Numerical modeling of size effect in shear strength of FRP-reinforced concrete beams

Structures 20 (2019) 237–254

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Numerical modeling of size effect in shear strength of FRP-reinforced concrete beams

T

Ghazi Bahroz Jumaaa, , Ali Ramadhan Yousifb ⁎

a b

Building and Construction Engineering Department, University of Garmian, Kurdistan Region, Iraq Department of Civil Engineering, Salahaddin University-Erbil, Kurdistan Region, Iraq

ARTICLE INFO

ABSTRACT

Keywords: FEA model Size effect Shear strength Fiber-reinforced polymer

This study presents the numerical modeling of size effect in shear strength of FRP-reinforced concrete (RC) beams with and without stirrups by finite element analysis (FEA) model. The size effect was incorporated in the model by controlling the mesh size, softening tensile stress-strain response, and dilation angle. A new modification was suggested to represent the softening tensile strength of concrete for different size of beams. The calibrated FEA model for beams with and without stirrups showed great power and ability in simulating the experimental results in terms of ultimate shear capacity, ultimate deflection, flexural and stirrup strains, and crack patterns. Additionally, the calibrated model was then verified by thirteen experimental results from four different studies, as well as a parametric study was performed.

1. Introduction The shear strength and behavior of reinforced concrete (RC) members, particularly members reinforced with fiber reinforced polymers (FRP) bars, is an interesting area of study and concern of researches as the shear failure is sudden and catastrophic as well as exhibits complex behavior. There are many factors affect the shear strength and behavior of FRP-RC members without stirrups among them the size effect is the least studied parameter in light of the difficulties in preparing and testing large beams as well as such tests are generally expensive [1]. Many experimental studies [2–5] have reported that the shear strength of FRP-RC members without stirrups is significantly reduced in large beams, that is called the size effect. The size effect phenomenon is more significant and is of fundamental and practical relevance in the design of FRP-RC members than that of steel-RC members; owing to the relatively low elastic modulus of FRP bars, the cracks will be deeper and wider, the dowel action will be reduced, and the radius of the flexural reinforcement influence zone will be smaller. A substantial size effect has been observed in the shear strength of large FRP-RC beams, even if a reasonable amount of FRP flexural reinforcement was provided [2–5]. The size effect on the shear strength should be a specific concern for high-strength concrete (HSC) members [6], as both the aggregate interlock shear resistance and the depth of the top compression zone are smaller in HSC members than in normal-strength concrete members. Recently, the finite element analysis (FEA) has become a more

⁎

promised tool for research and design. A detailed analysis can be a powerful entity for different engineering applications with the condition that the user has a proper understanding of modeling. Throughout modeling, using an incorrect aspect or incorrect technique can lead to a wrong solution. Therefore, to have full confidence in any FEA, experimental results or a closed form solution needs to be presented to validate the results. The chief advantage of the FEA is the ability to simulate expensive experiments and provides natural solutions to complicated problems. The FEA applies to a wide range of various engineering applications by utilizing different element types, different materials properties, various loading, and boundary conditions. Therefore, the utilization of FEA modeling for studying the size effect is the best tool to overcome the difficulties of preparing and testing large beams and performing expensive tests. The combination of numerical modeling with the experimental results, i.e., calibration of a numerical model by experimental results, will not only yield timeconsumption and financial benefits to the research but will also enhance the efficiency of the experimental studies. Then, the calibrated model can be used to accurate prediction of beam response and to propose future studies that have been designed with a higher technical understanding, thus providing more accurate outcomes [7]. Many studies have been performed on simulating the shear strength of FRP and steel reinforced concrete beams. However, only several numerical studies on size effect in steel-RC beams were performed [8–12]. While studies on numerical simulation of size effect in FRP-RC

Corresponding author. E-mail address: [email protected] (G.B. Jumaa).

https://doi.org/10.1016/j.istruc.2019.04.008 Received 29 January 2019; Received in revised form 24 March 2019; Accepted 10 April 2019 2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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G.B. Jumaa and A.R. Yousif

beams have not been investigated so far; therefore this study was motivated. Furthermore, for the first time, the mesh sizes, the softening tensile stress-strain response of concrete, and dilation angle were calibrated to simulate the size effect. In this study, the FEA software ABAQUS [13] was used for the simulation of shear strength and behavior of FRP-RC beams with and without stirrups [14,15]. ABAQUS is an FEA software that has become popular in academic and research institutions due to its extensive library of materials and elements and its capability to simulate multi-dimensional problems. For this purpose, an FEA model was proposed and calibrated using twelve FRP-RC beams without stirrups including the consideration of size effect and six FRPRC beams with stirrups. To measure the model accuracy, it was used to predict the shear strength of thirteen specimens from four different studies from the literature. Furthermore, a parametric study was performed to investigate how the proposed model will consider the effect of beam depth, shear span to depth ratio and concrete compressive strength on the shear strength of FRP-RC members.

t

t

t

for

t

cr

0.4

for

t

t

>

cr

(1b)

= ft

cr t

n

for

t

>

cr

(2)

As the presence of steel reinforcement has considerable effects on the tension stiffening of reinforced concrete members and the mechanical properties of FRP bars are significantly different than that of steel bars. Therefore, to simulate the size effect mechanism in shear strength of FRP-RC members, different tension stiffening model is required. The tension stiffening presented in Eq. (2) was adjusted to be used as a unique model for the total depth of members including the RC zone and PC zone. The tension stress-strain response in post-peak stage should be modified such that to simulate the rapid rate of decrease of tension stiffening whenever the beam depth increased. For this purpose, the tension softening equation was calibrated based on the experimental work results of beams of different sizes.

To a rational and proper simulation of size effect in shear strength of FRP-RC beams, it is required fully understand the mechanism of size effect. The size effect might be caused principally by an increase in crack width whenever the beam effective depth is increased, which prompts a diminishment in both aggregate interlock and residual tensile strength mechanisms [4,16]. In large size beams, the reason of occurrence of wider crack size about mid-height of the beam may be attributed to that the length of the flexural-shear cracks could be substantially longer than the radius of the reinforcement influence zone, which unable to restrain all crack extension. This explanation is by the observation that the presence of either skin reinforcement or transverse reinforcement substantially eliminates the size effect [17]. The size effect of concrete can be adequately explained by the energy release caused by micro-crack growth [18]. The tension-softening characteristic of plain concrete is regarded as the core factor for simulating the size effect of concrete members. Therefore, the concrete tensile model is utilized to model the reinforced concrete members under low confinement pressure. The energy stored in the concrete will transform to fracture energy when the cracks happened and propagated which makes the concrete soften rapidly. While with the presence of reinforcement, it stiffens the concrete between the cracks by bond stress which enhances the softening behavior and makes the tension stress to decay gradually; this phenomenon is called tension stiffening. Many studies have been performed to propose models for tension softening in plane concrete (PC) members as well as for tension stiffening for reinforced concrete members. The presence of reinforcement produces the tension stiffening, but its effect is only in a limited zone of surrounded concrete. In large beams, the effect of reinforcement is unable to stiffening the concrete at mid-height zone and the tensile stress will rapidly decay which causes the size effect. Therefore, to properly simulate the size effect, both the tension stiffening at the RC zone, and tension softening at the PC zone should be considered. An et al. [12] proposed a model to simulate the size effect by separating the tension response in reinforced concrete members into two zones; PC zone and RC zone. They reported that the size effect is based on tension softening and shear softening of plain concrete and on reinforcement bond performance. To predict the depth of the RC zone, they suggest a model based on bar diameter, concrete tensile strength, the yield strength of steel bars, beam width and concrete cover. To specify the tension stiffening Equ.1 was used in RC zone by Ana et al. [12] which also was used by many other researchers [19–21], this equation was first proposed by Tamai [22]:

= Ec

cr

where εt is the concrete tensile strain, εcr is the concrete strain at peak stress (at cracking), and ft is the tensile strength of the concrete (peak stress). Moreover, the fracture energy and crack band width usually are used to represent the softening tensile stress-strain relation. However, this softening model should be adjusted to account for the size of elements in terms of fracture energy. Some studies [12,23] utilized a modified version of the softening post-peak response as that in Equ.2, in which the value of n controls the rate of tensile strength softening. Ana et al. suggested that for PC zone the n value should be taken higher than (0.6).

