analytical model of reinforced cement concrete

International Journal of Advanced Engineering Technology

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

Research Article

ANALYTICAL MODEL OF REINFORCED CEMENT CONCRETE

BEAM USING GLASS FIBRE REINFORCED POLYMER *1Prof. Parikh Kaushal. B. ,

2Dr. Modhera Chetan. D.

Address for correspondence *1Department of Applied Mechanics, Government Engineering College, Surat, Gujarat,

India & Research scholar, Department of Applied Mechanics, SVNIT, Surat

E-mail: [email protected] 2Department 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.

INTRODUCTION

Glass fiber reinforced polymer laminates

are increasingly being applied for the

rehabilitation and strengthening of

infrastructure in lieu of traditional repair

techniques such as steel plates bonding.

FRP plates have many advantages over

steel plates in this application, and their

use can be extended to situations where

it would be impossible or impractical to

use steel. For example, FRP plates are

lighter than steel plates of equivalent

strength, which eliminates the need for

temporary support for the plates while

the adhesive gains strength. Also, since

FRP plates used for external bonding are

relatively thin, neither the weight of the

structure nor its dimensions are

significantly increased. The latter may be

important for bridges and tunnels with

limited headroom, or when

strengthening in two directions. In

addition, FRP plates can easily be cut to

length on site. These various factors in

combination make installation much

simpler and quicker than when using

steel plates. This is particularly

advantageous for bridges due to the high



costs of lane closures and possession

times on major highways and railway

lines.

Equally important is the fact that the

materials used to manufacture FRP

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

if correctly specified, and hence

requirements for maintenance are low. If

the materials are damaged in service, it

is relatively simple to repair them, by

bonding an additional layer. In addition

to plates, various types of fibres are

available in the form of fabrics, which

can be bonded to the concrete surface.

The chief advantage of fabrics over

plates is that they can be wrapped

around curved surfaces, for example

around columns and chimneys, or

completely around the sides and soffit of

beams. Experience has shown that

exhaustive testing is a very expensive

and time-consuming process and in

recent years more emphasis has placed

on numerical simulation complement

testing. The development of high speed

computers and more sophisticated non-

linear constitutive material models

capable of simulating exactly what

happens experimentally has helped to

make this transition. This paper presents

an analytical model of reinforced cement

concrete beam wrapped by glass

reinforced polymer sheet by using

ATENA software. This research article

has been published for pursing Ph.D of

first author.

MATERIAL MODELLING

Concrete

In ATENA, concrete can be modelled as

3DNonlinear Cementitious. In this set of

parameters is generated based on codes

and recommendations. This Fracture-

plastic model combines constitutive

models for tensile (fracturing) and

compressive (plastic) behavior. The

fracture model is based on the classical

orthotropic smeared crack formulation

and crack band model. The material

CC3DNonLinCementitious2 assumes a

hardening regime before the

compressive strength is reached and

purely incremental formulation is used.

Concrete in compression is considered to

be a strain softening material. Any

parameter can be changed by editing the

contents of its numerical field. The

nonlinear behavior of concrete in the

biaxial stress state is described by means

of the so-called effective stress σcef, and

the equivalent uniaxial strain εeq .The

effective stress is in most cases a

principal stress. The numbers of the

diagram parts in Fig. 1 (material state

numbers) are used in the results of the



analysis to indicate the state of damage

of concrete.

Fig. 1: Uniaxial stress-strain law of

concrete

The above defined stress-strain relation

is used to calculate the elastic modulus

for the material stiffness matrices. The

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. 2: Exponential crack opening law

The softening model is used as

exponential crack opening law as shown

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

opening, Wc is the crack opening at the

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 descending. The fictitious

compression plane model is used which

based on the assumption, that

compression failure is localized in a

plane normal to the direction of

compressive principal stress.

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

indirectly defined.

The material stiffness matrix for the

uncracked concrete has the form of an

elastic matrix of the isotropic material. It



is written in the global coordinate system

x and y.

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

The stresses in concrete are obtained

using the actual secant component

material stiffness matrix

Where is the secant material

stiffness matrix for the uncracked or

cracked concrete depending on the

material state.

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

used to simulate contact between two

materials such as concrete and glass

fiber reinforced polymer sheet.

The interface material is based on Mohr-

Coulomb criterion with tension cut off.

The constitutive relation is given in



terms of tractions on interface planes and

relative sliding and opening

displacements.

Linear bond-slip relationship for the

interface is assumed in both tangential

and normal directions as shown in fig.

5(a) and (b).

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

ATENA uses the following formulas

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 material model

Reinforcement is modeled as smeared.

The smeared reinforcement is a

component of composite material and

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

reinforcements. Here the bilinear stress-

strain is assumed for all reinforcement as

shown in the fig. 6.

Fig. 6: the bilinear stress-strain law

for reinforcement.

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

between concrete and reinforcement.

And

Glass fibre polymer sheet (GFRP)

model

Here GFRP material is modelled as 3D

elastic isotropic i.e. FRP plate was

assumed to behave elastically up to

rupture, the idealized stress-strain curve

is presented in Fig. 7.

Fig. 7: linear stress – strain relation

for GFRP.



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



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

Author(s) Index L

(mm)

l

(mm)

b

(mm)

h

(mm)

Asc

(mm2)

Ast

(mm2)

Asv

(mm2)

Sv

(mm)

N. Dash F1 2300 2000 200 250 56.6 226.2 56.6 150

F2 2300 2000 200 250 56.6 226.2 56.6 150

F3 2300 2000 200 250 56.6 226.2 56.6 115

A. Parghi et.

al 1 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

Sing-Ping

Chiew et. al 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

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



Continue Table 2…………









sheet.

