9-3-5.pdf

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.


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




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


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)


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.

References

[1] Saadatmanesh, H. and Ehsani, M. R., RC Beams St-

rengthened with GFRP Plates. I. Experimental Study,

Journal of Structural Engineering, Vol. 117, pp. 3417

3433 (1991).

[2] Saadatmanesh, H. and Ehsani, M. R., Flexural Stren-

gth of Externally Reinforced Concrete Beams, Pro-

ceedings of the First Material Engineering Congress,

pp. 11521161 (1990).

[3] Saadatmanesh, H., Fiber Composites for New and Exi-

sting Structures, ACI Structural Journal, Vol. 91, pp.

346354 (1994).

[4] Jin, L., Saadatmanesh, H. and Ehsani, M. R., Seismic

Retrofit of Existing Reinforced Concrete Columns by

Glass-fiber Composites, Proceedings of the Materials

Engineering Conference, pp. 758763 (1994).

[5] Saadatmanesh, H., Ehsani, M. R. and Jin, L., Seismic

Strengthening of Circular Bridge Pier Models with Fi-

ber Composites, ACI Structural Journal, Vol. 93, pp.

639647 (1996).

[6] Tannous, F. E. and Saadatmanesh, H., Environmental

Effects on the Mechanical Properties of E-glass FRP

Rebars, ACI Materials Journal, Vol. 95, pp. 87100

(1998).

[7] Albrecht, P., Li, W. and Saadatmanesh, H., Fatigue

Strength of Prestressed Composite Steel-concrete

Beams, Journal of Structural Engineering, Vol. 121,

pp. 18501856 (1995).

[8] Saadatmanesh, H. and Ehsani, M. R., Fiber Compos-

ite Plates Can Strengthen Beams, Concrete Interna-

tional: Design and Construction, Vol. 12, pp. 6571

(1990).

[9] Saadatmanesh, H., Ehsani, M. R. and Li, M. W., St-

rength and Ductility of Concrete Columns Externally

Reinforced with Fiber Composite Straps, ACI Struc-

tural Journal, Vol. 91, pp. 434447 (1994).

[10] Ehsani, M. R., Saadatmanesh, H. and Tao, S., Bond

Behavior of Deformed GFRP Rebars, Journal of Com-

230 Ming-Hung Hsu

Figure 8. The frequencies of the reinforced concrete beamsstrengthened by epoxy bonded glass fiber reinfor-ced plastic plates with different thicknesses.

posite Materials, Vol. 31, pp. 14131430 (1997).

[11] Saadatmanesh, H., Ehsani, M. R. and Jin, L., Seismic

Retrofitting of Rectangular Bridge Columns with Com-

posite Straps, Earthquake Spectra, Vol. 13, pp. 281

304 (1997).

[12] An, W., Saadatmanesh, H. and Ehsani, M. R., RC

Beams Strengthened with FRP Plates. II. Analysis and

Parametric Study, Journal of Structural Engineering,

Vol. 117, pp. 34343455 (1991).

[13] Hamoush, S. A. and Ahmad, S. H., Debonding of Steel-

plate-strengthened Concrete Beams, Journal of Stru-

ctural Engineering, Vol. 116, pp. 356371 (1990).

[14] Adhikary, B. B., Mutsuyoshi, H. and Sano, M., Shear

Strengthening of Reinforced Concrete Beams Using

Steel Plates Bonded on Beam Web: Experiments and

Analysis, Construction and Building Materials, Vol.

14, pp. 237244 (2000).

[15] Adhikary, B. B. and Mutsuyoshi, H., Numerical Sim-

ulation of Steel-plate Strengthened Concrete Beam by

a Non-linear Finite Element Method Model, Const-

ruction and Building Materials, Vol. 16, pp. 291301

(2002).