2. Simulation of size effect mechanism

t

= ft

3. Concrete constitutive models ABAQUS provides three constitutive models for concrete, including Smeared Crack Model, Brittle Cracking Model, and Concrete Damaged Plasticity Model (CDPM). All models can be used for plain concrete and can be used for the analysis of reinforced concrete structures. CDPM is utilized in this study as this model has the potential to simulate complete inelastic behavior of concrete, include tensile cracking and compressive crushing with damage properties. The CDPM is effective for static, cyclic, and dynamic loading under low confining pressures. Furthermore, it offers five different parameters that used to define characteristics like; yield surface, brittleness of concrete, viscoplastic regularization and non-associated flow rule with multi-variable hardening plasticity. So the CDPM is entirely versatile in modeling concrete under various loading conditions. The CDPM in ABAQUS employs the principle of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity to depict the inelastic characteristics of concrete [13]. 4. Material properties used in modeling 4.1. Concrete properties The concrete is modeled by using the 3D stress element C3D8R in ABAQUS (2009) which is an 8-node linear brick, reduced integration, hourglass control. The concrete compressive strength and modulus of elasticity used in FEA are obtained from experimental tests as given in Table 1. The tensile strength is strongly influenced by the type of aggregates, compressive strength of the concrete, and the stress acting transversely to the tensile stress. The tensile strength is also strongly affected by the type of test used to determine it [24]. The experimentally performed tests to determine the tensile strength property are indirect tests. Many researchers [7,25–27] used predicted equations for direct tensile strength, ft, as 0.33√fc′ for finite element modeling. While others [28] took it as 0.42√fc′. In this study, it was found that the 0.33√fc′ gives lower results than the experimental result; therefore the

(1a)

238

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G.B. Jumaa and A.R. Yousif

as the ratio of the concrete tensile stress after cracking to the ultimate tensile stress σto, which can be expressed as follows [32]:

Table 1 Concrete properties used in the FEA. Concrete property

High strength concrete beams

Concrete compressive strength; fc′, MPa Concrete tensile strength; ft, MPa Modulus of Elasticity; Ec, MPa Poisson's ratio, v Ultimate strain; ε0

Normal strength concrete beams

73.4

43.2

3.6 38,110 0.18 0.0025

2.7 28,592 0.18 0.0025

dc = 1

dt = 1

=

R=

RE =

E0 =

1 + (R + RE

2)

RE (R 1) (R 1)2

1 R

() c

o

Ec

c

(2R

1)

() c

2

o

+R

3

(3a)

o

(5)

BFRP rebars are assumed to be linear-elastic until failure; their properties are shown in Table 2. The poison's ratio is considered equal to 0.3 [33]. The BFRP rebars were modeled by using discrete truss sections (T3D2) which is a 2-node linear 3D truss element that can only transmit axial forces. The rebars were defined as deformable “wire” type parts. A wire is depicted as a line in ABAQUS/CAE and is used to model a solid that has a thickness and depth that are significantly smaller relative to its length. These individual bars are then incorporated into the model by being embedded into the concrete through the “Embedded Region” constraint.

(3b)

Ec E0

(3c)

fc

(3d)

0

where; Rσ = 4 and Rε = 4 may be used [31], ε0 is the concrete strain at peak stress which taken as 0.0025, Ec is the initial modulus of elasticity. In the analysis, Eqs. (3a), (3b), (3c) and (3d) are taken as the equivalent uniaxial stress-strain curve for concrete as shown in Fig. 1. When concrete is stressed beyond the elastic region and develops plastic strains, subsequent unloading of this concrete will exhibit a degraded or damaged elastic modulus. Therefore, to increase the accuracy of the concrete modeling, damage parameters dc and dt may be specified for both compression and tension, respectively. In this study, the compression damage parameters, dc is defined as the ratio of the concrete compression stress after crushing to the initial yield stress σco. Similarly, the tension damage parameters, dt is defined

5. Beam modeling The details of the analyzed beams with and without stirrups used in this study are given in Fig. 2 which include 12 beams without stirrup [14] and six beams with stirrups [15]. To make benefit from the symmetrical condition of the beams; half of the full beams without stirrup and one-quarter of beams with stirrup were employed for modeling with proper boundary conditions. This method substantially reduces analysis time and size of computer disk needed. The perfect bond strength between the concrete and FRP bars was assumed. The finite element model, boundary conditions, loading regions, and reinforcement details are shown in Fig. 3. Convergence problems associated with concrete cracking are usually encountered when a general static method is utilized to solve nonlinear structures. A useful feature of the dynamic process is that the convergence difficulties are overcome as well as the static solution can be obtained with sufficient accuracy [34]. Therefore, the most suitable approach for conducting the FEA of the current study using ABAQUS/ CAE is to utilize the “Displacement control” loading application. The loading was defined by specifying a line load at the middle of the loading steel plate by assigning a specified displacement to the line load. The specified displacement was established by using amplitude

80 70 60 50 Stress , MPa

t to

4.2. BFRP-material properties

() c

(4)

where; dc or dt = 1 is a state of the material wholly damaged and defines the complete local fracture, and dc or dt = 0 is a non-damaged material. If damage parameters are not specified, the model behaves as a plasticity model. In this study, both compression and tension damage parameters were used in the FEA model. The values for stiffness recovery factors are also used. The compressive stiffness recovery factor, wc = 0.8 is used only for beams with the stirrup, assuming that compressive stiffness is mostly recovered upon crack closure as the load changes from tension to compression. The tensile stiffness recovery factor, wt = 0 is used for all the beams, assuming that tensile stiffness is not recovered as the load changes from compression to tension once crushing of concrete is initiated.

tensile strength was taken as 0.42√fc′. The Poisson's ratio of concrete as recommended by many studies ranges from 0.14 to 0.26. Chen, [29] stated that the Poisson's ratio for concrete ranges from 0.15 to 0.22, with a representative range being 0.18 to 0.20. Therefore, a value of 0.18 was selected for all beam simulations performed in this study. The stress-strain relationship proposed by Saenz [30] has been widely used as the uniaxial compressive stress-strain curve for concrete, and it has the following form: c

c co

40 30 20

Table 2 Properties of basalt FRP rebars.

10

Size mm

0 0

0.0025

0.005

0.0075

0.01

0.0125

0.015

Strain, mm/mm 8 16 19

Fig. 1. Uniaxial stress-strain curve for HSC beams.

