日本建築学会大会学術講演梗概集	 （関東）	 2015 年	 9	 月	 

23020

Crack Width Evaluation of RC Members Reinforced with Braided AFRP Bars
(Part 2: Model for Bond Constitutive law and Crack Width Prediction Method)
Regular member
Regular member
Regular member
Crack width
Bond stress
Slippage
AFRP bars
Tri-linear
Evaluation
1.

Introduction

Toshiyuki KANAKUBO **Hitomi OKAZAKI***τmax

In part 1, both pullout bond test and tensile bond test were

Bond stresscarried out to obtain the bond constitutive law and measured
crack width. In this part, tri-linear models of bond constitutivelaw both for normal surface type and sand-coated type AFRPbars are proposed. Using the tri-linear models, solving ank3

τ2k2

τ1

Test result

equation of the relationship between reinforcement strain andModelk1

slip at the loaded end1) can obtain crack width prediction

0s1

formulas for RC members reinforced with AFRP bars.
2.

○Shuai HAO *Tri-linear Models2

Slips max

Fig.1. Tri-linear model for normal surface type

As there is a large difference between the bond behaviors oftwo types of AFRP surface, two types of tri-linear models areτmax
τ1and the other is for the sand-coated type (RA7S, RA13S).
2.1 Tri-linear model for normal surface typeFig. 1 shows the tri-linear model for normal surface type. The
definitions and mathematical expressions are given as follows.

k3

k2

Bond stressintroduced. One is for the normal surface type (RA7, RA13)

τu

Test result
Model

k1

τmax = maximum bond stress, smax = slip at τmax
τ2 = 2/3·𝜏max , s2 = slip at 𝜏2 , τ1 = k1 ·s1

0 s1 smax

6 · Gf2 − 2 · s2 · τmax
s1 =3 · k1 · s2 − 2 · τmax

SlipsuFig.2. Tri-linear model for sand adhesion type

Gf2 = fracture energy during bond stress from 0 to τ2
τ2 − τ1
τmax − τ2
k1 = initial stiffness, k2 =,k =s2 − s1 3 smax − s2wcr  2 

2 ct Ac 1  np  ct Ac  l 
k1b 2
Eb Ab The maximum bond stress of the RA13 specimens was not(1)( wcr  2  s1 )obtained due to concrete splitting. Those values are assumed
as the average value of RA7 specimens.
2.2 Tri-linear model for sand-coated typeTri-linear model for sand-coated type is shown in Fig. 2. The
definitions and mathematical expressions are given as follows.
τu = minimum value after slipping bond stress, su = slip at 𝜏uτmax = slipping bond stress , smax = slip at τmax , τ1 = k1 ·s1
s1 =

2 · Gf2 − smax · τmax
k1 · smax − τmax

Gf2 = fracture energy during bond stress from 0 to τ2
τ2 − τ1τu − τmax
k1 = initial stiffness, k2 =,k =s2 − s1 3 su − smax
2 ct Ac 
k1  np  ct Ac 2 k1  ( k1  k 2 ) wcr  2  (1  1 )  s1  (2) l   s1 2k2
k2b 
2Eb Ab k2
k1(k2  0, 2  s1  wcr  2  s2 , wcr  2  (1  )  s1 )k2
2 ct Ac 
k1  np  ct Ac 
2 k1  ( k1  k 2 ) wcr  2  (1  1 )  s1  (3) l   s1 2k2k2b 2
Eb Ab k2k1
(k2  0, 2  s1  wcr  2  s2 , wcr  2  (1  )  s1 )
k2
 1
 A 
1  np  ct Ac  wcr  2    s1  ct c   l 
2ks2
Eb Ab  
1 b 1 (4)
( k2  0, 2  s1  wcr  2  s2 )Crack Width Evaluation of RC Members Reinforced with Braided AFRP Bars
(Part 2: Model for Bond Constitutive law and Crack Width Prediction Method)― 39 ―

Shuai HAO, Toshiyuki KANAKUBO,Hitomi OKAZAKI

The stiff resistance is provided by bearing between sand andconcrete in the case of the sand-coated surface of bars. When
this resistance fails, the bar starts slipping with a sudden fall ofbond stress. The bond stress at where the bar starts slipping iscalled “slipping bond stress”.
Theoretical Calculation Formulas

The following equation gives the relationship between
reinforcement strain and slip at the loaded end 1).

