Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates

Abstract

In this paper an analysis model is presented for reinforced concrete beams externally reinforced with fiber reinforced polymer (FRP) laminates using finite elements method adopted by ANSYS. The finite element models are developed using a smeared cracking approach for concrete and three dimensional layered elements for the FRP composites. The results obtained from the ANSYS finite element analysis are compared with the experimental data for six beams with different conditions from researches (all beams are deficient shear reinforcement). The comparisons are made for load-deflection curves at mid-span; and failure load. The results from finite element analysis were calculated at the same location as the experimental test of the beams. The accuracy of the finite element models is assessed by comparison with the experimental results, which are to be in good agreement. The load-deflection curves from the finite element analysis agree well with the experimental results in the linear range, but the finite elements results are slightly stiffer than that from the experimental results. The maximum difference in ultimate loads for all cases is 7.8%.

Full text

European Journal of Scientific Research
ISSN 1450-216X Vol.30 No.4 (2009), pp.526-541
© EuroJournals Publishing, Inc. 2009
http://www.eurojournals.com/ejsr.htm
Finite Element Modeling of Reinforced Concrete Beams
Strengthened with FRP Laminates
Amer M. Ibrahim
Asst. prof, College of engineering
Diyala University, Iraq
Mohammed Sh. Mahmood
Asst. lecturer, College of engineering
Diyala University, Iraq
Abstract
In this paper an analysis model is presented for reinforced concrete beams
externally reinforced with fiber reinforced polymer (FRP) laminates using finite elements
method adopted by ANSYS. The finite element models are developed using a smeared
cracking approach for concrete and three dimensional layered elements for the FRP
composites. The results obtained from the ANSYS finite element analysis are compared
with the experimental data for six beams with different conditions from researches (all
beams are deficient shear reinforcement). The comparisons are made for load-deflection
curves at mid-span; and failure load. The results from finite element analysis were
calculated at the same location as the experimental test of the beams. The accuracy of the
finite element models is assessed by comparison with the experimental results, which are to
be in good agreement. The load-deflection curves from the finite element analysis agree
well with the experimental results in the linear range, but the finite elements results are
slightly stiffer than that from the experimental results. The maximum difference in ultimate
loads for all cases is 7.8%.
Keywords: Finite Element Modeling; Reinforced Concrete Beams; FRP Laminates
Introduction
Externally bonded FRP laminates and fabrics can be used to increase the shear strength of reinforced
concrete beams and columns. Figure1 shows examples of possible FRP shear strengthening
configurations. It can be seen that the shear strength of columns can be easily improved by wrapping
with a continuous sheet of FRP to form a complete ring around the member. Shear strengthening of
beams, however, is likely to be more problematic when they are cast monolithically with slabs. This
increases the difficulty of anchoring the FRP at the beam/slab junction and increases the risk of
debonding failure. Nevertheless, bonding FRP on either the side faces, or the side faces and soffit, will
provide some shear strengthening for such members. In both cases, it is recommended that the FRP is
placed such that the principal fiber orientation,
, is either 45º or 90º to the longitudinal axis of the
member. There is some evidence that the shear resistance of beams can be further improved by
bonding additional sheets with their fibers orientated at right angles to the principal fiber direction. In

