# Ductility Estimation for Flexural Concrete Beams Longitudinally Reinforced with Hybrid FRP–Steel Bars

^{*}

## Abstract

**:**

_{f}/A

_{s}) increased. This study also showed that the developed numerical model could predict the flexural behavior of beams with hybrid reinforcement with reasonable accuracy. Based on the test results, parametric analysis, and data obtained from the literature, the use of the neutral axis angle and displacement index value to evaluate the ductility of cross-sections with hybrid reinforcement is proposed.

## 1. Introduction

_{f}/A

_{s}, must be designed in the range of 1–2.5 to ensure that the beam retains strength with sufficient ductility and stiffness after exceeding the elastic phase.

_{f}/A

_{s}reinforcement ratio is small.

_{s}E

_{s}/A

_{f}E

_{f}). They also reported that the addition of axial stiffness is not proportional to the increase in moment capacity.

_{f}/A

_{s}) increases, the determination of the ductility becomes more complex because the yield position of the steel is not clear. Although previous researchers have investigated many attributes of the flexural behavior of hybrid reinforced beams, some features of the performance of these beams are still unclear. In particular, the effects of the hybrid reinforcement ratio (A

_{f}/A

_{s}) and the arrangement of the hybrid reinforcements on the flexural performance of reinforced concrete beams with hybrid FRP–steel bars have not been thoroughly investigated.

## 2. Experimental Study

_{y}) of 375 MPa, GFRP bars with a diameter of 13 mm, an ultimate tensile strength (f

_{fu}) of 788 MPa, and a modulus of elasticity (E

_{f}) of 43.9 GPa, and CFRP bars with a diameter of 13 mm, a maximum tensile strength (f

_{fu}) of 2070 MPa, and a modulus of elasticity (E

_{f}) of 124 GPa. The ratios of FRP reinforcement to steel reinforcement were 0.5 and 2.0. The GFRP and CFRP rods used, shown in Figure 2a, were supplied by FYFE Co. LLC from the USA. The mechanical properties of the FRP rods used in this study were obtained from the leaflet issued by the manufacturer. Transverse reinforcements with a diameter of 10 mm and yield stress of 454 MPa were used for all specimens. The properties of the materials used in this study as determined via experimental studies, parametric studies, and data adopted from the literature are shown in Table 1. Fresh concrete was ordered from a ready-mix company and the compressive strength (f

_{c}’) of the concrete at 28 days was 20 MPa.

## 3. Analytical Study

_{i}) can then be calculated using the distance from each element to the top of the cross-section (y

_{i}) and by assuming curvature (μ) (Equation (1)).

_{i}) is calculated using the stress on (σ

_{i}) and the area of (A

_{i}) each element. The stress is obtained from the nonlinear material stress–strain relationship inputted in the previous step. The material stress–strain models used in this study are shown in Figure 4.

_{o}) on the centroid axis. After the equilibrium condition is met, the moment (M) at each load step is obtained by multiplying the obtained internal forces by the distance from each element to the top of the cross-section (Equation (4)).

_{f}/A

_{s}) data that could not be determined from experimental studies or the literature. There were 12 specimens in the parametric study, with hybrid reinforcement ratios ranging from 1.3 to 4.3. With these data, it was expected that the behavior of cross-sections with higher hybrid reinforcement ratios could be represented. Based on the number of layers of tensile reinforcement, the beam cross-sections in the parametric study consisted of the same two types as in the experimental research, namely, Type I and Type II. This parametric study used two FRP materials (GFRP and CFRP).

## 4. Results and Discussion

#### 4.1. Crack Patterns and Failure Modes of the Tested Beams

_{f}/A

_{s}). The higher the hybrid reinforcement ratio, the lower the flexural cracking in the constant moment zone, as shown in Figure 5j,n. As the load increased, flexural cracks spread to the shear span zone, developing into shear cracks. Inclined cracks were dominant in beams with higher hybrid reinforcement ratios. When inclined cracks are formed and propagate towards the load position, the stress at the top of the compression zone increases until the beams reach their failure condition. A summary of the failure modes of the tested beams is shown in Table 2. Different reinforcing materials (GFRP and CFRP) caused significant differences in crack patterns. Cracks in beams with GFRP reinforcement were wider and higher than beams with CFRP reinforcement, due to the lower modulus of elasticity of GFRP.

