# Effects of Internal Force Redistribution on the Limit States of Continuous Beams with GFRP Reinforcement

^{*}

## Abstract

**:**

_{b}was investigated to assess adhesion conditions for GFRP reinforcement and concrete. The results showed that moment redistribution in continuous beams with GFRP reinforcement happens with slippage between the reinforcement and concrete in the middle support without the load capacity being reduced. A modified model was suggested for better deflection prediction of continuous beams reinforced with GFRP bars. Based on deformability factors, the tested continuous beams, although containing GFRP reinforcement that has brittle behavior, showed a certain kind of ductile behavior.

## 1. Introduction

_{b}, which depends on the characteristics of the FRP bars and the surrounding concrete [12,13,14,15,16], of which the crack width in the member is significantly dependent.

_{b}are determined for different critical sections of the beam. The ductility behavior of the beams with GFRP bars is presented and discussed.

## 2. Experimental Program

#### 2.1. Materials

#### 2.2. Experimental Setup and Instrumentation

## 3. Results and Discussion

#### 3.1. Crack Propagation and Modes of Failure

#### 3.2. Load Capacity and Moment Redistribution

_{n}is calculated by the following equations when the reinforcement ratio ρ

_{f}is greater than the balanced reinforcement ratio ρ

_{fb}:

_{f}is the stress in the tension FRP reinforcement; E

_{f}is the modulus of elasticity of the FRP reinforcement; ε

_{cu}is the ultimate strain of concrete; f

_{c}is concrete compressive strength; b and d are the width and effective depth of the beam, respectively; α

_{1}and β

_{1}are the strength reduction factors for concrete:

#### 3.3. Load–Crack Width Relationship

_{k}as the product of the maximal spacing of the cracks s

_{r,max}and the average strain, which represents the difference in the average strain in the reinforcement, including the tension stiffening of the concrete ε

_{fm}and the average strain in the concrete between the cracks ε

_{cm}:

_{1}is a coefficient taking into account the bond properties (0.8 for high bond bars and 1.6 for smooth bars); k

_{2}is a coefficient taking into account the distribution of the strain (0.5 for bending and 1 for pure tension); Ø is the bar diameter, σ

_{f}is the stress in the tension reinforcement; E

_{f}is the modulus of elasticity of the FRP reinforcement; α

_{e}is the modular ratio E

_{f}/E

_{c}; k

_{t}is a coefficient taking into account the load duration (0.6 for short term loads and 0.4 for long term loads); ρ

_{p,eff}is the effective reinforcement ratio; A

_{f}is the area of reinforcement; b, h, and d are the width, height, and effective depth of the beam, respectively; and x is the neutral axis depth.

_{k}as the product of the average distance between cracks s

_{rm}and the average strain accounting for tension stiffening ε

_{fm}:

_{1}is a coefficient taking into account the bond properties (1.6 for the FRP bars), σ

_{f}is the stress in the tension reinforcement of the cracked cross-section, σ

_{fr}is the stress in the tension reinforcement of the cracked cross-section when the first crack is observed, β is a coefficient relating the average crack width to the characteristic value (1.7 for cracking due to loads), β

_{1}is a coefficient taking into account the bond properties (0.5 for the FRP bars), and β

_{2}is a coefficient taking into account the load duration (1.0 for short-term loads and 0.5 for long-term loads).

_{f}is the stress in the tension reinforcement, E

_{f}is the modulus of elasticity of the FRP reinforcement, β = (h − x)/(d − x), d

_{c}is the concrete cover from the center of the reinforcing bar, and s is the bar spacing.

_{b}for the bond between the concrete and FRP reinforcement. For different types of FRP reinforcement, resin formulations, and surface treatments, ACI represents the values of the coefficient k

_{b}, ranging from 0.6 to 1.72, with an average of 1.10. ACI-06 [34] and ACI-15 [29] suggest a conservative k

_{b}value of 1.4, an ISIS-07 [35] value of 1.2, and a CSA-12 [30] value of 1.0 for deformed FRP bars. For FRP bars with similar adhesion conditions using concrete containing steel bars, a k

_{b}value of 1.0 is considered.

