Eindhoven University of Technology MASTER Shear failure of … · Darwish, 1987). The shear capacity of SFR-RC is currently included in some design codes, like RILEM TC 162-TDF, and

Eindhoven University of Technology

MASTER

Shear failure of reinforced concrete beams with steel fibre reinforcement

Krings, H.

Award date:2014

Link to publication

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https://research.tue.nl/en/studentTheses/969fad31-114e-4632-959f-97273ba62b95


Master’s Graduation Thesis

Hilde Krings

April 2014

Graduation committee Prof. dr. ir. T.A.M. (Theo) Salet Ing. O. (Ostar) Joostensz Ir. F.J.M. (Frans) Luijten

– TU/e – ABT – TU/e

A-2014.57


Master’s graduation thesis – A-2014.57

by

H. (Hilde) Krings, BSc – 0614545

In partial fulfilment of the requirements for the degree of

Master of Science

Eindhoven University of technology

Faculty Architecture Building and planning

Unit Structural Design

Chair in Concrete structures

Eindhoven, April 2014

Graduation committee

Prof. Dr. Ir. T.A.M. (Theo) Salet – Chairman

Professor Concrete Structures in Eindhoven University of Technology

Ing. O. (Ostar) Joostensz

Specialist Civil Engineering at ABT

Ir. F.J.M. (Frans) Luijten

Assistant Professor Concrete Structures in Eindhoven University of Technology

5 Preface

Preface The Master Architecture, Building and Planning at the Eindhoven University of Technology contains

multiple majors, which all use their own interpretation of the final project. The choice for the major

structural design leads to two options: a research or a design project. I chose a research project into

steel fibre reinforced concrete structure, because it gave me the opportunity to specialise in a

structural material.

This research led to an exploration of the subject steel fibre reinforced concrete, as well as the

subjects shear failure, concrete failure, and finite element methods among others. Of course, this

Master’s thesis does not contain my entire research. A lot of work contributed to my professional

knowledge, but did not contribute to the heart of the matter. However, this is not something I

regret, since it provides sufficient food for thought and matter for conversation. Thus, feel free to

discuss this thesis or related subjects with me.

To all my family and friends, which have often listened to my in-depth stories, had to beat their brain

about the content and who often heard that I had almost finished: Thanks to you, now I actually

have.

Kind regards,

Hilde Krings

[email protected]

mailto:[email protected]

6 Shear failure of reinforced concrete beams with steel fibre reinforcement

7 Summary

Summary The shear capacity of SFR-RC is currently included in some design codes, like RILEM TC 162-TDF, and

is based on experimental research. The aim of this Master’s thesis is to obtain insight about the

structural behaviour of SFR-RC beams subjected to shear failure. A beam model, based on the

moment-area method and the multi-layer model, is used to examine the hypothesis that

‘The shear capacity of a reinforced concrete element subjected to a flexural load

increases due to the contribution of steel fibre reinforcement in the tensile zone and

in the compressive zone.’

For this research the beam model is extended in two ways. First, shear failure is incorporated in the

beam model. For this purpose, the tensile principle stresses are evaluated because shear failure is

caused by the development of tensile stresses in the uncracked zone (Kotsovos, 1999). Second, the

contribution of steel fibre reinforcement is incorporated in the beam model. To include the

contribution of steel fibres in the cracks of the tensile zone, the residual tensile strength is added to

the bond theory. The bond theory is used to determine the tensile stress-strain relation of concrete.

Additionally, a contribution of the steel fibres in the compressive zone to the shear capacity is

incorporated.

From a comparison between the shear capacity according to the extended beam model,

experimental tests, and a formula of Dupont and Vandewalle (2002), is concluded that the extended

beam model confidently predicts the ultimate shear capacity. Thereupon, the effect of steel fibre

reinforcement is analysed using the extended beam model. The hypothesis that the shear capacity

increases due to addition of steel fibre reinforcement is validated. The conclusion is drawn that the

shear capacity increases due to the contribution of the steel fibre reinforcement in the effective

shear height.


9 Notations

Notations

Abbreviations

SFRC Steel Fibre Reinforced Concrete

FRC Fibre Reinforced Concrete

SFR-RC Reinforced Concrete with Steel Fibre Reinforcement

(Steel Fibre Reinforced-Reinforced Concrete)

CMOD Crack Mouth Opening Displacement

a/d Shear span to effective depth ratio

Latin letters

a Shear span

cA Cross-sectional area of the concrete

sA or slA Cross-sectional area longitudinal reinforcement

b Beam width

d Effective beam depth

cmE Young’s modulus concrete

sE Young’s modulus steel reinforcement bars

f Fibre volume ratio

ckf Characteristic compressive concrete strength

cmf Mean compressive concrete strength

ctf Axial tensile concrete strength

ctmf Mean tensile concrete strength

;fct Lf Flexural tensile strength of SFRC at the limit of proportionality

;4Rf Flexural residual tensile concrete strength at an CMOD of 3.5mm

resf Residual tensile concrete strength

yf Yield strength reinforcement bars

ywf Yield strength shear reinforcement

h Beam height

ih Height of layer i

ch or xh Compressive height

h Effective shear height

i Number of layers

fL Fibre length

tl or stl Transition length

m Number of load steps for a moment-curvature calculation

externM External bending moment

internM Internal bending moment

n number of segments of segment number


iN Normal force at layer i

cN Normal force in the compressive zone

sN Normal force in the steel reinforcement bars

p Number of load steps for a load-deflection calculation

cV Shear capacity of the concrete

exV External shear force

fV Shear capacity of the steel fibre reinforcement or Fibre volume ratio

inputV Inserted shear force

RdV Design shear capacity

uV Ultimate shear capacity

iy Distance from top of the cross-section to the centroid of layer i

Greek letters

Angle first principle stress

i Strain at layer i

c Maximum compressive concrete strain

ct (Maximum) tensile concrete strain

ctm Concrete strain which belongs to the mean tensile concrete strength

ctu Ultimate tensile concrete strain

cr Strain which belongs to the critical concrete strength

1cm of 1gem Average steel strain between cracks for a maximum crack distance

2cm Average steel strain between cracks for a minimum crack distance

cm Average steel strain between cracks for a average crack distance

s Steel strain

y Yielding strain steel

u Ultimate strain

i Shear stress in layer i

max Maximum shear stress

or l Longitudinal reinforcement

i Stress at layer i

c (Maximum) compressive concrete stress

ct (Maximum) tensile concrete stress

cr Critical concrete strength

cm Average concrete stress between cracks for a average crack distance

1cm Average concrete stress between cracks for a maximum crack distance

11 Notations

2cm Average concrete stress between cracks for a minimum crack distance

s Steel stress

,s cr Steel stress when critical concrete strength is reach

x Normal stress in x-direction

y Normal stress in y-direction

1 First principle stress

2 Second principle stress

or Diameter reinforcement


Table of Contents Preface ..................................................................................................................................................... 5

Summary ................................................................................................................................................. 7

Notations ................................................................................................................................................. 9

Table of Contents .................................................................................................................................. 12

1 Introduction ................................................................................................................................... 15

2 Literature ....................................................................................................................................... 17

2.1 Beam model .......................................................................................................................... 17

2.1.1 Moment-area method ................................................................................................... 17

2.1.2 Multi-layer model .......................................................................................................... 18

2.1.3 Numeric model .............................................................................................................. 19

2.1.4 Tension stiffening effect ................................................................................................ 22

2.2 Shear failure .......................................................................................................................... 25

2.2.1 Shear failure according to Kotsovos .............................................................................. 25

2.2.2 Shear failure according to Pruijssers ............................................................................. 28

2.2.3 Principle tensile stresses ............................................................................................... 29

2.3 Steel fibre reinforced concrete ............................................................................................. 32

2.3.1 Post-cracking behaviour of steel fibre reinforced concrete .......................................... 32

2.3.2 Addition of the fibre pull-out forces along the inclined crack ...................................... 33

2.3.3 Addition of shear capacity due to steel fibres ............................................................... 35

2.4 Conclusion and hypotheses ................................................................................................... 38

3 Extended beam model .................................................................................................................. 39

3.1 Extension I: Shear failure ....................................................................................................... 39

3.1.1 Shear stresses ................................................................................................................ 39

3.1.2 Principle stresses ........................................................................................................... 43

3.2 Extension II: Contribution steel fibre reinforcement ............................................................ 45

3.2.1 Contribution in the tensile zone .................................................................................... 45

3.2.2 Contribution in the compressive zone .......................................................................... 46

3.3 Limitations ............................................................................................................................. 47

4 Verification .................................................................................................................................... 49

4.1 Benchmarks extended beam model ...................................................................................... 49

4.1.1 Longitudinal reinforcement ratio .................................................................................. 50

4.1.2 Residual tensile strength ............................................................................................... 52

4.1.3 Shear span to depth ratio a/d ...................................................................................... 52

13 Table of Contents

4.1.4 Concrete strength .......................................................................................................... 54

4.1.5 Overview relations ........................................................................................................ 55

4.2 Verification steel fibre reinforcement in the tensile zone .................................................... 56

4.2.1 Verification tension stiffening module .......................................................................... 56

4.2.2 Verification contribution tensile zone ........................................................................... 58

4.3 Verification extended beam model ...................................................................................... 60

4.3.1 Qualitative verification .................................................................................................. 60

4.3.2 Quantitative verification ............................................................................................... 64

4.3.3 Comparison between the model and Kani .................................................................... 67

5 Conclusion and discussion ............................................................................................................. 69

5.1 Conclusion ............................................................................................................................. 69

5.2 Discussion and recommendations ........................................................................................ 70

References ............................................................................................................................................. 71

Appendices ............................................................................................................................................ 73


15 Introduction

1 Introduction In the past, safety was the main requirement for a structural design. Nowadays, a structural design

depends also on economic and sustainability requirements resulting in an optimisation of the

structural dimensions. To attain optimal dimensions, knowledge of material and structural behaviour

is necessary. Additionally, innovations are necessary to achieve improvements in material and

structural behaviour.

One of the innovations in structural materials is steel fibre reinforced concrete (SFRC). Previously, a

literature survey of SFRC was conducted (Aa et al, 2013). This literature survey contains the

interaction between fibre and concrete, testing methods for post-cracking behaviour, regulations,

and numerical modelling. Aa et al (2013) concluded that steel fibres improve the post-cracking

behaviour compared to plain concrete. The improved post-cracking behaviour of SFRC results in a

more ductile material compared to plain concrete.

In particular, the combination of steel fibre reinforced concrete and steel reinforcement bars (SFR-

RC) can provide new opportunities and applications (Walraven, 2011). For instance, SFR-RC beams

have smaller crack distances and crack widths preventing corrosion of the steel reinforcement bars.

Also, the shear capacity of beams improves through the addition of steel fibres (Narayanan and

Darwish, 1987). The shear capacity of SFR-RC is currently included in some design codes, like RILEM

TC 162-TDF, and is based on experimental research. Unfortunately, these empirical formulas provide

only a quantitative insight into the behaviour, not a qualitative insight. In other words, they explain

little or nothing about the real behaviour.

The aim of this Master’s thesis is to obtain insight into the structural behaviour of SFR-RC beams

subjected to shear failure. This aim is pursued by implementing theoretical knowledge of the

material and structural behaviour into a numerical model. This model is a beam model based on the

moment-area method and the multi-layer model . The beam model is used to examine the

hypothesis that




Chapter 2 provides an overview of the relevant literature about the beam model, shear failure, and

SFRC. After that, chapter 3 describes how the beam model is transformed to the ‘extended beam

model’ by the implementation of shear failure and steel fibre reinforcement. This ‘extended beam

model’ is benchmarked and verified in chapter 4. The results of the benchmarks and verification are

discussed in chapter 5.


17 Literature

2 Literature As mentioned in the introduction, the aim of this Master’s thesis is to obtain insight in the structural

behaviour of SFR-RC beams subjected to shear failure. This aim is pursued by implementing

theoretical knowledge of the material and structural behaviour of SFR-RC into a beam model. This

chapter describes the literature studies which are conducted to obtain the necessary knowledge.

First, the literature about the beam model is presented in paragraph 2.1. Second, paragraph 2.2

documents the relevant literature about shear failure of concrete beams. Finally, the contribution of

steel fibre reinforcement is described in paragraph 2.3. An general literature survey about steel fibre

reinforced concrete (SFRC) is separately documented (Aa, et al., 2013). Chapter 2 ends in paragraph

2.4 with a conclusion and from this the research questions.

2.1 Beam model

The beam model is based on the moment-area method and a multi-layer model. The moment-area

method is applied to calculate the deflection of a beam. This method is suitable for a variable

bending stiffness, so for a reinforced concrete beam after cracking. The moment-area method is

explained as well as the similarity between the moment-area method and the Euler-Bernoulli beam

theory. After that, the multi-layer model is described. The multi-layer model is applied to calculate

the moment-curvature relation and is suitable for a non-linear stress-strain relation, thus for

reinforced concrete after cracking.

2.1.1 Moment-area method

The moment-area method is applied to calculate the deflection of a reinforced concrete beam. This

method is suitable for a variable bending stiffness because the deflection is calculated from the

curvatures along the beam. To apply the moment-area method the beam is segmented in a finite

number of segments n. Due to the external load, an external moment is generated along the beam.

Thereupon, the moment-curvature relation is used to define the curvature along the beam. Figure

2.1 illustrates the segmentation of the beam and the relation between the external moment and

curvature along the beam.

Figure 2.1: Segmentation of the beam; external moment and curvature along x and per segment; moment-curvature relation.


From the curvature along the beam the deflection of the beam is calculated in four steps: first, per

segment the changes in slope is calculated from the curvature:

d x x dx ( 2-1 )

Second, the rotation along the beam is calculated from the change in slope:

x

x

x d ( 2-2 )

x =distance from 0 to the centroid of the curvature diagram

Third, per segment the change in deflection is calculated from the rotation:

du x x dx ( 2-3 )

Finally, the deflection along the beam is calculated by summing the change in deflection:

0

x

u x du x ( 2-4 )

2.1.2 Multi-layer model

The previous section explains how the deflection can be calculated from a moment-curvature

relation. The moment-curvature relation is calculated using the multi-layer model (Hordijk, 1991).

The multi-layer model divides the height of a cross-section into a finite number of layers i (fig. 2.2).

Every layer is subjected to a strain i which is defined by the (linear) strain flow. Due to the stress-

strain relation of the concrete, the stresses per layer i can be generated. Additionally, the stress-

strain relation of the steel reinforcement bars defines the steel stress s . The stress distribution

should result in an equilibrium of the internal forces:

0i i s sN bh A ( 2-5 )

Figure 2.2: Division of the cross-section in layers and strains and stresses per layer.

19 Literature

When the internal forces are in equilibrium, the curvature of the layered cross-section can be

calculated from the compressive strain at the top c and the tensile strain at the bottom ct :

c ct

h

( 2-6 )

Additionally, the internal moment can be calculated from the internal forces:

intern i i s sM N y N y ( 2-7 )

As a result, the moment-curvature relation can be calculated by a stepwise increment of c until the

failure strain of the concrete is reached.

Notice in the above described calculation procedure that the stress-strain relation plays a key role in

the relation between the internal moment and the curvature because the internal moment depends

on the stress flow and the curvature on the strain flow. The calculation procedure is further clarified

with a flow chart in appendix A.

