The effect of reinforcement configuration on crack widths ...1108075/FULLTEXT01.pdf · The effect of reinforcement configuration on crack widths in ... crack widths, crack ... En

The effect of reinforcement configuration

on crack widths in concrete deep beams

RAHIMEH HOSSEINI AND ANITA NOLSJÖ

Master of Science Thesis

Stockholm, Sweden 2017

TRITA-BKN. Master Thesis 506, 2017

ISSN 1103-4297

ISRN KTH/BKN/EX--506--SE

KTH School of ABE

SE-100 44 Stockholm

SWEDEN

© Rahimeh Hosseini and Anita Nolsjö 2017

KTH Royal Institute of Technology

Department of Civil and Architectural Engineering

Division of Concrete Structures

i

Abstract

Reinforced concrete deep beams are known for applications in tall buildings, foundations and

offshore structures. Deep beams are structural elements with length and height within the same

magnitude and have significantly smaller thickness compared to a conventional concrete beam.

Deep beams in bending have non-linear strain distribution compared to conventional beams

where Bernoulli’s hypothesis is valid.

Crack formation is a common problem in reinforced concrete structures, which reduce the

durability of the structure. Once the concrete cracks the tension reinforcement carry the tensile

forces instead of the concrete. Therefore, the design of tension reinforcement is important since

the serviceability should be retained even after the structure cracks. The crack widths can be

limited by using proper reinforcement and one alternative is to combine tensile reinforcement

with crack reinforcement. The function of the reinforcement is to distribute the cracks over the

cross section which leads to that many smaller cracks occur instead of fewer, wider cracks.

Small cracks are seen as less of a problem compared to large cracks since larger cracks reduce

the durability significantly. For deep beams, there is at the present no well-substantiated

analysis model for how crack widths shall be calculated when having reinforcement in multiple

layers with different diameters. The use of crack reinforcement in the outer bottom layer has by

tradition been considered as a cost efficient way to achieve small crack widths.

In this work the crack width in deep beams have been analysed using the finite element program

Atena 2D. The numerical results have been verified by analytical calculations based on

Eurocode 2. The aim is to achieve reduced crack widths by analysing the combination of crack-

and tensile reinforcement compared to the case with tensile reinforcement only. Tensile

reinforcement has a larger diameter, for example ø25 mm, and crack reinforcement has smaller

diameters, often between ø10 and ø16 mm.

The result from the calculations with Atena showed that there was an improvement regarding

the reduction of crack widths when using crack reinforcement in combination with tensile

reinforcement compared to using tensile reinforcement only. However, this improvement

decreased by using reinforcement in multiple layers since a tensile reinforcement bar 1ø25 mm

needed to be replaced by approximately six crack reinforcement bars 6ø10 mm in order to

achieve the same total reinforcement area. The main disadvantage was that more space was

required to place all reinforcement bars in the cross section, which reduced the lever arm. The

reduction of the lever arm resulted in a reduced capacity for the reinforcement and the cracks

might unintentionally become wider than expected. Furthermore, significant reduction of both

crack widths and reinforcement stresses were obtained when the total area for a case with 7ø25

mm was increased to 9ø25 mm. The increased total area of only tensile reinforcement ø25 mm

reduced the crack width more compared to using a combination of crack- and tensile

reinforcement, which could simplify the construction work at building sites and minimize time

consumption.

Keywords: crack widths, crack development, conventional beams, deep beams,

reinforcement stress, tensile- and crack reinforcement, minimum reinforcement,

reinforcement configuration, external loads, non-linear analysis

iii

Sammanfattning

Armerade höga betongbalkar är kända för tillämpningar i höga byggnader, grundsulor och

offshore konstruktioner. Höga balkar är konstruktionselement med längd och höjd i samma

storleksordning och har betydligt mindre tjocklek jämfört med en konventionell betongbalk.

Höga balkar i böjning har en icke-linjär töjningsfördelning jämfört med konventionella balkar

där Bernoullis hypotes gäller.

Sprickbildning är ett vanligt problem i armerade betongkonstruktioner, vilket minskar

beständigheten hos konstruktionen. När betongbalken spricker kommer armeringen att ta upp

dragkraften istället för betongen därför är utformningen av böjarmering viktig eftersom

bruksgränstillståndet bör behållas även efter att konstruktionen spricker. Sprickvidderna kan

begränsas genom att använda korrekt armering och ett alternativ är att kombinera kraftarmering

med sprickarmering. Armeringens funktion är att sprida ut sprickorna över tvärsnittet vilket

leder till att många små sprickor uppkommer i stället för färre, bredare sprickor. Små sprickor

ses som ett mindre problem jämfört med stora sprickor eftersom större sprickor minskar

beständigheten avsevärt. För höga balkar finns det för närvarande ingen välunderbyggd

analysmodell för hur sprickvidder ska beräknas när de har armering i flera lager och med olika

diametrar. Användningen av sprickarmering har traditionellt ansetts vara ett kostnadseffektivt

sätt att uppnå små sprickvidder.

I detta arbete har sprickvidden i höga balkar analyserats med hjälp av finita elementprogrammet

Atena 2D. De numeriska resultaten har verifierats med analytiska beräkningar baserade på

Eurokod 2. Syftet är att uppnå reducerade sprickvidder genom att analysera kombinationen av

sprick- och kraftarmering jämfört med fallet med endast kraftarmering. Kraftarmeringen har en

större diameter, till exempel ø25 mm och sprickarmering har mindre diametrar, ofta mellan ø10

och ø16 mm.

Resultaten från beräkningarna i Atena visade att sprickvidderna minskade vid användning av

sprickarmering i kombination med kraftarmering jämfört med användning av endast

kraftarmering. Denna förbättring minskade emellertid i och med användning av armering i flera

lager. En kraftarmeringsstång 1ø25 mm behöver ersättas med ungefär sex

sprickarmeringsstänger, 6ø10 mm, för att uppnå samma totala armeringsarea. Den största

nackdelen var att det krävdes mer utrymme för att placera alla sprickarmeringsstänger i

tvärsnittet, vilket minskade hävarmen. Minskningen av hävarmen medförde en reducerad

kapacitet i armeringen och sprickorna blev bredare än förväntat. Vidare erhölls signifikant

reduktion av både sprickvidder och armeringsspänningar när den totala arean för ett fall med

7ø25 mm ökades till 9ø25 mm. Den ökade totalarean av endast kraftarmeringsstänger ø25 mm

minskade sprickvidden mer jämfört med att använda en kombination av sprick- och

kraftarmering vilket skulle kunna förenkla byggarbetet på byggarbetsplatser och minimera

tidsförbrukningen.

Nyckelord: sprickvidder, ordinära balkar, höga balkar, armeringsspänning, kraft- och

sprickarmering, minimiarmering, armeringsutformning, externa laster, icke

linjär analys

v

Preface

This master thesis is written for the Department of Civil and Architectural Engineering at KTH

Royal Institute of Technology. The work has been performed in collaboration with WSP

Construction Design during the spring term 2017.

We would like to thank our supervisors Tech. Dr. Kent Arvidsson for his guidance and interest

throughout the work, Professor Anders Ansell for his support and advice and Adjunct Professor

Mikael Hallgren for his useful advices on the finite element software Atena.

Stockholm, June 2017

Rahimeh Hosseini & Anita Nolsjö

vii

Symbols

Latin upper case letters

𝐴𝑎 The area of the tensile reinforcement

𝐴𝑒𝑓 Effective concrete area

𝐴𝑐𝑡 The area of concrete within the tensile zone

𝐴𝑝′ Area of tendons (pre or post-tensioned)

𝐴𝑐,𝑒𝑓𝑓 Effective area of concrete in tension

𝐴𝑠 Reinforcement area

𝐴𝑠,𝑚𝑖𝑛 Minimum area of reinforcing steel within the tensile zone

𝐸 Modulus of elasticity

𝐸𝑎 Modulus of elasticity

𝐸𝑐𝑚 Mean elastic modulus of concrete

𝐸𝑠 Elastic modulus of reinforcement steel

𝐹 Force

𝐹𝑐 Concrete compressive force

𝐹𝑠 Reinforcement tension force

𝐺𝐹 Fracture energy

𝐿 Length

𝐿𝑡′ The failure bands for tension

𝑀 Moment

𝑅 Reaction force

𝑉 Shear force

𝑊𝑏 The plastic bending resistance

viii

Latin lower case letters

𝑐 Concrete cover

𝑐1 Constant that describes the curve of Hordjik’s exponential model for describing

the fracture energy

𝑐2 Constant

𝑐𝑐𝑙𝑒𝑎𝑟 Clear distance between the ribs

𝑑 The diameter of the reinforcement bar

𝑑𝑚𝑎𝑥 Maximum aggregate size

𝑒 Mathematical constant

𝑓𝑐𝑘 Characteristic concrete compressive strength

𝑓𝑐𝑚 Mean concrete compressive strength

𝑓𝑐𝑢 Mean concrete cube compressive strength

𝑓𝑐𝑡,𝑒𝑓𝑓 Mean value of the tensile strength of the concrete

𝑓𝑐𝑡ℎ High tension strength for the concrete

𝑓𝑐𝑡𝑚 Mean concrete characteristic compressive strength

𝑓𝑡 Effective concrete tensile strength

ℎ Height

𝑘 Constant

𝑘𝑐 Coefficient taking the stress distribution in the cross section before cracking in to

account and the change of level arm

𝑘𝑡 Factor dependent on the load

𝑘1 Coefficient that takes the bond properties into account

𝑘2 Coefficient that takes the distribution of strain into account

𝑘3𝑐 7ø (according to national annex)

𝑘4 0.4254

𝑙𝑠,𝑚𝑎𝑥 The length over which slip between steel and concrete occurs

𝑞 Distributed load

𝑠1 Sliding when the shear stress 𝜏𝑚𝑎𝑥 is achieved with description of adhesion

ix

𝑠2 Sliding when the shear stress begins to decline after 𝜏𝑚𝑎𝑥 is reached at description

of adhesion

𝑠3 Sliding when the shear stress drops to 𝜏𝑚𝑎𝑥 with description of adhesion

𝑠𝑟𝑚 The mean value of the crack spacing

𝑠𝑟,𝑚𝑎𝑥 Maximum crack spacing

𝜈 Coefficient that accounts for the tensioned concrete between cracks

𝑤 Crack opening

𝑤𝑐 Crack opening at complete release of stress

𝑤𝑘 Characteristic crack width

𝑤𝑚 Mean crack width

𝑧 Lever arm

Greek lower case letters

𝛼 Constant when calculating the adhesion of the reinforcement

𝛼𝑒 𝐸𝑠/𝐸𝑐𝑚

𝛽 Coefficient that accounts for the duration of the load

𝛿 The nominal maximum crack width

휀𝑐𝑟 The crack opening

휀𝑠ℎ Shrinkage

휀𝑐𝑠 Drying and autogenous shrinkage

휀𝑠𝑚 Mean strain in the reinforcement

휀𝑐𝑚 Mean strain in the concrete between cracks

𝜅1 Coefficient that accounts the adhesion of the reinforcement

𝜉1 Adjusted ratio of bond strength taking into account the different diameters of

prestressing and reinforcing steel

𝜎 Stress

𝜎𝑎 Reinforcement stress assuming the cracked concrete cross section

𝜎𝑠 Stress in the tension reinforcement when assuming a cracked concrete section

x

𝜎𝑠2 Reinforcement stress at the crack

𝜎𝑠𝐸 Reinforcement stress at the point of zero slip

𝜎𝑠𝑟 The stress in the reinforcement in the crack directly after cracking

𝜌 Reinforcement content

𝜏 Bond stress

𝜏𝑏𝑘 Lower fractile value of the mean bond stress

𝜏𝑓 Bond stress when adhesion failure occurs

𝜏𝑚𝑎𝑥 Maximum bond stress

𝜏𝑢,𝑠𝑝𝑙𝑖𝑡 Peak local bond resistance

𝜋 Mathematical constant

ø Reinforcement diameter

ø𝑒𝑞 Equivalent reinforcement diameter

ø𝑠 Reinforcement diameter

xi

Contents

1 Introduction ..................................................................................................................1

1.1 Background ..........................................................................................................1

1.2 Aim and scope ......................................................................................................2

1.3 Limitations ...........................................................................................................2

2 Reinforced concrete structures ....................................................................................3

2.1 Concrete strength and deformation in Eurocode....................................................3

2.2 Shrinkage .............................................................................................................3

2.3 Theory of concrete cracking and fracture energy ..................................................5

2.4 Reinforced concrete and reinforcement bond ........................................................8

3 Deep beams ................................................................................................................. 13

3.1 Theoretical background ...................................................................................... 13

3.2 Stress distribution in deep beams ........................................................................ 13

3.3 Experimental investigations of deep beams ........................................................ 16

3.3.1 Deep beams loaded at the upper edge ..................................................... 16

3.3.2 Deep beams loaded at the lower edge ..................................................... 17

3.4 Design of deep beams ......................................................................................... 19

3.4.1 Calculation model .................................................................................. 19

3.4.2 Truss models ......................................................................................... 20

4 Method ........................................................................................................................ 23

4.1 Cracks according to Eurocode 2 ......................................................................... 23

4.1.1 Minimum reinforcement ........................................................................ 23

4.1.2 Flexural crack widths ............................................................................. 24

4.1.3 Concrete Section (WIN-Statik) .............................................................. 25

4.2 Cracks according to BBK04 ............................................................................... 25

4.2.1 Minimum reinforcement for crack control .............................................. 25

4.2.2 Calculations of crack width .................................................................... 25

4.3 Calculations of the crack width according to B7 ................................................. 26

4.4 Cracks according to Model Code 1990 ............................................................... 27

4.4.1 Minimum reinforcement for crack control .............................................. 27

4.4.2 Calculations of crack width .................................................................... 28

xii

4.5 Atena 2D ............................................................................................................ 29

4.5.1 Smeared crack model ............................................................................. 29

4.5.2 Cracks according to Atena ..................................................................... 30

4.5.3 Newton-Raphson method ....................................................................... 31

5 Prerequisite for analysed examples ........................................................................... 33

5.1 Geometry ........................................................................................................... 33

5.2 Mesh .................................................................................................................. 33

5.3 Loads ................................................................................................................. 34

5.4 Concrete properties ............................................................................................ 34

5.5 Reinforcement and position ................................................................................ 35

5.6 Theory behind the reinforcement distribution ..................................................... 36

6 Results ......................................................................................................................... 39

6.1 Verification of the results in Atena ..................................................................... 39

6.2 Deep beam with distributed load ........................................................................ 42

6.3 Deep beam with concentrated load ..................................................................... 44

6.4 Relationship between equivalent diameter and crack width ................................. 46

6.4.1 Concrete Section .................................................................................... 46

6.4.2 Atena ..................................................................................................... 47

6.5 The effect of the reinforcement diameter ............................................................ 49

6.6 A comparison of different calculation codes ....................................................... 50

6.7 Additional tensile reinforcement ......................................................................... 51

7 Discussion and conclusions ........................................................................................ 53

7.1 Verification of the results in Atena ..................................................................... 53

7.2 Load impact and crack development ................................................................... 53

7.3 Equivalent diameter ............................................................................................ 54

7.4 The relation between reinforcement and crack width .......................................... 54

7.4.1 Position of the bending reinforcement .................................................... 54

7.4.2 Minimum reinforcement ........................................................................ 55

7.5 The effect of increased total area ........................................................................ 55

7.6 Further studies .................................................................................................... 56

Bibliography ....................................................................................................................... 57

Appendix ............................................................................................................................. 59

A Control calculation ..................................................................................................... 59

xiii

B Calculation of the fracture energy ............................................................................. 65

C Calculations of the crack widths in Concrete Section and Atena ............................. 67

1.1. BACKGROUND

1

1 Introduction

Concrete deep beams are common as wall constructions in many types of buildings, but often

with significant cracks shown in the walls. The cracks are problematic because of increased

risk for moisture problems and reinforcement corrosion, which leads to reduced load carrying

capacity of the structure. When the concrete cracks the durability of the structure also reduces.

