# Numerical Investigation of the Nonlinear Composite Action of FRP-Concrete Hybrid Beams/Decks
## 2 Construction of Governing Equation  
Assuming that (a) only shear connectors provide composite action between the FRP and concrete components; (b) the concrete slab and FRP profile can be simplified as two components discontinuously joined (see Figure 1), the longitudinal shear flow across the interface is transferred at a discrete number of points by the action of the shear connectors; and (c) there is no separation along the normal direction of the interface, the interfacial shear force flow can be obtained as  
\[v(x)=nQ(s)/p \tag{1}\]  
where \(v(x)\) is the distributed interfacial shear force, \(n\) is the connector number of the connector in the lateral direction, \(Q(s)\) is load-slip relationship per connector, and \(p\) is the longitudinal space between connectors. For a simply supported beam, equilibrium of the finite length \(dx\) (Figure 1) in the horizontal direction gives  
\[\frac{dF_{NC}(x)}{dx}=\frac{dF_{NF}(x)}{dx}=-v(x) \tag{2}\]  
where \(F_{NC}(x)\) and \(F_{NF}(x)\) are the axial force in concrete and FRP, respectively. Equilibriums in the vertical direction of the shear force and moment give  
\[V_{c}(x)+V_{F}(x)=V(x) \tag{3}\]  
\[\frac{dM_{c}(x)}{dx}-v(x)\cdot\frac{h_{c}}{2}+\frac{r(x)dx}{2}+V_{c}(x)=0 \tag{4}\]  
\[\frac{dM_{F}(x)}{dx}-v(x)\cdot\frac{h_{F}}{2}-\frac{r(x)dx}{2}+V_{F}(x)=0 \tag{5}\]  
where \(V_{c}(x)\) and \(V_{F}(x)\) are the shear force carried by concrete and FRP, respectively, \(V(x)\) is the sectional shear force. \(M_{c}(x)\) and \(M_{F}(x)\) are flexural moment carried by concrete and FRP, respectively; \(h_{c}\) is the distance from the bottom of concrete to its neutral axis; \(h_{F}\) is the distance from the top of FRP to its neutral axis, and \(r(x)\) is distributed normal force along FRP-concrete interface. Considering assumption (v), the curvature compatibility between the concrete and FRP gives  
\[\phi(x)=\frac{M_{c}(x)}{E_{c}I_{c}}=\frac{M_{F}(x)}{E_{Fx}I_{Fx}} \tag{6}\]  
where \(E_{Fx}\) is the elastic modulus of FRP; \(I_{Fx}\) is moment inertia of FRP; \(E_{c}\) is the elastic modulus of concrete; \(I_{c}\) is moment inertia of concrete, and \(\phi\) is the curvature of the beam. FRP-concrete hybrid beams/decks are modeled elastically since FRP materials are inherent linear elastic and the concrete stays at a low stress level before the failure of FRP webs. The orthotropic  
Figure 1: Model of sectional analysis of section \(dx\). mechanical properties of the FRP composite material are the same in the web and flanges, however, in this study, the longitudinal modulus of FRP is employed to compute the sectional rigidity. Therefore, strains of concrete \(\varepsilon_{c}(x,y)\) and FRP \(\varepsilon_{c}(x,y)\) are calculated from the moment and axial force as  
\[\varepsilon_{c}(x,y)=\frac{M_{c}(x)(h_{c}/2-y)}{E_{c}I_{c}}-\frac{F_{NC}(x)}{E _{c}A_{c}},0\leq y\leq h_{c} \tag{7}\]  
\[\varepsilon_{F}(x,y)=-\frac{M_{F}(x)(h_{F}/2+y)}{E_{Fx}I_{Fx}}+\frac{F_{NF}(x)} {E_{Fx}A_{F}},-h_{F}\leq y\leq 0 \tag{8}\]  
where \(y\) is the vertical coordinate that has an original point at FRP-concrete interface and has a direction upward. Equations (7) and (8) give the strains at the interface respectively as  
\[\varepsilon_{cb}(x)=\varepsilon_{c}(x,0)=\frac{M_{c}(x)h_{c}}{2E_{c}I_{c}}- \frac{F_{NC}(x)}{E_{c}A_{c}} \tag{9}\]  
\[\frac{dF_{NC}(x)}{dx}=\frac{dF_{NF}(x)}{dx}=-v(x) \tag{2}\]0  
where \(A_{C}\) and \(A_{F}\) are the area of concrete and FRP, respectively.