1
Shear Strength Model for FRP-Strengthened RC Beams with
Adverse FRP-Steel Interaction
G. M. Chen
1
; J. G. Teng
2
; and J. F. Chen
3
Abstract: RC beams shear-strengthened with externally-bonded FRP U-strips or side
strips usually fail due to debonding of the bonded FRP shear reinforcement. Because
such debonding usually occurs in a brittle manner at relatively small shear crack widths,
some of the internal steel stirrups intersected by the critical shear crack may not have
reached yielding at beam shear failure. Consequently, the yield strength of internal steel
stirrups in such a strengthened RC beam cannot be fully utilized. This adverse shear
interaction between the internal steel shear reinforcement and the external FRP shear
reinforcement may significantly reduce the benefit of the shear-strengthening FRP but
has not been considered explicitly by any of the shear strength models in the existing
design guidelines. This paper presents a new shear strength model considering this
adverse shear interaction through the introduction of a shear interaction factor. A
comprehensive evaluation of the proposed model, as well as three other shear strength
models, is conducted using a large test database. It is shown that the proposed shear
strength model performs the best among the models compared, and the performance of
the other shear strength models can be significantly improved by including the proposed
shear interaction factor. Finally, a design recommendation is presented.
CE Database subject headings: Fiber reinforced polymer; Reinforced Concrete;
Concrete beams; Bonding; Shear failures; Shear resistance; Shear strength.
_______________________
1
Postdoctoral Fellow, Department of Civil and Structural Engineering, The Hong Kong Polytechnic
University, Hong Kong, China
2
Chair Professor of Structural Engineering, Department of Civil and Structural Engineering, The Hong
Kong Polytechnic University, Hong Kong, China (Corresponding author); Email: cejgteng@polyu.edu.hk
3
Reader, Institute for Infrastructure and Environment, School of Engineering, The University of Edinburgh,
Edinburgh, U.K.
This is the Pre-Published Version.
2
INTRODUCTION
The external bonding of fiber-reinforced polymer (FRP) to reinforced concrete (RC)
structures has become a popular strengthening technique in the past decade; the
technique has also received much research attention (Bank 2006; Hollaway and Teng
2008; Oehlers and Seracino 2004; Teng et al. 2002). In particular, the shear resistance of
RC beams can be enhanced by bonding FRP shear reinforcement in the forms of
complete wraps, U-jackets and side strips (Chen and Teng 2003a, b; Teng et al. 2002).
Without loss of generality, the FRP shear reinforcement is assumed herein to be in the
form of discrete strips for ease of discussion; a continuous sheet with fibres oriented in a
single direction can be treated as discrete strips in the fibre direction with a zero net gap
between strips.
Existing research has established a general picture of the structural behaviour of RC
beams shear-strengthened with FRP and led to a number of shear strength models for
them (Chen and Teng 2003a, b; Khalifa et al. 1998; Monti and Liotta 2007; Triantafillou
1998; Triantafillou and Antonopoulos 2000); the more reliable of these shear strength
models have been adopted by design guidelines (ACI-440.2R 2008; CNR-DT200 2004;
fib 2001; HB305 2008). A comprehensive review of existing work (Chen 2010), however,
reveals that several aspects of the behaviour of such strengthened beams are still not well
understood. In particular, the adverse interaction between the different components of
shear resistance (Ali et al. 2006; Chen et al. 2010; Pellegrino and Modena 2002, 2006,
2008) has been identified as a major issue that requires further research.
This paper deals with the effect of interaction between the internal steel shear
reinforcement (only stirrups are considered to simplify the problem) and the external
FRP shear reinforcement in RC beams shear-strengthened with FRP U-strips or side
strips. Such strengthened beams commonly fail due to the debonding of FRP strips from
the beam sides (Chen and Teng 2003b; Teng and Chen 2009). This failure mode is
usually brittle so that the width of the critical shear crack is limited when FRP debonding
occurs (Ali et al. 2006; Chen et al. 2010; Pellegrino and Modena 2008). As a result, at
the instance of debonding failure, the component of shear resistance from concrete is
likely to be well maintained (e.g. Bousselham and Chaallal 2008), but the component of
shear resistance from steel may be significantly below what is expected in a conventional
RC beam because not all steel stirrups in an FRP-strengthened RC beam intersected by
the critical shear crack can reach yielding at the shear failure of the beam (Ali et al. 2006;
Chen et al. 2010; Deniaud and Cheng 2001; Li et al. 2002; Monti and Liotta 2007;
Pellegrino and Modena 2008; Teng et al. 2002; Teng et al. 2004). It may be noted that
this adverse shear interaction effect has not been duly considered in any of the existing
design guidelines (Chen 2010).
