Cracking Behaviour of Concrete Subjected to Restraint Forces

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Cracking Behaviour of Concrete Subjected

to Restraint Forces

Finite Element Analyses of Prisms with Different
Cross-Sections and Restraints
Master’s Thesis in the International Master’s programme Structural Engineering
HELENA ALFREDSSON
JOHANNA SPÅLS


Department of Civil and Environmental Engineering
Division of Structural Engineering
Concrete Structures
CHALMERS UNIVERSITY OF TECHNOLOGY
Göteborg, Sweden 2008
Master’s Thesis 2008:24





MASTER’S THESIS 2008:24
Cracking Behaviour of Concrete Subjected
to Restraint Forces
Finite Element Analyses of Prisms with Different Cross-Sections and Restraints
Master’s Thesis in the International Master’s programme Structural Engineering
HELENA ALFREDSSON & JOHANNA SPÅLS













Department of Civil and Environmental Engineering
Division of Structural Engineering
Concrete Structures
CHALMERS UNIVERSITY OF TECHNOLOGY
Göteborg, Sweden 2008



Cracking Behaviour of Concrete Subjected to Restraint Forces
Finite Element Analyses of Prisms with Different Cross-Sections and Restraints
Master’s Thesis in the International Master’s programme Structural Engineering
HELENA ALFREDSSON & JOHANNA SPÅLS

© HELENA ALFREDSSON & JOHANNA SPÅLS, 2008


Master’s Thesis 2008:24
Department of Civil and Environmental Engineering
Division of Structural Engineering
Concrete Structures
Chalmers University of Technology
SE-412 96 Göteborg
Sweden
Telephone: + 46 (0)31-772 1000













Cover:
Schematic view of how the concrete stress varies over the cross-section at different
distances from the load, Figure 3.32.

Chalmers Reproservice/ Department of Civil and Environmental Engineering
Göteborg, Sweden 2008




I
Cracking Behaviour of Concrete Subjected to Restraint Forces
Finite Element Analyses of Prisms with Different Cross-Sections and Restraints
Master’s Thesis in the International Master’s programme Structural Engineering
HELENA ALFREDSSON & JOHANNA SPÅLS
Department of Civil and Environmental Engineering
Division of Structural Engineering
Concrete Structures
Chalmers University of Technology

ABSTRACT
The knowledge of how to control cracking in concrete due to restraint forces is today
insufficient. The aim of this project was therefore to increase the knowledge of the
cracking behaviour of concrete subjected to restraint forces. To reach this aim finite
element (FE) analyses of concrete prisms, with different cross-sections and restraints,
were performed.
A parametric study was made to see what influence different parameters have on the
size of the effective concrete area. To calculate the effective area FE analyses of
concrete before cracking were used and it was found that the bond transfer between
concrete and reinforcement was a decisive parameter. The results from one of the
analysed sections were also compared to three expressions used in codes for
calculations on minimum amount of needed reinforcement and crack widths.
FE analyses of the cracking process were also performed. In these analyses concrete
prisms, with restraint at the short ends or continuous along the length, were subjected
to a temperature decrease. In all analyses it could be seen that skew crack indications
appeared close to free short edges or through cracks. Furthermore, it was found that
the crack distribution in the prism greatly depended on the bond transfer between
concrete and reinforcement and at the joint interface respectively.
For prisms with restraint at the short end analyses with stochastic variation of
non-uniform material properties were made and it was found that the overall cracking
behaviour was similar, independent of the randomly spread material properties.
Key words: bond transfer, continuous restraint, cracking process, effective concrete
area, finite element analyses, restraint forces, stochastic variation of
material properties, thermal strain, transmission length.


II
Sprickbeteende hos Betong som Utsätts för Tvångskrafter
Finita Elementanalyser av Prismor med Olika Tvärsnitt och Randvillkor
Examensarbete inom det internationella mastersprogrammet Structural Engineering
HELENA ALFREDSSON & JOHANNA SPÅLS
Institutionen för bygg- och miljöteknik
Avdelningen för Konstruktionsteknik
Betongbyggnad
Chalmers tekniska högskola

SAMMANFATTNING
Kunskapen om hur sprickbildning i betong, orsakad av tvångskrafter, ska beaktas är
idag otillräcklig. Målet med detta examensarbete var därför att öka kunnandet inom
detta område. För att nå målet har finita elementanalyser (FE-analyser) utförts på
betongprismor med olika tvärsnitt och olika typer av tvång.
En studie genomfördes för att undersöka hur storleken på den effektiva betongarean
påverkas av olika parametrar. För att beräkna den effektiva arean gjordes FE-analyser
av osprucken betong. Studien visade att den avgörande parametern var vidhäftningen
mellan betong och armering. Vidare jämfördes utryck från BBK 04 och Eurocode 2
som används vid beräkning av minsta armeringsbehov och sprickbredder med
resultaten från en av de analyserade tvärsnittskonfigurationerna.
FE-analyser utfördes även för att undersöka hela sprickbildningsprocessen. I dessa
analyser studerades betongprismor, som antingen hade tvång på båda kortsidorna eller
längs med ena långsidan, vilka utsattes för en temperatursänkning. I alla analyser
påträffades sneda sprickindikationer nära den fria kortsidan eller intill en
genomgående spricka. Vidare kunde det även ses att fördelningen av sprickor i
betongprismorna var starkt beroende av vidhäftningen mellan betong och armering
respektive i gjutfogen mellan ny och gammal betong.
Analyser gjordes även på betongprismor med slumpmässigt utspridda
materialegenskaper och tvång på båda kortsidorna. En jämförelse mellan dessa
analyser och analysen av ett betongprisma med homogent material visade att det
senare är tillräckligt vid analys av det övergripande sprickbeteendet.
Nyckelord: effektiv betongarea, finita elementanalyser, slumpvis positionerade
materialegenskaper, sprickbildning, temperaturtöjning, tvångskrafter,
vidhäftning, överföringssträcka.
CHALMERS Civil and Environmental Engineering, Master’s Thesis 2008:24

III
Contents
ABSTRACT I
SAMMANFATTNING II
CONTENTS III
PREFACE VII
NOTATIONS VIII
1 INTRODUCTION 1
1.1 Background 1
1.2 Aim 1
1.3 Method 1
1.4 Limitations 2
1.5 Outline of the thesis 2
2 MATERIAL BEHAVIOUR AND CRACKING PROCESS 3
2.1 Orientation 3
2.2 Concrete 3
2.3 Steel 5
2.4 Interaction between concrete and reinforcement 5
2.5 Interaction between concrete and concrete 6
2.6 Restraint forces 7
2.6.1 Introduction 7
2.6.2 Thermal strain 7
2.6.3 External and internal restraints 8
2.6.4 Short end and continuous edge restraints 9
2.6.5 Restraint degree 11
2.7 Cracking stages 12
2.8 Effective area 14
3 LINEAR ANALYSIS OF PRISMS WITH SHORT END RESTRAINT 18
3.1 Orientation 18
3.2 Type 1 19
3.2.1 FE analysis 21
3.2.1.1 Geometry 21
3.2.1.2 Material model 21
3.2.1.3 Boundary conditions and load 24
3.2.1.4 Mesh 24
3.2.1.5 Method 25
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

IV
3.2.2 Results 26
3.2.2.1 Stress distribution 26
3.2.2.2 Height of the effective area 29
3.2.3 Concluding remarks 35
3.3 Type 2 35
3.3.1 FE analysis 37
3.3.2 Results 38
3.3.2.1 Stress distribution 38
3.3.2.2 Height of the effective area 41
3.3.3 Concluding remarks 44
3.4 Comparison between the configurations 44
3.5 Comparison with the codes 47
4 NON-LINEAR ANALYSIS OF PRISMS WITH END RESTRAINT 52
4.1 Orientation 52
4.2 FE analysis 53
4.2.1 Geometry 53
4.2.2 Material model 54
4.2.3 Boundary conditions and load 56
4.2.4 Mesh 56
4.2.5 Method 57
4.3 Results 58
4.3.1 Uniform material properties 58
4.3.2 Random material properties 64
4.3.3 Comparisons 70
4.4 Concluding remarks 72
5 NON-LINEAR ANALYSIS OF PRISMS WITH CONTINUOUS EDGE
RESTRAINT 73
5.1 Orientation 73
5.2 FE analysis 74
5.2.1 Geometry 74
5.2.2 Material model 75
5.2.3 Boundary conditions and load 76
5.2.4 Mesh 77
5.2.5 Method 77
5.3 Results 77
5.3.1 Orientation 77
5.3.2 Symmetry line 78
5.3.3 Reference case 79
5.3.4 Length of the specimen 84
5.3.5 Bond transfer at joint interface 86
5.3.6 Infinite length 88
5.3.7 Influence of reinforcement 91
5.4 Concluding remarks 92
CHALMERS Civil and Environmental Engineering, Master’s Thesis 2008:24