2.1.3 Numeric model

The moment-area method and the multi-layer model were previously implemented into a numeric

model by three Master students of the faculty 'Architecture, Building an Planning’ at the Eindhoven

University of Technology, Thomas Paus, Reno Couwenberg, and Patrick Marinus. Their numeric

model, which is developed in a Microsoft Excel environment, is used for this Master’s thesis. Marinus

(2013) annexed a user manual of the numeric model and formulated the following principles:

- The numeric model functions in Excel from one ‘overview’ worksheet. Parameters can be

changed in the overview. Also, the output is as much as possible showed in the overview.

- The numeric model is modular so modules can be added in the future. Also, existing modules

can be altered or extended. The user chooses the modules which are necessary for his

calculation. (The model currently contains two modules the moment-curvature calculation

module and the deflection calculation module.)

- The model calculates an moment-curvature diagram, a rotation diagram and a deflection

diagram. Also, the stress-strain relation of the concrete and reinforcement bars are displayed

in diagrams.

- The module is suitable for a simple supported beam with rectangular cross-section and a

uniform line load.

- Material properties can be entered using a linear and/or bi-linear stress-strain relation.

- Two layers of reinforcement can be entered. More layers of reinforcement or shear

reinforcement cannot be entered.

The moment-area method is similar to Euler-Bernoulli beam theory. As a result, the problem could

also be described with a differential equation. First the Euler Bernoulli beam theory is explained and

second the similarity between the moment-area method and the Bernoulli beam theory.

The Euler-Bernoulli beam theory is a simple tool which enables the development of a one-

dimensional model to analyse a three-dimensional structure. To do so, the Euler-Bernoulli beam

theory has two key assumptions:


- The beam has a linear elastic material behaviour according to Hooke’s law,

- Plane sections remain plane and perpendicular to the neutral axis.

These key assumptions are used to describe the equilibrium, constitutive and kinematic condition of

an infinitesimally small beam element. The three conditions result in a differential equation that

describes the structural behaviour of whole beam. The equilibrium condition of a beam element dx

subjected to a line load q (fig. 2.3) defines the interdependence between q and the internal moment

M as

2

2

d Mq

dx ( 2-8 )

Additionally, the constitutive law for a beam subjected to bending can be expressed with the

moment-curvature relation:

M x EI x ( 2-9 )

Finally, the kinematic condition of the infinitesimally small beam segment (fig. 2.4) for small

displacement is defined as

2

2

d d u

dx dx

( 2-10 )

Figure 2.3: Infinitesimally small beam element dx subjected to a line load q.

Figure 2.4: Bending deformation of a infinitesimally beam element.

21 Literature

As a result of equation 2-8, 2-9, and 2-10, a fourth order differential equation is formulated:

2 2

2 2

d d uEI q

dx dx

( 2-11 )

This differential equation can be solved with four boundary conditions. Given the line load q the

deflection u can now be calculated for every location x along the beam length.

Similarity numeric model and Euler-Bernoulli beam theory

According to the moment-area method the deflection u (eq. 2-4) can also be expressed as

0

x x

x

u x x dxdx ( 2-12 )

When the integral is differentiated twice, the curvature is formulated as the differential equation

2

2

d u xx

dx

This differential equation is equal to the kinematic condition of the Euler-Bernoulli beam theory (eq.

2-10).

In case of a linear elastic stress-strain relation, the internal moment and the curvature of the multi-

layer model (eq. 2-6 and 2-7) are

21intern 6 cM E bh

2 c

h

Substituting c by 12

h the internal moment becomes

31intern 12

M E bh EI

This equation is equal to the constitutive law of the Euler-Bernoulli beam theory (eq. 2-9).

The external moment within the numeric model is defined by the user considering a moment-

equilibrium of the beam. In case of a simply supported beam with a line load q and a length L:

21 12 2

M x qxL qx

The second derivative of this equation is

2

2

d M xq

dx

This equation is equal to the equilibrium condition of the Euler-Bernoulli beam theory.

The mathematical similarity between the moment-area method and the Euler-Bernoulli beam theory

is demonstrated; therefore, a numeric model based on differential equation of the Euler-Bernoulli

beam theory is similar to the numeric model applied in this research. Due to this similarity, the


material and structural behaviour of SRF-RC beams described in this Master’s thesis can also be used

in a finite difference method. Paus (2014) already did this for the development of a numeric model

using the finite difference method.

2.1.4 Tension stiffening effect

The beam model calculates the critical cross-section from a cross-sectional equilibrium reducing the

relevant height of the concrete to the uncracked height. However, not every cross-section in a

reinforced concrete beam is a critical cross-section because between cracks the full concrete height

is present. The concrete between the cracks contributes to the bending stiffness of a reinforced

concrete beam. The phenomenon of extra stiffness due to the remaining concrete between cracks is

called the tension stiffening effect.

Figure 2.5 shows the test results of Gribniak et al. (2012) (solid black and grey line) and the output of

Marinus’ numeric model (yellow line). The difference between these results is e.g. due to the fact

that the numeric model does not include tension stiffening.

Bruggeling and Bruijn (1986) present a procedure to include tension stiffening in the tensile stress-

strain relation of concrete. This procedure is based on a bond theory for deformed reinforcing bars in

concrete. The bond theory describes the bond between the concrete and the reinforcement. When a

crack occurs, this bond fails at the crack; however, next to the crack the bond is disrupted but not

failed. The length along which the crack causes a disruption of the bond is called the transition

length. In addition to the bond theory, Bruggeling and Bruijn (1986) describe a procedure to translate

results from the bond theory into a post-cracking stress-strain relation for concrete. Appendix C

explains the bond theory and translation into a post-cracking stress-strain relation. Figure 2.6

presents the stress-strain relation that is defined in appendix C. This stress-strain relation can be

inserted in the beam model to include tension stiffening.

Figure 2.5: Moment-curvature relations of test results (continuous grey and black line) and a calculation without taken tension stiffening into account (yellow) (Gribniak et al., 2012).

23 Literature

Figure 2.6: Tensile stress-strain relation of concrete including tension stiffening (red). A: cracking point; B: average concrete stress and steel strain for the average crack distance; C: ending of tension stiffening at yielding.

Figure 2.7: Parabolic tensile load along the length.

Figure 2.8: 1 Point load and moment along the length.

Figure 2.9: Line load and bending moment along the length Figure 2.10: : 2 Point loads and moment along the length.


Discussion

Parameters

The bond between concrete and a rebar is influenced by the reinforcement ratio and the

reinforcement diameter (Bruggeling and Bruijn, 1986). The reinforcement ratio influences the steel

strain and stress because they depend on the reinforcement area. The reinforcement diameter

influences the traction between concrete and rebar because this traction depends on contact area

between the concrete and rebar. The traction affects the crack width and crack distance but not

significantly the average concrete stresses. As a result, the stress-strain relation is mainly affected by

the reinforcement ratio.

A flexural loaded beam

The developed stress-strain relation is based on an axial tensile load. Hence, application to a flexural

loaded beam is possible if the cracking pattern is similar to the crack pattern of an axial loaded bar

(Hamelink, 1989). Hamelink (1989) and Salet (1991) theoretically analysed the crack pattern of a

parabolic tensile load (fig. 2.7). Hamelink draws four conclusions:

1. The first crack occurs at the maximum tensile force

2. The load has to increase to cause new cracks

3. The transition length depends on the magnitude of the load and the load type

4. The tension stiffening effect is noticeable in the case of a low reinforcement ratio and crack

development along a large part of the length

Due to these conclusions, the crack pattern of a flexural loaded beam with a varying bending

moment probably differs from the axial loaded bar. Therefore, the defined stress-strain relation (fig.

2.6) might not be suitable for a flexural loaded beam with a varying bending moment such as a

simple supported beam subjected to one point load (fig. 2.8) or a simply supported beam subjected

to a line load (fig. 2.9). If the crack distance of these two load types can be represented by the

minimum crack distance instead of the average crack distance, point B of the post-cracking branch

could be adjusted so that point B describes the average stress and strain for a minimum crack

distance. In this case point B is equal to point 2 of the stress-strain diagrams in appendix C.

In case of a simply supported beam subjected to two point loads, the bending moment between the

loads is constant resulting in a constant tensile force between these point loads (fig. 2.10). The

assumption is made that the tensile behaviour between the cracks is similar to the behaviour of an

axial loaded bar. Consequently, the stress-strain relation should be applicable for a simply supported

beam subjected to two point loads.

25 Literature

2.2 Shear failure

From the previously presented beam model the internal stresses in a cross-section due to a bending

moment can be calculated. Besides these stresses, also the internal stresses due to shear forces are

present in a flexural loaded beam. In case of linear elastic behaviour, these stresses can easily be

determined with classical mechanics and evaluated with a failure criteria. In case of cracked

concrete, however, the behaviour is more complicated.

When shear reinforcement is applied, the stut-and-tie method, or truss analogy, is a generally

accepted method to calculated the shear capacity. When only longitudinal reinforcement is applied,

Kani’s comb-analogy is well-known (Kani, 1966). This analogy compares the concrete between cracks

with teeth of a comb. The fixation of a tooth in the uncracked arch of the beam is evaluated;

however, this results in a underestimation of the shear capacity. This underestimation is attributed to

aggregate interlock and dowel action (Pruijssers, 1986). Walraven performed an extensive research

into the contribution of aggregate interlock (Walraven and Reinhardt, 1981).

However, Kotsovos questions the existence of aggregate interlock and dowel action. He investigated

the internal stresses in concrete and demonstrated a multi-axial stress condition in reinforced

concrete beams (Kotsovos, 1987a) and states that concrete failure is a result of tensile stresses in the

compressive zone. The first section of this paragraph goes into the theory of Kotsovos.

The second section considers the research of Pruijssers (1986) and goes into a cross-sectional

approach of the shear stresses in a cracked reinforced concrete beam. Thereupon, the third section

explains the principle stresses in a beam, which can be defined by combining the normal stresses and

shear stresses.

2.2.1 Shear failure according to Kotsovos Kotsovos (1987b) performed research on shear failure of reinforced concrete beams and showed

that “the causes of shear failure are associated with the development of tensile stresses in the region

of the path along which the compressive force is transmitted to the supports and not, as is widely

considered, the stress conditions in the region below the neutral axis” (Kotsovos, 1988 : 68). As a

result of this research, the compressive force path concept was proposed by Kotsovos (1988) and

four mechanisms were identified that may give rise to tensile stresses in the uncracked concrete.

Figure 2.11 illustrates the compressive force path, which is in the uncracked part of the beam, and

the tensile stresses which could occur within. The four causes of tensile stresses in the uncracked

zone are explained and discussed below.

1. T2 are transverse tensile stresses due to the volume dilation of the compressive zone. These

tensile stresses are associated with flexural failure and known as the cause of concrete

crushing. The crushing of concrete is already incorporated in the compressive stress-strain

relation of concrete.

2. T1 is a tensile stress resultant due to the change in path direction which is necessary for an

equilibrium of the compressive forces path. Also, the change in path direction results from a

variable bending moment since the height of the uncracked zone depends on the magnitude

of the bending moment. Because a variable bending moment introduces shear forces, these

tensile stresses can be associated with shear failure.


3. T’ are tensile stresses at the interface between uncracked and cracked concrete. Kotsovos

(1999) obtains these stresses by considering a concrete tooth or cantilever. Figure 2.12

shows this concrete cantilever, which is the concrete between two cracks fixed at the

compressive zone. At the fixed end of the cantilever normal stresses and shear stresses occur

due to the bending moment and the shear force at the fixed end (fig. 2.13). These normal

stresses and shear stresses cause tensile stresses at the neutral axis. The tensile stresses are

likely to exceed the tensile concrete strength at E1 and E2. Besides horizontal shear stresses,

vertical shear stresses ought te be present. The vertical shear stresses and vertical shear

force are interdependent; consequently, shear results in vertical shear stresses which could

exceed the tensile concrete strength at the neutral axis.

4. T is not a tensile force, but symbolises the effect of bond failure. Due to bond failure, the

equilibrium condition changes, resulting in the previously described failure types occurring.

However, bond failure only occurs if the anchorage of the reinforcement bar is insufficient,

so a properly designed reinforced beam should not lead to bond failure.

Figure 2.11: Compressive force path concept with the four mechanisms that may give rise to tensile stresses. T1: change in path direction; T2: volume dilation of concrete; T’: interface of uncracked and cracked concrete; T: bond failure. R: reaction force; C: compressive force. (Kotsovos, 1999)

Figure 2.12: Left: concrete tooth or cantilever fixed at the compressive zone of the beam; middle: normal stresses due to bending moment at the fixed end of the cantilever; right: shear stresses due to the shear force at the fixed end of the cantilever. (Kotsovos, 1999)

27 Literature

In summary, Kotsovos showed that tensile stresses could rise in the uncracked height of the beam.

Four causes of tensile stresses were identified: volume dilation of the compressive zone T2, the

change in path direction T1, the interface between uncracked and cracked concrete T’, and bond

failure T.

Figure 2.13: Critical locations E1 and E2. Left: normal stress on element E1; right: shear stresses on element E2. (Kotsovos, 1999)

Figure 2.14: Concrete cantilever, or tooth, with shear stresses in the crack due to aggregate interlock; hτ: effective shear.height; T: tensile force (Pruijssers, 1986).

Figure 2.15: Representation of the effective shear height hτ (Pruijssers, 1986).


Discussion

T2 , the volume dilation of the compressive zone, is already a part of the basic beam model due to the

compressive stress-strain relation of concrete. T1, change in path direction, is already represented by

the moment-area method because this method calculates the cross-sectional equilibrium – thus the

uncracked height – for every segment. However, the shear force due to the variable bending

moment is not incorporated. T’, interface between uncracked and cracked concrete, results in tensile

stresses along the neutral axis which are not implemented in the basic beam model. T, bond failure,

does not have to be implemented when a properly designed beam is assumed. To conclude, the

extended beam model should implement shear forces and tensile stresses at the neutral axis.

2.2.2 Shear failure according to Pruijssers Pruijssers (1986) performed a theoretical study into the shear strength of reinforced concrete beams.

He achieved his aim, the formulation of a mechanism which causes shear failure, by extending Kani’s

comb-analogy. This extension considered the application of a so-called effective shear depth, which

is defined by the shear stiffness of not only the uncracked zone but also the cracked zone. The

contribution to the shear stiffness of the cracked zone is based on the micro-cracking of concrete.

Figure 2.14 shows two concrete teeth of Kani’s comb-analogy and the effective shear depth h .

Figure 2.15 illustrates the definition of the effective shear depth by showing the strains, normal

stress en shear stress along the effective shear depth. Included is the compressive zone xh and the

tension softening zone. Figure 2.15 presents the ‘real’ shear stress, which includes a shear-softening

zone, and a parabolic shear stress. The ‘real’ shear stresses depend on the deformation of the

tension-softening zone.

The tension softening zone is defined using the ultimate tensile strain ctu , which is estimated to be

eleven times the maximum elastic concrete strain ctm :

11 11 ctmctu ctm

cm

f

E ( 2-13 )

As a result, the effective shear height can be described as

11c ctmh h

( 2-14 )

Discussion

The mechanism of Pruijssers could be applied in the beam model because the effective shear depth

is based on the normal strains. The beam model could calculate the effective shear depth because

the model calculates the strains in a cross-section

The real shear stress depends on the deformation of the tension-softening zone. However, the beam

model does not calculate the shear deformation of the beam. Therefore, the real shear stress can

probably not accurately be determined. Furthermore, the definition of the real shear stress flow

would be complicated. The application of a parabolic shear stress flow would be more practical.

