Finite element analysis of end cover separation in RC beams strengthened in flexure with FRP

Engineering Structures 75 (2014) 550–560

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Finite element analysis of end cover separation in RC beams strengthened in flexure with FRP S.S. Zhang, J.G. Teng ⇑ Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China

a r t i c l e

i n f o

Article history: Received 23 March 2014 Revised 18 June 2014 Accepted 19 June 2014

Keywords: FRP Strengthening Bonded reinforcement Finite element (FE) modelling Cover separation Radial stress

a b s t r a c t The use of externally-bonded (EB) or near-surface mounted (NSM) FRP reinforcement in the strengthening of reinforced concrete (RC) beams in flexure has become increasingly popular in recent years. Such beams are likely to fail by end cover separation in which a major crack in the concrete initiates at a cut-off point of the FRP reinforcement and propagates along the level of steel tension bars, leading to the detachment of the FRP reinforcement together with the cover concrete. Due to the complexity of this failure mode, no reliable finite element (FE) approach for its accurate prediction has been published despite many previous experimental and theoretical studies on the problem. This paper presents a novel FE approach for predicting end cover separation failures in RC beams strengthened in flexure with either externally bonded or near-surface mounted FRP reinforcement. In the proposed FE approach, careful consideration is given to the constitutive modelling of concrete and interfaces. Furthermore, the critical debonding plane at the level of steel tension bars is given special attention: the radial stresses exerted by the steel tension bars onto the surrounding concrete are identified to be an important factor for the first time ever and are properly included in the FE approach. The proposed FE approach is shown to provide accurate predictions of test results, including load–deflection curves, failure loads and crack patterns. Ó 2014 Published by Elsevier Ltd.

1. Introduction The flexural strength of reinforced concrete (RC) beams can be effectively increased using either externally bonded (EB) [1,2] or near-surface mounted (NSM) fibre-reinforced polymer (FRP) reinforcement [3]. In such FRP-strengthened RC beams, plate-end/ bar-end debonding (referred to collectively as end debonding) has been widely recognized as one of the most common failure modes [1,3–8]. End debonding is related to the existence of large interaction forces (stresses) between FRP and concrete near a cut-off point of the bonded FRP reinforcement in such strengthened RC beams [9,10]. For ease of presentation, the discussions of the paper are limited to simply-supported beams. For RC beams strengthened with an externally bonded FRP plate, a systematic classification of the common failure modes has been given by Teng and Chen [8]. Two of the common failure modes, namely plate-end cover separation and plate-end interfacial debonding, involve the initiation of cracks near a cut-off point of the bonded FRP reinforcement and the propagation of cracks towards the beam mid-span. Although these two debonding failure

⇑ Corresponding author. Tel.: +852 2766 6012; fax: +852 2766 1354. E-mail address: [email protected] (J.G. Teng). http://dx.doi.org/10.1016/j.engstruct.2014.06.031 0141-0296/Ó 2014 Published by Elsevier Ltd.

modes both initiate at a plate end, their failure mechanisms differ. Plate-end interfacial debonding is induced by the high interfacial shear and normal stresses near the plate end, with the failure plane in the concrete being only a few millimetres away from the FRP plate; by contrast, in plate-end cover separation, a major crack in the concrete initiates at a cut-off point of the FRP reinforcement and propagates along the level of steel tension bars, leading to the detachment of the FRP reinforcement together with the cover concrete [8]. In RC beams strengthened with NSM FRP bars, barend debonding failures also include bar-end interfacial debonding [11] and bar-end cover separation [12–15]. The mechanisms of the two bar-end debonding failure modes are quite similar to those of the two plate-end debonding failures respectively. Both bar-end interfacial debonding and bar-end cover separation in RC beams strengthened with NSM FRP are illustrated in Fig. 1. Whether the controlling failure mode is end cover separation or end interfacial debonding depends on many factors. For RC beams strengthened with externally bonded FRP reinforcement, end cover separation has been much more common than end interfacial debonding in existing tests; the latter one is only likely to happen when the plate width is much smaller than the beam width [8]. Similarly, for RC beams strengthened with NSM FRP reinforcement, end cover separation has been found to be the main end debonding failure mode [3]. A major reason that end cover separation is much

S.S. Zhang, J.G. Teng / Engineering Structures 75 (2014) 550–560

Load

Load End cover separation

End interfacial debonding

Bottom steel

551

NSM FRP

(a) End interfacial debonding

(b) End cover separation Failure plane 2

Failure plane 1

(cover separation)

(interfacial debonding)

(c) Possible failure planes on the cross section Fig. 1. End debonding failure modes of RC beams strengthened with NSM FRP bars: (a) end interfacial debonding; (b) end cover separation; and (c) possible failure planes on the cross section.

more likely than end interfacial debonding is the significant radial stresses generated by the steel tension bars on the surrounding concrete, making the plane of steel tension bars (i.e. failure plane 2 in Fig. 1) a more critical plane than a plane near the adhesiveconcrete bi-material interface (i.e. failure plane 1 in Fig. 1). This aspect is further examined later in the paper. The present paper is concerned with the FE analysis of FRPstrengthened RC beams failing by end cover separation. A novel 2-D FE approach for the accurate prediction of end cover separation failures in RC beams strengthened in flexure with either EB or NSM FRP reinforcement is proposed. In the proposed approach, the cracking of concrete, the bond behaviour between steel bars and concrete as well as that between FRP and concrete, and the radial stresses exerted by steel tension bars onto the surrounding concrete are all carefully considered. The last aspect has never been considered in previous studies on end cover separation. It should be noted that various empirical expression for concrete-related properties derived from experimental data are used in this paper. An examination of existing information revealed that these expressions can be taken to be valid for concrete with cube compressive strengths within the range from 25 MPa to 80 MPa (or slightly beyond) unless otherwise specified. For concrete strengths significantly outside the range, the conclusions reached in the present paper need to be treated with due caution.

