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
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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
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.
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
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.
General Technical Description for Bridges. In Swedish). Swedish Road
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
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.
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.
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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.
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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.
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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.