Shear cracks in reinforced concrete in serviceability limit state

serviceability limit state

Alexander Chemlali Rickard Norberg

June 2015

TRITA-BKN, Examensarbete 449, Betongbyggnad ISSN 1103-4297

ISRN KTH/BKN/EX–449–SE

Master Thesis in Concrete Structures

©Alexander Chemlali, Rickard Norberg Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering Division of Concrete Structures

Stockholm, Sweden, 2015

Shear cracks are formed when high oblique tensile stresses, e.g. in thin webs, exceed the tensile strength. A known example of this phenomenon is the extensive shear cracks that were found on the box-girder bridges Gröndal and Alvik, which were mainly caused by insufficient amount of shear reinforcement. In order to avoid this incident (inadequate amount of shear reinforcement), the reinforcement stress is often being assumed as a ultimate limit load in order to fulfill requirements regarding crack control in the service- ability limit state (SLS). This method has led to a overestimation of the reinforcement amount in bridge-design. The aim of this master thesis is therefor to study the shear crack phenomenon and investigate if the amount of shear reinforcement in bridges can be reduced. The first part of this thesis studies the shear cracking behavior in concrete in a plane stress state, while the second part focus how design standards as well as manuals treats shear cracks.

Shear cracking in the reinforced concrete panels has been studied with non-linear finite element analysis and compared to experimental testings performed by the University of Toronto. Three different loading conditions for the panels has been analyzed: pure shear, compression or tension combined with shear. The panels are to represent parts of a web in a box-girder bridge that are subjected to in-plane stresses. The non-linear finite element analysis was performed in the FE-program Atena where mainly the crack propagation and crack pattern were studied. The material model in Atena is a smeared crack model with either fixed or rotated crack direction. The panel analysis, in SLS, gave various results. For loading conditions pure shear and tension/shear, the response of the FE-analysis gave a similar result regarding crack pattern but differed in size of crack width. For compression/shear, only micro-cracks developed and did not reflect the result from the real panel tests. This may be the consequence of a too stiff FE-model and the fact that, in the real tests, some cracks occurred due to out-of-plane bending.

With methods described in Eurocode 2 and the Swedish handbook for EC2, a shear crack

calculation model was created in order to determine the reinforcement stress and crack

width. As a reference for the shear crack calculations, a wing structure (1 m strip) has

been used which is part of a railway bridge located in Abisko. These calculations were

done in order to investigate if the amount of shear reinforcement could be reduced and

at the same time fulfill crack control demands in SLS. The bridge department at Tyréns

AB concluded, according to a truss model, that the wing section should be reinforced

with a amount of 14.1 cm

/m

while our model showed that the crack width demand

could be fulfilled with a equivalent amount of 9.82 cm

/m

, i.e. a reduction around

30%.

Sammanfattning

Skjuvsprickor bildas när höga sneda dragspänningar, t.ex. i tunna balkliv, överstiger draghållfastheten. Ett känt fall av detta var de omfattande skjuvsprickorna som hit- tades i lådbalkbroarna Gröndal och Alvik, som huvudsakligen var orsakat av otillräcklig mängd tvärkraftsarmering. För att undvika detta (otillräcklig mängd tvärkraftsarmer- ing) så antas ofta armeringsspänningen till en brottlast för att uppfylla sprickkraven i bruksgränstillstånd. Detta har dock lett till att armeringsinnehållet av tvärkraftsarmer- ing överskattas när man dimensionerar broar. Målet är därmed med detta examensar- bete att studera fenomenet skjuvsprickor samt undersöka ifall tvärkraftsarmeringen i broar kan reduceras. Första delen av detta examensarbete studerar skjuvspricksbeteen- det vid plant spänningstillstånd, medan den andra delen fokuserar på hur standarder samt handböcker behandlar skjuvsprickor.

Skjuvuppsprickning i armerade betongpaneler har studerats med icke-linjär finita ele- ment analyser och jämförts med experiment gjorda vid universitetet i Toronto. Tre olika lastfall har blivit analyserade: ren skjuvning, tryck eller drag tillsammans med skjuvn- ing. Panelerna ska representera olika delar av ett balkliv i en lådbalkbro som är belastat i plant spänningstillstånd. Den icke-linjära analysen har gjorts i FE-programmet Atena, där sprickmönstret samt sprickutbredningen har varit i fokus. Materialmodellen i Atena representerar sprickor som är utsmetade, där sprickriktningen är antingen fixerad eller roterande. Panelanalysen, i bruksgränstillstånd, gav varierande resultat. Lastfallen ren skjuvning samt biaxiellt drag/skjuvning gav FE-analysen en god överensstämmelse gällande sprickmönstret, men en skillnad i sprickvidd. För lastfallet biaxiellt tryck/skju- vning utvecklades endast mikrosprickor och reflekterade inte de riktiga panel-testen lika väl. Detta kan vara på grund av en för styv FE-modell och det faktum att några sprickor bildats utav böjning ut ur planet i de riktiga testen.

Med hjälp av metoder beskrivna i Eurokod 2 och Svenska handboken till Eurokod 2, kunde en beräkningsmodell skapas för att bestämma armeringsspänningar samt sprick- vidd. En ving-struktur (1 m strimla) har använts som referens för dessa beräkningar.

Vingen är en del av en järnvägsbro som finns i Abisko. Beräkningarna gjordes för

att undersöka möjligheten att reducera tvärkraftsarmeringen samtidigt som sprickbred-

dskravet i bruksgränstillstånd uppfylldes. Broavdelningen på Tyréns AB kom med

hjälp av en fackverksmodell fram till att vingen behövde armeras med innehållet 14.1

cm

/m

, medan våra beräkningar visade att sprickviddskrav kunde uppfyllas med en

armeringsmängd på 9.82 cm

/m

. En reducering på cirka 30%.