2. Existing finite element (FE) studies While experimental studies are essential in understanding debonding failure mechanisms in FRP-strengthened RC beams, there are many aspects that cannot be well understood using laboratory tests alone. As a result, researchers have explored the use of FE analysis in predicting debonding failures in these beams. An accurate FE approach not only offers a powerful tool to gain in-depth understanding of the problem but also allows the efficient generation of extensive data for use in the formulation of a relatively simple debonding strength model for use in practical design. All existing FE studies on debonding failures in RC beams strengthened in flexure with FRP have been concerned with externally bonded FRP reinforcement; that is, no FE study on end debonding failures in RC beams strengthened in flexure with NSM FRP reinforcement has previously been attempted. Arduini et al. [16] presented an FE study of end cover separation in

FRP-strengthened RC beams in which a perfect bond was assumed between FRP and concrete as well as between steel and concrete. The predicted load–deflection curves were acceptable, but the crack patterns inferred from strain distributions at failure did not allow the identification of the cover separation failure mode. Rahimi and Hutchinson [17] adopted the isotropic damage model proposed by Oliver et al. [18] in their FE simulation of FRP-strengthened RC beams. In their work, the adhesive layer was represented using plane stress elements, and the steel bars were smeared into the concrete elements directly. Supaviriyakit et al. [19] used the smeared crack model to simulate FRP-strengthened RC beams. In their study, the FRP-to-concrete interface was assumed to be perfectly bonded, and the steel bars were distributed uniformly in the concrete elements without any additional nodes and elements. The crack pattern predicted by their FE approach was somewhat obscure. To bypass the complexity of modelling the shear behaviour of cracked concrete, Pham and AlMahaidi [20] proposed an FE approach in which the rotating smeared crack model was used for the concrete under the level of steel tension bars but the fixed smeared crack model was used for the concrete above it. Although predictions from their FE analysis showed close agreement with the test results of selected specimens, they adopted a constant shear retention factor for the fixed smeared concrete model, which is unjustifiable. Yang et al. [21] adopted a discrete crack model based on linear elastic fracture mechanics for the concrete and employed a special treatment for modelling the bond–slip behaviour between steel bars and concrete to simulate the failure process of FRP-strengthened RC beams. They presented their predictions for two test beams, and showed close agreement between the FE predictions and the test results for these two test beams. Their approach required a remeshing process which is time-consuming and complicated. Camata et al. [22] presented an FE approach for FRP-strengthened RC beams in which a combination of the smeared crack model and the discrete interface crack model were used, with the former for concrete cracking modelling and the latter for cracks observed in tests and for possible cracks near the steel-to-concrete and the FRP-to-concrete interfaces. The key issue of this approach is that the possible crack paths need to be known a priori and properly pre-defined in the FE model. The specific features of the previous FE approaches for end cover separation of RC beams strengthened with externally bonded FRP are summarized in Table 1 to highlight their differences and

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weaknesses. As can be seen from this summary, none of the existing FE approaches includes accurate modelling of cracked concrete (particularly with regard to shear behaviour) or the bond–slip behaviour between steel and concrete. The modelling of the FRPto-concrete bond–slip behaviour has also been inadequate although this aspect is believed to be less critical to the accurate prediction of bar-end cover separation. Besides, in some of these studies, clear evidence for the correct prediction of end cover separation was not available [16,17]. Finally, Yang et al.’s approach [21] requires time-consuming and complicated re-meshing while Camata et al.’s approach [22] is not truly predictive as the experimentally-observed crack pattern needs to be known a priori for the FE analysis to give good predictions. It can therefore be concluded that a reliable FE approach for end cover separation failures in RC beams strengthened with externally bonded FRP is not yet available. Furthermore, no FE study has been found in the published literature on end cover separation failures in RC beams strengthened with NSM FRP. The present paper therefore presents the first reliable nonlinear FE approach for predicting end cover separation failures in RC beams strengthened with either externally bonded or NSM FRP reinforcement based on the general purpose FE program MSC.MARC [23].

3. Test database A total of 44 RC beams strengthened in flexure with FRP and failing by end cover separation were collected from 17 existing studies. These tests were chosen because sufficient geometric and material properties had been provided. None of the selected beams was subjected to preloading before the bonding of the FRP reinforcement, and the FRP was neither prestressed nor anchored using any means. Among the selected beams, 37 specimens are RC beams strengthened with externally bonded FRP, and 7 specimens are RC beams strengthened with NSM FRP. Details of the specimens are given in Tables 2–5. In the test database, the forms of externally bonded FRP reinforcement include FRP plates and FRP sheets, but for NSM FRP reinforcement, only beams strengthened with CFRP strips were included. Other forms of NSM FRP reinforcement were not considered in the study as reliable bond–slip models have not been developed for them for use in FE simulation. The distance between the ends of FRP reinforcement and the nearer support ranges from 0 mm to 1200 mm, so the end force combinations (combinations of beam cross-sectional shear force and moment at the plate

