly shown in Figure 18 (b). This means that the splitting stresses due to the bond

action do not start to increase again until the slip is back to zero. The relation between

the tractions t and the relative displacement u in the elastic range was accordingly

changed from equation (1) in Paper I to (1) in Paper III:

���

�

����

���

�

�

=��

��

�

t

nt

t

t

n

uu

D

Dtt

Dtt

0 22

1211 (6)

was changed to

25

���

�

����

���

�

�

=��

��

�

t

nt

t

t

n

uu

D

Duu

Dtt

0 22

1211 . (7)

Also, the plastic potential function G was changed slightly, from

0)( 0 =−+= nnt tttG η (8)

to

0=+= ntt

t ttuu

G η . (9)

With these changes, the splitting stress and the bond stress decrease until the

reinforcement is back in its original position, see Figure 19.

-60

-40

-20

0

20

40

-0.1 0 0.1 0.2 0.3 0.4 0.5

Load [kN]

Strain [‰](a)

0

0.1

0.2

0.3

0.4

0.5

-2 -1 0 1 2

Strain [‰]

Slip [mm](b)

Figure 18 Results from the first load cycles in the steel-encased pull-out test

No. C-0.5b: (a) Load versus tangential strain in the steel tube, and

(b) Tangential strain in the steel tube versus slip.

26

ut

tt

ut

tt

tn

tt

tn

tt

(a)

(b)

Figure 19 Comparison of results for the two versions of the model, at unloading

back to the original position: (a) The first version, and (b) the second

version.

4.2.3 The apex of the yield lines

In the first version of the bond model, the apex of the yield lines was moved in the

direction of the loading, see Figure 20. The main reason for this was that the increase

of the splitting stress within the damaged deformation zone led to an increase of this

stress for each successive load cycle. With this large splitting stress, there could also

be a large bond stress, when the apex of the yield lines remained at the origin. To

avoid this large bond stress, which did not correspond with experimental results, the

apex of the yield surface was moved. When, in the second version of the model, the

splitting stress decreased until the slip was zero, this stress no longer increased for

every load cycle. This seems more reasonable physically. Also, it allows the bond

capacity to be reasonably large without moving the apex of the yield lines. The apex

therefore remains at the origin in the second version of the model.

27

1

tn

tt

loading pathµ

elasticunloading

initialyield line

Figure 20 The apex of the yield lines was moved in the direction of the loading in

the first version of the model.

4.2.4 The parameters µµµµ and ηηηη within the damaged deformation zone

In the first version of the model, the coefficient of friction, µ, and the parameter η

were assumed to have constant values within the damaged deformation zone. The

parameter η within the damaged deformation zone, ηd, was set so low that the bond

stress was almost constant in this zone. When the undamaged deformation zone was

reached, a steep increase was obtained, see Figure 21 (a). A cyclic pull-out test by

Balázs and Koch (1995) was analysed in which the force was applied on one end of

the reinforcement bar, so the slip was not constant along the reinforcement bar. This

variation of the slip along the bar made the load increase in the analysis slightly less

abrupt than the increase in local bond stress, although this load increase was not as

gradual as was observed in their tests, see Figure 7 in Paper I.

ut

tt

ut

tt

(a) (b)

Figure 21 Bond stress versus slip: results from (a) the first version, and (b) the

second version of the model.

28

However, the steel-encased pull-out tests were loaded in a rigid frame, so that both

ends of the reinforcement bar were active. The variation of the slip along the bar was

thus very close to zero. In the analysis, the abrupt increase of the bond stresses when

reaching the undamaged deformation zone therefore gave a corresponding abrupt

increase in the load versus slip curve. Since this was not the case for the measured

results, a revision of the model was indicated.

In the first version of the model, it was assumed that there were “empty holes” in the

concrete in the range of the passed slip. In the second version of the model, the

concrete that is crushed in front of the ribs was taken into account. While the crushed

concrete can still have some capacity to carry compressive load, it has no capacity at

all in tension. Consequently, the friction was assumed to vary in the damaged zone

according to whether loading was applied in the direction away from, or towards, the

original position. It was assumed to drop immediately to a low value at load reversal,

and to keep this value until the original position was reached. For further loading,

away from the original position, the friction was assumed to increase gradually, until

the undamaged zone was reached, when the normal value was used again. To describe

this gradual increase, an equation of the second degree was chosen. In Figure 22, a

comparison of the two versions of the model is shown.

µ,=η

utmin

µd0 , ηd0

first version

second version

utmax ut

Figure 22 The coefficient of friction, µ, and the parameter η in the damaged

deformation zone. Comparison of the two versions of the model.

29

4.3 Calibration of the Model

The different versions of the model were calibrated against pull-out tests found in the

literature; the second version was also calibrated against the steel-encased pull-out

tests that are presented in Paper II. In order to investigate whether the model could

also describe the loss of bond when the reinforcement was yielding, a degree project

was carried out, see Haga and Olausson (1998), in which the first version of the model

was used. Since the calibration of the model was not quite finished, the input

parameters used were slightly different from the ones described in Paper I. Some

changes in the input assumptions were also made for the second version of the model,

compare Paper I with Paper III. In Figure 23 it can be seen that the coefficient of

friction was set slightly lower in the second calibration, to match the large tangential

strains that were measured in the steel-encased pull-out tests. Also the other

parameters were subjected to minor changes, for example the parameter η was

changed from 0.05 to 0.04.

Another, and perhaps more significant, change between the calibrations of the two

versions is that, for the second version, the stiffnesses in the stiffness matrix, D, were

assumed to be determined by the modulus of elasticity of the concrete rather than by

the compressive strength. The reason for this was further consideration about what the

stiffnesses physically described, and how they can be derived. The stiffnesses in the

elastic stiffness matrix, D, shall describe how the concrete between the ribs acts for

elastic conditions. In Appendix A it is shown how these stiffnesses were derived.

00.20.40.60.81.01.21.41.6

0 0.005 0.01 0.015 0.02

µ

κ [m]

µd

first version

second version

Figure 23 The coefficient of friction versus the hardening parameter: input chosen.

30

4.4 General Remarks

The new bond model was calibrated for reinforcement bars K500 φ 16 and normal

strength concrete (cylinder compressive strength about 30 MPa). However, the

calibration was made in such a way that the stiffnesses and the strength were

expressed in terms of modulus of elasticity and strength of the concrete. After this

calibration, Magnusson (2000) used the model in analyses of tests for which the same

type of reinforcement was used, although the concrete was a high strength one with

cylinder compressive strength of about 100 MPa. Since the analyses showed good

agreement with the tests, it seems as if the calibration is also applicable to concrete of

other qualities. The main reason for this is that the parameters are physically

meaningful, not chosen arbitrarily. Nevertheless, it must be emphasised that the way

the surrounding structure is modelled is critical. If splitting of the concrete dominates

the failure mode, parameters such as the fracture energy and the tensile strength of the

concrete are crucial.

Concerning other types of reinforcement bars, it is not very likely that the same

calibration will give good results. The stiffnesses D11 and D22 were derived for the

geometry of a reinforcement bar K500 φ 16, see Appendix A. However, if the same

derivations are made for the geometry of another kind of reinforcement bar, they can

probably be used. The input of the coefficient of friction will most likely also change

if the reinforcement type is changed. If the type of reinforcement is completely

different, new comparisons with tests would need to be done, preferably steel-encased

pull-out tests for which the tangential strains can be measured.

The model was calibrated with tests that were selected to show five different types of

failure; i.e. pull-out failure, splitting failure, pull-out failure after yielding of the

reinforcement, rupture of the reinforcement bar, and cyclic loading. The results show

that the model is capable of dealing with all these kinds of failure modes in a

physically meaningful way, and reasonably good agreement between analyses and

experimental results was found, see Paper III. On the other hand, there are still other

parameters that are known to influence the bond action. Two such parameters are the

presence of outer pressure, and shrinkage of the concrete; although the model was not

specifically calibrated with any tests for these two parameters, the behaviour of the

model was observed in relation to their presence or absence.

31

4.4.1 Outer pressure

Pull-out tests with short embedment length, Magnusson (1997), were analysed

without any outer pressure, for Paper III. Here, an outer pressure of 5 MPa was

applied, and kept constant while the pull-out force was applied. The results are

compared with results from the analysis without outer pressure, see Figure 24. While

the outer pressure was applied, the radial deformation between the reinforcement bar

and the concrete decreased, which implies a normal stress tn, see Figure 25. This

means that, when slipping between the concrete and the reinforcement began, some

normal stresses were already present. Therefore, the first part of the loading was

elastic, until the yield line was reached. Thus, the load versus slip starts with a stiff,

elastic part. The capacity is, however, not influenced, since the failure mode is pull-

out failure in both cases; the pull-out failure in the model is governed by the upper

limit in the form of the yield line, F2, which is determined from the compressive

strength of the concrete. Test results of Robins and Standish (1984) indicate that this

is a correct behaviour. They carried out cube pull-out tests with deformed bars with

lateral pressure varying from 0 to 28 MPa. They concluded that the maximum

capacity was increased for low levels of confinement, since the failure mode was

changed from splitting failure to pull-out failure. On the other hand, further increase

of the lateral confinement had no influence on the maximum capacity.

0

5

10

15

20

25

30

0 5 10 15

Load [kN]

Slip [mm]

00.51.01.52.02.53.03.54.0

0 0.01 0.02 0.03 0.04

Outer pressure 5 MPa

Without outer pressure

Figure 24 Comparison of results from analyses of a pull-out test where pull-out

failure is limiting, with and without an outer pressure.

32

initial yield linesyield lines afterhardeningloading path

tn

tt

(b)

(b)(a)

Figure 25 The effect of either outer pressure or shrinkage of the concrete, in the

stress space: (a) Without outer pressure and shrinkage of the concrete, and

(b) With either an outer pressure or shrinkage of the concrete taken into

account.

There are tests described in the literature that report a higher capacity due to outer

pressure. However, when these references were read more thoroughly, it appeared that

splitting cracks were present, Untrauer and Henry (1965), Eligehausen et al. (1983).

As these splitting cracks had probably reduced the capacity, the presence of an outer

pressure would have a beneficial effect. This also reflects the behaviour of the model

presented. The bar pull-out splitting test without spiral reinforcement carried out by

Noghabai (1995), see Paper I, was analysed both with and without a confining outer

pressure. In the analysis without outer pressure, failure was due to splitting of the

concrete. As can be seen in Figure 26, an outer pressure then increased the capacity.

In this example, the applied outer pressure was great enough to prevent the

development of splitting cracks; thus, the capacity was increased to the level of a pull-

out failure. For a low confining pressure, the formation of the splitting cracks would

only have been delayed, meaning that the capacity would have been greater than for

the unconfined specimen, although not enough to lead to a pull-out failure.

33

Load [kN]

Slip [mm]

0

50

100

150

200

250

300

0 5 10 15

Outer pressure 5 MPa

Without outer pressure

Figure 26 Comparison of results from analyses of a pull-out test where splitting

failure is limiting, with and without an outer pressure.

Magnusson (2000) has applied the model in some analyses of beam ends. The beam

ends were either supported at their lower edge, so that the support reaction gave

confinement to the reinforcement anchored over the support, or they were hung, so

that the support reaction acted over the reinforcement bars, i.e. there was no

confinement. It appeared from the analyses that the model could describe the

behaviour accurately, and reasonably good agreement was found between the analyses

and the test results. When no confinement was present, splitting failure occurred,

which reduced the anchorage capacity in both the analyses and the tests. The

confinement made it possible to obtain a pull-out failure in the analyses, i.e. the

capacity was increased by about as much as in the tests. From these tests and analyses,

it seems as if the model can also describe the effect of outer pressure in a reasonable

way. The results indicate that outer pressure can increase the bond capacity to the

limit of the pull-out failure, although no further.

34

4.4.2 Shrinkage

The adhesion between the concrete and the reinforcement bar is assumed to be

negligible in the new bond model. On the other hand, in pull-out tests it is usual to

have a first part of the load versus slip curve that is very stiff; this part is usually said

to be due to the adhesion. However, a part of it may be caused by shrinkage of the

concrete. When the concrete around the reinforcement bar shrinks, there are normal

stresses between the concrete and the reinforcement bar before slipping starts. This

resembles the situation with outer pressure discussed before, see Figure 25. Yet there

is a difference which is that the shrinkage of the concrete also causes tensile stresses

around the reinforcement bar, so that splitting cracks could appear. This is in contrast

to the application of outer pressure which does not give rise to any tensile stresses.

The pull-out tests with short embedment length, Magnusson (1997), were analysed

both with and without shrinkage of the concrete being taken into account. A shrinkage

strain of -1.1·10-5 was then applied, calculated from CEB (1993), taking into account

how the test specimens were stored. The results are compared in Figure 27. As can be

seen, the first part is stiffer when shrinkage is taken into account. However, for larger

values of the slip, there is no difference between the two analyses.

0

5

10

15

20

25

30

0 5 10 15

Load [kN]

Slip [mm]

With shrinkage

Without shrinkage

00.51.01.52.0

2.53.03.54.0

0 0.01 0.02 0.03 0.04

Figure 27 The results from analysis of a pull-out test, with and without shrinkage of

the concrete taken into account.

35

5 FRAME CORNERS SUBJECTED TO CLOSING

MOMENTS

Frame corners have been investigated by several researchers. Experimental work, for

example Mayfield et al. (1971) and Nilsson (1973), has shown that frame corners

subjected to opening moments are more sensitive to the method of detailing in the

reinforcement than those subjected to closing moments. Hence, most publications for

the past few decades have concentrated on opening moments. In the Swedish

Standards, Boverket (1994), it is recommended not to splice the reinforcement within

a corner region and, until recently, this has not been allowed by the Swedish Road

Administration, see Vägverket (1994). The reason for this was that for opening

moments the behaviour of the corner is sensitive to the detailing of the reinforcement.

Although corners subjected to closing moments were less well investigated, splices

were not allowed for this type either. The aim of this work was to investigate whether

splicing of the reinforcement can be allowed, at least for closing moments. In this

section, the structural behaviour of frame corners subjected to closing moments is

discussed. For a literature survey of work carried out on frame corners, see

Nilsson (1973) which treats work done before 1973 and Karlsson (1999) for later

work, or Johansson (2000).

5.1 Internal Forces in a Corner Subjected to Closing Moment

The internal forces in a corner subjected to a closing moment are shown in Figure 28.

After cracking of the concrete, the tensile forces are carried by the reinforcement, as

shown in Figure 28 (b). If the corner is well-designed, failure will be due to bending

in the sections adjacent to the corner, with yielding of the reinforcement. According to

Stroband and Kolpa (1983), there are three possible failure modes that will cause

premature failure of the corner.

36

(a) (b)

Figure 28 The internal forces in a corner subjected to a closing moment:

(a) Uncracked corner, and (b) corner with bending cracks. From Stroband

and Kolpa (1983).

• Crushing of the concrete in the compressive zone

For elastic materials, there are stress concentrations at corners. For a corner

subjected to closing moment, this leads to large compressive stress at the inner

part of the corner. However, when the concrete reaches the plastic stage, this

stress concentration is no longer so pronounced. Furthermore, there will be a

biaxial compressive state, due to the compressive stresses from both sides of the

corner, or even a triaxial stress state if lateral deformations are restricted. Thanks

to this bi- or triaxial compressive stress state, the concrete will have a greater

capacity, and also more ductility. However, as shown in Paper IV, the tested

corner specimens were very close to this failure mode.

• Crushing of the concrete in the compressive diagonal

In Figure 28, where the internal forces in a corner subjected to a closing moment

are shown, it can be seen that the compressive zones from each part of the corner

are balanced by a compressive diagonal. If the stress in this compressive diagonal

becomes large, crushing of the concrete might occur.

• Bearing failure at the bend of the reinforcement

When a reinforcement bar is bent, radial compressive stresses are present, see

Figure 29 (a). When these compressive stresses spread, as shown in Figure 29 (b),

tensile stresses act out of the plane of the bar curvature. If these tensile stresses

37

become too large, splitting cracks will appear. At first, this type of failure was

thought to be important in combination with reinforcement splices. Splicing the

reinforcement also causes splitting stresses, and it was believed that the

combination of these effects could cause splitting cracks that would reduce the

bond capacity. Nevertheless, the tests and analyses presented in Paper IV show

that this did not happen. Note, however, that bearing failure at the bend of the

reinforcement is more likely to occur near a reinforcement bar close to a free edge.

The main interest of this study is corners in bridges. Here, the corners have a long

extension with a large number of reinforcement bars. Furthermore, the edges are

usually not free; they are connected to other parts of the structure. Accordingly,

the failure mode with a splitting side cover is of no special interest in this study.

For the corners of beams, in particular when only two reinforcement bars are

present, the effect of the edges is of course much greater. Splitting of the side

cover must then be prevented, in order to avoid premature failure of the corner.

