full-range frp failure behaviour in rc beams shear

Accepted Manuscript

Full-range FRP failure behaviour in RC beams shear-strengthenedwith FRP wraps

G.M. Chen , S.W. Li , D. Fernando , P.C. Liu , J.F. Chen

PII: S0020-7683(17)30336-0DOI: 10.1016/j.ijsolstr.2017.07.019Reference: SAS 9665

To appear in: International Journal of Solids and Structures

Received date: 30 November 2016Revised date: 15 May 2017Accepted date: 20 July 2017

Please cite this article as: G.M. Chen , S.W. Li , D. Fernando , P.C. Liu , J.F. Chen , Full-range FRPfailure behaviour in RC beams shear-strengthened with FRP wraps, International Journal of Solids andStructures (2017), doi: 10.1016/j.ijsolstr.2017.07.019

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http://dx.doi.org/10.1016/j.ijsolstr.2017.07.019

http://dx.doi.org/10.1016/j.ijsolstr.2017.07.019

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Highlights

Full-range failure behaviour of FRP wraps for shear-strengthening RC beams

Progressive failure process involving both debonding and rupture of FRP

An analytical solution, validated by finite element analysis, is developed

Development of shear contribution by FRP with crack opening quantified

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Full-range FRP failure behaviour in RC beams

shear-strengthened with FRP wraps

G.M. Chen a, S.W. Li

b, D. Fernando

c, P.C. Liu

a and J.F. Chen

d,e*

a School of Civil and Transportation Engineering, Guangdong University of

Technology, Guangzhou, China.

b Guangzhou Power Supply Bureau, China Southern Power Grid Co. Ltd., Guangzhou,

China.

c School of Civil Engineering, Faculty of Engineering, Architecture and Information

Technology, The University of Queensland.

d College of Civil Engineering and Architecture, Wenzhou University, China.

e School of Natural and Built Environment, Queen‟s University Belfast, Belfast BT9

5AG, UK.

*Corresponding author. Tel.: +44 28 9097 4184; fax: +4428 9097 4278;

E-mail address: [email protected] (J.F. Chen)

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Abstract: Reinforced concrete (RC) beams can be strengthened in shear by externally

bonded (EB) fibre reinforced polymer (FRP) composites in the forms of side-bonded

FRP strips, FRP U-jackets or FRP wraps. The shear failure of almost all RC beams

shear-strengthened with FRP wraps and some of the beams strengthened with FRP

U-jackets, is due to the rupture of FRP. For an FRP wrapped beam with FRP bonded

to the beam surface, debonding of EB FRP from concrete substrate usually precedes

the ultimate FRP rupture failure, therefore the failure process of the beam is

associated with both FRP debonding and rupture failures. Despite extensive research

in the past decade, there is still a lack of understanding of how the failure of FRP

wraps in such an FRP-strengthened beam progresses and how it affects the shear

behaviour of the beam. This paper presents an analytical study on the progressive

failure of FRP wraps in such strengthened beams. In this study, the debonding and the

subsequent rupture processes are derived and the FRP contribution to the shear

capacity of the beam is quantified. The analytical solution is verified by comparing its

predictions with the predictions of a finite element model. An additional merit of the

analytical solution is that the development of FRP shear contribution with the opening

of the shear crack can be quantitatively determined, providing a useful tool for further

investigation on the shear interaction among different components (FRP, shear

reinforcements, steel shear reinforcements, and concrete) in RC beams

shear-strengthened with FRP.

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Keywords: Fibre reinforced polymers (FRP); Reinforced concrete beams;

Strengthening; Shear failure; Shear strength; FRP Debonding; FRP Rupture; Stress

distribution; Shear interaction

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1. Introduction

Externally bonded (EB) fibre-reinforced polymer (FRP) reinforcement has been

increasingly used to enhance the shear resistance of reinforced concrete (RC) beams

in the past two decades due to its unique advantages including high strength-to-weight

ratio, excellent corrosion resistance, and ease of installation (Bank, 2006; Hollaway

and Teng, 2008; Oehlers and Seracino, 2004). Shear strengthening of RC beams using

EB FRP can be achieved in different ways by: (1) bonding FRP laminates around the

entire beam section (referred to as “FRP wraps” hereafter), (2) bonding FRP laminates

to the two sides as well as the tension face of the beam (referred to as “U-strips”

hereafter), and (3) bonding FRP laminates to the two sides of the beam only (referred

to as “side strips” hereafter) (Teng et al., 2002). In RC beams strengthened with EB

FRP, FRP debonding failure is a typical failure mode, where FRP is detached from the

concrete substrate with a thin layer of concrete bonded to the debonded surface of

FRP. RC beams shear-strengthened with FRP side strips, or strengthened with U-strips

predominantly fail due to the debonding of FRP strips from the beam sides (Chen and

Teng, 2003a, b; Teng and Chen, 2009). However, often the failure of RC beams shear

strengthened with FRP wraps (Jirawattanasomkul et al., 2013; Teng et al., 2009) and

in some cases with FRP U-jackets [e.g. FRP U-jackets with anchorage (Chen et al.,

2016; Kim et al., 2014; Mofidi et al., 2012)], are due to the rupture of FRP, usually

preceded by FRP debonding (Cao et al., 2005; Teng et al., 2009).

Extensive studies carried out over the past two decades on RC beams

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shear-strengthened with EB FRP (Bousselham and Chaallal, 2004, 2006a, b, 2008,

2009; Carolin and Taljsten, 2005a; Islam et al., 2005; Khalifa et al., 1998; Khalifa and

Nanni, 2000, 2002; Leung et al., 2007; Matthys, 2000; Monti and Liotta, 2007;

Pellegrino and Modena, 2002, 2006, 2008; Taljsten, 2003; Triantafillou, 1998) have

led to many design models (Carolin and Taljsten, 2005b; Chaallal et al., 1998; Chajes

et al., 1995; Chen and Teng, 2003a, b; Khalifa et al., 1998; Mofidi and Chaallal, 2011;

Monti and Liotta, 2007; Pellegrino and Modena, 2002, 2006, 2008; Taljsten, 2003;

Triantafillou, 1998; Triantafillou and Antonopoulos, 2000; Ye et al., 2005), some of

which have been adopted in design guidelines (ACI-440.2R, 2008; CNR-DT-200R1,

2013; Concrete-Society, 2012; fib, 2001; GB50608-2010, 2010; HB305, 2008; JSCE,

2001). The majority of these existing design models have assumed that the shear

capacity of the EB FRP shear-strengthened beam (Vu ) is a sum of the shear

contributions of concrete (Vc), steel shear reinforcements (Vs ) and shear strengthening

FRP (Vf) based on simple superposition (i.e. Vu = Vc + Vs + Vf). However, all three

components may not reach their peak values (Vc , Vs , Vf ) simultaneously and adverse

shear interaction may exist among them during the failure process of the strengthened

beam, thus resulting in a shear strength less than the summation of the maximum

shear contribution of each component (Ali et al., 2006; Bousselham and Chaallal,

2004, 2006a, b, 2008; Chen, 2010; Chen et al., 2012, 2013; Chen et al., 2010; Denton

et al., 2004; Khalifa and Nanni, 2000, 2002; Li et al., 2001, 2002; Park et al., 2001;

Pellegrino and Modena, 2002, 2006, 2008; Teng et al., 2002; Teng et al., 2004).

In RC beams shear strengthened using EB FRP and failing due to FRP debonding,

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an adverse shear interaction usually exists between the external FRP (Vf) and internal

steel shear reinforcements (Vs) (Ali et al., 2006; Chen, 2010; Chen et al., 2012, 2013;

Chen et al., 2010) as explain next. Due to the brittle nature of debonding failures, the

width of the critical shear crack is usually small upon the occurrence of FRP

debonding failure; as a result, some of the internal steel shear reinforcements

intersected by the critical shear crack may not reach yielding (as demonstrated in

Chen et al. (2010)) as assumed in most of the existing studies except a few which

duly considered the shear interactions (Ali et al., 2006; Chen et al., 2012, 2013;

Pellegrino and Modena, 2002, 2006, 2008). Among the few studies considering the

shear interaction, Chen et al. (2012) presented a closed-form solution for the

full-range debonding process of FRP side strips and U strips, allowing different stages

of the shear debonding failure to be understood and the shear contribution of FRP at

different stages quantitatively determined. By evaluating the development of shear

resistance contributed by both external shear-strengthening FRP (based on the

solution of Chen et al. (2012)) and internal steel shear reinforcements as the critical

shear crack widens, Chen et al. (2013) developed a new shear strength model

considering the adverse shear interaction between the internal steel shear

reinforcement and the external FRP; the model provides improved predictions of the

shear strength of FRP shear-strengthened RC beams where FRP debonding governs.

