Fourth International Conference on FRP Composites in Civil Engineering (CICE2008) 22-24July 2008, Zurich, Switzerland

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1 INTRODUCTON

Structures may need to be strengthened for different reasons, among which a change in function, implementation of additional services or to repair damage. Different strengthening techniques exist. Often applied is externally bonded reinforcement (EBR), based on fibre reinforced poly-mer (FRP), the so-called FRP EBR. By applying FRP as external flexural strengthening, addi-tional failure modes, have to be taken into account. Following fib (2001) four different mecha-nisms can be distinguished, among which one of them is debonding by crack bridging. Investigating debonding by crack bridging, also called intermediate crack debonding, the crack spacing srm is one of the influencing parameters (Oller et al. (2007); Teng et al. (2003)). Here-with it is important to estimate the crack spacing as exact as possible. Regarding to this aspect, different approaches are evaluated and further compared to experimental observations.

2 THEORETICAL EVALUATION OF CRACK SPACING

2.1 Crack spacing and transfer length

Generally the maximum crack spacing is denoted as the length over which slip between the reinforcement and the concrete occurs. Hence, the length which contributes to the width of the crack. This length equals two times the transfer length tl . According to MC 90 (CEB (1995)), the mean crack spacing at stabilized cracking srm for steel reinforced structures can be assumed 2/3 srmax.

tmaxrrm 3

4s

3

2s l== (1)

The transfer length tl is the length over which the force in the reinforcement is transmitted to the concrete through bond interaction. This basically relates to the shear stress (τ) – slip (s) relation at the bond interface. For the τ-s relationship, different models are proposed, among

Influence of externally bonded reinforcement on the crack spacing

L. Vasseur, S. Matthys & L. Taerwe Department of Structural Engineering, Ghent University, Magnel Laboratory for Concrete Research, Ghent, Belgium

ABSTRACT: Typical for the use of Fiber Reinforced Polymer – Externally Bonded Reinforce-ment (FRP-EBR) for flexural strengthening is the possible occurrence of different types of debonding mechanisms. Especially in the case of intermediate crack debonding, the crack spac-ing is one of the influencing parameters. In literature it is generally assumed that the crack spac-ing stays constant over the depth of the beam. Herewith, cracks arise in the tension zone of the beam, are bridged by the internal and external reinforcement, and extend until the neutral axis of the beam. Based on microscopic analysis, a number of smaller cracks have been observed in the concrete cover near the laminate, in stead of one major crack. Hence the assumption of constant crack spacing over the depth of the beam appears questionable. Taking into account this obser-vation may result in more precise debonding modeling.

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which the linear elastic relation (Bresson (1971)), the non-linear relation (Wicke & Pichler (1991)) and the bilinear relation (Ranisch (1982)). In the following a linear τ-s relation is con-sidered, as often assumed for serviceability limit state conditions. Given or assuming the τ-s re-lationship, different approaches for modeling of the transfer length are possible, as discussed hereafter.

2.2 First approach based on pure shear model

The first model used in this paper to approach the transfer length is based on the simple pull test of FRP-to-concrete bonded joints (Fig. 1a). Note that this model only focuses on the behaviour between the external applied FRP reinforcement and the concrete, without any interaction with internal steel reinforcement.

ττ

Figure 1a. Pure shear model; 1b. Differential element

By analyzing a differential element taken from this model (Fig. 1b), the following differential

equation can be derived (e.g. Holzenkämpfer (1997)):

( )0)s(f

tE

1s f

ff

ff''

f =ρα+

− (2)

with αf = Ef/Ec, ρf = Af/Ac, τf = f(sf) and sf equals the slip between the FRP and the concrete. Herewith E and A are the elasticity modulus and the cross section area, respectively.

To solve Equation 2, boundary conditions with respect to the normal force in the concrete Nc and the FRP Nf (Eq. 3) have to be considered, which results in the following expression for τf:

;F)x(N:2BC

;0)0x(N:1BC

c

c

−====

l

;F)x(N

;0)0x(N

f

f

====

l (3)

)sinh(b

)xcosh(F)x(

f

flω

ωω=τ with

( )ffa

ffa2

tEt

1G ρα+=ω (4)

with F the applied tensile force, Ga the shear modulus of the adhesive, ta the thickness of the ad-hesive, tf the thickness and bf the width of the FRP laminate (Fig. 1).

