Proceedings of the International Symposium on Bond Behaviour of FRP in Structures (BBFS 2005) Chen and Teng (eds)

© 2005 International Institute for FRP in Construction

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A NUMERICAL PREDICTION METHOD FOR FLEXURAL BEHAVIOR OF RC BEAMS REINFORCED WITH FRP SHEET

G.F. Zhang 1, N. Kishi 1, H. Mikami 2, and M. Komuro 1 1 Dept. of Civil Engineering and Architecture, Muroran Institute of Technology, Japan

Email:zhang@news3.ce.muroran-it.ac.jp 2 Sumitomo Mitsui Construction Co., Ltd., Japan.

ABSTRACT Nonlinear finite element (FE) method incorporating with the discrete crack model is a rational research approaches for predicting the load-carrying capacity and failure behavior of reinforced concrete (RC) beams reinforced in flexure with bonding fibre reinforced polymer (FRP) sheet on the tension-side surface. In case applying this approach to simulate the geometrical discontinuities such as crack opening and FRP sheet debonding, the profile and location of discrete cracks must be decided at first. However, it is impossible to decide the profile and location of discrete cracks for RC beams, for which no cracking information is available beforehand. In this paper, in order to develop a simply discredizing method to model the real cracking condition for FE analysis, four-point loading tests for twenty-three RC beams were conducted and a general discrete cracks pattern was proposed. Applying the discrete cracks pattern proposed here, it was confirmed that the load-carrying capacity and failure mode of the RC beams can be better predicted numerically. KEYWORDS FRP, RC beams, flexural behavior, failure mode, nonlinear FE analysis, discrete crack model. INTRODUCTION Fibre reinforced polymer (FRP) sheet bonding method has been well known as one of the useful reinforcing method for the existing reinforced concrete (RC) members. The load-carrying capacity of the RC members can be easily enhanced in flexural capacity by bonding FRP sheet on to the tension-side surface. In case of applying this reinforcing method, RC beams reach the ultimate state with the following failure modes corresponding to the reinforcing volume and length of the FRP sheet such as: 1) sheet rupture; 2) concrete compression failure; 3) FRP sheet peel-off failure due to widening of a critical diagonal crack (CDC) initiated in the equi-shear span near loading point; and 4) concrete cover delamination failure from the sheet-end along the main rebar (Triantafillou and Plevris 1992; Buyukozturk and Hearing 1998; Kishi et al. 2001; Teng et al. 2004). These failure modes are illustrated as shown in Fig. 1. When the RC beams reach the ultimate state with the first two failure modes, load-carrying capacity of the beams can be easily estimated by using the conventional analysis method assuming a plane conservation rule and FRP sheet being bonded perfectly up to the ultimate state. However, when the RC beams reach ultimate state with the other two failure modes, the conventional analysis method can not be applied because FRP sheet will be debonded and/or the lower concrete cover will be delaminated before concrete in compression side reaching the ultimate state.

Figure 1 Schematic diagram for failure modes

In order to establish a rational design procedure for reinforcing method with FRP sheet, an evaluation method for the load-carrying capacity of FRP sheet reinforced RC beams must be specified for the case reaching the