Capital Roman Symbols

V

shear capacity N

A

cross-sectional area of the shear reinforcement m

E

elastic modulus of steel P a

E

elastic modulus of concrete P a

E

mean elastic modulus of concrete P a

A

area of the tendons (pre- or post-tensioned) m

A

effective area of concrete in tension surrounding the reinforcement m

G

fracture energy -

G

coefficient that depends on aggregate size -

M bending moment Nm

V shear force N

Lower case Roman Symbols

s spacing between reinforcement bars m

z inner lever arm m

b

minimum width of the tension and compression chord m v

strength reduction factor for cracked concrete in shear %

f

design compressive strength P a

s

maximum crack spacing m

w crack width m

f

mean value of tensile strength when first crack opens P a

f

mean compressive strength P a

k

factor dependent on the duration of the load -

c concrete cover m

k

coefficient of bond properties -

k

coefficient of distribution of strain -

k

given in national annex -

k

given in national annex -

iv

f

uniaxial cylinder compressive strength P a

f

uniaxial cylinder tensile strength P a

G

fracture energy -

G

coefficient that depends on aggregate size -

f

constant in fracture energy calculation P a

s

crack shear factor %

c

tension stiffening factor %

W

critical compressive displacement m

r

coefficient of compressive strength reduction %

v

shear stress in xy-plane P a

v poissons’ ratio -

b

width of concrete strip m

Greek Symbols

σ

reinforcement stress in x-direction P a

σ

reinforcement stress in y-direction P a

σ

concrete stress in x-direction P a

σ

concrete stress in y-direction P a

σ

reinforcement stress in tension assuming cracked state P a

σ

, σ

principal stresses in concrete P a

τ

shear stress in the xy-plan P a

ρ

reinforcement ratio in x-direction %

ρ

reinforcement ratio in y-direction %

ρ

effective reinforcement area %

ρ

density of reinforcement N/m

ρ

density of concrete N/m

θ inclination of compressive stresses to x-direction ◦

θ

inclination of principal stress to y-direction ◦



strain in reinforcement in the x-direction -



strain in reinforcement in the y-direction -



mean strain in the reinforcement -



strain in concrete in the x-direction -



strain in concrete in the y-direction -



mean strain in the concrete between cracks -



strain at compressive peak -

γ

shear strain in the xy-plane -

α

coefficient of the stress state in the compression chord -

α

ratio between the elastic modulus of the reinforcement and concrete -

α

coefficient of thermal expansion -

ω

characteristic expression for crack width m

ξ

adjusted ratio of bond strength taking into account the

different diameters of pre-stressing and reinforcing steel - ξ ratio of bond strength of pre-stressed steel and reinforcement steel -

φ reinforcement bar diameter m

φ

largest bar diameter of the reinforcement m

φ

equivalent diameter of tendon m

φ

equivalent bar diameter if a mixture of bar diameters is used m

Abbreviations

ACI American concrete institute CSA Canadian standard association DOF Degrees of freedom

EC2 Eurocode 2 (Part 1) EC2-2 Eurocode 2 (Part 2)

F CM Fixed crack model coefficient F EA Finite element analysis F EM Finite element method M C90 Model code 1990 M C10 Model code 2010

M CF T The modfied compression field theory SLS Serviceability limit state

T RV F S Trafikverkets författningssamling

U LS Ultimate limit state

Abstract i

Sammanfattning ii

Acknowledgements iii

Notations iv

1 Introduction 1

1.1 Background . . . . 1

1.2 Problem . . . . 3

1.3 Method . . . . 3

1.4 Limitations and assumptions . . . . 4

2 Concrete in the serviceability limit state 5 2.1 Cracks in reinforced concrete . . . . 5

2.1.1 Shear cracks . . . . 6

2.1.2 Crack width . . . 11

2.1.3 Crack spacing . . . 12

2.2 Modified compression field theory . . . 13

3 Shear crack calculations 17 3.1 Reinforcement stresses according to the Swedish handbook for EC2 . . . . 17

3.1.1 Reinforcement stresses out of equilibrium with concrete stress . . . 18

3.1.2 Reinforcement stress according to truss model . . . 21

3.1.3 The most suitable method . . . 22

3.2 Crack width according to EC2 . . . 23

3.3 Crack spacing according to EC2 . . . 25

4 Finite element analysis 27 4.1 Finite element analysis . . . 27

4.2 Non-linear finite element analysis . . . 29

4.3 Concrete model . . . 30

4.3.1 Uniaxial stress state . . . 30

4.3.2 Multiaxial stress state . . . 32

4.3.3 Crack model . . . 34

4.3.4 Tension softening . . . 36

4.4 Non-linear analysis in ATENA 3D . . . 38

vii

4.4.1 Concrete model . . . 38

4.5 Reinforcement model . . . 41

4.6 Solution methods . . . 42

4.6.1 Convergence criteria . . . 44

5 Analysis of reinforced concrete panels 45 5.1 Toronto tests . . . 46

5.1.1 Vecchio and Collins (1986) . . . 46

5.1.2 Xie (2009) . . . 47

5.2 Non-linear analysis of panels in ATENA 3D . . . 50

5.2.1 Geometry . . . 50

5.2.2 Concrete model . . . 52

5.2.3 Reinforcement model . . . 53

5.2.4 Load cases . . . 54

5.2.5 Crack setting . . . 54

5.2.6 Solution methods . . . 55

5.2.7 Mesh convergence . . . 55

5.2.8 Results . . . 58

5.2.9 Parametric study . . . 64

5.3 Shear crack calculations . . . 72

5.3.1 Pure shear . . . 72

5.3.2 Compression/shear . . . 73

5.3.3 Tension/shear . . . 73

6 Analysis of railway bridge in Abisko 75 6.1 The bridge system . . . 75

6.2 Shear crack calculations . . . 78

7 Discussion 81 7.1 Reinforced concrete panels . . . 81

7.1.1 Parametric study . . . 82

7.1.2 Results . . . 83

7.2 Wing wall . . . 85

8 Conclusions and further research 87 8.1 Conclusions . . . 87

8.2 Further research . . . 88

A ATENA analysis calculations 93

B Shear crack calculations 97

C ATENA analysis 107

Introduction

1.1 Background

For large and slender concrete structures the formation of shear cracks in the service- ability limit state (SLS), developed due to large shear forces, have caused problems that have led to expensive repair work. A noted and famous example of this problem was the extensive cracks that were found on the light-rail commuter line bridges, Gröndal and Alvik (illustrated in figure 1.1), in Stockholm. The bridges were closed, from fall 2002 to the summer of 2003, for reparation and strengthening which caused large expenses.

