Degree Project in Civil and Architectural Engineering

Second Cycle, 30 Credits

Stockholm, Sweden 2019

The effect of pre-stressing location

on punching shear capacity of

concrete flat slabs

Author: Saeed Vosoughian

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT

The effect of pre-stressing

location on punching shear

capacity of concrete flat slabs

SAEED VOSOUGHIAN

Master of Science Thesis

Stockholm, Sweden 2019

TRITA-ABE-MBT-19686. Master Thesis

2019.

KTH School of ABE

SE-100 44 Stockholm

SWEDEN

© S. Vosoughian 2019

Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering

Division of Concrete Structures

i

Abstract

Implementing pre-stressing cables is a viable option aiming at controlling deformation and

cracking of concrete flat slabs in serviceability limit state. The pre-stressing cables also

contribute to punching shear capacity of the slab when they are located in vicinity of the column.

The positive influence of pre-stressing cables on punching capacity of the concrete slabs is

mainly due to the vertical component of inclined cables, compressive in-plane stresses and

counter acting bending moments near the support region. The method presented in Eurocode 2

to determine the punching capacity of the pre-stressed concrete flat slabs considers the in-plane

compressive stresses but totally neglects the effect of counter acting moments. The effect of

vertical forces introduced by inclined cables is only considered when they are within the

distance 2d from the face of the column. This area is called basic control area in the Eurocode

2.

In this master thesis nonlinear finite element analysis is carried out to study the effect of pre-

stressing cables on punching shear capacity of concrete slabs respecting the distance of cables

from the face of the column. To attain this objective, the concrete damage plasticity model is

implemented to model the concrete. The results indicate that until the distance of 6d from the

face of the column the contribution of pre-stressing cables in punching shear capacity of slabs

is significant. Furthermore, comparing the numerical results with the punching shear capacity

of slabs predicted by Eurocode 2 reveals that Eurocode tremendously underestimates the

punching shear capacity when the cables are located outside the basic control area.

Keywords: Concrete flat slabs, Concrete damage plasticity model, Eurocode 2,

Nonlinear finite element analysis, Punching capacity, Pre-stressing cables.

iii

Sammanfattning

Inläggning av spännkablar är ett möjligt alternativ som syftar till att begränsa deformation

och sprickbildning i plana, pelarunderstödda betongplattor i bruksgränstillståndet.

Spännkablarna bidrar också till stanskapaciteten hos plattan när de är belägna i närheten av

pelaren. Spännkablarnas positiva påverkan på stansningskapaciteten hos betongplattorna beror

främst på den vertikala komponenten i lutande kablar, tryckspänningar i planet och

motverkande böjmoment nära stödområdet. Den metod som finns i Eurokod 2 för att bestämma

stansningskapaciteten hos förspända betongplattor beaktar tryckspänningarna i planet men

negligerar helt effekten av motverkande moment. Effekten av vertikala krafter som ges av

lutande spännkablar beaktas endast när kablarna är placerade inom avståndet 2d från pelarens

yta. Detta område begränsas av det s.k. grundkontrollsnittet enligt Eurokod 2.

I detta examensarbete genomförs icke-linjär finitelementanalys för att studera effekten av

spännkablar på betongplattors stansförmåga med avseende på kablarnas avstånd från pelarens

yta. För att uppnå detta mål implementeras en icke-linjär plasticitetsmodell för betongens

brottstadium (CDP, concrete damage plasticity model). Resultaten indikerar att fram till

avståndet 6d från pelarens yta är bidraget från spännkablarna till betongplattans stanskapacitet

betydande. En jämförelse av de numeriska resultaten med stansningskapaciteten hos plattor

som beräknas med Eurokod 2 visar att Eurokod kraftigt underskattar stansningskapaciteten

när kablarna är belägna utanför grundkontrollsnittet.

Nyckelord: Betongplattor, betongskadplastisitetsmodell, Eurocode 2, Icke-linjär analys

av finit element, stansningskapacitet, förspänningskablar.

v

Preface

This master thesis is initiated by the department of Civil and Architectural Engineering at

Royal Institute of Technology (KTH) and is result of collaboration with CBI Betonginstitutet

AB.

In advance, I would like to express my deepest gratitude to my supervisors Professor Johan

Silfwerbrand at KTH and Mr. Ghassem Hassanzadeh Lic. Tech., for all their support, guidance

and valuable comments throughout my work. Their dedication and constant encouragement

were inspirational while keeping me on the right track to attain determined objectives.

Secondly, I wish to express my sincere appreciation for Adj. Prof. Mikael Hallgren. He

contributed with valuable advice and generously shared his experience and knowledge in the

field.

Furthermore, I want to thank Dr. Richard Malm for providing worthwhile comments and help

about the finite element method.

Last but not the least, I want to send tons of love and appreciation to my family for their

support throughout my entire study at KTH.

Stockholm, October 2019

Saeed Vosoughian

Contents

Abstract................................................................................................................................. i

Sammanfattning ................................................................................................................. iii

Preface ..................................................................................................................................v

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

1.1 Problem definition ................................................................................................1

1.2 Background ..........................................................................................................2

1.3 Aim ......................................................................................................................3

1.4 Scope and delimitations ........................................................................................3

2 Punching shear models ...............................................................................................5

2.1 Kinnuen and Nylander model ...............................................................................5

2.2 Andersson model ..................................................................................................8

2.3 Hallgren model ................................................................................................... 10

2.3.1 The model.............................................................................................. 10

2.3.2 Failure criterion ..................................................................................... 15

2.4 Menétrey model ................................................................................................. 17

2.4.1 General assumptions .............................................................................. 17

2.4.2 Concrete tensile force ............................................................................ 18

2.4.3 Dowel effect .......................................................................................... 19

2.4.4 Shear reinforcement contribution ........................................................... 19

2.4.5 Pre-stressing tendon contribution ........................................................... 21

2.4.6 Relation between punching and flexural capacity ................................... 21

2.5 Georgopoulos model .......................................................................................... 21

2.6 Bond model for concentric punching shear ......................................................... 22

2.7 Muttoni model .................................................................................................... 23

2.8 Discussion about mechanical models .................................................................. 25

3 Effect of pre-stressing on punching capacity of flat slabs ......................................... 27

3.1 Effect of pre-stressing......................................................................................... 27

3.2 Eurocode 2, approach ......................................................................................... 29

4 Methodology ............................................................................................................... 31

vii

4.1 Introduction ........................................................................................................ 31

4.2 Concrete material behaviour ............................................................................... 33

4.2.1 Compressive behaviour .......................................................................... 33

4.2.2 Tensile behaviour .................................................................................. 34

4.2.3 Multiaxial behaviour .............................................................................. 37

4.3 Concrete model .................................................................................................. 38

4.3.1 Yield surface ......................................................................................... 38

4.3.2 Defining damage in CDP model............................................................. 42

4.3.3 Longitudinal bars and pre-stressing tendons ........................................... 43

4.4 Finite element model .......................................................................................... 46

4.5 Verification ........................................................................................................ 48

5 Result and Discussion ................................................................................................. 51

5.1 Results ............................................................................................................... 51

5.2 Discussion .......................................................................................................... 52

5.3 Future investigations .......................................................................................... 53

Bibliography ....................................................................................................................... 55

1.1. PROBLEM DEFINITION

1

1

Introduction

1.1 Problem definition

Concrete flat slab supported by columns is a widely used structural element in buildings and

bridges. It has the edge over the other forms of slab systems from both aesthetic and economic

points of view. This is due to flexibility of planning and elimination of beams and girders which

result in creating additional floor space for a certain height of the building. However, flat slabs

are more prone for brittle punching shear failure either next to the column or below a

concentrated force compared to beam-slab systems.

In design of flat slabs to avoid punching failure, usually huge amount of reinforcement is

required in critical zones. It goes without saying that the high amount of flexural and shear

reinforcement makes casting concrete cumbersome. Beside the need to have opening in vicinity

of columns, which especially is required when reconstructing old buildings to install new

apparatus, motivates to find an alternative method to normal reinforcement in order to increase

punching shear capacity of the flat slabs.

Previous researches have shown that the idea of pre-stressing is a suitable technique to reduce

the amount of reinforcement for flat slabs with large spans or slabs subjected to huge

concentrated forces. This is because:

i. Pre-stressing exerts in-plane compressive stresses on concrete which leads to increase

in punching shear capacity.

ii. Bending moments caused by eccentricity of the tendons counteract those of external

forces leading to smaller crack openings in failure region and thus increase of capacity

of the concrete to carry shear force.

iii. Inclined pre-stressed tendons introduce a vertical force in the punching failure surface

opposing the forces caused by external loads.

Reflecting on the formulas presented by design codes such as ACI 318 [1] and Eurocode 2 [2]

indicates that they attempt to acknowledge the effect of pre-stressing on punching shear capacity

based on empirical investigations and only account for in-plane stresses and deviation force.

Accordingly, the bending moment which is introduced due to pre-stressing eccentricity is not taken

into account [3] furthermore, despite both numerical and empirical investigations conducted in the

past, the contribution of active known pre-compression force is still not well understood.

Chapter

CHAPTER 1.

Considering these facts necessitates more detailed and accurate research in this field to attain

reliable design criteria.