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 527
FRP-strengthened beams failure may occur due to beam shear, flexural compression, FRP rupture, FRP
debonding or concrete cover ripping
[1]
.
Figure 1: FRP shear strengthening configurations
(a) Vertical strips
(b) Inclined strips
(c) Continuous
A concrete structure may need strengthening for many reasons:
• To increase live-load capacity, e.g. of a bridge subject to increased vehicle loads or a
building the use of which is to change from residential to commercial.
• To add reinforcement to a member that has been under designed or wrongly constructed.
• To improve seismic resistance, either by providing more confinement to increase the strain
capacity of the concrete, or by improving continuity between members.
• To replace or supplement reinforcement, e.g. damaged by impact or lost due to corrosion.
• To improve continuity, e.g. across joints between precast members.
In most cases it is only practical to increase the live-load capacity of a structure. However, in
some situations it may be possible to relieve dead load, by jacking and propping, prior to the
application of the additional reinforcement. In these cases, the additional reinforcement will play its
part in carrying the structures dead load. Three basic principles underlie the strengthening of concrete
structures using fiber composite materials, which are the same irrespective of the type of structure:
• Increase the bending moment capacity of beams and slabs by adding fiber composite
materials to the tensile face.
• Increase the shear capacity of beams by adding fiber composite materials to the sides in
the shear tensile zone.
• Increase the axial and shear capacity of columns by wrapping fiber composite materials
around the perimeter.
In the last decade, fiber reinforced polymer FRP composites have been used for strengthening
structural members of reinforced concrete bridges, which are deficient or obsolete due to changes in
their use or consideration of increased loadings
[2]
. Many researchers have found that FRP composites
applied to the reinforced concrete members provide efficiency, reliability and cost effectiveness in
rehabilitation
[3-4-5]
.
A large number of available software like sap2000, LUSAS, and ANSYS etc incorporate finite
elements based analysis. In this paper an attempt has been made with ANSYS (version 10)
[6]
software
to bring into focus the versatility and powerful analytical capabilities of finite elements technique by
objectively modeling the complete response of test beams. The finite elements model uses a smeared
cracking approach to model the reinforced concrete and three dimensional layered elements to model
the fiber reinforced polymer FRP composites. This model can help to confirm the theoretical
calculations as well as to provide a valuable supplement to the laboratory investigation of behavior.

528 Amer M. Ibrahim and Mohammed Sh. Mahmood
Finite Element Modeling
The finite elements analysis calibration study included modeling a reinforced concrete beams with the
dimensions and properties corresponding to beams tested in previous researches
[7-8]
.
Concrete
Solid65 element was used to model the concrete. This element has eight nodes with three degrees of
freedom at each node – translations in the nodal x, y, and z directions. This element is capable of
plastic deformation, cracking in three orthogonal directions, and crushing. A schematic of the element
is shown in Figure2
[6]
. Smeared cracking approach has been used in modeling the concrete in the
present study
[9]
.
Figure 2: Solid65 element geometry.
The following properties must be entered in ANSYS:
• Elastic modulus (E
c
).
• Ultimate uniaxial compressive strength ( ).
• Ultimate uniaxial tensile strength (modulus of rupture, f
r
)
[10]
• Poisson’s ratio (ν) = 0.2.
• Shear transfer coefficient (β
t
) which is represents conditions of the crack face. The value of
β
t
ranges from 0.0 to 1.0, with 0.0 representing a smooth crack (complete loss of shear
transfer) and 1.0 representing a rough crack (no loss of shear transfer)
[6]
. The shear transfer
coefficient used in present study varied between 0.3 and 0.4
• Compressive uniaxial stress-strain relationship for concrete.
The present study assumed that the concrete is a homogeneous and initially isotropic. The
compressive uniaxial stress-strain relationship for concrete model is obtained by using the following
equations to compute the multilinear isotropic stress-strain curve for the concrete is as shown in
Figure3.
(1)
2
1
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
o
c
c
E
f
ε
ε
ε
for
°
≤
≤
ε
ε
ε
1
(2)
'
cc
ff =
for
cu
ε
ε
ε
≤
≤
°
(3)

