Analysis of cracking behaviour and tension stiffening in FRP reinforced concrete tensile elements

Composites: Part B 45 (2013) 1360–1367

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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb

Analysis of cracking behaviour and tension stiffening in FRP reinforced concrete tensile elements M. Baena ⇑, Ll. Torres, A. Turon, C. Miàs Analysis and Advanced Materials for Structural Design (AMADE), Polytechnic School, University of Girona, Campus Montilivi s/n, 17071 Girona, Spain

a r t i c l e

i n f o

Article history: Received 3 May 2012 Received in revised form 5 July 2012 Accepted 16 July 2012 Available online 24 July 2012 Keywords: A. Glass fibres/carbon fibres C. Analytical modelling D. Mechanical testing E. Surface treatments

a b s t r a c t The differences in mechanical and bond properties of FRP bars when compared to those of traditional steel reinforcement for reinforced concrete (RC) structures may affect the cracking and deformability behaviour of FRP RC members. To analyse this cracking behaviour, the reinforcement strain profile along a reinforcing bar has been experimentally recorded by using a specially manufactured reinforced concrete element in tension, in which the FRP reinforcement was internally strain gauged. Furthermore, a general procedure derived from a cracking analysis based on slip and bond stresses has been used to study the deformability of FRP reinforced concrete elements under tension. The tension stiffening effect is included via the experimental nonlinear bond–slip law obtained from pull-out test of an experimental programme previously published. The comparison between numerical predictions of the reinforcement strain profile along the reinforcing bar during a tensile test and experimental data confirms that the bond-based model adequately reproduces the redistribution of stresses after crack formation. To further prove the model reliability, numerical predictions of the load–mean strain relationship are compared to previously reported experimental data on FRP RC tensile elements and predictions furnished by the models usually adopted for the analysis of FRP reinforced concrete structures. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction To enhance the durability of reinforced concrete (RC) structures, fibre reinforced polymer (FRP) materials have been increasingly used in recent decades. The differences in their mechanical and bond properties, when compared to those of traditional steel reinforcement, may affect the serviceability design of FRP RC structures. Therefore, the correct modelling of their cracking and deformability behaviour is of major importance. The structural behaviour of reinforced concrete members is affected by the interaction between the reinforcing bar and the surrounding concrete. This interaction allows the concrete between cracks to contribute to carry tensile stresses and provides stiffness, which is known as the tension stiffening effect. Therefore, the response of reinforced concrete is strongly dependent on the bond forces that appear between the concrete and the reinforcement, and how these forces are transferred from one material to the other. This bond behaviour is commonly described through a relationship between the bond stress and the relative slip of the bar, which is known as the bond–slip law. Contrary to what happens with steel, commercially available FRP rebars present different geometrical (and mechanical) properties and a different interaction mechanism between ⇑ Corresponding author. Tel.: +34 972 41 95 17; fax: +34 972 41 80 98. E-mail address: [email protected] (M. Baena). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2012.07.026

FRP reinforcement and concrete. As a consequence, the bond between FRP bars and concrete is affected, yielding to different bond–slip laws [1–3]. The cracking behaviour of reinforced concrete structures has been widely studied for traditional steel-reinforced concrete structures [4]. Some models evaluate the deformability of the structure using an effective cross-sectional area (effective moment of inertia for flexural elements) [5,6] which is determined by a combination of the gross area and the cracked area. As an alternative, some other models modify the constitutive equation of either the reinforcing material ([7,8] for steel reinforcement) or the concrete [9–13]. These models are usually dependent on geometrical and mechanical parameters such as reinforcement ratio and modular ratio, among others. However, they cannot be directly extended to FRP RC structures, particularly in the serviceability analysis where FRP materials’ higher deformability influences the RC mechanical behaviour. Furthermore, the nature of these models does not allow the singularities of the interaction between the reinforcing bar surface and the concrete (bond–slip) to be specifically accounted for. However, this interaction can be taken into account using models where member deformability is evaluated by a nonlinear analysis using a specific bond–slip law. Some of these bond models are based on the solution of the system of differential equations, which arise from the analytical study of the problem [14–16], while others use a numerical procedure to solve

