# Estimation of the Ultimate Strength of FRP Strips-to-Masonry Substrates Bond
## 2 Research Objectives
### Overview of Existing Formulas  
In order to present the existing formulas for assessing the ultimate bond strength (\(P_{max}\)) for members with materials that failed in a brittle manner and were strengthened with FRP sheets, the overview was executed, and Table 1 is presented. Some of the existing formulae (i.e., [PERSON] et al. [7], [PERSON] [3], and [PERSON] et al. [52]) were valid for the masonry materials. The others were valid for the concrete substrate, which had a similarly brittle manner; therefore, they could be compared with the considered dataset. There were some differences between masonry and concrete substrates. Nonetheless, all of these formulas were studied in previous research to show the viability of the proposed approach [7; 19]. The origin of some differences could be found in compressive strength levels, microscopic behaviors, and the brittle failure surface by considering internal reinforcement ([PERSON] and [PERSON] [3]). It was noted that by considering these references, the formulas proposed for the concrete substrates were utilized by researchers for the masonry substrates [7]. Most of these formulas were obtained via empirical methods that lead to lower accuracy (see Section 4). Therefore, soft computing techniques were used to obtain new formulas with higher accuracy. An investigation of the available formulas proved that all of the existing formulas were functions of the thickness of the FRP strip (\(t_{p}\)), the width of the FRP strip (\(b_{p}\)), the elasticity modulus of the FRP strip (\(E_{p}\)), the width of the masonry block (\(b_{m}\)), the bonded length (\(l_{b}\)), and the tensile strength of the masonry block (\(f_{ut}\)), respectively. Figure 1 displays a schematic of a masonry block strengthened with a FRP strip in the form of a pull-out test [19]. In [3; 7], some configurations of the test setup, measurement protocols, etc., were experimentally illustrated in more detail. Almost all of the pull-out test in a direct shear test manner was used to study the bond response of FRP to brittle substrates (e.g., masonry blocks) [53]. During test procedures, strains and surface displacements were obtained from optical techniques, such as digital image correlation (DIC) or electromechanical gases, in the form of strain gages and Linear Variable Deformational Transformers (LVDTs). In the test setup, the FRP sheets were mounted on the masonry blocks using epoxy, and the tensile loading was applied to the FRP sheet to pull it out. Consequently, by considering supports applied perpendicular to the direction along the loaded FRP, the stability of the setup was satisfied. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model’s Name** & **References** & **Existing Equations** \\ \hline Khalifa & [54] & \(P_{\text{max}}=110.2\times 10^{-6}\times(\frac{f_{\text{max}}^{\prime}}{42})^{( \frac{1}{6})}\times E_{p}\times t_{p}\times l_{e}\times b_{p}\) \\  & \(l_{e}=e^{6.134-0.58\times\ln(E_{p}\times l_{p})}\)unit of \(E_{p}\) is GPa \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Existing formula for forecasting ultimate bond strength of masonry members strengthened using FRP sheets (some of formulas presented here are extracted directly from [19]). Figure 1: A scheme of a masonry block strengthened with a FRP strip.