International Journal of Advanced Engineering Technology

Research Article

ANALYTICAL MODEL OF REINFORCED CEMENT CONCRETE BEAM USING GLASS FIBRE REINFORCED POLYMER *1

Prof. Parikh Kaushal. B. , 2Dr. Modhera Chetan. D.

Address for correspondence Department of Applied Mechanics, Government Engineering College, Surat, Gujarat, India & Research scholar, Department of Applied Mechanics, SVNIT, Surat E-mail: [email protected] 2 Department of Applied Mechanics, Sardar Vallabhbhai National Institute of Technology, Surat, India Email: [email protected] ABSTRACT World wide, a great deal of research is currently being conducted concerning the use of fiber reinforced laminates/sheets in the repair and strengthening of reinforced concrete members. Fiber reinforced polymer (FRP) application is a very effective way to repair and strengthen structures that have become structurally weak over their life span. FRP repair systems provide an economically viable alternative to traditional repair system and materials. Analytical investigations on the flexural behaviour of RC beams strengthened using continuous glass fiber reinforced polymer sheets are carried out by using ATENA software. The effect of number of layers of sheet on ultimate load carrying capacity and failure mode of the beams are investigated. KEYWORDS Beam, Glass fiber reinforced polymer sheet; reinforced cement concrete beam, finite element modelling; ATENA. *1

INTRODUCTION

the adhesive gains strength. Also, since

Glass fiber reinforced polymer laminates

FRP plates used for external bonding are

are increasingly being applied for the

relatively thin, neither the weight of the

rehabilitation

structure

and

strengthening

of

nor

its

dimensions

are

infrastructure in lieu of traditional repair

significantly increased. The latter may be

techniques such as steel plates bonding.

important for bridges and tunnels with

FRP plates have many advantages over

limited

steel plates in this application, and their

strengthening in two directions. In

use can be extended to situations where

addition, FRP plates can easily be cut to

it would be impossible or impractical to

length on site. These various factors in

use steel. For example, FRP plates are

combination make installation much

lighter than steel plates of equivalent

simpler and quicker than when using

strength, which eliminates the need for

steel

temporary support for the plates while

advantageous for bridges due to the high

IJAET/Vol. I/ Issue I/April-June, 2010/46-58

headroom,

plates.

This

or

is

when

particularly

International Journal of Advanced Engineering Technology

costs of lane closures and possession

reinforced polymer sheet by using

times on major highways and railway

ATENA software. This research article

lines.

has been published for pursing Ph.D of

Equally important is the fact that the

first author.

materials used to manufacture FRP

MATERIAL MODELLING

plates (i.e., fibres and resin) are durable

Concrete

if

hence

In ATENA, concrete can be modelled as

requirements for maintenance are low. If

3DNonlinear Cementitious. In this set of

the materials are damaged in service, it

parameters is generated based on codes

is relatively simple to repair them, by

and recommendations. This Fracture-

bonding an additional layer. In addition

plastic model combines constitutive

to plates, various types of fibres are

models for tensile (fracturing) and

available in the form of fabrics, which

compressive (plastic) behavior. The

can be bonded to the concrete surface.

fracture model is based on the classical

The chief advantage of fabrics over

orthotropic smeared crack formulation

plates is that they can be wrapped

and crack band model. The material

around curved surfaces, for example

CC3DNonLinCementitious2 assumes a

around columns and chimneys, or

hardening

completely around the sides and soffit of

compressive strength is reached and

beams. Experience has shown that

purely incremental formulation is used.

exhaustive testing is a very expensive

Concrete in compression is considered to

and time-consuming process and in

be a strain softening material. Any

recent years more emphasis has placed

parameter can be changed by editing the

on numerical simulation complement

contents of its numerical field. The

testing. The development of high speed

nonlinear behavior of concrete in the

computers and more sophisticated non-

biaxial stress state is described by means

linear

models

of the so-called effective stress σcef, and

capable of simulating exactly what

the equivalent uniaxial strain εeq .The

happens experimentally has helped to

effective stress is in most cases a

make this transition. This paper presents

principal stress. The numbers of the

an analytical model of reinforced cement

diagram parts in Fig. 1 (material state

concrete

numbers) are used in the results of the

correctly

specified,

constitutive

beam

and

material

wrapped

by

glass

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

regime

before

the

International Journal of Advanced Engineering Technology

analysis to indicate the state of damage

opening, Wc is the crack opening at the

of concrete.