To derive the transfer length tl , Equation 4 has to be solved for x = 0 (τ(0) = 0), which is theoretically obtained for l = ∞ . For practical calculations τ(0) = 2ζf ≈ 0 can be assumed, with

2/e)sinh( l

lω≈ω ( lω > 0) and Fmax = (τf,max bf)/ω (Matthys (2000), Deuring (1993)):

ζτ

ω=

ζω

ω≈

f

max,f

ff

maxt ln

1

b

Fln

1l (5)

where a boundary condition ζf = 0,0002 N/mm² is proposed.

2.3 Second approach based on crack formation in a tensile member

The second approach to evaluate the transfer length is based on the model illustrated in Fig-ure 2. Herewith interaction between FRP and concrete is assumed over the entire length of a tensile member which creates the possibility of multiple crack formation. This causes multiple

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peak stress zones in the laminate, and high shear stresses in the bond interface at both sides of every crack (Fig. 2).

By evaluating this model a similar differential element as in Figure 1b is obtained, from which the same differential equation (Eq. 2) can be derived. Nevertheless other boundary condi-tions apply in this model (Eq. 6), what results following Holzenkämpfer (1997) in a specific solution for sf(x) (Eq. 7).

τ

τ

Figure 2. Model based on crack formation in a tensile member

0fxf

f

s)x(s:2BC

0)0x(:1BC

====τ

l (6)

)sinh(

)xsinh(s)x(s

x

0fflω

ω= (7)

where sf0 equals the slip when debonding initiation between the laminate and the concrete is ob-served.

Holzenkämpfer (1997) defined the transfer length tl following (Eq. 8). Herewith a simplifi-cation is assumed expressed by a constant shear stress τfm over the transfer length (Fig. 2).

fm

fft

t.τ

σ∆=l (8)

with τfm = 1,25fct (Holzenkämpfer (1997))

2.4 Third approach based on the crack formation in a mixed reinforced tensile member

The third approach to evaluate the transfer length is based on the model presented in Figure 3 (MC90 CEB (1995), Matthys (2000)). Hereby, the different bond behaviour of the mixed rein-forcement is taken into account.

By analyzing a differential element from this model, two coupled differential equations can be derived:

( )0)s(f

dE

4)s(f

tE

1s s

ss

ssf

ff

ff''

f =ρα

−ρα+

− ; ( )

0)s(ftE

4)s(f

dE

1s f

ff

ffs

ss

ss''

s =ρα

−ρα+

− (9)

with αf = Ef/Ec, αs = Es/Ec, ρf = Af/Ac, ρs = As/Ac, τf = f(sf), τs = f(ss) and s equals the slip be-tween the reinforcement and the concrete (subscript f: FRP and s: steel).

To evaluate the transfer length, two different stadia are considered. Firstly there is the single crack stadium (Fig. 3a), herewith different transfer lengths sl and fl are obtained for the steel rebars and the FRP respectively:

ssm

sss u

A

τσ

=l ; ffm

fff u

A

τσ

=l with τsm = 1,80 fct (CEB (1995)) and τfm = 1,25 fct (10)

with us and uf the perimeter of the reinforcement (us=πds (with ds, the diameter of the steel bar) and uf = b (width of laminate))

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σσ

∆σ∆σ

σσ

Figure 3. Model based on the crack formation in a mixed reinforced tensile member

Remark the similarity between fl in Equation 10 and tl in Equation 8. When an equal stress for the steel and the FRP is assumed (σs = σf), as well as equal reinforcement cross-sectional ar-eas (As and Af), the transfer lengths are equivalent with the inverse of the contact surfaces mul-tiplied by the factor 1,80 and 1,25, respectively for steel and FRP reinforcement. In Figure 4 these factors are plotted in function of the applied reinforcement area (As and Af). Remark the three different steel reinforcement curves, which represent 1 bar, 2 equal bars and 3 equal bars respectively. Herewith a minimum diameter of 6 mm for each bar is assumed. For a reinforce-ment area of 150 mm² the transfer length of the FRP is 2,4 times smaller in comparison with the single steel bar reinforcement, 1,7 times smaller in comparison with 2 steel rebars and 1,4 times smaller in comparison with 3 steel rebars (Fig. 4). Herewith it can be concluded, that the trans-fer length for FRP tends to be smaller than the transfer length for steel rebars.

