INFLUENCE OF LOCAL BOND CHARACTERISTICSIN FRP-CONCRETE BOND BEHAVIORToshiyuki Kanakubo1, Zhi-shen Wu2 and Tamon Ueda3
1

Institute of Engineering Mechanics and Systems, University of Tsukuba, Ibaraki, Japan2

3

Dept. of Urban & Civil Engineering, Ibaraki University, Ibaraki, Japan

Division of Structural and Geotechnical Engineering, Hokkaido University, Hokkaido, Japan

ABSTRACT
Macro-level bond behavior between FRP laminates/sheet and concrete is influenced by local bondcharacteristics such as bond stress (τ) – slip (s) relationships. In this study, effect of local bondcharacteristics to macro-level bond behavior is investigated by numerical analysis based on thesequential integration method. Investigated influence factors are τ – s relation shapes, delamination
fracture energy and FRP stiffness. Analytical results are concluded that influence of differences of τ –s relation shapes is relatively small, and macro-level bond strength is mainly influenced by fractureenergy and FRP stiffness. To increase the macro bond strength, it is effective to increase fractureenergy. However, when the fracture energy varies, the effective bond length can vary too. It isessential that the bond length should be greater than the effective bond length. When the ratio of local
bond strength to ultimate slip is constant, the effective bond length remains the same, and macro bondstrength and loaded-end slip are proportional to square root of fracture energy.

1 INTRODUCTION
As element-level macro bond characteristics between FRP laminates/sheet andconcrete are influenced by micro bond characteristics (local bond characteristics) andvarious other factors (influencing factors). Macro characteristics as represented by themaximum load and the amount of slip (loaded-end slip) are subject to change. Micro bondcharacteristics and various influencing factors are closely related to each other, and thereinforcing material properties (e.g., stiffness) have especially strong influence on macrobond characteristics. On the other hand, there are cases in which macro bond characteristics
do not change noticeably even when the bond stress (τ) – slip (s) relationship changes by
various influencing factors which can have significant influence on micro bondcharacteristics. It is difficult to clearly distinguish between the influence of various
influencing factors on micro bond characteristics and the influence of those factors on
macro bond characteristics. Besides, even if those factors influence only micro bondcharacteristics, their influence is not always reflected directly in macro bond characteristics.
In this study, local bond stress – slip relationship and delamination fracture energy(one of the micro bond characteristics) and the reinforcing material stiffness (one of theinfluencing factor) as the factors that influence macro bond characteristics are adopted. The
influences of those factors on macro bond characteristics are assumed that it changes
independently of the other factors.
2 ANALYSIS METHOD
The bond characteristic analyzed is only shear bond characteristic (Mode II), and theanalytical method used is the sequential integration method in which numerical calculations

are performed based on force equilibrium and deformation compatibility conditions [1].
The specifications of the other calculating factors are as shown below. A concrete
block having a 100 mm x 50 mm cross section (contains deformed bar D22 in the center) is
bonded with FRP laminate (width: 50 mm) on one side. This gives the image of shear bondcharacteristic when the FRP laminate is subjected to an in-plane shear (Figure 1).
Concrete: Sectional area, Ac＝100 x 50mm Elastic modulus, Ec＝24GPa
Reinf. bar: Sectional area, As＝387mm2 Elastic modulus, Es＝206GPa
FRP laminate : Bonding width, bf＝50mm
It is assumed that the behavior of the concrete, reinforcing bar, and FRP laminate
within a finite bond length show only elastic behavior. This is different from the actual
element-level behavior and the behavior within a member in terms of the occurrence of acrack in concrete and the presence or absence of an out-of-plane deformation.
sisi-1
PfAdhesive