#### 4.2. Effect of Longitudinal Hybrid Reinforcement Ratio (A_{f}/A_{s})

_{f}/A

_{s}). The highest ductility was seen in beams with a small hybrid reinforcement ratio. By contrast, ductility decreased with a higher hybrid reinforcement ratio, even though the steel yielded.

#### 4.3. Effect of Reinforcement Position

_{f}/A

_{s}) and the effect of the position of the reinforcement can be seen in Table 2 and plotted in Figure 8. This study used the secant method to calculate the flexural stiffness from the load–deflection curve before yielding. It can be seen from Figure 8 that cross-sections with Type I reinforcement positions showed higher stiffness values than those with Type II. In addition, a smaller hybrid reinforcement ratio resulted in a higher stiffness value. Furthermore, cross-sections with hybrid reinforcement using CFRP exhibited higher stiffness values than those using GFRP.

#### 4.4. Strain Distribution in Cross-Sections with Hybrid Reinforcement

_{s}) exceeded the yield strain of steel (ε

_{y}).

_{f}) in all beam sections did not exceed the ultimate tensile strength of the material (ε

_{fu}). This result agrees with the beam test results obtained from experiments. In cross-sectional analysis via the fiber element method, the maximum compressive strain entered into the computer program was 0.003, which agreed with the experimental results, where all beams were crushed in the compression zone.

#### 4.5. Neutral Axis Growth in Cross-Sections with Hybrid Reinforcement

_{u}). These comparisons show that the neutral axis curves calculated with Equation (5) are very close to the results obtained using other softwares. This proves that a neutral axis curve calculated with Equation (5) can be used to determine the yield point of the steel reinforcement in a reinforced concrete cross-section with hybrid reinforcement.

#### 4.6. Ductility of Cross-Sections with Hybrid Reinforcement

_{y}) visually when the hybrid reinforcement ratio was large, the positions of P

_{y}for beam cross-sections with large hybrid reinforcement ratios (BHG-2, BHG-4, BHC-2, and BHC-4) were determined using the neutral axis curves shown in Figure 12 above. The yield load positions for cross-sections with large hybrid reinforcement ratios correspond to point C on the neutral axis curve. The calculated ductility values are presented in Table 3.

_{f}/A

_{s}) on the ductility (δ) of the tested beams, along with comparable data from the parametric study and obtained from the literature. The hybrid reinforcement ratios used in the specimens for experimental and parametric testing and derived from data obtained from the literature ranged from 0.3 to 4.3, with three types of FRP bars (AFRP, CFRP, and GRP). The red line shows the data trend, which indicated a decrease in ductility with increases in the hybrid reinforcement ratio. The vertical shaded area in Figure 15 indicates the limitation of the hybrid reinforcement ratio (1 to 2.5) recommended to ensure that the beam retains sufficient ductility and stiffness after exceeding the elastic phase [14].

_{f}/A

_{s}) and the ductility (δ). The results showed a moderate correlation with an R of −0.38, where the minus sign means that the value of the hybrid reinforcement ratio increases as the ductility value decreases.

#### 4.7. Effect of FRP Type

#### 4.8. Neutral Axis Angle (α) and Displacement Index (δ_{N})

_{N}) obtained from the neutral axis curve profile were introduced as an alternative method for evaluating the ductility of concrete sections with hybrid reinforcement. Figure 17a,b illustrate the moment–curvature curve and the corresponding neutral axis curve obtained from cross-section analysis. The definition of the neutral axis angle in this paper was the angle formed at Point C (see Figure 17b), following the yield point of the steel reinforcement. The neutral axis angle can be calculated using Equation (9) and with the help of Figure 17b. The value of this angle indicates the change in stiffness of the section after the yield point of the steel reinforcement. A positive angle value means that Point D is above Point E and vice versa. The greater the neutral axis angle value, the greater the ductility value.

_{N}) value can be calculated using Equation (10). This value denotes the propagation of the neutral axis from Point A to Point D. An index value greater than one indicates that Point D is above Point E, whereas an index value smaller than one means that Point D is below Point E. The greater the displacement index value, the greater the ductility value. Neutral axis angles and displacement index values calculated from experimental data, parametric studies, and the literature are shown in Table 3 and Figure 18 and Figure 19.

_{f}/A

_{s}) in Figure 18. It can be seen that the neutral axis angle decreased with increases in the hybrid reinforcement ratio. This result is in accordance with the observation in the previous section that ductility decreases with increases in the hybrid reinforcement ratio.

_{f}/A

_{s}) and the neutral axis displacement index (δ

_{N}). The data have a relatively strong negative correlation value of −0.83: the value of the neutral axis displacement index decreased as the hybrid reinforcement ratio increased.