0

50

100

150

200

250

300

350

400

0 0,01 0,02 0,03 0,04

1/(1,25.us ) , 1/(1,80.u f) [1/mm]

As ,

Af

[mm

²]

FRP laminate

steel bars - 1 bar

steel bars - 2 bar

steel bars - 3 bar

Figure 4. Factors to calculate transfer lengths (sl and fl ) in function of applied reinforcement section

In a second stadium a stabilized crack situation is observed, herewith a similar transfer length

n,crl is observed for the steel and the FRP (Fig. 3b).

ffbss

ffb

ffm

cr

ffbss

ss

ssm

crn,cr

AEAE

AE

u

N

AEAE

AE

u

N

ξ+ξ

τ=

ξ+τ=l

with sffsm

fssfmb uAE

uAE

ττ

=ξ (11)

a

b

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3 EXPERIMENTAL EVALUATION OF THE CRACK WIDTH AND CRACK SPACING

In the framework of a test programme on flexural strengthening of continuous girders (Vasseur et al. (2007)) microscopic investigations on the crack spacing have been conducted. Hereby, cy-lindrical specimens have been taken out of a tested girder. This girder has a cross-section of 198 mm x 400 mm and a total length of 10 m (two spans of 5 m). The beam was strengthened with a FRP laminate (section 100 mm x 1,2 mm) at the top of the beam at the mid-support (ρf,support = 0,17 %) with Ef = 189900 MPa. The flexural tensile strength and the modulus of elas-ticity of the concrete are 2,45 N/mm² (fct) and 32000 N/mm² (Ec).

After testing of the continuous girder, three cylinders of 100 mm diameter have been drilled at the top of the beam around the mid-support (Fig. 5). The first cylinder is drilled at a location where the FRP laminate has debonded from the concrete top layer. Cylinders 2 and 3 at the other hand are drilled in a zone where no debonding is noticed. The cylinders are drilled through the FRP laminate as well as through the internal steel stirrup reinforcement, until a depth of about 170 mm into the concrete. The location of the cylinders is chosen in order to cover each time a major crack and to investigate the crack pattern near to the FRP laminate.

cylinder 1

cylinder 2

cylinder 3

Figure 5. Drawing and photograph of location of drilled cylinders at the mid-support.

After drilling, the cylinders are sawed in two halves, corresponding to the longitudinal axis of

the continuous girder. Further the cross-section of one halve is polished and finally a micro-scopic investigation on the polished surfaces is executed. A graphical representation of all cracks visible in the cross-section of the cylinders is made (Fig. 6). Notice the missing of the FRP laminate at top of the first cylinder (cylinder 1 in Fig. 6), caused by the fact the first cylin-der is drilled in a debonded zone.

Figure 6. Visualisation of cracks in the cross sections of the cylinders

Major cracks, which are visible at the outside of the beam (Fig. 5), are also visible in the

cross-section of the cylinders (Fig. 6). However in the top layer of the concrete, just beneath the FRP laminate additional cracks are observed. Observations by means of the microscopic study demonstrated that these additional cracks are smaller in comparison with the major (or so-called macro) cracks (Table 1). Whereas the mean crack width of the macro cracks after failure equals 0,97 mm (mean value of all major cracks at the mid support of the continuous girder), the mean crack width of the micro cracks measures 0,16 mm (mean value of micro cracks found in cross sectional areas of the cylinders). This results in a ratio of 16 %.

macro crack

stirrup

laminate

micro crack

cylinder 1 cylinder 2 cylinder 3

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A comparison of the macro crack spacing (mean value of spacing between all major cracks at the mid support) and the micro crack spacing (mean value of spacing between micro cracks found in cross sectional areas of the cylinders) is given in Table 2. Herewith the smaller crack spacing in the concrete layer nearby the FRP bond interface can be noticed (more, yet smaller cracks). The ratio between macro and micro crack spacing equals 23 %.

