Composites in Construction 2005 – Third International Conference
Lyon, France, July 11 – 13, 2005
FINITE ELEMENT MODELING OF REINFORCED CONCRETE BEAMS STRENGTHENED WITH
CARBON FIBER COMPOSITES
C. M. Paliga and A. Campos Filho
Graduate Program in Civil Engineering, Federal University of Rio Grande do Sul
Osvaldo Aranha Avenue, Porto Alegre, Brazil
[email protected] [email protected]
M. V. Real
Graduate Course in Ocean Engineering, Federal University of Rio Grande
Itália Avenue, Rio Grande, Brazil
[email protected]
R. H. F. Souza
Department of Civil Engineering, Federal Fluminense University
Miguel de Frias Street, Niterói, Brazil
[email protected]
J. A. S. Appleton
Department of Civil Engineering, Technical University of Lisbon
Rovisco Pais Avenue, Lisbon, Portugal
[email protected]
ABSTRACT: This paper presents the results of a comparison between a finite element model and
experimental data of reinforced concrete beams strengthened with carbon fiber reinforced plastics.
For the experimental program, simply supported beams, with 12 x 20 cm cross section and 2.25 m
long span were tested. These beams have been submitted to short-term static loading tests from
which it has been possible to evaluate the stresses and strains, the displacements, the crack
widths and the failure mode. Strains gauges glued to the composite surface allowed the analysis of
the behavior at the interface. The tensile and shear stresses at interface could be estimated. The
numerical model consisted of a nonlinear finite element model for the flexure-shear response. The
concrete was represented through plane stress isoparametric, eight nodes, finite elements. The
concrete two-dimensional constitutive law was based on the orthotropic model proposed by
Darwin. The concept of uniaxial equivalent strain and the two-dimensional failure criterion of Kupfer
and Gerstle were adopted. The reinforcement was represented through an embedded model. Each
steel bar was considered as a more rigid line inside of the concrete element, which just resists to
axial efforts. The model includes a special interface element to simulate the bond between
concrete and the external composite plate which is represented by truss elements.
1.
INTRODUCTION
Every day more concrete structures have been presenting the need of rehabilitation and
repair due to overload problems, use change, cracking, reinforcement bars corrosion, concrete
deterioration, etc. Then, retrofit and repair systems are searched as a necessary alternative.
Various materials and techniques are today available for the repair design and it is up to the
engineer to choose the most appropriated for the case. The method of strengthening concrete
structures with externally epoxy-bonded carbon fiber reinforced plastics (CFRP) is having an
increasingly utilization. It offers the advantages of high strength, light weight, immunity to corrosion
and efficiency of application. In this study, the Freyssinet System of CFRP for structural repair is
evaluated. This is a composite made of a carbon fiber woven fabric impregnated with an epoxy
resin. The woven fabric is bidirectional, with 70% of the carbon fibers in the principal direction and
30% in the transversal direction. The numerically obtained results by a finite element model are
compared with the experimental data from tests carried out by Souza and Appleton [1, 2]. The
1
reinforced concrete beams - F1 and F5 - were tested with concentrated loads, accordingly the
scheme presented in Fig. 1. The F2 beam was tested in such a way to simulate uniformly
distributed loads, accordingly the scheme presented in Fig 2.
Figure 1 – Tests program scheme: F1 – F5 beams
Figure 2 – Tests program scheme: F2 beam
The reinforced concrete beams have a 12 x 20 cm cross section and a span 2.25 m long. Figure 3
shows the beams details. The beams characteristics and the materials properties are specified in
Tables 1 and 2, respectively.
2Ø6mm
P/2
70
70
20
70
P/2
2Ø8mm
7,5
18
12
7,5
210
dimensions in cm
10
23Ø6mm c/10cm - 63cm
Figure 3 – Tested beams details
Table 1 – Tested beams characteristics
Beam
Applied CFRP composite
F1
one composite layer at the bottom face
F2
one composite layer at the bottom face
F5
two composite layers at the bottom face
Loading type
concentrated loads at the thirds
uniformly distributed load
concentrated loads at the thirds
According to the Freyssinet manufacturer's indications [3], the physical and mechanical properties
of the carbon fiber composite are:
o Composite thickness: 0.43 mm;
o Composite width: 75 mm;
o Ultimate bond stress: 1.50 MPa;
o Epoxy resin tensile strength: 29.30 MPa;
o Tensile rupture strain: 1.37%;
o Tensile rupture stress: 1400 MPa;
o Tensile Young’s modulus: 105,000 MPa.