_{f}/A

_{s}), neutral axis angle (α), neutral axis displacement index value (δ

_{N}), and ductility (δ) are shown in Figure 20, with correlation values of −0.1, −0.06, 0.06, and 0.58 respectively.

_{N}, δ, and DF for cross-sections with hybrid reinforcement. The dotted circle indicates the range of recommended values for an over-reinforced cross-section with hybrid reinforcement (1–2.5).

_{N}) and ductility (δ) values that meet the recommended range are >1 and >4, respectively, while the deformation factor (DF) value that meets the recommended range is >6. A summary of the recommended values for α, δ

_{N}, δ, and DF in hybrid reinforced beams with adequate ductility is found in Table 4.

## 5. Conclusions

- (1).
- The reinforcement ratio (A
_{f}/A_{s}) strongly influenced the capacity and cross-sectional ductility of hybrid reinforced concrete. The higher the hybrid reinforcement ratio, the greater the capacity, but at the cost of ductility, which decreased with increases in the hybrid reinforcement ratio. - (2).
- The position of the reinforcement slightly affected the capacity and ductility of the beam. Differences in slope in the post-elastic area were seen in beams with lower hybrid reinforcement ratios. However, beams with higher hybrid reinforcement ratios did not show a significant difference in the slope of the load–deflection curve in the post-elastic region.
- (3).
- Cross-sections with Type I reinforcement positions showed higher flexural stiffness values than with Type II positions. A smaller hybrid reinforcement ratio resulted in a higher flexural stiffness value. Moreover, cross-sections with hybrid reinforcement using CFRP exhibited higher flexural stiffness values than those using GFRP.
- (4).
- The type of material used for FRP reinforcement (GFRP or CFRP) significantly affected the profile of the neutral axis curve, capacity, and ductility of hybrid reinforced cross-sections.
- (5).
- There were three regions on the neutral axis curve for cross-sections with hybrid reinforcement, i.e., the region before cracking, the region after cracking, and the region after yielding. The inclination of the neutral axis angle (α) after reinforcement yield depended on the type of FRP and the hybrid reinforcement ratio used.
- (6).
- The ductility of hybrid reinforced beams increased as the neutral axis angle increased. There were significant correlations between the neutral axis angle, neutral axis deformation index value, ductility, and deformation factor. The deformation factor increased with increasing neutral axis angles and deformation index values.
- (7).
- This result suggests that the neutral axis angle (α) and the deformation index value (δ
_{N}) proposed in this paper can be used to evaluate the ductility of cross-sections with hybrid reinforcement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic pictures of the tested beams and their identifications: (

