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Idealized tension stiffening model for finite element analysis of glass fibre reinforced polymer (GFRP) reinforced concrete members
Structures 24 (2020) 351–356
Contents lists available at ScienceDirect
Structures journal homepage: www.elsevier.com/locate/structures
Idealized tension stiffening model for finite element analysis of glass fibre reinforced polymer (GFRP) reinforced concrete members
T
⁎
M.S. Alama, , A. Husseinb a b
Department of Civil Engineering, University of Bahrain, P. O. 32038, Bahrain Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL A1B3X5, Canada
A R T I C LE I N FO
A B S T R A C T
Keywords: Reinforced concrete Shear strength GFRP bar FEA Tension stiffening
The capacity of concrete in carrying tension between two cracks is defined as tension stiffening. This paper presents the development and validation of tension stiffening model for finite element analysis (FEA) of glass fiber reinforced polymer (GFRP) reinforced concrete beams. The model was derived from an existing model for steel reinforced members using the same strain energy density. The proposed model was employed in a commercial finite element analysis program. The FEA results were compared with the test results in terms of loaddeflection behavior, crack patterns, ultimate loads. It was observed that the proposed tension stiffening model was capable of predicting load deflection of shear critical GFRP reinforced concrete beams with reasonable degree of accuracy.
1. Introduction Cracking of concrete has an effect on the behavior of reinforced concrete members. The concrete is assumed to carry no tension force at the cracks. However, in between cracks, the concrete is assumed to carry tension load through bond between the concrete and the reinforcing bars. The ability of concrete to carry tension between cracks and to provide additional stiffness is called tension stiffening [1]. This is an important issue in serviceability limit analysis and plays an important role in nonlinear analysis of reinforced concrete members. If the tension stiffening effect is neglected, the stiffness of a reinforced concrete bar or a structural member is under estimated [2] and this will results in overestimation of deflection. Usually the tension stiffening was overlooked in finite element analysis of concrete structures. Nonetheless, with time it was found that adding tension stiffening effect for reasonable calculation by finite element analysis is essential for actual forecast of the behavior of concrete structures [3]. Tension stiffening of GFRP reinforced concrete members is different from that of steel reinforced concrete members due to their different surface and bond characteristics, low modulus of elasticity. It was found that GFRP reinforced concrete showed more tension stiffening than the steel reinforced concrete [4]. Research have shown that the existing equations to model the tension stiffening effect is inaccurate in predicting the tension stiffening effect of GFRP reinforced concrete elements [5]. Therefore, there is a great need to develop tension stiffening
⁎
model for GFRP reinforced concrete members. Different approaches have been used to propose tension stiffening models. These are based on experimental results [6–8] and based on fracture mechanics principles [9,10]. In the current study, post failure stress-strain relation was used to develop the tension stiffening model. The proposed model was implemented in a finite element program ABAQUS [11], for simulating the shear behavior of full scale GFRP reinforced concrete beams. The FEA results were compared with the experimental data. 2. Background Besides, reinforced concrete members contain transverse reinforcement (stirrups) to resist shear. However, there are some members; such as slabs, footings, abutments, where there is no transverse reinforcement. These members resist shear by concrete. FEA of GFRP reinforced members containing stirrups have shown better performance than without stirrups. Furthermore, it was reported that GFRP reinforced concrete members have more tension stiffening than steel reinforced concrete members [4]. Fig. 1 shows the comparative results obtained from ABAQUS [11] and experiments for both steel and GFRP reinforced concrete beam in shear without stirrups. It should be mentioned here that the same tension stiffening model was used for both steel and GFRP reinforced members. All other modeling parameters were the same. Since the existing tension stiffening model works well for steel
Corresponding author. E-mail address: [email protected] (M.S. Alam).
https://doi.org/10.1016/j.istruc.2020.01.033 Received 7 August 2019; Received in revised form 23 January 2020; Accepted 23 January 2020 2352-0124/ Crown Copyright © 2020 Published by Elsevier Ltd on behalf of Institution of Structural Engineers. All rights reserved.