V
6 FINAL REMARKS 96
6.1 Conclusions 96
6.1.1 Linear analysis 96
6.1.2 Non-Linear analysis 97
6.2 Further investigations 97
7 REFERENCES 98

APPENDIX A CONVERGENCE STUDY OF MESH SIZE 100
APPENDIX B RESULTS FROM LINEAR ANALYSES 102
APPENDIX C COMPARISON BETWEEN TEMPERATURE- AND
DISPLACEMENT LOAD 124
APPENDIX D CHANGE FROM QUADRATIC TO TRIANGULAR MESH 127
APPENDIX E RANDOMLY SELECTED MATERIAL PROPERTIES 129
APPENDIX F RESULTS FROM NON-LINEAR ANALYSES 138
APPENDIX G INPUT FILES FOR ADINA 196


CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

VI
CHALMERS Civil and Environmental Engineering, Master’s Thesis 2008:24

VII
Preface
In this masters project finite element analyses have been performed to investigate the
cracking behaviour of concrete subjected to restraint forces. The study was carried out
from September 2007 to February 2008 and was a co-operation between Reinertsen
Sverige AB and the Division of Structural Engineering, Concrete Structures, at
Chalmers University of Technology, Sweden.
We wish to thank our supervisors Ph.D. Morgan Johansson, at Reinertsen Sverige
AB, and Professor Björn Engström for their excellent guidance and comments
throughout the work of this thesis. Björn Engström was also the examiner. This
project was a continuation of the master thesis made by Johan Nesset and Simon
Skoglund. We would like to greatly thank them for sharing their experience, pictures
and FE files with us.
Our opponents Martin Cagner and Daniel Thorell have given us feedback and support
during the work of this project. We would like to thank them for that. Furthermore we
would like to give our gratitude to all the staff at Reinertsen Sverige AB, Göteborg,
for support and for providing a good working climate.
Finally, we would like to thank Jim Brouzoulis, Stephan Bösch and David Sjödin for
their support.
Göteborg February 2008
Helena Alfredsson & Johanna Spåls
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

VIII
Notations
Roman upper case letters
A
c
Concrete area
A
ct
Concrete area within the tensile zone before cracking
A
ef
Effective area
A
net
Net concrete area
A
s
Steel area
A
s,min
Minimum area of reinforcement steel
E
c
Modulus of elasticity for concrete
E
cm
Modulus of elasticity for concrete, mean value
E
s
Modulus of elasticity for steel
F Force, Spring force
F
node
Reaction force in the node where the load is applied
G
f
Fracture energy
H Total height of Type 2
N Normal force
N
cr
Load at cracking
N
y
Load at reinforcement yielding
R Reaction force, Restraint degree
S Total stiffness of the support
T Temperature

Roman lower case letters
a
s
Distance from concrete edge to centre of reinforcement
b Width of the cross-section
f Frequency
f
cd
Concrete compression strength, design value
f
cm
Concrete compression strength, mean value
f
ct
Concrete tensile strength
f
ct,ef
Concrete tensile strength, mean value effective at the time when the
cracks may first be expected to occur
f
cth
Concrete tensile strength, high value
f
ctk
Concrete tensile strength, characteristic value
f
ctk0.05
Concrete tensile strength, lower characteristic value
f
ctk0.95
Concrete tensile strength, upper characteristic value
f
ctm
Concrete tensile strength, mean value
f
y
Reinforcement yield strength
f
yd
Reinforcement yield strength, design value
h Height of the cross-section
h
ef
Height of the effective concrete area
k, k
c
Coefficient
k
t
Factor dependent on the duration of the load
l Length
l
cr
Critical length
l
el
Element length
l
t
Transmission length
l
t,max
Transmission length, maximum
CHALMERS Civil and Environmental Engineering, Master’s Thesis 2008:24

IX
s Slip, Spacing
s
rm
Crack spacing, mean value
s
r,max
Crack spacing, maximum
t Thickness
u Displacement
w Crack width
w
k
Crack width, characteristic value
w
u
Ultimate crack opening

Greek lower case letters
α
cT
Coefficient for thermal expansion of concrete
α
e
The ratio E
s
/E
c

ε Strain
ε
c
Concrete strain
ε
c1
Uniaxial concrete strain corresponding to σ
c1
ε
c2
Ultimate uniaxial compressive concrete strain corresponding to σ
c2
ε
cm
Strain in the concrete between cracks, mean value
ε
cT
Thermal strain
ε
ct
Concrete tensile strain
ε
c,tot
Total concrete strain
ε
cu
Ultimate concrete strain
ε
s
Steel strain
ε
sm
Steel strain, mean value
φ Bar diameter
κ
1
Coefficient
ρ Reinforcement ratio in analyses
ρ
ef
The ratio A
s
/A
ef

ρ
r
Reinforcement ratio in codes
σ Stress
σ
c
Concrete stress
σ
c1
Maximum uniaxial compressive stress
σ
c2
Ultimate uniaxial compressive stress
σ
cd
Average normal stress
σ
ct
Concrete tensile stress
σ
s
Steel stress
τ
b
Bond stress
τ
max
Bond stress, maximum
τ
fd
Mobilized shear stress
τ
fu,d
Mobilized shear stress at s = 2.0 mm

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

1
1 Introduction
1.1 Background
When movements due to intrinsic deformation of concrete are hindered restraint
forces appear. These forces are internal forces and may cause tensile stresses in the
concrete and if the stress reaches the tensile capacity of the concrete the concrete will
crack. When an open crack appears, the reinforcement becomes exposed to the
environment and may start to corrode. With time corrosion of the reinforcement
results in a decrease of the load bearing capacity of the structure, and hence, it is
important to control the cracking process since uncontrolled cracks may cause
considerable damage.
The knowledge of how to treat internal forces is limited among engineers of today and
the design procedures are mainly based on external forces. To increase the knowledge
of concrete’s cracking behaviour due to restraint forces two Master Theses,
Hirschhausen (2000) and Nesset and Skoglund (2007), have been carried out at the
Division of Structural Engineering at Chalmers. This thesis is a continuation of their
work.
1.2 Aim
The main aim of this project was to increase the knowledge of the cracking behaviour
of concrete subjected to restraint forces. To reach this main aim the project was
divided into smaller parts. The aim of the different parts was to study:
• How different parameters influence the so called effective concrete area of the
cross-section and to compare the results with existing codes and guidelines.
• The cracking behaviour of a concrete prism, with short end restraints, that is
subjected to a temperature decrease.
• What influence it has on the cracking behaviour when the prism has a
stochastic variation of non-uniform material properties.
• The cracking behaviour of a concrete prism, with continuous edge restraint,
that is subjected to a temperature decrease and to compare the results with
existing literature.
1.3 Method
The project started with a literature study of the cracking behaviour of concrete,
especially due to restraint forces. Further, different methods used in codes for
calculations of crack widths and minimum area of reinforcement were studied.
Finite element (FE) analyses were made on three different concrete prisms. How
different parameters influence the results was investigated and the results were then
compared to each other, existing codes and literature.
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

2
1.4 Limitations
All specimens in this thesis were modelled with full strength concrete (28-days
strength) and without initial stresses and strains. No time dependent effects such as
creep and shrinkage were considered and all analyses were carried out in the service
state. Further, the specimens were only subjected to restraint loading, i.e. no external
loads were considered.
1.5 Outline of the thesis
The first part, Chapter 2, gives a theoretical background to the subject. It includes a
short description of restraint forces, cracking stages and the material behaviour of
concrete and steel. In this chapter also the background to the effective concrete area is
described and how this is treated in the Swedish concrete handbook BBK 04,
Boverket (2004), and in Eurocode 2, CEN (2004).
Chapter 3 presents FE analyses of reinforced uncracked concrete prisms with two
different configurations, Type 1 and Type 2. The prisms have end restraints. In these
analyses non-linear springs are used but the title, Linear Analyses of Prism with End
Restraints, refers to the linear elastic material model used for the concrete. The
FE analyses are made with an imposed elongation and how different parameters
influence the effective area is studied. The effective area for Type 1 and Type 2 is
compared to each other and the later is also compared to BBK 04 and Eurocode 2.
FE analyses of Type 2, with a non-linear material model of concrete, are presented in
Chapter 4. In these analyses the prism is subjected to a temperature decrease instead
of an imposed elongation as used in the analyses in Chapter 3. The stresses, strains,
crack indications and the distribution of fully open cracks for both a prism with
uniform material properties and for prisms with non-uniform material properties are
studied.
In Chapter 5 a concrete specimen with continuous edge restraint, exposed to a
temperature decrease, is studied. A small parametric study is made, where the
influence of the length of the prism, the edge restraint, the reinforcement and the end
restraints are studied. The overall cracking behaviour is studied and figures of the
principal stresses, crack indications and distribution of fully open cracks are shown.
Also a comparison to existing literature is made.
Finally, in Chapter 6, conclusions of the project and suggestions of further research
are presented.
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

