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INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING

Volume 2, No 2, 2011

© Copyright 2010 All rights reserved Integrated Publishing services

Research article ISSN 0976 – 4399

Received on November, 2011 Published on November 2011 579

Theoretical study on the behaviour of Rectangular Concrete Beams

reinforced internally with GFRP reinforcements under pure torsion Prabaghar.A

1, Kumaran.G

2

1-Associate Professor, Structural Engineering Wing, DDE

2- Professor, Department of Civil and Structural Engineering

Annamalai University, Annamalai nagar, 600 002, Tamilnadu, India

[email protected]

doi:10.6088/ijcser.00202010134

ABSTRACT

Finite Element Modelling and analysis of rectangular concrete beams reinforced internally

with Glass Fibre Reinforced Polymer (GFRP) reinforcements under pure torsion is carried

out in this study. Different parameters like grade of concrete, beam longitudinal

reinforcement ratio and transverse stirrups spicing are considered. The basic strength

properties of concrete, steel and GFRP reinforcements are determined experimentally.

Theoretical torque verses twist relationship is established for various values of torque and

twist using elastic, plastic theories of torsion. Finally the ultimate torque is determined using

space truss analogy for different parameters considered in this study. Based on this study, a

good agreement is made between finite element analysis and the theoretical behaviour.

Keywords: pure torsion; beam; GFRP reinforcements; steel; theoretical model

1. Introduction

Fibre Reinforced Polymer (FRP) materials are becoming a new age material for concrete

structures. Its use has been recommended in ACI codes. But in India its applicability is rare in

view of the few manufacturers and lacking in commercial viability. The advantages of the

FRP materials lie in their better structural performance especially in aggressive

environmental conditions in terms of strength and durability (Machida 1993; ACI 440R-96

1996; Nanni 1993). FRP materials are commercially available in the form of cables, sheets,

plates etc. But in the recent times FRPs are available in the form of bars which are

manufactured by pultrusion process which are used as internal reinforcements as an alternate

to the conventional steel reinforcements. These FRP bars are manufactured with different

surface imperfections to develop good bond between the bar and the surrounding concrete.

Fibre reinforcements are well recognised as a vital constituent of the modern concrete

structures. FRP reinforcements are now being used in increasing numbers all over the world,

including India. FRP reinforcements are preferred by structural designers for the construction

of seawalls, industrial roof decks, base pads for electrical and reactor equipment and concrete

floor slabs in aggressive chemical environments owing to their durable properties.

Due to the advantages of FRP reinforcements in mind, many research works have been

carried out throughout the world on the use of FRP reinforcing bars in the structural concrete

flexural elements like slabs, beams, etc. (Nawy et al., 1997; Faza and GangaRao, 1992;

Benmokrane, 1995; Sivagamasundari et al., 2008; Deiveegan et al., 2010; Saravanan et al.,

2011). Therefore the present study discusses mainly on the behaviour of beams reinforced

internally with Glass Fibre Reinforced Polymer (GFRP) reinforcements under pure torsion.

The scope of the present study is restricted to with the GFRP reinforcements because of their

availability in India. First part of this study covers the theoretical analysis based on the

Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP

reinforcements under pure torsion

Prabaghar. A, Kumaran. G

International Journal of Civil and Structural Engineering

Volume 2 Issue 2 2011

580

existing using space truss formulation for conventionally reinforced beams. Second part of

this study is related to the finite element modelling and analysis of GFRP/Steel reinforced

concrete beams. Finite element modelling is done with the help of ANSYS software with

different parametric conditions (Sivagamasundari et al., 2008; Deiveegan et al., 2010;

Saravanan et al., 2011). Finally, the results are summarised based on the Finite element

analysis and the theoretical analysis and are validated with the existing theories.

2. Materials

2.1 Concrete

Normal Strength Concrete (NSC) of grades M20 and M30 are used in this study. Ordinary

Portland cement is used to prepare the concrete. The maximum size of aggregate used is 20

mm and the size of fine aggregate ranges between 0 and 4.75 mm. After casting, the

specimens are allowed to cure in real environmental conditions for about 28 days so as to

attain strength. The test specimens are generally tested after a curing period of 28 days. The

strength of concrete under uni-axial compression is determined by loading ‗standard test

cubes‘ (150 mm size) to failure in a compression testing machine, as per IS 516 - 1959. The

modulus of elasticity of concrete is determined by loading ‗standard cylinders‘ (150 mm

diameter and 200 mm long) to failure in a compression testing machine, as per IS 516: 1959.

The mix proportions of the NSC are carried out as per Indian Standards (IS) 10262-1982 and

the average compressive strengths are obtained from laboratory tests (Sivagamasundari et al.,

2008; Deiveegan et al., 2010; Saravanan et al., 2011) and are depicted in Table 1(a).

Table 1(a): Properties of Concrete

Description M20 grade

(m1) M30 grade (m2) M40 grade (m3)

Ratio 1:1.75:3.75 1:1.45:2.85 1:1.09:2.42

W/C Ratio 0.53 0.45 0.4

Average Compressive

Strength of cubes 28.14 MPa 39.5 MPa 49.8 MPa

2.2 Reinforcements

The mechanical properties of all the types of GFRP reinforcements as per ASTM-D 3916-84

Standards and steel specimens as per Indian standards are obtained from laboratory tests and

the results are tabulated in Table 1(b). The tensile strength of steel reinforcements (S) used in

this study, conforming to Indian standards and having an average value of the yield strength

of steel is considered for this study. GFRP reinforcements used in this study are

manufactured by pultrusion process with the E-glass fibre volume approximately 60% and

these fibres are reinforced with epoxy resins. Previous studies were carried out with three

different types of GFRP reinforcements (grooved, sand sprinkled & threaded) (ACI 440R-96;

Muthuramu , et al., 2005; Sivagamasundari et al., 2008; Deiveegan et al., 2010; Saravanan et

al., 2011) with different surface indentations and are designated as Fg, Fs and Ft. In this study

threaded type GFRP reinforcement is used in place of conventional steel. The diameters of

the longitudinal and transverse reinforcements are 12 mm and 8 mm respectively. The

standard minimum diameters of the reinforcements as per Indian standards are adopted in this

study. The tensile strength properties are ascertained as per ACI standards shown in Table






