Concrete Beams Strengthened with Externally
Bonded Glass Fiber Reinforced Plastic Plates
Ming-Hung Hsu
Department of Electrical Engineering, National Penghu University,
Penghu, Taiwan 880, R.O.C.
Abstract
In this work, the flexural behavior of reinforced concrete beams strengthened by epoxy bonded
glass fiber reinforced plastic plates is investigated through the finite element method. The finite
element models based on the widely used package ABAQUS are employed in simulating the behavior
of reinforced concrete beams strengthened by externally bonded glass fiber reinforced polymer. The
numerical results of four-point bending test of reinforced concrete beams strengthened by externally
bonded glass fiber reinforced plastic plates in comparing with the experimental results show satisfactory
agreement. The results indicate that the flexure strength of reinforced concrete beams can be
significantly increased by externally bonded glass fiber reinforced plastic plates. The dynamic behavior
of reinforced concrete beams strengthened by epoxy bonded glass fiber reinforced plastic plates is
investigated through the differential quadrature method. In this approach, only eleven sample points
are required to obtain convergence.
Key Words: Glass Fiber Reinforced Plastic Plate, Reinforced Concrete Beam, Finite Element Method,
Differential Quadrature Method
1. Introduction
Existing cracking reinforced concrete beams may
require increasing the shear capacity and cracking.
The glass fiber reinforced plastic plate is used to bond
the beam in order to increase flexural capacity. Sa-
adatmanesh et al. [112] experimentally investigated
the static strength of reinforced concrete beams and
column strengthened by gluing glass fiber reinforced
plastic plates. High strength fiber reinforced plastic
straps are wrapped around the column in the potential
plastic hinge region to increase confinement and to
improve the behavior under seismic forces. Hamoush
and Ahmad [13] investigated the strength of steel-pla-
te-strengthened concrete beams. Adhikary et al. [14,15]
investigated the shear strengthening of reinforced con-
crete beams using steel plates bonded on beam web
through experiments and numerical analysis. Garden
et al. [16,17] experimentally studied the failure mo-
des of reinforced concrete beams strengthened with
pre-stressed carbon composite plates and the anchor-
age length of carbon fiber composite plates used to
strengthen reinforced concrete beams. Kankam [18]
studied the flexural strength and deformation charac-
teristics of concrete beams reinforced with threaded
steel bars that were tensioned against steel plates bear-
ing on the concrete ends by means of tighten nuts.
Subedi and Baglin [19] analyzed the ultimate load of
plate reinforced concrete beams. Picard et al. [20] in-
vestigated the influence of various design parameters
such as cube strength, size and position of main em-
bedded tensile bars, width of the beam, plate thick-
ness, and ratio of plate width on the magnitude of ulti-
mate plate peeling moment. Nguyen et al. [21] inves-
tigated an analytical model for reinforced concrete
beams with bolted side plates accounting for longitu-
dinal and transverse partial interaction. Shen et al.
[22] presented an analytical-numerical procedure for
Tamkang Journal of Science and Engineering, Vol. 9, No 3, pp. 223232 (2006) 223
*Corresponding author. E-mail: [email protected]
the bending and transverse vibration analysis of mod-
erately thick plates strengthened by a bonded thin com-
posite plate under different kinds of boundary condi-
tions. Ascione and Feo [23] developed a finite ele-
ment model for predicting shear and normal stresses
in the adhesive layer of plated reinforced concrete beams.
Raoof et al. [24] investigated the theoretical and ex-
perimental study on externally plated reinforced con-
crete beams. Barnes et al. [25,26] solved the transfer
of stress through steel to concrete adhesive bond and
analyzed the external steel plate systems for the shear
strengthening of reinforced concrete beams based on
the equilibrium of forces along the critical section. Sa-
pountzakis et al. [27,28] studied the interface forces and
stresses in composite steel-concrete structure. Wang [29]
investigated the analytical and experimental study on se-
ismic performance of reinforced concrete T-beams with
design deficiency in steel bar curtailment.
ABAQUS standard finite element analysis code is
used to model the reinforced concrete beams bonding
glass fiber reinforced plastic plates to the tension sur-
face. ABAQUS offers the facility of nonlinear model-
ing for the reinforced concrete beam. The differential
quadrature method (DQM) is used to form the vibration
problems of reinforced concrete beams strengthened by
epoxy bonded glass fiber reinforced plastic plates in
matrix form. To the authors knowledge, very few pub-
lished papers in the literature have presented the dy-
namic analysis of reinforced concrete beams strength-
ened by epoxy bonded glass fiber reinforced plastic pla-
tes using the DQM.