239

Cross sectional area mm2

Density g/cm3

Ultimate tensile strength ffu, MPa

50 180 278

2.1 2.1 2.1

1100 1089 996

Elastic modulus Ef, GPa 56 58 58

Ultimate strain mm/mm 0.019 0.019 0.017

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G.B. Jumaa and A.R. Yousif

Fig. 2. Details of the analyzed beams.

function (smooth step) with uniform distribution in the boundary condition. The analysis was conducted in step-1 (Dynamic, Explicit), which is a step after the initial step. Automatic increment and a total of one second “Time period” were selected.

6. Calibration model for beams without stirrups The first step in any finite-element modeling is the calibration of the parameters based on the experimental results. The FEA of beams with different sizes required that the size effect phenomenon is considered in the modeling. Various parameters were investigated to find their effect

Fig. 3. Boundary conditions and reinforcement details of the simulated beams. 240

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G.B. Jumaa and A.R. Yousif

160

M-H-R3

SHR1

350

140 300 120 250

Load, kN

Load, kN

100 80 60

SHR1(Exp) M20 M30 M40 M50 M60

40 20

200

100 50 0

0 0

5

10 15 20 Mid-Span Deflec"on, mm

25

MHR3(Exp) M20 M35 M40 M60 M80 M100

150

0

30

5

10

15 20 25 Mid-Span Deflec"on, mm

30

LHR2

350 300 250

Load, kN

200 LHR2(EXP) M20 M33 M47 M60 M80 M100

150 100 50 0 0

5

10 15 20 25 Mid-Shear Span Deflec"on, mm

30

Fig. 4. Mesh sensitivity for small, medium and large beams.

3

LHR3 n=0.40 2.5

450

n=0.46

400 350 300

1.5

Load, kN

Stress , MPa

n=0.61 2

1

0.5

0 0

0.002

0.004 0.006 Strain, mm/mm

0.008

250 200 Exp (LHR3) 150

n=0.40

100

n=0.44 n=0.52 (Eq. 6)

0.01

50

n=0.6

0

Fig. 5. Modeling of tension stress-strain relationship for concrete.

0

5

10

15

20

Mid-Span Deflec"on, mm

on the results. The settings that influence the size effect modeling are: mesh size, softening stress-strain response of concrete and dilation angle. To find the effect of mesh size systematic mesh sensitivity was performed for the three sizes of beams (SHR1, MHR3, and LHR2) by

Fig. 6. Sensitivity analysis of n value for beam LHR3.

241

25

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G.B. Jumaa and A.R. Yousif

Small Beams

Medium Beams 350

250

300 200 250 200 SNR1 EXP SHR1 EXP SHR2 EXP SHR3 EXP SNR1 FEA SHR1 FEA SHR2 FEA SHR3 FEA

100

50

Load, kN

Load, kN

150

MNR1 EXP MHR1 EXP MHR2 EXP MHR3 EXP MNR1 FEA MHR1 FEA MHR2 FEA MHR3 FEA

150 100 50

0

0 0

5

10 15 20 Mid-Span Deflec"on, mm

25

30

0

5

10 15 20 Mid-Span Deflec"on, mm

25

Large Beams

350 300

Load, kN

250 200 LNR1 EXP LHR1 EXP LHR2 EXP LHR3 EXP LNR1 FEA LHR1 FEA LHR2 EFA LHR3 FEA

150 100 50 0 0

5

10

15

20

25

Mid-Span Deflec"on, mm Fig. 7. Load versus deflection from experimental and FEA for beams without stirrup.

considering mesh size from 20 mm to 100 mm. The effects of mesh size on the load-deflection response are shown in Fig. 4. The three sizes of beams showed different results against mesh sensitivity. This variation in mesh sensitivity may be attributed to using stress-strain tension modeling as this method may lead to unreasonable mesh sensitivity in regions of concrete that contain little or no reinforcement [7]. After numerous trials on mesh sensitivity, it was found that the most reasonable mesh sizes which give better results for small, medium and large beams were 30 mm, 35 mm, and 47 mm respectively. According to these mesh sizes, the depth of beams is divided into 10, 14 and 15 elements for small, medium and large beams respectively. The mesh size in the critical region is related to fracture energy dissipation during brittle failure [35]. The fracture energy dissipation is reduced when the mesh size is decreased. Therefore, the results were more accurate when the number of elements increased for medium and large beams to 14 or 15 elements within the beam depth. Concluding that, for future studies, the mesh size can be taken about 10% of the beam depth for beams with depth smaller than or equal to 300 mm while for larger beams with depth of about 700 mm and 1000 mm, the mesh size can be taken as 7% and 5% of beam depth, respectively.

Another important parameter is the softening tensile stress-strain response or the stress-strain curve in the post-cracking stage which can be represented by n value in Eq. (2). As n increases, the rate of decay of the tensile capacity increases which represent less tension stiffening. Therefore, the value of n increased with increasing the member depth. As it is experimentally found that the cause of size effect is mainly attributed to that, in large beams, the cracks are wider, and the aggregate interlock mechanism is reduced rapidly. Then, the tensile strength through the cracks is decayed quickly after the formation of the cracks. The modeling of different sizes of beams was calibrated to determine their sensitivity to n value. The best values of n which accurately simulate the effect of beam depth were determined such that; for small beams (h ≤ 300 mm) n was taken equal to 0.4 and for larger beams can be finding from the following equation:

n = 0.40 + 0.3

h

300 1000

(6)

The effect of n value is depicted in Fig. 5 for three values of 0.4, 0.46 and 0.61 which are for beams of a depth of 300, 500 and 1000 mm respectively. 242

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G.B. Jumaa and A.R. Yousif

Small Beams

Medium Beams

700

450 400

600

350 500 Load, kN

Load, kN

300 250 200 150

SHR3S1 EXP SHR3S2 EXP SHR3S1 EFA SHR3S2 FEA

100 50

400 300 MHR3S1 EXP MHR3S2 EXP MHR3S1 FEA MHR3S2 FEA

200 100

0

0 0

10 20 Mid-Span Deflec"on, mm

30

0

10 20 30 Mid-Span Deflec"on, mm

40

Large Beams

900 800 700

Load, kN

600 500 400 LHR3S1 EXP

300

LHR3S2 EXP 200 LHR3S1 FEA 100

LHR3S2 FEA

0 0

10

20 30 40 Mid-Span Deflec"on, mm

50

60

Fig. 8. Load versus deflection from experimental and FEA for beams with stirrup.

this study, after calibration, the dilation angle was taken as 300 for beams of a depth of about 300 mm, 250 for beams of a depth of about 500 mm and 150 for beams of a depth of about 1000 mm. These values were selected based on the fact that whenever the dilation angle decreased the brittleness of concrete increased which is in agreement with that confirmed by experimental works on size effect. On the other hand, the default value for each of σbo/σco, Kc, ϵ and μ of CDPM material parameters were used which are; 1.16, 0.667, 0.1 and 0.0 respectively.

