post-cracking modelling of concrete beams reinforced with synthetic

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post-cracking modelling of concrete beams reinforced with synthetic
6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
POST-CRACKING MODELLING OF CONCRETE BEAMS
REINFORCED WITH SYNTHETIC FIBERS
Byung Hwan Oh, Ji Cheol Kim, Young Cheol Choi,Dae Gun Park
Dept. of Civil Engineering, Seoul National University, Korea
Abstract
The post-cracking behavior of newly-developed structural synthetic fiber reinforced
concrete beams is analyzed. The pullout behavior of single fiber is identified by tests and
employed in the model to predict the post-cracking behavior. A probabilistic approach is
used to calculate the effective number of fibers across crack faces and to calculate the
probability of non-pullout failure of fibers. The proposed theory is compared with test
data and shows good correlation. The proposed theory can be efficiently used to predict
the load-deflection behavior, moment-curvature relation, load-crack opening width
relation of synthetic fiber reinforced concrete beams.
1. Introduction
Most important application of fibers is to prevent or control the tensile cracking
occurring in concrete structures [1-11]. Recently, structurally-efficient synthetic fibers
have been developed by authors and coworkers [9]. These synthetic fibers have
advantages compared to steel or other fibers in that they are corrosion-resistant and
exhibit high energy-absorption capacity. The purpose of the present study is to explore
experimentally and theoretically the cracking resistance and post-cracking behavior of
newly-developed structural synthetic fiber reinforced concrete beams.
2. Models for post-cracking behavior
2.1 Concept of analysis
A fiber reinforced concrete beam as shown in Fig. 1 has been considered for the analysis
of post-cracking behavior. Fig. 1 shows the failure mode of a beam with the crack mouth
opening displacement. Fig. 2 shows the strain and stress distributions for cracked section
of fiber reinforced concrete (FRC) beam. In order to obtain the post-cracking behavior of
FRC beams, the stress-strain relations of concrete in compression and tension, and the
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6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
stress-crack width relation after cracking must be properly defined.
2.2 Stress-strain relation of concrete in compression
The following commonly-used compressive stress-strain equations were employed.
Hc
H
( c ) 2 ] for 0 d H c d H 0
H0 Ho
fc
f c ' [2
fc
f c ' [1 (1)
0.15
(H c H 0 )] for H 0 d H c d 0.003
0.004 H 0
(2)
in which fcː = compressive strength of concrete, H0 = the strain at peak stress. The force
C in compression zone can be written as follows.
C Ef c ' bc
(3)
where Eҏ= the factor for average stress, b = width of beam, and c = the depth of neutral
axis from the top face of a beam. The value of E can be obtained by equating the area
under actual stress-strain curve to the area under rectangular stress block as follows[10]
H cf
³ f dH
H cf
³
c
Df c ' H cf ; Then E
f c dH c
(4)
f c ' H cf
0
P/2
c
0
P/2
L
L/3
L/3
h=L/3
'0
P/2
P/2
c
dT
dG
dT
h
yi
wi
CMOD
Vc
f1 2T
f2
f3
f4
fi
fN
dT
c c'
force
axial strain of
top fiber(Hx)
displacement
x=L
Hcf
Vc = Vc(Hc)
stress
x=0
'c
H0
strain
L/3
fi = f(wi , Di , li)
dCMOD
dT
dT
Fig. 2 Schematic view of forces and stresses
acting on cracked section of FRC beam
The location of the compression force, Jc, from the top fiber can be obtained as
Fig. 1 Failure mode of FRC beam
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6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
H cf
³H
1
J
f dH c
c c
0
(5)
H cf
H cf
³ f dH
c
c
0
2.3 Stress-strain relation of concrete in tension
2.3.1 Post-cracking behavior of concrete after tensile strength
The cracking starts to occur right after the tensile stress reaches the tensile strength and
the tensile stress decreases as the crack width increases. The strain at first cracking, Hcr,
can be obtained as follows.G
fr
(6)
Ect
in which fr = tensile strength of concrete, Ect = elastic modulus before cracking.
The stresses after cracking depend on the widths of cracks. It is reasonably written here
based on the Gopalotratnam & Shah’s model [5].
H cr
V ct
e
fc
kwO
(7)
in which Vct = the tensile stresses after cracking, k = empirical constant = 60.8, w =
crack width(in mm), O= empirical constant = 1.01.
2.4 Calculation of sectional forces and deflections of FRC beams
The deformation of concrete beams without reinforcing bars is usually localized at
central position as shown in Fig. 1. The cracked portion at central location acts as a
plastic hinge. The displacement 'n at compression face can be obtained from the
compressive strain distribution Hx (see Fig. 1).G
'0
L
³ H x dx
0
H cf
2
L
(8)
The slope(rotation angle) of beam, dT, may also be obtained from Fig. 1.G
'0
(9)
2c
The deflection at central position is obtained from the slope of the beam as follows.
dT |
dG
dT
L
2
(10)
The crack mouth opening displacement (CMOD) at the bottom surface is also written as
dCMOD 2[dT (h c)]
(11)
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6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
The internal resisting moment, Me, of the beam can be derived from the stress
distributions of concrete in compression and tension zones and also the pullout forces of
all the fibers across the crack plane as shown in Fig. 2.
c
N
(12)
³ V c (bd y ) ¦ fi 0
i 1
0
N
³ V c bdy ˜ y ¦ ( fi yi )
0
i 1
c
Me
(13)
The pullout force fi of the fiber of ith layer in Equation 13 should be obtained from the
relation of bond stress and bond slip of a fiber and they directly depend on the crack
widths of the beam(see Equation 14).
