FEM study on the bond behaviors between sprayed fiber reinforced
plastics (FRPs) and concrete by the use of discrete elements
T. Kojima
Professor, College of Architecture and Urban Design, Ritsumeikan University, Shiga, Japan
K. Hibino
Incorporated Administrative Agency, Japan Nuclear Energy Safety Organization, Tokyo, Japan
M. Mizuta
Research Associate, College of Architecture and Urban Design, Ritsumeikan University, Shiga, Japan
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ABSTRACT: In this study, the FEM analysis using discrete crack and bond elements is developed to apply to
the test results by Banthia et al on the plain concrete beam reinforced by the FRPs subjected to bending. These
discrete elements developed by the authors are consisted of the isoparametric quadratic contact element.
Because of the discrete behavior of cracking and bond slip, it seems more reasonable to represent them by the
proposed discrete elements. But occurrence of crack and bond slip makes the computer program more
complicate because FEM mesh must be modified in each step whenever new crack or bond slip occurs, or any
crack or bond crack develops. By using this FEM analysis, it is possible to represent the development of
flexural crack and bond crack, and stress distribution of bond and FRPs. It is also possible to clarify the bond
capacity of the beam by this FEM analysis.
1 INSTRUCTION
It is basically important how to introduce cracking
and bond slip into analysis when analyzing the
mechanical properties of RC structures. In the finite
element method (FEM) analysis, there are two
approaches: (1) to represent cracking and bond slip
as the change of element characteristics of concrete,
"smeared element idea"; and (2) to introduce special
element such as bond links element, "discrete
element idea." The approach (1) is widely used. On
the other hand, the approach (2) is easy to analyze a
structure, which is already cracked. However it is
complicated when analyzing developing cracks,
because it is necessary to have a re-meshing program
due to developing cracks.
The authors have already developed a computer
program to analyze developing cracks by introducing
isoparametric crack and bond elements. These
elements are expressed as the isoparametric
quadratic contact element, and both sides of crack or
bond element has same nodes of adjacent concrete or
reinforcement elements. By this program, it is
possible to analyze crack development in RC
structures.
In this study, the FEM analysis using discrete
crack and bond elements is developed to apply to the
plain concrete beam reinforced by the FRPs
subjected to bending. Flexural capacity of the plain
concrete beam reinforced by the FRPs is affected by
the bond characteristics between FRPs and concrete.
In the analysis, the characteristic of bond is
represented as the bond stress-slip relation of the
bond element.
By using this FEM analysis, it is possible to
represent the development of flexural crack and
bond crack, and stress distribution of bond and FRPs.
It is also possible to clarify the bond capacity of the
beam.
2 CHARACTERISTICS OF ELEMENTS
2.1 Discrete crack element
A crack element is inserted at the point, where the
maximum principal tensile stress occurs, and to the
direction perpendicular to it. Generally normal stress
σn and shear stress τt are acting across a crack
plane, and normal and tangential displacement δn
and δt occur as shown in Fig. 1. The relationship
τt (MPa)
 t   max  1  t  ty  tu  ty  3
τmax
between these stresses and displacements can be
expressed
as
Equation
1
(3) unloadingreloading path
 t   d t td 2
(δtd , τd )
δty
0
δtu
δt (mm)
(1) plastic (2) softening region
region
(a)
τmaxin
(MPa)
 max   maxin tu   n  tu
τmax
0 δn
δtu δt (mm)
(b)
Figure 3. Shear stress – displacement diagram
crack
(1)
y
(3)
concrete
(4)
s (2)
(3)
(1)
(6)
(5)
(6)
(4)
concrete -1
(2)
0
1
(5)
(c)
local
coordinate
0
x
(a) global coordinate Fy1
fn1
Fy4
θ1
(1) ft1F
δn1
x1 s
ft1θ
t
1
Fx4
(3)
(4)
δ
t1
(6)
n
δn3
fn1
δt3 y
ds
dy
dx (5)
(2)
x
δ
t2
δn2
(b) details
ξ
Figure 4. Crack element
.
 n  C nn C nt   n 
(1)

    Dc  
  t   Ctn Ctt   t 
The relationship between σ n and δ n in the
   
crack element was assumed as that shown in Fig. 2.
