Analytical assessment and modeling of RC beam-column connections strengthened with CFRP composites

This paper presents an analytical study on the modeling of exterior reinforced concrete (RC) beam-col- umn connections strengthened using carbon fiber reinforced polymer (CFRP) composites subjected to lat- eral loading. To simulate the overall connection behavior reasonably well, the developed analytical model takes into account joint shear behavior, bond slip of longitudinal beam reinforcement, and effects of var- ious configurations of CFRP sheets. In particular, effects of anchorage at the ends of the attached CFRP sheets, which have never been modeled in previous analytical studies to date, were incorporated into the developed model. The results from analytical and experimental studies for seven beam-column con- nection specimens tested by the authors were compared in terms of initial stiffness, maximum strength, stiffness degradation, strength degradation, and energy dissipation. The comparison indicates that the analytical results showed a good agreement with the experimental results. Therefore, the developed con- nection model, which is a macro-scale model with a few elements, can be used for performance assess- ments of RC structures having CFRP-strengthened beam-column connections with an adequate accuracy and simplicity

Analytical assessment and modeling of RC beam-column connections

strengthened with CFRP composites

Kien Le-Trunga, Kihak Leea,⇑, Myoungsu Shinb,1, Jaehong Leea,2

a

Department of Architectural Engineering, Sejong University, 98 Gunja-Dong, Gwangjin-Gu, Seoul 143-747, Republic of Korea

b

School of Urban and Environmental Engineering, Ulsan National Institute of Science and Technology (UNIST), 100 Banyeon-Ri, Eonyang-Eup, Ulju-Gun, Ulsan

Metropolitan City 689-798, Republic of Korea

a r t i c l e i n f o

Article history:

Received 26 September 2010

Received in revised form 1 June 2011

Accepted 3 July 2011

Available online 13 July 2011

Keywords:

A Carbon fiber

B Debonding

C Analytical modeling

E Joints

Bond-slip

a b s t r a c t

This paper presents an analytical study on the modeling of exterior reinforced concrete (RC) beam-col-umn connections strengthened using carbon fiber reinforced polymer (CFRP) composites subjected to lat-eral loading To simulate the ovlat-erall connection behavior reasonably well, the developed analytical model takes into account joint shear behavior, bond slip of longitudinal beam reinforcement, and effects of var-ious configurations of CFRP sheets In particular, effects of anchorage at the ends of the attached CFRP sheets, which have never been modeled in previous analytical studies to date, were incorporated into the developed model The results from analytical and experimental studies for seven beam-column con-nection specimens tested by the authors were compared in terms of initial stiffness, maximum strength, stiffness degradation, strength degradation, and energy dissipation The comparison indicates that the analytical results showed a good agreement with the experimental results Therefore, the developed con-nection model, which is a macro-scale model with a few elements, can be used for performance assess-ments of RC structures having CFRP-strengthened beam-column connections with an adequate accuracy and simplicity

Ó2011 Elsevier Ltd All rights reserved

1 Introduction

The observations from recent earthquakes show that many RC

structures have failed in the brittle behavior of beam-column

con-nections due to the deficiency of seismic details in the joint

re-gions Most of these buildings were designed and constructed not

meeting the recent design and construction requirements set forth

by modern seismic design codes such as ACI 318-08[1]or

Euro-codes[2,3]

In order to upgrade the seismic performance of old existing

beam-column joints, the use of carbon fiber reinforced polymer

(CFRP) materials have been popularly considered by structural

engineers This is due to the fact that CFRP materials have many

advantages such as high strength and stiffness-to-weight ratios,

excellent fatigue behavior, and strong corrosion resistance [4]

Additionally, the executing work for CFRP composites is known

to be a simple application and requires short construction time

To date, the performance of CFRP-strengthened beam-column

joints have been investigated mostly by means of experimental

testing The previous studies were performed by Pantelides et al

Ghoba-rah and Said[9], Yao et al.[10]and Le-Trung et al.[11] A common conclusion was that using even very low quantities of CFRP mate-rials increased the shear capacity of RC beam-column joints con-siderably and improved the overall connection damage tolerance Due to the variety and complexity of the failure mechanisms of