To illustrate the effect of the n value on the FEA results of the shear strength and behavior of FRP-RC beams a sensitivity analysis was performed. Fig. 6 shows the results of the load-deflection behavior of experimental and FEA for beam LHR3. Four different values of n were taken which are; 0.4, 0.44, 0.52 and 0.60; the n value of 0.52 is that calculated by Eq. (6) for the beam LHR3. It is clear that the load and ultimate deflection increased when the n value decreased. The most definite trend to the experimental result was that for n = 0.52 while other values yielded much larger or lower load-deflection behavior. Therefore this confirms that the proposed equation predicts well the decay response of post-peak concrete tensile strength. The third influencing parameter is the dilation angle, ψ, which is usually calibrated based on the experimental results, as various studies used different ψ values. Malm [36] noted that small dilation angles (100) produced very brittle beam responses, whereas larger angles (> 400) produced responses that exhibited higher ductility and achieved larger peak loads. Also, Malm found that dilation angles between 300 and 400 provided the best agreement with experimental data. In the material model proposed by many studies [7,37] the dilation angle is defined as 310. While others [34,38,39], suggested a dilation angle in the range of (350–380). Stoner [7] used dilation angle of 300 for beams with no stirrup and 500 for beams with the stirrup. However, the default value of the dilation angle is 150 in ABAQUS. In

7. Calibration model for beams with stirrups The finite-element modeling of beams with stirrup is different than that without stirrup, as the stirrups entirely confine the concrete. More significantly, these stirrups provide passive confinement to the concrete-bound within the stirrups as a result of the inelastic expansion, or dilation, that concrete experiences under high compressive loading. This confinement strengthens the beam response by allowing the concrete to carry higher stresses and undergoes more considerable ductility. Therefore, the calibration of the dilation angle is necessary. As it is mentioned before, higher dilation angle values give more ductility results; therefore, higher values were used, and the most accurate results were obtained by using dilation angle of 450. The other parameters of CDPM were taken as default values. 243

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Medium Beams

Small Beams

350

350

SHR3 EXP SHR2 EXP SHR1 EXP SNR1 EXP SHR3 FEA SHR2 FEA SHR1 FEA SNR1 FEA

300 250

300 250 200

100

100

50

50

Load, kN

150

Load, kN

200

MHR3 EXP MHR2 EXP MHR1 EXP MNR1 EXP MHR3 FEA MHR2 FEA MHR1 FEA MNR1 FEA

150

0

0 0

2000

4000

6000

8000

10000

12000

0

2000

FRP Micro Strain

Large Beams

350

10000

12000

LHR3 EXP LHR2 EXP LHR1 EXP LNR1 EXP LHR3 FEA LHR2 FEA LHR1 FEA LNR1 FEA

300 250 Load, kN

4000 6000 8000 FRP Micro Strain

200 150 100 50 0 0

2000

4000

6000

8000

10000

12000

FRP Micro Strain Fig. 9. Experimental and the FEA main BFRP flexural strains for beams without stirrup.

On the other hand, the presence of stirrups interacts with the concrete and significantly influences the structure's behavior. These stirrups act to carry tensile loads across cracks, thus preventing brittle shear failures. Therefore, a model that can capture the ductility of the concrete reinforced with stirrups and shift the failure mode from a brittle tension fracture to a more ductile crushing failure is desired. The influence of the concrete reinforcement interaction that governs the effects of tension stiffening was therefore incorporated into the concrete modeling and not the reinforcement modeling. Thus the tension stressstrain modeling was kept like that for a small beam without stirrup, i.e., n was taken equal to 0.40. The mesh sensitivity for beams with stirrups was also performed by taking mesh size from 20 mm to 100 mm. It was found that there is no much discrepancy in the results of beams with various mesh sizes. The most accurate result was observed in beams with a mesh size of 30 mm; therefore, a mesh size of 30 mm was adopted in the FEA for all the beams with stirrups.

patterns. 8.1. Load deflection curves Figs. 7 and 8 compare the load-deflection curves obtained from the experimental tests and the FEA for small, medium and large beams without and with stirrup respectively. The curves show good agreement in the FEA model with the experimental results throughout the entire range of behavior. Comparable results can be observed in all sizes of beams in terms of linear stage, nonlinear stage, ultimate load, and ultimate deflection. These results indicate that the ABAQUS FEA can simulate the size effect in shear strength and behavior of FRP-RC beams. It can be observed that, in some beams, the FEA showed stiffer trends than the experimental trends, particularly in the linear stage. These stiffer trends can be caused by some differences in the FEA modeling and original beams. The bond between the concrete and FRP rebar is assumed to be perfect (no slip) in the FEA, while actually, this assumption is not perfect; small slip may occur, and therefore the composite action between the concrete and FRP rebar reinforcing is not perfect in the original beams. Additionally, the micro-cracks produced by drying shrinkage and that happened between coarse aggregate and mortar are exist in the concrete. These would reduce the stiffness of the original beams, but in the FEA modeling, perfect homogeneous materials were assumed.

8. Results and discussion of FEA model The following sections compare the results from the ABAQUS FEA with the experimental data for all the tested beams. The following comparisons were made: load-deflection behavior, load-flexural strain behavior, load-stirrup strain behavior, ultimate loads and crack 244

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Small Beams

350

Load, kN

250 200 150

MHR3 EXP MHR2 EXP MHR1 EXP MNR1 EXP MHR3 FEA MHR2 FEA MHR1 FEA MNR1 FEA

300 250 200 Load, kN

300

Medium Beams

350

SHR3 EXP SHR2 EXP SHR1 EXP SNR1 EXP SHR3 FEA SHR2 FEA SHR1 FEA SNR1 FEA

150

100

100

50

50

0

0 0

1000 2000 3000 Concrete Micro Strain

4000

0

4000

Large Beams

350

LHR3 EXP LHR2 EXP LHR1 EXP LNR1 EXP LHR3 FEA LHR2 FEA LHR1 FEA LNR1 FEA

300 250 Load, kN

1000 2000 3000 Concrete Micro Strain

200 150 100 50 0 0

1000

2000 3000 Concrete Micro Strain

4000

Fig. 10. Experimental and the FEA mid-span concrete strains for beams without stirrup.

8.2. Load-flexural strain behavior

stiffness degradation cannot be adequately modeled in the FEA.

• In original beams, due to localized tensile strains cracks occurred

The load versus main BFRP flexural reinforcement strain curves and the load versus concrete strain curves from the FEA model and experimental results at a mid-span for small, medium and large beam are shown in Figs. 9 and 10 respectively. The trends depict acceptable agreement in strains from the FEA and the experimental results for the tested beams. Generally, the strains in the FRP rebars for the FEA were compatible with the experimental results in the linear stage. However, in the nonlinear phase, the strains in the FRP rebars for the FEA were smaller than those for the experimental beams. While the ultimate strains from the FEA were close to the corresponding experimental results. The trends of load-concrete strains from the FEA showed more discrepancy compared with that obtained from strain gauges at midspan of the tested beams. For small beams, the patterns are in good agreement, while for medium and large beams, the strains from the FEA were lower than that from the experimental tests. These small discrepancies in load-strain behavior for both BFRP and concrete may be caused by the following points [40]:

•

and the tensile force is redistributed and partly resisted by tension stiffening. While in the FEA this behavior is simulated by tension stress-strain modeling which does not represent the actual cracks and tension stiffening. This simulated behavior, in turn, leads to differences in strains in FRP and concrete between tested beams and the FEA results. The steel plates used at point loads and supports were modeled in the FEA by tying it to the concrete elements which imply that both of adjacent concrete and steel elements share the same nodes. This increased the stiffness of the concrete elements in these regions. This simulation is different than the actual case as the steel plates were only in simple contact with the concrete surface.