581

1(b) and are validated by conducting the tensile tests at SERC, Chennai. The compressive

modulus of elasticity of GFRP reinforcing bars is smaller than its tensile modulus of elasticity

(ACI 440R-96; Lawrence C Bank 2006; Sivagamasundari et al., 2008). It varies between 36-

47 GPa which is approximately 70% of the tensile modulus for GFRP reinforcements. Under

compression GFRP reinforcements have shown a premature failures resulting from end

brooming and internal fibre micro-buckling. In this study, GFRP stirrups are manufactured

by Vacuum Assisted Resin Transfer Moulding process, using glass fibres reinforced with

epoxy resin (ACI 440R-96; Sivagamasundari et al., 2008; Deiveegan et al., 2010; Saravanan

et al., 2011). Based on the experimental study, it is observed that the strength of GFRP

stirrups at the bend location (bend strength) is as low as 50 % of the strength parallel to the

fibres. However, the stirrup strength in straight portion is comparable to the yield strength of

conventional steel stirrups. Therefore, in this study, GFRP stirrups strength is taken as 30%

of its tensile strength i.e 150 MPa

Table 1: Properties of reinforcements

Properties Threaded GFRP (Ft) Steel Fe 415 (S)

Tensile strength (MPa) 525 475

Longitudinal modulus (GPa) 63.75 200

Strain 0.012 0.002

Poisson’s ratio 0.22 0.25 – 0.3

Figure 1: GFRP reinforcements with end anchorages for tensile tests

Figure 2: Failure of GFRP reinforcements tensile test






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Figure 3: Stress-Strain curve for all the reinforcements involved in the present study

3. Theoretical Investigation

Theoretical torque verses twist relationship is established for various values of torque and

twist using elastic ,plastic theories of torsion and also the ultimate torque is determined using

space truss analogy (Hsu 1968; MacGregor et al., 1995; Rasmussen et al., 1995; Ashar et al.,

1996; Khaldown et al., 1996; Liang et al., 2000; Trahait 2005; Luis et al., 2008; Chyuan

2010). The theoretical investigation consists different rectangular beams and are designated

are as follows; Bp1m1Fe s1 ; Bp1m1Ft s1; Bp1m2Fe s1; Bp1m2Ft s1; Bp2m1Fe s1; Bp2m1Ft s1;

Bp2m2Fe s1 ; Bp2m2Ft s1, Bp1m3F1 s2, Bp2m3F1s2, Bp3m3F1s2. These beams are reinforced

internally with threaded type Glass Fibre Reinforced Polymer Reinforcements and

conventional steel reinforcements with different grades of concrete and steel reinforcement

ratio under pure torsion is considered in this study. The entire concrete beam is supported on

saddle supports which can allow rotation in the direction of application of torsion.

Table 2: Various Parameters involved

Parameters Description Designation

Types of

reinforcements

Threaded GFRP Ft

Conventional Fe

Concrete grade M 20, M30 & M40 m1 , m2 & m3

Beam size 160 x 275 mm B

Reinforcement

ratios

1. 0.56% (2-12 mm bars top & Bottom)

2. 0.85% (3-12 mm bars top & Bottom)

3. 1.4 % (5-12 mm bars top & Bottom)

p1, p2 & p3

Spacing of

stirrups 75 mm S1 & S2

3.1 Theoretical idealisation of T– curve






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The general theoretical torque twist curve T– curves are plotted for three stages and are

defined by their ( ;T) coordinates (Hsu 1968; Collins 1972; Bernardo. F.A et at., 1995).

These coordinates are shown in Figure 3.10.

Stage 1 represents the beam‘s behaviour before cracking. The slope of the curve represents

the elastic St. Venant stiffness (GC)I). In this stage the curve can be assumed as a straight line

with origin in the point (0;0) and end in ( el ;Tel). The theoretical model considered in this

study for this stage is based on Theory of Elasticity.

Le=3000 mm

e

W

W

e = 250 mm

Support

Saddle support

Beam

Support

Saddle support

Strain gauge

T/2

T=W × e

T=W × e

T/2

Twisting Moment diagram

Figure 4: Beam supported on saddle support

After cracking, the beams suffers a sudden increase of twist after what it resets the linear

behaviour. This stage is identified as Stage 2. It starts at ( cr ;Tcr) and ends at a certain level

of twist ( cr ). The slope of stage-2 represents the torsional stiffness in cracked stage (GC)II.

The model considered for stage-2 is based on the space truss analogy with 45° inclined

concrete struts and linear behaviour for the materials.

The points from T– curve which the non linear behaviour is defined by means of two

different criteria. The first one corresponds to finding the point for which at least one of the

torsion reinforcements (longitudinal or transversal) reaches the yielding point. The second

criterion corresponds to finding the point for which the concrete struts starts to behave non

linearly, due to high levels of loading (this situation may occur before any reinforcement bar

yields).






584

Stage 3 of the curve was plotted with non linear behaviour of the materials and considering

the Softening Effect. In this study, space truss analogy is used to the locate the following

coordinates (ty ;Tty) & ( ul ;Tul). The space truss model with softening effect is not

considered since it involves iterative procedure (Vasanth et al., 2010; Balajiponraj et al 2011).

The three stages are identified in the T– curve of Figure 5.2 are characterized separately.

Figure 5: Typical T- curve for a reinforced concrete beam under pure torsion

Where, Tcr = Cracking torque; cr = Twist corresponding to Tcr for the stage 1 (limit for

linear elastic analysis in non cracked and cracked stage);Tly = Torque corresponding to

yielding of longitudinal reinforcement; ly = Twist corresponding to Tly;Tty = Torque

corresponding to yielding of transversal reinforcement; ty = Twist corresponding to Tty;Tul

= Ultimate (maximum) torque; ul = Maximum twist at beam‘s failure.(GC)I = Torsional

stiffness of Zone 1 (for linear elastic analysis in non cracked stage);(GC)II = Torsional

stiffness of Zone 2.b (for linear elastic analysis in cracked stage).