RC beam-column connections strengthened with FRP materials [12], analytical studies to simulate the experimental results have not been much found in the literature The CFRP sheets were often modeled as shell elements using finite element software In 2000, Parvin and Granata [13] developed a three-dimensional model using the ANSYS finite element program to investigate the effects

of fiber composites attached on RC beam-column joints Antonop-oulos and Triantafillou[14]proposed a connection model capable

of estimating joint shear stresses and strains at the various stages

of the response of CFRP-strengthened beam-column joints until the ultimate capacity is reached Their conclusion was that the analytical predictions for the joint shear strength were in good agreement with the test results Also, Parvin and Wu[15] per-formed a finite element analysis to investigate the effects of CFRP ply angle on the joint shear capacity and overall ductility of beam-column connections strengthened with CFRP wraps under combined axial and cyclic lateral loads The tested beam-column connections were simulated using the Marc MentatTM 2001 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd All rights reserved.

⇑Corresponding author Tel.: +82 2 3408 3286; fax: +82 2 3408 3671.

E-mail addresses: ltkxd2@yahoo.com (K Le-Trung), kihaklee@sejong.ac.kr

(K Lee), msshin@unist.ac.kr (M Shin), jhlee@sejong.ac.kr (J Lee).

1

Tel.: +82 52 217 2814; fax: +82 52 217 2809.

2 Tel.: +82 2 3408 3287; fax: +82 2 3408 3331.

Contents lists available atScienceDirect Composites: Part B

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m p o s i t e s b

software They concluded that the proposed model offered a high

degree of accuracy for predicting the behavioral characteristics of

the corresponding physical beam-column connections Recently,

Lee et al.[16]developed an analytical model to predict the column

shear and joint shear strength of beam-column connections

strengthened with CFRP composites The developed model was

based on Shiohara’s model [17] with some modifications The

authors concluded that the proposed model could be used to

accu-rately predict the column shear and joint shear strengths of

CFRP-strengthened connections

However, the aforementioned studies were conducted using

micro-scale models such as finite element methods or a set of

equi-librium and compatibility equations, so that they may be too much

time-consuming or complicated for general use by practical

struc-tural engineers For the purpose of applicability and simplicity, this

study developed and proposed a macro-scale model with a few

ele-ments to simulate the behavior of RC beam-column connections

strengthened with CFRP composites

The modeling of a beam-column connection strengthened with

CFRP materials can be conducted in two parts The first part is the

modeling of the RC beam-column connection without CFRP effects

The second part incorporates the effects of the CFRP elements

at-tached around the beam-column joint

For the behavior of RC beam-column connections that are not

strengthened with CFRP materials, many analytical models have

been proposed [18–29] Lowes and Altoontash [26] proposed a

joint shear panel model having four interface-shear springs and

eight bar-slip springs (one interface-shear spring and two bar-slip

springs at each rigid boundary of the joint), specifically for older

non-ductile frames without joint transverse reinforcement Shin

and LaFave[27]developed a model of an interior connection using

the nonlinear structural analysis program DRAIN-2DX[30] This

model explicitly incorporated hysteretic joint shear behavior as

well as other inelastic behaviors (such as bond slip, plastic hinge)

occurring in and around the joint Mitra and Lowes[28]developed

a model by modifying the previously proposed model of Lowes and

Altoontash[26] The model was capable of accurately predicting

the response of a wide range of joints with various design

param-eters through the use of more than 30 elements Finally, Favvata

et al [29] developed a beam-column joint model using an

ad-vanced program for nonlinear static and dynamic analysis of

struc-tures ADAPTIC[31] The model was capable of describing the main

characteristics of the actual response of RC joints under cyclic

load-ing such as initial elastic stiffness, ultimate strength, post-yield

re-sponse with strength degradation and pinching effects for the

hysteretic joint response Other approaches taken for the previous

studies have included applying finite element methods [32,33],

and using a continuum-type model having a 12-node joint element

and four ten-node transient elements[34]

