# ANN-Based Model for the Prediction of the Bond Strength between FRP and Concrete

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{u}) which allows computation of the bond strength in terms of ultimate tangential stresses (τ

_{f}).

_{u}), calibrated by using an artificial neural networks (ANN) method, between an FRP plate and a concrete rigid substrate. It should be evidenced that the proposed formulation is based on an up-dated large database, analyzed by a modern and well recognized data-mining technique, such as the ANN. In this context a previous attempt was found in [20], where a database of fifty-nine experimental tests of FRP-strengthened specimens was used to study the debonding failure. Starting from that analysis, the present work aims to assess a new ANN predictive model, based on an updated database made up of more than 350 samples, tested in the single lap shear test configurations. In summary, the result consists in the calibration of relationships able to predict the ultimate delamination force P

_{u}. The main goal is to perform a novel analytical expression which may answer to the requirement of improved simplicity and accuracy with respect to the available analytical models as demonstrated in the next sections.

## 2. A Background of the Bond Behavior between FRP and Concrete Substrate

_{u}is measured. The failure will occur along the weakest plane of the system: the FRP laminate, the adhesive or the substrate. In fact, three different failure modes have been commonly recognized (excluding the tensile breakage of the FRP itself): a concrete cover delamination, for which the concrete thickness is detached by remaining bonded to the FRP; an adhesive failure in which failure occurs within the resin (in this case the concrete substrate remains un-cracked) and a mixed fracture, namely both cohesive and adhesive. Referring to the Figure 1, the FRP strip/plate transmits the tensile force to the concrete through the tangential stresses τ

_{f}; which arise at the interfaces. These stresses are not uniformly distributed since they are maximum at the loaded end, and decrease along the reinforcement, according to [13] and early in [22,23]. Based on non-linear fracture mechanics, the interfacial energy Gf is computed as the area below the bond stress–slip curve.

_{f}, and the role of the effective bond length. This last work inspired more recent studies, such as those reported in [33,34,35,36,37]. Most of the proposed relationships to determine the bond strength consist in a piecewise defined function (see [28,29,30,34,35,36,38,39,40]) depending on the anchorage length of the FRP sheet. In 1997 Maeda et al. [26] developed an empirical model in which, for the first time, a value of the effective bond length is proposed as a function of the properties of the FRP. Khalifa et al. [27] modified Maeda et al. model by including the concrete compressive strength, when evaluating the maximum bond stress (τ

_{a}). The reported formulations are applicable to both hand lay-up and prefabricated FRP-systems bonded to concrete substrates. It can be noticed that most of the models in Table 1 refers to an empirical calibration. Commonly, the experimental relationship between the bond shear stress and the distance from the applied force point was experienced non-linear (generally hyperbolic or trigonometric shape). In addition, the geometrical and the mechanical properties are generally considered by introducing them as dimensionless.

## 3. The Artificial Neural Networks (ANN)

- x
_{i}is the input data of the generic i-input-node; - w
_{i}is the weight of a generic node in the hidden layer; - b is the bias;
- y is the value of the output node;
- T is the target;
- K is a shape factor.

## 4. Experimental Database for the Formulation of the Theoretical Model

- t
_{f}, the thickness of the FRP sheet (mm); - b
_{f}, the width of the FRP sheet (mm); - E
_{f}, the Young’s modulus of the FRP sheet (GPa); - L
_{f}, the bond length of the FRP sheet (mm); - f
_{c}, the compressive strength of the concrete (MPa); - b
_{c}, the width of the tested concrete element (mm).

_{f}is referred to the thickness of the dry fiber unidirectional sheet, in the case of pultruded plates t

_{f}is the total thickness of the FRP plate. The ultimate force P

_{u}(kN), namely the bond strength, was imposed as the target of the analysis.

_{f}and b

_{f}(i.e., larger IQR); on the other hand, the dimension of the substrate, b

_{c}, is generally the same in the experimental programs (i.e., IQR = 0.12) as well as the length (L

_{f}) of the FRP bonded on the concrete block (i.e., IQR = 0.21). Finally, the mechanical proprieties of the FRP (E

_{f}) and the concrete (f

_{c}) results were explored; in fact, an IQR equal to 0.34 and 0.23 was reached for the compressive strength of the concrete and the elastic modulus of the FRP, respectively.

## 5. ANN Proposed Model

_{u}. Therefore, the outcome analytical model may assume a simple formulation according to Equation (3). Moreover, known parameters are introduced, namely b

_{f}, t

_{f}and f

_{c}; which are the width and the thickness of the FRP-plate and the concrete compressive strength, respectively. The main idea consists in computing the equivalent bond strength between the FRP and the concrete substrate by considering a reduction of the compressive strength of the concrete, i.e., f

_{c}× α

^{−}^{1}. Therefore, the ultimate bond force can be easily calculated by multiplying the bond strength with the net-area of the FRP (=b

_{f}× t

_{f}).

_{f}is considered in GPa while fc is expressed in MPa.

#### 5.1. Model Evaluation

_{u}was then calculated by Equation (3). The frequency of the ratio between the experimental results and the predicted values is illustrated in Figure 5, where the normal distribution of the probability frequency (blue line) and the cumulative probability (red line) were both reported.

^{2}) was calculated as equal to 0.95 in the box Q1–Q3 (see Figure 6 and Table 3) where more than 190 specimens are placed (Figure 7).

_{u}may be appreciated for the whole database. In particular, the specimens, with the worst forecast (i.e., larger scatter from the experimental relative result) are in the conservative side and represent the 15% of the whole population.

#### 5.2. Robustness Analysis

_{u}prediction with respect to the f

_{c}and f’

_{ct}(compressive and tensile strength of the concrete respectively) input has been investigated according to Figure 11. A proportional pseudo linear correlation was found, with a gradient that slightly increases for high levels of concrete compressive strength (45–60 MPa) according to the tensile strength (2.5–3.0 MPa). Whereby, the analytical results are reliable with experimentations.

- SE is the square error between the experimental and theoretical values;
- A is the SE higher response integer;
- n total number of respondents.

## 6. Comparison with Existing Analytical Models

- K > 0: the curve is defined leptokurtic, i.e., more “pointed” of a normal;
- K < 0: the curve is defined platykurtic, that is “flatter” than a normal;
- K = 0 the curve is defined normocurtica, i.e., “flat” as a normal.

- Mo is the mode.
- μ is the average value.
- σ is the standard deviator.

^{2}.

- n is the number of samples in the input database,
- i is the general sample,
- y
_{i}is the experimental value,

^{2}is reported in Figure 15. A deep evaluation and comparison of the literature analytical models and of the proposed one can be carried out. In fact, the predictive performance can be evaluated in terms of the best agreement of precision, accuracy and linear correlation between the experimental and theoretical values (see [46]). From this perspective, the precise model corresponds to the nearness of the average, the median and the mode (to each other) of the frequency of the ratio between the experimental and the theoretical outcomes. The accuracy is measured by the proximity of the average, the median and the mode to 1 and, at the same time, the lower value of the MAPE. The correlation is commonly computed by the R

^{2}factor (the more R

^{2}is close to 1 the higher is the correlation between variables). According to Figure 15, the models [24,25,26,27] give the worst predictions of the ultimate bond force; the reason could be related to the low weight attributed to the elastic modulus of the reinforcement, which resulted to affect mostly the accurate prediction of the target (see Figure 12). Controversially, the proposed model and the proposal reported in [35] have demonstrated to provide the best predictions of the failure of the FRP-to-concrete in pull-pull shear test.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

References | Specimen Label | t_{f} (mm) | b_{f} (mm) | L_{f} (mm) | E_{f} (GPa) | b_{c} (mm) | f_{c} (MPa) | P_{u} (kN) |
---|---|---|---|---|---|---|---|---|