Stage: 1 Linear Elastic Torque (Tel, el )

For rectangular sections using St. Venant theory, the maximum torsional shear stress occurs

at the middle of the wider face (Hsu 1968), and has a value given by

Db

Tt 2max,

(1)

where T is the twisting moment (torque), b (160 mm) and D (275 mm) are the cross-sectional

dimensions (b being smaller), and is a St. Venant coefficient whose value depends on the

D/b ratio; lies in the range 0.21 to 0.29 for D/b varying from 1.0 to 5.0 respectively.

Therefore =0.2243; max,t (MPa units) of about ckf2.0






585

(a) part section of beam

(b) torsional shear stress distributions

(c) degrees of plastic behaviour

potential tensile crack

45o

45o

ft t

b

D 0 B B 0

elastic

inelastic

plastic

y

z

x

T A

t,max

Figure 6: Torsional shear stresses in a beam of rectangular section

Elastic torque is given by

max,

2 tel DbT (2)

Using Torque-Twist relationship based on linear elastic analysis is given by

l

G

C

T elel (3)

From the above, the twist el per unit length of a beam can be expressed as

CG

lT elel

(4)

where elT is the elastic torque, GC is the torsional rigidity, obtained as a product of the shear

modulus G and the geometrical parameter C of the section. Since G (shear modulus) is equal

to CE /[2(1 + )], where CE is the Young Modulus of concrete and is the Poisson

Coefficient, and = 0.25. Therefore G=0.4 CE . The stiffness factor C (for a plain

rectangular section of size b D, with b < D), based on ‗St.Venant theory is given by the

following expression

DbC 3 (5)

where is a constant which may be calculated,

363.01

D

b= 3

275

16063.01

=0.211

Stage: 2 First crack Torque ( cr1T , cr1 ): (Linear Elastic Analysis in cracked phase)






586

The strength of a torsionally reinforced member at torsional cracking Tcr is practically the

same as the failure strength of a plain concrete member under pure torsion. Although several

methods have been developed to compute Tcr , the plastic theory approach based on Indian

standards is described here. The cracking Torque is given by,

32

2

max,bD

bT tcr (6)

Studies show that the torsion reinforcement has a negligible influence on the torsional

stiffness. However the presence of torsion reinforcement does delay the cracking point. Hsu,

1968 showed that the effective cracking moment, Tcr,ef, may be computed by:

crtefcr TT 41, (7)

Where, Al = total area of the longitudinal reinforcement; At = area of one leg of the transversal

reinforcement; s = spacing of stirrups; ls = perimeter of the centre line of the stirrups.

Using Torque-Twist relationship based on linear elastic analysis is given by

l

G

C

T effcreffcr ,, (8)

From the above, the twist effel , per unit length of a beam can be expressed by,

I

effel

effcrGC

lT

,

, (9)

Torsional stiffness is given by, IGC)( = I

tK = C50004.0 cuf (10)

Stage: 3 (a) Ultimate Torque (Tul, ul )

The use of the thin-walled tube analogy (or) space truss analogy the shear stresses are treated

as constant over a finite thickness t around the periphery of the member, allowing the beam to

be represented by an equivalent hollow beam of uniform thickness. Within the walls of the

tube, torque is resisted by the shear flow q, which has units of force per unit length. In the

analogy, q is treated as a constant around the perimeter of the tube. To predict the cracking

behaviour, the concrete tube may be idealized through the special truss analogy proposed by

Rausch. The space truss analogy is essentially an extension of the plane truss analogy used

to explain flexural shear resistance. The ‗space-truss model‘ is an idealisation of the effective

portion of the beam, comprising the longitudinal and transverse torsional reinforcement and

the surrounding layer of concrete. It is this ‗thin-walled tube‘ which becomes fully effective

at the post-torsional cracking phase. The truss is made up of the corner longitudinal bars as

stringers, the closed stirrup legs as transverse ties, and the concrete between diagonal cracks

as compression diagonals. Assuming torsional cracks are (under pure torsion) at 45o to the

longitudinal axis of the beam, and considering equilibrium of forces normal to section AB. It

is this ‗thin-walled tube‘ which becomes fully effective at the post-torsional cracking phase.

The truss is made up of the corner longitudinal bars as stringers, the closed stirrup legs as

transverse ties, and the concrete between diagonal cracks as compression diagonals (Mattock

et al., 1968; Niema 1993; Unnikrishna pillai et al 2003).






587

110 22 db

T

A

Tq uu (11)

11dbAo (12)

where Ao is the area enclosed by the centre line of the thickness; b1 and d1 denote the centre-

to-centre distances between the corner bars in the directions of the width and the depth

respectively Assuming torsional cracks (under pure torsion) at 45o to the longitudinal axis of

the beam, and considering equilibrium of forces normal to section, the total force in each

stirrup is given by qsv tan 45o = qsv where sv is the spacing of the (vertical) stirrups. Further,

assuming that the stirrup has yielded in tension at the ultimate limit state or (Design stress =

fy ; = partial safety factor for steel=0.87; fy yield strength of steel; Design stress for GFRP

reinforcements = fGFRP ; = strength reduction factor for GFRP reinforcements =0.80; fGFRP

tensile strength of GFRP reinforcements); it follows from force equilibrium that

vGFRPyt qsfA ) ( / (13)

where At is the cross-sectional area of the stirrup (equal to Asv /2 for two legged stirrups).

