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

Composites: Part B 42 (2011) 1786–1798 Contents lists available at ScienceDirect Composites: Part B journal homepage: www.elsevier.com/locate/compositesb Analytical assessment and modeling of RC beam-column connections strengthened with CFRP composites Kien Le-Trung a, Kihak Lee a,⇑, Myoungsu Shin b,1, Jaehong Lee a,2 a Department of Architectural Engineering, Sejong University, 98 Gunja-Dong, Gwangjin-Gu, Seoul 143-747, Republic of Korea 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 b a r t i c l e i n f o Article history: Received 26 September 2010 Received in revised form June 2011 Accepted 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-column connections strengthened using carbon fiber reinforced polymer (CFRP) composites subjected to lateral 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 various 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 connection 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 connection model, which is a macro-scale model with a few elements, can be used for performance assessments of RC structures having CFRP-strengthened beam-column connections with an adequate accuracy and simplicity Ó 2011 Elsevier Ltd All rights reserved Introduction The observations from recent earthquakes show that many RC structures have failed in the brittle behavior of beam-column connections due to the deficiency of seismic details in the joint regions 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 Eurocodes [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 ⇑ Corresponding author Tel.: +82 3408 3286; fax: +82 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) Tel.: +82 52 217 2814; fax: +82 52 217 2809 Tel.: +82 3408 3287; fax: +82 3408 3331 1359-8368/$ - see front matter Ó 2011 Elsevier Ltd All rights reserved doi:10.1016/j.compositesb.2011.07.002 testing The previous studies were performed by Pantelides et al [5,6], Gergely et al [7], Antonopoulos and Triantafillou [8], Ghobarah 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 materials increased the shear capacity of RC beam-column joints considerably 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 Antonopoulos 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] performed 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 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 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 accurately predict the column shear and joint shear strengths of CFRPstrengthened connections However, the aforementioned studies were conducted using micro-scale models such as finite element methods or a set of equilibrium and compatibility equations, so that they may be too much time-consuming or complicated for general use by practical structural engineers For the purpose of applicability and simplicity, this study developed and proposed a macro-scale model with a few elements 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 attached 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 parameters through the use of more than 30 elements Finally, Favvata et al [29] developed a beam-column joint model using an advanced program for nonlinear static and dynamic analysis of structures ADAPTIC [31] The model was capable of describing the main characteristics of the actual response of RC joints under cyclic loading such as initial elastic stiffness, ultimate strength, post-yield response 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 CFRPstrengthened RC beam-column connections is the bond-slip behavior 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-strengthened beam-column joints has not been conducted to date This was likely because of the complexity of force-resisting mechanisms 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 connections strengthened with CFRP composites, subjected to cyclic lateral loading The developed model is capable of taking into account 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 1787 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 Element 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 connections strengthened with CFRP sheets subjected to cyclic loading 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 shows the details of the specimen NS The specimen NS had a lack of seismic 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 created from the specimen NS with various ways of wrapping CFRP sheets as shown in Fig The CFRP sheets with a thickness of 0.33 mm including T-shaped, L-shaped, and X-shaped configurations were used to evaluate effects of the different wrapping approaches 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 