Lack of sufficient methods for analysing cracks in concrete deep beams makes it hard to

design for prevention of crack propagation (Ansell, et al., 2014).

1.1 Background

Reinforced concrete structures are relative brittle and therefore usually contain tensile

reinforcement. When the concrete cracks the tension reinforcement carry the tensile forces

instead of the concrete. With higher reinforcement content, the difference between the cracking

load and the ultimate load is expected to increase (Malm & Holmgren, 2008a). Therefore, the

design of tension reinforcement is important since the serviceability should be retained even

after the structure cracks. Cracks increase the risk for penetration of dangerous substances

which accelerate the degradation of both concrete and reinforcement (Bertolini, et al., 2013). It

is difficult to avoid the appearance of the cracks, however they can be limited e.g. by using

proper reinforcement. One alternative is to combine the tensile reinforcement with crack

reinforcement to minimize the crack width. The function of the reinforcement is to distribute

the cracks over the cross section; many smaller cracks occur instead of fewer, wider cracks.

In general, a deep beam is defined as “a structure with length and height in the same order of

magnitude” (Ansell, et al., 2014) and are an example of typical concrete structures where a

combination of tensile- and crack reinforcement is used. However, different guidelines define

deep beams differently. The ACI guideline (ACI, 1999) uses span/depth ratio less than 5 to

define concrete deep beams while Model Code 1990 (CEB-FIP, 1993) uses a span/depth ratio

less than 2.5. Eurocode 2 (SS-EN 1992-1-1, 2005) defines the deep beams using length/depth

ratio less than 3. Several researchers have investigated the behaviour of the deep beams

considering shear cracks and shear strength. In the report, Shear strength of RC deep beams

(Appa Rao, et al., 2017), the shear behaviour of deep beams has been studied. The investigation

shows that the increased depth of the beam increases the crack width, which can be reduced by

increasing the amount of shear reinforcement. However, for deep beams, there is at the present

no well-substantiated analysis model for how bending crack widths shall be calculated when

having reinforcement in multiple layers and with different diameters. The use of crack

reinforcement in the outer bottom layer has by tradition been considered as a cost efficient way

to achieve small crack widths. The outer layer consists of crack reinforcement and the layer

next to it of tensile reinforcement. The purpose is to achieve reduced crack widths compared to

with only tensile reinforcement. However, there is no standard for how to perform the

calculation in accordance with Eurocode 2 (SS-EN 1992-1-1, 2005), which is the design code

of concrete structures. Tensile reinforcement has a larger diameter, around ø25-ø32 mm, and

crack reinforcement has a smaller diameter, around ø10-ø16 mm. The content of this study

focuses on how the bending reinforcement affects the crack pattern of the beams. To optimize

1. THE EFFECT OF REINFORCEMENT CONFIGURATION ON CRACK WIDTHS IN CONCRETE DEEP BEAMS

2

this, the flexural shear force was minimized by using horizontal and vertical minimum

reinforcement, which in turn minimized the brittleness of the concrete (Arvidsson, 2016).

In Sweden, the dimensioning code for concrete structures is the Eurocode 2 in which the

formula for crack width calculations is presented. The prerequisite for Eurocode 2 is that

Bernoulli’s hypothesis is valid, i.e. the linear distribution of strain and stress in the cross-section

are retained during loading. However, for deep beams the hypothesis is not valid since larger

height in relation to the span, the larger the non-linear strain and stress distribution become. To

investigate the crack width of the deep beams, calculations were here done both numerically

and analytically. The numerical calculations were done using a finite element method program,

Atena (Cervenka Consulting, 2011), and the analytical calculations were based on Eurocode 2

using the program Concrete Section (StruSoft AB, 2015).

1.2 Aim and scope

The aim of this work is to study and explain the development of cracks in concrete deep beams

in order to investigate the most effective reinforcement configuration. Furthermore, studies are

performed to investigate how the use of crack reinforcement affects the crack behaviour, i.e. if

smaller crack widths occur for a combination of tensile reinforcement and crack reinforcement

compared to using tensile reinforcement only. The optimization of the amount of reinforcement

will improve the efficiency of construction work at building sites. The aim is also to suggest an

alternative solution that could work better for reducing the crack widths. It is not possible to

fully eliminate the appearance of cracks, but a more accurate reinforcement dimensioning could

lead to reduced risk of large cracks.

1.3 Limitations

As for all master projects, also this work had to be limited. Only cracking caused by external

loads in the serviceability limit state is included, analysed based on non-linear finite element

modelling in Atena 2D and calculations according to Eurocode 2 in Concrete Section. The study

included deep beams with dimensions 5.0×2.5×0.4 m where the cross section varied

considering reinforcement bars. The beams were considered as simply supported and cast from

concrete of class C30/37 with an aggregate size of 12 mm. These configurations were chosen

to represent a typical design for a wall structure. Furthermore, the beams have been analysed

with simple boundary conditions in an isolated environment without stress concentrations from

other surrounding structural parts that is normally present in a structure. The effect of shrinkage

and temperature loading on the material parameters were not considered. Additional parametric

studies were made regarding the best placement of the reinforcement for reducing cracks and

to find the most profitable element size. The element size of the model was selected based on a

mesh convergence analysis to a size that provides sufficiently good results without being

smaller than the biggest ballast grain. Furthermore, a convergence analysis was made using

both deep beams and conventional beams in order to investigate the different performances of

the beams regarding the crack widths.

2.1. CONCRETE STRENGTH AND DEFORMATION IN EUROCODE

3

2 Reinforced concrete structures

Concrete is a composite material that consists of cement, water, aggregate and additives. The

combination of cement and water, the cement paste, is the binder in the concrete matrix. The

aggregates consist of sand, shingle and stones, which are the filling materials. Additives are

often used to improve or achieve certain properties of the concrete. For instance,

superplasticizers are used to reduce the amount of water to achieve higher strength and lower

shrinkage without resulting in a more rigid concrete (Ansell, et al., 2014). The concrete strength

is highly dependent on the water-cement ratio. The concrete strength decreases with high water-

cement ratio and the concrete becomes more fluid while low water-cement ratio increase both

the concrete strength and the stiffness (Burström, 2006).

2.1 Concrete strength and deformation in Eurocode

Concrete is often characterized by its compressive strength which is significantly higher than

its tensile strength. The concrete’s brittleness is also the reason for the limited possibility of

fully using the material tensile strength. Reinforcement is normally used to compensate for the

poor tensile strength in concrete (Ansell, et al., 2014). Concrete is often considered as close to

fully cured after 28 days from casting, when stored at 20 °C. This is a practical limitation often

given in standards and guidelines for when the compressive strength of concrete should be

measured. However, in reality, concrete slowly continue to develop its strength for many years

after casting (Ansell, et al., 2014).

Concrete is divided into certain strength classes in the Eurocode (SS-EN 1992-1-1, 2005). One

example of a strength class is C20/25, where the first number represents the characteristic

cylinder compressive strength (at 5 % fractile) 28 days from casting and the second number the

corresponding characteristic cube compressive strength. Some examples of the different

concrete classes are listed in Table 2.1.

2.2 Shrinkage

Concrete shrinkage is a volume reduction phenomenon, which take place without any influence

from external loads. There are several types of shrinkage, namely plastic shrinkage, drying

shrinkage, autogenous shrinkage and carbonation shrinkage. Concrete behaves in a ductile

manner the first 24 hours after casting. Drying out during this time can lead to large movements,

which increase the risk of deep cracks to arise since the concrete at this stage has a low ability

to carry loads. The magnitude of the plastic shrinkage depends e.g. on the temperature, wind

speed and composition of concrete. The most common type of shrinkage is drying shrinkage

which is the result from that water evaporates from the concrete pore system causing a volume

reduction. This is usually as most notable around a month after casting and the magnitude of

the volume reduction here depends on the amount of water in the concrete and the relative

humidity of the surrounding environment. Another type of shrinkage is autogenous shrinkage

which occurs due to hydration of the cement paste. High cement content and low water cement


4

ratio increase the autogenous shrinkage but often in practice, the autogenous shrinkage can be

neglected. Furthermore, carbonation shrinkage occurs when cement reacts with water that leads

to the development of calcium hydroxide which reacts with carbon dioxide from the air, leading

to a volume reduction of the concrete. Autogenous shrinkage and carbonation shrinkage are

normally small compared to plastic shrinkage and drying shrinkage (Ansell, et al., 2014).

Table 2.1. Concrete parameters from Eurocode 2 (SS-EN 1992-1-1, 2005).

Strength class Relation

C20/25 C30/37 C40/50

𝒇𝒄𝒌 [MPa] Characteristic

Compressive

Strength

20 30 40

𝒇𝒄𝒎 [MPa] Mean

Compression

Strength

28 38 48 𝑓𝑐𝑘 + 8

𝒇𝒄𝒖 [MPa] Mean Cube Compression

Strength

33 45 56 * 𝑓𝑐𝑚/0.85

𝒇𝒄𝒕𝒎 [MPa] Mean

Characteristic

Compression

Strength

2.2 2.9 3.5 0.3𝑓𝑐𝑘2/3

𝑬𝒄𝒎 [GPa] Modulus of

Elasticity 30 33 35 22 (

𝑓𝑐𝑚

10)

0.3

*According to Model Code 1990

When the concrete structure shrinks, the movement is often restrained to different degrees and

stresses will occur. A concrete structure can have either no restraint, incomplete restraint or

complete restraint. An element that has no restraints will shrink without developing stresses due

to shrinkage, so called free shrinkage. This means that the concrete can move unhindered and

the shrinkage corresponds to the amount of volume reduction, i.e. the shrinkage times the length

휀𝑠ℎ𝐿, see Figure 2.1. An incompletely restrained concrete structure will both develop shrinkage

and tensile stresses, developed because of the prevented volume changes. However, an

incomplete restrained concrete will not shrink as much as a free (not restrained) concrete

structure and it will not develop as high stresses as a completely restrained concrete structure.

A totally restrained concrete element will develop high stresses but no global shrinkage (Ansell,

et al., 2014). According to Eurocode 2, the total stress in the concrete element is correspond to

the total shrinkage (drying- and autogenous shrinkage) times the E-modulus 𝜎 = 휀𝑐𝑠𝐸 (SS-EN

1992-1-1, 2005). In Figure 2.1 restrained and unrestrained shrinkage is shown.

2.3. THEORY OF CONCRETE CRACKING AND FRACTURE ENERGY

5

Figure 2.1. Deformations and stresses caused by shrinkage. a) Incomplete shrinkage, b)

incomplete restraint and c) complete restraint (Ansell, et al., 2014).

2.3 Theory of concrete cracking and fracture energy

Once the concrete stresses exceed the concrete strength, the concrete will crack. Cracking will

often occur where there is a natural weakness, for example air pores or existing micro cracks.

In concrete cracking theory there is a stage between uncracked and cracked concrete which is

called the “Fracture Process Zone” (FPZ). This is a transit zone between intact continuous

material and open discontinuous cracks and consists of micro cracks which are situated near

the crack tip (see Figure 2.2). When the crack propagates, the micro cracks merge into a single

micro crack (Kumar & Barai, 2011). In the FPZ, the material behaves nonlinear which is

difficult to take into account when performing a finite element analysis. The partially damaged

zone, FPZ, still has some stress-transferring capabilities through aggregate interlocking and

micro cracking activities. When the stress reaches zero, the crack is considered to be

continuously open. The material properties change with the crack propagation and there is

thereby a change in material behaviour as the crack propagates (Shi, 2009).

Figure 2.2. Schematic illustration of The Fracture Process Zone (FPZ) (Shi, 2009).


6

There is a common way of modelling this behaviour, namely the smeared crack approach, in

which the inelastic deformations in FPZ is distributed over a certain width and thereby modelled

as a continuous field. The material stiffness and strength are reduced according to a strain-

softening relation (Shi, 2009). The energy needed to propagate a tensile crack of unit area is the

fracture energy, 𝐺𝑓 (Malm & Holmgren, 2008b). There are several ways to calculate this

energy, however there are no method for calculation of fracture energy given in the Eurocode.

This complicates finite element analysis of concrete structures that should be modeled in

accordance with the Eurocode. The fracture energy is estimated from Eq. (2.1), according to

the Model Code 1990 (CEB-FIP, 1993) which is an international guideline that is the basis for

several national codes, including the Eurocode. This way of estimating the fracture energy is

also used in a finite element program called Atena (Červenka, et al., 2011) which is here later

used for calculation of crack widths. The way of calculating the fracture energy in Model Code

1990 is accepted according to the Swedish Concrete Association’s manual to Eurocode 2

(Svenska Betongföreningen, 2010). The fracture energy according to this formula depends on

the concrete compressive strength and the maximum aggregate size, see Eq. (2.1) and Table

2.2 (CEB-FIP, 1993). Cracks in concrete with coarser aggregates are more uneven and thereby

the fracture energy also becomes higher. This expression is generally accepted for use when

performing finite element analyses.

𝐺𝐹 = 𝐺𝐹0 (𝑓𝑐𝑚

𝑓𝑐𝑚0)

0.7

(2.1)

where 𝑓𝑐𝑚0 = 10 𝑀𝑃𝑎

Table 2.2. Basic values of fracture energy based on the maximum aggregate size.

dmax [mm] GF0 [Nmm/mm2]

8 0.025

16 0.030

32 0.058

According to the latest edition, Model Code 2010 (CEB-FIP, 2012), the method for calculation

of the fracture energy has changed and should now be estimated from Eq (2.2). It is now

dependent on concrete compressive strength but not on the aggregate size as in the previous

Model Code 1990 (CEB-FIP, 2012).

𝐺𝐹 = 73𝑓𝑐𝑚0.18 (2.2)

2.3. THEORY OF CONCRETE CRACKING AND FRACTURE ENERGY

7

In Table 2.3 results based on the two methods of estimating the fracture energy are compared.

The size of the fracture energy is different depending on which code that is used. As can be

seen, it is mainly for small aggregate sizes where the difference between Model Code 1990 and

2010 is the largest.

Table 2.3. Fracture energy for concrete, according to the Model Code.