The Gröndal bridge and the Alvik bridge are both box-girder bridges, which are made of concrete. The cracking were found in the webs of the hollow box-girder sections during various inspections between 2000 and 2002. The first inspection detected shear cracks and it were, in later inspections, confirmed that the cracks had propagated in numbers.

It was also noticed that the crack width had an increasing tendency and the bridges were closed due to the risk of shear failure. One reason for the formation of shear cracks are when high oblique tensile stresses in the thin webs exceed the tensile strengths. Other reasons for the excessive cracking on these bridges were high permanent loads combined with the effect of temperature and insufficient amount of shear reinforcement which did not manage to control the cracks in the serviceability limit state (Hejll and Täljsten, 2005).

1

(c) Alvik

(d) Gröndal

Figure 1.1: Shear Cracks on the Gröndal and Alvik bridge (Hejll and Täljsten, 2005)

Design methods that are available for analyzing the initiation of shear cracks and its propagation are therefor of interest. In most design codes, in the serviceability limit state, it are a necessity to either predefine a value for the reinforcement stress or limit the crack width (Malm, 2009). The design standard Eurocode 2 (part 1) for large concrete structures has clear methods for estimating crack widths and crack spacing, but regarding shear cracks there are no particular defined design method for how to calculate the reinforcement stress (EC2, 2004). The lack of an adequate evaluation method, and a method to avoid shear cracks, has led to overly safe and overestimated amounts of shear reinforcement when designing bridges in order to avoid similar consequences as in the bridges Gröndal and Alvik. This has also led to an increase of costs.

Another method for analyzing shear crack initiation and propagation is the method

with finite element analysis (FEA), where it is important to know the different limits

and conditions for each model. This is because of the complex non-linear behavior of

reinforced concrete structures. To obtain a more realistic model, the use of non-linear

FEA is more appropriate in order to analyze the change of behavior in both the ultimate

limit state and the serviceability limit state when concrete cracks (Betongföreningen,

2010).

1.2 Problem

It is rare that large concrete structures collapse because of an exceeded ultimate load ca- pacity, it is however more common that the durability of the structure is affected due to non-satisfying requirements in the serviceability state regarding crack widths and deflec- tions (Malm, 2009). Every day design work requires adequate methods and assumptions that can be easily applied and ensure the structures behavior for serviceability design.

When designing bridges and concrete structures generally in the serviceability limit state, the standard Eurocode 2 provides satisfying methods for calculating crack widths and crack spacing concerning cracks. The problem when designing for shear cracks is the inability of determine the reinforcement stress, which affects the magnitude of the crack width and the amount of shear reinforcement in the structure. A consequence is that reinforcement stresses are set to high (i.e. safe) and constant value which therefor leads to a reinforcement amount that is heavily overestimated.

By constructing different models with varying loading conditions in the non-linear finite element software Atena and applying methods described in Eurocode 2 together with the handbook for EC2 from the Swedish concrete association, the aim is to investigate and find values of the reinforcement stress which will reduce the amount of shear re- inforcement in structures. Since the reinforcement stress has a big effect on the crack width, a control of the crack width will be performed so they satisfy established regula- tions, i.e. Eurocode 2 (part 2) and the national annex to EC2 of the Swedish transport administration, for bridges. The aim of this master thesis is to show that a decrease of shear reinforcement is possible without an increased risk of shear cracks when designing in the serviceability limit state.

A second subject of this master thesis is to get an advanced understanding of the phe- nomenon of shear cracks as well as getting a deeper knowledge on how to model and analyze concrete structures, e.g. bridges, with non-linear finite element programs.

1.3 Method

In order to reach our aims of this master thesis, the following methodology were used:

To understand the formation of cracks in reinforced concrete and the phenomenon

of shear cracks a literature study was performed which focuses on different factors

affecting the response of the reinforced concrete. Due to the complex behavior

of reinforced concrete, the Modified Compression Theory by Vecchio and Collins

was also studied.

Since the study was performed with the non-linear finite element program Atena, numerical models for concrete as well as a constitutive overview of finite element analysis and non-linear finite element analysis in general are presented.

An investigation of how regulations, Eurocode 2 Part 1-1 (Concrete structures) and Part 2 (Bridges), handle crack calculations was performed. This was done in order to create our own model of crack control. The reinforcement stresses were determined in accordance with described and recommended methods in the handbook for EC2 of the Swedish concrete association.

Tests on concrete panels have been done by the University of Toronto in order to capture the shear behavior of reinforced concrete elements. Simulations of these panel tests were performed with the finite element software Atena 3d and the results were compared with the real tests. This was done to investigate if we could simulate the same crack patterns and crack widths as the Toronto tests. The aim of simulating these panel tests were also to obtained a better knowledge of how to model non-linear problems in the software Atena 3d.

The last stage of this master thesis was to analyze a railway-bridge in Abisko.

This was done to see if methods described in the handbook for EC2 could be implemented in order to reduce the amount of shear reinforcement. The railway bridge is a project that Tyréns’ bridge department has re-calculated.

1.4 Limitations and assumptions

This thesis was limited to study the phenomenon of shear cracks induced by stresses in a plane stress state, thus has no other possible source of cracking been studied. Such sources are e.g. the effect of temperature, plastic shrinkage and creep. The calculations were only performed in the serviceability limit state with long-term loading conditions.

Material properties and loading conditions for the panel analysis were defined in accor- dance to a previously done experimental study by the University of Toronto. In this previous experimental study, six panels with different loading conditions were tested.