end) cover nearly pure shear states (high shear force and nearly no moment), pure bending states as well as states between the two extremes. All the tests were conducted under either threepoint bending or four-point bending. In FE analyses and result comparisons, different forms of FRP reinforcement (plates, sheets and strips), different strengthening methods (externally bonded and NSM FRP), and different end force combinations are not individually differentiated as the fundamental failure mechanism is the same. 4. Proposed Fe approach 4.1. General considerations To accurately predict end cover separation, the following aspects need to be properly modelled: (1) the tensile and shear behaviour of cracked concrete; (2) the bond–slip response between steel bars and concrete and that between FRP and concrete; and (3) the behaviour of the critical debonding plane at the level of steel tension bars. Accurate modelling of cracked concrete, particularly under shear stresses, is still a major challenge. Accurate modelling of cracked concrete is a pre-requisite for the accurate prediction of end cover separation which involves the formation and propagation of major cracks. The bond–slip behaviour between steel bars and concrete also has a significant bearing on the predicted crack pattern [24], which indirectly affects the cover separation failure process. At the plane of steel tension bars, the total actual width of concrete is smaller than the beam width as part of the width is taken up by the steel bars. This is one of the weakening effects at the critical debonding plane. More importantly, radial stresses may be induced by the steel tension bars onto the surrounding concrete when significant slips between concrete and steel bars develop, further weakening the critical plane. Therefore, this effect should also be reflected in the FE model. 4.2. FE meshes In the proposed FE approach, the concrete is modelled using 4-node plane stress elements and both the steel bars and the FRP reinforcement are modelled using 2-node beam elements. The bond behaviour between FRP and concrete as well as that between steel bars and concrete are modelled using 4-node cohesive

Table 1 Existing FE studies on end cover separation in RC beams strengthened in flexure with externally bonded FRP reinforcement. Reference

[16] [17]

[19]

[20] [21] [22] a

Modelling of concrete cracking

Modelling of bond behaviour

Crack model

Tension

Shear

FRP-to-concrete

Steel-to-concrete

Smeareda Smeared, isotropic damage model by Oliver et al. [18] Fixed smeared

Not clearb Isotropic damage model by Oliver et al. [18]

Not mentioned NA

Perfect bond Rebars smeared into concrete elements

Tension-stiffening model by Okamura and Maekawa [34] Tension-softening model by Hordijk [33] NA Tension-softening model

Shear transfer model by Okamura and Maekawa [34] Constant shear retention factor NA NA

Perfect bond Connected by plane stress elements for an elastic adhesive Perfect bond

Rotating smeared and fixed smearedc Discrete, LEFM Rotating smeared and discretee

Bilinear modeld Crudely modelled Model by Cervenka et al. [43]

Rebars smeared into concrete elements with a tension-stiffening model Perfect bond Crudely modelled Perfect bond

It is not clear from the paper whether a fixed or rotating smeared crack model was used. The post-cracking behaviour of concrete in tension is not clearly described in the paper. The rotating smeared crack model was used for the concrete below the tension reinforcement, but the fixed smeared crack model was used for the rest. d Parameters in the bond–slip model were determined from shear tests of bonded joints. e The discrete crack model (using interface elements) was used for modelling the observed cracks, and the rotating smeared crack model was used for the remaining concrete. b

c

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S.S. Zhang, J.G. Teng / Engineering Structures 75 (2014) 550–560 Table 2 Details of RC beams strengthened with externally bonded FRP reinforcement. Source

Specimen

b (mm)

h (mm)

h0 (mm)

a (mm)

av (mm)

L (mm)

fc (MPa)

Ec (GPa)

[44] [45]

M B2 B4 A2b A2c 1Au 1Bu 1B2u 2Bu 1Cu 2Cu 2Au 3Au 3Bu 3Cu B1u, 1.0 B2u, 1.0 B1u, 2.3 A3 A8 C2 5 DF.2 DF.3 DF.4 AF4 3 4 5 B5 B6 1B 3B E5b 1T6LN 2T6LN 2T4LN

152 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 130 150 150 150 150 125 125 125 125 115 115 115 200 200 154 151 140 150 150 150

305 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 230 300 300 300 250 225 225 225 225 150 150 150 150 150 250 250 260 200 200 200

251 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 206 250 250 250 205 193 193 193 193 119 119 119 115 115 215 215 224 162 162 162

0 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 20 40 0 0 0 50 50 50 50 50 75 75 75 85 85 25 50 150 20 20 20

914 300 300 300 300 300 300 300 340 300 340 340 400 400 400 300 300 844 1065 1065 1065 500 500 500 500 500 500 500 500 750 750 500 500 700 500 500 500

2438 900 900 900 900 900 900 900 900 900 900 900 900 900 900 900 900 2200 2130 2130 2130 1500 1500 1500 1500 1500 1350 1350 1350 2100 2100 1500 1500 2300 1500 1500 1500

43 42.4 42.4 33.6 33.6 47.3 47.3 47.3 47.3 47.3 47.3 47.3 47.3 47.3 47.3 43.2 37.6 37.6 51.7 51.7 51.7 40.6 46 46 46 41 30.3 30.3 30.3 49.2 49.2 31.5 45.3 53.7 47.8 62.1 62.1

31 30.8 30.8 27.4 27.4 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 32.5 31.1 29 29 34 34 34 30.2 32.1 32.1 32.1 30.3 26 26 26 33.2 33.2 26.6 31.8 34.7 32.7 37.3 37.3

[46] [47] [48]

[49]

[50]

[51]

[52] [42]

[17] [53] [54] [55]

Note: b = width of beam; h = height of beam; h0 = effective depth of beam; a = distance from the plate end to the nearest support; av = shear span of beam; L = span of beam; fc = cylinder compressive strength of concrete; and Ec = elastic modulus of concrete.