For these types of failure, the strength of the concrete is critical. In the first and

second failure types discussed, premature crushing of the concrete, it is the

compressive strength that is decisive. Also the amount of reinforcement is important:

the larger the amount of reinforcement, the greater the forces the concrete must be

able to carry. Stroband and Kolpa (1983) derived an analytical expression for how

much reinforcement can be allowed; this was to avoid the premature failure of the

concrete in the compressive diagonal. In the third type of failure, bearing failure at the

bend of the reinforcement, it is mainly the tensile strength that has an influence on the

result, and also the thickness of the concrete cover.

(a) (b)

Figure 29 Bent reinforcement bar causing (a) radial compressive stresses and

(b) splitting stresses out of the plane of the bar curvature.

38

For a spliced corner, there is also a fourth type of premature failure:

• Anchorage of the reinforcement

If the anchorage of the reinforcement is not adequate, premature failure of the

corner will occur. To avoid this, a minimum splice length is required. The tests

and analyses presented in Paper IV show that the splice lengths required by the

existing codes are sufficient.

Altogether, this shows that splicing the reinforcement within a corner region does not

seem to have any negative effect on the behaviour of the corner, since the behaviour

of the spliced corner differs only a little from a corresponding unspliced corner. The

same conclusion has been drawn in Stroband and Kolpa (1983),

Plos (1994a, 1994b, 1995), Johansson (1995, 1996a, 1996b), Lundgren and

Plos (1996), and Olsson (1996). However, it is worth noting that, for certain

conditions, the capacity of a corner (spliced or unspliced) is less than the capacity of

the adjoining sections. For corner regions with free edges, splitting of the side cover

must be prevented, either with a sufficient thickness of the concrete cover or with

confining reinforcement. When this is done, or if the corner region does not have free

edges, the capacity of the corner is greater than the capacity of the adjoining sections

for the concrete qualities and amount of reinforcement that are usually used today. For

the low concrete qualities that were used some years ago (with a compressive strength

as low as about 15 MPa), premature failure of the corner might occur. Also, if the

capacity of the steel were dramatically increased beyond what is normal today, or

very large amounts of reinforcement were used, premature failure of the corner might

occur.

5.2 Frame Corners Subjected to Cyclic Loading

Frame corners subjected to closing moments and cyclic loading have been

investigated by Plos (1994b, 1995). In that investigation, spliced and unspliced

specimens were compared. Even though all of the reinforcement was spliced in the

same cross-section, no disadvantage in splicing the reinforcement could be found. All

of the tests resulted in fatigue of the reinforcement.

The results from the static analyses and tests indicate that reinforcement splices in a

corner region behave in about the same way as reinforcement splices in beams.

39

Probably, the same is valid also for cyclic loading. Therefore, there is little reason to

believe that it would be more dangerous to splice the reinforcement within the corner

region than outside it. The behaviour of lap splices in beams subjected to cyclic

loading has been examined by many researchers. A summary of the results can be

found in ACI (1992). By following the design rules for splices subjected to cyclic

loading, a sufficient level of safety can be obtained.

5.3 Tests and Analyses of Frame Corners

Tests and analyses within an earlier part of the project, “Detailing of frame corners in

concrete bridges”, did not reveal any disadvantages in splicing the reinforcement

within a frame corner. Hence, it was decided to investigate the worst case. Here, a

brief summary of the study is given; for more details see Paper IV or

Lundgren (1999b). Four frame corners with differing detailing were subjected to a

closing moment, combined with shear and a normal force. The detailing of the main

reinforcement is shown in Figure 30. One corner had unspliced reinforcement, (a),

while another had spliced reinforcement with the splice length required by the

codes, (b). The two last frame corners, (c) and (d), had spliced reinforcement with a

splice length that was less than half of that required by the codes. All of the test

specimens had a relatively high amount of reinforcement; the main reinforcement was

placed in one layer, and the distance between the main reinforcement bars was the

minimum distance allowed according to the Swedish Standards, Boverket (1994). The

reason for these choices was to investigate what was considered to be the worst case.

40

100

a) Test No. 1U,unspliced reinforcement.

10=φ 16 s56Transverse reinforcement φ10 s100

10φ 8 s56

360

53100

360r =125

b) Test No. 2L,long splice (560 mm).

c) Test No. 3S,short splice (250 mm).

250

d) Test No. 4Ss, symmetricalshort splice (250 mm).

205 155

205

155

Figure 30 Detailing of the reinforcement in the corners of the test specimens.

Measurements, compressive and transverse reinforcement in (b), (c) and

(d) were the same as in (a). Dimensions given in mm.

The tests with unspliced reinforcement and with a long splice showed very similar

behaviour, with the maximum capacity determined by the bending capacity of the

adjoining cross-sections. After yielding of the reinforcement, concrete began to spall

off at the inner part of the corner, in the compressive zone. Thereafter, inclined cracks

in the concrete led to a sudden failure in the test with unspliced reinforcement; the test

with the long splice was interrupted before this stage.

In one of the tests with a short splice length, centred in the corner, only slightly less

capacity than in the unspliced test was obtained. The failure here was caused by the

rather sudden appearance of an inclined crack in the concrete, after yielding of the

reinforcement. In the other test with a short splice length, with the splice placed

41

outside the bend of the reinforcement, fracture of the splice limited the capacity,

which was then only about half of the capacity of the unspliced corner.

Furthermore, detailed three-dimensional non-linear finite element analyses of the

corner regions of the frame corners tested were carried out. The second version of the

developed model was used to describe the bond mechanism; thus, the splitting stresses

resulting from the anchorage were taken into account. The results from these analyses

show that the overall behaviour of the specimens could be quite well described; in all

of the analyses the failure mode was the same as in the tests. In particular, it was

noted that the fracture of the splice in the specimen with the splice situated outside the

bend of the reinforcement was described realistically in the analyses.

Frame corners in large portal bridges have considerably larger dimensions than the

specimens tested. Therefore, a large frame corner was also analysed. It had

dimensions large enough for one splice, with a splice length as required in the codes,

along one of the sides of the corner. The analysis showed that the maximum capacity

was determined by the bending capacity of the adjoining cross-sections; i.e. the

capacity of the splice was enough so that it was not limiting.

In conclusion, the tests and analyses show that splicing the reinforcement in the

middle of the corner has advantages over splices placed outside the bend of the

reinforcement. They also indicate, in agreement with the previous analyses and tests,

that provided the splice length is as long as required in the codes, there are no

disadvantages in splicing the reinforcement within the corner of a frame.

42

6 CONCLUSIONS AND SUGGESTIONS FOR

FUTURE RESEARCH

From the work that has been carried out, the following conclusions can be drawn. The

first version of the proposed bond model could describe the behaviour of the bond

mechanism relatively well. The results from the steel-encased pull-out tests, however,

provided new information about the bond mechanism, in particular for cyclic loading.

Consequently, some drawbacks to the first version of the model became apparent.

This method of combining theoretical modelling with experimental work is believed

to give better results and, perhaps most valuable, to give a deeper understanding of the

problem studied than if the work were limited to only one of these aspects. The bond

model could thereby be further developed, and the second version of the model is

believed to reflect reality quite closely. Analyses of pull-out tests with differing

geometries and with both monotonic and cyclic loading showed that the new model is

capable of dealing with a variety of failure modes, such as pull-out failure, splitting

failure, and the loss of bond when the reinforcement is yielding, as well as dealing

with cyclic loading in a physically reasonable way. Results from Magnusson (2000)

also indicate that the effect of outer pressure is well described by the improved

version of the model.

The refined model was used in detailed three-dimensional analyses of frame corners,

to investigate the effect of splices within the corner region. When compared with

results from tests on the frame corners, it was found that the analyses could describe

the test results in a reasonable way. In particular, it was noted that the fracture of the

splice was described closely in the analyses. The tests and analyses showed that

splicing the reinforcement in the middle of the corner has advantages over splicing

placed outside the bend of the reinforcement. They also indicate, in agreement with

previous analyses and tests, Plos (1995), that provided the splice length is as long as

required by the codes, there are no disadvantages in splicing the reinforcement within

the corner of a frame subjected to closing moments.

The proposed bond model can also be used in other analyses where the bond

mechanism plays an important role. It is believed to be a powerful tool for parameter

studies of, for example, the effect of transverse reinforcement, anchorage at end

43

supports under different conditions, and rotation capacity. Such parameter studies can

serve as a basis for design codes. The model is calibrated for normal strength

concrete, but analyses by Magnusson (2000) show that this calibration also gives

satisfactory results for high strength concrete. Hence, it is likely that the calibration

would also be useful for other types of concrete, e. g. light weight concrete or fibre

reinforced concrete. Nevertheless, for each new application, it is recommended that

analyses be compared with experimental data first. This is recommended especially if

the model is intended to be used for other types of reinforcement bars. Since changing

the geometry of the ribs would most definitely affect the friction between the concrete

and the reinforcement bar, the calibration of the model would need to be revised for

this.

The finite element analyses were all carried out using the finite element program

DIANA. These analyses show clearly the advantage of using a rotating crack model

instead of fixed crack directions. In some of the analyses, both types of material

models gave the same result, while in the analyses where the direction of the principal

stress was changed after cracking had occurred, the rotating crack model gave results

that corresponded more closely to the measured response. Even though DIANA is

believed to have the best material models for concrete among commercial programs

today, the material models used to describe concrete still need to be improved. For

example, when the material model used is subjected to triaxial compressive stress

states, it does give an increase in capacity that seems to correspond well with the

measured one, but the increase in ductility appears to be too low for some stress

states. Another problem is how to take into account the effect of localisation in

compression. Although some research exists in this field, still more needs to be done,

especially when combining localisation with triaxial stress states. Furthermore, it

ought to be possible to take cyclic loading into account in a more generalised way. At

present, the only material models available that can deal with cyclic loading are one-

dimensional. The establishment of three-dimensional material models that can cope

with cyclic loading would be most useful.

44

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Administration, Borlänge, 1994.

Åkesson, M. (1993): Fracture Mechanics Analysis of the Transmission Zone in

Prestressed Hollow Core Slabs. Licentiate Thesis. Division of Concrete Structures,

Chalmers University of Technology, Publication 93:5, Göteborg, 1993.

50

Åkesson, M. (1996): Implementation and Application of Fracture Mechanics Models

for Concrete Structures. Ph.D. Thesis. Division of Concrete Structures, Chalmers

University of Technology, Publication 96:2, Göteborg, 1996.

A1

APPENDIX A

DERIVATION OF THE ELASTIC STIFFNESSES

IN THE ELASTIC STIFFNESS MATRIX

The stiffnesses in the elastic stiffness matrix, D, describe how the concrete between

the ribs behaves under elastic conditions. The dimensions of the ribs on several

reinforcement bars K500 φ 16 were measured in Al-Fayadh (1997). Here, the average

of the measured values are used, see Fig. A-1.

5.80

β

A’sin β

9.24

15.65

17.66

A’ = 20.32 mm2

β = 58.6°

[mm]

Fig. A-1 Dimensions of the ribs of reinforcement bars K500 φ 16. Values are

average values from measurements on several bars in Al-Fayadh (1997).

A2

The Stiffness D22

The stiffness D22 is the relation between the elastic part of the slip, ute, and the bond

stress, tt. An upper limit of D22 can be estimated by assuming that all of the bond

stress is carried by one rib, and that the next rib acts as a support, see Fig. A-2.

βπσεsin'2Adlt

El

AF

Ell

Elu kt

ccc

et ====

( ) cc

kce

t

t

EE

ldlAE

utD

⋅≈⋅+⋅⋅⋅⋅⋅⋅

°⋅⋅⋅=

===

−−−−

−

10

21080.51024.91024.91016

6.58sin1032.202

sin'2

3333

6

22

π

πβ

F

l

lk

Fig. A-2 Assumptions used to estimate the upper limit of the stiffness D22.

The stiffness D22 is also recognised as the stiffness of the first part, or the unloading

stiffness, in a bond-slip curve which can be measured experimentally. Since it is

difficult to measure the small deformations of the first part, the unloading stiffness

was used, see Fig. A-3. Balázs and Koch (1995) measured a value of about

4·1011 N/m3 for concrete with a wet cube compressive strength of about 30 MPa. This

corresponds to about 13·Ec. In the cyclically loaded steel-encased pull-out tests, the

stiffness was approximately 8·1010 N/m3 for concrete with a wet cylinder compressive

strength of about 35 MPa, which corresponds to about 2.5·Ec. The stiffness was

chosen to be somewhere between the two measured results, and below the upper limit

in the first equation:

-1222222 m 0.6 , =⋅= KEKD c . (A-1)

A3

-10

-5

0

5

10

15

20

25

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Bond stress [MPa]

Slip [mm]

D22

1

Fig. A-3 The stiffness D22 is the unloading stiffness in a bond-slip curve.

The Stiffness D11

The stiffness D11 describes the relation between the elastic part of the radial

deformation, une, and the splitting stress, tn. This stiffness was estimated by examining

the concrete between the ribs. The geometry was approximated as a thin ring with an

inner radius the same as the smallest radius of the reinforcement bar (without the

ribs). The outer radius was determined by the condition that the cross-sectional area of

the ring should equal the cross-sectional area of the ribs projecting from the bar core,

compare Fig. A-1 and Fig. A-4.

rb

tn

ra

1-A Fig. see mm, 82.72

== ia

dr

( )

( )

mm 50.8

1082.76.58sin1032.202

sin'2

sin'2

236

2

22

=

=⋅+°⋅⋅=

=+=

−=

−−

π

πβ

πβ

ab

ab

rAr

rrA

Fig. A-4 Approximated geometry to estimate the stiffness D11.

A4

The outer edges of the ring were assumed to be free, i.e. only the structural behaviour

of the ring itself was taken into account. The deformation at the distance r from the

centre of a ring is, according to Chen and Han (1987),

( )( )

( )( )

�

��

�

�+

+⋅−⋅

−⋅+

=r

rrrrE

tru b

abc

naen

2

22

2

1211

υυυ

which gives the stiffness D11 as

( )( )

( )( )

( )( )

( )( )

c

c

b

a

abce

n

n

E

E

rrr

rrrE

utD

⋅≈

≈�

���

�+

+⋅⋅−

⋅+−⋅=

=���

��

�

�+

+⋅−

⋅+−

⋅==

11

008.000850.0

15.01008.015.021

00782.015.0100782.000850.0

121

1

2

2

22

2

2

22

11 υυ

υ

It was also noted that the larger the D11 chosen, the more variation there was of the

stresses along the reinforcement bar. This variation arises from differences in the

strength of the structure modelled, as for example when stirrups are taken into

account. Since the derived value of D11 gave a physically reasonable variation, D11

was designated

-1111111 m 0.11 , =⋅= KEKD c . (A-2)

A5

The Stiffness D12

The stiffness D12 describes the relation between the elastic part of the slip, ute, and the

splitting stress, tn. Thus, it describes how much splitting stress will be caused by a

given slip. Since the calibration of the coefficient of friction derives from

experimental results, the model is expected to work in such a way that loading occurs

along the yield line. Therefore, the elastic loading ought to cause a larger bond stress

than that given by the yield line. From Fig. A-5, it follows that

ttn duDduDduD 221211 <⋅+ µ .

To be sure that this condition is fulfilled, the stiffness D12 is chosen so that

µ

2212

DD < .

The value of the stiffness D12 determines how large a part of the splitting stress

remains after unloading. The larger the value of |D12| chosen, the smaller the splitting

stress will be after unloading. By comparison with results from experiments, and

taking the previous derived expression into account, the D12 chosen was

max

2212 9.0

µDD −= . (A-3)

tn

tt

F1F2

µ1

te = Ddu

|D11 dun + D12 dut|

D22 dut

|D11 dun + D12 dut|·µ

Fig. A-5 The trial stress ought to cause a larger bond stress than is given by the yield

line.

I-1

Fracture Mechanics of Concrete StructuresProceedings FRAMCOS-3AEDIFICATIO Publishers, D-79104 Freiburg, Germany

MODELLING SPLITTING AND FATIGUE EFFECTS OF BOND

K. Lundgren and K. GylltoftDivision of Concrete Structures, Chalmers University of TechnologySweden

AbstractWhen deformed bars are anchored in concrete, this causes not only bondstresses, but also splitting stresses that are usually not taken into accountin FE-analyses of reinforced concrete structures. Therefore, a model hasbeen developed which takes the three-dimensional splitting effect, andalso the effect of cyclic loading and changing slip direction into account.Bar pull-out tests with various geometry and with both monotonic andcyclic loading have been analysed. With the same input parameters,various bond-slip curves were obtained, depending on the modelledgeometry and strength of the surrounding concrete. The results show thatthe new model is capable of predicting splitting failures, and of dealingwith cyclic loading in a physical reasonable way.Key words: Bond, splitting failure, cyclic loading, non-linear fracturemechanics, finite element analyses

1 Introduction

When modelling reinforced concrete structures with the finite elementmethod, it is common to assume either perfect bond between thereinforcement bars and the surrounding concrete, or a bond-slip relationfor an interface layer. However, none of these methods take the three-dimensional splitting effect into account, which can be of importance

I-2

when for example the concrete cover is insufficient and spalling willoccur. Also, the effect of cyclic loading with varying slip direction isimportant for the bond resistance. Therefore, a model has been developedwhich takes the three-dimensional splitting effect, and the effect of cyclicloading into account. The model is presented here, together with resultsfrom finite element analyses of pull-out tests.