In RC beams shear strengthened using EB FRP and failing due to FRP rupture, the

adverse shear interaction exists mainly between the external FRP (Vf ) and concrete

(Vc), which can be qualitatively explained as follows. As debonding of FRP strips on

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the beam sides generally precedes the eventual FRP rupture failure as experimentally

observed by Cao et al. (2005) and Teng et al. (2009), it can be reasonably assumed

that the average strain developed in the FRP along the critical shear crack is

proportional to the maximum shear crack width. Based on this assumption, it can be

easily inferred that the average strain in FRP is also affected by the beam size: the

larger the beam size the larger the shear crack width at the FRP rupture failure, and

thus the larger degradation of concrete shear resistance mechanism (i.e. larger

reduction of Vc). Note that the average strain in FRP is usually termed the effective

FRP strain following Chen and Teng (2003a,b). Given the relatively large failure

strain of FRP composites compared to the yield strain of normal steel (Teng et al.

2002), the crack width at FRP rupture failure may be large enough to cause the

yielding of steel shear reinforcements but compromise the shear resistance mechanism

in concrete including the friction along the cracked concrete surfaces, concrete

aggregate interlock and dowel action, and thus cause a degradation in cV (Teng et al.,

2002; Triantafillou and Antonopoulos, 2000). This phenomenon was first observed in

FRP-wrapped columns under cyclic loading (Priestley et al., 1996; Priestley and

Seible, 1995). More recently, the existence of this adverse shear interaction was

clearly observed by Teng et al. (2009) and Jirawattanasomkul Jirawattanasomkul et al.

(2013) in RC beams shear-strengthened by FRP wraps and by Chen et al. (2016) in

RC beams shear-strengthened with well-anchored FRP U-strips (failing by FRP

rupture). Using accurately measured FRP strains, it was shown that the concrete shear

contribution started to decrease while that of FRP was still increasing. In particular, at

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the ultimate state of the strengthened beams, the effective strain in FRP intersected by

the critical shear crack was found to be 24%-56% of FRP rupture strain (Chen et al.

2016). When FRP with large rupture-strain (e.g. >6%) was used, shear contribution of

concrete was found to degrade by 056.4%, depending on the amount of FRP used,

shear-span to beam depth ratio, and the member size (Jirawattanasomkul et al., 2013).

Based on the limited test results of small specimens (Khalifa et al., 1998; Priestley

and Seible, 1995), existing guidelines normally recommend a strain limit in the range

of 0.004 0.006 (e.g. 0.004 in ACI 440-2R (2008) and 0.006 in fib (2001)), to avoid a

significant degradation of the concrete shear resistance. Whilst this limit provides a

convenient design provision, it does not necessarily prevent the development of wide

cracks (see Denton et al., 2004), especially for large beams as discussed above.

Therefore, quantitative evaluation of the shear contribution of FRP throughout the

failure process of RC beams shear-strengthened with EB FRP is important for

achieving a better understanding of the shear resistant mechanism of EB FRP,

especially for developing a shear strength model capable of accurately considering the

shear interaction among concrete, internal steel and EB FRP. For FRP debonding

failure (of FRP side strips and U strips), such research had been done by the authors

using an analytical solution in Chen et al. (2012). This paper presents a closed-form

solution for the entire failure process of FRP wraps used to enhance the shear

resistance of RC beams where FRP rupture failure usually governs, allowing the

different stages of failure to be clearly explained and understood. A particularly

important outcome of the solution is the explicit expressions that describe the

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development of shear contribution of externally bonded FRP wraps as the critical

shear crack widens. An additional merit of the closed-form solution is that it can be

directly employed in investigating the effect of shear interaction on the shear strength

of FRP shear-strengthened RC beams where FRP rupture failure governs. It should be

noted that although the solution is developed specifically for FRP wraps, it is also

applicable to other FRP shear-strengthening schemes failing by FRP rupture, such as

RC beams shear-strengthened with well-anchored FRP U-strips (Chen et al., 2016;

Kim et al., 2014; Mofidi et al., 2012). In the following sections, the basic assumptions

of the solution is first presented, followed by solutions for different stages during the

failure process which are then validated by comparing the predictions of the solutions

with finite element (FE) predictions.

2. Basic assumptions

The basic assumptions adopted herein are exactly the same as those assumed for

FRP U strips and side strips which are detailed in Chen et al. (2012). As in (Chen et

al., 2012), the external FRP shear reinforcement is assumed to be in the form of

discrete strips for consistency without loss of generality. This means that the

analytical solution is exact for FRP shear reinforcement in the form of discrete narrow

FRP strips with zero net gaps, but is approximate for discrete FRP strips with

non-zero gaps or continuous FRP sheets as the approach neglects the interactions

between the bonded fibres. It is further assumed that the failure process of the FRP

shear-strengthened beam is dominated by the initiation and widening of a single

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dominant shear crack (termed as “critical shear crack” hereafter) although other

secondary cracks may exist in practice which are usually less critical as shown in

(Chen et al., 2007; Teng et al., 2006). It is also assumed that the bond-slip behaviour

of the FRP-to-concrete bonded interface is governed by the linearly softening

bond-slip model (Fig. 3(a)). The solution presented here as well as that in Chen et al.

(2012) are different from the solution of Monti et al. (2004) [see also Liotta (2006) for

details] in a number of significant aspects as explained in Chen et al. (2012). The

essential difference lies in that the present solution aims to find the full-range failure

behaviour of FRP while the critical shear crack widens, while Monti et al. (2004) only

attempted to obtain the maximum effective stress in FRP strips and hence the FRP

contribution to the shear capacity of the strengthened beam at the ultimate state.

3. Modelling the behaviour of shear-strengthening FRP using FRP-to-concrete

bonded joints

The present analytical solution is developed based on analytical solutions for the

full-range behaviour of FRP-to-concrete bonded joints with either free (type A joint)

and fixed (type B joint) far end as presented in Chen et al. (2012), with only the bond

behaviour in the fibre direction of the FRP considered (Fig. 1). As explained in Chen

et al. (2012), an FRP U-strip in a shear-strengthened RC beam can be represented

with a type A joint representing the FRP-to-concrete bonded interface above the

effective shear crack, connected at the location of the effective shear crack to a type B

joint representing the interface below the effective shear crack (Figs 1 and 2).

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Similarly, an FRP wrap can be well represented by two type B joints above and below

the effective shear crack respectively, with their far ends fixed at the beam top and

bottom, and connected with each other at the location of the effective shear crack

(Figs 1 and 2). If an FRP U-strip is well-anchored (i.e. fixed) at the upper end, it can

also be represented by two type B joints above and below the effective shear crack as

with the FRP wrap except that the fixed upper end of FRP is usually not at the beam

top. For the convenience of understanding, a summary of the solution of the full-range

behaviour of type B joint is presented next. Further details of the solution can be

found in Chen et al. (2012).

The full-range load-displacement response of a type B joint can be characterized

by key points O, B‟, D‟ and P‟ when the FRP bond length L is small [i.e. uL a ,

where 2

ua

is the effective bond length of FRP,

1f

f f fE t

is a

dimensionless constant defined to qualify the effective bond length, f and f are

the maximum interfacial shear stress and maximum interfacial slip of the

FRP-to-concrete bonded interface, as shown in Fig. 3(a)], and by key points O, A, B

and P when the bond length is large ( uL a ) as schematically shown in Fig. 3(b).

Accordingly, the full range load-displacement response (i.e. - P ) can be divided

into three segments, namely segments 'OB , ' 'B D , ' 'D P for uL a and OA , AB

and BP for uL a . The interfacial shear stress distributions along the

FRP-to-concrete interface experienced by a type B joint during the loading process

are schematically shown in Fig. 4 for uL a and in Fig. 5 for uL a , corresponding

to the different stages of the load-displacement curve shown in Fig. 3(b). In Figs 4

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and 5, the letters I, S and D stand for intact [i.e. “rigid part” in Chen et al. (2012)],

softening and debonded zones of the interface respectively.