Table 3 Details of RC beams strengthened with NSM FRP reinforcement. Source

Specimen

b (mm)

h (mm)

h0 (mm)

a (mm)

av (mm)

L (mm)

fc (MPa)

Ec (MPa)

[13]

B500 B1200 B1800 S2 S3 B2 B5

150 150 150 120 120 150 150

300 300 300 170 170 250 250

256 256 256 146 146 206 206

1200 900 600 60 60 100 100

1200 1200 1200 300 300 950 950

3000 3000 3000 900 900 2500 2500

35.2 35.2 35.2 52.2 52.2 50 50

28.1 28.1 28.1 34.2 34.2 33.4 33.4

[14] [15]

elements. All the elements employ a full Gauss integration. A typical mesh of an FRP-strengthened RC beam is shown in Fig. 2. Only one half of the test beam is modelled by taking advantage of symmetry. The side length of most concrete elements is around 15 mm based the results of a convergence study, with appropriate subdivisions in the vertical direction for the concrete under the level of steel tension bars to ensure that at least three elements are employed for the concrete cover. Horizontal displacements on the line of symmetry and vertical displacements at the location of support are prevented. The load is applied using displacement control. 4.3. Constitutive modelling of concrete 4.3.1. Modelling of cracking In the proposed FE approach, the cracking of concrete is simulated using the orthogonal fixed smeared crack model available

in the general-purpose FE package MSC.MARC [23], which allows the formation of two orthogonal cracks in two-dimensional elements. The crack band concept [25] is also adopted with the fracture energy being that given by CEB-FIP [26]. 4.3.2. Yield and failure surfaces The yield surface proposed by Buyukozturk [27] (Eq. (1)) with the associated flow rule is adopted in the proposed FE approach. The failure surface adopted is a combination of Buyukozturk’s model [27] (Eq. (1)) and the maximum tensile stress criterion (Eq. (2)). The above yield surface and failure surface are available in the general purpose FE package MSC.MARC [23] and are given by the following equations:

pffiffiffi 3vr I1 þ nI21 ¼ ðr Þ2 pffiffiffipffiffiffiffi f ðI1 ; J 2 ; hÞ ¼ 2 3 J 2 cos h þ I1  3f t ¼ 0

3J 2 þ

ð1Þ ð2Þ

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Table 4 Details of steel bars and FRP plates in RC beams strengthened with externally bonded FRP reinforcement. Source

[44] [45] [46] [47]

[48]

[49]

[50] [51]

[52] [42]

[17] [53] [54] [55]

Specimen

M B2 B4 A2b A2c 1Au 1Bu 1B2u 2Bu 1Cu 2Cu 2Au 3Au 3Bu 3Cu B1u, 1.0 B2u, 1.0 B1u, 2.3 A3 A8 C2 5 DF.2 DF.3 DF.4 AF4 3 4 5 B5 B6 1B 3B E5b 1T6LN 2T6LN 2T4LN

Ef (GPa)

ff (MPa)

tf (mm)

bf (mm)

Eten (GPa) s

117.9 49 49 49 49 111 111 111 111 111 111 111 111 111 111 111 111 115 203.3 203.3 203.3 19.7 240 240 240 240 230 230 230 127 127 271 257 213 235 235 235

1489 1078 1078 1078 1078 1273 1273 1273 1273 1273 1273 1273 1273 1273 1273 1414 1414 1284 3400 3400 3400 259 3500 3500 3500 3500 3400 3400 3400 1532 1532 3720 4519 3900 4200 4200 4200

1.29 1.2 1.6 1.2 1.2 0.5 0.7 0.7 0.7 1 1 0.5 0.5 0.7 1 0.82 0.82 1.28 0.495 0.99 0.495 2.64 0.334 0.501 0.668 0.334 0.222 0.333 0.444 1.2 1.2 0.33 0.33 1.17 0.66 0.66 0.44

152 80 60 80 80 90 65 65 65 45 45 90 90 65 45 67 67 90 150 75 150 150 75 75 75 75 115 115 115 150 150 150 147 100 150 150 150

200 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 220 207 207 207 231 185 185 185 185 183.6 183.6 183.6 200 200 207 207 192 200 200 200

fyten

Aten s (mm2)

(n  /)ten

(GPa) 414 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 556 427 427 427 537 568 568 568 568 534 534 534 575 575 506 506 504 460 460 460

63.3 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 84.8 236 99.3 99.3 99.3 157 151 151 151 101 236 236 236 157 157 157 157 339 157 157 157

2–6.4 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–6 3–10 2–8 2–8 2–8 2–10 3–8 3–8 3–8 2–8 3–10 3–10 3–10 2–10 2–10 2–10 2–10 3–12 2–10 2–10 2–10

Ecom s (GPa)

fycom

Acom s (mm2)

(n  /)com

(GPa)

Evc (GPa)

fvy (GPa)

Avc (mm2)

sv (mm)

200 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 220 207 207 207 231.0 195.0 195.0 195.0 195.0 183.6 183.6 183.6 200.0 200.0 207.0 207.0 192.0 200.0 200.0 200.0

NA 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 556 427 427 427 537 553 553 553 533 534 534 534 575 575 506 506 504 460 460 460

0.0 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 84.8 84.8 157 397 397 397 157 56.5 56.5 56.5 56.5 157 157 157 101 101 157 157 226 101 101 101

NAa 2–6 2–6 2–6 2–6 2–6 2–6 2–6 2–6 2–6 2–6 2–6 2–6 2–6 2–6 3–6 3–6 2–10 2–15.9 2–15.9 2–15.9 2–10 2–6 2–6 2–6 2–6 2–10 2–10 2–10 2–8 2–8 2–10 2–10 2–12 2–8 2–8 2–8