[47] | C1 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 36.1 | 8.462 |

C2 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 47.1 | 9.931 | |

C3 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 47.1 | 10.683 | |

C4 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 47.1 | 10.683 | |

C5 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 43.6 | 10.531 | |

C7 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 43.6 | 9.61 | |

C8 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 43.6 | 10.518 | |

C9 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 43.6 | 11.199 | |

C10 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 24 | 9.869 | |

C11 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 28.9 | 9.343 | |

C12 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 43.7 | 11.204 | |

C13 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 36.4 | 8.094 | |

C14 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 36.4 | 12.811 | |

C15 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 36.4 | 11.917 | |

C16 | 1.016 | 25.4 | 76.2 | 108.478 | 228.6 | 36.4 | 11.57 | |

[48] | 1_11 | 0.167 | 40 | 100 | 230 | 100 | 28.88 | 8.75 |

1_12 | 0.167 | 40 | 100 | 230 | 100 | 26.66 | 8.85 | |

1_21 | 0.167 | 40 | 200 | 230 | 100 | 28.88 | 9.3 | |

1_22 | 0.167 | 40 | 200 | 230 | 100 | 26.66 | 8.5 | |

1_31 | 0.167 | 40 | 300 | 230 | 100 | 28.88 | 9.3 | |

1_32 | 0.167 | 40 | 300 | 230 | 100 | 26.66 | 8.3 | |

1_41 | 0.167 | 40 | 500 | 230 | 100 | 28.88 | 8.05 | |

1_42 | 0.167 | 40 | 500 | 230 | 100 | 28.88 | 8.05 | |

1_51 | 0.167 | 40 | 500 | 230 | 100 | 26.47 | 8.45 | |

1_52 | 0.167 | 40 | 500 | 230 | 100 | 26.47 | 7.3 | |

2_11 | 0.167 | 40 | 100 | 230 | 100 | 24.99 | 8.75 | |

2_12 | 0.167 | 40 | 100 | 230 | 100 | 24.99 | 8.85 | |

2_13 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 7.75 | |

2_14 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 7.65 | |

2_15 | 0.167 | 40 | 100 | 230 | 100 | 24.4 | 9 | |

2_21 | 0.167 | 40 | 100 | 230 | 100 | 24.99 | 12 | |

2_22 | 0.167 | 40 | 100 | 230 | 100 | 24.99 | 10.8 | |

2_31 | 0.167 | 40 | 100 | 230 | 100 | 24.99 | 12.65 | |

2_32 | 0.167 | 40 | 100 | 230 | 100 | 24.99 | 14.35 | |

2_41 | 0.167 | 40 | 100 | 230 | 100 | 24.4 | 11.55 | |

2_42 | 0.167 | 40 | 100 | 230 | 100 | 24.4 | 11 | |

2_51 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 9.85 | |

2_52 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 9.5 | |

2_61 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 8.8 | |

2_62 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 9.25 | |

2_71 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 7.65 | |

2_71 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 6.8 | |

2_81 | 0.167 | 40 | 100 | 230 | 100 | 49.97 | 7.75 | |

2_82 | 0.167 | 40 | 100 | 230 | 100 | 49.97 | 8.05 | |

2_91 | 0.167 | 40 | 100 | 230 | 100 | 24.4 | 6.75 | |

2_92 | 0.167 | 40 | 100 | 230 | 100 | 24.4 | 6.8 | |

2_101 | 0.167 | 40 | 100 | 230 | 100 | 24.99 | 7.7 | |

2_102 | 0.167 | 40 | 100 | 230 | 100 | 26.17 | 6.95 | |

[49] | I-1 | 0.165 | 25 | 75 | 256 | 150 | 23 | 4.75 |

I-2 | 0.165 | 25 | 85 | 256 | 150 | 23 | 5.69 | |

I-3 | 0.165 | 25 | 95 | 256 | 150 | 23 | 5.76 | |

I-4 | 0.165 | 25 | 95 | 256 | 150 | 23 | 5.76 | |

I-5 | 0.165 | 25 | 95 | 256 | 150 | 23 | 6.17 | |

I-6 | 0.165 | 25 | 115 | 256 | 150 | 23 | 5.96 | |

I-7 | 0.165 | 25 | 145 | 256 | 150 | 23 | 5.95 | |

I-8 | 0.165 | 25 | 190 | 256 | 150 | 23 | 6.68 | |

I-9 | 0.165 | 25 | 190 | 256 | 150 | 23 | 6.35 | |

I-10 | 0.165 | 25 | 95 | 256 | 150 | 23 | 6.17 | |

I-11 | 0.165 | 25 | 75 | 256 | 150 | 23 | 5.72 | |

I-12 | 0.165 | 25 | 85 | 256 | 150 | 23 | 6 | |

I-13 | 0.165 | 25 | 95 | 256 | 150 | 23 | 6.14 | |

I-14 | 0.165 | 25 | 115 | 256 | 150 | 23 | 6.1 | |

I-15 | 0.165 | 25 | 145 | 256 | 150 | 23 | 6.27 | |

I-16 | 0.165 | 25 | 190 | 256 | 150 | 23 | 7.03 | |

II-1 | 0.165 | 25 | 95 | 256 | 150 | 22.9 | 5.2 | |

II-2 | 0.165 | 25 | 95 | 256 | 150 | 22.9 | 6.75 | |

II-3 | 0.165 | 25 | 95 | 256 | 150 | 22.9 | 5.51 | |

II-4 | 0.165 | 25 | 190 | 256 | 150 | 22.9 | 7.02 | |

II-5 | 0.165 | 25 | 190 | 256 | 150 | 22.9 | 7.07 | |

II-6 | 0.165 | 25 | 190 | 256 | 150 | 22.9 | 6.98 | |

III-1 | 0.165 | 25 | 100 | 256 | 150 | 27.1 | 5.94 | |

III-2 | 0.165 | 50 | 100 | 256 | 150 | 27.1 | 11.66 | |

III-3 | 0.165 | 75 | 100 | 256 | 150 | 27.1 | 14.63 | |

III-4 | 0.165 | 100 | 100 | 256 | 150 | 27.1 | 19.07 | |

III-7 | 1.27 | 25 | 100 | 225 | 150 | 27.1 | 4.78 | |

