New robust formulations for bond strength of FRP reinforcements externally glued on masonry units
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Abstract
The bond interface between fiber-reinforced polymer (FRP) and masonry units is one of the weakest links in tensile strengthened masonry structures. Several empirical formulae have been proposed for estimation of bond strength between FRP reinforcements and masonry units. However, the accuracy of existing formulae for predicting bond strength seems to be significantly limited. In the present study, the M5´ and Multivariate adaptive regression splines approaches are employed to predict the bond strength between FRP reinforcement and masonry units. To develop new models, a comprehensive database including 575 test series (230 distinctive specimens) is collected from different sources in the literature. The newly proposed formulations consider several preeminent parameters involved in the debonding process including the reinforcement width, the ratio between widths of FRP reinforcement and masonry unit, the tensile strength of substrate, the axial strength of reinforcement, and bond length. A comparative study is conducted to evaluate the performances of the developed models against the well-known equations. Results indicated that the proposed models remarkably outperform the existing equations in terms of accuracy. Furthermore, a sensitivity analysis is done to determine the most important parameters in predicting the bond strength. Finally, the safety of different methods is evaluated based on the demerit point classification scale.
Keywords
FRP EBR Masonry Bond strength M5′ MARSAbbreviations
- FRP
Fiber Reinforced Polymer
- EBR
Externally Bonded Reinforcement
- MARS
Multivariate Adaptive Regression Splines
- DPC
Demerit Points Classification
- GCV
Generalized Cross-Validation
- BF
Basis Function
- MAE
Mean Absolute Error
- RMSE
Root Mean Square Error
- C
Carbon
- G
Glass
- B
Basalt
- S
Steel
- CB
Clay Brick
- B-old
Ancient Brick
- B-new
Recent Brick
- NS-tuff
Tuff Natural Stones
- YT
Yellow Tuff
- GT
Gray Tuff
- LS
Lime Stones
- NS-Limes
Lime Natural Stones
- CS
Calcareous Stones
1 Introduction
Fiber reinforced polymer (FRP) sheets or laminates have been successfully applied as externally bonded reinforcement (EBR) systems for quite some time in civil engineering constructions. In particular, the use of FRP EBR system for strengthening and retrofitting the existing masonry buildings like monumental heritage is significantly increased. The reason for increasing the use of FRP materials for retrofitting masonry buildings is due to the favorable properties of FRP EBR techniques such as high strength and resistance to corrosion, high durability, non-magnetic, high fatigue resistance, and no increment in mass and stiffness of the structure [1]. In common practice, the FRP EBR technique is designed based on several empirical and semi-empirical design equations. The design procedure of FRP EBR technique including installation, loads, safety requirements and acceptable features of materials has been coded in several guidelines. Different in-plane or out-of-plane failure modes of retrofitted masonry units have also been investigated in recent years [2, 3].
The bond interface between FRP and masonry units is often one of the weakest links in tensile strengthened masonry structures, and debonding at this interface is one of the critical failure modes of FRP EBR systems. During the design procedure, bond strength at the interface level must be taken into account because of its sudden and brittle failure. Debonding process is very complex due to several preeminent parameters involved in this process including mechanical properties of masonry blocks, mortar joints, adhesive, and FRP reinforcement. This mechanism involves the presence of cracks a few millimeters underneath the bond line inside the masonry units [4]. In fact, FRP debonding will initiate in the substrate if the bond strength directly depends on the tensile strength of masonry units.
Various experimental studies have been conducted to investigate the effect of different influential parameters on debonding failure and a number of empirical and analytical prediction models have been developed. As examples of such studies, one can refer to [5, 6]. However, due to the complexity and the brittle characteristics of debonding failure mechanism, a remarkable scatter has been observed between predicted maximum bond strength and the measured ones [7]. Many of these models are highly empirical, and their predictive abilities are limited by the corresponding data sets from which they were derived and do not provide a reliable prediction of maximum transferable load. Recently, soft computing methods have been successfully used to develop predictive models for different problems in civil engineering [8, 9, 10, 11]. In particular, Mansouri and Kisi [7] evaluated the applications of neuro-fuzzy and neural network approaches for estimation of debonding strength for masonry elements retrofitted with FRP composites using eight available experimental datasets consisting of altogether 109 data points. Results showed that approaches can be successfully used to predict the bond strength. However, these methods do not give sufficient insight into the generated models and are not as easy to use as the empirical formulas. Gene expression programming has been used by Mansouri et al. [12] to predict the debonding strength of retrofitted masonry members using ten available experimental databases [5] consisting of 134 data points. Although the approach has successfully been used to predict the bond strength and results in an explicit formulation of maximum load but it is well known that compiling more comprehensive database is requisite in generating exact predictive models.
