## 4 Discussion
### Failure Mechanism and Proposed Rigid Block Model  
Figure 11 presents, for comparison with Figure 6, a failed 40 MPa concrete cylinder of 150 mm \(\times\) 300 mm with one CFRP wrap from the cylinders used to calibrate the model (refer to Section 2.3 and Figure 2). Figure 10: Confinement efficiency factor for various confinement levels and H/D ratios. The failure planes, wedge mechanism and the CFRP failure pattern that can be observed in Figure 8 are all well captured by the numerical results presented in Figure 5. The published literature contains a wealth of photos of similar experiments, showing the localized rupture of the FRP wraps and the development of a failure mechanism in the concrete cylinder, such as in [PERSON] and [PERSON] [10], [PERSON] and [PERSON] [78], [PERSON] [79], [PERSON] et al. [20], [PERSON] et al. [80] and [PERSON] and [PERSON] [29]. The response and failure of the confined concrete cylinders is characterized by the localization of failure and the formation of one or more inclined shear failure planes. This results in the separation of the concrete core into two or more wedges or blocks that form a failure mechanism. The movement of the broken wedges is laterally restrained in the presence of a lateral confinement device, such as hydraulic pressure, FRP wraps and steel stirrups. When the effect of lateral restraint is added up to the de-cohesion/localization response of the concrete, it results in a stable post peak behavior of the axially loaded concrete cylinder, such as indicated by the rising branch of the bi-linear stress-strain response in Figures 3 and 4. This post peak response is critical for the evaluation of the load-displacement behavior of confined concrete and to predict its failure load. A simple mechanistic model considering externally applied loads and internal reactions acting on an inclined crack was proposed by [PERSON] et al. (2002), similar to the mechanism shown in Figure 12 where the response of the model is controlled by the shear-friction on the sliding plane and the inclination of the failure angle [26; 27; 35; 63; 81; 82]. Following [PERSON] et al. [83], [PERSON] and [PERSON] [81], [PERSON] et al. [63] and [PERSON] and [PERSON] [26], an equation can be derived relating the increase in normalized residual axial stress, \(\Delta\overline{\sigma}_{1}\), over the unconfined compressive strength, \(f^{\prime}_{co}\), to the normalized confining stress, \(\overline{\sigma}_{3}\), using the coefficient of internal friction, \(\mu\), and the angle of inclination of the sliding plane, \(\varphi\), as  
\[\Delta\overline{\sigma}_{1}=\frac{sin\varphi+\mu\ cos\varphi}{(cos\varphi-\mu \ sin\varphi)tan\varphi}\overline{\sigma}_{3} \tag{1}\]  
The normalized axial stress in the confined cracked concrete can then be found by  
\[\overline{\sigma}_{1}=1+\Delta\overline{\sigma}_{1}=1+\frac{sin\varphi+\mu\ cos \varphi}{(cos\varphi-\mu\ sin\varphi)tan\varphi}\overline{\sigma}_{3} \tag{2}\]  
Figure 11: Photograph of a failed \(150\) mm \(\times\) 300 mm cylinder with 40 MPa concrete and one CFRP wrap. The coefficient of friction is a property of the concrete [81]; in the numerical model presented, the angle of friction was taken equal to \(35^{\circ}\), hence \(\mu=\tan\) (\(35\)) = 0.7. The failure angle can be measured from Figures 5-8, and it varies depending on the geometry of the failure mechanism. For L\({}_{c}\)/\(D\) = 1.5, \(\varphi\) = 33.6\({}^{\circ}\). Hence, Equation (2) can be calculated numerically as \(\overline{\sigma}_{1}\) = 1 + 3.84 \(\overline{\sigma}_{3}\) for the example at hand. Equation (2) follows the same shape as the one proposed by ACI 440.2R-17 [1].