IV-1 | 0.165 | 25 | 95 | 256 | 150 | 18.9 | 5.86 | |

IV-2 | 0.165 | 25 | 95 | 256 | 150 | 18.9 | 5.9 | |

IV-3 | 0.165 | 25 | 95 | 256 | 150 | 19.8 | 5.43 | |

IV-4 | 0.165 | 25 | 95 | 256 | 150 | 19.8 | 5.76 | |

IV-5 | 0.165 | 25 | 95 | 256 | 150 | 18.9 | 5 | |

IV-6 | 0.165 | 25 | 95 | 256 | 150 | 19.8 | 7.08 | |

IV-7 | 0.165 | 25 | 95 | 256 | 150 | 18.9 | 5.5 | |

IV-8 | 0.165 | 25 | 95 | 256 | 150 | 19.8 | 5.93 | |

IV-9 | 0.165 | 25 | 95 | 256 | 150 | 18.9 | 5.38 | |

IV-10 | 0.165 | 25 | 95 | 256 | 150 | 19.8 | 6.6 | |

IV-11 | 0.165 | 25 | 95 | 256 | 150 | 18.9 | 5.51 | |

IV-12 | 0.165 | 25 | 95 | 256 | 150 | 19.8 | 5.67 | |

IV-13 | 0.165 | 25 | 95 | 256 | 150 | 18.9 | 6.31 | |

IV-14 | 0.165 | 25 | 95 | 256 | 150 | 19.8 | 6.19 | |

V-1 | 0.165 | 15 | 95 | 256 | 150 | 21.1 | 3.81 | |

V-2 | 0.165 | 15 | 95 | 256 | 150 | 21.1 | 4.41 | |

V-3 | 0.165 | 25 | 95 | 256 | 150 | 21.1 | 6.26 | |

V-4 | 0.165 | 50 | 95 | 256 | 150 | 21.1 | 12.22 | |

V-5 | 0.165 | 75 | 95 | 256 | 150 | 21.1 | 14.29 | |

V-6 | 0.165 | 100 | 95 | 256 | 150 | 21.1 | 15.58 | |

VI-1 | 0.165 | 25 | 95 | 256 | 150 | 21.9 | 6.01 | |

VI-2 | 0.165 | 25 | 95 | 256 | 150 | 21.9 | 5.85 | |

VI-3 | 0.165 | 25 | 145 | 256 | 150 | 21.9 | 5.76 | |

VI-4 | 0.165 | 25 | 145 | 256 | 150 | 21.9 | 5.73 | |

VI-5 | 0.165 | 25 | 190 | 256 | 150 | 21.9 | 5.56 | |

VI-6 | 0.165 | 25 | 190 | 256 | 150 | 21.9 | 5.58 | |

VI-7 | 0.165 | 25 | 240 | 256 | 150 | 21.9 | 5.91 | |

VI-8 | 0.165 | 25 | 240 | 256 | 150 | 21.9 | 5.05 | |

VII-1 | 0.165 | 25 | 95 | 256 | 150 | 24.9 | 6.8 | |

VII-2 | 0.165 | 25 | 95 | 256 | 150 | 24.9 | 6.62 | |

VII-3 | 0.165 | 25 | 145 | 256 | 150 | 24.9 | 7.33 | |

VII-4 | 0.165 | 25 | 145 | 256 | 150 | 24.9 | 6.49 | |

VII-5 | 0.165 | 25 | 190 | 256 | 150 | 24.9 | 7.07 | |

VII-6 | 0.165 | 25 | 190 | 256 | 150 | 24.9 | 7.44 | |

VII-7 | 0.165 | 25 | 240 | 256 | 150 | 24.9 | 7.16 | |

VII-8 | 0.165 | 25 | 240 | 256 | 150 | 24.9 | 6.24 | |

[50] | I-3 | 0.825 | 50 | 100 | 110 | 200 | 17 | 11.64 |

I-4 | 0.99 | 50 | 100 | 110 | 200 | 17 | 12.86 | |

II-1 | 0.495 | 50 | 100 | 110 | 200 | 46.2 | 12.55 | |

II-2 | 0.66 | 50 | 100 | 110 | 200 | 46.2 | 14.25 | |

II-3 | 0.825 | 50 | 100 | 110 | 200 | 46.2 | 17.72 | |

II-4 | 0.99 | 50 | 100 | 110 | 200 | 46.2 | 18.86 | |

III-1 | 0.495 | 50 | 100 | 110 | 200 | 61.5 | 13.24 | |

III-2 | 0.66 | 50 | 100 | 110 | 200 | 61.5 | 15.17 | |

III-3 | 0.825 | 50 | 100 | 110 | 200 | 61.5 | 18.86 | |

III-4 | 0.99 | 50 | 100 | 110 | 200 | 61.5 | 19.03 | |

[51] | PG1-11 | 0.169 | 50 | 130 | 97 | 100 | 37.6 | 7.78 |

PG1-12 | 0.169 | 50 | 130 | 97 | 100 | 37.6 | 9.19 | |

PG1-1W1 | 0.169 | 75 | 130 | 97 | 100 | 37.6 | 10.11 | |

PG1-1W2 | 0.169 | 75 | 130 | 97 | 100 | 37.6 | 13.95 | |

PG1-1L11 | 0.169 | 50 | 100 | 97 | 100 | 37.6 | 6.87 | |

PG1-1L12 | 0.169 | 50 | 100 | 97 | 100 | 37.6 | 9.2 | |

PG1-1L21 | 0.169 | 50 | 70 | 97 | 100 | 37.6 | 6.46 | |

PG1-1L22 | 0.169 | 50 | 70 | 97 | 100 | 37.6 | 6.66 | |

PG1-21 | 0.338 | 50 | 130 | 97 | 100 | 37.6 | 10.49 | |

PG1-22 | 0.338 | 50 | 130 | 97 | 100 | 37.6 | 11.43 | |

PC1-1C1 | 0.111 | 50 | 130 | 235 | 100 | 37.6 | 9.97 | |

PC1-1C2 | 0.111 | 50 | 130 | 235 | 100 | 37.6 | 9.19 | |

NJ2 | 0.083 | 100 | 100 | 240 | 150 | 20.5 | 11 | |

NJ3 | 0.083 | 100 | 150 | 240 | 150 | 20.5 | 11.25 | |

NJ4 | 0.083 | 100 | 100 | 240 | 150 | 36.7 | 12.5 | |

NJ5 | 0.083 | 100 | 150 | 240 | 150 | 36.7 | 12.25 | |

NJ6 | 0.083 | 100 | 150 | 240 | 150 | 36.7 | 12.75 | |

[52,53] | DLUT5-2G | 0.507 | 20 | 150 | 83.03 | 150 | 28.7 | 5.81 |

DLUT5-5G | 0.507 | 50 | 150 | 83.03 | 150 | 28.7 | 10.6 | |

DLUT5-7G | 0.507 | 80 | 150 | 83.03 | 150 | 28.7 | 18.23 | |

DLUT30-1G | 0.507 | 20 | 100 | 83.03 | 150 | 45.3 | 4.63 | |

DLUT30-2G | 0.507 | 20 | 150 | 83.03 | 150 | 45.3 | 5.77 | |

DLUT30-3G | 0.507 | 50 | 60 | 83.03 | 150 | 45.3 | 9.42 | |

DLUT30-4G | 0.507 | 50 | 100 | 83.03 | 150 | 45.3 | 11.03 | |

DLUT30-6G | 0.507 | 50 | 150 | 83.03 | 150 | 45.3 | 11.8 | |

DLUT30-7G | 0.507 | 80 | 100 | 83.03 | 150 | 45.3 | 14.65 | |

DLUT30-8G | 0.507 | 80 | 150 | 83.03 | 150 | 45.3 | 16.44 | |

DLUT50-1G | 0.507 | 20 | 100 | 83.03 | 150 | 55.5 | 5.99 | |

DLUT50-2G | 0.507 | 20 | 150 | 83.03 | 150 | 55.5 | 5.9 | |

DLUT50-4G | 0.507 | 50 | 100 | 83.03 | 150 | 55.5 | 9.84 | |

DLUT50-5G | 0.507 | 50 | 150 | 83.03 | 150 | 55.5 | 12.28 | |

DLUT50-6G | 0.507 | 80 | 100 | 83.03 | 150 | 55.5 | 14.02 | |

DLUT50-7G | 0.507 | 80 | 150 | 83.03 | 150 | 55.5 | 16.71 | |

DLUT15-2C | 0.33 | 20 | 150 | 207 | 150 | 28.7 | 5.48 | |