#### 3.4. Bond-Dependent Coefficient k_{b} Prediction

_{b}. The coefficient k

_{b}was calculated for the beams at the midspan and middle support for 35 kN, 50 kN, 70 kN, and 90 kN loads for beams series 1; and 25 kN, 35 kN, 45 kN, 60 kN, and 75 kN loads for beams series 2 using Equations (14) and (15), in accordance with ACI-06 [34] and ISIS-07 [35], respectively. These load levels included a wide range of service loads, ranging from 22–72% of the failure load, or 2–6 times the first crack load. Table 3 gives the maximal values of coefficient k

_{b}for different load levels for the measured widths of three cracks in every critical section, in both the midspan and middle support sections. Additionally, Figure 10 shows the average values of the coefficient k

_{b}for the measured widths of three cracks in every critical section for different load levels of each beam, using Equation (14) in accordance with ACI-06 [34].

_{b}are obtained for the midspan for all beams series 1 using the applied codes. The values for coefficients k

_{b}are significantly less than the recommended values for the chosen codes. Additionally, these values are less than 1.0, the recommended value for the steel reinforcement, which indicates very good adhesion conditions between ribbed GFRP bars and the surrounding concrete. In Figure 10, uniform average coefficient values for different load levels can also be observed, except for beam G1-0, where the coefficient value increased with the 90 kN load. It can be concluded that for beams series 1, the designed internal force redistribution did not affect the variation of coefficient k

_{b}in the midspan. As expected, higher k

_{b}values were obtained for beams series 2, which also indicates poor adhesion conditions between wrapped GFRP bars and concrete. These values are greater than those recommended by the codes. Additionally, it can be observed that these coefficients are higher for beams G2-15 and G2-25, which were designed to achieve internal force redistribution, compared to beam G2-0.

_{b}values are significantly different for the tested beams series 1. The highest values were obtained for the beam G1-25 (1.01 and 1.33), which had the smallest amount of reinforcement in the middle support section, while the smallest values were obtained for the beam G1-0 (0.55 and 0.65), which had the highest amount of reinforcement. For the beam G1-0, coefficient k

_{b}gave similar values in the midspan and middle support sections. For beams G1-15 and G1-25, the k

_{b}values increase in the support section, indicating that the internal force redistribution in the continuous GFRP-reinforced beams occurs with the slippage between bars and concrete in the middle support section. Additionally, in these beams, the slippage of the reinforcement practically started after the cracks occurred in the middle support area, as the k

_{b}values were uniform for different load levels (Figure 10). Considering beams series 2, similar conclusions can be drawn for the coefficients k

_{b}values for the middle support section, but with significantly higher values. Additionally, uneven values for average coefficients for different load levels are observed in Figure 10. For beam G2-0, these values decrease, while for beam G2-25 coefficient k

_{b}values increase as the load increases.

#### 3.5. Load–Deflection Relationship

_{c}is the modulus of elasticity of concrete, and I

_{e}represents the effective moment of inertia.

_{e}based on Branson’s pattern:

_{cr}is the cracking moment, M

_{a}is the applied moment, I

_{g}is the moment of inertia of the gross concrete section, and I

_{cr}is the moment of inertia of the cracked section. Factor β

_{d}is represented as:

_{f}is the FRP reinforcement ratio and ρ

_{fb}is the balanced FRP reinforcement ratio.

_{G}= 0.6 is a reduction factor for the state after the cracks occur.

_{cr}were used for all models. The specific load level deflections obtained through experiment are significantly higher than calculated values. Surely, the main reason for the mismatched diagrams is the fact that models used to calculate deflection are mainly based on research studies that were enforced on simple beams, which are calculated by the elastic analysis for uniform stiffness along the beam. The only match is for experimental and calculated deflections for the lower loads levels. The suggested model by Habeeb and Ashour [23] shows much better fitting of deflection with experimental results, with slightly higher values. Within load levels close to failure, exceptions occur when the experimental deflection curve falls further. Similar observations and comparisons regarding deflection values for tested beams were also made El-Moggy in his experimental research [19]. Considering the experimental beams of a single series, deflections are fairly uniform, even for the different reinforcement axial stiffness in the middle of the beam span. This is a consequence of internal force redistribution, which occurs in the critical sections after formation of cracks.

