# Universal Bond Models of FRP Reinforcements Externally Bonded and Near-Surface Mounted to RC Elements in Bending

## Abstract

**:**

## 1. Introduction

## 2. Bond Model Based on the Fracture Mechanics of Solids

_{at}is the tensile strength of an adhesive on the concrete/FRP surface and f

_{ct}is the tensile strength of concrete.

_{f}is the modulus of elasticity of the FRP reinforcement; shape factor k

_{b}and calibration coefficient k

_{m}were neglected and taken equal to one; and length factor ${\beta}_{l}$ is calculated using Equation (6). The thickness of the FRP reinforcement in EBR t

_{f}can be changed into equivalent thickness:

_{f}is the area of the FRP reinforcement, u

_{f}is the bond perimeter of the FRP reinforcement (see Figure 3), and l

_{cr}is the crack spacing, the maximum bond length (l

_{b.}

_{max}) can be estimated as follows:

_{cr}is calculated according to the provisions of [48]. The average crack spacing takes into account the effects of both internal and external reinforcement.

_{b}is a bond parameter (Equation (10)); and ${A}_{ct.eff}$ is the concrete effective area in tension.

_{b}is the bond length.

_{cr}is the concrete crack depth; and d

_{1.eff}is the depth resultant of the internal and external reinforcements:

_{cr}and M

_{a}are concrete cracking and acting bending moments respectively and ${\epsilon}_{f}$ is FRP strain, the difference of FRP and tensile concrete strain between the cracks could expressed as follows:

_{c.uncr}is the second moment of area of the uncracked concrete section.

_{v}and F

_{h}are the vertical and horizontal forces of the bond, respectively (if unknown, the angle between them can be considered equal to $\alpha \approx {45}^{0}$) (see Figure 6).

_{max}is the maximum aggregate size and W/C is the ratio of water to cement. If the composition of the concrete is unknown, then the fracture energy can be calculated as proposed in [47]:

_{f}is the coefficient of the parabola equation (Equation (27)).

## 3. Bond Model Based on Built-Up Bars Theory

_{f}. For the beam loaded with two concentrated forces in the pure bending zone, it will be as follows [35]:

_{f}is the width of the FRP to the concrete bond (or the perimeter in the case of NSM FRP bars), E

_{f}A

_{f}is the axial stiffness of the FRP, E

_{c}A

_{c.eff}is the axial stiffness of the cracked concrete section, and EI is the composite flexural stiffness of two elements:

_{c}≈ a and t

_{eff}≈ t

_{a}, while the adhesive layer is completely deformed together with the FRP reinforcement. The adhesive layer recommended by the manufacturers is between 1–4 mm, but for safety reasons, it should be taken equal to 1 mm. L

_{cr}is the spacing of the cracks and G

_{c}is the shear modulus of the concrete. The graphic illustration of the strengthened member’s strain profile of the strengthened member taking into account a concrete–FRP partial shear connection is presented in Figure 7.

## 4. Load-Bearing Capacity

_{y}and f

_{cm}are the yield strength of the tensile steel reinforcement and the mean value of the compressive strength of concreate, respectively, η and λ are the reduction factors of the compressive strength of concrete and the compressive zone height, respectively (following [45]: η = 1.0, λ = 0.8 for concrete strength f

_{ck}< 50 MPa). The FRP stress of the ith step can be found assuming the linear elastic stress–strain relationship, but it must be lower than the design strength:

_{0}is the initial strain at the FRP level, and ${\epsilon}_{f.i}$ is a strain at the FRP level neglecting the prestressing and initial strains:

_{cu}can be taken as 3.5‰ when f

_{ck}< 50 MPa. Repeat iterations until the equilibrium condition is reached:

## 5. Validation of the Results

_{p}is FRP prestressing stress), as presented in Table 1.