DLUT15-5C | 0.33 | 50 | 150 | 207 | 150 | 28.7 | 10.02 | |

DLUT15-7C | 0.33 | 80 | 150 | 207 | 150 | 28.7 | 19.27 | |

DLUT30-1C | 0.33 | 20 | 100 | 207 | 150 | 45.3 | 5.54 | |

DLUT30-2C | 0.33 | 20 | 150 | 207 | 150 | 45.3 | 4.61 | |

DLUT30-4C | 0.33 | 50 | 100 | 207 | 150 | 45.3 | 11.08 | |

DLUT30-5C | 0.33 | 50 | 100 | 207 | 150 | 45.3 | 16.1 | |

DLUT30-6C | 0.33 | 50 | 150 | 207 | 150 | 45.3 | 21.71 | |

DLUT30-7C | 0.33 | 80 | 100 | 207 | 150 | 45.3 | 22.64 | |

DLUT50-1C | 0.33 | 20 | 100 | 207 | 150 | 55.5 | 5.78 | |

DLUT50-4C | 0.33 | 50 | 100 | 207 | 150 | 55.5 | 12.95 | |

DLUT50-5C | 0.33 | 50 | 150 | 207 | 150 | 55.5 | 16.72 | |

DLUT50-6C | 0.33 | 80 | 100 | 207 | 150 | 55.5 | 16.24 | |

DLUT50-7C | 0.33 | 80 | 150 | 207 | 150 | 55.5 | 22.8 | |

[54] | Ueda_A1 | 0.11 | 50 | 75 | 230 | 100 | 29.74 | 6.25 |

Ueda_A2 | 0.11 | 50 | 150 | 230 | 100 | 52.31 | 9.2 | |

Ueda_A3 | 0.11 | 50 | 300 | 230 | 100 | 52.31 | 11.95 | |

Ueda_A4 | 0.22 | 50 | 75 | 230 | 100 | 55.51 | 10 | |

Ueda_A5 | 0.11 | 50 | 150 | 230 | 100 | 54.36 | 7.3 | |

Ueda_A6 | 0.165 | 50 | 65 | 372 | 100 | 54.36 | 9.55 | |

Ueda_A7 | 0.22 | 50 | 150 | 230 | 100 | 54.75 | 16.25 | |

Ueda_A8 | 0.11 | 50 | 700 | 230 | 100 | 54.75 | 11 | |

Ueda_A9 | 0.11 | 50 | 150 | 230 | 100 | 51.03 | 10 | |

Ueda_A10 | 0.11 | 10 | 150 | 230 | 100 | 30.51 | 2.4 | |

Ueda_A11 | 0.11 | 20 | 150 | 230 | 100 | 30.51 | 5.35 | |

Ueda_A12 | 0.33 | 20 | 150 | 230 | 100 | 30.51 | 9.25 | |

Ueda_A13 | 0.55 | 20 | 150 | 230 | 100 | 31.67 | 11.75 | |

Ueda_B1 | 0.11 | 100 | 200 | 230 | 500 | 31.67 | 20.6 | |

Ueda_B2 | 0.33 | 100 | 200 | 230 | 500 | 52.44 | 38 | |

Ueda_B3 | 0.33 | 100 | 200 | 230 | 500 | 58.85 | 34 | |

[55] | D-CFS-150-30 a | 0.083 | 100 | 300 | 230 | 100 | 58.85 | 12.2 |

D-CFS-150-30 b | 0.083 | 100 | 300 | 230 | 100 | 73.85 | 11.8 | |

D-CFS-150-30 c | 0.083 | 100 | 300 | 230 | 100 | 73.85 | 12.25 | |

D-CFS-300-30 a | 0.167 | 100 | 300 | 230 | 100 | 73.85 | 18.9 | |

D-CFS-300-30 b | 0.167 | 100 | 300 | 230 | 100 | 73.85 | 16.95 | |

D-CFS-300-30 c | 0.167 | 100 | 300 | 230 | 100 | 73.85 | 16.65 | |

D-CFS-600-30 a | 0.333 | 100 | 300 | 230 | 100 | 73.85 | 25.65 | |

D-CFS-600-30 b | 0.333 | 100 | 300 | 230 | 100 | 73.85 | 25.35 | |

D-CFS-600-30 c | 0.333 | 100 | 300 | 230 | 100 | 73.85 | 27.25 | |

D-CFM-300-30 a | 0.167 | 100 | 300 | 390 | 100 | 73.85 | 19.5 | |

D-CFM-300-30 b | 0.167 | 100 | 300 | 390 | 100 | 73.85 | 19.5 | |

S-CFS-400-25 a | 0.222 | 40 | 250 | 230 | 100 | 73.85 | 15.4 | |

S-CFS-400-25 b | 0.222 | 40 | 250 | 230 | 100 | 73.85 | 13.9 | |

S-CFS-400-25 c | 0.222 | 40 | 250 | 230 | 100 | 73.85 | 13 | |

S-CFM-300-25 a | 0.167 | 40 | 250 | 390 | 100 | 73.85 | 12 | |

S-CFM-300-25 b | 0.167 | 40 | 250 | 390 | 100 | 73.85 | 11.9 | |

S-CFM-900-25 a | 0.5 | 40 | 250 | 390 | 100 | 73.85 | 25.9 | |

S-CFM-900-25 b | 0.5 | 40 | 250 | 390 | 100 | 73.85 | 23.4 | |

S-CFM-900-25 c | 0.5 | 40 | 250 | 390 | 100 | 73.85 | 23.7 | |

[56] | C165-100 | 1.2 | 50 | 100 | 165 | 100 | 29.7 | 18.25 |

C165-130 | 1.2 | 50 | 130 | 165 | 100 | 29.7 | 24.5 | |

C165-150 | 1.2 | 50 | 150 | 165 | 100 | 29.7 | 28.44 | |

C165-175 | 1.2 | 50 | 175 | 165 | 100 | 29.7 | 32 | |

C165-200 | 1.2 | 50 | 200 | 165 | 100 | 29.7 | 34.22 | |

C165-250 | 1.2 | 50 | 250 | 165 | 100 | 29.7 | 33.14 | |

C165-300 | 1.2 | 50 | 300 | 165 | 100 | 29.7 | 34.24 | |

CFRP-C210 | 1.2 | 50 | 150 | 210 | 100 | 35.8 | 30.4 | |

C210-180 | 1.2 | 50 | 180 | 210 | 100 | 35.8 | 34 | |

C210-190 | 1.2 | 50 | 190 | 210 | 100 | 35.8 | 36 | |

C210-200 | 1.2 | 50 | 200 | 210 | 100 | 35.8 | 36.02 | |

C210-230 | 1.2 | 50 | 230 | 210 | 100 | 35.8 | 37.02 | |

C210-255 | 1.2 | 50 | 255 | 210 | 100 | 35.8 | 36.8 | |

CFRP-C300 | 1.2 | 50 | 160 | 300 | 100 | 29.7 | 38.02 | |

C300-180 | 1.2 | 50 | 180 | 300 | 100 | 29.7 | 41.15 | |

C300-200 | 1.2 | 50 | 200 | 300 | 100 | 29.7 | 46.35 | |

C300-250 | 1.2 | 50 | 250 | 300 | 100 | 29.7 | 45.5 | |

C300-300 | 1.2 | 50 | 300 | 300 | 100 | 29.7 | 45.95 | |

[57] | - | 1.4 | 10 | 50 | 152.2 | 200 | 30 | 5.15 |

- | 1.4 | 10 | 100 | 152.2 | 200 | 30 | 7.55 | |

- | 1.4 | 10 | 150 | 152.2 | 200 | 30 | 7.7 | |

- | 1.4 | 10 | 200 | 152.2 | 200 | 30 | 7.9 | |

- | 1.4 | 10 | 250 | 152.2 | 200 | 30 | 6.25 | |

- | 1.4 | 10 | 300 | 152.2 | 200 | 30 | 7.58 | |

- | 1.4 | 10 | 50 | 152.2 | 200 | 40 | 5.1 | |

- | 1.4 | 10 | 100 | 152.2 | 200 | 40 | 6.85 | |

- | 1.4 | 10 | 150 | 152.2 | 200 | 40 | 6.35 | |

- | 1.4 | 10 | 200 | 152.2 | 200 | 40 | 6.95 | |

- | 1.4 | 10 | 250 | 152.2 | 200 | 40 | 6.8 | |