**a**) beam dimensions and loading position, (

**b**) BFS-2, (

**c**) BFS-4, (

**d**) BFG-1, (

**e**) BFG-2, (

**f**) BFC-1, (

**g**) BFC-2, (

**h**) BHC-1, (

**i**) BHC-2, (

**j**) BHC-3, (

**k**) BHC-4, (

**l**) BHG-1, (

**m**) BHG-2, (

**n**) BHG-3, and (

**o**) BHG-4.

**Figure 2.**(

**a**) GFRP and CFRP bars used in this study, (

**b**) reinforcement cages before concrete casting, and (

**c**) experimental setup and equipment used in beam test.

**Figure 3.**Analytical model using the fiber element model: (

**a**) reinforced concrete cross-section, (

**b**) fiber element model, and (

**c**) strain distribution.

**Figure 4.**Stress–strain models used in fiber element analysis: (

**a**) concrete, (

**b**) bilinear model for steel reinforcement, and (

**c**) linear model for CFRP and GFRP bars.

**Figure 5.**Crack patterns and failure modes of the tested beams: (

**a**) BFS-2, (

**b**) BFS-4, (

**c**) BFG-1, (

**d**) BFG-2, (

**e**) BFC-1, (

**f**) BFC-2, (

**g**) BHG-1, (

**h**) BHG-2, (

**i**) BHG-3, (

**j**) BHG-4, (

**k**) BHC-1, (

**l**) BHC-2, (

**m**) BHC-3, and (

**n**) BHC-4.

**Figure 6.**The load–deflection curves of the tested beams relative to the effect of the hybrid reinforcement ratio (

**a**) BFS-2, BFG-1, and BFC-1, (

**b**) BFS-4, BFG-2, and BFC-2, (

**c**) BFS-2, BHG-1, and BHG-2, (

**d**) BFS-2, BHC-1, and BHC-2, (

**e**) BFS-4, BHG-3, and BHG-4, (

**f**) BFS-4, BHC-3, and BHC-4, (

**g**) BFG-1, BHG-1, and BHG-2, (

**h**) BFG-2, BHG-3, and BHG-4, (

**i**) BFC-1, BHC-1, and BHC-2, (

**j**) BFC-2, BFC-3, and BHC-4.

**Figure 7.**The load–deflection curves of the tested beams relative to the effect of the reinforcement position (

**a**) BFS-2 and BFS-4, (

**b**) BFG-1 and BFG-2, (

**c**) BFC-1 and BFC-2, (

**d**) BHG-1 and BHG-3, (

**e**) BHG-2 and BHG-4, (

**f**) BHC-1 and BHC-3, (

**g**) BHC-2 and BHC-4.

**Figure 9.**Strain distribution of the tested beams, obtained analytically: (

**a**) BFS-2, (

**b**) BFC-1, (

**c**) BFG-1, (

**d**) BFS-4, (

**e**) BFC-2, (

**f**) BFG-2, (

**g**) BHG-1, (

**h**) BHG-2, (

**i**) BHG-3, (

**j**) BHG-4, (

**k**) BHC-1, (

**l**) BHC-2, (

**m**) BHC-3, and (

**n**) BHC-4.

**Figure 10.**Verification of calculated neutral axis curve using results from ATENA software [28] for specimens: (

**a**) BFS-2 and (

**b**) BFS-4.

**Figure 11.**Verification of calculated neutral axis curve using results from Response 2000 software [29] for specimens (

**a**) BFS-2 and (

**b**) BHG-1.

**Figure 12.**Neutral axis curves of the tested beams, obtained analytically: (

**a**) BFS-2, (

**b**) BFC-1, (

**c**) BFG-1, (

**d**) BFS-4, (

**e**) BFC-2, (

**f**) BFG-2, (

**g**) BHG-1, (

**h**) BHG-2, (

**i**) BHG-3, (

**j**) BHG-4, (

**k**) BHC-1, (

**l**) BHC-2, (

**m**) BHC-3, and (

**n**) BHC-4.

**Figure 13.**Typical neutral axis curves of reinforced concrete beam cross-sections with (

**a**) steel reinforcement, (

**b**) FRP reinforcement, (

**c**) hybrid steel–GFRP reinforcement, and (

**d**) hybrid steel–CFRP reinforcement.

**Figure 14.**Ductility of tested beams determined using the load–deflection curve: (

**a**) BFS-2, (

**b**) BFS-4, (

**c**) BHG-1, (

**d**) BHG-2, (

**e**) BHG-3, (

**f**) BHG-4, (

**g**) BHC-1, (

**h**) BHC-2, (

**i**) BHC-3, and (

**j**) BHC-4.

**Figure 15.**Relationship between hybrid reinforcement ratio (A

_{f}/A

_{s}) and ductility (δ) for experimental results, parametric study results, and data obtained from the literature.

**Figure 17.**(

**a**) The moment–curvature curve obtained from cross-section analysis and (

**b**) the corresponding neutral axis curve showing the points used for the calculation of the neutral axis angle (α) and displacement index (δ

_{N}) parameters.

**Figure 19.**Relationships between (

**a**) hybrid reinforcement ratio and neutral axis angle, (

**b**) hybrid reinforcement ratio and displacement index value, (

**c**) neutral axis angle and displacement index value, and (

**d**) displacement index value and ductility, for experimental results, parametric study results, and data obtained from the literature.

**Figure 20.**Relationships between (

**a**) hybrid reinforcement ratio and deformation factor, (

**b**) neutral axis angle and deformation factor, (

**c**) neutral axis deformation index value and deformation factor, and (

**d**) ductility and deformation factor for experimental results, parametric study results, and data obtained from the literature.

**Figure 21.**Relationships between (

**a**) α vs. A

_{f}/A

_{s}, (

**b**) δ

_{N}vs. A

_{f}/A

_{s}, (

**c**) δ vs. A

_{f}/A

_{s}, (

**d**) DF vs. A

_{f}/A

_{s}, (

**e**) DF vs. α, and (

**f**) DF vs. δ

_{N}for cross-sections with hybrid reinforcement obtained from the parametric study.

**Table 1.**Material properties of the tested beams and the beams used for parametric study, and corresponding data adopted from the literature.