Structures 24 (2020) 351–356
M.S. Alam and A. Hussein
Nomenclature a b h d a/d k n ρf fc ft fc’ ft’ Af Es Ef
L P V Vexp VFEA ε ε0= εc εm εcr εy εu εpl eel etotal δexp δFEA
shear span width of beam depth of beam effective depth of beam shear span to depth ratio stress decay factor curve fitting factor reinforcement ratio compressive stress of concrete tensile stress of concrete compressive strength of concrete tensile strength of concrete area of reinforcement modulus of elasticity of steel modulus of elasticity of FRP
span length of beam Applied load shear strength experimental shear strength predicted shear strength tensile strain concrete strain at fc’ compressive strain member strain strain at peak tensile stress (cracking strain) yield strain of steel ultimate strain of GFRP plastic strain elastic strain total strain deflection measured from experiment deflection measured from FEA
Fig. 1. Load-displacement behavior of (a) Steel and (b) GFRP reinforced concrete beam.
reinforced members, the same model was used to justify the need (if any) to develop a tension stiffening model for GFRP reinforced beam. Although it was expected that there will be differences in the response of steel and GFRP beams, it is not clearly known what the differences in the responses are. The observation revealed that the GFRP reinforced concrete beam showed poor performance than steel reinforced concrete beam both in cracking load and ultimate load. The underlying cause of this behavior was attributed to the low elastic modulus, differences in bond properties between GFRP and concrete as well as different tension stiffening behavior. Hence, proper modeling of tension stiffening can improve the behavior of concrete beam in shear reinforced with GFRP bars.
tension stiffening factor to incorporate the different bond characteristics. It was also reported that the tension stiffening factor should remain the same for different surface pattern of the bars. This model illustrate that the softening part of concrete after cracks depends essentially on the modulus of elasticity of the reinforcing bar. It is independent of reinforcing ratio and concrete strength. The authors also reported that the transverse cracking in GFRP reinforced concrete stabilizes at higher values of axial strain (Fig. 2). Hence, the existing models of the tension stiffening that limit the descending part of the stress-strain response to a fixed multiple of the concrete cracking strain should be changed [4]. The strain energy density (u) under the stress strain curve of this model can be written as:
3. Development of tension stiffening model
u=
As recommended by Prakhya and Morley [12], softening part of tensile stress-strain behavior of bare concrete can be utilized with some adjustments to replicate the tension stiffening behavior of concrete members reinforced with steel. The tension stiffening model proposed by some researchers looks like this behavior [4,5]. Bischoff and Paixao [4] recommended a combined tension stiffening model for both steel and FRP reinforced member. Based on this model, the response after peak, which is based on the curve fitting of stressstrain behavior of members loaded uni-axially, is given by Eq. (1).
In developing the new tension stiffening model, the same strain energy density relationship will be maintained. Based on t Bischoff and Paixao [4] recommendation, a tension stiffening model with bilinear descending branch (Fig. 3) assumed by
Ef ⎤ ⎞ ft = ft′ exp ⎡−1100(εm − εcr ) ⎛ ⎢ 200 ⎝ ⎠⎥ ⎣ ⎦ ⎜
∫0
εu
ft dε
⎟
(1)
where, ft = uniaxial tensile stress of concrete, ft’ = direct tensile strength, εcr = strain at peak tensile stress, Ef = modulus of elasticity of GFRP bars, and εm = member strain. Eq. (1) was developed from the test results by deriving a normalized
Fig. 2. FRP reinforced member tension stiffening model [4]. 352
(2)
Structures 24 (2020) 351–356
M.S. Alam and A. Hussein
Cope et al. [13] for members reinforced with steel was used for FRP reinforced members maintaining the same strain energy density. According to Bischoff and Paixao [4], k2 should be multiplied by the ratio of the modulus of elasticity of steel and GFRP, Es/Ef that is roughly equal to 4.0 and k3 value was determined equating the strain energy density of Bischoff and Paixao [4] model and the proposed model (Fig. 4) using Eq. (2). No change was made for the value of k1. After these modifications, the model shown in Fig. 5 is recommended for use in FE analysis of GFRP reinforced concrete members. This updated model is shown in Eq. (3). ε ft′ ε for ε≺εcr ⎧ cr ⎪ ⎪ ft′ ft = 10 (11 - ε εcr ) for εcr ≺ε≺ε1 ⎨ ⎪ ft′ ε − ε1 + 1 for ε ≺ε≺ε 1 2 ⎪ 2 ε 1 − ε2 ⎩
Fig. 3. Steel reinforced member tension stiffening model [13].