- | 1.4 | 10 | 300 | 152.2 | 200 | 40 | 6.4 | |

- | 1.4 | 10 | 50 | 152.2 | 200 | 50 | 4.55 | |

- | 1.4 | 10 | 100 | 152.2 | 200 | 50 | 7.1 | |

- | 1.4 | 10 | 150 | 152.2 | 200 | 50 | 7.78 | |

- | 1.4 | 10 | 200 | 152.2 | 200 | 50 | 7.65 | |

- | 1.4 | 10 | 250 | 152.2 | 200 | 50 | 6.8 | |

- | 1.4 | 10 | 300 | 152.2 | 200 | 50 | 7.25 | |

- | 1.4 | 30 | 50 | 152.2 | 200 | 30 | 9.3 | |

- | 1.4 | 30 | 100 | 152.2 | 200 | 30 | 16.25 | |

- | 1.4 | 30 | 150 | 152.2 | 200 | 30 | 16.2 | |

- | 1.4 | 30 | 200 | 152.2 | 200 | 30 | 22.1 | |

- | 1.4 | 30 | 250 | 152.2 | 200 | 30 | 15.6 | |

- | 1.4 | 30 | 300 | 152.2 | 200 | 30 | 15.85 | |

- | 1.4 | 30 | 50 | 152.2 | 200 | 40 | 9.15 | |

- | 1.4 | 30 | 100 | 152.2 | 200 | 40 | 14.9 | |

- | 1.4 | 30 | 150 | 152.2 | 200 | 40 | 16.05 | |

- | 1.4 | 30 | 200 | 152.2 | 200 | 40 | 16.15 | |

- | 1.4 | 30 | 250 | 152.2 | 200 | 40 | 16.11 | |

- | 1.4 | 30 | 300 | 152.2 | 200 | 40 | 16.9 | |

- | 1.4 | 30 | 100 | 152.2 | 200 | 50 | 17.8 | |

- | 1.4 | 30 | 150 | 152.2 | 200 | 50 | 15.22 | |

- | 1.4 | 30 | 200 | 152.2 | 200 | 50 | 18.5 | |

- | 1.4 | 30 | 250 | 152.2 | 200 | 50 | 19 | |

- | 1.4 | 30 | 300 | 152.2 | 200 | 50 | 17.71 | |

- | 1.4 | 50 | 50 | 152.2 | 200 | 30 | 13.3 | |

- | 1.4 | 50 | 100 | 152.2 | 200 | 30 | 26 | |

- | 1.4 | 50 | 150 | 152.2 | 200 | 30 | 27.8 | |

- | 1.4 | 50 | 200 | 152.2 | 200 | 30 | 27.2 | |

- | 1.4 | 50 | 250 | 152.2 | 200 | 30 | 24.84 | |

- | 1.4 | 50 | 300 | 152.2 | 200 | 30 | 23 | |

- | 1.4 | 50 | 100 | 152.2 | 200 | 40 | 24.5 | |

- | 1.4 | 50 | 150 | 152.2 | 200 | 40 | 27.75 | |

- | 1.4 | 50 | 200 | 152.2 | 200 | 40 | 19.3 | |

- | 1.4 | 50 | 250 | 152.2 | 200 | 40 | 21.9 | |

- | 1.4 | 50 | 300 | 152.2 | 200 | 40 | 27.3 | |

- | 1.4 | 50 | 100 | 152.2 | 200 | 50 | 16 | |

- | 1.4 | 50 | 150 | 152.2 | 200 | 50 | 21.25 | |

- | 1.4 | 50 | 200 | 152.2 | 200 | 50 | 25 | |

- | 1.4 | 50 | 250 | 152.2 | 200 | 50 | 24.9 | |

- | 1.4 | 50 | 300 | 152.2 | 200 | 50 | 34 | |

[58] | T1a | 0.352 | 100 | 60 | 209 | 140 | 55.6 | 20 |

T1b | 0.352 | 100 | 60 | 209 | 140 | 55.6 | 18.8 | |

T2a | 0.352 | 100 | 80 | 209 | 140 | 55.6 | 25.8 | |

T2b | 0.352 | 100 | 80 | 209 | 140 | 55.6 | 25.2 | |

T3a | 0.352 | 100 | 100 | 209 | 140 | 55.6 | 25.8 | |

T3b | 0.352 | 100 | 100 | 209 | 140 | 55.6 | 27.3 | |

T4a | 0.352 | 100 | 140 | 209 | 140 | 55.6 | 26.7 | |

T4b | 0.352 | 100 | 140 | 209 | 140 | 55.6 | 25.9 | |

T5a | 0.352 | 100 | 180 | 209 | 140 | 55.6 | 27.8 | |

T5b | 0.352 | 100 | 180 | 209 | 140 | 55.6 | 31.7 | |

T6a | 0.352 | 100 | 220 | 209 | 140 | 55.6 | 31.7 | |

T6b | 0.352 | 100 | 220 | 209 | 140 | 55.6 | 28.6 | |

T7a | 0.352 | 100 | 100 | 209 | 140 | 55.6 | 33 | |

T7b | 0.352 | 100 | 100 | 209 | 140 | 55.6 | 26.9 | |

T8a | 0.352 | 100 | 100 | 209 | 140 | 55.6 | 28.5 | |

T8b | 0.352 | 100 | 100 | 209 | 140 | 55.6 | 29.8 | |

T9a | 1.056 | 100 | 100 | 209 | 140 | 55.6 | 28.4 | |

T9b | 1.056 | 100 | 100 | 209 | 140 | 55.6 | 29.8 | |

T10a | 1.056 | 100 | 140 | 209 | 140 | 55.6 | 37.4 | |

T10b | 1.056 | 100 | 140 | 209 | 140 | 55.6 | 33.3 | |

T11a | 1.056 | 100 | 180 | 209 | 140 | 55.6 | 42.8 | |

T11b | 1.056 | 100 | 180 | 209 | 140 | 55.6 | 39 | |

T12a | 0.352 | 70 | 100 | 209 | 140 | 55.6 | 21.1 | |

T12b | 0.352 | 70 | 100 | 209 | 140 | 55.6 | 24.2 | |

C150_1 | 0.165 | 100 | 150 | 230 | 150 | 35 | 18.97 | |

C150_2 | 0.165 | 100 | 150 | 230 | 150 | 35 | 16.51 | |

C150_3 | 0.165 | 100 | 150 | 230 | 150 | 35 | 14.26 | |

C100_1 | 0.165 | 100 | 100 | 230 | 150 | 35 | 13.63 | |

C100_2 | 0.165 | 100 | 100 | 230 | 150 | 35 | 13.36 | |

C100a_1 | 0.33 | 50 | 150 | 230 | 150 | 35 | 15.24 | |

C100a_2 | 0.33 | 50 | 150 | 230 | 150 | 35 | 18.19 | |

C100a_3 | 0.33 | 50 | 150 | 230 | 150 | 35 | 20.53 | |

C100a_4 | 0.165 | 100 | 150 | 230 | 150 | 35 | 19.5 | |

[59] | C-60-1 | 1.3 | 60 | 300 | 175 | 160 | 19 | 19.5 |

C-602 | 1.3 | 60 | 300 | 175 | 160 | 19 | 19.5 | |

C-60-3 | 1.3 | 60 | 300 | 175 | 160 | 19 | 19.5 | |

C-100-1 | 1.6 | 100 | 300 | 109 | 160 | 19 | 19.5 | |

C-100-2 | 1.6 | 100 | 300 | 109 | 160 | 19 | 19.5 | |

C-100-3 | 1.6 | 100 | 300 | 109 | 160 | 19 | 19.5 | |

C-100-4 | 1.2 | 100 | 300 | 166 | 160 | 19 | 19.5 | |

[60,61] | C-1.3 × 60-1 | 1.3 | 60 | 300 | 175 | 160 | 19 | 33.18 |

C-1.3 × 60-2 | 1.3 | 60 | 300 | 175 | 160 | 19 | 29.86 | |

C-1.3 × 60-3 | 1.3 | 60 | 300 | 175 | 160 | 19 | 31.88 | |

C-1.6 × 100-1 | 1.6 | 100 | 300 | 109 | 160 | 19 | 41.41 | |

C-1.6 × 100-2 | 1.6 | 100 | 300 | 109 | 160 | 19 | 39.87 | |