3
2 Material Behaviour and Cracking Process
2.1 Orientation
This chapter gives a short introduction to the cracking behaviour of a reinforced
concrete structure. The concepts; different cracking stages in concrete, restraint forces
and the difference between external load and imposed elongation, are treated. The
chapter also includes a description of the effective area of concrete and presents how
this is calculated according to different codes.
2.2 Concrete
The material behaviour of concrete is greatly dependent on whether it is subjected to
compression or tension, see Figure 2.1. The load cases studied in this thesis will
mainly cause tensile stresses and therefore the tensile side is of most interest. In
tension, generally, the concrete is considered as linear elastic before it cracks. After
cracking, though, the behaviour is non-linear and a material model based on fracture
mechanics is normally used, Plos (2000).
σ
c

ε
c
compression
tension
ε
c
t
ε
cu
σ
ct


Figure 2.1 Material model for concrete.
A mean stress-displacement relation for concrete subjected to tension is shown in
Figure 2.2a. This relation is subdivided into a stress-strain and a stress-crack opening
relation, see Figure 2.2b-c, since a stress-strain relation for the whole sequence would
be different for specimens of various lengths.

σ
c

∆l
σ
c
σ
c
ε
c

w
ε
c

w
ε
c
∙l
w
+
w
u

G
f

(
a
)

(
b
)
(
c
)

Figure 2.2 (a) Mean stress-strain relation, (b) stress-strain relation and (c) stress-
crack opening relation. Based on Plos (2000).
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

4
The area under the curve in Figure 2.2c represents the energy that is transformed
when the concrete cracks. This energy is called the fracture energy G
f
and is an
important parameter to describe the fracture behaviour of concrete.
A schematic view over the fracture development of a concrete member subjected to
tension is shown in Figure 2.3. As seen in Figure 2.3b, microcracks start to form at
local weak points when the specimen is subjected to a tensile stress. If the stress
reaches the tensile strength, microcracks connect to each other at the weakest section,
see Figure 2.3c. After the tensile stress have reached its maximum value it starts to
decrease and so does also the strain outside the fracture zone, but the deformation in
the zone increases, see Figure 2.2 and Figure 2.3d. The deformation increases until the
member is separated, and thereafter no more stresses can be transferred, see
Figure 2.2c and Figure 2.3e.
w
l + w
(
e
)

σ
c
< f
ct

l
(a)
σ
c
= 0
l + ε
c
∙l
(b)
l + ε
c
∙l + w
(d)
0 < w< w
u

σ
c
= f(w)
σ
c
= 0
w ≥ w
u

σ
c
= f
ct

w = 0
l + ε
c
∙l
(c)

Figure 2.3 Schematic view over the fracture development in a concrete specimen
subjected to tension. Based on Johansson (2000).
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

5
2.3 Steel
A general stress-strain relation of steel is shown in Figure 2.4. As seen in the figure
steel is normally regarded as linear elastic until it reaches its yielding strength f
y
.
When the yielding strength is reached the material starts to plasticize and harden for
increased imposed strain. However the studies in this thesis were carried out in the
service state, which means that only the first elastic part was of interest.
f
y

ε

σ

Figure 2.4 A general material model of steel.
2.4 Interaction between concrete and reinforcement
To be able to simulate the performance of a reinforced concrete structure it is
important to understand the behaviour of the interaction between reinforcement and
concrete. The bond stress τ
b
acting on the surface area of the reinforcement depends
on the slip s of the reinforcement bar. A general bond stress-slip relation can be seen
in Figure 2.5a. In calculations a simplified curve is used and a schematic view of such
a curve is illustrated in Figure 2.5b.
frictional phase
adhesion

crack

softening

shear-keys broken
(b)
(a)
τ
b
τ
b
s
1
τ
max
s
3
s
2
s

s

Figure 2.5 (a) General bond stress-slip relation. (b) Schematic relationship
between bond stress and slip according to CEB (1993).
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

6
In the service state it is normally only the first part (s<s
1
) of the curve that needs to be
considered. This part of the curve is described by Equation (2.1) according to
Jaccoud (1997). Only the service state is considered in this thesis.
21.0
22.0)( sfs
cmb
⋅⋅=τ
1
ss

(2.1)
where
cm
f = concrete compression strength, mean value
s = slip, should be inserted in mm
2.5 Interaction between concrete and concrete
When concrete is cast against concrete, a certain shear resistance is possible at the
interface due to concrete-to-concrete friction. This shear resistance for rough
interfaces with a shear slip approximately equal to 2.0 mm is according to
CEB (1993) described by Equation (2.2).
3/13/2
,
)(40.0
ydrcdcddfu
ff ⋅+⋅⋅= ρστ
(2.2)
where =
dfu,
τ
m潢楬楺敤⁳h敡爠獴牥獳⁡琠 s = 2.0 mm
=
cd
f concrete compression strength, design value
=
cd
σ
average normal stress

=
r
ρ
reinforcement ratio
=
yd
f design yield stress of the reinforcement which perpendicularly
intersects the interface
When the shear slip s is less than 2.0 mm Equation (2.3) and Equation (2.4) can be
used according to CEB (1993).
s
dfufd
⋅⋅=
,
5
τ
τ
=
1.0
<
s
mm (2.3)
where =
fd
τ
mobilized shear stress

=s
slip should be inserted in mm
03.03.05.0
3
,
4
,
−⋅=

















s
dfu
fd
dfu
fd
τ
τ
τ
τ

0.21.0
<

s
mm (2.4)
where
=s
slip should be inserted in mm
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

7
2.6 Restraint forces
2.6.1 Introduction
Since concrete has a relatively low tensile strength, cracks appears in almost all
concrete structures during the service state. There are two different types of load cases
that cause tensile stresses in a concrete structure; external loads and restraint forces. In
this thesis only cracks caused by restraint forces are considered.
Restraint forces are often hard to predict and may cause damage if they are not
considered when designing the structure. These forces appear when the concrete has a
need to move but is unable to move freely. The need for movement of concrete can
for example be caused by temperature changes or shrinkage. In this thesis thermal
strain and imposed deformation are the causes of movement.
2.6.2 Thermal strain
The temperature distribution over a concrete section depends on many different
variables. Ghali, et al. (2002) have listed some of these:


Geometry of the cross-section


Material properties


Weather conditions
A temperature change causes thermal strains in the structure and a corresponding need
of movement is formed. If the structure is hindered to move, stresses occur. How the
thermal strain is distributed over the cross-section can vary, see Figure 2.6.
∆ε
cT
∆ε
cT
∆ε
cT
(a)
(b)
(c)

Figure 2.6 Thermal strain distribution over the cross-section, (a) constant,
(b) linear and (c) non-linear.
Thermal strain
cT
ε
is defined according to Equation (2.5).
T
cTcT


=
α
ε
(2.5)
where =
cT
α
coefficient for thermal expansion of concrete
∆T = change of temperature
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

8
The coefficient for thermal expansion α
cT
is in Eurocode 2, CEN (2004), given as
10·10
-6
1/˚C, independently of which aggregate material that is used. For steel the
same coefficient is normally used but a more correct value would according to
Engström (2007) be 11.5·10
-6
1/˚C. In this thesis a value of 10·10
-6
1/˚C is used for
both concrete and steel.
2.6.3 External and internal restraints
There are two different kinds of restraints, external and internal, see Figure 2.7.
External restraint is when a structure is hindered to move because of its boundary
conditions. The most common case of internal restraint stresses, also called
eigenstresses, appears because of the interaction between concrete and reinforcement.
Internal restraint also occurs when the temperature or shrinkage varies non-linearly
over the section or when the cross-section consists of materials with different
properties.
F
(
a
)

(
b
)
(
c
)

(d
)
(
e
)


Figure 2.7 Example of different restrains, (a), (b) and (c) are external restraints
caused by boundary conditions, (d) is an internal restraint caused by
the interaction between concrete and reinforcement and (e) is an
internal restraint caused by different material properties.
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

9
2.6.4 Short end and continuous edge restraints
The effect of the restraint depends on where it is located. In this thesis two different
restraints are treated, a restraint at the short ends and a continuous edge restraint along
the length of a prism, see Figure 2.8. The height h of the cross-section is defined
according to Figure 2.8.
(b)
(a)
h
h