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ultimate state with any failure mode described above. For this purpose, Kishi et al. (2002) proposed empirical equations for judging the failure mode between concrete compression failure mode and FRP sheet peel-off failure mode and have investigated to develop a restraining method for FRP sheet debonding experimentally. On the other hand, Pesic and Pilakoutas (2003) and Kishi et al. (2005) have tried to analyze RC beams reinforced with FRP sheet in flexure for investigating the failure behavior and estimating the load-carrying capacity. The former and latter have analyzed the RC beams by means of two- and three-dimensional nonlinear finite element (FE) analysis method, respectively. In both researches, smeared and discrete cracking models are employed for considering the effects of cracks. From these results, it is seen that the analytical results give a good agreement with the experimental one. However, in these analyses, dominant cracks were discretized according to the crack patterns obtained from the experiments. Therefore, to really predict the load-carrying behavior of the RC beams, the location of the discrete cracks must be determined under the conditions with no crack information beforehand. From this point of view, in this paper, in order to provide a general discredizing method for finite element analysis to model the real cracking condition with the discrete crack approach in predicting the flexural behavior of RC beams reinforced with FRP sheet in flexure, a general discrete cracks pattern was studied based on the static four-point loading test results for twenty-three RC beams, in which stirrup ratio and stirrup spacing, axial stiffness of FRP sheet and thickness of the lower concrete cover were taken as variables. An applicability of the proposed general discrete cracks pattern was discussed comparing the three-dimensional nonlinear FE analysis results with the experimental ones. DEVELOPMENT OF A GENERAL DISCRETE CRACKS PATTERN

Specimen Details In order to investigate the crack patterns of the RC beams reinforced in flexure with bonding FRP sheet on to the tension-side surface, total twenty-three RC beams were prepared taking three main parameters as variables. These are of three series of experiments: one series by varying stirrup ratio and stirrup spacing (Series A); one series by varying axial stiffness of FRP sheet (Series B); and one series by varying thickness of the lower concrete cover (Series C). Each parameter was varied as listed in Table 1. In Series B, four kinds of FRP sheet were used and the material properties are listed in Table 2. Figure 2 shows the configurations of the RC beams used in this study. All RC beams were reinforced with a 130 mm-wide unidirectional FRP sheet leaving 100 mm between the supporting point and the sheet end.

Table 1 Summary of specification of specimens Dimensions

w h d c ls lm ls/d Beams (mm) (mm) (mm) (mm) (mm) (mm) (mm)

Stirrups (mm)

Fiber type, plies

Axial stiffness

Ef Af (MN) A-1 150 340 300 40 1,200 600 4.0 D10 at 200 AK, 2-plies 8.78 A-2 150 340 300 40 1,200 600 4.0 D10 at 150 AK, 2-plies 8.78 A-3 150 340 300 40 1,200 600 4.0 D10 at 100 AK, 2-plies 8.78 A-4 150 340 300 40 1,200 600 4.0 D10 at 50 AK, 2-plies 8.78 A-5 150 340 300 40 1,200 600 4.0 D13 at 200 AK, 2-plies 8.78 A-6 150 340 300 40 1,200 600 4.0 D6 at 50 AK, 2-plies 8.78

B-1 150 250 210 40 1,050 500 5.0 D10 at 100 C1, 1-ply 10.58 B-2 150 250 210 40 1,050 500 5.0 D10 at 100 C2, 1-ply 4.99 B-3 150 250 210 40 1,050 500 5.0 D10 at 100 AK, 1-ply 4.39 B-4 150 250 210 40 1,050 500 5.0 D10 at 100 AT, 1-ply 2.57 B-5 150 400 360 40 1,050 500 2.9 D10 at 100 C1, 2-plies 21.16 B-6 150 400 360 40 1,050 500 2.9 D10 at 100 C2, 2-plies 9.99 B-7 150 400 360 40 1,050 500 2.9 D10 at 100 AK, 2-plies 8.78 B-8 150 400 360 40 1,050 500 2.9 D10 at 100 AT, 2-plies 5.14

C-1 150 235 210 25 650 500 3.1 D10 at 100 AK, 2-plies 8.78 C-2 150 250 210 40 650 500 3.1 D10 at 100 AK, 2-plies 8.78 C-3 150 270 210 65 650 500 3.1 D10 at 100 AK, 2-plies 8.78 C-4 150 235 210 25 1,050 500 5.0 D10 at 100 AK, 2-plies 8.78 C-5 150 250 210 40 1,050 500 5.0 D10 at 100 AK, 2-plies 8.78 C-6 150 270 210 65 1,050 500 5.0 D10 at 100 AK, 2-plies 8.78 C-7 150 235 210 25 1,450 500 6.9 D10 at 100 AK, 2-plies 8.78 C-8 150 250 210 40 1,450 500 6.9 D10 at 100 AK, 2-plies 8.78 C-9 150 270 210 65 1,450 500 6.9 D10 at 100 AK, 2-plies 8.78

