Shear strength of strengthened RC beams with FRPs in shear

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Construction and Building

MATERIALS

Construction and Building Materials 22 (2008) 1261–1270

www.elsevier.com/locate/conbuildmat

Shear strength of strengthened RC beams with FRPs in shear Gyuseon Kim a, Jongsung Sim b, Hongseob Oh b

c,*

a Korea Infrastructure Safety and Technology Corporation, 2311 Daehwa-dong, Goyang 411-758, Republic of Korea Department of Civil and Environmental Engineering, Hanyang University, 1271 Sa1-dong, Ansan 425-791, Republic of Korea c Department of Civil Engineering, Jinju National University, 150 Chilam-dong, Jinju 660-758, Republic of Korea

Received 20 April 2006; received in revised form 7 November 2006; accepted 31 January 2007 Available online 2 April 2007

Abstract We propose a predictive model for shear strength of RC beams strengthened by fiber material. It consists of a plasticity model for web crushing, a truss model for diagonal tension, and a simple flexural theory based on the ultimate strength method. To analyze the shearstrengthening effect of the fiber, the model considers the interfacial shear-bonding stress between base concrete and the fiber. This reflects that the primary cause of shear failure in strengthened RC beams is rapid loss of load capacity due to separation of the strengthening fibers from the base material. The predictive model can estimate load capacities of each failure mode, and is compared to tested specimen data including extreme load failure. The analysis matches well with the experiments concerning load capacity and failure mode. Also, the experimental results of other published data are compared to the predictive model to evaluate its application. The results show that the predictive model has good adaptability and high accuracy.  2007 Elsevier Ltd. All rights reserved. Keywords: Shear-strengthening material; Predictive model; Plasticity model; Truss model; Flexural capacity, Interfacial shear-bonding stress

1. Introduction The most efficient technique for improving the shear strength of deteriorated RC members is to externally bond fiber-reinforced polymer (FRP) plates or sheets [1–3]. Steel plates have traditionally been the most common strengthening material, but its use in concrete structures has gradually declined because of its poor workability in-situ and mechanical characteristics, including corrosion and heavy weight, even though its material properties are similar to steel bars. For the last 20 years, researchers have concentrated on developing FRP as an alternative strengthening material to steel plates [1–5]. External bonding techniques for RC beams are useful for rehabilitating flexural and/or shear strength. However, while appropriate analytic methods to analyze flexural stress

*

Corresponding author. Tel.: +82 55 751 3299; fax: +82 55 751 3209. E-mail address: [email protected] (H. Oh).

0950-0618/$ - see front matter  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.conbuildmat.2007.01.021

of FRP-strengthened beams have been widely developed, theoretical approaches for shear strength of strengthened beams have not yet been sufficient to predict shear strength of strengthened RC beam in shear. This is due to the complex relationship between strengthening parameters, such as shear span-to-depth ratio (a/d), the bond–slip relationship, and the inclination of FRP [1–4]. In addition, previously proposed models have left much to be desired regarding failure by either flexural-shear or compressive-shear. 2. Research significance We address the structural response of shear-strengthened beams with various strengthening variables under monotonic loading. A plastic analytical model for shear-strengthened beams is proposed to predict shear strength and failure modes; this proposed model is a modification of the plastic model by Neilsen and Braestrup [6–8] and the truss model [9], updated to consider the strengthening effect of FRPs. The flexural strength of shear-strengthened beams is ana-

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Nomenclature a/d af As A0s Av B C D d0 fc0 fy Fp hs ts T

shear span-to-depth ratio depth of equivalent stress block area of tensile reinforcement area of compressive reinforcement area of stirrups width of beam compressive force of concrete effective depth to tensile reinforcement effective depth to compression reinforcement compressive strength of concrete yield strength of rebar tensile force of shear-strengthened FRP height of FRP thickness of FRP tensile force