C-1.6 × 100-3 | 1.6 | 100 | 300 | 109 | 160 | 19 | 47.72 | |

C-1.2 × 100-1 | 1.2 | 100 | 300 | 166 | 160 | 19 | 49.85 | |

C-1.2 × 100-2 | 1.2 | 100 | 300 | 166 | 160 | 19 | 48.08 | |

C-1.2 × 100-3 | 1.2 | 100 | 300 | 166 | 160 | 19 | 52.6 | |

C-1.25 × 100-1 | 1.25 | 100 | 300 | 171 | 160 | 19 | 41.25 | |

C-1.25 × 100-2 | 1.25 | 100 | 300 | 171 | 160 | 19 | 38.14 | |

C-1.25 × 100-3 | 1.25 | 100 | 300 | 171 | 160 | 19 | 32.68 | |

C-1.7 × 100-1 | 1.7 | 100 | 300 | 221 | 160 | 19 | 54.79 | |

C-1.7 × 100-2 | 1.7 | 100 | 300 | 221 | 160 | 19 | 51.41 | |

C-1.7 × 100-3 | 1.7 | 100 | 300 | 221 | 160 | 19 | 54.57 | |

5 (25) | 1.4 | 50 | 250 | 140 | 150 | 37.55 | 39.78 | |

11 (25) | 1.4 | 50 | 250 | 140 | 150 | 35.7 | 31 | |

17 (25) | 1.4 | 50 | 200 | 140 | 150 | 32.78 | 35.65 | |

[62] | - | 1.02 | 25 | 203 | 108.38 | 228.6 | 36.4 | 11.57 |

- | 1.2 | 50 | 400 | 165 | 150 | 52.6 | 23 | |

- | 1.2 | 80 | 400 | 165 | 150 | 52.6 | 36.75 | |

- | 1.2 | 50 | 200 | 165 | 150 | 52.6 | 19.8 | |

- | 1.2 | 80 | 200 | 165 | 150 | 52.6 | 33 | |

- | 1.2 | 80 | 355 | 195.7 | 150 | 52.6 | 34.5 | |

- | 1.2 | 80 | 355 | 195.7 | 150 | 52.6 | 33.5 | |

- | 1.2 | 80 | 355 | 197.63 | 150 | 52.6 | 37.6 | |

- | 1.2 | 80 | 355 | 197.63 | 150 | 52.6 | 39.1 | |

- | 1.2 | 80 | 355 | 195.46 | 150 | 52.6 | 41 | |

- | 1.2 | 80 | 355 | 195.46 | 150 | 52.6 | 38 | |

- | 0.13 | 80 | 355 | 283.653 | 150 | 52.6 | 16.5 | |

- | 0.13 | 80 | 355 | 283.653 | 150 | 52.6 | 17.4 | |

- | 0.13 | 80 | 355 | 291.024 | 150 | 52.6 | 14.4 | |

- | 0.13 | 80 | 355 | 291.024 | 150 | 52.6 | 14.6 | |

V12A | 1.2 | 80 | 400 | 180 | 150 | 26 | 40 | |

V9A | 1.2 | 80 | 400 | 180 | 150 | 26 | 37 | |

V13A | 1.2 | 80 | 400 | 180 | 150 | 26 | 37.5 | |

V16A | 0.166 | 100 | 400 | 241 | 150 | 26 | 25.1 | |

V14A | 0.166 | 100 | 400 | 241 | 150 | 26 | 24.27 | |

V17A | 0.166 | 100 | 400 | 241 | 150 | 26 | 25.19 | |

V14B | 0.166 | 100 | 100 | 241 | 150 | 26 | 27 | |

V16B | 0.166 | 100 | 100 | 241 | 150 | 26 | 21 | |

V15B | 0.166 | 100 | 100 | 241 | 150 | 26 | 21.5 | |

V11A | 1.2 | 80 | 400 | 180 | 150 | 26 | 32.77 | |

V7A | 1.2 | 80 | 400 | 180 | 150 | 26 | 35.01 | |

V8A | 1.2 | 80 | 400 | 180 | 150 | 26 | 29.15 | |

V24A | 0.166 | 100 | 400 | 241 | 150 | 26 | 25.39 | |

V26A | 0.166 | 100 | 400 | 241 | 150 | 26 | 21.71 | |

V25A | 0.166 | 100 | 400 | 241 | 150 | 26 | 29.09 | |

V24B | 0.166 | 100 | 100 | 241 | 150 | 26 | 20.45 | |

V25B | 0.166 | 100 | 100 | 241 | 150 | 26 | 21.22 | |

V26B | 0.166 | 100 | 100 | 241 | 150 | 26 | 21.45 | |

V21b | 0.166 | 100 | 400 | 241 | 150 | 26 | 20.82 | |

V22b | 0.166 | 100 | 400 | 241 | 150 | 26 | 18.97 | |

V23b | 0.166 | 100 | 400 | 241 | 150 | 26 | 20.14 | |

V21a | 0.166 | 100 | 100 | 241 | 150 | 26 | 16.85 | |

V23a | 0.166 | 100 | 100 | 241 | 150 | 26 | 19.4 |

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**Figure 9.**Influence of the Young’s modulus of the FRP and the active bond length on the debonding force.

**Figure 10.**Influence of the width and the compressive strength of the concrete on the debonding force.

**Figure 11.**Influence of the concrete compressive (

**left**) and tensile (

**right**) strength on the debonding force.

Reference | Year | Model | Description |
---|---|---|---|

[22] | 1980 | ${P}_{u}=0.5{b}_{f}L{f}_{ctm}$ | Based on an assumed triangular bond-stress distribution. |

[23] | 1994 | ${P}_{u}={b}_{f}\sqrt{{E}_{f}{t}_{f}{G}_{f}}$ where: ${G}_{f}={c}_{f}{f}_{ctm}$ with c _{f} = 0.204 mm | Introduction of the fracture energy in a Ph.D thesis. |

[24] | 1996 | ${P}_{u}={\tau}_{a}{b}_{f}L$ where: ${\tau}_{a}=6.13-lnL$ | Analytical linear interpretation of the bond-stress based on experimental data. |

[25] | 1997 | ${P}_{u}={\tau}_{a}{b}_{f}L$ where: ${\tau}_{a}=5.88{L}^{-0.669}$ | Analysis of debonding in RC members strengthened with FRP. |

[26] | 1997 | ${P}_{u}={\tau}_{a}{b}_{f}{L}_{e}$ where: ${L}_{e}={e}^{\left[6.13-0.58ln\left({E}_{f}{t}_{f}\right)\right]}$ ${\tau}_{a}=110.2\times {10}^{-6}{E}_{f}{t}_{f}$ | First definition of the effective bond length calibrated with non-linear regression analysis based on experimental data. A linear bond-stress distribution is assumed. |

[27] | 1998 | ${P}_{u}={\tau}_{a}{b}_{f}{L}_{e}$ where: ${L}_{e}={e}^{\left[6.13-0.58ln\left({E}_{f}{t}_{f}\right)\right]}$ ${\tau}_{a}=110.2\times {10}^{-6}{E}_{f}{t}_{f}{\left(\frac{{f}_{c}}{42}\right)}^{2/3}$ | Effective bond length calibration with non-linear regression analysis based on experimental data. Non-linear bond-stress distribution is assumed |