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the system of differential equations [17–20]. Although these models include the bond phenomenon, no experimental bond–slip law has been used in their validation. In addition, although internal strain and stress redistributions caused by cracking can be predicted in these later models, no comparison to experimental results is available. In this paper, a general procedure derived from a cracking analysis based on slip and bond stresses is presented. As the model is developed in a bond framework instead of as an extension of a steel RC-aimed model, elements with non-conventional reinforcement can be studied and any user-defined bond–slip law can be assumed. In this paper, bond–slip laws obtained from experimental pull-out tests [1] are used in the validation of the model. Real-time monitoring of strain distribution along the FRP reinforcing bar during an experimental tensile test is used to validate the model. A reinforcement strain monitoring system, with no mechanical interference on the bond zone [21], has been introduced for this kind of non-metallic reinforcement. The experimental strain distributions are thereafter compared to predictions of the presented model. To further prove the reliability of the model, numerical predictions on the load–mean strain relationship, P–em, are compared to those obtained from the models usually adopted for the analysis of FRP structures and from experimental data [22,23]. 2. Nonlinear model for FRP reinforced tensile members 2.1. Cracking behaviour of RC tensile elements Before cracking, stresses and strains in an RC element in tension are uniform along the length of the member. The equilibrium of forces and the compatibility of strains are linked together by assuming linear elastic material behaviour for both the concrete and the reinforcement and perfect bond. Therefore, the applied load is shared between the concrete and the reinforcement in relation to their respective rigidities. At this uncracked stage, no differences between reinforcement and concrete strain exist and therefore no slip occurs. Once concrete tensile strength, fct, is reached, a first primary crack appears. Due to crack formation, the tensile concrete stress at the crack location drops to zero, and a redistribution of reinforcement and concrete strains takes place, and strain compatibility is lost. The applied axial load can be further increased until concrete tensile strength is again attained at any other section and new cracks appear. The process continues until final crack stabilisation.

P

0

srm /2

According to the cracking phenomenon explained above, the theoretical study of RC ties beyond the cracking load is related to the analysis of concrete and reinforcement strain distribution along the RC member, which in turn is based on the solution of the differential equation that describes the cracking behaviour. The rational derivation of the differential equation is thus presented, where the following assumptions have been considered:  The rebar and concrete follow Hooke’s law (serviceability conditions).  No tensile softening behaviour is considered in the constitutive equations for concrete.  Strains, stresses and displacements are examined in the longitudinal direction of the composite element, and these values are assumed not to vary with the radial coordinate. Let us consider a block of an RC member subjected to a uniaxial tensile force P as shown in Fig. 1a.

P srm /2

(a) Block of RC member in tension P

F c /2 Fr F c /2 srm /2-x

(b) Arbitrary section F c /2 F c /2

τ

F c /2+dF c /2

τ

Fr

τ

F c /2+dF c /2

F r+dF r

dx

dx

(c) Concrete and reinforcement interfaces Fig. 1. Free-body diagrams of RC tie.

The x-axis has its origin midway between the two cracks. Due to symmetry, only one half-member is analysed. Let us consider a section at a distance x from the origin, so that the free-body diagram of the right-hand side is shown in Fig. 1b. The equilibrium equation for the block reads:

P ¼ rr ðxÞAr þ rc ðxÞAc

ð1Þ

where P is the applied tensile load, r(x) is the tensile stress at a distance x from the origin and A is the cross-sectional area. Subscripts c and r refer to concrete and reinforcement, respectively. The reinforcement and concrete interfaces related to an infinitesimal element of length dx are shown in Fig. 1c. The equilibrium equations read:

dF c ðxÞ ¼ Ac drc ðxÞ ¼ pr sðxÞdx

ð2Þ

dF r ðxÞ ¼ Ar drr ðxÞ ¼ pr sðxÞdx

ð3Þ

where pr is the perimeter of the bar, s(x) is the bond stress, and dFr(x) and dFc(x) are the axial force in the concrete and in the rebar at a distance x from the origin, respectively. The slip at the reinforcement–concrete interface is defined by:

sðxÞ ¼ ur ðxÞ  uc ðxÞ 2.2. Bond problem formulation

x

ð4Þ

where s(x) is the slip, and ur(x) and uc(x) are the displacements along the x-axis of the reinforcement and concrete, respectively. Assuming a linear elastic constitutive behaviour for both reinforcement and concrete, and substituting Eqs. (2) and (3) into the second derivative of Eq. (4) yields: 2