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Figure 2 Configurations of RC beams

Table 2 Mechanical properties of FRP sheets

FRP Sheet Debonding Behavior and General Discrete Cracks Pattern Developments of cracks for all RC beams were investigated in detail. As one example, Photo 1 shows the distribution of cracks developed around the loading point at three levels of mid-span deflection (hereinafter, deflection) for beam A-1. From this photo, it is observed that: (1) at the deflection level of 19.5 mm, flexural crack F1 is initiated around the location of stirrup; (2) at deflection level of 36.85 mm, CDC is initiated and is widened clearly; and (3) at the deflection level of 45.86 mm, FRP sheet is peeled off due to widening of the CDC. These results suggest that the widening of CDC exerts a critical influence on debonding of FRP sheet. The similar FRP sheet debonding behavior was observed in the other beams in spite of the magnitude of stirrup ratio and stirrup spacing, axial stiffness of FRP sheet and thickness of the lower concrete cover.

(1) At deflection of 19.50 mm (2) At deflection of 36.85 mm (3) At deflection of 45.86 mm

Photo 1 Crack distribution at three deflection for beam A-1

Figure 3 Proposed general discrete cracks pattern Figure 4 FE analysis model (A series) Based on these experimental observations, a general discrete cracks pattern for FE analysis was proposed as shown in Fig. 3, in which cracks (i), (ii) and (iii) are specified as: a CDC caused in the equi-shear span near the loading point; a diagonal crack caused in the sheet-end; and splitting crack caused along the main rebar, respectively. Crack (i) is simply defined as that the origin point is fixed at the intersection point P1 between arranging line of stirrup nearest the loading point in the equi-shear span and the centerline in the beam height, and then the crack develops declining in the direction of an angle of 45 degree toward the lower edge. Cracks (ii) and (iii) are not observed in the above experiments but considered to be able to rationally analyze the

Fiber type Mass per unit area(g/m2)

Thicknesstf (mm)

Tensile strengthσf (GPa)

E-modulus Ef (GPa)

Ultimate elongationεfu (%)

AT (Aramid AT-90) 525 0.378 2.35 78.5 2.99 AK (Aramid AK-60) 415 0.286 2.06 118.0 1.75 C2 (Carbon UT70-30) 300 0.167 3.40 230.0 1.48

C1 (Carbon FTS-EA82-2) 340 0.185 2.40 440.0 0.55

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concrete cover delamination failure from the sheet-end along the main rebar. Crack (ii) is developed with an angle of 45 degree from the sheet-end toward the centreline in the beam height. Crack (iii) is distributed along the bottom surface of the main rebar between diagonal cracks (i) and (ii). OUTLINE OF FE ANALYSIS

FE Model Figure 4 shows the mesh geometry for beam A-1 as one example. In this model, a quarter of the RC beam was three-dimensionally modeled with respect to the two symmetrical axes. Main rebar and FRP sheet were modeled using eight-node solid elements and concrete was modeled using eight-node and/or six-node solid elements. Stirrups were modeled using embedded rebar elements assuming perfectly bonded between stirrup and concrete (DIANA, 2000). Discrete cracks were prepared under the consideration of the proposed general discrete cracks pattern. In addition, discrete cracks were also placed around the rebar elements along the whole span to take the slippage of rebar into account. Modeling of Geometrical Discontinuities In order to consider the effects of the geometrical discontinuities such as opening of cracks, slipping of rebar and debonding of FRP sheet, interface elements were applied, which is in pairs of overlapping nodes with a zero thickness. Failure of the interface element is considered by applying a stress-relative displacement relation to the interface element. In this study, three stress-relative displacement models named as discrete cracking model, bond-slip model, and FRP sheet debonding model as shown in Fig. 5 were adopted to consider the effects of the opening of dominant cracks, the slipping of rebar, and the peel-off of FRP sheet, respectively.