Substituting Eq. 12 & 13 in Eq. 11, the following expression is obtained for the ultimate

strength Tu = TuR in torsion:

vGFRPytuR sfdbAT ) (2 /11 (14)

Further, assuming that the longitudinal steel (symmetrically placed with respect to the beam

axis) has also yielded at the ultimate limit state, it follows from longitudinal force equilibrium

that (Figure 7):

)(245tan

) ( 11/ dbq

fAoGFRPyl (15)

where Al the total area of the longitudinal steel/GFRP reinforcements and fyl yield strength

of steel; fGFRP tensile strength of GFRP reinforcements. Substituting Eq. 11 in 15 , the

following expression is obtained for the ultimate strength Tu = TuR in torsion:

)() ( 11/11 dbfdbAT GFRPyluR (16)

The two alternative expressions for TuR viz. Eq. 14 and Eq. 16, will give identical results only

if the following relation between the areas of longitudinal steel and transverse steel (as

torsional reinforcement) is satisfied:

GFRPy

tGFRPyf

vtGFRPtl

fs

dbAA

/

/11/

)(2

(17)






588

sv

d1

AGFRP× fGFRP

do

b1

Tu

A

B

(a) space-truss model

(b) detail ‘X’ (c) thin-walled tube (box section)

potential cracks

X

q = t do

1

q / tan

sv

longitudinal bar (area AGFRP)

closed stirrup (area AGFRP-S)

sv

q

q tan q

Tu

b1

d1

q = Tu / (2Ao)

Ao = b1d1 = area enclosed by centreline of tube

AGFRP× fGFRP

AGFRP-S× fGFRP-S

AGFRP-S× fGFRP-S

AGFRP× fGFRP

AGFRP× fGFRP

AGFRP× fGFRP

Figure 7: Space-truss model for GFRP reinforced beams

where At cross sectional area of the 2 legged stirrups; AGFRP-t cross sectional area of the 2

legged GFRP stirrups; fyt yield strength of steel; fGFRP-t tensile strength of GFRP stirrups.

If the relation given by the Eq. 17 is not satisfied, then TuR may be computed by combining

Eq. 14 and Eq. 16, taking into account the areas of both transverse and longitudinal

reinforcements:

87.0)(2

211

11

db

fA

s

fAdbT

yll

v

yt

uR (18)

To ensure that the member does not fail suddenly in a brittle manner after the development of

torsional cracks, the torsional strength of the cracked reinforced section must be at least equal

to the cracking torque Tcr (computed without considering any safety factor).

The ultimate torque Tul may be computed by considering the contributions of both transverse

and longitudinal reinforcements:

)(22

11

////11

db

fA

s

fAdbT GFRPGFRP

v

GFRPyGFRPtul (20)






589

To predict the cracking behaviour, the concrete tube may be idealized through the spaces

truss analogy proposed by Rausch in 1929. Based on Hsu, the following equation for the

torsional stiffness (GC)II of rectangular sections:

tl

sII

Ddb

mBDdb

BDdbEGC

11

)(

2)(

)(

11

2

11

2

1

2

1 (21)

Transverse GFRP reinforcement alone

Considering shear –torsion interaction with Vu=0, which corresponds to space truss

formulation, considering the contribution of the transverse reinforcement alone,

vtGFRPtty sfdbAT )(11 (22)

Parameters considered in this study

B = 160 mm; D = 275 mm; b1 = 102 mm; d1 = 217 mm; Es = 212500 N/mm2; Ec =

ckf5000 ; m1 = 28.14 MPa; m2 = 39.5 MPa; m3 = 49.8 MPa; ; m = Es/Ec ; Al = 113 4=

452 mm2

(2 Nos. of top and bottom); Al = 113 6= 678 mm2

(3 Nos. of top and bottom); Al

= 113 10= 1,130 mm2

(5 Nos. of top and bottom); At = 2 50.3 = 100.6 mm2; fy = 475 MPa;

s1 = 75 mm; fGFRP-l = 525 MPa; s2 = 50 mm; ; fGFRP-t = 150 MPa.

Table 3:Torque verses twist for various Parameters involved

Tel

(kNm) el

(deg)

Tcr

(kNm) cr

(deg)

Teff

(kNm) eff

(deg)

Tyt

(kNm) yt

(deg)

Tul

(kNm) ul

(deg)

Bp1m1FeS1 1.67 0.057 3.01 0.102 3.37 0.114 14.10 4.22 19.12 5.72

Bp1m1Ft S1 1.67 0.057 3.01 0.102 3.37 0.114 4.45 4.28 11.29 10.86

Bp1m2Fe

S1 1.98 0.057 3.57 0.102 3.99 0.114 14.11 4.18 19.12 5.67

Bp1m2Ft S1 1.98 0.057 3.57 0.102 3.99 0.114 4.45 4.27 11.3 10.83

Bp2m1Fe

S1 1.67 0.057 3.01 0.102 3.43 0.116 14.1 3.35 23.42 5.56

Bp2m1Ft S1 1.67 0.057 3.01 0.102 3.43 0.116 4.45 3.36 13.84 10.45

Bp2m2Fe

S1 1.98 0.057 3.57 0.102 4.06 0.117 14.10 3.312 23.42 5.50

B2p2m2Ft

S1 1.98 0.057 3.57 0.102 4.06 0.117 4.453 3.352 13.84 10.42

Bp1m3Ft S2 2.2 0.057 3.95 0.102 4.58 0.118 6.68 5.67 13.84 11.75

Bp2m3Ft S2 2.2 0.057 3.95 0.102 4.66 0.12 6.68 4.29 16.95 10.87

Bp3m3Ft S2 2.2 0.057 3.95 0.102 4.82 0.125 6.68 3.18 21.88 10.44

4. Finite Element Modelling and Analysis

4.1 Finite Elements used in this study

The present study, discrete modelling approach is adoted to model the torsional behaviour of

GFRP and Steel Reinforced Concrete (RC) beams. Two different types of element are used

to model each of the reinforced concrete beams; the first one is the Solid65 concrete brick

element which is used for 3D modeling of concrete with reinforcing bars. This element has






590

eight nodes with three degrees of freedom per node-translations in the global x, y and z

directions. The element is capable of handling plastic deformation, cracking in three

orthogonal directions and crushing. Both standard and nonstandard elements can be refined

with additional nodes. These refined elements are of interest for more accurate stress analysis.

The second element type is the Link8 element which was used to model the steel/GFRP

reinforcement. This element is a 3D spar element and it has two nodes with three translational

degrees of freedom per node. This element is also capable of handling plastic deformation

and all are perfectly connected to the nodes of the concrete element. The connectivity

between a concrete node and a reinforcing steel/GFRP node can be achieved by the following

method.