The main purpose of this study is to employ M5′ and Multivariate adaptive regression splines (MARS) algorithms to develop transparent models to predict maximum bond strength and determine the most effective parameters. To achieve this aim, a new comprehensive database collected from different sources in the literature (230 data points from 23 available experimental studies). The M5′ model tree as a robust data-based method provides understandable formulas that allow users to have more insight in the physics of the phenomenon [13, 14]. The MARS algorithm is also known as a self-organized predictive approach that can discover complex behaviors between input and output parameters and determine the most effective parameters for predicting the maximum bond strength [15]. Five different predictive variables that characterize the mechanical and geometrical properties related to the FRP rods and the substrate including: (1) the reinforcement width, (2) ratio between the widths of FRP reinforcement and masonry unit, (3) tensile strength of substrate, (4) axial strength of reinforcement and (5) bond length are considered as input variables. A comparative study is implemented to evaluate the performances of the developed models against the most common design equations in literature. In addition, the safety analysis based on the Demerit Points Classification (DPC) scale has been also done to measure the reliability of the proposed formulations. It is demonstrated that the M5′ and MARS algorithms can successfully be used as reliable alternative approaches to predict the bond strength of FRP EBR systems.
2 Background
3 Materials and methods
The methodology adopted in this study is based on two well-known and practical decision tree algorithms namely the M5´ model tree and multivariate adaptive regression splines (MARS) approach. The distinctive features of the M5′ and MARS algorithms are employed to investigate the shear behavior of FRP EBR systems. The details of algorithms are presented as follows:
3.1 M5′ algorithm
4 Multivariate adaptive regression splines algorithm
5 Model development
5.1 Influential parameters
5.2 Dataset description
Details of database and range of input and output variables
Experimental program | No. of tests | Masonry | FRP | L_{b} (mm) | b_{p} (mm) | E_{p} (GPa) | T_{p} (mm) | F_{cm} (MPa) | F_{mt} (MPa) | B_{m} (mm) | B _{ p} /b _{ m} | F_{max,exp} (KN) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
[23] | 2 | NS-tuff | C | 200 | 100 | 230 | 0.165 | 2 | – | 206 | 0.48543 | 12.5 |
[24] | 17 | NS_limes. | C | 100–150 | 50 | 230 | 0.13 | 2.2–4.4 | 0.2 | 100 | 0.5 | 3.3–4.9 |
[25] | 5 | NS_limes. & NS-tuff | C | 150 | 80 | 234–246 | 0.165–0.185 | 5_26 | 1–2 | 100 | 0.8 | 8.1–9.25 |
[26] | 7 | YT | C | 150–200 | 100 | 230 | 0.165 | 3.2–3.7 | 0.3–0.4 | 200–250 | 0.4–0.5 | 7.96–20 |
27 | LS | C&G | 150 | 80 | 81.4–246 | 0.165–0.23 | 24 | 2.4 | 100–200 | 0.333–0.8 | 6.22–12.85 | |
8 | YT | C&G | 150 | 80 | 81.4–246 | 0.165–0.23 | 5–5.5 | 0.5–0.6 | 100–200 | 0.4–0.8 | 7.8–9.84 | |
