# Making a Case for Hybrid GFRP-Steel Reinforcement System in Concrete Beams: An Overview
## 3 Analytical Models for Hybrid GFRP-Steel Reinforced Beams
### Models Considering Mechanical Effects
#### 3.1.1 Ductility Analysis
[97] propose Equation (1) that expresses ductility as the ratio of the total energy to the elastic energy at the failure state of a beam. This approach utilises the elastic energy (\(E_{ela}\)) prior to failure. The total energy (\(E_{tot}\)) is taken as the area under the moment-curvature or load-deflection curve and \(E_{ela}\) as the elastic energy. The ductility index \(\mu_{en}\) was expressed as [97]:  
\[\mu_{en}=0.5\bigg{(}\frac{E_{tot}}{E_{ela}}+1\bigg{)} \tag{1}\]  
A new term \"deformability\" has been defined in Equation (2) [103]. It is intended to be used as a replacement of the conventional term ductility in design equations. This approach proposes an equivalent deformability factor as the ratio of the beam's equivalent deformation (\(\Delta_{t}\)) of the uncracked section to the actual deformation (\(\Delta_{u}\)) observed at the ultimate state of the section. This model gives a \(\mu\) value that is three times larger than the value given by the conventional ductility index:  
\[\mu=\frac{\Delta_{u}}{\Delta_{t}} \tag{2}\]  
A different approach to defining ductility in GFRP-RCs includes the consideration of strains at curvature. This model, proposed by [PERSON] et al. [104] was developed based on the failure of concrete crushing. As shown in Equation (3), it is an expression of the product of the ratio of curvature at ultimate state to the curvature when a strain of 0.001 is exhibited at the extreme compression edge of the RC section, i.e., \(\varepsilon_{c}\) = 0.001, and the ratio of the ultimate moment to that of the corresponding moment at \(\varepsilon_{c}\) = 0.001. \[\mu_{m}=\bigg{(}\frac{\varphi_{u}}{\varphi_{0.001}}\bigg{)}\bigg{(}\frac{M_{u} }{M_{0.001}}\bigg{)} \tag{3}\]where \(\varphi_{u}\) is the curvature at ultimate state; \(\varphi_{0.001}\) corresponds to the curvature when \(\varepsilon_{c}=0.001;M_{u}\) is the ultimate moment; and \(M_{0.001}\) corresponds to the moment at \(\varepsilon_{c}=0.001\). A comprehensive performance factor \(J\), combining deformability and strength, can be defined by the ratio of two energy quantities associated with the ultimate limit state condition and the proportional limit of the extreme compression zone [105] of the RC section. \(J\) must be a minimum of 4.0 for rectangular sections and at least 6.0 for T-sections, noting these design criteria have been included in the Canadian Highway Bridge Design Codes [106]. The \(J\)-factor is expressed as [105]:  
\[J=\left(\frac{M_{ult}\Psi_{ult}}{M_{c}\Psi_{c}}\right) \tag{4}\]  
where \(M_{ult}\) expresses the ultimate moment capacity of the section; \(M_{c}\) is the moment at \(\varepsilon_{c}=0.001;\Psi_{ult}\) is the curvature at \(M_{ult}\) and \(\Psi_{c}\) is the curvature at \(M_{c}\). The overall deformability factor \(Z\) considers the cracking behaviour in the ductility model. As expressed in Equation (5) [99], the effect of cracking is accounted for with the ratios of deflections at ultimate state and cracking initiation and of the ultimate and cracking moment of the RC section. The equation considers the products of both ratios. \[Z=\left(\frac{\Delta_{u}}{\Delta_{cr}}\right)\left(\frac{M_{u}}{M_{cr}}\right) \tag{5}\]  
where \(\Delta_{u}\) is the deflection at ultimate; \(\Delta_{u}\) corresponds to the deflection at cracking initiation; \(M_{cr}\) corresponds to the cracking moment and \(M_{u}\) is the ultimate moment. [PERSON] and [PERSON] [83], in experimentation with hybrid GFRP-steel RC, have partially introduced the conventional term of ductility index. In this approach, they redefined the term yield point (\(\Delta_{y}\)), which refers to the nonlinear point of the beam's load-deflection curve. [PERSON] and [PERSON] [83] employed a new term \(\mu\) as a dimensionless parameter given as:  
\[\mu=\frac{\Delta_{u}}{\Delta_{y}} \tag{6}\]  
where \(\Delta_{u}\) and \(\Delta_{y}\) are the midspan deflection at ultimate state and linear limit respectively.