Assumptions made in this study

The assumptions considered in this are as follows:

1. The entire length of the beam is considered for the finite element analysis. The

boundary conditions adopted at the supports to allow rotation in the direction of

application of torque and axial displacement along the length of beam and all other

degrees of freedom restrained and at centre of the beam (symmetry point) all the

displacement degrees of freedom are restrained allowing rotational degrees of

freedom only in order to alleviate the problem of rigid body displacement which can

provide a equilibrium.

2. Torque is applied by imposing a line load along the width of beam on either side of

the beam end at a equal distance from centre (eccentricity e = 280 mm), obviously this

loading system provide a pure and uniform torque.

3. GFRP reinforcement are modelled as one dimensional element, only a one-

dimensional stress-strain relation is adopted. GFRP reinforcements show their

ultimate tensile strength without exhibiting any material yielding. Unlike steel, the

tensile strength of GFRP reinforcements shown a liner relationship.

4. The nonlinearity derived from the nonlinear relationships in material models is

considered and the effect of geometric nonlinearity is not considered.

4.2 Material modelling

4.2.1 Concrete

The cracking behaviour in concrete elements needs to be modelled in order to predict

accurately the torque-twist behaviour of reinforced concrete members. At the cracked section

all tension is carried by the reinforcement. Tensile stresses are, however, present in the

concrete between the cracks, since some tension is transferred from steel to concrete through

bond. The magnitude and distribution of bond stresses between the cracks determine the

distribution of tensile stresses in the concrete and the reinforcing steel between the cracks.

Since cracking is the major source of material nonlinearity in the serviceability range of

reinforced concrete structures, realistic cracking models need to be developed. The selection

of a cracking model depends on the purpose of the finite element analysis (Filippou 1990;

Sivagamasundari et al, 2009; Deiveegan et al., 2010; Saravanan et al., 2011). A simplified

averaging procedure is adopted in this study, which assumes that cracks are distributed across

a region of the finite element. In this model, cracked concrete is supposed to remain a

continuum and the material properties are then modified to account for the damage induced in






591

the material. After the first crack has occurred, the concrete becomes orthotropic with the

material axes oriented along the directions of cracking.

The response of concrete under tensile stresses is assumed to be linearly elastic until the

fracture surface is reached. This tensile type of fracture or cracking is governed by a

maximum tensile stress criterion (tension cut-off). Cracks are assumed to form in planes

perpendicular to the direction of maximum tensile stress, as soon as this stress reaches the

specified concrete modulus of rupture ft. In the present study the ability of concrete to transfer

shear forces across the crack interface is accounted for by using two different shear retention

factor (β) for cracked shear modulus, it was assumed equal to 0.3 for opened crack and 0.7

for closed crack. The elasticity modulus and the Poisson‘s ratio are reduced to zero in the

direction perpendicular to the cracked plane, and a reduced shear modulus is employed.

Before cracking, concrete is assumed to be an isotropic material, with the stress - strain

relationship considered in this study. The time dependent effects of creep, shrinkage and

temperature variation are not considered in this study. o is the effective stress obtained from

a uni-axial compression test and ft is the modulus of rupture of concrete. The concrete

crushing condition is a strain controlled phenomenon. Once crushing has occurred the

concrete is assumed to lose all its characteristics of stiffness at the point under consideration.

Therefore the corresponding elasticity constitutive relation matrix is considered as a null

matrix and the vector of total stresses is reduced to zero.

Figure 8: Stress-strain relation of Concrete

In the present study, Hognestad model is used after some modifications. These modifications

are introduced in order to increase the computational efficiency of the model, in view of the

fact that the response of reinforced concrete member is much more affected by the

compressive behaviour than by the tensile behaviour of concrete. (Filippou 1990; Mohamad

Najim 2007; Sivagamasundari et al., 2008; Deiveegan et al., 2010; Saravanan et al., 2011).

4.2.2 Behaviour of sSteel reinforcements in tension and compression

In this study, a bilinear kinematics hardening is adopted to represent the behaviour of steel

reinforcements under tension and compression.






592

Figure 9: Stress- strain curve for Steel

Figure 4.3 shows the typical stress-strain curve used in this study, which exhibits an initial

linear elastic portion, a yield plateau, a strain hardening and, finally, a range in which the

stress drops off until fracture occurs. The extent of the yield plateau is a function of the

tensile strength of steel (Deiveegan et al., 2010; Saravanan et al., 2011). Es1 is Young's

modulus of reinforcing steel, Es2 is strain hardening modulus. Steel reinforcements exhibit the

same stress-strain curve in compression as in tension. The following parameters considered

in this study; fy=400 MPa; Es1 = 400 MPa; ful=475 MPa; Es2 = 50 MPa and Poisson's ratio

0.3.

4.2.3 Behaviour of GFRP reinforcements in tension and compression

In this study the GFRP reinforcing bars are modelled as a linear elastic material with

reinforcement rupture stress of fGFRP and elastic modulus of EGFRP as shown in Figure 4.4.

Grooved GFRP reinforcements are made from unidirectional polyester-glass materials having

a modulus of elasticity 59.5GPa and ultimate rupture stress of 580 MPa in compression.

Figure 10: Stress strain curve for GFRP reinforcements






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When FRP components are loaded in longitudinal compression, the failure of the composites

is associated with micro-buckling or kinking of the fibre within the restraint of matrix

material. Accurate experimental values for the compressive strength are difficult to obtain

and they are highly dependent on specimen geometry and the testing method. The mode of

failure depends on the properties of constituents (fibres and resin) and the fibre volume

fraction. From the literature it is observed that compressive strength of FRPs is lower than the

tensile strengths. Based on the routine compression test as applicable to concrete specimens,

the GFRP reinforcements show a similar trend than that of tensile stress-strain relationship.

The compressive strength of GFRP reinforcements shows 50% reduction than the tensile

strength (Sivagamasundari et al, 2009; Deiveegan et al., 2010; Saravanan et al., 2011). But

due to non availability of testing procedure, similar stress-strain behaviour is considered in

compression with a reduced tensile modulus (50% reduction).