4 | GT | C&G | 150 | 80 | 81.4–234 | 0.165–0.23 | 4.1 | 0.4 | 200 | 0.4 | 10.98–12.76 | |
[28] | 25 | CS | C | 150 | 50 | 230 | 0.35 | 2.2–11.3 | 0.2–1.1 | 100 | 0.5 | 3.58–5.65 |
[28] | 11 | CS | C | 100–150 | 50 | 230 | 0.35 | 2.9–4.2 | 0.3–0.4 | 100 | 0.5 | 3.28–4.93 |
[29] | 14 | B-old & B-new | C | 120–290 | 50 | 240 | 0.17 | 17.4–40.2 | 1.9 | 120–140 | 0.357–0.416 | 7.4–10.2 |
[29] | 10 | CB | C | 150–290 | 50 | 240 | 0.17 | 24.9–40.2 | 2.5–4 | 120–140 | 0.357–0.417 | 14.8–20.4 |
[27] | 9 | NS_limes. & NS-tuff | C&G | 150 | 80 | 81–234 | 0.165–0.37 | 5.5–24 | 0.4–3.3 | 100 | 0.8 | 6.2–12.8 |
[30] | 10 | CB | C&G | 200 | 50 | 65–230 | 0.165–0.23 | 42.3 | 4.2 | 120 | 0.417 | 13.34–18.1 |
[31] | 45 | CB | C | 150–240 | 40–80 | 230 | 0.165 | 17.4 | 1.7 | 120 | 0.333–0.667 | 10.03–19 |
[32] | 3 | CB | C&G | 250 | 50 | 73–240 | 0.097–0.17 | 7.3 | 0.7 | 120 | 0.417 | 5.25–13 |
[32] | 3 | CB | G | 250 | 50 | 73 | 0.12 | 7.25 | 0.7 | 120 | 0.417 | 10.5 |
[32] | 9 | B-old | C&G&S | 250 | 50 | 73–240 | 0.097–0.17 | 7.3 | – | 140 | 0.357143 | 5.266–11.5 |
[33] | 4 | B-new | C | 120–160 | 25 | 160 | 1.2 | 38.5 | – | 120 | 0.208333 | 8.3–10.4 |
[33] | 13 | B-new | S | 160 | 25 | 190 | 0.227 | 27.3–38.5 | – | 120 | 0.208333 | 6.24–12.04 |
[34] | 9 | CB | C&G | 150–200 | 25 | 80–215 | 0.117–0.149 | 8.8 | 0.9 | 130 | 0.192 | 4.34–4.91 |
[35] | 4 | B-new | C | 100 | 12 | 230 | 0.167 | 35.6 | – | 100 | 0.12 | 5.575 |
[36] | 8 | YT | C&G | 300 | 100 | 80.7–230 | 0.166–0.48 | 2.1–3.8 | 0.2–0.4 | 250–252 | 0.397–0.4 | 14.23–18.51 |
[36] | 5 | NS-tuff | C&G | 300 | 100 | 81–230 | 0.164–0.48 | 3.8 | 0.4 | 250 | 0.4 | 19.96–20.95 |
[37] | 30 | NS_limes. & NS-tuff & B-new | C&G | 230–245 | 51–129 | 81–230 | 0.164–0.342 | 2.3–70 | 0.3–5.7 | 118–129 | 0.425–1 | 11.5–36.4 |
[38] | 10 | CB | C | 200 | 50 | 240 | 0.17 | 9.9 | 1 | 120 | 0.417 | 6.28–8.97 |
[39] | 7 | B-old | G | 200 | 50 | 124 | 0.172 | 16.3–32.9 | 1.8–4.9 | 110–127 | 0.393–0.454 | 4.45–7.2 |
[39] | 8 | CB | C | 200 | 50 | 132 | 0.172 | 16.4–31.8 | 1.6–3.2 | 110–130 | 0.385–0.455 | 4.54–7.36 |
[40] | 1 | NS_limes. | C | 150 | 80 | 245 | 0.165 | 31 | 3.9 | 100 | 0.8 | 9.8 |
[41] | 6 | CB | C&B&G | 200 | 50 | 73–240 | 0.12–0.165 | 19.21 | 1.9 | 120 | 0.417 | 4.93–6.81 |
[41] | 23 | CB | G | 160 | 50 | 84 | 0.12 | 19.8 | 2 | 120 | 0.417 | 3.88–5.09 |
[41] | 62 | CB | C&B | 160 | 50 | 88.397–233 | 0.14–0.17 | 19.8–20.8 | 2–2.1 | 120–121 | 0.413–0.417 | 4.61–9.07 |
[41] | 163 | B-new | C&B&G&S | 160 | 50 | 84–234 | 0.12–0.231 | 19.8 | 1.8 | 120 | 0.416667 | 4.04–12.24 |
[42] | 5 | B-old | G | 250 | 50 | 70 | 0.12 | 15.4 | 1.6 | 120 | 0.416667 | 5.98 |
[42] | 8 | B-old | C | 150 | 35 | 390 | 0.23 | 11.4–17.2 | 1.6–1.9 | 131–147 | 0.238–0.267 | 5.6–10.433 |
To develop new models based on M5′ and MARS algorithms, the whole dataset is randomly divided into two parts such that 70% (161 data vectors) were used for the learning process and 30% (69 data vectors) were employed to test the developed models. The training and testing databases are presented in Tables 8 and 9, respectively, in the “Appendix” section.