29 Literature

2.2.3 Principle tensile stresses The basic beam model calculates the normal stresses in a beam. However, in reality a beam is just a

beam subjected to principle stresses. The magnitude of a principle stress can be compressive or

tensile; moreover, principle stresses are a vector which defines their direction and magnitude in a

beam. Figure 2.16 shows an example of the trajectories of the compressive principles stresses (solid

lines) and tensile principle stresses (dotted lines) in a beam subjected to bending. In case of a two-

dimensional representation of the beam, two principle stresses are present, with a direction

perpendicular to each other.

The principle stresses in an element can be calculated from the normal and shear stresses with

Mohr’s circle. Figure 2.17 presents Mohr’s cycle, which describes the interdependence between the

shear stresses ν, normal stresses σx and σy, and principle stresses σ1 and σ2. In accordance with

Mohr’s circle, figure 2.18 illustrates the definition of the principle stresses on an element.

Figure 2.16: Trajectories of the principle stresses. The solid lines represent the vectors of the compressive principle stresses and the dotted lines the tensile principle stresses (Kotsovos, 1999).

Figure 2.17: Mohr’s circle which describes the interdependence between the shear stresses ν, normal stresses σx and σy, and principle stresses σ1 and σ2. α is the angle of the first principle stress.

Figure 2.18: Definition of the principle stresses from the shear stresses and normal stresses according to Mohr’s circle


The magnitude of a first principle stress is expressed as

2

21

2 2

x y x y

( 2-15 )

The magnitude of a first principle stress is primarily determined by the magnitude of the normal

stress in x-direction. Therefore, the first principle stresses in the tensile zone will be tensile and the

first principle stresses in the compressive zone will be compressive. The normal stress in y-direction

and the shear stresses define whether the first principle stress is smaller or larger than the normal

stress in x-direction.

The magnitude of the second principle stress is expressed as

2

22

2 2

x y x y

( 2-16 )

The magnitude of second principle stress is strongly defined by the magnitude of the normal stress in

y-direction and the magnitude of the shear stress. That is why, the second principle stress in the

tensile zone can be compressive and the second principle stresses in the compressive zone can be

tensile. Furthermore, the magnitude of the second principle stress is much smaller than the

magnitude of the first principle stress.

The direction of the first principle stress is expressed by the angle :

2

tan 2x y

( 2-17 )

When the shear stress is small compared to the normal stress, the angle will be close to zero. This is

the case at the top and the bottom of the beam. On the other hand, when the normal stress is small

compared to the shear stress, the angle will be close to 45 degrees. This is the case at the neutral axis

of the beam.

Discussion

Figure 2.16 presented the trajectories of the principle stresses. From theses trajectories can be

concluded that the vectors of principle stresses are not necessarily horizontal and vertical. However,

the basic beam model divides a beam vertically and horizontally in respectively layers and segments.

This orthogonal orientation of elements and axes requires an application of shear stresses and

normal stresses. Subsequently, the principle stresses van be calculated from the shear stresses and

normal stresses. Since the shear stresses are not present in the basic beam model, they have to be

added to create an ability of calculating the principle stresses.

The magnitude of the first principle stress can be larger than the magnitude of the normal stress. For

instance, when the normal stress in y-direction is considered zero and a shear stress is present.

Although the first principle stresses are much larger than the second principle stresses, the second

principle stresses can still be important. For instance, when the normal stress in the y-direction is

considered zero, the second principle stress works in the opposite direction of the first principle

stress. Thus, the second principle stress is tensile in the compressive zone and compressive in the

31 Literature

tensile zone. Figure 2.19 illustrates these principle stresses, in the case that the normal stress in the

y-direction is considered zero. Since the tensile strength of concrete is much less than the

compressive strength of concrete, the second principle stresses can exceed the tensile strength in

the compressive zone.

Figure 2.19: Normal stresses σx, shear stresses ν, first principle stresses σ1 and second principle stresses σ2 along the height. The normal stress in y-direction σy is considered zero.

Figure 2.20: Schematic description of the effect of fibres on the fracture process in uni-axial tension (Löfgren, 2005).

Figure 2.21: Characterization of the tensile behaviour of SFRC.


2.3 Steel fibre reinforced concrete

First, the post-cracking behaviour of steel fibre reinforced concrete is introduces. The translation of

the post-cracking behaviour to a stress-strain relation of SFRC is described in appendix B. Second,

two approaches are described to define the shear capacity of SFR-RC. Both approaches extend the

shear capacity with a contribution of the steel fibres. The first approach is a contribution of the steel

fibres within a crack to the cracking behaviour. The second approach is an experimentally defined

contribution of the steel fibres to the shear capacity.

2.3.1 Post-cracking behaviour of steel fibre reinforced concrete Steel fibres increase the bridging and branching effect on the fracture process of concrete (fig. 2.20)

causing a more ductile post-cracking behaviour of SFRC compared to plain concrete. This post-

cracking behaviour can be defined as the force necessary to cause a ‘crack mouth opening

displacement’ (CMOD), which simply is the crack width. Subsequently, this force- CMOD relation can

be converted to a stress-strain relation. Appendix B describes how the post-cracking behaviour of

SFRC can be determined and translated in a stress-strain relation according to the RILEM TC 162-TDF

(2003).

A post-cracking stress-strain relation can be classified as strain hardening or strain softening (fig.

2.21). Strain hardening has an ascending post-cracking branch due to the development of multiple

cracks. On the contrary, strain softening has a descending post-cracking branch due to the

development of a single crack. The post-cracking behaviour – thus whether strain hardening or

softening occurs – depends mostly on

- the fibre volume ratio fV , which is the fibre to matrix ratio;

- the aspect ratio f fL d , which is the fibre length to fibre diameter ratio;

- the distribution and orientation of the fibres;

- and the pull-out force, which is the force necessary to extract a fibre from the matrix.

The previously conducted literature survey (Aa, et al., 2013) contains more extensive and detailed

information about SFRC.

This research focuses on standard concrete strengths and customary fibre volumes and sizes which

result in a strain softening response. A customary steel fibre reinforced mixture roughly has

- a fibre volume between 25 and 45 kg/m3

- a fibre volume ratio between 0.5 and 1.5%

- a fibre length between 50 and 60 mm

- a fibre diameter between 0.8 and 1.0 mm

- a residual tensile stress between 0 and the tensile strength

33 Literature

2.3.2 Addition of the fibre pull-out forces along the inclined crack Narayanan and Darwish (1987) present a contribution of steel fibres to the shear capacity. Figure

2.22 shows the part of the beam at the left of an inclined crack and illustrates the contribution of

- aggregate interlocking aV

- steel fibres bV

- the compressive zone cV

- dowel action dV

Narayanan and Darwish (1987) based the contribution of the steel fibres to the shear capacity on the

fibre pull-out forces along the inclined crack. The determination of fibre pull-out forces along the

inclined crack starts with the average number of fibres in a cross-section fmn according to Romuladi

et al. (1964):

2

1.64 f

fm

f

Vn

d ( 2-18 )

fV is the fibre volume ratio and fd is the fibre diameter

As a result, the total number of fibres at an inclined cracked section fn of the SFRC beam are

2

1.64

sin sin

f

f fm

f

Vjd jdn n b b

d

( 2-19 )

sin

jd

is the length of the inclined crack (fig. 2.22)

Figure 2.22: Shear capacity of a beam due the compression zone Vc, aggregate interlocking Va, dowel action Vd, and steel fibres Vb; C= compressive force, T= tensile force, V= shear force (Narayanan and Darwish, 1987).


Each fibre is embedded in the concrete matrix and the total bond area of the fibres across the

inclined cracked section is

0.414 sin

f f

b f f

f

L VjdA n d b L

d

( 2-20 )

fL is the fibre length

Where 4

fL is the average pull-out length since the pull-out length may range between 0 fL and

0.5 fL . Assuming a pull-out force perpendicular to the crack and an average fibre matrix interfacial

bond stress , the total force in the steel fibres bF will be

0.41sin

f

b b f

f

VjdF A b L

d

( 2-21 )

The contribution of the steel fibres to the shear capacity bV is the vertical component of the fibre

pull-out forces along the inclined crack:

cos 0.41 cotf

b b f

f

VV F b jd L

d ( 2-22 )

The term f

f

f

VL

dcan be replaced by the fibre factor fF which also includes the fibre shape fc :

f

f f f

f

VF L c

d ( 2-23 )

In case the inclined crack has an angle of 45 degrees, the maximum fibre pull-out stresses b along

the inclined crack are

0.41b fF ( 2-24 )

Finally, the ultimate shear stress u is presented by

' 'u sp b

de A f B

a

( 2-25 )

spf is the concrete splitting strength, the reinforcement ratio, a the shear span, and d the

effective depth.

From a regression analysis of experimental data A’ and B’ were defined as respectively 0.24 and 80

N/mm2 and e as

1.0 when a/d > 2.8

2.8 when a/d 2.8

e

de

a

35 Literature

Discussion

The analytical definition of the maximum fibre pull-out stress starts with an average number of steel

fibres in a cross-section and takes into account the efficiency of the fibre length. Unclear is whether

the efficiency of the fibre orientation is included. Furthermore, the efficiency of a fibre is a complex

three-dimensional problem (Kooiman, 2000).

The analytical definition is based on the suggestion that the contribution of the fibres is similar to the

contribution of the aggregate interlock (fig. 2.22). However, the contribution of aggregate interlock is

argued by Kotsovos (1983) as is the dowel action. Questioning the contribution of aggregate interlock

to the shear capacity, the contribution of the steel fibres as presented by Narayanan and Darwish

(1987) can be argued.

Although the predictions of the ultimate shear stress by Narayanan and Darwish (1987) are

satisfying, the correctness of the contribution of the steel fibres is not necessarily correct because of

the empirical definition of the ultimate shear strength.

2.3.3 Addition of shear capacity due to steel fibres The RILEM technische commissie 162-TDF (Vandewalle et al., 2003) proposed a section design

method based on Eurocode 2 (1991) and added a contribution due to the steel fibres fV (note that

here fV is not the fibre volume ratio). Vandewalle and Dupont (2002) modified the design shear

capacity to the ultimate shear capacity by using the original formulas without safety factor. The

differences between the two formulas are expressed in red. Vandewalle and Dupont compared the

ultimate shear capacity with experiments.

RILEM: Design shear capacity

( a ) Vandewalle and Dupont: Ultimate shear capacity

( b )

Rd cd wd fdV V V V ( a ) u cu wu fu

V V V V ( b ) ( 2-26 )

Shear capacity due to the concrete and the longitudinal reinforcement

1 3

1000.12cd l ck

fV k bd ( a )

1 330.1 05 103

c l mu cV k f bd

d

a

The term 0.12 is replaced by the original definition. Instead of the characteristic compressive strength the mean value is used.

( b ) ( 2-27 )

0.02sl

l

A

bd

2.5 1d a

Shear capacity due to the shear reinforcement

0.9wd

swywdV df

A

s ( a ) 0.9

wu

swywmV df

A

s

Instead of the characteristic compressive strength the mean value is used.

( b ) ( 2-28 )


Shear capacity due to the steel fibre reinforcement

0.7fd f fd

kV k bd ( a ) fu fl fuV k bdk

The term 0.7k is replaced by the original

definition lk .

( b )

( 2-29 )

,40.12fd Rkf ( a ) ,40.5 Rfu m

df

a

The term 0.12 is replaced by the original definition. Instead of the characteristic residual strength, the mean residual strength is used.

( b ) ( 2-30 )

size effect200

1 2;kd

16001

1000l

dk

for a rectangular cross-section1;fk

Discussion

Both formulas are defined from experimental research and do not clarify the actual mechanisms that

lead to failure. Notwithstanding, they illustrate the most important parameters for a SFR-RC beam

without shear reinforcement. Besides the cross-sectional dimensions, these parameters are the

longitudinal reinforcement ratio, the concrete compressive strength, and the residual tensile

strength. The definition of the ultimate shear also illustrates that the shear capacity depends on the

shear span to effective depth ratio (a/d).

The examination of the elimination of a/d in the design shear strength led to the following

observations. The term 3 3d

a

in equation 2-27b and the term 0.5d

a in equation 2-30b are

noticeable because these terms introduce an (extra) dependency on the shear span to effective

depth ratio a/d. Table 2.1 shows the influence of a/d on the term 30.15 3d

a

and makes clear that

the term 0.12 in equation 2-30a is based on an a/d of 6 while the capacity of most beams with an

a/d of 6 is defined by the moment capacity. The influence of a/d on the term 0.5d

ais also shown in

table 2.1 and the difference with the term 0.12 in equation 2-27a is significant.

Furthermore, two other observations were made. First, equation 2-28 is only valid for vertical applied

stirrups. Second, the term 0.5 in equation 2-30b is added to convert the flexural tensile strength into

the axial tensile strength. However, for the ultimate limit state for bending and axial forces

Vandewalle et al. (2003) use the term 0.37 (appendix B).

37 Literature

Table 2.1: Influence of a/d in equation 2-27a, 2-27b, 2-30a, and 2-30b.

a/d 1 1.5 2 2.5 3 4 5 6

30.15 3d

a

0.22 0.19 0.17 0.16 0.15 0.14 0.13 0.12

30.15 3 0.12d

a

182% 159% 144% 134% 126% 114% 106% 99%

0.5d

a 0.50 0.33 0.25 0.20 0.17 0.13 0.10 0.08

0.5 0.12d

a 417% 278% 208% 167% 139% 104% 83% 69%


2.4 Conclusion and hypotheses

Paragraph 2.1 presents a beam model which is only suitable for flexural deformation and flexural

failure. However, this beam model should incorporate shear failure and steel fibre reinforcement for

simulation of SFR-RC beam subjected to shear failure. Paragraph 2.2 presents the theoretical

knowledge which can be used to incorporate shear failure in the beam model. Paragraph 2.3

introduces two options of incorporating steel fibre reinforcement. The approach of the RILEM TC

162-TDF is preferred because this approach makes a clear distinction between the contribution of

the reinforced concrete, the steel fibre reinforcement, and the shear reinforcement. Furthermore,

this approach uses a cross-sectional approach as does the beam model. Unfortunately, the RILEM TC

162-TDF provides no insight into the actual contribution of steel fibre reinforcement because a

constant contribution along the effect depth is formulated. Therefore, the following hypothesis is

examined:




This hypothesis is examined with the effect of steel fibre reinforcement on the shear capacity. On the

basis of the literature study the following effects are expected:

‘The shear capacity for an increasing residual tensile strength’

‘The shear capacity increases due to steel fibres in the cracks of the tensile zone.’

‘The shear capacity increases due to steel fibres in the compressive zone.’

Due to the extension of the beam model, these hypotheses can not only be examined but

also theoretically underpinned. As a result, the aim of this research, obtaining more insight

into the behaviour of SFR-RC beam subjected to shear failure, should be achieved.

39 Extended beam model

3 Extended beam model The basic beam model as presented in paragraph 2.1 only includes the bending behaviour of

reinforced concrete beams. To obtain insight about the behaviour of SFR-RC beams subjected to

shear failure , the basic beam model is extended, resulting in the ‘extended beam model’. Three

extensions can be distinguished.