200 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 215 220 207 207 207 231 185 185 185 185 211 211 211 200 200 207 207 204 200 200 200

414 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 427 427 427 537 553 553 553 553 365 365 365 250 250 506 506 334 250 250 250

99.0 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 14.1 56.5 143 143 143 157 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 157 157 157 101 101 101

102 50 50 100 100 51 51 51 51 51 51 51 51 51 51 51 51 150 125 125 250 100 100 100 100 71 60 60 60 75 75 100 100 125 75 75 75

Note: Ef = elastic modulus of FRP; ff = tensile strength of FRP; tf = thickness of FRP plate; bf = width of FRP plate; Eten = elastic modulus of steel tension bars; fyten = yield stress of s com steel tension bars; Aten = elastic modulus of steel compression bars; s = total cross-sectional area of steel tension bars; (n  /)ten = (number-diameter) of steel tension bars; Es fycom = yield stress of steel compression bars; Acom = total cross-sectional area of steel compression bars; (n  /) = (number-diameter) of steel compression bars; Evc = elastic com s modulus of stirrups; fvy = yield stress of stirrups; Avc = total cross-sectional area of stirrups; sv = spacing of stirrups. a No compressive steel reinforcement was used.

Table 5 Details of steel bars and FRP strips in RC beams strengthened with NSM FRP reinforcement. Source Specimen Ef (GPa)

ff tf hf (MPa) (mm) (mm)

Eten s (GPa)

fyten

Aten s (mm2)

(n  /)ten Ecom s (GPa)

fycom (GPa)

Acom s (mm2)

(n  /)com

(GPa)

Evc (GPa)

fvy (GPa)

Avc (mm2)

sv (mm)

[13]

2068 2068 2068 2740 2740 2580 2500

210 210 210 200 200 203 203

532 532 532 627 627 530 530

226 226 226 66.4 99.5 226 226

2–12 2–12 2–12 2–6.5 3–6.5 2–12 2–12

375 375 375 627 627 530 530

101 101 101 66.4 66.4 226 226

2–8 2–8 2–8 2–6.5 2–6.5 2–12 2–12

210 210 210 200 200 200 200

375 375 375 540 540 530 530

101 101 101 56.5 56.5 101 101

100 100 100 80 80 100 100

[14] [15]

B500 B1200 B1800 S2 S3 B2 B5

151 151 151 158.8 158.8 157 153

4 4 4 2.8 4.2 6 5

16 16 16 9.6 9.6 15 20

210 210 210 200 200 203 203

Note: hf = height of NSM FRP strip; tf = thickness of NSM FRP strip.

where I1 and J2 are the first invariant of stresses and the second * invariant of deviatoric stresses pffiffiffi respectively; n, v and r are material constants and equal to 0.2, 3 and r^ =3 respectively; r^ is taken to be f3c according to Ref. [27] when Eq. (1) is used to define the yield surface and taken to be the cylinder compressive strength of concrete, fc, when Eq. (1) is used to define the failure surface; ft is the tensile strength of concrete; and h is the angle of similarity.

4.3.3. Uniaxial compressive stress–strain curve The uniaxial compressive stress–strain curve which was modified by Elwi and Murray [28] from Saenz’s curve [29] is employed in the proposed FE approach:

rc ¼



1 þ R þ EE0s  2

  e e0

E0 e  ð2R  1Þ

 2 e e0

þR

 3 e e0

ð3Þ

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Imposed displacement Horizontal restraint

Vertical restraint

Cohesive element and

Special cohesive-element-pair,

Cohesive element and

FRP element

as shown in Fig. 6

stirrup element

Fig. 2. Typical FE mesh of a beam.

in which

R¼

E0 =Es ðr0 =ru  1Þ ðeu =e0  1Þ2



1 ðeu =e0 Þ

ð4Þ

where r0 and e0 are the maximum stress and the corresponding strain respectively, and are assumed to be the compressive cylinder concrete strength and 0.002 respectively following Chen [30]; and eu and ru are the ultimate strain and the corresponding stress respectively, and are assumed to be 0.0038 and 0.85fc respectively following Hognestad [31]. When the compressive strain of concrete reaches 0.0038, the concrete fails by compressive crushing and the stress drops immediately to zero. E0 is the initial elastic pffiffiffiffi modulus and is defined according to ACI-318 [32]: E0 ¼ 4730 fc when both E0 and fc are in MPa; and Es is the secant modulus at the maximum stress: Es = r0/e0. In the present study, the cylinder compressive strength fc is estimated from the cube compressive strength fcu using Eq. (5) when fc is not directly found from tests:

fc ¼ 0:8f cu

ð5Þ

4.3.4. Tension-softening curve The tensile strength of concrete ft (MPa) is calculated from the cylinder compressive strength fc (MPa) using the formula proposed by CEB-FIP [26] (Eq. (6)).

 2 fc  8 3 ft ¼ 1:4 10

ð6Þ

The exponential tension-softening curve of concrete in tension proposed by Hordijk [33] (given by Eq. (7) and shown in Fig. 3) is used to represent the tensile behaviour of concrete.