This relationship is referred by tension softening
curve of concrete. On the other hand, the
relationship between τt and δt was assumed as
that shown in Fig. 3. In Fig. 3 (a), the τmax is the
maximum shear stress. In addition, the coefficient
τ max can be determined from the relationship
between τmax and δn in the crack element (see
Fig. 3(b)). The relationship between σn-δt is not
so important. In the analysis, a constant value of
Cnt=Ctn=0 was assumed.
The proposed crack element is a quadratic contact
element with 6 nodes as shown in Fig. 4. Both sides
of crack element have same nodes of adjacent
concrete elements. When the element is expressed in
local coordinate, the stiffness matrix [Kc] of the
crack element can be obtained by the numerical
integration of the following Equation 2.
K c    B T Dc Bt  ds
When the element is expressed in local coordinate,
the stiffness matrix [KB] of the bond linkage element
can be obtained by the numerical integration of the
following Equation 4.
K B    B T DB Bt  ds
s
  B  DB B t
1
T
1
dx
d   dy d  d
2
2
(4)
where [B] is matrix between displacement and
relative displacement, t is thickness of displacement.
δt
tention
internal crack
δn
σn
τt
reinforcement
tention
s

B T Dc Bt dx d 2  dy
1
1
concrete
d  d
2
main crack
Figure 5. Bond performance
(2)
where [B] is matrix between displacement and
relative displacement, t is thickness of concrete
where crack occurs.
τt (MPa)
   max 1  1    y 10 
  10   y  max
τmax
2.2 Discrete bond element
Generally, normal stress σn and bond stress τt are
defined as surface traction acting on a surface
between concrete and reinforcement, and normal and
tangential displacement δn and δt are defined as
shown in Fig. 5. The relationship between these
stresses and displacement can be expressed as
Equation 3.
B   
   B
(3)
    n    nn nt   n   DB  
  t   Btn Btt   t 
In the bond test, the relationship between bond
stressτt and displacement δt is dominating. The
coefficient Btt can be determined from Fig. 6. This
relationship is determined by the coefficients of arb
and δy. In the analysis, the coefficients are used as
the parameters. On the other hand, the relationship
between σn and δn is not so important. In the
analysis, a constant value of Bnn=1.0×104 N/mm3,
Bnt=Btn=0 was assumed.
The proposed bond linkage element is a quadratic
contact element with 6 nodes as shown in Fig. 7.
Both sides of bond linkage element have same nodes
of concrete and reinforcement elements respectively.
 max  arb ft
 u   y   max 10
0
δy
δu
δ (mm)
Figure 6. Bond stress – bond slip relationship
y
reinforcement
(1)
s
(3)
concrete
(2)
(2)
(4)
(6)
(5)
(5) concrete
-1
(3)
(1)
(6)
0
(4)
1
(c) local coordinate
x
0
(a) global coordinate F
y1
fn1
Fy4
θ1
(1) ft1F
δn1
x1 s
ft1θ
t
1
F
(3)
x4
(4)
δt1
(6)
n
δn3
fn1
δt3 y
ds
dy
dx (5)
(2)
x
δn2 δt2
(b) details
Figure 7. Bond element
ξ
εty
0
(εp, fp)
Table 1. Mechanical properties of concrete
f’c
ft
Ec*1
GF*2
(MPa)
(MPa)
(MPa)
(N/mm)
40
3.5
30000
0.08
*1
: Assumed value *2: Fracture energy
(a) Stress - strain relationship
shear compression
τ
failure
compression
tension
failure
failure
f’c
y
x
FRPs
concrete
Figure 10. Beam size and loading condition
f’c
3ft
P/2
50
CL
εcy
equation (6) equation (5) σ (MPa)
-ft
25 100
100
3 100
ε
ft
equation (7)
εcu
Table 2. Mechanical properties of FRPs
EF
fe
fu
εe
εy
(MPa)
(MPa)
(MPa)
(μ)
(μ)
4010
20
27
5100
8200
*
: Maximum strain εu is assumed to be twice εy.
