## 4 Discussion
### Failure Mechanism and Proposed Rigid Block Model
The normalized confining stress of an FRP wrap is given by  
\[\overline{\sigma}_{3}=\frac{2t}{D}\frac{E_{FRP}}{f^{\prime}_{co}}\varepsilon_ {FRPave} \tag{3}\]  
where \(t\) is the total thickness of the FRP wrap, \(D\) is the diameter of the concrete cylinder, \(E_{FRP}\) is the modulus of elasticity of the FRP wrap, \(f^{\prime}_{co}\) is the unconfined compressive strength of the concrete and \(\varepsilon_{FRPave}\) is the average hoop strain in the FRP wrap. Equations (2) and (3) can then be used to predict the stress-strain response as well as the failure load of the confined concrete. With reference to the mechanism shown in Figure 13 for a confined cylinder, and making the following simplifying assumptions--(i) all displacements are small; (ii) the wedges are relatively rigid, so elastic deformations can be neglected; and (iii) the shear bands' width at the localized failure zones is very small--then, the vertical shortening, u, can be related to geometry, \(v\), the radial bulging through \(v=u\)\(\tan\varphi\). The axial strain in the cylinder is then given by \(\varepsilon_{1}=2u\)/L\({}_{c}\), where L\({}_{c}\) is the characteristic length. The radial strain in the confined cylinder is given by \(\varepsilon_{3}=2v\)/\(D\), where \(D\) is the diameter of the cylinder. Noting that the radial strain and the hoop strain in the FRP are equal, we have \(\varepsilon_{\text{FRPave}}=\varepsilon_{3}\) and combining the previous geometric relationships:  
\[\varepsilon_{\text{FRPave}}=2v\text{/}D=2u.\tan\varphi\text{/}D=\tan\varphi \text{. }(\text{L}_{c}\text{/}D).\varepsilon_{1} \tag{4}\]  
Combining Equations (2)-(4):  
\[\overline{\sigma}_{1}=1+\frac{sin\varphi+\mu\ cos\varphi}{(cos\varphi-\mu\ sin \varphi)tan\varphi}\frac{2t}{D}\frac{E_{FRP}}{f^{\prime}_{co}}tan\varphi \text{.}(\text{L}c/D).\varepsilon_{1} \tag{5}\]  
The proposed model presents a formal mechanical confirmation for the bi-linear model proposed in ACI 440.2R-17 based on the work of [PERSON] and [PERSON] [5; 7] and offers a procedure for the calculation of the model parameters that are usually experimentally calibrated for each set of results. The normalized parameter of the bi-linear modulus of elasticity \(\overline{E}_{2}\) can be calculated as  
\[\overline{E}_{2}=\frac{sin\varphi+\mu\ cos\varphi}{(cos\varphi-\mu\ sin \varphi)tan\varphi}\frac{2t}{D}\frac{E_{FRP}}{f^{\prime}_{co}}tan\varphi \text{.}(\text{L}c/D) \tag{6}\]  
Figure 12: Shear-friction plane mechanism. The normalized maximum confined compressive strength, \(\overline{\sigma}_{CC}\), can also be calculated when \(\varepsilon_{\text{FRPave}}=\kappa\varepsilon_{\text{FRPmax}}\) using the confinement efficiency factor, \(\kappa\), presented in Figure 5.0  
Figure 5.1  
The calculation of \(\overline{\sigma}_{CC}\), uses material properties and the final state of the strains in the wrap prior to failure; hence, producing a reliable value. Further work is needed for the fine tuning of the proposed stress-strain model and its application to the extensive data available in the literature, especially with respect to the simplifying assumptions made for its derivation [84,85].