d sðxÞ 2

dx

¼

sðxÞpr Ar Er

ð1 þ nqÞ

ð5Þ

where n is the modular ratio and q is the reinforcement ratio. Eq. (5) is a differential equation with two unknown functions, s(x) and s(x), which are related through the bond–slip relationship. 2.3. Numerical solution Depending on the complexity of the bond–slip law, a straightforward solution of the differential equation is not always possible. In fact, the analytical solution of the bond problem has been presented previously for those cases assuming a constant bond–slip law [14,24], a linearly dependent bond–slip law [14,25,26] or an

M. Baena et al. / Composites: Part B 45 (2013) 1360–1367

MC 90 ascending branch bond–slip law [15]. As an alternative, a mathematical nonlinear model similar to existing previous works [17–19] is presented in this paper. The stress and strain conditions of a reinforced concrete tie before first cracking can be modelled as in the reinforced concrete block shown in Fig. 2. As long as the applied tensile load P is lower than the first-cracking load, and assuming that the transfer length is lower than half the length of the tie, composite action between the midway section and section C is assured. As the load is increased, the concrete strain increases up to a point where the concrete tensile strength is reached. Due to non-homogeneity in concrete material, a crack can appear in any section between section C and the halfway section. Within the numerical procedure presented, the following assumptions have been considered:  Concrete tensile strength fct is assumed to be constant along the reinforced tie.  Cracks always appear at the halfway section between two cracks. The numerical procedure implemented in the nonlinear model analyses the previous situation as follows. The tie is subdivided in n subintervals with length Dx. The procedure starts at the cracking load level P = Pcrack, when concrete tensile strength fct has been reached (see Fig. 3). Stress and strain at the halfway section are evaluated by the equilibrium equation (Eq. (1)) and by imposing symmetry conditions. Using Eqs. (1), (3) and the first derivative of Eq. (4), the stress and strain distributions at the end of the first subinterval of length Dx, measured from the halfway section, are obtained. The procedure is extended to every contiguous subinterval until the crack section, where rc = 0. Based on stress and strain distributions, transfer length lt can be defined as the distance between the crack section and the section where the composite action is recovered. At this point two situations can be found. If lt < LB/2, the first transversal crack appears and composite action is still assured in the two new blocks. Therefore, new cracking will again take place without an increase in the applied load. However, if lt > LB/2, the first transversal crack forms but composite action is no longer assured. In either case, the reinforced element is defined by a number of identical reinforced concrete blocks, where composite action is no longer possible. Therefore, further analysis is conducted on a representative block. The displacements and strain and stress distributions along the reinforced concrete element will be formed through the concatenation of the concrete block results. For each concrete block, the new origin is located at the halfway section and LB is updated to represent the concrete block halflength. From this point on, load is further increased until rc reaches

P

P

ε r(x)

C

ε c(x) 0

x

P

P

f ct

σ c(x) x

0

lt LB

Fig. 3. Concrete stress distribution when fct is attained at halfway section (unknown lt).

the concrete tensile strength fct at the midway section and new cracks can form. This sequence is successively reproduced until the stabilised cracking phase is reached and no more cracks appear. The characteristic shape of the load–mean strain (P–em) curve of an RC tie obtained using the presented numerical model is shown in Fig. 4. Under load control test conditions, the formation of cracks is represented by a horizontal line in the P–em curve. 2.4. Model capabilities To demonstrate the capabilities of the proposed bond-based model, numerical results of a reinforced concrete tension member are analysed. Consider a 750 mm length RC tie, in which a centrally embedded rebar is located. The rebar diameter is 12 mm and the reinforcing ratio is q = 1.78%. The concrete tensile strength and compressive strength are 3.1 MPa and 22.95 MPa, respectively. The elastic moduli of the reinforcing bar and the concrete are 210,000 MPa and 29,000 MPa, respectively. The BEP bond–slip model [27] is used for the bond–slip law. According to this model, the bond stress–slip of reinforcing bars shows four different branches (see Fig. 5a): initial ascending branch up to the peak bond strength (s1) for s 6 s1, a second branch with a constant bond (s = s1) up to slip s = s2, a linearly descending branch from (s2, s1) to (s3, s2) and a horizontal branch for s > s3, with a value of s due to the development of friction (s = s3). The BEP model expresses the ascending branch of bond–slip relationship as follows:

s ¼ sm



s sm

a ð6Þ

Different values of a, si and si have been observed in experimental results [1,3]. Values a = 0.4, s1 = 1 mm, s2 = 3 mm, s3 = 4 mm (rib spacing), s1 = 2.5(fck)0.5 and s2 = 0.4s1 have been assumed to perform a demonstrative example. These values can be assumed to be representative for good bond conditions. As a direct result of the numerical model, concrete and reinforcement strain distributions along the composite system are

Applied load P

1362

Numerical model Bare bar

s(x) lt

0

x

Pcrack Mean strain ε m

Fig. 2. Strain and slip distribution along a concrete block.

Fig. 4. Load–mean strain curve obtained using the presented numerical model.

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60

τ (MPa)

τ2 s1

s2

s (mm)

s3

Axial Load P (kN)

50

τ1

40 30 20

RC member Bare bar

10 0 0

0.4

Before 1st cracking P=22.26 kN After 1st cracking P=22.26 kN After 2nd cracking P=23.51 kN

0.6 0.4 0.2 300

450

600

1 0.8 0.6 0.4 0.2 0

0

150

300

450

600

750

Distance along tie x (mm) (d) Concrete strain distribution

Distance along tie x (mm) (c) Reinforcement strain distribution

Before 1st cracking P=22.26 kN After 1st cracking P=22.26 kN After 2nd cracking P=23.51 kN

Before 1st cracking P=22.26 kN After 1st cracking P=22.26 kN After 2nd cracking P=23.51 kN

0.06

5

0.03 0 -0.03 -0.06

1.6

1.2

750

Bond stress τ (MPa)

Concrete-rebar slip s (mm)

0.8

150

1.2

Before 1st cracking P=22.26 kN After 1st cracking P=22.26 kN After 2nd cracking P=23.51 kN Concrete strain εc x10-4

1

r

Reinforcement strain ε x10-3

(a) BEP Bond-slip law

0 0

0.8

Elongation δ (mm) (b) Load-elongation relationship

0

150

300

450

600

750

Distance along tie x (mm) (e) Slip distribution

0

-5 0

150

300

450

600

750

Distance along tie x (mm) (f) Bond stress distribution

Fig. 5. (a) BEP bond–slip law; (b) load–elongation relationship of an RC member; (c) reinforcement strain distribution (at cracking load levels); (d) concrete strain distribution (at cracking load levels); (e) slip distribution (at cracking load levels); (f) bond stress distribution (at cracking load levels).

obtained. The integral of the reinforcing bar strain along the length of the tie gives the rebar elongation. The load–rebar elongation relationship of the RC member is shown in Fig. 5b, along with the bare bar response. Due to the transmission of bond forces, a high initial stiffness is obtained at the early cracking stages. However, the stiffness diminishes throughout the cracking process. The first cracking load level is 22.26 kN. According to the numerical model assumptions, the first crack appears at the midway section and divides the member into two submembers of half length. The second cracking level is attained at 23.51 kN. At this load, two more cracks appear at the midway section of every submember. Reinforcing bar strain distribution along the length of the tie before and after the first cracking load and after the second cracking load is shown in Fig. 5c. Likewise, concrete strain distribution, relative slip distribution and bond stress distribution are shown in Fig. 5d–f, respectively. After any crack formation, forces and strains are redistributed so that the reinforcement strain reaches its upper limit value at

every cracked section. Similarly, the concrete strain reaches its lower limit value (ec = 0) at the same cracked sections. As a result of the force redistribution process, there is a relative slip between the concrete and the rebar. Because of the bond–slip behaviour (shown in Fig. 5a), bond forces appear wherever slip exists. Maximum values of both slip and bond stress take place at the cracked section and switch direction at the midway section. 3. FRP reinforcement strain monitoring system The numerical procedure is validated through the comparison with experimental results on the reinforcement strain profile. For that purpose, a tensile test on an FRP RC element was performed (reference 16-170-3N in [22]), where monitoring of the strain distribution along the reinforcing bar was possible. A specially manufactured internally strain gauged reinforcing FRP bar was considered (Fig. 6). The original bars were cut into two halves and smoothed. On one half, strain gauges were placed every 50 mm centres. On