complete release of stress, f is the normal stress in the crack (crack cohesion). Gf is the fracture energy needed to create a unit area of stress-free crack, ft’ is the effective tensile strength derived from a failure function. The softening law in compression is linearly

Fig. 1: Uniaxial stress-strain law of concrete

is used to calculate the elastic modulus for the material stiffness matrices. The

The

fictitious

compression plane model is used which based

The above defined stress-strain relation

descending.

on

the

assumption,

that

compression failure is localized in a plane

normal

to

the

direction

of

compressive principal stress.

secant modulus is calculated as . The behavior of concrete in tension without cracks is assumed linear elastic. A fictitious crack model based on a crack-opening law and fracture energy is used for crack opening. Fig. 3: softening displacement law in compression. In case of compression, the end point of the softening curve is defined by means of the plastic displacement wd. In this way, the energy needed for generation of a unit area of the failure plane is Fig. 2: Exponential crack opening law

indirectly defined.

The

as

The material stiffness matrix for the

exponential crack opening law as shown

uncracked concrete has the form of an

in the Fig. 2, where, W is the crack

elastic matrix of the isotropic material. It

softening

model

is

used

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

International Journal of Advanced Engineering Technology

is written in the global coordinate system

Where

x and y.

stiffness matrix for the uncracked or

is the secant material

cracked concrete depending on the material state. In the above E is the concrete elastic modulus derived from the equivalent uniaxial law. The Poisson's ratio ν is constant.

Fig. 4: Failure surface of interface element For the cracked concrete the matrix has the form of the elastic matrix for the orthotropic material. The stiffness matrix has given by

Fig. 5: Typical interface model behavior in (a) shear and (b) tension Following are the parameters have been used for the constitutive model for the generation of the model. The formulas for these functions are taken from the CEB-FIP Model Code 90. Interface material model Here interface material model can be

The stresses in concrete are obtained using the actual secant component material stiffness matrix

used to simulate contact between two materials such as concrete and glass fiber reinforced polymer sheet. The interface material is based on MohrCoulomb criterion with tension cut off. The constitutive relation is given in

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

International Journal of Advanced Engineering Technology

terms of tractions on interface planes and

reinforcements. Here the bilinear stress-

relative

strain is assumed for all reinforcement as

sliding

and

opening

shown in the fig. 6.

displacements.

Linear bond-slip relationship for the interface is assumed in both tangential and normal directions as shown in fig. Fig. 6: the bilinear stress-strain law

5(a) and (b).

for reinforcement.

The ktt and knn denote the initial elastic normal and shear stiffness respectively. The contact between surface and glass fiber

reinforced

polymer

sheet

considered as 3D interface having zero thickness. To estimate the stiffness value

The initial elastic part has the elastic modulus of steel Es. The second line represents the plasticity of the steel with hardening and its slope is the hardening modulus Esh. The CEB-FIB model code 1990, bond slip law is used for the bond

ATENA uses the following formulas

between concrete and reinforcement. And Where E and G is minimal elastic modulus and shear modulus respectively of the surrounding material, t is the width of the interface zone.

Reinforcement is modeled as smeared. smeared

reinforcement

model Here GFRP material is modelled as 3D elastic isotropic i.e. FRP plate was

Reinforcement material model

The

Glass fibre polymer sheet (GFRP)

is

a

component of composite material and

assumed to behave elastically up to rupture, the idealized stress-strain curve is presented in Fig. 7.

can be considered either as a single (only one-constituent) material in the element under consideration or as one of the more such constituents. The smeared reinforcement can be an element with concrete

containing

one

or

more

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

Fig. 7: linear stress – strain relation for GFRP.

International Journal of Advanced Engineering Technology

Table: 1 Parameters to be used in the model Parameter

Formula

Cylinder Strength Tensile strength Initial elastic modulus Poisson's ratio Softening compression Type of tension softening Compressive strength in cracked concrete Tension stiffening stress Shear retention factor

variable

Tension-compression function type

linear

Fracture energy Gf according to VOS 1983 Orientation factor for strain localization

Fig. 8:Geometry of (a) ccisobrick elements. and (b) ccisotetra elements.