Our analysis were limited to three of these cases. The three cases were chosen due to their distinctly defined loading cases.

For the railway bridge in Abisko, consideration was only made to the wing wall and

therefor calculations have only been made for this part of the bridge structure. To limit

this thesis, used material properties and input values for the wing structure have been

provided by a internal report from Tyrens’ bridge department.

Concrete in the serviceability limit state

2.1 Cracks in reinforced concrete

Cracking in reinforced concrete is very hard to avoid and may be inevitable and by keeping the cracks and deformations within reasonable limit, the impairment of the ser- viceability and durability of the structure is limited. However, cracks do not heavily jeopardize the structures serviceability and durability if requirements regarding mini- mum reinforcement are satisfied as well as reinforcement stresses are restricted. Cracks appear whenever the tensile strength capacity in the concrete is lower than the ten- sile stresses where tensile stresses occur due to mechanical effects which are induced by loads. Tensile stresses can also happen due to imposed deformations which are caused by creep, settlements and so on. All the tensile forces, in a cracked cross-section, are balanced by the reinforcement in the structure and are transferred between cracks (steel bars to surrounding concrete) through bond forces. This leads to an increase, due to the contribution of the concrete, of the tensile reinforcement stiffness (i.e. tension stiff- ening)(MC10, 2010).

The need of crack control is important for mainly two reasons; the appearance and durability. For the purpose of appearance it is generally considered that cracks give reason for concern and should be avoided, it will also give the structure an esthetically unappealing appearance. Steel reinforcement needs to be protected in order to ensure the durability of the structure. When cracks form on the concrete cover it may lead to ingress of corrosive factors such as oxygen, water and chlorides. This may lead to corrosion of the steel reinforcement through carbonization and chloride penetration (Balázs, 2012).

5

According to MC10 (2010), the idealized behavior of reinforced concrete is illustrated in figure 2.1.

Figure 2.1: Behavior in tensile loaded reinforced concrete (MC10, 2010)

Under increased tensile strain (elongation) four stages are distinguish; the uncracked stage (a), the crack formation stage (b), the stabilized cracking stage (c) and the steel yield stage (d). The uncracked stage is the stage before the tensile strength is reached.

Tensile forces, in the crack formation stage, increases very little due to the occurrence of new cracks. When the amount of cracks has increased so there are no undisturbed areas (i.e. areas between cracks) left the tensile strength of the concrete cannot be achieved.

This will lead to formation of additional cracks at lower tensile stress levels. In the stabilized cracking stage, no new cracks occur but existing cracks will widen and finally the reinforcement steel will start to yield.

2.1.1 Shear cracks

Shear cracks are caused either by shear forces or by torsional moments, which could give rise to shear stresses. Cracks in concrete occur when it is subject to stresses that are larger than the materials tensile capacity, f

. Typical for shear cracks is that they are often inclined to the longitudinal axis of the member.

When concrete cracks, the material directly seeks for a new stress equilibrium. To achieve equilibrium there must be longitudinal and transverse reinforcement or a good interlocking of the aggregates between two crack faces to absorb the shear stresses. If these parts cannot be fulfilled, shear cracks can result in three different failure modes;

sliding between crack faces, crushing of the intact concrete between cracks or crushing

of the concrete in the compressive zone (Broo et al., 2008).

Members without shear reinforcement

In an un-cracked concrete element with a constant cross-section that is subjected to a shear force (V

) at a level y from the neutral layer, the shear stress τ

is produced.

According to shear equilibrium, the horizontal and the vertical shear stresses is equal in magnitude at that point. The shear stress is given by:

τ

= V

· S

b · I

(2.1)

where

V

is the shear force [N]

S

is the static moment with respect to the neutral plane [m

] I

is the moment of inertia with respect to the neutral plane [m

] b is the width of the cross-section [m]

Shear stresses arise when the moment and the axial forces varies along the length of the member in the tension and compression zone (Ansell et al., 2013).

The concrete between two flexural shear cracks is called a beam lamella. In a crack, all the tensile forces are absorbed by the tension reinforcement. The lamella beam is in equilibrium when the cracks can take the shear stresses (τ ). When the crack has grown to a certain width, the concrete loses the ability to take shear stresses due to the loss of friction between crack faces.

Concrete subjected to external loading can, when its capacity has been reached, crack in different ways. At first flexural cracks or bending cracks occur in the tension zone.

Then when the cracks reaches the compression zone, a change of direction could occur and the cracks moves towards the loading point with an inclined angle compared to the longitudinal direction. Last type of crack is the web shear crack that occurs in the uncracked web around the neutral axis.

Flexural shear cracks occur from flexural bending cracks. The bending cracks arise in

areas where the magnitude of the moment is largest (normally mid-span for an idealized

beam) and vertical in shape. When the failure load is reached, it forms a diagonal crack

from the tensile reinforcement in the lower edge of the member and develops towards

the compression zone where it later flattens out, see figure 2.2. In the region where the

shear crack intersects, the tensile reinforcement has a tendency of form horizontal cracks

along the reinforcement.

Web shear cracks are normally formed inside webs in the area of the neutral axis, where the tensile stress reaches the tensile capacity of the concrete. These cracks are then propagating towards the free edges in the cross section. Their failure mode is more direct and hard to observe since the crack could form inside a structure (Ansell et al., 2013).

A difference from flexural shear cracks is that the web shear crack is not bound to areas with flexural cracks. It also requires that there are no flexural shear cracks in any cracked part of the beam. Normally, when designing, it is the flexural shear crack failure that is used but in pre-tensioned beams, which can take high loads without getting any flexural cracks, the web shear cracks could be decisive.

Figure 2.2: Illustration of different crack modes

In the tensile area of the beam, the shear force could be absorbed by the tensile rein- forcement in the longitudinal direction. The reinforcement works as a joining dowel that takes the forces induced by variable deflection of each face of a crack, see figure 2.3 (c).

If there is transverse reinforcement there could not be any variable deflection or sliding before the reinforcement starts to yield (Broo, 2008).