2 Modelling of the interface

In the finite element program DIANA, there are interface-elementsavailable, which can be used to model the bond-slip behaviour betweenreinforcement bars and concrete. The element describes a relationbetween the tractions t and the relative displacements u in the interface.The physical interpretation of the variables tn, tt, un and ut is shown inFig. 1.

2.1 Elasto-plastic formulationÅkesson (1993) has developed a frictional model for anchorage ofstrands, using elasto-plastic theory to describe the relations between thestresses and deformations to include the splitting effects. The model wasintended to be used only for monotonic loading. Therefore, a new modelhas been developed, still using elasto-plastic theory. The splitting effectsare included, and the model is capable of dealing with cyclic loading andvarying slip direction. The relation between the tractions t and therelative displacement u is in the elastic range:

tt

Dtt

D

D

uu

n

t

t

tn

t� � =

�

�

��

���

�

��

��

11 12

220(1)

where D12 normally is negative, meaning that slip in either direction willcause negative tn; i. e. compressive forces directed outwards in theconcrete.

ut

tntt

tttntn = normal stresstt = bond stressun = relative normal displacement in the layer (not shown in the figure)ut = slip

Fig. 1 Physical interpretation of the variables tn, tt, un and ut.

I-3

The model is further equipped with yield lines, flow rules, andhardening laws. The yield lines are two yield functions; one describingthe friction F1, and one describing the upper limit, a cap F2.

F t t tt n n1 0 0= + − =µ( ) (2) F t t ct n2 0= − − = (3)

For plastic loading along the yield line describing the upper limit, F2, anassociated flow rule is assumed. For the yield line describing the friction,F1, a non-associated flow rule is assumed, where the plastic part of thedeformations are

d d G G t t tpt n nu

t= = + − =λ ∂

∂η, ( ) 0 0 . (4)

The yield lines, together with the direction of the plastic part of thedeformations are shown in Fig. 2. At the corners, a combination of thetwo flow rules is used.

2.2 Damaged / undamaged deformation zonesA typical response for bond with varying slip direction is with a steepunloading and then an almost constant, low bond stress until the originalmonotonic curve is reached. To make the model describe this typicalresponse, a new concept, called damaged / undamaged deformation zone,is used. The idea is that when the reinforcement is slipping in theconcrete, the friction will be damaged in the range of the passed plasticslip. As shown in Fig. 3, slipping of the reinforcement in one direction

d d G d Fu

t tp = +λ ∂

∂λ ∂

∂1 22

d d Gu

tp = λ ∂

∂

d d Fut

p = λ ∂∂

2

F2

tn

tt

F1

G

Fig. 2 The yield lines with an associated flow rule at the yield linedescribing the upper limit, F2, and a non-associated flow rule at the yieldline describing the friction, F1.

I-4

compressedconcrete

reinforcement bar

”emptyhole”

ut

concrete

tt

ut

(a)

damageddeformationzone

undamageddeformationzone

undamageddeformationzone

(b)

Fig. 3 (a) Slipping of the reinforcement in one direction willtheoretically cause compressed concrete in front of the ribs, and ”emptyholes” behind the ribs; and (b) The range of the slip where plasticdeformations have occurred is called the damaged deformation zone.

will theoretically cause compressed concrete in front of the ribs, and”empty holes” behind the ribs. When the loading is reversed, first of allthe elastic part of the slip will cause unloading. For further unloading inthe damaged deformation zone a low coefficient of friction, µd, isassumed until the interface is back in the undamaged deformation zoneagain. Also the parameter η has a lower value in the damageddeformation zone, ηd, physically corresponding to that the increase in thestresses is lower than in the undamaged deformation zones.

2.3 HardeningFor the hardening rule of the model, a hardening parameter κ isestablished. It is defined as

d du dunp

tpκ = +2 2 in the undamaged deformation zones, and (5)

d du dudnp

tpκ

ηη

= +2 2 in the damaged deformation zones. (6)

It can be noted that for monotonic loading are dunp and the elastic part of

the slip very small compared to the plastic part of the slip, dutp; therefore

the hardening parameter κ will be almost equivalent to the slip, ut. Thevariables µ and c in the yield functions are assumed to be functions of κ. Also, the apex of the yield surface is moved in the direction of theloading, see Fig. 4. This can be compared with a kinematic hardening.However, for further loading in one direction, this movement will have noeffect on the yield line. Therefore, the apex is moved partwise when theinterface returns to the undamaged deformation zone, after being in thedamaged deformation zone. In Fig. 4, an example of how the apex tn0 ismoved is shown.

I-5

New apex is calculated at E:

( ) ( )( )t t

t u ta

t uun n

t t t t t

t0 = −

−+max max

maxµ

F a

1

A

tt

ut

B

C

D

E

F

(b)

1

tn

tt

loading pathµ

elasticunloading

(a)

yield lines in the damaged deformation zone

(c)

yield linesat:

C B A

a1

µ(utmax)

B

E A

1

D

tn

tt

C

tt(utmax)

DE

µ(utmax)1

F

Fig. 4 (a) The apex of the yield lines is moved in the direction of theloading; (b) bond-slip curve; and (c) the corresponding load cycle in thestress space, showing how the apex is moved partwise.

3 Comparison with pull-out tests

The new interface model has been used in FE-analyses to modelexperiments found in literature. In all analyses, the concrete wasmodelled with a constitutive model based on non-linear fracturemechanics with a combined Drucker-Prager and Rankine elasto-plasticmodel. The FE-models were axisymmetric; the localization of the defor-mations due to cracking of the concrete was then smeared out over theconcrete elements assuming that there were four radial cracks in acylinder. Yielding was modelled using associated flow and isotropichardening. The hardening of the Drucker-Prager yield surface wasevaluated from the shape of the uniaxial stress-strain relationship incompression, and was chosen to match typical test data presented byKupfer and Gerstle (1973), see Fig. 5b. From the various measuredcompressive strengths, an equivalent compressive cylinder strength, fcc,was evaluated. Other necessecary material data for the concrete was

I-6

calculated according to the expressions in CEB (1993) from fcc, and isshown in Table 1. The constitutive behaviour of the reinforcement steelwas modelled by the von Mises yield criterion with associated flow andisotropic hardening. The elastic modulus of the reinforcement wasassumed to be 200 GPa.

Table 1. Material data of the concrete in the analysed pull-out tests

Compressive tests Material data used in the analysesReference

Test specimenfcc

[MPa]fcc

[MPa]fct

[MPa]Ec

[GPa]GF

[N/m]Noghabai 150 mm cube, wet 47.6 35.7 2.7 32.9 73

Magnusson 150 mm cyl., wet 27.5 27.5 2.2 30.0 90Balázs and Koch 150 mm cube, wet 28-32 25.5 2.0 29.4 58

3.1 Input parameters for the interfaceRequired input data for the interface is the elastic stiffness matrix D inequation (1), the initial apex of the yield lines tn0 in equation (2), theparameter η defined in equation (4), and for loading in the damageddeformation zone the parameters ηd and µd. Furthermore, the functionsµ(κ) and c(κ) must be chosen. First of all, the stiffness D22 in the elastic stiffness matrix D isrecoqnized as the stiffness of the first part, or at unloading, in a bond-slipcurve. This stiffness was in the tests of Balázs and Koch (1995) about4·1011 N/m3, when the concrete compressive strength was 27.5 MPa.Since this stiffness depends on the concrete quality, it was thereforechosen to:

D fcc22314 5 10= ⋅. (7)

Next, the stiffnesses D11 and D12 were determined. To make the modeldescribe a bond-slip curve with an initial stiffness of about D22, and thendecreasing, these parameters were chosen to be:

DD

12220 98= − .

maxµ(8)

D fcc1131 7 10= ⋅. (9)

The adhesional strength between the reinforcement bar and the concretewas assumed to be negligible; i. e. the initial apex of the yield lines tn0was chosen to be zero. The parameter η is chosen in order to obtain adecreasing bond stress when the concrete around the bar splits, without

I-7

elastic unloading. Through calibration, η was chosen to be 0.05. Forloading in the damaged deformation zones, ηd was chosen to be η/400,and the coefficient of friction µ d was 0.3. The function µ(κ) describes how the relation between the bond stressand the normal splitting stress depends on the hardening parameter.Tepfers and Olsson (1992) performed “ring tests”; pull-out tests inconcrete cylinders confined by thin steel tubes. They measured the strainin the steel ring and used this to evaluate the normal stress. Also in someof the pull-out tests in Noghabai (1995), concrete cylinders were confinedby steel tubes, and the measured steel strains have been used to evaluatethe splitting stress in Lundgren and Gylltoft (1997). The results, togetherwith the chosen input for the analyses are shown in Fig. 5a. The variable c represents the upper limit of the stresses tn and tt andcombinations of them as shown in Fig. 2. This upper limit shallrepresent the case with a pull-out failure. A theoretical consideration of acase with zero bond stress will then lead to a limit of the normal splittingstress about the compressive strength of the concrete. The function c(κ)was therefore chosen to be the same as the uniaxial compression curve ofthe concrete, only with a factor between the plastic strain and thehardening parameter κ as shown in Fig. 5b.

3.2 Monotonic pull-out testsBar pull-out splitting tests performed by Noghabai (1995), Magnusson(1997) and Balázs and Koch (1995) have been analysed. In Noghabai(1995), the test specimens consisted of concrete cylinders with diameter313 mm, with deformed reinforcement bars, φ32 mm Ks 400. Theembedment length was 120 mm. In two of the three performed tests, the

(a)

0

1

2

0 5 10 15 20

chosen inputN: Ks400 φ32N: Ks400 φ32T: Ks400T φ16T: Ks600 φ16T: Ks400I φ16

slip; κ [mm]

µmax

µd

1

0

0.5

5 10 150 52.5 7.5

c/fcc; σ/ fcc

κ [mm]εp [‰]

(b)

µ

Fig. 5 (a) The coefficient of friction as a function of the slip evaluatedfrom tests, N: Noghabai (1995), T: Tepfers and Olsson (1992), togetherwith the chosen function µ(κ); and (b) Compressive stress versus plasticstrain, and the function c(κ).

I-8

concrete cylinders were reinforced with spiral reinforcement, φ6 mmKs 400 with a radius of 40.5 mm, with varying pitches s14 and s28 mm.This spiral reinforcement was modelled as embedded reinforcement,meaning that complete interaction between the steel and the concrete wasassumed. Noghabai (1995) also performed other pull-out tests with thesame reinforcement bars, φ32 mm Ks 400, with concrete cylindersconfined by 10 mm thick steel tubes. The diameter of the concretespecimen was 219 mm and the embedment length was 80 mm. Magnusson (1997) and Balázs and Koch (1995) have performed pull-out tests with deformed reinforcement bars, φ16 mm K 500. Magnussonhad concrete cylinders with a diameter of 300 mm and an embedmentlength of 40 mm; Balázs and Koch had concrete specimens with aquadratic cross-section 160 ·160 mm and an embedment length of 80 mm.In both cases, the concrete specimens were large enough to preventsplitting failure; thus, pull-out failures were obtained. To reduce thenumerical difficulties, the quadratic cross-section was approximated asaxisymmetric. The calculated load versus slip for these tests are shown in Fig. 6,together with results from the experiments. Especially in Fig. 6a it canbe seen that even with the same embedment length, and when excactly thesame input parameters were given for the interface, different load-slipcurves were obtained depending on the modelled structure. Comparingwith the measured response, the agreement is rather good, especiallywhen considering the large scatter that is always obtained in pull-outtests. Another important thing to compare is the failure mode, which iscorrect in all cases; splitting failure in Noghabai’s test without spiralreinforcement, a combination when spiral reinforcement with pitchs28 mm was used, and pull-out failure in the other cases. In Fig. 6d, thedeformed FE-mesh and the tangential stresses at maximum load is shownfor Noghabai’s test without spiral reinforcement. There it can be seenthat the maximum load is obtained when the crack reaches the outer edge.The elements inside this line are already cracked, and the stresses are onthe descending branch.

3.3 Cyclic pull-out testsBalázs and Koch (1995) have performed large investigations of pull-outtests loaded with cyclic loading. The test specimens had the samegeometry as in their monotonic tests, described in the previous section.One test, with cyclic load varying from -25% to 25% of the maximumload in the monotonic tests have been analysed with the same finiteelement model as in the monotonic tests, using the new model. Resultsfrom the experiments, together with results from the analyses are shownin Fig. 7.

I-9

050

100150200250

0 5 10 15

s28, exp.

s28, FEA

ref., exp.

ref., FEA

P [kN]

active slip [mm]0 5 10 15 20

s14, exp.s14, FEA

steel tube, exp.

steel tube, FEA

active slip [mm]

P [kN]

(a) Noghabai (1995)

2.001.821.641.451.27

axis of rotationalsymmetry

[MPa]

(d)

0 5 10 15passive slip [mm]

(c) B: Balázs and Koch(1995) M: Magnusson (1997)

0

20

40

60

M: FEA

B: exp.

M: exp

B: FEA

P [kN](b) Noghabai (1995)

050

100150200250

Fig. 6 (a), (b), and (c) Load versus slip in monotonic pull-out-tests; and(d) The deformed FE-mesh and the tangential stresses (in the directionout of the plane) at maximum load in Noghabai’s test without spiralreinforcement.

-16-12-8-4048

1216

-0.1 -0.05 0 0.05 0.1

n = 1, 10, 50, 100P [kN]

slip [mm]

experiment analysis

Fig. 7 Load versus slip in cyclic pull-out-tests, Balázs and Koch (1995).

I-10

4 Conclusions

A new model describing bond between deformed reinforced bars andconcrete has been developed. Since the model takes the three-dimensional splitting effect into account, the same input parameters willresult in different load-slip curves depending on the geometry andloading conditions of the concrete specimen. The model can alsodescribe the behaviour in cyclic loading in a realistic way, and reasonablegood agreement with experiments was found.

References

Balázs, G. and Koch, R. (1995) Bond Characteristics Under ReversedCyclic Loading, Otto Graf Journal, 6, 47-62.

CEB (1993) CEB-FIP Model Code 1990, CEB Bulletin d’InformationNo. 213/214, Lausanne.

Lundgren, K. and Gylltoft, K. (1997) Three-Dimensional Modelling ofBond, in Advanced Design of Concrete Structures (eds K. Gylltoft,B. Engström, L-O. Nilsson, N-E. Wiberg, and P. Åhman), CIMNE.Barcelona, 65-72.

Magnusson, J. (1997) Bond and Anchorage of Deformed Bars inHigh-Strength Concrete. Licentiate Thesis. Division of ConcreteStructures, Chalmers University of Technology, Publication 97:1.

Noghabai, K. (1995) Splitting of Concrete in the Anchoring Zone ofDeformed Bars; A Fracture Mechanics Approach to Bond.Licentiate Thesis. Division of Structural Engineering, Luleå Universityof Technology, 1995:26L.

Tepfers, R. and Olsson, P-Å. (1992) Ring Test for Evaluation of BondProperties of Reinforcing Bars, in Bond in Concrete Proceedings(CEB), Riga, 1-89 - 1-99.

Åkesson, M. (1993) Fracture Mechanics Analysis of the TransmissionZone in Prestressed Hollow Core Slabs. Licentiate Thesis. Divisionof Concrete Structures, Chalmers University of Technology,Publication 93:5.

II-1

Pull-out tests of steel-encased specimens

subjected to reversed cyclic loading

K. Lundgren

Division of Concrete Structures

Chalmers University of Technology

SE-412 96 Göteborg, Sweden

ABSTRACT

When deformed bars are anchored in concrete, this gives rise to not only bond

stresses but also splitting stresses. To measure the splitting stresses, tests were carried

out in which a reinforcement bar was pulled out of a concrete cylinder surrounded by

a thin steel tube. The tangential strains in the steel tube were measured, together with

the applied load and slip. In five tests, specimens were loaded by monotonically

increasing the load, while nine other tests were subjected to reversed cyclic loading.