(a) Discrete cracking model (b) Bond-slip model (c) FRP sheet debonding model

Figure 5 Stress-relative displacement relations for interface elements Discrete cracking model shown in Fig. 5(a) defines the relationship between normal stress and relative displacement regarding two surfaces of the interface element. In this model, once normal stress exceeds the ultimate tensile strength fct, the stress transmitted through the interface element is released and a crack is initiated. The relationship between tangential stress and relative displacement was assumed to be linear with a stiffness kt as the same as kn until the crack initiated. After the crack opened, the tangential stiffness was assumed to be zero ignoring the interlocking between two surfaces for simplicity. This discrete cracking model was applied to the interface elements placed at the location of cracks (i), (ii), and (iii), as defined above. Bond-slip model was applied to the interface elements arranged around the main rebar elements along the whole span to consider the slippage between rebar and concrete. The relationship between bonding stress τb and relative displacement S for this model was defined with reference to the CEB-FIP model code, 1990, as shown in Fig. 5(b). Relationship between normal stress and normal relative displacement is defined to be linear with a stiffness of 100 MPa/mm, which is the same value as that of the discrete cracking model. FRP sheet debonding model used to simulate debonding behavior of FRP sheet is the model proposed based on the Coulomb friction concept in authors’ previous researches (Kishi et. al. 2005). This model ignores interaction effects between normal tensile stress σd and plane shear stress τd. The criteria for FRP sheet debonding in the normal and plane shear directions are independently defined as follows:

2 2,max ,max d d d d a d t dσ σ τ τ τ τ− −> = + > (1)

where, σd,max is interfacial tensile strength, σd is applied normal stress, τd,max is interfacial shear strength, τd-a and τd-t are applied plane shear stress in axial and transverse direction, respectively. Interfacial tensile strength σd,max

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was assumed to be equal to the tensile strength of concrete ft because of bonding strength is stronger than the tensile strength of concrete, and normal stiffness kn,d was set to be 100 MPa/mm similarly to that for the discrete cracking model mentioned above. Shear stiffness kt,d is assumed to be equal to the normal stiffness kn,d and interfacial shear strength τd,max was estimated using the following equation (Chajes et al.1996):

,max

' (MPa)0.92 d cfτ = (2) Modeling of Materials For the compression region of concrete, a parabola-rectangle model was adopted up to 0.35% strain. In the post peak region, soften branch was defined as linearly declining with 5% of initial stiffness until stress reaching to 0.2 f'c. The von Mises yield criterion with an associated flow rule was adopted. In the tensile region, a linear tension-softening model was applied. The smeared crack model was applied to the concrete elements to incorporate the effects of cracking of concrete into the numerical analysis. Shear modulus of the concrete element after crack opening was assumed to be 1% of the initial shear modulus G of concrete. The stress-strain relation for main rebar and stirrup was assumed as following a bilinear isotropic-hardening model. Here, plastic hardening coefficient was assumed to be 1% of the Young's modulus. Yield of rebar and stirrup was controlled following the von Mises yield criterion. FRP sheet was assumed to be linear elastic material until rupture. Here, unidirectional FRP sheet is assumed to be an isotropic and homogeneous body, because influence of the stiffness in the transverse direction on the flexural behavior of beams is negligible.

Table 3 Comparisons between analytical results and experimental results

APPLICATIONS AND DISCUSSIONS All the beams listed in Table 1 were numerically analyzed using the proposed general discrete cracks pattern. Comparisons between applied load and deflection obtained from the numerical analyses and those from experimental results are listed in Table 3, in which, Py and δy are applied load and deflection at the rebar yielding point, Pmax and δmax are maximum applied load and deflection at the maximum loading point. As one example, the comparison of load-deflection curves for beam A-1 is shown in Fig. 6. From Table 3, it is seen that maximum errors of the predicted numerical results with reference to the experimental ones among all the beams are 12%, 15%, 13% and 25% for Py, δy, Pmax and δmax, respectively. Here, maximum error for δmax is almost twice of that for Pmax. It implies that the maximum deflection is hardly predicted with good agreement because debonding of FRP sheet has already been initiated near ultimate state and the flexural stiffness is decreased to nearly zero level as shown in Fig. 6.