The solutions for segments OA and AB for the case of uL a and 'OB for

the case of uL a (Fig. 3(b)) of the load-displacement response are the same as

those for a type A joint detailed in Chen et al. (2012). For segments ' 'B D of a type

B joint with a short bond length (L ≤ au), the solution is

max

tanP = + sinf

f f f

LL

E t b

(1)

max = 1 cosf L (2)

where P and are the force and slip at the loaded end of FRP respectively;

1f

f f fE t

is a dimensionless constant defined to qualify the effective bond

length of FRP ua ; f and f are the maximum interfacial shear stress and

maximum interfacial slip of the FRP-to-concrete bonded interface (Fig. 3(a)); fb , ft

and fE are width, thickness and modulus of elasticity of the FRP bonded to the

substrate concrete respectively; L is the bond length of FRP. At the end of this stage

(at point 'D ), the displacement ∆ and the corresponding force P at the loaded end are

f (3)

f

cos ( ) 1P = sin( )+

tan( ) sin

f f

f f f

bLE t b L

L L

(4)

The solution for segment BP of a type B joint with a long bond length ( uL a )

is the same as that for segment ' 'D P for a type B joint with a short bond length

( uL a ) and is given by

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1

P = sin '

f fb

a

(5)

'

- ' =

sin( )

f

f

L a

a

(6)

where 'a is the length of the softening part of the FRP-to-concrete interface at this

stage (Figs 4(e) and 5(d)) which can be related to the debonded length of the interface

d ( 0 d L ) through

dLa ' (7)

When the FRP-to-concrete interface approaches the state of complete debonding,

'a approaches zero (i.e. d approaches L ). Accordingly, the slope of the

load-displacement curve is

f f f

P

E b tK

L (8)

The maximum stress in FRP is bounded by the tensile strength of FRP ( ff ); in

Fig. 3(b), this is schematically marked as 'P for uL a and P for uL a .

When the debonding front (where interfacial slip 0 and interfacial

stress f ) does not reach the fixed end, the maximum stress in FRP is controlled

by the bond strength ( b ) of the FRP-to-concrete bonded interface above or below

the critical shear crack determined by the solution of a type A joint (Chen et al. 2012),

depending on which one is more critical (i.e. shorter). For instance, if uL a , the

bond strength is given by

2 =

f f

b f f

f

G Et

t (9)

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When the bond strength b is reached, the FRP slip at the crack is greater than

f and smaller than f f uL a for uL a (i.e. it can be anywhere between

A and B in Fig. 3(b)); for L < au

= 1 cos 1 sin tan( ) f fL L L (10)

which is marked as 'C in Fig. 3(b).

4. Shear contribution of shear-strengthening FRP

Similar to Chen et al. (2012), only the shear tension failure is considered herein

where the shear failure process of an RC beam shear-strengthened with FRP is

assumed to be dominated by the development of a single critical shear crack at an

angle from the beam longitudinal axis (Fig. 6). In practice, while additional shear

cracks are likely to occur, the assumption of a single diagonal crack is generally

conservative for predicting the FRP shear contribution for two reasons: a) the

FRP-concrete bond strength is usually higher when there are multiple cracks

compared with the case with a single crack (Chen et al., 2007; Teng et al., 2006); and

b) more distributed multiple cracks mean that the crack widths are smaller than that of

a single crack, implying less degradation of concrete shear resistance mechanism such

as aggregate interlock. Following Chen and Teng (2003a, b), only the contribution of

the FRP strips intersected by the critical shear crack is considered, with the “crack

tip” located at 0.1d (d is the effective depth of the beam) from the compression face of

the beam at failure and the “crack end” located at the centroid of the steel tension

reinforcement (i.e. intersection of crack and the centroid of steel tension

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reinforcement) which is ch above the beam bottom, as shown in Fig. 6. The shear

crack between this crack end and the crack tip is termed the “effective shear crack”

hereafter. The vertical distance from the upper edge of FRP strips to the crack tip is

th while that from the lower edge of FRP strip to the crack end is termed as bh , and

vertical distance from the crack tip to the crack end is ,f eh . For FRP wrap, the

bonded FRP strips cover the full height of the beam so that: , 0.9f eh d , 0.1th d

and b ch h ; for fully anchored FRP U-strips or FRP side strips, the upper and lower

edges of the FRP strips may not be extended to the beam top and bottom, as a result,

the value of th ( 0.1th d ) and bh ( b ch h ) should be determined according to the

location where the FRP strips are fully-anchored.

Based on the principle of force equilibrium in the vertical direction, the shear

contribution of shear-strengthening FRP is given by (Chen and Teng, 2003a, b):

,

,

cot cot sin2

f e

f f e f f

f

hV f t w

s

(11)

where ,f ef is the effective (average) stress in the FRP strips intersected by the

effective shear crack; fw is the width of an individual FRP strip perpendicular to the

fibre direction (all FRP strips are assumed to have the same fw ); fs is the

centre-to-centre spacing of FRP strips measured along the longitudinal axis (the FRP

strips are assumed to be evenly distributed); ft is the thickness of the FRP strips;

and are respectively the angles of the effective shear crack and the fibre direction

with respect to beam longitudinal axis. Note that for a continuous FRP sheet/plate (or

FRP strips with zero net gaps):

sinf fw s (12)

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From Eq. (11) the development of the shear contribution of FRP strips ( fV ) with

the opening-up of a shear crack can be determined if the development of effective

stress of FRP ( ,f ef ) with the widening of the shear crack is known.

The variation of the width of the shear crack along the crack length may take

various forms such as different parabolic forms controlled by a parameter C as

proposed by Chen and Teng (2003a). Li (2015) investigated the effect of the crack

width variation and concluded that a linear crack width variation (C=0 in Chen and

Teng‟s (2003a) model) is the most critical as shown in Fig. 7. Therefore, in this study

the shear crack width is assumed to increase linearly from zero at the crack tip to we at

the crack end (termed as “crack end width” for simplicity hereafter):

,

e

f e

zw z w

h (13)

where z is the vertical downward-coordinate starting from the crack tip (Fig. 6).

Following Chen and Teng‟s (2003a, b) definition, the effective stress in FRP ,f ef

is

,

0,

,

f eh

f e

f e

z dz

fh

(14)

where z is the stress in an FRP strips intersected by the critical shear crack at a

coordinate z . If discrete FRP strips are smeared and treated as an equivalent FRP

continuous sheet/plate, Eq. (14) is applicable to beams strengthened with either FRP

discrete strips or FRP continuous sheets or plates.

In the following section, two different solutions are given for ,f ef for two cases:

thick concrete cover with cosecbh > ua (i.e. ua is effective bond length of FRP)

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thin concrete cover with cosecbh ≤ ua . It should be noted that in this study the

bond length of FRP strips above the crack end is assumed to be larger than both that

below the crack end and the effective bond length of FRP, that is,

,( + )cosec cosect f e bh h h and ,( + )cosect f e uh h a (Fig. 6). This is believed to be

satisfied for the vast majority (if not all) of practical FRP configurations and beam

sizes.

As mentioned earlier, the FRP-to-concrete bonded interface (Figs 4 and 5) can be

classified as three typical parts according to the different stress states: intact part

(labelled by “I”), softening part (labelled by “S”) and debonded part (labelled by “D”).

Accordingly, the FRP bond area can thus be divided into the following three zones

during the loading process (schematically shown in Figs 8 and 11):

(1) Intact zone (I-zone), where the interface between FRP and concrete is not yet

stressed, which is defined as the “inactive zone” in Chen et al. (2012);

(2) Softening zone (S-zone), where the interface is in a softening state, which is

defined as “mobilized zone” in Chen et al. (2012);

(3) Debonded zone (D-zone), where the interface has debonded completely.

As schematically shown in Figs 8 and 11, I-zone and S-zone are separated by the

“softening front” where m and 0 (also see Figs 4 and 5). S-zone and

D-zone are separated by the “debonding front” where = 0 and = f (also see Figs 4

and 5).

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5. Development of FRP shear contribution with crack width

When FRP rupture is the governing failure mode, the failure process of FRP in an

RC beam shear-strengthened with FRP can be divided into the following four stages

(Figs 8 and 11):

(1) Softening stage (S-stage) during which no debonding has taken place;

(2) Debonding stage (D-stage) during which partial FRP debonding occurs both

below and above or just above the critical shear crack, depending on the concrete

cover thickness relative to the effective bond length (characterised by effective

bond length of FRP ua ) and the sequence of debonding;

(3) Hardening stage (H-stage) during which the upper fixed end of the FRP wrap

starts to develop a reaction force so that the FRP stress around the critical shear

crack increases to values above the bond strength (i.e. b ) of FRP-to-concrete

bonded interface;

(4) Rupture stage (R-stage) during which the tensile strength (i.e. ff ) of FRP is

reached.