σ
(b) Mohr – Coulomb criterion
Figure 8. Characteristic of concrete
concrete
P/2
2.3 Characteristics of other elements
notch
2.3.1 Concrete element
In the FEM analysis, a concrete specimen was
divided into finite element mesh as a two
dimensional plane stress problem. Quadratic
quadrilateral elements with 8 nodes were used for
concrete. The relationship between stress and strain
of concrete was assumed as shown in Fig. 8(a).
Mohr-Coulomb criterion was used as the failure
criteria of concrete element as shown in Fig. 8(b).
  f c1  1    y 2
(5)
  f c1     y   u   y 2
  f p   p 2
(6)
(7)
where f’c is compressive strength of concrete.
2.3.2 FRPs element
Quadratic linear elements with 3 nodes were used
for FRPs. The relationship between stress and strain
of FRPs was assumed as shown in Fig. 9.
3 OUTLINE OF THE EXPERIMENT
Sprayed fiber reinforced plastic (FRPs) is a novel
technique of repair using a thin coating of glass fiber
y
x
bond crack
FRPs
Figure 11. Analytical model
reinforced plastics composite produced through
spraying. Beams were sprayed on the tensioning
side with a 3 mm thick coating of glass fiber
reinforced plastic composite with a random
distribution of chopped fibers at an estimated fiber
content of 8 % by volume. The test results by
Banthia et al on the plain concrete beam reinforced
by the FRPs subjected to bending were referred.
Two kinds of the beams reinforced by the FRPs
were tested. One has a notch with 33 mm high and 2
mm wide and the other has no notch at the mid-span.
The size of the beam and the loading condition are
shown in Fig. 10. The size of the beam was 100×
100×350 mm. Its span length was 300 mm. The
beams were assumed to be simply supported and
subjected to two-point concentrated loading. The
mechanical properties of concrete are listed in Table
1. These values were given on the supposition that
target compressive strength of concrete was 40
N/mm2. The mechanical properties of FRPs are
listed in Table 2.
The analytical results are compared with
experimental results, in order to exam the
application of this analytical method. In the analysis,
the half of the beam was analyzed by taking the
advantage of symmetry (see Fig. 11) to decrease the
number of the element. Proposed bond elements
were inserted between the concrete and FRPs
elements. A crack element was inserted
perpendicular to the direction of the maximum
tensile stress at the position of that occurrence. Load
was kept constant and it was adjusted as the tensile
stress of crack element became the tensile strength
of concrete after inserting one crack element.
5 RESULTS AND DISCUSSIONS
Load and mid-span deflection diagrams obtained by
both the experiment and analysis are shown in Fig.
12. In this figure, load I to IV given by the analysis
load (kN)
30
20
--- Plain (Exp)
― FRPs (Exp)
II
I
10
- ○- Plain (FEM)
- ▲- FRPs (FEM)
III
IV
0
0
40
load (kN)
4 OUTLINE OF THE ANALYSIS
40
0.4
0.6
0.8
deflection (mm)
(a) Beam without a notch
--- Plain (Exp)
― FRPs (Exp)
30
- ○- Plain (FEM)
- ▲- FRPs (FEM)
I
III
10
0
0.2
0.4
0.6
0.8
deflection (mm)
(b) Beam with a notch
Figure 12. Load – mid span
deflection diagram
Exp
σ
fu
fe
FEM
0
εe εy
ε
Figure 9. Stress – strain relationship of FRPs
correspond to the loads at the quick change and
failure. The results on a plain concrete beam
subjected to bending are also shown for reference.
Fig. 13 demonstrates both the experimental and
analytical crack patterns of the beams. The
experimental results are crack patterns of the whole
beams with the abbreviation of "Exp" and a half of
the beam with "FEM" was obtained by the analysis.