S sp V Vc Vs Vp b h v q qmax sult sp

lyzed by traditional beam theory, and the theoretical shear strength of beams strengthened by various methods are compared to test results from previous studies [1,10].

spacing of stirrup spacing of shear-strengthened FRP shear strength shear strength of concrete shear strength of stirrups shear strength of FRP angle of FRP in strip type and fiber orientation of FRP strengthened in wing type angle of crack effective strength coefficient reinforcement ratio maximum reinforcement ratio ultimate bond strength average bond strength of FRP

P

P

a 3cm

2D13 22cm 4D16

3. Experimental study LVDT

3.1. Experimental program Eleven RC beams were prepared for an experiment to assess the strengthening effect of the strengthening materials, and define the shear failure characteristic. A coarse aggregate of normal Portland cement with a maximum size of 25 mm was used. Table 1 summarizes the concrete mixture proportions. The ultimate compressive strength of the concrete after 28 days was 34.7 MPa. The beams had a rectangular cross-section of 25 · 25 cm and effective spans of 140 cm and 200 cm, respectively. Carbon fiber-reinforced polymer plastic (CFRP) and carbon fiber sheets (CFS) were used as the shear-strengthening material. The reinforcement details and loading scheme are summarized in Fig. 1 and Table 3, respectively. To prevent flexural failure, four 16-mm-diameter deformed rebars were used for tensile reinforcement and two 13-mm-diameter deformed rebars were used for compressive reinforcement. The tensile reinforcement ratio (q = 0.014) of beams was only 54.9% of the maximum reinforcement ratio (qmax = 0.0266). The material properties of the rebar, CFS, and CFRP are summarized in Table 2. The properties of the strengthening materials were provided by the manufacturers. A summary of the experimental variables is shown in Table 1 Mix proportion of concrete

Mixture proportion

25cm

L = 140 or 200 cm

W/C

Cement (kgf/cm3)

Sand (kgf/cm3)

Aggregate (kgf/cm3)

Admixture (kgf/cm3)

49%

371

182

909

1.11

Fig. 1. Reinforcement details and loading scheme. Table 2 Material properties of reinforcing materials

Rebar CFS CFRP

Yield strength (MPa)

Ultimate strength (MPa)

Elastic modulus (MPa)

Ultimate strain

400

550 3550 3160

2.00 · 105 2.35 · 105 1.58 · 105

– 0.015 0.019

Table 3. The position of the steel strain gauges, and the linear variable differential transformer LVDT used to measure strains and deflections, are depicted in Table 3 and Fig. 1, respectively. A concrete strain gauge was attached to the top concrete surface of the mid-span. The strain gauge length for steel and FRP was 5 mm and that of concrete was 90 mm. Their strain limits were 3% and 2%, respectively. The capacity of the LVDT was 50 mm and the non-linearity was 0.1%. During the test, the displacements and strains of the strengthening material and reinforcing bars were measured and automatically recorded by a data acquisition system (see Fig. 2). Two different strengthening materials (CFRP and CFS) and two strengthening methods (strip type and wing type in either a vertical direction or a diagonal direction, respectively) were used in the experiment (Table 1). Two 13mm-diameter strengthening rebars were embedded in an overlaying layer prior to fiber reinforcement.

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Table 3 Details and designation of beams tested a/d Strengthening material Strengthening spacing CON-2 CON-3 CP2-VW

1.7 2.2 1.7

CFRP

– – –

Strengthening method – – Wing type

– –

P

CP2-1VS

1.7

CFRP

1S (10 cm)

P

90strip type

37. 4

65 .2

5

25

Gage

10

CP2-1DS

1.7

CFRP

1S (10 cm)

45strip type

CP3-VW

2.2

CFRP

–

Wing type

10

10

10

140cm

P

CP3-1VS

2.2

CFRP

1S (10 cm)

90strip type

P 103.2

5

48 .4

Gage

10

10

10

10

10

10

25

200 cm

P

P

65.2

CS2-VW

1.7

CFS

–

37.4

Wing type (90) Gage

25

22

CS2-DW

1.7

CFS

–

Wing type (45)

(continued on next page)

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Table 3 (continued)

CS3-VW

a/d

Strengthening material

Strengthening spacing

2.2

CFS

–

Strengthening method Wing type (90)

P

P 103.2

Gage

48.4

25

66

200 cm

CS3-DW

2.2

CFS

–

Wing type (45)

Fig. 2. Typical failure of beams.