Model Code 1990 Model Code 2010

𝑑𝑚𝑎𝑥 = 8 𝑚𝑚 𝑑𝑚𝑎𝑥 = 16 𝑚𝑚 𝑑𝑚𝑎𝑥 = 32 𝑚𝑚

C20/25 51 Nm/m2 62 Nm/m2 119 Nm/m2 133 Nm/m2

C30/37 64 Nm/m2 76 Nm/m2 148 Nm/m2 141 Nm/m2

C40/50 75 Nm/m2 90 Nm/m2 174 Nm/m2 147 Nm/m2

There are different models for expressing the fracture energy; linear, exponential, bi- or multi

linear, the method mentioned and used in this report is the exponential model. In most cases,

the shape of the curve has a low influence of the result. The fracture energy is described as the

relationship between the tension stress, 𝑓𝑡 and the crack opening at complete release of

stress, 𝑤𝑐. It is the area under the curve that represents the fracture energy, 𝐺𝑓. The relationship

between the parameters is calculated according to Eqs. (2.3-2.4), the corresponding curve to

this equation is shown in Figure 2.3. The exponential model also describes how the concrete

reacts to loading when cracks are formed. An advantage of using fracture mechanics when

modelling the concrete’s properties is that the finite element analysis is becoming nearly

independent of the mesh size (Svenska Betongföreningen, 2010).

𝜎

𝑓𝑡= [1 + (𝑐1

𝑤

𝑤𝑐)3] 𝑒

(−𝑐2𝑤

𝑤𝑐)

−𝑤

𝑤𝑐(1 + 𝑐1

3)𝑒−𝑐2 (2.3)

𝑤𝑐 = 𝑘𝐺𝐹

𝑓𝑡 (2.4)

where 𝑘 = 5.14 according to the Atena manual. The other parameters are:

𝑐1 = 3

𝑐2 = 6.93

𝑓𝑡 Effective tensile strength

𝐺𝐹 Fracture energy

𝜎 Normal stress in the crack

𝑤 Crack opening

𝑤𝑐 Crack opening at complete release of stress


8

Figure 2.3. The relationship between tensile stress 𝑓𝑡 and crack opening displacement 𝑤𝑐. The

area under the curve is the fracture energy 𝐺𝐹 (Shi, 2009).

2.4 Reinforced concrete and reinforcement bond

To increase the load carrying capacity and durability of a concrete structure, steel reinforcement

is often used. The reinforcement has much higher compressive and tensile strength than

concrete. The mean value of reinforcement compressive and tensile strength is around 500 MPa

and in comparison to the above mentioned concrete strength classes, the reinforcement has 10-

20 times higher compressive strength and 140-230 times higher tensile strength than concrete.

This makes steel a good material to use as reinforcement in concrete, to reduce cracks and to

achieve a more durable concrete structure. Furthermore, concrete has low tensile strength

compared to compressive strength therefore more reinforcement is needed where tensile forces

are expected. A prerequisite for reinforcement to work in a proper way is that it should be well

anchored in the concrete which enables the materials to transfer forces between each other and

allows the capacity of the concrete and reinforcement to be fully utilized. Otherwise the

concrete would fail after cracking (Ansell, et al., 2014).

The anchorage reinforcement bond is built up by different components (Ansell, et al., 2014):

- Cement as glue (adhesion)

- Grip between unevenness in the reinforcement surface and concrete (mechanical grip)

- Grip between surface pattern of the reinforcement bar and concrete (mechanical grip)

- Sliding friction.

Figure 2.4 illustrates the bond stress, 𝜏, that is evenly distributed along the mantle surface of

the reinforcement bars. The bond balances a change of the force ∆𝐹, in a reinforcement bar

according to:

𝜏𝜋ø ∆𝑥 = ∆𝐹 (2.5)

2.4. REINFORCED CONCRETE AND REINFORCEMENT BOND

9

Figure 2.4. Bond reinforcement (Ansell, et al., 2014).

In Model Code 1990 the reinforcement bond is treated with a model that describes the

relationship between bond stress and slip. This accounts for confinement, bond condition and

concrete strength, as also demonstrated in Table 2.4. Unconfined concrete leads to failure by

splitting of the concrete and confined concrete leads to failure by shearing of the concrete

between the ribs.

Table 2.4. Parameters defining the mean bond stress-slip relationship according to Model

Code 1990.

Unconfined concrete Confined concrete

Good bond

conditions

All other bond

conditions

Good bond

conditions

All other bond

conditions

𝑠1

0.6 mm

0.6 mm

1.0 mm

1.0 mm

𝑠2 0.6 mm 0.6 mm 3.0 mm 3.0 mm

𝑠3 1.0 mm 2.5 mm Clear rib spacing Clear rib spacing

𝛼 0.4 0.4 0.4 0.4

𝜏𝑚𝑎𝑥 2.0√𝑓𝑐𝑘 1.0√𝑓𝑐𝑘 2.5√𝑓𝑐𝑘 2.5√𝑓𝑐𝑘

𝜏𝑓 0.15𝜏𝑚𝑎𝑥 0.15𝜏𝑚𝑎𝑥 0.40𝜏𝑚𝑎𝑥 0.40𝜏𝑚𝑎𝑥

Figure 2.5 is derived from the below mentioned equations. The first part of the graph represents

the stage where ribs penetrate into the mortar matrix. This is distinguished by local crushing

and micro cracking. The second, horizontal part only occurs for confined concrete which is

characterized by crushing and shearing off of the concrete between the reinforcement. The third

part shows the reduction of bond resistance due to the presence of splitting cracks along the

reinforcement bars. The forth, lower horizontal part represents the remaining reinforcement

bond capacity (CEB-FIP, 1993). See Eqs. (2.6-2.9).


10

𝜏 = 𝜏𝑚𝑎𝑥 [𝑠

𝑠1]

𝛼

𝑓𝑜𝑟 0 ≤ 𝑠 ≤ 𝑠1 (2.6)

𝜏 = 𝜏𝑚𝑎𝑥 𝑓𝑜𝑟 𝑠1 < 𝑠 ≤ 𝑠2 (2.7)

𝜏 = 𝜏𝑚𝑎𝑥 − (𝜏𝑚𝑎𝑥 − 𝜏𝑓) [𝑠−𝑠2

𝑠3−𝑠2] 𝑓𝑜𝑟 𝑠2 < 𝑠 ≤ 𝑠3 (2.8)

𝜏 = 𝜏𝑓 𝑓𝑜𝑟 𝑠3 < 𝑠 (2.9)

Figure 2.5. Bond stress 𝜏 – slip 𝑠 relationship (CEB-FIP, 1993).

In the latest version, Model Code 2010, the bond and force distribution is slightly different.

Model Code 2010 is based on Model Code 1990, but there are more opportunities to describe

the reinforcement and the concrete around it. Eqs. (2.6-2.9) are also used in Model Code 2010;

however in this code different values are used when calculating the input values. Model Code

2010 also account for the difference between pull-out failure and splitting failure, see Table

2.5 and Figure 2.6.

2.4. REINFORCED CONCRETE AND REINFORCEMENT BOND

11

Table 2.5. Parameters defining the bond stress-slip relationship according to Model Code

2010.

Pull-out (PO) Splitting (SP)

휀𝑠 < 휀𝑠,𝑦 휀𝑠 < 휀𝑠,𝑦

Good

bond

condition

All other

bond

conditions

Good bond conditions Good bond conditions

Unconfined Stirrups Unconfined Stirrups

𝜏𝑚𝑎𝑥 2.5√𝑓𝑐𝑚 1.25√𝑓𝑐𝑚 2.5√𝑓𝑐𝑚 2.5√𝑓𝑐𝑚 1.25√𝑓𝑐𝑚 1.25√𝑓𝑐𝑚

𝜏𝑢,𝑠𝑝𝑙𝑖𝑡 _ _ 7.0 (

𝑓𝑐𝑚

25)

0.25

8.0 (𝑓𝑐𝑚

25)

0.25

5.0 (𝑓𝑐𝑚

25)

0.25

5.5 (𝑓𝑐𝑚

25)

0.25

𝑠1 1.0 mm 1.8 mm 𝑠(𝜏𝑢,𝑠𝑝𝑙𝑖𝑡 ) 𝑠(𝜏𝑢,𝑠𝑝𝑙𝑖𝑡 ) 𝑠(𝜏𝑢,𝑠𝑝𝑙𝑖𝑡) 𝑠(𝜏𝑢,𝑠𝑝𝑙𝑖𝑡)

𝑠2 2.0 mm 3.6 mm 𝑠1 𝑠1 𝑠1 𝑠1

𝑠3 𝑐𝑐𝑙𝑒𝑎𝑟 𝑐𝑐𝑙𝑒𝑎𝑟 1.2𝑠1 0.5𝑐𝑐𝑙𝑒𝑎𝑟 1.2𝑠1 0.5𝑐𝑐𝑙𝑒𝑎𝑟

𝑎 0.4 0.4 0.4 0.4 0.4 0.4

𝜏𝑓 0.40𝜏𝑚𝑎𝑥 0.40𝜏𝑚𝑎𝑥 0 0.4𝜏𝑢,𝑠𝑝𝑙𝑖𝑡 0.4 0.4𝜏𝑢,𝑠𝑝𝑙𝑖𝑡

*𝑐𝑐𝑙𝑒𝑎𝑟 is the clear distance between the ribs and 𝜏𝑢,𝑠𝑝𝑙𝑖𝑡 denote the peak local bond resistance

Figure 2.6. Bond stress 𝜏 – slip 𝑠 relationship (CEB-FIP, 2012).


12

3.1. THEORETICAL BACKGROUND

13

3 Deep beams

3.1 Theoretical background

Reinforced concrete deep beams are known for applications in tall buildings, foundations and

offshore structures (Winter & Nilson , 1989). Deep beams are structural elements with length

and height within the same magnitude and have significantly smaller thickness compared to a

conventional concrete beam. A deep beam is loaded in a similar way as a conventional beam

and transfers the load perpendicular to its own plane to the supports. The geometrical

relationship makes these structural elements to behave two-dimensional compared to normal

beams, which have one-dimensional behaviour. Another reason for this behaviour of the deep

beams is their high stiffness and the significant deflection at ultimate limit state (Ansell, et al.,

2014).

Reinforced concrete beams are generally designed according to the Bernoulli’s Hypothesis

which allows a linear stress distribution over the cross section and the flatness of the cross-

section remains during the loading. However, the Bernoulli’s hypothesis is not valid in a deep

beam since it behaves differently compared to a conventional beam. In a deep beam it is

assumed that the entire load is carried by arch action which means that the forces in the

reinforcement are transferred to the concrete beam along the entire length. Hence, a number of

arches appear between the cracks as each one is loaded with downwards forces which results

in that the arches are compressed (Cederwall, et al., 1990).

3.2 Stress distribution in deep beams

The load case in Figure 3.1 shows how the distribution of horizontal normal stresses varies in

a deep beam compared to in an ordinary beam according to the theory of elasticity. The stress

distribution is linear in the un-cracked state of the ordinary beam but increasing height in

relation to the span increases the deviancy of the stress distribution from the conventional stress

line. The distance between the tension- and compression force resultants defines the lever arm.

According to Figure 3.1 this distance increases with increased height of the beam but the

increment is not proportional and the effect is less notable after a particular height. This means

that a height larger than the span does not affect the level arm which stays at approximatively

70 % of the span, hence the use of a larger height of a deep beam is not effective. The relation

between decreased lever arm and height, when the ratio height/span increases, can be indicated

in the Figure 3.1. The same result is achieved even though the internal lever arm is related to

the span instead so that there is no advantage of a larger height than the span. Figure 3.2

indicates the stress distribution at a support, which is presented as an effect combination by the

reaction force R, and the bending moment M (Ansell, et al., 2014).


14

Figure 3.1 Distribution of horizontal normal stresses in the centre of a simply supported beam

with varying ratio height/span according to theory of elasticity (Ansell, et al., 2014).

3.2. STRESS DISTRIBUTION IN DEEP BEAMS

15

Figure 3.2 For a deep beam, the normal stress distribution over the support can be considered

as an action, which is a combination of moment and support reaction (Ansell, et al., 2014).

According to the assumptions of un-cracked elastic conditions, the moment M leads to a linear

stress distribution with a maximum value for an evenly distributed load which is proportional

to the ratio (l/h)2. As seen in Figure 3.2 the moment M is proportional to q×l2, and further the

stress distribution σ to M/h2. A linear stress distribution is common in ordinary beams due to

the relatively large (l/h) ratio compared to deep beams as described by Bernoulli’s hypothesis.

The influence of the reaction force, R, is indicated as a stress distribution shown in Figure 3.2;

which has its greatest value for an evenly distributed load with the proportionality to l/h, where

R is proportional to q×l and the stress distribution σ ≈ R/h. Deep beams have smaller l/h

compared to ordinary beams therefore the influence of the force R can dominate over the

influence of M which results in that the midpoint of the normal stresses moves downwards and

the lever arm reduces. When the cracks occur a redistribution of stresses appear, namely tensile

forces in the reinforcement replace tensile stresses in the concrete. The reinforcement should

be placed to prevent the appearance of coarse cracks and achieve the corresponding stress

distribution as stated in the theory of elasticity. When the cracks appear, the resulting

compression forces moves upwards and the internal lever arm increase. Figure 3.3 shows how

cracks in the concrete affects the distribution of horizontal normal stresses in a deep beam. The

appearance of the cracks are important for the stress distribution since the crack pattern

performs differently depending on how the load is applied, at the upper or lower edge of the

deep beam (Ansell, et al., 2014).

Figure 3.3. The left deep beam indicates the distribution of horizontal normal stresses

according to theory of elasticity. The right deep beams indicates the stress distribution after

cracking in the concrete (Ansell, et al., 2014).


16

3.3 Experimental investigations of deep beams

3.3.1 Deep beams loaded at the upper edge

The cause of failure for two simply supported concrete deep beams with different reinforcement

configurations is discussed by Ansell, et al (2014). According to Figure 3.4 the cause of failure

in the first beam with only one reinforcement layer is that the tension reinforcement reaches its

yield limit. For the second beam, with two reinforcement layers, failure occur for 8% higher

load, as a compression failure in the concrete at one of the supports, as shown in Figure 3.5.

Compression of concrete in support regions with small dimensions can be critical for deep

beams since a large load is carried with a large height in relation to the span, section width and

reinforcement content.

Figure 3.4. Simply supported deep beam, loaded at the upper edge. The cause of the failure is

the yield limit in the reinforcement. Reinforcement content ρ=0.134 % and failure load 1.2 MN

(Walther & Leonhardt, 1966).

3.3. EXPERIMENTAL INVESTIGATIONS OF DEEP BEAMS

17

Figure 3.5. Simply supported deep beam, loaded at the upper edge. The cause of failure is the

high compressive stresses at the support. Reinforcement content ρ= 0.268 % and failure load

1.29 MN (Walther & Leonhardt, 1966).

3.3.2 Deep beams loaded at the lower edge

In this case, the previous deep beams with the same amount of reinforcement are used with the

difference that the load is now applied at the lower edge of the beams. To be able to place the

load in a proper way, special reinforcement is used from the lower edge and upwards, which

means that when the reinforcement is strained cracks appears in the concrete. The compressive

stresses that are transferred downwards towards the supports influence the shape of the cracks

so that the pattern become arch shaped instead of horizontal. The failure load of a deep beam

with small amount of reinforcement is shown in Figure 3.6, which is 16% lower than the

equivalent beam loaded at the upper edge. The beam with larger amount of reinforcement

shown in Figure 3.7 has a failure load that is 9% lower than the beam loaded at the upper edge.