Shear strength models in existing guidelines are based on the simple additive
approach that the shear resistance of a shear-strengthened RC beam can be found from
the following equation::
ucsf
VVVV=++
(1)
where V
c
, V
s
and V
f
are the components contributed by the concrete, the steel shear
reinforcement, and the FRP shear reinforcement respectively. The values of V
c
and V
s
are
generally evaluated using provisions in existing design codes for RC structures, while
various expressions have been proposed for V
f .
Eq. (1) implies that the three shear
resistance components reach their ultimate values simultaneously in a real beam, which
is over-optimistic and un-conservative.
A number of studies have been conducted to consider the shear interaction issue (e.g.
Ali et al. 2006; Li et al. 2001; Pellegrino and Modena 2002, 2006, 2008), leading to
several shear strength models that consider the shear interaction effect (Li et al. 2001;
3
Pellegrino and Modena 2002, 2006, 2008). These models, however, have been developed
on the basis of limited experimental results and thus suffer from inevitable limitations.
Recently, Modifi and Chaallal (2011) proposed a new shear strength model that accounts
for this adverse shear interaction effect by introducing the so-called cracking
modification factor (β
c
) which is related to the rigidities of both steel shear reinforcement
and FRP shear reinforcement; the expression of β
c
was determined by curve-fitting of the
experimental results for V
f
. These authors have shown that the inclusion of β
c
can
improve the performance of the proposed shear strength model as well as some other
shear strength models. Whilst the work represents a valuable step forward in
understanding and modeling the FRP-steel interaction effect, their model requires
improvement, especially for beams with FRP U-strips where the FRP shear contribution
(V
f
) is significantly overestimated for a large number of specimens.
To understand the interaction between the three components of shear resistance in Eq.
(1), it is necessary to investigate how each of them develops during the loading process.
If these components are quantified during the loading process, the shear resistance of the
beam can also be quantified throughout the loading process and its ultimate value can be
obtained by finding the maximum of the sum of the three components as schematically
shown in Fig. 1. The authors have recently employed a theoretical approach to establish
the development of shear contributions from the FRP (Chen et al. 2011) and the steel
stirrups (Chen et al. 2010) throughout the loading process as characterized by the critical
shear crack width. This paper first presents a shear strength model for FRP debonding
failure considering the adverse FRP-steel shear interaction developed based on the work
presented in Chen et al. (2010; 2011). Its performance is then assessed using a large test
database collected from the literature. A simplified design recommendation is finally
presented.
SHEAR STRENGTH MODEL ACCOUNTING FOR FRP-STEEL INTERACTION
As with most of the shear strength models in existing guidelines, the proposed shear
strength model is based on the assumption that the shear failure of an FRP
shear-strengthened RC beam is dominated by a single critical shear crack as
schematically shown in Fig. 2, and the shear contributions of both FRP strips and steel
stirrups can be evaluated by truss analogy.
As discussed earlier, the contributions from the concrete, internal steel stirrups and
external FRP strips develop gradually during the loading process (Fig. 1). For FRP
debonding failure, it may be assumed that the contribution of concrete to the shear
capacity of the beam (V
c
) is the same as that in an un-strengthened RC beam because the
width of the critical shear crack is likely to be small when the beam fails due to FRP
debonding (Bousselham and Chaallal 2008). Therefore, the shear resistance of the beam
can be expressed as [instead of Eq. (1)]:
,,uc ssp ffp
VVKV KV=+ + (2)
where
,
s
p
V
and
,
f
p
V
are the maximum shear contributions of steel stirrups and FRP strips
respectively,
K
s
and K
f
are mobilization factors for the steel stirrups and FRP strips
respectively which have been defined by Chen et al. (2010) as:
,
/
s
se y
Kf
σ
= (3)
,,
/
f
fe fe
Kf
σ
= (4)
in which
σ
s,e
and σ
f,e
are respectively the average stress in the steel stirrups and FRP strips
intersected by the critical shear crack,
f
y
is the yield strength of the steel stirrups, and f
f,e
is the effective (average) stress in the FRP intersected by the critical shear crack when
V
f
peaks (i.e.
,
f
fp
VV= , which does not necessarily correspond to the ultimate state of the
4
beam as shown in Fig. 1).
K
s
and K
f
are respectively proportional to the average stress in the steel stirrups and
that in the FRP strips, which are in turn directly related to the shear crack width
w.
Clearly,
K
s
and K
f
reflect the degree of mobilization of the steel stirrups and that of the
FRP strips respectively in resisting shear at a given load level or a shear crack width and
capture the interaction between steel stirrups and FRP strips in resisting shear.