One of the important factors affecting the performance of

CFRP-strengthened RC beam-column connections is the bond-slip

behav-ior between the retrofitting CFRP sheets and the concrete surfaces

Many bond-slip models simulating FRP-strengthened beam tests

or pure tension tests were developed elsewhere[35–46] However,

analytical modeling for the bond-slip behavior of

CFRP-strength-ened beam-column joints has not been conducted to date This

was likely because of the complexity of force-resisting

mecha-nisms in the joint area, including the possible detachment of the

CFRP composites

In this study, a macro-scale model was developed for predicting

the nonlinear hysteretic behavior of exterior RC beam-column

con-nections strengthened with CFRP composites, subjected to cyclic

lateral loading The developed model is capable of taking into

ac-count the bond-slip behavior between the attached CFRP sheets

and the concrete, as well as the effect of anchorage conditions at

the ends of the CFRP sheets Nonlinear inelastic behaviors such

as joint shear behavior, bond slip behavior of beam longitudinal reinforcement and plastic hinge development in the beam were also considered Strength degradation, stiffness degradation, and pinching effects were also considered using the DRAIN-2DX Ele-ment 10 developed by Shi and Foutch[47] The analytical results then were compared with the tests of seven 1/3 scale RC exterior beam-column connection specimens reported by Le-Trung et al [11] It was shown that the developed computer model was able

to simulate closely the hysteretic behaviors of the tested connec-tions strengthened with CFRP sheets subjected to cyclic loading

2 Specimen configuration The test specimens used for the analytical study were reported

by Le-Trung et al.[11] Eight 1/3-scale specimens including one non-seismic (NS) specimen, one seismic specimen (SD) and six CFRP-retrofitted specimens were designed and tested.Fig 1shows the details of the specimen NS The specimen NS had a lack of seis-mic details with no transverse reinforcements in the joint region, relatively large stirrup spacing in the beam, and downward anchorage of the beam bars away from the joint This design did not clearly satisfy the requirements for intermediate moment frames of the ACI 318-08[48] Six retrofitted specimens were cre-ated from the specimen NS with various ways of wrapping CFRP sheets as shown in Fig 2 The CFRP sheets with a thickness of 0.33 mm including T-shaped, L-shaped, and X-shaped configura-tions were used to evaluate effects of the different wrapping ap-proaches Five of six retrofitted specimens were strengthened by one layer of CFRP sheets (from RNS-1 to RNS-5) The last specimen (RNS-6) was retrofitted by two layers of CFRP sheets to evaluate the effect of the thickness of CFRP layers The 50 mm wide CFRP strips were applied to the beam and/or column in three specimens (RNS-2, RNS-5, and RNS-6) to prevent the debonding of the retro-fitting CFRP sheets The dimensions of CFRP sheets are shown later,

column of the specimens with a maximum drift up to 0.10 The loading history was controlled by a 500 kN actuator as shown in

were provided inTables 1 and 2, respectively More details for the test specimens can be found in Le-Trung et al.[11]

3 Analytical modeling

A macro-scale model was developed for simulating the seismic behavior of exterior RC beam-column connections strengthened with CFRP sheets, which was advanced from Shin and LaFave

DRAIN-2DX (nonlinear frame analysis software [30]) Details of the model are described in the following:

 The RC joint region is presented by four rigid link elements located along the joint edges and four rotational springs (Ele-ment 10 in DRAIN-2DX program) embedded in one of the four hinges connecting adjacent rigid elements Three springs con-nected in parallel are used to represent the nonlinear hysteretic joint shear behavior The forth spring, which was also connected

in parallel with the previous springs, was added to simulate the effect of CFRP sheets (T- or X-shaped sheets) on the joint shear behavior The input parameters for this spring was determined based on the load carrying capacities and the corresponding deformations for each sheet wrapped on the connection (see