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 529
c
'
c
o
E
f2
=
ε
4)
The simplified stress-strain curve for each beam model is constructed from six points connected
by straight lines. The curve starts at zero stress and strain. Point 1, at , is calculated for the
stress-strain relationship of the concrete in the linear range (must satisfy Hooke’s law). Points 2, 3, and
4 are obtained from Equation 2, in which ε
o
is calculated from Equation 4. Point 5 is at ε
o
and . The
behavior is assumed to be perfectly plastic after point 5.
Figure 3: Simplified compressive uniaxial stress-strain curve for concrete.
ε
1
ε
2
ε
3
ε
4
ε
5
-
ε
+
ε
Reinforcing steel
Modeling of reinforcing steel in finite elements is much simpler than the modeling of concrete. A
Link8 element was used to model steel reinforcement. This element is a 3D spar element and it has two
nodes with three degrees of freedom – translations in the nodal x, y, and z directions. This element is
also capable of plastic deformation. This element is shown in Figure4
[6]
. A perfect bond between the
concrete and steel reinforcement considered. However, in the present study the steel reinforcing was
connected between nodes of each adjacent concrete solid element, so the two materials shared the same
nodes. The same approach was adopted for FRP composites.
Figure 4: Link8 element geometry.
Steel reinforcement in the experimental beams was constructed with typical steel reinforcing
bars. Elastic modulus and yield stress for the steel reinforcement used in this FEM study follow the
design material properties used for the experimental investigation. The steel for the finite element
models is assumed to be an elastic-perfectly plastic material and identical in tension and compression
as shown in Figure5. A Poisson’s ratio of 0.3 is used for the steel reinforcement.

530 Amer M. Ibrahim and Mohammed Sh. Mahmood
Figure 5: Stress-strain curve for steel reinforcement
Steel plate
Steel plates were added at support and loading locations in the finite element models (as in the actual
beams) in order to avoid stress concentration problems. An elastic modulus equal to 200,000 N/mm
2
and Poisson’s ratio of 0.3 were used for the plates. The steel plates were assumed to be linear elastic
materials. A Solid 45 element was used to model steel plates. The geometry and node locations for this
element type are shown in Figure 6
[6]
.
Figure 6: Solid 45 element geometry.
FRP Laminates
FRP composites are materials that consist of two constituents. The constituents are combined at a
macroscopic level and are not soluble in each other. One constituent is the reinforcement, which is
embedded in the second constituent, a continuous polymer called the matrix. The reinforcing material
is in the form of fibers, i.e., carbon and glass, which are typically stiffer and stronger than the matrix.
The FRP composites are orthotropic materials; that is, their properties are not the same in all directions.
Figure 7 shows a schematic of FRP composites.

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 531
Figure 7: Schematic of FRP composites.
A Solid 46 layered element was used to model FRP composites. The high strength of the epoxy
used to attach FRP sheets to the experimental beams supported the perfect bond assumption. The
geometry and node locations for this element type are shown in Figure 8
[6]
.
Figure 8: Solid 46 layered element geometry.
In the present study linear elastic properties of FRP composites are assumed as shown in Figure
9. A summary of material properties for FRP composites used for the finite elements modeling of the
strengthened beams in the present study is shown in Table 1.

532 Amer M. Ibrahim and Mohammed Sh. Mahmood
Figure 9: Stress-strain curves for the FRP composites in the direction of the fibers.
Table 1: Summary of material properties for FRP composite.
FRP composite Elastic modulus N /mm
2
Major Poisson’s ratio Shear modulus N /mm
2
Carbon fiber reinforced
polymer CFRP
Glass fiber reinforced
polymer GFRP
Numerical Analysis
In order to validate the numerical representation of the reinforced concrete beams strengthening with
fiber reinforced polymer composites, the finite elements representation using ANSYS program has
been applied to practical sections and the results will be compared with the experimental results
reported by previous researches
[7-8]
.
Geometry and materials properties.
Six beams with different conditions (all beams are deficient shear reinforcement) will be analyzed
using the proposed ANSYS finite elements model. Table2 shows all beams evaluated in the present
study.
Table 2: Summary of beams evaluated in the present study.
Symbol Description FRP Laminates thickness (mm)
B1 As built beam (control beam)
[7]
. ----
B1C-90
Strengthen by one layer of unidirectional transverse carbon/epoxy
laminates CFRP inclined at an angle of 90º to the longitudinal axis
[7]
.
1.6
B1G-90
Strengthen by two layers of unidirectional transverse E-glass/epoxy
laminates GFRP inclined at an angle of 90º to the longitudinal axis
[7]
.
2.1
B2 As built beam (control beam)
[8]
. ----
B2C-90
Strengthen by warping with one layer of CFRP inclined at an angle of
90º to the longitudinal axis
[8]
.
0.18
B2C-90-0
Strengthen by warping with one layer of CFRP inclined at an angle of
90º with an additional layer of CFRP on both sides of the web
inclined at an angle of 0o to the longitudinal axis
[8]
.
0.18