All of the tests resulted in pull-out failures. The results from the monotonic tests

indicate that the splitting stresses decreased after the maximum load had been

obtained, however not as much as the load decreased. The results from the cyclic tests

show a typical response for bond in cyclic loading. When there was almost no bond

capacity left the measured strain in the steel tubes stabilised and remained more or less

unaffected by the last load cycles. The test results give valuable information about the

splitting stresses that result from anchorage of reinforcement bars in concrete. These

test results can be useful as a reference when calibrating models of the bond

mechanism, and give a better understanding of the bond mechanism.

II-2

1. BACKGROUND AND AIM

When deformed bars are anchored in concrete, this gives rise to not only bond

stresses but also splitting stresses, see Fig. 1. Although many experiments have been

carried out to investigate the bond stresses, the splitting stresses are not so well

investigated. Tepfers and Olsson [1] have conducted “ring tests” in which a

reinforcement bar was pulled out of a concrete cylinder surrounded by a thin steel

tube. By measuring the tangential strains in the steel tube, the splitting stresses could

be evaluated. A few other researchers have also carried out tests, trying to solve the

problems of measuring the splitting stresses, for example Malvar [2]. The effect on

bond of cyclic loading has been investigated by, among others, Eligehausen et al. [3]

and Balázs and Koch [4] who have conducted large programmes of pull-out tests with

cyclically loaded specimens. However, no tests have been found in the literature that

show the effect of splitting stresses measured during cyclic loading.

When analysing concrete structures with the finite element method, the effect of

the splitting stresses can be taken into account, provided a suitable model for the

interaction between the concrete and the reinforcement bars is used. One model that

takes the splitting stresses into account is that of Åkesson [6]. Another model that

includes these splitting stresses, and also covers the effect of reversed cyclic loading,

is presented in Lundgren and Gylltoft [7]. Both of the models need the coefficient of

friction as input, to describe the relation between the bond stresses and the splitting

stresses. Other input parameters are also needed, which can be found by calibrating

the models against experiments. Accordingly, these tests were carried out: (1) to give

reference information when calibrating models of the bond mechanism, and (2) to

improve knowledge of the splitting stresses and how they are affected by reversed

cyclic loading.

II-3

Stress on the concreteand its components

(b)

P

Stress on thereinforcing bar

(a)

Fig. 1 – Bond and splitting stresses between a deformed bar and the surrounding

concrete. From Magnusson [5].

2. TEST PROGRAM

The aim of the tests was to study the splitting stresses in the bond mechanism

and, in particular, how the splitting stresses are affected by cyclic loading. Hence,

pull-out tests in which a reinforcement bar was pulled out of a concrete cylinder,

surrounded by a thin steel tube, were carried out. The effect of the splitting stresses is

studied by measuring the tangential strains in the steel tube. The test specimens were

carefully designed to give tangential strains in the steel tube that were large enough to

be measurable, but not large enough to cause yielding of the steel. The steel tubes had

a diameter of 70 mm, a height of 100 mm, and a thickness of 1.0 mm. The embedment

length of the reinforcement bars was 50 mm. The geometry of the test specimens is

shown in Fig. 2.

A total of fourteen tests were carried out. Three tests were loaded monotonically

in one direction; two other tests were loaded in the same way in the opposite direction.

Nine tests were subjected to cyclic loading, see Table 1. The load cycles had a

symmetric deformation interval growing larger for each load cycle, as shown in Fig. 3.

The reinforcement in all specimens was of type K500ST φ 16, and the concrete had a

compressive strength of 36 MPa (tested on wet stored 150 mm cylinders, 300 mm

high).

II-4

20

This side was upwhen grouted

[mm]t = 1.070

φ = 16

15

50

35

45

25

Fig. 2 – The geometry and cross-section of the test specimens.

Table 1 – Experimental Programme

Test No. monotonic / cyclic s1 [mm] (see Fig. 3) Age at testing [days]

M1a monotonic - 28

M1b monotonic - 28

M1c monotonic - 28

M2a monotonic - 31

M2b monotonic - 31

C–2.0a cyclic 2.0 31

C–2.0b cyclic 2.0 32

C–1.0a cyclic 1.0 29

C–1.0b cyclic 1.0 29

C–0.5a cyclic 0.5 29

C–0.5b cyclic 0.5 30

C–0.5c cyclic 0.5 30

C–0.25a cyclic 0.25 30

C–0.25b cyclic 0.25 31

II-5

-2s1

Slip

3s1

2s1

-3s1

s1

-s1Time

Fig. 3 – Deformation control of the cyclically loaded specimens.

3. TEST ARRANGEMENTS

The test specimens were cast at the laboratory of the Department of Structural

Engineering. The steel tubes were used as forms, with plastic tops and bottoms which

also steady the reinforcement bars. In the parts near the ends of the cylinder, bonding

between the reinforcement bar and the concrete was prevented by plastic tubes. The

top and bottom of the forms were removed after one day, together with the plastic

tubes preventing the bond between the reinforcement bar and the concrete. The

specimens were thereafter stored in water until testing began. The concrete surface

that faced upward when the specimens were grouted was in some specimens rather

rough; the concrete surfaces that faced downward when the specimens were grouted

were smoother. The concrete at the ends protruded about half a millimetre beyond the

steel tubes, so that the supports acted only on the concrete.

The tests were carried out when the test specimens were 28 to 32 days old. The

position of the test specimens was the reverse of the position when grouted. The test

specimens were in a rigid frame when the load was applied, so that both ends of the

reinforcement bars were active. Before testing, the reinforcement bar was screwed to

the frame at both ends. Steel plates, with a hole (φ 23 mm) for the reinforcement bar,

were used as supports against the concrete surfaces. These steel plates were kept in

II-6

place by four threaded bars. The threaded bars were not intended to apply any outer

pressure on the test specimen, only to keep the steel plates in place. Therefore, the

nuts were tightened by hand when testing began. In this way, only very limited outer

pressure on the bolts can have been applied, however still, the measurements do not

indicate any play. The test set-up is shown in Fig. 4(a), together with the positive

loading and deformation direction chosen. The loading conditions are shown in

Fig. 4(b). The tangential strains in the steel tube were measured with nine strain

gauges in each test; there were three gauges on three levels, as shown in Fig. 5.

Positiveloadingdirection

Passive part ofthe machine

steel plates

Active part ofthe machine

(a)

[mm]

45

25

3.5friction

steel - concrete

(b)

Fig. 4 – (a) The set-up of the pull-out tests. (b) The loading conditions.

II-7

3, 6, 9

[mm]

1, 2, 34, 5, 6

7, 8, 9

1525

35

251, 4, 72, 5, 8

Fig. 5 – Placement, on the steel tubes, of the strain gauges for measuring the tangential

strains.

4. TEST RESULTS

4.1 Monotonically loaded tests

Due to the confinement of the steel tubes, all of the tests resulted in pull-out

failures with rather high bond stresses, see Table 2. No cracks were visible in the test

specimens, even though they were very carefully examined. The load versus the slip in

the monotonically loaded tests is shown in Fig. 6. It can be seen that the scatter in the

results is rather small. Note that the capacity was greater in the loading direction here

defined as negative, in tests M2a and M2b, than for positive loading in tests M1a,

M1b, and M1c. The positive loading and slip direction is defined in Fig. 4. In Fig. 7,

which shows the first part of the load versus the slip in the monotonic loads, it can be

seen that the stiffness is larger as well for loading in the negative direction. Possible

explanations for these differences could be that the concrete surfaces which were up

when the specimens were grouted were not as smooth as the surfaces that were down,

the grouting direction, and/or the asymmetry of the test specimens.

II-8

Table 2 – Results from the monotonic tests.

Test No. Maximum load

[kN]

Maximum bond

stress [MPa]

Slip at maximum

load [mm]

M1a 48.5 19.3 1.52

M1b 47.6 18.9 1.45

M1c 46.7 18.6 1.56

M2a -52.1 -20.7 -0.85

M2b -53.0 -21.1 -0.89

-60

-40

-20

20

40

60

-10 -5

5 10 15

average

Load [kN]

Slip [mm]

M1aM1b

M1c

M2bM2a

Fig. 6 – Load versus slip in the monotonic tests.

average

Slip [mm]

-60

-40

-20

20

40

60

-2 -1 1 2

Load [kN]

Fig. 7 – Load versus slip in the monotonic tests; an enlargement of the first part in

Fig. 6.

II-9

In Fig. 8, the slip versus the average of the measured strains at each height in

each experiment is shown together with the total average strains at each height. It can

be seen that in the measured strains, the scatter is also rather small. The average

strains from all of the monotonic tests are shown in Fig. 9.

-100

0

100

200

300

400

500

600

700

800

-10 -5 0 5 10 15

Strain [microstrain]

Slip [mm]

2, 5, 8

1, 4, 7

3, 6, 9

2, 5, 8

3, 6, 9

1, 4, 7

averageabc

Fig. 8 – Strain versus slip in the monotonic tests.

3, 6, 9

-60

-40

-20

0

20

40

60

-100 0 300100 200 400 500 600 700Strain [microstrain]

Load [kN]

2, 5, 81, 4, 7

2, 5, 81, 4, 7 3, 6, 9

Fig. 9 – Load versus average strains in all of the monotonic tests.

II-10

4.2 Cyclically loaded tests

Some examples of results from the cyclic tests are shown in Figs. 10 - 12.

Results from all of the experiments can be found in Lundgren [8]. In Fig. 10, the load

versus the slip in tests C–1.0b and C–0.25b are shown. They show a typical response

for bond in cyclic loading: a steep unloading and then an almost constant, low load

until the original monotonic curve is reached. This constant, low load is slightly

larger, however, in the cycle where the maximum capacity has been obtained than in

the load cycles before and after.

-60

-40

-20

0

20

40

60

-8 -6 -4 -2 0 2 4 6 8

Load [kN]

Slip [mm]

(a)

-60

-40

-20

0

20

40

60

-3 -2 -1 0 1 2 3

Load [kN]

Slip [mm]

(b)

Fig. 10 – Load versus slip in tests (a) C–1.0b; and (b) C–0.25b.

II-11

In Fig. 11, the strains measured at the various points along one test specimen are

shown. It can be seen that the strains increase when the load is directed towards the

part of the specimen where the strain gauges are situated; this is due to the inclined

compressive forces of the anchorage, see Fig. 1. The strains in the middle decrease

during unloading, until the slip is zero, after which they start to increase again. At all

three levels, the strain remains the same when there is only little bond capacity left,

and is almost unaffected by the last load cycles. These residual strains from all tests

are shown in Table 3. There it can be seen that the residual strain level is about the

same value in all of the cyclic tests. For the strains in the middle of the zone with bond

(2, 5 and 8), this residual strain is also about the same value in the monotonic tests

after unloading. However, the residual strains in the monotonic tests, at the upper and

lower parts of the test specimens, are influenced by the direction of the monotonic

load. This is because the inclined compressive struts (that result from the anchorage)

have acted in only one direction during the whole test.

2, 5, 8

-100

100300

500

700Strain [microstrain]

-100

100

300

500

700

-100

100

300

500

700

-8 -6 -4 -2 0 2 4 6 8Slip [mm]

1, 4, 7

3, 6, 9

Positiveloadingdirection

Fig. 11 – Average of the strains at each level of the test specimen in test C–1.0b.

II-12

Table 3 – Residual strains after the tests.

Test No. strain gauge 1, 4, 7

[microstrain]

strain gauge 2, 5, 8

[microstrain]

strain gauge 3, 6, 9

[microstrain]

M1a 180 320 75

M1b 200 260 70

M1c 200 280 70

M2a1) (110) (380) (260)

M2b1) (90) (380) (250)

C–2.0a 90 230 60

C–2.0b 130 330 110

C–1.0a 110 220 110

C–1.0b 140 260 100

C–0.5a 80 180 50

C–0.5b 110 190 100

C–0.5c 120 200 90

C–0.25a 80 190 80

C–0.25b 110 230 1201) The values from tests M2a and M2b are put in parentheses, since the strains were

not measured after the tests were unloaded.

In Fig. 12, the load is plotted versus the average of the measured strains in strain

gauges 2, 5 and 8 (in the middle of the part with bond between the concrete and the

reinforcement). There it can be seen that also the strains had a rather steep unloading

when the load was reduced to zero. The strains were still decreasing when the load

was almost constant, after which they started to increase again when the load was

increased in the other direction.

II-13

-60

-40

-20

0

20

40

60

0 200 400 600 800

Load [kN]

Strain [microstrain]

Decreasing strain

(a)

-60

-40

-20

0

20

40

60

0 200 400 600 800

Load [kN]

Strain [microstrain]

(b)

Fig. 12 – Load versus the average of the measured strains in strain gauges 2, 5 and 8

in tests (a) C–1.0b; and (b) C–0.25b. Note that the strain was decreasing

when the load was almost constant, here especially marked in (a) in the first

load cycle.

II-14

5. RESULTS FROM ANALYSES OF THE TESTS

The pull-out tests have been analysed with finite element models, see Lundgren

and Gylltoft [7] or Lundgren [9]. The concrete was then modelled with a constitutive

model based on non-linear fracture mechanics. The bond between the reinforcement

bar was described by a new model, which includes the splitting stresses. Some results

about the magnitude of the splitting stress deriving from the analyses of these

experiments are presented here. The coefficient of friction is then used:

stress splittingstress bond=µ

By analysing the tests with finite element analyses, and comparing the measured

tangential strain in the steel tubes with the ones obtained in the analyses, it was found

that the coefficient of friction started at about 1.0, and then decreased with increasing

slip to about 0.4, see Fig. 13. For the cyclically loaded specimens, the coefficient of

friction was found to be about 0.4 in the regions after unloading when the bond stress

was almost constant, until the monotonic curve was reached again. In Fig. 14, the

bond stress versus the splitting stress are shown that in a finite element analysis of test

No. C–1.0b gave results that corresponded well with the measured ones.

Slip [mm]

µ

00.20.40.60.8

11.2

0 5 10 15

Fig. 13 – The coefficient of friction as a function of the slip; input in finite element

analyses of the tested specimens that gave the best correspondance with the

test results.

II-15

Splitting stress [MPa]

-20-15-10-505101520

-25 -20 -15 -10 -5 0

Bond stress [MPa]

Fig. 14 – The bond stress versus the splitting stress that in a finite element analysis of

test No. C–1.0b gave results that corresponded well with the measured

ones.

6. CONCLUSIONS

Pull-out tests were carried out, five loaded monotonically and nine loaded with

reversed cyclic loading. The results from the monotonic tests indicate that the splitting

stresses decreased after the maximum bond stress had been obtained, however not as

much as the bond stresses decreased. The results from the cyclic tests show a typical

response for bond in cyclic loading: a steep unloading and then an almost constant,

low load until the original monotonic curve was reached. The strains in the middle of

the steel tubes also had a rather steep unloading when the load was reduced to zero;

they then decreased even more when the load was almost constant, and started to

increase again when the load increased in the other direction. The strain remained at

the same level after about three to seven load cycles, depending on the deformation

interval. The strain was almost unaffected by the last load cycles. This residual strain

was about the same in all of the cyclic tests. For the strains in the middle of the test

II-16

specimens, this residual strain was also about the same value as in the monotonic tests

after unloading.

The test results provide valuable information about the splitting stresses that

result from anchorage of reinforcement bars in concrete. The effect of reversed cyclic

loading on both the bond stresses and the splitting stresses was investigated. These

test results can be a reference when calibrating models of the bond mechanism and,

hopefully, give a better understanding of the total bond mechanism.

REFERENCES

[1] Tepfers, R. and Olsson, P-Å., ‘Ring Test for Evaluation of Bond Properties of

Reinforcing Bars’, in ‘Bond in Concrete’ Proceedings (CEB), Riga, 1992,

pp. 1-89 - 1-99.

[2] Malvar, L. J., ‘Confinement Stress Dependent Bond Behavior, Part I:

Experimental Investigation’, in ‘Bond in Concrete’ Proceedings (CEB), Riga,

1992, pp.1–79 - 1–88.

[3] Eligehausen, R., Popov, E. P., and Bertero, V. V., ‘Local Bond Stress-Slip

Relationships of Deformed Bars Under Generalized Excitations’, v.4, Proceedings

of the Seventh European Conference on Earthquake Engineering, Athens, Sept.

1982, pp. 69-80.

[4] Balázs, G. and Koch, R., ‘Bond Characteristics under Reversed Cyclic Loading’,

Otto Graf Journal, 6, 1995, pp. 47-62.

[5] Magnusson, J., ‘Bond and Anchorage of Deformed Bars in High-Strength

Concrete’, Licentiate Thesis. Division of Concrete Structures, Chalmers

University of Technology, Publication 97:1, Göteborg, 1997.