M. Baena et al. / Composites: Part B 45 (2013) 1360–1367

the other half, a groove was cut to accommodate the wiring of the strain gauges. Additional holes were drilled every 100 mm centres to allow the gauge wiring to come out of the bar. After protecting the gauges from possible humidity, the bar was closed, with the two halves glued together, to give the appearance of a normal solid round bar. The bar surface was not modified; therefore bond performance was not altered due to the instrumentation of the bar. Three notches were created to induce cracks at specific locations where the reading of the reinforcement strain was feasible. Experimental reinforcement strain distribution along the tie before and after each crack opening is shown in Fig. 7. Every symbol represents the reading from a strain gauge. Before the first cracking, a peak in reinforcement strain distribution can be found at both ends of the RC tensile element. The value of this peak strain corresponds to that of a fully cracked section. Both ends of the RC tensile element can be analysed as cracked sections and, therefore, no concrete area contributes to the axial stiffness. From this section on, reinforcement strain decreases until composite action is attained. At this point, the reinforcement strain value equals that of an uncracked section. For the applied load (P = 20.60 kN), it is expected that the first crack will appear at x = 350 mm. The first crack is completely created at P = 20.63 kN, and a maximum peak in reinforcement strain distribution is displayed at x = 350 mm. From this load level on, second and third cracking, occurring at P = 26.73 kN and P = 35.16 kN, are captured in reinforcement strain distribution with new peak values at x = 650 mm and x = 950 mm, respectively. This is finally reflected in Fig. 7, where the cracking process and experimental distribution on strains along the member are represented for different levels of load. 4. Comparison with test data 4.1. FRP reinforcement strain distribution The presented model is used to reproduce the cracking behaviour of the aforementioned 16-170-3N specimen [22]. Two simulations were conducted that assumed the bond–slip law for the

4.5

Reinforcement strain εm x10-3

1364

before 1st cracking 20.60kN

after 1st cracking 20.63kN

4

before 2nd cracking 26.73kN

after 2nd cracking 26.73kN

3.5

before 3rd cracking 35.16kN

after 3rd cracking 35.16kN

3 2.5 2 1.5 1 0.5 0

0

350

650

950

1300

Distance along the tie (mm) Fig. 7. Experimental reinforcement strain distribution during crack formation.

loaded and unloaded end (hereafter referred to as BSLE and BSUE, respectively) obtained in the pull-out experimental campaign [1] (see Fig. 8). As the aim is to compare the strain profile along the reinforcing bar, the numerical model has been slightly modified to trigger the cracks at the same location as they appear in the experimental test. In Fig. 9a–f, the reinforcement strain distributions along the tie obtained from the numerical simulations and the experimental test are compared. The agreement between numerical predictions and experimental data proves the validity of the proposed method to predict the contribution of each component (reinforcement and concrete) before and after cracking, therefore describing satisfactorily the cracking behaviour in RC tensile members. 4.2. FRP RC tensile response The numerical model is compared to experimental results obtained from an experimental test programme carried out on GFRP reinforced concrete ties [22], where different reinforcement ratios were used (Table 1). The GFRP reinforcing bars had a helical wrapped surface and some sand coating. Mean values of the mechanical properties obtained from the uniaxial tension tests are shown in Table 2. Ready-mix concrete, with a maximum aggregate size of 20 mm, was used to cast the specimens. The compressive strength and modulus of elasticity were determined from control cylinders (150  300 mm) according to standards. The tensile strength was computed from tests on the GFRP RC tension members. Concrete mechanical properties are summarised in Table 3. The bond–slip laws used in the numerical predictions of the global tensile behaviour are presented in Fig. 10. These bond–slip laws

Bond stress τ (MPa)

18

12

Experimental loaded end (BSLE) 6

0

Experimental unloaded end (BSUE)

0

4

8

12

16

Slip s (mm) Fig. 6. Internally strain gauged reinforcing bar.

Fig. 8. Experimental bond–slip law derived from pull-out test and considered in the validation of the numerical procedure.