(
)
(3)
where,
ε1 = 6εcr ε2 = k3 εu, and εu=Ultimate strain of GFRP bars
4. FE model of GFRP reinforced beam The developed model is employed in ABAQUS [11] finite element program for simulating the shear behavior of full scale GFRP reinforced beams. Concrete damage plasticity model for concrete with no stiffness degradation was used in this analysis. The model is based on the model proposed by Lubliner et al. [14] and incorporates the modifications proposed by Lee and Fenves [15] to account for different evolution of strength under tension and compression. Concrete was modeled using three-dimensional (3D) eight-node solid element with reduced integration points to escape the shear locking effect. Each node has three degrees of freedom. However, to overcome the negative effect of reduced integration element (hour glassing) and to obtain better results, a reasonably fine mesh were used. The reduced integration technique has been adopted by several investigators [16–19] and has shown considerable improvement in the capability of modeling. The compressive behavior of concrete is modeled using the uniaxial stress-strain relationship proposed by Collins and Mitchell [20]. This relationship is given by Eq. (3) and implemented by some piecewise linear parts as shown in Fig. 6. As shown in figure, the response is linear until the value of initial yield in uniaxial compression. Beyond the ultimate strength, the response is usually described by stress hardening trail by strain softening. Compressive stress and plastic strain data were obtained from the uni-axial compressive stress-strain curve.
Fig. 4. Equating strain energy density.
Fig. 5. Tension stiffening model proposed for GFRP reinforced concrete beams.
fc = f ′c
() n−1+( ) n
εc ε0
εc nk ε0
(3)
where: ε0 = concrete strain at fc’, n = curve fitting factor = 0.08 + fc’/17 k = decay factor for stress, assumed 1.0 for (εc/ε0 < 1.0) and taken as greater than 1.0 for (εc/ε0 > 1.0) The incremental elastic-plastic concrete model which is available in the finite element package is utilized. According to this model, the elastic compressive behavior of concrete is assumed for the duration of a yield point is attained beyond that un-recoverable plastic strain exists. For this reason, the strain rate is divided into elastic and plastic strain rates. These are trailed by flow and hardening rule as per the following principle: dε = dεpl = dεel where dε is the overall strain rate, dεpl is the plastic strain rate, and dεel is the elastic strain rate.
Fig. 6. Compressive stress-strain curve of concrete.
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Fig. 7. Typical finite element mesh and boundary conditions.
Fig. 8. Geometry and loading condition of the beams. Table 1 Geometry and other properties of the beams. Beam ID
f′c (MPa)
h (mm)
d (mm)
b (mm)
L (mm)
a/d
ρf (%)
ρf /ρbal
G-2.5 GN-2 GN-3
39.8 44.6 43.6
350 400 400
305 326 326
250 250 250
2400 3250 3250
2.5 3.1 3.1
0.86 1.22 1.70
1.34 2.13 3.04
Table 2 GFRP bar properties. Bar #
Diameter (mm)
Cross-sectional area (mm2)
Tensile strength (MPa)
Modulus of elasticity (GPa)
#4 #5
12.7 15.9
127 198
786 751
46.3 48.2
GFRP bars (V-ROD) were modeled as linear elastic material until failure. These were incorporated in the model utilizing 2-node bar formulation embedded in the iso-parametric concrete elements. This formulation is suitable for modeling rebar in different ways without interrupting the desired mesh size and the bars are considered as an integral part of the concrete elements. Thus the combined inner resisting forces are constituent of both concrete and reinforcement. For simplicity, perfect bond is assumed between reinforcing bars and concrete. One half of the beam was modeled due to symmetry of the beams.
Fig. 9. Comparative load-deflection behaviors of test results and FEA.
This also reduces the computational time. Therefore, the beam center was modeled using symmetrical boundary condition and at a distance of 220 mm from the end of the beam, roller support condition was modeled. The displacement-controlled load was applied at the same location of the test beam (Fig. 7).
354
Structures 24 (2020) 351–356
M.S. Alam and A. Hussein
good accuracy. The comparative crack patterns and principal tensile stress (red line) and principal compressive stress (light blue line) directions for beam G-2.5 are shown in Fig. 10. It is known that the formation of cracks will occur when the principal tensile stress surpasses the tensile strength of concrete. Hence, the orientation of this stress will be perpendicular to the crack direction. At bottom of the beam, the cracks start to form in vertical direction due to flexural stress and hence the stress directions are horizontal. In shear span zone, the stress direction rotates and matches with the diagonal direction of cracks due to the combined action of shear and flexural stresses. The light blue line shows the direction of compressive stress to form a compression fan. Hence, it can be argued that the paths of cracks fairly follow the perpendicular direction of principal tensile stress direction.
Table 3 Comparative test and FEA results. Beam ID
G-2.5 GN-2 GN-3
Test Results
FEA Results
Vexp (kN)
δexp (mm)
VFEA (kN)
δFEA (mm)
Vexp/VFEA
δexp/δFEA
61.2 59.5 77.5
17.0 13.8 13.9
69.3 65.0 83.9
17.1 13.1 14.9
0.88 0.92 0.92
0.99 1.05 0.93
5. Detail of test beam Experimental details of the beam that were used in this investigation are given below. The thicknesses of the beams were 350 and 400 mm. The beams are reinforced in longitudinal direction with no top or transverse reinforcement. All beams are simply supported with 4point loading. The geometry and loading conditions of the beams are shown in Fig. 8. The beam dimensions and other relevant data are provided in Table 1 whereas Table 2 confirms the properties of GFRP bars. More detail on experimental investigation can be found in Alam [21]. Two of the beams GN2 and GN3 were used from El-Sayed et al. [22] to see the performance of the model.