Bond stress and bond slip relationship (see Fig. 6)
was obtained by trial and error so that the analytical
results of the load and mid-span deflection diagram
and crack pattern were fitted to the experimental
data. Bond stress and bond slip relationship involves
the following equations.
1
II
20
0
0.2
(a) Beam without a notch
Exp
FEM
(b) Beam with a notch
Figure 13. Crack patterns
1
Stress of FRPs
(N/mm2)
30
20
- ○- ◆-
was distinguished in both the analysis and
experiment. The maximum load was equal to load II
and the beam failed by tearing of FRPs at load III.
Stress distributions of the bond and FRPs at load I
to IV are shown in Fig. 14. These results are
obtained by the analysis of the beam without a notch.
The results of the beam with a notch showed a
similar tendency to these results without a notch. At
load I, the large contribution of the concrete to
tensile stress brought the less stress of the bond and
FRPs. As the flexural crack rapidly reached to the
near upper fiber, the stress of the bond and FRPs in
the maximum moment region much increased at
load II to III. On the other hand, the stress of FRPs
was approximately equal to zero in shear span and
the stress distribution of the bond in shear span
uniformly increased. Since bond performance was
considered in this analysis, the effect of the concrete,
FRPs and bond was clearly shown at each loading
step.
- ▲- II
- ■- IV
I
III
10
0
Bond stress
-10
0 20 40 60 80 100120140160180
x coordinate (mm)
2
(N/mm ) (a) Stress of FRPs element
30
20
10
0
- ○- ◆-
- ▲- II
- ■- IV
I
III
-10
0 20 40 60 80 100120140160180
x coordinate (mm)
(b) Stress of bond element
Figure 14. Stress distribution
(Beam without a notch)

  7 1  1    y 
10
  10   y   7

0   u  0.1
0.1   u  0.8
6 CONCLUSIONS
(5)
The coefficient of arb and δ y is 2.0 and 0.1,
respectively. In case of the experiment, the
maximum load of the beam without a notch was a
little larger than that with a notch and the analysis
could estimate the maximum loads by the use of the
suitable bond stress and bond slip relationship. Also,
the analysis with the proposed bond element could
capture the tendency for the maximum deflection of
the beam without a notch to be much larger than that
with a notch.
When the analytical applied load of the beam
without a notch was suddenly decreased after load I
(Fig. 12(a)), the flexural crack rapidly developed at
the near top of the beam. This result suggests that a
concrete cannot sufficiently resist the tensile stress
and that the tensile stress applied to FRPs much
increases. Also, the analysis could represent the
development of the bond crack between the concrete
and FRPs near the flexural crack as same as the
experimental result (Fig. 13(a)). After the gradual
increase in deflection, the beam reached to failure by
tearing of FRPs at load IV.
In case of the beam with a notch in Fig. 13(b), the
flexural crack propagated around the top of the beam
before load I. Only the flexural crack at the mid-span
In this study, the following conclusions were
obtained.
1. Through the use of crack and bond elements, the
characteristics of bond failure in plain concrete
beam reinforced by FRPs can be simulated by the
FEM analysis.
2. By applying the re-meshing program, the crack
propagation can be represented and bond failure
can be estimated.
REFERENCES
Banthia, N., Yan, C. & Nandakumar, N. 1996.,
Advanced composite materials in bridges and
structures, Sprayed fiber reinforced plastics
(FRPs) for repair of concrete structures; Proc.
second CSCE intern. conf., Montreal, Quebec,
Canada, 537-545
Hibino, K., Kojima, T., Takagi, N., Matsuo, M. (now
change to Mizuta) & Yokoyama, F. 1999., F.E.M.
study on the flexural crack behaviors of concrete
by the use of the crack element; Proc. seventh
east Asia-Pacific conf. on structural engineering
& construction (EASEC-7), Kochi, Japan, V-2,
1151-1156
Kojima, T., Takagi, N. & Hibino, K. 2001.,
Development of a discrete bond linkage element
between concrete and reinforcing bar; Proc.
second intern. conf. on engineering materials, Vol.
1, 315-326