In the case of CFRP strengthening, the concrete surfaces were carefully prepared prior to bonding both the concrete and the steel plate. The steel plates were degreased and then grit blasted to provide a suitable surface for bonding. The sides of the concrete beams were also grit blasted to remove the surface laitance. The epoxy resin used as the adhesive, manufactured by Shobond Korea, is commonly used for steel plate bonding and consists of bisphenol A (standard BPA epoxy). The viscosity of the two-component solvent-free epoxy

resin was 600 cps at 20 C. The curing agent used was phthalic anhydride. To control the final adhesive thickness, plastic spacers were placed within the adhesive layer between the concrete and steel plate. The plate was lifted into position, and pressure was applied through the use of a handheld injector; the plate was gently agitated to remove the excess adhesive. The adhesive was allowed to cure for 14 days at room temperature prior to testing. The strengthening material was N200 carbon fiber sheets produced in Korea by SK Chemical; these are polyacryloni-

G. Kim et al. / Construction and Building Materials 22 (2008) 1261–1270

trile (PAN)-based carbon fibers. The carbon fiber sheets were bonded to the beams in an upside-down position, as follows. To remove all laitance and smoothen the surface, the deck area was ground by hand and then cleaned with pressurized air. A resin was applied as a primer by roller and cured for 24 h with a protective cover to keep it dry. After blending the epoxy adhesive in a suitable container, it was spread evenly over the bottom surface of the beam by a roller. The epoxy resin used as the adhesive was SKPN, which is designed for moderate temperatures and is manufactured by SK Chemical; it consists of bisphenol A (standard BPA epoxy). The viscosity of the two-component solvent-free epoxy resin was 1300 cps at 20 C. It developed a bond strength of 2.5 MPa at 20 C within 7 days. The curing agent used was phthalic anhydride. Single-ply carbon fiber sheets were attached to the epoxy-coated surface, with the protective paper still in place, by pressing them into the epoxy with a rubber putter. After removing the protective paper, the sheets were further pressed into the epoxy coating with a screw type roller until they were completely immersed without air voids between the concrete and the sheets. After that, another CFS sheet was attached to the former CFS sheet using the epoxy adhesive for impregnation. About 30 min to 3 h after the second epoxy application, another layer of epoxy was applied to assure that the carbon fiber sheets were completely immersed within the epoxy. This resin used for impregnation of CFS was standard SKRN two-component solvent-free epoxy resin with a viscosity of 5000 cps at a temperature of 20 C. It had a compressive strength of 100 MPa and a bond strength of 2.5 MPa

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(7 days) at 20 C within a week. The strengthened beams were then cured for at least 10 days in air before testing. 3.2. Test results and discussion All CFS-strengthened beams in shear failed by typical shear cracking at the loading point, and the support as carbon fibers bonded to the beam side grabbed concrete fragments and then split the FRP composites, as shown in Fig. 2. Beams strengthened with CFRP (except CP3-VW applied by wing-type strengthening) developed a brittle shear failure caused by delamination of the FRP. In particular, CP2-DS-strengthened CFRP strips whose inclination was 45 failed by vertical cracking, because the CFRP strips prevented a diagonal tension crack from developing at the support and loading points. The load–displacement relationships of strengthened beams are shown in Fig. 3. For all beams, the load suddenly decreased after the peak load with the exception of CP3-VW, which developed a flexural failure after the tensile rebars yielded. Shear strength increased, due to FRP, by about 50–70%. The shear strength of beam CP2-1DS strengthened with inclined CFRP strips was 14% higher than that of beam CP2-1VS strengthened with vertical CFRP strips. Beams strengthened with CFS showed the same trend; the strength increase due to the 45 inclined CFS sheets ranged from 8% (beam CS3-DW) to 12% (beam CS2-DW) greater than those of beam-strengthened vertical sheets. In the case of a/d = 1.7, the strength of beams CS2-DW and CP2-1DS were 178 and 182 kN higher, respectively, whereas beams strengthened with vertical strips or vertical