5.3 The M5′ model
For example, the developed equations show that the maximum bond strength decreases as the ratio between b_{p} and b_{m} (b_{p}/b_{m}) increases. This can be justified that as the b_{p}/b_{m} increases, the effects of three-dimensional shear transfer increases. When the b_{p}/b_{m}< 1, the larger failure surface involved in the bond mechanism, and consequently, the bond stresses can spread laterally over the bond width (b_{p}). On the contrary, when the ratio b_{p}/b_{m} tends to 1 (corresponding to plane strain conditions), the fracture energy per unit of FRP-width is smaller than in case of b_{p}/b_{m}< 1. As seen in Eq. (15), the M5′ algorithm correctly captured this underlying physical concept. Furthermore, according to the developed tree in Fig. 5, the prediction of maximum bond strength for smaller FRP-width (< 50.5 mm) is more complex and the algorithm divided the space of problem into three subspaces. This is due to this fact that considering the effects of three-dimensional shear transfer in real problems is a quite complex task, which is also addressed in [44].
In general, most results derived from M5′ algorithm are based on information inherent in the collected database. As stated, the obtained results physically sound but some nonlinear relationships between input and output parameters may be different in comparison with other existing equations. For example, the relationship between L_{b} (or L_{e}, effective length) and F_{max} in most equations in literature are linear while this linear relation was found by M5′ algorithm only in two classes. However, it is shown that the M5′ results are more compatible with experimental observations.
5.4 The MARS model
The basis functions (BFs) of the developed MARS model
Basis function number | Equation |
---|---|
BF _{1} | max(0, 60–b_{p}) |
BF _{2} | max(0, 3.3–f_{mt}) |
BF _{3} | max(0, 160–L_{b}) |
BF _{4} | max(0, 38–E_{p}t_{p}) |
BF _{5} | BF_{1} × max(0, 0.36–b_{p}/b_{m}) |
BF _{6} | BF_{3}× max(0, b_{p}–35) |
BF _{7} | max(0, b_{p}–60) × max(0, 0.4–f_{mt}) |
BF _{8} | max(0, L_{b}–160) × max(0, f_{mt}–1.9) × max(0, b_{p}/b_{m}–0.45) |
BF _{9} | max(0, L_{b}–160) × max(0, f_{mt}–1.9) × max(0, 0.45–b_{p}/b_{m}) × max(0, E_{p}t_{p}–21) |
BF _{10} | max(0, 0.36–b_{p}/b_{m}) |
BF _{11} | BF_{10} × max(0, b_{p}–40) |
BF _{12} | BF_{10}× max(0, 40–b_{p}) |
BF _{13} | BF_{3} × max(0, 0.27–b_{p}/b_{m}) |
BF _{1 4} | max(0, 1.1–f_{mt}) × max(0, b_{p}–50) |
6 Results and discussions
6.1 Performance analysis
The performances of the developed models
Models | MAE | RMSE | SI (%) | R | R^{2} |
---|---|---|---|---|---|
M5′ | |||||
Training | 1.8191 | 3.0476 | 31.4 | 0.8606 | 0.7094 |
Testing | 1.3317 | 1.9082 | 32.76 | 0.8524 | 0.7137 |
Total | 1.6749 | 2.7458 | 31.79 | 0.8536 | 0.7133 |
MARS linear | |||||
Training | 1.2510 | 1.7449 | 19.70 | 0.9455 | 0.8934 |
Testing | 1.3658 | 1.8632 | 22.94 | 0.9157 | 0.8315 |
Total | 1.2854 | 1.7812 | 20.62 | 0.9377 | 0.8793 |
MARS cubic | |||||
Training | 1.4040 | 1.9887 | 22.45 | 0.9283 | 0.8616 |
Testing | 1.3389 | 1.9722 | 24.28 | 0.9017 | 0.8112 |
Total | 1.3845 | 1.9837 | 22.97 | 0.9221 | 0.8503 |
Most previous studies applied the correlation coefficient (R) to measure the correlation between observed and predicted values. Smith [46] suggested that if |R| > 0.8, there is a strong correlation between measured and predicted values. However, R cannot necessarily be considered as an indicator for the goodness of correlation between observed and predicted values; particularly, when data range is very wide and the data points distributed about their mean. Therefore, in the present study, the R^{2} parameter is employed as an unbiased estimate and also a better measure for evaluating the correlation between observed and predicted values. The MAE and RMSE are also used to measure the absolute difference between predicted and measured values. These values must be near to zero for having a close match between observed and predicted values. As shown in Table 3, the MARS model constructed based on piecewise linear basis segments outperformed the models of MARS with cubic segments and M5′ in terms of accuracy for both training and testing data sets. For example, it decreases the RMSE value by 11% and 35%, respectively, and increases the R^{2} values by 3.4% and 23.2% in respect to MARS model with cubic segment and M5′ model, respectively. It should be noted that the performances of the developed MARS with cubic segments and M5′ models are also acceptable and MARS with cubic segments also outperforms the M5′ model.