The other interesting issue is that the influence of pre-stressing on punching failure of a flat

slab is taken into account by conventional design codes only if the pre-stressed cables are

stretched in a certain area around the perimeter of the column. Series of experimental

investigations have been done in the past studying this phenomenon [4,5]. However numerical

models are more flexible in dimensioning and alteration of the position of pre-stressed cables.

Thereby, with a reliable numerical analysis, a more profound comprehension of influence of

cable arrangement is achieved.

1.2 Background

It seems first Elstner and Hognestad [6] noticed the shear capacity of thin concrete slabs. They

conducted an experiment and studied the influence of concrete strength, tension and

compression reinforcement amount as well as position and amount of shear reinforcement on

capacity of reinforced concrete flat plates. They concluded that increasing tension and

compression reinforcement over the column does not affect the load bearing capacity of the

slab. In contract with longitudinal reinforcement, they observed shear reinforcement contributes

to load capacity of the slab and could enhance it up to 30%. Nevertheless, they realized that

although achieving flexural failure rather than shear failure is desirable, it is impractical by

increasing shear reinforcement. Accordingly, they proposed to have small amount of shear

reinforcement by choosing appropriate values for concrete strength, slab thickness and column

stiffness as well.

Kinnunen and Nylander [7] in 1960 developed a model for punching capacity of flat slabs. To

attain this model, they performed series of experiments and studied the concrete segment

between radial flexural cracks and conical shear crack. Based on this model, punching failure

occurs when simultaneously, radial inclined compression stress and tangential compression

strain tend to critical values.

In 1961, Johannes Moe [8] conducted a thorough experiment to investigate the effect of hole

in vicinity of column, effect of eccentric loading, effect of shear and tensile reinforcement and

some other variables on load bearing capacity of flat slabs. In this research he also shed more

light on the concept of eccentricity of shear force. His design provisions for openings in vicinity

of columns and for eccentric loading are still implemented by building codes.

Sundquist [9] in 1978 investigated the effect of dynamic loads and presented a theoretical

model for capacity of flat plates when they are subjected to impulsive loads.

Franklin & Long [10] investigated the punching behaviour of unbounded post tensioned

concrete flat plates in 1982. They presented the test results for seven specimens with internal

columns and compared it with American design recommendations. Boundary conditions and

amount of eccentricity were the main variables in their study. The positive influence of pre-

stressing on punching behaviour of flat slabs also reported by number of studies, for example

the tests carried out by Ramos et al. [11]. Furthermore, Mostafaei et al. [12] indicated that

Design Code procedures are conservative in calculating punching shear capacity of pre-stressed

slabs.

1.3. AIM

3

The size effect was studied by Tolf [13]. He concluded that the increase in specimen size leads

to considerable decrease in punching shear stress.

Hallgren in 1996 [14] conducted a comprehensive study about punching shear behaviour of

high strength concrete flat slabs. For this purpose, he did both empirical investigations and

created numerical models as well. He improved a model presented by Kinnunen and Nylander

[7] for punching shear capacity of flat slabs implementing linear fracture mechanics approach.

The model presented by Kinnuen and Nylander was improved by Broms [15] taking into

account the size effect and brittleness of the concrete. He showed that the critical value for

concrete compression strain at which the punching failure occurs is significantly less than the

generally accepted 0.0035 for one way structural elements under bending. He also prescribed a

novel reinforcement concept through which punching shear failure is avoided and slab acquires

ductile behaviour.

Muttoni [16] in 2008 implemented critical shear crack theory to describe punching shear

phenomenon. He developed a criterion for punching shear on the basis of rotation of the slab

and verified it with experimental testing. Furthermore, he came up with an interesting result

asserting that despite to the conventional belief, punching shear capacity of a flat slab is more

dependent on the length of the span rather that slab thickness.

In 1998, Hassanzadeh [4] conducted an experiment to investigate the influence of pre-

stressing cables on punching shear capacity of the concrete flat plates. His research indicated a

large discrepancy between theoretical and practical results. He asserted that theoretical methods

presented by design codes for calculating punching shear capacity of pre-stressed slabs are

strongly conservative.

1.3 Aim

The aim of this master thesis is to investigate the effect of pre-stressing on punching shear

capacity of flat slabs. Furthermore, the influence of cable arrangement is studied. What is meant

by the term cable arrangement is distance of cables from the column. In other words, the area

around a column in which pre-stressing cables contribute to enhancement of punching shear

capacity is determined.

1.4 Scope and delimitations

The outline of the thesis falls into two categories. The first section sheds more light on

punching shear phenomenon and surveys the literature. It is worth mentioning that in this part

the treatment prescribed by ACI 318 [1] and Eurocode 2 [2] to calculated punch shear capacity

and the effect of pre-stressing is discussed.

In the second part a parametric study is done to investigate the effect of pre-stressed cables on

punching shear capacity of flat slabs when they are stretched outside the control section. To

attain this purpose, numerical study is done by creating finite element model of flat slabs in

Abaqus software. It goes without saying that, internal validity of numerical models is of

paramount importance in order to have reliable result. This characteristic of the created

CHAPTER 1.

numerical model is going to be satisfied by doing a verification with outcome of the empirical

study conducted by Hassanzadeh at KTH university [4].

While creating numerical model, the nonlinear behaviour of concrete is modelled by concrete

damage plasticity (CDP) model and both reinforcement and pre-stressed cables are assumed to

be fully embedded in the concrete.

2.1. KINNUEN AND NYLANDER MODEL

5

2

Punching shear models

Over the years, researchers have attempted to explain punching shear phenomena in concrete

slabs. They implemented theoretical concepts such as fracture mechanics, conducted

experimental investigations and created numerical modes to demystify punching shear

behaviour. Some of these investigators developed theoretical models through which a more

profound comprehension of the issue is possible. In this chapter more light is going to be shed

on some of these models.

2.1 Kinnuen and Nylander model

Conducting series of experiments on polar-symmetrical concrete slabs without shear

reinforcement Kinnuen and Nylander [7] observed that the concrete segment inscribed by

conical and radial cracks rotates like a rigid body. With regard to this fact, they satisfied the

equilibrium of forces for the segmental part of the concrete bounded between two radial cracks

with the angle of Δϕ and one shear crack, see Figure 2.1. Furthermore, the root of the shear

cracks was assumed as the centre of the rotation for the considered segment. The forces

affecting the mentioned concrete segment are illustrated in Figure 2.1(b). Where 𝑃∆∅/2𝜋 is the

fraction of the external force acting on the segment, R2 and R4 are resultants from the forces in

radial reinforcement and in tangential reinforcement respectively. The ultimate load Pu for a

two-way reinforced slab without shear reinforcement is achieved considering the equation of

vertical projection

2

2)()(

1

21

1.1 dtf

B

y

B

y

d

yBPu

(2.1)

Where y is the distance between root of the shear crack and bottom surface of the slab, α is

inclination of the conical shell, B is diameter of the column, d is effective depth of the slab and

σ(t) is the stress in conical shell. Kinnuen and Nylander took into account the effect of dowel

action by introducing the factor 1.1 in Eq. (2.1). Function f (α) is defined as

2tan1

tan1tan

f (2.2)

Chapter

CHAPTER 2.

The angle α can be achieved from the following equation:

d

y

d

Bd

c

B

yK y

5.0

5.0

ln135.22

1

tan1

tan11tan

2

(2.3)

Ky is a dimensionless parameter which is expressed as

3

25.0

yd

Bc

K y

(2.4)

In the Eq. 2.4, c stands for diameter of the area above the column in which the radial bending

moment is negative.

Figure 2-1: Mechanical model by Kinnuen and Nylander [7].

2.1. KINNUEN AND NYLANDER MODEL

7

Considering the experimental results, Kinnuen and Nylander proposed an approximated value

for the stress in conical segment, σ(t). It depends on tangential strain at the horizontal radius

r = B/2 + y and is expressed as

rcTcEt ,35.2 (2.5)

The failure criterion in this model is the tangential strain at the distance of B/2 + y from centre

of the column reaching a critical value as calculated below

For B/d ≤ 2

d

ByBrcT 22.010035.02/, (2.6a)

For B/d > 2

0019.02/, yBrcT (2.6b)

It is worth mentioning that beside Eq. (2.1), the equation of moments with regard to the point

Q1 illustrated in Figure 2-1 also yields ultimate capacity of the slab

For rs ≤ c0

Bc

yd

c

crdfP ssyu

35.0

ln1.41.10

(2.7a)

For rs> c0

Bc

yd

r

crdfP

s

ssyu

35.0

ln1.41.1 (2.7b)

Where ρ is mean ratio of reinforcement and fsy is yield stress of the reinforcement.

Furthermore, rs manifests the radius of the area in which the reinforcement reaches yield stress

and c0 stands for the distance between centre of the column to the concentric shear crack. rs and

c0 are approximated to

dBc 8.15.00 (2.8)

dd

y

f

Er

sy

ss

1 (2.9)

The angle ψ indicates the rotation of the slab outside the shear crack when ultimate load is

reached and can be calculated as

ydBrcTy

)/(,

5.01 (2.10)

CHAPTER 2.

As mentioned before y is the distance between root of the shear crack and bottom surface of

the slab which is calculated by the Eq. (2.4).