7. Conclusions The purpose of this paper was to generate a tension stiffening model for finite element analysis of shear critical concrete members reinforced with GFRP bars without transverse reinforcement. The model was employed in ABAQUS [11] finite element program. The predicted load deflection behavior using the proposed model was compared with the test results. Based on the investigation, following conclusions can be made: Finite element result of GFRP reinforced beam is poor than steel reinforced beam. This was attributed to the proper tension stiffening model for GFRP reinforced members. The proposed tension stiffening model improves the behavior of shear critical GFRP reinforced beams. The model predicts the load deflection behavior and crack patterns of the beams with a reasonable degree of accuracy. The predictions of load deflection behavior for beams with different depths and reinforcement ratios are close to the experimental results. The proposed models are validated by simulating the load deflection behavior of two GFRP reinforced beams from the literature. It can be argued that the models are robust to predict the behavior of GFRP reinforced beams and can be implemented for other GFRP reinforced members for prediction of load-deflection behavior numerically. The validity of the model should be verified for other FRP reinforced members to predict the load deflection and other behavior numerically.
6. Comparison of FEA results with test results The FE results of GFRP reinforced beams are compared with the experimental results of the beams given in Table 1. The load-deflection behavior of GFRP reinforced beams at mid-span for different beams are shown in Fig. 9. For all beams FE results follow almost the same loaddeflection behavior of the test results except beam GN-2, where the post cracking stiffness was a bit higher than the test results. The FE results showed a good prediction for both deflection and failure load of GFRP reinforced beams. Table 3 shows the comparative test and predicted shear capacities and mid-span deflections from numerical analysis. The ratios of experimental and predicted shear strength (V) range from 0.88 to 0.92 while the ratios for center deflections corresponding to the maximum shear force ranges from 0.93 to 1.05. Shear strength is defined as half of the maximum load carried by the beams. The variations of the results are within −12 to 5% of the experimental results. These variations are within the usual range of the variation of experimental results of shear critical beams. Hence, it can be concluded that the proposed tension stiffening model is capable to capture the load deflection behavior with
Declaration of competing interest This research did not receive any specific grant from funding
Fig. 10. Comparative crack patterns and stress directions for Beam G-2.5. 355
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M.S. Alam and A. Hussein
agencies in the public, commercial, or not-for-profit sectors. The authors also declare that there is no conflict of interest.
[11] ABAQUS. User’s Manual, Version 6.7, Hibbitt, Karlsson & Sorensen Inc, Providence, RI, USA; 2007. [12] Prakhya GKV, Morley CT. Tension stiffening and moment curvature relations of reinforced concrete elements. ACI J 1990;87(5):597–605. [13] Cope RJ, Rao PV, Clark LA. Nonlinear design of concrete bridge slabs using finite element procedures. In: Cohn MZ, editor. Nonlinear Design of Concrete Structures. Canada: University of Waterloo; 1979. [14] Lubliner J, Oliver J, Oller S, Onate EA. Plastic damage model for concrete. Int J Solids Struct 1989;25(3):299–326. [15] Lee J, Fenves G. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech, ASCE 1998;124(8):892–900. [16] Zienkiewicz OC, Taylor RL, Too JM. Reduced integration technique in general analysis of plates and shells. Int J Numer Methods Eng. 1971;3(2):275–90. [17] Hughes TJR, Cohen M, Haroun M. Reduced and selective integration techniques in finite element analysis of plates. Nucl Eng Des 1978;46(1):203–22. [18] Noor AK, Peters JM. Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams. Int J Numer Methods Eng 1981;17(4):615–31. [19] Garcıa-Vallejo D, Mikkola AM, Escalona JL. A new locking-free shear deformable finite element based on absolute nodal coordinates. Nonlinear Dyn 2007;50(12):249–64. [20] Collins MP, Mitchell D. Pre-Stressed Concrete Structures. Toronto, Canada: Response Publications; 1997. [21] Alam MS. Influence of different parameters on shear strength of FRP reinforced concrete beams without web reinforcement. St. John’s, NL, Canada: Faculty of Engineering and Applied Science, Memorial University of Newfoundland; 2010. (Ph. D. dissertation). [22] El-Sayed AK, El-Salakawy EF, Benmokrane B. Shear strength of FRP-reinforced concrete beams without transverse reinforcement. ACI Struct J 2006;103(2):235–43.