[6] Åkesson, M., ‘Fracture Mechanics Analysis of the Transmission Zone in

Prestressed Hollow Core Slabs’, Licentiate Thesis. Division of Concrete

Structures, Chalmers University of Technology, Publication 93:5, Göteborg, 1993.

[7] Lundgren, K. and Gylltoft, K., ‘Bond Modelling in Three-Dimensional Finite

Element Analyses’, provisionally accepted for publication in Magazine of

Concrete Research.

II-17

[8] Lundgren, K., ‘Steel-Encased Pull-Out Tests Subjected to Reversed Cyclic

Loading’, Chalmers University of Technology, Division of Concrete Structures,

Report 98:9, Göteborg, 1998.

[9] Lundgren, K., ‘Modelling of Bond: Theoretical Model and Analyses’, Chalmers

University of Technology, Division of Concrete Structures, Report 99:5,

Göteborg, 1999.

III-1

BOND MODELLING IN THREE-DIMENSIONAL

FINITE ELEMENT ANALYSES

Karin Lundgren

Research Assistant

Division of Concrete Structures

Chalmers University of Technology

SE-412 96 Göteborg, Sweden

Telephone: +46 31 772 22 56

Fax: +46 31 772 2260

Mail: karin.lundgren@con.chalmers.se

Kent Gylltoft

Professor

Division of Concrete Structures

Chalmers University of Technology

SE-412 96 Göteborg, Sweden

Telephone: +46 31 772 22 45

Fax: +46 31 772 2260

Mail: kent.gylltoft@con.chalmers.se

Abstract

The bond mechanism between deformed bars and concrete is known to be

influenced by a number of parameters, such as the strength of the surrounding

structure, the occurrence of splitting cracks and yielding of the reinforcement.

However, when reinforced concrete structures are analysed with finite element models

it is quite common to assume that the bond stress depends solely on the slip. A new

model has been developed, which is specially suited for detailed three-dimensional

analyses. In this new model, the splitting stresses of the bond action are included, and

the bond stress depends not only on the slip, but also on the radial deformation

between the reinforcement bar and the concrete. Bar pull-out tests with various

geometry and with both monotonic and cyclic loading have been analysed. The results

show that the new model is capable of predicting splitting failures, the loss of bond if

the reinforcement is yielding, as well as simulating cyclic loading in a physically

reasonable way.

III-2

Introduction

When modelling reinforced concrete structures with the finite element method,

the most common method has been to use a bond-slip relation for the interaction

between the reinforcement bars and the surrounding concrete. However, bond action

causes not only bond stresses, but also splitting stresses as was shown by Tepfers1

among others. If the concrete surrounding the reinforcement bar is well-confined,

meaning that it can withstand these splitting stresses, a pull-out failure is obtained; see

Fig. 1. If the concrete cannot withstand these splitting stresses, splitting cracks in the

concrete will cause a decrease of the bond capacity. Also if the reinforcement starts

yielding, the bond capacity will decrease; see Engström2. A disadvantage with using

predefined bond-slip relations as input for analyses is that these circumstances must

be known in advance: if the concrete will split, or if the reinforcement will yield, so

that the correct bond-slip relation can be given. Cox3 and Åkesson4 have developed

other bond models that include the splitting stresses. None of them has, however,

been shown to describe the loss of bond when the reinforcement starts yielding, and

Åkesson’s model is not suited for cyclic loading. In the applications of the model we

had in mind, cyclic loading often prevails.

Bondstress

Slip

(a)(b)

Fig. 1 Schematic bond-slip relationship: (a) pull-out failure; (b) splitting failure, or

loss of bond due to yielding of the reinforcement.

III-3

A new model has therefore been developed, which includes the splitting

stresses. With the same input parameters, this new model results in various bond-slip

curves, depending on the confinement of the surrounding structure, and on whether

the reinforcement is yielding or not. The effect of cyclic loading with varying slip

direction is also important for the bond resistance. Therefore, this effect has also been

included in the model. The model is presented here, together with results from finite

element analyses of pull-out tests. For more details, see Lundgren5.

Theoretical model

In the finite element program DIANA, there are interface elements available,

which describe a relation between the tractions t and the relative displacements u in

the interface. These interface elements are used at the surface between the

reinforcement bars and the concrete. The physical interpretations of the variables tn, tt,

un and ut are shown in Fig. 2. The interface elements have, initially, a thickness of

zero.

reinforcementbar

uttn

tt

un

���

�=�

���

�=

��

��

�=�

���

�=

sliplayer in thent displaceme normal relative

stress bondstress normal

t

n

t

n

uu

tt

u

t

Fig. 2 Physical interpretation of the variables tn, tt, un and ut.

III-4

Elasto-plastic formulation

The model is a frictional model, using elasto-plastic theory to describe the

relations between the stresses and the deformations. The relation between the

tractions t and the relative displacements u is in the elastic range:

���

�

����

���

�

�

=��

��

�

t

n

t

t

t

n

uu

D

Duu

Dtt

22

1211

0(1)

where D12 normally is negative, meaning that slip in either direction will cause

negative tn, i.e. compressive forces directed outwards in the concrete. Furthermore,

the model has yield lines, flow rules, and hardening laws. The yield lines are

described by two yield functions, one describing the friction, F1, assuming that the

adhesion is negligible:

0=+= nt1 ttF µ (2)

The other yield line, F2, describes the upper limit at a pull-out failure. This is

determined from the stress in the inclined compressive struts that result from the bond

action, see Fig. 3.

0222 =⋅++= nnt tcttF (3)

-tndl

rdϕ

c

α

tt

dlsinα

Equilibrium gives:

0

sin

sin

22

22

22

=⋅++

+

−=

⋅⋅=⋅⋅+

nnt

nt

n

nt

tctt

tt

trddlcrddltt

α

ϕαϕ

Fig. 3 The stress in the inclined compressive struts determines the upper limit.

III-5

For plastic loading along the yield line describing the upper limit, F2, an

associated flow rule is assumed. For the yield line describing the friction, F1, a non-

associated flow rule is assumed, where the plastic part of the deformations is

0 , =+== ntt

t ttuu

GGdd η∂∂λ

tup (4)

where dλ is the incremental plastic multiplier. The yield lines, together with the

direction of the plastic part of the deformations, are shown in Fig. 4. At the corners, a

combination of the two flow rules is used; this is known as the Koiter rule.

For the hardening rule of the model, a hardening parameter κ is established. It is

defined by

22 pt

pn dudud +=κ . (5)

It can be noted that for monotonic loading, dunp and the elastic part of the slip

are very small compared to the plastic part of the slip, dutp; therefore the hardening

parameter κ will be almost equivalent to the slip, ut. The variables µ and c in the yield

functions are assumed to be functions of κ.

tup

∂∂

= 2Fdd λ

tn

tt

F1

G

F2t

up

∂∂= Gdd λ

ttup

∂∂+

∂∂= 2

21FdGdd λλ

µη 1

1

Fig. 4 The yield lines. The plastic part of the deformations, dup, is given by an

associated flow rule at the yield line describing the upper limit, F2, and a

non-associated flow rule at the yield line describing the friction, F1.

III-6

Three-dimensional modelling

For three-dimensional modelling, a third component is added: the stress acting

in the direction around the bar. This is assumed to act independently of the other

components; i.e. the equation for the elastic stage is then assumed to be:

�

���

�

�

�������

������

�

�

=����

���

�

�

r

t

nt

t

r

t

n

uuu

DD

DuuD

ttt

33

22

1211

0000

0

(6)

The main objective with the stiffness D33 is that it prevents the bar from rotating

in the concrete. The traction tr has no influence on the yield lines.

Damaged / undamaged deformation zones

A typical response for bond with varying slip direction is with a steep unloading

and then an almost constant, low bond stress until the original monotonic curve is

reached; see Fig. 5(a). To make the model describe this typical response, a new

concept, called damaged / undamaged deformation zone, is used. The idea is that

when the reinforcement slips in the concrete, the friction will be damaged in the range

of the passed slip. This is a simplified way to describe the damage of the cracked and

crushed concrete. In Fig. 5(b), the reinforcement is back in its original position after

slipping in both directions. The concrete will then be crushed in the range of the

passed slip. This crushed concrete still has some capacity to carry compressive load,

but no capacity at all in tension. The friction is therefore assumed to vary in the

damaged zone depending on whether loading is applied in the direction away from, or

towards, the original position, as shown in Fig. 5(c) and (d). It is assumed to

immediately drop to a low value µd0 at load reversal, and to keep this value until the

original position is reached. For further loading, away from the original position, the

friction is assumed to gradually increase, until the undamaged zone is reached and the

normal value of µ is used again. To describe this gradual increase, an equation of the

second degree has been chosen.

The parameter η also has a lower value in the damaged deformation zone,

varying in the same way as just described about the coefficient of friction. This lower

III-7

value physically corresponds to the fact that the increase in the stresses is lower than

in the undamaged deformation zone.

“crushedconcrete”

utmax

concrete

reinforcement bar

(b)

utmin

tt

ut

damageddeformationzone

undamageddeformationzone

undamageddeformationzone

(a)

utmax

utmin

µ, η

ututmaxutmin

µd0, η d0

(c)

µ, η

ututmaxutmin

µd0, η d0

(d)

loadingdirection

loadingdirection

Fig. 5 (a) One load cycle with varying slip directions. (b) The reinforcement bar is

back in its original position, after slipping in both directions. Maximum and

minimum values of the slip are especially marked. (c) and (d) The

parameters µ and η vary within the damaged deformation zone depending on

whether the loading is directed towards or away from the original position.

III-8

Discussion of the model

The most important feature of the model is that it applies both the bond stress

and the splitting stress, thus describing the inclined struts that result from the bond

action. By modelling the surrounding structure, it is possible to obtain for example

splitting cracks in the concrete, or cone cracks close to free edges. One of the main

ideas of the model is that the complex crushing and cracking of the concrete close to

the reinforcement bar is described in a simplified way by decreasing the friction

between the reinforcement bar and the concrete. The model is sensitive to the

resistance of the surrounding structure, so that if the pressure around the bar for some

reason is lost, the bond stress will decrease.

To better understand how the model works, a monotonic loading of a pull-out

test will be discussed. When the loading starts, there are no initial stresses. This

means that the loading will start at point A in Fig. 6. When a tensile force is applied,

loading along yield line F1 will occur. For further loading, the coefficient of friction

decreases due to the hardening in the model. The loading will therefore follow a

curved line as shown in Fig. 6. For the loading along yield line F1, an equivalent

stiffness matrix Deq can be deduced:

)( pe uuDuDt dddd −==

where, by equation (4):

�

���

�

�

==

t

t

uudGddη

λ∂∂λ

tup

Loading along the yield line implies that F1 equals zero both before and after the

loading step, so

0=1dF

By assuming that the change in µ is negligible, we thus obtain

0 01 =+= ntt

t dtdttt

dF µ∂∂ t

t

Loading with a positive slip, and a positive bond stress, then gives:

���

���

��

�

−−

++=�

���

�

t

n

t

n

dudu

DDDDD

dtdt

ηµµη

µηµ1

121122

2211 (7)

III-9

This means that the bond stress, tt, will depend on both the slip, ut, and the

relative normal displacement between the reinforcement bar and the concrete, un. In a

pull-out test, this normal displacement will increase with increasing load. This is

because the concrete will move away from the reinforcement bar due to the normal

stresses, and the radius of the reinforcement bar will decrease due to the Poisson

effect. Hence, in equation (7), the parameter η must be chosen so that the bond stress

will increase for normal loading, i.e. nt dudu >η . If, however, the concrete splits or

the reinforcement starts yielding, this normal displacement will start to increase faster.

If the parameter η then is properly calibrated, the bond stress will decrease, i.e.

nt dudu <η . The loading will thus follow a path as indicated in Fig. 6 with (b). If

neither the concrete splits, nor the reinforcement starts yielding, the upper limit in the

form of the cap F2 will be reached. The hardening of this function is chosen to

describe the pull-out failure, and the loading path will be as indicated in Fig. 6 with

(a). Between these two extreme cases are, of course, many other possible loading

paths, where perhaps the initial yield line F2 is reached and, after some hardening,

splitting occurs.

The discussion above shows two important criteria to take into account when

the value of the parameter η is chosen. Another important criterion is that the

dissipated energy must be positive, i.e. that the model cannot store any more energy

than the elastic one. This criterion leads to a condition for the selected input variables:

that -µ ≤η ≤ µ. The model must also give a unique solution for each point. For this

criterion to be fulfilled at the corner, .2

12

µµη −≥

initial yield linesyield lines afterhardening

tn

tt

A(a)

(b)

F1

F2F1

F2

Fig. 6 Various loading paths depending on the failure mode: (a) pull-out failure;

(b) splitting failure, or loss of bond due to yielding of the reinforcement.

III-10

Input parameters for the interface

Required input data for the interface are the elastic stiffness matrix D in

equation (1), the parameter η defined in equation (4), and for loading in the damaged

deformation zone the parameters ηd0 and µd0. Furthermore, the functions µ(κ) and

c(κ) must be chosen. The model is calibrated for reinforcement bars K500 φ 16 and

normal strength concrete. Before using the values recommended here on other

concrete strengths or reinforcement qualities, analyses and comparisons with

experimental results are recommended. However, generally speaking, a smother

surface of the reinforcement bar would give lower friction and lower values of the

stiffnesses in the elastic stiffness matrix D. More specific background for the choice

of the input parameters can be found in Lundgren5.

First of all, the stiffness D22 in the elastic stiffness matrix D was recognised as

the stiffness of the first part, or at unloading, in a bond-slip curve. By assuming that

this is proportional to the modulus of elasticity of the concrete, and by comparing

with results from experiments, this stiffness was chosen to be:-1

222222 m 0.6 , =⋅= KEKD c (8)

Next, the stiffnesses D11 and D12 were determined. The larger D11 is chosen to

be, the more variation of the stresses along the reinforcement bar is obtained. This

variation depends on variations of the strength of the modelled structure, as for

example when stirrups are taken into account. To obtain a physically reasonable

variation, D11 was chosen to be:-1

111111 m 0.11 , =⋅= KEKD c (9)

The stiffness D12 determines how large a part of the normal stresses will remain

after unloading. By comparing with results from experiments, D12 was chosen to be:

max

2212 9.0

µDD −= (10)

For three-dimensional modelling, the stiffness D33 in equation (6) is also

required. A value of 1010 N/m3 was used in all analyses, since it was found that this

was enough to prevent the bar from rotating in the concrete.

The parameter η is chosen in order to obtain a decreasing bond stress when the

concrete around the bar splits, without elastic unloading; see the discussion in the

previous section. Through calibration, η was chosen to be 0.04. For loading in the

III-11

damaged deformation zones, ηd0 was chosen to be 0.004, and the coefficient of

friction µ d0 was 0.4.

The function µ(κ) describes how the relation between the bond stress and the

normal splitting stress depends on the hardening parameter. Tepfers and Olsson6

performed “ring tests”: pull-out tests in concrete cylinders confined by thin steel

tubes. They measured the strain in the steel ring and used this to evaluate the normal

stress. The results, together with the chosen input for the analyses, are shown in

Fig. 7(a). The chosen input is also selected to match the experimental data of steel-

encased pull-out tests carried out by the authors7; see the next section. Since a unique

solution should be given at the corner, the coefficient of friction was not assumed to

be larger than 1.0.

The variable c represents the stress in the inclined compressive struts as shown

in Fig. 3. The function c(κ) was therefore chosen to be the same as the uniaxial

compression curve of the concrete, only with a factor between the plastic strain and

the hardening parameter κ as shown in Fig. 7(b).

1.0c/fcc ; σ/fcc

0.5

0

µdmax

(a)

chosen inputT: Ks400T φ16T: Ks600 φ16T: Ks400I φ16

Slip; κ [mm]

µd0

µ

00.20.40.60.81.01.21.4

0 5 10 15

5 100 52.5 7.5

κ [mm]εp [‰]

(b)

Fig. 7 (a) The coefficient of friction as a function of the slip evaluated from tests,

T: Tepfers and Olsson6, together with the chosen function µ(κ). (b)

Compressive stress versus plastic strain, and the function c(κ).

III-12

Comparison with pull-out tests

Pull-out tests of various kinds have been analysed with finite element models.

The tests have been selected to show various types of failure: pull-out failure, splitting

failure, pull-out failure after yielding of the reinforcement, rupture of the

reinforcement bar, and cyclic loading. In all tests, the reinforcement is of type

K500 φ 16, and the concrete is of normal strength (the compressive cylinder strength

varies from 25 to 35 MPa).