As discussed earlier, the FRP wrap in an FRP shear-strengthened RC beam can be

represented by two type B joints representing the interfaces above and below the

effective shear crack respectively, connected with each other at the location of the

effective shear crack (Fig. 1(d)), with their ends fixed at the beam top and beam

bottom respectively. In terms of boundary conditions and the bonded interfaces, an

FRP wrap differs from an FRP U strip only in that the FRP wrap is fixed at the upper

end of the FRP at the beam top (Fig. 1(d)) while for the FRP U-strip, the upper end of

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FRP is free to move (Fig. 1(c)). As a result, the failure process of an FRP wraps are

exactly the same as that of an FRP U-strip until the upper fixed end of the FRP wrap

at beam top is mobilized to affect the failure process. In particular, for the four failure

stages of FRP wrap mentioned above, the first two stages (i.e. softening stage and

debonding stage) are exactly the same for the FRP warp and the FRP U-strip, and the

latter two stages (hardening stage and rupture stage) are unique for the FRP wrap.

Next, solutions of the four stages are presented with respect to the thickness of the

concrete cover relative to the effective bond length of FRP (i.e. au ) as in Chen et al.

(2012). Since the solutions for the first two stages (S-stage and D-stage) are exactly

the same as those for the FRP U-strips presented in Chen et al. (2012), only the

solutions for these later two stages (H-stage and R-stage) are presented in detail here

and those for the first two stages are only briefly introduced for completeness

(detailed solutions can be found in Chen et al. (2012)).

5.1 Thick concrete cover with cosecb uh a (Fig. 8)

For a thick concrete cover with cosecb uh a , the failure process of an FRP wrap

can be divided into four successive stages including: 1)softening stage

with 0 m uL a (Fig. 8(a)); 2) partial debonding stage with

,( )cosecu m f e ta L h h (Figs. 8(b)-8(d)); 3) hardening stage with 0dbh (Figs.

8(e) and 8(f)); and 4) FRP rupture stage with 0rh (Fig. 8(g)).

5.1.1 Softening stage with 0 m uL a (Fig. 8(a))

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The softening stage (Fig. 8(a)) is characterized by a very small crack end width

and a very small mobilized softening zone quantified by the mobilized length L z

in the fibre direction. This stage ends when the maximum mobilized bond length

above the shear crack mL at the crack end reaches au. From Chen et al. (2012), the

effective stress in FRP ,f ef is:

, = f e b frpf D (15)

where frpD is the stress distribution factor as defined in Chen and Teng (2003b)

m m m

m

sin( )cos( )

2 2cos( )frp

L L LD

L

(16)

and b is the bond strength of the type A joint for uL a :

2 =

f f f

b

f f

G E

t t

(17)

The corresponding shear crack width at the crack end is

m1 cos = 2

sin( )e f

Lw

(18)

5.1.2 Partial debonding stage with ,( )cosecu m f e ta L h h (Figs 8(b)-8(d))

This stage can be further divided into two successive sub-stages: (a) two-way

partial debonding stage (I) with cosu m ba L h ec (Fig. 8b), and (b) two-way

partial debonding stage (II) with ,cosec ( )cosecb m f e th L h h (Figs. 8(c) & 8(d)).

The partial debonding stage (I) starts when the most critical FRP fibre (leftmost in

Figs 8(b)-8(d)) intersected by the critical shear crack starts to debond so that the

maximum mobilized length above the shear crack mL is equal to effective bond

length of FRP (i.e. m uL a ) there and the vertical distance from the crack tip to the

intersection of debonding front and the effective crack (marked as point D in Figs

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8(b)-8(d)) is ,df f eh h . This stage ends, and the two-way partial debonding stage (II)

starts when the softening front below the critical shear crack reaches the beam bottom

(i.e. cosecm bL h ) where FRP is assumed fixed (Fig. 8(b)), and ends when the

softening front of the leftmost FRP fibre reaches the upper edge of the FRP wrap

(i.e. , cosecm f e tL h h , see Fig. 8(d)), which is corresponding to the ultimate state

of FRP U-strip. At the end of the two-way partial debonding stage (II): ,df df uh h and

,e e uw w at which complete debonding of FRP occurs there for FRP U-strips.

During the partial debonding stage (I), the boundary condition of fixed ends does

not affect the bond behaviour of the interface, i.e., the lower softening front is above

the lower fixed boundary. Consequently, the solution for this stage is exactly the same

as that for the two-way partial debonding stage with cosecu m ba L h for FRP

side strips as presented in Chen et al. (2012), which is:

, ,

14

df df

frp

f e f e

h hD

h h

(19)

f2 [1 ( )]

= sin

m ue

L aw

(20)

,=

1 ( )

f e

df

m u

hh

L a (21)

At the end of the partial debonding stage (I), the position of D point (when the

leftmost softening front just reaches the bottom fixed boundary) can be determined by

substituting cosm bL h ec into Eq. (21).

During the partial debonding stage (II), Eq. (19) still apply although point D

moves upwards along the effective shear crack (Figs. 8(c) and 8(d)). Immediately

after the start of the partial debonding stage (II), the maximum mobilized bond length

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above the crack end mL is larger than that below it ( bL ) at the crack end (i.e.

cosm b bL L h ec ), so Eq. (20) does not apply anymore. However, the crack end

width ew can determined using the compatibility condition that the crack end width

is equal to the sum of crack widths arising from the relative slip between the FRP and

concrete right below the critical shear crack and that right above it:

+

sin( )

t bew

(22)

where f [1 ( )]t m uL a and f [1 ( )]b b uL a are the slips of the leftmost

FRP fibre (along the fibre direction) at the location of crack right above and below the

effective shear crack.

Noting that at dfz h , ( ) 2 sin( )fw z , the following expression can be

obtained for dfh from Eq. (13):

f ,2

= sin

f e

df

e

hh

w

(23)

At the end of the partial debonding stage (II), the maximum mobilized bond

length above the crack end ,( )cosecm f e tL h h while that below it is still

cosb bL h ec ; as a result, the crack end width can be expressed as [according to Eq.

(22)]:

f f ,

,

2 + cos 2=

sin( )

f e t b u

e u

h h h ec aw

(24)

5.1.3 Hardening stage with ,0 db db rh h (or , ,e u e e rw w w ) (Figs 8(e) and 8(f))

The hardening stage starts when the softening front of the leftmost (also the most

critical) fibre reaches the upper edge of the FRP wrap, i.e. , cosecm f e tL h h (see

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Fig. 8(d)) which corresponds to the ultimate state of an FRP U-strip with ,df df uh h

and ,e e uw w , and ends when the leftmost FRP fibre begins to rupture at which the

crack end width ,e e rw w (Fig. 8(f)). For consistency with Chen et al. (2012), the

intersection of the rightmost fibre in hardening and the effective shear crack is marked

as “H”; the vertical distance from point H to the crack end is defined as dbh (Figs. 8

(e) and 8(f)). With these definition, the hardening stage is also characterized by a

portion of FRP in hardening state in the range of , ,f e db f eh h z h . For the FRP in

hardening state, the total bond length and the length of FRP-to-concrete bonded

interface in softening (referred to as “remaining softening bond length” hereafter) are

termed as zL and ( )a z at the coordinate of z . Consequently, the P

response of the FRP in hardening (i.e. load-slip relationship) can be determined by

Eqs. (5)-(6) ,which is corresponding to segment BP for uL az or segment

' 'D P for z uL a , as shown in Fig. 3(b). If the remaining softening bond length

at the crack end is termed as ea (i.e. ,

( )f e

e z ha a z

), then ea are equal to ua and

ra at the start and end of the hardening stage respectively (i.e. e ua a for ,e e uw w

and e ra a for ,e e rw w ).

For the FRP in hardening state, as the maximum stress above the effective shear

crack is equal to that below the crack based on the principle of force equilibrium, the

remaining softening bond length ( )a z above the effective shear crack is equal to that

below the crack according to Eq. (5). As a result, ( )w z (the crack width at z ),

which is the sum of the crack widths arising from the FRP slip below [i.e.

f f ( ) sin( )b bL a z a z according to Eq. (6)] the effective shear crack

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and that above it [ f f ( ) sin( )b tL a z a z according to Eq. (6)], can be

expressed as follows (with reference to Eq. (22)):

f f ,2 + cos 2 ( ) sin( ( ))( )=

sin( )

f e t bh h h ec a z a zw z

(25)

According to Eq. (5), the maximum stress of the FRP in hardening state can be

expressed as:

1

(z) = sin ( )

ba z

(26)

where b is the bond strength of the FRP-to-concrete bonded interface without

anchorage (i.e. type A joint) for uL a (see Eq. (17)) .