Beam Notation | Width | Overall Depth | Clear Span Length | f_{c}’ | f_{fu} | f_{y} | d_{f} | d_{s} | E_{f} |
---|---|---|---|---|---|---|---|---|---|

(mm) | (mm) | (mm) | (MPa) | (MPa) | (MPa) | (mm) | (mm) | (GPa) | |

Data from this study [Experimental] | |||||||||

BFS-2 | 125 | 250 | 2000 | 20 | - | 375 | - | 13 | - |

BFS-4 | 125 | 250 | 2000 | 20 | - | 375 | - | 13 | - |

BFG-1 | 125 | 250 | 2000 | 20 | 788 | - | 13 | - | 43.9 |

BFG-2 | 125 | 250 | 2000 | 20 | 788 | - | 13 | - | 43.9 |

BFC-1 | 125 | 250 | 2000 | 20 | 2070 | - | 13 | - | 124 |

BFC-2 | 125 | 250 | 2000 | 20 | 2070 | - | 13 | - | 124 |

BHG-1 | 125 | 250 | 2000 | 20 | 788 | 375 | 13 | 13 | 43.9 |

BHG-2 | 125 | 250 | 2000 | 20 | 788 | 375 | 13 | 13 | 43.9 |

BHG-3 | 125 | 250 | 2000 | 20 | 788 | 375 | 13 | 13 | 43.9 |

BHG-4 | 125 | 250 | 2000 | 20 | 788 | 375 | 13 | 13 | 43.9 |

BHC-1 | 125 | 250 | 2000 | 20 | 2070 | 375 | 13 | 13 | 124 |

BHC-2 | 125 | 250 | 2000 | 20 | 2070 | 375 | 13 | 13 | 124 |

BHC-3 | 125 | 250 | 2000 | 20 | 2070 | 375 | 13 | 13 | 124 |

BHC-4 | 125 | 250 | 2000 | 20 | 2070 | 375 | 13 | 13 | 124 |

Data from this study [Parametric] | |||||||||

BHG-5 | 125 | 250 | 2000 | 20 | 788 | 375 | 16 | 13 | 43.9 |

BHG-6 | 125 | 250 | 2000 | 20 | 788 | 375 | 16 | 13 | 43.9 |

BHG-7 | 125 | 250 | 2000 | 20 | 788 | 375 | 19 | 13 | 43.9 |

BHG-8 | 125 | 250 | 2000 | 20 | 788 | 375 | 19 | 13 | 43.9 |

BHG-9 | 125 | 250 | 2000 | 20 | 788 | 375 | 16 | 10 | 43.9 |

BHG-10 | 125 | 250 | 2000 | 20 | 788 | 375 | 16 | 10 | 43.9 |

BHC-5 | 125 | 250 | 2000 | 20 | 2070 | 375 | 16 | 13 | 124 |

BHC-6 | 125 | 250 | 2000 | 20 | 2070 | 375 | 16 | 13 | 124 |

BHC-7 | 125 | 250 | 2000 | 20 | 2070 | 375 | 19 | 13 | 124 |

BHC-8 | 125 | 250 | 2000 | 20 | 2070 | 375 | 19 | 13 | 124 |

BHC-9 | 125 | 250 | 2000 | 20 | 2070 | 375 | 16 | 10 | 124 |

BHC-10 | 125 | 250 | 2000 | 20 | 2070 | 375 | 16 | 10 | 124 |

Aiello et al. [1] | |||||||||

A1 | 150 | 200 | 2700 | 45.7 | 1674 | 465 | 7.5 | 8 | 49 |

A2 | 150 | 200 | 2700 | 45.7 | 1366 | 465 | 10 | 8 | 50.1 |

A3 | 150 | 200 | 2700 | 45.7 | 1366 | 465 | 10 | 12 | 50.1 |

C1 | 150 | 200 | 2700 | 45.7 | 1674 | 465 | 7.5 | 8 | 49 |

Qu et al. [3] | |||||||||

B3 | 180 | 250 | 1800 | 33.10 | 782 | 363 | 12.7 | 12 | 45 |

B4 | 180 | 250 | 1800 | 33.10 | 755 | 336 | 15.9 | 16 | 41 |

B5 | 180 | 250 | 1800 | 34.40 | 778 | 336 | 9.5 | 16 | 37.7 |

B6 | 180 | 250 | 1800 | 34.40 | 782 | 336 | 12.7 | 16 | 45 |