2. Flexural Strength of Reinforced Concrete
Beams Strengthened by Epoxy Bonded Glass
Fiber Reinforced Plastic Plates
Figure 1 shows the geometry of a reinforced concrete
beam strengthened by epoxy bonded a glass fiber rein-
forced plastic plate for four-point bending test. L is the
length of the reinforced concrete beam. An epoxy layer is
applied to the straps while wrapping for inter laminar
bond. The finite element models based on the widely used
package ABAQUS are employed in simulating the be-
havior of reinforced concrete beams strengthened by ex-
ternally bonded glass fiber reinforced plastic plates. We
use the method for measuring the shear strength to get
the simple compressive strength and the simple tensile
strength by Mohr-Columb criterion. Meshes with trian-
gle type elements are created with an automatic mesh ge-
nerator. In this two dimensional non-linear finite element
model, the boundary nodes are considered simple sup-
ported. The modeling approach performed in this study
is based on finite element method to describe the total
strain. ABAQUS provides an elastic plastic damage the-
ory; including tension cracking, compression crushing,
and post crack response. In this study of FEM model, the
elastic plastic damage theory is adopted to solve the stress
distribution.
224 Ming-Hung Hsu
Figure 1. Schematic view of the reinforced concrete beam strengthened by epoxy bonded a glass fiber reinforced plastic platealong the tension face.
Figure 2 shows the central deflection of the reinfor-
ced concrete beam. The beam size used is 0.20 m (b)
0.30 m (h) 3.00 m (L). b is the width of the reinforced
concrete beam. h is the thickness of the reinforced con-
crete beam. The beams are reinforced with two D6 bars
in the compression zone, whereas two D9 bars are pro-
vided in the tension. The diameter of D6 bar is 6.35 mm.
The diameter of D9 bar is 9.53 mm. Youngs modulus of
the concrete is 21000 MPa. The experimental ultimate
load is 91.9 kN [30]. The analytical ultimate load is 98.9
kN. The agreement between numerical and experimentcurves is satisfactory.
Figure 3 shows the central deflection of the reinfor-
ced concrete beam strengthened by epoxy bonded a 2 mm
thick glass fiber reinforced plastic plate along the tension
face. Youngs modulus of the glass fiber reinforced plas-
tic plate is 37230 MPa. The experimental ultimate load is
172.4 kN [30]. The analytical ultimate load is 165.5 kN.
The results solved using FEM and experimental results
are put together for comparison. Results indicate that the
central deflection calculated using the FEM agreed very
well with the measured deflection. Figure 4 shows the
central deflection of the reinforced concrete beam stren-
gthened by epoxy bonded a 4 mm thick glass fiber rein-
forced plastic plate along the tension face. The experi-
mental ultimate load is 182.5 kN [30]. The analytical ul-
timate load is 220.0 kN. The numerical results and the re-
sults measured experimentally correspond closely. Fig-
ure 5 shows the central deflection of the reinforced con-
crete beam strengthened by epoxy bonded a 6 mm thick
glass fiber reinforced plastic plate along the tension face.
The experimental ultimate load is 255.7 kN [30]. The an-
alytical ultimate load is 248.4 kN. The agreement between
numerical and experiment curves is satisfactory.
Figure 6 displays the central deflection of the rein-
forced concrete beam strengthened by epoxy bonded a 6
mm thick glass fiber reinforced plastic plate along the
Concrete Beams Strengthened with Externally Bonded Glass Fiber Reinforced Plastic Plates 225
Figure 2. Central deflection of the reinforced concrete beam.
Figure 3. Central deflection of the reinforced concrete beamstrengthened by epoxy bonded a 2 mm thick glassfiber reinforced plastic plate along the tension face.