The cause of failure is due to yielding of the bending reinforcement, as shown in Figure 3.6.

The lower failure load in this case is caused by the internal lever arm, which decreases with

16% for a load applied at the lower edge. Furthermore, the application of the load has no

influence on the yield force. For the beam in Figure 3.7 the reinforcement does not reach the

yield stress but a compression failure appears in the support zone. The failure load in the beams

with larger amount of reinforcement depends slightly on where the load is applied, i.e. at the

lower or upper edge (Ansell, et al., 2014).


18

Figure 3.6. Simply supported deep beam, loaded at the lower edge. The deep beam is tested up

to ultimate limit state. Reinforcement content ρ=0.134 % and failure load 1.01 MN (Walther &

Leonhardt, 1966).

Figure 3.7. Simply supported deep beam, loaded at the lower edge. The deep beam is tested up

to ultimate limit state. The cause of the failure is the hight compressive stresses at the support.

Reinforcement content ρ= 0.268 % and failure load 1.17 MN (Walther & Leonhardt, 1966).

3.4. DESIGN OF DEEP BEAMS

19

3.4 Design of deep beams

3.4.1 Calculation model

Basic model- arch with tendon

The behaviour of deep beams differs from conventional beams for which the internal lever arm

is constant while the tension force varies with the moment. Furthermore, for the conventional

beams the influence of shear forces and sloped cracks are not usually considered in the design.

In deep beams, the internal lever arm varies with the moment curve but the tension force is

constant, resulting in that the deep beams perform as an arch with a tendon, as shown in Figure

3.8. The arch action is assumed to carry the full load of the deep beams, which results in three

significant consequences; that the reinforcement must not be shortened and that the

reinforcement must be end-anchored for the maximum tensile force at the supports and that

there is no need to design for shear failure (Ansell, et al., 2014).

Figure 3.8. Mode of action in a deep beam-“arch with tendon”. The force in the reinforcement

is constant but the internal lever arm, z, varies. The reinforcement must not be shortened and

must be end anchored for the whole force Fs (Ansell, et al., 2014).

Load application at the upper and lower edge

The load is transferred to the supports in different ways depending on if the load is applied at

the lower or upper edge, as indicated in Figure 3.9. From the upper edge, the load is transferred

to the supports by concrete compression only and from the lower edge, the load must first

transferred via reinforcement to the top of the deep beam before being distributed to the

supports. The load applied at the lower edge thus requires special suspension reinforcement and

gives a smaller internal lever arm. Compression arches within the beam are loaded by the

suspension reinforcement, which is strained and anchored in the concrete (Ansell, et al., 2014).


20

Figure 3.9. Principle concept of the compression line for load at the upper edge, zö, and load

at the lower edge, zu, respectively (Ansell, et al., 2014).

3.4.2 Truss models

The design of deep beams is not described in detail in Eurocode 2 (SS-EN 1992-1-1, 2005),

more than that they should be treated as discontinuity regions, D-regions, which have nonlinear

strain distribution. For the design of these zones, the use of truss models are recommended

where external loads are transferred through compressed concrete struts and tendons of

reinforcement. Previous Swedish guidelines (Boverket, 2004) have been used for determination

of lever arms in deep beams, where a difference was made between loading at the upper and

lower edge, which made that different designers of deep beams obtained the same results. The

deep beams designed according to Eurocode 2 and truss models give larger variations in results,

which are all accurate according to the code. The model must fulfil the condition of equilibrium

and ensure the impression of the natural behaviour at failure of the structure. Significant

simplifications of the model increase the risk of excluding any possible failure mechanism and

therefore the model should not deviate from the elastic behaviour of the structure. This is

important regarding the behaviour of the structure in the serviceability limit state where cracks

occur. If these conditions are satisfied the reinforcement will be placed where the main tensile

stresses occur before the concrete cracks, which also means that wide cracks can be avoided

(Ansell, et al., 2014).

Simply supported deep beam

In Figure 3.10 a truss model for a simply supported deep beam is shown. The model is created

using the force line method, which consists of the resultants of support reactions and the

external loads in various regions to keep the structure in equilibrium. These regions are divided

by lines in the middle where the shear force is zero. The deep beam shown is divided into two

regions associated with support 1 and 2 as shown in Figure 3.10. The external load is divided

in two forces corresponding to each support. The load Q1 is pushed down to the left in the upper

part of the deep beam using horizontal compressive stresses in the concrete with the resultant

Fc. Furthermore, the force transfer down and reaches the main reinforcement which must be

connected to the tension force in the reinforcement Fs. Otherwise, there is a risk for

fragmentation of the structure while cracking. The force continues to transfer down towards the

supports, where it reaches equilibrium with the reaction force R1. The truss model is given by

the study of resultants of different forces for both external load and the internal forces for

example the horizontal compressive force in the concrete. The difference between this truss

model and a model based on the principle arch with tendon is that in the truss model the

3.4. DESIGN OF DEEP BEAMS

21

resultants of the different forces are included while for an arch with tendon, the real distribution

of the external load is accounted for (Ansell, et al., 2014).

Figure 3.10. Force lines and the corresponding truss model for a simply supported deep beam

(Ansell, et al., 2014).


22

4.1. CRACKS ACCORDING TO EUROCODE 2

23

4 Method

In order to investigate which of the suggested combinations of amount, position and size of

reinforcement bars that is the most efficient for reducing crack widths, also accounting for

practical aspects, e.g. how easy it is to implement the proposed reinforcement layout on a

building site. To investigate the crack widths, the finite element program Atena was used. The

calculations was also made according to Eurocode 2 using a program called Concrete Section

as well as hand calculations. In the method chapter, different codes are presented for how to

calculate the required area of reinforcement and crack width. The mentioned codes are

Eurocode 2 (SS-EN 1992-1-1, 2005), BBK 04 (Boverket, 2004), B7 (Humble, 1969) and Model

Code 1990 (CEB-FIP, 1993). The codes BBK 04 and B7 are two old Swedish codes that were

replaced by Eurocode 2.

4.1 Cracks according to Eurocode 2

4.1.1 Minimum reinforcement

Both minimum- and tension (flexural) reinforcement are important when analysing crack

widths. The simplified formula for calculating minimum reinforcement area is described as:

𝐴𝑠,𝑚𝑖𝑛𝜎𝑠 = 𝑘𝑐𝑘𝑓𝑐𝑡,𝑒𝑓𝑓𝐴𝑐𝑡 (4.1)

where:

𝐴𝑠,𝑚𝑖𝑛 The minimum area of the reinforcement within the tensile zone

𝐴𝑐𝑡 The area of concrete within the tensile zone

𝜎𝑠 Yield strength of the reinforcement

𝑓𝑐𝑡,𝑒𝑓𝑓 Mean value of the tensile strength of the concrete

𝑘 Coefficient which allows for the effect of non-uniform self-equilibrating stresses


account and the change of level arm, 𝑘𝑐 = 1 for rectangular cross sections

Eurocode 2 (SS-EN 1992-1-1, 2005) has a prerequisite for the minimum reinforcement for deep

beams. According to chapter 9.7 in Eurocode 2, the recommended minimum reinforcement

ratio is 0.1 % of the total area of concrete. Therefore, this value is implemented in the models

used for the analyses.


24

4.1.2 Flexural crack widths

The tension reinforcement is important when calculating crack widths due to flexural, i.e.

bending. According to Eurocode 2 (SS-EN 1992-1-1, 2005), the characteristic crack width 𝑤𝑘

is calculated based on the difference between the reinforcement and concrete elongation as well

as the maximum crack spacing, following:

𝑤𝑘 = 𝑠𝑟,𝑚𝑎𝑥(휀𝑠𝑚 − 휀𝑐𝑚) (4.2)

휀𝑠𝑚 − 휀𝑐𝑚 =𝜎𝑠−𝑘𝑡

𝑓𝑐𝑡,𝑒𝑓𝑓

𝜌𝑝,𝑒𝑓𝑓(1+𝛼𝑒𝜌𝑝,𝑒𝑓𝑓)

𝐸𝑠≥ 0.6

𝜎𝑠

𝐸𝑠 (4.3)

𝑠𝑟,𝑚𝑎𝑥 = 𝑘3𝑐 + 𝑘1𝑘2𝑘4ø (4.4)

where:


𝑠𝑟,𝑚𝑎𝑥 Maximum crack spacing

휀𝑠𝑚 Mean strain in the reinforcement

휀𝑐𝑚 Mean strain in the concrete between cracks

𝜎𝑠 Stress in the tension reinforcement when assuming a cracked concrete section

𝛼𝑒 𝐸𝑠/𝐸𝑐𝑚

𝜌𝑝,𝑒𝑓𝑓 (𝐴𝑠 + 𝜉12𝐴𝑝

′ )/𝐴𝑐,𝑒𝑓𝑓 , where 𝐴𝑝′ is the area of tendons (pre or post-tensioned)

𝑘𝑡 Factor dependent on the load

= 0.6 for short term loading

= 0.4 for long term loading

𝑘1 Coefficient that takes the bond properties into account

= 0.8 for high bond bars

= 1.6 for bars with an effectively plain surface

𝑘2 Coefficient that takes the distribution of strain into account

= 0.5 for bending

= 1.6 for pure tension

𝑘3𝑐 7ø (according to national annex)

𝑘4 0.4254

𝑐 Concrete cover

ø Reinforcement dimension

4.2. CRACKS ACCORDING TO BBK04

25

4.1.3 Concrete Section (WIN-Statik)

Concrete Section is an analytical program (StruSoft, 2015) for beam concrete sections in the

Ultimate and Serviceability Limit States. The program takes bending, shearing, axial forces and

torque into account. It also calculates the crack width 𝑤𝑘 as described in Eqs. (4.1-4.4) and thus

follows the Eurocode 2. The Swedish national standard for Eurocode 2 is also considered in the

program.

4.2 Cracks according to BBK04

In Boverkets handbok om betongkonstruktioner (Boverket, 2004) the calculation of required

reinforcement area and crack width are described. The formula for calculation of the minimum

reinforcement is similar to the expression in Eurocode 2.

4.2.1 Minimum reinforcement for crack control

The expression for minimum reinforcement described in section 4.5.6 in BBK04 (Boverket,

2004) is based on the principal for balanced reinforcement where the minimum reinforcement

𝐴𝑠 is calculated based on a high tension strength for the concrete 𝐴𝑒𝑓, effective concrete area

and reinforcement stress 𝜎𝑠, following:

𝐴𝑠𝜎𝑠 ≥ 𝐴𝑒𝑓𝑓𝑐𝑡ℎ (4.5)

where:

𝐴𝑠 Reinforcement area

𝐴𝑒𝑓 Effective concrete area

𝜎𝑠 Tension stress in the reinforcement, maximum 420 MPa or 𝑓𝑦𝑘 it is lower

𝑓𝑐𝑡ℎ High tension strength for the concrete

4.2.2 Calculations of crack width

There is (Boverket, 2004) a relationship between the average crack width 𝑤𝑚 and the

characteristic crack width 𝑤𝑘 . The method can be applied if the reinforcement direction

deviates at a maximum of 15° from the principle stress direction. A prerequisite for using this

method is that the previously described minimum reinforcement in BBK 04 is used. The

characteristic crack width is described as 1.7 times higher than the average crack width

according to:

𝑤𝑘 = 1.7𝑤𝑚 (4.6)

𝑤𝑚 = 𝜈𝜎𝑠

𝐸𝑠𝑠𝑟𝑚 (4.7)


26

𝜈 = 1 −𝛽

2.5𝜅1∙

𝜎𝑠𝑟

𝜎𝑠 if 𝜈 ≥ 0.4 (4.8)

where:

𝑤𝑚 Mean crack width


𝐸𝑠 The modulus of elasticity of reinforcement (200 GPa)

𝑠𝑟𝑚 The mean value of the crack spacing

𝛽 Coefficient that accounts for the duration of the load

𝜅1 Coefficient that accounts the adhesion of the reinforcement

𝜈 Coefficient that accounts for the tensioned concrete between cracks

𝜎𝑠 The stress in the reinforcement in the crack

𝜎𝑠𝑟 The stress in the reinforcement in the crack directly after cracking

The expression for the crack width is not explicitly described in Eurocode 2. However,

according to the Swedish Concrete Association's Guide to Eurocode 2 (Svenska

Betongföreningen, 2010) the factor 1.7 is included in the expression for the crack

distance 𝑠𝑟,𝑚𝑎𝑥, see Eq. (4.4).

4.3 Calculations of the crack width according to B7

The crack width due to pure bending is calculated according to B7 Bestämmelser för

betongkonstruktioner, which is an old Swedish guideline for concrete structures. The guideline

B7 was valid long before BBK 04.

The nominal maximum crack width 𝛿 in the same height with the tensile reinforcement at pure

bending should be calculated based on the formula according to Jonsson, Osterman, Wästlund,

see Eq. (4.9) (Humble, 1969).

𝛿 = 𝑘 ∙ 𝑑 (𝑊𝑏

𝐴𝑎∙ℎ∙

𝜎𝑎

𝐸𝑎)

23⁄

(4.9)

where:

𝑘 Coefficient according to Table 4.1

𝑑 The diameter of the reinforcement bar

4.4. CRACKS ACCORDING TO MODEL CODE 1990

27

𝑊𝑏 The plastic bending resistance of the concrete cross section considering the

tensioned edge without taking the reinforcement into account and with the

prerequisite that the entire cross section is un-cracked.

𝐴𝑎 The area of the tensile reinforcement

ℎ The effective height

𝜎𝑎 Reinforcement stress assuming the cracked concrete cross section

𝐸𝑎 Modulus of elasticity

Table 4.1. The coefficient k depending on reinforcement type.

Reinforcement type Group 1 Group 2

Smooth bars 0.15 0.23

Bundled smooth bars 0.14 0.21

Profiled bars 0.13 0.20

Bundled profiled bars 0.12 0.18

Ribbed bars 0.11 0.15

Bundled ribbed bars 0.10 0.13

The k-values for Group 2 are designed to include the accumulating impact on the crack width

due to repeated loading and unloading for long term. The values for Group 1 are designed for

cases where such impacts are not needed to be considered, namely loading and unloading occur

rarely.

It should be observed that if no crack reinforcement is inserted in the deep beam’s web, there

is risk for appearance of larger cracks in the web than at the level of the tensile reinforcement

(Humble, 1969).

4.4 Cracks according to Model Code 1990

The Model Code describes in detail the theoretical background behind cracking theory. The

description and theoretical background are more detailed in Model Code 1990 (CEB-FIP, 1993)

compared to Eurocode 2. Model Code 1990 for instance describes how to calculate the

minimum reinforcement 𝐴𝑠,𝑚𝑖𝑛 and the crack width 𝑤𝑘 .