8.3. Load-stirrup strain behavior The load versus BFRP-stirrup strain curves from the FEA and experimental results at mid-height of maximum stressed stirrup for small, medium and large beam are shown in Fig. 11. Generally, the trends indicate acceptable agreement in strains from the FEA and the experimental results for the tested beams. It can be observed that the load at which the strains initiate to increase are in good agreement with the experimental results. However, in the next part of the curves, the stirrup-strains for the experimental tests were greater than that from the FEA results. This may be attributed to that in the tested beams due to

• The bond between the concrete and FRP rebar is assumed to be

perfect (no slip) in the FEA, while this assumption is not accurate; small slip occurs. Therefore, the composite action between the concrete and FRP rebar reinforcing is not perfect in the original beams due to the local slip of FRP at the crack locations. This local 245

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G.B. Jumaa and A.R. Yousif

Small Beams

450

Medium Beams

700

400

600

350 500 300 400 Load, kN

Load, kN

250 200 150

50

MHR3S1 EXP MHR3S1 FEA MHR3S2 EXP MHR3S2 FEA

200

SHR3S1 EXP SHR3S1 FEA SHR3S2 EXP SHR3S2 FEA

100

300

100 0

0 0

0

2000 4000 6000 8000 10000 12000 14000 BFRP S!rrup Micro-Strain

2000

4000 6000 8000 10000 BFRP S!rrup Micro-Strain

12000

Large Beams

900 800 700 600 Load, kN

500 400 300 LHR3S1 EXP LHR3S1 FEA LHR3S2 EXP LHR3S2 FEA

200 100 0 0

2000

4000

6000

8000

10000 12000 14000

BFRP S!rrup Micro-Strain Fig. 11. Load versus stirrup strain from experimental and FEA. Table 3 Experimental and FEA results of beams without stirrup. No.

Specimen designation

1 2 3 4 5 6 7 8 9 10 11 12

SNR1 MNR1 LNR1 SHR1 MHR1 LHR1 SHR2 MHR2 LHR2 SHR3 MHR3 LHR3

FEA ultimate load PFEA, kN 110.4 122.5 162.8 129.6 161.2 206.3 165.3 178.4 246.7 208.5 259 316.2

Exp. ultimate load PExp, kN 107.4 121.6 136.6 132.4 151.4 185.2 172.4 190.2 230.7 213.1 277.5 331.7 Mean SD COV%

PExp/PFEA

FEA ultimate deflection yFEA, mm

0.97 0.99 0.84 1.02 0.94 0.90 1.04 1.07 0.94 1.02 1.07 1.05 0.987 0.073 7.34

the crack occurrence, sometimes the strain of stirrup was increased rapidly, and this can be seen in the curves of large beams. More significantly, the strains at ultimate load from the FEA were close to the corresponding experimental results.

16.1 12.3 13.7 16.2 15.9 13.5 15.8 13.9 11.7 8.6 9.2 9.8

Exp. ultimate deflection yexp, mm 16.5 11.5 10.5 17.4 14.2 13.4 17.6 12.95 11.8 10.4 9.95 8.9

yExp/yFEA 1.02 0.93 0.77 1.07 0.89 0.99 1.11 0.93 1.01 1.21 1.08 0.91 0.995 0.118 11.84

8.4. Failure load Table 3 and Table 4 present the ultimate loads and ultimate deflections of the beams obtained from the FEA simulations and the 246

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G.B. Jumaa and A.R. Yousif

Table 4 Experimental and FEA results of beams with stirrup. No.

Specimen designation

1 2 3 4 5 6

SHR3S1 MHR3S1 LHR3S1 SHR3S2 MHR3S2 LHR3S2

FEA ultimate load PFEA, kN

Exp. ultimate load PExp, kN

349.5 543.2 753.8 405.9 558.2 801.8

334.5 555 776 378.8 581.4 797 Mean SD COV%

PExp/PFEA

FEA ultimate deflection yFEA, mm

0.96 1.02 1.03 0.93 1.04 0.99 0.996 0.043 4.33

Exp. ultimate deflection yexp, mm

18.5 26.6 35.7 23.2 26.5 35.2

17.85 23.1 37.75 19.5 22.7 37.48

yExp/yFEA 0.96 0.87 1.06 0.84 0.86 1.06 0.942 0.102 10.82

Fig. 12. Comparison between tensile damage from FEA and crack patterns from experiments. Table 5 Details of the selected database and the results of the FEA model. Authors

Beam #

Bentz 2010

L-05 M-05 S-05 G800 G650 G500 G350 S1-0.12-1A S3-0.12-1A S6-0.12-2A B400-2 B300-2 B200-2

Alam and Hussein 2013

Matta, 2013 Ashour and Kara 2014

d, mm

bw, mm

a/d

fc′, MPa

ρf%

Ef, Gpa

ffu, MPa

Vexp, kN

VFEA, kN

VExp/VFEA

937 438 194 734 584 440 305 883 292 146 370 270 170

450 450 450 300 300 250 250 457 114 229 200 200 200

3.2 3.5 3.9 2.5 2.5 2.5 2.4 3.11 3.11 3.11 2.7 3.6 5.9

46 35 35 41.8 37 37.4 39.8 29.5 32.1 59.7 22 28.7 23.7

0.5 0.5 0.5 0.9 0.91 0.9 0.86 0.59 0.59 0.59 0.16 0.22 0.35

37 37 37 48 48 48 48 41 43 43 141 141 141

397 397 474 751 751 751 751 476 849 849 1100 1100 1100

135 86 54.5 129 104 77.2 61 154 19.2 28.6 32.8 32.8 17.6

158 102 60.7 128.8 104 77 66.6 155.6 19.9 28.6 34 27.8 15.8

0.852 0.843 0.898 0.994 0.994 1.003 0.916 0.947 0.965 1.00 0.965 1.18 1.114 0.978 0.094 9.60

247

Mean SD COV%

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G.B. Jumaa and A.R. Yousif

180

G350-EXP G350-FEA Series6 G500-FEA G650-EXP G650-FEA G800-EXP G800-FEA

160 140

Shear Load, kN

120 100 80 60 40 20 0 0

5

10 15 Mid-Span Deflec"on, mm

20

25

Fig. 13. Experimental and FEA model shear load-deflection curves for beams of Alam and Hussein 2013. Table 6 Details and results of beams used in parametric study. Beam #

h, mm

d, mm

bw, mm

L, mm

a/d

fc′, MPa

ρf%

Ef, Gpa

ffu, MPa

VFEA, kN

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

250 500 1000 1500 250 500 1000 1500 250 500 1000 1500 250 500 1000 1500 250 500 1000 1500

200 440 920 1380 584 440 305 883 292 146 370 270 200 440 920 1380 200 440 920 1380

400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400

1600 2320 3760 5140 1900 2980 7210 6520 2200 3640 5140 9280 1600 2320 3760 5140 1600 2320 3760 5140

3 3 3 3 4.5 4.5 4.5 4.5 6 6 6 6 3 3 3 3 3 3 3 3

40 40 40 40 40 40 40 40 40 40 40 40 80 80 80 80 120 120 120 120

1.13 1.13 1.13 1.15 1.13 1.13 1.13 1.15 1.13 1.13 1.13 1.15 1.13 1.13 1.13 1.15 1.13 1.13 1.13 1.15

60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60

1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

111.6 195.3 311.9 384 77 137.9 240 278 64.3 109.8 186.4 225 124 249 413 543 148.3 281.4 456 618

experimental results for beams without stirrup and with stirrup respectively. The comparisons are made in the term of ultimate load (PExp/PFEA) and ultimate deflections (yExp/yFEA). It can be seen that the FEA is accurately predicted the ultimate loads and ultimate deflections as the average ratio of (PExp/PFEA) and (yExp/yFEA) for the tested beams are 0.987 and 0.995 for beams without stirrup and are 0.996 and 0.942 for beams with stirrup respectively. The FEA model results confirm the power and ability of the calibrated model for predicting the shear failure in high strength concrete beams reinforced with BFRP bars without and with shear reinforcement. The importance of FEA as a simulation tool is that it can provide insight into shear failure and crack formation and allows for parametric studies, which cannot be obtained through experimental investigations. The presented analyses indicate that the proposed calibrated model can be used in parametric studies on different aspects influencing shear strength and size effect in FRP-RC beams without stirrups.