retrofitting CFRP sheets The dimensions of CFRP sheets are shown later, in Fig 14 The cyclic loading (Fig 3) was applied at the top of the 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 Fig The material properties of steel, concrete and CFRP materials were provided in Tables and 2, respectively More details for the test specimens can be found in Le-Trung et al [11] 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 [27] Fig illustrates the developed connection model for use in 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 (Element 10 in DRAIN-2DX program) embedded in one of the four hinges connecting adjacent rigid elements Three springs connected 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 Tables and 4)  Each column is modeled using Element 02, which consists of an elastic-perfectly plastic component and a strain-hardening component in parallel 1788 φ4@134 φ4@67 φ4@87 134 end mid TOP : 4-D10 TOP : 2-D10 BOT : 2-D10 BOT : 3-D10 STIRRUP: φ4@87 STIRRUP : φ4@87 34 67 67 167 93 172 134 26 φ42 φ17 200 44 87 200 φ4@134 384 18 1142 φ4@67 384 124 114 172 φ42 18 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 167 REBAR : 6-3-D10 HOOP BAR: φ4@134 Fig Reinforcement details of the non-seismic specimen NS (unit: mm) (a) NS (c) RNS-2 (b) RNS-1 (e) RNS-4 (f) RNS-5 (d) RNS-3 (g) RNS-6 12 10 -2 -4 -6 -8 -10 -12 500 kN Actuator Global Displacement 1142 mm 968 mm Story drift (%) Fig Description of all test specimens 20 40 60 80 100 120 Flexual and Shear Deformations 140 Step Fig Cyclic loading history  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 connected in series (one of rotational springs presents fixed end Fig Illustration of the test setup 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 1789 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 sponding deformation, which are presented in Tables and 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 Table Properties of concrete and reinforcing bars Strength of concrete (MPa) Strength of reinforcements (MPa) Type Compression Tension Type Yield strength Cured in water Cured in the air 33.8 36.5 4.0 3.8 D10 U4 324.0 459.0 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 in Fig This is a relatively simple inelastic element that can be used for the modeling of structural connections with rotational and/or translational flexibility The input data for the Element 10 include: Table Properties of CFRP composites Type Carbon FRP (CF720) Top coat (CLR67) Primer (CLR67) Tensile strength (MPa) Elastic modulus (GPa) 4965.8 240.5 59.5 3.7 56.8 3.7      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- Initial stiffness: k1 Strain hardening ratio: k2/k1  Positive and negative yield moments: Mỵ y , My Strength degradation factor: sdf  Positive and negative pinching moments: Mỵ g , Mg The strength degradation factor is defined as the ratio of the present to the previously adjacent cycle moment at the maximum Cyclic loading Column: Element 02 (Elastic-plastic element) Four rotational springs in parallel: (Element 10) (1): joint shear behavior (three springs) (2): CFRP effect on the joint (one spring) Joint rigid links connected by hinges Two rotational springs in series: (Element 10) (1): Bond slip (2): Plastic hinge Beam depth Beam: Element 02 (Elastic element) Column depth One rotational spring: (Element 10) for CFRP effect on the beam Fig DRAIN-2DX model for an external beam-column connection Table Axial load capacities of CFRP sheets Sheet no b1 (mm) 134 134 200 167 145 t1 (mm) 0.33 0.33 0.33 0.33 0.33 b2 (mm 134 134 200 167 193 t2 mm) 200 137 134 167 167 bw (–) 0.75 0.75 0.75 0.75 0.87 tf (MPa) 4.47 4.47 4.47 4.47 5.19 Gf (–) 0.3 0.3 0.3 0.3 0.5 k (–) 0.016 0.016 0.016 0.016 0.014 Le (mm) 98 98 98 98 114 L (mm) Pmax (kN) Pf (kN) No anchor Anchor No anchor Anchor 300 300 384 400 – 275 275 384 375 167 57.3 57.1 85.3 71.3 – 57.3 57.1 85.3 71.3 72.0 219.6 219.6 327.7 273.7 237.6 1790 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 Table Calculation of CFRP spring parameters Sheet No Configuration ΔL Spring moment (M) CFRP L P hb/2 θ beam Spring rotation (h) Spring stiffness (k1) b M ¼ Ph Mmax ¼ Pmax2hb h¼ DL hb =2 k1 ¼ kp h4b c M ¼ Ph L h ¼ hDc =2 k1 ¼ kp h4c L h ¼ hDb =2 b k1 ¼ kp jdh L h ¼ hDc =2 c k1 ¼ kp jd h a h ¼ DLh cos =2 b k1 ¼ kp jdh a h0 ¼ DhL csin =2 c k1 ¼ kp jd h 2 hb/2 joint/beam intersection hc /2 hc /2 P ΔL θ L column joint/column intersection CFRP Mmax ¼ Pmax2hc ΔL L M¼ hb/2 θ P V fjh ¼ P CFRP V fjh  jd ¼ P  jd Mmax ¼ P max  jd hb/2 hc/2 hc /2 beam V fjv ¼ P CFRP column hc /2 hc/2 M ¼ V fjv  jd ¼ P  jd 0 P ΔL θ hb/2 hb/2 L Mmax ¼ P max  jd V fjh ¼ P h ; V fjv ¼ P v ; column α L ΔL