 Each column is modeled using Element 02, which consists of an elastic-perfectly plastic component and a strain-hardening component in parallel

 Each beam is modeled using Element 02 for the elastic part and

Element 10 for each of the two nonlinear rotational springs

located at the beam/joint interface The three elements are

con-nected in series (one of rotational springs presents fixed end

rotations arising at the joint interface due to bond slip and yielding of longitudinal beam bars in the joint, while the other represents plastic hinge rotations near the end of the beam) One more spring is used at the joint/beam interface to consider

Fig 2 Description of all test specimens.

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

12

Step

Fig 3 Cyclic loading history.

1142 mm

Flexual and Shear Deformations

500 kN

Fig 4 Illustration of the test setup.

9

167

1142

φ4@87

φ17

mid TOP : 2-D10 BOT : 3-D10 STIRRUP : φ4@87 REBAR : 6-3-D10

134

134 26

end TOP : 4-D10 BOT : 2-D10 STIRRUP: φ4@87

44 87

93

φ42 φ42

Fig 1 Reinforcement details of the non-seismic specimen NS (unit: mm).

the effect of CFRP sheets (L- and/or T-shaped sheets) on beam

flexural performance This spring is in parallel with the two

springs presenting the beam plastic hinge and the bond-slip

behavior of beam longitudinal bars The input data for this

spring (initial stiffness and yield moments) was determined

based on the calculation of load carrying capacity and

corre-sponding deformation, which are presented inTables 3 and 4 Neither pinching nor strength degradation was assigned for this spring

 The modeling of the bond-slip behavior between CFRP sheets and concrete surfaces and anchorage conditions of the CFRP sheets is discussed later in detail

 The modeling of the effect of CFRP sheets on the flexural strength of the column is discussed later in detail

The input parameters for Element 02 are simply determined based on the DRAIN-2DX guidelines[30]

The typical behavior of Element 10, which was developed by Shi and Foutch[47], is illustrated inFig 6 This is a relatively simple inelastic element that can be used for the modeling of structural connections with rotational and/or translational flexibility The in-put data for the Element 10 include:

 Initial stiffness: k1

 Strain hardening ratio: k2/k1

 Positive and negative yield moments: Mþy, M

y

 Strength degradation factor: sdf

 Positive and negative pinching moments: Mþ

g

The strength degradation factor is defined as the ratio of the present to the previously adjacent cycle moment at the maximum

Table 1

Properties of concrete and reinforcing bars.

Strength of concrete (MPa) Strength of reinforcements

(MPa) Type Compression Tension Type Yield strength

Cured in the air 36.5 3.8 U4 459.0

Table 2

Properties of CFRP composites.

(CF720)

Top coat (CLR67)

Primer (CLR67)

Beam depth

Beam: Element 02 (Elastic element)

Two rotational springs in series: (Element 10) (1): Bond slip

(2): Plastic hinge

Four rotational springs in parallel:

(Element 10) (1): joint shear behavior (three springs) (2): CFRP effect on the joint (one spring)

Column depth

Cyclic loading Column: Element 02

(Elastic-plastic element)

One rotational spring: (Element 10) for CFRP effect on the beam

Joint rigid links connected by hinges

Fig 5 DRAIN-2DX model for an external beam-column connection.

Table 3

Axial load capacities of CFRP sheets.