The shear force could also be taken up via the so-called aggregate interlocking, see figure 2.3 (b), between the crack surfaces. This ability is dependent on the aggregate size, the width of the crack and the magnitude of the compressive force keeping the crack together.

All of these factors are interacting, in a structure, when "carrying" a load. If their ability

to carry load has been consumed, the compression zone takes all of the shear force above

the flexural cracks, which could be seen in figure 2.3 (a). This could cause a crushing

failure of the concrete in the compressive zone (Ansell et al., 2013).

When the reinforcement content is increased, the longitudinal reinforcement influences the load carrying capacity at flexural shear failure in three different ways: the height of the compressive zone increases, which increases the ability of the beam to take the shear component in the compressive zone (a). More reinforcement also prevents the crack to open, which increases the influence of friction between the aggregates in the crack (b).

The ability of the reinforcement to prevent transverse displacement increases the shear force in the reinforcement (c). This is illustrated in figure 2.3.

Figure 2.3: Shear force absorbed by (a) compressive zone, (b) aggregate interlock and (c) tensile reinforcement, reproduction from Ansell et al. (2013)

The shear capacity of a concrete member, i.e. expressed as the variable V

, is a common estimate of the strength of a member without shear reinforcement. In design, it is dependent on the shear strength which is determined by semi-empirical expressions that vary between design codes. The shear strength is dependent on factors such as height of the cross section, loading or the amount of longitudinal reinforcement content (Broo, 2008).

Shear slenderness is a ratio defined as a/d, where a is the distance from the support that a point load is acting on a beam and d is the effective height at the position of a. In tests performed by Leonhardt and Walter (1962) it were proven that beams with ratio a/d > 3 showed a crack pattern typical of a long beam, and that the shear force at failure was the same for all beams with a/d > 3.

Beams with a/d > 6 showed flexural failure when the moment was close to the moment

capacity of the beam. But for a/d > 2.5, the shear force increased rapidly with a

decreasing a/d.

The reason why shorter beams can take larger shear forces is that the forces in the beam readjusts and redistribute, which creates an arch effect that carries the load. During this arch effect, the tension reinforcement is loaded under a constant force from one support to the other. It is therefor important with a proper bond between concrete and reinforcement.

When uniform load, the shear slenderness is defined as L/d, where L is the length of the span and d is still the effective height. To get an comparison with the point load case, a/d = 2.5 can be set equal to L/(4d) = 2.5. The shear force for ratios L/d > 9 is close to constant and then increase rapidly for ratios lower than that.

For a flexural shear failure to take place there must be a bond between the reinforcement and the concrete. Bars that are ribbed or profiled create such a bond that could lead to a flexural shear failure, while smooth bars cannot create this effect. Smooth bars with end anchorage would though increase the ability to carry load until a flexural shear failure. This type of arrangement is however costly.

Test by Leonhardt and Walter (1962) of two beams with ribbed bars and two with smooth bars showed that the ones with smooth bars and end anchorage could carry higher loads. They produced few, but large cracks. The ones with ribbed produced more but finer cracks, which is preferred in practice. The smaller and more spread cracks and the positive economical character of ribbed bars make them more common in practice (Ansell et al., 2013).

Tests have shown that samples ability to carry loads decreases with increased cross- sectional height according to the fundamental formula for shear strength:

f

= V

b · d (2.2)

where

V is the shear force [N]

b is the width of the cross-section [m]

d is the height of the cross-section [m]

This is a common behavior of brittle materials. When it comes to shear failures, it could

be concluded that the crack widths increase with the cross-sectional height, which lead

to a decrease in friction between the crack surfaces.

Beams with shear reinforcement

In order to determine the required amount of shear reinforcement, a truss model was developed by Ritter and Mörsch already in 1899 and 1902 respectively. In the first model, the inclination of the compressed diagonals was determined to 45

but after extensive research it was concluded that the 45

truss model was too conservative. This led to that the model got reworked and gave rise to a new model called the addition principle model. This model added the shear capacity of a beam without shear reinforcement with the capacity of a truss.

In the latest version of Eurocode 2 (EC2, 2004), the addition principle model has been implemented to the truss model. Later on, the new truss model was further developed by Canadian scientists into the modified compression field theory (MCFT). The MCFT is a more complete theory that also considers strain compatibility and friction in cracks in an explicit way.

The truss model is reinforced with a system resembling a normal truss. The tension ties are made up of the flexural reinforcement in the tension zone and the vertical shear bars. Compressed parts are the concrete compressive zone and compressed diagonals, also known as compression struts. The inclination of the compression struts can be smaller than 45

in the new truss model. In EC2 (2004) the angles are specified in a range within 21.8

and 45

which corresponds to the trigonometric value of cotθ = 2.5 and cotθ = 1.0. The longitudinal reinforcement in the tension zone and concrete in the compression zone constitute a frame. The compressed part of the frame can also be inclined towards the supports to capture the arch action.

2.1.2 Crack width

Calculation of crack width is one of the serviceability requirements in structural concrete standards since excessive crack widths may lead to a reduction of service life for rein- forced concrete structures, especially for those structures that are exposed to aggressive environments. Such reduction is mainly a result of the penetration of corrosive factors (oxygen, water and chlorides).

Long-term effects, i.e. repeated loading or long-term loading, has the tendency to in- crease crack widths in the serviceability limit state. This is the result of a decrease in material stiffness and an increase of concrete deformations. These long-term effects will lead to a growth of reinforcement stresses in the cracks.

Generally the crack width widens between the reinforcement bar and the concrete cover

but it is, however, assumed that the crack width is constant over the concrete cover and

measured at the concrete surface. Too meet specific demands, an appropriate limitation

of crack widths has to be set which can be met by either performing a calculation procedure or an analytical application (i.e. crack control without calculation). In order to meet requirements regarding functionality, durability and/or appearance of crack widths of a concrete structure the following condition has to be satisfied;

w

≤ w

where w

denotes the design crack width while w

is the nominal limit value of crack width. The nominal limit value represents specified values for cases where cracking is expected (MC10, 2010).