Fig. 9:Geometry of ccisogap elements for interface elements IJAET/Vol. I/ Issue I/April-June, 2010/47-59

International Journal of Advanced Engineering Technology

Table 2 : Geometrical and Mechanical data of the experimental R/C beam L

l

b

Author(s)

Index

N. Dash

F1

2300

2000

200

F2

2300

2000

F3

2300

1

A. Parghi et. al

Sing-Ping Chiew et. al

h

Asc

Ast

Asv

Sv

2

2

2

250

56.6

226.2

56.6

150

200

250

56.6

226.2

56.6

150

2000

200

250

56.6

226.2

56.6

115

1200

1000

150

200

100.5

100.5

56.6

115

2

1200

1000

150

200

100.5

100.5

56.6

115

3

1200

1000

150

200

100.5

100.5

56.6

115

4

1200

1000

150

200

100.5

100.5

56.6

115

A1

2800

2600

200

350

157

402.0

157

150

A2

2800

2600

200

350

157

402.0

157

150

A3

2800

2600

200

350

157

402.0

157

150

A4

2800

2600

200

350

157

402.0

157

150

A5

2800

2600

200

350

157

402.0

157

150

A6

2800

2600

200

350

157

402.0

157

150

B1

2800

2600

200

350

157

402.0

157

150

B2

2800

2600

200

350

157

402.0

157

150

B3

2800

2600

200

350

157

402.0

157

150

B4

2800

2600

200

350

157

402.0

157

150

B5

2800

2600

200

350

157

402.0

157

150

B6

2800

2600

200

350

157

402.0

157

150

(mm) (mm) (mm) (mm) (mm ) (mm ) (mm ) (mm)

Where L = total length of beam, l = effective span of beam, b = width of beam, h = depth of beam, Ast = Area of tension reinforcement, Asc = Area of compression reinforcement, Asv = Area of vertical stirrups, Sv = spacing of stirrups, fy1 = yield strength of main, reinforcement, fy2 = yield strength of stirrups, Es1 = young modulas of main, reinforcement, Es2 = young modulas of stirrups, fck = compressive strength of concrete, l1 = length between two loading point, l2 = length from loading point to support, l3 = length from loading point to laminate, t = thickness of glass fiber reinforced polymer sheet, Eg = young modulas of glass fiber reinforced polymer sheet IJAET/Vol. I/ Issue I/April-June, 2010/47-59

International Journal of Advanced Engineering Technology

Continue Table 2…………

Author(s)

Index

fck (MPa)

fy1 (MPa)

Es1 (MPa)

fy2 (MPa)

Es2 (MPa)

N. Dash

F1

31

437

2.10 x 105

240

2.10 x 105

F2

31

437

2.10 x 105

240

2.10 x 105

F3

31

437

2.10 x 105

240

2.10 x 105

1

29

415

2.10 x 105

250

2.10 x 105

2

29

415

2.10 x 105

250

2.10 x 105

3

29

415

2.10 x 105

250

2.10 x 105

4

29

415

2.10 x 105

250

2.10 x 105

A1

41.4

516

2.06 x 105

560

2.03 x 105

A2

41.4

516

2.06 x 105

560

2.03 x 105

A3

41.4

516

2.06 x 105

560

2.03 x 105

A4

41.4

516

2.06 x 105

560

2.03 x 105

A5

41.4

516

2.06 x 105

560

2.03 x 105

A6

41.4

516

2.06 x 105

560

2.03 x 105

B1

41.4

516

2.06 x 105

560

2.03 x 105

B2

41.4

516

2.06 x 105

560

2.03 x 105

B3

41.4

516

2.06 x 105

560

2.03 x 105

B4

41.4

516

2.06 x 105

560

2.03 x 105

B5

41.4

516

2.06 x 105

560

2.03 x 105

B6

41.4

516

2.06 x 105

560

2.03 x 105

A. Parghi et al.

Sing-Ping Chiew et. al

Where L = total length of beam, l = effective span of beam, b = width of beam, h = depth of beam, Ast = Area of tension reinforcement, Asc = Area of compression reinforcement, Asv = Area of vertical stirrups, Sv = spacing of stirrups, fy1 = yield strength of main, reinforcement, fy2 = yield strength of stirrups, Es1 = young modulas of main, reinforcement, Es2 = young modulas of stirrups, fck = compressive strength of concrete, l1 = length between two loading point, l2 = length from loading point to support, l3 = length from loading point to laminate, t = thickness of glass fiber reinforced polymer sheet, Eg = young modulas of glass fiber reinforced polymer sheet.