The model developed by Kinnuen and Nylander was a base for later important investigations.

For instance, Andersson [17] modified this model to take into account the effect of shear

reinforcement. The model developed by Andersson is presented in the following.

2.2 Andersson model

Andersson conducted a series of experimental tests to study the punching behaviour of slabs

with shear reinforcement. He realized that adding shear reinforcement alters the behaviour of

the slab at failure load in a way that its deflection varies linearly from the column to edge of the

slab. Accordingly, he proposed a mechanical model to take into account the effect of shear

reinforcement by modifying the Kinnuen Nylander model. His model is illustrated in Figure

2-2. Going into the depth, his model assumes a bi-linear and perfectly elasto-plastic behaviour

for the concrete in tangential direction. The failure criterion in this model is that the failure

occurs when the strain at a certain distance cpl from the centre of the column reaches the limit

of plastification εpl. This distance is determined as

For B/d ≤ 2.5

dd

Bc pl

45.174.0 (2.11a)

For B/d >2.5

dd

Bc pl

05.25.0 (2.11b)

There is an upper limit for cpl depending on the type of shear reinforcement. This value for

slabs in which longitudinal bars are bent to function as shear reinforcement is from the centre

of the column to outermost part of the shear bar. While the upper limit of cpl for slabs with

stirrups is the distance between the centre of the column and 3d/4 outside the most far stirrups.

In failure stage, when εpl tends to 0.002 at the point of cpl, the stress of concrete in tangential

direction σcT and the stress in conical shell σt are approximated to

153.035.025

,cubc

cT

f [MPa] (2.12)

cTt 9.1 (2.13)

Looking at the isolated segment of the slab depicted in Figure 2.2(b) the resultant force in

tangential direction is expressed as

82ln1 1321

syv

sh

s

ssy

fA

r

crdfRRR (2.14)

2.2. ANDERSSON MODEL

9

Where, fsyv and Ash are yield stress and the cross-sectional area of shear reinforcement

intersecting the tangential flexural crack. Δϕ stands for the angle illustrated in Figure 2.2.

Furthermore, rs is determined by Eq. (2.9). However, it should be noticed that the angle of

rotation ψ for a segment of the concrete which behaves like a rigid body while failure is

occurring, in Andersson’s model is different from the value according to the Kinnuen model

and is expressed as

y

c pl

pl (2.15)

Figure 2-2 Mechanical model by Andersson [17].

In Eq. (2.15), κ1 is reduction factor taking into account the effect of reinforcement when the

slab is not polar symmetrical. κ1 is function of 2rs/c and is determined through the graph

presented in Figure 2.3.

The force R4 indicated in Figure 2.2 is the projection of the sum of the compressive stresses

in concrete beneath the neutral surface, in radial direction. The amount of this force can be

calculated as

plplpl

plcTc

c

c

B

c

BcyR

2ln

8

15.15.0

2

4 (2.16)

CHAPTER 2.

The influence of the conical shell on the rigid segment as shown in Figure 2.2 is the

compression force T which can be calculated as below

cos5.0 tyBT (2.17)

Establishing the equations of the equilibrium for the isolated rigid body yields three equations

from which three unknowns including the ultimate load Pu, distance y and angle α shown in

Figure 2.2 are acquired

2sin

cos

72.022cos

2

3214

4

u

u

PT

RRRRT

Bc

ydR

Bc

ydTP

(2.18)

Figure 2-3 Reduction factor κ [17].

2.3 Hallgren model

2.3.1 The model

Hallgren [14] conducted experimental investigations to study the punching shear capacity of

high strength concrete slabs. His research revealed, the higher the strength of the concrete, the

higher the punching shear capacity. However, he observed that the increase in punching shear

capacity is not proportional to the increase in concrete strength. He concluded that the

brittleness of the concrete which is higher for high strength concrete is the fact reducing the

boost rate of punching shear capacity. Accordingly, he modified the punching shear model

proposed by Kinnuen and Nylander [7] to take into account the size effect and concrete

brittleness. The failure criterion for both models is the same where the failure is assumed to

occur when the strain of the concrete in tangential direction reaches its limit. However, despite

2.3. HALLGREN MODEL

11

the Kinnuen and Nylander model which implements semi-empirical expressions, Hallgren used

fracture mechanics considering size effect and concrete brittleness.

Hallgren model is illustrated in Figure 2.4 in which 1.5α is the angle of shear crack and c0 is

the distance between the center of the column and the shear crack at the level of the flexural

reinforcement. The amplification factor of 1.5 which is used for α is achieved based on test

observations and finite element analysis results and c0 is determined as

5.1tan20

xdx

Bc

(2.19)

Figure 2-4 Hallgren model [14].

As shown in Figure 2.5 test results and numerical analysis unveiled the fact that strain in

tangential direction for the concrete and steel, respectively on compression side and tension

side of the slab are inversely proportional to the distance from the center of the slab.

Furthermore, the tangential compression zone x is calculated based on the state of the material

as below

If concrete and steel are elastic

dE

Ek

Ek

Ex

s

cE

cE

s

1

21

(2.20)

Where kE is a factor accounting increase of concrete stiffness under bi-axial compression and

is determined as

12

2

21

Ek (2.21)

CHAPTER 2.

In the case that steel is in the elastic state while concrete has yielded, x is determined as

dE

f

f

Ex

cTs

ccc

ccc

cTs

1

41

2 0

0

0

0

(2.22)

0

0 1cT

cc

(2.23)

Where concrete strain in tangential direction at the point c0 and at ultimate load is expressed

as

0

0

2

c

yB

cTucT

(2.24)

Where εcTu is the ultimate strain of concrete in tangential direction. Hallgren considered this

parameter as failure criterion and achieved it by fracture mechanics.

Figure 2-5 Concrete and steel strain in tangential direction [14].

For the condition in which at the point c0 concrete is in elastic state while steel has been yielded

x is given by

cEcT

sy

Ek

fdx

0

2

(2.25)

If both concrete and steel yield

2.3. HALLGREN MODEL

13

ccc

sy

f

fdx

0

(2.26)

Considering compressive stress distribution of the concrete in tangential direction which is

shown in fig. 2.6 the projection of the resultant force of concrete compressive stresses in

tangential direction RcT acting on the rigid wedge as shown in Figure 2.4 is calculated as

If concrete is in elastic state

yB

cy

BxER cTuccT ln

22

1 (2.27)

If concrete has yielded

2

24

1

22ln

2

1

4

3y

B

ry

B

r

crxfR

cc

ccccT (2.29)

rc indicates the yield radius and is given as

y

Br

cy

cTuc

2

(2.30)

Figure 2-6 Variation of stress in concrete according to Hallgren [14].

Distribution of stress in reinforcement at the state of punching failure is depicted in Figure

2.7. If the flexural bars are in elastic state, the force caused by flexural reinforcement acting on

rigid segment RsT in tangential direction is calculated as

02ln

21

c

cy

B

x

dEdR cTussT (2.31)

In the case that flexural bares are yielded

CHAPTER 2.

0

2ln1 c

r

crfdR

s

ssysT (2.32)

Where rs indicates a radius along which steel has plastic behaviour and is determined as

y

B

x

dr

sy

cTus

21

(2.33)

Figure 2-7 Steel stress distribution of flexural reinforcement in tangential direction [14].

To determine the radial force caused by reinforcement which is denoted by RsR in Figure 2.4,

the strain of steel in radial direction at the point intersecting the shear crack needs to be known.

Hallgren assumes it to be equal to the strain of steel in tangential direction and proposes the

following set of equation to determine RsR

If rs < c0

y

B

x

dEdR cTussR

212 (2.34)

If rs ≥ c0

sysR fcdR ..2 0 (2.35)

In Hallgren’s model the dowel force effect D is taking into account by implementing semi-

experimental expression presented by Hamadi and Regan [18] as

3/1

03

2

6.119.27 ccfd

cD

[N] (2.36)

2.3. HALLGREN MODEL

15

Where Φ is the diameter of the steel bars in [mm]. It worth mentioning that the dowel force

significantly decreases when the steel is yielded. Thereby and for the sake of simplicity this

force is neglected if reinforcement bars are yielded at r = c0.

Finally establishing the equilibrium equations for the rigid segment illustrated in Figure 2.4

leads to two equations presented below through which the punching capacity of the slab is

determined. This process is iterative where the angle α is iteration variable. The failure load is

achieved when the compressive strain of the concrete at distance y from the column as shown

in Figure 2.4 reaches the critical value of εcTu. This is the failure criterion in Hallgren’s model

and is explained in the following.

DRRRP cTsTsRu tan2 (2.37)

xBc

xR

xRRRx

BcDxdRR

P

cT

cTsTsR

sRsT

u

5.0

3

22

cos2

2

22

20

(2.38)

2.3.2 Failure criterion

Numerical studies done by Hallgren [14], [19] revealed that at the column-slab root the

concrete is under tri-axial compression while at r=B/2+y, and in the compression zone it is

under biaxial compression, see Figure 2.8. By increasing the external load, the stress σ111

decreases and leads to an unstable state for tri-axial compressive stress. At this moment the

shear crack propagates through the compression zone resulting in punching failure.