Comparison of the developed models with the most common equations
Model | MAE | RMSE | SI (%) | R | R^{2} |
---|---|---|---|---|---|
Tanaka [46] | 3.60 | 4.42 | 51.27 | 0.53 | 0.25 |
Sato [47] | 13.35 | 17.97 | 208.17 | 0.49 | − 11.29 |
Iso [47] | 21.82 | 27.65 | 320.24 | 0.79 | − 28.09 |
Yang et al. [48] | 2.91 | 4.25 | 49.3 | 0.69 | 0.31 |
Neubauer and Rostasy [49] | 3.56 | 4.60 | 53.27 | 0.67 | 0.19 |
Willis et al. [6] | 5.29 | 6.85 | 79.33 | 0.57 | − 0.78 |
Kashyap et al. [5] | 8.04 | 12.70 | 147.12 | 0.38 | − 5.14 |
Maeda et al. [50] | 5.89 | 6.98 | 80.88 | 0.51 | − 0.85 |
Khalifa et al. [51] | 3.10 | 4.52 | 0.5242 | 0.66 | 0.22 |
De Lorenzis et al. [52] | 10.66 | 12.40 | 143.63 | 0.47 | − 4.85 |
ACI 440.2R-08 | 8.57 | 10.82 | 125.63 | 0.47 | − 3.47 |
CNR. DT 200/2012 | 2.88 | 3.73 | 43.21 | 0.80 | 0.47 |
M5′ (preset study) | 1.67 | 2.74 | 31.79 | 0.85 | 0.71 |
MARS (present study) | 1.28 | 1.78 | 20.62 | 0.93 | 0.87 |
6.2 Sensitivity analysis
The analysis of variances (ANOVA) decomposition of the developed MARS models
Function | STD | GCV | Variable(s) | ||
---|---|---|---|---|---|
Linear | Cubic | Linear | Cubic | ||
1 | 2812.925 | 49.643 | 11,572,307.35 | 3607.591 | b _{ p} /b _{ m} |
2 | 1.242 | 2.122 | 12.99 | 33.324 | f _{ mt} |
3 | 0.759 | 0.961 | 5.415 | 8.992 | E _{ p} t _{ p} |
4 | 5.647 | 5.213 | 105.579 | 87.188 | b _{ p} |
5 | 4.032 | 3.169 | 32.75 | 29.965 | L _{ b} |
6 | 2817.19 | 52.778 | 11,038,785.57 | 3922.107 | b_{p}/b_{m}, b_{p} |
7 | 1.904 | 1.454 | 9.351 | 14.589 | b_{p}/b_{m}, L_{b} |
8 | 1.111 | 1.505 | 6.018 | 10.967 | f_{mt}, b_{p} |
9 | 4.832 | 4.555 | 47.506 | 54.292 | f_{mt}, L_{b} |
10 | 2.038 | 1.848 | 10.11 | 12.049 | f_{mt,} b_{p}/b_{m}, L_{b} |
11 | 1.35 | 0.384 | 6.929 | 7.618 | f_{mt,} b_{p}/b_{m}, L_{b}, E_{p}t_{p} |
6.3 Safety analysis
To have a safe and economical design of EBR FRP systems, the reliability of developed models for prediction of maximum bond strength between FRP reinforcements and masonry units must be investigated. To achieve this, the box plot of discrepancy ratio between observed and predicted maximum bond strength (DR) is used to measure the reliability and uncertainty of the existing and developed models. Box plot is a convenient graphical way to illustrate data points through their quartiles. The variations in samples of a statistical population are monitored without making any prior assumptions about the underlying distribution. The space between the different parts of the box can be assumed as an indicator for the degree of dispersion (spread) or skewness of data points. According to the statistical analysis of previous sections, it can be expected that the developed M5′ and MARS models are more reliable than other existing equations. However, the uncertainty/safety factor of existing models and the developed ones cannot be determined. To mitigate this limitation, a safety factor can be attributed to different models based on the acceptable level of risk.