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

International Journal of Advanced Engineering Technology

Author(s)

Index

N. Dash

F1

A. Parghi

Sing-Ping Chiew et. al

t l1 l2 l3 Eg (mm) (mm) (mm) (mm) (MPa) -667 667 ---

Remarks Control Beam

F2

2.2

667

667

667

11310

F3

2.2

667

667

667

11310

1

--

333

333

333

---

Wrapping on bottom Wrapping on bottom & side up to NA Control Beam

2

1.2

333

333

333

---

Single layer

3

2.4

333

333

333

---

Two layer

4

3.6

333

333

333

---

Three layer

A1

--

1000

800

--

--

Control Beam

A2

1.7

1000

800

750

27000

Single layer

A3

3.4

1000

800

750

27000

Two layer

A4

5.1

1000

800

750

27000

A5

1.7

1000

800

600

27000

A6

1.7

1000

800

450

27000

B1

--

400

1100

--

--

Three layer Single layer with less length of wrapping Single layer with less length of wrapping Control beam

B2

1.7

400

1100

1050

27000

Single layer

B3

3.4

400

1100

1050

27000

Two layer

B4

5.1

400

1100

1050

27000

B5

1.7

400

1100

900

27000

B6

1.7

400

1100

750

27000

Three layer Single layer with less length of wrapping Single layer with less length of wrapping

Where L = total length of beam, l = effective span of beam, b = width of beam, h = depth of beam, Ast = Area of tension reinforcement, Asc = Area of compression reinforcement, Asv = Area of vertical stirrups, Sv = spacing of stirrups, fy1 = yield strength of main, reinforcement, fy2 = yield strength of stirrups, Es1 = young modulas of main, reinforcement, Es2 = young modulas of stirrups, fck = compressive strength of concrete, l1 = length between two loading point, l2 = length from loading point to support, l3 = length from loading point to laminate, t = thickness of glass fiber reinforced polymer sheet, Eg = young modulas of glass fiber reinforced polymer sheet.

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

International Journal of Advanced Engineering Technology

Fig. 9: Geometry of ccisogap elements for interface elements.

Fig. 10: Typical finite element model of gfrp sheet strength beam FINITE ELEMENT:

fiber reinforced polymer sheet. The

Here in concrete, support, loading steel

validation of this model has been carried

plates and glass fibre sheet brick element

by

as well as tetra element is used from the

experimental data. The geometrical and

ATENA

mechanical

library.

For

the

interface

various

available

data

of

literature

experimental

element Gap element is used from the

reinforced concrete beam of various

ATENA library as shown in fig. 8 and

researches are shown in table 2.

fig. 9. FINITE ELEMENT MODEL FOR BEAM Using finite element programme of non linear

analysis

ATENA

software,

analytical model for beam having glass fiber reinforced polymer has been developed. Fig. 10 shows typical finite element model of beam with using glass IJAET/Vol. I/ Issue I/April-June, 2010/47-59

Fig.11:Graph of load v/s deflection of beam [Nishikant Dash]

International Journal of Advanced Engineering Technology

Table 3: Comparison of results of analytical model with available experimental results Test results /Ultimate Load (KN)

Relative Error δ (%)

Author(s)

Index

Model results /Ultimate load (KN)

N. Dash

F1

79.5

78

1.92

F2

97.5

104

-6.25

F3

110.3

112

-1.52

1

63.4

60

5.67

Control Beam

2

90.8

88

3.18

Single layer

3

108.9

100

8.90

Two layer

4

126.8

120

5.67

Three layer

A1

159

163

-2.45

Control Beam

A2

200.6

203.5

-1.43

Single layer

A3

219

219.3

-0.14

Two layer

A4

236.2

238.5

-0.96

A5

190.4

196

-2.86

A6

192.5

204.8

-6.00

B1

118

122

-3.28

Three layer Single layer with less length of wrapping Single layer with less length of wrapping Control beam

B2

156

146.2

6.70

Single layer

B3

163

152

5.90

Two layer

B4

187

176.9

5.70

B5

140.8

144

-2.22

B6

136.7

145.6

-6.11

Three layer Single layer with less length of wrapping Single layer with less length of wrapping

A. Parghi et al.

Sing-Ping Chiew et. al

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

Remarks Control Beam Wrapping on bottom Wrapping on bottom & side up to NA

International Journal of Advanced Engineering Technology

RESULTS AND DISCUSSION Using the finite element model of beam the following results and graphs were obtained. The graphs are as shown in fig. 11 to14.

