Nonlinear finite element analysis of high and ultra-high strength concrete beams reinforced with FRP bars

ABSTRACT A finite element (FE) model is carried out to study the bending behavior of concrete beams constructed of high and ultra-high strength concrete (HSC/UHSC) under three-point bending. The simulated beams are reinforced with longitudinal reinforcement bars from fiber reinforced polymer (FRP) material. The ABAQUS FE software is used in the modeling. The FE model was calibrated using the experimental data of the specimens carried out by M.W. Goldston et al. (2017). Twenty-eight validated FE models were performed to study the effects of concrete compressive strength (high and ultra-high), type of FRP longitudinal bars (AFRP, BFRP, CFRP and GFRP), main FRP reinforcement ratio (0.74%, 1.16% and 1.67%), compression FRP reinforcement ratio (As’ %; 25%, 40%, 100%), and both of beam width and depth (b and d). From the results; it is found that the concrete strength has a small effect on the ultimate capacity. However, the deflection is higher for beams with lesser strength. For all the cases, CFRP reinforced beams showed higher capacity and lesser ductility, while GFRP reinforced beams showed lesser capacity and higher ductility. As’ ratio has insignificant effect on load capacity, but it can improve deflection; it is recommended that As’ did not exceed 40% from AFRP. The beam width has insignificant effect on carrying capacity, but it decreases the corresponding deflection, while increasing the beam depth led to increase the maximum load and decrease the corresponding deflection. Comparable error is generally obtained when using the both equations of Issa and Issa (2017) and ACI440.1 (2015).


Introduction
Steel corrosion is one of the primary reasons of structural collapse.When exposed to harsh conditions, corrosion of steel bars increases, causing degradation of the structure's integrity and limiting its service life.Conventional reinforced concrete (RC) buildings are often affected by corrosion due to the presence of chloride ions.In addition, the high alkalinity of sea water can cause severe damage to concrete structures.Reinforced bars made from fiber polymer (FRP) bars are utilized to substitute steel bars in many RC constructions, due to FRP perfect corrosion resistance, electrical insulation, high tensile strength, and weight to strength ratio.FRP bars are frequently utilized in concrete structures as longitudinal or transverse reinforcement include aramid FRP (AFRP), basalt FRP (BFRP), carbon FRP (CFRP) and glass FRP (GFRP) [1][2][3][4][5][6][7][8][9][10][11][12][13].
The finite element method is a powerful tool for analyzing reinforced concrete structures.It can take into account the material behavior of concrete and reinforcement, as well as cracking and bond-slip.However, solutions made using the finite element method can be time-consuming, and expert evaluation of the results may be necessary.Thus, the finite element approach is generally used to increase the range of experimental data and design unusual structures.The finite element method is mainly used for threedimensional (3D) analysis, rather than two-dimensional (2D) analysis.This is because 2D analysis only takes into account plane strain and plane stress, which are not always valid.Three-dimensional analysis requires more computational power than two-dimensional analysis, but it is more accurate.When modeling the nonlinear behavior of concrete in compression, plasticity or damage (or both) are typically used.Plasticity assumes that deformations are permanent after loads are removed.However, damage only reduces the stiffness of the material [14][15][16][17][18][19][20][21].
In order to develop reliable codes for fiber reinforced polymer bars (FRP) as flexural reinforcement, we need to study how concrete beams react when reinforced with these bars.The current research is addressing some of the parameters that affect the bending behavior of beams reinforced with FRP bars utilizing the ABAQUS software [21].A parametric study using calibrated FE model for the effect of; concrete strength, both of FRP bars types and ratios, and width to depth ratio (b/d) on the bending behavior of beams constructed of high and ultra-high strength concrete and reinforced with FRP bars will be performed.This will be followed by the application of two analytical methods, (the equations of Issa and Issa (2017) [9] and the equations of ACI440.1 (2015) [8]) for calculating the load-deflection curves of FRP bars reinforced concrete beams; which will be compared with the FE results.