References [1] ACI Committee 224. Cracking of Concrete Members in Direct Tension (ACI 224.2R92, Reapproved 2004). American Concrete Institute, Detroit; 1992. [2] CEB-FIP Model Code 90. Design Code, Thomas Thelford, London; 1993. [3] ASCE Committee 447. State of the Art Report on Finite Element Analysis of Reinforced Concrete; 1982. [4] Bischoff PH, Paixao R. Tension stiffening and cracking of concrete reinforced with glass fiber reinforced polymer (GFRP) bars. Can J Civ Eng 2004;31(4):579–88. [5] Sooriyaarachchi H, Pilakoutas K, and Byars E. Tension stiffening behavior of GFRPreinforced concrete. In: 7th International Symposium on Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures (FRPRCS-7). New Orleans, Louisiana, USA; 2005. [6] Gilbert RI, Warner RF. Tension stiffening in reinforced concrete slabs. J Struct Div, ASCE 1978;104(2):1885–900. [7] Damjanic F, Owen DRJ. Practical considerations for modeling of post cracking concrete behavior for finite element analysis of reinforced concrete structures. In: Proc. I. Conf. on Computer Aided Analysis of the design of concrete structures. Pineridge, Swansea, UK; 1984. [8] Cope RJ. Nonlinear analysis of reinforced concrete slabs. Computer modeling of RC Structures, Pineridge, Swansea, UK; 1986: 3–43. [9] Hilleborg A. The influence of tensile toughness of concrete on the behavior of reinforced concrete structures. Proc. 9th FIP Congress, Wexham Springs. 1982. p. 57–164. [10] Bažant ZP, Oh BH. Deformation of progressively cracking reinforced concrete beams. ACI J 1984;81(3):268–78.
356
Contents lists available at ScienceDirect
Structures journal homepage: www.elsevier.com/locate/structures
Idealized tension stiffening model for finite element analysis of glass fibre reinforced polymer (GFRP) reinforced concrete members
T
⁎
M.S. Alama, , A. Husseinb a b
Department of Civil Engineering, University of Bahrain, P. O. 32038, Bahrain Faculty of Engineering and Applied Science, Memorial University of Newfoundland, St. John’s, NL A1B3X5, Canada
A R T I C LE I N FO
A B S T R A C T
Keywords: Reinforced concrete Shear strength GFRP bar FEA Tension stiffening
The capacity of concrete in carrying tension between two cracks is defined as tension stiffening. This paper presents the development and validation of tension stiffening model for finite element analysis (FEA) of glass fiber reinforced polymer (GFRP) reinforced concrete beams. The model was derived from an existing model for steel reinforced members using the same strain energy density. The proposed model was employed in a commercial finite element analysis program. The FEA results were compared with the test results in terms of loaddeflection behavior, crack patterns, ultimate loads. It was observed that the proposed tension stiffening model was capable of predicting load deflection of shear critical GFRP reinforced concrete beams with reasonable degree of accuracy.
1. Introduction Cracking of concrete has an effect on the behavior of reinforced concrete members. The concrete is assumed to carry no tension force at the cracks. However, in between cracks, the concrete is assumed to carry tension load through bond between the concrete and the reinforcing bars. The ability of concrete to carry tension between cracks and to provide additional stiffness is called tension stiffening [1]. This is an important issue in serviceability limit analysis and plays an important role in nonlinear analysis of reinforced concrete members. If the tension stiffening effect is neglected, the stiffness of a reinforced concrete bar or a structural member is under estimated [2] and this will results in overestimation of deflection. Usually the tension stiffening was overlooked in finite element analysis of concrete structures. Nonetheless, with time it was found that adding tension stiffening effect for reasonable calculation by finite element analysis is essential for actual forecast of the behavior of concrete structures [3]. Tension stiffening of GFRP reinforced concrete members is different from that of steel reinforced concrete members due to their different surface and bond characteristics, low modulus of elasticity. It was found that GFRP reinforced concrete showed more tension stiffening than the steel reinforced concrete [4]. Research have shown that the existing equations to model the tension stiffening effect is inaccurate in predicting the tension stiffening effect of GFRP reinforced concrete elements [5]. Therefore, there is a great need to develop tension stiffening
⁎
model for GFRP reinforced concrete members. Different approaches have been used to propose tension stiffening models. These are based on experimental results [6–8] and based on fracture mechanics principles [9,10]. In the current study, post failure stress-strain relation was used to develop the tension stiffening model. The proposed model was implemented in a finite element program ABAQUS [11], for simulating the shear behavior of full scale GFRP reinforced concrete beams. The FEA results were compared with the experimental data. 2. Background Besides, reinforced concrete members contain transverse reinforcement (stirrups) to resist shear. However, there are some members; such as slabs, footings, abutments, where there is no transverse reinforcement. These members resist shear by concrete. FEA of GFRP reinforced members containing stirrups have shown better performance than without stirrups. Furthermore, it was reported that GFRP reinforced concrete members have more tension stiffening than steel reinforced concrete members [4]. Fig. 1 shows the comparative results obtained from ABAQUS [11] and experiments for both steel and GFRP reinforced concrete beam in shear without stirrups. It should be mentioned here that the same tension stiffening model was used for both steel and GFRP reinforced members. All other modeling parameters were the same. Since the existing tension stiffening model works well for steel
Corresponding author. E-mail address: [email protected] (M.S. Alam).