Substituting Eq. (26) into Eq. (25) gives:

f ,

( ) 22 + cos arcsin

( )( )=

sin( )

f

f e t b

f f

zh h h ec

E t zw z

(27)

where ( )b rz , where r is the FRP rupture stress (i.e. r ff where ff

is the material strength of FRP).

At the end of hardening stage ( e ra a , ,e e rw w ), ,f e

fz hz f

, so the crack end

width can be obtained by substituting ,f e

fz hz f

into Eq. (27):

f ,

,

22 + cos arcsin

= sin( )

f bf e t b

f f

e r

fh h h ec

E fw

(28)

where arcsin b

ff

should be in the range of 0 and 2

.

For the hardening stage (i.e. , ,e u e e rw w w ), Eq. (14) can be rewritten as follows

to determine the effective stress in FRP strips intersected by the effective shear crack:

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, ,

,

,

, 0

1sin[ ( )]

sin[ ( )]

df f e db f e

df f e db

h h h h

bf e b frp

f e h h h

f L z dz dz dz Dh a z

(29)

where

,

,, , ,

1 11

4 sin[ ( )]

f e

f e db

h

df df db

frp

f e f e f e h h

h h hD dz

h h h a z

(30)

Solving the last term in the right hand side of Eq. (30) requires the relationship

between ( )a z and z . Substituting Eq. (13) into Eq. (25) gives:

f f ,,2 + cos 2 ( ) sin( ( ))

z= sin( )

f e t bf e

e

h h h ec a z a zh

w

(31)

which is a transcendental relationship with no closed-form expression of ( )a z in

terms of z . Within the range of ( )r ua a z a , the ( ) ( )z w z relationship can be

plotted with ( )w z and ( )z determined from Eqs (25) and (26) respectively.

Typical ( ) ( )z w z curves are shown in Fig. 9 for the following typical cases: (a)

tf=0.11mm, hb=50 mm; (b) tf=0.11 mm, hb= 80 mm; (c) tf=0.44 mm, hb= 50 mm; (d)

tf=0.44 mm, hb= 80 mm; (e) tf=0.88 mm, hb=50 mm. The results in Fig. 9 show that

the relationship between ( )z and ( )w z is almost linear (further information can

be found in (Li, 2015)), as schematically shown in Fig. 10. Nothing that

,( ) e uw z w , ( ) bz when ( ) ua z a , and ,( ) e rw z w , r( ) fz f when

( ) ra z a (see Eqs (25) and (26)) , this approximate linear relationship can be

expressed as:

,( ) ( ( ) )b b e uz k w z w (32)

, ,

r bb

e r e u

kw w

(33)

where , ,( )e u e rw w z w , ( )b rz , 2

= f f

b

f

G E

t , r ff ; ,e uw and ,e rw

are crack end widths at the start and the end of the hardening stage respectively, which

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can be determined by Eqs (24) and (28) respectively.

From Eqs (26) and (32), the last term in the right hand side of Eq. (30) can be

rewritten as:

, ,

, ,

,

, ,

, 2

, ,

1 1 1

sin[ ( )]

1( ( )) ( )

2

f e f e

f e db f e db

e

e u

h h

f e f e bh h h h

w

e e u bb b e u e e u

b e e b ew

zdz dz

h a z h

w w kk w w dw w w

w w w

(34)

Substituting Eq. (34) into Eq. (30), the expression of frpD can be obtained as:

, 2

,

, ,

1 ( )4 2

df df db e e u bfrp e e u

f e f e e b e

h h h w w kD w w

h h w w

(35)

During the hardening stage ( , ,e u e e rw w w ), the height of the softening FRP dfh

can be obtained using Eq. (23). Given that,

,( )f e db

e uz h hw z w

, the crack end width

ew can be obtained according to Eq. (13) from:

,

,

,

f e

e e u

f e db

hw w

h h

(36)


2,

, , , ,

14 ( ) 2

df df b e udbfrp

f e f e f e f e db b

h h k whD

h h h h h

(37)

where ,0 db db rh h and ,db rh is the vertical distance from point H to the crack end

(also vertical length of FRP in hardening) at the end of the hardening stage.

Eq. (36) can be alternatively expressed as follows to give dbh :

,

,(1 )e u

db f e

e

wh h

w (38)

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At the end of the hardening stage, the value of ,db rh can be obtained by setting

,db db rh h and ,e e rw w in Eq. (38) (shown in Fig. 8(f)), as follows:

,

, ,

,

(1 )e u

db r f e

e r

wh h

w (39)

At the end of the hardening stage, the vertical distance from point D to the crack

tip (also the vertical length of FRP in softening) ,df rh can be obtained using Eq. (23)

by replacing ew with ,e rw , as follows:

f ,

,

,

2=

sin

f e

df r

e r

hh

w

(40)

Accordingly, frpD at the end of the hardening stage can be obtained by

substituting ,df rh and ,db rh into Eq. (37):

2

, , , ,

,

, , , , ,

14 ( ) 2

df r df r db r b e u

frp r

f e f e f e f e db r b

h h h k wD

h h h h h

(41)

5.1.4 Rupture stage with 0rh (or ,db db rh h and ,e e rw w ) (Fig. 8(g))

This stage is featured by a portion of ruptured FRP. For convenience of

description, the intersection of rightmost fibre of ruptured FRP and the effective shear

crack is defined as point “R” with a vertical distance rh to the crack end (Fig. 8(g))，

so FRP have ruptured within the range of , ,f e r f eh h z h . As the crack widths at

the location of ,f e r dbz h h h and ,f e rz h h are equal to ,e uw and ,e rw

which can be determined from Eqs (24) and (28) respectively, the crack end with ew

during this stage can be expressed as follows (according to Eq. (13)):

, ,

, ,

, ,

f e f e

e e u e r

f e r db f e r

h hw w w

h h h h h

(42)

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During this rupture stage, the effective FRP stress can be expressed as follows

(according to Eq. (14)):

, ,

,

,

, 0

1sin[ ( )]

sin[ ( )]

df f e r db f e r

df f e r db

h h h h h h

bf e b frp

f e h h h h

f L z dz dz dz Dh a z

(43)

where

, , 2

, ,

, ,

1 ( )4 2

df df db r e r e u bfrp e r e u

f e f e e b e

h h h h w w kD w w

h h w w

(44)


2,

, , , ,

14 ( ) 2

df df r b e udbfrp

f e f e f e f e r db b

h h h k whD

h h h h h h

(45)

Based on Eq. (42), rh , dbh and dfh in the rupture stage can be expressed as:

,

,1e r

r f e

e

wh h

w

(46)

, ,

,

e r e u

db f e

e

w wh h

w

(47)

f ,2

sin

f e

df

e

hh

w

(48)

where , ,e e r e uw w w during the rupture stage and ,e e rw w at the start of the

rupture stage (i.e. at the end of hardening stage).

5.2 Thin concrete cover with cosecb uh a

Similar to the case of thick concrete cover with cosecb uh a , the failure process

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for the case of thin concrete cover with cosecb uh a can be divided into four

successive stages: 1) softening stage ( 0 m uL a ) (Figs 11(a) and 11(b)), 2) partial

debonding stage [ ,( )cosecu m f e ta L h h ] (Figs 11(c)-11(d), 3) hardening stage

( 0dbh ) (Figs. 11(e) and 11(f)) , and 4) FRP rupture stage ( 0rph ) (Fig. 11(g)). The

solutions for these stages are presented as follows.

5.2.1 Softening stage with 0 m uL a (Figs 11(a) and 11(b))

This stage can be further divided into two sub-stages: 1) softening stage (I) with

0 cosecm bL h (Figs 11(a)) and 2) softening stage (II) with cosecb m uh L a

(Figs 11(b)). For the softening stage (I), the softening front of the leftmost FRP does

not reach the fixed end below the crack end until the end of this stage

when cosecm bL h . Solution of this sub-stage is exactly the same as that of

softening stage of for the strengthened beam with a thick cover, with ,f ef , frpD and

ew determined from Eqs. (15), (16) and (18) respectively. At the start of the softening

stage (II), the softening front of the leftmost FRP fibre below the effective shear crack

has already reached the fixed end. The maximum mobilized bond length of FRP

below the crack cosecb bL h while the softening front of the leftmost FRP fibre

above the crack continues to move upwards, leading to a maximum mobilized bond

length mL larger than bL . Eqs. (15) and (16) can be used to calculate ,f ef and frpD

while ew is determined from:

+

sin( )

t bew

(49)

where t and b are the slips of the leftmost FRP fibre in the fibre direction at

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the location of crack above and below the effective shear crack respectively.