Details of simulated beams
ABAQUS/CAE-(2021) [21] software was used to investigate the bending behavior of high and ultra-high strength concrete beams reinforced with FRP bars.The simulated model was developed to validate the test results obtained by M.W. Goldston et al. (2017) [1].The specimens were 2400 mm overall span, 1000 mm shear span, and their cross-sectional dimensions were 100 mm width and 150 mm height.The beam extends 200 mm from each side beyond the supports to allow for the development length of the FRP bars.The clear concrete cover to the stirrup from each side is 15 mm.All specimens have 2 FRP longitudinal reinforcement bars in both tension and compression zones, and 4 mm diameter steel stirrups every 50 mm in the shear zone; see Figure 1.

Materials of the FE model
3D FE models were introduced using ABAQUS software version 2021 [21].The modeling materials are discussed as follows:

Concrete
The damage plasticity model of concrete in ABAQUS was used to simulate concrete behavior.Compression crushing and tensile cracking are two failure mechanisms considered in this model [21].A linear relationship between stress and strain is considered up to 40% of peak strength (f c ') for uniaxial compression behavior of high strength concrete.Furthermore, the modified formula by Wee et al. (1996) [19] was used to evaluate compressive stress-strain curve of both the high-strength and ultra-high-strength concrete material.The stress versus inelastic strain curves of concrete are given in Figure 2.
The tensile behavior of concrete is considered to be linear-elastic until the crack initiation, which corresponds to the tensile strength (ft').The bilinear model was used to define the concrete constitutive relationship in tension (Coronado and Lopez, 2006) [20].The tensile stress-cracking displacement curves are presented in Figure 3.

Steel and FRP Reinforcement
The relationship between stress and strain for the mild-steel bars was considered to be elastic-perfectly plastic.While for FRP bars the stress-strain curve was considered to be linear up to rupture strain, where the loading capacity was lost, as illustrated in Figure 4.

Beam model
A 3D ABAQUS/Standard was used to simulate the FRP reinforced HSC/UHSC beams.Figure 5.a.presents the meshes, and geometry of the modeled beams, while the shear and longitudinal reinforcement, loading plate, and support plates of the same beam are shown in Figure 5.b.One support was considered a hinge, while the other was considered a roller.As part of a displacement control analysis, the top of the loading plate in the opposite of Y direction was used to apply displacement load.The concrete and plates  were selected as a reduced integration C3D8R element type.While, both the steel and FRP reinforcement are modeled as a 2-node linear 3-D truss element (T3D2).A full bond between all reinforcement and concrete was considered.

Validation of FE model
The proposed model was calibrated based on the experimental results obtained from M.W. Goldston et al. (2017) with respect to failure-mode, ultimate capacity, and deflection at ultimate capacity.The mode of failure for FE model beams was compatible with that obtained from experimental test.Also, the numerical results were found to agree with the experimental ones.The FE model can be used to evaluate the bending behavior of HSC/ UHSC beams reinforced with any type of FRP bars.Both numerical and experimental results of comparison are given in both Table 1 and Figure 6.

Parametric studies using validated model
The validated FE model was used to investigate the influence of concrete compressive strength (40 MPa, 80 MPa, and 120 MPa), FRP types (AFRP, BFRP, CFRP, and GFRP) and ratio of FRP reinforcement on the bending behavior of HSC/UHSC beams reinforced with FRP bars.Twenty-eight reinforced concrete beams are modeled.Twelve of them were of concrete cube compressive strength equal to 120 MPa, twelve were of 80 MPa and four were of 40 MPa.
For each strength, the beams were consisting of three groups of four beams each.These four beams were reinforced with either AFRP, BFRP, CFRP or GFRP.The three groups differ in the reinforcement ratios, which were two bars of diameter 8 mm, 10 mm or 12 mm.Beam 120-2#8-AFRP, for example, is of concrete cube compressive strength = 120 MPa, and contains two AFRP bars of 8 mm diameter at both top and bottom.The equal top and bottom reinforcement simulates the experiment used to calibrate the finite element models and also controls the deflection.The transverse reinforcement consists of 4 mm diameter of mild-steel stirrups.This transverse reinforcement was spaced at 50 mm to guarantee flexural failure.The FE model results of the parametric study are illustrated in Table 2.