First, shear failure is incorporated in the beam model. For this purpose, the tensile principle stresses

are evaluated because Kotsovos (1999) showed that shear failure is caused by the development of

tensile stresses in the uncracked zone. Paragraph 3.1 presents how these principle stresses are

determined. Second, the contribution of steel fibre reinforcement is incorporated. This extension is

described by paragraph 3.2 and divided into a contribution in the tensile zone and a contribution in

the compressive zone and. The last paragraph, 3.3, describes the limitations of the model .

Besides the three mentioned extensions, the original beam model made by Couwenberg, Marinus

and Paus was adjusted to overcome (practical) limitations. The most important one is the addition of

the ‘tension stiffening’ module, which calculates the bond between the reinforcement bar and the

concrete as described in section 2.1.4. The output of this module is used to define the tensile stress-

strain relation for concrete. Furthermore, a load module and a load-displacement module are added.

The load module generates an external moment from an external load and load type. This external

moment is linked to the deflection module; as a result, the module is suitable for several load types.

The load-displacement module executes a deflection calculation for multiple loads to define a load-

displacement diagram. An extended description of these extra modules is annexed in appendix D.

3.1 Extension I: Shear failure

To include shear failure, the tensile principle stresses are evaluated because Kotsovos (1999) showed

that shear failure is caused by the development of tensile stresses in the uncracked zone. However,

the beam model originally only calculates the normal stresses. Therefore, shear stresses are added.

From the normal stresses and shear stresses the principle stresses can be calculated and evaluated.

Because the beam model is a cross-sectional approach, the shear stresses in a cross-section should

be defined. This is done in the first section. The principle stresses are calculated according to section

2.2.3. The evaluation of the principle stresses, is explained in the second section of this paragraph.

3.1.1 Shear stresses Using classical mechanics the shear stresses in a concrete beam can be calculated when the beam

behaves linear elastic. That is when the beam is not cracked. However, when the reinforced concrete

beam is cracked in the tensile zone, the definition of the shear stress flow is a problem. Due to

presence of reinforcement bars, the shear stresses cannot be calculates from the difference between

the normal stresses. Furthermore, the questions rises whether the tensile zone contributes to the

shear stiffness, thus transferring shear stresses.

First, the shear stresses according to classical mechanics are described. Second, an assumption about

the shear stress flow in a cracked cross-section is made.


Uncracked cross-section

In case of uncracked plain concrete, the beam behaves linear elastic and the shear stresses can easily

be derived from the horizontal equilibrium of a beam segment (fig. 3.1):

0

y

xy xb dx d dA ( 3-1 )

This equilibrium equation can be rewritten to define the interdependence of the shear force yV and

the shear stresses xy :

y yc xxy

x

V VdN SdM

b dx z b dx z b z b S

y x

xy

x

V S

bI ( 3-2 )

Cracked cross-section

In case of cracked cross-section, the cross-sectional capacity decreases as do the first moment of

area yS and the moment of inertia

yI . Applying these effects to equation 3-2 result in increasing

shear stresses.

When only the uncracked zone contributes to the shear capacity, which is states by Kotsovos, the

shear stresses should be distributed along the uncracked height. Figure 3.2 shows four shear stress

flows that are considered. The first shear flow (A) is a constant shear stress along the uncracked

height. The second shear flow (B) is a parabolic flow with a maximum shear stress halfway the

uncracked height. Also the third shear flow (C) is a parabolic flow; however, this flow has a maximum

shear stress at the neutral axis. The last shear flow (D) is defined with equation 3-2 using the

uncracked height to define the first moment of area and the moment of inertia.

Figure 3.1: Left: segment dx subjected to a bending moment M and a shear force V at x and x+dx; mid: normal stress distribution at x and x+dx and a shear stress at y; right: shear stresses in segment dx.

A shear flow should meet two keynotes. First, the maximum shear stress should be at the neutral

axis. Second, no shear stresses should be present at the edges of the cross-section. The constant


flow would be very convenient but neglects both keynotes. The standard parabolic flow meets the

second keynote but not the first if the neutral axis is not halfway the uncracked height. The adjusted

parabolic flow meets both keynotes. The shear stress flow according to equation 3-2 also meets both

keynotes but the shear stress jumps from one value to another at the neutral axis. This jump is a

result of the difference between the static moment of the compressive zone and the uncracked

tensile zone. This difference is a consequence of the difference between the compressive and tensile

concrete strength. In conclusion, the adjusted parabolic flow and the flow according to equation 3-2

could be feasible shear stress flows. The distinction between these flows is the contribution of the

tensile zone.

Pruijssers (1986) investigated the contribution of the tensile zone, which is presented in section

2.2.2, and defined an effective shear height h . This effective shear height is applied in the extended

beam model. Furthermore, the parabolic shape of shear flow C is applied because this flow is more

straightforward and easier to calculate than shear flow D. Figure 3.3 presents the parabolic shear

stress flow applied along the effective shear height.

Figure 3.2: Four different shear stress flows along the uncracked height of a cross-section.

Figure 3.3: Normal strains, normal stresses, and shear stresses along the height of a cross-section.


The application of the parabolic shear stress flow is based on the equilibrium between the external

shear force exV and the integral of the shear stress. This equilibrium can be expresses as

max

2

3exV h b

( 3-3 )

The load module of the extended beam model calculates the external shear force per segment.

Furthermore, the effective shear height is calculated from the cross-sectional equilibrium using

equation 2-14. Subsequently, the maximum shear stress max can be calculated and used to calculate

the shear stresses per layer i along the compressive height ch :

2max

max2i i c

c

y hh

( 3-4 )

Figure 3.4 shows these shear stress per layer in the compressive zone.

Figure 3.4: Shear stresses per layer in the extended beam model

Figure 3.5: Mohr’s circle which describes the interdependence between the shear stresses ν, normal stresses σx and σy, and principle stresses σ1 and σ2. α is the angle of the first principle stress.


3.1.2 Principle stresses

Failure criteria

The previous section clarified how the shear stresses are implemented in the beam model. The beam

model can now calculate the normal stresses in x-direction and shear stresses in every layer and

segment of the beam. The normal stresses in y-direction are considered zero. Subsequently, the

normal stresses and shear stresses per layer and segment are translated to principle using Mohr’s

circle (fig. 3.5). The tensile principle stresses are evaluated because Kotsovos (1999) showed that

shear failure is caused by the development of tensile stresses in the uncracked zone.

The discussion in section 2.2.3 explained that when the normal stresses in y-direction are considered

zero, the second principle stresses in the compressive zone is tensile. Therefore, the second principle

stress is evaluated (fig. 3.6). Failure is assumed when the second principle stress exceeds the tensile

concrete strength resulting in the failure criteria

2 ctmf ( 3-5 )

Tensile zone

Also the influence of the tensile principle stresses in the tensile zone is investigated. A first principle

stress in the tensile zone is larger than the normal stress when a shear stress is present. As a result,

this principle stress can exceed the tensile concrete strength even though the normal stress does not.

If this is the case, the actual cracked height will be larger and the cross-sectional equilibrium is

Figure 3.6: Flow chart evaluation second principle stress in the compressive zone.


invalid. By increasing the tensile strain a valid equilibrium can be established, in which the

reinforcement bars incorporate the eliminated tensile force of the concrete. This process can be

iterated until all boundary conditions are met (fig. 3.7). This process can physically be interpreted as

the development of an inclined crack.

As a result of this iteration, the author expected an increase of the strains whereby the compressive

zone reduces. The reduction of the compressive zone should more rapidly lead to crushing and

failure of the concrete. However, the compressive zone is not significantly reduced because the

increase of the tensile strain is small. The reason or this small increment is the small value of the

eliminated tensile concrete force. Moreover, the longitudinal reinforcement has a large Young’s

modulus which causes a significant increase of the normal force when the strain slightly increases.

To summarize, the iterative process does not result in premature failure of the beam because a small

increase of the tensile strain is but necessary to achieve a valid equilibrium. He calculation time, on

the other hand, increase and the accuracy decreases; that is why, the first principle stresses are not

evaluated and this iterative process is not implemented in the beam model.

Figure 3.7: Flow chart iteration process in the tensile zone.


3.2 Extension II: Contribution steel fibre reinforcement

The contribution of steel fibre reinforcement is investigated by analysing the contribution in the

tensile zone and the compressive zone. First, the contribution in the tensile zone is described and

after that the contribution in the compressive zone.

3.2.1 Contribution in the tensile zone Section 2.3.1 already explained the increased cohesion of cracked steel fibre reinforced concrete

(SFRC) compared to plain concrete due to the increased bridging and branching effect on the

fracture process of concrete. This cohesion can be expresses as the residual tensile strength of the

SFRC mixture (appendix B). The contribution of the steel fibres in the cracks is combined with the

bond theory. The bond theory describes the bond between a reinforcement bar and concrete

subjected to a tensile force. The relevant theory is already explained in section 2.1.4 and appendix C.

On the left figure 3.8 illustrates the bond of a reinforcement bar and plain concrete. According to the

bond theory, the concrete stress is zero at the crack and increases along the transition length due to

traction between the concrete and the reinforcement bar. On the right figure 3.8 illustrates the

cohesion between the concrete parts beside the crack. To what extent the concrete is kept together

at a crack depends on the post-cracking behaviour of the SFRC, which is can be expressed by the

residual tensile strength. As a result, steel fibre reinforcement extends the concrete stress at the

crack from zero to the residual tensile strength of the SFRC. The diagrams of the concrete stress an

steel strain in figure 3.8 illustrate the effect of the residual tensile strength. The cracking stress cr is

the maximum concrete stress and the cracking strain cr is the strain of the steel when the concrete

reaches the cracking stress but is not yet cracked.

The residual tensile strength is implemented into the beam model as a boundary condition of the

tension stiffening module. The tension stiffening model calculates the concrete stresses and steel

strains along the transition length such as the stresses and strains in figure 3.8. The average values of

these stresses and strain are used to define the tensile stress-strain relation of concrete. Appendix C

explains in detail the functioning of the tension stiffening module and translation into a stress-strain

relation.

Figure 3.8: Left: tensile bar with plain concrete; right: tensile bar with SFRC; top: traction between reinforcement bar and concrete; mid: concrete stresses along the transition length; bottom: steel strains along the transition length.


Figure 3.9 presents the tensile stress-strain relation of SFR-RC. Compared to reinforced concrete,

point B moves up to the top. Point C represents the assumption that the tension stiffening effect

stops when the yield strain y is reached. On the contrary, the effect of the steel fibres is noticeable

until a large crack width is reached. That is why, the concrete stress equals the residual tensile

strength after the yield strain is reached. Assumed is that a very large strain is required to attain a

large crack width; therefore, the concrete stress is kept constant after reaching the yield strain until

an ultimate strain u of 2.5%.

3.2.2 Contribution in the compressive zone Besides a contribution in the tensile zone, the hypothesis is that also the steel fibre reinforcement in

the compressive zone does contribute to the shear capacity. In section 3.1.3 is assumed that failure

occurs when the second principle stress exceeds the tensile concrete strength. The contribution of

steel fibre reinforcement in the compressive zone is based on the assumption that the steel fibres

are activated in the compressive zone because the failure criteria is based on the development of

tensile stresses in the compressive zone. However, the model is not suitable for the implementation

of a post-cracking stress-strain relation for the second principle stresses. Therefore, the post-cracking

behaviour is implemented by a reduction fV of the external shear force exV . Subsequently, the

inserted shear force inputV , which is the shear force distributed along the effective shear height, is

input ex fV V V ( 3-6 )

fV is defined by the residual tensile strength resf along the effective shear height h :

f resV f h b ( 3-7 )

The shear stresses per layer are calculated from the maximum shear stress (eq. 3-4 ). In section 3.1.1

the maximum shear stress was defined from the external shear force (eq. 3-3 ). However, the

external shear force is not necessarily equal to the inserted shear force (eq. 3-6 ). Therefore, the

maximum shear stress is redefined:

max

2

3inputV h b

( 3-8 )

Figure 3.9: Tensile stress-strain relation for SFR-RC. Light red represents normal reinforced concrete.


3.3 Limitations

This paragraph highlights four limitations of the basic beam model and extended beam model. These

limitations are intrinsic to the design of the model. In chapter 5 the functioning of the model is

discussed and recommendations concerning the input and design are suggested.

1. The design of the basic beam model does not include shear deformation. In case of flexural

failure, the shear deformation is mostly quite small compared to the deformation due to

bending. However, this might not be the case for beams subjected to shear failure.

Therefore, the extended beam model might not be suitable for simulating the deformation

of a beam subjected to shear failure.

2. The moment-curvature calculation is displacement-controlled, since the compressive strain

at the top is stepwise increased until the failure strain of the concrete is reached. However,

the deflection calculation is load-controlled, since the internal moment should be equal to

the external moment. Therefore, the force-displacement diagram cannot include a

decreasing branch after the maximum force is reached.

3. The extended beam model is not suitable for beams with a shear span to effective depth a/d

ratio smaller than 2.5. Although these beams fail due to the development of tensile stresses

in the compressive zone, a complex tri-axial compressive stress condition is present

(Kotsovos, 1987b). As a result of this tri-axial compressive stress condition, a higher shear

stress is necessary to develop tensile principle stresses in the compressive zone. The

extended beam model considers only an uni-axial normal stress; therefore, shear failure

cannot be simulated for beams with a/d smaller than 2.5.

4. The extended beam model is not suitable for beams with shear reinforcement. The addition

of shear reinforcement changes the structural behaviour of a beam. The extended beam

model focuses on behaviour due to the interaction between steel fibre reinforcement and

reinforced concrete, thus not on the structural behaviour due to shear reinforcement.


49 Verification

4 Verification The previous chapter described the extended beam model, which incorporates shear failure and

steel fibre reinforcement. This chapter examines whether the extended beam model correctly

simulated the structural behaviour and presents the executed benchmarks and verification. The

inserted material properties and applied calculation settings are introduced in appendix E.

4.1 Benchmarks extended beam model

The output of the extended beam model is examined by analysing the effects of four parameters on

the shear capacity. The analysis of each parameter can be found in the following sections:

4.1.1 Longitudinal reinforcement ratio,

4.1.2 Residual tensile strength of the SFRC, resf

4.1.3 Shear span to effective depth ratio, a d

4.1.4 Concrete strength, expresses in cmf

The height, width and length of the beam are kept constant to minimize the number of parameters.

Furthermore, the beams are simply supported and loaded by two point loads (fig. 2.7). Table 4.1

shows these properties, which are chosen similar to the geometric properties of the test beams of

Dupont and Vandewalle (2002). The material properties of the steel reinforcement bars (table 4.2)

and the range of the analysed parameters (table 4.3) are also based on the tests of Dupont and

Vandewalle.

The reinforcement ratios represent a wide range of application. The residual tensile strengths are

chosen smaller than the critical tensile strength because this research focuses on a declining tensile

strength after cracking. The smallest value of a d , 2.5, is based on the field of application of the

extended beam model (paragraph 3.3). The largest value of a d is 4 because higher values are likely

to result in flexural failure, making the data unusable. The concrete strengths represent a range of

normal strength concrete.

Table 4.1: Geometric properties used in all benchmark simulations.

Height, h Effective depth, d Width, b Length, l

300 mm 260 mm 200 mm 2300 mm

Table 4.2: Material properties of the steel reinforcement bars used in all benchmarks simulations.

Yield strength, yf Young’s modulus, sE

560 N/mm2 200 000 N/mm2

Table 4.3: Range of the analysed parameters. The reinforcement area As is based on the reinforcement ratio. In grey the inserted type of reinforcement bars, which is used in the tension stiffening module.