Sheet no. b1 (mm) t1 (mm) b2 (mm t2 mm) b w(–) t f(MPa) G f(–) k (–) L e(mm) L (mm) Pmax (kN) P f(kN)

No anchor Anchor No anchor Anchor

rotation reached during the previous cycle (for example, M9/M2in

extent of pinching in the middle part of each hysteretic loop, by designating the direction of reloading branches in conjunction with the maximum rotations reached during the previous cycle The extent of stiffness degradation during reloading is determined from assigning pinching moments, while stiffness during

unload-ing is kept as a constant value equal to the initial stiffness k1

3.1 Parameters for rotational springs simulating joint shear behavior (without CFRP effects)

At first, the envelope joint shear force (V j) vs joint shear strain (c) curve was determined from the envelope joint shear stress– strain (sj–c) curve obtained using the Modified Compression-Field Theory (MCFT) developed by Vecchio and Collins[49].Fig 7shows

a flowchart of the MCFT method Because of the paper length, some details and notations were not presented here This information can be found in Shin and LaFave[27]

The joint shear stress vs joint shear strain relationships ob-tained from the test and the analysis for the specimen NS are shown inFig 8 The shear stress–strain curve can be simplified

as four linear segments, starting from the origin and connecting

three key points, so-called as joint shear cracking (ccr,scr ), reinforce-ment yielding (cy,sy ), and joint shear strength (cm,sm) As the figure shows, the joint shear stress values from the MCFT were little

Table 4

Calculation of CFRP spring parameters.

Sheet No Configuration Spring moment (M) Spring rotation (h) Spring stiffness (k1 )

1

P L

θ ΔL

CFRP

beam joint/beam intersection

M ¼ Ph b

2 4

2

M ¼ Ph c

2

c

4

2

3

P

CFRP

beam θ

h b=2 k1¼ k p jdh2b

M ¼ V f jh  jd ¼ P  jd

Mmax¼ Pmax jd

4

h b

h b

V f

2

M ¼ V f

v jd0¼ P  jd0

Mmax¼ Pmax jd0

5

P

α column

ΔL

L b f

h b

α

V f jh ¼ P h;V fv¼ Pv ; h ¼DL cosa

h b=2 k1¼ k p jdh2b

M ¼ V f jh  jd; M0¼ V fv jd0

Mmax¼ Pmax jd  cosa; h 0 ¼DL sina

1¼ k p jd

2

M0 max¼ Pmax jd0 sina;

k p = E1A1/L e is the axial stiffness of the CFRP sheet (where A1= b1 t1 is the cross section area of the CFRP sheet).

V jh f and V iv

f are the contributions to horizontal and vertical shear forces of the joint from the CFRP sheet, respectively.

jd and jd0 are the lever arms of the beam and the column, respectively,ais the angle between the fiber and the horizontal directions.

Rotation

Moment

1

18

4 12

5 6

14

7

8 9

13

15

19 16

17

22

20 21 Mg+

My+

Mg-

My-k1 k2

k2

larger than those from the test at the same levels of joint shear

strain values Furthermore, the MCFT is only able to give a

sym-metric curve of joint shear response in positive and negative sides

while the real joint shear response from the test did not present a

symmetric curve The maximum of negative stress was about 1.5

times larger than the maximum of positive joint shear stress for

the specimen NS These are limitations of the MCFT method Thus,

in order to get a more realistic analytical result, the result from the

MCFT was modified to capture the asymmetric joint response in

the developed model

The total behavior of joint shear (without the CFRP effects) can

be obtained from the combination of three springs as shown in

k2/k1values set to zero The strength degradation factor for these

springs is specified as a value of 0.95 and the pinching moments

are assumed as one-fifth of the yielding moment of each spring

The third spring has a negative second slope equal to that of the

fourth linear segment in the quad-linear envelope M–h curve

ob-tained from converting the s–c curve Neither pinching nor

strength degradation is considered for this spring Then, the joint shear force can be calculated as

wheresj is the joint shear stress, d c is the column depth, and b efis the effective joint width (average of the beam and column widths)

Finally, the hysteretic moment (M) vs rotation (h) curve to be

expressed by the combination of the three joint springs can be determined based on the following relationships:

where jd is assumed to be the average of the positive and negative

beam moment arms at the beam/joint interface

3.2 Bond-slip behavior of beam longitudinal reinforcement

The bond-slip model of longitudinal beam bars was mainly based on the model proposed by Morita and Kaku[50], with some modifications for an exterior beam-column joint (Fig 10) The pro-cedure for calculating the moment vs rotation relationship of the bond slip spring is presented as follows

The beam rotation (hs) occurring at the beam/joint interface due

to the beam bar slip can be estimated, neglecting the push-in of reinforcing bars in compression, as:

Fig 7 Flowchart of MCFT method for specimen NS.

