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Eindhoven University of Technology
MASTER
Shear failure of reinforced concrete beams with steel fibre reinforcement
Krings, H.
Award date:2014
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Shear failure of reinforced concrete beams with steel fibre reinforcement
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
Shear failure of reinforced concrete beams with steel fibre reinforcement
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
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.
8 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
10 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
12 Shear failure of reinforced concrete beams with steel fibre 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
14 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
‘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.’
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.
16 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
18 Shear failure of reinforced concrete beams with steel fibre reinforcement
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:
20 Shear failure of reinforced concrete beams with steel fibre reinforcement
- 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
22 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
24 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
26 Shear failure of reinforced concrete beams with steel fibre reinforcement
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).
28 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
30 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
32 Shear failure of reinforced concrete beams with steel fibre reinforcement
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).
34 Shear failure of reinforced concrete beams with steel fibre reinforcement
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 )
36 Shear failure of reinforced concrete beams with steel fibre reinforcement
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%
38 Shear failure of reinforced concrete beams with steel fibre reinforcement
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:
‘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.’
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.
40 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
41 Extended beam model
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.
42 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
43 Extended beam model
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.
44 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
45 Extended beam model
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.
46 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
47 Extended beam model
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.
48 Shear failure of reinforced concrete beams with steel fibre 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 -
50 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
52 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
54 Shear failure of reinforced concrete beams with steel fibre reinforcement
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
.
56 Shear failure of reinforced concrete beams with steel fibre reinforcement
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).
58 Shear failure of reinforced concrete beams with steel fibre reinforcement
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).
60 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
62 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
64 Shear failure of reinforced concrete beams with steel fibre reinforcement
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)
Dimensions [mm] Material properties [N/mm2]
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
66 Shear failure of reinforced concrete beams with steel fibre reinforcement
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).
Dimensions [mm] Material properties [N/mm2]
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.
68 Shear failure of reinforced concrete beams with steel fibre reinforcement
69 Conclusion and discussion
5 Conclusion and discussion
5.1 Conclusion
Subsequent to the literature study, the main hypothesis of this research is formulated:
‘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.’
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.
70 Shear failure of reinforced concrete beams with steel fibre reinforcement
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.
72 Shear failure of reinforced concrete beams with steel fibre reinforcement
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)
76 Shear failure of reinforced concrete with steel fibre reinforcement
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.
78 Shear failure of reinforced concrete with steel fibre reinforcement
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).
80 Shear failure of reinforced concrete with steel fibre reinforcement
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.
82 Shear failure of reinforced concrete with steel fibre reinforcement
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
84 Shear failure of reinforced concrete with steel fibre reinforcement
Second generation cracks
85 Appendix D: Flow charts additional modules beam model
D.2 Flow chart shear module
86 Shear failure of reinforced concrete with steel fibre reinforcement
D.3 Flow chart load module
87 Appendix D: Flow charts additional modules beam model
D.4 Flow chart load-deflection module
Load-deflection
diagram
88 Shear failure of reinforced concrete with steel fibre reinforcement
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.
90 Shear failure of reinforced concrete with steel fibre reinforcement
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).
91 Appendix E: Material properties and calculation settings
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.
92 Shear failure of reinforced concrete with steel fibre reinforcement
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 )
93 Appendix E: Material properties and calculation settings
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.
94 Shear failure of reinforced concrete with steel fibre reinforcement
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).
95 Appendix E: Material properties and calculation settings
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.
96 Shear failure of reinforced concrete with steel fibre reinforcement
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.
97 Appendix E: Material properties and calculation settings
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).
98 Shear failure of reinforced concrete with steel fibre reinforcement
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.3: SFRC with a fibre volume ratio Vf of 1.0%. 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.4: SFRC with a fibre volume ratio Vf of 1.5%. 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.
100 Shear failure of reinforced concrete with steel fibre reinforcement
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
Figure F.7: SFRC with a fibre volume ratio Vf of 1.0%. 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.8: SFRC with a fibre volume ratio Vf of 1.5%. 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.
102 Shear failure of reinforced concrete with steel fibre reinforcement
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 .
104 Shear failure of reinforced concrete with steel fibre reinforcement
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 .