Figure 2.8 Two different types of restraints, (a) at the short ends and
(b) continuous along a long edge of the prism.
In earlier studies performed by Hirschhausen (2000) and Nesset and Skoglund (2007)
prisms with short end restraints were used, see Figure 2.8a.
Hirschhausen (2000) presents the differences in the cracking behaviour for prisms
with low or high sections. According to Hirschhausen (2000), Helmus (1990) says
that in a low member the concrete cross-section has a uniform stress distribution and
after the first through crack a new through crack appears, see Figure 2.9a. In higher
members the cross-section has a non-uniform stress distribution after a crack has gone
through the member. Instead of a new through crack skew cracks appear, see
Figure 2.9b. This because the direction of the crack surface must be perpendicular to
the principle stresses and therefore the direction of the next crack is skew against the
first through crack. Hirschhausen (2000) finally stated that the cracking process in
especially thick reinforced concrete members are a complex problem and modelled
roughly in the codes.
(a)
(b)

Figure 2.9 Development of crack pattern in concrete members with low or high
sections, Hirschhausen (2000).
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

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In Nesset and Skoglund (2007) a parametric study of a reinforced concrete member
with a low cross-section was performed. In this thesis both analytical methods and a
non-linear FE model were used. Nesset and Skoglund (2007) came to the conclusion
that the model, for external loads in BBK 04, often used by Swedish designers today
regarding the number of cracks and crack distribution in concrete structures subjected
to restraint situations, is not suitable for restrained forces.
In ACI (2007) the sequence of cracks for a prism with continuous edge restraint is
described according to Figure 2.10. As seen in the figure the first crack appears in the
middle of the prism and splits it in two parts. Thereafter the following cracks develop
in the middle of the new parts, and so on.
l/8
l/4
l/2
1
2 2
33
3
3
4 4
4 4 4 4 44
Crack sequence
l/2
l/16

Figure 2.10 Sequence of cracks for a prism with continuous base restraint
according to ACI (2007).
A similar model is presented in Pettersson (2000), see Figure 2.11. This model should
represent a wall structure and includes effect of reinforcement in the horizontal
direction. The model is subjected to a temperature change distributed according to
Figure 2.11. The numbers in this figure represents the sequence of the cracks.
l
1
2
3
4


∆T

Figure 2.11 Sequence of the cracks for a prism with continuous base restraint
according to Pettersson (2000).
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2.6.5 Restraint degree
How effectively the restraint prevents the concrete from moving is called the restraint
degree R and is defined according to Equation (2.6) to Equation (2.9),
Engström (2007). Equation (2.8) is used when there is a restraint from the short ends
and Equation (2.9) when the restraint is from embedded reinforcement. In this thesis
the specimens with short end restraint were fully restrained, i.e. R = 1.
restraint full of casein strain imposed
strain imposed actual
degreerestraint = (2.6)
cT
c
R
ε
ε
=
(2.7)









+
=
lS
A
E
E
R
c
c
c
c
c
1
σ
σ
(2.8)
where σ
c
= concrete stress
E
c
= modulus of elasticity for concrete
A
c
= area of concrete section
S = total stiffness of the supports S = N / u
N = normal force
u = total displacement of the supports
l = length of element
s
net
s
c
A
A
E
E
R
+
=
1
1
(2.9)
where E
s
= modulus of elasticity for steel
A
net
= A
c
– A
s

A
s
= steel area

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

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For a structure with continuous base restraint the restraint degree varies within the
member, see Figure 2.12. At the bottom the member shown in Figure 2.12 is fully
restrained and at the top it is more or less free to move. How the restraint degree
varies within the member depends on the relationship between the length and the
height. The longer the member is in relation to its height, the higher restraint degree it
will be within the structure.

10%
20%
40%
60%
80%
90%
90%

Figure 2.12 Variation of restraint degree within a member with a fixed bottom edge,
Jonasson et al. (1994).
2.7 Cracking stages
Cracking of a concrete element can be described in three different stages: uncracked,
crack formation and stabilised cracking, see Figure 2.13.

N

state I
state II
N
c
r

tension stiffening effect
crack formation
stabilised cracking
uncracked stage
N
y

ε

Figure 2.13 Global average response of a concrete element at various cracking
stages.
The concrete element is uncracked if the stresses has not reached the concrete tensile
strength f
ct
. During this stage both the concrete and the reinforcement have linear
elastic response and sectional analysis is carried out in state I. When the tensile stress
in the element reaches the tensile strength, the crack formation stage starts. In this
stage analyses of cracked sections are performed in state II. Since there is a
contribution to the global stiffness by the uncracked concrete between the cracks the
global stiffness is in reality higher than a state II analysis. This phenomenon is called
the tension stiffening effect, see Figure 2.13.
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Within a concrete strength class there will always be a variation of material
properties. A frequency curve of how the tensile strength varies in a concrete class
with a mean tensile strength f
ctm
is shown in Figure 2.14. The figure also shows the
lower (5% fractile) characteristic value of the tensile strength f
ctk0.05
and the upper
(95% fractile) characteristic value of the tensile strength f
ctk0.95
.

f
f
ctm
f
ctk
0
.95
f
c
t
f
ctk
0
.05

Figure 2.14 Frequency curve of tensile strength in concrete.
There is a difference between the behaviour of the structure depending on whether it
is subjected to an external force or an imposed elongation. For an external force the
elongation increases instantaneously when a crack appears, see Figure 2.15a. When a
crack appears in an element subjected to an imposed elongation, the force instead
decreases rapidly, see Figure 2.15b. For both cases the stiffness decreases after each
crack.

u
F
u
(a) (b)
F
F
F
u

Figure 2.15 Response of a reinforced concrete element subjected to (a) external
load and (b) imposed elongation.
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When a crack appears in a reinforced concrete specimen, the concrete is no longer
able to carry the force. Instead the force is taken by the reinforcement. Some part of
the force is then by bond transferred from the reinforcement bar to the concrete
between the cracks. The distance it takes from the crack until the reinforcement has
finished its transfer of force to the concrete is called the transmission length l
t
. After
this distance the force in the concrete becomes constant. A new crack develops if the
stress in the concrete at least one transmission length from the crack reaches the
tensile capacity of concrete. If the distance from one crack to another is less than two
transmission lengths, theoretically no more cracks can appear between them, see
Figure 2.16. Stabilised cracking is reached when the distances between all the cracks
in the specimen are smaller than two transmission lengths. In this stage further
elongations are taken by already existing cracks, which results in an increase of the
crack widths of existing cracks.
σ
c
f
ct
l
t,max
l
t,max
F

F

Figure 2.16 Theoretical distribution of stresses in concrete after cracking.
2.8 Effective area
Before cracking a concrete member subjected to uniform tension, with an equal
amount of reinforcement in the upper and lower part, has a uniformly distributed
tensile stress, see Figure 2.17a. After the first crack some part of the tensile stress in
the reinforcement steel is transferred to the adjacent concrete, because of the bond. It
takes a distance from the crack until the tensile stress is uniform again. If the length
between two cracks is too small, the stresses will be non-uniform in the concrete
between the cracks, see Figure 2.17b.

(
a
)

σ
ct
(
b
)
σ
ct

Figure 2.17 Distribution of tensile stresses in a reinforced concrete member
(a) before cracking and (b) after cracking.
Since the tensile stresses are non-uniform, it would be wrong to consider the concrete
in state I. Instead the cross-section between the cracks should be considered as
state II, but the contribution from the concrete around the reinforcement should be
included. To predict the area of the concrete that contributes to the overall behaviour
i.e. the effective area A
ef
is hard. How this is treated in the Swedish concrete handbook
BBK 04, Boverket (2004), and in Eurocode 2, CEN (2004), is described below.
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In the Swedish concrete handbook BBK 04 the effective area A
ef
is defined according
to Figure 2.18 and in Eurocode 2 according to Figure 2.19. The definition of the
effective area according to Eurocode 2 is in this thesis referred to as EC2-1. The
distance from the edge to the gravity centre of the reinforcement is defined as a
s
.
φ
⋅≤16
h
a
s

)
2
,2min(
h
a
s

)
2
,2min(
h
a
s


Figure 2.18 Definitions of effective area according to BBK 04 for a member
subjected to tension. Only the case studied in this thesis is illustrated.
h
)
2
,5.2min(
h
a
s

a
s

)
2
,5.2min(
h
a
s


Figure 2.19 Definitions of effective area according to Eurocode 2 for a member
subjected to tension. Only the case studied in this thesis is illustrated.
The effective area A
ef
, illustrated in Figure 2.18, is in BBK 04 used for calculations of
the mean crack spacing s
rm
, see Equation (2.10).
r
rm
s
ρ
φ
κ
⋅⋅+=
1
25.050
(2.10)
where
1
κ
㴠〮㠬⁦潲⁡⁨楧栠扯湤⁲敩湦潲捥=敮琠扡爠e
=
φ
㴠扡爠摩慭整敲Ⱐ獨潵汤⁢攠楮獥牴敤⁩渠浭=
ρ
r
= A
s
/ A
ef