 2 A k1s2 1  np  ct Ac  11wcr  2   ct c   l )   s2  s1  (1 2EA22
 2 b b  b 2 ( k3  0, wcr  2  s2 )2Bond stress(N/mm )where,εbl ∶ strain of reinforcement at loaded end
ϕb ∶ perimeter of bar, σct ∶ tensile strength of concrete
Eb ∶ elastic modulus of reinforcement, sl ∶ slip at loaded end

Test result
Model0n ∶ elastic modulus ratio = Eb /Ec

1245

the loaded end gives a half of crack width, theoretical
calculation formulas for crack width Eq.(1)~Eq.(7) can beobtained.

Test result
Model

12455
Test result
Model060.5

1

Loaded end slip(mm)1.5 80x80mm
RA7S

Reinf.strain(%)

Reinf.strain(%)

10.5

10.5

Calculated
Test result

0formulas. The experimental curves of the specimens
reinforced with normal surface type (RA7 or RA13) show

Calculated
Test result

123
Crack width (mm)01.5 80x80mm
Reinf.strain(%)

RA13S

10.5

that perpendicular cracks in tensile bond test affect theexperimental measured crack width in those specimens.
Tri-linear models for bond constitutive law are proposed based

10.5Calculated
Test result

0Conclusions

12
Crack width (mm)1.5 80x80mm

RA13

Reinf.strain(%)

little deviation from the experimental curves. It is considered

1.580x80mm
RA13S

RA7

experimental results and calculated curves by the proposed

curves for the sand-coated type (RA7S or RA13S) show a3

Loaded end slip(mm)1.5 80x80mm

the average values of three specimens according to thegood agreements with calculated ones. However, calculated

1Fig.3. Models of bond constitutive law

Fig.3. The characteristic points of the models are obtained byshown in Fig.4 for specimens of 80 x 80mm, comparing

0.5

Loaded end slip(mm)25

Examples of the models of bond constitutive law are shown in

between reinforcement strain and measured crack width are

Test result
Model

100

definitions of tri-linear models. Examples of relationships

(7)

1510

Adaptability of Proposed Method

50680x80mm
RA132

Bond stress(N/mm )Using the tri-linear models and an assumption that the slip at5.

3

Loaded end slip(mm)15p ∶ reinforcement ratio=Ab /Ac(6)105Ec ∶ elastic modulus of concrete, τb ∶ bond stress(5)80x80mm
RA7S

1510Ac ∶ sectional area of concrete,Ab ∶ sectional area of bar

4.

80x80mmRA7

15

Bond stress(N/mm2)sl

1+n·p σct · Ac
∫ τb · ds + (εbl =)σct · Ac 0
2
Eb · Abϕb
2 ct Ac 
1  np  ct Ac   2 2 k1s1s2   2 s2   2 s1 wcr  2   s2  2  l k3k3b 2
Eb Ab  k32k3k2( k3  0, wcr  2  s2 , wcr  2(1  )  s2 )k3

Bond stress(N/mm )3.

2 ct Ac 
1  np  ct Ac   2 2 k1s1s2   2 s2   2 s1 wcr  2   s2  2  l 
kk
2Eb Ab  k32
k3
33 b


k2
( k3  0, wcr  2  s2 , wcr  2  (1  )  s2 )k3

123
Crack width (mm)Calculated
Test result0

1
2
Crack width (mm)3Fig.4.Crack width – reinforcement strain relationshipon the experimental results of pullout bond test both for normal surface type and sand-coated type AFRP bars. Based on the bondtheory which gives the relationship between reinforcement strain and slip at the loaded, theoretical calculation formulas to predict
crack width of RC members for AFRP bars are proposed. The predicted crack widths show a good agreement with test results.
References
1)

Kanakubo T., Yamato N., Crack Width Prediction Method for Steel and FRP Reinforcement Based on Bond Theory, Journal ofAdvanced Concrete Technology, Vol. 12(2014), No. 9, pp. 310-319* Dept. of Engineering Mechanics and Energy, Univ. of Tsukuba, Graduate Student** Dept. of Engineering Mechanics and Energy, Univ. of Tsukuba, Assoc. Professor*** Dept. of Engineering Mechanics and Energy, Univ. of Tsukuba, Graduate Student― 40 ―