## 3 Expression for Effective FRP Strain of Interior Joints Derivation
### Developing an Equation for FRP Effective Strain  
Initially, the experimental effective FRP strain is derived using the available experimental database. Then, the power-type equation is derived in terms of elastic modulus and area of FRP fibers and concrete strength. Then, by using a similar equation, the shear strength of the FRP-strengthened joints are determined and compared with that of the experimentally determined shear strength of the joints. The procedure is briefly explained below. * Equation of Effective FRP strain from experimental data: \[\epsilon_{f,e}^{exp}=\frac{p_{t,f}^{exp}\cdot b_{c}\cdot h_{c}}{A_{f,eq}\cdot E_{ f}\cdot sin\theta}\] (9)
* Calculation of parameters of Equation (9): (a) The input of FRP to principal tensile stress as measured experimentally (\(p_{t,f}^{exp}\)): \[p_{t,f}^{exp}=\left(p_{t,tot}^{exp}\right)-\left(p_{c,t}\right)\] (10)  
where \(p_{t,f}^{exp}\) is the total principal tensile stress of the interior beam-column joints strengthened with FRP, and \(p_{c,t}\) refers to an increase in the principal tensile stress of the joint panel due to concrete. Total principal tensile stress \(\left(p_{t}=p_{t,tot}^{exp}\right)\) is determined by using [PERSON]'s circle approach with the following equation:  
\[p_{t,tot}^{exp}=\frac{-\sigma_{c}}{2}\cdot\sqrt{\left(\frac{\sigma_{c}}{2} \right)^{2}+v_{jh}{}^{2}} \tag{11}\]  
Contribution of concrete in principle tensile stress for the deformed bar, interior BCJ is determined by using:  
\[p_{t,c}=\text{k}\cdot\sqrt{f_{c}} \tag{12}\]  
When deformed bars are utilized (during the joint's initial breaking), the numerical coefficient \(k\) is taken as 0.29 and when 0.42 is used (at the peak strength). At \(k=0.20\), maximum peak strength and joint breaking are achieved for smooth internal reinforcement ([PERSON] 1997; Eurocode, [PERSON] 2004; [PERSON] et al. 2015). Numerous authors such as [PERSON] (1997) and [PERSON] et al. (2002) suggested limiting the compression principal stresses (\(p_{c}\)) such as average tensile principal stresses in tension. As per their experimental findings, the compression principal stresses (\(p_{c}\)) must be restricted to values proportionate to the compressive strength of concrete as given in Equation (13). \[p_{c}=\frac{\sigma_{c}}{2}\cdot\sqrt{\left(\frac{\sigma_{c}}{2}\right)^{2}+v_ {jh}{}^{2}}\ \leq 0.5f_{c} \tag{13}\]  
Note: In some papers, \(p_{t,tot}^{exp}\) is denoted by \(f_{1}\). Likewise, \(\sigma_{c}\) ([20]) is also denoted by \(f_{a}\) ([19]) in some research articles. \(\sigma_{c}=F_{a}=\) axial compressive stress = (axial force)/(joint x-area). (b) Principal compressive stress's direction, \(\theta\):  
The equations suggested by [46] (also applied by [19; 20] are used as given below:  
* Constant direction: \[\theta=const=a\ tan\left(\frac{h_{b}}{h_{c}}\right)\] (14)
* Varying direction: \[\theta^{exp}=varying=\frac{1}{2}\left[\ \pi-atan\left(\frac{v_{jh}^{exp}}{f_{a}/2}\right)\right]\] (15) where \(f_{a}\) (also denoted as \(\sigma_{c}\)) = axial stresses = \(N\)/\(A_{col}\), \(v_{jh}^{exp}\) = joint horizontal shear stress = \(V_{jh}{}^{exp}/A_{col}\), \(N\) = axial load on column, and \(A_{cal}=\) X-area of column = \(b_{c}\times h_{c}\)  
(c) Equivalent area of FRP, (\(A_{f,eq}\)):  
The equivalent area of the FRP, \(A_{f,eq}\), is calculated using the available simple equations for the most common applications of FRP sheets [19]. Equations (16)-(18) are used for calculating the area of continuous FRP sheets as given below:  
Uniaxial fabric with fibers in the direction of beam axis (\(0^{\circ}\)) or column axis (\(90^{\circ}\)):  
\[\left[ \begin{array}{cccc}A_{f,eq}=n_{l}\cdot n_{s}\cdot t_{f}\cdot h_{b} \cdot sin\theta&for&\beta=0^{\circ}\\ A_{f,eq}=n_{l}\cdot n_{s}\cdot t_{f}\cdot h_{b}\cdot cos\theta&for&\beta=90^{ \circ}\end{array} \right.