According to Chen et al. (2012), t can be determined from:

= 1 cost f mL (50)

Substituting the force in FRP at the softening stage (i.e. sin( )

P =f f mb L

, see

Chen et al. (2012) for details) into Eq. (1) gives:

= 1 cos sin sin tan( )b f b f m b bL L L L (51)

Substituting Eqs (50) and (51) into Eq. (49) gives:

2 cos cos sin sin tan( ) =

sin( )

f b m f m b b

e

L L L L Lw

(52)

The expression of Eq. (52) is quite complicated. Chen et al. (2012) has shown that

the prediction of Eq. (52) is close to those of Eq. (18) in predicting the f eV w curve,

Eq. (18) is used to determine the value of ew for simplicity in the rest of this paper if

not otherwise specified. As a result, for the whole softening stage with 0 m uL a ,

Eqs. (15)-(18) can be used to calculate ,f ef , frpD and ew respectively.

5.2.2 Partial debonding stage with ,( )cosecu m f e ta L h h (Figs 11(c)-11(d))

This stage starts when the leftmost FRP strips begins to debond (i.e. m uL a ), and

ends when the softening front of the leftmost FRP fibre reaches the top fixed end of

the beam (i.e. ,( )cosecm f e tL h h ). During this stage, a portion of FRP near the

crack end are partially debonded as with the partial debonding stage for thick concrete

cover (i.e. cosecb uh a ) (Figs 11(c) and 11(d)). In this case Eqs (15) and (19) can be

used to determine ,f ef and frpD respectively, and crack end width ew can be

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determined by Eq. (49) except that t and b should be determined from (see Chen

et al. (2012)):

= 1t f m uL a (53)

= 1 cos 1 sin tan( )b f b f b bL L L (54)

where cosb bL h ec .


1 cos 1 sin tan( )

sin( )

f f m u f b f b b

e

L a L L Lw

(55)

At the end of this stage, the crack end width ,e uw can be obtained by substituting

, cosm f e tL h h ec and cosb bL h ec into Eq. (55):

,

,

[ csc ] 1 cos csc 1 sin csc tan( csc )

sin( )

f f f e t u f b f b b

e u

h h a h h hw

(56)

5.2.3 Hardening stage with ,0 db db rh h (or , ,e u e e rw w w ) (Figs. 11(e) and 11(f))

This stage starts when the softening front of the leftmost FRP fibre reaches the top

fixed end of the beam (i.e. ,( )cosecm f e tL h h ), and ends when the leftmost FRP

fibre ruptures (i.e. e ra a , ,db db rh h and ,e e rw w ). Similar to the hardening stage

for a thick concrete cover (i.e. cosecb uh a ), this stage is characterized by a portion

of FRP in hardening state (i.e. , ,f e db f eh h z h ). Eqs. (15) and (37) can be used to

determine ,f ef and frpD respectively. As the maximum mobilized FRP bond length

of FRP below the crack end cosecb bL h is less than or equal to ua (i.e. b uL a ),

the remaining softening bond length at the crack end ea above the effective shear

crack (termed as ,e ta ) is not equal to that below the crack (termed as ,e ba ) until

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,e t ba L although the maximum stress above the effective shear crack is always

equal to that below the crack to satisfy force equilibrium during the whole stage.

Consequently, this stage can be subdivided into two sub-stages: 1) hardening (I) with

,e t ba L ; and 2) hardening (II) with ,e t ba L .

For hardening (I) stage, the P responses of the FRP above the crack end and

that below the crack end correspond to segment BP of type B joint with bond

length uL a and segment ' 'C D of type B joint with bond length uL a

respectively, as shown in Fig. 3. From Eqs (1) and (6), the FRP slip at the crack end

for FRP below the crack (i.e. b ) and that above the crack (i.e. t ) can be determined

from:

tan

= 1 cos + sinbe

b f b f b

f

LL L

E

(57)

,

,

- =

sin( )

f m e t

t f

e t

L a

a

(58)

where ,

( )f e

e z hz

is the FRP stress at the crack end; cosb bL h ec and

,( )cosm f e tL h h ec are maximum mobilized FRP bond lengths (including both

softening and debonded parts) above and below the crack end respectively.


,

,

- tan1 cos + sin

sin( ) =

sin

f m e t bef f b f b

e t f

e

L a LL L

a Ew

(59)

where cosb bL h ec and ,( )cosm f e tL h h ec .

For hardening stage (II), the P responses of the FRP above the crack end and

that below the crack end correspond to segment BP of type B joint with bond

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length uL a and segment ' 'D P of type B joint with bond length uL a

respectively (see Fig. 3). According to Eq. (6), the FRP slips at the crack end for FRP

below ( b ) and above the crack ( t ) can be determined as:

,

,

- =

sin( )

f b e b

b f

e b

L a

a

(60)

,

,

- =

sin( )

f m e t

t f

e t

L a

a

(61)

where cosb bL h ec and ,( )cosm f e tL h h ec .

According to Eq. (5), it can be inferred that the remaining softening bond length

, ,e b e ta a because the maximum FRP stress ( ,e t ) above the effective shear crack is

always equal to that below the crack ( ,e b ) due to force equilibrium.


2 - 2 sin( ) =

sin

f f m b e e

e

L L a aw

(62)

where , ,e e b e ta a a is the remaining softening bond length of the leftmost FRP

fibre at crack end .

Given that cosb bL h ec and ,( )cosm f e tL h h ec , Eq. (62) is essentially

the same as Eq. (25) because ,f e

e z ha a z

(i.e. ea is the remaining softening bond

length of the leftmost FRP fibre at the crack end ) . Therefore, for hardening stage (II)

the solutions of ,f ef , frpD as well as ew are the exactly same as the hardening

stage for the case with a thick concrete cover. As a result, Eqs (15), (37) and (25) can

be used to determine ,f ef , frpD and ew respectively; and Eqs. (23), (38), (24), (28)

can be used to determine dfh , dbh ,e uw and

,e rw respectively.

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Comparing Eqs (57)-(59) and Eqs (60)-(62), it can be found that the difference in

the expressions of the crack end width ew between the hardening stages (I) and (II)

only lies in the part of b . For the hardening stage (I), the e ew curves predicted

by Eq. (59) and Eq. (62) are shown in Fig. 12 (where e is the FRP stress at the

crack end) for 0.25 0.9u b ua L a . Fig. 12(a) shows that the difference between the

e ew curves are nearly negligible, with the crack end width ew being slightly

underestimated by Eq. (62). It is noted that in Fig. 12(a) the upper and lower bounds

of e in each curves are b and r respectively, corresponding to the start of the

hardening stage (i.e. the end of debonding stage) and the end of the hardening stage.

Fig. 12(b) shows a close-up of the lower part of the e ew curves shown in Fig.

12(a), it can be seen that for the same e , the difference between the values of ew

predicted by Eq. (59) and (62) significantly decreases when bL increases and

becomes negligible when bL approaches the effective bond length of FRP

(e.g. 0.9 ua ); and for the same bL , the difference of ew significantly decreases when

e increases. This trend is more clearly demonstrated in Fig. 12(c) where e is

plotted against ew , the difference between the values of ew predicted by Eqs (59)

and (62). Fig. 12(c) suggests that the difference between e ew curves predicted by

Eqs (59) and (62) is very limited, especially for the later phase of the hardening stage

when e is approaches r ( r 3900 MPa for the examples shown in Fig. 12). It

is also clear in Fig. 12(c) that the difference between e ew curves predicted by

Eqs (59) and (62) becomes negligible when bL approaches the effective bond length

of FRP ua (e.g. 0.9 ua ). Based on these observations, Eq. (62) is be used to determine

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the value of ew for the whole hardening stage (i.e. both hardening stage (I) and (II))

in the remainder of this paper for simplicity if not otherwise specified).

5.2.4 Rupture stage with 0rh (or ,db db rh h and ,e e rw w ) (Fig. 11(g))

Similar to the case with a thick concrete cover (i.e. cosecb uh a ), this stage is

characterized by a portion of ruptured FRP characterised by rh , the vertical distance

from point R to crack end. The solution of this stage is exactly the same as that for the

rupture stage of a thick concrete cover (i.e. cosecb uh a ), so Eqs (15) , (45) and (42)

can be used to determine ,f ef , frpD and ew respectively, except that in Eqs (42)

and (45), ,e uw is determined according to Eq. (56) instead of Eq. (24). Nevertheless,

for simplicity, ,e rw can still be determined by Eq. (28) without significant loss of

accuracy, as discussed above in Section 5.2.3.