Fig. 14: Finite element model - graph of load v/s deflection of beam [SingPing chiew et al.]

The following table 3 shows the Fig. 12: Finite element model - graph of load v/s deflection of beam

from the finite element model and

[A Parghi et. al] It is very much clear from the graphs that glass fiber reinforced polymer sheet enhances

the

flexural

strength

comparison of ultimate load received

of

reinforced concrete beam.

available experimental researches. From the above table it is very much clear that the generated model accurately accesses the flexural strength of beam wrapped with glass fiber reinforced polymer sheet. CONCLUSION This

paper

presents

a

numerical

modelling technique for FRP plate strengthened RC beams by using type of 3D interface element in a standard finite

element

analysis of ATENA

software. It is assumed that the bond development along the interface is Fig. 13: Finite element model - graph

related to the relative slip between the

of load v/s deflection of beam

concrete surface and the FRP plate.

[Sing-Ping chiew et al.]

Comparison of the analytical results with the published experimental data shows

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

International Journal of Advanced Engineering Technology

that the proposed finite element model

perspective,

with interface element can predict the

January 21-22, 2010, pp. 1-8

load

deflection

response

of

the

strengthened beam reasonably well, and is less sensitive to variation of concrete

SVIT,

Vasad

(India),

[5] W.F. Wong, S.P. Chiew and Q. Sun, “Flexural

Strength

of

RC

Beams

Strengthened with FRP Plate”, FRP Composites of Civil Engineering, Vol. 1,

tensile strength.

J.G. Tang (Ed), 2001, pp. 633-640.

REFRENCES

[6] Sing-Ping Chiew, Qin Sun and Yi Yu, “flexural Strength of RC Beams with

[1] K.B.

Parikh

and

C.D.

Modhera,

GFRP laminates”, Journal of composites

“Application of glass fibre reinforced

for Construction, Vol. 11, No. 5,

polymer to structural components – A

October 2007, pp. 497-506.

state of art review”,

International

[7] Nishikant

Dash,

“Strengthening

of

Conference on Advances in Concrete,

Reinforced Concrete Beams using Glass

Structural

Fiber Reinforced Polymer composites”,

and

Engineering,

BITS,

Geotechnical Pilani

(India),

October 25-27, 2009, pp. 1-10

M.Tech Thesis, NIT, Rourkela (India), 2009, pp. 1-145.

[2] C.Arya, J.L. Clarke, E.A. Kay and P.D.

[8] K.J. Bathe, “Finite Element Procedures

O’Regan, “TR 55 : Design Guidance for

In Engineering Analysis”, Prentice-

Strengthening Concrete Structures Using

Hall, Inc.

Fibre Composite Materials: A Review”,

[9] O.C. Zienkiewicz and R.L. Taylor, “The

Engineering Structures, Vol. 24, 2002,

Finite Element Method”, McGraw-Hill

pp. 889-900.

Book Company.

[3] K.B. Parikh, M.M. Shirgar, K.M. Shiraj

[10]

CEB-FIP Model Code 1990, “First

and C.D. Modhera, “Analytical Work on

Draft, Comitte Euro-International du

Beam by Using GFRP Laminates”,

Beton”, Bulletin d’information,

International conferences in advances in

195-196.

materials

and

engineering,

techniques

VLBJECT,

in

civil

Coimbatore

(India), January 07-09, 2010, pp. 67-79. [4] K.B. Parikh, M.M. Shirgar, K.M. Shiraj and C.D. Modhera, “Experimental Work on Beam by using GFRP Laminates”, A

[11] for

V. Cervenka, “Constitutive Model Cracked

Reinforced

Vol. 82, No. 6, Nov. – Dec. 1985, pp. 877-882. [12]

V. Cervenka and j. Cervenka,

“ATENA

research and development in civil and

available with software.

IJAET/Vol. I/ Issue I/April-June, 2010/47-59

Concrete”,

American concrete Institute Journal,

national conference on current trends on environment engineering – An Indian

No.

Theory”,

Documentation