14osios Lietuvos jaunųjų mokslininkų konferencijos
„Mokslas – Lietuvos ateitis“
2011 metų temins konferencijos straipsnių rinkinys
ISSN 20297149 online
STATYBA
ISBN 9789955289296
© Vilniaus Gedimino technikos universitetas
http://dspace.vgtu.lt 1
DISCRETE CRACK MODEL OF STEEL FIBRE REINFORCED CONCRETE MEMBERS
SUBJECTED TO TENSION
Adas Meskenas
1
,
Darius Ulbinas
2
Vilnius Gediminas technical university, Sauletekis al. 11
1
adasmes@gmail.com;
2
Darius.Ulbinas@vgtu.lt
Abstract. This paper presents discrete crack model analysis of steel fibre reinforced concrete tensile members. The crack model uses
residual stresses of stressstrain relationships obtained from two different models: simplified model, proposed in the literature and the
Layer section model. The latter model is based on the stressstrain relationships obtained from the test data of steel fibre reinforced
concrete (SFRC) beams by the inverse technique. Crack width and average strain were investigated performing a comparative
analysis of different prediction methodologies, significance of influence of residual stresses to the results is shown.
Keywords: discrete crack, steel fibre, tensile members, crack model, simplified model, Layer section model, inverse technique, crack
width.
Introduction
In last decades, fibre reinforcement is widely used in
many countries as additive for concrete and cement
mortar mixture for production of structures. Fibre
reinforced concrete is being increasingly used in
pavements, offshore platforms, water retaining
structures, bridges, tunnels, etc. (Reinhardt and Naaman
1999; Rossi and Chanwillard 2000; König et al. 2002).
The effectiveness of steel fibres in enhancing the
mechanical properties, in particular toughness, of a quasi
brittle material as concrete, is primarily controlled by the
processes of transferring stresses from the matrix to the
fibre and by the bridging effect of the fibres across the
cracks (Bentur and Mindess 1998). The addition of steel
fibres to conventional reinforced concrete beams greatly
improves the cracking strength, restricts the growth of
cracks, and reduces the tensile strains in steel
reinforcement bars, thereby resulting in smaller crack
widths (Yang et al., 2009).
Discrete crack model was introduced by D. Ngo.
A.C. Scordel and Y.R. Raschid (1967). This model shows
the real behaviour of element, when with increasing
loading all discrete cracks, opening in construction, are
taken in consideration. Discrete crack model is used for
calculation of crack widths and evaluation of strain and
stress distribution between the cracks.
In this paper, calculations of the crack width and
average strain were investigated performing a
comparative analysis of different methodologies. In the
discrete crack model residual stresses of stressstrain
relationships obtained from two different models 
simplified model, proposed in the literature and the Layer
section model – are used. The latter model is based on the
stressstrain relationships obtained from the test data of
steel fibre reinforced concrete (SFRC) beams using the
inverse technique.
Simplified model
Campione (2008) proposed the Simplified model for the
calculation of the loaddeflection curves of simply
supported reinforced concrete beams in the presence of
fibre and transverse stirrups under flexure and shear. In
the case of flexural failure, under the hypothesis of a
perfect bond of steel bars and concrete, the plane section
theory is considered, including the strength contribution
due to the residual tensile strength of fibrous concrete. In
this section approaches for calculations of residual
stresses used in Simplified model by
Campione (2008) are
presented.
In Simplified model, for the tensile response of
fibrous concrete simple trilinear stressstrain curve is
considered. It is suggested by Mansur et al. (1981) and
considers a stressstrain relationship constituted by three
branches, as shown in Fig. 1.
Fig. 1. Stressstrain relationship for fiber reinforced concrete in
direct tension
2
The whole expression of the relationship is presented
below:
( )
( )
,
,1
,
t ctf t ctf
t ctf
t ctf r r ctf t ctu
ctu t
r t ctu
E
f f f
f
ε ε ε
ε ε
σ ε ε ε
ε ε
ε ε
⋅ <
−
= − ⋅ + < <
−
>
here
ctf
f
,
ctf
ε
 the maximum tensile strength and
corresponding strain, respectively;
r
f
,
ctu
ε
 residual
strength and corresponding tensile strain, respectively;
ctf
E
 elastic modulus in tension.