https://doi.org/10.1016/j.istruc.2020.01.033 Received 7 August 2019; Received in revised form 23 January 2020; Accepted 23 January 2020 2352-0124/ Crown Copyright © 2020 Published by Elsevier Ltd on behalf of Institution of Structural Engineers. All rights reserved.
Structures 24 (2020) 351–356
M.S. Alam and A. Hussein
Nomenclature a b h d a/d k n ρf fc ft fc’ ft’ Af Es Ef
L P V Vexp VFEA ε ε0= εc εm εcr εy εu εpl eel etotal δexp δFEA
shear span width of beam depth of beam effective depth of beam shear span to depth ratio stress decay factor curve fitting factor reinforcement ratio compressive stress of concrete tensile stress of concrete compressive strength of concrete tensile strength of concrete area of reinforcement modulus of elasticity of steel modulus of elasticity of FRP
span length of beam Applied load shear strength experimental shear strength predicted shear strength tensile strain concrete strain at fc’ compressive strain member strain strain at peak tensile stress (cracking strain) yield strain of steel ultimate strain of GFRP plastic strain elastic strain total strain deflection measured from experiment deflection measured from FEA
Fig. 1. Load-displacement behavior of (a) Steel and (b) GFRP reinforced concrete beam.
reinforced members, the same model was used to justify the need (if any) to develop a tension stiffening model for GFRP reinforced beam. Although it was expected that there will be differences in the response of steel and GFRP beams, it is not clearly known what the differences in the responses are. The observation revealed that the GFRP reinforced concrete beam showed poor performance than steel reinforced concrete beam both in cracking load and ultimate load. The underlying cause of this behavior was attributed to the low elastic modulus, differences in bond properties between GFRP and concrete as well as different tension stiffening behavior. Hence, proper modeling of tension stiffening can improve the behavior of concrete beam in shear reinforced with GFRP bars.
tension stiffening factor to incorporate the different bond characteristics. It was also reported that the tension stiffening factor should remain the same for different surface pattern of the bars. This model illustrate that the softening part of concrete after cracks depends essentially on the modulus of elasticity of the reinforcing bar. It is independent of reinforcing ratio and concrete strength. The authors also reported that the transverse cracking in GFRP reinforced concrete stabilizes at higher values of axial strain (Fig. 2). Hence, the existing models of the tension stiffening that limit the descending part of the stress-strain response to a fixed multiple of the concrete cracking strain should be changed [4]. The strain energy density (u) under the stress strain curve of this model can be written as:
3. Development of tension stiffening model
u=
As recommended by Prakhya and Morley [12], softening part of tensile stress-strain behavior of bare concrete can be utilized with some adjustments to replicate the tension stiffening behavior of concrete members reinforced with steel. The tension stiffening model proposed by some researchers looks like this behavior [4,5]. Bischoff and Paixao [4] recommended a combined tension stiffening model for both steel and FRP reinforced member. Based on this model, the response after peak, which is based on the curve fitting of stressstrain behavior of members loaded uni-axially, is given by Eq. (1).