2
The mechanical characteristics of the materials were tested according to the normalized standards.
The obtained values of these tests are presented at Table 2.
Table 2 – Materials properties
Material
Property
Description
fcm = 33 MPa
Mean compressive strength
Concrete
fctm = 3.10 MPa Mean tensile strength
Ecm = 30,500 MPa Mean Young’s modulus
fsy = 486 MPa
Yielding stress
8 mm bar
Es = 210,000 MPa Mean Young’s modulus
Steel
f = 557 MPa
Yielding stress
6mm bar sy
Es = 210,000 MPa Mean Young’s modulus
fu
Composite
2.
= 1425 MPa
Tensile rupture stress
= 0.66 %
Tensile rupture strain
Ef = 138,000 MPa Tensile Young’s modulus
fu
FINITE ELEMENT MODEL
The concrete is represented by two-dimensional, eight node, isoparametric, plane stress finite
elements in the theoretical analysis. The concrete biaxial constitutive model involves the uniaxial
equivalent strain concept of Darwin [4] and the biaxial failure criteria for concrete of Kupfer and
Gerstle [5]. The model for the tensile behavior of concrete includes a descending stress-strain
curve branch after cracking to incorporate the tension-stiffening effect.
The steel reinforcement is represented by an embedded model, based on the work of Elwi and
Hrudey [6]. Each steel bar is considered as a more rigid material line inside the concrete element
which resists only the axial force in the bar direction. Perfect adherence is assumed between the
steel reinforcement and the concrete that involves it. In this way, the steel reinforcement stiffness
matrix has the same dimensions as the concrete element stiffness matrix. The steel stress-strain
relationship is bilinear.
A three-node truss element is used to model the CFRP composite. Due to efforts transference
between the composite and the concrete, bond stresses appear at the interface of these two
materials. These bond stresses may cause the premature debonding of the composite leading to
the structure failure. A six node interface element, with quadratic shape functions, is employed for
the determination of the bond stresses. This interface element is based on the formulation
presented by Adhikary and Mutsuyoshi [7], according to Fig. 4.
6
5
4
0
1
3
2
Gauss point
Figure 4 – Interface finite element
The slip “s” between the composite and concrete can be evaluated using the following equation
s = N1 ( u6
u1 ) + N2 ( u5
3
u2 ) + N3 ( u4
u3 ) ,
(1)
where Ni( ) are the shape functions given in reference [7] and ui are the nodal displacements of
the interface element in the horizontal direction.
The bond stress ( ) between concrete and the CFRP composite can be evaluated as a function of
the relative slip “s” according to the CEB-FIP 1990 Model Code [8] bond-slip law, which is given by
the following equations
= max
s
s1
O
for 0
for s1 < s
= max
= max –
(
max
- f)
s
s1
(2)
s2
(3)
s s2
s3 s2
for s2 < s
s3
(4)
for s3 < s
= f
(5)
The adopted parameters for the evaluation of the bond stress between the concrete and the CFRP
composite are given in Table 3, according Silva [9] and Aurich [10].
Table 3 – Parameters for the bond stress evaluation
S1 (mm) S2 (mm) S3 (mm)
0.08
3.
0.08
0.65
max
0.6
(MPa)
f
(MPa)
0.1
1.5
max
FINITE ELEMENT DISCRETIZATION
All the analyses were performed using a mesh of 6x2 =12 elements for concrete, 6 elements for
the interface and 12 truss elements for the CFRP composite. Due to geometry and loading
symmetry only half of the beam was analyzed. The mesh employed in the analyses is showed in
Fig. 5. The loading showed in this figure is for F1 and F5 beams, while in the F2 beam, the loading
is uniformly distributed.
P/2
8
5
4
3
2
13
16
2
12
4
7
11
15
1
10
3
1
24
21
29
32
RC element
6
28
8
36
19
23
27
31
35
39
18
5
26
7
34
9
20
10
6
9
14
17
22
25
30
33
38
6
9
14
17
22
25
30
33
38
48
45
40
37
44
43
53
12
52
47
51
50
11
42
49
46
41
Interface element
1
13
54
54
19
14
55
56
57
55 20
56 21
57
15
59
58
22
58
16
23
59
24
60
61
60 25
61
17
62
26
62
CFRP element
4
18
63
27
63
49
46
41
28
64
66
65
64
29
65
30
66
Figure 5 – Finite elements mesh
4.
RESULTS AND DISCUSSIONS
In the tests, it was observed a sudden failure of the beams when the composite suffered
debonding and/or rupture. The composite debonding or rupture occurred always under a bending
crack with a large opening.