By neglecting the effect of shear deformation Hallgren realized that at the point r=B/2+y, the

tangential strain εcT and the radial strain εcR are equal. Furthermore, considering the experimental

investigation conducted by Kupfer et al. [20], when a concrete specimen is under biaxial

compression, at the level of ultimate stress where the macro cracks begin to develop, the

compressive strains are equal to the transverse tensile strain. Based on these facts and with

regard to the fictitious crack model proposed by Hillerborg et al. [21] ultimate tangential strain

at the point r=B/2+y in Hallgren’s model is defined as

x

wccTu (2.39)

Where wc is critical crack width.

CHAPTER 2.

Figure 2-8 State of stress [14].

Hallgren implemented bi-linear tension-softening curve proposed by Petersson [22] to

calculate the critical crack width, see Figure 2.9, which gives

ct

Fc

f

Gw 6.3 (2.40)

Figure 2-9 Bi-linear tension-softening curve [21].

Based on previous studies the fracture energy GF is size dependent which for an element with

very large size is considered as the area under the tension-softening curve illustrated in Figure

2.9. thereby, Hallgren used the multifractal scaling law to take into account the size effect,

which gives

2/1.

1

x

dGG aF

FF

(2.41)

Where dα is the maximum size of the aggregate in concrete, αF is an empirical factor which with

regard to the empirical investigations [23] can be assumed to be equal 13.

2.4. MENÉTREY MODEL

17

Based on RILEM Draft Recommendation [24] setting α equal to 13 achieves fracture energy for

infinite size as

2/113

1

R

aR

FFd

dGG (2.42)

Where dR is the height of the beam and R

FG is the fracture energy according to RILEM.

Accordingly, considering Eqs. (2.39), (2.40), (2.41) and (2.42) the failure criterion for

Hallgren’s model is written as

2/113

1.

6.3

x

d

fx

G a

ct

FcTu (2.43)

2.4 Menétrey model

2.4.1 General assumptions

Menétrey did a comprehensive study to present an analytical model for calculating punching

shear capacity of flat slabs [23]. He took into account the effect of shear reinforcement, pre-

stressing and attempted to link the flexural capacity to the punching shear capacity with regard

to the crack inclination as well. Menétrey used the strut-and-tie model to study the punching

phenomenon and concluded that punching failure corresponds to concrete tie failure.

Furthermore, he asserts that punching crack inclination varies between 25̊ to 35̊ and α=30̊ will

be the best approximation. Thereby, he claimed that punching capacity of a concrete flat plate

is equal to the sum of concrete tensile force along the shear crack. Based on this hypothesis,

Menétrey simply added the effect of any counteracting forces crossing the shear crack to include

influence of shear reinforcement and pre-stressing. Thus the shear capacity of the slab is defined

as

pswdowctu FFFFP (2.44)

Where Fct stands for the sum of the concrete tensile stress, Fdow takes into account the effect

of flexural reinforcement, Fsw is the counteracting force due to shear reinforcement and Fp

represents the effects of pre-stressing. Figure 2.10 illustrates the effect of different components

of punching capacity of the slab in Menétrey model [24,25].

CHAPTER 2.

Figure 2-10 Representation of punching shear capacity of a general reinforced slab [23].

2.4.2 Concrete tensile force

While studying the influence of concrete on punching shear capacity Menétrey neglected the

effect of friction which arises due to aggregate interlock along the shear crack. Furthermore, he

implemented non-linear fracture mechanics to calculate tensile stress in the concrete and

assumed at ultimate limit state as depicted in Figure 2.10 an inclined crack formed at a distance

r1 and r2 which are given as

tan10

11

drr s (2.45)

tan2

drr s (2.46)

Menétrey et al. studied tensile force created in concrete while cracking by non-linear fracture

mechanics and realized that 3/2

ctct fF [26]. What is more, in the Menétrey model the effect of

longitudinal reinforcement, slab thickness and the radius of punching crack initiation on tensile

force of the concrete are taking into account by introducing parameters ξ, μ and η respectively.

So the tensile force is expressed as

3/222

1221 )9.0()( ctct fdrrrrF (2.47)

Where

02.087.0

02.0035.046.01.0 2

(2.48)

2.4. MENÉTREY MODEL

19

2/1

16.1

ad

d (2.49)

Where da is maximum aggregate size.

5.2625.0

5.25.025.15.01.0

2

h

r

h

r

h

r

h

r

s

sss

(2.50)

Where h is slab thickness and rs is radius of the column.

2.4.3 Dowel effect

The resisting force produced by longitudinal reinforcement is known as dowel effect. This

force significantly enhances the punching shear capacity of the concrete flat slabs. As the study

done by Regan and Braestrup [26] indicates, up to 34% of the punching capacity of reinforced

slabs is due to dowel force contribution. In the Menétrey model this force is calculated as

sin)1(2

1 22

isyic

bars

i

idow ffF (2.51)

Where Φi is diameter of the reinforcement crossing the shear crack and ζi is given as

sysi f/ (2.52)

Stress in the reinforcement which is denoted by σs is determined by dividing horizontal

component of the force in compressive strut, illustrated in Figure 2.10, by the total area of the

longitudinal reinforcement crossing the punching crack which gives

bar

i

si

u

s

A

P

tan (2.53)

It is worth mentioning that in Eq. (2.51) the reduction factor of 1/2 is an approximation taking

into account the fact that the reinforcement bars are orthogonal and as a consequence don’t

cross the punching crack at right angle.

2.4.4 Shear reinforcement contribution

Shear reinforcements such as stirrups and bolts are used to increase the punching capacity of

slabs. Generally speaking, there are three distinct failure mechanisms for concrete slabs with

shear reinforcement as are shed on light below and illustrated in Figure 2.11.

CHAPTER 2.

i. In the case that a punching crack occurs between the column face and the shear

reinforcement the ultimate load is calculating by Eq. (2.57) where the interaction of

punching and flexural stiffness is considered. What is more, the inclination of the crack

in this failure mode is defined as

h

rr sswi1tan (2.54)

Where, as illustrated in Figure 2.11., rswi is the distance between the shear stud and the

center of gravity of the column and s is inclined length of the punching crack which is

determined as

22

12 9.0 drrs (2.55)

ii. In the case that punching crack happens outside the shear reinforcement as depicted in

Figure 2.11 b), the punching capacity of the slab is determined as for the case without

shear reinforcement except the radius of column rs, is considered rsc, which is the radius

of the punching crack initiation and calculated by Eq. (2.45). To elucidate on, in this

case the punching capacity of the slab increases due to having a more stiff core around

the column.

iii. When the punching crack crosses through the shear reinforcement the bond properties

are decisive in the failure mechanism. So that, for the cases with poor bond strength

shear reinforcement doesn’t contribute to the punching capacity of the slab. However,

the added punching capacity due to shear reinforcement for the case with sufficient bond

strength is determined as

cswswsw FAF sin (2.55)

αc is the angle between the shear stud and the horizontal direction shown in Figure 2.11

c).

Figure 2-11 Three different punching failure mechanisms considering the location a strut and

crack initiation point.

2.5. GEORGOPOULOS MODEL

21

2.4.5 Pre-stressing tendon contribution

Menétrey takes into account the effect of pre-stressing simply by considering the vertical

component of force in tendons crossing the punching crack. It is expressed as

)sin( p

tendons

i

pp AF (2.56)

Where β is the inclination of the tendon at the place it intersects the punching crack.

Menétrey asserts that when tendons are bonded to the surrounding concrete even additional

enhancement of the punching capacity. It is due to the fact that bonded tendons in addition to

applying counter acting moment contributing to the flexural stiffness of the slab function in the

same way as longitudinal reinforcement. Accordingly, in this case while calculating ζ by Eq.

(2.48) the area of pre-stressing tendons intersecting punching cracks should be added to the area

of longitudinal reinforcement.

2.4.6 Relation between punching and flexural capacity

It goes without saying, of paramount importance is the geometry of the failure surface in

determining the failure load for a slab. This issue has been studied experimentally by Menétrey

[26]. He concluded that the inclination of the crack α is a key parameter through which flexural

and punching failure loads are linked to each other. He suggested the following expression

9030452

3sin)(

2/1

uflexufail PFPF (2.57)

Contemplating on Eq. (2.57) indicates that for α=30̊, Ffail=Pu, while for α=90̊, Ffail=Fflex. It is

worth mentioning that, for design purpose, Pflex can be computed by yield-line theory.

2.5 Georgopoulos model

Georgopoulos [27] developed a method to predict the inclination of the punching crack and

the punching capacity of flat slabs without shear reinforcement. In his model the mechanical

reinforcement ratio and the tensile strength of the concrete are the main variables. In this model

the inclination of the crack is determined as

3.0056.0

tan

(2.58)

Where ω is mechanical reinforcement ration and is defined as

cubck

yk

f

f

,

(2.59)

CHAPTER 2.

The model presented by Georgopoulos is illustrated in Figure 2.12 where the punching

capacity of the slab is approximated to

cot35.020.0

2cot13.4 2

d

ddP s

u (2.60)

Where ds is diameter of the column, and σ is maximum principle tensile stress of the concrete

in the punching crack which is determined by the following expression

3/2

,17.0 cubckf (2.61)

Figure 2-12 Load bearing model of flat slab developed by Georgopoulos [27].