Displacement (mm) a) a/d=1.7 (CFRP)

Displacement (mm) b) a/d=2.2 (CFRP)

Displacement (mm)

Displacement (mm)

c) a/d=1.7 (CFS)

d) a/d=2.2 (CFS)

Fig. 3. Load–displacement relationship.

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Table 4 Comparisons between predicted and experimental results Designation CON-2 CON-3 CP2-VW CP2-1VS CP2-1DS CP3-VW CP3-1VS CS2-VW CS2-DW CS3-VW CS3-DW a b

Experimental result (kN) 105.0 62.5 173.0 163.0 178.0 154.0 94.5 170.0 182.0 108.0 110.0

Failure mode (exp.) S S S S S M S S S S S

Predictional result (kN) 103.8 82.4 160.3 164.9 190.1 – 71.8 178.6 208.4 88.6 124.4

Failure mode (pre.) a

S Sb Sa Sa Sb – Sb Sa Sa Sb Sb

Exp./pre. 1.01 0.76 1.08 0.99 0.94 – 1.32 0.95 0.87 1.22 0.88

Web crushing failure. Diagonal tension failure.

sheets (CP2-VW and CS2-VW) showed a similar strength. Even though the strengthening ability of beam CP2-1DS was only half of that of beam CP2-VW, its shear strength was higher, suggesting that inclined strips more effectively control shear cracks. The ultimate strengths of the CP3 series (excluding CP3-VW) with regard to flexural failure were approximately the same regardless of fiber direction and strengthening scheme, such as strips or wing jacketing. Table 4 summarizes the failure loads and patterns of the beams. 4. Analytical model for shear strength The strengthening effect varied according to the shear span-to-depth ratio (a/d) of the beams. The ultimate strength of strengthened beams in shear was assessed using various methods depending on the failure mode. We used a plastic model of upper bound theory when the beam suffered a shear-compressive failure, a truss model of lower bound theory when the beam suffered a shear-tension failure with diagonal tension cracks, and finally a flexural analysis based on traditional beam theory in the case of flexural failure. Shear strength by strengthening materials was considered bonding stress between concrete and FRP because, as observed in previous research, shear failure of strengthened beams was caused by the debonding of FRP, not fracture of FRP.

4. Shear force in beams is resisted by stirrups, and the stress of stirrups approaches fy. 5. FRP-strengthening material also resists shear force, and its strength is governed by shear bond stress sp. Fig. 4 depicts the failure mechanism of strengthened RC beams in shear. Here, the yield line consists of a narrow crack across the beam and the compressive zone, and has an inclination h to the x-axis. Beam failure was caused by a yield of stirrups and compressive failure of concrete, not a yield of tensile rebars. FRP-strengthened materials showed a similar pattern, and developed relative displacement (d). The equilibrium equation for the yield line of a strengthened beam in shear can be derived as follows: V ¼VcþVsþVp 1 b  d Av fy þ d cot h þ V p ; ¼ vf 0c ð1  cos hÞ 2 sin h s

ð1Þ

where V is the shear strength, Vc is the shear strength of concrete, Vs is the shear strength of stirrups, Vp is the shear strength of FRP, b is the width of the beam, d is the effective depth of the beam section, h is the crack angle, Av is the area of stirrups, fy is the yield strength of rebar, and s is the spacing of stirrups.