## 6. Conclusions

- In this paper, two different universal models are presented for the assessment of the bond between concrete and FRPs, both of which assess the behaviour of the bond between two cracks. The first model is based on the fracture mechanics of solids, distinguishing different stages of failure development and distribution over the length of the joint. The second one is the fully analytical model based on the built-up bars theory, considering the joint as a single unit.
- In both cases, the load-bearing capacity of the member’s normal section is determined very accurately with a low mean error (5% and 9%), a low random error (0.15 in both cases), and a high correlation (0.97 or 0.96). The results of the calculation have been validated with 77 beam tests carried out by different researchers. The beams were strengthened using both EBR and NSM methods, with strong variations in performance. Both the prestress force and the initial stress state before strengthening were evaluated.
- The first approach, based on the fracture mechanics of solids, has advantages over the second approach in that it allows for a complete analysis of the behaviour of the joint, the development, and the propagation of rupture. However, the second approach is well suited to the calculation of the load-bearing capacity, requires much less computation, and can be fully exploited where it is sufficient to treat the joint as a unit, without subdividing.
- The most common description of a concrete–FRP bond found in the literature is based on some specific testing, or it is greatly simplified, resulting in a number of limitations in the application and a number of aspects that are not assessed. The significance of this paper lies in the fact that the proposed models are universal, not tied to specific tests, suitable for different strengthening methods, do not use any major simplifications, and the only limitation is the normal section of the bending element.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Bond perimeter of FRP (fibre reinforced polymer) and steel reinforcements: (

**a**) FRP rods; (

**b**) FRP strips; (

**c**) EBR (externally bonded reinforcement).

**Figure 4.**The effective area of concrete in tension (

**a**) EBR (externally bonded reinforcement); (

**b**) NSM (near surface mounted) [39].

**Figure 7.**Strain profile [41].

**Table 1.**The properties of the analysed beams (EBR is externally bonded reinforcement, NSM is near surface mounted reinforcement).

Ref. | A_{s}_{1}/bd (%) | f_{y} (MPa) | A_{f}/bd_{f} (%) | f_{f} (MPa) | E_{f} (GPa) | σ_{p} (MPa) | EBR/NSM |
---|---|---|---|---|---|---|---|

[16] | 0.85 | 400 | 0.11 | 3100 | 165 | 1000 | EBR |

[16] | 0.85 | 400 | 0.13–0.14 | 2068 | 131 | 0–1000 | NSM |

[17] | 0.40 | 426 | 0.04–0.12 | 2453–3479 | 165–230 | 0 | EBR |

[17] | 0.40 | 426 | 0.04–0.11 | 1878–2453 | 121–165 | 0 | NSM |

[43] | 0.45 | 436 | 0.04–0.22 | 1500–2483 | 100–167 | 0 | NSM |

[25] | 0.29–1.19 | 466–501 | 0.08 | 2850 | 165 | 0–1323 | EBR |

[44] | 0.50–0.75 | 525–531 | 0.11 | 3263 | 251 | 0 | EBR |

[23] | 0.58 | 545 | 0.12–0.26 | 1350–2350 | 64–170 | 0 | NSM |

[24] | 0.58 | 540 | 0.13–0.26 | 1350–2500 | 64–170 | 0 | NSM |

[20] | 0.77 | 475 | 0.08 | 2167 | 130 | 0–1241 | NSM |

[21] | 0.54–0.94 | 730 | 0.16–0.24 | 2740 | 159 | 0 | NSM |

[22] | 0.39 | 585 | 0.06 | 1922 | 164 | 0–823 | NSM |

[39] | 0.68–1.13 | 318–569 | 0.08–0.32 | 2334–4800 | 213–230 | 0–120 | EBR |

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**MDPI and ACS Style**

Slaitas, J.
Universal Bond Models of FRP Reinforcements Externally Bonded and Near-Surface Mounted to RC Elements in Bending. *Materials* **2024**, *17*, 493.
https://doi.org/10.3390/ma17020493

**AMA Style**

Slaitas J.
Universal Bond Models of FRP Reinforcements Externally Bonded and Near-Surface Mounted to RC Elements in Bending. *Materials*. 2024; 17(2):493.
https://doi.org/10.3390/ma17020493

**Chicago/Turabian Style**

Slaitas, Justas.
2024. "Universal Bond Models of FRP Reinforcements Externally Bonded and Near-Surface Mounted to RC Elements in Bending" *Materials* 17, no. 2: 493.
https://doi.org/10.3390/ma17020493