Parameter Range

[%] 0.5 1.15 1.81 2.45

Type 3*φ10 3*φ16 3*φ20 4*φ20

resf [N/mm2] 0 0.56 1.11 1.67

a d 2.5 3.25 4 -

cmf [N/mm2] 50 40 30 -


The effect of a parameter is clarified by analysing the stresses in the critical cross-section. The critical

cross-section for this load type, two point loads, is located at the point load because at this location

the bending moment and shear force are maximal. Thus, the normal stresses and shear stresses are

maximal at this location of the beam.

4.1.1 Longitudinal reinforcement ratio

For this analysis a d is assumed 2.5 and the concrete strength is assumed 50 N/mm2. Figure 4.1

shows the simulated ultimate shear capacity for the reinforcement ratios 0.5%, 1.15%, 1.81%, and

2.45%. According to the extended beam model, the beams with a reinforcement ratio of 0.5% fail

due to flexure, not shear, which is represented in grey data in figure 4.1.

Figure 4.1 illustrates an increasing ultimate shear capacity Vu when the reinforcement ratio

increases. Furthermore, the increase of Vu is on average not linear but declining. The relation

between the reinforcement ratio and the ultimate shear capacity can be clarified by analysing the

stresses in the critical cross-section.

Figure 4.2 shows the normal stress distribution in case of a top strain of 0.5‰ and illustrates an

increasing compressive zone when the reinforcement ratio increases. Since the effective shear height

is proportional to the compressive height, the increase of Vu can be explained by the increment of

the compressive height. The on average declining increase of Vu correlates with the declining

increase of the compressive height. Figure 4.3 clarifies the effect of an increasing reinforcement ratio

on Vu, from which the relations a, b, c and d are explained below.

a. The reinforcement ratio ρ influences the strain flow along the height, as well as the effective

shear height hτ.

b. The maximum shear stress νmax depends on the effective shear height (eq. 3-3).

c. The maximum tensile stress in the compressive zone σ2, max is located at the neutral axis so it

is equal to the maximum shear stress (fig. 2.19).

d. When the maximum second principle stress exceeds the tensile concrete strength, the

ultimate shear capacity Vu is reached.

Figure 4.1: Ultimate shear capacity Vu calculated by the extended beam model. fres in N/mm

2. The data in grey

represent flexural failure instead of shear failure.

51 Verification

Figure 4.2: Normal stresses along the height of the critical cross-section when the strain at the top is 0.5‰ and the residual tensile strength is 0.

Figure 4.3: Effect increment longitudinal reinforcement ratio ρ on the effective shear height hτ, maximum shear stress νmax, the maximum second principle stress in the compressive zone σ2,max, and the ultimate shear capacity Vu.

Figure 4.4: Normal stresses and shear stresses along the height of the critical cross-section when the strain at the top is 0.5‰.

Figure 4.5: Effect increment residual tensile strength fres on the effective shear height hτ, reduction Vf , inserted shear force Vin, maximum shear stress νmax, the maximum second principle stress in the compressive zone σ2,max,and the ultimate shear capacity Vu.

> ρ a > hτ b < νmax c < σ2,max d > Vu

> fres

> hτ

< νmax < σ2,max > Vu

> Vf g < Vin

h


4.1.2 Residual tensile strength

For this analysis a d is assumed 2.5 and the concrete strength is assumed 50 N/mm2. Figure 4.1

shows the simulated ultimate shear capacity Vu for the residual tensile strengths 0 N/mm2, 0.56

N/mm2, 1.11 N/mm2, and 1.67 N/mm2.

Figure 4.1 illustrates an increasing ultimate shear capacity Vu when the residual strength fres

increases. Furthermore, the increase of Vu is on average constant. The relation between fres and Vu

can be clarified by analysing the stresses in the critical cross-section.

Figure 4.4 shows the normal stress and shear stress distribution in case of a top strain of 0.5‰. The

shear stress flow illustrates a slightly increasing effective shear height when fres increases.

Additionally, the shear stress flow illustrates a smaller maximum shear stress when fres increases.

The on average constant increase of Vu correlates with the decrement of the shear stresses.

Furthermore, the slightly increasing effective shear height has only a minor effect on Vu. Figure 4.5

clarifies the effect of an increasing fres on Vu, from which the relations b, c and d are explained in the

previous section and relation e, f ,g , and h below.

e. The residual tensile strength fres influences the strain flow along the height, as well as the

effective shear height hτ.

f. The reduction Vf depends on the residual tensile strength (eq. 3-7).

g. The inserted shear force Vin is determined by reducing the external shear force Vex by Vf

(eq.3-6).

h. The maximum shear stress νmax is calculated from the inserted shear force (eq. 3-8).

4.1.3 Shear span to depth ratio a/d For this analysis the concrete strength is assumed 50 N/mm2. Figure 4.6 shows the simulated

ultimate shear capacity Vu for the shear span to depth ratios 2.5, 3.25, and 4. Vu is expressed as a

correction to the previously calculated Vu, for which a/d is 2.5 was assumed.

Figure 4.6: Ultimate shear capacity Vu calculated by the extended beam model. The data in grey represent flexural failure instead of shear failure.

53 Verification

Figure 4.6 illustrates a smaller ultimate shear capacity Vu when the a/d increases. Furthermore, the

decrease of Vu is gradual in case of flexural failure (grey data), but scatters in case of shear failure

(green data). The relation between a/d and the ultimate shear capacity can be clarified by analysing

the stresses in the critical cross-section.

Figure 4.7 shows the normal stress and shear stress distribution in case of an external shear force of

58kN. The shear stress flow illustrates a decreasing effective shear height when a/d increases.

Additionally, the shear stress flow illustrates a larger maximum shear stress when a/d increases.

Figure 4.5 clarifies the effect of an increasing a/d on Vu, from which the relations b, c and d are

explained in the first section and relation i and j below.

i. When the external shear force Vex is kept constant, the bending moment M depends on the

shear span a, because for the used load type applies exM V a .

j. The bending moment influences the strain flow along the height, as well as the effective

shear height hτ.

Figure 4.7: Normal stresses and shear stresses along the height of the critical cross-section when the external force Vex is 58kN.

Figure 4.8: Effect increment shear span a on the bending moment M, effective shear height hτ, maximum shear stress νmax, the maximum second principle stress in the compressive zone σ2,max, and the ultimate shear capacity Vu.

Figure 4.9: Normal stresses and shear stresses along the height of the critical cross-section when the external force (Vex) is 62kN and the bending moment 40kNm.

> a i > M j < hτ b > νmax c > σ2,max d < Vu


4.1.4 Concrete strength The concrete strength is expressed as the compressive concrete strength fcm in the graphs. Figure

4.6 shows the simulated ultimate shear capacity Vu for the compressive concrete strengths 50

N/mm2, 40 N/mm2, and 30 N/mm2. Vu is expressed as a correction to the previously calculated Vu,

for which fcm=50 N/mm2 was assumed.

Figure 4.10 illustrates on average a smaller ultimate shear capacity (Vu) when the concrete strength

increases. Furthermore, the Vu does not on average change in case of flexural failure (grey data) and

scatters in case of shear failure (puple data). The relation between the concrete strength and the

ultimate shear capacity can be clarified by analysing the stresses in the critical cross-section.

Figure 4.9 shows the normal stress and shear stress distribution in case of an external shear force of

62kN. The shear stress flow illustrates a smaller effective shear height when the concrete strength

decreases. Additionally, the shear stress flow illustrates a larger maximum shear stress when the

concrete strength decreases. Figure 4.11 clarifies the effect of a decreasing concrete strength on Vu,

from which the relations b, c and d are explained in the first section and relation k, l, and m below.

k. The tensile strength fctm is related to the compressive strength fcm.

l. The ultimate tensile strain that defines the effective shear height hτ depends on the strain at

cracking, so on the tensile strength (eq. 2-13).

m. The tensile strength influences the ultimate shear capacity (Vu) directly, due to the failure

criteria, which states that failure occurs when 2 ctmf .

Figure 4.10: Ultimate shear capacity Vu calculated by the extended beam model. The data in grey represent flexural failure instead of shear failure.

Figure 4.11: Effect decrement compressive strength fres on the tensile strength fctm, effective shear height hτ, maximum shear stress νmax, the maximum second principle stress in the compressive zone σ2,max,and the ultimate shear capacity Vu.

< fcm k < fctm

< hτ b > νmax c > σ2,max

< Vu m

55 Verification

4.1.5 Overview relations Figure 4.12 illustrates the relations between the examined parameters and the ultimate shear

capacity Vu. The reinforcement ratio ρ, residual tensile strength fres, and concrete strength fcm are

positively correlated to Vu. Contrarily, the shear span to effective depth ratio a/d is negatively

correlated to Vu. The relations ‘a’ until ‘m’ are described in the previous sections. Relation n has not

been mentioned but is describe below. All relations together illustrate a more complex effect of the

parameters. The complexity of this effect probably clarifies the scattered data in figure 4.6 and 4.10.

n. The reduction of Vf depends on the effective shear height hτ (eq. 3-7)

Figure 4.12: Relation between the analysed parameters and the ultimate shear capacity described in the previous sections.

n

.


4.2 Verification steel fibre reinforcement in the tensile zone

Paragraph 3.2.1 explained how the contribution of the steel fibres in cracks is combined with the

bond theory. As a result, the beam model is extended with a tension stiffening module, which the

combination of these two in the tensile zone. The results of this tension stiffening module are

compared with results of Bruggeling and Bruijn (1986), since the module was based on their

description of the bond theory.

Bruggeling and Bruijn only analysed tensile loaded bars without steel fibre reinforcement. Therefore,

the functioning of the tension stiffening module within the extended beam model is investigated by a

comparison with experiments of Gribniak et al. (2012). Gribniak et al. defined a stress-strain relation

to represent the post-cracking behaviour of SFR-RC beams and provided a value for the residual

tensile stress. Ergo, their research is particularly suitable for comparison.

4.2.1 Verification tension stiffening module Bruggeling and Bruijn (1986) performed tensile tests on reinforced bars for comparison with their

determined formulas. Their test program contained twelve reinforced tensile bars with varying

reinforcement (tab. 4.5). Figure 4.13 is a schematisation of the test set-up. The dimensions and

material properties of the test pieces are shown in table 4.4.

Appendix D contains a flow chart which illustrates how the tension stiffening module functions. The

maximum number of steps (100) is applied. The traction between the reinforcement bar and the

concrete τ is described by

cm cmx a f b f u x dx

This equation is suggested by Bruggeling and Bruijn (1986). Hamelink (1989) determined a=0.14 and

b=0.36, these values are inserted in the tension stiffening module.

Table 4.5 presents the results of the three methods: the tension stiffening module, the prediction by

Bruggeling and Bruijn, and the experiments. The predictions of the tension stiffening module are

nearly equal to the predictions of Bruggeling and Bruijn. Furthermore, the predicted number of

cracks correspond approximately to the number of cracks in the experiments.

The difference between the tension stiffening module and Bruggeling and Bruijn can be attributed to

the elastic deformations of the concrete, which is not included by Bruggeling and Bruijn but is in the

tension stiffening module. Furthermore, the dissimilarity between the test and the predictions can be

attributed to the concrete cover, since this is not included in the prediction. Because the differences

are limited and explicable, the conclusion is drawn that the tension stiffening module confidently

predicts the tension stiffening effect of a reinforced concrete bar subjected to a tensile load.

Figure 4.13: Schematisation of a reinforced concrete bar subjected to a tensile load.

57 Verification

Table 4.4: Dimensions and material properties of the test pieces (Bruggeling and Bruijn, 1986).

Dimensions [mm] Material properties [N/mm2]

Width b 100 concrete Compressive strength fcm 38

Height h 100 Tensile strength fctm 3.8

Length L 1000 Young’s modules Ec 33000

steel Yield strength fy 400

Young’s modules Es 205000

Table 4.5: Results test program. The crack distance Lcr is an average value as is the crack width wcr.

Parameters

Ρ [%] 1.13 2.01 3.14 4.52

Φ [mm] 12 6 16 8 20 10 24 12

ns 1 4 1 4 1 4 1 4

c [mm] 44 32 22 42 31 21 40 30 20 38 29 19

Prediction tension stiffening module

# cracks 5 9 6 11 7 12 7 14

Lcr [mm] 177 102 158 89 142 79 129 71

wcr [mm] 0.127 0.074 0.068 0.039 0.042 0.024 0.029 0.016

Prediction Bruggeling and Bruijn

# cracks 5 9 6 11 7 13 7 14

Lcr [mm] 177 98 150 84 132 73 119 66

wcr [mm] 0.142 0.080 0.072 0.040 0.042 0.024 0.028 0.016

Experiments

# cracks 4 8 12 6 10 12 7 9 14 7 9 13

Figure 4.14: Test set-up. A defines the mid span of the beam were de deflections and strains are measured (Gribniak et al., 2012).


4.2.2 Verification contribution tensile zone Gribniak et al. (2012) performed four bending tests on large scale SFR-RC beams with varying fibre

volume ratios. Figure 4.14 shows the test set-up, which is a simply supported beam subjected to two

point loads. The benchmarks of the previous paragraph are performed with a similar test set-up.

Table 4-6 and 4-7 present the properties of the test pieces. In the mid span of the beam the

deflections and strains are measured. Gribniak et al. determined the moment-curvature relations

from the experimental data. Furthermore, the concrete stress-strain relations are determined

through an inverse technique. Table 4.8 shows the cracking strength, residual tensile strength and

Young’s modules, which are determined by Gribniak et al. These material properties are entered in

the extended beam model.

Figure 4.15 shows the stress-strain relations which are calculated by the tension stiffening module.

These stress-strain relations are used in the extended beam model to describe the behaviour of

concrete in tension. Figure 4.16 presents the moment-curvature diagrams obtained from the

simulations of the extended beam model (red lines) and the experiments (grey lines). The simulations

of the extended beam model are in good agreement with the experiments. Therefore, the conclusion

is drawn that the incorporation of the residual tensile strength into the tension stiffening module

confidently predicts the contribution of the tensile zone to the stiffness of a beam subjected to

flexural loading.

Appendix F presents the moment-curvature diagrams in separate diagrams . Furthermore, the stress-

strain determined by Gribniak et al. are annexed in appendix F.

Table 4.6: Main dimensions test beams (Gribniak et al., 2012)

Table 4.7: General dimensions test beams.

Length L [mm] 3000

Reinforcement ratio ρ [%] 0.31

Reinforcement diameter Φ [mm] 10

Number of bars ns 3

Table 4.8: Material properties inserted in the simulations

Cracking strength σcr [N/mm2]

Residual tensile strength fres [N/mm2]

Young’s modules Ec [N/mm2]

S3-2-3 2.70 0 40000

S3-1-F05 3.20 1.25

S3-1-F10 3.20 2.00

S3-1-F15 3.00 2.70

59 Verification

Figure 4.15: Stress-strain relations calculated by the tension stiffening module and used for calculating the moment-curvature diagrams. fres in N/mm

2.

Figure 4.16: Moment-curvature diagrams for a residual tensile strength fres of 0 N/mm2, 1.25 N/mm

2, 2.0 N/mm

2, and

2.7 N/mm2. The red lines represent the simulations by the extended beam models. The grey and black lines represent the

curves determined by Gribniak et al. (2012).


4.3 Verification extended beam model

Paragraph 4.1 presented and explained the effect of four parameters on the ultimate shear capacity.