For the calculation of the residual stress, as pointed
out by Campione (2008), its value depends on the
effective content of fibres in the composite, and that it
also depends on several factors, such as the orientation of
fibres in the composite, on the expected pullout length
ratio, and the group reduction factor associated with the
number of fibres pulling out per unit area. To take these
phenomena into account, a simple analytical expression
for residual stress, modified by Naaman (2003), Valle and
Buyukozturk (1993) is proposed:
( )
'
0.2 2
r c
f f F= ⋅ ⋅
here
'
c
f
 compressive strength of plain concrete;
F
 the
fiber factor, taking into account the effect the fiber
characteristics and calculated as:
( )
3
f
f
f
L
F V
d
β= ⋅ ⋅
here
f
L
– length of fibre;
f
d
 equivalent diameter;
f
V

volume percentage;
β
 a bond factor assumed according
to Campione et al. (2006) equal to 1.0 for hooked fibers.
This equation already tested in Campione et al. (2006), is
continuously assumed in the Simplified model.
Layer Section model
For determination of average stressstrain (
σ ε
−
)
relation for concrete in tension from experimental
momentcurvature diagrams of flexural SFRC members
the Inverse analysis can be imposed.
The applied technique is based on the following
approaches and assumptions on the behaviour of flexural
reinforced concrete members subjected to shortterm load
(Kaklauskas, Ghaboussi 2001):
1. The constitutive model is based on a smeared crack
approach, i.e. average stresses and strains are used.
2. The Bernulli Hypothesis of beam bending is adopted
implying a linear distribution of strain within the
depth of the beam section.
3. Perfect bond between reinforcement and concrete is
assumed. Reinforcement slippage occurring at
advanced stressstrain states is included into
σ ε
−
diagram of tensile concrete.
4. All fibres in the tensile concrete zone follow a uniform
stressstrain law.
The Layer Section model, proposed by G.
Kaklauskas and J. Ghaboussi (2001), is employed for
computation of the internal forces in the crosssection.
Analysis is performed with incrementally increasing
bending moment. The two equilibrium equations of
internal forces and bending moments are solved for each
loading stage yielding a solution for the coordinate of
neutral axis and the concrete stress in the extreme tension
fibre. Since the extreme fibre has the largest strain, other
tension fibres of concrete have smaller strains falling
within the portion of the stressstrain diagram, which had
already been determined. However, there are many layers
in both the tensile and compressive concrete zones and
the variation in stresses in each of these zones is not
known. The assumption that the same stressstrain
relations apply to all layers allows the reduction of the
number of unknowns to one for each of the tensile and
compressive concrete zones.
Computation is performed for incrementally
increasing load. During the first load stage, tensile
stresses corresponding to the strains in the extreme layers
are computed. These stresses are then used in the
equilibrium equations for the second load stage when new
stresses corresponding to larger extreme layer strains are
determined. In this way, stressstrain curves for the tensile
concrete are progressively obtained from all previous
stages and used in the next load stage.
The discussed method is illustrated in Fig. 2. It is
assumed that the “experimental” momentaverage strain
curves for extreme fibres of the tensile concrete are given,
as in Fig. 2 (a). Circled points in these curves correspond
to the experimental data. Here the momentstrain curves
were numerically smoothed, resulting in the solid lines,
with on purpose to avoid slope discontinuities in the
momentstrain curves, which can lead to jumps in the
computed material stressstrain diagrams. Computation is
performed for n load increments. In order to avoid
oscillations in the computed material stressstrain curves,
n has to be sufficiently large. For most practical cases,
n ≥ 50 is sufficient.
Discrete crack model
Discrete crack model shows the real behaviour of the
concrete element, when with increasing loading all
discrete cracks, opening in construction, are evaluated.
Considering SFRC element in uniaxial tension a
schematic load displacement relationship according to this
model is plotted in Fig.3 (b), in which can be founded that
the response of a SFRC element can be subdivided into
four stages:
1. The first stage represents elastic deformations of the
member, the concrete tensile stress increases with load up
to start of cracking.
3
2. The second stage covers the region between the first
and the final primary cracks. When the stress in the
concrete first reaches the tensile strength at a particular
section, cracking occurs at the applied load P
cr
.
