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1.
INTRODUCTION
Extensive research has been conducted on the strengthening of reinforced
concrete (RC) beams with externally bonded fibre reinforced polymer (FRP)
composites [1-16]. RC beams can be strengthened through complete
wrapping, U jacketing or bonding on their sides only. The main shear
failure modes in shear-strengthened beams are FRP rupture and FRP debonding
[14, 15]. In both failure modes, the stress (or strain) distribution
in the FRP at the ultimate state is non-uniform and a key factor influencing
the contribution of the FRP to the shear capacity of the beam for all
three strengthening schemes mentioned above [2, 16]. Based on a
number of simple but rational assumptions, Chen and Teng [2,16] proposed
methods for predicting the maximum values of the FRP stress and the stress
distribution factors at the ultimate limit state for both failure modes.
Among the above three strengthening schemes, U jacketing is probably
the most commonly used in practice. Complete wrapping, although
most effective, is often difficult to implement in application (e.g. difficult
to apply to T beams unless holes are cut through the flange), whilst side
bonding is the least effective [13]. Both FRP rupture and FRP debonding
can occur in U jacketed beams. This paper presents a simple theoretical
study on the stress distribution in the FRP along the critical shear crack
at debonding failure in U jacketed beams. A recently developed FRP-to-concrete
bond-slip model [17] is used in the analysis. Several idealised crack
width variations, which are closely related to the FRP slip field due
to the critical shear crack, are investigated. The results are compared
with Chen and Teng¡¯s [2] simple predictive model. For convenience of description,
the bonded FRP reinforcement is assumed to be in the form of strips, with
a continuous sheet being a special case or a smeared equivalent representation
of strips. Depending on the context, either strips or sheets may be used
to refer to the bonded FRP reinforcement.
2.
EXISTING SHEAR STRENGTH MODELS FOR FRP
DEBONDING
A number of models for the shear strength of FRP-strengthened RC beams
have been proposed for design use [2, 5, 9, 11, 12, 16]. In most
of these proposals, the shear strength of an FRP-strengthened RC beam
Vn is evaluated by assuming that the contributions of the concrete
Vc, the steel shear reinforcement Vs and the bonded
FRP reinforcement Vf are additive, i.e.
Vn = Vc +
Vs + Vf
(1)
Because it is usually proposed
that Vc and Vs should be calculated according to
provisions in existing design codes, the main differences between the
available proposals lie in the evaluation of the FRP contribution Vf.
Steel stirrups have a small
elastic strain limit but excellent ductility, so all stirrups intersected
by the critical shear crack can reach their yield stress at the shear
failure of the RC beam as a result of stress re-distribution. This means
that their yield strength is fully utilized and the stress distribution
in the steel stirrups along the intersecting critical shear crack is uniform.
By contrast, both FRP rupture and FRP debonding are brittle processes
that allow little or limited stress redistribution, so the stress distribution
in the FRP along the shear crack at the ultimate state is non-uniform
and can significantly affect the FRP contribution to the shear capacity
of the beam.
Chen and Teng [1, 15] were
probably the first to propose the explicit inclusion of the effect of
non-uniform stress distribution in the FRP on the shear capacity. The
FRP contribution in their model [2, 16] is given by:
(2)
where q is the critical shear crack angle, hfe
is the effective height of the FRP, wf is the width of the
FRP strip, sf is the centre to centre spacing of FRP strips
along the beam longitudinal axis, and the effective FRP stress ffe
is defined as the maximum FRP stress sf,max times the stress distribution factor Df
(3)
For debonding failures, the
maximum FRP stress sf,max is calculated according to Chen and Teng¡¯s
bond strength model [18], which is given as:
 ,
 ,
(4a,b,c)
 ,
 ,
(4d,e,f)
Assuming that all bonded FRP strips intersected by the
critical shear crack can develop the full bond strength at the ultimate
limit state, Chen and Teng [1, 2] deduced the FRP stress distribution
factor as
(5)
Because both Df
and bL are functions of the normalised maximum bond
length l, the following modified FRP stress factor DFL is defined here for
convenience of comparison with numerical results presented later:
(6)
Using Eq. 6, the effective
FRP stress of Eq. 3 can alternatively be expressed as:
(7)
where sf,inf is
the FRP debonding stress when the FRP bond length is infinite:
(8)
Furthermore, for ease of comparison
for different forms of crack width variations and for both U jacketing
and side bonding, a normalised bond length at the middle of the critical
shear crack l¡¯ is defined here, which can be related to the normalised maximum bond length
l for U jacketing as follows:
(9)
Clearly, the assumption adopted
by Chen and Teng [2] for deriving the FRP stress distribution factor that
all FRP strips intersected by the critical shear crack can reach the full
bond strength at the ultimate limit state is simplistic. The aim of this
study is to assess the validity of this assumption.
3.
Idealisation of FRP slip field along critical
shear crack
Shear cracking in concrete beams is a complex phenomenon and there is
still a lack of accurate methods for predicting the initiation and propagation
of these cracks. Depending on many factors such as loading, amount
and detailed arrangement of internal reinforcement and the properties
of concrete, there may be one or more diagonal cracks. However,
unless the shear failure is due to diagonal compression which often occurs
in beams with very small shear span-to-depth ratios, there is usually
a critical shear crack which dominates the shear failure process.