B7 | 180 | 250 | 1800 | 40.65 | 778 | 363 | 9.5 | 12 | 37.7 |

B8 | 180 | 250 | 1800 | 40.65 | 755 | 336 | 15.9 | 16 | 41 |

Lau & Pam [4] | |||||||||

G0.3-MD1.0-A90 | 280 | 380 | 4200 | 41.3 | 588 | 336 | 19 | 25 | 39.5 |

G1.0-T0.7-A90 | 280 | 380 | 4200 | 39.8 | 582 | 597 | 25 | 20 | 38.0 |

G0.6-T1.0-A90 | 280 | 380 | 4200 | 44.6 | 588 | 550 | 19 | 25 | 39.5 |

Yinghao & Yong [5] | |||||||||

S2 | 150 | 250 | 1800 | 80.1 | 1301 | 374.5 | 12 | 24 | 75.98 |

S3 | 150 | 250 | 1800 | 80.1 | 1301 | 374.5 | 12 | 24 | 75.98 |

S4 | 150 | 250 | 1800 | 80.1 | 1301 | 374.5 | 12 | 24 | 75.98 |

Refai et al. [8] | |||||||||

2G12-1S10 | 230 | 300 | 3700 | 40 | 1000 | 520 | 12 | 10 | 50 |

2G12-2S10 | 230 | 300 | 3700 | 40 | 1000 | 520 | 12 | 10 | 50 |

2G12-2S12 | 230 | 300 | 3700 | 40 | 1000 | 520 | 12 | 12 | 50 |

2G16-2S10 | 230 | 300 | 3700 | 40 | 1000 | 520 | 16 | 10 | 50 |

2G16-2S12 | 230 | 300 | 3700 | 40 | 1000 | 520 | 16 | 12 | 50 |

2G16-2S16 | 230 | 300 | 3700 | 40 | 1000 | 520 | 16 | 16 | 50 |

Beam Notation | A_{f}/A_{s} | First Crack Load | Stiffness | Type of Reinforcement | Reinforcement Ratio | Failure Mode |
---|---|---|---|---|---|---|

(kN) | (kN/mm) | |||||

BFS-2 | - | 3.6 | 3.91 | Steel | Under Reinforced | SY, CC |

BFS-4 | - | 5.3 | 3.71 | Steel | Under Reinforced | SY, CC |

BFG-1 | - | 4.6 | 1.33 | GFRP | Over Reinforced | CC |

BFG-2 | - | 6.2 | 1.27 | GFRP | Over Reinforced | CC |

BFC-1 | - | 3.7 | 2.24 | CFRP | Over Reinforced | CC |

BFC-2 | - | 3.9 | 2.21 | CFRP | Over Reinforced | CC |

BHG-1 | 0.5 | 5.5 | 3.16 | Steel and GFRP | Over Reinforced | SY, CC |

BHG-2 | 2.0 | 3.5 | 2.23 | Steel and GFRP | Over Reinforced | SY, CC |

BHG-3 | 0.5 | 5.1 | 2.79 | Steel and GFRP | Over Reinforced | SY, CC |

BHG-4 | 2.0 | 3.6 | 1.89 | Steel and GFRP | Over Reinforced | SY, CC |

BHC-1 | 0.5 | 5.3 | 3.71 | Steel and CFRP | Over Reinforced | SY, CC |

BHC-2 | 2.0 | 3.7 | 2.45 | Steel and CFRP | Over Reinforced | SY, CC |

BHC-3 | 0.5 | 3.2 | 3.17 | Steel and CFRP | Over Reinforced | SY, CC |

BHC-4 | 2.0 | 6.3 | 2.39 | Steel and CFRP | Over Reinforced | SY, CC |

**Table 3.**Ductility (δ), neutral axis angle (α), yield moment, ultimate moment, and deformability factor (DF) of cross-sections with hybrid reinforcement.

Beam Notation | A_{f}/A_{s} | δ | α | δ_{N} | M_{y} | μ_{y} | M_{u} | μ_{u} | DF | Type of FRP | Type of Cross-Section |
---|---|---|---|---|---|---|---|---|---|---|---|