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In Eurocode 2 the effective area is used for calculations of the crack width, see
Equation (2.11).
)(
max,cmsmrk
sw
ε
ε
−⋅=
(2.11)
where
max,r
s
= maximum crack spacing

sm
ε
= mean strain in the reinforcement

cm
ε
= mean strain in the concrete between cracks

s
s
s
efe
ef
efct
ts
cmsm
EE
f
k
σ
ρα
ρ
σ
εε ⋅≥
⋅+⋅⋅−
=− 6.0
)1(
,


s
σ
= stress in the reinforcement in the crack
k
t
= a factor dependent on the duration of the load

efct
f
,
= concrete mean tensile strength effective at the time when the
cracks may first be expected to occur

ef
ρ
= the ratio A
s
/ A
ef

e
α
= the ratio E
s
/ E
cm
A
ef
= effective area according to Figure 2.19
In BBK 04 the effective area is also used in the estimation of the minimum amount of
needed reinforcement in a structure exposed to restraint forces, see Equation (2.12).
cthefss
fAA ⋅≥⋅
σ
(2.12)
where A
ef
= the effective concrete area according to Figure 2.18
f
cth
= a high value for the tensile concrete strength, 1,5 f
ctk

CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

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In Eurocode 2 the effective area defined in Figure 2.19 is not used in the estimation of
the minimum amount of needed reinforcement. Instead an approach according to
Equation (2.13) is used.
A
s,min
ctefctcs
Afkk ⋅⋅⋅
=

,
σ

㈮ㄳ⤠
睨敲攠w A
s.min
= minimum area of reinforcement steel
k
c
= 1.0 for pure tension
k = a coefficient which allows for the effect of non-uniform self-
equilibrating stresses, which lead to a reduction of restrain
forces. k=1.0 for webs with h

㌰ね3⁡湤㴰⸶㔠景爠
h≥800mm, intermediate values may be interpolated
A
ct
= concrete area within the tensile zone before cracking
If Equation (2.13) from Eurocode 2 is compared to Equation (2.12) from BBK 04, an
alternative expression can be found for the effective area A
ef
, see Equation (2.14).
ctcef
AkkA ⋅

= (2.14)
If this expression is used the effective area varies between 65%-100% of the total
concrete area. When this expression is used in this thesis, it is denoted EC2-2.
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3 Linear Analysis of Prisms with Short End
Restraint
3.1 Orientation
As described in Section 2.6.4 the earlier studies performed by Nesset and
Skoglund (2007) considered low cross-sections. In low cross-sections the stresses are
uniformly distributed and the whole section cracks at once. In high cross-sections
however, the stresses are non-uniformly distributed and therefore an effective area
needs to be defined in order to make an analytical analysis of the cracking
performance. How the effective area ought to be defined for high cross-sections was
analysed in the present project by using the FE method performed in the commercial
general FE software ADINA (2006). ADINA stands for Automatic Dynamic
Incremental Non-linear Analysis. The results from the FE analyses were also
compared to the rules given in the Swedish handbook BBK 04, Boverket (2004), and
Eurocode 2, CEN (2004).
In the analyses the concrete area around one reinforcement bar was modelled. The
definitions used in the studies are presented in Figure 3.1. Hence, the width b of the
prism is the same as the spacing s between the reinforcement bars. The distance from
a free edge to the centre of a reinforcement bar is denoted a
s
.
b
s
h
a
s

Figure 3.1 Definitions for spacing s, width b, height h and the edge distance a
s
.
The analyses were performed for two different configurations. The first configuration,
denoted Type 1, was used for a parametric study. With the results from Type 1 a more
realistic configuration, denoted Type 2, was created. Type 2 was used in both a small
parametric study and in a comparison with the codes. Also a comparison was made
between Type 1 and Type 2.
The analysis in this chapter is denoted as linear analysis. Hence, concrete is
considered as linear elastic until it cracks. Although the analysis performed in this
chapter was not linear, since the interface between concrete and reinforcement was
modelled with non-linear springs.
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3.2 Type 1
Configuration Type 1 was based on the model used by Nesset and Skoglund (2007)
but with various heights of the cross-section, even high sections. In their case they
always studied a rather small cross-section that should represent a part of a larger
structure, see Figure 3.2. Type 1 was studied to get a better understanding of how
different parameters influence the effective area of the concrete section.


Figure 3.2 The case studied in Nesset and Skoglund (2007).
Type 1 was modelled to simulate a reinforced concrete prism with fixed boundaries at
the short ends that was subjected to a uniform temperature decrease. The temperature
decrease was modelled as an imposed displacement. To be able to study the effective
area of the specimen a crack must appear. When a crack appears all stresses in this
section have to be taken by the reinforcement. Therefore the specimen was modelled
with one fixed edge and one free edge with the imposed displacement applied to the
reinforcement, see Figure 3.3, i.e. in this model it was assumed that the first crack had
already appeared.


Figure 3.3 Modelled case.
In Type 1 the reinforcement bar is always centrically placed. Hence, if the height of
the cross-section is varied, the thickness of the concrete cover will change. For high
cross-sections this leads to a model that is hard to compare with a real structure.
However, the aim is not to simulate a real structure, but rather to study and better
understand the effects different parameters have on the stress distribution prior to
cracking.
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The material properties and geometry of the cross-section for the reference case is
defined in Figure 3.4.
b
h
φ

l
Material:
Concrete C30/37:
f
ctm
= 2.9 MPa,
E
cm
= 33 GPa

Reinforcing steel B500B:
f
y
= 500 MPa
Dimensions:
A
c
= h ∙ b = 0.80 ∙ 0.08 m
2

φ
‽‱㘠浭=
l = 1 m
A
c

Figure 3.4 Geometry and material properties for Type 1 (reference case).
The following parameters were varied in the parametric study with underlined values
denoting the reference case:


Height h of the cross-section.
- 0.10, 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80
, 0.90, 1.00 m


Width b of the cross-section, i.e. spacing s of the reinforcement bars.
- 60, 80
, 100 mm


Concrete strength class.
- C30/37
, C40/50


Reinforcement ratio ρ.
- was kept constant for each height by varying the width of the section


Diameter φ of the reinforcement bars.
- 12, 16
, 20 mm


Modulus of elasticity for concrete E
cm
.
- 33
, 43 GPa
Many of the parameters above have an influence on the reinforcement ratio. The
reinforcement ratio decreases for example when the width of the cross-section
increases or when the diameter of the reinforcement bar decreases. Another important
parameter is the bond between reinforcement and concrete. This parameter depends
on the concrete strength class and the diameter of the reinforcement bar. A change of
concrete strength class results in that both the modulus of elasticity and the tensile
strength of concrete changes.
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3.2.1 FE analysis
3.2.1.1 Geometry
Because of symmetry it was sufficient to model only one fourth of the cross-section in
the analyses, see Figure 3.5. This was made to utilize the computer capacity fully. The
geometry used in the FE analyses for the specimen can be seen in Figure 3.6.
z
x


Figure 3.5 Modelling of the cross-section.
y
Reinforcement

Figure 3.6 Geometry of the specimen.
3.2.1.2 Material model
The behaviour of the concrete was modelled to be linear-elastic. This was made since
the specimen was only analysed until the first crack appeared and, as described in
Section 2.2, the response of concrete is considered as linear-elastic until it cracks. The
concrete was modelled with four node 2D-solid plane stress elements.
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

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The reinforcing steel was modelled as bilinear, see Figure 3.7, and two node truss
elements were used. A minor strain hardening of the steel material was included to
avoid numerical problems.
10; 0
12.5; 502
0; 0
2.5; 500
0
100
200
300
400
500
600
0 2 4 6 8 10 12 14
Steel strain, ε
s
[‰]
Steel stress, σs
[MPa]

Figure 3.7 Material model for reinforcing steel B500B.
To describe the bond behaviour between the reinforcement and the concrete in the
studied specimen a non-linear spring was used. The behaviour of the spring was based
on the bond stress-slip relation described in Section 2.4. The spring force F can be
derived from the bond stress that acts on the interface area, see Equation (3.1). The
interface area depends on the bar circumference and the element length l
el
. The
springs nearest to the short edges were modelled with one half of the spring force,
since they were only affected by half of the element length.
elb
lF ⋅

⋅=
4
φ
π
τ (3.1)
where τ
b
= bond stress
l
el
= element length
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In analysis of Type 1 the spring force was divided by four since only one fourth of the
reinforcement bar was modelled in ADINA. How the force depends on the slip in
Type 1 is seen in Figure 3.8. The bond behaviour was assumed to be the same in
tension and compression, and the springs were therefore modelled with the same
properties irrespective of the loading direction.
1.00; 2.63
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.2 0.4 0.6 0.8 1.0
Slip, s [mm]
Force, F [kN]
7.00; 1.05
1.00; 2.63 3.00; 2.63
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 2 4 6 8 10
Slip, s [mm]
Force, F [kN]