Figure 4. Central deflection of the reinforced concrete beamstrengthened by epoxy bonded a 4 mm thick glassfiber reinforced plastic plate along the tension face.
tension face. The beam size used is 0.205 m (b) 0.565
m (h) 4.875 m (L). The beams are reinforced with two
D13 bars in the compression zone, whereas three D25
bars are provided in the tension. The diameter of D13 bar
is 12.7 mm. The diameter of D25 bar is 25.4 mm. The ex-
perimental ultimate load is 325.0 kN [30]. The analytical
ultimate load is 352.0 kN. The agreement between nu-
merical and experiment results is satisfactory. Figure 7
reveals the central deflection of the reinforced concrete
beam strengthened by epoxy-bonding a 6 mm thick glass
fiber reinforced plastic plate along the tension face. The
beams are reinforced with two D13 bars in the compres-
sion zone, whereas two D25 bars are provided in the ten-
sion. The experimental ultimate load is 255.0 kN [30].
The analytical ultimate load is 263.0 kN. The numerical
results of four-point bending test of reinforced concrete
beams strengthened by externally bonded glass fiber re-
inforced plastic plates in comparing with the experimen-
tal results show satisfactory agreement.
3. Dynamic Analysis of Reinforced Concrete
Beams Strengthened by Epoxy Bonded Glass
Fiber Reinforced Plastic Plates
This work demonstrates the dynamic behavior of re-
inforced concrete beams strengthened by epoxy bonded
glass fiber reinforced plastic plates using the DQM. The
influences of glass fiber reinforced plastic plate thickness
on the dynamic behavior of reinforced concrete beams
strengthened by epoxy bonded glass fiber reinforced plas-
tic plates are investigated. The DQM is used to form the
reinforced concrete beams strengthened by epoxy bond-
ed glass fiber reinforced plastic plates problems in matrix
form. The calculated accuracy and integrity of this prob-
lem is demonstrated by a series of case studies using the
226 Ming-Hung Hsu
Figure 5. Central deflection of the reinforced concrete beamstrengthened by epoxy bonded a 6 mm thick glassfiber reinforced plastic plate along the tension face.
Figure 6. Central deflection of the reinforced concrete beamstrengthened by epoxy bonded a 6 mm thick glassfiber reinforced plastic plate along the tension face.
Figure 7. Central deflection of the reinforced concrete beamstrengthened by epoxy bonded a 6 mm thick glassfiber reinforced plastic plate along the tension face.
DQM. There are many computational methods available
for vibration analysis. Bellman et al. [31,32] introduced
the DQM. DQM reduces the partial differential equations
into a set of algebraic equations and has been used exten-
sively to solve a variety of problems in different fields of
science and engineering [3348].
The efficiency and accuracy of the DQM are not de-
pendent on the number and accuracy of the selected com-
parison functions. The essence of the DQM is that the de-
rivative of a function at a sample point is approximated
by a weighted linear sum of the functional values at all
the sample points in the domain. DQM approximates the
mth order partial derivate of f(z) with respect to z as:
for i, j = 1, 2, ,N
(1)
where f(zi) is the functional value at the sample point zi,
and Dijm( )
are the DQ coefficients of the mth order differ-
entiation attached to these functional values.
The DQ coefficients are difficult to obtain, because
they may be ill conditioned. To overcome the numerical
ill-conditioning in determining the DQ coefficients Dijm( )
,
Quan and Chang [42,43] established a set of algebraic
expressions to calculate the DQ coefficients, namely:
(2)
where
,
for i = 1, 2, , N
Substituting equation (2) into equation (1) yields:
for i, j = 1, 2, ,N and i j
(3)
and
for i = 1, 2, , N (4)
The second-order and higher-order derivatives of the DQ
coefficients can also be obtained by matrix multiplica-
tion [33,34], thus:
for i, j = 1, 2, , N (5)
for i, j = 1, 2, , N (6)
for i, j = 1, 2, , N (7)
for i, j = 1, 2, , N (8)
The selection of sample points always plays an im-
portant role in the solution accuracy of the DQM. The
unequally spaced sample points of each beam using the
Chebyshev-Gauss-Lobatto distribution in the present com-
putation are chosen as the following equation:
for i, j = 1, 2, , N (9)
By assuming the reinforced concrete beam strength-
ened by epoxy bonded a glass fiber reinforced plastic
plate is long, the strain energy of the reinforced concrete
beam strengthened by epoxy bonded a glass fiber rein-
forced plastic plate can be simplified as:
(10)
where E1 is Youngs modulus of the reinforced concrete
beam and E2 is Youngs modulus of the glass fiber rein-
forced plastic plate. u1 is the transverse displacement of
the reinforced concrete beam and u2 is the transverse de-
flection of the glass fiber reinforced plastic plate. I1 is
the 2nd moment area of concrete beam section and I2 is
Concrete Beams Strengthened with Externally Bonded Glass Fiber Reinforced Plastic Plates 227
1 1
2 2m
m
ijm
N N
f z f z
f z f zD
z
f z f z
1 1
N
i
i i i
M zf z f z
z z M z
1
N
j
j
M z z z
1
1,
N
i i j
j j i
M z z z
1(1)
1
i
ij
j i j
M zD
z z M z
1 1
1,
N
ii ij
j j i
D D
(2) (1) (1)
1
N
ij ik kj
k
D D D
(3) (1) (2)
1
N
ij ik kj
k
D D D
(4) (1) (3)
1
N
ij ik kj
k
D D D
( ) (1) ( 1)
1
Nm m
ij ik kj
k
D D D
i
i 11z 1 cos
2 N 1
2 22 2
L Le 1 21 1 2 22 20 0
u u1 1U E I dz E I dz
2 2z z
L 2
1 20
1k u u dz
2
the 2nd moment area of glass fiber reinforced plastic plate
section. k is the stiffness of the epoxy adhesive. The ki-
netic energy of the reinforced concrete beam strength-
ened by epoxy bonded a glass fiber reinforced plastic
plate is
(11)
where 1 is the density of the material of the concrete
beam and !2 is the density of the material of the glass fi-
ber reinforced plastic plate. The cross-section area of the
concrete beam is A1 and the cross-section area of the
glass fiber reinforced plastic plate is A2. Substituting equ-
ations (10) and (11) into Hamiltons equation,
(12)
the differential equations governing the dynamics of re-
inforced concrete beam strengthened by epoxy bonded
a glass fiber reinforced plastic plate are:
(13)
(14)
The corresponding boundary conditions of the reinfor-
ced concrete beam strengthened by epoxy bonded a glass
fiber reinforced plastic plate are:
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
Consider the displacements to be of the form
(23)
(24)
Equations (13) and (14) can be simplified to
(25)
(26)
The corresponding boundary conditions of the reinfor-
ced concrete beam strengthened by epoxy bonded a glass
fiber reinforced plastic plate are:
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
By employing the DQM, the equations of motion of the
reinforced concrete beam strengthened by epoxy bond-
ed a glass fiber reinforced plastic plate can be discre-
228 Ming-Hung Hsu
2 2L Le 1 2
1 1 2 20 0
u u1 1T A dz A dz
2 t 2 t! !
2
1
t e e
tT U dt 0" "
( )2 22
1 11 1 1 1 1 22 2 2
u uA E I k u u 0
t z z!
( )2 22
2 22 2 2 2 2 12 2 2
u uA E I k u u 0
t z z!
,1u 0 t 0
21
1 1 2
0,0
u tE I
z
,1u L t 0
21
1 1 2
,0
u L tE I
z
,2u 0 t 0
22
2 2 2
0,0
u tE I
z
,2u L t 0
22
2 2 2
,0
u L tE I
z
1 1, expu z t U z i t#
2 2, expu z t U z i t#
1U 0 0
1U L 0
2U 0 0
,2U L t 0
21
1 1 2
00
d UE I
dz
21
1 1 20
d U LE I
dz
22
2 2 2
00
d UE I
dz
22
2 2 20
d U LE I
dz
( )22
2 11 1 1 1 1 1 22 2
d UdAU E I k U U
dz dz# !
( )22
2 22 2 2 2 2 2 12 2
d UdA U E I k U U
dz dz# !