  3 #  6:93ww w w ð6:93Þ 0 e ¼ 1 þ 3:0 e  10 w0 w0 ft "

rt

ð7Þ

where w is the crack width (i.e. crack opening displacement), and w0 is the crack width at the complete release of stress or fracture energy. The value of w0 (lm) can be related to the tensile fracture energy Gft (N/m) as w0 = 5.4Gft/ft for the exponential softening curve, based on the following relationship:

Z

rt dw ¼ Gft

ð8Þ

The tensile fracture energy Gft (Eq. (9)) proposed by CEB-FIP [26] is used in this study:

  f 0:7 c Gft ¼ 0:0469D2a  0:5Da þ 26 10

ð9Þ

where Da (mm) is the maximum aggregate size and fc is in MPa. The unloading and reloading curves of the tensile stress–crack opening displacement relationship follow the linear path that passes through the origin of the coordinate system as shown in Fig. 3. 4.3.5. Shear retention factor model The shear retention factor reflects the shear stress–strain (or slip) relationship after the cracking of concrete and significantly influences the predicted behaviour of cracked concrete. In the present study, the shear modulus Gcr (MPa) of concrete after cracking is deduced from the shear stress–slip relationship of Okamura and Maekawa [34] (Eq. (10)) with the shear retention factor bs as defined by Eq. (11). This shear stress–slip relationship was shown to be the best among the six models studied by Teng et al. [35] for the modelling of NSM FRP-to-concrete bonded joints. 2

Gcr ¼

bs ¼ Fig. 3. Exponential tension-softening curve for concrete.

bcr ecr nt w2

þ D2

Gcr G0 þ Gcr

3:8fc1=3

ð10Þ

ð11Þ

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cr where fc is in MPa; D ¼ bcr ecr nt is the crack slip; w ¼ bcr enn is the pcrack ffiffiffi width; bcr is the crack band width which was chosen to be 2e in the present study (e is the square root of the element area); ecr nt and ecr nn are the shear strain and the normal strain of the crack respectively; and G0 (MPa) is the shear modulus of un-cracked concrete.

s1 = s2 = 0.6 mm and s3 = 1.0 mm for deformed steel pffiffiffiffiffi bars; s1 = s2 = and s3 = 0.1 mm for plain steel bars; ssmax ¼ 2 fck (MPa) p ffiffiffiffiffi ssf ¼ 0:5ssmsx (MPa) for deformed steel bars; and ssf ¼ ssmsx ¼ 0:3 fck (MPa) for plain steel bars. For RC beams strengthened with externally bonded FRP, Lu et al.’s simplified bond–slip model [36] between FRP and concrete is employed:

4.4. Constitutive modelling of FRP and steel bars

sEB

ffi 8 EB qffiffiffiffi s s > > < max sEB 0   ¼ > s 1 a EB > : s 0 sEB max e

for s 6 sEB 0 ð13Þ

The FRP is modelled as an elastic isotropic brittle material with its stress–strain curve as shown in Fig. 4. The steel bars including the steel tension bars, steel compression bars and the stirrups are modelled as an elastic-perfectly plastic material, with its stress–strain curve as shown in Fig. 5.

sEB max ¼ 1:5bw ft

ð14Þ

4.5. Bond–slip models for interfacial cohesive elements

sEB 0 ¼ 0:0195bw ft

ð15Þ

The bond–slip model between steel bars and concrete proposed by CEB-FIP [26] is employed in the proposed FE approach:

8  u s s > > > smax s1 > > < s

for s 6 s1

1

a¼

GEB f

s

EB sEB max 0

ð16Þ  23

2 GEB f ¼ 0:308bw

for s1 < s 6 s2 s ¼ smax s s s ss2 > > s  ð s  s Þ for s2 < s 6 s3 > max max f s3 s2 > > : s sf for s > s3 s

ð12Þ

where ss (MPa) is the local shear bond stress; s (mm) is the slip; u = 0.4 for deformed steel bars and 0.5 for plain steel bars; σ σ max

pffiffiffi ft

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u2:25  bbf c bw ¼ t b 1:25 þ bcf

ð17Þ

ð18Þ

EB where sEB max (MPa) and s0 (mm) are the maximum shear bond stress and the corresponding slip respectively; GEB f (N/mm) is the interfacial fracture energy for externally bonded FRP; bf and bc are the width of FRP plate and the beam respectively; and bw is referred to as the width ratio factor. For beams strengthened with NSM CFRP strips, the following bond–slip relationship proposed by Zhang et al. [37] (Eq. (19)) is used:

sNSM ¼ A ε

for s >

sEB 0

 2   2B  s p 2B  s with s 6 2B sin  B B 2

ð19Þ

A ¼ 0:72c0:138 fc0:613

ð20Þ

B ¼ 0:37c0:284 fc0:006

ð21Þ

NSM

− σ max

Fig. 4. Stress–strain curve for FRP.

σ σy

ε

where s (MPa) and s (mm) are the bond stress and the slip between FRP and concrete respectively; and c is the groove height-to-width ratio. In the proposed FE approach, the bond–slip models given above for interfacial shear behaviour are represented using cohesive elements. The behaviour of the cohesive elements between FRP and concrete in the normal direction are defined as follows: (1) if the force between FRP and concrete is compressive, the stiffness of the cohesive elements in the normal direction is infinite; (2) if the force between FRP and concrete is tensile and debonding in the shear direction has not happened, the stiffness of the interfacial elements in the normal direction is infinite; and (3) if the force between the FRP and the concrete is tensile but debonding in the shear direction has happened, the stiffness of the cohesive elements in the normal direction is zero. 4.6. Cohesive-element-pairs

−σy Fig. 5. Stress–strain curve for steel bars.