Figure 3.8 Bond force relationship for element length 25 mm and reinforcement
bar φ16.
In ADINA non-linear springs should act in the direction of the loading. To be able to
use the springs in the analyses, the truss elements were displaced at a small distance in
the longitudinal direction compared to the 2D-solid elements, see Figure 3.9.

truss element
spring element
2D-solid
element
element length, l
el


Figure 3.9 Schematic view over the different elements in the FE model. Note that
the truss elements were only displaced in the horizontal direction. The
vertical displacement was only made in this figure to better visualize
the spring elements. Nesset and Skoglund (2007).
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3.2.1.3 Boundary conditions and load
The boundary at the short end of the specimen was assumed to be fully fixed. A stress
concentration might appear when the size of the cross-section changes due to
deformations, see Figure 3.10a. To avoid this stress concentration the specimen was
modelled as fully fixed for the reinforcement and the first node of the concrete. The
rest of the concrete was free to move in the z-direction and only locked in the
y-direction, see Figure 3.10b. In order to get symmetry the reinforcement and all the
nodes at the top of the concrete were locked in the z-direction.
y
z
(a)
(
b
)
Stress concentration

Figure 3.10 Different behaviour due to the chosen boundary conditions. Alternative
(b) was chosen.
To be able to simulate how the specimen acts when it is subjected to restraint forces
an imposed elongation was used as load. This displacement was applied to the
reinforcement, see Figure 3.6, and increased until the concrete in one point reached its
tensile capacity f
ctm
. The imposed elongation u was expected to correspond to a
temperature decrease and was calculated according to Equation (3.2).
lu
cT
⋅=
ε
(3.2)
where
cT
ε
is calculated according to Equation (2.5)
3.2.1.4 Mesh
The ADINA version used for these analyses has a limit of 900 nodes. This had to be
taken into consideration in the choice of which element size that should be used. For
high cross-sections (h > 0.800 m) the limit of 900 nodes means that the smallest
element size possible was 0.040
×
0.040 m.
A convergence study was performed as a basis for the choice of an appropriate size of
the mesh, see APPENDIX A. This study showed that the mesh size 0.040
×
0.040 m
was small enough. However to run the analyses with a finer mesh was not more time
consuming and therefore an element size of 0.025
×
〮〲㔠m⁷慳⁵獥搠景爠瑨攠
捲潳猭獥捴楯湳⁷桥牥⁴n楳⁷慳⁰潳ii扬攠⡦潲b h


0.800 m), see Figure 3.11. In all
analyses the lengths of the elements were kept constant to 0.025 m or 0.040 m. In
sections where it was not possible to use quadratic elements the heights of the
elements were adjusted.
CHALMERS, Civil and Environmental Engineering, Master’s Thesis 2008:24

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Figure 3.11 Mesh of the reference specimen.
3.2.1.5 Method
In this study static analysis with displacement controlled procedure was used. The
iteration method Broyden, Fletcher, Goldfarb, Shanno (BFGS-Method), was chosen
since this method was used later on in the non-linear analysis.
The analyses were made for an increasing deformation until the specimen somewhere
reached the concrete tensile strength f
ctm
. At this moment the force in the node F
node

where the load was applied was determined. The effective area A
ef
was evaluated
according to Equation (3.3). The height of the effective area h
ef
was then calculated
according to Equation (3.4).
bhA
efef

= (3.3)
where h
ef
= height of the effective area
b = width of section
bf
F
h
ctm
node
ef

=
(3.4)
where F
node
= reaction force in the node where the load is applied when
cracking is reached
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3.2.2 Results
3.2.2.1 Stress distribution
When the elongation was applied to the reinforcement bar, the stresses dispersed from
the reinforcement to the concrete because of the bond. How the stress varied along the
reinforcement bar and in the concrete at the reinforcement level is for the reference
case illustrated in Figure 3.12. The numbers in the figure indicate the positions of the
cross-sections that later on will be studied in this chapter.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Horizontal distance, y [m]
Concrete stress, σc
[MPa]
0
50
100
150
200
250
300
350
400
Steel stress, σs
[MPa]
Concrete
Reinforcement
y
Reinforcement
z
1
6
5
4
3
2

Figure 3.12 Variation of stresses along the length in the reinforcement bar and in
the concrete at the level of the reinforcement (z=0) for the reference
case. The numbers are used later in this chapter.
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As seen in Figure 3.12, the concrete reached its maximum stress 0.35 m from the free
edge (point 3). The cross-section where the maximum concrete stress is reached is in
this thesis called the critical section and the distance to this section from the loaded
end is denoted the critical length l
cr
. After the critical distance the transfer of stresses
from the reinforcement to the concrete is smaller than the spread of stresses within the
concrete, and therefore the concrete stress at the level of the reinforcement bar starts
to decrease. The distance where the stress transfer between the reinforcement and the
concrete takes place is the transmission length l
t
, see Section 2.7. The transmission
length is for low cross-sections the same as the critical length, but as seen in
Figure 3.12 the distances differ somewhat for high cross-sections. Figure 3.12 also
shows that the stresses in the reinforcement and in the concrete do not reach a
constant level in the same section. This is an effect of the fact that a high cross-section
was used, and therefore the concrete stress depends not only on the horizontal
distance but also on the vertical level of the specimen, see Figure 3.13.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Horizontal distance, y [m]
Concrete stress, σc
[MPa]
z = 0.00
z = 0.05
z = 0.10
z = 0.20
z = 0.30
z = 0.40
y
Reinforcemen
t
z

Figure 3.13 Variation of stresses in the concrete along the length, at different
horizontal levels z for the reference case.
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Figure 3.14 shows a schematic view of how the stress in the concrete varied over the
cross-section at different vertical sections. The values of the stresses and how the
variation over the cross-section changed for the different sections can be seen in
Figure 3.15.
(a) (b) (c) (d)
(e)
(f)

Figure 3.14 Schematic view of how the concrete stress varied over the cross-section
at the distance (a) 0.05 m, (b) 0.15 m, (c) 0.35 m, (d) 0.40 m, (e) 0.50 m,
(f) 0.90 m from the loaded end.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Concrete stress, σ
c
[MPa]
Horizontal level, z [m]
y = 0.05
y = 0.15
y = 0.35
y = 0.40
y = 0.50
y = 0.90
1 26 5 4 3
y
Reinforcement
z

Figure 3.15 Comparison of stress distribution curves (a)-(f) in Figure 3.14. The
value of the concrete stress at number 1-6 refers to Figure 3.12 that
shows how the stress varies along the length. The lines are solid until
the concrete stress reached its maximum value.
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Close to the short end where the load was applied the concrete stresses were high next
to the reinforcement bar but small further down in the cross-section, see Figure 3.14a.
In Figure 3.14b-c the stresses start to spread over the cross-section but were still
significantly high close to the reinforcement bar. In Figure 3.14c the stresses in the
reinforcement have almost reached a constant level and only a small amount of
stresses still transfers to the concrete, see Figure 3.12 at y = 0.35 m. The concrete
stresses spread until the distribution was uniform over the cross-section, see
Figure 3.14d-f. This means that the concrete stress decreases at the level of the
reinforcement bar, which explains the behaviour shown in Figure 3.12.
The critical length was found to be longer for high cross-sections than for low ones,
but it did not change when the stiffness of the specimen was varied by increasing the
modulus of elasticity, see Figure 3.16. This indicates that the differences of the critical
length for high and low cross-sections did not depend on different concrete stiffness,
but was rather due to a better possibility to distribute the stresses in a specimen with a
higher cross-section. This means that the force needed in the active end of the bar to
introduce a second crack increased for higher cross-sections.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Horizontal distance, y [m]
Concrete stress, σc
[MPa]
h = 600 mm
E = 33 GPa
h = 800 mm
E = 33 GPa
h = 800 mm
E = 43 GPa
y
Reinforcement
z
h=600 mm
E
cm
=33 GPa
h=800 mm
E
cm
=33 GPa
h=800 mm
E
cm
=43 GPa

Figure 3.16 Concrete stress along the specimen for different cross-sectional heights.
3.2.2.2 Height of the effective area
The results and conclusions in this section are based on many analyses. In this section
some representative results are shown. For more results, see APPENDIX B. In some
of the analyses the steel yielded before the concrete reached its tensile capacity.
Therefore no results can be presented for these cases. Some of the results may also
differ slightly since the analysed values were taken for the load step closest to when
cracking was reached. All results though, were taken in the span f
ctm
± 0.05MPa.
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The height of the effective area was in the analyses defined around one reinforcement
bar according to Figure 3.17.