This paragraph investigates whether these effects are correct. In the first section a qualitative

analysis is performed. In the second section a quantitative analysis is performed.

4.3.1 Qualitative verification The simulations of the extended beam model are compared to the ultimate shear capacity according

to the formula of Dupont and Vandewalle (2002), which is presented in section 2.2.3. Paragraph 4.1

already presented the effects of four parameters on the ultimate shear capacity in figure 4.1, 4.6,

and 4.9. These three diagrams are combined into one diagram from which the ultimate shear

capacity can be read. This diagram of the extended beam model is annexed as figure G.1 and the

diagram of the formula as figure G.2 in appendix G. Comparing both diagram is difficult because of

the scattering of the data; therefore, the average values are calculated. These average values are

determined though a linear or parabolic trend line and are shown in the diagrams.

Figure 4.17 presents the average values of the extended beam model (red lines) and the formula of

Dupont and Vandewalle (dotted grey lines). The ultimate shear capacity Vu can be read from this

diagram in three steps, which are explained below the diagram. The formula shows the same trends

as the extended beam model. Two dissimilarities are noticeable. First, at the top right the curves of

the model are higher than the curves of the formula. Second, at the bottom left the curves for

fcm=40 and fcm=30 of the model are higher than the curves of the formula causing a larger

correction.

The first dissimilarity is mainly caused by a different prediction of the shear capacity for reinforced

concrete beams without steel fibre reinforcement fres=0. This difference could be the result of an

overestimation by the model. Therefore, the application of a smaller tensile concrete strength is

analysed.

61 Verification

Figure 4.17: Effect of the reinforcement ratio ρ, residual tensile strength fres, shear span to effective depth ratio a/d, and the compressive concrete strength fcm according to the extended beam model (using the mean tensile strength). fres and fcm in N/mm

2.

Rea

din

g st

eep

s:

Firs

t, s

tart

at

the

top

rig

ht

wit

h t

he

rein

forc

em

ent

rati

o ρ

an

d r

esid

ual

ten

sile

str

engt

h f

res,

her

e is

ass

um

ed t

hat

a/d

is 2

.5 a

nd

th

e c

om

pre

ssiv

e co

ncr

ete

stre

ngt

h f

cm is

50

N/m

m2 .

Seco

nd

, rea

d f

rom

rig

ht

to le

ft t

he

corr

ecti

on

fo

r

shea

r sp

an t

o e

ffec

tive

dep

th r

atio

a/d

. Th

ird

, rea

d f

rom

to

p t

o b

ott

om

th

e co

rrec

tio

n f

or

com

pre

ssiv

e c

on

cret

e

stre

ngt

h f

cm. F

rom

left

to

rig

ht

no

w t

he

ult

imat

e sh

ear

cap

acit

y V

u c

an b

e re

ad.


Initially the mean tensile strength is entered because the ultimate shear capacity is simulated, not

the design. However, failure occurs at the weakest point. That is why, also a simulation is made with

the characteristic tensile strength instead of the mean tensile strength. The results of the simulation

with the characteristic tensile strength are annexed as figure G.3 in appendix G. Again the average

values are compared with the formula of Dupont and Vandewalle.

Figure 4.18 presents the average values of the extended beam model (red lines)and the formula of

Dupont and Vandewalle (dotted grey lines). The ultimate shear capacity Vu can be read from this

diagram in three steps, which are explained below the diagram. The extended beam model shows

the same trends for this simulation as the formula and the simulation with the mean tensile strength.

Noticeable at the top right of figure 4.18 is that the curve of the model almost coincides with the

formula when no steel fibre reinforcement is applied fres=0. However, the curves for SFR-RC beams

of the model lie lower than the curves of the formula. Furthermore, the correction for a/d of this

simulation is a bit smaller than the simulation with the mean tensile strength. The correction for the

compressive strength of both simulations are similar.

The conclusion is drawn that the ultimate shear capacity according to the extended beam model

corresponds more closely to the ultimate shear capacity according to the formula of Dupont and

Vandewalle when the characteristic tensile strength is used instead of the mean tensile strength.

Furthermore, the simulation with the characteristic tensile strength results in a smaller ultimate

shear capacity than the simulation with the mean tensile strength.

63 Verification

Figure 4.18: Effect of the reinforcement ratio ρ, residual tensile strength fres, shear span to effective depth ratio a/d, and the compressive concrete strength fcm according to the extended beam model (using the characteristic tensile strength). fres and fcm in N/mm

2.

Rea

din

g st

eep

s:

Firs

t, s

tart

at

the

top

rig

ht

wit

h t

he

rein

forc

em

ent

rati

o ρ

an

d r

esid

ual

ten

sile

str

engt

h f

res,

her

e is

ass

um

ed t

hat

a/d

is 2

.5 a

nd

th

e c

om

pre

ssiv

e co

ncr

ete

stre

ngt

h f

cm is

50

N/m

m2.

Seco

nd

, rea

d f

rom

rig

ht

to le

ft t

he

corr

ecti

on

fo

r

shea

r sp

an t

o e

ffec

tive

dep

th r

atio

a/d

. Th

ird

, rea

d f

rom

to

p t

o b

ott

om

th

e co

rrec

tio

n f

or

com

pre

ssiv

e co

ncr

ete

stre

ngt

h f

cm. F

rom

left

to

rig

ht

no

w t

he

ult

imat

e sh

ear

cap

acit

y V

u c

an b

e re

ad.


4.3.2 Quantitative verification Whether the extended beam model provides an upper bound or a lower bound is investigated in this

section. For this purpose, the extended beam model is compared with experiments performed by

Dupont and Vandewalle (2002).

Figure 4.19 shows the test set-up, which is a simply supported beam subjected to two point loads.

This set-up is similar to the one in the previous section and paragraph. The general dimensions and

material properties of the test program are shown in table 4.9. The test program contains three

series. Per series three beams with a different fibre volume ratio Vf are tested. The first series, beam

16 till 18, has a reinforcement ratio ρ of 1.15% and an shear span to effective depth ratio a/d of 2.5.

The second series, beam 19 till 21, has a higher reinforcement ratio than the first series being 1.81%.

The third series, beam 22 till 24, has a higher a/d than the second series being 4. The specific values

per beam are included in table 4-10. Furthermore, Vandewalle and Dupont determined the mean

residual flexural strength 4Rmf , which is translated to a residual tensile strength according to

Vandewalle et al. (2003):

40.37res Rmf f

Table 4-10 shows the test program and the ultimate shear capacity Vu according to the test, the

simulations of the extended beam model using the mean tensile strength fctm and the characteristic

tensile strength fctk, and the formula of Dupont and Vandewalle. Additionally, the ratio between the

Vu of the test and the Vu of the model and formula is presented. The results in proportion to each

other are also illustrated by figure 4.20.

All results show a larger Vu when the residual tensile strength increases. The Vu according to the

model with fctm is both smaller and larger than the Vu of the tests. Conversely, the Vu according to

the model with fctk is larger than the Vu of the tests. The Vu according to the formula is in general

also larger than the Vu of the tests.

The difference between the tests and the model with fctm is probably due to erratic character of

shear failure. This erratic character causes a large variance of test result.

The conclusion is drawn that the extended beam model provides an lower bound when the

characteristic tensile strength is used. When the mean tensile strength is used, the extended beam

model provides an average value of the ultimate shear capacity. The formula of Dupont and

Vandewalle provides a lower bound.

65 Verification

Figure 4.19: Test set-up experiments Dupont and Vandewalle (2002).

Table 4.9: General dimensions and material properties of the experiments performed by Dupont and Vandewalle (2002)


Width, b 200 concrete Compressive strength, fcm ≈40

Effective height, d

260 Mean residual flexural strength, fRm4

0.0 – 4.7

Height, h 300 Steel Yield strength, fy 560

Length, L 2300 Young’s modules, Es 200000

Table 4-10: Results experiments, extended beam model, and empirical formula.

Beam 16 17 18 19 20 21 22 23 24

ρ 1.15% 1.15% 1.15% 1.81% 1.81% 1.81% 1.81% 1.81% 1.81%

fcm -39.6 -39.1 -38.6 -39.6 -39.1 -38.6 -40.3 -40.7 -42.4

Vf 0 % 0.5% 1.5% 0% 0.5% 1.5% 0% 0.5% 1.5%

fres 0.00 0.41 1.30 0.00 0.41 1.30 0.00 0.56 1.74

a 650 650 650 650 650 650 1050 1050 1050

a/d 2.5 2.5 2.5 2.5 2.5 2.5 4 4 4

Test

Vu 53.5 82.5 108 95.5 108 144 75 82.5 117

Model with fctm

Vu 75 86 106 88 100 123 76 91 122

test/model 0.72 0.96 1.02 1.09 1.08 1.17 0.99 0.90 0.96

Model with fctk

Vu 52 63 84 61 74 97 53 67 98

test/model 1.03 1.30 1.29 1.56 1.46 1.48 1.42 1.24 1.20

Formula Dupont and Vandewalle

Vu 56 61 73 65 70 82 55 60 71

test/formule 0.96 1.35 1.48 1.48 1.54 1.75 1.35 1.37 1.64


Figure 4.20: Ultimate shear capacity Vu of the SFR-RC beams tested by Dupont and Vandewalle (2002) (black); according to the simulations by the extended beam model using the mean tensile strength fctm (dark bleu) and the characteristic tensile strength fctk (light blue); and according to the formula of Dupont and Vandewalle (2002) (red). Fres: residual tensile strength, ρ: reinforcement ratio, fcm: compressive concrete strength.

67 Verification

4.3.3 Comparison between the model and Kani The benchmark in section 4.1.3. showed the effect of the a/d ratio on the shear capacity. Kani (1967)

investigated the influence of the a/d ratio on the failure of reinforced concrete beams. The extended

model is compared with experiments performed by Kani.

The experimental results are clarified by expressing the ultimate moment uM with respect to the

moment capacity of the cross-section flM . This u flM M ratio of a reinforced concrete beam is

plotted against the a/d ratio and the reinforcement ratio of the beam resulting in a two-dimensional

diagram. This diagram is called Kani’s valley. The extended beam model is compared with the valleys

of three reinforcement ratios” 0.80%, 1.18%, and 2.80%. Table 4.7 presents the normalized

properties of the experiments. The test set-up is a simply supported beam with two point loads.

Figure 4.21 presents valleys with the results from Kani’s tests (blue lines) and the extended beam

model (red lines). In case of a/d greater than 2.5 the results of extended beam model fairly fit the

experiments. In case of a/d smaller than 2.5 the Mu/Mfl ratio of Kani’s results increases. This

increase cannot me simulated by the extended beam model. The reason of this limitation is

explained in paragraph 3.3.

Table 4.11: Dimensions and material properties of the test beams (Kani, 1967).


Width b 152.4 concrete Compressive strength fcm 26.2

Height h 304.8 Steel Yield strength fy 344.7

Young’s modules Es 200000

Figure 4.21: ‘Valleys’ of tests Kani (blue), model (red) for three reinforcement ratios ρ; vertical axis: ultimate moment Mu to the moment capacity of the cross-section Mfl; horizontal axis: shear span to effective depth ratio a/d.


69 Conclusion and discussion

5 Conclusion and discussion

5.1 Conclusion

Subsequent to the literature study, the main hypothesis of this research is formulated:




An contribution of steel fibre reinforcement in the tensile zone and compressive zone is presented in

chapter 3. These contributions are implemented in a beam model, as is shear failure, resulting in the

extended beam model. This extended beam model is designed to predict shear failure of SFR-RC

beams. Chapter 4 shows that the extended beam model confidently predicts the ultimate shear

capacity.

In addition to the main hypothesis, three sub hypotheses are formulated:

‘The shear capacity increases for an increasing residual tensile strength’

‘The shear capacity increases due to steel fibres in the cracks of the tensile zone.’

‘The shear capacity increases due to steel fibres in the compressive zone.’

These hypotheses can be confirmed with the benchmarks of section 4.1.2. Figure 4.1 validates that

the shear capacity increases for an increasing residual tensile strength. This effect is also confirmed

by the simulation in section 4.3.2.

The normal stress flows in figure 4.4 show that the residual tensile strength increases the

contribution of the tensile concrete zone to the cross-sectional equilibrium. As a result, the effective

shear height becomes larger. However, the increment of the effective shear height is minimal.

Hence, the increment of the shear capacity due to steel fibres in the cracks of the tensile zone is

minimal. The contribution of the steel fibres in the cracks of the tensile zone probably represents the

increment of the moment capacity.

The shear stress flows in figure 4.4 show that the residual tensile strength strongly effects the

maximum shear stress in the cross-section. This effect is caused by the reduction of the external

shear force, which depends on the residual tensile strength. Section 3.2.2 explains that this reduction

is based on the assumption that the steel fibres in the compressive zone are activated when a tensile

stress in the compressive zone exceeds the tensile concrete strength. Thus, the hypothesis that the

shear capacity increases due to steel fibres in the compressive zone is validated. However, the

reduction of the external shear force is related to the effective shear height, not the compressive

height.

In conclusion, the shear capacity of a reinforced concrete element subjected to a flexural load

increases due to the contribution of the steel fibre reinforcement in the effective shear height.


5.2 Discussion and recommendations

The performed research is based on the extended beam model . Consequently, this model defines

the outcome of this research. That is why, the three assumptions for the design of the extended

beam model are reviewed.

1. In section 2.1.4 is assumed that the stress-strain relation representing the tension stiffening

effect is applicable for a simply supported beam subjected to two point loads. The effect of

tensile stress-strain relation on the shear capacity, however, is minimal. Therefore, the shear

capacity of other load types is most likely also possible. This only concerns determined

structures.

2. The definition of the effective shear height is based on an research of Pruijssers (1986),

which investigated reinforced concrete beams, thus without steel fibre reinforcement.

Section 4.1.2 illustrated that the residual tensile strength influences the effective shear

height. However, whether steel fibre reinforcement influences the definition of the effective

shear height is unknown.

3. The shear stress flow is assumed parabolic. As a result, the contribution of the tensile zone is

overestimated, which is illustrated by figure 2.15. The magnitude of this overestimation is

unknown. However, the overestimation is expected to be limited because the extended

beam model provides an quite accurate prediction of the shear capacity.

Although the main hypothesis only concerns the contribution of the steel fibre reinforcement, three

other parameters are also investigated. Paragraph 4.1 explains how and why the residual tensile

strength, the reinforcement ratio, shear span to effective depth ratio, and the concrete strength

effect the shear capacity. Not only the residual tensile strength but also the other three parameters

influences the effective shear height. Since the contribution of steel fibre reinforcement to the shear

capacity strongly depends on the effective shear height, this contribution is effected by the

reinforcement ratio, the shear span to depth ratio and the concrete strength. This shows an complex

interference between parameters, which can more thoroughly be investigated.

This research considers a beam which is simply supported and has normal strength concrete, a

residual tensile strength lower than the critical strength, and a shear span to depth ratio larger than

2.5. Whether the conclusions are also valid for other boundary condition should be investigated.

The extended beam model can be used to analyse beams or slabs of a determined structure. Other

material properties can be inserted in the extended beam model. However, the model currently

calculates several material and structural properties from a few inserted properties. For instance, the

tensile concrete strength is calculated from the compressive concrete strength. These internal

calculation possibly have to be adjusted. Inserting a residual tensile strength higher than the critical

strength is possible but this significantly increases the calculation time of the tension stiffening

module. Furthermore, the tension stiffening effect disappears in this case.