_{e}for continuous beams should be calculated by the expression:

_{em}and I

_{ec}are the effective moment of inertia in the midspan and middle support sections, respectively.

_{e}is less than the cracked moment of inertia I

_{cr}. Using the coefficient P, the proposed model shows a good development of deflection for higher load levels. Immediately after cracks occur, a large increase in deflections in accordance with ISIS-07 [35] is noticed. This sudden reduction in stiffness was not observed during testing of the beams series 1, as can be seen in Figure 11, which again indicates very good adhesion conditions between ribbed GFRP bars and concrete. Further experimental results would better confirm the proposed model.

#### 3.6. Experimental Service Load

#### 3.7. Ductility and Deformability

_{ult}, φ

_{ult}, and Δ

_{ult}are the moment, curvature, and deflection under the ultimate limit state, respectively; and M

_{ser}, φ

_{ser}, and Δ

_{ser}are the moment, curvature, and deflection under the serviceability limit state, respectively.

_{ult}/Δ

_{ser}) compared to strength factors (M

_{ult}/M

_{ser}) indicate that this ductile behavior of the beams is mainly a consequence of the high deformation capacity, which provides sufficient warning before failure.

_{ult}/M

_{ser}) is the strength factor and (Δ

_{ult}/Δ

_{ser}) is the deflection factor. In Figure 12, Figure 13 and Figure 14, the strength, deflection, and deformability factors are given, respectively, where moment and midspan deflection values under the serviceability limit state are defined for the serviceability criteria limitations given in Section 3.6.

## 4. Conclusions

- For beams with ribbed GFRP bars with epoxy, the values of the maximal crack widths were less than those predicted by actual codes, while for beams with wrapped GFRP reinforcement with polyester they were higher than predicted. As the crack width largely depends on the degree of adhesion between the GFRP reinforcement and the concrete, the reliability of the chosen code in calculating the crack width depends on the prescribed coefficient k
_{b}. - The bond dependent coefficient k
_{b}determined for the ribbed GFRP bars for crack widths in the midspan of the beams indicated very good bond strength between GFRP bars and concrete. With the reduction of the reinforcement over the middle support, the k_{b}increased, indicating that the internal force redistribution in the GFRP-reinforced continuous beams occurred with the slippage between bars and concrete in the middle support section, without decreasing the load-carrying capacity. - Actual codes for elements with FRP reinforcement underestimate the deflection of continuous beams with GFRP reinforcement. Therefore, a modified model for effective moment of inertia I
_{e}for deflection prediction of GFRP-reinforced continuous beams was suggested, using non-linear parameter P for higher load levels. The proposed model for deflection calculation, when the effective moment of inertia I_{e}is less than the cracked moment of inertia I_{cr}, gives very good prediction of the obtained experimental results for all load levels. - The design of continuous concrete structures with GFRP reinforcement is governed by different criteria imposed for the serviceability limit state, as a low percentage of the service load is obtained in relation to the ultimate load. For service criteria limitations, with the increase of the designed moment redistribution, the level of service load decreases in relation to the ultimate load.
- Although the FRP reinforcement showed linear elastic behavior and the beams experienced concrete compression failure, obtaining deformability factors higher than 4, as recommended, indicated that continuously supported GFRP-reinforced beams showed a certain kind of ductile behavior, giving sufficient warning before failure.
- Continuous beams that reached the designed moment redistribution showed greater deformability factors, meaning that they exhibited a greater degree of ductile behavior compared to the beams that were designed based on elastic analysis.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**Load–moment redistribution relationship in the middle support section for experimental beams: (