6. Verification of the closed-form solution

To verify the closed-form solution presented above, its predictions are compared

with the numerical results of the FE model presented in Chen et al. (2010) which was

successfully used to predicted the progressive debonding behaviour of FRP side strips

and U-strips (Chen et al., 2012). In the FE model, the continuous FRP sheet is

represented by 20 discrete FRP strips with each FRP strip simulated using 2-D truss

elements (T2D2) in ABAQUS; and the bond-slip behaviour between the FRP and

concrete is represented by the nonlinear spring element Spring2. As in Chen et al.

(2012b), an element size of 1 mm was adopted for the truss elements representing the

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FRP strips and compatible number of spring elements simulating the bond-slip

behaviour were used based on a mesh convergence study. In the FE models, the

bond-slip behaviour of the FRP-to-concrete bonded interface was defined using either

the linearly softening bond-slip model (Fig. 3(a)) or the more accurate nonlinear

bond-slip model of Lu et al. (2005). The results of the former are used to verify the

accuracy of the analytical solution presented in this study, and those of the latter are

used mainly to assess the effect of adopting the linearly softening bond-slip model on

the full-range behaviour of FRP wrap in shear-strengthened RC beams, in particular

on the full-range -f eV w responses. The following parameters were used in both

closed-form solution and the FE analyses unless otherwise stated: concrete cylinder

compressive strength fc’ = 30 MPa (with an equivalent cube strength of 37 MPa

according to CEB-FIP (1993)); elastic modulus of FRP Ef = 2.3105 MPa; and tensile

strength of FRP ff = 3900 MPa. In the closed-form solution, the maximum interfacial

shear stress f and interfacial facture energy fG were determined according to Lu

et al.‟s (2005) bond-slip model. The maximum interfacial slip f was then

calculated from 2f f fG (see Fig. 3(a)). The shear crack end was assumed to be

at 50 mm from the beam soffit where FRP is assumed to be fixed (i.e. 50 mmbh )

which is in the practical range as explained in Chen et al. (2010). It was further

assumed that the beam sides are fully covered with FRP (thus ,t b f eh h h h or

0.1th d and b ch h see Fig 6) with all fibres oriented vertically (= 90) and the

angle of the shear crack is = 45.

Analyses were conducted for a series of practical cases in which the beam height

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varies so that the vertical distance from the crack tip to the crack end ,f eh is in the

range of ,300 600f eh mm, the FRP thickness varies in the range of

0.11-0.88ft mm, and the concrete cover thickness 50 80b ch h mm. Figs 13

and 14 compare the -f eV w curves for , 300f eh mm and , 600f eh mm

respectively for nine typical cases: (1) ft = 0.11 mm, bh = 30 mm; (2) ft = 0.11 mm,

bh = 50 mm; (3) ft = 0.11 mm, bh = 80 mm; (4) ft = 0.44 mm, bh = 30 mm; (5) ft =

0.44 mm, bh = 50 mm; (6) ft = 0.44 mm, bh = 80 mm; (7) ft = 0.88 mm, bh = 30

mm; (8) ft = 0.88 mm, bh = 50 mm; (9) ft = 0.88 mm, bh = 80 mm .

It is noted that the FE model predicts stepwise drops after the -f eV w curve peaks

because each drop represents the rupture of an individual FRP strip. The size of the

stepwise drop would naturally reduce if the continuous FRP sheet is divided into more

„strips‟ in the FE model, and eventually converge to the analytical solution. The

predictions of closed-form solution, which is valid for an equivalent continuous FRP

sheet, generally passes through the mid-point of each stepwise drop of the FE

predicted -f eV w curve. It should also be noted that the following method was used

to determine the peak FRP shear contribution ,f pV and the corresponding crack end

width ,e rw in FE predictions: the descending branch of the FE predicted -f eV w

curve was assumed to pass through the mid-point of each step drop and its

intersection with the ascending branch represents the peak point of the -f eV w curve.

Clearly the -f eV w curves predicted by the closed-form solution are in close

agreement with the FE predictions for all the nine typical cases mentioned above.

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Table 1 lists the percentage differences of the FRP shear contribution ( ,f pV ) and

crack end width ( ,e rw ) at the peak FRP shear contribution ( ,f pV ) between the

predictions of the closed-form solution and the FE analyses. It can be seen that for the

cases examined, ,f pV and ,e rw are very small, with the absolute maximum

values being 1.23% and 1.92% respectively. This also implies that the linearized

approximation of the ( )z w z curve for FRP in hardening (Eqs (32) and (33))

and the simplification of ew for b uL a (i.e. use Eq (62) instead of Eq (59) for

evaluating ew ), have introduced little errors.

Figures 13 and 14 also show that the FE prediction based on Lu et al.‟s (2005)

nonlinear FRP-concrete bond-slip model is nearly identical to that based on the

simplified linearly softening bond-slip model, although the latter usually predicts a

slightly stiffer response in the initial stage of the -f eV w curve. According to

numerical results not shown here, the percentage differences between the nonlinear

bond slip model and linearly softening bond-slip mode are within 0.2% for both ,e rw

and ,f pV , demonstrating that the shape of the bond-slip curve has insignificant effect

on the shear behaviour in FRP wrapped beams.

7. Concluding remarks

This paper has presented a closed-form solution for the whole failure process of

FRP wraps in RC beams shear-strengthened with wraps. It has been developed based

on the assumptions of a linear critical shear crack shape and the full-range behaviour

of FRP-to-concrete bonded joints based on a linearly softening bond-slip model. The

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closed-form solution has been validated by comparing its predictions with finite

element predictions. The solution is also applicable to configurations of FRP

side-strips or U-strips when a reliable anchorage system is deployed at the FRP ends

such that FRP rupture (instead of FRP debonding) dominates the final failure.

The closed-form solution is a powerful tool for understanding the behaviour of

FRP wraps. One of the major benefits is that it explicitly describes the development of

shear contribution of externally bonded FRP shear reinforcement to the shear

resistance of the RC beam as the critical shear crack widens. It may further be used

directly to evaluate the effect of the possible shear interaction between external FRP

shear reinforcement, concrete and internal steel stirrups on the shear strength of RC

beams shear-strengthened with FRP.

For RC beams shear-strengthened with FRP wraps or well-anchored FRP U- or

side strips where adverse shear interaction exists mainly between the external FRP (Vf )

and concrete (Vc) as aforementioned, a new shear strength model considering the

shear interaction between FRP and concrete can be developed based on the solution

presented in this paper. As the direct relationship between effective FRP strain and

maximum width of the critical shear crack can be established (e.g. Eqs (15) and (35)),

it can be further used to develop a new criterion of FRP strain limit by specifying a

reasonable crack width for serviceability consideration and to avoid a significant

degradation of the concrete shear resistance. A salient feature of the shear strength

model developed following this approach is that the effect of beam size can be

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automatically considered. Such a shear strength model is being developed and will be

published in due course.

Acknowledgements

The authors acknowledge the financial support received from the National Natural

Science Foundation of China (Project Nos. 51108097, 51378130, 51578423) and

Guangdong Natural Science Foundation (Project No. S2013010013293). The authors

are also grateful for the financial support received from the Department of Education

of Guangdong Province for Excellent Young College Teacher of Guangdong Province

(Project No. Yq2013056). The first author would like to thank the China Scholarship

Council (CSC) for the scholarship awarded to him as a visiting scholar (File No.

201408440320) in the Department of Civil and Environmental Engineering,

University of California, Berkeley, USA.