Factors that mainly influence an increase of crack widths is reinforcement content, the design of reinforcement bars, creep, loading time and load level which influence the structures crack formation stage (b in figure 2.1)(Malm, 2006).

2.1.3 Crack spacing

The crack width is the product of the maximum crack spacing multiplied with the mean strain difference between the concrete and the reinforcement.

Crack spacing is mainly influenced by the size of reinforcement bars diameter, the cover to the longitudinal and vertical reinforcement, the distribution of strains, the bond properties of bonded reinforcement and so on (EC2, 2004).

A concrete structure that is experiencing repeated loading or long-term loading may

show a decrease in crack spacing, i.e. the number of cracks increase, during the crack

formation stage which is the period when first crack (point R, in figure 2.1) appear to

the last crack (point S, in figure 2.1) (Malm, 2006).

2.2 Modified compression field theory

The modified compression field theory (MCFT) is an analytical model which is capable of determine the response of concrete elements subjected to in-plane stress, i.e. shear and normal stresses. Due to the complexity of analyzing reinforced concrete elements, Vecchio and Collins (1986) developed the MCFT. It was developed from the former model Compression-field theory, were the difference is that MCFT takes the tensile stresses in the concrete between cracks into account. Otherwise, both models treat the cracked concrete as a new material with its own stress-strain properties and formulate equilibrium, compatibility as well as stress-strain relationships in terms of mean stresses and mean strains. This is summarized in figure 2.4.

Figure 2.4: Modified Compression Field Theory for Reinforced Concrete (Vecchio and Collins, 1986)

The modified compression field theory makes it possible to follow and analyze an element from loading to failure. The stresses in the reinforcement will vary along the reinforce- ment bars and will have its maximum at the crack locations. The concrete element will be capable of transferring shear and compression stresses across the crack surface by aggregate interlock, illustrated in figure 2.7, but will not be capable of transmit tension.

Tensile stresses will, however, exist in the un-cracked concrete between cracks. The uniaxial behavior in compression and tension is shown in figure 2.5 and in figure 2.6.

The principal compressive stress in the concrete (σ

) is a function of both the principal compressive strain (

) and co-existing principle tensile strain (

). By increasing 

a reduction of the maximum compressive stress (σ

) will occur, which is illustrated in figure 2.5 (Vecchio and Collins, 1986).

Figure 2.5: Mean stress-strain relationship in compression (Vecchio and Collins, 1986) reproduced by Malm (2006)

Prior to cracking; the relationship between the mean principal tensile stress in the con-

crete (σ

) and the mean tensile strain 

is almost linear. After cracking; by increasing

the mean principal strain will lead to a decrease of the mean principal stress, which is

illustrated in figure 2.6 (Vecchio and Collins, 1986).

Figure 2.6: Mean stress-strain relationship in tension (Vecchio and Collins, 1986) reproduced by Malm (2006)

Even though the experiment (Vecchio and Collins, 1986) showed that the directions of principal strains in the concrete differed from the directions of principal stresses in the concrete, less than 10

for elements with transverse (y-axis) and longitudinal (x-axis) reinforcement, it remained reasonable to assume that the principal axes for strains and stresses coincide (Malm, 2006).

The average crack width w over the crack surface is calculated as the product of the mean principle tensile strain and the inclined crack spacing.

Figure 2.7: Compression stresses σ^ci = fci and shear stresses τci = vci at a crack (Vecchio and Collins, 1986)

The modified compression field theory has also proved to be applicable for predicting the

response of beams loaded in shear, flexure and axial loads. The modified compression

field theory is also often the basis for non-linear finite element analysis programs (Malm,

2006).

Shear crack calculations

Crack control is a serviceability requirement for concrete structures given in design codes. Today’s design standard Eurocode 2 is divided into two parts; Part 1-1 (EC2, 2004) applies for concrete structures in general, while Part 2 (EC2-2, 2004) is specified for concrete bridges. Regarding crack control, Part 2 refers to the procedure and formulas that is described in Part 1-1. The difference, when designing bridges in Sweden, is that regulations set by the Swedish transport administration and given as a national annex to EC2 have to be followed.

In this chapter the procedure in order to calculate cracks, mainly crack widths as well as spacing, and the different parameters needed will be described in accordance to set regulations in Eurocode 2 (concrete structures and bridges) and by the Swedish transport administration.

Since there is also no defined way to determine the reinforcement stresses, it will be described with the help of the handbook for EC2 developed by the Swedish concrete association (Betongföreningen, 2010) (i.e. the Swedish handbook for EC2).

3.1 Reinforcement stresses according to the Swedish hand- book for EC2

When calculating shear cracks in a plane stress state, there exist adequate formulas in EC2 part 1-1 for obtaining both the crack width (3.8) and the equivalent crack spacing (3.10), in combination with (3.12) since the crack angle of the shear crack often is inclined. In addition, a calculation has to be done for the reinforcement stress which is used in order to obtain the mean strain difference between the reinforcement and the concrete (3.9). However, no instructions are given in Eurocode 2 for solving this.

17

In the Swedish handbook for EC2 (Betongföreningen, 2010), two methods are described for calculation of the reinforcement stresses in the serviceability limit state. The first method is based on stresses in the un-cracked concrete. The corresponding tensile strength is then, when the concrete cracks, absorbed by the reinforcement. The other method calculates the reinforcement stress by using a truss model.

3.1.1 Reinforcement stresses out of equilibrium with concrete stress

Reinforcement stresses, at a plane stress state, are calculated so they are in equilibrium with the compressive stresses (σ

, σ

and τ

) in the concrete at a un-cracked stage (see (3.1a), (3.1b)). This is illustrated in figure 3.1. Normally the reinforcement stresses are calculated with the prerequisite that the concrete, after cracking, only can take compression while the reinforcement absorbs all the tensile strength.