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 10

5

F3 31 437 2.10 x 105 240 2.10 x 10

5

A. Parghi et al. 1 29 415 2.10 x 105 250 2.10 x 10

5

2 29 415 2.10 x 105 250 2.10 x 10

5

3 29 415 2.10 x 105 250 2.10 x 105

4 29 415 2.10 x 105 250 2.10 x 10

5

Sing-Ping Chiew

et. al A1 41.4 516 2.06 x 10

5 560 2.03 x 10

5

A2 41.4 516 2.06 x 105 560 2.03 x 10

5

A3 41.4 516 2.06 x 105 560 2.03 x 10

5

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 10

5

B1 41.4 516 2.06 x 105 560 2.03 x 10

5

B2 41.4 516 2.06 x 105 560 2.03 x 10

5

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 10

5

B6 41.4 516 2.06 x 105 560 2.03 x 10

5



Author(s) Index t

(mm)

l1

(mm)

l2

(mm)

l3

(mm)

Eg

(MPa) Remarks

N. Dash F1 -- 667 667 -- -- Control Beam

F2 2.2 667 667 667 11310 Wrapping on bottom

F3 2.2 667 667 667 11310 Wrapping on bottom

& side up to NA

A. Parghi 1 -- 333 333 333 --- 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

Sing-Ping Chiew

et. al 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 Three layer

A5 1.7 1000 800 600 27000 Single layer with less

length of wrapping

A6 1.7 1000 800 450 27000 Single layer with less

length of wrapping

B1 -- 400 1100 -- -- 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 Three layer

B5 1.7 400 1100 900 27000 Single layer with less

length of wrapping

B6 1.7 400 1100 750 27000 Single layer with less

length of wrapping









sheet.



Fig. 9: Geometry of ccisogap<….> elements for interface elements.

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

FINITE ELEMENT:

Here in concrete, support, loading steel

plates and glass fibre sheet brick element

as well as tetra element is used from the

ATENA library. For the interface

element Gap element is used from the

ATENA library as shown in fig. 8 and

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

fiber reinforced polymer sheet. The

validation of this model has been carried

by various available literature

experimental data. The geometrical and

mechanical data of experimental

reinforced concrete beam of various

researches are shown in table 2.

Fig.11:Graph of load v/s deflection of

beam [Nishikant Dash]



Table 3: Comparison of results of analytical model with available experimental

results

Author(s) Index

Model results

/Ultimate load

(KN)

Test results

/Ultimate

Load (KN)

Relative

Error δ

(%)

Remarks

N. Dash F1 79.5 78 1.92 Control Beam

F2 97.5 104 -6.25 Wrapping on

bottom

F3 110.3 112 -1.52

Wrapping on

bottom & side up

to NA

A. Parghi et

al. 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

Sing-Ping

Chiew et. al 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 Three layer

A5 190.4 196 -2.86

Single layer with

less length of

wrapping

A6 192.5 204.8 -6.00

Single layer with

less length of

wrapping

B1 118 122 -3.28 Control beam

B2 156 146.2 6.70 Single layer

B3 163 152 5.90 Two layer

B4 187 176.9 5.70 Three layer

B5 140.8 144 -2.22

Single layer with

less length of

wrapping

B6 136.7 145.6 -6.11

Single layer with

less length of

wrapping



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. 12: Finite element model - graph

of load v/s deflection of beam

[A Parghi et. al]

It is very much clear from the graphs

that glass fiber reinforced polymer sheet

enhances the flexural strength of

reinforced concrete beam.


of load v/s deflection of beam

[Sing-Ping chiew et al.]


of load v/s deflection of beam [Sing-

Ping chiew et al.]

The following table 3 shows the

comparison of ultimate load received

from the finite element model and

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

related to the relative slip between the

concrete surface and the FRP plate.

Comparison of the analytical results with

the published experimental data shows



that the proposed finite element model

with interface element can predict the

load deflection response of the

strengthened beam reasonably well, and

is less sensitive to variation of concrete

tensile strength.

REFRENCES

[1] K.B. Parikh and C.D. Modhera,

“Application of glass fibre reinforced

polymer to structural components – A

state of art review”, International

Conference on Advances in Concrete,

Structural and Geotechnical

Engineering, BITS, Pilani (India),

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

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

O’Regan, “TR 55 : Design Guidance for

Strengthening Concrete Structures Using

Fibre Composite Materials: A Review”,

Engineering Structures, Vol. 24, 2002,

pp. 889-900.

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

and C.D. Modhera, “Analytical Work on

Beam by Using GFRP Laminates”,

International conferences in advances in

materials and techniques in civil

engineering, VLBJECT, 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

national conference on current trends on

research and development in civil and

environment engineering – An Indian

perspective, SVIT, Vasad (India),

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

[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,

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

[6] Sing-Ping Chiew, Qin Sun and Yi Yu,

“flexural Strength of RC Beams with

GFRP laminates”, Journal of composites

for Construction, Vol. 11, No. 5,

October 2007, pp. 497-506.

[7] Nishikant Dash, “Strengthening of

Reinforced Concrete Beams using Glass

Fiber Reinforced Polymer composites”,

M.Tech Thesis, NIT, Rourkela (India),

2009, pp. 1-145.

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

In Engineering Analysis”, Prentice-

Hall, Inc.

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

Finite Element Method”, McGraw-Hill

Book Company.

[10] CEB-FIP Model Code 1990, “First

Draft, Comitte Euro-International du

Beton”, Bulletin d’information, No.

195-196.

[11] V. Cervenka, “Constitutive Model

for Cracked Reinforced Concrete”,

American concrete Institute Journal,

Vol. 82, No. 6, Nov. – Dec. 1985, pp.

877-882.

[12] V. Cervenka and j. Cervenka,

“ATENA Theory”, Documentation

available with software.

analytical model of reinforced cement concrete

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