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(a)

1 BSLE BSUE exp.

Before crack 1 (8.64kN)

0.6 0.4 0.2

1.5 1 0.5

300

600

900

1200

300

x (mm)

(c)

3 Before crack 2 (26.73kN)

BSLE BSUE exp.

(d)

1200

3 After crack 2 (26.73kN)

BSLE BSUE exp.

2 ε m x10-3

ε m x10-3

900

2.5

2 1.5

1.5

1

1

0.5

0.5

300

600

900

1200

300

x (mm)

600

900

1200

x (mm)

4 Before crack 3 (35.16kN)

BSLE BSUE exp.

(f)

2

1

4 After crack 3 (35.16kN)

BSLE BSUE exp.

3

ε m x10-3

3

ε m x10-3

600

x (mm)

2.5

(e)

BSLE BSUE exp.

After crack 1 (20.63kN)

2 ε m x10-3

ε m x10-3

0.8

2.5

(b)

2

1

300

600

900

300

1200

600

900

1200

x (mm)

x (mm)

Fig. 9. Numerical and experimental reinforcement strain distribution at cracking load levels.

Table 1 Geometric characteristics of the experimental ties [22].

Table 2 GFRP rebars geometrical and mechanical properties [22].

Specimen

Experimental bar diameter, db (mm)

Tie width (mm)

Reinforcement ratio q (%)

13-170 16-170 19-170 16-110 16-170-3N

13.7 16.9 19.1 16.1 19.1

170 170 170 110 170

0.51 0.71 1.00 1.69 1.00

were obtained from an experimental campaign on pull-out tests [1]. Predictions of member tensile behaviour using the proposed numerical model and EC-2 are compared to the experimental responses in Fig. 11. A large plateau after Pcrack is observed in the numerical predictions. This plateau is related to the multicrack formation that takes place at Pcrack when the transfer length, lt, is shorter than LB/2, where LB reads for half the length of the tie, and more than one crack can form. This multicracking is therefore responsible for a significant underestimation of the tension stiffening effect at early cracking stages. Although the assumption of constant concrete tensile strength along the RC tie and crack formation prefixed at midway section between two existing cracks, numerical predictions follow the general trends of the experimental results. These numerical predictions fit with the limiting responses

Specimen

Nominal bar diameter, dn (mm)

Experimental bar diameter, db (mm)

Tensile strength ffu (MPa)

Axial stiffness, ErAr (kN)

13-170 16-170 19-170 16-110 16-170-3N

12.7 15.9 19.1 15.9 15.9

13.7 16.9 19.1 16.1 19.1

770 1030 637 751 917

5540 9362 11,680 7900 10,087

Table 3 Concrete mechanical properties [22]. Specimen

Compressive strength, fc (MPa)

Modulus of elasticity, Ec (MPa)

Tensile strength fct (MPa)

13-170 16-170 19-170 16-110 16-170-3N

48.4 48.1 56.2 56.2 46.6

27,315 27,315 33,275 33,275 34,514

1.75 2.58 2.10 2.34 1.22

obtained from EC-2, using the two limit values for bond coefficient (b1 = 1 for high bond bars and b1 = 0.5 for plain bars).

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M. Baena et al. / Composites: Part B 45 (2013) 1360–1367

20 Diameter 13mm

10

(a) 0 0

10

20

15 Bond stress τ (MPa)

Bond stress τ (MPa)

Bond stress τ (MPa)

20

Diameter 16mm

10

(b) 0 0

30

10

20

10 Diameter 19mm

5

0 0

30

(c) 10

20

Slip s (mm)

Slip s (mm)

Slip s (mm)

Axial Load P (kN)

(a)

180

120

Experimental Bare bar EC2-92 β1=0.5 EC2-92 β1=1 Numerical

60

(b)

200

Axial Load P (kN)

Fig. 10. Experimental bond–slip laws.