(kNm) | (kNm) | ||||||||||

Data from this study [Experimental] | |||||||||||

BFS-2 | − | 4.9 | 77.7 | 1.7 | 28.2 | 1.4 | 29.1 | 7.2 | 5.4 | - | Type I |

BFS-4 | − | 6.3 | 58.8 | 1.6 | 25.1 | 1.4 | 27.4 | 6.0 | 4.8 | - | Type II |

BHG-1 | 0.5 | 6.6 | 19.6 | 1.4 | 21.2 | 1.3 | 29.9 | 5.8 | 6.4 | GFRP | Type I |

BHG-2 | 2.0 | 6.3 | 3.6 | 1.1 | 14.2 | 1.2 | 30.4 | 5.7 | 9.9 | GFRP | Type I |

BHG-3 | 0.5 | 6.1 | 25.7 | 1.4 | 20.6 | 1.3 | 27.2 | 6.1 | 6.0 | GFRP | Type II |

BHG-4 | 2.0 | 5.6 | 2.8 | 1.1 | 13.6 | 1.5 | 28.9 | 5.7 | 8.2 | GFRP | Type II |

BHC-1 | 0.5 | 5.4 | 8.5 | 1.2 | 24.4 | 1.4 | 38.1 | 4.3 | 4.9 | CFRP | Type I |

BHC-2 | 2.0 | 3.7 | −2.4 | 0.9 | 21.2 | 1.3 | 41.4 | 3.9 | 5.7 | CFRP | Type I |

BHC-3 | 0.5 | 4.5 | 12.2 | 1.3 | 22.7 | 1.4 | 33.4 | 4.6 | 4.9 | CFRP | Type II |

BHC-4 | 2.0 | 4.0 | −3.6 | 0.9 | 23.1 | 1.7 | 39.9 | 3.9 | 3.9 | CFRP | Type II |

Data from this study [Parametric] | |||||||||||

BHG-5 | 3.0 | 3.9 | 1.5 | 1.0 | 15.9 | 1.2 | 34.6 | 4.9 | 8.7 | GFRP | Type I |

BHG-6 | 3.0 | 3.2 | 0.5 | 1.0 | 15.7 | 1.5 | 33.1 | 4.9 | 6.9 | GFRP | Type II |

BHG-7 | 4.3 | 3.3 | −0.8 | 1.0 | 18.4 | 1.3 | 38.3 | 4.3 | 7.0 | GFRP | Type I |

BHG-8 | 4.3 | 2.5 | −1.2 | 1.0 | 19.2 | 1.6 | 36.8 | 4.3 | 5.2 | GFRP | Type II |

BHG-9 | 1.3 | 5.1 | 7.0 | 1.2 | 14.7 | 1.2 | 28.7 | 6.1 | 9.6 | GFRP | Type I |

BHG-10 | 1.3 | 5.2 | 9.4 | 1.2 | 13.9 | 1.3 | 24.6 | 6.5 | 9.0 | GFRP | Type II |

BHC-5 | 3.0 | 2.3 | −4.8 | 0.9 | 26.8 | 1.4 | 45.9 | 3.3 | 4.0 | CFRP | Type I |

BHC-6 | 3.0 | 1.8 | −6.6 | 0.9 | 15.7 | 1.5 | 33.1 | 4.9 | 6.9 | CFRP | Type II |

BHC-7 | 4.3 | 2.0 | −7.1 | 0.8 | 33.6 | 1.6 | 49.6 | 2.9 | 2.7 | CFRP | Type I |

BHC-8 | 4.3 | 1.3 | −7.8 | 0.8 | 38.8 | 2.1 | 48.1 | 2.9 | 1.7 | CFRP | Type II |

BHC-9 | 1.3 | 3.0 | 0.7 | 1.0 | 19.7 | 1.3 | 38.9 | 4.2 | 6.5 | CFRP | Type I |

BHC-10 | 1.3 | 3.4 | 2.6 | 1.1 | 17.3 | 1.3 | 32.6 | 4.6 | 6.7 | CFRP | Type II |