In developing the new tension stiffening model, the same strain energy density relationship will be maintained. Based on t Bischoff and Paixao [4] recommendation, a tension stiffening model with bilinear descending branch (Fig. 3) assumed by
Ef ⎤ ⎞ ft = ft′ exp ⎡−1100(εm − εcr ) ⎛ ⎢ 200 ⎝ ⎠⎥ ⎣ ⎦ ⎜
∫0
εu
ft dε
⎟
(1)
where, ft = uniaxial tensile stress of concrete, ft’ = direct tensile strength, εcr = strain at peak tensile stress, Ef = modulus of elasticity of GFRP bars, and εm = member strain. Eq. (1) was developed from the test results by deriving a normalized
Fig. 2. FRP reinforced member tension stiffening model [4]. 352
(2)
Structures 24 (2020) 351–356
M.S. Alam and A. Hussein
Cope et al. [13] for members reinforced with steel was used for FRP reinforced members maintaining the same strain energy density. According to Bischoff and Paixao [4], k2 should be multiplied by the ratio of the modulus of elasticity of steel and GFRP, Es/Ef that is roughly equal to 4.0 and k3 value was determined equating the strain energy density of Bischoff and Paixao [4] model and the proposed model (Fig. 4) using Eq. (2). No change was made for the value of k1. After these modifications, the model shown in Fig. 5 is recommended for use in FE analysis of GFRP reinforced concrete members. This updated model is shown in Eq. (3). ε ft′ ε for ε≺εcr ⎧ cr ⎪ ⎪ ft′ ft = 10 (11 - ε εcr ) for εcr ≺ε≺ε1 ⎨ ⎪ ft′ ε − ε1 + 1 for ε ≺ε≺ε 1 2 ⎪ 2 ε 1 − ε2 ⎩
Fig. 3. Steel reinforced member tension stiffening model [13].
(
)
(3)
where,
ε1 = 6εcr ε2 = k3 εu, and εu=Ultimate strain of GFRP bars
4. FE model of GFRP reinforced beam The developed model is employed in ABAQUS [11] finite element program for simulating the shear behavior of full scale GFRP reinforced beams. Concrete damage plasticity model for concrete with no stiffness degradation was used in this analysis. The model is based on the model proposed by Lubliner et al. [14] and incorporates the modifications proposed by Lee and Fenves [15] to account for different evolution of strength under tension and compression. Concrete was modeled using three-dimensional (3D) eight-node solid element with reduced integration points to escape the shear locking effect. Each node has three degrees of freedom. However, to overcome the negative effect of reduced integration element (hour glassing) and to obtain better results, a reasonably fine mesh were used. The reduced integration technique has been adopted by several investigators [16–19] and has shown considerable improvement in the capability of modeling. The compressive behavior of concrete is modeled using the uniaxial stress-strain relationship proposed by Collins and Mitchell [20]. This relationship is given by Eq. (3) and implemented by some piecewise linear parts as shown in Fig. 6. As shown in figure, the response is linear until the value of initial yield in uniaxial compression. Beyond the ultimate strength, the response is usually described by stress hardening trail by strain softening. Compressive stress and plastic strain data were obtained from the uni-axial compressive stress-strain curve.
Fig. 4. Equating strain energy density.
Fig. 5. Tension stiffening model proposed for GFRP reinforced concrete beams.
fc = f ′c
() n−1+( ) n
εc ε0
εc nk ε0
(3)
where: ε0 = concrete strain at fc’, n = curve fitting factor = 0.08 + fc’/17 k = decay factor for stress, assumed 1.0 for (εc/ε0 < 1.0) and taken as greater than 1.0 for (εc/ε0 > 1.0) The incremental elastic-plastic concrete model which is available in the finite element package is utilized. According to this model, the elastic compressive behavior of concrete is assumed for the duration of a yield point is attained beyond that un-recoverable plastic strain exists. For this reason, the strain rate is divided into elastic and plastic strain rates. These are trailed by flow and hardening rule as per the following principle: dε = dεpl = dεel where dε is the overall strain rate, dεpl is the plastic strain rate, and dεel is the elastic strain rate.
Fig. 6. Compressive stress-strain curve of concrete.
353
Structures 24 (2020) 351–356
M.S. Alam and A. Hussein
Fig. 7. Typical finite element mesh and boundary conditions.
Fig. 8. Geometry and loading condition of the beams. Table 1 Geometry and other properties of the beams. Beam ID
f′c (MPa)
h (mm)
d (mm)
b (mm)
L (mm)
a/d
ρf (%)
ρf /ρbal
G-2.5 GN-2 GN-3
39.8 44.6 43.6
350 400 400
305 326 326
250 250 250
2400 3250 3250
2.5 3.1 3.1
0.86 1.22 1.70
1.34 2.13 3.04
Table 2 GFRP bar properties. Bar #
Diameter (mm)
Cross-sectional area (mm2)
Tensile strength (MPa)
Modulus of elasticity (GPa)
#4 #5
12.7 15.9
127 198
786 751
46.3 48.2
GFRP bars (V-ROD) were modeled as linear elastic material until failure. These were incorporated in the model utilizing 2-node bar formulation embedded in the iso-parametric concrete elements. This formulation is suitable for modeling rebar in different ways without interrupting the desired mesh size and the bars are considered as an integral part of the concrete elements. Thus the combined inner resisting forces are constituent of both concrete and reinforcement. For simplicity, perfect bond is assumed between reinforcing bars and concrete. One half of the beam was modeled due to symmetry of the beams.