4.1. Plastic model The plastic model used the following assumptions to predict the shear strength of strengthened beams. 1. Strength of concrete vf 0c , defined according to the failure criterion of maximum principal stress. Under this assumption, the tensile strength of concrete is assumed 0, v is the effective strength coefficient, and fc0 is the compressive strength of concrete. 2. Reinforcing steel bars deform perfectly plastically and resist only axial stresses. 3. The compressive force of concrete and tensile force are expressed as C and T, respectively, and T is Ty when the rebar yields.

Fig. 4. Failure mode with web crushing criterion in shear of RC beams shear-strengthened by FRP.

G. Kim et al. / Construction and Building Materials 22 (2008) 1261–1270

a

1267

b

Shear stress distribution in strip

Shear stress distribution in wing

Fig. 5. Calculation of the component of Vp: (a) strip type; (b) wing or jacket type.

Typical shear failure of beams strengthened in shear is caused by either rip-off or debonding of FRP, as previously reported. Shear resistance of FRP is governed by the bond strength of adhesive material and the bonding area of FRP, not the tensile strength of FRP (Fig. 5). Therefore, shear strength Vp can be calculated from the bond strength as follows: For the strip type in Fig. 5a: 2 Fp d cot hðsin b þ cos bÞ sp    2 sp ts 2hs ¼ d cot hðsin b þ cos bÞ sp

Vp ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ d 2  ð1  2uÞa a2 þ d 2  a for u < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ k 2h 2 a2 þ d 2 ð5aÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 uð1  uÞ for k 6 u 6 0 ¼ vf c 2 s 1 1 ¼ for u > vf 0c 2 2

ð2Þ

ð3Þ

ð4Þ

where u¼

ð5cÞ

Because stirrups and shear-reinforced FRP are assumed to be vertical struts in the truss model, the shear force can be evaluated from the axial forces of stirrups and shearreinforced FRP. In the under-reinforced beam section, the failure mode of the strengthened beam can be repre-

Eq. (1) can also be expressed as follows: s 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 1 þ cot2 h  cot hÞ þ u cot h ¼ mfc0 2

ð5bÞ

4.2. Truss model

For the wing or jacket type in Fig. 5b: V p ¼ 2 F p d cot hðsin b þ cos bÞ    t s  hs ¼ 2 sp d cot hðsin b þ cos bÞ 2

s ¼ vf 0c

Av  fy Vp 0 þ b  s  vf c b  vf 0c2

The effective strength coefficient v ranged from 0.0 to 1.0 due to concrete softening, caused by either internal cracking or sliding within the initial cracks. We used a value of 0.4 for v when the beam width was greater than the height, as proposed by Taylor [11], and 0.7 for other cases. That value is similar to that (0.72) proposed by Neilsen and Braestrup [7,12]. It is assumed that debonding of shear-strengthening FRP occurred when the maximum shear stress smax of FRP reached the ultimate bond strength sult, and then calculated Vp. sp is the average bond strength of FRP. b is the angle of FRP in strip type and fiber orientation of FRP strengthened in the wing style. If vfs 0 in Eq. (4) is minimized c with respect to cot h with the boundary condition 0 6 cot h 6 a/d, we get the following expression:

Fig. 6. Failure mode in diagonal tension of RC beams.

Fig. 7. Simplified sectional force equilibriums.

G. Kim et al. / Construction and Building Materials 22 (2008) 1261–1270

Plasticity

2-D13 (fy=400MPa)

f’ c =37.3MPa

truss

D10 @ 250 (300MPa)

flexural

t s=40 sp =210

, h s=300 , τ p =1.5MPa

300

Shear Stress Capacity - τ / U

200

250

1268

3- D22 (fy=400MPa)

Span/Depth Ratio -

Fig. 8. Analytical capacity curve derived for the beam shown.

a

b

c

d

e

f

Fig. 9. Predicted and experimentally measured capacity of test beams: (a) control specimens; (b) vertical strip strengthened specimens (CFRP); (c) diagonal strip strengthened specimen (CFRP); (d) wing type strengthened specimen (CFRP); (e) vertical wing strengthened specimens (CFS); (f) diagonal wing strengthened specimen (CFS).