Aiello et al. [1] | |||||||||||

A1 | 0.9 | 7.7 | 3.6 | 1.1 | 8.8 | 1.9 | 20.2 | 11.9 | 14.4 | AFRP | Type II |

A2 | 1.6 | 6.4 | 3.1 | 1.1 | 10.4 | 1.9 | 25.8 | 10.0 | 12.7 | AFRP | Type II |

A3 | 1.0 | 4.8 | 8.7 | 1.1 | 20.0 | 2.2 | 34.1 | 7.6 | 5.8 | AFRP | Type II |

C1 | 0.9 | 7.7 | 11.2 | 1.1 | 9.7 | 1.8 | 21.2 | 11.9 | 14.5 | AFRP | Type I |

Qu et al. [3] | |||||||||||

B3 | 1.1 | 4.0 | 6.9 | 1.2 | 20.7 | 1.2 | 46.0 | 7.5 | 14.0 | GFRP | Type I |

B4 | 2.0 | 2.9 | 4.9 | 1.1 | 19.5 | 1.1 | 49.9 | 6.7 | 16.0 | GFRP | Type I |

B5 | 0.4 | 4.8 | 21.5 | 1.4 | 28.4 | 1.1 | 43.4 | 8.4 | 11.4 | GFRP | Type I |

B6 | 0.6 | 4.5 | 14.7 | 1.3 | 30.4 | 1.2 | 51.9 | 6.6 | 9.5 | GFRP | Type I |

B7 | 1.3 | 8.5 | 7.3 | 1.2 | 10.7 | 1.1 | 30.8 | 10.5 | 27.0 | GFRP | Type I |

B8 | 0.3 | 3.1 | 38.3 | 1.4 | 77.6 | 1.6 | 87.3 | 3.7 | 2.7 | GFRP | Type II |

Lau & Pam [4] | |||||||||||

G0.3-MD1.0-A90 | 0.3 | 1.9 | 21.9 | 1.4 | 114.9 | 0.8 | 166.3 | 5.0 | 9.4 | GFRP | Type I |

G1.0-T0.7-A90 | 1.6 | 3.9 | 6.3 | 1.1 | 160.7 | 1.2 | 237.3 | 3.6 | 4.4 | GFRP | Type I |

G0.6-T1.0-A90 | 0.6 | 4.2 | 17.1 | 1.3 | 195.4 | 1.1 | 250.1 | 3.8 | 4.3 | GFRP | Type I |

Yinghao & Yong [5] | |||||||||||

S2 | 0.3 | 2.6 | 19.9 | 1.4 | 66.7 | 1.6 | 93.1 | 5.5 | 4.7 | GFRP | Type II |

S3 | 0.3 | 3.8 | 29.9 | 1.6 | 72.8 | 1.4 | 95.2 | 5.8 | 5.5 | GFRP | Type II |

S4 | 0.3 | 1.7 | 24.6 | 1.6 | 75.0 | 1.3 | 103.2 | 5.5 | 5.7 | GFRP | Type I |

Refai et al. [8] | |||||||||||

2G12-1S10 | 2.9 | 5.3 | 0.7 | 1.1 | 15.2 | 1.4 | 47.3 | 8.2 | 18.5 | GFRP | Type I |

2G12-2S10 | 1.4 | 3.0 | 3.2 | 1.1 | 25.9 | 1.3 | 58.4 | 7.1 | 12.3 | GFRP | Type I |

2G12-2S12 | 1.0 | 4.4 | 5.5 | 1.1 | 31.1 | 1.5 | 55.7 | 6.8 | 8.2 | GFRP | Type I |

2G16-2S10 | 2.6 | 3.8 | 1.3 | 1.0 | 31.0 | 1.4 | 71.4 | 5.8 | 9.8 | GFRP | Type I |

2G16-2S12 | 1.8 | 2.4 | 2.6 | 1.1 | 37.4 | 1.4 | 70.9 | 5.5 | 7.3 | GFRP | Type I |

2G16-2S16 | 1.0 | 4.0 | 7.1 | 1.1 | 56.5 | 1.6 | 81.4 | 4.7 | 4.4 | GFRP | Type I |

**Table 4.**Recommended parameter values of α, δ

_{N}, δ, and DF for cross-sections with hybrid reinforcement.

Parameter | Recommended Value |
---|---|

α | >0° |

δ_{N} | >1 |

δ | >4 |

DF | >6 |

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## Share and Cite

**MDPI and ACS Style**

Thamrin, R.; Zaidir, Z.; Iwanda, D. Ductility Estimation for Flexural Concrete Beams Longitudinally Reinforced with Hybrid FRP–Steel Bars. *Polymers* **2022**, *14*, 1017.
https://doi.org/10.3390/polym14051017

**AMA Style**

Thamrin R, Zaidir Z, Iwanda D. Ductility Estimation for Flexural Concrete Beams Longitudinally Reinforced with Hybrid FRP–Steel Bars. *Polymers*. 2022; 14(5):1017.
https://doi.org/10.3390/polym14051017

**Chicago/Turabian Style**

Thamrin, Rendy, Zaidir Zaidir, and Devitasari Iwanda. 2022. "Ductility Estimation for Flexural Concrete Beams Longitudinally Reinforced with Hybrid FRP–Steel Bars" *Polymers* 14, no. 5: 1017.
https://doi.org/10.3390/polym14051017