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 533
The geometry of all beams is shown in Figure 10, and the material properties adopted in the
analysis are given in Table 3.
Figure 10: Loading reigns and geometrical properties of analyzed beams.
0.37m
0.37m
1.7m
0.5
P
0.5
P
2.44m
3.62m
0.15m
2Φ10
2Φ13
Φ10@0.61
m
0.25m
(a) Dimension and reinforcement of as built beam B1.
0.37m
0.37m
1.7m
0.5
P
0.5
P
2.44m
3.62m
0.15m
0.41m
0.05m0.05m
0.41m
(b) Shear strengthening details for beams B1C-90, and B1G-90.
FRP
FRP
0.915m
P
1.83m
2.134m
0.23m
2Φ9
2Φ25
Φ9@0.3m
0.38m
(c) Dimension and reinforcement of as built beam B2.
0.915m

534 Amer M. Ibrahim and Mohammed Sh. Mahmood
Table 3: Summary of Material Properties of Selected Beams
B1, B1C-90 & B1G-90 B2, B2C-90 & B2C-90-0
Steel yield strength f
y
(N/mm
2
) 420 414
Steel modulus of elasticity E
s
(N/mm
2
) 200000 200000
Steel Poisson's ratio v
s
0.3 0.3
Concrete compressive strength (N/mm
2
)
27.54 31
Concrete Poisson's ratio v
c
0.2 0.2
Due to the symmetry in cross-section of the concrete beam and loading, symmetry was utilized
in the finite elements analysis; only one quarter of the beam was modeled. This approach reduced
computational time and computer disk space requirements significantly. The finite element mesh,
boundary condition and loading regions of all beams are shown in Figure11.
Figure 11: Finite element mesh, boundary condition and loading regions for a quarter beam model of all beams
a. Finite element modeling for B1C-90 & B1G-90
Loading steel plate
FRP composite
Supporting steel plate

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 535
c.Steel reinforcement for B1, B1C-90 & B1G-90
Stirrups Φ10@ 0.6m
Φ12 tension
reinforcement
Φ10 compression
reinforcement
d.Steel reinforcement for a B2, B2C-90 & B2C-90-0
Stirrups Φ9@ 0.3m
Φ25 tension
reinforcement
Φ9 compression
reinforcement
Discussion of Results
Load deflection curves
The experimental and numerical load-deflection curves obtained for the beams are illustrated in
Figure11. The curves show good agreement in finite element analysis with the experimental results
throughout the entire range of behavior and failure mode, for all beams the finite element model is
stiffer than the actual beam in the linear range. Several factors may cause the higher stiffness in the
finite element models. The bond between the concrete and steel reinforcing is assumed to be perfect
(no slip) in the finite element analyses, but for the actual beams the assumption would not be true slip
occurs, therefore the composite action between the concrete and steel reinforcing is lost in the actual
beams. Also the microcracks produced by drying shrinkage and handling are present in the concrete to
some degree. These would reduce the stiffness of the actual beams, while the finite element models do
not include microcracks due to factors that are not incorporated into the models. After the initiation of
flexural cracks, the beam stiffness was reduced and the linear load –deflection behavior ended when
the internal steel reinforcement began to yield.

536 Amer M. Ibrahim and Mohammed Sh. Mahmood
Figure11: Load deflection curves.
a. Load deflection curve for beam B1.
b. Load deflection curve for beam B1C-90.