In some beams, such as those with large shear span-to-depth ratios, the
critical shear crack may be the only crack intersecting the FRP strips
on the beam sides. It may be noted that the FRP-to-concrete bond strength
is increased if there are more than one crack intersecting the FRP strip
[19]. Therefore, it is conservatively assumed in this study that secondary
shear cracks are insignificant compared with the critical shear crack
intersecting the FRP strips in determining the interfacial behaviour between
the FRP strip and the concrete substrate.
Provided that the critical shear crack is predominant over other shear
cracks, the interfacial slips (and thus stresses) between an FRP strip
and the substrate concrete is then almost wholly induced by the widening
of this crack. Clearly, the sum of the interfacial slips on both
sides of the critical shear crack is equal to the crack width. This
means that the bond slip behaviour of the FRP-to-concrete interface is
predominately controlled by the width of the critical shear crack at the
location where it intersects the FRP strip, and that the variation of
this width determines the stress distribution in the FRP along the critical
shear crack.
Because of the lack of information on the precise shape of the critical
shear crack, four simple crack types as shown in Fig. 1 are considered
in this study. Note that the shape of the critical shear crack may
be close to Type A (Fig. 1a) in beams very heavily reinforced in flexure,
whilst it is expected to be closer to Type C (Fig. 1c) in beams with minimal
flexural reinforcement. The crack shape for most practical beams
is likely to lie between these two cases and may be close to Type B (Fig.
1b). If the FRP strips cover only part of the beam height, it is
possible that the critical shear crack is fairly uniform within the effective
FRP height (Fig. 1d). If it is further assumed that the slips between
FRP and concrete on the two sides of the critical shear crack are symmetrical,
which means that the distribution of slip along the critical shear crack
is assumed to follow the shape of the crack. The FRP slip fields
corresponding to the four crack shapes are termed here as Slip Fields
A ¨C D respectively (Fig. 1).
 
(a) Crack type A and Slip Field
A (b) Crack type B and Slip
Field B
 
(c) Crack type C and Slip Field
C (d) Crack type D and Slip
Field D
Figure 1 - Typical idealised
crack shapes and slip fields
4.
Computational models
For U-jacketed beams considered in this study, the FRP strips intersected
by the critical shear crack usually debond from the concrete in the area
above the crack as shown in Fig. 2. The critical area of FRP is
marked as ¡°effective FRP¡± in Fig. 2. In the following finite element
analysis, only the bond resistance of this critical FRP area is considered.
Based on the assumption that the interfacial slips on the two sides of
the critical shear crack are symmetrical, the interfacial slip of the
FRP in the critical area above the crack equals to half of the width of
the critical shear crack. As a result, four simplified computational models
(Fig. 2) can be established, corresponding to the four types of cracks
(Fig. 1). It may be noted that the assumption that the slip is symmetrical
on the two sides of the crack is not exactly valid, but numerical results
from more complete models considering the full height of the FRP intersected
by the crack show that the error resulting from this simplistic treatment
is insignificant.
 
(a) Slip Field A (U-jacketing)
(b) Slip Field B (U-jacketing)
 
(c) Slip Field C (U-jacketing)
(d) Slip Field D (U-jacketing)
Figure
2 - Computational models
5.
Finite element implementation
The idealised computational models shown in Fig. 2 were implemented
using ANSYS [20]. The FRP strips were modelled by a series of truss
elements, whilst the interfaces were modelled by a series of spring elements,
as shown in Fig. 3. The truss elements were connected to the rigid
substrate concrete (e.g. deformations of un-cracked concrete were ignored)
using the spring elements. The bond-slip relationship proposed by
Lu et al. [17] (Fig. 4) was used to derive the properties of the spring
elements. The LINK 1 element in ANSYS was used to model the FRP
truss elements and the COMBIN 39 element was used to model the spring
elements.
The loading process was effected by continuously increasing the displacements
of the truss elements at the boundary that represents the critical shear
crack, according to one of the four different slip fields (Fig. 2).
An analysis was terminated when the average stress in the FRP sf,ave was reduced to half of its peak value. The results
were then used to deduce the average FRP stress and the FRP stress distribution
factor at different slip values.
Figure 3 - Finite
element models (Slip Field C)
Figure 4 - Bond-slip model [17]
6.
Numeriacal results
6.1
Development of stresses in FRP
Figure 5 shows a typical example of the variation of the average FRP
stress against the increase of slip at the mid-height of the effective
FRP for a U jacked beam for different slip fields. Here the average
FRP stress sf,ave is normalised by sf,inf, and the slip at the mid-height
smid is normalised by the slip at the peak interfacial stress
s0 (Fig. 4). The following parameters were used in this
example: concrete tensile strength ft = 3MPa, axial stiffness
of FRP Eftf = 16GPa-mm, FRP strip width to spacing
ratio wf/sf = 0.5, normalised FRP bond length at
the mid-length of the crack (or mid-crack) l¡¯ = 1.5.