Development of the FRP Contribution K
f
V
f
with crack width
It shall be noted that in both the numerical study (Chen et al. 2010) and analytical
solution (Chen et al. 2011) on which the present study is based, it was assumed that the
width of the critical shear crack varies linearly from the crack tip to the crack end; this
assumption normally leads to conservative results for both FRP strips and steel stirrups
(Chen 2010). With this assumption, the maximum value of the shear crack width is
always at the crack end (Fig. 2); this value is referred to as the crack end width and is
represented by
e
w in this paper. In Chen et al. (2010, 2011), it was also assumed that
the upper end (i.e. the crack tip) of the critical shear crack at the ultimate state is located
at
0.1d from the compression face of the beam (see Fig. 2), with d being the effective
depth of the beam.
Based on the above assumptions and the full-range behaviour of FRP-to-concrete
bonded joints, Chen et al. (2011) developed closed-formed solutions for the development
of the shear contribution of FRP (
f
V
) with the crack end width (
e
w ) for both FRP
U-strips and side strips. Figure 3 shows example
f
e
Vw
−
curves, where the peak loads
are denoted by
u
P and
s
P respectively for FRP U-strips and side strips. From the
f
e
Vw− curve, the
f
e
Kw− curve can be easily obtained according to Eq. (3). Figure 4
shows the
f
e
Kw−
curves corresponding to the
f
e
Vw
−
curves in Fig. 3. Chen et al.
(2011) demonstrated that the
f
e
Kw
−
curve depends mainly on the FRP stiffness
f
f
Et
and the beam height (which can be represented by the effective height of FRP
,
f
e
h as
shown in Fig. 2).
The maximum FRP contribution
,
f
p
V
(the peak value on the
f
e
Vw−
curve) can be
obtained by setting
0
fe
Vw∂∂= based on the
f
e
Vw
−
relationship presented in Chen
et al. (2011). The general expression for
,
f
p
V is given as (Chen et al. 2011):
()
,
,,
cot cot sin
2
fe
fp fef f
f
h
Vftw
s
θ
ββ
+
=
(5)
,,max
f
ef frp
f
D
σ
= (6)
⎪
⎩
⎪
⎨
⎧
=
max,
max,
min
db
f
f
f
σ
σ
(7a)
=
max,db
σ
⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
f
ff
e
f
ff
t
GE
L
L
t
GE
2
2
sin
2
π
e
e
LL
LL
<
≥
max
max
(7b)
5
⎪
⎪
⎩
⎪
⎪
⎨
⎧
++
++
=
strips-for U
sin
strips sidefor
sin2
,
,
max
β
β
btef
btef
hhh
hhh
L
(7c)
where
,
f
e
f
is the effective (average) stress in the FRP intersected by the critical shear
crack;
f
w is the width of individual FRP strips perpendicular to the fiber direction (all
FRP strips are assumed to have the same
f
w );
f
s is the centre-to-centre spacing of FRP
strips measured along the longitudinal axis of the beam (the FRP strips are assumed to be
evenly distributed; and thus for an FRP continuous sheet,
sin
ff
sw
β
=
.);
f
t is the
FRP strip thickness;
θ
is the angle between the critical shear crack and the longitudinal
beam axis;
β
is the angle between the fiber direction and the longitudinal beam axis;
,max
f
σ
is the maximum stress in the FRP strips intersected by the critical shear crack;
max,db
σ
is the maximum stress in the FRP strips intersected by the critical shear crack as
governed by debonding failure;
max
L
is the maximum bond length of FRP strips
intersected by the critical shear crack;
e
L
is the effective bond length of FRP strips as
defined by Eq. (16);
b
h is the thickness of concrete cover (from the beam bottom to the
crack end) (see Fig 2);
t
h is the vertical distance from the top of FRP strips to the crack
tip (see Fig 2); and
f
rp
D is the stress/strain distribution factor determined according to
Chen et al. (2011) as follows.
For FRP side strips, the expression of
f
rp
D
is given by
,,
1(1 )
4
df
db
frp
f
efe
h
h
D
hh
π
=− − ⋅ − (8)
()
,
11
2
fe db
df
h
hh
h
k
π
−
=
+−
(9)
sin
db m b
hL h
β
=− (10)
mhe
LkL=⋅ (11)
,
()
22
11 1 1
4224sin
fe b
h
e
hh
k
L
ππππ
π
βπ
+
⎛⎞⎛⎞⎛⎞
=−−+− +−
⎜⎟⎜⎟⎜⎟
⎝⎠⎝⎠⎝⎠
(12)
For FRP U-strips, the expression of
f
rp
D is given by
,
1(1 )
4
df
frp
f
e
h
D
h
π
=− − ⋅
(13)
,
,
2
sin( )
fe
df f
ep
h
h
w
δ
θ
β
=⋅
+
(14)
,
,f
1`1
2sin
=
sin( )
fe
e
ep
h
L
w
π
β
δ
θβ
⎛⎞
+−
⎜⎟
⎝⎠
⋅
+
(15)
f
e
f
tf
L
Et
τ
δ
=
(16)
2
f
ff
G
δ
τ
=
(17)