2.6 Bond model for concentric punching shear

Alexander and Simonds [28] developed a mechanical model relying on radial arch action and

critical shear stress for beams. They assumed that punching capacity of the slab is mainly

yielded from the resistance of four strips branching out of the column and extended parallel to

the reinforcement direction as shown in Figure 2.13. The length of the strips in this model,

denoted by lw, is from the edge of the column to the point where shear force is zero. These

stirrups are assumed as cantilever beams having the bending capacity of Ms. Assuming q, as the

shear force that can be transferred by the quadrant of the slab adjacent to the cantilever strip,

Ms is determined by implementing equilibrium as

2

2 2

ws

qlM (2.62)

On the other hand, by isolating a single strip and writing the equilibrium equations the axial

force created in column due to the effect of one strip, P1 is determined as

wqlP 21 (2.63)

Considering Eqs. (2.62) and (2.63) and with regard to the fact that the column is affected by

four strips, the maximum axial load in the column is achieved

2.7. MUTTONI MODEL

23

qMP su 8 (2.64)

Looking at Eq. (2.64) indicates that by determining q, which is the maximum shear force that

the strips can bear, the punching capacity of the slab is calculated. Alexander and Simonds

claim that using one-way shear capacity which is known as inclined crack load to determine the

q leads to satisfactory results. Accordingly, they propose

cfdq 1667.0 [kN/m] (2.65)

Where fc is in MPa and d is in mm.

Figure 2-13 layout of radial strips in bond model [28].

2.7 Muttoni model

Muttoni applied critical shear crack theory to determine the punching failure capacity of

concrete slabs [29]. As illustrated in Figure 2.14, the inclined crack propagates through the slab

into the compression strut transferring the force to the column and disturbs its performance. As

the crack becomes wider the capacity of inclined concrete strut decreases which eventually

leads to punching failure. In other words, based on critical shear crack theory, for concrete

elements without shear reinforcement the roughness and width of the inclined crack developing

through the concrete compressive strut govern the shear strength.

Figure2-14 Influence of inclined crack on performance of compressive concrete strut [29].

CHAPTER 2.

Walraven et al. [30] indicated that roughness of the crack can be taken into consideration by

dividing the width of the inclined crack by the quantity (dg+16). Where dg and dg0 are maximum

aggregate size and the reference size of 16 mm, respectively. On the other hand, Muttoni and

Schwarts [31] showed that the width of the critical shear crack is proportional to product of slab

rotation ψ and effective height d as illustrated in Figure 2.14. Accordingly, Muttoni suggests

the punching failure criterion for flat slabs without shear reinforcement as

0

0 151

4/3

gg

c

u

dd

dfdb

P

[MPa, mm] (2.66)

Where b0 is the perimeter of the control area which is at the distance d/2 from the face of the

column. Furthermore, Muttoni by analytical approach indicated that the relationship between

load and rotation in flat slabs is expressed as

2/3

25.1

flexs

y

V

V

E

f

d

B (2.67)

Where Vflex is the flexural capacity of the slab and V is the acting shear force. By implementing

yield-line theory flexural capacity of uniformly reinforced slab loaded as Figure 2.15 is

determined as

cB

ccBB

cr

mV

q

rflex

4

8sin

8cos

4

22

(2.68)

Where mr is the moment capacity. In Muttoni model to calculated the punching shear capacity,

failure criterion expressed by Eq. (2.66) and load-rotation relationship presented by Eq. (2.67)

should be solved simultaneously, see Figure 2.16.

Figure 2-15 Loading setup and associated yield lines [29].

2.8. DISCUSSION ABOUT MECHANICAL MODELS

25

Figure 2-16 Load-rotation curve and failure criterion [29].

2.8 Discussion about mechanical models

Mechanical models which have been developed in history by studying the punching behaviour

of flat slabs are based on equilibrium and compatibility concepts. Each model has its own

limitations and best fits for special cases. Some don’t take into account the effect of

reinforcement and pre-stressing while some attempt to consider these factors. Furthermore,

failure criteria implemented in different models are of significant importance which sometimes

vary dramatically. In the author’s opinion it could be an interesting field of research to improve

existing mechanical models based on modifying the failure criterion. Alongside with that, the

role of concrete tensile strength is not acknowledged by all mechanical models. Generally

speaking, how mechanical models are taking into account the role of concrete tensile strength

in punching capacity of slabs fall into two categories as illustrated in Figure 2.17. Some models

neglect this effect while some models attempt to take it into account. However, implementing

nonlinear fracture mechanics and creating numerical models provide an opportunity through

which more comprehension of this issue could be grasped.

Figure 2-17 Two basic models for punching [27].

CHAPTER 2.

3.1. EFFECT OF PRE-STRESSING

27

3

Effect of pre-stressing on punching

capacity of flat slabs

3.1 Effect of pre-stressing

Concrete is a material with relatively high compressive capacity and low tensile strength.

Alongside with that, concrete has very low deformability and cracks in the case that shrinkage

is prevented. Accordingly, devising methods to tackle such drawbacks are quite natural.

Engineers came up with two solutions. They suggest either implementing pre-stressing

technique or reinforcing the concrete.

Generally speaking, pre-stressing in flat slabs leads to reduction of cracking and deformation

under serviceability loading. Furthermore, it gives an opportunity to reduce dead weight of the

slab having slender section which results in an economic design. However, in such slabs usually

punching failure is dominant in the ultimate limit state due to the limited thickness of the slab.

On the other hand, although punching failure is a local damage, it can yield in progressive

damage by overloading adjacent columns. These facts necessitate a deep and more detailed

reflection on the issue.

As mentioned in chapter one, both experimental and numerical studies have been conducted

over the years intending to unveil the influence of pre-stressing punching behaviour of slabs.

They stress some potential beneficial effects as

i. In vicinity of the column, implementing inclined tendons results in vertical forces

assisting punching resistance.

ii. Pre-stressing leads to increase of the compressive stress in the concrete which has been

reported to enhance the punching capacity.

iii. The counter acting moments produced by eccentricity of the tendons also have been

reported to boost punching capacity.

In 2012, Clément et al. [3] attempted to have a thorough look at the issue by applying the

critical shear crack theory. They considered the failure criterion proposed by Muttoni, presented

by Eq. (2.66) and tried to investigate the influence of pre-stressing. They assert that the vertical

component of the pre-stressed tendon reduces the shear force transferred by the concrete in

vicinity of the column so that

Chapter

CHAPTER 3.

PEPc VVVdAqRV (3.1)

Where q is the external load exerted on the punching cone, R is the reaction force, VE is thw

acting shear force and VP is the vertical component of the pre-stressing force at the place where

tendons intersect the punching crack, see Figure 3.1. Considering Eq. (2.66) punching failure

is reached when PU=Vc=VE-VP. It is all transparent that in this situation part of the load is carried

by the pre-stressing tendons resulting in achieving higher punching capacity.

Figure 3-1 Reduction of shear force due to tendon inclination [3].

Alongside with that, Clément et al. shed more light on the influence of pre-stressing on general

behaviour of the slab which is expressed by rotation of the slab, ψ. They concluded that pre-

stressing contributes to the amount of slab rotation through two distinct mechanisms. First one

is due to the effect of the normal compressive stresses and the second effect is caused by tendons

eccentricity. To elucidate on, as illustrated in Figure 3.2, normal compressive stresses increase

the rotational stiffness of the slab which leads to less ultimate rotation and higher punching

capacity.

Figure 3-2 Influence of in-plane force caused by pre-stressing on punching capacity [3].

As mentioned above the other phenomenon affecting the load-rotation of the slab is counter

acting moments resulted due to the eccentricity of the pre-stressing tendons. The effect of

moments opposing external loads is illustrated in Figure 3.3 where they postpone cracking of

the concrete and as a consequence based of critical shear crack theory, enhance the punching

capacity of the slab. However, looking at design codes indicates that the positive influence of

the pre-stressing on punching capacity of concrete slabs by altering its behaviour is usually

3.2. EUROCODE 2, APPROACH

29

neglected. While they only take into account the contribution of vertical components of pre-

stressing tendons. In the following the design approach proposed by Eurocode 1992 is

presented.

Figure 3-3 Influence of bending moment caused by pre-stressing on punching capacity [3].

3.2 Eurocode 2, approach

While determining the punching capacity of concrete flat slabs, Eurocode 2 [2] acknowledges

only the vertical components of tendons Vp,EC, stretching within the area at distance of 2d from

edge of the column, see Figure 3.4. and neglects the effect of eccentricity and in-plane

compressive stresses. The criterion presented by Eurocode 2, considering punching failure of

pre-stressed flat slabs is expressed as

dbfkVVV ECccECRECpE 1.010018.03/1

,, (3.2)

Where bEC as shown in Figure 3.4 is perimeter of the control area, k and ρ are size effect factor

and reinforcement ration, respectively, which are expressed as

2200

1 d

k (3.3)

02.0 lylx (3.4)

Figure 3-4 Control area for punching shear in Eurocode 2.

4.1. INTRODUCTION

31

4

Methodology

4.1 Introduction

In this chapter the process of creating numerical model of a pre-stressed concrete flat slab is

presented and the theoretical concept behind each step is discussed. Furthermore, validity of

the created numerical model is going to be controlled by experimental investigation conducted

by Hassanzadeh [4].