The reinforcement stress is obtained from the equilibrium condition, in the un-cracked stage, by:

σ

= σ

+ cotθ · τ

ρ

(3.1a)

σ

= σ

+ tanθ · τ

ρ

(3.1b)

where

σ

is the reinforcement stress in the x-direction σ

is the reinforcement stress in the y-direction

ρ

is the reinforcement ratio in the x-direction = A

/(b

· s

) ρ

is the reinforcement ratio in the y-direction = A

/(b

· s

)

θ is the compressive stresses inclination in relationship to the beam axis

(a) Un-cracked (b) Cracked

Figure 3.1: Equilibrium between concrete stresses and reinforcement stresses (Be- tongföreningen, 2010)

In figure 3.1, θ

is the principal stress σ

inclination towards the y-direction or corre- sponding crack angle against the x-direction. θ is the compressive stresses inclination in relationship to the x-direction at a cracked stage.

After cracking, the inclination θ will somewhat adapt to the reinforcement which will lead to a difference between θ

and θ. As a limit θ

= θ could be chosen, but it would be better to choose a angle where the steel strain corresponds to the principal strain.

The principal strain should fulfill equilibrium conditions described in equations (3.1a), (3.1b) and should be, after cracking, perpendicular to the inclined compressive stresses.

Strains, at a crack, can be seen in figure 3.2.

Figure 3.2: Strains at a crack, reproduction of Betongföreningen (2010)

The strain is defined as

 = ∆u u

where u is an arbitrary length with its corresponding length in the x and y-direction.

The reinforcement is assumed to stretch with an constant strain along these lengths.

From figure 3.2 the following relationships can be given:

















= ∆x

x = ∆u · sinθ

x = {∆u =  · u} =  · u · sinθ x



= ∆y

y = ∆u · cosθ

y = {∆u =  · u} =  · u · sinθ

y

If the reinforcement could stretch freely along the x and y-direction in figure 3.2, it would result in strain lengths x = u/sinθ and y = u/cosθ and therefor give the following relationship:



=  · sin

θ (3.2a)



=  · cos

θ (3.2b)

Stresses is, however, transmitted to the concrete due to bond strengths between the reinforcement and the concrete which results in a reduction of the reinforcements strain lengths. For simplicity the strain lengths will therefor be set to x = y = u, which gives:



=  · sinθ (3.3a)



=  · cosθ (3.3b)

Through the equilibrium and strain conditions, the relationship (3.4) is given which enables to get a solution for cotθ.

σ

σ

= 



= (cosθ)

(sinθ)

= (cotθ)

= ρ

ρ

·

σ

+ 1 cotθ · τ

σ

+ cotθ · τ

(3.4)

where

n = 1 if equations (3.3a) and (3.3b) are used

= 2 if equations (3.2a) and (3.2b) are used

An iterative solution can also be performed in order to obtain the solution for equation (3.4). Following definitions are defined: cotθ = µ and µ

is the start value or value obtained from the previous iteration-step, µ

.

µ

=





µ

τ

·

ρ

ρ

· σ

− µ

σ



+ ρ

ρ





1 n + 2

(3.5)

When a solution for cotθ is obtained, the reinforcement stresses can be solved through

equations (3.1a) and (3.1b).

3.1.2 Reinforcement stress according to truss model

The other possibility, in order to obtain the reinforcement stress, is to use the truss model which is normally used when designing shear reinforcement. This is illustrated in figure 3.3. However, it is important to understand that this model is based on the ultimate limit state (ULS) and is therefor questionable when designing in a serviceability state. It probably results in a overestimated and safe value if the strut inclination is chosen to follow the linear-elastic principal stresses in SLS.

Figure 3.3: Truss model when designing shear reinforcement (Betongföreningen, 2010). Dotted lines represent compression struts while solid lines are flexural and

shear reinforcement.

Through vertical equilibrium, the reinforcement stress may be calculated according to equation (3.6):

V

= A

s · z · σ

cotθ =⇒ σ

= V

· s

A

· z · cotθ (3.6)

where

V

is the shear capacity: V

≥ V

A

is the cross-sectional area of the shear reinforcement s is the spacing between reinforcement bars

z is the inner lever arm

= 0.9d for reinforced concrete without axial force σ

is the reinforcement stress

θ is the compression struts inclination in relationship to the beam axis

The shear capacity (V

) could, as an alternative, be replaced with the relevant shear force (V ) in the serviceability limit state.

According to the handbook for EC2 (Betongföreningen, 2010), the compression struts inclination in relationship to the x-direction (i.e. θ) can be determined in accordance to EC2’s instructions in chapter 6.8.2 equation (6.65) (EC2, 2004). A fatigue control of shear reinforcement, which is formulated as equation (3.7):

tanθ =

tanθ

(3.7)

where

θ

is the angle of concrete compression struts to the beam axis assumed in ULS design:

1 ≤ cotθ

≤ 2.5 for non pre-stressed constructions 1 ≤ cotθ

≤ 3.0 for pre-stressed constructions

By using (3.7) more suitable results are obtained since, in SLS, the angle θ should be chosen with a steeper value (i.e. smaller value on cotθ) as a upper limit.

3.1.3 The most suitable method

The method of obtaining the reinforcement stress in accordance of a truss model is an easier and simpler approach since calculations are performed without consideration of the compressive stresses in the concrete at a un-cracked stage. But since the method is based on vertical equilibrium, horizontal reinforcement will be hard to account for which is not preferable in order to minimize the shear reinforcement content.

According to recommendations and calculations from the Swedish handbook for EC2 (Betongföreningen, 2010), reinforcement stresses obtained from equilibrium with con- crete stresses and n = 1 is preferable. By using n = 1 consideration is taken that the reinforcement cannot strain freely in the concrete while n = 2 is based on that the reinforcement stretches freely along the strain lengths. In reality, reinforcement cannot strain freely due bond relations between the reinforcement and the concrete and therefor, by using n = 2, result in a less realistic value on cotθ.