M¼ Pv α bf  jd; M ¼ b V fjv  jd Mmax ¼ P max  jd  cos a; M0max ¼ P max  jd  sin a; Ph 0 hb P V fjh P hc kp = E1A1/Le is the axial stiffness of the CFRP sheet (where A1 = b1  t1 is the cross section area of the CFRP sheet) and V ifv are the contributions to horizontal and vertical shear forces of the joint from the CFRP sheet, respectively V jh f jd and jd0 are the lever arms of the beam and the column, respectively, a is the angle between the fiber and the horizontal directions rotation reached during the previous cycle (for example, M9/M2 in Fig 6) The positive and negative pinching moments determine the Moment k2 My+ 18 10 17 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 unloading 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) k1 Mg+ 15 21 16 11 22 19 20 Rotation 12 Mg- 13 k2 My- 14 Fig DRAIN-2DX Element 10 developed by Foutch and Shi [47] At first, the envelope joint shear force (Vj) 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 shows 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 obtained from the test and the analysis for the specimen NS are shown in Fig 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), reinforcement yielding (cy, sy), and joint shear strength (cm, sm) As the figure shows, the joint shear stress values from the MCFT were little 1791 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 Fig Flowchart of MCFT method for specimen NS σ 2.5 1.5 γcr γ y γm γ Joint shear strain + + Joint shear stress σ Joint shear stress Joint shear stress (MPa) γcr γy γm Joint shear strain Experiment 0.5 Fig Combination of three bilinear joint springs in parallel MCFT 0 0.01 0.02 0.03 0.04 Joint shear strain (rad) Fig Comparison of hysteretic joint shear responses from the test and the MCFT larger than those from the test at the same levels of joint shear strain values Furthermore, the MCFT is only able to give a symmetric 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 Fig Two of the springs are elastic and then perfectly plastic with k2/k1 values 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 obtained from converting the sj–c curve Neither pinching nor strength degradation is considered for this spring Then, the joint shear force can be calculated as V j ẳ sj  dc  bef 1ị where sj is the joint shear stress, dc is the column depth, and bef is 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: h ¼ c; M ¼ V j  jd ð2Þ 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 procedure 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: 1792 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 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 degradation is specified, and pinching moments are assumed as onefifth of the yielding moments, again based on experimental results Lcs σst Stress distr Esh σy Es 3.4 Modeling for effects of CFRP sheets εy Strain distr εst Ly Ls hc Fig 10 Stress and strain distributions of a longitudinal beam bar in an exterior joint hs ẳ Ds dd 3ị where (d  d0 ) is the vertical distance between top and bottom beam bars, and Ds is the amount of beam bar pullout slip at the interface, which can be calculated as follows: Ds ¼ Ds ¼ Z Ls es xịdx ẳ Z Ls Ls ey ỵ Ly ey ỵ est ị est Ls x dx ¼ Ls est ðbefore yieldingÞ ðafter yieldingÞ ð4Þ ð5Þ where es(x) is the strain distribution of the longitudinal beam bar in the joint, est is the reinforcing bar strain occurring at the beam/joint interface, ey is the yield strain of the longitudinal beam bar, Ly is the length within which beam bar yielding occurs in the joint (Ly Lcs), Lcs is the horizontal part of the anchorage, and Ls is the length of the bond slip region before yielding, which can be estimated as: Es db Ls ¼ 4a ð6Þ where Es and db is the elastic modulus and the diameter of longitudinal beam bar, respectively; and a is a factor computed by an empirical equation proposed by Morita and Kaku [50] q a MPaị ẳ 600 fc0 ðMPaÞ ð7Þ Finally, the initial stiffness of the bond slip rotational spring can be determined as the beam yield moment divided by the beam rotation occurring when the reinforcing bar yields at the beam/joint interface The post-yield stiffness of the bond slip spring can be estimated 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 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 carrying 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 in Fig 11 was used In this model, initial micro-cracks occur when the local bond stress reaches its peak value When the bond stress reduces to zero, the initial macrocracks occur and the CFRP sheet begins detaching from the concrete surface The bond stress (s) and slip (d) between the CFRP sheet and the concrete surface can be