First
cracking occurs at the weakest crosssection and this is
usually assumed to occur when the concrete tensile stress
reaches the lower characteristic value of the direct tensile
strength,
'
t
f
(Wu and Gilbert 2008). A relatively small
increase in load is causing a second crack to develop at a
cross section. At the end of this stage the SFRC member
becomes separated by the developed cracks into a number
of concrete blocks.
Fig. 2. a) Experimental momentcurvature diagram; b) computed stress  strain relationship for concrete in tension
3. After the primary crack pattern is established, further
increase in load results in further slip at the concretesteel
interface causing covercontrolled cracks to develop
between the primary cracks and a gradual breaking down
of the bond between the steel and the concrete,
collectively reducing tension stiffening ( Wu and Gilbert
2008). Such behavior of the member corresponds to the
third stage.
4. The fourth stage starts with yielding of reinforcement at
the applied load P
u
.
The model is based on the following assumptions:
1. Linearelastic properties were assumed for
reinforcement and concrete;
2. All cracks form at the cracking load dividing the beam
into a number of blocks;
3. For deformation and average crack width analysis the
crack spacing l
cr
is taken 1.5l
tr
, whereas 2l
tr
is used for
calculation of maximum crack width (Yang et al., 2009).
4. The bond stresses τ
a
, as reported by Fehling and König
(1988), Marti et al. (1998), Alvarez (1998) for the case of
simplicity is assumed τ
a
= 2f
ct
.
When cracking occurs, distribution of stresses in
fibre concrete and reinforcement changes with the length
of element, as shown in Fig. 3(c). Discrete crack model
evaluates that maximal stress in steel reinforcement
appears in the cracked cross section. FRC can exhibit
significant amounts of postcracking tensile resistance at a
crack, depending on the type and dosage of fibers used.
This results in reduced crack spacing, as shown in Fig.
3(a). The concrete tensile stress gradually increases as the
distance from the nearest crack increases, due to bond
stress that develops at the steelconcrete interface,
reaching a maximum midway between the two cracks.
cr
l
cr
l
cr
l
0
tr
l
N
N
(
)
s
N x
(
)
cf
N x
f
N
5
Calculation results
In this section calculation results obtained by Layer
section model and Simplified model are presented. After
performing the Inverse analysis stressstrain (
σ ε
−
)
relations for SFRC in tension from experimental moment
curvature diagrams of flexural SFRC members are
obtained. Acquired graphs are approximated with three
lines, as shown in Fig. 6.
6
Comparative analysis of loadstrain relationships
employing residual stresses obtained by two different
techniques is done. Results of calculated members S0, S
05, S10, S15 are presented in Figs. 7 – 9.
Fig. 7. Load–strain relationships of member S05 Fig. 8. Load–strain relationships of member S10
Fig. 9. Load–strain relationships of member S15
0
0,01
0,02
0,03
0,04
0,05
0,06
0 0,1 0,2 0,3 0,4 0,5
Load [MN]
Crack width [mm]
S0
S05 (Simplified model)
S05 (Layer section model)
0
0,01
0,02
0,03
0,04
0,05
0,06
0 0,1 0,2 0,3 0,4 0,5
Load [MN]
Crack width [mm]
S0
S10 (Simplified model)
S10 (Layer section model)
Fig. 10. Load – crack width relationships of member S05 Fig. 11. Load – crack width relationships of member S10
0
0,01
0,02
0,03
0,04
0,05
0,06
0 0,1 0,2 0,3 0,4 0,5
Load [MN]
Crack width [mm]
S0
S15 (Simplified model)
S15 (Layer section model)
Fig. 12. Load – crack width relationships of member S15
7
In Figs. 10–12 graphs of load–crack width relationships
are presented. As it can be seen, values of crack widths
significantly depend on the chosen calculation technique
of residual stresses. It can lead to the difference up to
330%, depending on the volume of fibres and giving
smaller value of crack width by using residual stresses
calculated by Layer section model. Results of average
strain can differ up to 280% respectively, as presented in
Fig. 7 – 9.
Conclusions
1. Layer section model and the Simplified model,
proposed in the literature, were employed for the
calculation analysis of residual stresses.
2. Application of Layer section model led to the
greater values of residual stresses varying up to 3 times
depending on the volume of fibres.
