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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
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
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
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
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
582
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
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
583
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).
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
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
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
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)
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
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).
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
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)
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
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)
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
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
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
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
Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP
reinforcements under pure torsion
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International Journal of Civil and Structural Engineering
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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.
Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP
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International Journal of Civil and Structural Engineering
Volume 2 Issue 2 2011
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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
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
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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
Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP
reinforcements under pure torsion
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International Journal of Civil and Structural Engineering
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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.
Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP
reinforcements under pure torsion
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International Journal of Civil and Structural Engineering
Volume 2 Issue 2 2011
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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
Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP
reinforcements under pure torsion
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Volume 2 Issue 2 2011
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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
Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP
reinforcements under pure torsion
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International Journal of Civil and Structural Engineering
Volume 2 Issue 2 2011
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Figure 17: Torque verses twist for Rectangular beam of grade of concrete
m1= 28.14 mPa; p1=0.56%
Figure 18: Torque verses twist for Rectangular beam of grade of concrete
m2= 39.5 mPa; p1=0.56%
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
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Figure 19: Torque verses twist for Rectangular beam of grade of concrete
m1= 28.14 mPa; p1=0.85%
Figure 20: Torque verses twist for Rectangular beam of grade of concrete
m2= 39.5 mPa; p1=0.85%
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
599
Figure 21: Torque verses twist for Rectangular beam of grade of concrete
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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reinforcements under pure torsion
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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.
Theoretical study on the behaviour of Rectangular Concrete Beams reinforced internally with GFRP
reinforcements under pure torsion
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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.
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