[28] | 2000 | ${P}_{u}=\{\begin{array}{c}0.78{b}_{f}\sqrt{{E}_{f}{t}_{f}{G}_{f}}seL\ge {L}_{e}\\ 0.78{b}_{f}\sqrt{{E}_{f}{t}_{f}{G}_{f}}\frac{L}{{L}_{e}}\left(2-\frac{L}{{L}_{e}}\right)seL{L}_{e}\end{array}$ where: ${L}_{e}=\sqrt{\frac{{E}_{f}{t}_{f}}{4{f}_{ctm}}}$ ${G}_{f}={c}_{f}{f}_{ctm}{\beta}_{w}^{2}$ ${\beta}_{w}=\sqrt{1.125\left(\frac{2-{b}_{f}/{b}_{c}}{1+{b}_{f}/400}\right)}$ | Introduction of the geometrical factor related to the width of the bonded plate and the width of the concrete member. |

[29] | 2001 | ${P}_{u}=\{\begin{array}{c}0.427{\beta}_{w}{b}_{f}{L}_{e}\sqrt{{f}_{c}}seL\ge {L}_{e}\\ 0.427{\beta}_{w}{b}_{f}{L}_{e}\sqrt{{f}_{c}}\mathrm{sin}\left(\frac{\pi L}{2{L}_{e\text{}}}\right)se\text{}L{L}_{e}\end{array}$ where: ${L}_{e}=\sqrt{\frac{{E}_{f}{t}_{f}}{\sqrt{{f}_{c}}}}$ ${\beta}_{w}=\sqrt{\frac{2-{b}_{f}/{b}_{c}}{1+{b}_{f}/{b}_{c}}}$ | A modified version of the model reported in [23], validated for both CFRP and steel plates. The shear-slip relationship is represented by a triangular shape. |

[30] | 2001 | ${P}_{u}=\{\begin{array}{c}0.64\alpha {\beta}_{w}{b}_{f}{k}_{c}\sqrt{{E}_{f}{t}_{f}{f}_{ctm}}seL\ge {L}_{e}\\ 0.64\alpha {\beta}_{w}{b}_{f}{k}_{c}\sqrt{{E}_{f}{t}_{f}{f}_{ctm}}\frac{L}{{L}_{e}}\left(2-\frac{L}{{L}_{e}}\right)seL{L}_{e}\end{array}$ where: ${\beta}_{w}=1.06\sqrt{\frac{2-{b}_{f}/{b}_{c}}{1+{b}_{f}/400}}$ ${L}_{e}=\sqrt{\frac{{E}_{f}{t}_{f}}{2{f}_{ctm}}}$ with α = 1 and kc = 1. | International code based on the proposals from [27,28]. |

[31] | 2001 | ${P}_{u}=\left(0.5+0.08\sqrt{\frac{{E}_{f}{t}_{f}}{100{f}_{ctm}}}\right){\tau}_{a}{b}_{f}{L}_{e}$ where: ${\tau}_{a}=0.5{f}_{ctm}$ Le = 100 mm | The effective bond length has been considered not affected by the compressive strength of the substrate and fixed to 100 mm based on authors experimentations. An empirical model has been calibrated. |

[32] | 2003 | ${P}_{u}={\tau}_{a}{b}_{f}{L}_{e}$ where: ${L}_{e}=0.125{\left({E}_{f}{t}_{f}\right)}^{0.57}$ ${\tau}_{a}=0.93{f}_{c}^{0.44}$ | Simplified empirical model based on literature database. A non-linear bond-stress distribution, along the FRP-length, has been assumed. |

[33] | 2004 | ${P}_{u}={b}_{f}\sqrt{2{E}_{f}{t}_{f}{G}_{f}}={b}_{f}\sqrt{2\frac{{f}_{ctm}}{8}{E}_{f}{t}_{f}}\phantom{\rule{0ex}{0ex}}=0.5{b}_{f}\sqrt{{E}_{f}{t}_{f}{f}_{ctm}}$ where: ${G}_{f}=\frac{{f}_{ctm}}{8}$ | Further empirical model based on literature database. |

[34] | 2005 | ${P}_{u}=\{\begin{array}{c}{b}_{f}\sqrt{2{E}_{f}{t}_{f}{G}_{f}}se{b}_{f}100\mathrm{mm}\\ \left({b}_{f}+2\Delta {b}_{f}\right)\sqrt{2{E}_{f}{t}_{f}{G}_{f}}se{b}_{f}\ge 100\mathrm{mm}\end{array}$ where: ${G}_{f}=0.514{f}_{c}^{0.236}$ $\Delta {b}_{f}=3.7\text{}\mathrm{mm}$ | Analytical model for defining the nonlinear bond stress–slip law by means of non-linear regression analysis (authors data consisting in 26 tests). |

[35] | 2005 | ${P}_{u}={\beta}_{l}{b}_{f}\sqrt{2{E}_{f}{t}_{f}{G}_{f}}$ where: ${\beta}_{l}=\{\begin{array}{c}1seL\ge {L}_{e}\\ \frac{L}{{L}_{e}}\left(2-\frac{L}{{L}_{e}}\right)seL{L}_{e}\end{array}$ ${L}_{e}=a+\frac{1}{2{\lambda}_{1}}ln\left[\frac{{\lambda}_{1}+{\lambda}_{2}tg\left({\lambda}_{2}a\right)}{{\lambda}_{1}-{\lambda}_{2}tg\left({\lambda}_{2}a\right)}\right]$ ${G}_{f}=0.308{\beta}_{w}^{2}\sqrt{{f}_{ctm}}$ ${\lambda}_{1}=\sqrt{\frac{{\lambda}_{max}}{{s}_{o}{E}_{f}{t}_{f}}}$ ${\lambda}_{2}=\sqrt{\frac{{\lambda}_{max}}{({s}_{f}-{s}_{o}){E}_{f}{t}_{f}}}$ ${\lambda}_{max}=1.5{\beta}_{w}{f}_{ctm}$ ${s}_{o}=0.0195{\beta}_{w}{f}_{ctm}$ ${s}_{f}=\frac{2{G}_{f}}{{\lambda}_{max}}$ ${\beta}_{w}=\sqrt{\frac{2.25-{b}_{f}/{b}_{c}}{1.25+{b}_{f}/{b}_{c}}}$ | A linear best-fit line between finite element predictions, data collected from the existing literature (253 tests) and theoretical outcomes has been proposed by regression analysis. |

[36] | 2009 | ${P}_{u}=\{\begin{array}{c}0.585{b}_{f}{\beta}_{w}{f}_{c}^{0.1}{\left({E}_{f}{t}_{f}\right)}^{0.54}seL\ge {L}_{e}\\ 0.585{b}_{f}{\beta}_{w}{f}_{c}^{0.1}{\left({E}_{f}{t}_{f}\right)}^{0.54}{\left(\frac{L}{{L}_{e}}\right)}^{1.2}seL{L}_{e}\end{array}$ where: ${\beta}_{w}=\sqrt{\frac{2.25-{b}_{f}/{b}_{c}}{1.25+{b}_{f}/{b}_{c}}}$ ${L}_{e}=\frac{0.395{\left({E}_{f}{t}_{f}\right)}^{0.54}}{{f}_{c}^{0.09}}$ | A bond strength model has been calibrated in order to reach an empirical formulation from the analysis of about 311 experimental data. |

[37] | 2010 | ${P}_{u}={\beta}_{w}{b}_{f}\sqrt{2\left(1+\frac{{\lambda}^{\prime}}{\Sigma}\right){E}_{f}{t}_{f}{G}_{cf}}$ where: ${\beta}_{w}=\sqrt{\frac{2-{b}_{f}/{b}_{c}}{1+{b}_{f}/{b}_{c}}}$ ${\lambda}^{\prime}=\frac{{t}_{d}}{{t}_{f}}$ with t _{d} = 3.5 mm $\Sigma =\frac{{E}_{f}}{{E}_{c}}$ G _{cf} = 0.17 N/mm | A finite element analysis has been performed to determine the fracture energy. |