ª CMOD º
(14)
« ( h c ) » yi
¼
¬
The fibers are randomly distributed at the crack plane and this effect must be considered
appropriately to calculate the fiber forces. Soroushian and Lee[11] proposed the number
of fibers per unit area, N1, as follows.
Vf
N1 D
(15)
Af
wi
where, Dҏ= fiber orientation factor, Vf = fiber content, and Af = cross-sectional area of
fibers.
If the boundary of the structure restrains the arbitrary orientation of the fibers, the
orientation factor for a specified direction becomes larger. Soroushian and Lee[11]
reported the orientation factor for two-side-restrained case as follows.
h/2
³ E1dy
D1
df /2
for h l f
h/2
(16)
lf /2
lf
D1
³ E1dy
df /2
hl f / 2
(17)
0.405(1 l f / h)
S /2 r
where E1
³ ³l
f
cos T cos\dTd\
0 r0
l f (S / 2)r
with r0 sin 1 ( d f / l f ) and r sin 1 (2 y / l f ) (18)
The tensile force(Fi) resisted by fibers at each layer of tensile zone of the beam can be
obtained by multiplying the fiber force fi of that layer by the number of fibers. The force
fi of single fiber can be derived from the bond stress-slip relation, which is also
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6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
dependent upon the crack width wi of ith layer(see Fig. 2). This bond stress-slip relation
of structural synthetic fiber will be directly obtained by tests, which will be described in
the following section. The development length Lt of a fiber required for not to be pulled
out at the crack plane may be derived as shown in Equation 19.
Lt
fi
(19)
60W u
in which 60 = perimeter of a fiber, and Wu = bond strength of a fiber. Therefore, the actual
embedment length of fibers should be larger than the required development length in
order not to be pulled out.
Fig. 3 summarizes the fiber forces, required anchorage lengths, and the probabilities of
non-anchorage failure for various layers of cracked section of a beam. The probability of
non-anchorage(or non-pullout) failure Pr can be derived as follows by two cases.
(1) Case 1 : Required anchorage length d the half of fiber length(Lf d lf/2)G
This is the case that the half of the actual fiber length is larger than the required
anchorage length. This can be written in a formalized equation as follows.
Pr
1
(h c) Lt / 2
(h c)l
f
/2
L
1
l
f
(20)
f
(2) Case 2 : Required anchorage length > the half of fiber length(Lf > lf/2)
This is the case that the required anchorage length is larger than the half of actual fiber
length. This can be expressed as the following equation.
(l
Pr
/ 2)k / 2
f
(h c)l / 2
f
l /2
1 f
2 Lt
l
1 f
4 Lt
(21)
in which k l f (h c) / Lt .
2
Fig. 3 Fiber force, required anchorage length and probability of non-anchorage failure
Finally, the tensile forces resisted by fibers can be obtained from the number of fibers at
each layer, effective orientation factor D, and the probability of non-anchorage failure Pr.
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6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
3. Tests for post-cracking behavior of synthetic FRC beams
3.1 Pullout test for single fiber
The authors developed recently new structural synthetic fibers which are of crimped type
with the length of 50mm. Fig. 4 shows the photograph for actual pullout test
arrangement.
Fig. 5 depicts the average pullout load versus slip relation obtained from the present tests.
The bond load-slip equation obtained from the tests may have the following form.
Fp
aS
(22)
b cS d
where a, b, c, and d are the constants to be obtained from test data, and Fp= pullout
load(kN), S = slip in mm unit.
Fig. 4 Photograph of pullout test
Fig. 5 Load-slip relation for crimped-type synthetic
fiber
3.2 Flexural tests for structural synthetic fiber reinforced concrete beams
The concrete beams reinforced with structural synthetic fibers have been tested to obtain
the flexural behavior including the load-deflection behavior, load-CMOD relations, and
moment-curvature relations. The water-cement ratio was 0.45 and the fiber content was
1 percent of total concrete volume. The dimension of the beam was 100u100u400mm
and the span length was 300mm. The load was applied in third point loading in
displacement control manner with the rate specified in the static testing standard.G
4. Analysis of test results and comparisons with theory
Fig. 6(a) shows the load-deflection curves obtained from the present tests for the beam
with structural synthetic fiber volume of 1 percent. Fig. 6(b) also compares the test data
with the theory proposed in the previous section. It can be seen that the theoretical
predictions fairly well agree with test data even after post-cracking ranges.