h
h
ef

b

Figure 3.17 Definition of the height of the effective area.
In all cases of the parametric study the same effect of the cross-section height could
be seen, i.e. the percentage of the concrete area that was effective, decreased for
increased cross-section height, see Figure 3.18. For the height 0.1 m, which was
studied by Nesset and Skoglund (2007), the entire area was always effective. In higher
cross-sections a tendency of convergence could be seen.
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Cross-section height, h [m]
Height of the effective area, hef
[m]
0%
20%
40%
60%
80%
100%
Ratio, hef
/ h
Height
of the
effective
area
hef/h

Figure 3.18 Height of the effective area for a cross-section with φ16, b=80 mm.
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To analyse what influence the bar diameter had on the effective area, the
reinforcement ratio ρ, see Equation (3.5), was kept constant for each height, by
changing the width b of the cross-section.
bh
A
A
A
s
c
s

==ρ
(3.5)
The results of these analyses showed that the effective area increased with increasing
bar diameter, see Figure 3.19. This behaviour was obtained since a small bar diameter
gives a higher bond and the transfer of the stresses was therefore more efficient than
for a larger bar diameter. When the transfer of stresses was strong the stresses could
not spread out in the section as much as when the bond transfer was weaker. Thus the
effective area became smaller.
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Cross-section height, h [m]
Height of the effective area, hef
[m]
φ20
b=125mm
φ16
b=80mm
φ12
b=45mm

Figure 3.19 Height of effective area for constant reinforcement ratio ρ for each
cross-section height.
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The heights of the effective area shown in Figure 3.19 are based on the reaction force
F
node
when the concrete stress somewhere in the specimen reached the tensile capacity
of concrete. The distances to the location of the highest concrete stress, the critical
length l
cr
, for the different heights of the cross-sections are shown in Figure 3.20.
Since the values of the maximum concrete stress were taken from the nodes the
critical length may vary with the size of one element (± 0.025 m). For cross-sections
with a height larger than 0.8 m the mesh with an element size of 0.040 m was used.
For this coarser mesh the critical length was even more approximate, and therefore the
critical lengths for the highest cross-sections are not presented. If Figure 3.20 is
compared to Figure 3.19, it is found that the critical length increases with the
cross-sectional height in a similar way as the height of the effective area.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Cross-section height, h [m]
Critical length, lcr
[m]
φ20
b=125mm
φ16
b=80mm
φ12
b=45mm

Figure 3.20 Critical length for constant reinforcement ratio ρ for each cross-section
height.
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Another way to study how the height of the effective area is influenced by the bond
between the reinforcement bar and the concrete is to change the concrete strength. In
Figure 3.21 the concrete strength class C30/37 that was used for the reference case is
compared to the stronger concrete strength class C40/50. The properties used for
C40/50 was: f
ctm
=3.5 MPa and E
cm
=35 GPa. As seen in Figure 3.21 the height of the
effective area is smaller for a higher concrete strength class. The modulus of elasticity
did not affect the concrete stress along the specimen, see Figure 3.16, and therefore
the result was caused by the higher concrete strength. According to Equation (2.1),
describing the bond, a higher concrete strength leads to a better bond. Hence, this also
supports the conclusion drawn earlier that a better bond results in a smaller height of
the effective area. The critical length corresponding to the height of the effective area
in Figure 3.21 is shown in Figure 3.22.
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Cross-section height, h [m]
Height of the effective area, hef
[m]
C30/37
C40/50

Figure 3.21 Height of the effective area for different concrete strength classes.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Cross-section height, h [m]
Critical length, lcr
[m]
C30/37
C40/50

Figure 3.22 Critical length for different concrete strength classes.
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Another parameter that influenced the height of the effective area was the width. The
results showed that a larger width, i.e. a wider spacing, gave an increased height of the
effective area, see Figure 3.23. This is logical since a thinner section gives a higher
reinforcement ratio and therefore also a lower capacity to spread stresses, which leads
to that the height of the effective area decreases. The critical length corresponding to
the height of the effective area in Figure 3.23 is shown in Figure 3.24.
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Cross-section height, h [m]
Height of the effective area, hef
[m]
b=100mm
b=80mm
b=60mm

Figure 3.23 Height of the effective area for different width, φ16.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Cross-section height, h [m]
Critical length, lcr
[m]
b=100mm
b=80mm
b=60mm

Figure 3.24 Critical length for different width, φ16.
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3.2.3 Concluding remarks
From the results described above, it was found that the bond behaviour is an important
factor to consider when deciding what height of the effective area that should be used
in the design. Parameters that influence the bond behaviour are the concrete strength,
the diameter of the reinforcement bar and the reinforcement ratio ρ. The results also
showed that the critical length increased with the height of the cross-section in a
similar way as the height of the effective area increased.
Type 1 was used in the first parametric study. The reinforcement was in this study
placed in the centre of the section. In a real structure the reinforcement is
asymmetrically placed. A specimen with a new configuration, denoted Type 2, was
created to better simulate this. For Type 2 the influence from the width will probably
be smaller since there is one more edge that hinders the stresses to spread.
3.3 Type 2
This specimen was based on a more realistic cross-section with a concrete cover
which was independent of the sectional height, see Figure 3.25 and Figure 3.26. Apart
from the change of the cross-section the conditions were the same as in Type 1, see
Section 3.2.


Figure 3.25 Modelled case for Type 2.
s
b
a
s
2h

Figure 3.26 Geometry of Type 2.
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The dimensions of the reference case for Type 2 are defined in Figure 3.27. The
height of the section was defined as 2h to simplify the comparison with Type 1.
A
c
φ

b
2h
a
s
l
Material:
Concrete C30/37:
f
ctm
= 2.9 MPa,
E
cm
= 33 GPa

Reinforcing steel B500B
:
f
y
= 500 MPa
Dimensions:
A
c
= 2h ∙ b
= 0.800 ∙ 0.080 m
2

a
s
= 50 mm
φ

= 16 mm
l = 1 m

Figure 3.27 Geometry and material properties for Type 2 (reference case).
The following parameters were varied in the parametric study with underlined values
denoting the reference case:


Height h of the cross-section.
- 0.125, 0.200, 0.250, 0.300, 0.350, 0.400



Width b of the cross-section, i.e. spacing s of the reinforcement bars.
- 80
, 100, 125 mm


Concrete strength class.
- C30/37
, C40/50


Reinforcement ratio ρ.
- was kept constant for each height by varying the spacing


Diameter φ of the reinforcement bars.
- 16
, 20 mm


Distance from concrete edge to centre of reinforcement a
s
- 50
, 75, 100 mm
As mentioned in Section 3.2 many of the parameters have an influence on the
reinforcement ratio.
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3.3.1 FE analysis
As in Type 1 only one fourth of the cross-section was used in the FE analyses to fully
utilize the computer’s capacity, see Figure 3.28. In Type 2 one half of a reinforcement
bar contributes in the analysed part which is different to Type 1, where only one
quarter of the reinforcement bar was included. This fact means that the bond acted on
a larger interface in Type 2 compared to Type 1.
z
x


Figure 3.28 Geometry used in the FE analysis.
The material properties for Type 2 were the same as for Type 1, see Section 3.2.1.2.
The bond behaviour was represented differently due to model geometry. As
mentioned above the properties of the springs differed, i.e. twice as stiff, since half of
a reinforcement bar was included in Type 2 instead of one fourth that was included in
Type 1.
The boundary conditions for Type 2 can be seen in Figure 3.29. At the right short end
almost the same boundary condition was used as for Type 1, which is described in
Section 3.2.1.3. The difference between the boundary conditions for the two
configurations is that the fixed node was moved from the level of the reinforcement
bar to the lower corner of the concrete. To get symmetry all the nodes at the bottom
side were locked in the z-direction. The reinforcement bar was free to move in the y-
direction and constrained to follow the concrete in the z-direction. The load case and
the mesh density was the same as in Type 1.
z
y


Figure 3.29 Chosen boundary conditions.
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The method used in the analyses was the same as in Type 1 except the definition of
the height of the effective concrete area. Equation (3.4), used for Type 1 was divided
with two since Type 2 includes two reinforcement bars, see Equation (3.6).
bf
F
h
ctm
node
ef
⋅⋅
=
2
(3.6)

3.3.2 Results
3.3.2.1 Stress distribution
In this section results are presented for the reference case. More results can be seen in
APPENDIX B. As seen in Figure 3.30 it takes for Type 2 a distance of 0.15 m from
the applied load until the concrete reaches its maximum stress. This distance was
considerably shorter for Type 2 than for Type 1, compare Figure 3.12 and
Figure 3.30. Because of the limiting upper edge in Type 2 a lower force was needed to
reach the concrete tensile capacity than for Type 1.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Horizontal distance, y [m]
Concrete stress, σc
[MPa]
0
50
100
150
200
250
300
Steel stress, σs
[MPa]
Concrete
Reinforcement
y
z

