# 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
#### 3.1.2 Crack Width Analysis  
The concrete structural elements are expected to develop cracks under service loads due to the concrete's inherent low tensile strength. As such, crack width calculation is one of the serviceability requirements of all structural concrete elements. The crack width evaluations are developed based on the durability of steel-reinforced concrete structures. Steel reinforcements embedded in the concrete structure are vulnerable to permeating electrolytes, and the crack networks are potential pathways to breach the barrier for transporting corroding agents [107]. Therefore, these crack limitation considerations are traditionally developed with a focus on steel-only RC elements.  
With GFRP-only reinforced beams, there is no risk of corrosion. The Japan Society of Civil Engineers [108] therefore proposed disregarding the crack width limitations in the GFRP reinforced structural elements. On the other hand, building standards such as ACI [109] advocate guidelines to crack control in GFRP-reinforced structures. Models that give consideration to crack development may be more suited for hybrid GFRP-steel reinforcements because of the presence of steel bars that requires protection from corrosion. In other words, determining the width and development of cracks in hybrid GFRP-steel RCs is critical to avoiding corrosion damage and ensuring their durability.  
The design recommendation by the American Association of State Highway and Transportation Officials (AASHTO) [110] gives limitations for crack width (\(w\)) as:  
\[w=\frac{2\,d_{c}\,f_{fs}\,\xi}{C_{b}E_{f}} \tag{8}\]  
Equation (8) establishes the crack limit is the product of the bond (reduction factor) (\(C_{b}\)) between GFRP reinforcing bars and surrounding concrete, the thickness of the concrete cover (\(d_{c}\)), the tensile modulus of the GFRP bar (\(E_{f}\)), the ratio of the distance from the neutral axis to extreme tensile zone (\(\xi\)), and calculated tensile stress in GFRP (\(f_{\text{S}}\)). Other design standards such as ACI and CSA also give limitations similar to AASHTO, but these guidelines did not include the crack width limitations for hybrid systems.  
The bond coefficient significantly influences and impacts on the accuracy of crack width predictions put forward by the design standards. Hence, researchers [110,111] developed predictive models that consider both GFRP and steel bars while designing hybrid RC elements.  
Equation (9) provided by ACI-440.1R-06 [112,113] gives the maximum crack width limit in GFRP-reinforced concrete beams. Where \(\sigma_{f}\) represents reinforcement stress, \(E_{f}\) represents the modulus of elasticity rebar, \(d_{c}\) is the distance from the tension face to the centre of the closest bar, and is the bar spacing. It also recommends \(k_{b}\) as the bond coefficient that is primarily taken as 1.40 for GFRP-reinforced beams.  
\[w=2\frac{\sigma_{f}}{Ef}\,\beta k_{b}\,\sqrt{d_{c}^{2}+\left(\frac{s}{2} \right)^{2}} \tag{9}\]  
Experimental studies by [PERSON] et al. [111] suggested the expression given in Equation (10) to estimate the bond coefficient, \(k_{b}\), for the hybrid GFRP-steel reinforced beams. It is evident from the work of [PERSON] et al. [111] that \(k_{b}\) depends on the area ratio of GFRP (\(A_{f}\)) to that of steel (\(A_{s}\)) and is given as:  
\[k_{b}=1.4\alpha\bigg{(}\frac{A_{f}}{A_{s}}\bigg{)}^{\frac{1}{3}} \tag{10}\]  
where \(\alpha\) accounts for the size effect of GFRP bars on the embedded concrete interface, \(A_{f}\) and \(A_{s}\) are the areas of GFRP and steel reinforcements, respectively.