To represent the effect of radial stresses generated by the steel tension bars on the surrounding concrete, a special cohesive-element-pair (CEP) is devised (Fig. 6). The CEP consists of two 4-node cohesive elements each of which connects two nodes of the plane stress element representing the adjacent concrete to the two nodes of the beam element representing the steel tension bar or bars

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Cohesive

Concrete

element pair

element 

Shear force (kN)

Steel element Fig. 6. Cohesive-element-pair (CEP).

located at the mid-height of the concrete element. The upper cohesive element is employed to simulate the shear bond behaviour between the steel tension bars and its surrounding concrete, while the lower cohesive element simulates the development of the vertical interfacial force fr invoked by the radial stresses, as expressed by Eq. (22). Before any deformation occurs, the CEP overlaps with the corresponding concrete element whose height is equal to the actual diameter of the steel tension bars which are represented by a beam element located at the mid-height of the concrete element. When slips between the steel bars and the surrounding concrete occur, the upper cohesive element will deform in shear to represent the shear bond–slip response and does not experience any deformation in the normal (vertical) direction. In the meantime, from the shear bond force generated in the upper cohesive element, the lower cohesive element will produce an associated normal (vertical) force that simulates the effect of radial stresses generated by the steel tension bars; the lower cohesive element does not possess any shear stiffness or strength. The choice of the angle hsr (Fig. 7) is an important aspect of the present FE approach. This angle has been studied by a number of researchers. Ferguson and Briceno [38] recommended a value of 45° for simplicity. Tepfers [39] found that the angle to vary between 30° and 45° from an FE study. Orangun et al. [40] concluded that the angle ranged from 38° to 53° by analysing test data. Tepfers and Olsson [41] conducted a batch of ring tests and reported that the angle was between 30° and 65°. In the proposed FE approach, the lower-bound value of 30° is used, considering that flexural cracks may weaken the integrity of the concrete cover thus the normal stresses generated by slip actions of the steel bars. As shown in Fig. 7, assuming that the radial stress and the shear bond stress are both uniformly distributed around the circumference of a steel bar, the interfacial force generated by a single steel bar in the vertical direction per unit length is

fr ¼ Dsr ¼ Dss tan hsr



160WHVWEHDP>@ 160EHDP)(0ZLWKWKHUDGLDOVWUHVVHIIHFW 160EHDP)(0ZLWKRXWWKHUDGLDOVWUHVVHIIHFW (%WHVWEHDP>@ (%EHDP)(0ZLWKWKHUDGLDOVWUHVVHIIHFW (%EHDP)(0ZLWKRXWWKHUDGLDOVWUHVVHIIHFW



 

θsr

τs

τr





5. Results and comparisons 5.1. Ultimate shear forces The FE modelling work began with a convergence study for two of the selected specimens. One beam was strengthened with externally bonded FRP (Beam 3 tested by Maalej and Bian [42]) and another beam with NSM CFRP strips (Beam-1800 tested by Teng et al. [13]). Three sizes of concrete elements were examined: 30 mm, 15 mm and 5 mm, with necessary subdivisions in the vertical direction for the concrete under the level of steel tension bars to ensure that at least three elements were employed across the thickness of the concrete cover. In the convergence study, the three finite element meshes examined were determined as follows: (1) the element width for concrete was equal to a chosen characteristic element width (30 mm, 15 mm or 5 mm) except for the column of elements nearest the vertical support for which a different element width had to be employed when the beam half-span was indivisible by the characteristic element width; (2) the element height for concrete was the same or similar to the element width except for the concrete occupying the same height as the longitudinal steel bars or in the cover layers; (3) the concrete occupying the same height as the longitudinal steel bars was represented



\ [ :LWKWKHUDGLDOVWUHVVHIIHFW :LWKRXWWKHUDGLDOVWUHVVHIIHFW



Test shear force (kN)

τt



Fig. 8. Shear force–deflection curves.

r

τr



Deflection at mid-span (mm)

ð22Þ

where D is the diameter of the steel bar, and s is radial normal stress generated by the steel bar. In the present study, the shear retention factor model mentioned above and the CEP were implemented as user subroutines of MSC.MARC [23].











 











Predicted shear force (kN) Fig. 7. Bond stresses between steel and concrete.

Fig. 9. Test versus predicted shear forces at failure.



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by a single row of elements with their height equal to the steel bar diameter; and (4) each concrete cover layer was divided by the chosen characteristic element width to find a suitable integer for the number of element rows, but at least 2 rows of elements were employed to represent each cover layer. From the convergence study, it was found that there were negligible differences among the selected element sizes in terms of load–deflection curves, indicating that the FE results were not sensitive to element sizes within the range. This was mainly due to the fact that the concrete fracture energy was taken into account while the tensile behaviour and the shear behaviour of cracked concrete were modelled based on the relationship between shear stresses and crack slips as well as crack widths but not the relationship between shear stresses and crack strains. Finally, the element size of 15 mm was selected for use in the subsequent FE analyses whose results are reported below, considering that it led to accurate load–deflection curves, without the need for excessive computing time. The shear force–deflection curves of the two selected specimens obtained from FE analyses and tests are compared in Fig. 8, while a

comparison of the failure shear forces between FE analyses and tests for all the collected specimens are given in Fig. 9 and Table 6. Fig. 8 shows that when the radial stress effect is included in the FE analysis, the predicted shear force–deflection curve agrees closely with the test results for both beams, with the failure shear force from FE analysis being a little higher. If the effect of radial stresses is ignored, the FE analysis significantly over-estimates the failure shear force and the corresponding deflection. This indicates that the radial stress effect is an important factor for obtaining accurate numerical predictions. As can be seen from Fig. 9 and Table 6, the FE models that account for the radial stress effect give close predictions of the ultimate shear forces from tests, with an average prediction-to-test ratio of 1.06, a standard deviation (STD) of 0.118, and a coefficient of variation (CoV) of 0.111 respectively. On the contrary, the FE models in which the radial stress effect is ignored overestimate the ultimate shear force substantially, with an average prediction-to-test ratio of 1.30, an STD of 0.275, and a CoV of 0.211 respectively. These comparisons further demonstrate the