[16] Garden, H. N. and Hollaway, L. C., An Experimental

Study of the Failure Modes of Reinforced Concrete

Beams Strengthened with Prestressed Carbon Compo-

site Plates, Composite Part B, Vol. 29B, pp. 411424

(1998).

[17] Garden, H. N., Quantrill, R. J., Hollaway, L. C., Thorne,

A. M. and Parke, G. A. R., An Experimental Study of

the Anchorage Length of Carbon Fibre Composite

Plates Used to Strengthen Reinforced Concrete Beams,

Construction and Building Materials, Vol. 12, pp. 203

219 (1998).

[18] Kankam, C. K., Structural Behavior of Bolted End-

plated Reinforced Concrete Beams, Materials & De-

sign, Vol. 24, pp. 367375 (2003).

[19] Subedi, N. K. and Baglin, P. S., Ultimate Load Analy-

sis of Plate Reinforced Concrete Beams, Engineering

Structures, Vol. 23, pp. 10681079 (2001).

[20] Picard, A., Massicotte, B. and Boucher, E., Stren-

gthening of Reinforced Concrete Beams with Compo-

site Materials: Theoretical Study, Composite Struc-

tures, Vol. 33, pp. 6375 (1995).

[21] Nguyen, N. T., Oehlers, D. J. and Bradford, M. A., An

Analytical Model for Reinforced Concrete Beams with

Bolted Side Plates Accounting for Longitudinal and

Transverse Partial Interaction, International Journal

of Solids and Structures,Vol. 38, pp. 69856996 (2001).

[22] Shen, H. S., Chen, Y. and Yang, J., Bending and Vi-

bration Characteristics of a Strengthened Plate under

Various Boundary Conditions, Engineering Structures,

Vol. 25, pp. 11571168 (2003).

[23] Ascione, L. and Feo, L., Modeling of Composite/

Concrete Interface of RC Beams Strengthened with

Composite Laminates, Composites: Part B, Vol. 31,

pp. 535540 (2000).

[24] Raoof, M., El-Rimawi, J. A. and Hassanen, M. A. H.,

Theoretical and Experimental Study on Externally

Plated R.C. Beams, Engineering Structures, Vol. 22,

pp. 85101 (2000).

[25] Barnes, R. A. and Mays, G. C., The Transfer of Stress

through a Steel to Concrete Adhesive Bond, Interna-

tional Journal of Adhesion & Adhesives, Vol. 21, pp.

495502 (2001).

[26] Barnes, R. A., Baglin, P. S., Mays, G. C. and Subedi,

N. K., External Steel Plate Systems for the Shear St-

rengthening of Reinforced Concrete Beams, Engine-

ering Structures, Vol. 23, pp. 11621176 (2001).

[27] Sapountzakis, E. J. and Katsikadelis, J. T., Interface

Forces in Composite Steel-concrete Structure, Inter-

national Journal of Solids and Structures, Vol. 37, pp.

44554472 (2000).

[28] Smith, S. T. and Teng, J. G., Interfacial Stresses in

Plated Beams, Engineering Structures, Vol. 23, pp.

857871 (2001).

[29] Wang, Y. C., Analytical and Experimental Study on

Seismic Performance of RC T-beams with Design De-

ficiency in Steel Bar Curtailment, Engineering Struc-

tures, Vol. 25, pp. 215227 (2003).

[30] Hsu, M. H., Quantitative Analysis of RC Beams St-

rengthened by Externally Bonded GFRP, Master The-

sis, Feng Chia University (1997).

[31] Bellman, R. E. and Casti, J., Differential Quadrature

and Long-term Integration, Journal of Mathematical

Analysis and Application,Vol. 34, pp. 235238 (1971).

[32] Bellman, R. E., Kashef, B. G. and Casti, J., Differen-

tial Quadrature: A Technique for Rapid Solution of

Nonlinear Partial Differential Equations, Journal Com-

putational Physics, Vol. 10, pp. 4052 (1972).