teized in a matrix form as the following equation:
(35)
where
(36)
The elements in the mass matrix are:
Mii = !1A1 for i = 3, 4, , N 2 (37)
Mii = !2A2 for i = N + 3, N + 4, , 2N 2 (38)
Mii = 0 for i = 1, 2,N 1,N,N + 1,N + 2, 2N 1, 2N
(39)
Mij = 0 for i j, i = 1, 2, , 2N and j = 1, 2, , 2N
(40)
The elements in the stiffness matrix are:
for i j, i = 3, 4, , N 2 and j = 1, 2, , N (41)
for i = 3, 4, , N 2 (42)
Cij = 0
for i j N, i = 3, 4, , N 2 and j = N + 1, N + 2,
2N (43)
Ci,i+N = -k
for i = 3, 4, , N 2 (44)
Cij = 0
For i j +N, i =N + 3,N + 4, , 2N 2 and j = 1, 2,
, N (45)
Ci,iN = -k
for i = N + 3, N + 4, 2N 2 (46)
for i j, i = N + 3, N + 4, , 2N 2
and j = N + 1, N + 2, , 2N (47)
for i = N + 3, N + 4, , 2N 2 (48)
The boundary conditions of the simple supported rein-
forced concrete beam strengthened by epoxy bonded a
glass fiber reinforced plastic plate can be rearranged into
the matrix form as
for j = 1, 2, ..., N (49)
for j = 1, 2, ..., N (50)
for j = 1, 2, ..., N (51)
for j = 1, 2, ..., N (52)
Concrete Beams Strengthened with Externally Bonded Glass Fiber Reinforced Plastic Plates 229
2[ ]{ ( )} [ ]{ ( )}ij j ij jC W z M W z#
1 1 1
2 1 2
1
1 2 1
2 2 2
2 2
N N
N
N
N N
W z U z
W z U z
W z U z
W z U z
W z U z
W z U z
2 32
1 1 1 1
2 2 32
ii
ij ij
ij
z zz z
D Dd E I z dE I zC
dzdz L L
4
1 1 4
ij
i
DE I z
L
2 2 3
1 1 1 1
2 2 32
ii
ii iiii
z zz z
d E I z dE I zD DC
dzdz L L
4
1 1 4
iii
DE I z k
L
22
,2 2 2 2
2 22
i Ni N
i N j N
ij
z zz z
Dd E I z dE I zC
dzdz L
3 4
, ,
2 23 4
i N j N i N j N
i N
D DE I z
L L
22
,2 2 2 2
2 22
i Ni N
i N i N
ii
z zz z
Dd E I z dE I zC
dzdz L
3 4
, ,
2 23 4
i N i N i N i N
i N
D DE I z k
L L
(2) (2)(2)(2) (2)11, 1 1,1311 12
2 2 2 2 2
1 0 0 0 00
{ ( )}0
jN NU zD DDD D
L L L L L
(2) (2)(2)(2) (2)21, 1 1,1311 12
2 2 2 2 2
1 0 0 0 00
{ ( )}0
jN NU zD DDD D
L L L L L
(2) (2)(2) (2) (2)1, 1 ,1 2 3
2 2 2 2 2
0 0 0 0 10
{ ( )}0
jN N N NN N NU zD DD D D
L L L L L
(2) (2)(2) (2) (2)2, 1 ,1 2 3
2 2 2 2 2
0 0 0 0 10
{ ( )}0
jN N N NN N NU zD DD D D
L L L L L
Figure 8 illustrates the frequencies of the reinforced
concrete beams strengthened by epoxy bonded glass fi-
ber reinforced plastic plates with different thicknesses.The beam size used is 0.20 m (b) 0.30 m (h) 3.00 m
(L). The beams are reinforced with two D6 bars in the
compression zone, whereas two D9 bars are provided in
the tension. The numerical results show that higher fre-
quencies are found for the beam with a higher thickness
glass fiber reinforced plastic plate. The frequency of the
reinforced concrete beam strengthened by epoxy-bond-
ing a 2 mm thick glass fiber reinforced plastic plate along
the tension face is greater than the frequency of the rein-
forced concrete beam strengthened by no epoxy-bonding
glass fiber reinforced plastic plate for mode 1. The fre-
quencies of the reinforced concrete beam strengthened
by epoxy-bonding a 2 mm thick glass fiber reinforced
plastic plate along the tension face is smaller than the fre-
quency of the reinforced concrete beam strengthened by
no epoxy bonded glass fiber reinforced plastic plate for
mode 2 to mode 6.
4. Conclusion
The results conduct that the flexure strength of rein-
forced concrete beams can be significantly increased by
externally bonded glass fiber reinforced plastic plates.
The calculated results agreed closely with the measured
results in the literature. This paper demonstrates the dy-
namic behavior of reinforced concrete beams strengthened
by epoxy bonded glass fiber reinforced plastic plates us-
ing the DQM. The results indicate that bonding compos-
ite plates to concrete beams increase the stiffness, yield
moment, and flexural strength of the beams. The simplic-
ity of this formulation makes it a desirable candidate for
modeling reinforced concrete beams strengthened by ep-
oxy bonded glass fiber reinforced plastic plates.
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Manuscript Received: Dec. 21, 2004
Accepted: Nov. 4, 2005
232 Ming-Hung Hsu
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