4.4.1 Minimum reinforcement for crack control

In Model Code 1990, a similar approach for calculating the minimum reinforcement as in

Eurocode 2 is used. The procedure described in Model Code is shown in Eq. (4.10) below, as

a simplified method that can be applied for the calculation of the required area of minimum


28

reinforcement within the tensioned concrete zone. The area 𝐴𝑠,𝑚𝑖𝑛 is considered dependent of

the tensile strength of concrete when the first crack occurs, for the stress 𝑓𝑐𝑡,𝑚𝑎𝑥. The area of

the tensioned concrete zone is 𝐴𝑐𝑡 and the concrete yield stress 𝜎𝑠2.

𝐴𝑠,𝑚𝑖𝑛 =𝑘𝑐∙𝑘∙𝑓𝑐𝑡,𝑚𝑎𝑥∙𝐴𝑐𝑡

𝜎𝑠2 (4.10)

where:

𝑓𝑐𝑡,𝑚𝑎𝑥 Upper fractile of the tensile strength of the concrete

𝐴𝑐𝑡 Tensile part of the concrete section

𝜎𝑠2 Maximum tensile stress of the reinforcement (can be considered to be equal to the

characteristic value of the yield stress)


account and the change of level arm, 𝑘𝑐 = 1 for rectangular cross sections

𝑘 Coefficient taking the non-uniform self-equilibrating stresses into account

4.4.2 Calculations of crack width

The basic formula for calculating crack width is presented in chapter 7.4.3.1.1 in Model Code

1990 (CEB-FIP, 1993). The formulas used are here described in Eq. (4.11-4.13) below. The

crack width 𝑤𝑘 depends on the steel and concrete strain as on the length of the slip between the

concrete and steel (𝑙𝑠,𝑚𝑎𝑥). Furthermore, the length 𝑙𝑠,𝑚𝑎𝑥 depends on if the stabilising cracking

condition is reached or if single cracks should be considered. Observe that longitudinal cracks

caused by corroded steel bars are not covered by the below mentioned formula. For all stages

of cracking, the crack width 𝑤𝑘 can be calculated according to:

𝑤𝑘 = 𝑙𝑠,𝑚𝑎𝑥(휀𝑠𝑚 − 휀𝑐𝑚 − 휀𝑐𝑠) (4.11)

For a stabilized cracking condition 𝑙𝑠,𝑚𝑎𝑥 is calculated as:

𝑙𝑠,𝑚𝑎𝑥 = 2𝜎𝑠2−𝜎𝑠𝐸

4𝜏𝑏𝑘ø𝑠 (4.12)

For single cracking conditions, 𝑙𝑠,𝑚𝑎𝑥 is calculated according to:

𝑙𝑠,𝑚𝑎𝑥 = 2𝜎𝑠2

4𝜏𝑏𝑠,𝑘ø𝑠 (4.13)

4.5. ATENA 2D

29

where:


𝑙𝑠,𝑚𝑎𝑥 The length over which slip between steel and concrete occurs

휀𝑠𝑚 Mean steel strain of the reinforcement within 𝑙𝑠,𝑚𝑎𝑥

휀𝑐𝑚 Mean strain of the concrete within 𝑙𝑠,𝑚𝑎𝑥

휀𝑐𝑠 Strain of the concrete due to shrinkage

𝜎𝑠2 Reinforcement stress at the crack

𝜎𝑠𝐸 Reinforcement stress at the point of zero slip

ø𝑠 Reinforcement diameter

𝜏𝑏𝑘 Lower fractile value of the mean bond stress. Assumed to be 1.8 ∙ 𝑓𝑐𝑡𝑚 for

stabilized cracking and single cracking for short term load. Assumed to be 1.35 ∙𝑓𝑐𝑡𝑚 for single cracking for long term load.

4.5 Atena 2D

The program Atena (Červenka, et al., 2011) is based on nonlinear finite element analysis, a

method that numerically solves differential equations. The structure is divided into elements

which are connected by nodes. Between those nodes a solution is interpolated using shape

functions that approximate the displacement between the nodes. The accuracy of the solution

depends on this interpolation, and thus also on the used mesh size, linear or quadratic

interpolation and size of the analysis time step. Concrete behaviour is modelled in a material

model called SBeta, in which a smeared crack approach is used, describing cracking of

tensioned and compressed concrete based on non-linear fracture mechanics. Many of the

calculations in Atena are based on Model Code 1990 (Alavala, 2008).

4.5.1 Smeared crack model

In Atena, there are two models of Smeared Crack type, namely, a fixed crack model and a

rotated crack model. For the fixed crack model the crack directions are given by the direction

the principal stresses of the model has at the moment of cracking and after the first crack the

direction is fixed. Rotated crack model means that the crack direction also rotates if the principal

strain axes rotates during loading (Červenka, et al., 2011). The two models are illustrated in

Figures 4.1 and 4.2.


30

Figure 4.1. Fixed crack model (Červenka, et al., 2011).

Figure 4.2. Rotated crack model (Červenka, et al., 2011).

4.5.2 Cracks according to Atena

As described earlier in Chapter 2, Reinforced concrete structures, the process of crack

formation is divided into three stages. Un-cracked stage where the concrete tensile strength is

not reached, fracture process zone where the tensile stress decreases in the concrete due to a

bridging effect and lastly the cracked zone where there is zero stress in the concrete (Červenka,

et al., 2011). The crack width is calculated as a total crack opening displacement within the

crack band according to Figure 4.3 and:

𝑤 = 휀𝑐𝑟𝐿𝑡′ (4.14)

where 휀𝑐𝑟 is the crack opening strain equal to the strain normal to the crack direction in cracked

state where there is zero stress in the concrete and 𝐿𝑡′ the failure bands for tension.

4.5. ATENA 2D

31

Figure 4.3. Stages of crack opening (Červenka, et al., 2011).

4.5.3 Newton-Raphson method

Atena contains a method called Newton-Raphson (see Figure 4.4), which is used to achieve

equilibrium in one load step. The structural behaviour is described by a nonlinear equation

where the stiffness and internal forces are based on the deformation in the previous load step.

The Newton-Raphson method is suitable for finding the maximum on a working curve but not

to describe the behaviour of a load- deformation curve in a downward going direction, a so

called “snap-back” behaviour. If this is the case, the Arc-length method can be used instead. In

this work all simulations was made using the Newton-Raphson method because the

serviceability limit state were of interest and thus on the upward going load- deformation curve

(Cook, et al., 2001).

Figure 4.4. Newton-Raphson method (Červenka, et al., 2011).


32

5.1. GEOMETRY

33

5 Prerequisite for analysed examples

The analysis consists of different simply supported deep beams in order to see the difference of

their behaviour using tensile and crack reinforcement. The beams were tested with different

reinforcement positions and diameters and were all exposed to external loads, either

concentrated or distributed loads. The concrete strength class used in the tests was C30/37. The

analyses were made according to Eurocode 2 (SS-EN 1992-1-1, 2005), with calculations made

using Concrete Section (StruSoft AB, 2015), and with the finite element program Atena

(Cervenka Consulting, 2011), which is mainly based on Model Code 1990 (CEB-FIP, 1993).

The study was done in the serviceability limit state since the cracks start to appear at this stage

and govern the durability of the reinforcement.

As mentioned earlier, the finite element software Atena, was used to model the concrete beams,

using quadrilateral isoparametric elements and the fixed crack model. The reaction forces were

added to the supports. Considering the material properties, the concrete was modelled in

compression based on nonlinear stress-strain curves. The material properties were chosen

according to the material behaviour in Atena denoted as SBeta, which uses a softening

coefficient to reproduce the stress-strain behaviour of concrete. The behaviour of the

reinforcement steel bars were assumed elastic-plastic and the bars were placed as low as

possible to utilize the capacity of the reinforcement. In order to avoid early punching failure

stiff steel plates were added to the supports of the model. A steel plate was also added at the

top of the beam where the applied concentrated load was distributed. Moreover, the horizontal

and vertical minimum reinforcement were used to carry the flexural shear force which in turn

minimized the brittleness of the concrete.

5.1 Geometry

The deep beams had dimensions 5×2.5×0.4 m. The geometry of the deep beam was chosen to

represent a normal wall construction, however the construction is thicker than average in order

to include enough reinforcement. Further, a convergence analysis was done testing

conventional beams by varying the length. The reason for testing both deep beams and

conventional beams was that the stress variation is different in those structures. The stress

variation before cracking is linear in a conventional beam and non-linear in a deep beam.

5.2 Mesh

A number of different models have been analysed to see which mesh is the most suitable,

regarding both computational time consumption and convergence i.e. how the accuracy of the

results changes. Generally, a coarser mesh gives a faster calculation process but sometimes a

more inexact result, in contrast to a more fine mesh that often gives a more exact result but a

slower calculation process. As described earlier, the mesh size should be larger than the largest

aggregate size which is here assumed to be 12 mm. If a mesh size smaller than the aggregate

size is used the risk for obtaining inaccurate results increases. The models with the same input


34

values have been tested using both larger and smaller mesh size than 100 mm. However, a

larger mesh than 100 mm gave imprecise results of the crack widths. It was also preferable that

the reinforcement bars would be placed almost in the middle of the element size which was

difficult to achieve by using small mesh sizes. Further, a smaller mesh size than 100 mm gave

approximately the same crack width as a 100 mm mesh and it was thereby unnecessary to use

a smaller mesh.

5.3 Loads

The analyses were made considering two different loads; the dead load of the concrete beam

and the external load. The magnitude of the external loads were mainly decided based on which

reinforcement stress that occurred when applying the load. In Atena the loads were gradually

increased step by step, which means that the analysis of the crack width was read at

reinforcement stresses, e.g. 200, 250, 300 and 400 MPa. The crack widths were read at the same

reinforcement stresses in the program Concrete Section as well as in Atena.

5.4 Concrete properties

The concrete used in the tests was C30/37. The general material properties used while

modelling in Atena are defined in accordance to Eurocode 2 and Model Code 1990, and were

as presented in Table 5.1.

Table 5.1. Strength properties of concrete used as input values of Atena.

Strength

class Relation

C30/37

𝒇𝒄𝒌 [MPa] Characteristic

Compressive

Strength

30

𝒇𝒄𝒎 [MPa] Mean

Compression

Strength

38 𝑓𝑐𝑘 + 8

𝒇𝒄𝒖 [MPa] Mean Cube

Compression

Strength

45 * 𝑓𝑐𝑚/0.85

𝒇𝒄𝒕𝒎 [MPa] Mean

Characteristic

Compression Strength

2.9 0.3𝑓𝑐𝑘2/3

𝑬𝒄𝒎 [GPa] Modulus of

Elasticity 33 22 (

𝑓𝑐𝑚

10)

0.3

*According to Model Code 1990

5.5. REINFORCEMENT AND POSITION

35

The fracture energy was calculated according to Model Code 1990 which depends on the

maximum aggregate size. The maximum aggregate size was assumed to be 12 mm which is

recommended for wall constructions such as deep beams. Since there is no information about

the fracture energy for aggregate size 12 mm, it is here assumed to be the mean value of the

fracture energy for 8 mm and for 16 mm, see Table 5.2. For concrete strength class C30/37 this

gives a fracture energy of 70 Nm/m2 according to calculations based on Model Code 1990.

Table 5.2. The fracture energy for C30/37 with maximum aggregate size of 8 mm and 16 mm.

Model Code 1990

𝑑𝑚𝑎𝑥 = 8 𝑚𝑚 𝑑𝑚𝑎𝑥 = 16 𝑚𝑚

C30/37 64 𝑁𝑚/𝑚2 76 𝑁𝑚/𝑚2

5.5 Reinforcement and position

Well-anchored reinforcement is a prerequisite for a concrete structure to work in an optimal

way. The anchorage enables forces to be transmitted between the concrete and reinforcement.

There were mainly two different reinforcement requirements to take into account during the

modelling. The first was the minimum reinforcement requirement according to Eurocode 2 and

the second was the bending reinforcement needed in the lower edge of the beam where tensile

forces were expected. The minimum reinforcement used was 0.1% horizontally and 0.1%

vertically. The percentage represents the reinforcement area relative to the concrete area. The

amount of bending reinforcement was held almost constant for all the tests, to facilitate

comparison between the cases. The reinforcement follows a linear-elastic approach up to 500

MPa after which the reinforcement yields and failure occurs, see

Figure 5.1.

Figure 5.1. Stress-strain relationship of the reinforcement.


36

Bond slip

The bond between concrete and reinforcement is an important aspect when calculating crack

widths. The bond model used in Atena follows the descriptions in Model Code 1990. Atena

presents several different settings for bond models and thereby a verification has been made in

order to see which bond model gives the best possible result, see Figure 5.2. All the bond

models showed almost similar results. It was assumed that the reinforcement had a good

adhesion and therefore the bond model “Confined Good” has been used for modelling the

beams.

Figure 5.2. A verification of bond model based on reinforcement stress and crack width, Atena.

5.6 Theory behind the reinforcement distribution

In order to investigate the relationship between reinforcement diameter and crack width

different types of cross sections with distributed reinforcement bars were used. The total

reinforcement area was chosen based on loads and reinforcement stresses distributed between

tensile and crack reinforcement. The tensile reinforcement in all beams consist of ø25 mm bars

and the crack reinforcement consist of ø10 mm bars in cross sections 2-5, and ø16 mm bars in

cross sections 6-10. The total reinforcement area was assumed to be constant and theoretically

distributed in different layers. An illustration of the cross sections are shown in Figure 5.3. The

reinforcement cover and bar spacing is chosen according to limitations in Eurocode 2, for use

with both Atena and Concrete Section. Eurocode 2 requires the minimum bar spacing 2𝑑 and

the reinforcement cover is as low as possible to optimize the effect of the crack reinforcement.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 100 200 300 400 500 600

Cra

ck w

idth

[m

m]

Reinforcement stress [MPa]

Model Code Confined Good

Model Code Confined Poor

Model Code Unconfined Good

Model Code Unconfined Poor

Perfect Connection

Bijay Good

5.6. THEORY BEHIND THE REINFORCEMENT DISTRIBUTION

37

Figure 5.3. Illustration of each cross section of the

deep beams 1-10. All cross sections consist of tensile

reinforcement ø25 mm. The cross sections 2-5

consist of crack reinforcement ø10 mm and the cross

sections 6-10 consist of crack reinforcement ø16 mm.

1 2 3

4 5 6

7 8 9

10


38

6.1. VERIFICATION OF THE RESULTS IN ATENA

39

6 Results

The investigation consisted of different beams, both deep and conventional, analysed

analytically and numerically. The analytical calculations were based on Eurocode 2 (SS-EN

1992-1-1, 2005) using the program Concrete Section (StruSoft AB, 2015) and the numerical

calculations on Model Code 1990 (CEB-FIP, 1993) using the finite element program Atena

(Cervenka Consulting, 2011). The main purpose of the analytical calculations was to verify the

numerical results.

6.1 Verification of the results in Atena

In order to verify the results from Atena, a comparison has been made between Atena and

Eurocode 2 using beams with different lengths but the same magnitude of height, see Figures

6.2-6.5. The cross sections in all beams have the same reinforcement configuration and

geometry; i.e. cross section 6 where the lower layer consists of crack reinforcement bars ø16

mm and the upper layer consists of tensile reinforcement bars ø25 mm, see Figure 5.3. Both

Atena and Eurocode 2 showed similar results for crack widths in the conventional beam with

L=15 m but the results began to differ with decreased length. This was because the program

Concrete Section, which is based on the Eurocode 2, assumed that the Bernoulli’s hypothesis

was valid in all beams regardless of the length. However, in Atena a decreased length and

nonlinear strain distribution in deep beams were taken into account. The load in a deep beam is

partly carried by arch action, which leads to smaller crack widths compared to results from

Eurocode 2. Furthermore, the results from Atena and Concrete Section showed almost the same

reinforcement stress for the same magnitude of crack width independent of the relation between

length and height of the beams. Figure 6.1 shows an illustration of the crack pattern for the

tested beams.