Py (kN) δy (mm) Pmax (kN) δmax (mm) Failure modeBeams Exp. Ana. (Ana./Exp.) Exp. Ana. (Ana./Exp.) Exp. Ana. (Ana./Exp.) Exp. Ana. (Ana./Exp.) Exp. Ana.

A-1 88.4 83.5 (0.94) 11.7 10.1 (0.86) 126.7 123.5 (0.97) 45.9 43.2 (0.94) a a A-2 87.5 85.1 (0.97) 11.1 10.6 (0.95) 127.0 119.0 (0.94) 42.2 39.9 (0.95) a a A-3 87.6 85.3 (0.97) 11.4 10.7 (0.94) 126.2 118.5 (0.94) 40.5 39.5 (0.98) a a A-4 88.2 84.9 (0.96) 11.1 10.6 (0.95) 131.6 123.5 (0.94) 39.8 38.7 (0.97) a a A-5 87.5 84.0 (0.96) 11.5 10.1 (0.88) 124.3 127.6 (1.03) 41.2 38.9 (0.94) a a A-6 87.3 83.3 (0.95) 10.9 10.1 (0.93) 124.2 123.0 (0.99) 37.6 37.7 (1.00) a a

B-1 65.8 68.2 (1.04) 10.8 11.7 (1.08) 74.5 76.8 (1.03) 15.9 17.0 (1.07) b b B-2 62.7 62.2 (0.99) 11.0 11.7 (1.06) 80.9 85.1 (1.05) 33.8 40.1 (1.19) a b B-3 62.6 61.4 (0.98) 10.6 11.7 (1.10) 84.2 89.7 (1.07) 50.9 53.1 (1.04) a a B-4 60.0 59.6 (0.99) 11.1 11.7 (1.05) 82.1 85.5 (1.04) 67.5 69.0 (1.02) a a B-5 133.6 134.3 (1.01) 6.6 6.9 (1.05) 174.4 170.7 (0.98) 13.0 12.8 (0.98) c c B-6 115.3 119.7 (1.04) 6.3 6.9 (1.10) 156.3 164.7 (1.05) 18.1 20.2 (1.12) a a B-7 114.9 115.8 (1.01) 6.3 6.4 (1.02) 159.2 165.5 (1.04) 21.4 24.8 (1.16) a a B-8 108.0 110.3 (1.02) 6.2 6.4 (1.03) 156.2 156.4 (1.00) 29.5 30.2 (1.02) a a

C-1 106.5 96.1 (0.90) 6.4 5.6 (0.88) 149.9 139.8 (0.93) 21.7 20.0 (0.92) a a C-2 111.2 99.1 (0.89) 6.5 5.6 (0.86) 151.9 140.2 (0.92) 19.5 16.3 (0.84) a a C-3 116.7 105.6 (0.90) 6.6 5.6 (0.85) 150.9 143.2 (0.95) 16.1 15.6 (0.97) c c C-4 66.7 61.2 (0.92) 11.5 10.6 (0.92) 90.5 94.9 (1.05) 32.4 40.4 (1.25) a a C-5 68.7 62.9 (0.92) 10.9 10.6 (0.97) 94.4 90.8 (0.96) 32.1 31.4 (0.98) a a C-6 73.7 64.9 (0.88) 11.5 10.1 (0.88) 97.0 93.8 (0.97) 30.8 28.8 (0.94) a a C-7 46.2 43.7 (0.95) 18.3 16.6 (0.91) 68.8 77.4 (1.13) 66.8 83.6 (1.25) a a C-8 47.9 45.4 (0.95) 17.1 16.6 (0.97) 68.0 76.4 (1.12) 62.2 69.6 (1.12) a a C-9 49.5 47.0 (0.95) 17.8 16.1 (0.90) 70.8 74.6 (1.05) 57.0 54.6 (0.96) a a Failure mode: a: Peel off b: Sheet rupture c: Concrete cover delamination and peel-off of FRP sheet