G. Kim et al. / Construction and Building Materials 22 (2008) 1261–1270

This truss model can be applied if either the beam failed by flexure or the beam did not have enough shear span-todepth ratio (a/d) so that the disturbed region (the so-called D-region) of stress did not develop in the support and loading points. 4.3. Assessment of flexural strength

to be 1.5 MPa. As depicted in Fig. 8, the shear strength of the beams varied from shear-compression (Eq. (5)), diagonal shear failure (Eq. (6)) to flexural failure (Eq. (8)) throughout the shear span-to-depth ratio. The theoretical results of strengthened beams tested in this paper are summarized in Fig. 9. The bond strengths used to predict the shear strength of FRP applied to other values according to the mechanical properties of strengthening material and epoxy and the ratio of strengthening width to height. The average bond strength of CFRP strips was assumed to be 1.3 MPa based on the test results and the average bond strength of CFRP in wing type was assumed to be 0.6 MPa. These values are based on the test results of Al-Sulaimani et al. [1]. Also, the average bond strength of CFS in the wing type was assumed to be 0.8 MPa.

The flexural strength of the RC beams was calculated according to the traditional ultimate strength design concept (Fig. 7), and according to the following equation:  af M u ¼ ðAs  A0s Þfy d  ð7Þ þ A0s fy ðd  d 0 Þ: 2 Therefore, the above equation can be solved for shear stress as follows: s Mu ¼ : ð8Þ vf 0c a  b  d  vf 0c 5. Verification of proposed equations The theoretical results calculated from Eqs. (5), (6), and (7) are shown in Fig. 8. In this analysis, shear-reinforced beams with FRP used CFS strips at an angle of 90 and a spacing of 21 cm. The bond strength of epoxy is assumed

Experimental failure bads (KN)

sented as depicted in Fig. 6. At the failure state, tensile rebar and stirrups yield, and then shear-strengthened FRP suffers a debonding failure caused by macro-diagonal shear cracks. The shear strength based on the lower bound solution proposed by Thurlimann [9] can be expressed as follows: s V ¼ 2bdvf 0c ðu1 þ u2 Þ or; ¼ 2ðu1 þ u2 Þ: ð6Þ vf 0c

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Predictional failure bads (KN) Fig. 10. Measured and predicted shear force at failure for test beams.

Table 5 Comparison of predicted failure loads and experimental results of other published data Part

Beam designation

a/d

Strengthening method

Ref. [1]

CO CP SO SP WO WP JO JP

3.54 3.54 3.54 3.54 3.54 3.54 3.54 3.54

Control Control (f) 90strip 90strip (f) Wing Wing (f) Jacket Jacket (f)

35.2 35.4 42.3 42.0 42.8 46.1 51.1 63.6

S S S S S S M M

Ref. [12]

S-0-0 S-50-0 S-100-0 S-0-50-UT S-0-75-UT S-0-100-UT S-0-125-UT S-50-50-BT S-50-50-DT S-50-50-UT S-50-75-UT

3 3 3 3 3 3 3 3 3 3 3

Control Control (s) Control (s) 90strip 90strip 90strip 90strip 90strip (s) 45strip (s) 90strip (s) 90strip (s)

81.5 134.5 151.0 88.3 106.0 122.8 133.0 145.0 149.5 141.5 152.3

S S M S S S S M S S M

f: Beams repaired in flexure. s: Beams with stirrup.

Experimental result (kN)

Failure mode (exp.)