150

Experimental Bare bar EC2-92 β1=0.5 EC2-92 β1=1 Numerical

100

50

13-170 ρ =0.51% 0.005

0.01

Member mean strain εm

0.015

0.009

0.0135

(d)

Experimental Bare bar EC2-92 β1=0.5 EC2-92 β1=1 Numerical

60

180

Experimental Bare bar EC2-92 β1=0.5

120

EC2-92 β1=1 Numerical

60

19-170 ρ =1.00% 0.005

0.01

Member mean strain εm

0.018

Member mean strain ε m

180

120

0.0045

Axial Load P (kN)

Axial Load P (kN)

(c)

16-170 ρ =0.71%

16-110 ρ =1.69% 0.005

0.015

0.01

0.015

Member mean strain ε m

Fig. 11. Comparison of P–em numerical predictions with code provisions and experimental results.

Table 4 Properties of RC ties tested in [23].

RS1 RS2 RS3

Rebar diameter dr (mm)

Reinforcement ratio, q (%)

Rebar elastic modulus, Er (GPa)

12.7 15.9 19.1

1.26 1.98 2.86

40.3 41.3 41.5

A comparison between the numerical predictions and experimental results of three GFRP RC tensile tests presented in [23] is also conducted. The ties were 1100 mm long and had a square section with 100 mm sides. The concrete tensile and compressive strengths were 2.6 and 48.8 MPa, respectively. The secant modulus of elasticity (Ecm = 37.28 GPa) has been estimated according to MC90 [28]. For the bond–slip law, the ascending branch of BEP bond– slip model [27] is adopted (see Eq. (6)). This well-known model has previously been successfully applied to FRP rebars [3,29]. In this paper, a = 0.4 and s1 = 1 mm have been assumed. Moreover, a bond strength value of s1 = 17.5 MPa has been adopted, according to: 0:5

s1 ¼ 2:5ðfck Þ

ð7Þ

Experimental

RS3

Numerical 135

Load P (kN)

Specimen

180

Bare bar

RS2 RS1

90

45

0 -2

0

2

4

6

8

10

12

14

16

Average strain x10-3 , εm Fig. 12. Comparison between experimental results presented in [23] and numerical predictions.

In Table 4 the rebar size, dr, of each test is presented, along with the reinforcement ratio, q, and the elastic modulus of the GFRP rebar, Er.

M. Baena et al. / Composites: Part B 45 (2013) 1360–1367

The numerically predicted P–em curves are compared to experimental responses in Fig. 12. An initial offset is applied to the experimental results according to Bischoff’s indication on initial shortening caused by shrinkage effects. It can be observed in Fig. 12 that the responses obtained from the numerical model accurately predict experimental behaviour at all stages of the cracking process.

5. Conclusions A numerical model based on a nonlinear finite difference method is proposed to simulate response of FRP reinforced concrete under uniaxial tension. Within the model, the tension stiffening effect is represented through the bond action appearing between the two materials, defined by a bond–slip law. Owing to the adopted numerical procedure, whatever material and geometrical properties, and nonlinear bond–slip law can be introduced. Furthermore, internal instrumentation along the reinforcing bar of a specially manufactured RC tensile element has demonstrated its efficiency in making reinforcement strain distribution data available throughout the test. With this system, readings on reinforcement strain distribution along the bar can be captured along the loading process of FRP RC tensile elements. In this sense, comparison between experimental data and numerical simulations proves the validity of the proposed model to describe the cracking behaviour of FRP RC tensile members. For the numerical simulation, experimental (pull-out) bond–slip laws have been used. Moreover, the model has been used to estimate the overall deformability of GFRP reinforced concrete ties. Predicted responses are compared to available experimental values and code provisions. Numerical responses in terms of load–mean strain satisfactorily reproduce the experimental results, and are similar to those predicted by EC-2 (CEN 1992). Acknowledgements The authors gratefully acknowledge the support provided by the Spanish Government (Ministerio de Ciencia e Innovación), Project Ref. BIA-2010-20234-C03-02. References [1] Baena M, Torres LI, Turon A, Barris C. Experimental study of bond behaviour between concrete and FRP bars using a pull-out test. Compos Part B-Eng 2009;40(8):784–97. [2] Lee J-Y, Kim T-Y, Kim T-J, Yi C-K, Park J-S, You Y-C, et al. Interfacial bond strength of glass fiber reinforced polymer bars in high-strength concrete. Compos Part B-Eng 2008;39(2):258–70. [3] Cosenza E, Manfredi G, Realfonzo R. Behaviour and modeling of bond of FRP rebars to concrete. J Compos Constr ASCE 1997;1(2):40–51.

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