Comparing the macro crack spacing at stabilized cracking (load level of about 1,3 Ncr) with the analytical models, the best correspondence is found for the mixed reinforced tensile member model (Model 3). Nevertheless, the micro crack spacing in the concrete layer nearby the FRP bond interface appears much lower (Table 2).

As reported in many research projects (e.g. Matthys (2000)), the (macro) crack spacing of flexural strengthened beams or girders is smaller compared to unstrengthened reference mem-bers. In this study the authors demonstrated, that additional crack formation occurs in the con-crete cover nearby the laminate, resulting in even smaller crack spacing at the location of the FRP laminate.

Table 1. Crack widths of macro and micro cracks cylinder 1 cylinder 2 cylinder 3 mean Mean crack width of micro cracks [mm] 0,24 0,17 0,13 0,16 Crack width of macro crack [mm] 0,95 1,75 1,15 0,97 Ratio 25 % 10 % 11 % 16 %

Table 2. Crack spacing of macro and micro cracks Experimental [mm] Analytical [mm] * After Nu At 1,3Ncr M1 (Eq. 5) M2 (Eq.8) M3 (Eq. 11) Crack spacing micro cracks 20 - - - - Crack spacing macro cracks 87 192 247 771 218 * For Model 1 and Model 2 srm equals 4/3l t (Eq. 1), for Model 3 srm equals 2l cr,n

4 CONCLUSIONS

Following a microscopic investigation of the crack pattern underneath the FRP laminate of a continuous girder strengthened in flexure, it has been observed that macro cracks visible at the sides of the beam distribute in multiple micro cracks. For the considered girder the width of these micro cracks equals 16 % of those of the macro cracks. The crack spacing of the micro cracks appeared equal to 23 % compared to the macro cracks. The depth over which the micro cracks expands, roughly corresponds to the concrete layer between the FRP laminate and the in-ternal steel reinforcement.

Analytical models to predict the crack spacing have been compared. The best prediction of the macro crack spacing of a strengthened beam is obtained for the model based on the crack forma-tion in a mixed reinforced tensile member. From observations made in this study it appears that there is no constant crack spacing over the depth of strengthened flexural members. Taking into account this observation may allow more refined modelling of FRP debonding. REFERENCES Bresson J. 1971. Nouvelles recherches at applications concernant l'utilisation des collages dans les structures. Béton plaqué. Annales

de l'ITBTP 278: 22-25. CEB 1995. New developments in non-linear analysis methods. Lausanne. 94 Deuring M. 1993. Verstärken von Stahlbeton mit gespannten Faserverbundwerkstoffen. EMPA, Dübendorf, Switzerland: fib 2001. fib bulletin 14, Externally bonded FRP reinforcement for RC structures. International federation for structural concrete,

Lausanne. Holzenkämpfer P. 1997. Ingenieurmodelle des Verbunds geklebter Bewehrung für Betonbauteile. Berlin: Matthys S. 2000. Structural behaviour and design of concrete members strengthened with externally bonded FRP reinforcement.

Ghent. 345 Oller E., Cobo D. and Mari A. R. 2007. A new design proposal to prevent failure in beams strengthened by plate debonding. Patras,

GREECE. Ranisch E.-H. 1982. Zur Tragfähigkeit von Verklebungen zwisschen Baustahl und beton - Geklebte Bewehrung. TU Braunschweig: Teng J. G., Smith S. T., Yao J. and Chen J. F. 2003. Intermediate crack-induced debonding in RC beams and slabs. Construction

and Building Materials 17: 447-462. Vasseur L., Matthys S. and Taerwe L. 2007. Load tests on 2-span reinforced concrete beams strengthened with fibre reinforced

polymer. Claverton Down, Bath, England. 422-429. Wicke M. and Pichler D. 1991. Geklebte Bewehrung - Endverankerung mit und ohne Anpressdruck - Bemessungskonzept. HILTI-

Konzern-Forschung.