[33] Bert, C. W., Jang, S. K. and Striz, A. G., Two New

Approximate Methods for Analyzing Free Vibration of

Structural Components, AIAA Journal, Vol. 26, pp.


612618 (1988).

[34] Bert, C. W. and Malik, M., Free Vibration Analysis of

Tapered Rectangular Plates by Differential Quadrature

Method: a Semi-analytical Approach, Journal of Sound

and Vibration, Vol. 190, pp. 4163 (1996).

[35] Bert, C. W. and Mailk, M., Differential Quadrature

Method in Computational Mechanics: A Review, Ap-

plied Mechanics Review, Vol. 49, pp. 128 (1996).

[36] Bert, C. W. and Malik, M., On the Relative Effects of

Transverse Shear Deformation and Rotary Inertia on

the Free Vibration of Symmetric Cross-ply Laminated

Plates, Journal of Sound and Vibration, Vol. 193, pp.

927933 (1996).

[37] Bert, C. W., Wang, X. and Striz, A. G., Differential

Quadrature for Static and Free Vibration Analysis of

Anisotropic Plates, International Journal of Solids and

Structures, Vol. 30, pp. 17371744 (1993).

[38] Bert, C. W., Wang, X. and Striz, A. G., Convergence

of the DQ Method in the Analysis of Anisotropic

Plates, Journal of Sound and Vibration, Vol. 170, pp.

140144 (1994).

[39] Jang, S. K., Bert, C. W. and Striz, A. G., Application

of Differential Quadrature to Static Analysis of Struc-

tural Components, International Journal for Numeri-

calMethods inEngineering,Vol. 28, pp. 561577 (1989).

[40] Wang, X. and Bert, C. W., A New Approach in Ap-

plying Differential Quadrature to Static and Free Vi-

brational Analyses of Beams and Plates, Journal of

Sound and Vibration, Vol. 162, pp. 566572 (1993).

[41] Malik, M. and Bert, C. W., Implementing Multiple

Boundary Conditions in the DQ Solution of Higher-

order PDEs Application to Free Vibration of Plates,

International Journal for Numerical Methods in Engi-

neering, Vol. 39, pp. 12371258 (1996).

[42] Quan, J. R. and Chang, C. T., New Insights in Solving

Distributed System Equations by the Quadrature Met-

hod-I. Analysis, Computers in Chemical Engineer-

ing, Vol. 13, pp. 779788 (1989).

[43] Quan, J. R. and Chang, C. T., New Insights in Solving

Distributed System Equations by the Quadrature Met-

hod-II. Numerical Experiments, Computers in Chem-

ical Engineering, Vol. 13, pp. 10171024 (1989).

[44] Sherbourne, A. N. and Pandey, M. D., Differential

Quadrature Method in the Buckling Analysis of Beams

and Composite Plates, Computers and Structures,Vol.

40, pp. 903913 (1991).

[45] Feng, Y. and Bert, C. W., Application of the Quadra-

ture Method to Flexural Vibration Analysis of a Geo-

metrically Nonlinear Beam, Nonlinear Dynamics, Vol.

3, pp. 1318 (1992).

[46] Chen, W. and Zhong, T., The Study on the Nonlinear

Computations of the DQ and DC Methods, Journal of

Numerical Method for Partial Differential Equations,

Vol. 13, pp. 5775 (1997).

[47] Chen, R. P., Zhou, W. H., Wang, H. Z. and Chen, Y. M.,

One-dimensional Nonlinear Consolidation of Multi-

layered Soil by Differential Quadrature Method, Com-

puters and Geotechnics, Vol. 32, pp. 358369 (2005).

[48] Tomasiello, S., Differential Quadrature Method: Ap-

plication to Initial-boundary-value Problems, Journal

of Sound and Vibration,Vol. 218, pp. 573585 (1998).

Manuscript Received: Dec. 21, 2004

Accepted: Nov. 4, 2005

232 Ming-Hung Hsu

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