8.5. Crack pattern The ABAQUS FEA program does not have a tool for displaying cracks propagation at the concrete integration points or centroid of elements. Other parameters can be regarded as an indication of crack development like tensile damage, compressive damage, plastic strain or logarithmic strain. Therefore, in this study, it was assumed that the cracks initiated at the elements where tensile damage happened. Fig. 12 shows both tensile damage diagram from the FEA and crack pattern from experimental tests of beams SHR2, MHR2, and LHR2. The red elements indicate that the tensile damage ratios are > 80% in the element. By comparing the results of crack pattern and tensile damage, it can be observed that both of them to a large extent are similar. This similarity is more noticeable where the mesh size is refined as that in tensile damage diagram of beam MHR2. This comparison indicates that the ABAQUS can simulate the fracture mechanism of shear and flexural crack in the concrete beams.

248

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G.B. Jumaa and A.R. Yousif

0.25

450

h=250mm

400

h=500mm

0.2

a/d=4. 5

h=1000mm

350

h=1500mm

Shear Capacity, kN

Normalized Shear Strength (VFEA/√fc' bw d)

a/d=3

0.15

0.1

300 250 200 150 100

0.05

50 0 0

500

1000

1500

0

2000

0

2

4

6

8

a/d

Beam Depth, mm

(a)

(b) 700

h=250mm h=500mm

Shear Capacity, VFEA, kN

600

h=1000mm h=1500mm

500 400 300 200 100 0 0

20

40

60

80

100

120

140

fc', MPa

(c) Fig. 14. FEA model results (a) Normalized shear versus beam depth (b) Shear capacity versus shear-span to depth ratio (c) Shear capacity versus concrete compressive strength.

9. Verification of the calibrated FEA model

criteria and the same criteria for n values of softening tensile stressstrain response. The simple equation of ACI 318–14 was used to calculate the modulus of elasticity of concrete (Ec = 4700√fc′) based on the concrete compressive strength, while the tensile strength of concrete was calculated by (ft = 0.42√fc′). The properties of the beams and the results of the FEA are shown in Table 5. It is clear that the FEA model showed good agreement with the experimental results. The mean and COV% for the ratio of VExp/VFEA are 0.979 and 9.6% respectively. This indicates the high accuracy results of the calibrated FEA model in spite of the variability in the materials and geometry of the considered beams. The experimental results of shear load-deflection behavior are only available for beams tested by Alam and Hussein [3]. Therefore, the comparison of shear load-deflection behavior is just presented for this study as shown in Fig. 13.

In the previous sections, an FEA model was developed and calibrated using the experimental results including the consideration of the size effect parameter. To further verify this model and measure its prediction accuracy, it was used to predict the shear capacity of beams from other studies and then it was compared with the experimental results. For this purpose, only that studies were selected which investigated the effect of beam depth on the shear strength of FRP-RC beams without stirrups. Therefore, the results of 13 beams with different sizes were selected from four available studies on size effect [2–4,41]. The same parameters and criteria of the calibrated FEA model were used as a generalized model which includes the same CDPM material parameters (σbo/σco, Kc, ϵ, μ and ψ), the same mesh size 249

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G.B. Jumaa and A.R. Yousif

3.00

ACI 440

2.50

2.50

2.00

2.00

VFEA/Vpred

VFEA/Vpred

3.00

1.50 1.00

CSA S806-12

1.50 1.00

0.50

0.50

0.00

0.00 0

500

1000

1500

0

500

d, mm 3.00

CSA S6S1

2.50

2.50

2.00

2.00

VFEA/Vpred

VFEA/Vpred

3.00

1.50 1.00

1.00

0.50

0.50

ISIS-07

0.00 0

500

1000

1500

0

d, mm 3.00

500

d, mm

3.00

Jumaa and Yousif 2018

2.50

2.50

2.00

2.00

VFEA/Vpred

VFEA/Vpred

1500

1.50

0.00

1.50

1000

1500

Razaqpur & Spadia 2010

1.50

1.00

1.00

0.50

0.50 0.00

0.00 0

500

d,mm

3.00

1000

0

1500

500

1000

1500

d, mm 3.00

Hegger et al 2009

2.50

2.50

2.00

2.00

VFEA/Vpred

VFEA/Vpred

1000 d, mm

1.50 1.00 0.50

Alam & Hussein 2013

1.50 1.00 0.50

0.00

0.00 0

500

1000

1500

0

500

1000

1500

d, mm

d, mm

Fig. 15. VFEA/Vpred for different models versus beam effective depth.

This figure shows a good simulation of the FEA results with that of the experimental results in terms of ultimate shear strength and deflection behavior. However, some deviation can be observed in beam G350 which may be due to the poor measurement of deflection during experimental testing. Finally, it can be concluding that the calibrated FEA model which considers the size effect showed sufficient accuracy and excellent performance in simulating the shear strength and behavior of FRP-RC beams.

10. Parametric study As one of the best advantages of FEA is the simulation of complicated and expensive tests and up to date there are no experimental tests on beams larger than 1000 mm and large beams with shear span to depth ratio > 3.2. Therefore, by using the numerical model described previously, a parametric study was carried out to investigate the shear capacity of large beams with a large shear-span-to-depth ratio. The details of the beams used for the parametric study are given in Table 6. 250

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G.B. Jumaa and A.R. Yousif

3.00

3.00

CSA S806-12

2.50

2.50

2.00

2.00 VFEA/Vpred

VFEA/Vpred

ACI 440

1.50 1.00 0.50

1.00 0.50

0.00

0.00 0

2

4 a/d

3.00

6

8

0

4 a/d

2.50

2.00

2.00

1.50

8

1.50

1.00

1.00

0.50

0.50 0.00 0

2

3.00

4 a/d

6

0

8

2

3.00

Jumaa and Yousif 2018

2.50

2.00

2.00

VFEA/Vpred

2.50

1.50 1.00

4 a/d

6

8

Razaqpur& Spadea 2010

1.50 1.00 0.50

0.50

0.00

0.00 0

2

4 a/d

6

0

8

2

4

6

8

a/d 3.00

3.00

Alam & Hussein 2013

Hegger et al 2009 2.50

2.50

2.00

2.00 VFEA/Vpred

VFEA/Vpred

6

ISIS-07

2.50

0.00

VFEA/Vpred

2

3.00

CSA S6S1

VFEA/Vpred

VFEA/Vpred

1.50

1.50 1.00 0.50

1.50 1.00 0.50

0.00

0.00 0

2

4

6

8

0

a/d

2

4 a/d

6

8

Fig. 16. VFEA/Vpred for different models versus shear span to depth ratio.