related by following equation s ẳ f dị ẳ ( sf  2Gsf f d when d df when d P df   ð8Þ where sf, df, and Gf is the local bond strength, slip at the point of initial macro-cracks, and the interfacial fracture energy equal to sf  df/2 (see Fig 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 (see Fig 12), Yuan [35] obtained the following governing equation: d d dx  2Gf s2f k2 f dị ẳ 9ị Also, the axial stress in the CFRP sheet (r1) can be calculated as r1 ¼ s2f dd 2Gf t1 k dx ð10Þ where k2 ẳ s2f  2Gf E1 t ỵ b1 b2 E2 t  ð11Þ Substituting Eq (8) into Eq (9), we have d d dx ỵ k2 d ẳ k2 2Gf 12ị sf where E1, t1, and b1 are the elastic modulus, thickness, and width of the CFRP sheet, respectively; and E2, t2, and b2 are the same parameters for the concrete structure τ 3.3 RC beam and column modeling The column strength input parameters for Element 02 are positive and negative yield moments, compression and tension yield forces, and positive and negative balanced points of the P–M interaction 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 Initial micro-cracks τf τ = f(δ) Initial macro-cracks Gf δf δ Fig 11 Bond-slip model of a CFRP sheet attached on an RC member K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 1793 Fig 12 Configuration of the analytical model r1 ¼ 0; d ¼ at x ¼ L  a The general solution of Eq (12) is d ẳ A sin kẵx  L  aị ỵ B cos kẵx  L  aị ỵ 2Gf tf ð13Þ 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 conditions at the ends of the CFRP sheet Two stages were considered as follows r1 ¼ P b1 t at x ¼ L Thus, the final solution of Eq (12) is given as dẳ 2Gf f1  cos kẵx  L  aÞgðL  a x LÞ tf ð14Þ Substituting Eq (14) into Eqs (8) and (10), s ¼ sf cos kẵx  L  aị r1 ẳ sf kt sin kẵx  L  aị 15ị 16ị Then, the applied load is Stage 1: P < Pmax (no macro-cracks) When P is small, the CFRP sheet can be divided into two segments 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 in Fig 13a Therefore, the following boundary conditions can be set Pẳ sf b k sinkaị 17ị From Eq (17), if the bonded length, L, is large enough, the maximum load occurs when sin (ka) = (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 13 Boundary conditions for different cases 1794 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 significant This value of a is called the effective bonded length, Le In this case, we have Le ¼ r1 ¼ p 2k Case 1: L P Le (with or without anchorage) sf b k Case 2: L < Le (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 (Fig 13c) Thus, we obtain sf b k d¼ P ; b1 t s ¼ at x ẳ L 2Gf ẵcotkLị sinkxị  coskxị ỵ 1 tf s ẳ sf ẵcoskxị  cotkLị sinkxị r1 ẳ In this case, the anchorage (if any) has no effect on the load carrying 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 Pmax ¼ at x ¼ The solution of Eq (12) is as follows: 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 Pmax ¼ d ¼ 0; sinkLị ẵcotkLị coskxị ỵ sinkxị 300 Pmax ẳ sf b1 k sinðkLÞ The value of Pmax in all the cases must be not larger than the tensile strength of the CFRP sheet That is Pmax Pf ¼ rf  t  b1 where rf is 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 For all of the CFRP sheets in this study, the bonded length (L) was larger than the effective bonded length (Le) Therefore, the axial load capacities for two cases (with anchorage and without anchorage) had the same values However, the anchorage affected to the ductility of the CFRP sheets in some cases After reaching to the maximum 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 sustain the load 3.5 Conversion from P–d to M–h relationship Fig 14 shows the configurations of the CFRP sheets attached on the connection specimens Assuming that influence of the CFRP α 5’ 16 968 200 1’ 300 300 968 50 2’ α 167 ð20Þ ð21Þ ð19Þ Finally, we obtain 50 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) sf kt ð18Þ 167 Fig 14 Configurations of CFRP sheets 1795 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 sheets on the shear capacities of beams and columns are neglected, sheets number and 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 was attached above sheet number Therefore, sheet number was considered as anchored at one end of the sheet For sheets number 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 Fig 15 shows conversion procedures from P–d to M–h relationships of different CFRP sheet configurations The formulae of the conversion for different types of attached CFRP sheets are presented in Table The sheet number was simulated using one spring in parallel with the beam rotational spring The effect of sheet number 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 effects 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 (see Fig 14) were similar to those for sheets number 1, 2, and 5, respectively Sheet number was assumed to be anchored at the column edges Neither 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, Mf, was assumed very small The values of k2/k1 for the CFRP springs were set to zero M The analytical and experimental results for overall load–displacement responses were presented in the same graphs shown in Fig 17 The initial stiffness of the lateral load–displacement curve of the specimen NS from the analysis and the test are somewhat different This is mainly because the specimen NS was initially 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 experimental 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 analytical and experimental results were not much different, as shown in Fig 18 This indicates that the analytical model was closely able P k1 δ Pmax Mmax k1 kp θ1 δ1 θ θ1 δ (a) For sheets number 1,2 and θ rotation axial deformation rotation axial deformation M Moment Mmax kp δ1 Analytical results Axial load Moment Pmax (b) For sheets number and Fig 15 Conversion from P–d to M–h relationship 3 NS -0.06 -0.04 -0.02 0.02 0.04 -1 -2 -3 Experiment Analysis -4 0.06 Joint shear stress (MPa) Joint shear stress (MPa) Axial load P Fig 16 shows joint shear responses of the two specimens NS and RNS-5 obtained from the test and the analysis The analytical results agreed well with the experiment results for these specimens, 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-column connections RNS-5 -0.06 -0.04 -0.02 0.02 0.04 -1 t -2 -3 Experiment Analysis -4 Joint shear strain (rad) Joint shear strain (rad) (a) Specimen NS (b) Specimen RNS-5 Fig 16 Comparison of hysteretic joint shear responses from the analysis and the test 0.06 1796 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 15 Lateral Load (kN) 10 NS -60 -40 -20 -5 20 40 60 -10 Experiment Analysis -15 -20 -25 Displacement (mm) (a) NS 15 15 10 -60 -40 -20 -5 10 RNS-2 20 40 60 -10 Experiment -15 Lateral Load (kN) Lateral Load kN) RNS-1 -60 -40 -20 -25 Displacement (mm) (c) RNS-2 15 10 10 RNS-4 -20 -5 20 40 60 -10 Experiment -15 Lateral Load (kN) Lateral Load (kN) 15 5 -60 -40 -20 -25 Displacement (mm) (d) RNS-3 (e) RNS-4 15 10 -5 20 40 60 -10 -15 Experiment Analysis -20 -25 Displacement (mm) (f) RNS-5 Lateral Load (kN) RNS-6 -20 60 Experiment Analysis -25 Displacement (mm) RNS-5 40 -20 10 Lateral Load (kN) 20 -15 Analysis 15 -40 -5 -10 -20 -60 60 -20 (b) RNS-1 -40 40 Experiment Analysis -15 Analysis -25 Displacement (mm) -60 20 -10 -20 RNS-3 -5 -60 -40 -20 -5 20 -10 -15 40 60 Experiment Analysis -20 -25 Displacement (mm) (g) RNS-6 Fig 17 Comparison of overall load–displacement responses from the analysis and the test to define and predict the local failure mechanisms and consequent overall hysteresis behavior of the test specimens For the specimen RNS-1, which was strengthened by T- and Lshaped CFRP sheets without any anchorage strip, a brittle load reduction appeared in the overall load–displacement response (see Fig 17b) This was due to the detachment of the CFRP sheets at a column tip displacement of about 25 mm However, this reduction was not found in the analysis result The reason was that the bonding strength, which was assumed in the analysis, was relatively large so that the analysis did not reach to that stage In spite of that, the developed model can well simulate the overall load– displacement response in terms of stiffness and lateral load carrying capacity The specimen RNS-2 had the same CFRP configurations as the specimen RNS-1, except the presence of the anchorage strips on the column As a result, the specimen RNS-2 showed a more ductile 1797 K Le-Trung et al / Composites: Part B 42 (2011) 1786–1798 Table Comparison of Pmax obtained from the analysis and the test Energy dissipation (kNmm) 8000 7000 Specimen Analytical results Pmax (kN) Test results Pmax (kN) Difference (%) NS RNS-1 RNS-2 RNS-3 RNS-4 RNS-5 RNS-6 8.95 9.69 10.32 10.55 10.36 9.39 10.90 8.56 10.10 9.87 10.06 9.90 9.52 11.27 4.6 4.1 4.6 4.9 4.6 1.4 3.3 Experiment 6000 Analysis 5000 4000 3000 2000 1000 NS RNS-1 RNS-2 RNS-3 RNS-4 RNS-5 RNS-6 Test specimens Table Comparison of P40 obtained from the analysis and the test Specimen Analytical results P40 (kN) Test results P40 (kN) Difference (%) NS RNS-1 RNS-2 RNS-3 RNS-4 RNS-5 RNS-6 6.02 6.21 9.25 9.87 9.48 8.15 9.97 5.50 5.82 8.70 9.90 9.30 7.70 10.60 9.5 6.7 6.3 0.3 1.9 5.8 5.9 Fig 18 Comparison of energy dissipations from the analysis and the test behavior than the specimen RNS-1 did (see Fig 17c) The anchorage strips on the column helped the effects of CFRP sheets to sustain longer on the performance