The method (section 3.1.1) has also been used for the internal reports concerning the

box-girder bridges Alvik and Gröndal. By putting the reinforcement stresses in equi-

librium with the concrete stress with n = 1, calculations were made to investigate if

the characteristic yield point of the reinforcement was reached and if the crack width

criteria was reached in the serviceability limit state (Westerberg, 2005).

3.2 Crack width according to EC2

In the serviceability limit state (SLS), crack width calculations are performed on concrete structures which are assumed to have formed stabilized crack pattern (Malm, 2006). The characteristic expression for crack width is calculated as equation (3.8):

w

= s

(

− 

) (3.8)

where

s

is the maximum crack spacing (see section 3.3)



is mean strain in the reinforcement under relevant combination of loads,

including the effect of imposed deformations and taking into account the effects of tension stiffening. Only the additional tensile strain beyond the state of zero strain of the concrete at the same level is considered



is the mean strain in the concrete between cracks

The mean strain difference between the reinforcement and the concrete, 

− 

, is calculated according to equation (3.9):



− 

= max



















σ

− k

f

ρ

(1 + α

ρ

) E

0.6 σ

E

(3.9)

where

σ

is the stress in the tension reinforcement assuming a cracked section. For pre- tensioned members, σ

may be replaced with 4σ

which represents the stress variation in the pre-stressing tendons from state of zero strain of the concrete at the same level

f

is the mean value of the tensile strength of the concrete at the time when the cracks may first be expected to occur:

= f

α

is the ratio between the elastic modulus of the reinforcement and concrete;

α

= E

E

ρ

is the effective reinforcement area;

ρ

= (A

+ ξ

A

) A

A

is the area of the tendons (pre or post-tensioned) within A

ρ

= A

A

if no pre or post-tensioned tendons exist

A

is the effective area of concrete in tension surrounding the reinforcement with the height h

where h

is the minimum value of 2.5(h − d), (h − x)/3 or h/2 ξ

is the adjusted ratio of bond strength taking into account the different diameters

of pre-stressing and reinforcing steel:

ξ

=

ξ · φ

φ

ξ

= √

ξ if only pre-stressed steel is used for crack control

ξ is the ratio of bond strength of pre-stressed steel and reinforcement steel, according to table 3.1

φ

is the largest bar diameter of the reinforcement φ

is the equivalent diameter of tendon

k

is a factor dependent on the duration of the load k

= 0.6 for short term loading

k

= 0.4 for long term loading

Table 3.1: Ratio of bond strength, ξ (EC2, 2004)

Pre-stressing steel Pre-tensioned Bonded, Post-tensioned

≤ C50/60 ≥ C70/85 Smooth bars/wires Not applicable 0.30 0.15

Strands 0.60 0.50 0.25

Indented wires 0.70 0.60 0.30

Ribbed bars 0.80 0.70 0.35

In bridge design; the Swedish transport administration (TRVFS, 2011) advices, when

determining an acceptable calculated crack width w

, that consideration should be

taken regarding the exposure class, load combination and design working life. These

limits, depending on the exposure class, can be seen in the table 3.2.

Table 3.2: Acceptable crack width, wk,max (TRVFS, 2011)

Reinforced members Prestressed members Exposure class and prestressed members with bonded tendons

with unbonded tendons

Quasi-permanent load comb. Frequent load comb.

L 100 L 50 L 20 L 100 L 50 L 20

X0, XC1 0.45

0.45

0.45

0.40 0.45 - XC2 0.40 0.45 - 0.30

0.40

0.45

XC3, XC4 0.30 0.40 - 0.20

0.30

0.40

XS1, XS2, XD1, XD2 0.20 0.30 0.40

Absence of tensile stresses

XS3, XD3 0.15 0.20 0.30

1)

For exposure classes X0 and XC1, the crack width has no effect on the resistance which is why this limit is set with the respect on the appearance of the structure.

2)

For these exposure classes the absence of tensile stresses should also be checked for quasi-permanent load combinations.

3.3 Crack spacing according to EC2

In situations where bonded reinforcement is fixed at reasonably close centres within the tension zone (≤ 5(c + φ/2)), the maximum crack spacing may be calculated from equation (3.10):

s

= k

c + k

k

k

φ

ρ

(3.10)

where

c is the concrete cover

k

is the coefficient that takes bond properties of the bonded reinforcement into account:

= 0.8 for high bond bars

= 1.6 for bars with effectively plain surfaces (e.g. pre-stressing tendons) k

is the coefficient which takes account of the distribution of strain:

= 0.5 for bending

= 1.0 for pure tension

For cases of eccentric tension or for local areas, intermediate values of k

should be calculated from the relationship:

k

= (

+ 

)

2

where 

≤ 

φ is the bar diameter. If a mixture of bar diameters is used in a section, the equivalent diameter φ

should be used. For a section containing n

bars of diameter φ

and n

bars of diameter φ

the following equation should be used:

φ

= n

φ

+ n

φ

n

φ

+ n

φ

According to Eurocode 2 (EC2, 2004), the values of coefficients k

and k

should be set according to each countries specific National Annex. Both BBK 04 (BBK04, 2004) and the Swedish transport administration (TRVFS, 2011) recommend these coefficients to be set as k

= 7φ/c, which makes this parameter dependent on the bar diameter instead of the concrete cover, and k

= 0.425 which is also the recommended value described in Eurocode 2.

In areas where the spacing of the bonded reinforcement > 5(c + φ/2) or where there is no bonded reinforcement within the tension zone, an upper bound to the crack width may be found by assuming a maximum crack spacing, illustrated in equation (3.11):

s

= 1.3(h − x) (3.11)

where x denotes the height of the compression zone.

Where the crack angle between the principal stress and the direction of the reinforcement (members with two orthogonal reinforced directions) deviates more than 15

, then the maximum crack spacing s

may be calculated from equation (3.12):

s

= 1

cosθ s

+ sinθ s

(3.12)

where

θ is the angle between the principal tensile stress and the reinforcement in the y-direction

s

is the crack spacing calculated in the y-direction according to equation (3.10)

s

is the crack spacing calculated in the x-direction according to equation (3.10)