[38] | 2012 | ${P}_{u}=0.5{b}_{f}{\beta}_{w}\sqrt{{E}_{f}{t}_{f}{f}_{ctm}}$ where: ${\beta}_{w}=1.06\sqrt{\frac{2-{b}_{f}/{b}_{c}}{1+{b}_{f}/400}}$ | Simplified model calibrated by existing literature. It indicates that limiting the longitudinal shear stress (at the ultimate limit state) to a value not greater than 0.8 MPa, premature peeling failure can be avoided. It is further recommended that a minimum anchorage length of 500 mm should be provided. |

[39] | 2013 | ${P}_{u}=\{\begin{array}{c}\frac{{b}_{f}}{{\gamma}_{Fd}}\sqrt{2{E}_{f}{t}_{f}{K}_{G}{\beta}_{w}\sqrt{{f}_{c}{f}_{ctm}}}seL\ge {L}_{e}\\ \frac{{b}_{f}}{{\gamma}_{Fd}}\sqrt{2{E}_{f}{t}_{f}{K}_{G}{\beta}_{w}\sqrt{{f}_{c}{f}_{ctm}}}\frac{L}{{L}_{e}}\left(2-\frac{L}{{L}_{e}}\right)seL{L}_{e}\end{array}$ where: ${L}_{e}=max\left\{\frac{1}{{\gamma}_{Rd}{f}_{bd}}\sqrt{\frac{{\pi}^{2}{E}_{f}{t}_{f}{K}_{G}{\beta}_{w}\sqrt{{f}_{c}{f}_{ctm}}}{2}\text{}};\text{}200\text{}\mathrm{mm}\right\}$ ${\beta}_{w}=\sqrt{\frac{2-{b}_{f}/{b}_{c}}{1+{b}_{f}/{b}_{c}}}$. se b _{f}/b_{c} < 0.25γ _{Rd} = 1.25${f}_{bd}=\frac{2{\Gamma}_{Fd}}{{s}_{u}}$ with su = 0.25 mm ${\Gamma}_{Fd}={K}_{G}{\beta}_{w}\sqrt{{f}_{c}{f}_{ctm}}$ with γ _{Fd} = 1.25 | Empirical model derived from available data and based on the evaluation of the specific fracture energy—${\Gamma}_{Fd}$. |

Parameter | t_{f} (mm) | b_{f} (mm) | L_{f} (mm) | E_{f} (GPa) | b_{c} (mm) | f_{c} (MPa) |
---|---|---|---|---|---|---|

Interquartile range (IQR) | 0.61 | 0.55 | 0.21 | 0.23 | 0.12 | 0.34 |

Max | 1.40 | 100 | 700 | 390 | 500 | 74 |

Min | 0.083 | 10 | 50 | 83 | 100 | 17 |

Quartile | Value |
---|---|

Q0 | 0.15 |

Q1 | 0.79 |

Q3 | 1.20 |

Q4 | 2.53 |

Me | 1.01 |

References | Year | IQR | K | as1 | as2 | as3 |
---|---|---|---|---|---|---|

[34] | 2005 | 0.23 | 2.88 | 0.00 | 0.00 | −0.11 |

[35] | 2005 | 0.23 | 3.75 | 0.00 | 0.00 | 0.25 |

[30] | 2001 | 0.24 | 3.28 | 0.00 | 0.00 | 0.09 |

[29] | 2001 | 0.25 | 4.49 | 0.50 | 1.50 | 0.50 |

[38] | 2012 | 0.27 | 3.24 | 0.50 | 1.50 | 0.08 |

[39] | 2013 | 0.30 | 3.09 | 0.00 | 0.00 | 0.34 |

[36] | 2009 | 0.35 | 2.45 | −0.50 | −1.50 | −0.25 |

[28] | 2000 | 0.38 | 3.17 | 0.50 | 1.50 | 0.11 |

Proposed Model [[–] | - | 0.41 | 4.86 | 0.00 | 0.00 | 0.00 |

[32] | 2003 | 0.42 | 4.69 | −0.50 | −1.50 | 0.11 |

[33] | 2004 | 0.44 | 2.64 | 0.00 | 0.00 | 0.01 |

[37] | 2010 | 0.45 | 4.02 | 0.00 | 0.00 | 0.72 |

[31] | 2001 | 0.45 | 2.70 | 0.50 | 1.50 | 0.24 |

[23] | 1994 | 0.49 | 2.64 | 0.00 | 0.00 | 0.01 |

[22] | 1980 | 1.14 | 7.76 | 0.50 | 1.50 | 0.52 |

[24] | 1996 | 1.85 | 7.43 | 0.50 | 1.50 | 1.02 |

[25] | 1997 | 6.29 | 3.55 | 0.50 | 1.50 | 0.90 |

[26] | 1997 | 15.92 | 3.16 | 0.00 | 0.00 | 0.36 |

[27] | 1998 | 33.46 | 2.17 | −0.50 | −1.50 | 0.09 |

Max | 33.46 | 7.76 | 0.50 | 1.50 | 1.02 | |

Min | 0.23 | 2.17 | 0.00 | 0.00 | 0.00 |

References | RRMSE (%) | MAE (kN) | MAPE (%) | R^{2} |
---|---|---|---|---|

[34] | 2.00% | 3.34 | 20.00% | 0.98 |

[35] | 0.00% | 0.46 | 0.60% | 0.99 |

[30] | 1.00% | 2.87 | 17.00% | 0.97 |

[29] | 0.00% | 0.46 | 1.10% | 0.99 |

[38] | 0.00% | 0.61 | 0.40% | 0.98 |

[39] | 1.00% | 2.71 | 15.00% | 0.96 |

[36] | 3.00% | 6.35 | 37.00% | 0.90 |

[28] | 1.00% | 2.26 | 15.00% | 0.98 |

Proposed Model [–] | 0.00% | 0.84 | 0.30% | 0.95 |

[32] | 1.00% | 1.09 | 1.80% | 0.95 |

[33] | 1.00% | 1.67 | 10.00% | 0.97 |

[37] | 0.00% | 0.95 | 1.10% | 0.94 |

[31] | 1.00% | 0.96 | 4.90% | 0.95 |

[23] | 1.00% | 2.29 | 14.00% | 0.97 |

[22] | 1.00% | 2.27 | 12.00% | 0.86 |

[24] | 2.00% | 4.05 | 24.70% | 0.80 |

[25] | 3.00% | 7.69 | 44.70% | 0.93 |

[26] | 4.00% | 8.43 | 49.10% | 0.98 |

[27] | 4.00% | 7.56 | 48.90% | 0.90 |

Max | 4.00% | 8.43 | 49.10 | 0.99 |

Min | 0.00 | 0.46 | 0.30% | 0.80 |

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Cascardi, A.; Micelli, F.
ANN-Based Model for the Prediction of the Bond Strength between FRP and Concrete. *Fibers* **2021**, *9*, 46.
https://doi.org/10.3390/fib9070046

**AMA Style**

Cascardi A, Micelli F.
ANN-Based Model for the Prediction of the Bond Strength between FRP and Concrete. *Fibers*. 2021; 9(7):46.
https://doi.org/10.3390/fib9070046

**Chicago/Turabian Style**

Cascardi, Alessio, and Francesco Micelli.
2021. "ANN-Based Model for the Prediction of the Bond Strength between FRP and Concrete" *Fibers* 9, no. 7: 46.
https://doi.org/10.3390/fib9070046