The salient feature of the post-cracking behavior of structural synthetic fiber reinforced
concrete beams is that the resisting load drops down right after first cracking, probably
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6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
due to initial slip of fibers at crack plane, and then starts to increase due to structurally
effective synthetic fibers in tensile region. Fig. 6(b) also exhibits the similar behavior of
structural synthetic fiber reinforced concrete beam for fiber volume of 1.5%. It is seen
again in Fig. 6(b) that the proposed theory agrees very well with the measured data, even
up to the large deflection of the beam.
20
20
Experimental (Vf = 1.5%)
Experimental (Vf = 1.0%)
16
Analytical (Vf = 1.0%)
Load (kN)
Load (kN)
16
12
8
4
Analytical (Vf = 1.5%)
12
8
4
0
0
0
0.5
1
1.5
2
0
2.5
0.5
1
1.5
2
2.5
Midpoint Deflection (mm)
Midpoint Deflection (mm)
(a) fiber content: Vf = 1.0% by volume
(b) fiber content: Vf = 1.5% by volume
Fig. 6 Load-deflection curve for synthetic FRC beams
Fig. 7 describes the load-CMOD relations for two different fiber volume contents. It is
noted here that, at the same loads after cracking, the FRC beam with larger volume of
fibers exhibits much smaller CMOD values. This is indeed a great beneficial effect of
structural synthetic fibers. Fig. 8 shows the relation between CMOD and central
deflection of FRC beam. These relations are almost similar for different fiber volume
contents. Therefore, this relation of CMOD versus central displacement may be regarded
as a material property for structural synthetic fiber reinforced concrete beam.
5
20
Vf = 1.0%
Vf = 1.0%
4
Vf = 1.5%
CMOD (mm)
Load (kN)
16
12
8
Vf = 1.5%
3
2
1
4
0
0
0
0.5
1
1.5
2
0
2.5
0.5
1
1.5
2
2.5
Midpoint Deflection (mm)
CMOD (mm)
Fig. 7 Effect of fiber content on loadCMOD relation
Fig. 8 Relation between CMOD and
deflection
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6th RILEM Symposium on Fibre-Reinforced Concretes (FRC) - BEFIB 2004
20-22 September 2004, Varenna, Italy
5. Conclusions
A realistic model for post-cracking behavior of structural synthetic fiber reinforced
concrete beams is developed in this study. The randomness of orientation of fibers and
the effective number of fibers at the crack plane were first considered. New concept of
the probability of non-pullout failure of fibers at the crack plane was then introduced and
derived in this study. The pullout tests for fibers were conducted and an appropriate
relation between bond forces and slips was derived in order to calculate the pullout
forces of structural synthetic fibers at the crack plane. The load-CMOD relation, CMODdeflection, and moment-curvature relation were also reasonably predicted by the
proposed theory. The present study also indicates that the relation between CMODs and
central displacements is almost same for the beams with different fiber volumes and,
therefore, this relation can be regarded as a material property for structural synthetic
fiber reinforced concrete members. The present study allows more realistic analysis and
application of recently-developed structural synthetic fiber reinforced concrete beams.
References
1. ACI Committee 544, ‘Design Considerations for Steel Fiber Reinforced Concrete’.
2. ASTM C 1018-89, ‘Standard Test Method for Flexural Toughness and First-Crack
Strength of Fiber Reinforced Concrete(Using Beam with Third-Point Loading)’,
Book of ASTM Standards, V. 04.02, ASTM, Philadelphia: (1991) 507-513.
3. Banthia, N. & Trottier, J., ‘Concrete Reinforced with Deformed Steel Fibers, Part I :
Bond-Slip Mechanism’, ACI Material Journal, 91 (5) (1994) 435-446.
4.
Ezeldin, A.S. & Balaguru, PN., ‘Normal and High Strength Fiber Reinforced
Concrete under Compression’, Journal of Materials in Civil Engineering, ASCE,
4(4) (1992) 415-427
5. Gopalaratnam, V.S. & Shah, S.P., ‘Softening Response of Plain Concrete in Direct
Tension’, ACI Journal, Proceeding 82 (3) (1985) 310-323
6. Leung, C., Geng, Y.P., ‘Micromechanical Modeling of Softening Behavior In Steel
Fiber Reinforced Cementitious Composites’, Int. J. Solids Structures, 35 (31-32)
(1998) 4205-4222.
7. Li, V.C., et al., ‘Influence of Fiber Bridging on Structural Size-effect’, Int. J. Solids
Structures, 35 (1998) 4223-4238
8. Oh, B.H., ‘Flexural analysis of reinforced concrete beams containing steel fibers’,
Journal of Structural Engineering, ASCE, 118(10) (1992) 2691-8.
9. Oh, B.H., ‘Cracking and flexural behavior of structural synthetic fiber reinforced
concrete beams’, KCI Journal, 14 (6) (2002) 900-909
10. Park, R. & Paulay. T., ‘Reinforced concrete structures’, John Wiley and Sons, Inc.,
New York, N.Y. (1975)
11. Soroushian, P. & Lee, C., ‘Distribution and Orientation of Fibers in Steel Fiber
Reinforced Concrete’, ACI Material Journal, 87 (5) (1990) 433-439
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