Failure mode
The failure shapes of GFRP reinforced concrete beams were due to concretecrushing for both concrete compressive strengths and GFRP reinforcement ratios.While for beams reinforced with CFRP bars, the failure mode was result of CFRP-rupture.In addition, the failure mode of beams reinforced with AFRP/ BFRP bars varies between FRP-rupture and concrete crushing depending on compressive strength of the concrete and ratios of AFRP/BFRP reinforcement.All the beams which failed due to FRP-rupture also reached concrete crushing.Figure 7 shows an example of the failure mode of beams reinforced with;

Effect of Compression FRP area ratio (As' %)
To study the effect of compression FRP area (As') ratio, as a percent from main FRP reinforcement area, on the ultimate capacity, deflection, and strain in both bottom and top FRP bars, three ratios of 25 %, 40% and 100% from the main reinforcement area were chosen.The concrete compressive strength was 80 MPa and main reinforcement ratio was 1.67%.The FE model results shows insignificant effect of As' ratio in ultimate capacity and deflection except the deflection of beam reinforced with CFRP bars decreases with   increasing As' ratio.Also, the top FRP bar strain decreases with increasing the As' ratio.In addition, for good performance and economic stage, the compression reinforcement ratio As' should not exceed 40% from the main reinforcement area.The FE model results of effect of As' are given in Table 3.

Influence of ratio of FRP reinforcement
To evaluate the effect of the ratio of FRP on both the load capacity and maximum deflection at midspan of the HS/UHS beams; the validated FE model was used.The FRP reinforcement ratios were 0.74%, 1.16% and 1.67%, for all types of FRP (AFRP, BFRP, CFRP and GFRP).The f cu of concrete were 80 MPa and 120 MPa.From the results shown in Figure 9, it can be noticed that the midspan deflection of the beams reinforced with CFRP had the smallest value, followed by BFRP and AFRP, while the highest value of midspan deflection was recorded for the beam reinforced with GFRP bars, as shown in Figure 9(a, b).While the rupture load values of the beams reinforced with CFRP or AFRP were higher than the beams reinforced with BFRP and GFRP, respectively, due to the effect of tensile stress and modulus of elasticity of FRP bars, as shown in Figure 9(c, d).

Effect of compressive strength of concrete
Figure 10 shows the effect of the f cu of the concrete on the behavior of concrete beams reinforced with two FRP bars of diameter 10 mm (AFRP, BFRP, CFRP and GFRP).Almost all the specimens attained a similar maximum load as the flexural capacity is dependent on the ratio and type of FRP reinforcement.However, beams with lesser concrete compressive strength attained larger deflection at the same load.

The influence of beam width to depth ratio
The validated FE model was used to study the effect of beam width to depth ratio of the concrete beam; three different beam width (b/d = 0.74, 1.1 and  4 and Table 5.

Equations used to predict the deflection
The following equation was used to estimate the immediate deflection of a simply supported RC beam loaded at mid span with concentrated load: Where M a = maximum service moment of the beam L = total span of the beam I e = effective moment of inertia of the beam The actual stiffness of the beam varies between the gross stiffness of the uncracked section and the stiffness of the cracked section.The ACI440.1 R-15 [10] gives the following equation for the (I e ) to calculate the immediate deflection.
Where, I g = gross moment of inertia of the uncracked cross-section I cr = moment of inertia of the cracked cross-section M cr = bending moment at cracking = f r * I g /y t f r = tensile strength f r ¼ 0:62 ffi ffi ffiffi f 0 c p y t = half the beam depth Issa and Issa (2017) [9] suggested another effective moment of inertia formula, which has the following form: Where ρ F = FRP reinforcement ratio

Evaluating the deflection equations against the finite element results
The two above-mentioned equations for calculating the deflections are evaluated against the finite element deflection curves obtained for the twentyeight beams.Table 6, gives the deflections at ultimate loads obtained when using each of the two equations for each beam.The method of Issa and Issa (2017) gives an average error, compared to the finite element, which is comparable to that obtained by the method of ACI440.1 (2015), compared to the finite element models.This error is explained by the approximations of the design codes.