**a**) series 1; (

**b**) series 2.

**Figure 9.**(

**a**) Experimental and predicted load–crack width relationships for tested beams series 1. (

**b**) Experimental and predicted load–crack width relationships for tested beams series 2.

**Figure 10.**Bond-dependent coefficient k

_{b}for different load levels of tested beams: (

**a**) midspan; (

**b**) middle support.

Beam | Series | Middle Support—Top Reinforcement | Midspan—Bottom Reinforcement | Concrete Compressive Strength f _{c} (MPa) | ||||
---|---|---|---|---|---|---|---|---|

Longitudinal Reinforcement | EA (kN) | Reinforcement Ratio ρ _{f}/ρ_{fb} | Longitudinal Reinforcement | EA (kN) | Reinforcement Ratio ρ _{f}/ρ_{fb} | |||

G1-0 | 1 | 4Ø12 | 17,654 | 3.35 | 2Ø10 + 2Ø9 | 10,734 | 2.27 | 50.2 |

G1-15 | 3Ø10 + 1Ø12 | 12,482 | 2.37 | 2Ø12 + 2Ø9 | 14,182 | 2.69 | 50.2 | |

G1-25 | 3Ø9 | 8033 | 1.70 | 3Ø12 + 1Ø10 | 15,930 | 3.02 | 50.2 | |

G2-0 | 2 | 3Ø14 | 20,318 | 3.01 | 2Ø12 + 1Ø10 | 13,558 | 1.75 | 42.2 |

G2-15 | 2Ø10 + 1Ø14 | 12,604 | 1.63 | 2Ø12 + 1Ø14 | 17,415 | 2.58 | 42.2 | |

G2-25 | 2Ø10 + 1Ø12 | 11,153 | 1.44 | 2Ø14 + 1Ø12 | 18,867 | 2.80 | 42.2 |

_{f}—fiber-reinforced polymer (FRP) reinforcement ratio; ρ

_{fb}—balanced FRP reinforcement ratio; EA—axial stiffness of glass FRP (GFRP) reinforcement.

Diameter | Tensile Strength f _{u} (MPa) | Modulus of Elasticity E _{f} (MPa) | Ultimate Strain ε _{u} (‰) |
---|---|---|---|

G1—Ø9 | 1170.4 | 50,235 | 23.3 |

G1—Ø10 | 1059.3 | 43,734 | 24.2 |

G1—Ø12 | 1060.4 | 48,182 | 22.0 |

G2—Ø8 | 714.8 | 42,640 | 16.8 |

G2—Ø10 | 703.1 | 41,300 | 17.0 |

G2—Ø12 | 865.9 | 45,832 | 18.9 |

G2—Ø14 | 813.5 | 44,324 | 18.4 |

Code | Beam | Midspan | Middle Support | Recommended Value |
---|---|---|---|---|

ACI-06 CSA-12 | G1-0 | 0.75 | 0.65 | (ACI)—1.4 (CSA)—1.0 |

G1-15 | 0.74 | 1.02 | ||

G1-25 | 0.75 | 1.33 | ||

G2-0 | 2.07 | 2.17 | ||

G2-15 | 2.51 | 3.59 | ||

G2-25 | 2.50 | 4.05 | ||

ISIS-07 | G1-0 | 0.63 | 0.55 | 1.2 |

G1-15 | 0.62 | 0.85 | ||

G1-25 | 0.62 | 1.01 | ||

G2-0 | 1.58 | 1.65 | ||

G2-15 | 1.91 | 2.73 | ||

G2-25 | 1.90 | 3.08 |

Beam | Experimental Service Load P _{ser} (kN) | Experimental Ultimate Load P_{ult} (kN) | P_{ser}/P_{ult} (%) |
---|---|---|---|

G1-0 | 63.2 | 125.2 | 50.5 |

G1-15 | 60.5 | 124.9 | 48.4 |

G1-25 | 64.6 | 137.8 | 46.9 |

G2-0 | 61.3 | 115.6 | 53.0 |

G2-15 | 65.7 | 115.2 | 57.0 |

G2-25 | 65.3 | 119.6 | 54.6 |

**Table 5.**Experimental service load for the tested beams at which the maximal crack width reaches 0.7 mm.

Beam | Experimental Service Load P_{ser} (kN) | Experimental Ultimate Load P_{ult} (kN) | P_{ser}/P_{ult} (%) | |||
---|---|---|---|---|---|---|