Fig. 9. Comparative load-deflection behaviors of test results and FEA.
This also reduces the computational time. Therefore, the beam center was modeled using symmetrical boundary condition and at a distance of 220 mm from the end of the beam, roller support condition was modeled. The displacement-controlled load was applied at the same location of the test beam (Fig. 7).
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good accuracy. The comparative crack patterns and principal tensile stress (red line) and principal compressive stress (light blue line) directions for beam G-2.5 are shown in Fig. 10. It is known that the formation of cracks will occur when the principal tensile stress surpasses the tensile strength of concrete. Hence, the orientation of this stress will be perpendicular to the crack direction. At bottom of the beam, the cracks start to form in vertical direction due to flexural stress and hence the stress directions are horizontal. In shear span zone, the stress direction rotates and matches with the diagonal direction of cracks due to the combined action of shear and flexural stresses. The light blue line shows the direction of compressive stress to form a compression fan. Hence, it can be argued that the paths of cracks fairly follow the perpendicular direction of principal tensile stress direction.
Table 3 Comparative test and FEA results. Beam ID
G-2.5 GN-2 GN-3
Test Results
FEA Results
Vexp (kN)
δexp (mm)
VFEA (kN)
δFEA (mm)
Vexp/VFEA
δexp/δFEA
61.2 59.5 77.5
17.0 13.8 13.9
69.3 65.0 83.9
17.1 13.1 14.9
0.88 0.92 0.92
0.99 1.05 0.93
5. Detail of test beam Experimental details of the beam that were used in this investigation are given below. The thicknesses of the beams were 350 and 400 mm. The beams are reinforced in longitudinal direction with no top or transverse reinforcement. All beams are simply supported with 4point loading. The geometry and loading conditions of the beams are shown in Fig. 8. The beam dimensions and other relevant data are provided in Table 1 whereas Table 2 confirms the properties of GFRP bars. More detail on experimental investigation can be found in Alam [21]. Two of the beams GN2 and GN3 were used from El-Sayed et al. [22] to see the performance of the model.
7. Conclusions The purpose of this paper was to generate a tension stiffening model for finite element analysis of shear critical concrete members reinforced with GFRP bars without transverse reinforcement. The model was employed in ABAQUS [11] finite element program. The predicted load deflection behavior using the proposed model was compared with the test results. Based on the investigation, following conclusions can be made: Finite element result of GFRP reinforced beam is poor than steel reinforced beam. This was attributed to the proper tension stiffening model for GFRP reinforced members. The proposed tension stiffening model improves the behavior of shear critical GFRP reinforced beams. The model predicts the load deflection behavior and crack patterns of the beams with a reasonable degree of accuracy. The predictions of load deflection behavior for beams with different depths and reinforcement ratios are close to the experimental results. The proposed models are validated by simulating the load deflection behavior of two GFRP reinforced beams from the literature. It can be argued that the models are robust to predict the behavior of GFRP reinforced beams and can be implemented for other GFRP reinforced members for prediction of load-deflection behavior numerically. The validity of the model should be verified for other FRP reinforced members to predict the load deflection and other behavior numerically.
6. Comparison of FEA results with test results The FE results of GFRP reinforced beams are compared with the experimental results of the beams given in Table 1. The load-deflection behavior of GFRP reinforced beams at mid-span for different beams are shown in Fig. 9. For all beams FE results follow almost the same loaddeflection behavior of the test results except beam GN-2, where the post cracking stiffness was a bit higher than the test results. The FE results showed a good prediction for both deflection and failure load of GFRP reinforced beams. Table 3 shows the comparative test and predicted shear capacities and mid-span deflections from numerical analysis. The ratios of experimental and predicted shear strength (V) range from 0.88 to 0.92 while the ratios for center deflections corresponding to the maximum shear force ranges from 0.93 to 1.05. Shear strength is defined as half of the maximum load carried by the beams. The variations of the results are within −12 to 5% of the experimental results. These variations are within the usual range of the variation of experimental results of shear critical beams. Hence, it can be concluded that the proposed tension stiffening model is capable to capture the load deflection behavior with
Declaration of competing interest This research did not receive any specific grant from funding
Fig. 10. Comparative crack patterns and stress directions for Beam G-2.5. 355
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agencies in the public, commercial, or not-for-profit sectors. The authors also declare that there is no conflict of interest.
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