# Numerical Investigation of the Nonlinear Composite Action of FRP-Concrete Hybrid Beams/Decks
## 6 Evaluation of Deflection Considering Slip Effect  
The total deflection can be computed by adding the additional deflection caused by slip to the deflection without slip [35].  
\[\delta(x)=\delta_{full}(x)+\delta_{slip}(x) \tag{29}\]  
where \(\delta(x)\) is the total deflection, \(\delta_{full}(x)\) is the deflection assuming that full composite action is ensured between FRP and concrete, and \(\delta_{slip}(x)\) is the deflection caused by the slip effect.  
Figure 8: Relationship between slip at beam end and the value of \(Q_{u}\) in Model_6.  
Figure 7: Slip along the half span for different values of \(Q_{u}\) in Model_6.  
The additional curvature due to the slip is a calculation from Equations (11) and (12) as  
\[\Delta\phi(x)=\frac{\varepsilon_{slip}(x)}{h_{0}}=\frac{s^{\prime}(x)}{h_{0}} \tag{30}\]  
Therefore, the deflection of the FCHB caused by slip effect can be computed as second order integration of Equation (30).  
\[\delta_{slip}(x)=\iint_{0}^{x}\Delta\phi(x)dx \tag{31}\]  
The mid-span deflection, \(\delta_{slip}(L/2)\) of the FCHB caused by slip effect for three-point bending scenario has been analytically given in [35; 36] as  
\[\delta_{slip}\left(\frac{L}{2}\right)=\frac{\beta P}{2h_{0}}\left[\frac{L}{2} +\frac{1}{\alpha}\left(\frac{1-e^{-\alpha L}}{1+e^{-\alpha L}}\right)\right] \tag{32}\]  
Herein, \(\delta_{slip}(L/2)\) is computed numerically using Equation (31) and analytically by Equation (32). The results are given in Table 3. It is found that with the increase of element numbers, the difference between the analytical and numerical result of \(\delta_{slip}(L/2)\) decreases. It should be noted that even using 10 elements in half span can lower the error of FDM to 0.225%.  
Figure 9 shows that the deflection distribution of FCHB using Model_1, Model_2, and Model_3. The deflections at the mid-span of Model_1, Model_2, and Model_3 are 0.133 mm, 0.130 mm and 0.124 mm, respectively. It is interesting to see that using these three connections leads to a difference of less than 7.1%, which means that all of these connections behave similarly in terms of deflection. The deflections at the mid-span of Model_4, Model_5, and Model_6 are 0.176 mm, 0.130 mm, and 0.125 mm, respectively. The difference between Model_5 and Model_6 is 4.04%, which means using high strength bolts is effective both for normal concrete and UHPC. Therefore, it is suggested that high strength bolts are beneficial for limiting deflection caused by slip and the UHPC is not that a significant factor.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Analysis**} & \multicolumn{8}{c}{**FDM**} \\ \cline{2-9}  & & \(N=10\) & \(N=20\) & \(N=40\) & \(N=60\) & \(N=80\) & \(N=100\) & \(N=1000\) \\ \hline \(\delta(L/2)\) (mm) & 0.13432 & 0.13462 & 0.13449 & 0.13441 & 0.13438 & 0.13437 & 0.13436 & 0.13433 \\ Error (\%) & & 0.225 & 0.128 & 0.067 Figure 10 shows the relationship between mid-span deflection considering slip effect and the value of slip modulus of connectors (\(K\)). It can be found that in with the increase of \(K\) from very low to 20 kN/mm, mid-span deflection decreases sharply to a relatively stable value. And when \(K\) decreases from 40 kN/mm to 100 kN/mm, the drop is gradual. Thus \(K>40\) kN/mm is suggested to be used in FCHB.  
A similar conclusion can be drawn in Figure 11. It can be found that in with the increase of \(Q_{u}\) from very low to 25 kN, mid-span deflection decreases sharply to a relatively stable value. \(Q_{u}\geq 25\) kN is suggested to be used in FCHB.  
Figure 11: Relationship between slip at beam end and the value of \(Q_{u}\) in Model_6.  
Figure 10: Relationship between mid-span deflection and slip modulus of connectors (\(K\)).