Classification by demerit points
F_{max,exp}/F_{max,predicted} | Classification | Demerit points |
---|---|---|
< 0.50 | Extremely dangerous | 10 |
[0.50–0.85) | Dangerous | 5 |
[0.85–1.15) | Appropriate and safe | 0 |
[1.15–2.00) | Conservative | 1 |
≥ 2.00 | Extremely conservative | 2 |
Classification of developed and design equations according to the criteria of collins
Models | DR < 0.5 | 0.5 < DR < 0.85 | 0.85 < DR < 1.15 | 1.15 < DR < 2 | DR > 2 | Total |
---|---|---|---|---|---|---|
Tanaka [46] | 20 | 115 | 41 | 39 | 15 | 844 |
20 × 10 | 115 × 5 | 41 × 0 | 39 × 1 | 15 × 2 | ||
Yang et al. [48] | 4 | 46 | 81 | 57 | 42 | 411 |
4 × 10 | 46 × 5 | 81 × 0 | 57 × 1 | 42 × 2 | ||
Neubauer and Rostasy [49] | 19 | 102 | 44 | 50 | 16 | 782 |
19 × 10 | 102 × 5 | 44 × 0 | 50 × 1 | 16 × 2 | ||
Khalifa et al. [51] | 1 | 24 | 74 | 77 | 54 | 315 |
1 × 10 | 24 × 5 | 74 × 0 | 77 × 1 | 54 × 2 | ||
ACI 440.2R-08 | 123 | 74 | 20 | 12 | 1 | 1614 |
123 × 10 | 74 × 5 | 20 × 0 | 12 × 1 | 1 × 2 | ||
CNR. DT 200/2012 | 7 | 90 | 63 | 49 | 21 | 611 |
7 × 10 | 90 × 5 | 63 × 0 | 49 × 1 | 21 × 2 | ||
MARS (present study) | – | 47 | 136 | 47 | – | 282 |
– | 47 × 5 | 136 × 0 | 47 × 1 | – | ||
M5′ (present study) | – | 47 | 136 | 45 | 2 | 284 |
– | 47 × 5 | 136 × 0 | 45 × 1 | 2 × 2 |
7 Conclusions
In this study, a comprehensive database of 575 measurements of bond strength between FRP reinforcements externally glued on masonry units was compiled for the first time from datasets published in the literature. New equations for predicting maximum bond strength based on M5′ and MARS algorithms were proposed. The final models were established using the reinforcement width (b_{p}), the ratio between FRP reinforcement and masonry width (b_{p}/b_{m}), the tensile strength of substrate (f_{mt}), the axial strength of reinforcement (E_{p}t_{p}) and the bond length (L_{b}) as input variables. The M5′ model as a rule-based method was employed to provide understandable formulas that allow users to have more insight into the physics of the phenomenon. The MARS algorithm was also used as a reliable predictive model to determine the most important parameters in predicting the bond strength.
The performances of some common empirical models and the proposed ones were evaluated based on the size of the errors and uncertainty. The prediction errors and uncertainties associated with the developed M5′ and MARS models were remarkably smaller than those associated with the most common existing models. The proposed M5′ and MARS models outperformed the CNR model as the best empirical model by improving the R^{2} value by 51% and 86%, respectively and the RMSE value by 42% and 55%, respectively. The results of sensitivity analysis based on MARS models showed that the width ratio between FRP reinforcement and masonry substrate (b_{p}/b_{m}) was the most important and the axial strength of FRP reinforcement (E_{p}t_{p}) was the least important parameter in predicting the bond strength. Furthermore, the safety analysis based on Collins criteria indicated that the developed MARS and M5′ models also remarkably outperformed the existing equations in terms of safety.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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