P f − ∆P f

τ b,iFRP

Pm +ΔPm

Pm

Concretedx

δm,iFigure 1 Image of analysis target
3 ANALYTICAL RESULTS AND DISCUSSION
3.1 Influence of shape of local bond stress – slip relationships
To investigate the influence of the shape of local bond stress (τ) – slip (s) relationship,
the four models shown in Figure 2 are used. They are: (a) perfectly rigid-plastic model, (b)perfectly elastic-plastic model, (c) bi-linear model (softening type), and (d) Popovics model[1]. Since the influence of only the shape of τ – s relationship is to be investigated, thedelamination fracture energy (area enclosed by τ – s relationship) and the FRP stiffness are
assumed to be the same for all the models. The values of analysis factors are as shown
below. It should be noted that for the Popovics model, the range of calculation of fractureenergy is 0 to 0.354 mm in terms of the amount of local slip.
Fracture energy
Gf＝1.145N/mm
FRP stiffness
t･E＝ (0.167mm) x (230GPa) ＝38.4kN/mm
Bond length
lb＝120mm
The relationship between loaded-end tensile load and loaded-end slip and the bond

stress distributions in the bonded region are shown in Figure 2. The bond stress distribution
is obtained one at the point marked with the circle on the tensile load – loaded-end slip
relationship.

00.2
Local slip (mm)10500.2 0.4 0.6 0.8
Loaded end slip (mm)0.2
Local slip (mm)6420 10 20 30 40 50 60 70 80 90 100 110 120
Free end
Distance from free end (mm) Loaded end

020 10 20 30 40 50 60 70 80 90 100 110 120
Free end
Distance from free end (mm) Loaded end(b) Elastic-plastic model

0.2Local slip (mm)10500.2 0.4 0.6 0.8
Loaded end slip (mm)500.2
Local slip (mm)10500.2 0.4 0.6 0.8
Loaded end slip (mm)8Bond stress (MPa)

8Tensile load (kN )515Local bond stress (MPa)Tensile load (kN )Local bond stress (MPa)0.2 0.4 0.6 0.8
Loaded end slip (mm)415

Bond stress (MPa)56(a) Rigid-plastic model

0108
Bond stress (MPa)

Bond stress (MPa)508Tensile load (kN )515Local bond stress (MPa)Tensile load (kN )Local bond stress (MPa)1564
20 10 20 30 40 50 60 70 80 90 100 110 120
Free end
Distance from free end (mm) Loaded end

64
20 10 20 30 40 50 60 70 80 90 100 110 120
Free end
Distance from free end (mm) Loaded end(c) Bi-linear model(d) Popovics model
Figure 2 Local τ – s relations and macro bond behavior
From the analysis results obtained by the each model, the bond stress distributiondiffers among the models. However, there is no remarkable difference in maximum tensile
strength among the models: they are in the range from 14.4kN to 15.1kN. With the
exception of the Popovics model, the loaded-end slip under maximum tensile load is lessthan 0.4 mm. Thus, as long as the fracture energy and FRP stiffness are the same, the shapeof the local bond stress – slip relationship, which is one of the micro bond characteristics,does not have significant influence on the macro bond characteristics.
3.2 Influence of delamination fracture energy
To investigate the influence of delamination fracture energy (Gf), an analysis isperformed by using the bi-linear model for local bond stress – slip relationship and thesame FRP stiffness. The fracture energy is assumed to be in the range from 0.045N/mm to2.045N/mm, and a similar bi-linear model shown in Figure 3 is used. The values of theother analysis factors are as shown below:
FRP stiffness t･E＝ (0.167mm) x (230GPa) ＝38.4kN/mm
Bond length lb＝120mm
The relationship between fracture energy and maximum tensile load obtained byanalysis and the relationship between fracture energy and loaded-end slip at maximum load
are shown in Figure 4. For the maximum tensile load, which is proportional to square rootof Gf, the approximate expression shown in the figure is obtained in this analysis. The
loaded-end slip at the maximum load is also proportional to square root of Gf.
Local bond stress (MPa)108Gf = 2.045N/mm

6420.0450Figure 3

0.1

0.2
0.3
Local slip (mm)Slip at max. load (mm)Maximum load (kN)

0.5

15
Pmax = 13.4 Gf0.5

5

00.5

Bi-linear model varied by Gf

20

10

0.4

0.5
1
1.5
Fracture energy (N/mm)

20.40.3sl,max = 0.335 Gf0.5

0.2
0.1
00.5
11.5
Fracture energy (N/mm)