4.2.4 Modelling of slip between the reinforcements and concrete

In the simplified analysis of reinforced concrete structures complete compatibility of strains

between concrete and steel is usually assumed, which implies perfect bond. In reality there is

no strain compatibility between reinforcing steel and surrounding concrete near cracks. This

incompatibility and the crack propagation give rise to relative displacements between steel

and concrete, which are known as bond-slip. The bond-slip relationship is established based

on the laboratory experiments, such as the standard pull-out test, the force is applied at the

projecting end of a bar which is embedded in a concrete cylinder. In the finite element model,

when the nodes of the truss element do not need to coincide with the nodes of the concrete

element, then there is a slip. This relative slip between the reinforcement and the concrete is

explicitly taken in this study. Steel/GFRP reinforcements exhibit a uniaxial response, having

strength and stiffness characteristics in the bar direction only. The behaviour is treated

incrementally as a one dimensional problem.

Figure 11: Discrete Reinforcement Model with Bond-slip Element

Figure 4.5 depicts the discrete reinforcements model with bond slip element. Therefore in this

study, the GFRP reinforcement is modelled by a one dimensional truss element which is

embedded in concrete, whose bond slip relationship is represented by a tri-linear relationship






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as shown in Figure 4.6. The parameters of the model are derived from the material properties

of each specimen through laboratory studies. The parameters include: ds is the bond-slip, Eb

is the initial slip modulus and, τ1 is the bond stress. When ds1 exceeds ds2, the value Eb1 is

replaced by Eb2. The parameters considered in this study for different GFRP reinforcements

are as follows: ds1 = 0.017 mm, ds2 = 0.4 mm, τ1 = 6.5 MPa and τ2 = 13.23 MPa (Ft

reinforcements); ds1 = 0.017 mm, ds2 = 0.3 mm, τ1 = 7.6 MPa and τ2 = 17.2 MPa (Fe

conventional). These values are derived from the previous studies (Sivagamasundari et al.,

2008; Deiveegan et al., 2010). The rigorous bond slip studies pertaining to the anchorage

type of failures are beyond the scope of this study.

Figure 12: Bond stress-slip relationship for finite element model

The longitudinal and transverse reinforcement namely, high yield strength deformed (HYSD)

are modelled using link 8/spring elements. Rate independent multi linear isotropic hardening

option with von-mises yield criterion has been used to define the material property of steel

rebar. The tensile stress strain response of steel/GFRP bars based on the test data has been

used in the analysis. The bond slip between reinforcement and concrete has been modelled

using CONBIM 39 Non-linear spring element. In the present analysis, CONBIM 39 connects

the nodes of link 8 and SOLID65 elements. The slip test data has observed that the

experimental study has used the load deformation characteristics. A linear variation without

tension cutoff has been used for steel reinforced concrete specimen. The transverse

reinforcements (stirrups) are assumed to be perfectly bonded to the surrounding concrete in

the present analysis.

5. Finite Element Implementation

5.1 Finite element discretization

An important step in finite element modelling is the selection of the meshing density.

Therefore, a finer mesh density with a total number of elements 1532 is considered for this

study.






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(a) (b)

Figure 13: Finite element model (a) Solid elements (b) Link elements

Figure 14: Finite Element Model with loading and boundary conditions

Table 4: Material properties of Steel/GFRP reinforcements

Properties Steel(S) GFRP (Ft)

Modulus of elasticity Es, (GPa) 212.5 63.75

Rebar size (mm) 12 12

Tensile strength, for main reinforcement 475 525

Tensile strength, for stirrup reinforcement 475 100.5

Compressive strength, fcGFRP (MPa) 415 250

Shear strength, (Mpa) 180 122

Poisson‘s ratio 0.15-0.3 0.22

Table 5: Material properties for concrete

Properties M20 Grade M30 Grade

Elastic modulus Ec (GPa) 26.52 31.42

Poisson‘s ratio 0.15 0.2

Ultimate compressive stress fcu (MPa) 28.14 39.5

Tension stiffening coefficient, α 0.6 0.6

Tension stiffening strain coefficient, εm 0.0015 0.0016






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5.2 Nonlinear analysis procedure

This study uses Newton-Raphson equilibrium iterations for updating the model stiffness.

Newton-Raphson equilibrium iterations provide convergence at the end of each load

increment within tolerance limits. This approach assesses the out-of-balance load vector,

which is the difference between the restoring forces (the loads corresponding to the element

stresses) and the applied loads prior to the application of load. Subsequently, the program

carries out a linear solution, using the out-of-balance loads, and checks for convergence. If

convergence criteria are not satisfied, the out-of-balance load vector is to be re evaluated, the

stiffness matrix is to be updated, and thus a new solution is attained. This iterative procedure

continues until the problem converges. In this study, convergence criteria are based on force

and displacement, and the convergence tolerance limits are initially selected by the program.

It is found that convergence of solutions for the models was difficult to achieve due to the

nonlinear behaviour of reinforced concrete. Therefore, the convergence tolerance limits are

increased to a maximum of 5 times the default tolerance limits (0.5% for force checking and

5% for displacement checking) in order to obtain convergence of the solutions.

For the nonlinear analysis, automatic time stepping is done in the program and it predicts that

it controls load step sizes. Based on the previous solution history and the physics of the

models, if the convergence behaviour is smooth, automatic time stepping will increase the

load increment up to a selected maximum load step size. If the convergence behaviour is

abrupt, automatic time stepping will bisect the load increment until it is equal to a selected

minimum load step size. The maximum and minimum load step sizes are required for the

automatic time stepping. The ultimate torque is the torque at which divergence occurred.

These analyses are carried out with high end system configuration with an integrated

environment for modelling and analysis ( Vasanth 2010 ). The results are also compared with

experimental observations.