The numerical and experimental results for beam F1 are compared in Figs. 6 and 7. Figure 6
presents the evolution of the midspan deflection with the increasing load, while Fig. 7 shows the
composite strains at the same section. A loading-deflection curve corresponding to the hypothesis
of perfect bond between the composite and the concrete beam was added to Fig. 6. Naturally,
when perfect bond was assumed, it was obtained a greater stiffness than when the slip between
the composite and concrete was considered.
55
50
50
45
45
40
40
35
load P (kN)
load P (k N)
35
30
25
20
25
20
15
15
10
10
Experimental
Numerical - Perfect bond
Numerical
5
0
30
0
0.25
0.5
0.75
1
1.25
1.5
1.75
midspan displacement (cm)
2
2.25
E xperimental
Numerical
5
0
2.5
0
0.1
Figure 6 – Load - deflection curve: beam F1
0.2
0.3
0.4
0.5
0.6
0.7
compos ite's strains (% )
0.8
0.9
1
Figure 7 – Composite strains: beam F1
The numerically obtained bond stresses between the CFRP composite and the concrete beam are
show in Fig. 8, for various load levels. The bond stress peaks occurred in a zone of intense
cracking. It was observed experimentally that the composite debonding took place in this same
zone. The bond stresses obtained in the numerical analysis are compared with those achieved in
the tests in Fig. 9, for two load levels.
1.75
1.2
P
P
P
P
P
1.5
=
=
=
=
=
10kN
20kN
30kN
40kN
46,5kN
Numerical - P=30kN
Experimental - P=34,2kN
Numerical - P=38kN
1
1.25
bond stress (M P a)
bond s tress (M Pa)
0.8
1
0.75
0.6
0.4
0.5
0.2
0.25
0
0
10
20
30
40
50
60
70
distance from support (cm)
80
90
0
100
Figure 8 – Bond stresses: beam F1
0
10
20
30
40
50
60
70
distance from support (cm)
80
90
100
Figure 9 – Bond stresses along the
composite: beam F1
5
110
0
0
Experimental - P=39.9kN
Numerical - P=40kN
0.1
0.1
c om pos ite´s s trains (% )
c ompos ite´s strains (% )
0.2
0.3
0.4
0.5
0.15
0.2
0.25
0.3
0.6
0.35
0.7
0.4
0.8
Experimental - q=23.12kN/m
Numerical - q=23kN/m
0.05
0.45
0
10
20
30
40
50
60
70
distance from support (cm)
80
90
100
110
0
Figure 10 – Strains evolution along the
composite: beam F1
10
20
30
40
50
60
70
distance from support (cm)
80
90
100
110
Figure 11 – Strains evolution along the
composite: beam F2
The Figs. 10 and 11 present a comparison between experimental and numerical strains in the
composite along F1 and F2 beams for one load level.
35
35
32.5
32.5
30
30
27.5
27.5
25
25
22.5
22.5
load q (kN/m )
load q (k N/m )
The numerical and experimental results of the greatest values of the deflections and strains at the
composite for beam F2 are shown in Figs. 12 and 13, respectively. It must be noted that beam F2
is identical to beam F1; the only difference is that beam F2 has a uniformly distributed load.
20
17.5
15
12.5
20
17.5
15
12.5
10
10
7.5
7.5
Experimental
Numerical - Perfect bond
Numerical
5
2.5
0
0
0.25
0.5
0.75
1
1.25 1.5 1.75
2
midspan displacement (cm)
2.25
2.5
2.75
5
Experimental
Numerical
2.5
0
3
0
0.1
0.2
0.3
0.4
0.5 0.6 0.7 0.8 0.9
composite's strains (%)
1
1.1
1.2
1.3
1.4
Figure 13 – Composite strains: beam F2
Figure 12 – Load - deflection curve: beam F2
The calculated bond stresses at the interface between the concrete beam and the CFRP
composite for various load levels are given in Fig. 14. It was again observed that the bond stresses
peaks occurred in a zone of intense cracking.
Figure 15 compares the numerical and experimental values of the bond stresses for different load
levels.
6
1.75
1
q=6kN/m
q=12kN/m
q=18kN/m
q=24kN/m
q=30kN/m
q=32,75kN/m
1.5
0.8
0.7
bond s tress (M Pa)
bond stress (M Pa)
1.25
Numerical - q=20kN/m
Experimental - q=23,2kN/m
Numerical - q=26kN/m
0.9
1
0.75
0.6
0.5
0.4
0.3
0.5
0.2
0.25
0.1
0
0
10
20
30
40
50
60
70
distance from support (cm)
80
90
0
100
0
Figure 14 – Bond stresses: beam F2
10
20
30
40
50
60
70
distance from support (cm)
80
90
100
110
Figure 15 – Bond stresses along the
composite: beam F2
The loading scheme of beam F5 was identical to that of beam F1. The only difference between the
two beams was the number of composite layers. The greatest values of displacements and strains
in the composite and the bond stresses are shown in Figs. 16 to 19. These results are similar to
those observed in the aforementioned beams.