Predictional result (kN)

Failure mode (pre.)

Exp./ pre.

31.4 31.4 49.1 49.1 55.0 55.0 41.0 67.4

S S S-M S-M S-M S M M

1.12 1.13 0.86 0.86 0.78 0.84 1.25 0.94

105.7 131.9 131.9 131.9 131.9 131.9 131.9 131.9 146.2 130.6 150.1

S S-M M S-M S-M S-M S-M S-M S S S-M

0.77 1.02 1.15 0.67 0.80 0.93 1.00 1.10 1.02 1.08 1.01

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Table 4 summarizes the test and theoretical results. The estimated shear strength and failure patterns of beams coincided well with the test results, because the shear strength of FRP-strengthened beams was governed by bond strength. The results of the proposed model were compare to those of Al-Sulaimani et al. [1] and Shin [10] (Table 5). The theoretical results showed reasonable accuracy with the actual test results (Fig. 10). 6. Conclusion We proposed an equation to predict shear strength and failure patterns of FRP-strengthened beams in shear. We used the plastic model and the truss model to consider the contribution of FRP in shear. Our proposed model can be used to predict with reasonable accuracy the load carrying capacity of strengthened beams in shear, even if these are strengthened with various FRP materials. The composite shear strength of strengthened beams is governed by the bonding characteristics as well as the strength of FRP. This proposed model for the actual shear strength of beams is governed by the bond stress, the strengthening scheme, and the strengthening direction. To determine the amount of strengthening needed to improve the load carrying capacity of beams, this theory is more effective than either traditional theory or FE analysis. Acknowledgements This work is supported by the R&D Program (C105B1030001-05B0303-000 and C105D2010001-05D0201-00230)

of the Korea Institute of Construction and Transportation Technology Evaluation and Planning (KICTTEP). References [1] Al-Sulaimani GJ, Sharif AM, Basunbul IA, Baluch MH, Ghaleb BN. Shear repair for reinforced concrete by fiberglass plate bonding. ACI Struct J 1994;91(3):458–64. [2] Chajes MJ, Januszka TF, Mertz DR, Thomson Jr TA, Finch Jr WW. Shear strengthening of reinforced concrete beams using externally applied composite fabrics. ACI Struct J 1995;92(3):295–303. [3] Khallifa A, Nanni A. Improving shear capacity of existing RC Tsection beams using CFRP composites. Cem Concr Compos 2000;22:165–74. [4] Triantafillou TC. Shear strengthening of reinforced concrete beams using epoxy-bonded FRP composites. ACI Struct J 1998;95(2): 107–15. [5] Triantafillou TC, Antonopoulos CP. Design of concrete flexural members strengthened in shear with FRP. ASCE J Compos Constr 2000;4(4):198–205. [6] Nielsen MP. Limit analysis and concrete plasticity. second ed. Boca Raton: CRC press; 1999. [7] Nielsen MP, Braestrup MW. Plastic Shear Strength of Reinforced Concrete Beams. Bygningsstalixke Meddelelser 1975;46(3):61–99. [8] Braestrup MW. Plastic Analysis of Shear in Reinforced Concrete. Mag Concr Res 1974;26(89, December):221–8. [9] Thurlimann B. Plastic Analysis of Reinforced Concrete Beams. IABSE Colloquium on Plasticity in Reinforced Concrete, Copenhagen, 1979, Introductory Report, vol. 26, No. 89, December 1974. p. 221–8. [10] Shin S, Bahn B, Lee. Structural behavior of shear strengthened beam with CFS. Technical Report, Hanyang University, 1998. [11] Taylor R. Some Aspects of the Problem of Shear in Reinforced Concrete Beams. Civil Public Works Rev 1963;58(May):629–32. [12] Lim TY. Shear and Moment Capacity of Reinforced Steel-FibreConcrete Beams. Mag Concr Res 1987;39(140, September):148–60.