From economical point of view pre-stressed flat slabs made of normal strength concrete are

suitable for medium spans where the advantage of implementing pre-stressing tendons

overweighs the self-weight of the slab. Accordingly, in this investigation diameter of the slab,

c, corresponding to the area around the column in which the radial bending moment is negative

is assumed to be 2.5 m which makes sense for medium length spans of about 10 m, see Figure

4-1. Going into the depth, in order to study the punching shear capacity of the slab resting on

group pf columns, only a circular part of the slab a long circumference of which the bending

moment is zero may be modelled. Alongside with this idealization, an appropriate boundary

condition should be assigned for the isolated circular part of the slab in vicinity of the column.

The slab can be considered to have simply supported boundary condition by accepting the

simplifying assumption that the vertical displacement of the slab would be zero at the same

place where negative radial bending moment vanishes. This assumption has been implemented

in this investigation.

Figure 4-1 Diameter of the modeled slab in relation to bending moment diagram [14].

Chapter

CHAPTER 4.

As presented in Table 4-1 properties of the concrete and reinforcement steel in the model is

assumed to be identical to concrete class C30/37and steel B500B based on Eurocode 2 [2].

Furthermore, pre-stressing cables consist of 7 wire strands of type Y1670S7 based on prEN

10138 [32]. Figure 4-2 illustrates the geometry of the slab which is going to be studied in this

investigation.

Table 4-1 Material properties

Density

(kg/m3

)

E

(GPa)

fcm

(MPa)

fctm

(Mpa)

fy

(MPa)

fu

(MPa)

εcu

(%)

εc1 (‰)

concrete 2400 33 38 2.9 - - 0.35 2.2

reinforcement 7850 200 - - 500 540 5 -

cable 7850 195 - - 1640 1860 2.5 -

Figure 4-2 Reinforcement and dimensions of the slab.

To study the punching behaviour of the slab a numerical model is created in Abaqus /CAE

software [33].

4.2. CONCRETE MATERIAL BEHAVIOUR

33

4.2 Concrete material behaviour

In this part, first the mechanical properties of the concrete is presented and then the

mathematical model implemented to define concrete in numerical analysis is discussed.

4.2.1 Compressive behaviour

Concrete has completely distinct response when it is subjected to compressive or tensile force.

Accordingly, two different stress-strain curves should be introduced to represent compressive

and tensile behaviour of the concrete.

Concrete shows highly non-linear behaviour while applying uniaxial compressive load so that

only up to around 40% of the ultimate compressive strength, fcm, the stress-strain curve remains

linear. At this point micro cracks appear and propagate due to increasing the load. This

phenomenon which results in gradual decrease of modulus of elasticity continues

approximately up to 70% of ultimate compressive strength. After this stage, bond cracks form

between cement paste and aggregates due to increasing the external load which leads to

dramatic reduction of the modulus of elasticity. Then, gradually the number of these bond

cracks increases and matrix cracking begins. This process is continued until the maximum

compressive strength is achieved. Finally, the softening stage initiates where stress-strain curve

descends and failure occurs at ultimate strain. Eurocode 1992 [2] presents an empirical formula

to determine nonlinear stress-strain curve of concrete when it is subjected to uniaxial

compressive force. Figure 4.3 illustrates the idealized uniaxial compressive behaviour of the

concrete introduced by Eurocode [2].

Figure 4-3 Uniaxial compressive of concrete [2].

21

2

k

k

f cm

c

(4.1)

Where

CHAPTER 4.

1c

c

(4.2)

cm

ccm

fEk 105.1

(4.3)

It is worth mentioning that Abaqus software assumes the stress at last point of the stress-strain

curve is maintained for strains higher than εcu1, which means neglecting the crushing of concrete.

Therefore, to have more realistic behaviour one point is added to stress-strain curve where stress

is equal to zero1. Figure 4-4 depicts the stress-strain curve for concrete C30/37 which is assumed

while creating numerical model.

Figure 4-4 Stress-strain curve for concrete C30/37.

4.2.2 Tensile behaviour

Concrete has brittle behaviour while subjected to tensile force. As illustrated in Figure 4-5 the

response of the concrete remains linear up to just below the tensile strength which is denoted

by point b. At this stage increasing the external load leads to formation of micro cracks resulting

in reduction of modulus of elasticity and as a consequence nonlinear behaviour. This continues

until point c which represents the tensile strength of the concrete. Before reaching the tensile

strength crack propagation is ceased if the external load remains constant. However, after this

point cracks becomes unstable meaning that the released strain energy is adequate to cause

crack propagation without increasing the external load. Accordingly, after point c, in order to

have stable crack growth the external load has to be decreased. Descending part of the stress-

1 While modeling nonlinear material abaqus software does not allow to introduce a point with zero stress after

yield point. Thus the stress at last point is considered around 1% of compressive strength.

0

5

10

15

20

25

30

35

40

0 0.001 0.002 0.003 0.004

Stre

ssM

pa

strain

uniaxial compressiveresponse of concreteC30/37introduced plastic curvein abaqus software fornonlinear analysis

4.2. CONCRETE MATERIAL BEHAVIOUR

35

strain curve confirms this fact. During this stage, micro cracks which are accumulated in limited

areas named fracture process zone are merged together forming macro cracks. This results in

softening behaviour of the concrete in tension [33].

Figure 4-5 Behaviour of concrete under uniaxial tensile loading [33].

In numerical analysis, to have a more realistic tensile behaviour of concrete especially in

descending part of uniaxial tensile response, introducing stress-strain is not an appropriate

format. This is due to the fact that in this stage total displacement of the concrete consists of

sum of elastic strain in un-cracked part of the concrete and displacement caused by increasing

of crack’s width. Accordingly, as illustrated in Figure 4-6, it is proper to divide the tensile curve

into two parts; un-cracked part and cracked part. the response of the un-cracked concrete is

elastic and defined by stress-strain curve while crack opening stage in which the width of the

crack starts to increase is more suitable to be defined in terms of stress-displacement (σ-w) [34].

This is because displacement resulted by crack opening is independent of specimen size and

rather is related to fracture energy which is a material property.

Figure 4-6 Crack propagation in concrete at uniaxial tensile loading [34].

It is worth mentioning that fracture energy which is equal to the area under stress-displacement

curve represents the amount of energy required to establish a stress free crack. As given in Eq.

(4.4) International concrete design standard model code 2010 [35] presents a formula in which

fracture energy of the concrete is only function of compressive strength.

18.073 cmf fG (4.4)

Where Gf stands for fracture energy in Nm/m2 and fcm should be taken into account in MPa.

CHAPTER 4.

There are several crack opening curves defining the uniaxial nonlinear tensile behaviour of

the concrete while crack propagation among which linear, bi-linear and exponential curves are

common, see Figure 4-7.

Figure 4-7 three common crack opening curves [36].

Deciding about which curve to be used in numerical analysis depends on the required

accuracy. As depicted in Figure 4.7 implementing linear curve overestimates the stiffness and

tensile capacity of the concrete. In other words, exponential and bi-linear curves result in

quicker depletion of strain energy just after crack initiation. At the same time Figure 4-7

indicates that the higher tensile strength results in more ductile behaviour. This can be realized

in Figure 4.7 by observing steeper response in unloading stage for concrete samples with higher

tensile strength.

Figure 4-8 Crack opening law and tensile strength effect on tensile response of concrete

[37].

In this study the exponential curve is chosen to define crack opening behaviour of the concrete.

Cornellissen et al. [38] suggested the formula given below to define the exponential curve

c

ct

wfw

wwf

f

(4.5)

4.2. CONCRETE MATERIAL BEHAVIOUR

37

Where

cc w

wc

w

wcwf 2

3

1 exp1 (4.6)

wc is amount of displacement which is required to occur in a crack to be considered as stress-

free and is approximated to

t

f

cf

Gw 14.5 (4.7)

Where Gf is fracture energy and is calculated by Eq. (4.4).

In Eq. (4.6) c1 and c2 are material constants and are considered equal to 3.0 and 6.93,

respectively.

Figure 4.9 depicts the crack opening curve which is introduced in Abaqus software for

concrete material in this investigation.

Figure 4-9 Crack opening curve for concrete material.

4.2.3 Multiaxial behaviour

While creating a numerical the method multiaxial behaviour of the concrete should not be

neglected. Concrete indicates more ductile behaviour when it is subjected to multiaxial

compression compered to uniaxial loading. In other words, the ability of the material for energy

consumption is increased. Furthermore, in the condition of multiaxial compression the

compressive strength of the concrete is increased. This is due to the phenomenon called

confinement. Figure 4-10 indicates the biaxial behaviour of the concrete.

The other feature of biaxial response of concrete which can be inferred by reflecting on Figure

4-10 is that applying tensile and compressive stress simultaneously, decreases the capacity of

the sample. Based on what alluded to above, about the multiaxial behaviour of concrete it is of

great importance to define appropriate material model in order to achieve a realistic behaviour

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

0 0.00005 0.0001 0.00015 0.0002 0.00025

stre

ss (

Mp

a)

displacement (m)

CHAPTER 4.

in numerical analysis. Conventionally, Von Mises theorem is used to define the behaviour of

elasto-plastic materials when they indicate analogous behaviour under tensile and compressive

stress which is not the case for concrete material the behaviour of which dramatically is distinct

under compression and tension. Accordingly, in this study concrete damage plasticity (CDP)

model is implemented to define behaviour of the concrete. In CDP model modified Drucker-

Prager strength hypothesis is taken into account to define yield surface of the material.