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Figure 6 Comparison of load-deflection curves (A-1) Figure 7 Comparison of failure conditions (B-5, B-8) Figure 7 compares the failure condition at the loading state of immediately before full peel-off of FRP sheet for beam B-5 and delaminating of the lower concrete cover for beam B-8. Here, contour level of 0.35% corresponds to the ultimate tensile strain and the dark gray region with a strain beyond this value represent the distributions of smeared cracks. From the result for Beam B-5, it is seen that CDC widened obviously and peel-off of FRP sheet initiated, as well as initiating of the concrete cover delamination from the sheet end. Result for Beam B-8 shows that peel-off of FRP sheet developed significantly toward to the supporting point at deflection of 30.2 mm. There is no tendency for the initiating of concrete cover delamination because axial strain occurred in the concrete around the sheet end is still far lower than the ultimate tensile strain. Each of the two analytical results shows a similar failure behavior as that obtained from the experiments. Numerically predicted failure modes for all the other beams are listed in Table 3. It can be confirmed that failure mode of the beams considered here were predicted well by applying the general discrete cracks pattern proposed above. CONCLUSIONS A general discrete cracks pattern was studied based on the four-point loading test results for twenty-three beams, to provide a general discredizing method for finite element analysis to model the real cracking condition with the discrete crack approach in predicting the flexural behavior of RC beams reinforced with FRP sheet in flexure. Three-dimensional elasto-plastic finite element analyses incorporating with this general discrete cracks pattern were performed for all the total twenty-three beams. It can be confirmed that the load-carrying capacity and failure mode could be better predicted by applying the proposed general discrete cracks pattern. REFERENCES Buyukozturk, O., and Hearing, B. (1998). “Failure behavior of precracked concrete beams retrofitted with FRP”,

Journal of Composites for Construction, Vol. 2, No. 3, 138-144. CEB-FIP model code 1990, design code. (1993). Thomas Telford, Lausanne, Switzerland. Chajes, M.J., Finch, W.W., Januszka T.F., and Thomson T.A. (1996). “Bond and force transfer of composite

material plates bonded to concrete”, ACI Structural Journal, Vol. 93, No. 2, 208-217. DIANA, release 7.2. (2000). Division of Engineering Mechanics and Information Technology, TNO Building

and Construction Research, Delft, The Netherlands. Kishi, N., Mikami, H., and Kurihashi, Y. (2002). “Experimental study for investigation of load-carrying

behavior and prediction of failure model of RC beams reinforced in flexure with bonding FRP sheet”, Journal of ST, JSCE, 771(V-56), 91-109 (in Japanese).

Kishi, N., Zhang, G.F., and Mikami, H. (2005). “Numerical cracking and debonding analysis of RC beams reinforced with FRP sheet”, Journal of Composites for Construction, Vol. 9, No. 6 (In press)

Pesic, N., and Pilakoutas, K. (2003). “Concrete beams with externally bonded flexural FRP-reinforcement: analytical investigation of debonding failure”, Composites Part B: Engineering, Vol. 34, 327-338.

Triantafillou, T. C., and Plevris, N. (1992). “Strengthening of RC beams with epoxy-bonded fibre-composite materials”, Materials and Structures, Vol. 25, 201-211.

Teng, J.G., Lu, X.Z., Ye, L.P., and Jiang, J.J. (2004) “Recent research on intermediate crack debonding in FRP-strengthened RC beams”, Proceeding of 4th International Conference on Advanced Composite Materials in Bridges and Structures (CD-Rom), July, 2004.