3. It is shown that residual stress is one of the
significant factors for the prediction of crack width and
average strain results by the Discrete crack model. The
difference up to 330% for the crack widths and up to
280% for average strain is obtained depending on the
volume of fibres and method of calculation of residual
stresses.
4. The scope of the discussed technique should be
further investigated considering the future experimental
results performed in VGTU.
References
ACI Committee. (2002). Building code requirements for
reinforced concrete. ACI No. 31802, American Concrete
Institute, Detroit.
Campione G. 2008. Simplified Flexural Response of Steel
FiberReinforced Concrete Beams, Journal of Materials in
Civil Engineering, 20(4).
Eligehausen, R.; Popov, E. P.; Bertero, V. V. 1983. Local bond
slip relationship of deformed bars under generalized
excitations, Earthquake Engineering Research Center,
Report UCB/EERC: 8319.
Kaklauskas, G.; Holschemacher, K.; Gribniak, V.; Bacinskas,
D.; Sokolov, A. 2009. A stressstrain constitutive
relationship of steel fiber reinforced concrete. Final Report
No. T101/09. Vilnius Gediminas Technical University. 57
p. (in Lithuanian).
Kaklauskas, G.; Ghaboussi, J. 2001. Stressstrain relations for
cracked tensile concrete from RC beam tests, ASCE Journal
of Structural Engineering 127(1): 64–73.
Lofgren, I. 2005. Fibrereinforced Concrete for Industrial
Construction  a fracture mechanics approach to material
testing and structural analysis. Chalmers University of
Technology. Göteborg, 2005 ISBN 9172916966.
Naaman, A. E. 2003. Engineered Steel Fibers with Optimal
Properties for Reinforcement of Cement Composites.
Journal of Advanced Concrete Technology, 1(3): 241252.
Nataraja, M. C.; Dhang, M.; Gupta, A. P. 1999. Stress strain
curve for steelfiber reinforced concrete under compression,
Cement & Concrete Composites 4(1): 383390.
Salys, D.; Kaklauskas, G.; Gribniak, V. 2009. Modelling
deformation behaviour of RC beams attributing tension
stiffening to tensile reinforcement, Statybines
konstrukcijos ir tehnologijos 1(3): 141147. (in Lithuanian)
Wu, H. Q.; Gilbert, R. I. 2008. An Experimental Study of
Tension Stiffening in Reinforced Concrete Members under
ShortTerm and LongTerm Loads. UNICIV Report No. R
449. Sydney: The University of South Wales. 32 p.
Wu, H. Q.; Gilbert, R. I. 2009. Modeling shortterm tension
stiffening in reinforced concrete prisms using continuum
based finite element model, Eng Struct 31: 2380–2391.
Yang, Y.; Walraven, J. C.; den Uijl, J. A. 2009. Combined effect
of fibers and steel rebars in high performance concrete,
HERON Vol. 54 (2/3): 205.
DISKREČIŲJŲ PLYŠIŲ MODELIO TAIKYMAS PLIENO
PLUOŠTU ARMUOTIEMS TEMPIAMIESIAMS
GELŽBETONINIAMS ELEMENTAMS
A. Meskenas,
D. Ulbinas
Santrauka
Šiame straipsnyje aptariamas tempiamų plieno pluoštu armuotų
gelžbetoninių elementų diskretusis pleišjimo modelis.
Plyšio pločio ir deformacijų skačiavimams reikšmingą įtaką turi
naudojami liekamųjų įtempių rezultatai, gauti pasitelkus du
skirtingus skaičiavimo modelius: supaprastintą modelį, pasiūlytą
Campione (2008) ir sluoksnių modelį. Pastarasis modelis yra
pagrįstas įtempiųdeformacijų priklausomybe gauta iš plieno
pluoštu armuotų lenkiamųjų elementų eksperimentinių
duomenų, sprendžiant atvirkštinį uždavinį. Straipsnyje atlikta
palyginamoji plyšio pločio ir vidutinių deformacijų skaičiavimų
analiz taikant diskrečiųjų plyšių modelį.
Reikšminiai žodžiai: diskretusis modelis, plieno pluoštas,
tempiamas elementas, sluoksnių modelis, atvirkštinis uždavinys,
plyšio plotis.
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