Figure 4-10 Failure envelope under biaxial loading [36].

4.3 Concrete model

4.3.1 Yield surface

As mentioned above CDP is used to define concrete behaviour. This model is a continuum

plasticity based damage mode for concrete. the main assumption in this model is that the

concrete fails either due to compressive crushing or tensile cracking. In this model the yield

function which was defined by Lubliner et al. [39] and later modified by Lee and Fenves [40]

is determined as

pl

cc

plpqF

~~31

1maxmax

(4.8)

Where max

is maximum eigenvalue of stress tensor and pl

c~ is compressive plastic strain.

1/2

1/

00

00

cb

cb

(4.9)

σb0, σc0 are compressive biaxial strength and uniaxial compressive strength, respectively, and

4.3. CONCRETE MODEL

39

the amount of σb0/σc0 is suggested to be considered equal to 1.16 [41].

is Macauley bracket, defined as

xxx 2

1 (4.10)

)1(1~

~~

pl

tt

pl

ccpl (4.11)

In Eq. (4.8) p indicates hydrostatic pressure and q stands for Von Mises equivalent stress

which are determined as

σ(3

1trp ) (4.12)

23Jq (4.13)

Where σ is stress tensor and J2 second invariant of deviatoric stress which is calculated as

SSj :2

12 (4.14)

The function (:) stands for doubly contracted product and S indicates deviatoric stress which

is defined as

IpS

I is identity tensor.

γ is an empirical parameter which is determined by conducting triaxial test and is equal to

12

)1(3

c

c

K

K (4.15)

Kc is interpreted as the ratio of distance between hydrostatic axis and respectively compression

meridian and tension meridian in the deviatoric cross section. However, in lack of experimental

data parameter Kc is suggested to be assumed as 2/3 in CDP model [41]. By calculating γ which

is function of Kc the shape of the yield surface presented in Eq. (4.8) is determined. As

illustrated in Figure 4.11 considering Kc=1 results in circular deviatoric cross section of yield

surface which corresponds to Drucker-Prager criterion. Also in Figure 4.11 the yield surface

for Kc=2/3 which is the case in CDP model is depicted [42]. Furthermore, Figure 4.12 illustrates

the yield surface of concrete achieved by implementing Eq. (4.8) and considering Kc=2/3 and

the yield surface in linear Drucker-Prager model. After forming failure surface the failure

criterion may be defined in meridian plane (the plane containing hydrostatic axis) and by the

equivalent stress, q .

CHAPTER 4.

Figure 4-13 illustrates failure curve in meridian plane for CDP model. Reflecting on figure

4-12 confirms the shape of the failure curve presented in Figure 4-13. With regard to Figure

4.13 in CDP model dilation angle ψ and eccentricity needs to be de determined to define failure

curve. The physical interpretation of dilation angle is concrete internal friction which is

recommended to be considered 4030 [41]. In CDP model as discussed before, yield

surface assumes the shape of hyperbola. Mathematically speaking, the eccentricity which needs

to be introduced in CDP for defining yield surface is distance between vertex of hyperbola with

its asymptote. Eccentricity can be defined as ratio of tensile strength to compressive strength

[42]. To have better sense about the physical interpretation of eccentricity the concept of flow

rate in CDP model is explained beforehand.

It is all transparent that the behaviour of material after yield point is of great importance in

nonlinear analysis. Variation of deviatoric cross section (the section resulted by intersection of

deviatoric plane which is a plane the normal vector of which is hydrostatic axis, with yield

surface) explains behaviour of material after yield point which is pointed out as flow rate, see

figure 4.14. Mathematically, flow rate is formulated as

gdd pl (4.16)

Where λ is plastic multiplier and is determined experimentally and g is plastic potential which

in CDP model is defined as

tantan 22pqg to (4.17)

Where ε is eccentricity parameter introduced above.

With regard to what explained above it can be concluded that ε indicates that at which rate

yield surface approaches linear flow rate. CDP model assumes non-associated flow rule which

means plastic potential as shown in E.q (4.17) is not only function of yield stress. It is worth

mentioning that for materials the plastic potential of which only depends on yield stress the

flow rule will be associated. Mathematically speaking, the difference between materials with

associated flow rule and non-associated flow rule is that in former, the vector of plastic strain

increment is normal to yield surface. See Figure 4.15.

Figure 4-11 Deviatoric cross section of yield surface in CDP model [42].

4.3. CONCRETE MODEL

41

Figure 4-12 Yield surface in CDP model and Linear Drucker-Prager model [41,42].

Figure 4-13 Failure curve in meridian plane [42].

Figure 4-14 Yielding curve in same deviatoric plane [41].

CHAPTER 4.

Figure 4-15 Plastic strain increment vector, on yield surface considering associated and non-

associated flow rule.

4.3.2 Defining damage in CDP model

In CDP model, damage index is introduced to assess the and quantify the amount of damage

occurred in concrete material. As illustrated in Figure 4-16 total strain in concrete, t , contains

two terms which are inelastic strain, in and initial elastic strain, 0 . In this method pl , is

defined as reference point beyond which the damage occurs in concrete and as a consequence

damage index has the value higher than zero. It goes without saying that the maximum amount

of damage is equal to 1 and it is reached when concrete is fully cracked under tension or fully

crushed under compression. However, in this investigation pl for compression is considered

equal to 1c see Table 4-1, and cracking displacement is assumed equal to wc, see Eq. (4.7).

Figure 4-16 Stress-strain curve for uniaxial tension and compression curve and associated

damage index.

4.3. CONCRETE MODEL

43

By defining pl , damage index, is calculated as

01)1(

c

c

cinc

d

d

(4.18)

0)1(

t

t

tincr w

d

dww

(4.19)

Where dc and dt are compressive and tensile damage index, respectively.

Finally, with regard to the theoretical concept of CDP model explained in this section the

parameters required to define concrete material model is given in Table 4.2.

Table 4.2 CDP model parameters

Parameter

Name

Dilation

angle

Eccentricity α K εpl wcr

(m)

Value 36̊ 0.1 1.16 0.667 0.0022 2.12×10-4

4.3.3 Longitudinal bars and pre-stressing tendons

As illustrated in Figure 4.2 bars with diameter of 10 mm at distance of 20 cm are used for

longitudinal reinforcement in two perpendicular directions. Properties of reinforcement bars are

presented in Table 4.2. Eurocode 2 asserts that in lack of test data a bilinear stress-strain curve

can be assumed [2]. However, in Abaqus software stresses and strains for steel should be

introduced in terms of Cauchy stress (true stress) and logarithmic strain. It is worth mentioning

that actual stresses and strains are determined by considering the actual cross-section and length

of the element while loading. Nominal stress, σnom, is converted to true stress as follows

)1( nomnomtrue (4.20)

Logarithmic plastic, pl

ln strain is calculated as

E

truenom

pl )1ln(ln

(4.21)

Figure 4-17 and Figure 4-18 show the uniaxial stress-strain curve for implemented

reinforcement bars and pre-stressing cables respectively. Since steel has the same tensile and

compressive yield strength Abaqus software considers Von Mises criterion for nonlinear

behaviour of steel materials. However, in this investigation cables and reinforcement bars are

modeled as truss elements and as a consequence they response only when they are subjected to

axial forces. The reason for this assumption is that the flexural stiffness of reinforcement bars

and pre-stressing cables compared to their axial stiffness is negligible.

CHAPTER 4.

The influence of cable arrangement on punching capacity of the flat slabs is the main objective

of this study. Accordingly, 7 alternatives are considered for cable locations, see Figure 4-19.

Figure 4-17 Stress-strain curve of reinforcement bars.

Figure 4-18 Stress-strain curve of pre-stressing cables.

0

100

200

300

400

500

600

0 0.02 0.04 0.06

Stre

ss (

Mp

a)

Strain

nominal

0

500

1000

1500

2000

2500

0 0.005 0.01 0.015 0.02 0.025 0.03

Stre

ss (

Mp

a)

Strain

nominal

true-log

inelastic

4.3. CONCRETE MODEL

45

Figure 4-19 Alternative cable arrangements.

Tendons are placed in parabolic shape, which is common arrangement of pre-stressing cables

in continuous slabs and beams where the eccentricity of the tendons must be opposite in middle

of the span and over the support. Furthermore, implementing inclined cables results in cross-

sectional forces in vertical direction counteracting the shear forces. Figure 4-20 depicts the

parabolic shape of the cables.

Figure 4-20 Parabolic shape of pre-stressing cables.

Pre-stressing force, P, is determined based on Eurocode 1992 [2] as

kNP

Mpa

fkfk kppkp

2051467139

147616409.0;18608.0min

;min

max

1.021max,

(4.22)

So P is considered equal to 200 kN.

CHAPTER 4.