80
75
70
70
65
60
60
55
50
45
load P (kN)
load P (kN)
50
40
35
30
40
30
25
20
20
15
Experimental
Numerical - Perfect bond
Numerical
10
5
0
10
0
0
0.25
0.5
0.75
1
1.25 1.5 1.75
2
midspan displacement (cm)
2.25
2.5
2.75
3
Experimental
Numerical
0
0.1
Figure 16 – Load - deflection curves: beam F5
0.7
0.8
0.9
1.75
P
P
P
P
P
P
1.5
=
=
=
=
=
=
10kN
20kN
30kN
40kN
50kN
55,5kN
Experim ental - P= 55kN
Numerical - P =55kN
1.5
1.25
bond stress (M P a)
1.25
bond s tress (M Pa)
0.3
0.4
0.5
0.6
composite's strains (% )
Figure 17 – Composite strains: beam F5
1.75
1
0.75
1
0.75
0.5
0.5
0.25
0.25
0
0.2
0
0
10
20
30
40
50
60
70
distance from support (cm)
80
90
100
Figure 18 – Bond stresses: beam F5
0
10
20
30
40
50
60
70
distance from support (cm )
80
90
100
Figure 19 – Bond stresses along the
composite: beam F5
7
110
5.
CONCLUDING REMARKS
The most relevant aspect of this study is the possibility of a numerical prediction of the failure
mode of CFRP composite strengthened beams by a finite element model. As experimentally
verified, the rupture condition is reached in an intense cracking region, where the ultimate value of
the bond stress between the concrete beam and the composite is attained.
The beams F1 and F2, with only one composite layer, achieved almost the same rupture bending
moment, although they were submitted to different loading schemes. However, it was verified that
the bond stresses distribution between the concrete beam and the composite were different in
these two beams. From Fig. 8, it can be observed that the bond stress peak moves towards the
supports in the beam F1. This fact could not be noted in the beam F2.
By comparing the results obtained for the beams F1 and F5, it was noted that the addiction of one
more CFRP composite layer enhanced the beam loading capacity. Nevertheless, it must be
observed that the bond stresses distribution was not modified. A rapid growth of the bond stresses
was observed after the concrete tensile strength was surpassed.
6.
ACKNOWLEDGEMENTS
The authors are grateful to the CNPq (Brazilian Council of Research) for the financial support to
the realization of this research.
7.
REFERENCES
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strengthened with carbon fiber composites. Composites in Construction, Italy, 2003.
[2] Souza RHF, Appleton JAS. Estudo experimental sobre o reforço de vigas de concreto armado
com tecido compósito de fibras de carbono. 45o Congresso Brasileiro do Concreto. 2003.
[3] Freyssinet. Cahier des clauses techniques – Renforcement du beton par collage de tissu de
fibres de carbone procede TFC. 1997, p. 85-107.
[4] Darwin D, Pecknold D. Nonlinear biaxial stress-strain law for concrete. Journal of Engineering
Mechanics Division, 1977;103;229:241.
[5] Kupfer HB, Gerstle KH. Behavior of concrete under biaxial stresses. Journal of Engineering
Mechanics Division, 1973; 99;853:866.
[6] Elwi AE, Hrudey TM. Finite Element model for curved embedded reinforcement. Journal of
Engineering Mechanics Division, 1989; 115;740:745.
[7] Adhikary BB, Mutsuyoshi H. Numerical simulation of steel-plate strengthened concrete beam by
a nonlinear finite element method model. Construction and Building Materials, 2002; 16; 291:301.
[8] Comité Euro-International du Béton. CEB-FIP Model Code 1990. Design Code. London:
Thomas Telford Services; 1993.
[9] Silva PASCM. Modelação e análise de estruturas de betão reforçadas com FRP. Faculdade de
Engenharia da Universidade do Porto. 254p. MSc. Thesis; 1999.
[10] Aurich M. Modelo da ligação entre concreto e armadura na análise de estruturas de concreto
pelo método dos elementos finitos. Universidade Federal do Rio Grande do Sul. 115p. MSc.
Thesis; 2001.
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