## 3 Expression for Effective FRP Strain of Interior Joints Derivation
### Experimental Joint Shear Strength
#### 3.1.1 Horizontal Shear (\(V_{ji}\)) over the Joint Core's Mid-Depth  
Equilibrium considerations are used to determine the horizontal shear acting at the joint's center, \(V_{ji}\). The free-body of an inner beam-column sub-assembly between its points of contra flexure is shown in Figure 1a, which is typical of test specimens often used in laboratory studies for idealizing RC frame joints. At the beam tip, the beam is subjected to a reversed cyclic load (\(P_{b1}\)). The internal forces acting on the face of the joint core because of the applied seismic forces can be demonstrated, as given in Figure 1b. In Figure 1b, the \"\(T_{s2}\)\" is the compression force (\(C\)) in the beam section on the opposite side of the joint, but using the horizontal equilibrium equation to the beam cross-section gives \(C^{\prime}=T_{s2}\)[43]. Therefore, \(T_{s2}\) is directly put in the equation by replacing \(C^{\prime}\) with \(T_{s2}\). The tension force in the beam bars framing into the joint (one or two forces depending on the exterior or interior joints) and the shear force from the column, \(V_{col}\), acting in the opposite direction, creates the shear. This is in line with EC8 [44] and ACI 352R-02 [45] for joint shear demand evaluation:  
\[V_{jh}=T_{s1}\ (+T_{s2})-V_{col} \tag{1}\]  
Figure 1: Joint shear of interior joint: (**a**) loading condition; (**b**) free body diagram at the mid-height of joint core. Forces in the Steel of the Beam (\(T_{s1}\), \(T_{s2}\))  
The tension in the bars, \(T_{s1}\) and \(T_{s2}\), at the ultimate load (\(P_{i}\)) at the tip of the beam or column, may be calculated by dividing the moment in the beam at the beam-joint interface, \(M_{bi}\), by the lever (\(j_{d}\)) between the centroid of compression forces and the tension forces operating in the joint. Where \(j_{d}=d_{b}\) is the distance between the tension and compression resultants in the beam at the beam joint interface, estimated to be \(0.75h_{b}\) (\(h_{b}\) = beam height) in the lack of a more in-depth examination of all the possible joint set-ups in the literature. \(J_{d}=0.75h_{b}\) is also used by [19,20]. \[T_{s1}=\frac{M_{b1}}{j_{d1}} \tag{2}\]  
\[T_{s2}=\frac{M_{b2}}{j_{d2}} \tag{3}\]  
\[\Rightarrow V_{jh}=\frac{M_{b1}}{j_{d1}}\bigg{(}+\frac{M_{b2}}{j_{d2}}\bigg{)} -V_{col} \tag{4}\]  
For symmetrical interior joints, \(M_{b1}\) and \(M_{b2}\) are considered to be equal, and \(M_{b1}\) may be determined from:  
\[M_{b1}=P_{b1}\cdot l_{b1} \tag{5}\]  
The peak force applied at the beam tip is \(P_{b1}\), and the distance from the point on which the force acts and the joint interface is \(l_{b1}\). Thus, if a load is exerted at the column, for example, \(P_{b1}\) may be calculated from \(P_{c}\), the ultimate force at the column tip, as follows:  
\[P_{b1}=P_{c}\cdot l_{c}/W_{b} \tag{6}\]  
where \(l_{c}\) refers to the entire test set-up's height (i.e., from the bottom column pin to the top column pin), and \(W_{b}\) denotes the whole set-up's width (i.e., one beam length and column's half-width for exterior joints or both beams length and column width for interior joints). Shear Force in Column (\(V_{col}\))  
If the force is acting at the column tip, the horizontal shear force \(V_{col}\) is taken as \(P_{c}\); if the force is exerted at the beam tip, equilibrium conditions may be used to calculate the force:  
\[V_{col}=P_{b1}\cdot W_{b}/l_{c} \tag{7}\]  
Horizontal Shear Stress (\(v_{jh}^{exp}\)) over the Joint Core's Mid-Depth  
The following equation is used for calculating the horizontal shear stress of the joint core. \[v_{jh}^{exp}=\frac{V_{jh}}{A_{col}}=\frac{V_{jh}}{b_{c}\cdot l_{c}}=\frac{ \frac{V_{jh}}{j_{d1}}\bigg{(}+\frac{V_{b2}}{j_{d2}}\bigg{)}-V_{col}}{b_{c} \cdot l_{c}} \tag{8}\]  
\(j_{d}=d_{b}\) is the distance from the tension and compression resultants in the beam at the beam joint interface, which is considered \(0.75h_{b}\) (\(l_{b}\) = beam height). \(j_{d}=0.75h_{b}\) is also used by [19,20].