Paragraph 3.3 explained that the beam model is only suitable for a shear span to effective depth

ratio larger than 2.5. Beams with a smaller ratio develop a multi-axial stress condition, which is not

incorporated in the extended beam model. When an extra normal stress, in y-direction, would be

defined, it could be possible to implement the multi-axial stress condition. However, this could be

very complex and other models might be more suitable to define complex stress conditions.

71 References

References Aa, P.J., Krings, H., Paus, T.J. and Wijffels, M.J.H., 2013. Steel fibre reinforced concrete, Literature

Survey. Master. Eindhoven University of Technology.

Bruggeling, A.S.G. and Bruijn, W.A. de, 1986. Theorie en praktijk van het gewapend beton. ’s Hertogenbosch: Vereniging Nederlandse Cementindustrie.

European standard with Dutch national annex, 2011. NEN-EN 1992-1-1, eurocode 2: design of concrete structures, Part 1: general rules and rules for buildings. Nederlands normalisatie-instituut [online] Available through: BRIS bv <http://www.briswarenhuis.nl> [August 2013].

Gribniak, V., Kaklauskas, G., Kwan, A.K.H., Bacinskas, D. and Ulbinas, D., 2012. Deriving stress-strain

relationships for steel fibre concrete in tension from tests of beams with ordinary reinforcement.

Engineering structures, 42, pp. 387-395.

Hamelink, S.A., 1989. Ontwikkeling van een trekstaaf-analogie (toepassing op een buigligger van

gewapend beton). Master. Eindhoven University of Technology.

Hordijk, D., 1991. Local approach to fatigue of concrete. PhD. Delft University of Technology.

Kani, G.N.J., 1967. How safe are our large reinforced concrete beams? ACI Journal, 64(4), pp. 128-

141.

Kani, G.N.J., 1966. How Basic facts concerning shear failure. ACI Journal, 63(6), pp. 675-692.

Kooiman, A.G., 2000. Modelling steel fibre reinforced concrete for structural design. PhD. Delft

University of Technology.

Kotsovos, M.D., 1987a. Consideration of triaxial stress conditions in design: a necessity. ACI Structural

Journal, 84 (3), pp. 266-273.

Kotsovos, M.D., 1987b. Shear failure of reinforced concrete beams. Engineering Structures, 9 (1), pp.

32-38.

Kotsovos, M.D., 1988. Compressive force path concept: Basis for reinforced concrete ultimate limit

state design. ACI Structural Journal, 85(1), pp.68-75.

Kotsovos, M.D. and Pavlovic, M.N., 1999. Ultimate limit-state design of concrete structures, a new

approach. Londen: Thomas Telford Ltd.

Marinus, P.C.P., 2013. Onderzoeksproject beton, non-ferro wapening als alternatief voor staal in

gewapend betonconstructies. Master. Eindhoven University of Technology

Narayanan, R. and Darwish, I.Y.S., 1987. Use of steel fibres as shear reinforcement. ACI Structural

Journal, 84(3), pp. 216-227.

Paus, T.J., 2014. Steel fibre reinforced concrete in flat slaps, by using the finite difference method for

design analysis. Master. Eindhoven University of Technology.

Pruijssers, A.F., 1986. Shear resistance of beams based on the effective shear depth. Delft: Stevin-

reports.


Salet,T.A.M., 1991. Structural analysis of sandwich beams composed of reinforced concrete faces and

a foamed concrete core. PhD. Eindhoven University of Technology.

Vandewalle, L. and Dupont, D., 2002. Dwarskrachtcapaciteit van staalvezel-betonbalken. Cement,

2002(8), pp. 92-96.

Vandewalle, L., 2003. RILEM TC 162-TDF: ‘Test and design methods for steel fibre reinforced

concrete’ σ-ε-design method final recommendation. Materials and Structures, 36, pp. 560-567.

Walraven, J., (2011). Van exotisch naar volwassen product. Cement, 2011(3), pp. 4-9.

Walraven, J.C. and Reinhardt, H.W., (1981). Theory and experiments on the mechanical behaviour of

cracks in plain and reinforced concrete subjected to shear loading. HERON, 26 (1A).

73 Appendix A: Functioning beam model

Appendices

A. Functioning beam model 72

A.1 Flow chart calculation moment-curvature diagram (multi-layer model) 72

A.2 Flow chart calculation deflection diagram (multi-layer model and moment-area method) 73

B. Determination post-cracking behaviour of SFRC 75

C. Tension stiffening effect 77

C.1 Bond theory 77

C.2 Reinforced concrete bar 78

C.3 Impact area 80

D. Flow charts additional modules beam model 81

D.1 Flow charts tension stiffening module 81

D.2 Flow chart shear module 83

D.3 Flow chart load module 84

D.4 Flow chart load-deflection module 85

E. Material properties and calculation settings 86

E.1 Inserted material properties 86

E.2 Applied calculation settings 88

E.3 Inserted compressive behaviour 89

E.4 Critical concrete stress 90

E.5 Influence number of layers, segments and load steps 92

F. Additional diagram verification contribution tensile zone 96

G. Combination diagrams effect parameters 100

74 Shear failure of reinforced concrete with steel fibre reinforcement

A. Functioning beam model

A.1 Flow chart calculation moment-curvature diagram (multi-layer model)

75 Appendix A: Functioning beam model

A.2 Flow chart calculation deflection diagram (multi-layer model and moment-area method)


77 Appendix B: Determination post-cracking behaviour of SFRC

B. Determination post-cracking behaviour of SFRC

To determine the post-cracking behaviour of a SFRC-mixture, a three-point-bending test is proposes

by the ‘International Union of Laboratories and Experts in Construction Materials, Systems and

Structures’ Technical Committee 162-TDF (RILEM TC 162-TDF). Among others the RILEM TC 162-TDF

conducted research on steel fibre reinforced concrete.

To execute the three-point-bending test, RILEM TC 162-TDF prescribes a small beam with a cross-

section of 150mm by 150 mm and a span of 500mm (fig. B.1). The specimen is supported by two

rolling pins and subjected to a point load, which transferred to the specimen by a rolling pin. In the

middle of the beam a notch of 25mm is made because this guaranties a crack location in the middle.

The CMOD of this notch is measured by a device during the loading. From this test a load-CMOD

diagram is obtained (fig. B.2) in which five points are indicated. The first point represents the Limit Of

Proportionality (LOP or L), which is the end of the linear elastic branch. The second till fifth point

represent respectively a CMOD of 0.5, 1.5, 2.5, and 3.5 mm.

As mentioned before, this force-CMOD relation can be converted to a stress-strain relation. The

RILEM TC 162-TDF proposes a stress-strain relation (fig. B.3)which is obtained in two steps. First, the

forces of each point are converted to stresses

;

3

2L

fct L

el sp

F lMf

W bh ( B-1 )

;;

3

2R i

R i

el sp

F lMf

W bh ( B-2 )

In case of the residual tensile stresses ;R if the use of the elastic section modulus is noticeable

because the behaviour is not elastic after cracking.

Next, only three stresses are used: the stress at cracking ;fct Lf , the first residual tensile stress ;1Rf ,

and the forth residual tensile stress ;4Rf . Thus, the second and third residual tensile stresses, ;2Rf

and ;3Rf , are omitted. ;1Rf and ;4Rf are multiplied with respectively the factors 0.45 and 0.37 to

correct the use of the elastic section modulus instead of the plastic section modulus. Furthermore,

the CMOD’s corresponding to the three stresses ;fct Lf , ;1Rf ,and ;4Rf are replaced by three strains.

The stress-strain diagram derived from the load-CMOD relation in accordance with RILEM TC 152-

TDF (2003) is shown in figure B.3.


Figure B.1: Three-point-bending test to determine the flexural behaviour of a SFRC-mixture according to the RILEM TC 162-TDF. In red the device that measures the CMOD during loading. Measurements in mm.

Figure B.2: Load-CMOD diagram of a three-point bending test (Vandewalle, 2003)

Figure B.3: Stress-strain diagram derived from the load-CMOD diagram according to RILEM (Vandewalle, 2003)

a size-factor

1 ;

2 ;1

3 ;4

1 1

0002 1

0003

0.45

0.37

:

0.1

25

fct L

R

R

c

f k

f k

f k

k

E

79 Appendix C: Tension stiffening effect

C. Tension stiffening effect

The first paragraph explains the bond theory. After that, the second paragraph describes how the

bond theory is used to derive a post-cracking stress-strain relation for a reinforced concrete bar

subjected to a tensile load. Finally, the third paragraph clarifies the concrete area which is affected

by the bond between concrete and reinforcement bar.

C.1 Bond theory The bond between concrete and reinforcement is explained with a tension bar. Figure C.1 shows the

impact of a crack on the traction between the steel rebar and surrounding concrete and the impact

on the stress in the steel and concrete. At the crack, which is located at the right, the tensile force is

entirely carried by the rebar ( cr sN N ). A transition length away from the crack, x=0, the bond

between the concrete and rebar is entirely intact; as a result, the tensile force is carried by both the

steel and the concrete ( cr s cN N N ). Along the transition length the situation gradually changes.

This change is defined by the relative displacement of the steel and the traction-separation relation

between the concrete and rebar.

Figure C-.C.1: Along the transition length from top to bottom: bond between concrete and rebar; displacement of the rebar relative to the concrete; traction between concrete and rebar; stress in the steel rebar; stress in the concrete. Right: relation between the traction and relative displacement. (Bruggeling and Bruijn, 1986).


C.2 Reinforced concrete bar This bond theory can be applied on a reinforced concrete bar. When the cracking moment is reached

in a reinforced bar, the development of cracks can be described by two phases. During the first phase

the crack pattern develops until the crack distance crl is twice the transition length tl (fig. C.2).

Because the concrete stress at the end of transition length is equal to the cracking stress, secondary

cracks will develop between the primary cracks (fig. C.3). As a result, the transition lengths of

adjacent cracks overlap. For second generation cracks the steel stress along the beam is defines by

superposition of the steel stresses of each transition length. Due to the increase of the steel stress,

the concrete stresses decrease. The strains in the rebar and concrete change similar to the stresses.

However, the real crack pattern shows a erratic course due to stochastic effects. As a result, the

minimum crack distance occurs when two transition lengths completely overlap ( cr tl l ) and the

maximum crack distance occurs when two transition lengths do not overlap ( 2cr tl l ).

The contribution of the concrete between cracks to the bearing of the tensile load results in a stiffer

reaction of the reinforced bar compared to an uncovered rebar. This contribution of the concrete can

be included in definition of the tensile behaviour of concrete through the addition of a post-cracking

branch to the tensile stress-strain relation of concrete. Bruggeling and Bruijn (1986) propose a

bilinear post-cracking branch (fig. C.4).

Point 1 and 2 are respectively defined by the average concrete stress and steel strain for the

maximum cracks distance and the minimum crack distance. Point B is based on the logical

assumption that the average crack distance is 1.5 times the transition length:

;min ;max

;

21.5

2 2cr cr t t

cr average t

l l l ll l

( C-1 )

Considering the above definition of the average crack distance, the average steel strain along the bar

is

1 22

3sm sm

sm

( C-2 )

And the average concrete stress along the bar is

1 22

3cm cm

cm

( C-3 )

For point C the logical assumption is made that concrete does not further contribute to the stiffness

when the reinforcement yields.

81 Appendix C: Tension stiffening effect

Figure C.2: First generation cracks in a tension bar (top); concrete stress along the length (mid); steel stress along the length (bottom); in red the averages stress in the bar.

Figure C.3: Second generation cracks in a tension bar (top); concrete stress along the length (mid); steel stress along the length (bottom); in light grey the stresses related to each transition length; the black curve is the actual stress defined by superposition; in red the averages stress in the bar.

Figure C.4: Tensile stress-strain relation of concrete including tension stiffening (red). A: cracking point; B: decrease of concrete contribution after cracking; C: ending of tension stiffening at yielding; 1: average concrete stress and steel strain for first generation cracks; 2: average concrete stress and steel strain for second generation cracks.


C.3 Impact area Finally, the impact area, which is the concrete area that contributes to the bond with the rebar, has

to be known to calculate the transition length. This area is equal to the concrete cross-section in case

of an axial tensile load; however, in case of bending not the whole tensile zone contributes. The

impact area in this case is defined with the effective concrete height ,c effh and the width. The

horizontal equilibrium of the compressive force (eq. C-4) and the tensile force (eq. C-5) is used to

formulate the effective height (eq. C-6).

, ,

;2

c cr c cr

c cr

bhN

( C-4 )

; ; 1t cr cr c eff e effN h b ( C-5 )

with ,

seff

c eff

A

bh and s

e

cm

E

E

, ,

;2

c cr c cr scr e

c eff

cr

h A

bh

( C-6 )

83 Appendix D: Flow charts additional modules beam model

D. Flow charts additional modules beam model

D.1 Flow charts tension stiffening module

First generation cracks


Second generation cracks


D.2 Flow chart shear module


D.3 Flow chart load module


D.4 Flow chart load-deflection module

Load-deflection

diagram


E. Material properties and calculation settings

The first paragraph describes which material properties are inserted in the beam model. All

calculations are performed with these material properties unless states otherwise. The applied

calculations settings are specified in paragraph E.2. The underlying research of the compressive

behaviour and the critical concrete strength is presented in respectively the third and fourth

paragraph. Finally, paragraph E.5 contains the research regarding the influence of the number of

layers, load steps and segments.

E.1 Inserted material properties

Steel fibre reinforced concrete

The tensile behaviour of a SFR-RC beam is presented in section 3.2.1. The compressive behaviour and

linear-elastic tensile behaviour is based on Eurocode 2 (European standard with Dutch national

annex, 2011). This research is aimed at normal strength concrete and the real material behaviour;

therefore, safety factors are omitted and mean values are applied. The European Standard

formulates the mean compressive and tensile strength and Young’s modulus as

(MPa)8cm ckf f ( E-1 )

2/30.30ctm ckf f ( E-2 )

[MPa]0.3

22 10cm cmE f ( E-3 )

Figure E.1 shows the applied stress-strain relation for steel SFRC. The compressive behaviour is

described by a bilinear stress-strain relation. This relation is a non-conservative an realistic

assumption based on the parabolic-linear relation of the European standard (2011). Details of this

assumption can be found in paragraph E.3. A parabolic behaviour would be more realistic than a

bilinear behaviour; however, this results in higher calculation time because an equilibrium is harder

to gain (paragraph E.5).

The critical concrete strength cr is not equal to the mean tensile strength because the tensile

strength decreases with the load time. For comparison with experiments, the load time is assumed

rapidly (first cracks appear after a load time of a few minutes) resulting in a reduction of 0.7:

0.7cr ctmf ( E-4 )

The influence of the load time and the reduction for the critical strength is described paragraph E.4.

The applied reduction of 0.7 is in good agreement with experimental results (paragraph E.4).

Steel reinforcement bars

Figure E.2 shows the applied behaviour of the reinforcement steel. The stress-strain relation is

elastic-plastic and uses the mean yielding strength:

if

if

if

0

s y s y

s s s y

y y s

f

E

f

( E-5 )

89 Appendix E: Material properties and calculation settings

with y

s

s

f

E

An ascending branch after yielding would be a more realistic approach; however, the ultimate

strength is mostly not given in the references. An ultimate steel strain is not entered because failure

of the concrete is assumed. The relations between the reinforcement ratio s , the reinforcement

area sA , the diameter of a reinforcement bar , and the number of reinforcement bars sn is

described as

s sA bd and 0.25

s

s

A

n

Figure E.1: The bilinear stress-strain relation assumed for compressive behaviour of concrete.