In all analyses, the concrete was modelled with a constitutive model based on

non-linear fracture mechanics. Three slightly different material models have been

used: a combined Drucker-Prager and Rankine elasto-plastic model, a Drucker-Prager

together with a fixed crack model, and a rotating crack model based on total strain;

see TNO8. One main difference between these material models is their behaviour for

cyclic loading in tension. While the Rankine model gives unloading according to the

elastic modulus, and then compressive stresses already when the strain is positive, the

crack models instead unload back to zero again; see Fig. 8. In the following sections,

the results from analyses with the combined Rankine and Drucker-Prager model are

presented if nothing else is mentioned. When the different material models gave

different results, this is discussed in the text.

ε

σ

(b)

(a)

Fig. 8 Unloading / reloading in the material models used: (a) crack models and

(b) Rankine model.

III-13

Most of the finite element models were axisymmetric; the only exception is the

eccentric pull-out tests of Magnusson9 (see section “Splitting failure”). The main

advantage when using axisymmetric models is that the calculation time required for

the analyses is dramatically decreased. One disadvantage of axisymmetric models is

that a certain number of discrete radial cracks must be assumed. Close to the

reinforcement bar, there are probably, at least in the beginning, rather many cracks.

Further out from the reinforcement bar, the number of cracks depends on the

structure. In the following, the results from analyses with four radial cracks are

shown, and when the assumed number of cracks has any influence on the results, it is

discussed in the text.

Yielding was modelled using associated flow and isotropic hardening. The

hardening of the Drucker-Prager yield surface was evaluated from the shape of the

uniaxial stress-strain relationship in compression, and was chosen to match typical

test data presented by Kupfer and Gerstle10. From the various measured compressive

strengths, an equivalent compressive cylinder strength, fcc, was evaluated. Other

necessary material data for the concrete were calculated according to the expressions

in CEB11 from fcc, and are shown in Table 1. The constitutive behaviour of the

reinforcement steel was modelled by the Von Mises yield criterion with associated

flow and isotropic hardening. The elastic modulus of the reinforcement was assumed

to be 200 GPa when it had not been measured.

Table 1. Material data of the concrete in the analysed pull-out tests

Compressive tests Material data used in the analysesReference

Test specimenfcc

[MPa]fcc

[MPa]fct

[MPa]Ec

[GPa]GF

[N/m]Lundgren7 150 mm cyl., wet 35.6 35.6 2.7 32.9 61

Magnusson9

emb. length 40emb. length 220emb. length 360

eccentrically reinf.

150 mm cyl., wet27.530.627.629.4

27.530.627.629.4

2.22.42.22.3

30.031.330.230.9

90969094

Balázs and Koch12 150 mm cube, wet 28-32 25.5 2.0 29.4 58

III-14

Pull-out failure

In the tests carried out by the authors7, reinforcement bars were pulled out of

concrete cylinders surrounded by steel tubes. The steel tubes had a diameter

of 70 mm, a height of 100 mm, and a thickness of 1.0 mm. The embedment length of

the reinforcement bars was 50 mm. The tangential strains in the steel tubes were

measured at three heights, together with the applied load and slip. Five tests were

carried out, three in one direction and two in the other. When these tests were

analysed, friction between the edges of the concrete and the support plates was

introduced, assuming the coefficient of friction to be 0.4. The friction at the supports

did not influence the achieved load versus slip; however, the tangential strains in the

analyses were slightly influenced (increasing the friction at the supports led to larger

strain in the middle of the zone with bond, and lower strains close to the supports).

Results from the analyses with four assumed radial cracks, together with the

finite element mesh used, are shown in Fig. 9. As can be seen, the choice of material

model strongly affects the tangential strains after the maximum load. This is

reasonable, since when the load decreases, the cracked concrete is unloaded. Since the

Rankine model uses the modulus of elasticity for unloading, quite large residual

strains are obtained, while the crack models that unload back to zero again give only

small residual strains. As can be seen in Fig. 9, the experimental results lie in between

these two extreme cases. When the assumed number of radial cracks was two instead

of four, the load versus slip was not affected. As can be expected, the tangential strain

was, however, slightly higher when only two cracks were assumed. Instead of a

maximum of about 0.50 ‰, the highest strain was about 0.55 ‰.

III-15

-60

-40

-20

0

20

40

60

-100 0 100 200 300 400 500 600 700Strain [microstrain]

Load [kN]

Positive loadingdirection

2

3

FEA

Exp.

-60

-40

-20

0

20

40

60

-10 -5

5 10 15

Load [kN]

Slip [mm]

1

3

21

1 32

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Strain [‰]

FEA,crack model,fixed androtating

Exp.FEA,Rankine

Fig. 9 Comparison between test results and results from the analyses of the

monotonic loaded steel-encased pull-out tests. In the load versus slip, the

different material models gave the same results, while the tangential strains

differed.

Pull-out tests carried out by Magnusson9 and Balázs and Koch12 have been

analysed. Magnusson had concrete cylinders with a diameter of 300 mm and an

embedment length of 40 mm; Balázs and Koch had concrete specimens with a

quadratic cross-section 160 ·160 mm and an embedment length of 80 mm. In both

cases, the concrete specimens were large enough to prevent splitting failure; thus,

pull-out failures were obtained. Results from the analyses are compared with

experiments in Fig. 10. As can be seen, a reasonably good agreement was obtained.

III-16

0

10

20

30

40

50

60

70

0 5 10 15

B: Exp.B: FEAM :Exp.M: FEA

Load [kN]

Passive slip [mm]

Fig. 10 Load versus slip in pull-out tests with short embedment length. B: Balázs

and Koch12 and M: Magnusson9.

Splitting failure

Magnusson9 has also carried out pull-out tests on eccentrically reinforced

specimens with varying stirrup configurations. The different stirrup configurations

(without stirrups, with two and with four stirrups along the embedment length) led to

splitting failures at various levels. In the test specimen with four stirrups, the stirrups

gave enough confinement to obtain a ductile failure after splitting. In the analyses of

these experiments, the stirrups were modelled as embedded reinforcement, meaning

that complete interaction between the stirrups and the concrete was assumed. The

finite element model, together with the boundary conditions, is shown in Fig. 11(a).

Since a smeared crack model was used, the input of a characteristic length was

needed. This length should be related to the size of one element. This is based on an

assumption that a crack will localise in one element. In these analyses, however, the

crack localised in two elements. The characteristic length was therefore estimated to

be 40 mm, based on the size of the area where the cracks localised; see Fig. 11(b).

III-17

~40 mm

~40 mm

crackedelements

(a) (b)

Fig. 11 (a) The finite element model of the pull-out tests on eccentrically reinforced

specimens of Magnusson9. (b) Localisation of the main radial cracks in a

cross-section.

When these tests were analysed with Drucker-Prager combined with the fixed

crack model, it was noted that the bond action caused splitting cracks that reached the

outer surface of the concrete at about the maximum load that was measured in the

tests. In the analyses, however, it was possible to further increase the load, which does

not seem reasonable, especially not in the case without stirrups. This load increase

was possible because the direction of the cracks was locked in the material model.

Consequently, large stresses, about five times the tensile strength, could be

transferred. With the use of rotating cracks, these large tensile stresses were

prevented, and the correlation between the analyses and the experiments was

improved, as can be seen in Fig. 12. It can be noted that even with the same

embedment length, and when exactly the same input parameters were given for the

interface, different load-slip curves were obtained depending on the modelled

structure, in this case the number of stirrups. Comparing with the measured response,

the agreement is good, especially when considering the large scatter that is always

obtained in pull-out tests.

III-18

0

20

40

60

80

100

0 2 4 6 8 10

Load [kN]

Active slip [mm](a) Without stirrups

Active slip [mm](c) With four stirrups

Load [kN]

0

20

40

60

80

100

0 2 4 6 8 10

FEA, rotating crack

Experimental9

FEA, fixed crack

0

20

40

60

80

100

0 2 4 6 8 10

Load [kN]

Active slip [mm](b) With two stirrups

Fig. 12 Load versus slip in eccentrically reinforced pull-out tests.

Yielding of the reinforcement

Magnusson9 has also carried out pull-out tests where the reinforcement had an

embedment length long enough to give yielding of the reinforcement. Two of these

tests have been analysed, where the reinforcement was centrically placed in a concrete

specimen of dimensions 400 ·400 mm. In one of the experiments, with an embedment

length of 220 mm, a pull-out failure after yielding of the reinforcement was obtained;

and in the other one, with an embedment length of 360 mm, rupture of the

reinforcement bar occurred. As can be seen in Fig. 13, the same results were obtained

in the analyses. In Fig. 14, the bond–slip resulting from the analyses at various levels

along the bar is shown. It can be seen that the bond stress decreased drastically when

the reinforcement reached the yield plateau. This is because the normal stress

decreased when the radius of the reinforcement bar decreased. When the

reinforcement began to harden again, a small bond capacity was obtained. This was

possible since the decrease of the radius of the reinforcement was lower when the

reinforcement hardened, and thus, normal stresses could be built up again.

III-19

Load [kN]

020406080

100120140

0 5 10 15 20

Passive slip Active slip

FEAExp.9

020

4060

80100120140

0 10 20 30 40Deformation [mm]

Load [kN]Passive slip

Active slip

FEAExp.9

Deformation [mm]

(a) Embedment length 220 mm (b) Embedment length 360 mm

Fig. 13 Load versus slip in pull-out tests with long embedment length. Experimental

results from Magnusson9.

0

5

10

15

0 1 2 3 4 5 6

0

5

10

15

0 0.2 0.4 0.6 0.8

tt [MPa]

tt [MPa]

ut [mm]

ut [mm]

σs

εs

Fig. 14 Bond stress versus slip at various integration points along the bar in pull-out

tests with embedment length 360 mm. The reinforcement elements that are

marked grey reached yielding.

III-20

Cyclic pull-out tests

The steel-encased pull-out tests carried out by the authors7, briefly described in

the section “Pull-out failure”, have also been carried out with reversed cyclic loading.

The results from analyses of these experiments, using the fixed crack model for the

concrete, are shown in Fig. 15. When the elasto-plastic Rankine material model is

-60

-40

-20

0

20

40

60

-5 -4 -3 -2 -1 0 1 2 3 4 5

FEAExp.

Slip [mm] (a)

Load [kN]

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-5 -4 -3 -2 -1 0 1 2 3 4 5

Strain [‰]

Slip [mm] (b)

FEAExp.

Fig. 15 Comparison between test results and results from the analysis of one

cyclically loaded steel-encased pull-out test. (a) Load versus slip.

(b) Tangential strain in the steel tube in the middle of the zone with bond.

III-21

used for the concrete instead, only the tangential strains are affected, especially after a

few load cycles. Instead of a residual value of about 0.05 ‰, about 0.4 ‰ is obtained.

This can be compared to what is measured in the tests: about 0.25 ‰. This shows that

in order to improve the calibration of the bond model for cyclic loading, a concrete

material model better suited for cyclic loading ought to be used.

Conclusions

A new model describing bond between deformed reinforcement bars and

concrete has been developed. Since the model takes the three-dimensional splitting

effect into account, the same input parameters will result in different load-slip curves

depending on whether the reinforcement starts yielding, and on the geometry and

strength of the surrounding structure. Steel-encased pull-out tests, where the

tangential strain in the steel tubes had been measured, were used to calibrate the

model. The analyses show that the behaviour of the concrete material model for cyclic

loading, or even for the first unloading, gives a quite large influence on the tangential

strains. In order to improve the calibration of the bond model for cyclic loading, a

concrete material model better suited for cyclic loading ought to be used. There is,

however, no such model that is possible to use in these analyses implemented in the

finite element program today.

Depending on which material model was used for the concrete, slightly different

results were obtained. The use of a fixed crack model, in analyses where splitting

cracks determined the failure, resulted in larger loads than was measured in

experiments. Since the direction of the cracks was locked in the material model, large

stresses, about five times the tensile strength, could be transferred. With the use of

rotating cracks, these large tensile stresses were prevented, and the correlation

between the analyses and the experiments was improved.

Comparing with the measured response from different experiments, the

agreement is rather good. The failure mode is the same as in experiments in all of the

analyses carried out. The bond model can also describe the behaviour for cyclic

loading in a realistic way, and reasonably good agreement with experiments was

found.

III-22

References

1. TEPFERS R. A Theory of Bond Applied to Overlapped Tensile Reinforcement

Splices for Deformed Bars. Division of Concrete Structures, Chalmers University

of Technology, Publication 73:2, Göteborg 1973.

2. ENGSTRÖM B. Ductility of Tie Connections in Precast Structures. Chalmers

University of Technology, Division of Concrete Structures, Publication 92:1,

Göteborg 1992.

3. COX J. V. Development of a Plasticity Bond Model for Reinforced Concrete –

Theory and Validation for Monotonic Applications. Naval Facilities Engineering

Service Center, Port Hueneme, USA 1994.

4. ÅKESSON M. Fracture Mechanics Analysis of the Transmission Zone in

Prestressed Hollow Core Slabs. Licentiate Thesis. Division of Concrete

Structures, Chalmers University of Technology, Publication 93:5, Göteborg 1993.

5. LUNDGREN K. Modelling of Bond: Theoretical Model and Analyses. Chalmers

University of Technology, Division of Concrete Structures, Report 99:5,

Göteborg 1999.

6. TEPFERS R. and OLSSON P-Å. Ring Test for Evaluation of Bond Properties of

Reinforcing Bars. in Bond in Concrete Proceedings (CEB), Riga 1992,

pp. 1-89 - 1-99.

7. LUNDGREN K. Steel-Encased Pull-Out Tests Subjected to Reversed Cyclic

Loading. Chalmers University of Technology, Division of Concrete Structures,

Report 98:9, Göteborg 1998.

8. TNO Building and Construction Research. DIANA Finite Element Analysis,

User’s Manual release 7, Hague 1998.

9. MAGNUSSON J. Bond and Anchorage of Deformed Bars in High-Strength

Concrete. Licentiate Thesis. Division of Concrete Structures, Chalmers University

of Technology, Publication 97:1, Göteborg 1997.

10. KUPFER H.B. and GERSTLE K.H. Behaviour of Concrete Under Biaxial Stresses.

Journal of the Engineering Mechanics Division, ASCE, Vol. 99, 1973,

pp. 853-866.

11. CEB. CEB-FIP Model Code 1990, CEB Bulletin d’Information No. 213/214,

Lausanne 1993.

12. BALÁZS G. and KOCH R. Bond Characteristics Under Reversed Cyclic Loading,

Otto Graf Journal, Vol. 6, 1995, pp. 47-62.

IV-1

STATIC TESTS AND ANALYSES OF FRAME CORNERS

SUBJECTED TO CLOSING MOMENT

K. Lundgren1

ABSTRACT: Until recently, splicing of the reinforcement in frame corners had not

been allowed by the Swedish Road Administration. However, previous tests and

analyses have not revealed any disadvantages in splicing the reinforcement within a

corner. Therefore, what was considered to be the worst case was investigated here.

Tests on four frame corners with differing detailing were carried out. Furthermore,

detailed three-dimensional non-linear finite element analyses of corners were

conducted, taking into account the splitting stresses resulting from the anchorage. A

parameter study of the importance of the loading conditions was made with two-

dimensional models. The tests and analyses show that splicing the reinforcement in

the middle of the corner has advantages over placing splices outside the bend of the

reinforcement. They also indicate, in agreement with the previous analyses and tests,

that provided the splice length is as long as required by the codes, there are no

disadvantages in splicing the reinforcement within the corner of a frame.

Key words: Reinforced concrete, frame corners, splicing of reinforcement, detailing,

three-dimensional analysis, non-linear fracture mechanics, finite element analysis,

bond, splitting effects.

1) Research Assistant, Division of Concrete Structures, Chalmers University of

Technology, S-412 96 Göteborg, Sweden.

IV-2

BACKGROUND AND AIM

Until recently, splicing of the reinforcement in frame corners had not been

allowed by the Swedish Road Administration, see Vägverket (1994). For portal frame

bridges with long spans, this had led to complicated reinforcement layouts that were

hard to realise on site. Therefore, the effect of reinforcement splices within corner

regions has been studied in a project known as “Application of fracture mechanics to

concrete bridges”. Tests and analyses within that project, and also within a closely

related project “Detailing of frame corners in civil defence shelters”, have not shown

any disadvantages in splicing the reinforcement within a corner, see Plos (1995),

Johansson (1996), and Olsson (1996). In the work presented here, the aim was to

investigate what was considered to be the worst case. Tests and analyses of frame

corners with differing detailing were devised and carried out. The tests are presented

briefly here; for details see Lundgren (1999b). Since the frame corners in large portal

bridges have considerably larger dimensions than the specimens tested, a frame corner

with dimensions large enough for one splice along one of the sides of the corner was

also analysed.