Figure 15: Stress Contours obtained from analysis

T T

Locations to measure

twist and torque

Saddle support

Torque T= W × e

Figure 16: Finite Element model of beam with helical crack






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Figure 17: Torque verses twist for Rectangular beam of grade of concrete

m1= 28.14 mPa; p1=0.56%


m2= 39.5 mPa; p1=0.56%






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m1= 28.14 mPa; p1=0.85%


m2= 39.5 mPa; p1=0.85%






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m3= 49.8 MPa; p1=0.56%; p1=0.85%; p1=1.4%;

Table 6: Results obtained from theoretical and FEM values

Beam (Tcr-Sp-Thu/Tcr-

FEM) (θcr-TH/θcr-FEM)

(Tul-TH/Tul-FEM) (θul-TH/θul-FEM)

Bp1m1 Fe 0.95 1.02 0.9 1.65

Bp1m1 Ft 0.95 1.02 0.86 0.98

Bp1m2 Fe 1.27 2.02 0.9 1.23

Bp1m2 Ft 1.27 2.03 0.89 1.14

Bp2m1 Fe 0.88 0.63 0.9 1.31

Bp2m1 Ft 0.88 0.63 1.15 1.17

Bp2m2 Fe 0.8 1.26 0.9 1.22

Bp2m2 Ft 0.8 1.14 0.92 2.3

Bp1m3 Ft 0.98 2.28 0.95 1.175

Bp2m3 Ft 0.98 2.28 0.94 1.21

Bp3m3 Ft 0.98 2.28 0.96 1.1

Table 6: Ratio of ultimate to first crack torsional strengths from theoretical and FEM values

Beam (Tul-Sp-Thu/Tcr- Sp-Thu) (Tul- FEM /Tcr-FEM)

Bp1m1 Fe 2.8 6.6

Bp1m1 Ft 2.4 4.1

Bp1m2 Fe 2.8 7.5

Bp1m2 Ft 2.25 4.5

Bp2m1 Fe 2.66 5.75

Bp2m1 Ft 2.2 3.5

Bp2m2 Fe 2.66 4

Bp2m2 Ft 1.8 3.75

Bp1m3 Ft 3.02 3.5

Bp2m3 Ft 4.28 4.5

Bp3m3 Ft 4.7 5.75






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4. Conclusions

The results of the finite element study are compared with theoretical study and for better

improvement in the finite element modelling. The torque and twist are measured in finite

element model near the support locations at the mid depth of the beams.

1. Each beam is analyzed under increasing incremental torque equal to 0.5 kNm up to

failure. The torque is induced by applying equal and opposite eccentric load on either

end of the beam in opposite face of the beam. The ultimate torque is assumed to be

reached when the solution is not converged.

2. The torque-twist behaviour of concrete beams reinforced with GFRP reinforcements are

similar to that of concrete beams reinforced with steel reinforcements until the

formation of the first torsional crack. The value of first crack (Tcr ) is insensitive to the

type of reinforcement used. When cracking occurs, there is a increase in twist under

nearly constant torque, due to a drastic loss of torsional stiffness. Beyond this,

however, the strength and behaviour depend on the amount and type of longitudinal and

torsional reinforcement present in the beam.

3. Torsional strength and angle of twist increases with the increase in increase grade of

concrete and percentage of longitudinal and transverse reinforcements. But GFRP

reinforced concrete beams showed higher angle of twist than the conventional

reinforcements (Figures 17 to 20). This fact is primarily due to lower elastic modulus

and higher tensile strain values for GFRP reinforcements than the steel

reinforcements.

4. It is also noted that the replacing main and transverse steel reinforcements by an equal

percentage of GFRP reinforcements, reduced their torsional capacities by 30% for

lower grade concrete beams but their increase is reduced by 20 % for higher grade

concrete and higher percentage of steel. To compensate the reduction of torsional

strengths, the grade of concrete and spacing of stirrups are modified for the beams

designated as Bp1m3S1Ft, Bp2m3S1Ft and Bp2m3S1Ft, these beam shown that the

torsional strength values are improved.

5. GFRP reinforcements do not have definite yield point, and its stress- strain response

shows linear-elastic response up to failure, therefore no GFRP reinforced beams

exhibit a failure point before concrete fibre reaches its limiting strain 0.0035. But

none of beams failed due to rupture of GRFP reinforcements prior to concrete strain

reaches ultimate. It is probably due to the fact that the ultimate tensile strains of GFRP

reinforcements are greater than the ultimate compressive strains of concrete.

6. The torque twist plots show good agreement with the theoretical data. However, the

finite element models show higher torques than the experimental study (Figures 17 to

20). The ratios of the first crack torque from theoretical to the FEM for the lie

between 0.8 to 1.27 and the ratios of the ultimate crack torque from experiment to the

FEM for the lie between 0.86 to1.15. Ultimate torques observed from finite element

model provides a better representation of crack formation during ultimate pure torque

than at first crack (Figures 17 to 20).

7. The ratios of the angle of twist at first crack from theoretical to the FEM lie between

0.63 to 2.28. Similarly the ratios of the angle of twist at ultimate from experiment to

the FEM lie between 0.98 to 2.3. This is attributed to higher stiffness of the finer

finite element meshing strategy.






601

8. The angle of twist observed from FEM is lower than the conventionally reinforced

beams and is probably due to difference in elastic torsional stiffness for GFRP

reinforced concrete beams.

9. Also the ratio of ultimate to cracking torsional strengths of various beams are as

follows, (Tul/Tcr )sp-Thu= 1.8 to 4.7 and (Tul/Tcr )sp-Thu= 3.5 to 7.5. The values prescribed

by Rasmussen & Baker‘s (1995) are 2.61 for normal strength concrete beams. This is

probably due to the overestimation of ultimate torsional strengths based on space truss

analogy and FEM

10. The softening branch of the torsion verses twist, energy dissipated during the

softening process, post peak behaviour of beams, ductility index ratio and the size

effects in the softening region are however the quantification of this phenomenon and

is beyond the scope.

5. References

1. ACI 440R-96, ―State-of-the-Art Report on Fiber Reinforced Plastic (FRP)

Reinforcement for Concrete Structures‖, Reported by ACI Committee 440.

2. Asghar.M Bhatti and Anan Almughrabi (1996), "Refined Model to Estimate

Torsional Strength of Reinforced Concrete Beams, "ACI Structural Journal, 95(5) pp

614622,.

3. Balajiponraj. G (2011), ―Experimental Study on the Behaviour of Concrete Beams

Reinforced Internally With GFRP Reinforcements under Pure Torsion‖, M.E Thesis,

Department of Civil and Structural Engineering, Annamalai University,.

4. Benmokrane. B, Chaallal.O and Masmoudi. R (1995), flexural response of concrete

Beams Reinforced with FRP Reinforcing Bars, ACI Materials Journal, 91(2), pp 46-

55.