Figure E.2: Elastic-plastic behaviour assumed for the behaviour of reinforcement steel.


E.2 Applied calculation settings Four settings can be adjusted for a calculation:

1. the number of layers ‘i’

2. the number of segments ‘n’

3. the number of load steps

a. for the moment-curvature diagram ‘m’

b. for the load-deflection diagram ‘p’

The maximum number for these settings is 100. Paragraph E.5 contains an underlying research on

the influence of the numbers. The applied calculation settings are affect by the following conclusions:

- The calculation time does not decrease in proportion to a decrease of layers.

- A very low number of layers (n=10) results in an inaccurate calculation.

- De calculation time and preciseness of the moment-curvature diagram is strongly influenced

by the number of load steps.

- The load-deflection diagram is considered equally affected as the moment-curvature

diagram.

- The number of segments logically influences the calculation time and accuracy due to the

segmentation of the external moment.

- In case of one point load in the middle, the deflection is slightly underestimated for an even

number and (strongly) overestimated for an uneven number.

In accordance with the underlying research, 50 layers, 50 segments, and 100 moment-curvature

diagram load steps are applied. In case of a shear failure analysis, 100 layers are used to obtain a

precise calculation of the stresses.

Figure E.3: stress-strain relations for compressive behaviour of concrete. Dark, mid tone and light blue: respectively equation E-6, E-7 and E-8 according to the European standard (2011).


E.3 Inserted compressive behaviour The behaviour of concrete in compression can be specified by different stress-strain relations. The

European standard (2011) specifies three non-linear options which are discussed in this paragraph.

The first one is a parabolic relation, which is described by the following stress-strain relation:

2

1 2c cm

kf

k

( E-6 )

with 1

c

c

,

11.05 cm c

cm

Ek

f

, 2.0n , 0.31

1 0.7c cmf , and 0001 3.5cu

The second one is a parabolic-linear relation, describe by

if 0

if

2

2

2 2

1 1

n

cc cd c c

c

c cd c c cu

f

f

( E-7 )

with 0002 2.0c and 0

002 3.5cu

The third one is a bilinear relation:

if 0

if

3

3

3 3

cdc c c c

c

c cd c c cu

f

f

( E-8 )

with 0003 1.75c and 0

003 3.5cu

The parabolic relation (eq. E-6) should be suitable for short-term uni-axial loading; however, the

crushing strain 1c and failure strain 1cu are nominal and not real values. The parabolic-linear

relation (eq. E-7) is prescribed for a cross sectional calculation with design (or characteristic) value of

the compressive strength; therefore, this relation is not suitable for a realistic approach. The

European Standard (2011) allows other relations for a cross-sectional calculation if they are more

conservative than the parabolic-linear relation. The bilinear relation (eq. E-8) is an example of a more

conservative approach and is frequently used for ultimate limit strength calculations. Figure E.3

presents these three relations using the mean compressive strength as the maximum concrete

stress.

The red curve in figure E.3 presents the applied bilinear stress-strain relation, which is defined by

if 0

if

c c cm c cm

c cm cm c cmu

E

f

( E-9 )

with cmcm

cm

f

E and 2cmu cm

The applied stress-strain relation is a non-conservative approach of de parabolic-linear relationship;

it uses the mean compressive strength and the mean Young’s modulus. A smaller ultimate strain is

assumed because experiments are a rapid load type resulting is a less ductile behaviour.


E.4 Critical concrete stress When post-cracking behaviour is not taken into account, the cracking stress is assumed to be the

mean tensile strength of the concrete ctmf . However, post-cracking behaviour is taken into account

and the tensile strength of concrete decreases with the load time, shown in figure E.4. That is why, a

reduction of the mean tensile strength should be used to define the critical concrete stress.

Bruggeling and Bruijn (1986) distinguished three reduction factor to define the critical concrete

stress:

1. Long-term load (load time is endless)

,00.6cr ctmf

1. Slowly increased deformation (first cracks after a load time of a few days)

,00.75cr ctmf

2. Rapidly increased deformation (first cracks after a load time of a few minutes)

,00.9cr ctmf

,0ctmf Is the short-term mean tensile strength for concrete; however, nowadays only the mean

tensile strength ctmf is used. Therefore, ,0ctmf should be converted into ctmf .

According to Bruggeling and Bruijn (1986), the short-term mean tensile strength can be expressed as

,0 ,0

,0 ,

,0 ,

1.451.26 1 0.05

0.87 1 0.05

ctm ct

ctm ck cube

ct ck cube

f ff f

f f

( E-10 )

According to the eurocode (2011), the mean tensile strength can be expressed as

2/30.30ctm ckf f

( E-11 )

Figure E.5 illustrates that 2/3

,0.30 1 0.05ck ck cubef f , resulting in

The definition of cracking stress can now be expressed as a function of the mean tensile strength for

three different load types:

1. long-term load (load time is endless):

0.6 0.79 0.5cr ctm ctmf f

2. slowly increased deformation (first cracks after a load time of a few days)

0.75 0.79 0.6cr ctm ctmf f

3. rapidly increased deformation (first cracks after a load time of a few minutes)

0.9 0.79 0.7cr ctm ctmf f

Experiments are mostly rapidly enforced deformation; therefore, in combination with tension

stiffening a reduction of 0.7 is applied in this research.

The influence of the critical tensile strength is illustrated in figure E.6. The yellow line represents a

concrete without a post-cracking behaviour and with ctmf as the maximum tensile stress. The blue

lines represent a concrete with a post-cracking behaviour that includes the tension stiffening effect.

,0

,00.791.26ctm

ctm ctm

ff f

( E-12 )


Figure E.4: Decrease of tensile strength with load time (Bruggeling and Bruijn, 1986).

Figure E.5: Comparison of two formulas for the mean tensile strength.

Figure E.6: The influence of the critical strength on the tension stiffening effect.


E.5 Influence number of layers, segments and load steps Expected is a more accurate output and an increase of the calculation time when the number of the

settings increase

- the number of layers ‘i’

- the number of segments ‘n’

- the moment-curvature diagram load steps ‘m’

- the load-deflection diagram load steps ‘p’

Firstly, the accuracy and calculation time of the moment-curvature diagram is tested for a variation

of layers and load steps. Secondly, the accuracy and calculation time of a deflection diagram is

tested. At last, a comparison is made between a bilinear and a parabolic compressive concrete

behaviour.

Layers

The influence of the number of layers ‘i’ is tested with i=100, i=50, i=20, and i=10 for m=100. Table

E.1 shows that the decrease of the calculation time is not proportional to de decrease of the number

of layers. Furthermore, table E.1 shows that the output of 20 layers is only a bit less accurate and

that the output of 10 layers is inaccurate. The latter is also shows in figure E.7. As a result, 100 or 50

layers is recommended.

Table E.1: Results test for varying number of layers ‘I’ and load steps ‘m’. Reference test (100%): m=100; 1=100; bilinear.

Figure E.7: Moment-curvature diagrams for different number of layers ‘i' created by 100 load steps (large) and connecting the cracking, yielding, crushing and failure moment (small).


Load steps

Table E.1 shows a logical decrease in calculation time for a lower number of load steps ‘m’. The

number of load steps also influences the accuracy of the moment-curvature diagram as shown in

figure E.8. The best choice depends on the required accuracy and calculation time. The number of

load steps ‘p’ is not analyses because the findings are assumed similar to the ‘m’ tests.

Segments

The number of segments logically influences the calculation time because for every segment an

equilibrium has to be solved (tab. E.2). The accuracy of the deflection is influenced due to the

segmentation of the external moment. In addition to the number of segments, the load type also

influences the maximum deflection. For example, when a point load and an even number of

segments are entered, the maximum moment will not be reached (fig. E.9). Consequently, the

maximum deflection will be an underestimation. However, when applying a point load and an

uneven number of segments (fig. E.10), the maximum moment is reached in the middle segment.

Because the middle segment is not infinite small, the maximum deflection will be an overestimation.

Although an overestimation is saver, the effect of overestimation seems to be much bigger than the

effect of underestimation (fig. E.11). Thus, an even number of segments is recommended.

Figure E.8: Moment-Curvature diagram for different numbers of load steps ‘m’.

Table E.2: Results test for varying number of segments ‘n’. Reference test (100%): i=50; n=100.


Figure E.9: External moments in case of 20 segments Figure E.10: External moments in case of 21 segments

Figure E.11: Deflection along the length of a beam with one point load in the middle.

Table E.3: Results test for varying number of layers ‘I’ and load steps ‘m’. Reference test (100%): m=100; 1=100; bilinear.


Parabolic compressive behaviour

The calculation time for a parabolic stress-strain relation is longer than for a bilinear stress-strain

relation (tab. E.3). Therefore, a bilinear compressive behaviour is recommended. However, when a

parabolic compressive behaviour is chosen, the calculation time for 100 layers is significantly longer

than for 50, 20 and 10 layers. Additionally, the output of 20 layers is only a bit less accurate and the

output of 10 layers is inaccurate. As a result, 50 layers is recommended for a parabolic compressive

behaviour. The inaccuracy of 10 layers is also illustrated by figure E.12.

Figure E.12: Moment-curvature diagrams for different number of layers ‘i' created by 50 load steps (large) and connecting the cracking, yielding, crushing and failure moment (small).


F. Additional diagram verification contribution tensile zone

Figure F.1: Concrete without steel fibre reinforcement, Vf=0%. Stress-strain relations calculated by the tension stiffening module (layered mode, red line) and derived from two experiments by Gribniak et al. (2012) (experiment 1 and 2, black and grey line). The dotted line represents the stress-strain relation Gribniak et al. inserted in a finite element method (FEM) calculation.

Figure F.2: SFRC with a fibre volume ratio Vf of 0.5%. Stress-strain relations calculated by the tension stiffening module (layered mode, red line) and derived from two experiments by Gribniak et al. (2012) (experiment 1 and 2, black and grey line). The dotted line represents the stress-strain relation Gribniak et al. inserted in a finite element method (FEM) calculation.

99 Appendix F: Additional diagram verification contribution tensile zone




Figure F.5: Concrete without steel fibre reinforcement, Vf=0%. Moment-curvature diagram calculated by the tension stiffening module (layered mode, red line) and derived from two experiments by Gribniak et al. (2012) (experiment 1 and 2, black and grey line). The dotted line represents the stress-strain relation Gribniak et al. inserted in a finite element method (FEM) calculation.

Figure F.6: SFRC with a fibre volume ratio Vf of 0.5%. Moment-curvature diagram calculated by the tension stiffening module (layered mode, red line) and derived from two experiments by Gribniak et al. (2012) (experiment 1 and 2, black and grey line). The dotted line represents the stress-strain relation Gribniak et al. inserted in a finite element method (FEM) calculation.

101 Appendix F: Additional diagram verification contribution tensile zone




G. Combination diagrams effect parameters

Figu

re G

.1:

Dia

gram

s w

ith

th

e r

esu

lts

of

the

ext

en

de

d b

eam

mo

de

l usi

ng

the

me

an t

en

sile

str

en

gth

. Th

e u

ltim

ate

sh

ear

cap

aci

ty (

Vu

) ca

n b

e

read

fro

m t

his

dia

gram

in t

hre

e s

tep

s. F

irst

, sta

rt a

t th

e r

igh

t to

p w

ith

th

e r

ein

forc

em

en

t ra

tio

(ρ

) an

d r

esi

du

al t

en

sile

str

en

gth

(fr

es)

, he

re i

s

assu

me

d t

hat

a/d

is 2

.5 a

nd

th

e c

om

pre

ssiv

e c

on

cre

te s

tre

ngt

h (

fcm

) is

50

N/m

m2 .

Seco

nd

, re

ad f

rom

rig

ht

to le

ft t

he

co

rre

ctio

n f

or

she

ar

span

to

eff

ect

ive

de

pth

rat

io (

a/d

). T

hir

d, r

ead

fro

m t

op

to

bo

tto

m t

he

co

rre

ctio

n f

or

com

pre

ssiv

e c

on

cre

te s

tre

ngt

h (

fcm

). F

rom

left

to

rig

ht

no

w t

he

ult

imat

e s

he

ar c

apac

ity

(Vu

) ca

n b

e r

ead

. fre

s an

d f

cm in

N/m

m2 .

103 Appendix G: Combination diagrams effect parameters

Figu

re G

.2:

Dia

gram

s w

ith

th

e r

esu

lts

of

the

fo

rmu

la o

f D

up

on

t a

nd

Van

de

wal

le (

20

02

). T

he

ult

imat

e s

he

ar c

apac

ity

(Vu

) ca

n b

e r

ead

fro

m

this

dia

gram

in t

hre

e s

tep

s. F

irst

, sta

rt a

t th

e r

igh

t to

p w

ith

th

e r

ein

forc

em

en

t ra

tio

(ρ

) a

nd

re

sid

ual

te

nsi

le s

tre

ngt

h (

fre

s), h

ere

is

assu

me

d

that

a/d

is 2

.5 a

nd

th

e c

om

pre

ssiv

e c

on

cre

te s

tre

ngt

h (

fcm

) is

50

N/m

m2 .

Seco

nd

, re

ad f

rom

rig

ht

to le

ft t

he

co

rre

ctio

n f

or

she

ar s

pan

to

eff

ect

ive

de

pth

rat

io (

a/d

). T

hir

d, r

ead

fro

m t

op

to

bo

tto

m t

he

co

rre

ctio

n f

or

com

pre

ssiv

e c

on

cre

te s

tre

ngt

h (

fcm

). F

rom

left

to

rig

ht

no

w t

he

ult

imat

e s

he

ar c

apa

city

(V

u)

can

be

re

ad. f

res

and

fcm

in N

/mm

2 .


Figu

re G

.3:

Dia

gram

s w

ith

th

e r

esu

lts

of

the

ext

en

de

d b

eam

mo

de

l usi

ng

the

ch

arac

teri

stic

te

nsi

le s

tre

ngt

h.

The

ult

imat

e s

he

ar c

apac

ity

(Vu

)

can

be

re

ad f

rom

th

is d

iagr

am in

th

ree

ste

ps.

Fir

st, s

tart

at

the

rig

ht

top

wit

h t

he

re

info

rce

me

nt

rati

o (

ρ)

and

re

sid

ual

te

nsi

le s

tre

ngt

h (

fre

s),

he

re i

s as

sum

ed

th

at a

/d is

2.5

an

d t

he

co

mp

ress

ive

co

ncr

ete

str

en

gth

(fc

m)

is 5

0 N

/mm

2 . Se

con

d, r

ead

fro

m r

igh

t to

left

th

e c

orr

ect

ion

fo

r

she

ar s

pan

to

eff

ect

ive

de

pth

rat

io (

a/d

). T

hir

d, r

ead

fro

m t

op

to

bo

tto

m t

he

co

rre

ctio

n f

or

com

pre

ssiv

e c

on

cre

te s

tre

ngt

h (

fcm

). F

rom

left

to

righ

t n

ow

th

e u

ltim

ate

sh

ear

cap

acit

y (V

u)

can

be

re

ad. f

res

and

fcm

in N

/mm

2 .

Documents

Eindhoven University of Technology MASTER Shear failure of … · Darwish, 1987). The shear capacity of SFR-RC is currently included in some design codes, like RILEM TC 162-TDF, and