Three main variables were considered, the beam depth of 250, 500, 1000 and 1500 mm, the shear span to depth ratio of 3, 4.5 and 6 and the concrete compressive strength of 40, 80 and 120 MPa. The results of the FEA model for the selected three parameters are shown in Fig. 14. It is clear that there is a significant reduction in shear strength as the member depth increases which is averagely reduced about 20, 36 and 49% for beam effective depth of 440, 920 and 1380 mm respectively with respect to 200 mm beam effective depth. These results are in agreement with the results performed by [2–4,14]. Fig. 14(b) shows a considerable reduction in shear capacity as the shear span to depth ratio

increased. It was found that the shear capacity is proportional to a/d raised to a power of about (−2/3) which is more than the results of some previous studies. Some codes and proposed equations from the literature defined the effect of a/d raised to a power of (−1/3). While Hegger et al. [42] defined this power as (−1) and Alam and Hussein [3] defined it as (−2/3). Therefore, more studies are needed both experimentally and numerically to confirm these results. On the other hand, the effect of concrete compressive strength is shown in Fig. 14(c). It was found that the shear capacity is proportional to the concrete compressive strength raised to a power of about (1/3). The results are in 251

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G.B. Jumaa and A.R. Yousif

ACI 440

2.50

2.50

2.00

2.00

1.50

1.50

1.00

1.00

0.50

0.50

0.00

0.00 0

50

100

150

0

fc', MPa 3.00

VFEA/Vpred

CSA S806-12

3.00

VFEA/Vpred

VFEA/Vpred

3.00

50

2.50

2.50

2.00

2.00

VFEA/Vpred

1.50 1.00

150 ISIS 2007

1.50 1.00 0.50

0.00

0.00 0

50

100

150

0

50

fc', MPa 3.00

100

150

fc', MPa Razaqpur & Spadea 2010

3.00

Jumaa and Yousif 2018

2.50

2.50

2.00

2.00

VFEA/Vpred

VFEA/Vpred

100

3.00

CSA S6S1

0.50

1.50

1.50

1.00

1.00

0.50

0.50

0.00

0.00 0

50

fc', MPa

100

0

150

50

100

Alam & Hussein 2013

3.00 2.50

2.00

2.00

VFEA/Vpred

2.50

1.50 1.00 0.50

150

fc', MPa

Hegger et al 2009

3.00

VFEA/Vpred

fc', MPa

1.50 1.00 0.50

0.00

0.00 0

50

100

150

fc', MPa

0

50

fc', MPa

100

150

Fig. 17. VFEA/Vpred for different models versus shear span to depth ratio.

agreement with the most codes and proposed equations which defined this power as (1/5, 1/3 or 1/2). To further assess the FEA model, the results of the selected beams were compared with the results of four codes [43–46] and four proposed equations from the literature [3,42,47,48]. The comparison was made in the term of VFEA/Vpred versus beam effective depth, shear span to depth ratio and concrete compressive strength as shown in Figs. 15, 16, and 17 respectively. The ratio of VFEA/Vpred is significantly decreased with increasing the beam depth for ACI440-15 and to a lower

degree for CSA S806-12 as these two codes do not account for the effect of beam depth. While for all other codes and proposed equations the ratio of VFEA/Vpred is approximately constant throughout the different values of beam effective depth. The ratio of VFEA/Vpred is decreased with increasing the shear span to depth ratio for most models except for Hegger et al. [42] and Alam and Hussein [3] as these two equations are defined the effect of shear span to depth ratio raised to a power of (−1) and (−2/3) respectively. The reason behind this significant reduction in VFEA/Vpred is due to that 252

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G.B. Jumaa and A.R. Yousif

some of these codes and equations do not account for the effect of a/d and others defined its effect raised to a power in the range of (0.3–0.4). In contrary, the FEA model showed a significant effect of shear span to depth ratio on the results of shear capacity for the beams selected in this parametric study as illustrated in Fig. 16. On the other hand, the ratio of VFEA/Vpred is approximately constant throughout the different values of concrete compressive strength for all the codes and design equations. The ratios of VFEA/Vpred are larger than one for the majority of codes and design equations and for all selected parameters. In contrary, this ratio for large beams and large shear span to depth ratio are less than one for ACI 440-15 and CSA S806-12, in spite of the large margin of safety of ACI 440-15 design provision. These results are attributed to the combined effect of beam effective depth and shear span to depth ratio as these two factors are not considered in the ACI 440-15 design provision. While CSA S806-12 accounts only for the shear span to depth ratio as well as has less margin of safety in comparison to ACI 440-15. Finally it can be concluding that the parametric study for the proposed FEA model showed good performance in predicting the effect of different parameters on shear capacity of FRPRC beams; however, its performance in predicting the effect of shear span to depth ratio need more studies to confirm it.

[3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

11. Conclusions

[15]

Based on the outcomes of the numerical modeling of size effect in the shear failure of FRP-RC beams with and without stirrups and the results of the parametric study, the following points can be drawn:

[16]

1. The calibrated FEA model by ABAQUS for beams with and without stirrups showed great power and ability in simulating the overall experimental results including; ultimate shear capacity, ultimate deflection, load-deflection behavior and crack patterns. 2. The flexural strain behavior from the FEA showed close agreement with the experiments; however, the FEA trends were slightly less than that of the experiments which may be due to; perfect bond assumption, tying the steel plate elements with the concrete elements and the presence of micro-cracks in original beams. 3. The investigation of mesh size sensitivity showed that most accurate results could be obtained by meshing the depth of the beams into about 10, 14 and 16 elements for small, medium, and large beams. 4. The softening tensile stress-strain response was successfully calibrated to simulate the size effect such that in the actual experimental tests. A new model was suggested to represent the decay response of post-peak concrete tensile strength. 5. The calibrated FEA model was adopted to predict the shear capacity of (13) beams from four different sources. The comparison of the experimental to predicted results showed high accuracy and excellent performance in simulating the shear strength and behavior of FRP-RC beams. 6. A parametric study was conducted to validate the performance of the FEA model and the results were compared with design codes and proposed equations from the literature. The FEA model showed good performance and exceptional ability in predicting the effect of different parameters on shear capacity of FRP-RC beams, as well as it yields more accurate results than the design provisions.