Except for Model D, the FRP average stress reaches its peak value when
the normalised slip at mid-height is about 7.5 (corresponding to a maximum
crack width of about 0.88mm for Models A and B but 1.75mm for Model C).
sf,ave reduces rapidly after its peak value. For Slip Field
D (uniform crack width), the behaviour is quit different, where sf,ave reaches its peak value
when smid/s0 is only slightly grater than 1 (corresponding
to a maximum crack width of about 0.12mm) but its descending rate thereafter
is much smaller than other models. This is because the FRP strips
in this model debond from the concrete one by one at a regular and slow
pace. In Model C, the FRP strips debond from the concrete almost
simultaneously because the variation of the bond length in the critical
FRP area matches the distribution of the slip field. In general,
a model that leads to a higher peak FRP average stress has a steeper descending
branch, and predicts a failure process that is less ductile. It
is interesting to note that the differences between the peak FRP average
stresses from the four models are not large.
The distributions of the FRP
stress over the height from the four models with l¡¯=1.5 are shown in Fig. 6 at their corresponding
peaks of the FRP average stress. The distribution predicted by Chen
and Teng¡¯s model [1, 2] is also shown for comparison. It is seen
that Chen and Teng¡¯s prediction is in very close agreement with those
from Slip Fields B and C. In Model D, the peak average stress is
reached before the stress in any of the FRP strips reaches sf,inf because the strips near the crack tip (top) start to debond
from the concrete at a very early stage. For Model A, the stresses
in the FRP strips close to the beam bottom are not well developed because
the crack widens fastest at the mid-height of the beam.
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Figure 5 ¨C Variation of average FRP stress with slip
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Figure 6 - FRP stress distribution at peak average stress (¦Ë¡¯=1.5)
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6.2
Effect of beam size on FRP stress distribution
If the bonded FRP strips cover
the full height of the beam on both sides, the normalised FRP bond length
l¡¯ is proportional to the beam height. Therefore, the effect of the beam size
on the FRP stress distribution can be investigated by varying the normalised
FRP bond length l¡¯. For different l¡¯ values, Fig. 7 shows the FRP stress distributions
corresponding to the peak values of the FRP average stress for Model C.
The FRP stress distribution is close to the shape of the slip field (linear
in Model C) when l¡¯ is very small because the slip is the largest
at the bottom where the FRP bond length is also the largest. As
the normalised bond length increases(beam size increases or FRP axial stiffness decreases), the maximum FRP
stress also increases until the maximum bond length (at the bottom of
the crack) reaches the effective bond length. After that, the stress
distribution in the FRP becomes more and more uniform because the FRP
stress cannot exceed sf, inf.
6.3
Relationship between DfL and
l¡¯
As is evident from Eq. 7, the effective FRP stress can be easily obtained
if the modified FRP stress factor DfL is given. Furthermore,
the FRP contribution to the shear capacity is proportional to DfL.
The effect of the bond length (beam size) on DfL is investigated
here for the four slip fields (Fig. 8). Chen and Teng¡¯s [1,
2] prediction is also shown for comparison. Although the four crack
shapes are significantly different and they may represent some extreme
scenarios, the predicted FRP stress factors are not significantly different.
The maximum difference between the four models for l¡¯=1 is less than 15% and this difference reduces further as l¡¯ increases. It is shown that Chen and
Teng¡¯s prediction lies nicely between these four curves: it is generally
higher than those from Models A and D, lower than that from Model C, and
very close to but slightly lower than that from Model B. As noted
earlier, the shape of the critical shear crack for most practical beams
is expected to lie between those of Models A and C, which may explain
the close agreement between Chen and Teng¡¯s shear strength model and independent
test data [2
]. Therefore, it may be concluded Chen and Teng¡¯s model is
safe and accurate.
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Figure 7 ¨C FRP stress distributions at peak average stress (Slip Field C)
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Figure 8 - Dfl~¦Ë relationships from different models
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7.
Conclusions
This paper has presented a theoretical study on the stress distribution
in the FRP in RC beams shear-strengthened with FRP U jackets at debonding
failure. Four different idealised crack shapes for the critical
shear crack were considered. A rigorous FRP-to-concrete bond-slip
model was used to represent the interfacial behaviour between the FRP
and the substrate concrete. The finite element method was employed
to model the behaviour of an idealised system consisting of FRP strips
bonded to a rigid concrete prism. Although the different crack shapes
result in significantly different stress distributions in the FRP, their
effect on the stress distribution factor is much less significant.
The study confirms that Chen and Teng¡¯s [1, 2] simple assumption for the
stress distribution in the FRP results in satisfactory predictions of
the contribution of the FRP to the shear capacity of the beam.
8.
acknowledgements
The authors would like to
acknowledge financial support provided by the Royal Society through Royal
Society-NSFC UK-China Joint Project (Grant No. IS 16657), the Research Grants Council of the Hong Kong Special Administrative Region,
China (Project No: PolyU 5151/03E), and the National Natural Science Foundation
of China through a key project for FRP in construction (Project No. 50238030).
9.
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