# Numerical Investigation of the Nonlinear Composite Action of FRP-Concrete Hybrid Beams/Decks
## 5 Load-slip Models for Connection of FRP-concrete Hybrid Beams/Decks  
In this section, the choice of \(Q(s)\) models were made based on some formulas found in literatures [13; 14; 15]. The formulas have been given by the authors of this paper in previous literatures, which are presented in Table 2 and Figure 4. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multirow{2}{*}{**Equation (27)**} & \multicolumn{6}{c}{**FDM**} \\ \cline{3-8}  & & **10** & **20** & **40** & **60** & **80** & **100** & **1000** \\ \hline Beam end slip (mm) & 1.4853 & 1.4742 & 1.4828 & 1.4847 & 1.4850 & 1.4851 & 1.4852 & 1.4853 \\ Quarter span slip (mm) & 1.2707 & 1.3197 & 1.2958 & 1.2834 & 1.2792 & 1.2771 & 1.2758 & 1.2713 \\ Error at beam end (\%) & \(-0.745\) & \(-0.166\) & \(-0.039\) & \(-0.017\) & \(-0.010\) & \(-0.006\) & 0.000 \\ Error at quarter span (\%) & 3.850 & 1.975 & 0.998 & 0.667 & 0.501 & 0.401 & 0.040 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Errors of slip between FDM using different numbers of half span. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Name** & **Model Type** & **Formula** & **Equation** & **Parameters** \\ \hline Model\_1 & Linear & & & & \(K=7.27\) kN/mm \\ Model\_2 & Linear & & \(Q(s)=Ks\) & (28a) & \(K=11.58\) kN/mm \\ Model\_3 & Linear & & & & \(K=112.67\) kN/mm \\ \hline Model\_4 & Trilinear & \(Q(s)=Q_{u}\left\{\begin{array}{c}0.25s,\;0<s<0.8\\ 0.353s-0.082,0.8<s<2.5\\ 0.467+0.133s,2.5<s<4.0\end{array}\right.\) & (28b) & \(Q_{u}=28.5\) kN \\ Model\_6 & Trilinear & \(Q(s)=Q_{u}\left\{\begin{array}{c}0.8s,\;0<s<0.25\\ 0.133+0.27s,0.25<s<2.5\\ 0.467+0.133s,2.5<s<4.0\end{array}\right.\) & (28c) & \(Q_{u}=52.3\) kN \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Alternative load-slip models (s in unit of mm). Figure 3: The relationship between numbers of elements in half span and errors of slip. Model_1-3 are based on linear load-slip relationship and different values of \(K\) given by Table 2 in Reference [13]. Model_1 and Model_2 reflect the average slip modulus of normal and high strength steel bolts, respectively. Model_3 uses the value of \(K\) for perforated FRP ribs. Model_4-6 are based on nonlinear load-slip relationship given by Figure 11 in Reference [15] and can be seen in Figure 4 below. Model_5 is for the interface between FRP and normal concrete, and it consists of three parts: the compacting stage (0\(\sim\)0.2\(Q_{u}\)); the hardening stage (0.2\(Q_{u}\)-0.8\(Q_{u}\)), and the softening stage (0.8\(Q_{u}\)-\(Q_{u}\)). The compacting stage is caused by packing the initial clearance between the bolt and the predrilled hole as well as the torsion of the bolts. In the hardening stage, the shear resistance is provided by the elastic flexure and shear deformation of the bolt. Finally, in the softening stage, the large deformation of the bolts and the local FRP around the hole provide the shear resistance. Model_6 is for the high strength bolted connection between FRP and ultra-high performance concrete (UHPC) also consists of three parts, but they appear in a different pattern: the hardening stage (0\(\sim\)0.2\(Q_{u}\)) is first, followed by the compacting stage (0.2\(Q_{u}\)-0.8\(Q_{u}\)), and finally the softening stage (0.8\(Q_{u}\)-\(Q_{u}\)). Herein, according to the experimental data in Table 5 in Reference [15] and is also shown in Table 2 of this paper. The average load per bolts of ordinary strength bolts in the normal concrete is 28.5 kN. And the average loads of the high strength bolts in the normal strength concrete and the high strength bolts in UHPC are 40.8 kN and 52.3 kN, respectively. Figure 5a shows that the Model_3 has a very small slip level (less than 0.1 mm at the load of 500 kN), which shows that almost full composite action has been obtained, which indicates that this type of connection is suitable for FCHB system. Also, Model_1 leads to almost 50% larger slip than Model_2 which means that the interfacial slip is very sensitive to the value of slip modulus of bolts.