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LIST OF TABLES

Table 1. Comparison between closed-form solution and FE predictions.

hf,e

(mm)

hb

(mm)

tf (mm)

0.11 0.44 0.88

Δwe,r(%) ΔVf,p(%) Δwe,r(%) ΔVf,p (%) Δwe,r(%) ΔVf,p (%)

300

30 0.64 0.31 0.03 0.35 -1.77 0.62

50 0.69 0.69 -0.38 0.80 -1.55 0.15

80 0.73 1.23 -0.12 0.78 -1.92 0.84

600

30 1.02 -0.19 -0.55 -0.01 -1.79 -0.10

50 0.54 0.37 -0.23 -0.03 -1.82 -0.04

80 0.20 0.11 -0.65 0.26 -1.90 0.02

Note: Δ=(AS-FE)/NS×100%; AS=Analytical solution; FE= Finite element prediction;

Vf,p=Peak FRP shear contribution; we,r=Crack end width at Vf,p

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LIST OF FIGURES

Support

Crack tip

Crack end

tip

FRP strips

Shear crack

FRP strip

point

h

Load

Steel tension reinforcement

A

A

(a)

FRP U-strip

symmetry

h

Concrete

plate/str

ip

symmet

ry

Adhesive FRP side strip

h

Concrete

plate/str

ip

symmet

ry

Adhesive

Location of

shear crack

Concrete

plate/str

ip

symmet

ry

Concrete

plate/str

ip

symmet

ry

Type A joint

Type A joint

Type A joint

Type B joint

Type B joint

Type B joint

h

FRP wrap strip

symmetry

Adhesive

(b) (c) (d)

Fig. 1. RC beam shear-strengthened with FRP strips: (a) elevation; (b) cross-section

A-A for FRP side strip; (c) cross-section A-A for FRP U-strip; (d)

cross-section A-A for FRP wrap strip.

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FRP strip

symmetry

FRP strip

symmetr

y

tc

bc bf

tf

Concrete prism

FRP plate/strip

symmetry

Adhesive

L

P

P

Concrete prism

FRP plate/strip

symmetry

Free far end

Free far end Elevation prism FRP

plate/strip

symmetry

Plan prism

Loaded end

(a)

FRP strip

symmetr

y

FRP strip

symmetry

tc

bc bf

tf

Concrete prism

FRP plate/strip

symmetry

Adhesive

L

P

P

Concrete prism

FRP plate/strip

symmetry

Fixed far end

Fixed far end Elevation

prism FRP

plate/strip

symmetry

Plan

prism

Loaded end

(b)

Fig. 2. Two types of FRP-to-concrete bonded joints: (a) type A joint; (b) type B joint.

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f

f

- f

-f

Gf

Gf

(a)

Displacement at the loaded end ∆

Load

P

B'

D'

O

B

PP'

A

Failure load controled by FRP rupture

K P -∆

C'

K P -∆

L<a u L>a u

f f uL a

f

Failure load controlled by FRP rupture

Slip at the loaded end Δ

(b)

Fig. 3. Solution of an FRP-to-concrete bonded joint with a fixed far end (type B joint):

(a) linearly softening bond-slip model; (b) full-range load-displacement

response.

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(a)

(b)

(c)

(d)

(e)

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Fig. 4. Interfacial shear stress distributions at various stages for a type B joint

with uL a : (a) development of softening zone; (b) initiation of debonding at

the loaded end (point A in Fig. 3(b)); (c) propagation of debonding (AB in Fig.

3(b)); (d) softening front reaching the fixed end (point B in Fig. 3(b)); (e) final

stage of debonding ( BP in Fig. 3(b)).

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(a)

(b)

(c)

(d)

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Fig. 5. Interfacial shear stress distributions at various stages for a type B joint

with uL a : (a) development of softening zone (OB‟ in Fig. 3(b)); (b)

softening front reaching the fixed end (point B‟ in Fig. 3(b)); (c) initiation of

debonding at the loaded end (point D‟ in Fig. 3(b)); (d) final stage of

debonding (D‟P‟ in Fig. 3(b)).

Fig. 6. Notations for a general shear strengthening scheme.

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z/z

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Normalized crack width w/wmax

C=0

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Normalized crack width w/wmax

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FR

P抗

剪贡

献值

Vf（

kN）

裂缝末端宽度we(mm)

:C=0:C=0.25:C=0.5:C=0.75:C=0.9:C=1

hf,e=300mm; tf =0.11mm

Ef =230GPa; ff =3900MPa

Crack end width we (mm)

Sh

ear

forc

e V

f (k

N)

(b)

Fig. 7. Effects of crack shape on f eV w curves: (a) crack shape corresponding to

different values of C; (b) f eV w curves of different values of C.

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H

R

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Fig. 8. Failure process of FRP wrap for thick concrete cover (hbcscβ>au): (a)

0≤Lm<au ; (b) au≤ Lm <hbcscβ ; (c) hbcscβ≤Lm<(hf,e+ht)cscβ ; (d) Lm

=(hf,e+ht)cscβ（or we=we,u）; (e) hdb>0（or we,u<we<we,r）; (f) hdb= hdb,r

(we=we,r) ; (g) hr>0（or we>we,r）.

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Linearized Line(tf=0.11mm)





Real curve(hf,e=300mm)

Linearized line(hf,e=300mm)


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r

bσ

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Linearization (hf,e=300mm)





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(b)

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412.9MPaλt

τσ

f

fb

r

bσ

MPa3900=r







(e)

Fig. 9. Comparison between analytical solution and linearized FRP stress at crack end

vs. crack end width relationship for typical cases ( ' 30cf MPa, 230fE GPa,

, 3900f ef MPa): (a) tf=0.11mm, hb=50mm; (b) tf=0.11mm, hb=80mm; (c)

tf=0.44mm, hb=50mm; (d) tf=0.44mm, hb=80mm; (e) tf=0.88mm, hb=50mm.

Real Line

Linearized Line

σ

w

we,r we,u

σr

Analytical Solution

Linearization

σb

Fig. 10. Schematic diagram of linear approximation of ( )z w z curve.

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R

(g)

Fig.11. Failure process of FRP wraps for thin concrete cover (hbcscβ≤au): (a) 0≤Lm<

hbcscβ; (b) hbcscβ≤ Lm < au ; (c) au≤Lm<(hf,e+ht)cscβ ; (d) Lm =(hf,e+ht)cscβ（or

we=we,u）; (e) hdb>0（or we,u<we<we,r）(f) hdb= hdb,r (we=we,r) ; (g) hr>0（or

we>we,r）.

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True Line(L=0.25au)Simplified Line(L=0.25au)系列3系列4系列5系列6系列7系列8

Accurate solution (L=0.25au)

Simplified solution(L=0.25au)


Simplified solution (L=0.5au)





1165MPaλt

τσ

f

fb

r

bσ

MPa3900=r

Accurate solution (Lb=0.25au) Simplified solution(Lb=0.25au) Accurate solution (Lb=0.5au) Simplified solution (Lb=0.5au) Accurate solution (Lb=0.75au) Simplified solution (Lb=0.75au) Accurate solution (Lb=0.9au) Simplified solution (Lb=0.9au)

(a)

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Simplified solution(L=0.25au)







1165MPaλt

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f

fb

bσ

Accurate solution (Lb=0.25au) Simplified solution(Lb=0.25au) Accurate solution (Lb=0.5au) Simplified solution (Lb=0.5au) Accurate solution (Lb=0.75au) Simplified solution (Lb=0.75au) Accurate solution (Lb=0.9au) Simplified solution (Lb=0.9au)

(b)

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Δwe (mm)

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Lb=0.25au

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1165MPaλt

τσ

f

fb

bσ

(c)

Fig. 12. Comparison between simplified solution and accurate solution for σe -we

curves (, 300f eh mm, ' 30cf MPa, 230fE GPa, 0.11ft mm,

, 3900f ef MPa).

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: FEM Solution(Lu's model)

: FEM Solution(Linear softening model)

: Analytical Solution

FEM solution (Lu et al. (2005) ‟s model)

FEM solution (Linear softening model)

Analytical solution

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(e)

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rce

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(f)

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(g)

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(h)

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Analytical solution

(i)

Fig. 13. Closed-form solution versus FE predictions for hf,e=300 mm：(a) tf=0.11mm,

hb=30mm；(b) tf=0.11mm, hb=50mm；(c) tf=0.11mm, hb=80mm；(d) tf=0.44mm,

hb=30mm；(e) tf=0.44mm, hb=50mm；(f) tf=0.44mm, hb=80mm；(g) tf=0.88mm,

hb=30mm；(h) tf=0.88mm, hb=50mm；(i) tf=0.88mm, hb=80mm.

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(a)

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Analytical solution

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Analytical solution

(h)

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Analytical solution

(i)

Fig. 14. Closed-form solution versus FE predictions for hf,e=600 mm：(a) tf=0.11mm,

hb=30mm;(b) tf=0.11mm, hb=50mm;(c) tf=0.11mm, hb=80mm;(d) tf=0.44mm,

hb=30mm;(e) tf=0.44mm, hb=50mm;(f) tf=0.44mm, hb=80mm;(g) tf=0.88mm,

hb=30mm；(h) tf=0.88mm, hb=50mm；(i) tf=0.88mm, hb=80mm.

full-range frp failure behaviour in rc beams shear

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