Comparison between the capacities from FE and ACI440.1 (2015)
The bending capacities of the beams are calculated according to ACI440.1 (2015) and compared against the finite element value for the twenty-eight beams.Table 7, gives these values for each beam.Generally, the ACI440.1 (2015) values are on conservative side.This is explained by both the conservative nature of the design codes and the inherent approximations.

Conclusions
This study investigates the bending behavior of high and ultra-high concrete beams with two concrete compressive strengths of 80 MPa and 120 MPa, reinforced with the most commonly used FRP bars (AFRP, BFRP, CFRP and GFRP).The following conclusions based on finite element model results may be noted: 1 The investigation of the behavior of high and ultra-high strength concrete (HSC/UHSC) beams reinforced with FRP bars can be carried out using the FE model.
2. CFRP reinforced beams sustain the highest load and produce the lowest deflection, while BFRP reinforced beams sustain the lowest load.The maximum deflection is for beams reinforced by GFRP bars.
3. The increasing percentage of compression FRP ratio As' led to improve the deflection of the beam but it had insignificant effect of carrying capacity.It is recommended that As' did not exceed 40% from the main FRP reinforcement area.
4. The concrete compressive strength has a small effect on the maximum beam capacity; however, it affects the deflection.Beams with smaller strength deflect larger.
5. For the same FRP type, the FRP reinforcement ratios have a significant effect on failure load, midspan deflection and mode of failure of the beams.6. Changing the beam width (b) has insignificant effect on carrying capacity and decreases the corresponding deflection depending on the type of FRP.
7. The increasing depth (d) of high strength concrete beams increases their maximum load and decreases the corresponding deflection of FRP bars reinforced HSC/UHSC beams.Also increasing beam depth with constant other variables (L, As, As', and b) has a significant effect on the mode of failure of AFRP and BFRP reinforced HSC/UHSC beams.
8. The method of Issa and Issa (2017) for calculating the deflection gives comparable error to the ACI440.1 (2015) method based on the deflection obtained from the FE model.
a. Concrete Compressive Strength f'c = 32 MPa b.Concrete Compressive Strength f'c = 64 MPa c. Concrete Compressive Strength f'c = 96 MPa

Figure 8
Figure8shows the load-deflection relationships obtained by ABAQUS for beams made of concrete with 80 MPa and 120 MPa compressive strengths, and reinforced by two bottom and two top FRP bars of 10 mm diameter.Four beams are reinforced by FRP bars of the following types; AFRP, BFRP, CFRP and GFRP.At the beginning, the four curves are identical until cracking, after which they differ as the capacity is more dependent on FRP type.The beam with CFRP bars sustained the highest load, followed by the beam with AFRP bars with small difference.This is explained by the high and close values of the tensile strength of these two types of bars.The beams of GFRP and BFRP bars recorded a smaller capacity, as explained by their lower tensile strength.The deflection was the smallest for the beam with CFRP bars.This is explained by the high value of the elastic modulus for this type of bars.The highest value of deflection was recorded for the beam with GFRP bars as a reason for the lowest modulus of elasticity of this type of bars.Intermediate values of deflection were

Figure 8 .
Figure 8.Effect of FRP types on load-deflection of HS/UHS beams.

Table 1 .
Experimental results versus FE model results.
varying FRP bars types, FRP reinforcement ratio of 1.16%, and varying compressive strengths of concrete.

Table 3 .
Effect of As' ratio.
recorded for beams with BFRP and AFRP bars, as explained by the intermediate values of the elastic modulus of their reinforcements.

Table 4 .
FE model results for the influence of beam width (b).

Table 5 .
FE model results for the influence of beam depth (d).

Table 6 .
Deflection at ultimate load comparison.