0

0.5

1

1.5

2

2.5

3

Joint shear strain (rad)

Experiment MCFT

Fig 8 Comparison of hysteretic joint shear responses from the test and the MCFT.

σ

γ

γy γm γcr

Joint shear strain

σ

γy γm γcr

Joint shear strain

1 2 3

1 + 2 + 3

Fig 9 Combination of three bilinear joint springs in parallel.

hs¼ Ds

where (d  d0) is the vertical distance between top and bottom

beam bars, andDsis the amount of beam bar pullout slip at the

interface, which can be calculated as follows:

Ds¼

0

0

est

L s

x dx ¼ L sest

Ds¼L sey þ L yðeyþestÞ

wherees (x) is the strain distribution of the longitudinal beam bar in

the joint,estis the reinforcing bar strain occurring at the beam/joint

interface,ey is the yield strain of the longitudinal beam bar, L yis the

length within which beam bar yielding occurs in the joint (L y6Lcs),

L cs is the horizontal part of the anchorage, and L sis the length of the

bond slip region before yielding, which can be estimated as:

L s¼E s d b

where E s and d bis the elastic modulus and the diameter of

longitu-dinal beam bar, respectively; and a is a factor computed by an

empirical equation proposed by Morita and Kaku[50]

q

ð7Þ

Finally, the initial stiffness of the bond slip rotational spring can be

determined as the beam yield moment divided by the beam

rota-tion occurring when the reinforcing bar yields at the beam/joint

interface The post-yield stiffness of the bond slip spring can be

esti-mated based on the beam nominal moment and beam rotation after

yielding calculated above Pinching and strength degradation are

not considered in the bond-slip spring

3.3 RC beam and column modeling

The column strength input parameters for Element 02 are

posi-tive and negaposi-tive yield moments, compression and tension yield

forces, and positive and negative balanced points of the P–M

inter-action curve The beam (elastic part) strength input parameters for

Element 02 are simply the positive and negative yield moments,

which are set to very large values in order to lump all inelastic

deformations at the bond slip and plastic hinge rotational springs

For the beam plastic hinge spring, the initial stiffness is assigned

a large value in order to generate no rotation before yielding, and

the strain-hardening ratio is set to a value equivalent to 0.03 times the elastic stiffness of the beam The yield moments are then taken

as the positive and negative beam yield moments No strength deg-radation is specified, and pinching moments are assumed as one-fifth of the yielding moments, again based on experimental results

3.4 Modeling for effects of CFRP sheets

The effects of CFRP sheets depend on many parameters such as CFRP configurations, anchorage conditions, material properties, and bond slip Due to the possible debonding of CFRP sheets from concrete surfaces, the load capacities of the CFRP sheets may not reach their full strengths The procedure to determine the load

car-rying capacity (Pmax) of a CFRP sheet considering the bond-slip behavior is presented in the following

For simplicity, the bond-slip model that has only linearly descending branch as shown inFig 11was used In this model, ini-tial micro-cracks occur when the local bond stress reaches its peak value When the bond stress reduces to zero, the initial macro-cracks occur and the CFRP sheet begins detaching from the con-crete surface

The bond stress (s) and slip (d) between the CFRP sheet and the concrete surface can be related by following equation

(

ð8Þ

wheresf, df , and G fis the local bond strength, slip at the point of ini-tial macro-cracks, and the interfacial fracture energy equal to

sf df/2 (seeFig 11), respectively These values can be determined using the formulae developed by Lu et al.[39]

From the equilibrium equations applied for the analytical model

d2d

dx2

s2

f

Also, the axial stress in the CFRP sheet (r1) can be calculated as r1¼ s2

f

dd

where

f

1

E1t1þ b1

b2E2t2

ð11Þ

Substituting Eq.(8)into Eq.(9), we have

d2d

dx2þ k

2

sf

ð12Þ

where E1,t1, and b1are the elastic modulus, thickness, and width of

the CFRP sheet, respectively; and E2,t2, and b2are the same param-eters for the concrete structure

L s

h c

E s

Stress distr.