Table 6 Comparison of ultimate shear forces. Source

Specimen

Shear force from test (kN)

FE prediction without radial stresses (kN)

FE prediction/test

FE prediction with radial stresses (kN)

FE prediction/test

[44] [45]

M B2 B4 A2b A2c 1Au 1Bu 1B2u 2Bu 1Cu 2Cu 2Au 3Au 3Bu 3Cu B1u, 1.0 B2u, 1.0 B1u, 2.3 A3 A8 C2 5 DF.2 DF.3 DF.4 AF4 3 4 5 B5 B6 1B 3B E5b 1T6LN 2T6LN 2T4LN B500 B1200 B1800 S2 S3 B2 B5

72.1 17.0 17.5 18.4 18.7 19.8 18.3 18.2 17.0 16.0 17.8 19.3 19.5 17.3 15.4 18.3 16.0 50.2 86.1 98.2 79.3 79.4 60.3 60.0 62.8 55.5 43.0 41.0 39.5 34.9 34.8 66.8 65.3 63.2 58.1 68.0 66.7 23.9 31.6 45.9 46.7 48.3 67.1 68.1

89.9 26.9 27.5 28.6 28.6 30.7 28.1 28.1 26.0 25.0 30.3 32.9 25.4 31.1 22.6 24.6 24.6 63.3 98.6 115 90.9 97.5 71.2 70.5 69.9 72.1 49.4 49.7 50.3 46.6 46.6 82.2 85.3 79.6 66.8 73.1 72.0 27.5 40.4 53.6 57.0 59.3 76.0 79.2

1.25 1.58 1.57 1.55 1.53 1.55 1.54 1.54 1.53 1.56 1.70 1.70 1.30 1.80 1.47 1.34 1.54 1.26 1.15 1.17 1.15 1.23 1.18 1.18 1.11 1.30 1.15 1.21 1.27 1.34 1.34 1.23 1.31 1.26 1.15 1.08 1.08 1.15 1.28 1.17 1.22 1.23 1.13 1.16

74.5 20.5 20.2 17.9 17.9 18.5 19.0 19.0 17.8 18.1 19.2 21.1 20.8 18.5 16.6 18.6 18.6 49.5 89.9 103 76.5 80.8 60.4 59.5 58.9 57.8 45.4 46.3 47.1 36.4 36.4 74.2 69.3 69.2 60.6 69.9 67.2 26.2 33.8 48.8 49.8 52.3 61.8 63.4

1.03 1.21 1.15 0.973 0.957 0.934 1.04 1.04 1.05 1.13 1.08 1.09 1.07 1.07 1.08 1.02 1.16 0.986 1.04 1.05 0.965 1.02 1.00 0.992 0.938 1.04 1.06 1.13 1.19 1.04 1.05 1.11 1.06 1.09 1.04 1.03 1.01 1.10 1.07 1.06 1.07 1.08 0.921 0.931

[46] [47]

[48]

[49]

[50] [51]

[52] [42]

[17] [53] [54] [55]

[13]

[14] [15] [15] Statistic characteristics

Average = STD = CoV =

1.30 0.275 0.211

1.06 0.118 0.111

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559

Test

Without the radial stress effect

With the radial stress effect

(a) RC beam with externally bonded FRP reinforcement (Beam 3 in Ref. [42])

Test

Without the radial stress effect

With the radial stress effect

(b) RC beam with NSM FRP reinforcement (Beam-1800 in Ref. [13]) Fig. 10. Crack patterns at failure: (a) RC beam with externally bonded FRP; and (b) RC beam with NSM FRP.

importance of including the effect of radial stresses from steel tension bars.

In Fig. 10, crack patterns at the ultimate state of the two selected specimens obtained from FE analyses are compared with those from tests. For each specimen, both the crack pattern obtained from the FE model including the radial stress effect and that ignoring the radial stress effect are shown. It can be seen from Fig. 10 that the crack patterns are closely predicted by FE analysis when the radial stress effect is taken into account; by contrast, the crack patterns obtained from FE analysis ignoring the radial stress effect are far different from those from tests.

In the proposed FE approach, several important aspects for accurate predictions are considered: (1) the tensile and shear behaviour of cracked concrete are closely represented; (2) the bond–slip response between steel tension bars and concrete as well as that between steel stirrups and concrete are properly modelled; (3) the bond–slip responses between FRP and concrete in both the shear and the normal directions are properly considered; (4) the critical debonding plane at the level of steel tension bars is given special attention; and (5) most importantly, the radial stresses exerted by steel tension bars onto surrounding concrete are included in the FE approach. Numerical results obtained using the proposed FE approach suggest for the first time ever that these radial stresses play an important role in cover separation failure and need to be taken into account to achieve accurate predictions.

6. Concluding remarks

Acknowledgements

This paper has presented a novel 2-D nonlinear FE approach for predicting end cover separation failures in RC beams strengthened in flexure with either externally bonded or NSM FRP reinforcement.

The authors are grateful for the financial support received from the National Basic Research Program of China (‘‘973’’ Program) (Project No.: 2012CB026201) and the Research Grants Council of

5.2. Crack patterns

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the Hong Kong Special Administrative Region (Project Nos.: PolyU 5173/04E and PolyU 5315/09E). They are also grateful to Professor Jian-Fei Chen, Queens University Belfast, UK for his helpful comments on the work.

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