4.4 Finite element model

Finite element method (FEM) is a numerical and approximate solution for partial differential

equations by which physical phenomena are described. It has been used to solve differential

equations governing structural analysis in recent decades. The main idea in FEM is discretizing

the domain and implementing appropriate shape functions for each discrete part (structural

element). Abaqus software has the element library providing an opportunity through which the

structure is discretized. In this investigation truss elements are assigned for reinforcement bars

and pre-stressing cables. This is due to the fact that their flexural stiffness is negligible and thus

is not taken into account while creating numerical model. It is all transparent that concrete, in

punching failure is in multiaxial tress situation which necessitates implementing the continuum

(brick) elements for discretizing the slab.

The other important issue which needs to be considered in numerical modelling is interaction

between materials with different properties or different assigned element types. It is assumed

here that the reinforcement bars and pre-stressing cables are embedded in the concrete which is

an idealization for bonded pre-stressed cables. This assumption yields in tied degrees of

freedom for cables and bars with concrete at points where they contact with each other. This

concept is illustrated in Figure 4-21.

With regard to the symmetric condition, only one-fourth of the slab needs to be modelled.

However, appropriate boundary conditions should be assigned in symmetry plane. Figure 4-22

indicates the created model and the assigned boundary conditions as well.

Figure 4-21 Tying condition between truss and brick element.

4.4. FINITE ELEMENT MODEL

47

Figure 4-22 Boundary conditions.

After creating the model, non-linear static analysis is implemented to analyse the structure. In

this method the equilibrium equation governing the problem which is presented below is solved

uKP . (4.23)

Where P is load vector, K is stiffness matrix and u is displacement vector.

The equation above for nonlinear structures which is the case here cannot be solved directly.

This is because the constitutive matrix, the matrix containing properties of the material, which

is included in K matrix in Eq. (4.23) due to material nonlinearity is not constant. Accordingly,

it needs to be solved in an iterative process. The iteration is required in two stages; local iteration

and global iteration. Here each step is explained separately:

Local iteration: In finite element method the local stiffness matrix, k, of each element is given

as

dVBEBkt

(4.24)

Where the elements of matrix B is independent of material properties and contains derivations

of shape functions and E is constitutive matrix the elements of which is function of Poisson’s

ratio and modulus of elasticity. On the other hand, as explained before CDP model assumes

non-associative flow rule and the growth of yield surface after yielding is not linear. With regard

to this fact, determining modulus of elasticity for generating constitutive matrix requires an

iterative process, see Figure 4-23.

CHAPTER 4.

Figure 4-23 Local iterative process to capture the correct constitutive matrix.

Global Iteration: the global stiffness matrix can be generated by assembling local stiffness

matrices. To elucidate on, assembling process takes into account the geometry of the problem

by determining how elements are connected to each other. Assembling local stiffness matrices

each of which is achieved by assuming non-linear stress-strain relations results in forming

global stiffness matrix which itself is in nonlinear relation with the applied force on the structure

in global coordinate. Accordingly, it can be concluded that, the other iterative process is

required to solve Eq. (4.23). Figure 4-24 schematically illustrates two steps of iterative process

for solving equilibrium equation where Abaqus software implements mathematical methods

such as Newton-Raphson to minimize the residual force at each step. In this figure epA, epB and

epC are residual forces.

Figure 4-24 Newton Raphson iteration process.

4.5 Verification

Hassanzadeh conducted an experimental investigation at KTH university in 1998 to

investigate the effect of location of pre-stressing cables on punching capacity of flat slabs [4].

During his investigation he tested slabs with distinct mechanical characteristics and cable

arrangements among which the sample A1 is selected for verification of the numerical model

4.5. VERIFICATION

49

created in this thesis. As illustrated in Figure 4-25 the behaviour of the slab while calculating

punching capacity in both numerical analysis and experimental data follows the same path. The

final displacement for both experimental test and numerical analysis are close enough.

However, the numerical model has more stiff response compared to experimental sample. This

can be due to the fact that material properties such as stress-strain curve and fracture energy in

numerical model have been taken into account by approximate equations. The other important

factor which may affected the results is that the numerical model considers fully bonded

boundary conditions for pre-stressing cables while in experimental investigation the cables are

un-bonded.

Figure 4-25 force displacement curve for numerical analysis and experimental text.

0

100

200

300

400

500

600

700

800

-2 3 8 13 18

forc

e (M

pa)

displacement (mm)

experimental

Mesh size: 20mm

mesh size 15mm

5.1. RESULTS

51

5

Result and Discussion

5.1 Results

In this chapter first the results are presented and then in order to shed more light on the issue

they are going to be discussed thoroughly.

In Table 5.1 the punching shear capacity of the created models by both implementing

numerical analysis and the Eurocode procedure is presented. Figure 5.1 illustrates the force-

displacement curves achieved by conducting FEM. Figure 5.2 depicts the punching shear

capacity of the slab models with respect to location of pre-stressing cables.

Table 5.1 Punching shear capacity of the slab models.

Case Distance

from

face of

the

column

ρ (σc)ave

Mpa

Vp,EC

(each cable)

kN

Eurocode Numerical Numerical

Eurocode

V

V

VR,EC

kN

VR,EC

kN

Without

Cable

- 2.27×10-3 - - 182 218 1.2

A 0.5d 2.27×10-3 3.95 42 496 594 1.2

B 1.5d 2.27×10-3 3.78 42 460 570 1.24

C 2.5d 2.27×10-3 3.12 - 306 527 1.72

D 3.5d 2.27×10-3 2.70 - 288 501 1.74

E 5d 2.27×10-3 1.52 - 242 419 1.73

F 6d 2.27×10-3 0.33 - 195 288 1.47

G 7d 2.27×10-3 0.12 - 187 240 1.28

Chapter

Figure 5.1 Force-displacement curve for slab models.

Figure 5.2 Effect of location of pre-stressing cables on punching shear capacity of slab

models.

5.2 Discussion

As already explained, in the vicinity of the column implementing inclined pre-stressing cables

results in vertical forces assisting the punching shear resistance of the slab. Furthermore, pre-

stressing increases the compressive stress around the column which also leads to increase of

0

100

200

300

400

500

600

700

-2 -1 0 1 2 3 4 5 6 7 8

Pu

nch

ing

cap

acit

yM

Pa

deformation mm

0.5d 1.5d 2.5d 3.5d 5d 6d 7d without cable

0

100

200

300

400

500

600

700

0 1 2 3 4 5 6 7 8

Pu

nch

ing

shea

r ca

pac

ity

MP

a

distance

punching shear capacity of slab without pre-stressing cables, Eurocode 2

punching shear capacity of slab without pre-stressing cables, Numerical analysis

punching shear capacity of pre-stressed slabs, Eurocode 2

punching shear capacity of pre-stressed slabs, Numerical analysis

xd xd

5.3. FUTURE INVESTIGATIONS

53

the punching shear capacity. The other positive effect of pre-stressing is the counter acting

moments introduced via eccentricity of the tendons. Conducting numerical analysis takes into

account all of these positive influences of pre-stressing cables. Reflecting on Figure 5.1 reveals

that the effect of implementing pre-stressing cables until the distance of 6d (where d is thickness

of the slab) from the face of the column is considerable. Table 5.2 presents the percent of boost

in punching shear capacity respecting the location of pre-stressing cables from the face of the

column.

Table 5.2 amount of increase in punching capacity regarding the location of cables.

Distance of cable

from face of the

column

0.5d 1.5d 2.5 d 3.5 d 5d 6d 7d

Punching shear

capacity

enhancement

172% 161% 141% 130% 92% 33% 10%

Reflecting on the procedure presented by Eurocode 2 in order to determine the punching shear

capacity of concrete slabs equipped with pre-stressing cables indicates that the effect of counter

acting moment due to the eccentricity of pre-stressing is completely neglected and the vertical

components of pre-stressing cables is considered only when they are located in the area within

distance 2d from the face of the column. This area is called basic control area in Eurocode 2.

Comparing the numerical results with the amount of punching capacity predicted by Eurocode

2 which is presented in Table 5.1 reveals that inside the control area the predicted value for

punching capacity by Eurocode has almost acceptable accuracy (the difference with numerical

results is about 20 percent). However, outside the basic control area where Eurocode 2 only

takes into account the positive effect of compressive stresses introduced by pre-stressing the

difference between numerical results and Eurocode procedure are very high. In other words, in

the distance of 2d until 6d from face of the column where the effect of pre-stressing cables on

increase of punching capacity of the slab is still significant Eurocode 2 noticeably

underestimates the punching shear capacity of the slab, see Figure 5.2.

5.3 Future investigations

Further steps can be taken in continuation of current master thesis considering the suggestions

presented below:

1. In this thesis thickness of the slab is constant. So it is recommended to carry out the

investigation considering various thickness for the slab in order to study the size effect.

Eurocode1992 takes into account the size effect through the factor k while calculating

punching capacity of the slab.

2. Studying the effect of opening in vicinity of the column on punching capacity of pre-

stressed concrete flat slabs is of great interest.

3. Further investigations can be carried out to propose an analytical method which is able

to predict the punching shear capacity of the concrete flat slabs when the pre-stressing

cables are located outside the basic control area

55

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TRITA-ABE-MBT-19686. Master Thesis 2019.