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).

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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.

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

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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

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

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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).

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

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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

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

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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.

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

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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.

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