σy

Strain distr.

E sh

L y

σst

εy

εst

L cs

Fig 10 Stress and strain distributions of a longitudinal beam bar in an exterior

joint.

τf

τ

G f

0

Initial micro-cracks

Initial macro-cracks

τ = f(δ)

The general solution of Eq.(12)is

where L is the bonded length of the CFRP sheet and a is the length of

the micro-crack segment When subjected the applied loading, the

CFRP sheet can be divided into two segments One is micro-crack

segment with presence of interface slip between the CFRP sheet

and the concrete surface The other is the rest segment with no

interface slip The constants A and B were determined from the

boundary conditions of the CFRP sheet

The final solution of Eq.(12)depends on the boundary

condi-tions at the ends of the CFRP sheet Two stages were considered

as follows

Stage 1: P < Pmax(no macro-cracks)

When P is small, the CFRP sheet can be divided into two

seg-ments as mentioned above At the intersection point, the axial

stress (r1) in the CFRP sheet and the interface slip (d) are equal

to zero as shown inFig 13a Therefore, the following boundary

conditions can be set

r1¼ P

b1t1

at x ¼ L

Thus, the final solution of Eq.(12)is given as

t f

Substituting Eq.(14)into Eqs (8) and (10),

r1¼ sf

Then, the applied load is

P ¼sf b1

From Eq.(17), if the bonded length, L, is large enough, the maximum load occurs when sin (ka) = 1 (i.e a = amax=p/2k) If L is larger than

amax, then the effect of the bonded length on the maximum load is Fig 12 Configuration of the analytical model.

significant This value of a is called the effective bonded length, L e.

In this case, we have

L e¼ p

2k

Stage 2: P = Pmax(appearance of initial macro-cracks)

In order to consider the effect of the bonded length of CFRP

sheets and the effect of anchorage strips, three following cases

are investigated

Case 1: L P Le(with or without anchorage)

In this case, the anchorage (if any) has no effect on the load

car-rying capacity of the CFRP sheet when the initial debonding occurs

In fact, the anchorage will affect on the load capacity when the

bonded length is reduced to be less than the effective bonded

length (due to the propagation of the CFRP sheet) However, for a

simplicity, the propagation of CFRP sheets is not considered in this

study (Fig 13b)

From Eq.(17), we have

Pmax¼sf b1

k

Case 2: L < L e(without anchorage)

The maximum applied load is obtained when the shear stress at

the free end of the CFRP sheet reaches the bond strength At this

point, the axial stress in the CFRP sheet is zero After this stage,

the CFRP sheet cannot sustain the load anymore Therefore, the

boundary conditions are the same with those of Stage 1

Pmax¼sf b1

Case 3: L < Le(with anchorage)

Because of the presence of the anchorage, after reaching the

same state of Case 2, the CFRP sheet is still capable of sustaining

the load until the initial debonding occurs at the loaded end At

the loaded end, the bond stress is zero Therefore, the boundary

conditions in this case are as taken as (Fig 13d)

r1¼ P

The solution of Eq.(12)is as follows:

r1¼ sf

Finally, we obtain

Pmax¼sf b1

k

1

The value of Pmaxin all the cases must be not larger than the tensile strength of the CFRP sheet That is