Figure 6.1. View of test beams subjected to evenly distributed loads. The results from Atena

show how the length/height ratio varies and how the crack patterns change from conventional

to deep beams.


40

Figure 6.2. Relationship between crack width and bending moment for a conventional beam

with L=15 m, h= 2.5 m and ratio L/h=6.

Figure 6.3. Relationship between crack width and bending moment for a conventional beam

with L=10 m, h= 2.5 m and ratio L/h=4.

Figure 6.4. Relationship between crack width and bending moment for a deep beam with L=7.5

m, h= 2.5 m and ratio L/h=3.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1000 2000 3000 4000 5000 6000

Cra

ck w

idth

[m

m]

Bending moment [kNm]

Eurocode 2

Atena L= 15 m

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1000 2000 3000 4000 5000 6000

Cra

ck w

idth

[m

m]


Eurocode 2

Atena L= 10 m

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1000 2000 3000 4000 5000 6000

Cra

ck w

idth

[m

m]


Eurocode 2

Atena L= 7.5 m


41

Figure 6.5. Relationship between crack width and bending moment for a deep beam with L=5

m, h= 2.5 m and ratio L/h=2.

In theory, Atena and Concrete Section should show the same behaviour considering crack width

for conventional beams. As seen in Figure 6.2, the curves showed approximately the same crack

widths at the beginning but changed with incerased bending moment, i.e. the crack widths

calculated in Atena became smaller. In Figure 6.6, a demonstration is made in order to

investigate the reason behind the deviation of the crack widths in both programs, showing the

reinforcement stress plotted against bending moment. Both curves show similar reinforcement

stresses for the same bending moment up to approximately 3000 kNm, but then the deviation

increase for higher bending moments. This could be explained by the different assumptions that

each calculating program uses, e.g. the used lever arm.

Figure 6.6. Bending moment in relation to reinforcement stress, calculated with Atena and

Concrete Section.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1000 2000 3000 4000 5000 6000

Cra

ck w

idth

[m

m]


Eurocode 2

Atena L=5 m

wk=0.10

wk=0.15

wk=0.28

wk=0.42

wk=0.62

wk=0.11

wk=0.18

wk=0.28

wk=0.38

wk=0.54

0

100

200

300

400

500

600

0 1000 2000 3000 4000 5000 6000

Re

info

rce

me

nt

stre

ss [

MP

a]


Eurocode 2

Atena L= 15 m


42

6.2 Deep beam with distributed load

In Figures 6.8-6.11 it is illustrated how the crack width changes for deep beams with different

cross sections depending on bending moment and reinforcement stress. The crack- and tensile

reinforcements are placed in separate layers in Figures 6.8 and 6.10 and every other in Figures

6.9 and 6.11. The cross section of each beam is illustrated in Figure 5.3.

The results in Figures 6.8 and 6.10 show that the effect of the tensile reinforcement decreases

if placed further away from the lower edge of the beam. When replacing a tensile reinforcement

by crack reinforcement the total bar area increases, i.e. the bars need more space, which in turn

resulted in a reduced lever arm. As seen in Figure 6.8, the deep beam C2 with two layers of

reinforcement showed smaller crack widths compared to the deep beam C1 where only one

reinforcement layer was used. The reason for this could be the use of crack reinforcement in

deep beam C2. However, the effect of the crack reinforcement became smaller in deep beam

C3 where three layers of reinforcement were used. The decrement could be explained by a

decreased lever arm in deep beam C3, i.e. the effect of the crack reinforcement on the crack

width was counteracted by decreased lever arm. The same behaviour was obtained for the deep

beams illustrated in Figure 6.10, i.e. deep beam C8 with three layers of reinforcement showed

the largest crack width compared to deep beams C1, C6 and C7.

According to Figure 6.9, the deep beam C1 showed the smallest crack width related to bending

moment but the results changed when the crack width was related to the reinforcement stress

instead, i.e. deep beam C4 showed the smallest crack width. Figure 6.11 shows that deep beam

C9 had the smallest crack width in relation to both bending moment and reinforcement stress.

The placement of the crack- and tensile reinforcement as every other bar showed smaller crack

widths which indicated a proper interaction between the two reinforcement types, compared to

for deep beam C1 where only tensile reinforcement was used. Further, the amount of tensile

reinforcement in the lower layer of the deep beam C9 was larger compared to deep beam C10,

which resulted in smaller crack widths in deep beam C9. Figure 6.7 shows an illustration of a

deep beam with an evenly distributed load. As seen in the figure, additional cracks appeared

over the supports which could be explained by that the arch action cannot carry the entire effect

of the distributed load. In order to reduce these cracks more minimum reinforcement is required.

For the complete results inclusive crack pattern of each deep beam, see Appendix C.4.

Figure 6.7. The crack pattern in a simply supported deep beam with an evenly distributed load.

6.2. DEEP BEAM WITH DISTRIBUTED LOAD

43

Figure 6.8. Crack width variation with bending moment or reinforcement stress calculated with

Atena, for beams C1-C3 with a distributed load.

Figure 6.9. Crack width variation with bending moment or reinforcement stress calculated with

Atena, for beams C1, C4-C5 with a distributed load.

Figure 6.10. Crack width variation with bending moment or reinforcement stress calculated

with Atena, for beams C1, C6-C8 with a distributed load.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 2000 4000 6000

Cra

ck w

idth

[m

m]


C1

C2

C3

0

0,1

0,2

0,3

0,4

0,5

0,6

0 200 400 600

Cra

ck w

idth

[m

m]


C1C2C3

0

0,1

0,2

0,3

0,4

0,5

0,6

0 2000 4000 6000

Cra

ck w

idth

[m

m]


C1C4C5

0

0,1

0,2

0,3

0,4

0,5

0,6

0 100 200 300 400 500 600

Cra

ck w

idth

[m

m]


C1C4C5

0

0,1

0,2

0,3

0,4

0,5

0,6

0 2000 4000 6000

Cra

ck w

idth

[m

m]


C1C6C7C8

0

0,1

0,2

0,3

0,4

0,5

0,6

0 200 400 600

Cra

ck w

idth

[m

m]


C1C6C7C8


44


with Atena, for beams C1, C9-C10 with a distributed load.

6.3 Deep beam with concentrated load

In Figures 6.13-6.16 it is illustrated how the crack width changes for deep beams affected by a

concentrated load, depending on bending moment and reinforcement stress. All beams have

different cross sections, which are illustrated in Figure 5.3. For the presented results, the crack-

and tensile reinforcements are placed in separate layers in the beams shown in Figures 6.13

and 6.15 and as every other bar in Figures 6.14 and 6.16.

All deep beams D with concentrated load showed similar behaviour as in the corresponding

deep beams C with distributed load. The Figures 6.13 and 6.15 show that the use of crack

reinforcement and the placement of the reinforcement influence the size of the crack width. The

deep beams D2 and D6 had the smallest crack widths, with a combination of tensile and crack

reinforcement, compared to deep beam D1. Moreover, the deep beams D2 and D6 also had

larger lever arms compared to the other beams with crack and tensile reinforcement. Figure

6.14 shows varying results depending on if the crack width is related to bending moment or

reinforcement stress. Deep beam D1 shows the smallest crack width in relation to bending

moment and deep beam D4 the smallest crack width in relation to the reinforcement stress.

Furthermore, Figure 6.16 show that the deep beam D9 had the smallest crack width in relation

to both bending moment and reinforcement stress. Figure 6.12 shows an illustration of a deep

beam affected by a concentrated load at mid-span. For the complete results, including crack

patterns for deep beam, see Appendix C.5.

Figure 6.12. The crack pattern in a simply supported deep beam with a concentrated load at

mid-span.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 2000 4000 6000

Cra

ck w

idth

[m

m]


C1C9C10

0

0,1

0,2

0,3

0,4

0,5

0,6

0 200 400 600

Cra

ck w

idth

[m

m]


C1C9C10

6.3. DEEP BEAM WITH CONCENTRATED LOAD

45


with Atena, for beams D1, D2-D3 with a concentrated load at mid-span.





0

0,1

0,2

0,3

0,4

0,5

0 1000 2000 3000 4000 5000

Cra

ck w

idth

[m

m]


D1D2D3

0

0,1

0,2

0,3

0,4

0,5

0 100 200 300 400 500

Cra

ck w

idth

[m

m]


D1D2D3

0

0,1

0,2

0,3

0,4

0,5

0 1000 2000 3000 4000 5000

Cra

ck w

idth

[m

m]


D1D4D5

0

0,1

0,2

0,3

0,4

0,5

0 100 200 300 400 500

Cra

ck w

idth

[m

m]


D1D4D5

0

0,1

0,2

0,3

0,4

0,5

0 1000 2000 3000 4000 5000

Cra

ck w

idth

[m

m]


D1D6D7D8

0

0,1

0,2

0,3

0,4

0,5

0 100 200 300 400 500

Cra

ck w

idth

[m

m]


D1D6D7D8


46



6.4 Relationship between equivalent diameter and

crack width

According to Eurocode 2, when the cross section consists of two or more different

reinforcement diameters, an equivalent diameter should be calculated. The equivalent diameter

is a value used when a mixture of bar diameters exists in a section, this can be calculated

according to Eq. (6.1). In the formula, ø1 and ø2 are the reinforcement diameters and n1 and

n2 the amount of each reinforcement bar. In this part, the equivalent diameter was used for

evaluating how the crack width was affected by the diameter of the reinforcement bars. The

total area of the reinforcement was approximately the same for every beam analysed. The

equivalent diameters are calculated for all cross sections with different reinforcement

configuration, see Appendix C.2.

øeq =n1ø1

2+n2ø22

n1ø1+n2ø2 (6.1)

6.4.1 Concrete Section

In Figures 6.17-6.18, the relationship between crack width and equivalent diameter is shown,

having been analysed based on results obtained from Concrete Section. As seen in the figures

the crack width increased with increasing equivalent diameter and thereby the effect of the

equivalent diameter was proven to follow Eurocode 2.

0

0,1

0,2

0,3

0,4

0,5

0 1000 2000 3000 4000 5000

Cra

ck w

idth

[m

m]


D1D9D10

0

0,1

0,2

0,3

0,4

0,5

0 100 200 300 400 500

Cra

ck w

idth

[m

m]


D1

D9

D10

6.4. RELATIONSHIP BETWEEN EQUIVALENT DIAMETER AND CRACK WIDTH

47

Figure 6.17. Relationship between crack width and equivalent diameter. All beams with

distributed loads and modelled in Concrete Section. Beam A1 consists of only tensile

reinforcement ø25 mm.

Figure 6.18. Relationship between crack width and equivalent diameter. All beams with

distributed loads and modelled in Concrete Section. Beam A1 consists of only tensile

reinforcement ø25 mm..

6.4.2 Atena

To investigate the effect of using crack reinforcement in deep beams, the crack width has been

related to an equivalent diameter as shown in Figures 6.19-6.20. Both deep beams with

distributed and concentrated loads showed that deep beams with cross section 2 had the smallest

crack width in relation to the equivalent diameter, i.e. deep beams C2 and D2, see Figure 6.19.

Additional investigation has been made for the deep beams where crack reinforcement ø16 mm

was used, see Figure 6.20. The test showed that the deep beams with distributed and

concentrated load that consisted of cross sections 6, 9 and 10 had the same and thereby smallest

crack width compared to the other deep beams; i.e. C6, C9-C10 and D6, D9-D10 in Figure

6.20.

Less total reinforcement area and shorter lever arm were two factors that increased the crack

width. The difficulty of comparing crack- and tensile reinforcement was that a change of one

parameter (e.g. the reinforcement diameter) would unintentionally change other parameters

consequently. If a tension reinforcement bar ø25 mm was substituted with several crack

reinforcement bars, e.g. ø10 mm in order to correspond to the same total area as the tension

reinforcement bars, the lever arm changed since more space was needed to fit the crack

A1A2

A3

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]

Equivalent diameter [mm]

M = 2170 kNm

A1A4 & A5

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]


M = 2170 kNm

A1A6A7

A8

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]


M = 2170 kNm

A1A9

A10

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]


M = 2170 kNm


48

reinforcement. As previously described the total area varied slightly between different

reinforcement cases since the area of one tension reinforcement bar did not correspond to an

even number of crack reinforcement (one ø25 mm corresponds to 6.25ø10 mm bars). Overall,

the factors mentioned earlier, i.e. the total area of reinforcement and internal lever arm, might

affect the crack width more than the equivalent diameter. This could explain the non-linear

behaviour of the relationship between the crack width and equivalent diameter.

Figure 6.19. Relationship between crack width and equivalent diameter. All deep beams are

modelled in Atena, a distributed load is applied for the beams denoted C and a concentrated

load is applied at mid-span for the beams denoted D.

Figure 6.20. Relationship between crack width and equivalent diameter. All deep beams are

modelled in Atena, a distributed load is applied for the beams denoted C and a concentrated

load is applied at mid-span for the beams denoted D.

C1C2C3D1

D2

D3

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]


M = 3038 kNm

M = 3038 kNm

C1C4 & C5

D1D4

D5

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]


M = 3038 kNm

M = 3038 kNm

C1C6C7C8

D1D6D7

D8

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]


M = 3038 kNm

M = 3038 kNm

C1C9C10

D1D9D10

0

0,1

0,2

0,3

0,4

0,5

0,6

0 10 20 30

Cra

ck w

idth

[m

m]


M = 3038 kNm

M = 3038 kNm

6.5. THE EFFECT OF THE REINFORCEMENT DIAMETER

49

6.5 The effect of the reinforcement diameter

A theoretical comparison was made regarding the effect of the reinforcement bars diameter.

The comparison was between sets with 7ø25, 11ø20 and 17ø16 mm bars with the same total

area of reinforcement in all cases. All reinforcement bars were placed in one layer in the cross

section to avoid the effect of reduced lever arm. The test was performed both in Atena and in

Concrete Section, to be able to analyse the deep beams behaviour. Figure 6.21 shows the

relationship between the crack width with the bending moment and the reinforcement stresses

from Atena. The results showed that there were no difference in crack width between using

different reinforcement diameters. Moreover, when performing the same analysis in Concrete

Section the result showed that using bars with smaller diameter reduced the crack width, see

Figure 6.22. Furthermore, it was not sufficient to compare the results from the programs since

the results were calculated based on different guidelines and assumptions as described before.

Figure 6.21. Crack width variation with bending moment or reinforcement stress, calculated

with Atena for beams with reinforcement 7ø25, 11ø20 and 17ø16 mm.