Midspan | Middle Support | Midspan | Middle Support | Min | ||

G1-0 | 78.0 | 106.0 | 125.2 | 62.3 | 84.7 | 62.3 |

G1-15 | 90.0 | 66.0 | 124.9 | 72.1 | 52.8 | 52.8 |

G1-25 | 110.0 | 43.0 | 137.8 | 79.8 | 31.2 | 31.2 |

G2-0 | 28.0 | 35.0 | 115.6 | 24.2 | 30.3 | 24.2 |

G2-15 | 32.0 | 20.0 | 115.2 | 27.8 | 17.4 | 17.4 |

G2-25 | 29.0 | 20.0 | 119.6 | 24.2 | 16.7 | 16.7 |

**Table 6.**Experimental service load for tested beams at which stress in the GFRP reinforcement reaches 20% of tensile strength.

Beam | Experimental Service Load P_{ser} (kN) | Experimental Ultimate Load P_{ult} (kN) | P_{ser}/P_{ult} (%) | |||
---|---|---|---|---|---|---|

Midspan | Middle Support | Midspan | Middle Support | Min | ||

G1-0 | 41.7 | 48.5 | 125.2 | 33.3 | 38.7 | 33.3 |

G1-15 | 42.4 | 35.0 | 124.9 | 33.9 | 28.0 | 28.0 |

G1-25 | 42.4 | 30.4 | 137.8 | 30.8 | 22.1 | 22.1 |

G2-0 | 41.1 | 56.1 | 115.6 | 35.6 | 48.5 | 35.6 |

G2-15 | 51.6 | 33.3 | 115.2 | 44.8 | 28.9 | 28.9 |

G2-25 | 55.6 | 37.3 | 119.6 | 46.5 | 31.2 | 31.2 |

Beam | M_{ult} (kNm) | ∆_{ult}(mm) | M_{ser} (kNm) | ∆_{ser}(mm) | Strength Factor | Deflection Factor | J Factor |
---|---|---|---|---|---|---|---|

G1-0 | 32.63 | 24.06 | 16.39 | 7.19 | 1.99 | 3.35 | 6.7 |

G1-15 | 40.12 | 27.14 | 15.83 | 5.75 | 2.53 | 4.72 | 12.0 |

G1-25 | 46.21 | 31.2 | 23.90 | 10.07 | 1.93 | 3.10 | 6.0 |

G2-0 | 33.31 | 25.76 | 13.61 | 5.52 | 2.45 | 4.67 | 11.4 |

G2-15 | 38.67 | 26.81 | 21.26 | 7.48 | 1.82 | 3.58 | 6.5 |

G2-25 | 38.41 | 24.41 | 19.81 | 7.07 | 1.94 | 3.45 | 6.7 |

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**MDPI and ACS Style**

Baša, N.; Kopitović Vuković, N.; Ulićević, M.; Muhadinović, M.
Effects of Internal Force Redistribution on the Limit States of Continuous Beams with GFRP Reinforcement. *Appl. Sci.* **2020**, *10*, 3973.
https://doi.org/10.3390/app10113973

**AMA Style**

Baša N, Kopitović Vuković N, Ulićević M, Muhadinović M.
Effects of Internal Force Redistribution on the Limit States of Continuous Beams with GFRP Reinforcement. *Applied Sciences*. 2020; 10(11):3973.
https://doi.org/10.3390/app10113973

**Chicago/Turabian Style**

Baša, Nikola, Nataša Kopitović Vuković, Mladen Ulićević, and Mladen Muhadinović.
2020. "Effects of Internal Force Redistribution on the Limit States of Continuous Beams with GFRP Reinforcement" *Applied Sciences* 10, no. 11: 3973.
https://doi.org/10.3390/app10113973