Pmax6P f¼rf  t1 b1 whererfis the rupture stress of the CRFP sheet

The axial load capacities of the CFRP sheets considering the bond-slip effects were calculated and presented in Table 3 For

all of the CFRP sheets in this study, the bonded length (L) was lar-ger than the effective bonded length (L e) Therefore, the axial load capacities for two cases (with anchorage and without anchorage) had the same values However, the anchorage affected to the duc-tility of the CFRP sheets in some cases After reaching to the max-imum load, the CFRP sheet without anchorage strips assumed to fail in a brittle mode due to the debonding of the sheet On the other hand, the CFRP sheet with anchorage strips was able to sus-tain the load

3.5 Conversion from P–d to M–h relationship

the connection specimens Assuming that influence of the CFRP

300

50

167

167

1

3

2 4

5

5’

2’

1’

α

α

167

sheets on the shear capacities of beams and columns are neglected,

sheets number 1 and 2 mainly increased the flexural strength of

the beams and columns On the other hand, sheets number 3–5

mainly affected the joint shear strength

For sheets number 1–3, their effects on the performance of the

connections are assumed to be lost after the initial detachment of

the CFRP sheets It is noted that sheet number 3 was attached

above sheet number 4 Therefore, sheet number 3 was considered

as anchored at one end of the sheet For sheets number 4 and 5,

anchorage existed in both ends of the sheets, so that they were still

able to sustain the load after the initial detachment of the CFRP

sheets

relation-ships of different CFRP sheet configurations The formulae of the

conversion for different types of attached CFRP sheets are

pre-sented inTable 4 The sheet number 1 was simulated using one

spring in parallel with the beam rotational spring The effect of

sheet number 2 to the column flexural strength was incorporated

into the model by increasing the column moment strength by the

maximum moment of the CFRP sheet, Mmax In this case, the effect

of the CFRP sheet on the column stiffness was neglected The

ef-fects of the other sheets to the joint shear response were simulated

by rotational springs in parallel with three joint rotational springs

The calculations for sheets number 10, 20, and 50(seeFig 14) were

similar to those for sheets number 1, 2, and 5, respectively Sheet

number 5 was assumed to be anchored at the column edges

Nei-ther strength degradation nor pinching was considered for the

CFRP springs Element 10 with the elastic code of 3, which can

simulate the brittle behavior, was used for modeling the CFRP

springs The value of the post-failure moment, M f, was assumed

very small The values of k2/k1for the CFRP springs were set to

zero

and RNS-5 obtained from the test and the analysis The analytical results agreed well with the experiment results for these speci-mens, both of which underwent significant joint shear damage as illustrated by large inelastic joint shear deformations

The joint shear responses of the test specimens were complex due to cracking, bond slip of beam longitudinal reinforcement, and the detachment of the CFRP composites As reported elsewhere

by the authors[11], the specimens experienced different levels of joint shear distress Therefore, a macro-scale model such as the developed one in this study was not expected to well simulate these complex responses The expectation is that the developed model is able to well predict the overall responses of the beam-col-umn connections

4 Analytical results The analytical and experimental results for overall load–dis-placement responses were presented in the same graphs shown

curve of the specimen NS from the analysis and the test are some-what different This is mainly because the specimen NS was ini-tially damaged during the pretest loading (during the initial test setup, the column of the specimen NS was accidentally loaded in the longitudinal direction and resulted in minor cracks[11]) On the other hand, for the other specimens, the analytical and exper-imental responses were very similar in terms of the initial stiffness and the maximum strength Additionally, for all the specimens, the total energy dissipation (during the test) calculated from the ana-lytical and experimental results were not much different, as shown

δ

P

Pmax

axial deformation

M

Mmax

rotation

θ1

δ

P

axial deformation

M

rotation

θ1

Mmax

Pmax

Fig 15 Conversion from P–d to M–h relationship.

NS

-4 -3 -2 -1 0 1 2 3

Joint shear strain (rad)

RNS-5

-4 -3 -2 -1 0 1 2 3

Joint shear strain (rad)

t Experiment

Analysis

(b) Specimen RNS-5 (a) Specimen NS

Experiment Analysis