TEST PROGRAM AND ARRANGEMENTS

The aim of the tests was to investigate the effect of splicing the reinforcement

within a frame corner. Accordingly, tests on four frame corners, each with a different

kind of detailing were carried out. The geometry of the detailing in the corner regions

is shown in Fig. 1. All test specimens had a reinforcement ratio (the area of the main

reinforcement divided by the area of the concrete) of about 1 %, which can be

considered to be a rather large amount of reinforcement in a bridge. All of the main

reinforcement was placed in one layer, and the distance between the main

reinforcement bars was the minimum distance allowed according to the Swedish

Standards, Boverket (1994). The reason for these choices was to investigate what was

regarded as the worst case. One corner had unspliced reinforcement, another had

IV-3

spliced reinforcement with the same splice length as required by the codes, and the

last two frame corners had spliced reinforcement with less than half of the splice

length required by the codes.

100

a) Test No. 1U,unspliced reinforcement.

10=φ 16 s56Transverse reinforcement φ10 s100

10φ 8 s56

360

53100

360r =125

b) Test No. 2L,long splice (560 mm).

c) Test No. 3S,short splice (250 mm).

250

d) Test No. 4Ss, symmetricalshort splice (250 mm).

205 155

205

155

Figure 1 Detailing of the reinforcement in the corners of the test specimens.

Measurements, compressive and transverse reinforcement in (b), (c) and

(d) were the same as in (a). Dimensions given in mm.

IV-4

The main reinforcement in all specimens was of type K500ST φ 16, with a

yield strength of 603 MPa and an ultimate strength of 707 MPa. The concrete in the

first two specimens, denoted 1U and 2L, had a compressive strength of 37 MPa, while

in the last two specimens, 3S and 4Ss, it was 33 MPa. The compressive strength was

measured on wet stored 150 mm cylinders (300 mm high). The aggregate of the

concrete had a maximum size of 16 mm.

The test specimens were reinforced and cast at the laboratory of the

Department of Structural Engineering, Chalmers University of Technology, two at a

time. These specimens were placed on roller bearings on the floor, and a hydraulic

jack was used to apply the load to one of the legs, see the test set-up in Fig. 2. The

load was measured with a load cell, and the deformations at nine points were

measured with displacement transducers as shown in the figure. The strain in the

reinforcement was measured with 20 to 26 strain gauges, according to the type of

detailing in the reinforcement.

600

1266

Teflon

6

984

5

245 180

75

3

2

1

AA

Loadgauge

Hydraulicjack

Teflon

7

720 720

A-A

Figure 2 Test set-up and measurements of deformations.

IV-5

TEST RESULTS

The measured load versus deformation in all of the tests is compared in Fig. 3.

As can be seen, the tests with unspliced reinforcement and with a long splice obtained

almost the same maximum load, 307 and 312 kN. This corresponds to the bending

capacity of the adjoining cross-sections. In the tests with short splice length, the

maximum loads were 150 and 274 kN. In all four tests, the first bending cracks were

observed when the load was about 50 to 70 kN.

The tests with unspliced reinforcement and with a long splice showed very

similar behaviour. Large bending cracks were observed, and the reinforcement began

to yield at a load of about 290 to 295 kN. Concrete then began to spall off at the inner

part of the corner, in the compressive zone. After the maximum load had been

obtained, inclined cracks in the concrete led to a sudden failure in the test with

unspliced reinforcement; the test with the long splice was interrupted before this stage.

When examining the results from these two tests, no disadvantage can be found when

the reinforcement is spliced within the corner, since the overall behaviour was about

the same.

0

50

100

150

200

250

300

350

0 20 40 60 80 10

4Ss 1U

3S

2L

δ [mm]

Load [kN]

δ

Figure 3 Measured load versus deformation in the tests.

IV-6

In test No. 3S, where the short splice was situated outside the bend of the

reinforcement, fracture of the splice limited the capacity. Splitting cracks just at the

splice were observed at a load of about 135 kN, and cracks around the splice began to

grow very fast at a load of about 140 kN. A maximum capacity of 150 kN was

obtained. To obtain fracture of the splice was the aim of the test, the main reason for

which was to see whether the analyses could predict the behaviour correctly. As

expected, the test shows that failure of a splice within a corner region can be obtained,

which is why a minimum splice length is required.

For the other test with a short splice length, No. 4Ss where the splice was

centred in the corner, only slightly less capacity than for the unspliced specimen test

was obtained. The failure of No. 4Ss was caused by a rather sudden inclined crack in

the concrete, after yielding of the reinforcement. It is uncertain whether the

compressive zone was close to its maximum capacity in this case; no spalling of the

concrete was noted here before the inclined crack led to failure. This test does not

reveal any drawbacks to splicing the reinforcement within a corner either. One reason

for the slightly lower capacity than the unspliced specimen could be that the concrete

compressive strength in this specimen was lower, 33 MPa compared to 37 MPa. This

is further discussed in the section “Comparison between tests and analyses”. This test

shows that even though less than half of the splice length required by the codes was

used, the reinforcement had anchorage enough to reach yielding. However, whether

the anchorage was great enough to also retain this yield force is uncertain, since

inclined cracks in the concrete led to failure. It also shows that splicing the

reinforcement in the middle of the corner has great advantages over splicing placed

outside the bend of the reinforcement. Two reasons for this are that the stresses in the

reinforcement are lower in the bend, and that the bend itself has a positive influence

on the anchorage of the reinforcement.

IV-7

FINITE ELEMENT ANALYSES

Frame corners with various kinds of detailing were analysed using the non-

linear finite element program DIANA. Detailed three-dimensional modelling of the

four corners tested was carried out. Since the frame corners in large portal bridges

have considerably larger dimensions than the specimens tested, it is possible to have

two (or even more) splices in series within the corner region. These larger dimensions

are not easy to test. Hence, a frame corner with dimensions large enough for a splice

along one of the sides of the corner was analysed with a three-dimensional model, see

Fig. 4. Furthermore, two-dimensional analyses were used for a parameter study on the

importance of the loading conditions. For the analyses of the large frame corner and

the two-dimensional analyses, the same material parameters as for the unspliced

specimen were used.

In all of the analyses, the concrete was modelled with a constitutive model

based on non-linear fracture mechanics. A rotating crack model based on total strain

was used for the concrete, see TNO (1998). The deformation of one crack was

smeared out over one element; i.e. the size of the characteristic length was chosen to

be about the size of one element (40 mm in the analysis of the large frame corner, and

20 mm in the other analyses).

100

2φ16=s56

780

53

r = 125

Transverse reinforcement φ10 s100

φ16 s56

40

560780

Figure 4 Detailing of the reinforcement in the large frame corner analysed.

IV-8

The hardening in compression was described by the expression of Thorenfeldt

et al. (1987) in the three-dimensional analyses; in the two-dimensional models the

concrete was instead assumed to be elastic–perfectly plastic. The reason for this was

that, otherwise, the compressive region in the two-dimensional analyses would

become much too weak, and premature failure in the compressive region would limit

the capacity. In all analyses, the compressive behaviour was assumed to be

uninfluenced by lateral cracking. Lateral confinement of the concrete was assumed to

increase the compressive strength according to the Hsieh-Ting-Chen failure surface,

Chen (1982). Lateral confinement also increased the ductility according to the model

by Selby and Vecchio, as described in TNO (1998). Necessary material data for the

concrete was calculated, according to the expressions in CEB (1993), from the

compressive cylinder strength. For the tension softening, the curve by Hordijk et al.

was chosen, as described in TNO (1998). The constitutive behaviour of the reinforce-

ment steel was modelled by the von Mises yield criterion with associated flow and

isotropic hardening.

Since the models easily become very large, thus requiring much time for

analyses, only the parts of the test specimens close to the corner were analysed. The

corner region models were subjected to bending moment, normal force and shear

force acting on the end cross-sections of the models. These moments and forces were

calculated to represent those caused by the loads on the specimens, in the

corresponding cross-sections of the frames tested. The end cross-sections were

assumed to remain plane and the nodes in each of these cross-sections were forced to

remain in straight lines, at the same relative distance from each other, throughout the

analyses. The section moments and forces were applied as pairs of forces acting on the

outer nodes of the end cross-sections. Second order effects were taken into account by

approximation, using the measured deformed geometry at maximum load. This means

that the second order effects are somewhat overestimated before the maximum load is

reached, and somewhat underestimated after this point.

IV-9

Two-Dimensional Models

In the two-dimensional analyses, the concrete was modelled with plane stress

elements, and the reinforcement was modelled with truss elements. Special interface

elements were used to model the bond between the reinforcement and the concrete.

The bond action in these analyses was approximated with a bond-slip correlation; the

splitting stresses were not included in the analyses, and yielding of the reinforcement

did not cause any decrease in the bond. The bond-slip correlation used was the one

from CEB (1993), for unconfined concrete and “other” bond conditions.

The two-dimensional models were used for a parameter study on the

significance of the loading conditions. Analyses of the tested unspliced frame corner

No. 1U were compared with analyses of a specimen with a short splice, No. 3S. Four

load cases with different amounts of shear force were studied, see Fig. 5 and Table 1.

The first load case studied, denoted A, corresponds to the load in the test; for load

case B the loading is pure bending moment; the last two, C and D, have a larger shear

force, which, for D, is asymmetrically applied to the frame corner. The analyses of the

unspliced frame corner all resulted in bending failure, which was also accompanied by

inclined cracks when the corner was subjected to shear force. The analyses of the

spliced frame corner all resulted in failure of the splice. When the maximum obtained

loads were compared for each loadcase, see Table 1, it appeared that the effect of the

splice was about the same for all of the load cases studied; it was only slightly larger

for load case B, with pure bending moment, and for load case A, which was tested.

Hence, the load case selected for testing appears to be a rather good choice. The large

frame corner was analysed for pure bending moment.

Table 1 Load cases and the results of the two-dimensional analyses

M’max [kNm]Load case M [kNm]1)

M’[kNm]2)

V [kN]

1U 3S max1U

max3S

''

MM

A 1.11 1.00 0.42 37.0 25.0 0.675

B 1.00 1.00 0 33.9 22.9 0.675

C 1.13 1.00 0.75 37.6 25.5 0.678

D 1.04 1.00 /0.92

0.23 /0.69

37.4 25.3 0.679

1) Corner moment 2) Moment in the cross-sections adjoining the corner

IV-10

Load case ALoad case BLoad case CLoad case D

VM’

M

VM’

Figure 5 Bending moment distribution of the four load cases analysed with two-

dimensional models.

Three-Dimensional Models

In the three-dimensional analyses, the concrete and the main reinforcement

were modelled with eight-node solid elements. The compressive and the transverse

reinforcement were modelled as “embedded” reinforcement, meaning that complete

interaction between the concrete and the reinforcement was assumed. The bond

mechanism between the concrete and the main reinforcement bars was described with

special interface elements by a new model, developed by the author. The model

includes the splitting stresses that result from the bond action. It also describes the

loss of bond due to splitting cracks in the concrete or due to yielding of the

reinforcement. For more information about this model, see Lundgren (1999a). The

bends of the reinforcement bars in the corners generate splitting stresses perpendicular

to the plane of the bend, see for example Nilsson (1973). When the reinforcement is

spliced, additional splitting stresses arising from the anchorage of the reinforcement

could decrease the bond capacity. By using detailed three-dimensional models,

combined with the new model for the bond, these effects could be taken into account

in the analyses.

To limit the size of the models, as many symmetry planes as possible were

used. For the unspliced frame corner, this means a thin slice, with only half of a

reinforcement bar and extending halfway to the next reinforcement bar, was cut out,

IV-11

see Fig. 6. All nodes in these symmetry planes were supported for deformations

perpendicular to the plane, thus modelling a centre slice. Further, since there is

symmetry around the centre line of the corner, only one side of the corner was

modelled. For the frame corners with spliced reinforcement, there is no symmetry line

in the reinforcement bars; instead, a slice double the width of the unspliced frame

corner was used. For the analyses of 2L and 4Ss, the symmetry around the centre line

of the corner can still be used, although the main reinforcement bars are not

symmetrical here. Instead, the deformations of the reinforcement bars at the centre of

the corner were tied to each other as shown in Fig. 7. In frame corner No. 3S, and in

the large frame corner analysed, there is not any symmetry line in the centre of the

corner. Therefore, the whole corner region was modelled as shown in Fig. 8.

(a) (b)

Figure 6 Mesh used for the analysis of the frame corner No. 1U, with unspliced

reinforcement: (a) The whole mesh; grey marked elements are the main

reinforcement modelled with solid elements and the thick lines are the

compressive and transverse reinforcement which are modelled as

embedded reinforcement; (b) A part of a cross-section.

δcδc

(a) (b)

Figure 7 Mesh used for the analysis of the frame corner No. 2L, with a long splice:

(a) The reinforcement in the model; (b) The deformations of the

reinforcement bars at the centre of the corner are tied to each other.

IV-12

(a) (b)

Figure 8 Mesh used for the analysis of the frame corner No. 3s, with a short splice:

(a) The reinforcement in the model, and (b) The complete mesh.

COMPARISON OF TESTS AND ANALYSES

Comparisons of test results and analyses for the four tests conducted are

shown in Figs. 9, 10, 13 and 15. The load obtained from the analyses of thin slices

was multiplied by the number of such slices in the real test specimen, to enable the

comparisons. As can be seen, the overall behaviour of the specimens is quite well

described. All of the analyses showed the same failure mode as in the tests. There was

bending failure with yielding of the reinforcement followed by spalling of the concrete

in the compressive zone and inclined cracks causing collapse in 1U and 2L; fracture of

the splice in 3S; and yielding of the reinforcement followed by an inclined crack

causing collapse in 4Ss. Bending cracks were obtained at a load of about 55 kN in the

analyses, i.e. about the same as observed experimentally.

Fracture Caused by Inclined Crack

It was interesting to note that the inclined cracks that caused failure in

tests 1U, 2L, and 4Ss were obtained also in the analyses. Inclined cracking is known

to be difficult to obtain when crack models with fixed directions are used. To check

whether it was the rotating crack model that allowed this behaviour to be captured, an

analysis of the unspliced corner was made, using the same crack model but with fixed

IV-13

crack directions. In this analysis, no inclined crack was obtained; instead large tensile

stresses could be transferred. This clearly shows the benefits of a rotating crack model.

δl

Load [kN]

δl, δr [mm](a)

B, B

S I

Y

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12

Exp.FEA

720720

δr

Y

SI

(b) (c)

Y

S

YY

I

S

Figure 9 Comparison of test results and analysis for test No. 1U, with unspliced

reinforcement: (a) Load versus deformation; the letter designations are

B: Bending cracks, Y: Yielding of reinforcement, S: Spalling of concrete,

and I: Inclined crack; bold for the analysis and regular for the experiment.

(b) The test specimen after the test. (c) Deformed mesh just after the

maximum load, grey marked elements indicate that the principal strain is

greater than 0.001.

IV-14

Load [kN]

δl, δr [mm]

B, B

S I

Y

Exp.FEA

Y S

I

0

50

100

150

200

250

300

350

0 5 10 15 20

720720

δl δr

Figure 10 Comparison of test results and analysis for test No. 2L, with long

splice; load versus deformation: the letter designations are B: Bending

cracks, Y: Yielding of reinforcement, S: Spalling of concrete, and

I: Inclined crack; bold for the analysis and regular for the experiment.

In the analyses of tests 1U and 2L, yielding of the reinforcement started at a

load of about 300 kN; the yielding then penetrated the reinforcement so that the whole

bar was yielding when the maximum load of about 320 kN was reached, see Figs. 9

and 10. In the tests, yielding of the reinforcement was observed at a slightly lower

load, about 295 kN, and the maximum capacity was also slightly lower in the

experiments than in the analyses. The main reason is probably that the elements used,

eight-node isoparametric solid brick elements, behave rather poorly in bending.

DIANA, the finite element program used, has options available that improve the

behaviour in bending. Unfortunately, the elements then become deficient in describing

plastic behaviour. An option with incompatible bubble displacement modes was

tested. This means that an additional incompatible strain field is added to the the

compatible displacements, see TNO (1998). When this option was used in the analysis

of the unspliced specimen, the analysis became unstable for loads of as little as 90 kN,

when a few elements in the compressive zone reached the plastic region. Therefore,

this option was not used in the analyses.

IV-15

Another trend that can be noted in the analyses of tests 1U and 2L is that the

ductility is decreased, and the inclined crack which leads to failure is obtained earlier

(for smaller deformations) in the analyses than in the tests, see Figs. 9 and 10. The

main reason for this is probably that the compressive region was too weak in the

analyses. Close to the inner corner, a stress state with compression in all three

directions is obtained. Due to this lateral confinement, the concrete can carry

compressive stresses that are higher than the uniaxial compressive capacity, and it also

becomes more ductile. The material model used should be able to describe this. There

was, however, a small bug in the program, in the first version 7.1, that affected the

post-peak behaviour for compressed concrete with lateral confinement. In the next

version of the program, this was corrected. All of the analyses were conducted with

the first version 7.1; only one analysis of test 1U was done with the corrected version

in order to compare the results. Nevertheless, the difference between the two versions

appeared to be small for the rather low levels of confining stress that were present

here, see Fig. 11. This can also be seen in Fig. 12, where results from the two versions

are compared with the experimental results of Imran and Pantazopoulou (1996) for

triaxial stress states with about the same confinement level as in the frame corners. As

can be seen, both versions of the material model used give a reasonable increase of the

capacity, while the increase of the ductility for the triaxial compressive state here is

too low. This is most likely the primary reason why the compressive region was too

weak, which, in turn, caused too low ductility of the corners in the analyses of 1U

and 2L.