2

Figure 4 Fracture energy and macro bond behavior
3.3 Influence of FRP stiffness
To investigate the influence of FRP stiffness, an analysis is performed by using abi-linear model (τmax = 6.47MPa, smax = 0.065 mm, su = 0.354 mm) for the bond stress – slip
relationship. The stiffness of the FRP is assumed to be 2.51kN/mm to 1030kN/mm. The
values of the other analysis factors are as shown below.
Fracture energy
Gf＝1.145N/mm
Bond length
lb＝600mm
The relationship between FRP stiffness and maximum tensile load and therelationship between FRP stiffness and loaded-end slip at maximum load are shown inFigure 5. For the maximum tensile load, which is nearly proportional to t ⋅ E , theapproximate expression shown in the figure is obtained in this analysis. The loaded-end slip
at maximum load is almost constant.
0.560Pmax = 2.52 (t×E)0.475

40200200400
600
800
FRP stiffness (kN/mm)

Slip at max. load (mm)Maximum load (kN )

80

0.4
0.3
0.2
0.1

1000

0200
400
600
800
FRP stiffness (kN/mm)

1000

Figure 5 FRP stiffness and macro bond behavior
4 INFLUENCE FACTORS FOR MACRO BOND BEHAVIOR
When the bond length is sufficiently large and the concrete deformation is negligibly
small, there is the following correlation among bond strength, fracture energy, andreinforcing material stiffness [2, 3].
Pmax = b f ⋅ 2 ⋅ G f ⋅ t ⋅ E

where,Pmax
bfGft･E(1): bond strength: FRP width: delamination fracture energy: FRP stiffness

Substituting the values used in the analysis in Equation (1), Pmax with the analysis factors
shown in 3.2 becomes as follows:
Pmax = 13.9 ⋅ G f(2)

Similarly, with the analysis factors shown in 3.3, Pmax becomes as follows:
Pmax = 2.39 ⋅ t ⋅ E(3)The approximate expressions obtained in the analysis are close to the above expressions.
The slight difference between the approximate expression shown in 3.3 and the aboveexpression (3) is considered due to the fact that the difference in stiffness between the FRP
and concrete is not negligible. In practical use, however, Equation (1) is consideredsufficient.
To calculate effective bond length roughly, the following expression is consideredeffective.

l e = su ⋅2⋅t ⋅ E
Gfwhere,lesut･EGf(4): effective bond length: ultimate slip in local τ-s relationship (at point where bond stress = 0)
: FRP stiffness: delamination fracture energy

Equation (4) is derived from Equation (5) which represents the anchoring conditions of
FRP when the local bond stress – slip relationship is assumed as a perfectly rigid-plastic
model with bond stress τy and ultimate slip su.
su =τy2⋅t ⋅ E
G f = τ y ⋅ su

⋅ le

2(5)
(6)

5 SUMMARIES
Analytical results show that the influence of differences of local bond stress – slip
relation shapes is relatively small, and macro-level bond strength is mainly influenced byfracture energy and FRP stiffness. To increase the macro bond strength, it is effective toincrease the fracture energy. However, when the fracture energy varies, the effective bondlength can vary too. It is, therefore, essential that the bond length should be greater than theeffective bond length. Generally speaking, when the ratio of local bond strength to ultimate
slip is constant, i.e., the local bond stress – slip relationship is similar, the effective bondlength remains the same, and the bond strength and loaded-end slip are proportional toGf . Since the macro bond strength is proportional to t ⋅ E , the macro bond strength canbe increased by using a stiffer FRP. In this case, the effective bond length also increases inproportion to t ⋅ E , whereas the loaded-end slip remains the same.
Acknowledgement
This study was conducted under the research project in Japan Concrete InstituteTechnical Committee on Retrofitting Technology for Concrete Structures.
References[1] K. Nakaba, T. Kanakubo, T. Furuta, H. Yoshizawa : Bond Behavior between
Fiber-Reinforced Polymer Laminates and Concrete, ACI Structural Journal, Vol. 98,No. 3, pp.359-367, May-June 2001
[2] B. Täljsten : Strengthening of Concrete Prisms using the Plate-Debonding Technique,International Journal of Fracture, Vol.82, pp.253-266, 1996
[3] Z. S. Wu, H. Yuan and H. D. Niu : Stress Transfer and Fracture Propagation inDifferent Kinds of Adhesive Joints, Journal of Engineering Mechanics, ASCE, Vol.128,
No.5, pp.562-573, 2002.3