of the specimen RNS-2 The analysis results agreed well with the ductile experimental response The most stable and favorable behavior was found in the specimen RNS-3 The X-shaped CFRP sheets with the fiber direction close to the direction of the principal stress in the joint area improved the lateral load resistance of the specimen The highest energy dissipation with the least pinching was also found in this specimen The analytical and experimental results were very similar in both the hysteretic behavior (see Fig 17d) and the energy dissipation (see Fig 18) The specimen RNS-4 also showed a good performance (see Fig 17e), although it was not better than the specimen RNS-3 The L-shaped sheets increased the beam strength and stiffness; therefore, it caused more concentrated deformations in the joint area The energy dissipation of the specimen RNS-4 (see Fig 18) was less than that of the specimen RNS-3, which was well replicated by the analysis In short, the proposed model could predict well the overall hysteretic behavior of this specimen Comparing the test results of the specimens RNS-5 and RNS-2, was found the anchorage strip on the beam were not effective for enhancing the performance of the beam-column connection Due to the bonded length of the CFRP sheet was larger than its effective bonded length, the anchorage strip had no effect on the performance of the CFRP sheet, which was also identified in the analytical results (see Fig 17f) For the specimen RNS-6, in which two layers of CFRP sheets as well as the anchorage strips on the column were used, the energy dissipation of the connection (see Fig 18) much improved compared to the specimens RNS-1 and RNS-2 In addition, it was better than that of the specimen RNS-5 In general, the developed model can well predict the overall hysteretic response of this tested specimen (see Fig 17g) Table shows the maximum lateral loads obtained from the analysis and the test The discrepancies between the analytical and experimental results were calculated to be less than 5% for all specimens ranging from 1.4% to 4.9% It indicated that the proposed model was able to predict the lateral strengths of the test specimens with an acceptable accuracy To evaluate the performance of the tested beam-column connections in the inelastic region, the lateral loads at the column top displacement of 40 mm (approximately refers to a story drift value of 4%) from the analysis and the test were compared in Table The analytical values were quite close to the experimental values In general, the analytical model was able to simulate main force-resisting mechanisms of RC beam-column connections strengthened with CFRP composites, bond slip of longitudinal beam reinforcement, effects of different configurations of CFRP sheets, bond-slip between CFRP sheets and concrete surfaces, and effects of anchorage at the ends of CFRP sheets Since the developed macro-scale model consists of a few elements, it can be easily adopted and used by practicing structural engineers However, the proposed analytical model has some limitations in simulating other complex behaviors such as progressive detachment of CFRP sheets, shear contribution of CFRP sheets on beams and columns Conclusions In this study, a simple analytical model was developed to simulate the nonlinear hysteretic behavior of RC exterior beam-column connections strengthened using CFRP composites The developed model is a macro-scale model with a small number of elements using the DRAIN-2DX program, so that the analysis should be very efficient in term of time The model can adequately simulate main mechanisms such as joint shear behavior, bond slip of longitudinal beam reinforcement, effects of different configurations of CFRP sheets, bond-slip between CFRP sheets and concrete surfaces, and effects of anchorage at the ends of CFRP sheets With the simplicity and the adequate accuracy of the developed model, it is believed that the model can be used for performance assessments of RC buildings having beam-column connections strengthened with CFRP composites Acknowledgments This research was supported by a Grant (#06R&D B03) funded by the Ministry of Land Transport and Maritime Affairs and 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concrete (MPa)... E1, t1, and b1 are the elastic modulus, thickness, and width of the CFRP sheet, respectively; and E2, t2, and b2 are the same parameters for the concrete structure τ 3.3 RC beam and column modeling. .. bond slip of longitudinal beam reinforcement, effects of different configurations of CFRP sheets, bond-slip between CFRP sheets and concrete surfaces, and effects of anchorage at the ends of CFRP