Load [kN]

δl, δr [mm]

7209

720δl δr

050

100150200250300350

0 2 4 6

FEA corr.FEA

Figure 11 Comparison of results from analyses of test No. 1U, with the two different

versions of the material model.

IV-16

0

20

40

60

80

0 0.005 0.01 0.015Strain

0.02

Stress [MPa]8.6 MPa, Exp.8.6 MPa, FEA8.6 MPa, FEA corr.4.3 MPa, Exp.4.3 MPa, FEA4.3 MPa, FEA corr.Uniaxial

Confining stress

Figure 12 Compressive stress versus strain for unconfined concrete, and for two

levels of confinement. Experimental results from Imran and

Pantazopoulou (1996).

Another reason for the weak compressive region is that in compression, as

well as in tension, a failure will localise. While in tension, it seems reasonable to

assume that one crack will localise in one element (as long as bond between the

reinforcement and the concrete is taken into account); the same assumption is not so

obvious for compression. In the analyses of the frame corners tested, the compressive

failure localised in three element rows. These three rows have a total length of 75 mm.

The compressive stress versus strain curve used was from Thorenfeldt et. al (1987).

Since they have calibrated this with tests on cylinders, φ100 mm and 300 mm high,

see Tomaszewicz (1985), it would be reasonable to increase the strains in the stress-

strain diagram, after the maximum stress, by a factor of about 300/75. This would

have improved the results of the analyses. However, for this, some of the results need

to be known in advance, i.e. the number of elements where the compressive region

will localise. Moreover, it is not possible at present to combine an arbitrary stress

strain curve with the lateral confinement model by Selby and Vecchio in the finite

element program DIANA, unless a complete material model is programmed by the

user; this is why it was not done here.

When examining the results from the analysis of test No. 4Ss, the symmetric

short splice, the maximum load was again found to be overestimated, see Fig. 13,

probably due to the poor behaviour of the elements in bending. Still, the overall

IV-17

behaviour is the same as in the tests, with an inclined crack in the concrete and,

thereafter, spalling of the concrete at the inner part of the corner. The failure mode is

thus slightly different from that in tests 1U and 2L, where spalling of the concrete

reduced the cross-section, after which the inclined crack caused collapse. This

difference could be due either to the lower concrete strength in test No. 4Ss, or to the

shorter length of the lap splice. The slip at the free end of the reinforcement bar was

about 1.0 mm, which can be compared with about 0.7 mm in the analysis of test

No. 2L. The local bond versus slip at the end of the splice, obtained from the analysis

of No. 4Ss, is shown in Fig. 14. As can be seen, there was a rather large scatter in the

bond stresses, caused by the various levels of confinement around the bar. The main

part of the bond stresses was carried at the sides between the bars, since the

compressive struts of the bond action from the reinforcement bars nearby provided

enough lateral confinement to keep the structure together. A crack splitting the cross-

section into two parts at the level of the reinforcement reduced the capacity in the

other directions. This splitting crack, which started from the reinforcement bars,

reached the symmetry lines at about the maximum load. The same effects were noted

in the analysis of test No. 2L. These cracks were also observed in the experiments to

follow the reinforcement bars.

Exp.FEA

Load [kN]

δl, δr [mm]

B, B

Y

IS

I

050

100150200250300350

0 1 2 3 4 5

SY

720720

δl δr

Figure 13 Comparison of test results and analysis for test No. 4Ss, with a symmetric

short splice: (a) Load versus deformation; the letter designations are

B: Bending cracks, Y: Yielding of reinforcement, S: Spalling of concrete,

and I: Inclined crack; bold for the analysis and regular for the experiment.

IV-18

01020

0 0.5 1 1.5

01020

0 0.5 1 1.5

01020

0 0.5 1 1.501020

0 0.5 1 1.5

01020

0 0.5 1 1.5

01020

0 0.5 1 1.5

01020

0 0.5 1 1.501020

0 0.5 1 1.5(a)

0

4

8

12

16

0 0.5 1 1.5

Bond stress [MPa]

Slip [mm](b)

Figure 14 Results from the analysis of test No. 4Ss, with a short symmetric splice:

(a) Local bond stresses (in MPa) versus slip (in mm) close to the free end

of the spliced reinforcement bar; grey marked elements indicate that the

principal strain is larger than 0.001; (b) Average of the local bond stresses

around the reinforcement bar, grey marked region indicates the variation

in the results.

IV-19

Fracture of the Splice

In the analysis of test No. 3S, cracks splitting the cross-section into two parts

at the level of the reinforcement developed from the spliced reinforcement bars, and

reached the symmetry lines at a load about 120 kN. The load could, however, continue

to increase, while bond could still be carried at the sides between the reinforcement

bars in the same way as shown in Fig. 14. When a maximum load of about 200 kN

was obtained, the capacity of the splice was reached, see Fig. 15. In the experiment,

the cracks around the splice were noted first when the load was about 135 kN, and

maximum load was obtained at 150 kN. On the other hand, in the test there was a free

edge, on both sides of the test specimens, where it is not possible to carry bond

stresses in the directions between the reinforcement bars. Hence, another analysis was

done with the same mesh, but the boundary conditions were changed so that one of the

symmetry planes between two bars became a free edge. The splitting crack was then

formed at a load corresponding to about 110 kN for the whole specimen, i.e. a load

only slightly lower than that of the centre slice. The main difference between the two

analyses was that, when there was a free edge, the load could not be increased much

after the splitting crack had formed; the maximum load obtained was only 128 kN. As

can be expected, the test results lie between the two analyses, see Fig. 15, which

means that the fracture of the splice is well described by the analyses.

IV-20

0

50

100

150

200

250

0 5 10

Load [kN]

δl + δr [mm](a)

B, B

SS Exp.

FEA, edgeFEA, centre

720720

δl δr

(b) (c)

S

Figure 15 Comparison of test results and analysis for test No. 3S, with a short splice:

(a) Load versus deformation; the letter designations are B: Bending

cracks, and S: Splitting crack; bold for the analysis and regular for the

experiment. (b) The test specimen after the test. (c) Deformed mesh just

after the maximum load, grey marked elements indicate that the principal

strain is greater than 0.001. From the analysis of a centre part.

IV-21

Large Frame Corner

When assessing the results from the analysis of the large frame corner with

spliced reinforcement, the reinforcement was found to be yielding when the applied

bending moment was 140 kNm; the maximum bending moment obtained was

156 kNm for a slice 56 mm wide, see Fig. 16. This corresponds to the bending

capacity of the adjoining cross-sections. When the maximum load was obtained, an

inclined crack in the centre of the corner formed, as marked in Fig. 16(b). After that, it

was not possible to continue the analysis, since it became unstable.

Y

δl [mm](a)

0

40

80

120

160

0 1 2 3 4 5

Moment [kNm]

δl δr

1230 1230

B

I

(b)

I

Figure 16 Results from the analyses of the spliced large frame corner: (a) Load

versus deformation; the letter designations are B: Bending cracks, Y:

Yielding of reinforcement, and I: Inclined crack; (b) Deformed mesh at the

maximum load, where grey marked elements indicate that the principal

strain is larger than 0.001.

IV-22

To check whether the splice was limiting, the stress in the reinforcement at the

beginning of the splice was plotted versus the slip at the end of the splice, for all of the

analyses of spliced specimens, see Fig. 17. These results show the difference between

the analyses where the splice was limiting and the others. In specimen No. 2L, where

the splice was not limiting, the slip was decreasing while the stress in the

reinforcement was unloaded, due to the spalling of the compressive zone and the

inclined crack. In specimen No. 3S, where the splice was limiting, the slip continued

to increase when the reinforcement was unloaded, as expected. Studying the analysis

of test specimen No. 4Ss, shows that the results are not so clear. The slip first

continued to increase, while the reinforcement was unloaded. This was about when

the inclined crack appeared. Eventually, spalling of the concrete on the compressive

edge caused total collapse; the slip then decreased. To conclude, in test specimen

No. 4Ss, the inclined crack might have been caused by the splice, while the total

collapse was due to the spalling of the concrete at the compressive edge of the inner

part of the corner. For the large frame corner analysed, the slip decreased when the

reinforcement was unloaded. This shows that the splice was not limiting in the large

frame corner.

0

100

200

300

400

500

600

700

0 0.5 1 1.5 2

2LLarge, inner layerLarge, outer layer4Ss3S, centre3S, edge

Slip [mm]

Stress [MPa]

Figure 17 Results from the analyses of all the spliced frame corners: Stress in the

reinforcement at the beginning of the splice versus slip at the free end.

IV-23

CONCLUSIONS

Tests and analyses of frame corners with differing detailing subjected to

closing moments were carried out. The results from the finite element analyses show

that the overall behaviour of the specimens could be quite well described. The failure

mode in the analyses was the same as in the tests. However, some details are worth

noting. When the ductility of the specimen was determined by the compressive region,

the ductility was too low in the analyses, since the compressive region was too weak.

The triaxial compressive stress state at the inner corner increased the compressive

strength and ductility of the concrete. In the material model used, the increase of the

stress appeared to be reasonable, although the increase of the ductility seemed to be

lower than test results. Another problem was that the elements used behaved poorly in

bending. It was not possible to improve the behaviour in bending without causing

problems with the plastic behaviour. However, fracture of the splice was shown to be

described realistically in the analyses; the use of a rotating crack model even enabled

the capture of inclined cracking, which is known to be difficult when using crack

models with fixed directions.

The results of the combined tests and analyses indicated that the behaviour of

the spliced specimens was about the same as that of the corresponding unspliced

specimen, provided the splice was long enough. In conclusion, this work shows that

splicing the reinforcement in the middle of the corner has advantages over splicing

placed outside the bend of the reinforcement. It also indicates, in agreement with

previous analyses and tests, that there are no disadvantages in splicing the

reinforcement within the corner of a frame.

IV-24

APPENDIX. REFERENCES

Boverket (1994). Boverkets handbok om betongkonstruktioner BBK 94, Band 1 –

Konstruktion (Boverket’s Handbook for Concrete Structures BBK94, Vol. 1:

Design. In Swedish), Boverket, Byggavdelningen, Karlskrona, Sweden,

185 pp.

CEB (1993). CEB-FIP Model Code 1990. Bulletin d’Information 213/214, Lausanne,

Switzerland.

Chen, W.F. (1982). Plasticity in Reinforced Concrete. McGraw-Hill, New York.

Imran, I. and Pantazopoulou, S. J. (1996). “Experimental Study of Plain Concrete

under Triaxial Stress”. ACI Materials Journal, V. 93, No. 6, Nov.-Dec.1996,

pp. 589-601.

Johansson, M. (1996). “New Reinforcement Detailing in Concrete Frame Corners of

Civil Defence Shelters: Non-linear Finite Element Analyses and Experiments.”

Licentiate Thesis, Publication 96:1, Division of Concrete Structures, Chalmers

University of Technology, Göteborg.

Lundgren, K. (1999a). “Modelling of Bond: Theoretical Model and Analyses.”

Report 99:5, Chalmers University of Technology, Division of Concrete

Structures, Göteborg.

Lundgren, K. (1999b). “Static Tests of Frame Corners Subjected to Closing

Moments.” Report 99:2, Chalmers University of Technology, Division of

Concrete Structures, Göteborg.

Nilsson, I. (1973). “Reinforced concrete corners and joints subjected to bending

moment.” Ph.D. Thesis, Document D7:1973, National Swedish Institute for

Building Research, Stockholm.

Olsson, M. (1996). “Olinjär finit elementanalys av ramhörn i armerad betong” (Non-

linear Finite Element Analyses of Frame Corners in Reinforced Concrete. In

Swedish). Diploma work 96:6, Division of Concrete Structures, Chalmers

University of Technology, Göteborg.

Plos, M. (1995). “Application of Fracture Mechanics to Concrete Bridges - Finite

Element Analyses and Experiments”. Ph.D. Thesis, Publication 95:3, Division

of Concrete Structures, Chalmers University of Technology, Göteborg.

IV-25

Thorenfeldt, E., Tomaszewicz, A., Jensen, J. J. (1987). “Mechanical Properties of

High-Strength Concrete and Application in Design”. In: Proceedings

Utilization of High Strength Concrete, Symposium in Stavanger, Norway.

Tapir N-7034 Trondheim.

TNO Building and Construction Research (1998). DIANA Finite Element Analysis,

User’s Manual release 7, Hague.

Tomaszewicz. A., (1985). “Hoyfast betong arbetsdiagram (2)”. (High Strength

Concrete; Stress versus Strain Diagram. In Norwegian). From FCBs

Informasjonsdag, Trondheim, 25 October, 1985, Forskningsinstituttet for

Cement og Betong.

Vägverket (1994). Bro 94 – Allmän teknisk beskrivning för broar. (Bridge 94 –

General Technical Description for Bridges. In Swedish). Swedish Road

Administration, Borlänge.

Doctoral Theses and Licentiate Theses at the Division of Concrete

Structures, Chalmers University of Technology, 1990 - 1999

90:1 Stig Öberg: Post Tensioned Shear Reinforcement in Rectangular RC Beams.

Publication 90:1. Göteborg, April, 1990. 603 pp. (No. 1021). Doctoral Thesis.

90:2 Johan Hedin: Långtidsegenskaper hos samverkanskonstruktioner av stål och

betong (Long Time Behaviour of Composite Steel Concrete Structures).

Publication 90:2. Göteborg, August, 1990. 53 pp. (No. 1079). Licentiate

Thesis.

92:1 Björn Engström: Ductility of Tie Connections in Precast Structures.

Publication 92:1. Göteborg, October, 1992. 368 pp. (Nos. 936, 999, 1023,

1052). Doctoral Thesis.

93:1 Mario Plos: Shear Behaviour in Concrete Bridges - Full Scale Shear Test.

Fracture Mechanics Analyses and Evaluation of Code Model. Publication

93:1. Göteborg, April, 1993. 70 pp. (Nos. 1088, 1084). Licentiate Thesis.

93:2 Marianne Grauers: Composite Columns of Hollow Steel Sections Filled with

High Strength Concrete. Publication 93:2. Göteborg, June, 1993. 140 pp.

(No. 1077). Doctoral Thesis.

93:4 Li An: Load Bearing Capacity and Behaviour of Composite Slabs with

Profiled Steel Sheet. Publication 93:4. Göteborg, September, 1993. 134 pp.

(No. 1075). Doctoral Thesis.

93:5 Magnus Åkesson: Fracture Mechanics Analysis of the Transmission in Zone in

Prestressed Hollow Core Slabs. Publication 93:5. Göteborg, November, 1993.

64 pp. (No. 1112). Licentiate Thesis.

95:1 Christina Claeson: Behavior of Reinforced High Strength Concrete Columns.

Publication 95:1. Göteborg, June, 1995. 54 pp. (No. 1105). Licentiate Thesis.

95:2 Karin Lundgren: Slender Precast Systems with Load-Bearing Façades.

Publication 95:2. Göteborg, November, 1995. 60 pp. (No. 1098). Licentiate

Thesis.

95:3 Mario Plos: Application of Fracture Mechanics to Concrete Bridges. Finite

Element Analysis and Experiments. Publication 95:3. Göteborg, November,

1995. 127 pp. (Nos. 1067, 1084, 1088, 1106). Doctoral Thesis.

96:1 Morgan Johansson: New Reinforcement Detailing in Concrete Frame Corners

of Civil Shelters: Non-linear Finite Element Analyses and Experiments.

Publication 96:1. Göteborg, November, 1996. 77 pp. (No. 1106). Licentiate

Thesis.

96:2 Magnus Åkesson: Implementation and Application of Fracture Mechanics

Models for Concrete Structures. Publication 96:2. Göteborg, November, 1996.

159 pp. (No. 1112). Doctoral Thesis.

97:1 Jonas Magnusson: Bond and Anchorage of Deformed Bars in High-Strength

Concrete. Publication 97:1. Göteborg, November, 1997. 234 pp. (No. 1113.)

Licentiate Thesis.

98:1 Christina Claeson: Structural Behavior of Reinforced High-Strength Concrete

Columns. Publication 98:1. Göteborg, August, 1998. 167 pp. (No. 1105.)

Doctoral Thesis.