[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

Funding

[34]

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

[35]

References

[36] [37]

[1] Yu Q, Bažant ZP. Shear strength of reinforced concrete beams: size effect and its fracture-mechanics basis. Special Publication 2015;300:1–32. [2] Bentz EC, Massam L, Collins MP. Shear strength of large concrete members with FRP

[38]

253

reinforcement. Journal of Composites for Construction 2010;14(6):637–46. https:// doi.org/10.1061/(ASCE)CC.1943-5614.0000108. Alam MS, Hussein A. Size effect on shear strength of FRP reinforced concrete beams without stirrups. Journal of Composites for Construction 2013;17(4):507–16. https://doi.org/10.1061/(ASCE)CC.1943-5614.0000346. Matta F, E-S.A.K., Nanni Antonio, Benmokrane Brahim. Size effect on concrete shear strength in beams reinforced with fiber-reinforced polymer bars. ACI Structural Journal 2013;110-S50(4):617–28. Jumaa GB, Yousif AR. Size effect in shear failure of high strength concrete beams without stirrup reinforced with basalt FRP bars. KSCE Journal of Civil Engineering 2019;23(4):1–15. Thorenfeldt E, Drangsholt G. Shear capacity of reinforced high-strength concrete beams. Special Publication 1990;121:129–54. Stoner J. Finite element modelling of GFRP reinforced concrete beams. University of Waterloo; 2015. Syroka-Korol E, Tejchman J, Mróz Z. FE analysis of size effects in reinforced concrete beams without shear reinforcement based on stochastic elasto-plasticity with non-local softening. Finite Elements in Analysis and Design 2014;88:25–41. Yu Q, Bažant ZP. Can stirrups suppress size effect on shear strength of RC beams? Journal of Structural Engineering 2011;137(5):607–17. Sato Y, Tadokoro T, Ueda T. Diagonal tensile failure mechanism of reinforced concrete beams. J Adv Concrete Technol 2004;2(3):327–41. Kotsovos M, Pavlović M. Size effects in structural concrete: a numerical experiment. Comput Struct 1997;64(1–4):285–95. An X, Maekawa K, Okamura H. Numerical simulation of size effect in shear strength of RC beams. Doboku Gakkai Ronbunshu 1997;1997(564):297–316. DSS. D.S.S., ABAQUS [6.9-2], in Providence, RI, USA. 2009. Jumaa GB, Yuosif AR. Size effect in shear failure of high strength concrete beams without stirrups reinforced with basalt FRP bars. KSCE Journal of Civil Engineering 2019;23(4):1636–50. https://doi.org/10.1007/s12205-019-0121-3. Jumaa GB, Yousif AR. Size effect on the shear failure of high-strength concrete beams reinforced with basalt FRP bars and stirrups. Construct Build Mater 2019;209:77–94. Razaqpur AG, Shedid M, Isgor B. Shear strength of fiber-reinforced polymer reinforced concrete beams subject to unsymmetric loading. Journal of Composites for Construction 2011;15(4):500–12. https://doi.org/10.1061/(ASCE)CC.1943-5614. 0000184. Collins MP, Kuchma D. How safe are our large, lightly reinforced concrete beams, slabs, and footings? Structural Journal 1999;96(4):482–90. Bažant ZP, Ožbolt J, Eligehausen R. Fracture size effect: review of evidence for concrete structures. Journal of Structural Engineering 1994;120(8):2377–98. Belarbi A, Hsu TT. Constitutive laws of concrete in tension and reinforcing bars stiffened by concrete. Structural Journal 1994;91(4):465–74. Hsu TT, Zhang L-X. Tension stiffening in reinforced concrete membrane elements. Structural Journal 1996;93(1):108–15. Wang T, Hsu TT. Nonlinear finite element analysis of concrete structures using new constitutive models. Comput Struct 2001;79(32):2781–91. Tamai S. Average stress-strain relationship in post yield range of steel bar in concrete. Concrete Lib JSCE 1988(11):117–29. Kmiecik P, Kamiński M. Modelling of reinforced concrete structures and composite structures with concrete strength degradation taken into consideration. Archives of civil and mechanical engineering 2011;11(3):623–36. MacGregor JG, et al. Reinforced concrete: mechanics and design. vol. 3. Upper Saddle River, NJ: Prentice Hall; 1997. Genikomsou AS, Polak MA. Finite element analysis of punching shear of concrete slabs using damaged plasticity model in ABAQUS. Eng Struct 2015;98:38–48. Obaidat YT. Structural retrofitting of reinforced concrete beams using carbon fibre reinforced polymer. Department of Construction Sciences, Structural Mechanics; 2011. Hu H-T, Lin F-M, Jan Y-Y. Nonlinear finite element analysis of reinforced concrete beams strengthened by fiber-reinforced plastics. Composite Structures 2004;63(3–4):271–81. Zhang L, Sun Y, Xiong W. Experimental study on the flexural deflections of concrete beam reinforced with Basalt FRP bars. Materials and Structures 2015;48(10):3279–93. Chen W-F. Plasticity in reinforced concrete. J. Ross Publishing; 2007. Saenz LP. Equation for the stress-strain curve of concrete. ACIJour 1964;61(9):1229–35. Hu H-T, Schnobrich WC. Constitutive modeling of concrete by using nonassociated plasticity. Journal of Materials in Civil Engineering 1989;1(4):199–216. Malm R. Predicting shear type crack initiation and growth in concrete with nonlinear finite element method. KTH; 2009. Trotsek D. Flexural behavior of concrete using basalt FRP rebar. Florida Atlantic University; 2017. Chen G, Teng J, Chen J. Finite-element modeling of intermediate crack debonding in FRP-plated RC beams. Journal of Composites for Construction 2010;15(3):339–53. Slobbe A, Hendriks M, Rots J. Systematic assessment of directional mesh bias with periodic boundary conditions: applied to the crack band model. Engineering Fracture Mechanics 2013;109:186–208. Malm R. Shear cracks in concrete structures subjected to in-plane stresses. KTH; 2006. Lee J, Fenves GL. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 1998;124(8):892–900. Henriques J, Silva LS da, Valente IB. Numerical modeling of composite beam to reinforced concrete wall joints: part I: calibration of joint components. Eng Struct

Structures 20 (2019) 237–254

G.B. Jumaa and A.R. Yousif 2013;52:747–61. [39] Sümer Y, Aktaş M. Defining parameters for concrete damage plasticity model. Challenge Journal of Structural Mechanics 2015;1(3):149–55. [40] Ismail KS, Guadagnini M, Pilakoutas K. Numerical investigation of the shear strength of rc deep beams using the microplane model. Journal of Structural Engineering 2016;142(10):04016077. [41] Ashour AF, Kara IF. Size effect on shear strength of FRP reinforced concrete beams. Compos Part B Eng 2014;60:612–20. https://doi.org/10.1016/j.compositesb.2013. 12.002. [42] Hegger J, Niewels J, Kurth M. Shear analysis of concrete members with fiber-reinforced polymers (FRP) as internal reinforcement. FRPRCS-9, Sydney, Australia. 2009. [43] CSA. Code for the design and construction of building structures with fibre-reinforced polymers. CAN/CSA/S806-12. Toronto, Ontario, Canada: Canadian Standard Association; 2012. 198 pp.

[44] CSA, Canadian Highway Bridge Design Code. Supplement no. 1 to CAN/CSA S606(S6.1S1-10). Mississauga, Ontario, Canada: Canadian Standards Association (CSA); 2010. [45] ACI-440. Guide for the design and construction of structural concrete reinforced with fiber-reinforced polymer (FRP). Farmingtons, Hills, MI: ACI 440. 1R-15, American Concrete Institute; 2015. [46] ISIS. Reinforced concrete structures with fibre reinforced polymers – design manual no. 3. Manitoba, Canada: ISIS Canada Corporation: University of Manitoba; 2007. [47] Razaqpur A, Spadea S. Resistenza a taglio di elementi di calcestruzzzo reinforzatti e staffe di FRP. Paper presented at 39th AIAS National Conference. Italy: Maratea; 2010. [48] Jumaa GB, Yousif AR. Predicting shear capacity of FRP-reinforced concrete beams without stirrups by artificial neural networks, gene expression programming, and regression analysis. Advances in Civil Engineering 2018;2018.

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