Figure 6.22. Crack width variation with bending moment or reinforcement stress s, calculated

with Concrete Section (Eurocode 2) for beams with reinforcement 7ø25, 11ø20 and 17ø16 mm.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 2000 4000 6000

Cra

ck w

idth

[m

m]


7ø25 (C1)

11ø20

17ø16

0

0,1

0,2

0,3

0,4

0,5

0,6

0 200 400 600

Cra

ck w

idth

[m

m]


7ø25 (C1)11ø2017ø16

0

0,1

0,2

0,3

0,4

0,5

0,6

0 2000 4000 6000

Cra

ck w

idth

[m

m]


7ø25 (A1)

11ø20

17ø16

0

0,1

0,2

0,3

0,4

0,5

0,6

0 200 400 600

Cra

ck w

idth

[m

m]


7ø25 (A1)

11ø20

17ø16


50

6.6 A comparison of different calculation codes

The theory behind concrete cracking is complex and it is difficult to measure the crack widths

precisely. Therefore, different methods have been used for estimating the crack widths through

the years. In Figure 6.23 a comparison between the crack widths calculated with different

dimensioning guidelines are illustrated where the deep beam with cross section 1 is used. The

comparison was made between Eurocode 2 (SS-EN 1992-1-1, 2005), BBK 04 (Boverket,

2004), B7 (Humble, 1969) and Atena (Červenka, et al., 2011). Here, BBK 04 and B7 are two

old Swedish tentative standard specifications for limitation of crack widths in reinforced

concrete structures. The difference between the curves could be explained by which calibration

coefficient that is used in each particular design code. Eurocode 2 is the latest and currently the

code used in Sweden today. The results from Eurocode 2 were similar to the results from BBK

04 which can be explained by the fact that both design codes are calibrated in a similar way and

follow the same methodology and mechanical model. In some cases, Eurocode 2 can be

complemented by BBK 04 (Engström, et al., 2010). For example, the design of deep beams are

not well described in Eurocode 2 and therefore the designers can be referred to use the truss

model in BBK 04 when suitable.

The calculations according to B7 showed a much higher crack width than the three other

guidelines that might depend on previous material properties i.e. concrete strength class, which

have changed as the construction industry developed. The finite element program Atena, which

is based on Model Code 1990, gave the smallest crack widths compared to the results from

other guidelines. Atena seems to not consider any safety factors during calculation of the crack

width, which can explain the obtained small crack widths values.

Figure 6.23. A comparison between crack widths calculated according to Eurocode 2, BBK04,

B7 and Atena, performed considering reinforcement stresses of 200, 300 and 400 MPa. The

analysed deep beam consists of 7ø25 mm in one layer.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 100 200 300 400 500

Cra

ck w

idth

[m

m]


EC2BBK04B7Atena

6.7. ADDITIONAL TENSILE REINFORCEMENT

51

6.7 Additional tensile reinforcement

In previous parts, the effect of using crack reinforcement have been investigated in order to

optimize the reduction of the crack widths. All the deep beams with different cross sections

gave similar results for the crack widths. Nevertheless, with small variations the beams C2, C6

and C9 presented the smallest crack widths depending on tensile and crack reinforcement bars

and their position, two factors that could complicate the construction work. An additional test

has been made using only tensile reinforcement bars ø25 mm, where the total area of the

reinforcement bars was increased to 9ø25 mm and thereby being comparable to the beams C1

(7ø25 mm), C2, C6 and C9, see Figure 6.24. The results showed that the increased area with

only tensile reinforcement minimized the crack width sufficiently compared to other

reinforcement alternatives, according to investigations in Atena.

Figure 6.24. Crack width variation with bending moment for beam cross sections C1, C2, C6,

C9 and 9ø25, calculated with Atena.

0

0,1

0,2

0,3

0,4

0,5

0,6

0 1000 2000 3000 4000 5000 6000 7000

Cra

ck w

idth

[m

m]


9ø25C1C2C6C9


52


53

7 Discussion and conclusions

7.1 Verification of the results in Atena

The convergence verification made with the numerical and analytical programs showed

expected crack widths, with similar results for a conventional beam from both Atena and

Concrete Section. However, the results began to differ with decreased length of the beam, i.e.

with decreasing length/height ratio as the beams reached the geometry for a deep beam. The

calculated crack widths for the deep beams were much wider from Concrete Section than with

Atena, probably because the programs used different prerequisites.

Concrete Section follows the descriptions in Eurocode 2, which covers the calculations of crack

widths in deep beams briefly. Further, the numerical program does not take the actual length

into account, i.e. the Bernoulli’s hypothesis is assumed to be valid for all beams including deep

beams. The numerical calculation in Atena is based on Model Code 1990 and thereby can

forecast the crack width of deep beams more specifically. Atena considers the non-linear

behaviour of the stress distribution in a deep beam where the load is carried by arch action in

comparison to Concrete Section, where the stress distribution is assumed linear. Eurocode 2

generally uses more simplifications and higher safety factors which results in sufficient safety

margins compared to Atena which reflects the actual size of the crack widths, e.g. crack widths

measured in situ. According to Figure 6.1, where the crack width for a conventional beam

analysed with both Atena and Concrete Section is shown, the obtained crack widths at high

reinforcement stresses differ more between the programs. This could depend on the different

reinforcement stress distribution and lever arms that is assumed in each program.

7.2 Load impact and crack development

The crack development analysis was performed by assuming varying different external

concentrated and distributed loads. The behaviour of the concrete deep beam can be connected

to the load dependence and independence of the external load. The cracks were initiated when

the moment became large enough for developing tensile stresses greater than the concrete

tensile strength. Both the deep beams with evenly distributed loads and with a concentrated

load at mid-span had the maximum moment at their centre, i.e. where the maximum bending

cracks occurred.

The crack pattern in the deep beams varied depending on which external load that was applied.

The deep beams with the distributed load had a wider crack pattern over its entire length while

the deep beams with concentrated loads had a more concentrated crack pattern, towards the

middle of the beam. In both deep beams, the cracks were initiated at the middle of the lower

edge and propagated upwards, while new cracks appeared towards the supports. The deep

beams with evenly distributed loads showed that the cracks appeared at the supports while the

deep beams with concentrated loads had no such cracks. The bending reinforcement that takes

care of the tensile forces after concrete cracking prevents the beam from early failure. The

higher the load becomes, the higher the reinforcement stress will be. A conclusion drawn from


54

the tests with Atena and Eurocode 2 calculations was that the crack width would grow with

increasing load. This also means that the stresses in the reinforcement increase with increased

load. This relationship is set for concentrated loads, distributed loads and pure bending moment.

The corresponding moments from the concentrated loads were approximately in the same order

of magnitude as the moments from the distributed loads, for the same reinforcement stress. In

Eqs. (7.1) and (7.2) the bending moment is calculated for the distributed and concentrated loads

for the deep beams when reaching 300 MPa.

𝑀 =𝑞𝑙2

8=

1 195∙52

8= 3 734 kNm (7.1)

𝑀 =𝑃𝑙

4=

2 835∙5

4= 3 544 kNm (7.2)

7.3 Equivalent diameter

According to the results calculated with Concrete Section, the crack widths were clearly

dependent on the equivalent diameter, so that smaller bar diameters resulted in smaller crack

widths in the beam. Calculations with Concrete Section seemed to take the effect of each

reinforcement bars into account since it was possible to place the bars as desired, see Figure

6.22. However, Atena seemed to not consider the effect of the different diameters of the

reinforcement bars as much, which was particularly noticeable for the reinforcement bars with

different diameters shown in Figure 6.21 where the reinforcement bars were placed in one

layer. When the reinforcement bars were placed in different layers the program accounted for

the varying bar diameters more accurate, especially at low reinforcement stresses, see Figure

6.6. The difficulty of observing how Atena distributed the bars made it hard to know if they

were placed as desired. The algorithm in Atena might thus consider the total area of the

reinforcement bars as independent of the actual size of the bars during calculations of crack

width.

7.4 The relation between reinforcement and crack

width

7.4.1 Position of the bending reinforcement

According to test examples, the capacity of the reinforcement is highly dependent of its position

in the vertical direction. If the bending reinforcement is placed high up in the cross section of

the beam, the capacity of the reinforcement cannot be effectively utilized. The reduction of the

crack width depends on the lever arm from thle reinforcement (𝐹𝑠), to the resultant force of the

concrete (𝐹𝑐). This means that the reinforcement will be more effective in carrying the bending

moment (𝑀) the lower it is placed, see Figure 7.1.

7.5. THE EFFECT OF INCREASED TOTAL AREA

55

Figure 7.1. The cross section force from the reinforcement, Fs counteracts the bending moment,

M (Ansell, et al., 2014).

7.4.2 Minimum reinforcement

A high reinforcement content normally gives smaller crack widths and increases the ability to

redistribute the tensile stresses. Minimum reinforcement is preferred for control of cracking

although the concrete structure sometimes can withstand some of the tension forces without its

contribution. The reason for using minimum reinforcement was to minimize the size of shear

and bending cracks, which thereby increased the shear- and bending capacity. Another reason

for using minimum reinforcement in the deep beam was here to reduce the crack widths above

the bending reinforcement in order to minimize the risk that the cracks became wider than at

the level of the bending reinforcement. Although only the bending cracks have been considered

in this work, it can be noted that the horizontal minimum reinforcement ø10s100 mm had a

significant effect on reducing the size of both bending- and shear cracks.

7.5 The effect of increased total area

The main purpose of using crack reinforcement was to achieve many thinner cracks instead of

a few wider. Thin cracks are seen as less of a problem compared to wide cracks since these

reduce the durability significantly. The result from the calculations with Atena showed that

there was an improvement regarding the reduction of crack widths when using crack

reinforcement in combination with tensile reinforcement compared to using tensile

reinforcement only. However, this improvement decreased by using reinforcement in multiple

layers since a tensile reinforcement bar 1ø25 mm needed to be replaced by approximately six

crack reinforcement bars 6ø10 mm in order to achieve the same total reinforcement area. The

main disadvantage was that more space was required to place all reinforcement bars in the cross

section, which reduced the lever arm. As mentioned before, the reduction of the lever arm

resulted in a reduced capacity for the reinforcement and the cracks might unintentionally

become coarser than expected. The conclusion is that the crack reinforcement was not as

effective as it seemed at a first glance. Furthermore, the additional test where two tensile

reinforcement ø25 mm bars were added to the case with tensile reinforcement only, showed

significant reduction of both crack widths and reinforcement stresses. Another conclusion is

that increasing the total area by adding a few more tensile reinforcement ø25 mm bars

influenced the reduction of crack width more compared to using a combination of crack- and

tensile reinforcement. The suggested reinforcement configurations using crack- and tensile

reinforcement can be used for prefabricated concrete deep beams but they make in situ

construction work complicated, which in turn result in high building costs and high time


56

consumption. Therefore, the increased total area and the use of tensile reinforcement only could

be a simpler alternative.

Moreover, a comparison was made considering crack widths calculated using different

dimensioning guidelines. The compared codes were B7, BBK 04 and Eurocode 2. These

calculations reflected the development of the material properties and the crack width estimation

in reinforced concrete structures through the years. The guideline B7 presented almost 1.5 times

higher values of crack widths as in the other guidelines. Eurocode 2 and BBK 04 presented

similar results of the crack widths. Further, the finite program Atena gave the smallest crack

widths compared to the other three guidelines. The difference between the results could be

explained by the fact that each guideline is based on specific assumptions and thereby uses

different safety margin. This means that Atena represents the real crack widths which means

that the structure should not be dimensioned according to these results and thereby it is

important that accurate safety margins are considered.

7.6 Further studies

To achieve a complete overview of the cracking process, additional studies has to be made. The

program Atena 2D is not entirely reliable because the user does not have complete control of

the exact placement of the reinforcement bars. This means that it would be preferable to analyze

some beams with another finite element program to investigate if there are any differences

between the results. Moreover, it would also be interesting to analyze the beams in Atena 3D

(Cervenka Consulting, 2011) to see if it is possible to achieve better control of the positions of

the reinforcement bars. Another solution could also be to improve the finite element model with

more data about the steel reinforcement since it plays a central role in the development of

cracks.

Furthermore, additional studies could be made to investigate if there are other options for

reducing the crack widths other than increased reinforcement area and the use of the crack

reinforcement. One suggestion to a further study is to investigate the influence of the minimum

reinforcement considering crack width. This would probably not only be profitable for reducing

bending cracks, but also for reducing shear cracks.


57

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SS-EN 1992-1-1, 2005. Eurokod 2: Dimensionering av betongkonstruktioner-Del 1-1

Allmänna regeler och regeler för byggnader, Stockholm: SIS.

StruSoft AB, 2015. Concrete Section [Computer program]. [Online]

Available at: http://www.strusoft.com/products/win-statik

StruSoft, 2015. Theory Manual for Concrete Section. Stockholm: StruSoft.

Svenska Betongföreningen, 2010. Svenska Betongföreningens handbok till Eurokod 2.

Betongrapport nr 15 Volume I. Stockholm: Svenska Betongföreningen.

Walther, R. & Leonhardt, F., 1966. Wandartige Träger. Berlin: Deutscher Ausschuss für

Stahlbeton.

Winter, G. & Nilson , A. H., 1989. Design of Concrete Structure. 11th ed. New York:

McGraw-Hill Company.


59

Appendix

A Control calculation

A.1 Control calculations for moment in Concrete

Section

The calculations was used to control that Concrete Section and hand calculations gives the same

moment at the same stress. The example below shows the moment for the beams with a cross

section of 2500 ∙ 400 𝑚𝑚2. The calculation is performed in MathCad (PTC, 2015).

Appendix

A. THE EFFECT OF REINFORCEMENT CONFIGURATION ON CRACK WIDTHS IN CONCRETE DEEP BEAMS

60

A.2 Control calculations according to Eurocode 2

for crack widths in concrete section

Hand calculations according to Eurocode 2 and concrete section gives the same result. The

calculation is performed in MathCad (PTC, 2015).


61

The test verifies that concrete section can be used for calculations according to Eurocode 2

since they give almost the same result. Observe that the crack widths with Concrete Section for

this case is calculated without horizontal minimum reinforcement.

A.3 Crack width calculations according to BBK04

The calculation was made according to BBK04 and are performed in MathCad (PTC, 2015).


62

A.4 Crack width calculations according to B7

The calculation was made according to B7 and are performed in MathCad (PTC, 2015).


63


64

B. THE EFFECT OF REINFORCEMENT CONFIGURATION ON CRACK WIDTHS IN CONCRETE DEEP BEAMS

65

B Calculation of the fracture energy

The fracture energy is described in chapter 2.4 and was calculated according to the calculations

below.

Since the aggregate size 12 mm was used in Atena, the mean value of the fracture energy of

aggregate sizes 8 mm and 16 mm are used.

70𝑁𝑚

𝑚2 is the fracture energy used in Atena.

B. THE EFFECT OF REINFORCEMENT CONFIGURATION ON CRACK WIDTHS IN CONCRETE DEEP BEAMS

66

C. THE EFFECT OF REINFORCEMENT CONFIGURATION ON CRACK WIDTHS IN CONCRETE DEEP BEAMS

67

C Calculations of the crack widths in

Concrete Section and Atena

C.1 Presentation of the beams

The investigation consisted of different beams both deep beams that were analysed both

analytically and numerically. The analytical calculation was based on Eurocode 2 using the

program Concrete Section and the numerical calculation was based on finite element method

using Atena 2D. Each analyse is described in cases below.