Figure 3.30 Variation of stresses along the length in the reinforcement bar and in
the concrete at the level of the reinforcement (z=0), for the reference
case.
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If Figure 3.31 is compared to the corresponding one for Type 1, see Figure 3.13, it is
seen that the behaviour of the stress distributions at different levels below the
reinforcement bar are similar. In Figure 3.31 it can also be seen that the stresses are
highest in the concrete above the reinforcement, i.e. for negative z-values. This was
caused by the fact that the stress distribution in this part was influenced by the
thickness of the concrete cover.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Horizontal distance, y [m]
Concrete stress, σc
[MPa]
z =-0.050
z =-0.025
z = 0.000
z = 0.025
z = 0.050
z = 0.100
z = 0.350
y
z

Figure 3.31 Variation of stresses in the concrete along the length at different
horizontal levels z for the reference case. The thicker line (z=0)
represents the concrete at the same level as the reinforcement bar.
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Figure 3.32 illustrates how the stress varied over the height of the cross-section in
different vertical sections. The distribution of the stresses below the reinforcement
was similar as for Type 1. As mentioned earlier the stress was highest at the top of the
specimen. When the transfer of stresses from the steel to the concrete had finished, the
concrete stresses were uniformed successively over the cross-section, see
Figure 3.32d-f. The values of the stresses and a comparison of the stress distributions
at different vertical sections are found in Figure 3.33.
(
a
)

(
b
)

(
c
)

(
d
)
(
e
)
(
f
)

Figure 3.32 Schematic view of how the concrete stress varies over the cross-section
at the distance (a) 0.05 m, (b) 0.10 m, (c) 0.15 m, (d) 0.30 m, (e) 0.50 m,
(f) 0.90 m from the load.
-0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Concrete stress, σ
c
[MPa]
Horizontal level, z [m]
y = 0.05
y = 0.10
y = 0.15
y = 0.30
y = 0.50
y = 0.90
y
z

Figure 3.33 Comparison of the stress distribution curves (a)-(f) in Figure 3.32. The
lines are solid until the concrete stress reaches its maximum value.
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3.3.2.2 Height of the effective area
The height of the effective area was defined around one reinforcement bar, see
Figure 3.34. This means that half of the total height of Type 2 should be compared to
the height in Type 1.

h
h
ef

b

Figure 3.34 Definition of the effective area for Type 2.
As for Type 1 the height of the effective area in Type 2 in percentage of the height h
decreased for increased cross-sectional heights, see Figure 3.35. However, in
Figure 3.35 it can be seen that the height of the effective area converges to a value
around 0.17 m for cross-sections higher than 0.25 m. This convergence is probably an
effect of one more limiting edge in Type 2. The height of the effective area had a top
value for the cross-section with the height of 0.30 m. This maximum value is due to
that the maximum stress was found at the top of the specimen instead of at the level of
the reinforcement.
0.00
0.05
0.10
0.15
0.20
0.25
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Half cross-section height, h [m]
Height of the effective area, hef
[m]
0%
20%
40%
60%
80%
100%
Ratio, hef
/ h
Height of
the
effective
area
hef/h

Figure 3.35 Height of the effective area for a cross-section with φ16, b=80 mm.
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If the bar diameter was changed but the reinforcement ratio was kept constant by
varying the specimen width it was found that a larger bar diameter gave a larger
height of the effective area, see Figure 3.36. This is, as described in Section 3.2.2.2,
reasonable since the bond transfer, is more efficient with a small bar diameter than a
large. Therefore, the stresses in the concrete will not be able to spread as much. As
seen in Figure 3.36 the height of the effective area was still increasing for h greater
than 0.25 m but the change was small. The critical length corresponding to the height
of the effective area in Figure 3.36 converges to a constant value in a similar way as
the height of the effective area, see Figure 3.37.
0.00
0.05
0.10
0.15
0.20
0.25
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Half cross-section hei
g
ht, h
[
m
]
Height of the effective area, hef
[m]
φ20
b=125mm
φ16
b=80mm

Figure 3.36 Height of the effective area for constant reinforcement ratio for each
cross-section height.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Half cross-section height, h [m]
Critical length, lcr
[m]
φ20
b=125mm
φ16
b=80mm

Figure 3.37 Critical length for constant reinforcement ratio for each cross-section
height.
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In Figure 3.38 the width of the specimen is changed, while the diameter of the
reinforcement bar is kept constant, which means that the reinforcement ratio varies.
However the height of the effective area started to converge at a height h of 0.25 m
independently of the width, but it converged to different values. The height of the
effective area converges to a higher value for larger widths, which seems reasonable
since an increased width, i.e. a smaller reinforcement ratio, results in a larger
possibility for the stresses to spread. Also in this figure the maximum values were
caused by the fact that the maximum stress was found in different points. The critical
length corresponding to the height of the effective area in Figure 3.38 are shown in
Figure 3.39.
0.00
0.05
0.10
0.15
0.20
0.25
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Half cross-section height, h [m]
Height of the effective area, hef
[m]
b=100mm
b=80mm

Figure 3.38 Height of the effective area for different width, φ16.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Half cross-section height, h [m]
Critical length, lcr
[m]
b=100mm
b=80mm

Figure 3.39 Critical length for different width, φ16.
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An increase of the distance from the edge to the centre of the reinforcement a
s

resulted in a larger height of the effective area, see Figure 3.40. This result is
reasonable since an enlargement of the distance a
s
increases the possibility for the
stresses to spread.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Half cross-section height, h [m]
Height of the effective area, hef
[m]
as=100 mm
as=75 mm
as=50 mm

Figure 3.40 Height of the effective area for different distance a
s
.
3.3.3 Concluding remarks
From the results of Type 2 it was found that the height of the effective area converged
to a constant value when the height of the cross-section increased. This behaviour was
probably caused by the non symmetrical placement of the reinforcement bar, which
limited the possibilities of the stresses to spread. As for Type 1 the distance to the
critical section in Type 2 varied with the height of the cross-section in a similar way
as the height of the effective area.
3.4 Comparison between the configurations
To be able to compare the two different configurations the reinforcement ratio should
be equal. This was made by comparing the total height of Type 1 with half the height
of Type 2, see Figure 3.41.
h
h
(a) (b)

Figure 3.41 Definition of heights h used in the comparison where (a) is Type 1 and
(b) is Type 2.
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In Figure 3.42 the total height 0.4 m was used for Type 1 and 0.8 m (which according
to Figure 3.41 corresponds to h=0.4 m) was used for Type 2. This was made to be
able to compare the differences between the configurations of how the concrete stress
next to the reinforcement bar varied. The values for the concrete stress were taken
when the tensile strength of the concrete was reached somewhere in the specimen. To
reach the tensile strength a higher load was needed for Type 1 than for Type 2 and
because of this the average concrete stress became higher for Type 1. The distance,
between the load application and the point where the maximum stress was reached,
was shorter for Type 2 than for Type 1. This indicated that the next crack will appear
closer to the load for Type 2 than for Type 1. That it took a longer distance for Type 1
to reach the maximum value seems reasonable, since the stresses have a larger
possibility to spread.
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Horizontal distance, y [m]
Concrete stress, σc
[MPa]
Type 1
Type 2

Figure 3.42 Comparison between Type 1 and Type 2 of how the concrete stress next
to the reinforcement bar varied along the specimen when the tensile
capacity was reached. The horizontal distance is the distance from the
load.
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In the analyses of Type 2 the highest cross-section studied was 0.8 m which
corresponds to a maximum height h of 0.4 m in the comparison. As seen in
Figure 3.43 the height of the effective area for Type 2 converges, while the height of
the effective area of Type 1 continuous to increase. This can, as earlier mentioned, be
explained by the fact that Type 2 has one more edge that limits the possibility for the
stresses to spread.
0.0
0.1
0.2
0.3
0.4
0.0 0.1 0.2 0.3 0.4 0.5
Height, h [m]
Height of the effective area, hef
[m]
Type 1
Type 2
h
h

Figure 3.43 Comparison of how the height of the effective area varies for Type 1
and Type 2. Note that the height is defined different for the two
configurations.
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3.5 Comparison with the codes
In Figure 3.44 a comparison is made of how the height of the effective area varies
with the height of the cross-section for Type 2, the Swedish handbook BBK 04,
Boverket (2004), and Eurocode 2, CEN (2004). For the comparison a cross-section
with a bar diameter φ of 16 mm, a width b of 80 mm and a distance a
s
of 50 mm was
used. How the effective area is calculated according to the two codes is described in
Section 2.8. In EC2-2 the total height H of the section is considered. To be able to
compare the results, half of the total height of Type 2 was used. This calculation of
the height of the effective area does not consider the position of the reinforcement bar.