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 537
c. Load deflection curve for beam B1G-90.
d. Load deflection curve for beam B2C-90.

538 Amer M. Ibrahim and Mohammed Sh. Mahmood
e. Load deflection curve for beam B2C-90-0.
As shown in Figure11 a ,b, and c, the strengthened beams B1C-90 and B1G-90 are stiffer than
the control beam B1, but B1C-90 appear stiffer than B1G-90 which means that carbon fiber polymer is
better than glass fiber polymer in strengthening the reinforced concrete beams for shear. Figure11 d,
and e indicate that the using of additional layer of carbon fiber polymer composite to both side of the
beam web inclined at an angle of 0º to the longitudinal axis increase the stiffness of the beam by 2.3% ,
so that the additional layer is not sufficient to increase the beam stiffness.
Crack Pattern
The ANSYS program records a crack pattern at each applied load step. Figure12 shows evolutions of
crack patterns developing for each beam at the last loading step. ANSYS program displays circles at
locations of cracking or crushing in concrete elements. Cracking is shown with a circle outline in the
plane of the crack, and crushing is shown with an octahedron outline. The first crack at an integration
point is shown with a red circle outline, the second crack with a green outline, and the third crack with
a blue outline
[6]
.

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 539
Figure 12: Evolution of Crack Patterns.
B1
B1C-90
B1G-90
B2
B2C-90
B2C-90-0
The failure modes of the finite element models show good agreement with observations and
data from the experimental full-scale beams. The addition of FRP reinforcement to the control beam

540 Amer M. Ibrahim and Mohammed Sh. Mahmood
shifts the behavior of the beams from a shear failure near the ends of the beam to flexure failure at the
midspan.
Failure load
The failure load obtained from the numerical solution for all beams is slightly smaller than
experimental load. The final loads for the finite element models are the last applied load step before the
solution diverges due to numerous cracks and large deflections. Table4 shows comparison between the
ultimate loads of the experimental beams and the final loads from the finite element models, and the
ultimate capacity of the strengthened beams with ultimate capacity of the control beams.
Table 4: Comparsions between experimental and finite element ultimate loads, and ultimate capacity of the
strengthened beams with ultimate capacity of the control beams.
Beam
Experimental ultimate
load (kN)
Numerical ultimate
load (kN)
%
Difference
Increased in ultimate load of
strengthened
B1 69 66 4.3 1
B1C-90 125 119 4.8 1.6
B1G-90 116 107 7.8 1.8
B2 416 405 2.6 1
B2C-90 435 414 4.8 1.02
B2C-90-0 445 420 5.6 1.03
Conclusions
The numerical solution was adopted to evaluate the ultimate shear strength of the reinforced concrete
beams reinforced with FRP laminates in simple, cheap and rapid way compared with experimental full
scale test. The general behaviors of the finite element models show good agreement with observations
and data from the experimental full-scale beam tests. The addition of FRP reinforcement to the control
beam shifts the behavior of the control beams from shear failure near the ends of the beam to flexure
failure at the midspan. The results obtained demonstrate that carbon fiber polymer is efficient more
than glass fiber polymer in strengthening the reinforced concrete beams for shear. The present finite
element model can be used in additional studies to develop design rules for strengthening reinforced
concrete members using FRP laminates.

Finite Element Modeling of Reinforced Concrete Beams Strengthened with FRP Laminates 541
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Nomenclature
(N/mm
2
)
Ultimate uniaxial compressive strength
(N/mm
2
) Concrete elastic modulus E
c
(N/mm
2
) Steel elastic modulus E
s
(N/mm
2
) stress at any strain
ε
ƒ
c
(N/mm
2
) Concrete modulus of rupture f
r
Shear transfer coefficient β
t
Strain Ε
strain corresponding to (
)
ε
1
ultimate compressive strain ε
cu
Strain at the ultimate compressive strength
ε
o
Concrete Poisson’s ratio ν
c
Steel Poisson’s ratio ν
s