5. Bernardo. F.A et at., (1995), ―Behaviour of Concrete Beams Under Torsion: NSC

Plain and hollow beams‖, Journal of Material and Structures, 41, pp 1143-1167, 2008

6. Chyuan-Hwan Jeng, ―Simple Rational Formulas for cracking Torque and twist of

Reinforced Concrete Members‖, ACI Structural Journal, 107(2), pp 189-197, 2010.

7. Collins, M.P., The Torque-Twist Characteristics of Reinforced Concrete Beams,

Inelasticity and Nonlinearity in Structural Concrete, SM Study No. 8, University of

Waterloo Press, Waterloo, 1972, pp 211–232.

8. Deiveegan. A and Kumaran. G, ―Reliability Study of Concrete Columns internally

reinforced with Non-metallic Reinforcements‖, International Journal of Civil and

Structural Engineering, 1(2), pp 270-287, 2010.

9. Faza. S.S and Ganga Rao. H.V.S, (1992), bending and bond behavior of concrete

beams reinforced with fibre reinforced plastic rebars, WVDOH-RP-83 phase 1 report,

west Virginia University, Morgantown, pp 128-173.






602

10. Filippou. Kwak, H.G.FilipC (1990), finite element analysis of reinforced structures

under monotonic loads, A report on research conducted in Department of Civil

engineering, University of California, Berkeley.

11. Hsu (1968), T.T.C., Behaviour of Reinforced Concrete Rectangular Members, ACI

Publication SP – 18, ‗Torsion of Structural Concrete‘ American Concrete Institute,

Detroit, pp 261–306.

12. Hsu, T.T.C., Ultimate Torque of Reinforced Rectangular Beams, ASCE Journal of

Structural Engineering, 94(1), pp 485–510, 1968.

13. Hsu T.T.C (1984), Torsion of Reinforced Concrete, Van Nostrand Reinhold

Publishing Company, Inc, New York

14. Hsu T.T.C, Mo YL (1985), softening of concrete in torsional members—Design

recommendations, Journal of American Concrete Institute Proceedings, 82(4), pp

443–452, 1985.

15. IS: 10262 (1982), Guide lines for concrete mix design, Bureau of Indian Standards,

New Delhi.

16. IS: 516 (1959), Code of practice-Methods of test for strength of concrete, Bureau of

Indian standards, New Delhi.

17. Khaldoun.N et al.,―Simple Model for Predicting Torsional Strength of Reinforced and

Prestressed Concrete Sections‖, ACI Structural Journal, 93(6), pp 658-666, 1996.

18. Lawrence C. Bank (2006), ―Composite for Construction – Structural Design with FRP

Materials, John Wiley & Sons,.

19. Liang Jenq Leu and Yu Shu Lee, ― Torsion Design Charts for Reinforced Concrete

Rectangular Members, Journal of Structural Engineering, 126(2), pp 210-218, 2000.

20. Lu ıs F. A. Bernardo and Sergio M. R. Lopes (2007), behaviour of concrete

beams under tors ion , Journal of Materials and Structures, 2, pp 35-49, 2008.

21. Machida. A (1993), State-of-the-Art Report on Continuous Fiber Reinforcing

Materials, Society of Civil Engineers (JSCE), Tokyo, Japan,

22. MacGregor. J.G and Ghoneim.M.G (1995), ―Design for Torsion‖, ACI Structural

Journal, 92(2), pp 211-18

23. Mattock, A.H (1968), ―How to Design for Torsion‖, ACI Publication SP – 18,

‗Torsion of Structural Concrete‘, American Concrete Institute, March.

24. Mohammad Najim Mahmood (2007), Nonlinear Analysis of Reinforced Concrete

Beams Under Pure Torsion, Journal of Applied Sciences, 7(22), pp: 3524-3529.

25. Muthuramu. K.I, Palanichamy. M.S, Vandhiyan. R (2005), retrofitting of reinforced

concrete rectangular beam using E-glass fibre, Indian Concrete Journal, 1, pp 15-20.






603

26. Nawy. E.G and Neuwerth. G.E (1997), Fiberglass Reinforced Concrete Slabs and

Beams, Journal of Structural Division, ASCE, 103(2).

27. Nanni, A (1993), "Flexural Behaviour and Design of RC Members using FRP

Reinforcement, "Journal of the structural Engineering, ASCE, 119(11), pp 3344-3359.

28. Niema (1993), EI, Fiber Reinforced concrete Beams under Torsion, ACI Structural

Journal, 90(5).

29. Rahal K.N and Collins. MP (1995), ―Analysis of sections subjected to combined shesr

and torsion-A theoretical model‖, ACI Structural Journal, 92(4), pp 459-469.

30. Rasmussen. L.J and Basker.G (1995), Torsion in Reinforced Normal and High

Strength Concrete Beams-Part-2: Theory and Design, ACI Structural Journal, 92(1),

pp149-156.

31. Saravanan. J and Kumaran. G, 2011), ―Joint shear strength of FRP reinforced concrete

beam-column joints‖, Central European Journal of Engineering, 1(1), pp 89-102

32. Salih Elhadi. Moh.Ahmed and EL-Niema (1998), Fiber Reinforced Concrete Beams

Subjected to Combined Bending, Shear and Torsion, Sudan Engineering Society

Journal, 45(36).

33. Sivagamasundari, R (2008), ―Analytical and experimental study of one way slabs

reinforced with glass fibre polymer reinforcements‖, Ph.D. Thesis, Department of

Civil and Structural Engineering, Annamalai University.

34. Trahait. N.S (2005), ―Nonlinear Nonuniform Torsion‖, ASCE Journal of Structural

Engineering, 131(7), pp 1135-1142.

35. Unnikrishna Pillai and Devdas Menon, (2003), ―Reinforced Concrete Design‖, Tata

Mc Graw-Hill Publishing Company Limited, New Delhi.

36. Vasanth.V (2010), ―Torsional Behaviour of rectangular concrete beams reinforced

with Glass Fibre Polymer Reinforcements‖, M.E Thesis, Department of Civil and

Structural Engineering, Annamalai University.

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