Beam A represents deep beams calculated in Concrete Section according to Eurocode 2.

Beam C represents deep beams loaded with distributed load modelled in Atena.

Beam D represents deep beams loaded with concentrated load calculated in Atena.

Every category from Beams A, C and D consists of 10 different Cross Sections which are

illustrated in Figures 5.3. The amount of reinforcement area was the same for all deep beams.

To enable a good comparison the reinforcement diameters and positions were the same in case

A1 as case C1, D1.

C.2 Equivalent diameter

The concrete strength used in all the tests was C30/37. Since the diameter was varying, it was

difficult to achieve the same amount of area in all cases therefore the best approximate values

was used for every case. The total reinforcement area were approximately the same for every

case listed below. In all cases the crack width was calculated based on the actual size of the

reinforcement bars; the equivalent diameter was only used to compare how the crack width

depends on the equivalent diameter as:

øeq =n1ø1

2+n2ø22

n1ø1+n2ø2

Table C.1. Equivalent diameters of each beam.

Beam 1 2 3 4 5 6 7 8 9 10

øequ

[mm]

25 19.4 16.8 19.4 19.4 21.1 19.9 18.7 21.1 19.9


68

C.3 Deep beam in Concrete Section

𝜎𝑠 = 115 MPa → 𝑀 = 1 010 kNm

𝜎𝑠 = 200 MPa → 𝑀 = 1 735 kNm

𝜎𝑠 = 250 MPa → 𝑀 = 2 170 kNm

𝜎𝑠 = 300 MPa → 𝑀 = 2 605 kNm

𝜎𝑠 = 400 MPa → 𝑀 = 3 480 kNm

𝜎𝑠 = 500 MPa → 𝑀 = 4345 kNm

ℎ = 2500 mm 𝑡 = 400 mm

Table C.2. The reinforcement stresses for beams A1-A10.

Beam Reinforcement

σs1

[MPa]

σs2

[MPa]

σs3

[MPa]

σs4

[MPa]

σs5

[MPa]

σs6

[MPa]

A1 7ø25 116 200 250 300 400 500

A2 6ø25, 9ø10 114 196 246 295 394 492

A3 5ø25, 6ø10, 9ø10 117 201 251 302 403 500

A4 6ø25, 9ø10 113 194 242 291 388 485

A5 6ø25, 9ø10 113 195 244 293 391 488

A6 5ø25, 6ø16 113 194 243 292 390 487

A7 4ø25, 8ø16 115 198 247 297 396 495

A8 3ø25, 4ø16, 7ø16 113 195 244 292 391 488

A9 5ø25, 6ø16 112 192 241 289 386 482

A10 4ø25, 8ø16 114 197 246 295 394 492


69

Table C.3. The maximum bending crack widths at reinforcement stress 200, 300 and 400 MPa

for beams A1-A10.

Beam Reinforcement

Maximum bending crack [mm]

σs1 σs2 σs3 σs4 σs5 σs6

A1 7ø25 0.10 0.18 0.23 0.30 0.45 0.60

A2 6ø25, 9ø10 0.09 0.15 0.19 0.25 0.38 0.51

A3 5ø25, 6ø10, 9ø10 0.09 0.15 0.18 0.24 0.36 0.49

A4 6ø25, 9ø10 0.09 0.15 0.19 0.25 0.37 0.49

A5 6ø25, 9ø10 0.09 0.15 0.19 0.25 0.38 0.50

A6 5ø25, 6ø16 0.09 0.15 0.20 0.27 0.40 0.53

A7 4ø25, 8ø16 0.09 0.15 0.20 0.26 0.39 0.52

A8 3ø25, 4ø16, 7ø16 0.09 0.15 0.18 0.25 0.37 0.49

A9 5ø25, 6ø16 0.09 0.15 0.20 0.26 0.39 0.51

A10 4ø25, 8ø16 0.09 0.15 0.19 0.26 0.39 0.51


70

C.4 Deep beam in Atena with distributed load

The Tables C.4 and C.5 show the results from the numerical calculations in Atena. The mesh

size 0.05 m was used. In Atena a higher load is needed to achieve the same stress as in concrete

section. This is because Concrete Section is simplifying the calculations as in Eurocode 2.

𝜎𝑠1 = ~100 − 115 MPa → 𝑀 = 1600 kNm

𝜎𝑠2 = ~200 MPa → 𝑀 = 2 672 kNm

𝜎𝑠3 = ~250 MPa → 𝑀 = 3 188 kNm

𝜎𝑠4 = ~300 MPa → 𝑀 = 3 734 kNm

𝜎𝑠5 = ~400 MPa → 𝑀 = 4 750 kNm

𝜎𝑠6 = ~450 − 500 MPa → 𝑀 = 5281 − 5750 kNm

𝐿 = 5000 mm ℎ = 2500 mm 𝑡 = 400 mm

Table C.4. The reinforcement stress for beams C1-C10.

Beam

Reinforcement

σs1

[MPa]

σs2

[MPa]

σs3

[MPa]

σs4

[MPa]

σs5

[MPa]

σs6

[MPa]

C1 7ø25 104 200 250 300 400 453

C2 6ø25, 9ø10 116 206 257 311 407 500

C3 5ø25, 6ø10, 9ø10 117 212 265 315 415 500

C4 6ø25, 9ø10 153 265 320 379 475 500

C5 6ø25, 9ø10 140 237 286 341 446 500

C6 5ø25, 6ø16 112 202 248 300 393 470

C7 4ø25, 8ø16 109 201 250 304 400 491

C8 3ø25, 4ø16, 7ø16 115 214 261 310 401 500

C9 5ø25, 6ø16 112 198 250 300 396 460

C10 4ø25, 8ø16 116 211 250 315 412 483


71

Table C.5. The maximum crack width in bending for beams C1-C10.

Beam Reinforcement



C1 7ø25 0.09 0.20 0.25 0.30 0.40 0.46

C2 6ø25, 9ø10 0.10 0.18 0.23 0.29 0.39 0.49

C3 5ø25, 6ø10, 9ø10 0.11 0.22 0.27 0.33 0.44 0.53

C4 6ø25, 9ø10 0.12 0.22 0.27 0.33 0.43 0.50

C5 6ø25, 9ø10 0.12 0.22 0.27 0.33 0.43 0.50

C6 5ø25, 6ø16 0.09 0.19 0.24 0.30 0.40 0.48

C7 4ø25, 8ø16 0.10 0.20 0.25 0.30 0.40 0.49

C8 3ø25, 4ø16, 7ø16 0.11 0.22 0.27 0.33 0.44 0.54

C9 5ø25, 6ø16 0.10 0.19 0.24 0.29 0.38 0.44

C10 4ø25, 8ø16 0.10 0.20 0.24 0.30 0.40 0.47


72

Crack pattern from Atena

The deep beams C1-C10 are loaded with a distributed load of 1 200 kN/m over the entire deep

beam. The crack pattern and cross section of the deep beams are illustrated in Figures C.1-

C.10. The maximum crack width in bending of each deep beam is represented in Table C.5.

C1

Figure C.1. The crack pattern of beam C1 when applying a distributed load.

C2


C3



73

C4


C5


C6



74

C7


C8


C9



75

C10



76

C.5 Deep beam in Atena with concentrated load

Table C.7 shows the crack width at different reinforcement stresses calculated in Atena. The

mesh size 0.025 m was used.

𝜎𝑠1 = ~100 − 110 MPa → 𝑀 = 1 781 kNm

𝜎𝑠2 = ~200 MPa → 𝑀 = 2 513 kNm

𝜎𝑠3 = ~250 MPa → 𝑀 = 3 038 kNm

𝜎𝑠4 = ~300 MPa → 𝑀 = 3 544 kNm

𝜎𝑠5 = ~380 − 475 MPa → 𝑀 = 4 313 − 4 519 kNm

𝐿 = 5000 mm ℎ = 2500 mm 𝑡 = 400 mm

Table C.6. The reinforcement stress for beams D1-D10.

Beam Reinforcement

σs1

[MPa]

σs2

[MPa]

σs3

[MPa]

σs4

[MPa]

σs5

[MPa]

D1 7ø25 122 199 250 300 381

D2 6ø25, 9ø10 134 214 272 318 400

D3 5ø25, 6ø10, 9ø10 126 220 273 325 415

D4 6ø25, 9ø10 184 268 306 371 475

D5 6ø25, 9ø10 156 240 296 349 436

D6 5ø25, 6ø16 125 191 260 304 380

D7 4ø25, 8ø16 125 203 255 307 400

D8 3ø25, 4ø16, 7ø16 131 204 262 317 421

D9 5ø25, 6ø16 144 222 274 327 417

D10 4ø25, 8ø16 126 206 258 306 404


77

Table C.7. The maximum crack width in bending for beams D1-D10.

Beam Reinforcement


σs1 σs2 σs3 σs4 σs5

D1 7ø25 0.11 0.19 0.24 0.29 0.37

D2 6ø25, 9ø10 0.09 0.16 0.21 0.25 0.32

D3 5ø25, 6ø10, 9ø10 0.12 0.20 0.25 0.30 0.38

D4 6ø25, 9ø10 0.13 0.21 0.26 0.31 0.40

D5 6ø25, 9ø10 0.13 0.22 0.27 0.32 0.40

D6 5ø25, 6ø16 0.09 0.15 0.21 0.25 0.33

D7 4ø25, 8ø16 0.10 0.17 0.22 0.27 0.35

D8 3ø25, 4ø16, 7ø16 0.14 0.20 0.25 0.31 0.41

D9 5ø25, 6ø16 0.10 0.17 0.22 0.27 0.35

D10 4ø25, 8ø16 0.10 0.17 0.22 0.26 0.35


78

Crack pattern from Atena

The deep beams D1-D10 are loaded with a concentrated load at the centre. The crack pattern

and cross section of the deep beams are illustrated in Figures C.11-C.20. The maximum crack

width in bending for each deep beam is represented in Table C.7.

D1

Figure C.11. The crack pattern of beam D1 when applying a concentrated load.

D2



79

D3


D4


D5



80

D6


D7


D8



81

D9


D10



82

C.6 The convergence analysis of Atena

A convergence analyse was made of beam C6 to investigate how the crack width in Atena varies

for different lengths of the beam.

Table C.8. The reinforcement stress for beams A6 and 6 (with different lengths).

Beam Length

σs1

[MPa]

σs2

[MPa]

σs3

[MPa]

σs4

[MPa]

σs5

[MPa]

A6 - 113 194 292 394 487

C6 5.0 m 104 201 300 400 469

6* 7.5 m 117 197 299 399 500

6* 10.0 m 133 198 301 403 500

6* 15.0 m 129 192 293 397 500

Table C.9. The maximum crack width in bending for beams A6 and 6 (with different lengths).

Beam Length



A6 - 0.09 0.16 0.27 0.40 0.53

C6 5.0 m 0.08 0.19 0.30 0.41 0.48

6* 7.5 m 0.10 0.18 0.29 0.39 0.48

6* 10.0 m 0.12 0.19 0.29 0.39 0.50

6* 15.0 m 0.11 0.18 0.28 0.38 0.54

* The beam consists of Cross Section 6.


83

C.7 The effect changing reinforcement diameter

Atena

𝜎𝑠1 = ~100 − 118 MPa → 𝑀 = 1 594 kNm

𝜎𝑠2 = ~200 MPa → 𝑀 = 2 672 kNm

𝜎𝑠3 = ~250 MPa → 𝑀 = 3 188 kNm

𝜎𝑠4 = ~300 MPa → 𝑀 = 3 734 kNm

𝜎𝑠5 = ~400 𝑀𝑃𝑎 → 𝑀 = 4 750 kNm

𝜎𝑠6 = ~432 − 455 MPa → 𝑀 = 5 110 − 5 282 kNm

There is a relation between equivalent reinforcement diameter of the bars and the crack width

i.e. the crack width decreases with smaller bar diameter. In this test, the same total area of

reinforcement bars is used. The test was performed in Atena.

Table C.10. The reinforcement stress for beams with different reinforcement diameters.

Reinforcement

σs1

[MPa]

σs2

[MPa]

σs3

[MPa]

σs4

[MPa]

σs5

[MPa]

σs6

[MPa]

7ø25 103 200 251 300 399 453

11ø20 102 197 247 299 398 432

17ø16 118 204 256 309 408 455

Table C.11. The crack width in bending for different reinforcement stresses and different

diameters.

Reinforcement



7ø25 0.09 0.20 0.25 0.30 0.40 0.46

11ø20 0.09 0.20 0.25 0.30 0.40 0.43

17ø16 0.12 0.20 0.26 0.31 0.41 0.46


84

Concrete Section

The crack width is small for smaller diameters of the reinforcement bars, and increases with

larger diameters. As seen in the graph, the difference between crack widths in beams become

larger when the reinforcement stress increases.

𝜎𝑠1 = ~116 MPa → 𝑀 = 1 010 kNm

𝜎𝑠2 = ~200 MPa → 𝑀 = 1 735 kNm

𝜎𝑠3 = ~300 MPa → 𝑀 = 2 555 kNm

𝜎𝑠4 = ~400 MPa → 𝑀 = 3 500 kNm

𝜎𝑠5 = ~500 MPa → 𝑀 = 4 345 kNm

Table C.12. The reinforcement stress for beams with different reinforcement diameters.

Reinforcement

σs1

[MPa]

σs2

[MPa]

σs3

[MPa]

σs4

[MPa]

σs5

[MPa]

7ø25 116 200 300 400 500

11ø20 116 199 293 401 497

17ø16 117 211 295 405 600


diameters.

Reinforcement



7ø25 0.10 0.18 0.31 0.45 0.60

11ø20 0.09 0.15 0.25 0.39 0.51

17ø16 0.08 0.14 0.22 0.34 0.44


85

C.8 The effect of increased area

The increased area by adding two extra tensile reinforcement bars

𝜎𝑠1 = ~100 MPa → 𝑀 = 1 594 − 1 875 kNm

𝜎𝑠2 = 155 𝑎𝑛𝑑 200 MPa → 𝑀 = 2 672 kNm

𝜎𝑠3 = 195 𝑎𝑛𝑑 251 MPa → 𝑀 = 3 188 kNm

𝜎𝑠4 = 244 𝑎𝑛𝑑 300 MPa → 𝑀 = 3 734 kNm

𝜎𝑠5 = 317 𝑎𝑛𝑑 399 MPa → 𝑀 = 4 750 kNm

𝜎𝑠6 = 348 𝑎𝑛𝑑 453 𝑀𝑃𝑎 → 𝑀 = 5 172 − 5 282 kNm

Table C.14. The reinforcement stress for beams with different number of reinforcement bars.

Reinforcement

σs1

[MPa]

σs2

[MPa]

σs3

[MPa]

σs4

[MPa]

σs5

[MPa]

σs6

[MPa]

7ø25 (C1) 103 200 251 300 399 453

9ø25 100 150 195 244 317 348


number of reinforcement bars.

Reinforcement



7ø25 0.09 0.20 0.25 0.30 0.40 0.46

9ø25 0.09 0.15 0.19 0.24 0.32 0.35

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